# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1465913176 -3600
# Node ID e3cf792db636213a4feeb652dbb14e12a8bf2125
# Parent  5d8ec128518bb7c43e3aadd8a596b0c7b0f29e15
removed some files to attic

diff -r 5d8ec128518b -r e3cf792db636 Attic/CpsG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/CpsG.thy	Tue Jun 14 15:06:16 2016 +0100
@@ -0,0 +1,4669 @@
+theory CpsG
+imports PIPDefs
+begin
+
+section {* Generic aulxiliary lemmas *}
+
+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 rel_eqI:
+  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
+  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
+  shows "A = B"
+  using assms by auto
+
+section {* Lemmas do not depend on trace validity *}
+
+lemma birth_time_lt:  
+  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
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, 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 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)
+
+lemma runing_ready: 
+  shows "runing s \<subseteq> readys s"
+  unfolding runing_def readys_def
+  by auto 
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+  unfolding readys_def
+  by auto
+
+lemma wq_v_neq [simp]:
+   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
+  by (auto simp:wq_def Let_def cp_def split:list.splits)
+
+lemma runing_head:
+  assumes "th \<in> runing s"
+  and "th \<in> set (wq_fun (schs s) cs)"
+  shows "th = hd (wq_fun (schs s) cs)"
+  using assms
+  by (simp add:runing_def readys_def s_waiting_def wq_def)
+
+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
+
+lemma isP_E:
+  assumes "isP e"
+  obtains cs where "e = P (actor e) cs"
+  using assms by (cases e, auto)
+
+lemma isV_E:
+  assumes "isV e"
+  obtains cs where "e = V (actor e) cs"
+  using assms by (cases e, auto) 
+
+
+text {*
+  Every thread can only be blocked on one critical resource, 
+  symmetrically, every critical resource can only be held by one thread. 
+  This fact is much more easier according to our definition. 
+*}
+lemma held_unique:
+  assumes "holding (s::event list) th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+ by (insert assms, unfold s_holding_def, auto)
+
+lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits)
+
+lemma last_set_unique: 
+  "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
+          \<Longrightarrow> th1 = th2"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits dest:last_set_lt)
+
+lemma preced_unique : 
+  assumes pcd_eq: "preced th1 s = preced th2 s"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "th1 = th2"
+proof -
+  from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
+  from last_set_unique [OF this th_in1 th_in2]
+  show ?thesis .
+qed
+                      
+lemma preced_linorder: 
+  assumes neq_12: "th1 \<noteq> th2"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
+proof -
+  from preced_unique [OF _ th_in1 th_in2] and neq_12 
+  have "preced th1 s \<noteq> preced th2 s" by auto
+  thus ?thesis by auto
+qed
+
+lemma in_RAG_E:
+  assumes "(n1, n2) \<in> RAG (s::state)"
+  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
+      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
+  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
+  by auto
+
+lemma count_rec1 [simp]: 
+  assumes "Q e"
+  shows "count Q (e#es) = Suc (count Q es)"
+  using assms
+  by (unfold count_def, auto)
+
+lemma count_rec2 [simp]: 
+  assumes "\<not>Q e"
+  shows "count Q (e#es) = (count Q es)"
+  using assms
+  by (unfold count_def, auto)
+
+lemma count_rec3 [simp]: 
+  shows "count Q [] =  0"
+  by (unfold count_def, auto)
+
+lemma cntP_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"
+proof(cases e)
+  case (P th' pty)
+  show ?thesis
+  by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)", 
+        insert assms P, auto simp:cntP_def)
+qed (insert assms, auto simp:cntP_def)
+  
+lemma cntV_diff_inv:
+  assumes "cntV (e#s) th \<noteq> cntV s th"
+  shows "isV e \<and> actor e = th"
+proof(cases e)
+  case (V th' pty)
+  show ?thesis
+  by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)", 
+        insert assms V, auto simp:cntV_def)
+qed (insert assms, auto simp:cntV_def)
+
+lemma eq_dependants: "dependants (wq s) = dependants s"
+  by (simp add: s_dependants_abv wq_def)
+
+lemma inj_the_preced: 
+  "inj_on (the_preced s) (threads s)"
+  by (metis inj_onI preced_unique the_preced_def)
+
+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
+
+(* ccc *)
+
+section {* Locales used to investigate the execution of PIP *}
+
+text {* 
+  The following locale @{text valid_trace} is used to constrain the 
+  trace to be valid. All properties hold for valid traces are 
+  derived under this locale. 
+*}
+locale valid_trace = 
+  fixes s
+  assumes vt : "vt s"
+
+text {* 
+  The following locale @{text valid_trace_e} describes 
+  the valid extension of a valid trace. The event @{text "e"}
+  represents an event in the system, which corresponds 
+  to a one step operation of the PIP protocol. 
+  It is required that @{text "e"} is an event eligible to happen
+  under state @{text "s"}, which is already required to be valid
+  by the parent locale @{text "valid_trace"}.
+
+  This locale is used to investigate one step execution of PIP, 
+  properties concerning the effects of @{text "e"}'s execution, 
+  for example, how the values of observation functions are changed, 
+  or how desirable properties are kept invariant, are derived
+  under this locale. The state before execution is @{text "s"}, while
+  the state after execution is @{text "e#s"}. Therefore, the lemmas 
+  derived usually relate observations on @{text "e#s"} to those 
+  on @{text "s"}.
+*}
+
+locale valid_trace_e = valid_trace +
+  fixes e
+  assumes vt_e: "vt (e#s)"
+begin
+
+text {*
+  The following lemma shows that @{text "e"} must be a 
+  eligible event (or a valid step) to be taken under
+  the state represented by @{text "s"}.
+*}
+lemma pip_e: "PIP s e"
+  using vt_e by (cases, simp)  
+
+end
+
+text {*
+  Because @{term "e#s"} is also a valid trace, properties 
+  derived for valid trace @{term s} also hold on @{term "e#s"}.
+*}
+sublocale valid_trace_e < vat_es!: valid_trace "e#s" 
+  using vt_e
+  by (unfold_locales, simp)
+
+text {*
+  For each specific event (or operation), there is a sublocale
+  further constraining that the event @{text e} to be that 
+  particular event. 
+
+  For example, the following 
+  locale @{text "valid_trace_create"} is the sublocale for 
+  event @{term "Create"}:
+*}
+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"
+
+text {*
+  locale @{text "valid_trace_p"} is divided further into two 
+  sublocales, namely, @{text "valid_trace_p_h"} 
+  and @{text "valid_trace_p_w"}.
+*}
+
+text {*
+  The following two sublocales @{text "valid_trace_p_h"}
+  and @{text "valid_trace_p_w"} represent two complementary 
+  cases under @{text "valid_trace_p"}, where
+  @{text "valid_trace_p_h"} further constraints that
+  @{text "wq s cs = []"}, which means the waiting queue of 
+  the requested resource @{text "cs"} is empty, in which
+  case,  the requesting thread @{text "th"} 
+  will take hold of @{text "cs"}. 
+
+  Opposite to @{text "valid_trace_p_h"},
+  @{text "valid_trace_p_w"} constraints that
+  @{text "wq s cs \<noteq> []"}, which means the waiting queue of 
+  the requested resource @{text "cs"} is nonempty, in which
+  case,  the requesting thread @{text "th"} will be blocked
+  on @{text "cs"}: 
+
+  Peculiar properties will be derived under respective 
+  locales.
+*}
+
+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
+
+text {*
+  The following @{text "holder"} designates
+  the holder of @{text "cs"} before the @{text "P"}-operation.
+*}
+definition "holder = hd (wq s cs)"
+
+text {*
+  The following @{text "waiters"} designates
+  the list of threads waiting for @{text "cs"} 
+  before the @{text "P"}-operation.
+*}
+definition "waiters = tl (wq s cs)"
+end
+
+text {* 
+  @{text "valid_trace_v"} is set for the @{term V}-operation.
+*}
+locale valid_trace_v = valid_trace_e + 
+  fixes th cs
+  assumes is_v: "e = V th cs"
+begin
+  -- {* The following @{text "rest"} is the tail of 
+        waiting queue of the resource @{text "cs"}
+        to be released by this @{text "V"}-operation.
+     *}
+  definition "rest = tl (wq s cs)"
+
+  text {*
+    The following @{text "wq'"} is the waiting
+    queue of @{term "cs"}
+    after the @{text "V"}-operation, which
+    is simply a reordering of @{term "rest"}. 
+
+    The effect of this reordering needs to be 
+    understood by two cases:
+    \begin{enumerate}
+    \item When @{text "rest = []"},
+    the reordering gives rise to an empty list as well, 
+    which means there is no thread holding or waiting 
+    for resource @{term "cs"}, therefore, it is free.
+
+    \item When @{text "rest \<noteq> []"}, the effect of 
+    this reordering is to arbitrarily 
+    switch one thread in @{term "rest"} to the 
+    head, which, by definition take over the hold
+    of @{term "cs"} and is designated by @{text "taker"}
+    in the following sublocale @{text "valid_trace_v_n"}.
+  *}
+  definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+
+  text {* 
+  The following @{text "rest'"} is the tail of the 
+  waiting queue after the @{text "V"}-operation. 
+  It plays only auxiliary role to ease reasoning. 
+  *}
+  definition "rest' = tl wq'"
+
+end
+
+text {* 
+  In the following, @{text "valid_trace_v"} is also 
+  divided into two 
+  sublocales: when @{text "rest"} is empty (represented
+  by @{text "valid_trace_v_e"}), which means, there is no thread waiting 
+  for @{text "cs"}, therefore, after the @{text "V"}-operation, 
+  it will become free; otherwise (represented 
+  by @{text "valid_trace_v_n"}), one thread 
+  will be picked from those in @{text "rest"} to take 
+  over @{text "cs"}.
+*}
+
+locale valid_trace_v_e = valid_trace_v +
+  assumes rest_nil: "rest = []"
+
+locale valid_trace_v_n = valid_trace_v +
+  assumes rest_nnl: "rest \<noteq> []"
+begin
+
+text {* 
+  The following @{text "taker"} is the thread to 
+  take over @{text "cs"}. 
+*}
+  definition "taker = hd wq'"
+
+end
+
+
+locale valid_trace_set = valid_trace_e + 
+  fixes th prio
+  assumes is_set: "e = Set th prio"
+
+context valid_trace
+begin
+
+text {*
+  Induction rule introduced to easy the 
+  derivation of properties for valid trace @{term "s"}.
+  One more premises, namely @{term "valid_trace_e s e"}
+  is added, so that an interpretation of 
+  @{text "valid_trace_e"} can be instantiated 
+  so that all properties derived so far becomes 
+  available in the proof of induction step.
+
+  You will see its use in the proofs that follows.
+*}
+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
+
+text {*
+  The following lemma says that if @{text "s"} is a valid state, so 
+  is its any postfix. Where @{term "monent t s"} is the postfix of 
+  @{term "s"} with length @{term "t"}.
+*}
+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
+end
+
+text {*
+  The following locale @{text "valid_moment"} is to inherit the properties 
+  derived on any valid state to the prefix of it, with length @{text "i"}.
+*}
+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)
+
+locale valid_moment_e = valid_moment +
+  assumes less_i: "i < length s"
+begin
+  definition "next_e  = hd (moment (Suc i) s)"
+
+  lemma trace_e: 
+    "moment (Suc i) s = next_e#moment i s"
+   proof -
+    from less_i have "Suc i \<le> length s" by auto
+    from moment_plus[OF this, folded next_e_def]
+    show ?thesis .
+   qed
+
+end
+
+sublocale valid_moment_e < vat_moment_e!: valid_trace_e "moment i s" "next_e"
+  using vt_moment[of "Suc i", unfolded trace_e]
+  by (unfold_locales, simp)
+
+section {* Distinctiveness of waiting queues *}
+
+context valid_trace_create
+begin
+
+lemma wq_kept [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_kept [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_exit wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace_p 
+begin
+
+lemma wq_neq_simp [simp]:
+  assumes "cs' \<noteq> cs"
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_p wq_def
+  by (auto simp:Let_def)
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_p]
+  show ?thesis by (cases, simp)
+qed
+
+lemma th_not_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 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_kept [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  finite_threads:
+  shows "finite (threads s)"
+  using vt by (induct) (auto elim: step.cases)
+
+lemma finite_readys [simp]: "finite (readys s)"
+  using finite_threads readys_threads rev_finite_subset by blast
+
+lemma wq_distinct: "distinct (wq s cs)"
+proof(induct rule:ind)
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case 
+  proof(cases e)
+    case (Create th prio)
+    interpret vt_create: valid_trace_create s e th prio 
+      using Create by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept) 
+  next
+    case (Exit th)
+    interpret vt_exit: valid_trace_exit s e th  
+        using Exit by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept) 
+  next
+    case (P th cs)
+    interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) 
+  next
+    case (V th cs)
+    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) 
+  next
+    case (Set th prio)
+    interpret vt_set: valid_trace_set s e th prio
+        using Set by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) 
+  qed
+qed (unfold wq_def Let_def, simp)
+
+end
+
+section {* Waiting queues and threads *}
+
+context valid_trace_e
+begin
+
+lemma wq_out_inv: 
+  assumes s_in: "thread \<in> set (wq s cs)"
+  and s_hd: "thread = hd (wq s cs)"
+  and s_i: "thread \<noteq> hd (wq (e#s) cs)"
+  shows "e = V thread cs"
+proof(cases e)
+-- {* There are only two non-trivial cases: *}
+  case (V th cs1)
+  show ?thesis
+  proof(cases "cs1 = cs")
+    case True
+    have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
+    thus ?thesis
+    proof(cases)
+      case (thread_V)
+      moreover have "th = thread" using thread_V(2) s_hd
+          by (unfold s_holding_def wq_def, simp)
+      ultimately show ?thesis using V True by simp
+    qed
+  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+next
+  case (P th cs1)
+  show ?thesis
+  proof(cases "cs1 = cs")
+    case True
+    with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
+      by (auto simp:wq_def Let_def split:if_splits)
+    with s_i s_hd s_in have False
+      by (metis empty_iff hd_append2 list.set(1) wq_def) 
+    thus ?thesis by simp
+  qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+
+lemma wq_in_inv: 
+  assumes s_ni: "thread \<notin> set (wq s cs)"
+  and s_i: "thread \<in> set (wq (e#s) cs)"
+  shows "e = P thread cs"
+proof(cases e)
+  -- {* This is the only non-trivial case: *}
+  case (V th cs1)
+  have False
+  proof(cases "cs1 = cs")
+    case True
+    show ?thesis
+    proof(cases "(wq s cs1)")
+      case (Cons w_hd w_tl)
+      have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
+      proof -
+        have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
+          using  Cons V by (auto simp:wq_def Let_def True split:if_splits)
+        moreover have "set ... \<subseteq> set (wq s cs)"
+        proof(rule someI2)
+          show "distinct w_tl \<and> set w_tl = set w_tl"
+            by (metis distinct.simps(2) local.Cons wq_distinct)
+        qed (insert Cons True, auto)
+        ultimately show ?thesis by simp
+      qed
+      with assms show ?thesis by auto
+    qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
+  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+  thus ?thesis by auto
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+
+end
+
+lemma (in valid_trace_create)
+  th_not_in_threads: "th \<notin> threads s"
+proof -
+  from pip_e[unfolded is_create]
+  show ?thesis by (cases, simp)
+qed
+
+lemma (in valid_trace_create)
+  threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
+  by (unfold is_create, simp)
+
+lemma (in valid_trace_exit)
+  threads_es [simp]: "threads (e#s) = threads s - {th}"
+  by (unfold is_exit, simp)
+
+lemma (in valid_trace_p)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_p, simp)
+
+lemma (in valid_trace_v)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_v, simp)
+
+lemma (in valid_trace_v)
+  th_not_in_rest[simp]: "th \<notin> set rest"
+proof
+  assume otherwise: "th \<in> set rest"
+  have "distinct (wq s cs)" by (simp add: wq_distinct)
+  from this[unfolded wq_s_cs] and otherwise
+  show False by auto
+qed
+
+lemma (in valid_trace_v) distinct_rest: "distinct rest"
+  by (simp add: distinct_tl rest_def wq_distinct)
+
+lemma (in valid_trace_v)
+  set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
+proof(unfold wq_es_cs wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume "distinct x \<and> set x = set rest"
+  thus "set x = set (wq s cs) - {th}" 
+      by (unfold wq_s_cs, simp)
+qed
+
+lemma (in valid_trace_exit)
+  th_not_in_wq: "th \<notin> set (wq s cs)"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis
+  by (cases, unfold holdents_def s_holding_def, fold wq_def, 
+             auto elim!:runing_wqE)
+qed
+
+lemma (in valid_trace) wq_threads: 
+  assumes "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+  using assms
+proof(induct rule:ind)
+  case (Nil)
+  thus ?case by (auto simp:wq_def)
+next
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case
+  proof(cases e)
+    case (Create th' prio')
+    interpret vt: valid_trace_create s e th' prio'
+      using Create by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems by auto
+  next
+    case (Exit th')
+    interpret vt: valid_trace_exit s e th'
+      using Exit by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto 
+  next
+    case (P th' cs')
+    interpret vt: valid_trace_p s e th' cs'
+      using P by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems readys_threads 
+        runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv 
+        by fastforce 
+  next
+    case (V th' cs')
+    interpret vt: valid_trace_v s e th' cs'
+      using V by (unfold_locales, simp)
+    show ?thesis using Cons
+      using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
+  next
+    case (Set th' prio)
+    interpret vt: valid_trace_set s e th' prio
+      using Set by (unfold_locales, simp)
+    show ?thesis using Cons.hyps(2) Cons.prems vt.is_set 
+        by (auto simp:wq_def Let_def)
+  qed
+qed 
+
+section {* RAG and threads *}
+
+context valid_trace
+begin
+
+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
+
+section {* The change of @{term RAG} *}
+
+text {*
+  The following three lemmas show that @{text "RAG"} does not change
+  by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
+  events, respectively.
+*}
+
+lemma (in valid_trace_set) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s"
+   by (unfold is_set s_RAG_def s_waiting_def wq_def, simp add:Let_def)
+
+lemma (in valid_trace_create) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s"
+ by (unfold is_create s_RAG_def s_waiting_def wq_def, simp add:Let_def)
+
+lemma (in valid_trace_exit) RAG_unchanged[simp]: "(RAG (e # s)) = RAG s"
+  by (unfold is_exit s_RAG_def s_waiting_def wq_def, simp add:Let_def)
+
+context valid_trace_v
+begin
+
+lemma holding_cs_eq_th:
+  assumes "holding s t cs"
+  shows "t = th"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis
+  proof(cases)
+    case (thread_V)
+    from held_unique[OF this(2) assms]
+    show ?thesis by simp
+  qed
+qed
+
+lemma distinct_wq': "distinct wq'"
+  by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
+  
+lemma set_wq': "set wq' = set rest"
+  by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def)
+    
+lemma th'_in_inv:
+  assumes "th' \<in> set wq'"
+  shows "th' \<in> set rest"
+  using assms set_wq' by simp
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis by (cases, simp)
+qed
+
+lemma neq_t_th: 
+  assumes "waiting (e#s) t c"
+  shows "t \<noteq> th"
+proof
+  assume otherwise: "t = th"
+  show False
+  proof(cases "c = cs")
+    case True
+    have "t \<in> set wq'" 
+     using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
+     by simp 
+    from th'_in_inv[OF this] have "t \<in> set rest" .
+    with wq_s_cs[folded otherwise] wq_distinct[of cs]
+    show ?thesis by simp
+  next
+    case False
+    have "wq (e#s) c = wq s c" using False
+        by (unfold is_v, simp)
+    hence "waiting s t c" using assms 
+        by (simp add: cs_waiting_def waiting_eq)
+    hence "t \<notin> readys s" by (unfold readys_def, auto)
+    hence "t \<notin> runing s" using runing_ready by auto 
+    with runing_th_s[folded otherwise] show ?thesis by auto 
+  qed
+qed
+
+lemma waiting_esI1:
+  assumes "waiting s t c"
+      and "c \<noteq> cs" 
+  shows "waiting (e#s) t c" 
+proof -
+  have "wq (e#s) c = wq s c" 
+    using assms(2) is_v by auto
+  with assms(1) show ?thesis 
+    using cs_waiting_def waiting_eq by auto 
+qed
+
+lemma holding_esI2:
+  assumes "c \<noteq> cs" 
+  and "holding s t c"
+  shows "holding (e#s) t c"
+proof -
+  from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
+  from assms(2)[unfolded s_holding_def, folded wq_def, 
+                folded this, unfolded wq_def, folded s_holding_def]
+  show ?thesis .
+qed
+
+lemma holding_esI1:
+  assumes "holding s t c"
+  and "t \<noteq> th"
+  shows "holding (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms using holding_cs_eq_th by blast 
+  from holding_esI2[OF this assms(1)]
+  show ?thesis .
+qed
+
+end
+
+context valid_trace_v_n
+begin
+
+lemma neq_wq': "wq' \<noteq> []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x \<noteq> []" using rest_nnl by auto
+qed 
+
+lemma eq_wq': "wq' = taker # rest'"
+  by (simp add: neq_wq' rest'_def taker_def)
+
+lemma next_th_taker: 
+  shows "next_th s th cs taker"
+  using rest_nnl taker_def wq'_def wq_s_cs 
+  by (auto simp:next_th_def)
+
+lemma taker_unique: 
+  assumes "next_th s th cs taker'"
+  shows "taker' = taker"
+proof -
+  from assms
+  obtain rest' where 
+    h: "wq s cs = th # rest'" 
+       "taker' = hd (SOME q. distinct q \<and> set q = set rest')"
+          by (unfold next_th_def, auto)
+  with wq_s_cs have "rest' = rest" by auto
+  thus ?thesis using h(2) taker_def wq'_def by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
+  by (smt all_not_in_conv bot.extremum insertI1 insert_subset 
+      mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {(Cs cs, Th taker)}"
+  using next_th_taker taker_def waiting_set_eq 
+  by fastforce
+   
+lemma holding_taker:
+  shows "holding (e#s) taker cs"
+    by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, 
+        auto simp:neq_wq' taker_def)
+
+lemma waiting_esI2:
+  assumes "waiting s t cs"
+      and "t \<noteq> taker"
+  shows "waiting (e#s) t cs" 
+proof -
+  have "t \<in> set wq'" 
+  proof(unfold wq'_def, rule someI2)
+    show "distinct rest \<and> set rest = set rest"
+          by (simp add: distinct_rest)
+  next
+    fix x
+    assume "distinct x \<and> set x = set rest"
+    moreover have "t \<in> set rest"
+        using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto 
+    ultimately show "t \<in> set x" by simp
+  qed
+  moreover have "t \<noteq> hd wq'"
+    using assms(2) taker_def by auto 
+  ultimately show ?thesis
+    by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+     |    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
+  have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
+  hence "t \<noteq> taker" by (simp add: taker_def) 
+  moreover hence "t \<noteq> th" using assms neq_t_th by blast 
+  moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv) 
+  ultimately have "waiting s t cs"
+    by (metis cs_waiting_def list.distinct(2) list.sel(1) 
+                list.set_sel(2) rest_def waiting_eq wq_s_cs)  
+  show ?thesis using that(2)
+  using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto   
+qed
+
+lemma holding_esI1:
+  assumes "c = cs"
+  and "t = taker"
+  shows "holding (e#s) t c"
+  by (unfold assms, simp add: holding_taker)
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c = cs" "t = taker"
+      | "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from assms[unfolded True, unfolded s_holding_def, 
+             folded wq_def, unfolded wq_es_cs]
+  have "t = taker" by (simp add: taker_def) 
+  from that(1)[OF True this] show ?thesis .
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that(2)[OF False this] show ?thesis .
+qed
+
+end 
+
+
+context valid_trace_v_e
+begin
+
+lemma nil_wq': "wq' = []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x = []" using rest_nil by auto
+qed 
+
+lemma no_taker: 
+  assumes "next_th s th cs taker"
+  shows "False"
+proof -
+  from assms[unfolded next_th_def]
+  obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
+    by auto
+  thus ?thesis using rest_def rest_nil by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
+  using no_taker by auto
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
+  using no_taker by auto
+   
+lemma no_holding:
+  assumes "holding (e#s) taker cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_holding_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma no_waiting:
+  assumes "waiting (e#s) t cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma waiting_esI2:
+  assumes "waiting s t c"
+  shows "waiting (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms
+    using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto 
+  from waiting_esI1[OF assms this]
+  show ?thesis .
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from no_waiting[OF assms[unfolded True]]
+  show ?thesis by auto
+qed
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from no_holding[OF assms[unfolded True]] 
+  show ?thesis by auto
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that[OF False this] show ?thesis .
+qed
+
+end 
+
+  
+context valid_trace_v
+begin
+
+lemma RAG_es:
+  "RAG (e # s) =
+   RAG s - {(Cs cs, Th th)} -
+     {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+     {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R"
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_n.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      next
+        case 2
+        with waiting(1,2)
+        show ?thesis
+         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      qed
+    next
+      case True
+      interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_e.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      qed
+    qed
+  next
+    case (holding th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from holding(3)
+      show ?thesis
+      proof(cases rule:h_n.holding_esE)
+        case 1
+        with holding(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      next
+        case 2
+        with holding(1,2)
+        show ?thesis
+         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold holding_eq, auto)
+      qed
+    next
+      case True
+      interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+      from holding(3)
+      show ?thesis
+      proof(cases rule:h_e.holding_esE)
+        case 1
+        with holding(1,2)
+        show ?thesis
+        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
+             fold holding_eq, auto)
+      qed
+    qed
+  qed
+next
+  fix n1 n2
+  assume h: "(n1, n2) \<in> ?R"
+  show "(n1, n2) \<in> ?L"
+  proof(cases "rest = []")
+    case False
+    interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+    from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
+    have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
+                            \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or> 
+          (n2 = Th h_n.taker \<and> n1 = Cs cs)" 
+      by auto
+   thus ?thesis
+   proof
+      assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
+      with h_n.holding_taker
+      show ?thesis 
+        by (unfold s_RAG_def, fold holding_eq, auto)
+   next
+    assume h: "(n1, n2) \<in> RAG s \<and>
+        (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
+    hence "(n1, n2) \<in> RAG s" by simp
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from h and this(1,2)
+      have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
+      hence "waiting (e#s) th' cs'" 
+      proof
+        assume "cs' \<noteq> cs"
+        from waiting_esI1[OF waiting(3) this] 
+        show ?thesis .
+      next
+        assume neq_th': "th' \<noteq> h_n.taker"
+        show ?thesis
+        proof(cases "cs' = cs")
+          case False
+          from waiting_esI1[OF waiting(3) this] 
+          show ?thesis .
+        next
+          case True
+          from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
+          show ?thesis .
+        qed
+      qed
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from h this(1,2)
+      have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+      hence "holding (e#s) th' cs'"
+      proof
+        assume "cs' \<noteq> cs"
+        from holding_esI2[OF this holding(3)] 
+        show ?thesis .
+      next
+        assume "th' \<noteq> th"
+        from holding_esI1[OF holding(3) this]
+        show ?thesis .
+      qed
+      thus ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+   qed
+ next
+   case True
+   interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+   from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
+   have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)" 
+      by auto
+   from h_s(1)
+   show ?thesis
+   proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from h_e.waiting_esI2[OF this(3)]
+    show ?thesis using waiting(1,2)
+      by (unfold s_RAG_def, fold waiting_eq, auto)
+   next
+    case (holding th' cs')
+    with h_s(2)
+    have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+    thus ?thesis
+    proof
+      assume neq_cs: "cs' \<noteq> cs"
+      from holding_esI2[OF this holding(3)]
+      show ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    next
+      assume "th' \<noteq> th"
+      from holding_esI1[OF holding(3) this]
+      show ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+   qed
+ qed
+qed
+
+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_p
+begin
+
+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 
+
+lemma (in valid_trace_p) th_not_waiting: "\<not> waiting s th c"
+proof -
+  have "th \<in> readys s"
+    using runing_ready runing_th_s by blast 
+  thus ?thesis
+    by (unfold readys_def, auto)
+qed
+
+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_w
+begin
+
+lemma wq_s_cs: "wq s cs = holder#waiters"
+    by (simp add: holder_def waiters_def wne)
+    
+lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
+  by (simp add: wq_es_cs wq_s_cs)
+
+lemma waiting_es_th_cs: "waiting (e#s) th cs"
+  using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
+
+lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
+   by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
+
+lemma holding_esE:
+  assumes "holding (e#s) th' cs'"
+  obtains "holding s th' cs'"
+  using assms 
+proof(cases "cs' = cs")
+  case False
+  hence "wq (e#s) cs' = wq s cs'" by simp
+  with assms show ?thesis
+    using cs_holding_def holding_eq that by auto 
+next
+  case True
+  with assms show ?thesis
+  by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that 
+        wq_es_cs' wq_s_cs) 
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) th' cs'"
+  obtains "th' \<noteq> th" "waiting s th' cs'"
+     |  "th' = th" "cs' = cs"
+proof(cases "waiting s th' cs'")
+  case True
+  have "th' \<noteq> th"
+  proof
+    assume otherwise: "th' = th"
+    from True[unfolded this]
+    show False by (simp add: th_not_waiting)
+  qed
+  from that(1)[OF this True] show ?thesis .
+next
+  case False
+  hence "th' = th \<and> cs' = cs"
+      by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) 
+        set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
+  with that(2) show ?thesis by metis
+qed
+
+lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R" 
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:waiting_esE)
+      case 1
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case 2
+      thus ?thesis using waiting(1,2) by auto
+    qed
+  next
+    case (holding th' cs')
+    from this(3)
+    show ?thesis
+    proof(cases rule:holding_esE)
+      case 1
+      with holding(1,2)
+      show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  qed
+next
+  fix n1 n2
+  assume "(n1, n2) \<in> ?R"
+  hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
+  thus "(n1, n2) \<in> ?L"
+  proof
+    assume "(n1, n2) \<in> RAG s"
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from waiting_kept[OF this(3)]
+      show ?thesis using waiting(1,2)
+         by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from holding_kept[OF this(3)]
+      show ?thesis using holding(1,2)
+         by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+  next
+    assume "n1 = Th th \<and> n2 = Cs cs"
+    thus ?thesis using RAG_edge by auto
+  qed
+qed
+
+end
+
+context valid_trace_p
+begin
+
+lemma RAG_es: "RAG (e # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+                                                  else RAG s \<union> {(Th th, Cs cs)})"
+proof(cases "wq s cs = []")
+  case True
+  interpret vt_p: valid_trace_p_h using True
+    by (unfold_locales, simp)
+  show ?thesis by (simp add: vt_p.RAG_es vt_p.we) 
+next
+  case False
+  interpret vt_p: valid_trace_p_w using False
+    by (unfold_locales, simp)
+  show ?thesis by (simp add: vt_p.RAG_es vt_p.wne) 
+qed
+
+end
+
+section {* Finiteness of RAG *}
+
+context valid_trace
+begin
+
+lemma finite_RAG:
+  shows "finite (RAG s)"
+proof(induct rule:ind)
+  case Nil
+  show ?case 
+  by (auto simp: s_RAG_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+next
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case
+  proof(cases e)
+    case (Create th prio)
+    interpret vt: valid_trace_create s e th prio using Create
+      by (unfold_locales, simp)
+    show ?thesis using Cons by simp
+  next
+    case (Exit th)
+    interpret vt: valid_trace_exit s e th using Exit
+      by (unfold_locales, simp)
+    show ?thesis using Cons by simp
+  next
+    case (P th cs)
+    interpret vt: valid_trace_p s e th cs using P
+      by (unfold_locales, simp)
+    show ?thesis using Cons using vt.RAG_es by auto 
+  next
+    case (V th cs)
+    interpret vt: valid_trace_v s e th cs using V
+      by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt.finite_RAG_kept) 
+  next
+    case (Set th prio)
+    interpret vt: valid_trace_set s e th prio using Set
+      by (unfold_locales, simp)
+    show ?thesis using Cons by simp
+  qed
+qed
+end
+
+section {* RAG is acyclic *}
+
+text {* (* ddd *)
+  The nature of the work is like this: since it starts from a very simple and basic 
+  model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
+  For instance, the fact 
+  that one thread can not be blocked by two critical resources at the same time
+  is obvious, because only running threads can make new requests, if one is waiting for 
+  a critical resource and get blocked, it can not make another resource request and get 
+  blocked the second time (because it is not running). 
+
+  To derive this fact, one needs to prove by contraction and 
+  reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
+  named @{text "p_split"}, which is about status changing along the time axis. It says if 
+  a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
+  but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} 
+  in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history 
+  of events leading to it), such that @{text "Q"} switched 
+  from being @{text "False"} to @{text "True"} and kept being @{text "True"}
+  till the last moment of @{text "s"}.
+
+  Suppose a thread @{text "th"} is blocked
+  on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, 
+  since no thread is blocked at the very beginning, by applying 
+  @{text "p_split"} to these two blocking facts, there exist 
+  two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that 
+  @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} 
+  and kept on blocked on them respectively ever since.
+ 
+  Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
+  However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
+  in blocked state at moment @{text "t2"} and could not
+  make any request and get blocked the second time: Contradiction.
+*}
+
+
+context valid_trace
+begin
+
+lemma waiting_unique_pre: (* ddd *)
+  assumes h11: "thread \<in> set (wq s cs1)"
+  and h12: "thread \<noteq> hd (wq s cs1)"
+  assumes h21: "thread \<in> set (wq s cs2)"
+  and h22: "thread \<noteq> hd (wq s cs2)"
+  and neq12: "cs1 \<noteq> cs2"
+  shows "False"
+proof -
+  let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  from h11 and h12 have q1: "?Q cs1 s" by simp
+  from h21 and h22 have q2: "?Q cs2 s" by simp
+  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+  from p_split [of "?Q cs1", OF q1 nq1]
+  obtain t1 where lt1: "t1 < length s"
+    and np1: "\<not> ?Q cs1 (moment t1 s)"
+    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
+  from p_split [of "?Q cs2", OF q2 nq2]
+  obtain t2 where lt2: "t2 < length s"
+    and np2: "\<not> ?Q cs2 (moment t2 s)"
+    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
+  { fix s cs
+    assume q: "?Q cs s"
+    have "thread \<notin> runing s"
+    proof
+      assume "thread \<in> runing s"
+      hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and> 
+                 thread \<noteq> hd (wq_fun (schs s) cs))"
+        by (unfold runing_def s_waiting_def readys_def, auto)
+      from this[rule_format, of cs] q 
+      show False by (simp add: wq_def) 
+    qed
+  } note q_not_runing = this
+  { fix t1 t2 cs1 cs2
+    assume  lt1: "t1 < length s"
+    and np1: "\<not> ?Q cs1 (moment t1 s)"
+    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
+    and lt2: "t2 < length s"
+    and np2: "\<not> ?Q cs2 (moment t2 s)"
+    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
+    and lt12: "t1 < t2"
+    let ?t3 = "Suc t2" 
+    interpret ve2: valid_moment_e _ t2 using lt2
+     by (unfold_locales, simp)
+    let ?e = ve2.next_e
+    have "t2 < ?t3" by simp
+    from nn2 [rule_format, OF this] and ve2.trace_e
+    have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" and
+         h2: "thread \<noteq> hd (wq (?e#moment t2 s) cs2)" by auto
+    have ?thesis
+    proof -
+      have "thread \<in> runing (moment t2 s)"
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        have "?e = V thread cs2"
+        proof -
+          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
+              using True and np2  by auto 
+          thus ?thesis
+            using True h2 ve2.vat_moment_e.wq_out_inv by blast 
+        qed
+        thus ?thesis
+          using step.cases ve2.vat_moment_e.pip_e by auto 
+      next
+        case False
+        hence "?e = P thread cs2"
+          using h1 ve2.vat_moment_e.wq_in_inv by blast 
+        thus ?thesis
+          using step.cases ve2.vat_moment_e.pip_e by auto 
+      qed
+      moreover have "thread \<notin> runing (moment t2 s)"
+        by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
+      ultimately show ?thesis by simp
+    qed
+  } note lt_case = this
+  show ?thesis
+  proof -
+    { assume "t1 < t2"
+      from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
+      have ?thesis .
+    } moreover {
+      assume "t2 < t1"
+      from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
+      have ?thesis .
+    } moreover { 
+      assume eq_12: "t1 = t2"
+      let ?t3 = "Suc t2"
+      interpret ve2: valid_moment_e _ t2 using lt2
+        by (unfold_locales, simp)
+      let ?e = ve2.next_e
+      have "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and ve2.trace_e
+      have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" by auto
+      have lt_2: "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and ve2.trace_e
+      have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" and
+           h2: "thread \<noteq> hd (wq (?e#moment t2 s) cs2)" by auto
+      from nn1[rule_format, OF lt_2[folded eq_12], unfolded ve2.trace_e[folded eq_12]] 
+           eq_12[symmetric]
+      have g1: "thread \<in> set (wq (?e#moment t1 s) cs1)" and
+           g2: "thread \<noteq> hd (wq (?e#moment t1 s) cs1)" by auto
+      have "?e = V thread cs2 \<or> ?e = P thread cs2"
+        using h1 h2 np2 ve2.vat_moment_e.wq_in_inv 
+              ve2.vat_moment_e.wq_out_inv by blast
+      moreover have "?e = V thread cs1 \<or> ?e = P thread cs1"
+        using eq_12 g1 g2 np1 ve2.vat_moment_e.wq_in_inv 
+              ve2.vat_moment_e.wq_out_inv by blast
+      ultimately have ?thesis using neq12 by auto
+    } ultimately show ?thesis using nat_neq_iff by blast 
+  qed
+qed
+
+text {*
+  This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
+*}
+
+lemma waiting_unique:
+  assumes "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+  using waiting_unique_pre assms
+  unfolding wq_def s_waiting_def
+  by auto
+
+end
+
+lemma (in valid_trace_v)
+  preced_es [simp]: "preced th (e#s) = preced th s"
+  by (unfold is_v preced_def, simp)
+
+lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
+proof
+  fix th'
+  show "the_preced (V th cs # s) th' = the_preced s th'"
+    by (unfold the_preced_def preced_def, simp)
+qed
+
+
+lemma (in valid_trace_v)
+  the_preced_es: "the_preced (e#s) = the_preced s"
+  by (unfold is_v preced_def, simp)
+
+context valid_trace_p
+begin
+
+lemma not_holding_s_th_cs: "\<not> holding s th cs"
+proof
+  assume otherwise: "holding s th cs"
+  from pip_e[unfolded is_p]
+  show False
+  proof(cases)
+    case (thread_P)
+    moreover have "(Cs cs, Th th) \<in> RAG s"
+      using otherwise cs_holding_def 
+            holding_eq th_not_in_wq by auto
+    ultimately show ?thesis by auto
+  qed
+qed
+
+end
+
+
+lemma (in valid_trace_v_n) finite_waiting_set:
+  "finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
+    by (simp add: waiting_set_eq)
+
+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_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 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 
+  next
+    case (Exit th)
+    interpret vt: valid_trace_exit s e th using Exit
+      by (unfold_locales, simp)
+    show ?thesis using Cons by simp
+  next
+    case (P th cs)
+    interpret vt: valid_trace_p s e th cs using P
+      by (unfold_locales, simp)
+    show ?thesis
+    proof(cases "wq s cs = []")
+      case True
+      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
+      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 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 (Set th prio)
+    interpret vt: valid_trace_set s e th prio using Set
+      by (unfold_locales, simp)
+    show ?thesis using Cons by simp 
+  qed
+qed
+
+end
+
+section {* RAG is single-valued *}
+
+context valid_trace
+begin
+
+lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
+  apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
+  by(auto elim:waiting_unique held_unique)
+
+lemma sgv_RAG: "single_valued (RAG s)"
+  using unique_RAG by (auto simp:single_valued_def)
+
+end
+
+section {* RAG is well-founded *}
+
+context valid_trace
+begin
+
+lemma wf_RAG: "wf (RAG s)"
+proof(rule finite_acyclic_wf)
+  from finite_RAG show "finite (RAG s)" .
+next
+  from acyclic_RAG show "acyclic (RAG s)" .
+qed
+
+lemma wf_RAG_converse: 
+  shows "wf ((RAG s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+  from finite_RAG 
+  show "finite (RAG s)" .
+next
+  from acyclic_RAG
+  show "acyclic (RAG s)" .
+qed
+
+end
+
+section {* RAG forms a forest (or tree) *}
+
+context valid_trace
+begin
+
+lemma rtree_RAG: "rtree (RAG s)"
+  using sgv_RAG acyclic_RAG
+  by (unfold rtree_def rtree_axioms_def sgv_def, auto)
+
+end
+
+sublocale valid_trace < rtree_RAG: rtree "RAG s"
+  using rtree_RAG .
+
+sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
+proof -
+  show "fsubtree (RAG s)"
+  proof(intro_locales)
+    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
+  next
+    show "fsubtree_axioms (RAG s)"
+    proof(unfold fsubtree_axioms_def)
+      from wf_RAG show "wf (RAG s)" .
+    qed
+  qed
+qed
+
+
+section {* Derived properties for parts of RAG *}
+
+context valid_trace
+begin
+
+lemma acyclic_tRAG: "acyclic (tRAG s)"
+proof(unfold tRAG_def, rule acyclic_compose)
+  show "acyclic (RAG s)" using acyclic_RAG .
+next
+  show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+next
+  show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+qed
+
+lemma sgv_wRAG: "single_valued (wRAG s)"
+  using waiting_unique
+  by (unfold single_valued_def wRAG_def, auto)
+
+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)
+
+end
+
+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 < 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 tRAG_nodeE:
+  assumes "(n1, n2) \<in> tRAG s"
+  obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
+  using assms
+  by (auto simp: tRAG_def wRAG_def hRAG_def)
+
+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 subtree_nodeE:
+  assumes "n \<in> subtree (tRAG s) (Th th)"
+  obtains th1 where "n = Th th1"
+proof -
+  show ?thesis
+  proof(rule subtreeE[OF assms])
+    assume "n = Th th"
+    from that[OF this] show ?thesis .
+  next
+    assume "Th th \<in> ancestors (tRAG s) n"
+    hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+    hence "\<exists> th1. n = Th th1"
+    proof(induct)
+      case (base y)
+      from tRAG_nodeE[OF this] show ?case by metis
+    next
+      case (step y z)
+      thus ?case by auto
+    qed
+    with that show ?thesis by auto
+  qed
+qed
+
+lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
+proof -
+  have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*" 
+    by (rule rtrancl_mono, auto simp:RAG_split)
+  also have "... \<subseteq> ((RAG s)^*)^*"
+    by (rule rtrancl_mono, auto)
+  also have "... = (RAG s)^*" by simp
+  finally show ?thesis by (unfold tRAG_def, simp)
+qed
+
+lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
+proof -
+  { fix a
+    assume "a \<in> subtree (tRAG s) x"
+    hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
+    with tRAG_star_RAG
+    have "(a, x) \<in> (RAG s)^*" by auto
+    hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
+  } thus ?thesis by auto
+qed
+
+lemma tRAG_trancl_eq:
+   "{th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
+    {th'. (Th th', Th th)  \<in> (RAG s)^+}"
+   (is "?L = ?R")
+proof -
+  { fix th'
+    assume "th' \<in> ?L"
+    hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
+    from tranclD[OF this]
+    obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
+    from tRAG_subtree_RAG and this(2)
+    have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG) 
+    moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto 
+    ultimately have "th' \<in> ?R"  by auto 
+  } moreover 
+  { fix th'
+    assume "th' \<in> ?R"
+    hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
+    from plus_rpath[OF this]
+    obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
+    hence "(Th th', Th th) \<in> (tRAG s)^+"
+    proof(induct xs arbitrary:th' th rule:length_induct)
+      case (1 xs th' th)
+      then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
+      show ?case
+      proof(cases "xs1")
+        case Nil
+        from 1(2)[unfolded Cons1 Nil]
+        have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
+        hence "(Th th', x1) \<in> (RAG s)" 
+          by (cases, auto)
+        then obtain cs where "x1 = Cs cs" 
+              by (unfold s_RAG_def, auto)
+        from rpath_nnl_lastE[OF rp[unfolded this]]
+        show ?thesis by auto
+      next
+        case (Cons x2 xs2)
+        from 1(2)[unfolded Cons1[unfolded this]]
+        have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
+        from rpath_edges_on[OF this]
+        have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
+        have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+            by (simp add: edges_on_unfold)
+        with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
+        then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
+        have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+            by (simp add: edges_on_unfold)
+        from this eds
+        have rg2: "(x1, x2) \<in> RAG s" by auto
+        from this[unfolded eq_x1] 
+        obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
+        from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
+        have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
+        from rp have "rpath (RAG s) x2 xs2 (Th th)"
+           by  (elim rpath_ConsE, simp)
+        from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
+        show ?thesis
+        proof(cases "xs2 = []")
+          case True
+          from rpath_nilE[OF rp'[unfolded this]]
+          have "th1 = th" by auto
+          from rt1[unfolded this] show ?thesis by auto
+        next
+          case False
+          from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
+          have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
+          with rt1 show ?thesis by auto
+        qed
+      qed
+    qed
+    hence "th' \<in> ?L" by auto
+  } ultimately show ?thesis by blast
+qed
+
+lemma tRAG_trancl_eq_Th:
+   "{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
+    {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}"
+    using tRAG_trancl_eq by auto
+
+
+lemma tRAG_Field:
+  "Field (tRAG s) \<subseteq> Field (RAG s)"
+  by (unfold tRAG_alt_def Field_def, auto)
+
+lemma tRAG_mono:
+  assumes "RAG s' \<subseteq> RAG s"
+  shows "tRAG s' \<subseteq> tRAG s"
+  using assms 
+  by (unfold tRAG_alt_def, auto)
+
+lemma tRAG_subtree_eq: 
+   "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th'  \<in> (subtree (RAG s) (Th th))}"
+   (is "?L = ?R")
+proof -
+  { fix n 
+    assume h: "n \<in> ?L"
+    hence "n \<in> ?R"
+    by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) 
+  } moreover {
+    fix n
+    assume "n \<in> ?R"
+    then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
+      by (auto simp:subtree_def)
+    from rtranclD[OF this(2)]
+    have "n \<in> ?L"
+    proof
+      assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
+      with h have "n \<in> {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}" by auto
+      thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
+    qed (insert h, auto simp:subtree_def)
+  } ultimately show ?thesis by auto
+qed
+
+lemma threads_set_eq: 
+   "the_thread ` (subtree (tRAG s) (Th th)) = 
+                  {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
+   by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
+
+context valid_trace
+begin
+
+lemma RAG_tRAG_transfer:
+  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 -
+  { 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(1) have cs_in: "(Cs cs', Th th2) \<in> RAG s" by auto
+    from h(3) and assms(1) 
+    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(2) cs_in[unfolded this]
+        show ?thesis using 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(1), auto)
+      qed
+      ultimately show ?thesis by auto
+    next
+      assume eq_n: "(n1, n2) = (Th th, Th th'')"
+      from assms(1, 2) have "(Cs cs, Th th'') \<in> RAG s'" by auto
+      moreover have "(Th th, Cs cs) \<in> RAG s'" using assms(1) by auto
+      ultimately show ?thesis 
+        by (unfold eq_n tRAG_alt_def, auto)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+lemma subtree_tRAG_thread:
+  assumes "th \<in> threads s"
+  shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
+proof -
+  have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+    by (unfold tRAG_subtree_eq, simp)
+  also have "... \<subseteq> ?R"
+  proof
+    fix x
+    assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+    then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
+    from this(2)
+    show "x \<in> ?R"
+    proof(cases rule:subtreeE)
+      case 1
+      thus ?thesis by (simp add: assms h(1)) 
+    next
+      case 2
+      thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) 
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma 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
+
+end
+
+section {* Chain to readys *}
+
+context valid_trace
+begin
+
+lemma chain_building:
+  assumes "node \<in> Domain (RAG s)"
+  obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"
+proof -
+  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
+    ultimately show ?thesis by (auto simp:readys_def)
+  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
+  The following is just an instance of @{text "chain_building"}.
+*}                    
+lemma th_chain_to_ready:
+  assumes th_in: "th \<in> threads s"
+  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
+proof(cases "th \<in> readys s")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  from False and th_in have "Th th \<in> Domain (RAG s)" 
+    by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
+  from chain_building [rule_format, OF this]
+  show ?thesis by auto
+qed
+
+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 runing_unique:
+  assumes runing_1: "th1 \<in> runing s"
+  and runing_2: "th2 \<in> runing s"
+  shows "th1 = th2"
+proof -
+  from runing_1 and runing_2 have "cp s th1 = cp s th2"
+    unfolding runing_def 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
+  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
+  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
+  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 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
+    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
+    ultimately have "xs2 = xs1" by simp
+    from rpath_dest_eq[OF rp1 rp2[unfolded this]]
+    show ?thesis by simp
+  qed
+qed
+
+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
+
+end
+
+
+section {* Relating @{term cp} and @{term the_preced} and @{term preced} *}
+
+context valid_trace
+begin
+
+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 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
+
+
+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
+
+section {* Relating @{term cntP}, @{term cntV}, @{term cntCS} and @{term pvD} *}
+
+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 -
+  { 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
+       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
+  
+lemma (in valid_trace) finite_holdents: "finite (holdents s th)"
+  by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)
+
+context valid_trace_p 
+begin
+
+lemma ready_th_s: "th \<in> readys s"
+  using runing_th_s
+  by (unfold runing_def, auto)
+
+lemma live_th_s: "th \<in> threads s"
+  using readys_threads ready_th_s by auto
+
+lemma live_th_es: "th \<in> threads (e#s)"
+  using live_th_s 
+  by (unfold is_p, simp)
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+end
+
+context valid_trace_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
+      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 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 
+        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 
+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
+begin
+
+lemma th_not_waiting: 
+  "\<not> waiting s th c"
+proof -
+  have "th \<in> readys s"
+    using runing_ready runing_th_s by blast 
+  thus ?thesis
+    by (unfold readys_def, auto)
+qed
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+end
+
+context valid_trace_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
+      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
+        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 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_kept]
+  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_kept]
+  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_kept]
+  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_kept]
+  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_kept, 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_kept, 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_kept]
+    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_kept]
+         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_kept]
+  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_kept, 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_kept, 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_kept]
+    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_kept]
+         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_kept, 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_kept, 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_kept]
+    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_kept]
+         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 (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
+
+end
+
+section {* Corollaries of @{thm valid_trace.cnp_cnv_cncs} *}
+
+context valid_trace
+begin
+
+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"
+  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 eq_pv_children:
+  assumes eq_pv: "cntP s th = cntV 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)
+    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 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)
+
+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
+
+lemma count_eq_tRAG_plus:
+  assumes "cntP s th = cntV s th"
+  shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+  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"
+  shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+  using count_eq_RAG_plus[OF assms] by auto
+
+lemma count_eq_tRAG_plus_Th:
+  assumes "cntP s th = cntV s th"
+  shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+   using count_eq_tRAG_plus[OF assms] by auto
+
+end
+
+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
+
+section {* Recursive definition of @{term "cp"} *}
+
+lemma cp_alt_def1: 
+  "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
+proof -
+  have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
+       ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
+       by auto
+  thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
+qed
+
+lemma cp_gen_def_cond: 
+  assumes "x = Th th"
+  shows "cp s th = cp_gen s (Th th)"
+by (unfold cp_alt_def1 cp_gen_def, simp)
+
+lemma cp_gen_over_set:
+  assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
+  shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
+proof(rule f_image_eq)
+  fix a
+  assume "a \<in> A"
+  from assms[rule_format, OF this]
+  obtain th where eq_a: "a = Th th" by auto
+  show "cp_gen s a = (cp s \<circ> the_thread) a"
+    by  (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
+qed
+
+
+context valid_trace
+begin
+(* ddd *)
+lemma cp_gen_rec:
+  assumes "x = Th th"
+  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
+proof(cases "children (tRAG s) x = {}")
+  case True
+  show ?thesis
+    by (unfold True cp_gen_def subtree_children, simp add:assms)
+next
+  case False
+  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
+  note fsbttRAGs.finite_subtree[simp]
+  have [simp]: "finite (children (tRAG s) x)"
+     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 
+            rule children_subtree)
+  { fix r x
+    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
+  } note this[simp]
+  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
+  proof -
+    from False obtain q where "q \<in> children (tRAG s) x" by blast
+    moreover have "subtree (tRAG s) q \<noteq> {}" by simp
+    ultimately show ?thesis by blast
+  qed
+  have h: "Max ((the_preced s \<circ> the_thread) `
+                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
+        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
+                     (is "?L = ?R")
+  proof -
+    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
+    let "Max (_ \<union> (?h ` ?B))" = ?R
+    let ?L1 = "?f ` \<Union>(?g ` ?B)"
+    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
+    proof -
+      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
+      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
+      finally have "Max ?L1 = Max ..." by simp
+      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
+        by (subst Max_UNION, simp+)
+      also have "... = Max (cp_gen s ` children (tRAG s) x)"
+          by (unfold image_comp cp_gen_alt_def, simp)
+      finally show ?thesis .
+    qed
+    show ?thesis
+    proof -
+      have "?L = Max (?f ` ?A \<union> ?L1)" by simp
+      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
+            by (subst Max_Un, simp+)
+      also have "... = max (?f x) (Max (?h ` ?B))"
+        by (unfold eq_Max_L1, simp)
+      also have "... =?R"
+        by (rule max_Max_eq, (simp)+, unfold assms, simp)
+      finally show ?thesis .
+    qed
+  qed  thus ?thesis 
+          by (fold h subtree_children, unfold cp_gen_def, simp) 
+qed
+
+lemma cp_rec:
+  "cp s th = Max ({the_preced s th} \<union> 
+                     (cp s o the_thread) ` children (tRAG s) (Th th))"
+proof -
+  have "Th th = Th th" by simp
+  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
+  show ?thesis 
+  proof -
+    have "cp_gen s ` children (tRAG s) (Th th) = 
+                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
+    proof(rule cp_gen_over_set)
+      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
+        by (unfold tRAG_alt_def, auto simp:children_def)
+    qed
+    thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+  qed
+qed
+end
+
+section {* Other properties useful in Implementation.thy or Correctness.thy *}
+
+context valid_trace_e 
+begin
+
+lemma actor_inv: 
+  assumes "\<not> isCreate e"
+  shows "actor e \<in> runing s"
+  using pip_e assms 
+  by (induct, auto)
+end
+
+context valid_trace
+begin
+
+lemma readys_root:
+  assumes "th \<in> readys s"
+  shows "root (RAG s) (Th th)"
+proof -
+  { fix x
+    assume "x \<in> ancestors (RAG s) (Th th)"
+    hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+    from tranclD[OF this]
+    obtain z where "(Th th, z) \<in> RAG s" by auto
+    with assms(1) have False
+         apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
+         by (fold wq_def, blast)
+  } thus ?thesis by (unfold root_def, auto)
+qed
+
+lemma readys_in_no_subtree:
+  assumes "th \<in> readys s"
+  and "th' \<noteq> th"
+  shows "Th th \<notin> subtree (RAG s) (Th th')" 
+proof
+   assume "Th th \<in> subtree (RAG s) (Th th')"
+   thus False
+   proof(cases rule:subtreeE)
+      case 1
+      with assms show ?thesis by auto
+   next
+      case 2
+      with readys_root[OF assms(1)]
+      show ?thesis by (auto simp:root_def)
+   qed
+qed
+
+lemma not_in_thread_isolated:
+  assumes "th \<notin> threads s"
+  shows "(Th th) \<notin> Field (RAG s)"
+proof
+  assume "(Th th) \<in> Field (RAG s)"
+  with dm_RAG_threads and rg_RAG_threads assms
+  show False by (unfold Field_def, blast)
+qed
+
+lemma next_th_holding:
+  assumes nxt: "next_th s th cs th'"
+  shows "holding (wq s) th cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  thus ?thesis
+    by (unfold cs_holding_def, auto)
+qed
+
+lemma next_th_waiting:
+  assumes nxt: "next_th s th cs th'"
+  shows "waiting (wq s) th' cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  from wq_distinct[of cs, unfolded h]
+  have dst: "distinct (th # rest)" .
+  have in_rest: "th' \<in> set rest"
+  proof(unfold h, rule someI2)
+    show "distinct rest \<and> set rest = set rest" using dst by auto
+  next
+    fix x assume "distinct x \<and> set x = set rest"
+    with h(2)
+    show "hd x \<in> set (rest)" by (cases x, auto)
+  qed
+  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
+  moreover have "th' \<noteq> hd (wq s cs)"
+    by (unfold h(1), insert in_rest dst, auto)
+  ultimately show ?thesis by (auto simp:cs_waiting_def)
+qed
+
+lemma next_th_RAG:
+  assumes nxt: "next_th (s::event list) th cs th'"
+  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
+  using vt assms next_th_holding next_th_waiting
+  by (unfold s_RAG_def, simp)
+
+end 
+
+context valid_trace_p
+begin
+
+find_theorems readys th
+
+end
+
+end
diff -r 5d8ec128518b -r e3cf792db636 Attic/CpsG_1.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/CpsG_1.thy	Tue Jun 14 15:06:16 2016 +0100
@@ -0,0 +1,4403 @@
+theory CpsG
+imports PIPDefs 
+begin
+
+lemma Max_f_mono:
+  assumes seq: "A \<subseteq> B"
+  and np: "A \<noteq> {}"
+  and fnt: "finite B"
+  shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+  from seq show "f ` A \<subseteq> f ` B" by auto
+next
+  from np show "f ` A \<noteq> {}" by auto
+next
+  from fnt and seq show "finite (f ` B)" by auto
+qed
+
+(* I am going to use this file as a start point to retrofiting 
+   PIPBasics.thy, which is originally called CpsG.ghy *)
+
+locale valid_trace = 
+  fixes s
+  assumes vt : "vt s"
+
+locale valid_trace_e = valid_trace +
+  fixes e
+  assumes vt_e: "vt (e#s)"
+begin
+
+lemma pip_e: "PIP s e"
+  using vt_e by (cases, simp)  
+
+end
+
+locale valid_trace_create = valid_trace_e + 
+  fixes th prio
+  assumes is_create: "e = Create th prio"
+
+locale valid_trace_exit = valid_trace_e + 
+  fixes th
+  assumes is_exit: "e = Exit th"
+
+locale valid_trace_p = valid_trace_e + 
+  fixes th cs
+  assumes is_p: "e = P th cs"
+
+locale valid_trace_v = valid_trace_e + 
+  fixes th cs
+  assumes is_v: "e = V th cs"
+begin
+  definition "rest = tl (wq s cs)"
+  definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+end
+
+locale valid_trace_v_n = valid_trace_v +
+  assumes rest_nnl: "rest \<noteq> []"
+
+locale valid_trace_v_e = valid_trace_v +
+  assumes rest_nil: "rest = []"
+
+locale valid_trace_set= valid_trace_e + 
+  fixes th prio
+  assumes is_set: "e = Set th prio"
+
+context valid_trace
+begin
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes "PP []"
+     and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
+                   PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
+     shows "PP s"
+proof(induct rule:vt.induct[OF vt, case_names Init Step])
+  case Init
+  from assms(1) show ?case .
+next
+  case (Step s e)
+  show ?case
+  proof(rule assms(2))
+    show "valid_trace_e s e" using Step by (unfold_locales, auto)
+  next
+    show "PP s" using Step by simp
+  next
+    show "PIP s e" using Step by simp
+  qed
+qed
+
+end
+
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+  by  (unfold s_waiting_def cs_waiting_def wq_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def wq_def cs_holding_def, simp)
+
+lemma runing_ready: 
+  shows "runing s \<subseteq> readys s"
+  unfolding runing_def readys_def
+  by auto 
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+  unfolding readys_def
+  by auto
+
+lemma wq_v_neq [simp]:
+   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
+  by (auto simp:wq_def Let_def cp_def split:list.splits)
+
+lemma runing_head:
+  assumes "th \<in> runing s"
+  and "th \<in> set (wq_fun (schs s) cs)"
+  shows "th = hd (wq_fun (schs s) cs)"
+  using assms
+  by (simp add:runing_def readys_def s_waiting_def wq_def)
+
+context valid_trace
+begin
+
+lemma runing_wqE:
+  assumes "th \<in> runing s"
+  and "th \<in> set (wq s cs)"
+  obtains rest where "wq s cs = th#rest"
+proof -
+  from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
+    by (meson list.set_cases)
+  have "th' = th"
+  proof(rule ccontr)
+    assume "th' \<noteq> th"
+    hence "th \<noteq> hd (wq s cs)" using eq_wq by auto 
+    with assms(2)
+    have "waiting s th cs" 
+      by (unfold s_waiting_def, fold wq_def, auto)
+    with assms show False 
+      by (unfold runing_def readys_def, auto)
+  qed
+  with eq_wq that show ?thesis by metis
+qed
+
+end
+
+context valid_trace_p
+begin
+
+lemma wq_neq_simp [simp]:
+  assumes "cs' \<noteq> cs"
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_p wq_def
+  by (auto simp:Let_def)
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_p]
+  show ?thesis by (cases, simp)
+qed
+
+lemma th_not_waiting: 
+  "\<not> waiting s th c"
+proof -
+  have "th \<in> readys s"
+    using runing_ready runing_th_s by blast 
+  thus ?thesis
+    by (unfold readys_def, auto)
+qed
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+lemma th_not_in_wq: 
+  shows "th \<notin> set (wq s cs)"
+proof
+  assume otherwise: "th \<in> set (wq s cs)"
+  from runing_wqE[OF runing_th_s this]
+  obtain rest where eq_wq: "wq s cs = th#rest" by blast
+  with otherwise
+  have "holding s th cs"
+    by (unfold s_holding_def, fold wq_def, simp)
+  hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
+    by (unfold s_RAG_def, fold holding_eq, auto)
+  from pip_e[unfolded is_p]
+  show False
+  proof(cases)
+    case (thread_P)
+    with cs_th_RAG show ?thesis by auto
+  qed
+qed
+
+lemma wq_es_cs: 
+  "wq (e#s) cs =  wq s cs @ [th]"
+  by (unfold is_p wq_def, auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+proof(cases "cs' = cs")
+  case True
+  show ?thesis using True assms th_not_in_wq
+    by (unfold True wq_es_cs, auto)
+qed (insert assms, simp)
+
+end
+
+
+context valid_trace_v
+begin
+
+lemma wq_neq_simp [simp]:
+  assumes "cs' \<noteq> cs"
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_v wq_def
+  by (auto simp:Let_def)
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis by (cases, simp)
+qed
+
+lemma th_not_waiting: 
+  "\<not> waiting s th c"
+proof -
+  have "th \<in> readys s"
+    using runing_ready runing_th_s by blast 
+  thus ?thesis
+    by (unfold readys_def, auto)
+qed
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+lemma wq_s_cs:
+  "wq s cs = th#rest"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis
+  proof(cases)
+    case (thread_V)
+    from this(2) show ?thesis
+      by (unfold rest_def s_holding_def, fold wq_def,
+                 metis empty_iff list.collapse list.set(1))
+  qed
+qed
+
+lemma wq_es_cs:
+  "wq (e#s) cs = wq'"
+ using wq_s_cs[unfolded wq_def]
+ by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) 
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+proof(cases "cs' = cs")
+  case True
+  show ?thesis
+  proof(unfold True wq_es_cs wq'_def, rule someI2)
+    show "distinct rest \<and> set rest = set rest"
+        using assms[unfolded True wq_s_cs] by auto
+  qed simp
+qed (insert assms, simp)
+
+end
+
+context valid_trace
+begin
+
+lemma actor_inv: 
+  assumes "PIP s e"
+  and "\<not> isCreate e"
+  shows "actor e \<in> runing s"
+  using assms
+  by (induct, auto)
+
+lemma isP_E:
+  assumes "isP e"
+  obtains cs where "e = P (actor e) cs"
+  using assms by (cases e, auto)
+
+lemma isV_E:
+  assumes "isV e"
+  obtains cs where "e = V (actor e) cs"
+  using assms by (cases e, auto) 
+
+lemma wq_distinct: "distinct (wq s cs)"
+proof(induct rule:ind)
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case 
+  proof(cases e)
+    case (V th cs)
+    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) 
+  qed
+qed (unfold wq_def Let_def, simp)
+
+end
+
+context valid_trace_e
+begin
+
+text {*
+  The following lemma shows that only the @{text "P"}
+  operation can add new thread into waiting queues. 
+  Such kind of lemmas are very obvious, but need to be checked formally.
+  This is a kind of confirmation that our modelling is correct.
+*}
+
+lemma wq_in_inv: 
+  assumes s_ni: "thread \<notin> set (wq s cs)"
+  and s_i: "thread \<in> set (wq (e#s) cs)"
+  shows "e = P thread cs"
+proof(cases e)
+  -- {* This is the only non-trivial case: *}
+  case (V th cs1)
+  have False
+  proof(cases "cs1 = cs")
+    case True
+    show ?thesis
+    proof(cases "(wq s cs1)")
+      case (Cons w_hd w_tl)
+      have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
+      proof -
+        have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
+          using  Cons V by (auto simp:wq_def Let_def True split:if_splits)
+        moreover have "set ... \<subseteq> set (wq s cs)"
+        proof(rule someI2)
+          show "distinct w_tl \<and> set w_tl = set w_tl"
+            by (metis distinct.simps(2) local.Cons wq_distinct)
+        qed (insert Cons True, auto)
+        ultimately show ?thesis by simp
+      qed
+      with assms show ?thesis by auto
+    qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
+  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+  thus ?thesis by auto
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+
+lemma wq_out_inv: 
+  assumes s_in: "thread \<in> set (wq s cs)"
+  and s_hd: "thread = hd (wq s cs)"
+  and s_i: "thread \<noteq> hd (wq (e#s) cs)"
+  shows "e = V thread cs"
+proof(cases e)
+-- {* There are only two non-trivial cases: *}
+  case (V th cs1)
+  show ?thesis
+  proof(cases "cs1 = cs")
+    case True
+    have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
+    thus ?thesis
+    proof(cases)
+      case (thread_V)
+      moreover have "th = thread" using thread_V(2) s_hd
+          by (unfold s_holding_def wq_def, simp)
+      ultimately show ?thesis using V True by simp
+    qed
+  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+next
+  case (P th cs1)
+  show ?thesis
+  proof(cases "cs1 = cs")
+    case True
+    with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
+      by (auto simp:wq_def Let_def split:if_splits)
+    with s_i s_hd s_in have False
+      by (metis empty_iff hd_append2 list.set(1) wq_def) 
+    thus ?thesis by simp
+  qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+
+end
+
+
+
+context valid_trace
+begin
+
+
+text {* (* ddd *)
+  The nature of the work is like this: since it starts from a very simple and basic 
+  model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
+  For instance, the fact 
+  that one thread can not be blocked by two critical resources at the same time
+  is obvious, because only running threads can make new requests, if one is waiting for 
+  a critical resource and get blocked, it can not make another resource request and get 
+  blocked the second time (because it is not running). 
+
+  To derive this fact, one needs to prove by contraction and 
+  reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
+  named @{text "p_split"}, which is about status changing along the time axis. It says if 
+  a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
+  but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} 
+  in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history 
+  of events leading to it), such that @{text "Q"} switched 
+  from being @{text "False"} to @{text "True"} and kept being @{text "True"}
+  till the last moment of @{text "s"}.
+
+  Suppose a thread @{text "th"} is blocked
+  on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, 
+  since no thread is blocked at the very beginning, by applying 
+  @{text "p_split"} to these two blocking facts, there exist 
+  two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that 
+  @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} 
+  and kept on blocked on them respectively ever since.
+ 
+  Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
+  However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
+  in blocked state at moment @{text "t2"} and could not
+  make any request and get blocked the second time: Contradiction.
+*}
+
+lemma waiting_unique_pre: (* ddd *)
+  assumes h11: "thread \<in> set (wq s cs1)"
+  and h12: "thread \<noteq> hd (wq s cs1)"
+  assumes h21: "thread \<in> set (wq s cs2)"
+  and h22: "thread \<noteq> hd (wq s cs2)"
+  and neq12: "cs1 \<noteq> cs2"
+  shows "False"
+proof -
+  let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  from h11 and h12 have q1: "?Q cs1 s" by simp
+  from h21 and h22 have q2: "?Q cs2 s" by simp
+  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+  from p_split [of "?Q cs1", OF q1 nq1]
+  obtain t1 where lt1: "t1 < length s"
+    and np1: "\<not> ?Q cs1 (moment t1 s)"
+    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
+  from p_split [of "?Q cs2", OF q2 nq2]
+  obtain t2 where lt2: "t2 < length s"
+    and np2: "\<not> ?Q cs2 (moment t2 s)"
+    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
+  { fix s cs
+    assume q: "?Q cs s"
+    have "thread \<notin> runing s"
+    proof
+      assume "thread \<in> runing s"
+      hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and> 
+                 thread \<noteq> hd (wq_fun (schs s) cs))"
+        by (unfold runing_def s_waiting_def readys_def, auto)
+      from this[rule_format, of cs] q 
+      show False by (simp add: wq_def) 
+    qed
+  } note q_not_runing = this
+  { fix t1 t2 cs1 cs2
+    assume  lt1: "t1 < length s"
+    and np1: "\<not> ?Q cs1 (moment t1 s)"
+    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
+    and lt2: "t2 < length s"
+    and np2: "\<not> ?Q cs2 (moment t2 s)"
+    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
+    and lt12: "t1 < t2"
+    let ?t3 = "Suc t2"
+    from lt2 have le_t3: "?t3 \<le> length s" by auto
+    from moment_plus [OF this] 
+    obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+    have "t2 < ?t3" by simp
+    from nn2 [rule_format, OF this] and eq_m
+    have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+         h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+    have "vt (e#moment t2 s)"
+    proof -
+      from vt_moment 
+      have "vt (moment ?t3 s)" .
+      with eq_m show ?thesis by simp
+    qed
+    then interpret vt_e: valid_trace_e "moment t2 s" "e"
+        by (unfold_locales, auto, cases, simp)
+    have ?thesis
+    proof -
+      have "thread \<in> runing (moment t2 s)"
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        have "e = V thread cs2"
+        proof -
+          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
+              using True and np2  by auto 
+          from vt_e.wq_out_inv[OF True this h2]
+          show ?thesis .
+        qed
+        thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto
+      next
+        case False
+        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+        with vt_e.actor_inv[OF vt_e.pip_e]
+        show ?thesis by auto
+      qed
+      moreover have "thread \<notin> runing (moment t2 s)"
+        by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
+      ultimately show ?thesis by simp
+    qed
+  } note lt_case = this
+  show ?thesis
+  proof -
+    { assume "t1 < t2"
+      from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
+      have ?thesis .
+    } moreover {
+      assume "t2 < t1"
+      from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
+      have ?thesis .
+    } moreover {
+      assume eq_12: "t1 = t2"
+      let ?t3 = "Suc t2"
+      from lt2 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+      have lt_2: "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+           h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+      from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12]
+      have g1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+           g2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have "vt (e#moment t2 s)"
+      proof -
+        from vt_moment 
+        have "vt (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      then interpret vt_e: valid_trace_e "moment t2 s" "e"
+          by (unfold_locales, auto, cases, simp)
+      have "e = V thread cs2 \<or> e = P thread cs2"
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        have "e = V thread cs2"
+        proof -
+          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
+              using True and np2  by auto 
+          from vt_e.wq_out_inv[OF True this h2]
+          show ?thesis .
+        qed
+        thus ?thesis by auto
+      next
+        case False
+        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+        thus ?thesis by auto
+      qed
+      moreover have "e = V thread cs1 \<or> e = P thread cs1"
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        have eq_th: "thread = hd (wq (moment t1 s) cs1)" 
+              using True and np1  by auto 
+        from vt_e.wq_out_inv[folded eq_12, OF True this g2]
+        have "e = V thread cs1" .
+        thus ?thesis by auto
+      next
+        case False
+        have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
+        thus ?thesis by auto
+      qed
+      ultimately have ?thesis using neq12 by auto
+    } ultimately show ?thesis using nat_neq_iff by blast 
+  qed
+qed
+
+text {*
+  This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
+*}
+
+lemma waiting_unique:
+  assumes "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+  using waiting_unique_pre assms
+  unfolding wq_def s_waiting_def
+  by auto
+
+end
+
+(* not used *)
+text {*
+  Every thread can only be blocked on one critical resource, 
+  symmetrically, every critical resource can only be held by one thread. 
+  This fact is much more easier according to our definition. 
+*}
+lemma held_unique:
+  assumes "holding (s::event list) th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+ by (insert assms, unfold s_holding_def, auto)
+
+
+lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits)
+
+lemma last_set_unique: 
+  "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
+          \<Longrightarrow> th1 = th2"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits dest:last_set_lt)
+
+lemma preced_unique : 
+  assumes pcd_eq: "preced th1 s = preced th2 s"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "th1 = th2"
+proof -
+  from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
+  from last_set_unique [OF this th_in1 th_in2]
+  show ?thesis .
+qed
+                      
+lemma preced_linorder: 
+  assumes neq_12: "th1 \<noteq> th2"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
+proof -
+  from preced_unique [OF _ th_in1 th_in2] and neq_12 
+  have "preced th1 s \<noteq> preced th2 s" by auto
+  thus ?thesis by auto
+qed
+
+(* An aux lemma used later *) 
+lemma unique_minus:
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r"
+  and xz: "(x, z) \<in> r^+"
+  and neq: "y \<noteq> z"
+  shows "(y, z) \<in> r^+"
+proof -
+ from xz and neq show ?thesis
+ proof(induct)
+   case (base ya)
+   have "(x, ya) \<in> r" by fact
+   from unique [OF xy this] have "y = ya" .
+   with base show ?case by auto
+ next
+   case (step ya z)
+   show ?case
+   proof(cases "y = ya")
+     case True
+     from step True show ?thesis by simp
+   next
+     case False
+     from step False
+     show ?thesis by auto
+   qed
+ qed
+qed
+
+lemma unique_base:
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r"
+  and xz: "(x, z) \<in> r^+"
+  and neq_yz: "y \<noteq> z"
+  shows "(y, z) \<in> r^+"
+proof -
+  from xz neq_yz show ?thesis
+  proof(induct)
+    case (base ya)
+    from xy unique base show ?case by auto
+  next
+    case (step ya z)
+    show ?case
+    proof(cases "y = ya")
+      case True
+      from True step show ?thesis by auto
+    next
+      case False
+      from False step 
+      have "(y, ya) \<in> r\<^sup>+" by auto
+      with step show ?thesis by auto
+    qed
+  qed
+qed
+
+lemma unique_chain:
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r^+"
+  and xz: "(x, z) \<in> r^+"
+  and neq_yz: "y \<noteq> z"
+  shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
+proof -
+  from xy xz neq_yz show ?thesis
+  proof(induct)
+    case (base y)
+    have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
+    from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
+  next
+    case (step y za)
+    show ?case
+    proof(cases "y = z")
+      case True
+      from True step show ?thesis by auto
+    next
+      case False
+      from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
+      thus ?thesis
+      proof
+        assume "(z, y) \<in> r\<^sup>+"
+        with step have "(z, za) \<in> r\<^sup>+" by auto
+        thus ?thesis by auto
+      next
+        assume h: "(y, z) \<in> r\<^sup>+"
+        from step have yza: "(y, za) \<in> r" by simp
+        from step have "za \<noteq> z" by simp
+        from unique_minus [OF _ yza h this] and unique
+        have "(za, z) \<in> r\<^sup>+" by auto
+        thus ?thesis by auto
+      qed
+    qed
+  qed
+qed
+
+text {*
+  The following three lemmas show that @{text "RAG"} does not change
+  by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
+  events, respectively.
+*}
+
+lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+
+context valid_trace_v
+begin
+
+
+lemma distinct_rest: "distinct rest"
+  by (simp add: distinct_tl rest_def wq_distinct)
+
+definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis by (cases, simp)
+qed
+
+lemma holding_cs_eq_th:
+  assumes "holding s t cs"
+  shows "t = th"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis
+  proof(cases)
+    case (thread_V)
+    from held_unique[OF this(2) assms]
+    show ?thesis by simp
+  qed
+qed
+
+lemma th_not_waiting: 
+  "\<not> waiting s th c"
+proof -
+  have "th \<in> readys s"
+    using runing_ready runing_th_s by blast 
+  thus ?thesis
+    by (unfold readys_def, auto)
+qed
+
+lemma waiting_neq_th: 
+  assumes "waiting s t c"
+  shows "t \<noteq> th"
+  using assms using th_not_waiting by blast 
+
+lemma wq_s_cs:
+  "wq s cs = th#rest"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis
+  proof(cases)
+    case (thread_V)
+    from this(2) show ?thesis
+      by (unfold rest_def s_holding_def, fold wq_def,
+                 metis empty_iff list.collapse list.set(1))
+  qed
+qed
+
+lemma wq_es_cs:
+  "wq (e#s) cs = wq'"
+ using wq_s_cs[unfolded wq_def]
+ by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) 
+
+lemma distinct_wq': "distinct wq'"
+  by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
+  
+lemma th'_in_inv:
+  assumes "th' \<in> set wq'"
+  shows "th' \<in> set rest"
+  using assms
+  by (metis (mono_tags, lifting) distinct.simps(2) 
+        rest_def some_eq_ex wq'_def wq_distinct wq_s_cs) 
+
+lemma neq_t_th: 
+  assumes "waiting (e#s) t c"
+  shows "t \<noteq> th"
+proof
+  assume otherwise: "t = th"
+  show False
+  proof(cases "c = cs")
+    case True
+    have "t \<in> set wq'" 
+     using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
+     by simp 
+    from th'_in_inv[OF this] have "t \<in> set rest" .
+    with wq_s_cs[folded otherwise] wq_distinct[of cs]
+    show ?thesis by simp
+  next
+    case False
+    have "wq (e#s) c = wq s c" using False
+        by (unfold is_v, simp)
+    hence "waiting s t c" using assms 
+        by (simp add: cs_waiting_def waiting_eq)
+    hence "t \<notin> readys s" by (unfold readys_def, auto)
+    hence "t \<notin> runing s" using runing_ready by auto 
+    with runing_th_s[folded otherwise] show ?thesis by auto
+  qed
+qed
+
+lemma waiting_esI1:
+  assumes "waiting s t c"
+      and "c \<noteq> cs" 
+  shows "waiting (e#s) t c" 
+proof -
+  have "wq (e#s) c = wq s c" 
+    using assms(2) is_v by auto
+  with assms(1) show ?thesis 
+    using cs_waiting_def waiting_eq by auto 
+qed
+
+lemma holding_esI2:
+  assumes "c \<noteq> cs" 
+  and "holding s t c"
+  shows "holding (e#s) t c"
+proof -
+  from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
+  from assms(2)[unfolded s_holding_def, folded wq_def, 
+                folded this, unfolded wq_def, folded s_holding_def]
+  show ?thesis .
+qed
+
+lemma holding_esI1:
+  assumes "holding s t c"
+  and "t \<noteq> th"
+  shows "holding (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms using holding_cs_eq_th by blast 
+  from holding_esI2[OF this assms(1)]
+  show ?thesis .
+qed
+
+end
+
+context valid_trace_v_n
+begin
+
+lemma neq_wq': "wq' \<noteq> []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x \<noteq> []" using rest_nnl by auto
+qed 
+
+definition "taker = hd wq'"
+
+definition "rest' = tl wq'"
+
+lemma eq_wq': "wq' = taker # rest'"
+  by (simp add: neq_wq' rest'_def taker_def)
+
+lemma next_th_taker: 
+  shows "next_th s th cs taker"
+  using rest_nnl taker_def wq'_def wq_s_cs 
+  by (auto simp:next_th_def)
+
+lemma taker_unique: 
+  assumes "next_th s th cs taker'"
+  shows "taker' = taker"
+proof -
+  from assms
+  obtain rest' where 
+    h: "wq s cs = th # rest'" 
+       "taker' = hd (SOME q. distinct q \<and> set q = set rest')"
+          by (unfold next_th_def, auto)
+  with wq_s_cs have "rest' = rest" by auto
+  thus ?thesis using h(2) taker_def wq'_def by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
+  by (smt all_not_in_conv bot.extremum insertI1 insert_subset 
+      mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {(Cs cs, Th taker)}"
+  using next_th_taker taker_def waiting_set_eq 
+  by fastforce
+   
+lemma holding_taker:
+  shows "holding (e#s) taker cs"
+    by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, 
+        auto simp:neq_wq' taker_def)
+
+lemma waiting_esI2:
+  assumes "waiting s t cs"
+      and "t \<noteq> taker"
+  shows "waiting (e#s) t cs" 
+proof -
+  have "t \<in> set wq'" 
+  proof(unfold wq'_def, rule someI2)
+    show "distinct rest \<and> set rest = set rest"
+          by (simp add: distinct_rest)
+  next
+    fix x
+    assume "distinct x \<and> set x = set rest"
+    moreover have "t \<in> set rest"
+        using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto 
+    ultimately show "t \<in> set x" by simp
+  qed
+  moreover have "t \<noteq> hd wq'"
+    using assms(2) taker_def by auto 
+  ultimately show ?thesis
+    by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+     |    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
+  have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
+  hence "t \<noteq> taker" by (simp add: taker_def) 
+  moreover hence "t \<noteq> th" using assms neq_t_th by blast 
+  moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv) 
+  ultimately have "waiting s t cs"
+    by (metis cs_waiting_def list.distinct(2) list.sel(1) 
+                list.set_sel(2) rest_def waiting_eq wq_s_cs)  
+  show ?thesis using that(2)
+  using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto   
+qed
+
+lemma holding_esI1:
+  assumes "c = cs"
+  and "t = taker"
+  shows "holding (e#s) t c"
+  by (unfold assms, simp add: holding_taker)
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c = cs" "t = taker"
+      | "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from assms[unfolded True, unfolded s_holding_def, 
+             folded wq_def, unfolded wq_es_cs]
+  have "t = taker" by (simp add: taker_def) 
+  from that(1)[OF True this] show ?thesis .
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that(2)[OF False this] show ?thesis .
+qed
+
+end 
+
+
+context valid_trace_v_n
+begin
+
+lemma nil_wq': "wq' = []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x = []" using rest_nil by auto
+qed 
+
+lemma no_taker: 
+  assumes "next_th s th cs taker"
+  shows "False"
+proof -
+  from assms[unfolded next_th_def]
+  obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
+    by auto
+  thus ?thesis using rest_def rest_nil by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
+  using no_taker by auto
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
+  using no_taker by auto
+   
+lemma no_holding:
+  assumes "holding (e#s) taker cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_holding_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma no_waiting:
+  assumes "waiting (e#s) t cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma waiting_esI2:
+  assumes "waiting s t c"
+  shows "waiting (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms
+    using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto 
+  from waiting_esI1[OF assms this]
+  show ?thesis .
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from no_waiting[OF assms[unfolded True]]
+  show ?thesis by auto
+qed
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from no_holding[OF assms[unfolded True]] 
+  show ?thesis by auto
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that[OF False this] show ?thesis .
+qed
+
+end (* ccc *)
+
+lemma rel_eqI:
+  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
+  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
+  shows "A = B"
+  using assms by auto
+
+lemma in_RAG_E:
+  assumes "(n1, n2) \<in> RAG (s::state)"
+  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
+      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
+  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
+  by auto
+  
+context valid_trace_v
+begin
+
+lemma RAG_es:
+  "RAG (e # s) =
+   RAG s - {(Cs cs, Th th)} -
+     {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+     {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R"
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_n.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      next
+        case 2
+        with waiting(1,2)
+        show ?thesis
+         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      qed
+    next
+      case True
+      interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_e.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      qed
+    qed
+  next
+    case (holding th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from holding(3)
+      show ?thesis
+      proof(cases rule:h_n.holding_esE)
+        case 1
+        with holding(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      next
+        case 2
+        with holding(1,2)
+        show ?thesis
+         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold holding_eq, auto)
+      qed
+    next
+      case True
+      interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+      from holding(3)
+      show ?thesis
+      proof(cases rule:h_e.holding_esE)
+        case 1
+        with holding(1,2)
+        show ?thesis
+        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
+             fold holding_eq, auto)
+      qed
+    qed
+  qed
+next
+  fix n1 n2
+  assume h: "(n1, n2) \<in> ?R"
+  show "(n1, n2) \<in> ?L"
+  proof(cases "rest = []")
+    case False
+    interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+    from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
+    have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
+                            \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or> 
+          (n2 = Th h_n.taker \<and> n1 = Cs cs)" 
+      by auto
+   thus ?thesis
+   proof
+      assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
+      with h_n.holding_taker
+      show ?thesis 
+        by (unfold s_RAG_def, fold holding_eq, auto)
+   next
+    assume h: "(n1, n2) \<in> RAG s \<and>
+        (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
+    hence "(n1, n2) \<in> RAG s" by simp
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from h and this(1,2)
+      have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
+      hence "waiting (e#s) th' cs'" 
+      proof
+        assume "cs' \<noteq> cs"
+        from waiting_esI1[OF waiting(3) this] 
+        show ?thesis .
+      next
+        assume neq_th': "th' \<noteq> h_n.taker"
+        show ?thesis
+        proof(cases "cs' = cs")
+          case False
+          from waiting_esI1[OF waiting(3) this] 
+          show ?thesis .
+        next
+          case True
+          from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
+          show ?thesis .
+        qed
+      qed
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from h this(1,2)
+      have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+      hence "holding (e#s) th' cs'"
+      proof
+        assume "cs' \<noteq> cs"
+        from holding_esI2[OF this holding(3)] 
+        show ?thesis .
+      next
+        assume "th' \<noteq> th"
+        from holding_esI1[OF holding(3) this]
+        show ?thesis .
+      qed
+      thus ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+   qed
+ next
+   case True
+   interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+   from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
+   have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)" 
+      by auto
+   from h_s(1)
+   show ?thesis
+   proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from h_e.waiting_esI2[OF this(3)]
+    show ?thesis using waiting(1,2)
+      by (unfold s_RAG_def, fold waiting_eq, auto)
+   next
+    case (holding th' cs')
+    with h_s(2)
+    have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+    thus ?thesis
+    proof
+      assume neq_cs: "cs' \<noteq> cs"
+      from holding_esI2[OF this holding(3)]
+      show ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    next
+      assume "th' \<noteq> th"
+      from holding_esI1[OF holding(3) this]
+      show ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+   qed
+ qed
+qed
+
+end
+
+
+
+context valid_trace
+begin
+
+lemma finite_threads:
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+apply(simp add: Let_def cpreced_initial)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, auto)
+
+lemma wq_threads: 
+  assumes h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+
+
+lemma wq_threads: 
+  assumes h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+proof -
+ from vt and h show ?thesis
+  proof(induct arbitrary: th cs)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s
+      using vt_cons(1) by (unfold_locales, auto)
+    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+      and stp: "step s e"
+      and vt: "vt s"
+      and h: "th \<in> set (wq (e # s) cs)"
+    show ?case
+    proof(cases e)
+      case (Create th' prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    next
+      case (Exit th')
+      with stp ih h show ?thesis
+        apply (auto simp:wq_def Let_def)
+        apply (ind_cases "step s (Exit th')")
+        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
+               s_RAG_def s_holding_def cs_holding_def)
+        done
+    next
+      case (V th' cs')
+      show ?thesis
+      proof(cases "cs' = cs")
+        case False
+        with h
+        show ?thesis
+          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+          by (drule_tac ih, simp)
+      next
+        case True
+        from h
+        show ?thesis
+        proof(unfold V wq_def)
+          assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+          show "th \<in> threads (V th' cs' # s)"
+          proof(cases "cs = cs'")
+            case False
+            hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
+            with th_in have " th \<in> set (wq s cs)" 
+              by (fold wq_def, simp)
+            from ih [OF this] show ?thesis by simp
+          next
+            case True
+            show ?thesis
+            proof(cases "wq_fun (schs s) cs'")
+              case Nil
+              with h V show ?thesis
+                apply (auto simp:wq_def Let_def split:if_splits)
+                by (fold wq_def, drule_tac ih, simp)
+            next
+              case (Cons a rest)
+              assume eq_wq: "wq_fun (schs s) cs' = a # rest"
+              with h V show ?thesis
+                apply (auto simp:Let_def wq_def split:if_splits)
+              proof -
+                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
+                proof(rule someI2)
+                  from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
+                  show "distinct rest \<and> set rest = set rest" by auto
+                next
+                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+                    by auto
+                qed
+                with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
+                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+              next
+                assume th_in: "th \<in> set (wq_fun (schs s) cs)"
+                from ih[OF this[folded wq_def]]
+                show "th \<in> threads s" .
+              qed
+            qed
+          qed
+        qed
+      qed
+    next
+      case (P th' cs')
+      from h stp
+      show ?thesis
+        apply (unfold P wq_def)
+        apply (auto simp:Let_def split:if_splits, fold wq_def)
+        apply (auto intro:ih)
+        apply(ind_cases "step s (P th' cs')")
+        by (unfold runing_def readys_def, auto)
+    next
+      case (Set thread prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    qed
+  next
+    case vt_nil
+    thus ?case by (auto simp:wq_def)
+  qed
+qed
+
+lemma dm_RAG_threads:
+  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
+  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
+  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
+  hence "th \<in> set (wq s cs)"
+    by (unfold s_RAG_def, auto simp:cs_waiting_def)
+  from wq_threads [OF this] show ?thesis .
+qed
+
+
+lemma cp_le:
+  assumes th_in: "th \<in> threads s"
+  shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
+  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
+         \<le> Max ((\<lambda>th. preced th s) ` threads s)"
+    (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
+  proof(rule Max_f_mono)
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
+  next
+    from finite_threads
+    show "finite (threads s)" .
+  next
+    from th_in
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
+      apply (auto simp:Domain_def)
+      apply (rule_tac dm_RAG_threads)
+      apply (unfold trancl_domain [of "RAG s", symmetric])
+      by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
+  qed
+qed
+
+lemma le_cp:
+  shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+  show "Prc (priority th s) (last_set th s)
+    \<le> Max (insert (Prc (priority th s) (last_set th s))
+            ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
+    (is "?l \<le> Max (insert ?l ?A)")
+  proof(cases "?A = {}")
+    case False
+    have "finite ?A" (is "finite (?f ` ?B)")
+    proof -
+      have "finite ?B" 
+      proof-
+        have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
+        proof -
+          let ?F = "\<lambda> (x, y). the_th x"
+          have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+            apply (auto simp:image_def)
+            by (rule_tac x = "(Th x, Th th)" in bexI, auto)
+          moreover have "finite \<dots>"
+          proof -
+            from finite_RAG have "finite (RAG s)" .
+            hence "finite ((RAG (wq s))\<^sup>+)"
+              apply (unfold finite_trancl)
+              by (auto simp: s_RAG_def cs_RAG_def wq_def)
+            thus ?thesis by auto
+          qed
+          ultimately show ?thesis by (auto intro:finite_subset)
+        qed
+        thus ?thesis by (simp add:cs_dependants_def)
+      qed
+      thus ?thesis by simp
+    qed
+    from Max_insert [OF this False, of ?l] show ?thesis by auto
+  next
+    case True
+    thus ?thesis by auto
+  qed
+qed
+
+lemma max_cp_eq: 
+  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
+  (is "?l = ?r")
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  have "?l \<in> ((cp s) ` threads s)"
+  proof(rule Max_in)
+    from finite_threads
+    show "finite (cp s ` threads s)" by auto
+  next
+    from False show "cp s ` threads s \<noteq> {}" by auto
+  qed
+  then obtain th 
+    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
+  have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
+  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
+  proof -
+    have "?r \<in> (?f ` ?A)"
+    proof(rule Max_in)
+      from finite_threads
+      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
+    next
+      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
+    qed
+    then obtain th' where 
+      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
+    from le_cp [of th']  eq_r
+    have "?r \<le> cp s th'" by auto
+    moreover have "\<dots> \<le> cp s th"
+    proof(fold eq_l)
+      show " cp s th' \<le> Max (cp s ` threads s)"
+      proof(rule Max_ge)
+        from th_in' show "cp s th' \<in> cp s ` threads s"
+          by auto
+      next
+        from finite_threads
+        show "finite (cp s ` threads s)" by auto
+      qed
+    qed
+    ultimately show ?thesis by auto
+  qed
+  ultimately show ?thesis using eq_l by auto
+qed
+
+lemma max_cp_eq_the_preced:
+  shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
+  using max_cp_eq using the_preced_def by presburger 
+
+end
+
+lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
+  by (unfold preced_def, simp)
+
+lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
+proof
+  fix th'
+  show "the_preced (V th cs # s) th' = the_preced s th'"
+    by (unfold the_preced_def preced_def, simp)
+qed
+
+lemma step_RAG_v: 
+assumes vt:
+  "vt (V th cs#s)"
+shows "
+  RAG (V th cs # s) =
+  RAG s - {(Cs cs, Th th)} -
+  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+  {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof -
+  interpret vt_v: valid_trace_v s "V th cs"
+    using assms step_back_vt by (unfold_locales, auto) 
+  show ?thesis using vt_v.RAG_es .
+qed
+
+
+
+
+
+text {* (* ddd *) 
+  The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
+  with the happening of @{text "V"}-events:
+*}
+lemma step_RAG_v:
+assumes vt:
+  "vt (V th cs#s)"
+shows "
+  RAG (V th cs # s) =
+  RAG s - {(Cs cs, Th th)} -
+  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+  {(Cs cs, Th th') |th'.  next_th s th cs th'}"
+  apply (insert vt, unfold s_RAG_def) 
+  apply (auto split:if_splits list.splits simp:Let_def)
+  apply (auto elim: step_v_waiting_mono step_v_hold_inv
+              step_v_release step_v_wait_inv
+              step_v_get_hold step_v_release_inv)
+  apply (erule_tac step_v_not_wait, auto)
+  done
+
+text {* 
+  The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
+  with the happening of @{text "P"}-events:
+*}
+lemma step_RAG_p:
+  "vt (P th cs#s) \<Longrightarrow>
+  RAG (P th cs # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+                                             else RAG s \<union> {(Th th, Cs cs)})"
+  apply(simp only: s_RAG_def wq_def)
+  apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
+  apply(case_tac "csa = cs", auto)
+  apply(fold wq_def)
+  apply(drule_tac step_back_step)
+  apply(ind_cases " step s (P (hd (wq s cs)) cs)")
+  apply(simp add:s_RAG_def wq_def cs_holding_def)
+  apply(auto)
+  done
+
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, auto)
+
+context valid_trace
+begin
+
+text {*
+  The following lemma shows that @{text "RAG"} is acyclic.
+  The overall structure is by induction on the formation of @{text "vt s"}
+  and then case analysis on event @{text "e"}, where the non-trivial cases 
+  for those for @{text "V"} and @{text "P"} events.
+*}
+lemma acyclic_RAG:
+  shows "acyclic (RAG s)"
+using vt
+proof(induct)
+  case (vt_cons s e)
+  interpret vt_s: valid_trace s using vt_cons(1)
+    by (unfold_locales, simp)
+  assume ih: "acyclic (RAG s)"
+    and stp: "step s e"
+    and vt: "vt s"
+  show ?case
+  proof(cases e)
+    case (Create th prio)
+    with ih
+    show ?thesis by (simp add:RAG_create_unchanged)
+  next
+    case (Exit th)
+    with ih show ?thesis by (simp add:RAG_exit_unchanged)
+  next
+    case (V th cs)
+    from V vt stp have vtt: "vt (V th cs#s)" by auto
+    from step_RAG_v [OF this]
+    have eq_de: 
+      "RAG (e # s) = 
+      RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+      {(Cs cs, Th th') |th'. next_th s th cs th'}"
+      (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+    from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+    from step_back_step [OF vtt]
+    have "step s (V th cs)" .
+    thus ?thesis
+    proof(cases)
+      assume "holding s th cs"
+      hence th_in: "th \<in> set (wq s cs)" and
+        eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
+      then obtain rest where
+        eq_wq: "wq s cs = th#rest"
+        by (cases "wq s cs", auto)
+      show ?thesis
+      proof(cases "rest = []")
+        case False
+        let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+        from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
+          by (unfold next_th_def, auto)
+        let ?E = "(?A - ?B - ?C)"
+        have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
+        proof
+          assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
+          hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+          from tranclD [OF this]
+          obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
+          hence th_d: "(Th ?th', x) \<in> ?A" by simp
+          from RAG_target_th [OF this]
+          obtain cs' where eq_x: "x = Cs cs'" by auto
+          with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
+          hence wt_th': "waiting s ?th' cs'"
+            unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
+          hence "cs' = cs"
+          proof(rule vt_s.waiting_unique)
+            from eq_wq vt_s.wq_distinct[of cs]
+            show "waiting s ?th' cs" 
+              apply (unfold s_waiting_def wq_def, auto)
+            proof -
+              assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq_fun (schs s) cs = th # rest"
+              have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+              next
+                fix x assume "distinct x \<and> set x = set rest"
+                with False show "x \<noteq> []" by auto
+              qed
+              hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
+                set (SOME q. distinct q \<and> set q = set rest)" by auto
+              moreover have "\<dots> = set rest" 
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+              next
+                show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+              qed
+              moreover note hd_in
+              ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
+            next
+              assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+              have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" by auto
+              next
+                fix x assume "distinct x \<and> set x = set rest"
+                with False show "x \<noteq> []" by auto
+              qed
+              hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
+                set (SOME q. distinct q \<and> set q = set rest)" by auto
+              moreover have "\<dots> = set rest" 
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" by auto
+              next
+                show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+              qed
+              moreover note hd_in
+              ultimately show False by auto
+            qed
+          qed
+          with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
+          with False
+          show "False" by (auto simp: next_th_def eq_wq)
+        qed
+        with acyclic_insert[symmetric] and ac
+          and eq_de eq_D show ?thesis by auto
+      next
+        case True
+        with eq_wq
+        have eq_D: "?D = {}"
+          by (unfold next_th_def, auto)
+        with eq_de ac
+        show ?thesis by auto
+      qed 
+    qed
+  next
+    case (P th cs)
+    from P vt stp have vtt: "vt (P th cs#s)" by auto
+    from step_RAG_p [OF this] P
+    have "RAG (e # s) = 
+      (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else 
+      RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+      by simp
+    moreover have "acyclic ?R"
+    proof(cases "wq s cs = []")
+      case True
+      hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
+      have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
+      proof
+        assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
+        hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+        from tranclD2 [OF this]
+        obtain x where "(x, Cs cs) \<in> RAG s" by auto
+        with True show False by (auto simp:s_RAG_def cs_waiting_def)
+      qed
+      with acyclic_insert ih eq_r show ?thesis by auto
+    next
+      case False
+      hence eq_r: "?R =  RAG s \<union> {(Th th, Cs cs)}" by simp
+      have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
+      proof
+        assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
+        hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+        moreover from step_back_step [OF vtt] have "step s (P th cs)" .
+        ultimately show False
+        proof -
+          show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+            by (ind_cases "step s (P th cs)", simp)
+        qed
+      qed
+      with acyclic_insert ih eq_r show ?thesis by auto
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (Set thread prio)
+      with ih
+      thm RAG_set_unchanged
+      show ?thesis by (simp add:RAG_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "acyclic (RAG ([]::state))"
+      by (auto simp: s_RAG_def cs_waiting_def 
+        cs_holding_def wq_def acyclic_def)
+qed
+
+
+lemma finite_RAG:
+  shows "finite (RAG s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s using vt_cons(1)
+      by (unfold_locales, simp)
+    assume ih: "finite (RAG s)"
+      and stp: "step s e"
+      and vt: "vt s"
+    show ?case
+    proof(cases e)
+      case (Create th prio)
+      with ih
+      show ?thesis by (simp add:RAG_create_unchanged)
+    next
+      case (Exit th)
+      with ih show ?thesis by (simp add:RAG_exit_unchanged)
+    next
+      case (V th cs)
+      from V vt stp have vtt: "vt (V th cs#s)" by auto
+      from step_RAG_v [OF this]
+      have eq_de: "RAG (e # s) = 
+                   RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+                      {(Cs cs, Th th') |th'. next_th s th cs th'}
+"
+        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+      moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
+      moreover have "finite ?D"
+      proof -
+        have "?D = {} \<or> (\<exists> a. ?D = {a})" 
+          by (unfold next_th_def, auto)
+        thus ?thesis
+        proof
+          assume h: "?D = {}"
+          show ?thesis by (unfold h, simp)
+        next
+          assume "\<exists> a. ?D = {a}"
+          thus ?thesis
+            by (metis finite.simps)
+        qed
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (P th cs)
+      from P vt stp have vtt: "vt (P th cs#s)" by auto
+      from step_RAG_p [OF this] P
+      have "RAG (e # s) = 
+              (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else 
+                                    RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+        by simp
+      moreover have "finite ?R"
+      proof(cases "wq s cs = []")
+        case True
+        hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
+        with True and ih show ?thesis by auto
+      next
+        case False
+        hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
+        with False and ih show ?thesis by auto
+      qed
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio)
+      with ih
+      show ?thesis by (simp add:RAG_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "finite (RAG ([]::state))"
+      by (auto simp: s_RAG_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+text {* Several useful lemmas *}
+
+lemma wf_dep_converse: 
+  shows "wf ((RAG s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+  from finite_RAG 
+  show "finite (RAG s)" .
+next
+  from acyclic_RAG
+  show "acyclic (RAG s)" .
+qed
+
+end
+
+lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
+  by (induct l, auto)
+
+lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
+  by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+
+context valid_trace
+begin
+
+lemma wq_threads: 
+  assumes h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+proof -
+ from vt and h show ?thesis
+  proof(induct arbitrary: th cs)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s
+      using vt_cons(1) by (unfold_locales, auto)
+    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+      and stp: "step s e"
+      and vt: "vt s"
+      and h: "th \<in> set (wq (e # s) cs)"
+    show ?case
+    proof(cases e)
+      case (Create th' prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    next
+      case (Exit th')
+      with stp ih h show ?thesis
+        apply (auto simp:wq_def Let_def)
+        apply (ind_cases "step s (Exit th')")
+        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
+               s_RAG_def s_holding_def cs_holding_def)
+        done
+    next
+      case (V th' cs')
+      show ?thesis
+      proof(cases "cs' = cs")
+        case False
+        with h
+        show ?thesis
+          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+          by (drule_tac ih, simp)
+      next
+        case True
+        from h
+        show ?thesis
+        proof(unfold V wq_def)
+          assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+          show "th \<in> threads (V th' cs' # s)"
+          proof(cases "cs = cs'")
+            case False
+            hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
+            with th_in have " th \<in> set (wq s cs)" 
+              by (fold wq_def, simp)
+            from ih [OF this] show ?thesis by simp
+          next
+            case True
+            show ?thesis
+            proof(cases "wq_fun (schs s) cs'")
+              case Nil
+              with h V show ?thesis
+                apply (auto simp:wq_def Let_def split:if_splits)
+                by (fold wq_def, drule_tac ih, simp)
+            next
+              case (Cons a rest)
+              assume eq_wq: "wq_fun (schs s) cs' = a # rest"
+              with h V show ?thesis
+                apply (auto simp:Let_def wq_def split:if_splits)
+              proof -
+                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
+                proof(rule someI2)
+                  from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
+                  show "distinct rest \<and> set rest = set rest" by auto
+                next
+                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+                    by auto
+                qed
+                with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
+                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+              next
+                assume th_in: "th \<in> set (wq_fun (schs s) cs)"
+                from ih[OF this[folded wq_def]]
+                show "th \<in> threads s" .
+              qed
+            qed
+          qed
+        qed
+      qed
+    next
+      case (P th' cs')
+      from h stp
+      show ?thesis
+        apply (unfold P wq_def)
+        apply (auto simp:Let_def split:if_splits, fold wq_def)
+        apply (auto intro:ih)
+        apply(ind_cases "step s (P th' cs')")
+        by (unfold runing_def readys_def, auto)
+    next
+      case (Set thread prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    qed
+  next
+    case vt_nil
+    thus ?case by (auto simp:wq_def)
+  qed
+qed
+
+lemma range_in: "\<lbrakk>(Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+  apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
+  by (auto intro:wq_threads)
+
+lemma readys_v_eq:
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and not_in: "th \<notin>  set rest"
+  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
+proof -
+  from assms show ?thesis
+    apply (auto simp:readys_def)
+    apply(simp add:s_waiting_def[folded wq_def])
+    apply (erule_tac x = csa in allE)
+    apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
+    apply (case_tac "csa = cs", simp)
+    apply (erule_tac x = cs in allE)
+    apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
+    apply(auto simp add: wq_def)
+    apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
+    proof -
+       assume th_nin: "th \<notin> set rest"
+        and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+        and eq_wq: "wq_fun (schs s) cs = thread # rest"
+      have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+      proof(rule someI2)
+        from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
+        show "distinct rest \<and> set rest = set rest" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+      qed
+      with th_nin th_in show False by auto
+    qed
+qed
+
+text {* \noindent
+  The following lemmas shows that: starting from any node in @{text "RAG"}, 
+  by chasing out-going edges, it is always possible to reach a node representing a ready
+  thread. In this lemma, it is the @{text "th'"}.
+*}
+
+lemma chain_building:
+  shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
+proof -
+  from wf_dep_converse
+  have h: "wf ((RAG s)\<inverse>)" .
+  show ?thesis
+  proof(induct rule:wf_induct [OF h])
+    fix x
+    assume ih [rule_format]: 
+      "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow> 
+           y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
+    show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
+    proof
+      assume x_d: "x \<in> Domain (RAG s)"
+      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
+      proof(cases x)
+        case (Th th)
+        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
+        with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
+        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
+        hence "Cs cs \<in> Domain (RAG s)" by auto
+        from ih [OF x_in_r this] obtain th'
+          where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
+        have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
+        with th'_ready show ?thesis by auto
+      next
+        case (Cs cs)
+        from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
+        show ?thesis
+        proof(cases "th' \<in> readys s")
+          case True
+          from True and th'_d show ?thesis by auto
+        next
+          case False
+          from th'_d and range_in  have "th' \<in> threads s" by auto
+          with False have "Th th' \<in> Domain (RAG s)" 
+            by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
+          from ih [OF th'_d this]
+          obtain th'' where 
+            th''_r: "th'' \<in> readys s" and 
+            th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
+          from th'_d and th''_in 
+          have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
+          with th''_r show ?thesis by auto
+        qed
+      qed
+    qed
+  qed
+qed
+
+text {* \noindent
+  The following is just an instance of @{text "chain_building"}.
+*}
+lemma th_chain_to_ready:
+  assumes th_in: "th \<in> threads s"
+  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
+proof(cases "th \<in> readys s")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  from False and th_in have "Th th \<in> Domain (RAG s)" 
+    by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
+  from chain_building [rule_format, OF this]
+  show ?thesis by auto
+qed
+
+end
+
+
+
+lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
+  by (unfold s_holding_def cs_holding_def, auto)
+
+context valid_trace
+begin
+
+lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
+  apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
+  by(auto elim:waiting_unique holding_unique)
+
+end
+
+
+lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
+by (induct rule:trancl_induct, auto)
+
+context valid_trace
+begin
+
+lemma dchain_unique:
+  assumes th1_d: "(n, Th th1) \<in> (RAG s)^+"
+  and th1_r: "th1 \<in> readys s"
+  and th2_d: "(n, Th th2) \<in> (RAG s)^+"
+  and th2_r: "th2 \<in> readys s"
+  shows "th1 = th2"
+proof -
+  { assume neq: "th1 \<noteq> th2"
+    hence "Th th1 \<noteq> Th th2" by simp
+    from unique_chain [OF _ th1_d th2_d this] and unique_RAG 
+    have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
+    hence "False"
+    proof
+      assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+      from dd eq_n have "th1 \<notin> readys s"
+        by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
+      with th1_r show ?thesis by auto
+    next
+      assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+      from dd eq_n have "th2 \<notin> readys s"
+        by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
+      with th2_r show ?thesis by auto
+    qed
+  } thus ?thesis by auto
+qed
+
+end
+             
+
+lemma step_holdents_p_add:
+  assumes vt: "vt (P th cs#s)"
+  and "wq s cs = []"
+  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from assms show ?thesis
+  unfolding  holdents_test step_RAG_p[OF vt] by (auto)
+qed
+
+lemma step_holdents_p_eq:
+  assumes vt: "vt (P th cs#s)"
+  and "wq s cs \<noteq> []"
+  shows "holdents (P th cs#s) th = holdents s th"
+proof -
+  from assms show ?thesis
+  unfolding  holdents_test step_RAG_p[OF vt] by auto
+qed
+
+
+lemma (in valid_trace) finite_holding :
+  shows "finite (holdents s th)"
+proof -
+  let ?F = "\<lambda> (x, y). the_cs x"
+  from finite_RAG 
+  have "finite (RAG s)" .
+  hence "finite (?F `(RAG s))" by simp
+  moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>" 
+  proof -
+    { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
+      fix x assume "(Cs x, Th th) \<in> RAG s"
+      hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
+      moreover have "?F (Cs x, Th th) = x" by simp
+      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp 
+    } thus ?thesis by auto
+  qed
+  ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
+qed
+
+lemma cntCS_v_dec: 
+  assumes vtv: "vt (V thread cs#s)"
+  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
+proof -
+  from vtv interpret vt_s: valid_trace s
+    by (cases, unfold_locales, simp)
+  from vtv interpret vt_v: valid_trace "V thread cs#s"
+     by (unfold_locales, simp)
+  from step_back_step[OF vtv]
+  have cs_in: "cs \<in> holdents s thread" 
+    apply (cases, unfold holdents_test s_RAG_def, simp)
+    by (unfold cs_holding_def s_holding_def wq_def, auto)
+  moreover have cs_not_in: 
+    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
+    apply (insert vt_s.wq_distinct[of cs])
+    apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
+            auto simp:next_th_def)
+  proof -
+    fix rest
+    assume dst: "distinct (rest::thread list)"
+      and ne: "rest \<noteq> []"
+    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+    qed
+    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                     set (SOME q. distinct q \<and> set q = set rest)" by simp
+    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      fix x assume " distinct x \<and> set x = set rest" with ne
+      show "x \<noteq> []" by auto
+    qed
+    ultimately 
+    show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
+      by auto
+  next
+    fix rest
+    assume dst: "distinct (rest::thread list)"
+      and ne: "rest \<noteq> []"
+    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+    qed
+    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                     set (SOME q. distinct q \<and> set q = set rest)" by simp
+    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      fix x assume " distinct x \<and> set x = set rest" with ne
+      show "x \<noteq> []" by auto
+    qed
+    ultimately show "False" by auto 
+  qed
+  ultimately 
+  have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
+    by auto
+  moreover have "card \<dots> = 
+                    Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
+  proof(rule card_insert)
+    from vt_v.finite_holding
+    show " finite (holdents (V thread cs # s) thread)" .
+  qed
+  moreover from cs_not_in 
+  have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
+  ultimately show ?thesis by (simp add:cntCS_def)
+qed 
+
+lemma count_rec1 [simp]: 
+  assumes "Q e"
+  shows "count Q (e#es) = Suc (count Q es)"
+  using assms
+  by (unfold count_def, auto)
+
+lemma count_rec2 [simp]: 
+  assumes "\<not>Q e"
+  shows "count Q (e#es) = (count Q es)"
+  using assms
+  by (unfold count_def, auto)
+
+lemma count_rec3 [simp]: 
+  shows "count Q [] =  0"
+  by (unfold count_def, auto)
+
+lemma cntP_diff_inv:
+  assumes "cntP (e#s) th \<noteq> cntP s th"
+  shows "isP e \<and> actor e = th"
+proof(cases e)
+  case (P th' pty)
+  show ?thesis
+  by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)", 
+        insert assms P, auto simp:cntP_def)
+qed (insert assms, auto simp:cntP_def)
+  
+lemma cntV_diff_inv:
+  assumes "cntV (e#s) th \<noteq> cntV s th"
+  shows "isV e \<and> actor e = th"
+proof(cases e)
+  case (V th' pty)
+  show ?thesis
+  by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)", 
+        insert assms V, auto simp:cntV_def)
+qed (insert assms, auto simp:cntV_def)
+
+context valid_trace
+begin
+
+text {* (* ddd *) \noindent
+  The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} 
+  of one particular thread. t
+*} 
+
+lemma cnp_cnv_cncs:
+  shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) 
+                                       then cntCS s th else cntCS s th + 1)"
+proof -
+  from vt show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
+    assume vt: "vt s"
+    and ih: "\<And>th. cntP s th  = cntV s th +
+               (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
+    and stp: "step s e"
+    from stp show ?case
+    proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+        and not_in: "thread \<notin> threads s"
+      show ?thesis
+      proof -
+        { fix cs 
+          assume "thread \<in> set (wq s cs)"
+          from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
+          with not_in have "False" by simp
+        } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
+          by (auto simp:readys_def threads.simps s_waiting_def 
+            wq_def cs_waiting_def Let_def)
+        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+        have eq_cncs: "cntCS (e#s) th = cntCS s th"
+          unfolding cntCS_def holdents_test
+          by (simp add:RAG_create_unchanged eq_e)
+        { assume "th \<noteq> thread"
+          with eq_readys eq_e
+          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+            by (simp add:threads.simps)
+          with eq_cnp eq_cnv eq_cncs ih not_in
+          have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp
+          moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
+          moreover note eq_cnp eq_cnv eq_cncs
+          ultimately have ?thesis by auto
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread" 
+      and is_runing: "thread \<in> runing s"
+      and no_hold: "holdents s thread = {}"
+      from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+      from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+      have eq_cncs: "cntCS (e#s) th = cntCS s th"
+        unfolding cntCS_def holdents_test
+        by (simp add:RAG_exit_unchanged eq_e)
+      { assume "th \<noteq> thread"
+        with eq_e
+        have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+          (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+          apply (simp add:threads.simps readys_def)
+          apply (subst s_waiting_def)
+          apply (simp add:Let_def)
+          apply (subst s_waiting_def, simp)
+          done
+        with eq_cnp eq_cnv eq_cncs ih
+        have ?thesis by simp
+      } moreover {
+        assume eq_th: "th = thread"
+        with ih is_runing have " cntP s th = cntV s th + cntCS s th" 
+          by (simp add:runing_def)
+        moreover from eq_th eq_e have "th \<notin> threads (e#s)"
+          by simp
+        moreover note eq_cnp eq_cnv eq_cncs
+        ultimately have ?thesis by auto
+      } ultimately show ?thesis by blast
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+        and is_runing: "thread \<in> runing s"
+        and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
+      from thread_P vt stp ih  have vtp: "vt (P thread cs#s)" by auto
+      then interpret vt_p: valid_trace "(P thread cs#s)"
+        by (unfold_locales, simp)
+      show ?thesis 
+      proof -
+        { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
+          assume neq_th: "th \<noteq> thread"
+          with eq_e
+          have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
+            apply (simp add:readys_def s_waiting_def wq_def Let_def)
+            apply (rule_tac hh)
+             apply (intro iffI allI, clarify)
+            apply (erule_tac x = csa in allE, auto)
+            apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
+            apply (erule_tac x = cs in allE, auto)
+            by (case_tac "(wq_fun (schs s) cs)", auto)
+          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+            apply (simp add:cntCS_def holdents_test)
+            by (unfold  step_RAG_p [OF vtp], auto)
+          moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
+            by (simp add:cntP_def count_def)
+          moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
+            by (simp add:cntV_def count_def)
+          moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
+          moreover note ih [of th] 
+          ultimately have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          have ?thesis
+          proof -
+            from eq_e eq_th have eq_cnp: "cntP (e # s) th  = 1 + (cntP s th)" 
+              by (simp add:cntP_def count_def)
+            from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
+              by (simp add:cntV_def count_def)
+            show ?thesis
+            proof (cases "wq s cs = []")
+              case True
+              with is_runing
+              have "th \<in> readys (e#s)"
+                apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
+                apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
+                by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
+              moreover have "cntCS (e # s) th = 1 + cntCS s th"
+              proof -
+                have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
+                  Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
+                proof -
+                  have "?L = insert cs ?R" by auto
+                  moreover have "card \<dots> = Suc (card (?R - {cs}))" 
+                  proof(rule card_insert)
+                    from vt_s.finite_holding [of thread]
+                    show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
+                      by (unfold holdents_test, simp)
+                  qed
+                  moreover have "?R - {cs} = ?R"
+                  proof -
+                    have "cs \<notin> ?R"
+                    proof
+                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
+                      with no_dep show False by auto
+                    qed
+                    thus ?thesis by auto
+                  qed
+                  ultimately show ?thesis by auto
+                qed
+                thus ?thesis
+                  apply (unfold eq_e eq_th cntCS_def)
+                  apply (simp add: holdents_test)
+                  by (unfold step_RAG_p [OF vtp], auto simp:True)
+              qed
+              moreover from is_runing have "th \<in> readys s"
+                by (simp add:runing_def eq_th)
+              moreover note eq_cnp eq_cnv ih [of th]
+              ultimately show ?thesis by auto
+            next
+              case False
+              have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
+                    by (unfold eq_th eq_e wq_def, auto simp:Let_def)
+              have "th \<notin> readys (e#s)"
+              proof
+                assume "th \<in> readys (e#s)"
+                hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
+                from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
+                hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)" 
+                  by (simp add:s_waiting_def wq_def)
+                moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
+                ultimately have "th = hd (wq (e#s) cs)" by blast
+                with eq_wq have "th = hd (wq s cs @ [th])" by simp
+                hence "th = hd (wq s cs)" using False by auto
+                with False eq_wq vt_p.wq_distinct [of cs]
+                show False by (fold eq_e, auto)
+              qed
+              moreover from is_runing have "th \<in> threads (e#s)" 
+                by (unfold eq_e, auto simp:runing_def readys_def eq_th)
+              moreover have "cntCS (e # s) th = cntCS s th"
+                apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
+                by (auto simp:False)
+              moreover note eq_cnp eq_cnv ih[of th]
+              moreover from is_runing have "th \<in> readys s"
+                by (simp add:runing_def eq_th)
+              ultimately show ?thesis by auto
+            qed
+          qed
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_V thread cs)
+      from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
+      then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
+      assume eq_e: "e = V thread cs"
+        and is_runing: "thread \<in> runing s"
+        and hold: "holding s thread cs"
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+      have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+      proof(rule someI2)
+        from vt_v.wq_distinct[of cs] and eq_wq
+        show "distinct rest \<and> set rest = set rest"
+          by (metis distinct.simps(2) vt_s.wq_distinct)
+      next
+        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+          by auto
+      qed
+      show ?thesis
+      proof -
+        { assume eq_th: "th = thread"
+          from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
+            by (unfold eq_e, simp add:cntP_def count_def)
+          moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
+            by (unfold eq_e, simp add:cntV_def count_def)
+          moreover from cntCS_v_dec [OF vtv] 
+          have "cntCS (e # s) thread + 1 = cntCS s thread"
+            by (simp add:eq_e)
+          moreover from is_runing have rd_before: "thread \<in> readys s"
+            by (unfold runing_def, simp)
+          moreover have "thread \<in> readys (e # s)"
+          proof -
+            from is_runing
+            have "thread \<in> threads (e#s)" 
+              by (unfold eq_e, auto simp:runing_def readys_def)
+            moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
+            proof
+              fix cs1
+              { assume eq_cs: "cs1 = cs" 
+                have "\<not> waiting (e # s) thread cs1"
+                proof -
+                  from eq_wq
+                  have "thread \<notin> set (wq (e#s) cs1)"
+                    apply(unfold eq_e wq_def eq_cs s_holding_def)
+                    apply (auto simp:Let_def)
+                  proof -
+                    assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                    with eq_set have "thread \<in> set rest" by simp
+                    with vt_v.wq_distinct[of cs]
+                    and eq_wq show False
+                        by (metis distinct.simps(2) vt_s.wq_distinct)
+                  qed
+                  thus ?thesis by (simp add:wq_def s_waiting_def)
+                qed
+              } moreover {
+                assume neq_cs: "cs1 \<noteq> cs"
+                  have "\<not> waiting (e # s) thread cs1" 
+                  proof -
+                    from wq_v_neq [OF neq_cs[symmetric]]
+                    have "wq (V thread cs # s) cs1 = wq s cs1" .
+                    moreover have "\<not> waiting s thread cs1" 
+                    proof -
+                      from runing_ready and is_runing
+                      have "thread \<in> readys s" by auto
+                      thus ?thesis by (simp add:readys_def)
+                    qed
+                    ultimately show ?thesis 
+                      by (auto simp:wq_def s_waiting_def eq_e)
+                  qed
+              } ultimately show "\<not> waiting (e # s) thread cs1" by blast
+            qed
+            ultimately show ?thesis by (simp add:readys_def)
+          qed
+          moreover note eq_th ih
+          ultimately have ?thesis by auto
+        } moreover {
+          assume neq_th: "th \<noteq> thread"
+          from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" 
+            by (simp add:cntP_def count_def)
+          from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" 
+            by (simp add:cntV_def count_def)
+          have ?thesis
+          proof(cases "th \<in> set rest")
+            case False
+            have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+              apply (insert step_back_vt[OF vtv])
+              by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
+            moreover have "cntCS (e#s) th = cntCS s th"
+              apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
+              proof -
+                have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+                      {cs. (Cs cs, Th th) \<in> RAG s}"
+                proof -
+                  from False eq_wq
+                  have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
+                    apply (unfold next_th_def, auto)
+                  proof -
+                    assume ne: "rest \<noteq> []"
+                      and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                      and eq_wq: "wq s cs = thread # rest"
+                    from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                                  set (SOME q. distinct q \<and> set q = set rest)
+                                  " by simp
+                    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+                    proof(rule someI2)
+                      from vt_s.wq_distinct[ of cs] and eq_wq
+                      show "distinct rest \<and> set rest = set rest" by auto
+                    next
+                      fix x assume "distinct x \<and> set x = set rest"
+                      with ne show "x \<noteq> []" by auto
+                    qed
+                    ultimately show 
+                      "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
+                      by auto
+                  qed    
+                  thus ?thesis by auto
+                qed
+                thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+                             card {cs. (Cs cs, Th th) \<in> RAG s}" by simp 
+              qed
+            moreover note ih eq_cnp eq_cnv eq_threads
+            ultimately show ?thesis by auto
+          next
+            case True
+            assume th_in: "th \<in> set rest"
+            show ?thesis
+            proof(cases "next_th s thread cs th")
+              case False
+              with eq_wq and th_in have 
+                neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
+                by (auto simp:next_th_def)
+              have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+              proof -
+                from eq_wq and th_in
+                have "\<not> th \<in> readys s"
+                  apply (auto simp:readys_def s_waiting_def)
+                  apply (rule_tac x = cs in exI, auto)
+                  by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
+                moreover 
+                from eq_wq and th_in and neq_hd
+                have "\<not> (th \<in> readys (e # s))"
+                  apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
+                  by (rule_tac x = cs in exI, auto simp:eq_set)
+                ultimately show ?thesis by auto
+              qed
+              moreover have "cntCS (e#s) th = cntCS s th" 
+              proof -
+                from eq_wq and  th_in and neq_hd
+                have "(holdents (e # s) th) = (holdents s th)"
+                  apply (unfold eq_e step_RAG_v[OF vtv], 
+                         auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
+                                   Let_def cs_holding_def)
+                  by (insert vt_s.wq_distinct[of cs], auto simp:wq_def)
+                thus ?thesis by (simp add:cntCS_def)
+              qed
+              moreover note ih eq_cnp eq_cnv eq_threads
+              ultimately show ?thesis by auto
+            next
+              case True
+              let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
+              let ?t = "hd ?rest"
+              from True eq_wq th_in neq_th
+              have "th \<in> readys (e # s)"
+                apply (auto simp:eq_e readys_def s_waiting_def wq_def
+                        Let_def next_th_def)
+              proof -
+                assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                show "?t \<in> threads s"
+                proof(rule vt_s.wq_threads)
+                  from eq_wq and t_in
+                  show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
+                qed
+              next
+                fix csa
+                assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                  and neq_cs: "csa \<noteq> cs"
+                  and t_in': "?t \<in>  set (wq_fun (schs s) csa)"
+                show "?t = hd (wq_fun (schs s) csa)"
+                proof -
+                  { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
+                    from vt_s.wq_distinct[of cs] and 
+                    eq_wq[folded wq_def] and t_in eq_wq
+                    have "?t \<noteq> thread" by auto
+                    with eq_wq and t_in
+                    have w1: "waiting s ?t cs"
+                      by (auto simp:s_waiting_def wq_def)
+                    from t_in' neq_hd'
+                    have w2: "waiting s ?t csa"
+                      by (auto simp:s_waiting_def wq_def)
+                    from vt_s.waiting_unique[OF w1 w2]
+                    and neq_cs have "False" by auto
+                  } thus ?thesis by auto
+                qed
+              qed
+              moreover have "cntP s th = cntV s th + cntCS s th + 1"
+              proof -
+                have "th \<notin> readys s" 
+                proof -
+                  from True eq_wq neq_th th_in
+                  show ?thesis
+                    apply (unfold readys_def s_waiting_def, auto)
+                    by (rule_tac x = cs in exI, auto simp add: wq_def)
+                qed
+                moreover have "th \<in> threads s"
+                proof -
+                  from th_in eq_wq
+                  have "th \<in> set (wq s cs)" by simp
+                  from vt_s.wq_threads [OF this] 
+                  show ?thesis .
+                qed
+                ultimately show ?thesis using ih by auto
+              qed
+              moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
+                apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
+              proof -
+                show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
+                               Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
+                  (is "card ?A = Suc (card ?B)")
+                proof -
+                  have "?A = insert cs ?B" by auto
+                  hence "card ?A = card (insert cs ?B)" by simp
+                  also have "\<dots> = Suc (card ?B)"
+                  proof(rule card_insert_disjoint)
+                    have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)" 
+                      apply (auto simp:image_def)
+                      by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
+                    with vt_s.finite_RAG
+                    show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
+                  next
+                    show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
+                    proof
+                      assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
+                      hence "(Cs cs, Th th) \<in> RAG s" by simp
+                      with True neq_th eq_wq show False
+                        by (auto simp:next_th_def s_RAG_def cs_holding_def)
+                    qed
+                  qed
+                  finally show ?thesis .
+                qed
+              qed
+              moreover note eq_cnp eq_cnv
+              ultimately show ?thesis by simp
+            qed
+          qed
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_set thread prio)
+      assume eq_e: "e = Set thread prio"
+        and is_runing: "thread \<in> runing s"
+      show ?thesis
+      proof -
+        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+        have eq_cncs: "cntCS (e#s) th = cntCS s th"
+          unfolding cntCS_def holdents_test
+          by (simp add:RAG_set_unchanged eq_e)
+        from eq_e have eq_readys: "readys (e#s) = readys s" 
+          by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
+                  auto simp:Let_def)
+        { assume "th \<noteq> thread"
+          with eq_readys eq_e
+          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+            by (simp add:threads.simps)
+          with eq_cnp eq_cnv eq_cncs ih is_runing
+          have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          with is_runing ih have " cntP s th  = cntV s th + cntCS s th" 
+            by (unfold runing_def, auto)
+          moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
+            by (simp add:runing_def)
+          moreover note eq_cnp eq_cnv eq_cncs
+          ultimately have ?thesis by auto
+        } ultimately show ?thesis by blast
+      qed   
+    qed
+  next
+    case vt_nil
+    show ?case 
+      by (unfold cntP_def cntV_def cntCS_def, 
+        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma not_thread_cncs:
+  assumes not_in: "th \<notin> threads s" 
+  shows "cntCS s th = 0"
+proof -
+  from vt not_in show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e th)
+    interpret vt_s: valid_trace s using vt_cons(1)
+       by (unfold_locales, simp)
+    assume vt: "vt s"
+      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
+      and stp: "step s e"
+      and not_in: "th \<notin> threads (e # s)"
+    from stp show ?case
+    proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+        and not_in': "thread \<notin> threads s"
+      have "cntCS (e # s) th = cntCS s th"
+        apply (unfold eq_e cntCS_def holdents_test)
+        by (simp add:RAG_create_unchanged)
+      moreover have "th \<notin> threads s" 
+      proof -
+        from not_in eq_e show ?thesis by simp
+      qed
+      moreover note ih ultimately show ?thesis by auto
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+      and nh: "holdents s thread = {}"
+      have eq_cns: "cntCS (e # s) th = cntCS s th"
+        apply (unfold eq_e cntCS_def holdents_test)
+        by (simp add:RAG_exit_unchanged)
+      show ?thesis
+      proof(cases "th = thread")
+        case True
+        have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
+        with eq_cns show ?thesis by simp
+      next
+        case False
+        with not_in and eq_e
+        have "th \<notin> threads s" by simp
+        from ih[OF this] and eq_cns show ?thesis by simp
+      qed
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      and is_runing: "thread \<in> runing s"
+      from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      hence "cntCS (e # s) th  = cntCS s th "
+        apply (unfold cntCS_def holdents_test eq_e)
+        by (unfold step_RAG_p[OF vtp], auto)
+      moreover have "cntCS s th = 0"
+      proof(rule ih)
+        from not_in eq_e show "th \<notin> threads s" by simp
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+        and is_runing: "thread \<in> runing s"
+        and hold: "holding s thread cs"
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      from assms thread_V vt stp ih 
+      have vtv: "vt (V thread cs#s)" by auto
+      then interpret vt_v: valid_trace "(V thread cs#s)"
+        by (unfold_locales, simp)
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+      from not_in eq_e eq_wq
+      have "\<not> next_th s thread cs th"
+        apply (auto simp:next_th_def)
+      proof -
+        assume ne: "rest \<noteq> []"
+          and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
+        have "?t \<in> set rest"
+        proof(rule someI2)
+          from vt_v.wq_distinct[of cs] and eq_wq
+          show "distinct rest \<and> set rest = set rest"
+            by (metis distinct.simps(2) vt_s.wq_distinct) 
+        next
+          fix x assume "distinct x \<and> set x = set rest" with ne
+          show "hd x \<in> set rest" by (cases x, auto)
+        qed
+        with eq_wq have "?t \<in> set (wq s cs)" by simp
+        from vt_s.wq_threads[OF this] and ni
+        show False
+          using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)` 
+            ni vt_s.wq_threads by blast 
+      qed
+      moreover note neq_th eq_wq
+      ultimately have "cntCS (e # s) th  = cntCS s th"
+        by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
+      moreover have "cntCS s th = 0"
+      proof(rule ih)
+        from not_in eq_e show "th \<notin> threads s" by simp
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (thread_set thread prio)
+      print_facts
+      assume eq_e: "e = Set thread prio"
+        and is_runing: "thread \<in> runing s"
+      from not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] and eq_e
+      show ?thesis 
+        apply (unfold eq_e cntCS_def holdents_test)
+        by (simp add:RAG_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (unfold cntCS_def, 
+        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+  qed
+qed
+
+end
+
+
+context valid_trace
+begin
+
+lemma dm_RAG_threads:
+  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
+  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
+  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
+  hence "th \<in> set (wq s cs)"
+    by (unfold s_RAG_def, auto simp:cs_waiting_def)
+  from wq_threads [OF this] show ?thesis .
+qed
+
+end
+
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+thm cpreced_initial
+apply(simp add: Let_def cpreced_initial)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+
+context valid_trace
+begin
+
+lemma runing_unique:
+  assumes runing_1: "th1 \<in> runing s"
+  and runing_2: "th2 \<in> runing s"
+  shows "th1 = th2"
+proof -
+  from runing_1 and runing_2 have "cp s th1 = cp s th2"
+    unfolding runing_def
+    apply(simp)
+    done
+  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
+                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
+    (is "Max (?f ` ?A) = Max (?f ` ?B)")
+    unfolding cp_eq_cpreced 
+    unfolding cpreced_def .
+  obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
+  proof -
+    have h1: "finite (?f ` ?A)"
+    proof -
+      have "finite ?A" 
+      proof -
+        have "finite (dependants (wq s) th1)"
+        proof-
+          have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+              apply (auto simp:image_def)
+              by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
+            moreover have "finite \<dots>"
+            proof -
+              from finite_RAG have "finite (RAG s)" .
+              hence "finite ((RAG (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_RAG_def cs_RAG_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependants_def)
+        qed
+        thus ?thesis by simp
+      qed
+      thus ?thesis by auto
+    qed
+    moreover have h2: "(?f ` ?A) \<noteq> {}"
+    proof -
+      have "?A \<noteq> {}" by simp
+      thus ?thesis by simp
+    qed
+    from Max_in [OF h1 h2]
+    have "Max (?f ` ?A) \<in> (?f ` ?A)" .
+    thus ?thesis 
+      thm cpreced_def
+      unfolding cpreced_def[symmetric] 
+      unfolding cp_eq_cpreced[symmetric] 
+      unfolding cpreced_def 
+      using that[intro] by (auto)
+  qed
+  obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
+  proof -
+    have h1: "finite (?f ` ?B)"
+    proof -
+      have "finite ?B" 
+      proof -
+        have "finite (dependants (wq s) th2)"
+        proof-
+          have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+              apply (auto simp:image_def)
+              by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
+            moreover have "finite \<dots>"
+            proof -
+              from finite_RAG have "finite (RAG s)" .
+              hence "finite ((RAG (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_RAG_def cs_RAG_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependants_def)
+        qed
+        thus ?thesis by simp
+      qed
+      thus ?thesis by auto
+    qed
+    moreover have h2: "(?f ` ?B) \<noteq> {}"
+    proof -
+      have "?B \<noteq> {}" by simp
+      thus ?thesis by simp
+    qed
+    from Max_in [OF h1 h2]
+    have "Max (?f ` ?B) \<in> (?f ` ?B)" .
+    thus ?thesis by (auto intro:that)
+  qed
+  from eq_f_th1 eq_f_th2 eq_max 
+  have eq_preced: "preced th1' s = preced th2' s" by auto
+  hence eq_th12: "th1' = th2'"
+  proof (rule preced_unique)
+    from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
+    thus "th1' \<in> threads s"
+    proof
+      assume "th1' \<in> dependants (wq s) th1"
+      hence "(Th th1') \<in> Domain ((RAG s)^+)"
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+        by (auto simp:Domain_def)
+      hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      from dm_RAG_threads[OF this] show ?thesis .
+    next
+      assume "th1' = th1"
+      with runing_1 show ?thesis
+        by (unfold runing_def readys_def, auto)
+    qed
+  next
+    from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
+    thus "th2' \<in> threads s"
+    proof
+      assume "th2' \<in> dependants (wq s) th2"
+      hence "(Th th2') \<in> Domain ((RAG s)^+)"
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+        by (auto simp:Domain_def)
+      hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      from dm_RAG_threads[OF this] show ?thesis .
+    next
+      assume "th2' = th2"
+      with runing_2 show ?thesis
+        by (unfold runing_def readys_def, auto)
+    qed
+  qed
+  from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
+  thus ?thesis
+  proof
+    assume eq_th': "th1' = th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+    thus ?thesis
+    proof
+      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+    next
+      assume "th2' \<in> dependants (wq s) th2"
+      with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
+      hence "(Th th1, Th th2) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      hence "Th th1 \<in> Domain ((RAG s)^+)" 
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+        by (auto simp:Domain_def)
+      hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
+      from RAG_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th1, Cs cs') \<in> RAG s" by simp
+      with runing_1 have "False"
+        apply (unfold runing_def readys_def s_RAG_def)
+        by (auto simp:waiting_eq)
+      thus ?thesis by simp
+    qed
+  next
+    assume th1'_in: "th1' \<in> dependants (wq s) th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+    thus ?thesis 
+    proof
+      assume "th2' = th2"
+      with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
+      hence "(Th th2, Th th1) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      hence "Th th2 \<in> Domain ((RAG s)^+)" 
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+        by (auto simp:Domain_def)
+      hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
+      from RAG_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th2, Cs cs') \<in> RAG s" by simp
+      with runing_2 have "False"
+        apply (unfold runing_def readys_def s_RAG_def)
+        by (auto simp:waiting_eq)
+      thus ?thesis by simp
+    next
+      assume "th2' \<in> dependants (wq s) th2"
+      with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
+      hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      show ?thesis
+      proof(rule dchain_unique[OF h1 _ h2, symmetric])
+        from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
+        from runing_2 show "th2 \<in> readys s" by (simp add:runing_def) 
+      qed
+    qed
+  qed
+qed
+
+
+lemma "card (runing s) \<le> 1"
+apply(subgoal_tac "finite (runing s)")
+prefer 2
+apply (metis finite_nat_set_iff_bounded lessI runing_unique)
+apply(rule ccontr)
+apply(simp)
+apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
+apply(subst (asm) card_le_Suc_iff)
+apply(simp)
+apply(auto)[1]
+apply (metis insertCI runing_unique)
+apply(auto) 
+done
+
+end
+
+
+lemma create_pre:
+  assumes stp: "step s e"
+  and not_in: "th \<notin> threads s"
+  and is_in: "th \<in> threads (e#s)"
+  obtains prio where "e = Create th prio"
+proof -
+  from assms  
+  show ?thesis
+  proof(cases)
+    case (thread_create thread prio)
+    with is_in not_in have "e = Create th prio" by simp
+    from that[OF this] show ?thesis .
+  next
+    case (thread_exit thread)
+    with assms show ?thesis by (auto intro!:that)
+  next
+    case (thread_P thread)
+    with assms show ?thesis by (auto intro!:that)
+  next
+    case (thread_V thread)
+    with assms show ?thesis by (auto intro!:that)
+  next 
+    case (thread_set thread)
+    with assms show ?thesis by (auto intro!:that)
+  qed
+qed
+
+context valid_trace
+begin
+
+lemma cnp_cnv_eq:
+  assumes "th \<notin> threads s"
+  shows "cntP s th = cntV s th"
+  using assms
+  using cnp_cnv_cncs not_thread_cncs by auto
+
+end
+
+
+lemma eq_RAG: 
+  "RAG (wq s) = RAG s"
+by (unfold cs_RAG_def s_RAG_def, auto)
+
+context valid_trace
+begin
+
+lemma count_eq_dependants:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "dependants (wq s) th = {}"
+proof -
+  from cnp_cnv_cncs and eq_pv
+  have "cntCS s th = 0" 
+    by (auto split:if_splits)
+  moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
+  proof -
+    from finite_holding[of th] show ?thesis
+      by (simp add:holdents_test)
+  qed
+  ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
+    by (unfold cntCS_def holdents_test cs_dependants_def, auto)
+  show ?thesis
+  proof(unfold cs_dependants_def)
+    { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
+      then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
+      hence "False"
+      proof(cases)
+        assume "(Th th', Th th) \<in> RAG (wq s)"
+        thus "False" by (auto simp:cs_RAG_def)
+      next
+        fix c
+        assume "(c, Th th) \<in> RAG (wq s)"
+        with h and eq_RAG show "False"
+          by (cases c, auto simp:cs_RAG_def)
+      qed
+    } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
+  qed
+qed
+
+lemma dependants_threads:
+  shows "dependants (wq s) th \<subseteq> threads s"
+proof
+  { fix th th'
+    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
+    have "Th th \<in> Domain (RAG s)"
+    proof -
+      from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
+      hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
+      with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
+      thus ?thesis using eq_RAG by simp
+    qed
+    from dm_RAG_threads[OF this]
+    have "th \<in> threads s" .
+  } note hh = this
+  fix th1 
+  assume "th1 \<in> dependants (wq s) th"
+  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
+    by (unfold cs_dependants_def, simp)
+  from hh [OF this] show "th1 \<in> threads s" .
+qed
+
+lemma finite_threads:
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+lemma Max_f_mono:
+  assumes seq: "A \<subseteq> B"
+  and np: "A \<noteq> {}"
+  and fnt: "finite B"
+  shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+  from seq show "f ` A \<subseteq> f ` B" by auto
+next
+  from np show "f ` A \<noteq> {}" by auto
+next
+  from fnt and seq show "finite (f ` B)" by auto
+qed
+
+context valid_trace
+begin
+
+lemma cp_le:
+  assumes th_in: "th \<in> threads s"
+  shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
+  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
+         \<le> Max ((\<lambda>th. preced th s) ` threads s)"
+    (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
+  proof(rule Max_f_mono)
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
+  next
+    from finite_threads
+    show "finite (threads s)" .
+  next
+    from th_in
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
+      apply (auto simp:Domain_def)
+      apply (rule_tac dm_RAG_threads)
+      apply (unfold trancl_domain [of "RAG s", symmetric])
+      by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
+  qed
+qed
+
+lemma le_cp:
+  shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+  show "Prc (priority th s) (last_set th s)
+    \<le> Max (insert (Prc (priority th s) (last_set th s))
+            ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
+    (is "?l \<le> Max (insert ?l ?A)")
+  proof(cases "?A = {}")
+    case False
+    have "finite ?A" (is "finite (?f ` ?B)")
+    proof -
+      have "finite ?B" 
+      proof-
+        have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
+        proof -
+          let ?F = "\<lambda> (x, y). the_th x"
+          have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+            apply (auto simp:image_def)
+            by (rule_tac x = "(Th x, Th th)" in bexI, auto)
+          moreover have "finite \<dots>"
+          proof -
+            from finite_RAG have "finite (RAG s)" .
+            hence "finite ((RAG (wq s))\<^sup>+)"
+              apply (unfold finite_trancl)
+              by (auto simp: s_RAG_def cs_RAG_def wq_def)
+            thus ?thesis by auto
+          qed
+          ultimately show ?thesis by (auto intro:finite_subset)
+        qed
+        thus ?thesis by (simp add:cs_dependants_def)
+      qed
+      thus ?thesis by simp
+    qed
+    from Max_insert [OF this False, of ?l] show ?thesis by auto
+  next
+    case True
+    thus ?thesis by auto
+  qed
+qed
+
+lemma max_cp_eq: 
+  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
+  (is "?l = ?r")
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  have "?l \<in> ((cp s) ` threads s)"
+  proof(rule Max_in)
+    from finite_threads
+    show "finite (cp s ` threads s)" by auto
+  next
+    from False show "cp s ` threads s \<noteq> {}" by auto
+  qed
+  then obtain th 
+    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
+  have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
+  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
+  proof -
+    have "?r \<in> (?f ` ?A)"
+    proof(rule Max_in)
+      from finite_threads
+      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
+    next
+      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
+    qed
+    then obtain th' where 
+      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
+    from le_cp [of th']  eq_r
+    have "?r \<le> cp s th'" by auto
+    moreover have "\<dots> \<le> cp s th"
+    proof(fold eq_l)
+      show " cp s th' \<le> Max (cp s ` threads s)"
+      proof(rule Max_ge)
+        from th_in' show "cp s th' \<in> cp s ` threads s"
+          by auto
+      next
+        from finite_threads
+        show "finite (cp s ` threads s)" by auto
+      qed
+    qed
+    ultimately show ?thesis by auto
+  qed
+  ultimately show ?thesis using eq_l by auto
+qed
+
+lemma max_cp_readys_threads_pre:
+  assumes np: "threads s \<noteq> {}"
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq)
+  show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
+  proof -
+    let ?p = "Max ((\<lambda>th. preced th s) ` threads s)" 
+    let ?f = "(\<lambda>th. preced th s)"
+    have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
+    proof(rule Max_in)
+      from finite_threads show "finite (?f ` threads s)" by simp
+    next
+      from np show "?f ` threads s \<noteq> {}" by simp
+    qed
+    then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
+      by (auto simp:Image_def)
+    from th_chain_to_ready [OF tm_in]
+    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
+    thus ?thesis
+    proof
+      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
+      then obtain th' where th'_in: "th' \<in> readys s" 
+        and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
+      have "cp s th' = ?f tm"
+      proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
+        from dependants_threads finite_threads
+        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))" 
+          by (auto intro:finite_subset)
+      next
+        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+        from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
+        moreover have "p \<le> \<dots>"
+        proof(rule Max_ge)
+          from finite_threads
+          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        next
+          from p_in and th'_in and dependants_threads[of th']
+          show "p \<in> (\<lambda>th. preced th s) ` threads s"
+            by (auto simp:readys_def)
+        qed
+        ultimately show "p \<le> preced tm s" by auto
+      next
+        show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+        proof -
+          from tm_chain
+          have "tm \<in> dependants (wq s) th'"
+            by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
+          thus ?thesis by auto
+        qed
+      qed
+      with tm_max
+      have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+      show ?thesis
+      proof (fold h, rule Max_eqI)
+        fix q 
+        assume "q \<in> cp s ` readys s"
+        then obtain th1 where th1_in: "th1 \<in> readys s"
+          and eq_q: "q = cp s th1" by auto
+        show "q \<le> cp s th'"
+          apply (unfold h eq_q)
+          apply (unfold cp_eq_cpreced cpreced_def)
+          apply (rule Max_mono)
+        proof -
+          from dependants_threads [of th1] th1_in
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq> 
+                 (\<lambda>th. preced th s) ` threads s"
+            by (auto simp:readys_def)
+        next
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
+        next
+          from finite_threads 
+          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        qed
+      next
+        from finite_threads
+        show "finite (cp s ` readys s)" by (auto simp:readys_def)
+      next
+        from th'_in
+        show "cp s th' \<in> cp s ` readys s" by simp
+      qed
+    next
+      assume tm_ready: "tm \<in> readys s"
+      show ?thesis
+      proof(fold tm_max)
+        have cp_eq_p: "cp s tm = preced tm s"
+        proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+          fix y 
+          assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+          show "y \<le> preced tm s"
+          proof -
+            { fix y'
+              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
+              have "y' \<le> preced tm s"
+              proof(unfold tm_max, rule Max_ge)
+                from hy' dependants_threads[of tm]
+                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+              next
+                from finite_threads
+                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+              qed
+            } with hy show ?thesis by auto
+          qed
+        next
+          from dependants_threads[of tm] finite_threads
+          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
+            by (auto intro:finite_subset)
+        next
+          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+            by simp
+        qed 
+        moreover have "Max (cp s ` readys s) = cp s tm"
+        proof(rule Max_eqI)
+          from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
+        next
+          from finite_threads
+          show "finite (cp s ` readys s)" by (auto simp:readys_def)
+        next
+          fix y assume "y \<in> cp s ` readys s"
+          then obtain th1 where th1_readys: "th1 \<in> readys s"
+            and h: "y = cp s th1" by auto
+          show "y \<le> cp s tm"
+            apply(unfold cp_eq_p h)
+            apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
+          proof -
+            from finite_threads
+            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+          next
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
+              by simp
+          next
+            from dependants_threads[of th1] th1_readys
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) 
+                    \<subseteq> (\<lambda>th. preced th s) ` threads s"
+              by (auto simp:readys_def)
+          qed
+        qed
+        ultimately show " Max (cp s ` readys s) = preced tm s" by simp
+      qed 
+    qed
+  qed
+qed
+
+text {* (* ccc *) \noindent
+  Since the current precedence of the threads in ready queue will always be boosted,
+  there must be one inside it has the maximum precedence of the whole system. 
+*}
+lemma max_cp_readys_threads:
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis 
+    by (auto simp:readys_def)
+next
+  case False
+  show ?thesis by (rule max_cp_readys_threads_pre[OF False])
+qed
+
+end
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+  apply (unfold s_holding_def cs_holding_def wq_def, simp)
+  done
+
+lemma f_image_eq:
+  assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
+  shows "f ` A = g ` A"
+proof
+  show "f ` A \<subseteq> g ` A"
+    by(rule image_subsetI, auto intro:h)
+next
+  show "g ` A \<subseteq> f ` A"
+   by (rule image_subsetI, auto intro:h[symmetric])
+qed
+
+
+definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
+  where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
+
+lemma detached_test:
+  shows "detached s th = (Th th \<notin> Field (RAG s))"
+apply(simp add: detached_def Field_def)
+apply(simp add: s_RAG_def)
+apply(simp add: s_holding_abv s_waiting_abv)
+apply(simp add: Domain_iff Range_iff)
+apply(simp add: wq_def)
+apply(auto)
+done
+
+context valid_trace
+begin
+
+lemma detached_intro:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "detached s th"
+proof -
+ from cnp_cnv_cncs
+  have eq_cnt: "cntP s th =
+    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+  hence cncs_zero: "cntCS s th = 0"
+    by (auto simp:eq_pv split:if_splits)
+  with eq_cnt
+  have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
+  thus ?thesis
+  proof
+    assume "th \<notin> threads s"
+    with range_in dm_RAG_threads
+    show ?thesis
+      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
+  next
+    assume "th \<in> readys s"
+    moreover have "Th th \<notin> Range (RAG s)"
+    proof -
+      from card_0_eq [OF finite_holding] and cncs_zero
+      have "holdents s th = {}"
+        by (simp add:cntCS_def)
+      thus ?thesis
+        apply(auto simp:holdents_test)
+        apply(case_tac a)
+        apply(auto simp:holdents_test s_RAG_def)
+        done
+    qed
+    ultimately show ?thesis
+      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
+  qed
+qed
+
+lemma detached_elim:
+  assumes dtc: "detached s th"
+  shows "cntP s th = cntV s th"
+proof -
+  from cnp_cnv_cncs
+  have eq_pv: " cntP s th =
+    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+  have cncs_z: "cntCS s th = 0"
+  proof -
+    from dtc have "holdents s th = {}"
+      unfolding detached_def holdents_test s_RAG_def
+      by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
+    thus ?thesis by (auto simp:cntCS_def)
+  qed
+  show ?thesis
+  proof(cases "th \<in> threads s")
+    case True
+    with dtc 
+    have "th \<in> readys s"
+      by (unfold readys_def detached_def Field_def Domain_def Range_def, 
+           auto simp:waiting_eq s_RAG_def)
+    with cncs_z and eq_pv show ?thesis by simp
+  next
+    case False
+    with cncs_z and eq_pv show ?thesis by simp
+  qed
+qed
+
+lemma detached_eq:
+  shows "(detached s th) = (cntP s th = cntV s th)"
+  by (insert vt, auto intro:detached_intro detached_elim)
+
+end
+
+text {* 
+  The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived
+  from the concise and miniature model of PIP given in PrioGDef.thy.
+*}
+
+lemma eq_dependants: "dependants (wq s) = dependants s"
+  by (simp add: s_dependants_abv wq_def)
+
+lemma next_th_unique: 
+  assumes nt1: "next_th s th cs th1"
+  and nt2: "next_th s th cs th2"
+  shows "th1 = th2"
+using assms by (unfold next_th_def, auto)
+
+lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
+  apply (induct s, simp)
+proof -
+  fix a s
+  assume ih: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
+    and eq_as: "a # s \<noteq> []"
+  show "last_set th (a # s) < length (a # s)"
+  proof(cases "s \<noteq> []")
+    case False
+    from False show ?thesis
+      by (cases a, auto simp:last_set.simps)
+  next
+    case True
+    from ih [OF True] show ?thesis
+      by (cases a, auto simp:last_set.simps)
+  qed
+qed
+
+lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
+  by (induct s, auto simp:threads.simps)
+
+lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
+  apply (drule_tac th_in_ne)
+  by (unfold preced_def, auto intro: birth_time_lt)
+
+lemma inj_the_preced: 
+  "inj_on (the_preced s) (threads s)"
+  by (metis inj_onI preced_unique the_preced_def)
+
+lemma tRAG_alt_def: 
+  "tRAG s = {(Th th1, Th th2) | th1 th2. 
+                  \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
+ by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
+
+lemma tRAG_Field:
+  "Field (tRAG s) \<subseteq> Field (RAG s)"
+  by (unfold tRAG_alt_def Field_def, auto)
+
+lemma tRAG_ancestorsE:
+  assumes "x \<in> ancestors (tRAG s) u"
+  obtains th where "x = Th th"
+proof -
+  from assms have "(u, x) \<in> (tRAG s)^+" 
+      by (unfold ancestors_def, auto)
+  from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
+  then obtain th where "x = Th th"
+    by (unfold tRAG_alt_def, auto)
+  from that[OF this] show ?thesis .
+qed
+
+lemma tRAG_mono:
+  assumes "RAG s' \<subseteq> RAG s"
+  shows "tRAG s' \<subseteq> tRAG s"
+  using assms 
+  by (unfold tRAG_alt_def, auto)
+
+lemma holding_next_thI:
+  assumes "holding s th cs"
+  and "length (wq s cs) > 1"
+  obtains th' where "next_th s th cs th'"
+proof -
+  from assms(1)[folded eq_holding, unfolded cs_holding_def]
+  have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
+  then obtain rest where h1: "wq s cs = th#rest" 
+    by (cases "wq s cs", auto)
+  with assms(2) have h2: "rest \<noteq> []" by auto
+  let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+  have "next_th s th cs ?th'" using  h1(1) h2 
+    by (unfold next_th_def, auto)
+  from that[OF this] show ?thesis .
+qed
+
+lemma RAG_tRAG_transfer:
+  assumes "vt s'"
+  assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+  and "(Cs cs, Th th'') \<in> RAG s'"
+  shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
+proof -
+  interpret vt_s': valid_trace "s'" using assms(1)
+    by (unfold_locales, simp)
+  interpret rtree: rtree "RAG s'"
+  proof
+  show "single_valued (RAG s')"
+  apply (intro_locales)
+    by (unfold single_valued_def, 
+        auto intro:vt_s'.unique_RAG)
+
+  show "acyclic (RAG s')"
+     by (rule vt_s'.acyclic_RAG)
+  qed
+  { fix n1 n2
+    assume "(n1, n2) \<in> ?L"
+    from this[unfolded tRAG_alt_def]
+    obtain th1 th2 cs' where 
+      h: "n1 = Th th1" "n2 = Th th2" 
+         "(Th th1, Cs cs') \<in> RAG s"
+         "(Cs cs', Th th2) \<in> RAG s" by auto
+    from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
+    from h(3) and assms(2) 
+    have "(Th th1, Cs cs') = (Th th, Cs cs) \<or> 
+          (Th th1, Cs cs') \<in> RAG s'" by auto
+    hence "(n1, n2) \<in> ?R"
+    proof
+      assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
+      hence eq_th1: "th1 = th" by simp
+      moreover have "th2 = th''"
+      proof -
+        from h1 have "cs' = cs" by simp
+        from assms(3) cs_in[unfolded this] rtree.sgv
+        show ?thesis
+          by (unfold single_valued_def, auto)
+      qed
+      ultimately show ?thesis using h(1,2) by auto
+    next
+      assume "(Th th1, Cs cs') \<in> RAG s'"
+      with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
+        by (unfold tRAG_alt_def, auto)
+      from this[folded h(1, 2)] show ?thesis by auto
+    qed
+  } moreover {
+    fix n1 n2
+    assume "(n1, n2) \<in> ?R"
+    hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
+    hence "(n1, n2) \<in> ?L" 
+    proof
+      assume "(n1, n2) \<in> tRAG s'"
+      moreover have "... \<subseteq> ?L"
+      proof(rule tRAG_mono)
+        show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
+      qed
+      ultimately show ?thesis by auto
+    next
+      assume eq_n: "(n1, n2) = (Th th, Th th'')"
+      from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
+      moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
+      ultimately show ?thesis 
+        by (unfold eq_n tRAG_alt_def, auto)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+context valid_trace
+begin
+
+lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
+
+end
+
+lemma cp_alt_def:
+  "cp s th =  
+           Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
+proof -
+  have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
+        Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})" 
+          (is "Max (_ ` ?L) = Max (_ ` ?R)")
+  proof -
+    have "?L = ?R" 
+    by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
+    thus ?thesis by simp
+  qed
+  thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
+qed
+
+lemma cp_gen_alt_def:
+  "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
+    by (auto simp:cp_gen_def)
+
+lemma tRAG_nodeE:
+  assumes "(n1, n2) \<in> tRAG s"
+  obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
+  using assms
+  by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
+
+lemma subtree_nodeE:
+  assumes "n \<in> subtree (tRAG s) (Th th)"
+  obtains th1 where "n = Th th1"
+proof -
+  show ?thesis
+  proof(rule subtreeE[OF assms])
+    assume "n = Th th"
+    from that[OF this] show ?thesis .
+  next
+    assume "Th th \<in> ancestors (tRAG s) n"
+    hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+    hence "\<exists> th1. n = Th th1"
+    proof(induct)
+      case (base y)
+      from tRAG_nodeE[OF this] show ?case by metis
+    next
+      case (step y z)
+      thus ?case by auto
+    qed
+    with that show ?thesis by auto
+  qed
+qed
+
+lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
+proof -
+  have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*" 
+    by (rule rtrancl_mono, auto simp:RAG_split)
+  also have "... \<subseteq> ((RAG s)^*)^*"
+    by (rule rtrancl_mono, auto)
+  also have "... = (RAG s)^*" by simp
+  finally show ?thesis by (unfold tRAG_def, simp)
+qed
+
+lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
+proof -
+  { fix a
+    assume "a \<in> subtree (tRAG s) x"
+    hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
+    with tRAG_star_RAG[of s]
+    have "(a, x) \<in> (RAG s)^*" by auto
+    hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
+  } thus ?thesis by auto
+qed
+
+lemma tRAG_trancl_eq:
+   "{th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
+    {th'. (Th th', Th th)  \<in> (RAG s)^+}"
+   (is "?L = ?R")
+proof -
+  { fix th'
+    assume "th' \<in> ?L"
+    hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
+    from tranclD[OF this]
+    obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
+    from tRAG_subtree_RAG[of s] and this(2)
+    have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG) 
+    moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto 
+    ultimately have "th' \<in> ?R"  by auto 
+  } moreover 
+  { fix th'
+    assume "th' \<in> ?R"
+    hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
+    from plus_rpath[OF this]
+    obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
+    hence "(Th th', Th th) \<in> (tRAG s)^+"
+    proof(induct xs arbitrary:th' th rule:length_induct)
+      case (1 xs th' th)
+      then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
+      show ?case
+      proof(cases "xs1")
+        case Nil
+        from 1(2)[unfolded Cons1 Nil]
+        have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
+        hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
+        then obtain cs where "x1 = Cs cs" 
+              by (unfold s_RAG_def, auto)
+        from rpath_nnl_lastE[OF rp[unfolded this]]
+        show ?thesis by auto
+      next
+        case (Cons x2 xs2)
+        from 1(2)[unfolded Cons1[unfolded this]]
+        have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
+        from rpath_edges_on[OF this]
+        have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
+        have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+            by (simp add: edges_on_unfold)
+        with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
+        then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
+        have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+            by (simp add: edges_on_unfold)
+        from this eds
+        have rg2: "(x1, x2) \<in> RAG s" by auto
+        from this[unfolded eq_x1] 
+        obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
+        from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
+        have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
+        from rp have "rpath (RAG s) x2 xs2 (Th th)"
+           by  (elim rpath_ConsE, simp)
+        from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
+        show ?thesis
+        proof(cases "xs2 = []")
+          case True
+          from rpath_nilE[OF rp'[unfolded this]]
+          have "th1 = th" by auto
+          from rt1[unfolded this] show ?thesis by auto
+        next
+          case False
+          from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
+          have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
+          with rt1 show ?thesis by auto
+        qed
+      qed
+    qed
+    hence "th' \<in> ?L" by auto
+  } ultimately show ?thesis by blast
+qed
+
+lemma tRAG_trancl_eq_Th:
+   "{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
+    {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}"
+    using tRAG_trancl_eq by auto
+
+lemma dependants_alt_def:
+  "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
+  by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
+  
+context valid_trace
+begin
+
+lemma count_eq_tRAG_plus:
+  assumes "cntP s th = cntV s th"
+  shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+  using assms count_eq_dependants dependants_alt_def eq_dependants by auto 
+
+lemma count_eq_RAG_plus:
+  assumes "cntP s th = cntV s th"
+  shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+  using assms count_eq_dependants cs_dependants_def eq_RAG by auto
+
+lemma count_eq_RAG_plus_Th:
+  assumes "cntP s th = cntV s th"
+  shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+  using count_eq_RAG_plus[OF assms] by auto
+
+lemma count_eq_tRAG_plus_Th:
+  assumes "cntP s th = cntV s th"
+  shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+   using count_eq_tRAG_plus[OF assms] by auto
+
+end
+
+lemma tRAG_subtree_eq: 
+   "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th'  \<in> (subtree (RAG s) (Th th))}"
+   (is "?L = ?R")
+proof -
+  { fix n 
+    assume h: "n \<in> ?L"
+    hence "n \<in> ?R"
+    by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) 
+  } moreover {
+    fix n
+    assume "n \<in> ?R"
+    then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
+      by (auto simp:subtree_def)
+    from rtranclD[OF this(2)]
+    have "n \<in> ?L"
+    proof
+      assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
+      with h have "n \<in> {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}" by auto
+      thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
+    qed (insert h, auto simp:subtree_def)
+  } ultimately show ?thesis by auto
+qed
+
+lemma threads_set_eq: 
+   "the_thread ` (subtree (tRAG s) (Th th)) = 
+                  {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
+   by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
+
+lemma cp_alt_def1: 
+  "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
+proof -
+  have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
+       ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
+       by auto
+  thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
+qed
+
+lemma cp_gen_def_cond: 
+  assumes "x = Th th"
+  shows "cp s th = cp_gen s (Th th)"
+by (unfold cp_alt_def1 cp_gen_def, simp)
+
+lemma cp_gen_over_set:
+  assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
+  shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
+proof(rule f_image_eq)
+  fix a
+  assume "a \<in> A"
+  from assms[rule_format, OF this]
+  obtain th where eq_a: "a = Th th" by auto
+  show "cp_gen s a = (cp s \<circ> the_thread) a"
+    by  (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
+qed
+
+
+context valid_trace
+begin
+
+lemma RAG_threads:
+  assumes "(Th th) \<in> Field (RAG s)"
+  shows "th \<in> threads s"
+  using assms
+  by (metis Field_def UnE dm_RAG_threads range_in vt)
+
+lemma subtree_tRAG_thread:
+  assumes "th \<in> threads s"
+  shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
+proof -
+  have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+    by (unfold tRAG_subtree_eq, simp)
+  also have "... \<subseteq> ?R"
+  proof
+    fix x
+    assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+    then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
+    from this(2)
+    show "x \<in> ?R"
+    proof(cases rule:subtreeE)
+      case 1
+      thus ?thesis by (simp add: assms h(1)) 
+    next
+      case 2
+      thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) 
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma readys_root:
+  assumes "th \<in> readys s"
+  shows "root (RAG s) (Th th)"
+proof -
+  { fix x
+    assume "x \<in> ancestors (RAG s) (Th th)"
+    hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+    from tranclD[OF this]
+    obtain z where "(Th th, z) \<in> RAG s" by auto
+    with assms(1) have False
+         apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
+         by (fold wq_def, blast)
+  } thus ?thesis by (unfold root_def, auto)
+qed
+
+lemma readys_in_no_subtree:
+  assumes "th \<in> readys s"
+  and "th' \<noteq> th"
+  shows "Th th \<notin> subtree (RAG s) (Th th')" 
+proof
+   assume "Th th \<in> subtree (RAG s) (Th th')"
+   thus False
+   proof(cases rule:subtreeE)
+      case 1
+      with assms show ?thesis by auto
+   next
+      case 2
+      with readys_root[OF assms(1)]
+      show ?thesis by (auto simp:root_def)
+   qed
+qed
+
+lemma not_in_thread_isolated:
+  assumes "th \<notin> threads s"
+  shows "(Th th) \<notin> Field (RAG s)"
+proof
+  assume "(Th th) \<in> Field (RAG s)"
+  with dm_RAG_threads and range_in assms
+  show False by (unfold Field_def, blast)
+qed
+
+lemma wf_RAG: "wf (RAG s)"
+proof(rule finite_acyclic_wf)
+  from finite_RAG show "finite (RAG s)" .
+next
+  from acyclic_RAG show "acyclic (RAG s)" .
+qed
+
+lemma sgv_wRAG: "single_valued (wRAG s)"
+  using waiting_unique
+  by (unfold single_valued_def wRAG_def, auto)
+
+lemma sgv_hRAG: "single_valued (hRAG s)"
+  using holding_unique 
+  by (unfold single_valued_def hRAG_def, auto)
+
+lemma sgv_tRAG: "single_valued (tRAG s)"
+  by (unfold tRAG_def, rule single_valued_relcomp, 
+              insert sgv_wRAG sgv_hRAG, auto)
+
+lemma acyclic_tRAG: "acyclic (tRAG s)"
+proof(unfold tRAG_def, rule acyclic_compose)
+  show "acyclic (RAG s)" using acyclic_RAG .
+next
+  show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+next
+  show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+qed
+
+lemma sgv_RAG: "single_valued (RAG s)"
+  using unique_RAG by (auto simp:single_valued_def)
+
+lemma rtree_RAG: "rtree (RAG s)"
+  using sgv_RAG acyclic_RAG
+  by (unfold rtree_def rtree_axioms_def sgv_def, auto)
+
+end
+
+sublocale valid_trace < rtree_RAG: rtree "RAG s"
+proof
+  show "single_valued (RAG s)"
+  apply (intro_locales)
+    by (unfold single_valued_def, 
+        auto intro:unique_RAG)
+
+  show "acyclic (RAG s)"
+     by (rule acyclic_RAG)
+qed
+
+sublocale valid_trace < rtree_s: rtree "tRAG s"
+proof(unfold_locales)
+  from sgv_tRAG show "single_valued (tRAG s)" .
+next
+  from acyclic_tRAG show "acyclic (tRAG s)" .
+qed
+
+sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
+proof -
+  show "fsubtree (RAG s)"
+  proof(intro_locales)
+    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
+  next
+    show "fsubtree_axioms (RAG s)"
+    proof(unfold fsubtree_axioms_def)
+      from wf_RAG show "wf (RAG s)" .
+    qed
+  qed
+qed
+
+sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
+proof -
+  have "fsubtree (tRAG s)"
+  proof -
+    have "fbranch (tRAG s)"
+    proof(unfold tRAG_def, rule fbranch_compose)
+        show "fbranch (wRAG s)"
+        proof(rule finite_fbranchI)
+           from finite_RAG show "finite (wRAG s)"
+           by (unfold RAG_split, auto)
+        qed
+    next
+        show "fbranch (hRAG s)"
+        proof(rule finite_fbranchI)
+           from finite_RAG 
+           show "finite (hRAG s)" by (unfold RAG_split, auto)
+        qed
+    qed
+    moreover have "wf (tRAG s)"
+    proof(rule wf_subset)
+      show "wf (RAG s O RAG s)" using wf_RAG
+        by (fold wf_comp_self, simp)
+    next
+      show "tRAG s \<subseteq> (RAG s O RAG s)"
+        by (unfold tRAG_alt_def, auto)
+    qed
+    ultimately show ?thesis
+      by (unfold fsubtree_def fsubtree_axioms_def,auto)
+  qed
+  from this[folded tRAG_def] show "fsubtree (tRAG s)" .
+qed
+
+lemma Max_UNION: 
+  assumes "finite A"
+  and "A \<noteq> {}"
+  and "\<forall> M \<in> f ` A. finite M"
+  and "\<forall> M \<in> f ` A. M \<noteq> {}"
+  shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
+  using assms[simp]
+proof -
+  have "?L = Max (\<Union>(f ` A))"
+    by (fold Union_image_eq, simp)
+  also have "... = ?R"
+    by (subst Max_Union, simp+)
+  finally show ?thesis .
+qed
+
+lemma max_Max_eq:
+  assumes "finite A"
+    and "A \<noteq> {}"
+    and "x = y"
+  shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
+proof -
+  have "?R = Max (insert y A)" by simp
+  also from assms have "... = ?L"
+      by (subst Max.insert, simp+)
+  finally show ?thesis by simp
+qed
+
+context valid_trace
+begin
+
+(* ddd *)
+lemma cp_gen_rec:
+  assumes "x = Th th"
+  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
+proof(cases "children (tRAG s) x = {}")
+  case True
+  show ?thesis
+    by (unfold True cp_gen_def subtree_children, simp add:assms)
+next
+  case False
+  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
+  note fsbttRAGs.finite_subtree[simp]
+  have [simp]: "finite (children (tRAG s) x)"
+     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 
+            rule children_subtree)
+  { fix r x
+    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
+  } note this[simp]
+  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
+  proof -
+    from False obtain q where "q \<in> children (tRAG s) x" by blast
+    moreover have "subtree (tRAG s) q \<noteq> {}" by simp
+    ultimately show ?thesis by blast
+  qed
+  have h: "Max ((the_preced s \<circ> the_thread) `
+                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
+        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
+                     (is "?L = ?R")
+  proof -
+    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
+    let "Max (_ \<union> (?h ` ?B))" = ?R
+    let ?L1 = "?f ` \<Union>(?g ` ?B)"
+    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
+    proof -
+      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
+      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
+      finally have "Max ?L1 = Max ..." by simp
+      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
+        by (subst Max_UNION, simp+)
+      also have "... = Max (cp_gen s ` children (tRAG s) x)"
+          by (unfold image_comp cp_gen_alt_def, simp)
+      finally show ?thesis .
+    qed
+    show ?thesis
+    proof -
+      have "?L = Max (?f ` ?A \<union> ?L1)" by simp
+      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
+            by (subst Max_Un, simp+)
+      also have "... = max (?f x) (Max (?h ` ?B))"
+        by (unfold eq_Max_L1, simp)
+      also have "... =?R"
+        by (rule max_Max_eq, (simp)+, unfold assms, simp)
+      finally show ?thesis .
+    qed
+  qed  thus ?thesis 
+          by (fold h subtree_children, unfold cp_gen_def, simp) 
+qed
+
+lemma cp_rec:
+  "cp s th = Max ({the_preced s th} \<union> 
+                     (cp s o the_thread) ` children (tRAG s) (Th th))"
+proof -
+  have "Th th = Th th" by simp
+  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
+  show ?thesis 
+  proof -
+    have "cp_gen s ` children (tRAG s) (Th th) = 
+                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
+    proof(rule cp_gen_over_set)
+      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
+        by (unfold tRAG_alt_def, auto simp:children_def)
+    qed
+    thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+  qed
+qed
+
+end
+
+(* keep *)
+lemma next_th_holding:
+  assumes vt: "vt s"
+  and nxt: "next_th s th cs th'"
+  shows "holding (wq s) th cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  thus ?thesis
+    by (unfold cs_holding_def, auto)
+qed
+
+context valid_trace
+begin
+
+lemma next_th_waiting:
+  assumes nxt: "next_th s th cs th'"
+  shows "waiting (wq s) th' cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  from wq_distinct[of cs, unfolded h]
+  have dst: "distinct (th # rest)" .
+  have in_rest: "th' \<in> set rest"
+  proof(unfold h, rule someI2)
+    show "distinct rest \<and> set rest = set rest" using dst by auto
+  next
+    fix x assume "distinct x \<and> set x = set rest"
+    with h(2)
+    show "hd x \<in> set (rest)" by (cases x, auto)
+  qed
+  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
+  moreover have "th' \<noteq> hd (wq s cs)"
+    by (unfold h(1), insert in_rest dst, auto)
+  ultimately show ?thesis by (auto simp:cs_waiting_def)
+qed
+
+lemma next_th_RAG:
+  assumes nxt: "next_th (s::event list) th cs th'"
+  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
+  using vt assms next_th_holding next_th_waiting
+  by (unfold s_RAG_def, simp)
+
+end
+
+-- {* A useless definition *}
+definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+
+lemma "wq (V th cs # s) cs1 = ttt"
+  apply (unfold wq_def, auto simp:Let_def)
+
+end
+
diff -r 5d8ec128518b -r e3cf792db636 Attic/CpsG_2.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/CpsG_2.thy	Tue Jun 14 15:06:16 2016 +0100
@@ -0,0 +1,3557 @@
+theory CpsG
+imports PIPDefs
+begin
+
+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 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"
+
+locale valid_trace_e = valid_trace +
+  fixes e
+  assumes vt_e: "vt (e#s)"
+begin
+
+lemma pip_e: "PIP s e"
+  using vt_e by (cases, simp)  
+
+end
+
+locale valid_trace_create = valid_trace_e + 
+  fixes th prio
+  assumes is_create: "e = Create th prio"
+
+locale valid_trace_exit = valid_trace_e + 
+  fixes th
+  assumes is_exit: "e = Exit th"
+
+locale valid_trace_p = valid_trace_e + 
+  fixes th cs
+  assumes is_p: "e = P th cs"
+
+locale valid_trace_v = valid_trace_e + 
+  fixes th cs
+  assumes is_v: "e = V th cs"
+begin
+  definition "rest = tl (wq s cs)"
+  definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+end
+
+locale valid_trace_v_n = valid_trace_v +
+  assumes rest_nnl: "rest \<noteq> []"
+
+locale valid_trace_v_e = valid_trace_v +
+  assumes rest_nil: "rest = []"
+
+locale valid_trace_set= valid_trace_e + 
+  fixes th prio
+  assumes is_set: "e = Set th prio"
+
+context valid_trace
+begin
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes "PP []"
+     and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
+                   PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
+     shows "PP s"
+proof(induct rule:vt.induct[OF vt, case_names Init Step])
+  case Init
+  from assms(1) show ?case .
+next
+  case (Step s e)
+  show ?case
+  proof(rule assms(2))
+    show "valid_trace_e s e" using Step by (unfold_locales, auto)
+  next
+    show "PP s" using Step by simp
+  next
+    show "PIP s e" using Step by simp
+  qed
+qed
+
+lemma  vt_moment: "\<And> t. vt (moment t s)"
+proof(induct rule:ind)
+  case Nil
+  thus ?case by (simp add:vt_nil)
+next
+  case (Cons s e t)
+  show ?case
+  proof(cases "t \<ge> length (e#s)")
+    case True
+    from True have "moment t (e#s) = e#s" by simp
+    thus ?thesis using Cons
+      by (simp add:valid_trace_def valid_trace_e_def, auto)
+  next
+    case False
+    from Cons have "vt (moment t s)" by simp
+    moreover have "moment t (e#s) = moment t s"
+    proof -
+      from False have "t \<le> length s" by simp
+      from moment_app [OF this, of "[e]"] 
+      show ?thesis by simp
+    qed
+    ultimately show ?thesis by simp
+  qed
+qed
+
+lemma finite_threads:
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, auto)
+
+locale valid_moment = valid_trace + 
+  fixes i :: nat
+
+sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
+  by (unfold_locales, insert vt_moment, auto)
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+  by  (unfold s_waiting_def cs_waiting_def wq_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def wq_def cs_holding_def, simp)
+
+lemma runing_ready: 
+  shows "runing s \<subseteq> readys s"
+  unfolding runing_def readys_def
+  by auto 
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+  unfolding readys_def
+  by auto
+
+lemma wq_v_neq [simp]:
+   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
+  by (auto simp:wq_def Let_def cp_def split:list.splits)
+
+lemma runing_head:
+  assumes "th \<in> runing s"
+  and "th \<in> set (wq_fun (schs s) cs)"
+  shows "th = hd (wq_fun (schs s) cs)"
+  using assms
+  by (simp add:runing_def readys_def s_waiting_def wq_def)
+
+context valid_trace
+begin
+
+lemma runing_wqE:
+  assumes "th \<in> runing s"
+  and "th \<in> set (wq s cs)"
+  obtains rest where "wq s cs = th#rest"
+proof -
+  from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
+    by (meson list.set_cases)
+  have "th' = th"
+  proof(rule ccontr)
+    assume "th' \<noteq> th"
+    hence "th \<noteq> hd (wq s cs)" using eq_wq by auto 
+    with assms(2)
+    have "waiting s th cs" 
+      by (unfold s_waiting_def, fold wq_def, auto)
+    with assms show False 
+      by (unfold runing_def readys_def, auto)
+  qed
+  with eq_wq that show ?thesis by metis
+qed
+
+end
+
+context valid_trace_create
+begin
+
+lemma wq_neq_simp [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_create wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace_exit
+begin
+
+lemma wq_neq_simp [simp]:
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_exit wq_def
+  by (auto simp:Let_def)
+
+lemma wq_distinct_kept:
+  assumes "distinct (wq s cs')"
+  shows "distinct (wq (e#s) cs')"
+  using assms by simp
+end
+
+context valid_trace_p
+begin
+
+lemma wq_neq_simp [simp]:
+  assumes "cs' \<noteq> cs"
+  shows "wq (e#s) cs' = wq s cs'"
+    using assms unfolding is_p wq_def
+  by (auto simp:Let_def)
+
+lemma runing_th_s:
+  shows "th \<in> runing s"
+proof -
+  from pip_e[unfolded is_p]
+  show ?thesis by (cases, simp)
+qed
+
+lemma 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"
+  shows "actor e \<in> runing s"
+  using assms
+  by (induct, auto)
+
+lemma isP_E:
+  assumes "isP e"
+  obtains cs where "e = P (actor e) cs"
+  using assms by (cases e, auto)
+
+lemma isV_E:
+  assumes "isV e"
+  obtains cs where "e = V (actor e) cs"
+  using assms by (cases e, auto) 
+
+lemma wq_distinct: "distinct (wq s cs)"
+proof(induct rule:ind)
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case 
+  proof(cases e)
+    case (Create th prio)
+    interpret vt_create: valid_trace_create s e th prio 
+      using Create by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept) 
+  next
+    case (Exit th)
+    interpret vt_exit: valid_trace_exit s e th  
+        using Exit by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept) 
+  next
+    case (P th cs)
+    interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) 
+  next
+    case (V th cs)
+    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) 
+  next
+    case (Set th prio)
+    interpret vt_set: valid_trace_set s e th prio
+        using Set by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) 
+  qed
+qed (unfold wq_def Let_def, simp)
+
+end
+
+context valid_trace_e
+begin
+
+text {*
+  The following lemma shows that only the @{text "P"}
+  operation can add new thread into waiting queues. 
+  Such kind of lemmas are very obvious, but need to be checked formally.
+  This is a kind of confirmation that our modelling is correct.
+*}
+
+lemma wq_in_inv: 
+  assumes s_ni: "thread \<notin> set (wq s cs)"
+  and s_i: "thread \<in> set (wq (e#s) cs)"
+  shows "e = P thread cs"
+proof(cases e)
+  -- {* This is the only non-trivial case: *}
+  case (V th cs1)
+  have False
+  proof(cases "cs1 = cs")
+    case True
+    show ?thesis
+    proof(cases "(wq s cs1)")
+      case (Cons w_hd w_tl)
+      have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
+      proof -
+        have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
+          using  Cons V by (auto simp:wq_def Let_def True split:if_splits)
+        moreover have "set ... \<subseteq> set (wq s cs)"
+        proof(rule someI2)
+          show "distinct w_tl \<and> set w_tl = set w_tl"
+            by (metis distinct.simps(2) local.Cons wq_distinct)
+        qed (insert Cons True, auto)
+        ultimately show ?thesis by simp
+      qed
+      with assms show ?thesis by auto
+    qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
+  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+  thus ?thesis by auto
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+
+lemma wq_out_inv: 
+  assumes s_in: "thread \<in> set (wq s cs)"
+  and s_hd: "thread = hd (wq s cs)"
+  and s_i: "thread \<noteq> hd (wq (e#s) cs)"
+  shows "e = V thread cs"
+proof(cases e)
+-- {* There are only two non-trivial cases: *}
+  case (V th cs1)
+  show ?thesis
+  proof(cases "cs1 = cs")
+    case True
+    have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
+    thus ?thesis
+    proof(cases)
+      case (thread_V)
+      moreover have "th = thread" using thread_V(2) s_hd
+          by (unfold s_holding_def wq_def, simp)
+      ultimately show ?thesis using V True by simp
+    qed
+  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+next
+  case (P th cs1)
+  show ?thesis
+  proof(cases "cs1 = cs")
+    case True
+    with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
+      by (auto simp:wq_def Let_def split:if_splits)
+    with s_i s_hd s_in have False
+      by (metis empty_iff hd_append2 list.set(1) wq_def) 
+    thus ?thesis by simp
+  qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+
+end
+
+
+context valid_trace
+begin
+
+
+text {* (* ddd *)
+  The nature of the work is like this: since it starts from a very simple and basic 
+  model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
+  For instance, the fact 
+  that one thread can not be blocked by two critical resources at the same time
+  is obvious, because only running threads can make new requests, if one is waiting for 
+  a critical resource and get blocked, it can not make another resource request and get 
+  blocked the second time (because it is not running). 
+
+  To derive this fact, one needs to prove by contraction and 
+  reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
+  named @{text "p_split"}, which is about status changing along the time axis. It says if 
+  a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
+  but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} 
+  in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history 
+  of events leading to it), such that @{text "Q"} switched 
+  from being @{text "False"} to @{text "True"} and kept being @{text "True"}
+  till the last moment of @{text "s"}.
+
+  Suppose a thread @{text "th"} is blocked
+  on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, 
+  since no thread is blocked at the very beginning, by applying 
+  @{text "p_split"} to these two blocking facts, there exist 
+  two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that 
+  @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} 
+  and kept on blocked on them respectively ever since.
+ 
+  Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
+  However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
+  in blocked state at moment @{text "t2"} and could not
+  make any request and get blocked the second time: Contradiction.
+*}
+
+lemma waiting_unique_pre: (* ddd *)
+  assumes h11: "thread \<in> set (wq s cs1)"
+  and h12: "thread \<noteq> hd (wq s cs1)"
+  assumes h21: "thread \<in> set (wq s cs2)"
+  and h22: "thread \<noteq> hd (wq s cs2)"
+  and neq12: "cs1 \<noteq> cs2"
+  shows "False"
+proof -
+  let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  from h11 and h12 have q1: "?Q cs1 s" by simp
+  from h21 and h22 have q2: "?Q cs2 s" by simp
+  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+  from p_split [of "?Q cs1", OF q1 nq1]
+  obtain t1 where lt1: "t1 < length s"
+    and np1: "\<not> ?Q cs1 (moment t1 s)"
+    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
+  from p_split [of "?Q cs2", OF q2 nq2]
+  obtain t2 where lt2: "t2 < length s"
+    and np2: "\<not> ?Q cs2 (moment t2 s)"
+    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
+  { fix s cs
+    assume q: "?Q cs s"
+    have "thread \<notin> runing s"
+    proof
+      assume "thread \<in> runing s"
+      hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and> 
+                 thread \<noteq> hd (wq_fun (schs s) cs))"
+        by (unfold runing_def s_waiting_def readys_def, auto)
+      from this[rule_format, of cs] q 
+      show False by (simp add: wq_def) 
+    qed
+  } note q_not_runing = this
+  { fix t1 t2 cs1 cs2
+    assume  lt1: "t1 < length s"
+    and np1: "\<not> ?Q cs1 (moment t1 s)"
+    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
+    and lt2: "t2 < length s"
+    and np2: "\<not> ?Q cs2 (moment t2 s)"
+    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
+    and lt12: "t1 < t2"
+    let ?t3 = "Suc t2"
+    from lt2 have le_t3: "?t3 \<le> length s" by auto
+    from moment_plus [OF this] 
+    obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+    have "t2 < ?t3" by simp
+    from nn2 [rule_format, OF this] and eq_m
+    have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+         h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+    have "vt (e#moment t2 s)"
+    proof -
+      from vt_moment 
+      have "vt (moment ?t3 s)" .
+      with eq_m show ?thesis by simp
+    qed
+    then interpret vt_e: valid_trace_e "moment t2 s" "e"
+        by (unfold_locales, auto, cases, simp)
+    have ?thesis
+    proof -
+      have "thread \<in> runing (moment t2 s)"
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        have "e = V thread cs2"
+        proof -
+          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
+              using True and np2  by auto 
+          from vt_e.wq_out_inv[OF True this h2]
+          show ?thesis .
+        qed
+        thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto
+      next
+        case False
+        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+        with vt_e.actor_inv[OF vt_e.pip_e]
+        show ?thesis by auto
+      qed
+      moreover have "thread \<notin> runing (moment t2 s)"
+        by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
+      ultimately show ?thesis by simp
+    qed
+  } note lt_case = this
+  show ?thesis
+  proof -
+    { assume "t1 < t2"
+      from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
+      have ?thesis .
+    } moreover {
+      assume "t2 < t1"
+      from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
+      have ?thesis .
+    } moreover {
+      assume eq_12: "t1 = t2"
+      let ?t3 = "Suc t2"
+      from lt2 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+      have lt_2: "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+           h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+      from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12]
+      have g1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+           g2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have "vt (e#moment t2 s)"
+      proof -
+        from vt_moment 
+        have "vt (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      then interpret vt_e: valid_trace_e "moment t2 s" "e"
+          by (unfold_locales, auto, cases, simp)
+      have "e = V thread cs2 \<or> e = P thread cs2"
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        have "e = V thread cs2"
+        proof -
+          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
+              using True and np2  by auto 
+          from vt_e.wq_out_inv[OF True this h2]
+          show ?thesis .
+        qed
+        thus ?thesis by auto
+      next
+        case False
+        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+        thus ?thesis by auto
+      qed
+      moreover have "e = V thread cs1 \<or> e = P thread cs1"
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        have eq_th: "thread = hd (wq (moment t1 s) cs1)" 
+              using True and np1  by auto 
+        from vt_e.wq_out_inv[folded eq_12, OF True this g2]
+        have "e = V thread cs1" .
+        thus ?thesis by auto
+      next
+        case False
+        have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
+        thus ?thesis by auto
+      qed
+      ultimately have ?thesis using neq12 by auto
+    } ultimately show ?thesis using nat_neq_iff by blast 
+  qed
+qed
+
+text {*
+  This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
+*}
+
+lemma waiting_unique:
+  assumes "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+  using waiting_unique_pre assms
+  unfolding wq_def s_waiting_def
+  by auto
+
+end
+
+(* not used *)
+text {*
+  Every thread can only be blocked on one critical resource, 
+  symmetrically, every critical resource can only be held by one thread. 
+  This fact is much more easier according to our definition. 
+*}
+lemma held_unique:
+  assumes "holding (s::event list) th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+ by (insert assms, unfold s_holding_def, auto)
+
+lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits)
+
+lemma last_set_unique: 
+  "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
+          \<Longrightarrow> th1 = th2"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits dest:last_set_lt)
+
+lemma preced_unique : 
+  assumes pcd_eq: "preced th1 s = preced th2 s"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "th1 = th2"
+proof -
+  from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
+  from last_set_unique [OF this th_in1 th_in2]
+  show ?thesis .
+qed
+                      
+lemma preced_linorder: 
+  assumes neq_12: "th1 \<noteq> th2"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
+proof -
+  from preced_unique [OF _ th_in1 th_in2] and neq_12 
+  have "preced th1 s \<noteq> preced th2 s" by auto
+  thus ?thesis by auto
+qed
+
+text {*
+  The following three lemmas show that @{text "RAG"} does not change
+  by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
+  events, respectively.
+*}
+
+lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma (in valid_trace_set)
+   RAG_unchanged: "(RAG (e # s)) = RAG s"
+   by (unfold is_set RAG_set_unchanged, simp)
+
+lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma (in valid_trace_create)
+   RAG_unchanged: "(RAG (e # s)) = RAG s"
+   by (unfold is_create RAG_create_unchanged, simp)
+
+lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma (in valid_trace_exit)
+   RAG_unchanged: "(RAG (e # s)) = RAG s"
+   by (unfold is_exit RAG_exit_unchanged, simp)
+
+context valid_trace_v
+begin
+
+lemma distinct_rest: "distinct rest"
+  by (simp add: distinct_tl rest_def wq_distinct)
+
+lemma holding_cs_eq_th:
+  assumes "holding s t cs"
+  shows "t = th"
+proof -
+  from pip_e[unfolded is_v]
+  show ?thesis
+  proof(cases)
+    case (thread_V)
+    from held_unique[OF this(2) assms]
+    show ?thesis by simp
+  qed
+qed
+
+lemma distinct_wq': "distinct wq'"
+  by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
+  
+lemma 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
+    have "t \<in> set wq'" 
+     using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
+     by simp 
+    from th'_in_inv[OF this] have "t \<in> set rest" .
+    with wq_s_cs[folded otherwise] wq_distinct[of cs]
+    show ?thesis by simp
+  next
+    case False
+    have "wq (e#s) c = wq s c" using False
+        by (unfold is_v, simp)
+    hence "waiting s t c" using assms 
+        by (simp add: cs_waiting_def waiting_eq)
+    hence "t \<notin> readys s" by (unfold readys_def, auto)
+    hence "t \<notin> runing s" using runing_ready by auto 
+    with runing_th_s[folded otherwise] show ?thesis by auto
+  qed
+qed
+
+lemma waiting_esI1:
+  assumes "waiting s t c"
+      and "c \<noteq> cs" 
+  shows "waiting (e#s) t c" 
+proof -
+  have "wq (e#s) c = wq s c" 
+    using assms(2) is_v by auto
+  with assms(1) show ?thesis 
+    using cs_waiting_def waiting_eq by auto 
+qed
+
+lemma holding_esI2:
+  assumes "c \<noteq> cs" 
+  and "holding s t c"
+  shows "holding (e#s) t c"
+proof -
+  from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
+  from assms(2)[unfolded s_holding_def, folded wq_def, 
+                folded this, unfolded wq_def, folded s_holding_def]
+  show ?thesis .
+qed
+
+lemma holding_esI1:
+  assumes "holding s t c"
+  and "t \<noteq> th"
+  shows "holding (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms using holding_cs_eq_th by blast 
+  from holding_esI2[OF this assms(1)]
+  show ?thesis .
+qed
+
+end
+
+context valid_trace_v_n
+begin
+
+lemma neq_wq': "wq' \<noteq> []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x \<noteq> []" using rest_nnl by auto
+qed 
+
+definition "taker = hd wq'"
+
+definition "rest' = tl wq'"
+
+lemma eq_wq': "wq' = taker # rest'"
+  by (simp add: neq_wq' rest'_def taker_def)
+
+lemma next_th_taker: 
+  shows "next_th s th cs taker"
+  using rest_nnl taker_def wq'_def wq_s_cs 
+  by (auto simp:next_th_def)
+
+lemma taker_unique: 
+  assumes "next_th s th cs taker'"
+  shows "taker' = taker"
+proof -
+  from assms
+  obtain rest' where 
+    h: "wq s cs = th # rest'" 
+       "taker' = hd (SOME q. distinct q \<and> set q = set rest')"
+          by (unfold next_th_def, auto)
+  with wq_s_cs have "rest' = rest" by auto
+  thus ?thesis using h(2) taker_def wq'_def by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
+  by (smt all_not_in_conv bot.extremum insertI1 insert_subset 
+      mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {(Cs cs, Th taker)}"
+  using next_th_taker taker_def waiting_set_eq 
+  by fastforce
+   
+lemma holding_taker:
+  shows "holding (e#s) taker cs"
+    by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, 
+        auto simp:neq_wq' taker_def)
+
+lemma waiting_esI2:
+  assumes "waiting s t cs"
+      and "t \<noteq> taker"
+  shows "waiting (e#s) t cs" 
+proof -
+  have "t \<in> set wq'" 
+  proof(unfold wq'_def, rule someI2)
+    show "distinct rest \<and> set rest = set rest"
+          by (simp add: distinct_rest)
+  next
+    fix x
+    assume "distinct x \<and> set x = set rest"
+    moreover have "t \<in> set rest"
+        using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto 
+    ultimately show "t \<in> set x" by simp
+  qed
+  moreover have "t \<noteq> hd wq'"
+    using assms(2) taker_def by auto 
+  ultimately show ?thesis
+    by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+     |    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
+  have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
+  hence "t \<noteq> taker" by (simp add: taker_def) 
+  moreover hence "t \<noteq> th" using assms neq_t_th by blast 
+  moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv) 
+  ultimately have "waiting s t cs"
+    by (metis cs_waiting_def list.distinct(2) list.sel(1) 
+                list.set_sel(2) rest_def waiting_eq wq_s_cs)  
+  show ?thesis using that(2)
+  using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto   
+qed
+
+lemma holding_esI1:
+  assumes "c = cs"
+  and "t = taker"
+  shows "holding (e#s) t c"
+  by (unfold assms, simp add: holding_taker)
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c = cs" "t = taker"
+      | "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from assms[unfolded True, unfolded s_holding_def, 
+             folded wq_def, unfolded wq_es_cs]
+  have "t = taker" by (simp add: taker_def) 
+  from that(1)[OF True this] show ?thesis .
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that(2)[OF False this] show ?thesis .
+qed
+
+end 
+
+
+context valid_trace_v_e
+begin
+
+lemma nil_wq': "wq' = []" 
+proof (unfold wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest) 
+next
+  fix x
+  assume " distinct x \<and> set x = set rest" 
+  thus "x = []" using rest_nil by auto
+qed 
+
+lemma no_taker: 
+  assumes "next_th s th cs taker"
+  shows "False"
+proof -
+  from assms[unfolded next_th_def]
+  obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
+    by auto
+  thus ?thesis using rest_def rest_nil by auto 
+qed
+
+lemma waiting_set_eq:
+  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
+  using no_taker by auto
+
+lemma holding_set_eq:
+  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
+  using no_taker by auto
+   
+lemma no_holding:
+  assumes "holding (e#s) taker cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_holding_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma no_waiting:
+  assumes "waiting (e#s) t cs"
+  shows False
+proof -
+  from wq_es_cs[unfolded nil_wq']
+  have " wq (e # s) cs = []" .
+  from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
+  show ?thesis by auto
+qed
+
+lemma waiting_esI2:
+  assumes "waiting s t c"
+  shows "waiting (e#s) t c"
+proof -
+  have "c \<noteq> cs" using assms
+    using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto 
+  from waiting_esI1[OF assms this]
+  show ?thesis .
+qed
+
+lemma waiting_esE:
+  assumes "waiting (e#s) t c" 
+  obtains "c \<noteq> cs" "waiting s t c"
+proof(cases "c = cs")
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
+  from that(1)[OF False this] show ?thesis .
+next
+  case True
+  from no_waiting[OF assms[unfolded True]]
+  show ?thesis by auto
+qed
+
+lemma holding_esE:
+  assumes "holding (e#s) t c" 
+  obtains "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+  case True
+  from no_holding[OF assms[unfolded True]] 
+  show ?thesis by auto
+next
+  case False
+  hence "wq (e#s) c = wq s c" using is_v by auto
+  from assms[unfolded s_holding_def, folded wq_def, 
+             unfolded this, unfolded wq_def, folded s_holding_def]
+  have "holding s t c"  .
+  from that[OF False this] show ?thesis .
+qed
+
+end 
+
+lemma rel_eqI:
+  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
+  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
+  shows "A = B"
+  using assms by auto
+
+lemma in_RAG_E:
+  assumes "(n1, n2) \<in> RAG (s::state)"
+  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
+      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
+  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
+  by auto
+  
+context valid_trace_v
+begin
+
+lemma RAG_es:
+  "RAG (e # s) =
+   RAG s - {(Cs cs, Th th)} -
+     {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+     {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof(rule rel_eqI)
+  fix n1 n2
+  assume "(n1, n2) \<in> ?L"
+  thus "(n1, n2) \<in> ?R"
+  proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_n.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      next
+        case 2
+        with waiting(1,2)
+        show ?thesis
+         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      qed
+    next
+      case True
+      interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+      from waiting(3)
+      show ?thesis
+      proof(cases rule:h_e.waiting_esE)
+        case 1
+        with waiting(1,2)
+        show ?thesis
+        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      qed
+    qed
+  next
+    case (holding th' cs')
+    show ?thesis
+    proof(cases "rest = []")
+      case False
+      interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+      from holding(3)
+      show ?thesis
+      proof(cases rule:h_n.holding_esE)
+        case 1
+        with holding(1,2)
+        show ?thesis
+        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold waiting_eq, auto)
+      next
+        case 2
+        with holding(1,2)
+        show ?thesis
+         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
+             fold holding_eq, auto)
+      qed
+    next
+      case True
+      interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+      from holding(3)
+      show ?thesis
+      proof(cases rule:h_e.holding_esE)
+        case 1
+        with holding(1,2)
+        show ?thesis
+        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
+             fold holding_eq, auto)
+      qed
+    qed
+  qed
+next
+  fix n1 n2
+  assume h: "(n1, n2) \<in> ?R"
+  show "(n1, n2) \<in> ?L"
+  proof(cases "rest = []")
+    case False
+    interpret h_n: valid_trace_v_n s e th cs
+        by (unfold_locales, insert False, simp)
+    from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
+    have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
+                            \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or> 
+          (n2 = Th h_n.taker \<and> n1 = Cs cs)" 
+      by auto
+   thus ?thesis
+   proof
+      assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
+      with h_n.holding_taker
+      show ?thesis 
+        by (unfold s_RAG_def, fold holding_eq, auto)
+   next
+    assume h: "(n1, n2) \<in> RAG s \<and>
+        (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
+    hence "(n1, n2) \<in> RAG s" by simp
+    thus ?thesis
+    proof(cases rule:in_RAG_E)
+      case (waiting th' cs')
+      from h and this(1,2)
+      have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
+      hence "waiting (e#s) th' cs'" 
+      proof
+        assume "cs' \<noteq> cs"
+        from waiting_esI1[OF waiting(3) this] 
+        show ?thesis .
+      next
+        assume neq_th': "th' \<noteq> h_n.taker"
+        show ?thesis
+        proof(cases "cs' = cs")
+          case False
+          from waiting_esI1[OF waiting(3) this] 
+          show ?thesis .
+        next
+          case True
+          from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
+          show ?thesis .
+        qed
+      qed
+      thus ?thesis using waiting(1,2)
+        by (unfold s_RAG_def, fold waiting_eq, auto)
+    next
+      case (holding th' cs')
+      from h this(1,2)
+      have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+      hence "holding (e#s) th' cs'"
+      proof
+        assume "cs' \<noteq> cs"
+        from holding_esI2[OF this holding(3)] 
+        show ?thesis .
+      next
+        assume "th' \<noteq> th"
+        from holding_esI1[OF holding(3) this]
+        show ?thesis .
+      qed
+      thus ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+   qed
+ next
+   case True
+   interpret h_e: valid_trace_v_e s e th cs
+        by (unfold_locales, insert True, simp)
+   from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
+   have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)" 
+      by auto
+   from h_s(1)
+   show ?thesis
+   proof(cases rule:in_RAG_E)
+    case (waiting th' cs')
+    from h_e.waiting_esI2[OF this(3)]
+    show ?thesis using waiting(1,2)
+      by (unfold s_RAG_def, fold waiting_eq, auto)
+   next
+    case (holding th' cs')
+    with h_s(2)
+    have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+    thus ?thesis
+    proof
+      assume neq_cs: "cs' \<noteq> cs"
+      from holding_esI2[OF this holding(3)]
+      show ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    next
+      assume "th' \<noteq> th"
+      from holding_esI1[OF holding(3) this]
+      show ?thesis using holding(1,2)
+        by (unfold s_RAG_def, fold holding_eq, auto)
+    qed
+   qed
+ qed
+qed
+
+end
+
+lemma step_RAG_v: 
+assumes vt:
+  "vt (V th cs#s)"
+shows "
+  RAG (V th cs # s) =
+  RAG s - {(Cs cs, Th th)} -
+  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+  {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
+proof -
+  interpret vt_v: valid_trace_v s "V th cs"
+    using assms step_back_vt by (unfold_locales, auto) 
+  show ?thesis using vt_v.RAG_es .
+qed
+
+lemma (in valid_trace_create)
+  th_not_in_threads: "th \<notin> threads s"
+proof -
+  from pip_e[unfolded is_create]
+  show ?thesis by (cases, simp)
+qed
+
+lemma (in valid_trace_create)
+  threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
+  by (unfold is_create, simp)
+
+lemma (in valid_trace_exit)
+  threads_es [simp]: "threads (e#s) = threads s - {th}"
+  by (unfold is_exit, simp)
+
+lemma (in valid_trace_p)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_p, simp)
+
+lemma (in valid_trace_v)
+  threads_es [simp]: "threads (e#s) = threads s"
+  by (unfold is_v, simp)
+
+lemma (in valid_trace_v)
+  th_not_in_rest[simp]: "th \<notin> set rest"
+proof
+  assume otherwise: "th \<in> set rest"
+  have "distinct (wq s cs)" by (simp add: wq_distinct)
+  from this[unfolded wq_s_cs] and otherwise
+  show False by auto
+qed
+
+lemma (in valid_trace_v)
+  set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
+proof(unfold wq_es_cs wq'_def, rule someI2)
+  show "distinct rest \<and> set rest = set rest"
+    by (simp add: distinct_rest)
+next
+  fix x
+  assume "distinct x \<and> set x = set rest"
+  thus "set x = set (wq s cs) - {th}" 
+      by (unfold wq_s_cs, simp)
+qed
+
+lemma (in valid_trace_exit)
+  th_not_in_wq: "th \<notin> set (wq s cs)"
+proof -
+  from pip_e[unfolded is_exit]
+  show ?thesis
+  by (cases, unfold holdents_def s_holding_def, fold wq_def, 
+             auto elim!:runing_wqE)
+qed
+
+lemma (in valid_trace) wq_threads: 
+  assumes "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+  using assms
+proof(induct rule:ind)
+  case (Nil)
+  thus ?case by (auto simp:wq_def)
+next
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case
+  proof(cases e)
+    case (Create th' prio')
+    interpret vt: valid_trace_create s e th' prio'
+      using Create by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems by auto
+  next
+    case (Exit th')
+    interpret vt: valid_trace_exit s e th'
+      using Exit by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto 
+  next
+    case (P th' cs')
+    interpret vt: valid_trace_p s e th' cs'
+      using P by (unfold_locales, simp)
+    show ?thesis
+      using Cons.hyps(2) Cons.prems readys_threads 
+        runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv 
+        by fastforce 
+  next
+    case (V th' cs')
+    interpret vt: valid_trace_v s e th' cs'
+      using V by (unfold_locales, simp)
+    show ?thesis using Cons
+      using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
+  next
+    case (Set th' prio)
+    interpret vt: valid_trace_set s e th' prio
+      using Set by (unfold_locales, simp)
+    show ?thesis using Cons.hyps(2) Cons.prems vt.is_set 
+        by (auto simp:wq_def Let_def)
+  qed
+qed 
+
+context valid_trace
+begin
+
+lemma  dm_RAG_threads:
+  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
+  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
+  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
+  hence "th \<in> set (wq s cs)"
+    by (unfold s_RAG_def, auto simp:cs_waiting_def)
+  from wq_threads [OF this] show ?thesis .
+qed
+
+lemma 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)
+
+end
+
+
+
+
+lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
+  by (unfold preced_def, simp)
+
+lemma (in valid_trace_v)
+  preced_es: "preced th (e#s) = preced th s"
+  by (unfold is_v preced_def, simp)
+
+lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
+proof
+  fix th'
+  show "the_preced (V th cs # s) th' = the_preced s th'"
+    by (unfold the_preced_def preced_def, simp)
+qed
+
+lemma (in valid_trace_v)
+  the_preced_es: "the_preced (e#s) = the_preced s"
+  by (unfold is_v preced_def, simp)
+
+context valid_trace_p
+begin
+
+lemma not_holding_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)
+  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)
+    interpret vt: valid_trace_create s e th prio using Create
+      by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
+  next
+    case (Exit th)
+    interpret vt: valid_trace_exit s e th using Exit
+      by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt.RAG_unchanged)
+  next
+    case (P th cs)
+    interpret vt: valid_trace_p s e th cs using P
+      by (unfold_locales, simp)
+    show ?thesis using Cons using vt.RAG_es' by auto 
+  next
+    case (V th cs)
+    interpret vt: valid_trace_v s e th cs using V
+      by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt.finite_RAG_kept) 
+  next
+    case (Set th prio)
+    interpret vt: valid_trace_set s e th prio using Set
+      by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
+  qed
+qed
+
+lemma acyclic_RAG:
+  shows "acyclic (RAG s)"
+proof(induct rule:ind)
+  case Nil
+  show ?case 
+  by (auto simp: s_RAG_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+next
+  case (Cons s e)
+  interpret vt_e: valid_trace_e s e using Cons by simp
+  show ?case
+  proof(cases e)
+    case (Create th prio)
+    interpret vt: valid_trace_create s e th prio using Create
+      by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
+  next
+    case (Exit th)
+    interpret vt: valid_trace_exit s e th using Exit
+      by (unfold_locales, simp)
+    show ?thesis using Cons by (simp add: vt.RAG_unchanged)
+  next
+    case (P th cs)
+    interpret vt: valid_trace_p s e th cs using P
+      by (unfold_locales, simp)
+    show ?thesis
+    proof(cases "wq s cs = []")
+      case True
+      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
+      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 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 (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 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 < 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
+
+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: 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 max_cp_eq_the_preced:
+  shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
+  using max_cp_eq using the_preced_def by presburger 
+
+lemma wf_RAG_converse: 
+  shows "wf ((RAG s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+  from finite_RAG 
+  show "finite (RAG s)" .
+next
+  from acyclic_RAG
+  show "acyclic (RAG s)" .
+qed
+
+lemma chain_building:
+  assumes "node \<in> Domain (RAG s)"
+  obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"
+proof -
+  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
+    ultimately show ?thesis by (auto simp:readys_def)
+  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
+  The following is just an instance of @{text "chain_building"}.
+*}
+lemma th_chain_to_ready:
+  assumes th_in: "th \<in> threads s"
+  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
+proof(cases "th \<in> readys s")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  from False and th_in have "Th th \<in> Domain (RAG s)" 
+    by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
+  from chain_building [rule_format, OF this]
+  show ?thesis by auto
+qed
+
+end
+
+lemma count_rec1 [simp]: 
+  assumes "Q e"
+  shows "count Q (e#es) = Suc (count Q es)"
+  using assms
+  by (unfold count_def, auto)
+
+lemma count_rec2 [simp]: 
+  assumes "\<not>Q e"
+  shows "count Q (e#es) = (count Q es)"
+  using assms
+  by (unfold count_def, auto)
+
+lemma count_rec3 [simp]: 
+  shows "count Q [] =  0"
+  by (unfold count_def, auto)
+
+lemma cntP_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"
+proof(cases e)
+  case (P th' pty)
+  show ?thesis
+  by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)", 
+        insert assms P, auto simp:cntP_def)
+qed (insert assms, auto simp:cntP_def)
+  
+lemma cntV_diff_inv:
+  assumes "cntV (e#s) th \<noteq> cntV s th"
+  shows "isV e \<and> actor e = th"
+proof(cases e)
+  case (V th' pty)
+  show ?thesis
+  by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)", 
+        insert assms V, auto simp:cntV_def)
+qed (insert assms, auto simp:cntV_def)
+
+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)
+
+fun the_cs :: "node \<Rightarrow> cs" where
+  "the_cs (Cs cs) = cs"
+
+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
+
+lemma finite_holdents: "finite (holdents s th)"
+  by (unfold holdents_alt_def, insert 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 -
+  { 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
+       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
+      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 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 
+        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:
+  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
+      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
+        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 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 (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"
+  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)
+
+end
+
+
+
+end
+
diff -r 5d8ec128518b -r e3cf792db636 Attic/Moment.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Moment.thy	Tue Jun 14 15:06:16 2016 +0100
@@ -0,0 +1,105 @@
+theory Moment
+imports Main
+begin
+
+definition moment :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where "moment n s = rev (take n (rev s))"
+
+value "moment 3 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9::int]"
+value "moment 2 [5, 4, 3, 2, 1, 0::int]"
+
+lemma moment_app [simp]:
+  assumes ile: "i \<le> length s"
+  shows "moment i (s' @ s) = moment i s"
+using assms unfolding moment_def by simp
+
+lemma moment_eq [simp]: "moment (length s) (s' @ s) = s"
+  unfolding moment_def by simp
+
+lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"
+  by (unfold moment_def, simp)
+
+lemma moment_zero [simp]: "moment 0 s = []"
+  by (simp add:moment_def)
+
+lemma least_idx:
+  assumes "Q (i::nat)"
+  obtains j where "j \<le> i" "Q j" "\<forall> k < j. \<not> Q k"
+  using assms
+  by (metis ex_least_nat_le le0 not_less0) 
+
+lemma duration_idx:
+  assumes "\<not> Q (i::nat)"
+  and "Q j"
+  and "i \<le> j"
+  obtains k where "i \<le> k" "k < j" "\<not> Q k" "\<forall> i'. k < i' \<and> i' \<le> j \<longrightarrow> Q i'" 
+proof -
+  let ?Q = "\<lambda> t. t \<le> j \<and> \<not> Q (j - t)"
+  have "?Q (j - i)" using assms by (simp add: assms(1)) 
+  from least_idx [of ?Q, OF this]
+  obtain l
+  where h: "l \<le> j - i" "\<not> Q (j - l)" "\<forall>k<l. \<not> (k \<le> j \<and> \<not> Q (j - k))"
+    by metis
+  let ?k = "j - l"
+  have "i \<le> ?k" using assms(3) h(1) by linarith 
+  moreover have "?k < j" by (metis assms(2) diff_le_self h(2) le_neq_implies_less) 
+  moreover have "\<not> Q ?k" by (simp add: h(2)) 
+  moreover have "\<forall> i'. ?k < i' \<and> i' \<le> j \<longrightarrow> Q i'"
+      by (metis diff_diff_cancel diff_le_self diff_less_mono2 h(3) 
+              less_imp_diff_less not_less) 
+  ultimately show ?thesis using that by metis
+qed
+
+lemma p_split_gen: 
+  assumes "Q s"
+  and "\<not> Q (moment k s)"
+  shows "(\<exists> i. i < length s \<and> k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
+proof(cases "k \<le> length s")
+  case True
+  let ?Q = "\<lambda> t. Q (moment t s)"
+  have "?Q (length s)" using assms(1) by simp
+  from duration_idx[of ?Q, OF assms(2) this True]
+  obtain i where h: "k \<le> i" "i < length s" "\<not> Q (moment i s)"
+    "\<forall>i'. i < i' \<and> i' \<le> length s \<longrightarrow> Q (moment i' s)" by metis
+  moreover have "(\<forall> i' > i. Q (moment i' s))" using h(4) assms(1) not_less
+    by fastforce
+  ultimately show ?thesis by metis
+qed (insert assms, auto)
+
+lemma p_split: 
+  assumes qs: "Q s"
+  and nq: "\<not> Q []"
+  shows "(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
+proof -
+  from nq have "\<not> Q (moment 0 s)" by simp
+  from p_split_gen [of Q s 0, OF qs this]
+  show ?thesis by auto
+qed
+
+lemma moment_Suc_tl:
+  assumes "Suc i \<le> length s"
+  shows "tl (moment (Suc i) s) = moment i s"
+  using assms 
+  by (simp add:moment_def rev_take, 
+      metis Suc_diff_le diff_Suc_Suc drop_Suc tl_drop)
+
+lemma moment_Suc_hd:
+  assumes "Suc i \<le> length s"
+  shows "hd (moment (Suc i) s) = s!(length s - Suc i)"
+  by (simp add:moment_def rev_take, 
+      subst hd_drop_conv_nth, insert assms, auto)
+  
+lemma moment_plus:
+  assumes "Suc i \<le> length s"
+  shows "(moment (Suc i) s) = (hd (moment (Suc i) s)) # (moment i s)"
+proof -
+  have "(moment (Suc i) s) \<noteq> []" using assms 
+    by (simp add:moment_def rev_take)
+  hence "(moment (Suc i) s) = (hd (moment (Suc i) s)) #  tl (moment (Suc i) s)"
+    by auto
+  with moment_Suc_tl[OF assms]
+  show ?thesis by metis
+qed
+
+end
+
diff -r 5d8ec128518b -r e3cf792db636 CpsG.thy
--- a/CpsG.thy	Tue Jun 14 13:56:51 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,4669 +0,0 @@
-theory CpsG
-imports PIPDefs
-begin
-
-section {* Generic aulxiliary lemmas *}
-
-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 rel_eqI:
-  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
-  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
-  shows "A = B"
-  using assms by auto
-
-section {* Lemmas do not depend on trace validity *}
-
-lemma birth_time_lt:  
-  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
-
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
-  by (unfold s_RAG_def, 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 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)
-
-lemma runing_ready: 
-  shows "runing s \<subseteq> readys s"
-  unfolding runing_def readys_def
-  by auto 
-
-lemma readys_threads:
-  shows "readys s \<subseteq> threads s"
-  unfolding readys_def
-  by auto
-
-lemma wq_v_neq [simp]:
-   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
-  by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-lemma runing_head:
-  assumes "th \<in> runing s"
-  and "th \<in> set (wq_fun (schs s) cs)"
-  shows "th = hd (wq_fun (schs s) cs)"
-  using assms
-  by (simp add:runing_def readys_def s_waiting_def wq_def)
-
-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
-
-lemma isP_E:
-  assumes "isP e"
-  obtains cs where "e = P (actor e) cs"
-  using assms by (cases e, auto)
-
-lemma isV_E:
-  assumes "isV e"
-  obtains cs where "e = V (actor e) cs"
-  using assms by (cases e, auto) 
-
-
-text {*
-  Every thread can only be blocked on one critical resource, 
-  symmetrically, every critical resource can only be held by one thread. 
-  This fact is much more easier according to our definition. 
-*}
-lemma held_unique:
-  assumes "holding (s::event list) th1 cs"
-  and "holding s th2 cs"
-  shows "th1 = th2"
- by (insert assms, unfold s_holding_def, auto)
-
-lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
-  apply (induct s, auto)
-  by (case_tac a, auto split:if_splits)
-
-lemma last_set_unique: 
-  "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
-          \<Longrightarrow> th1 = th2"
-  apply (induct s, auto)
-  by (case_tac a, auto split:if_splits dest:last_set_lt)
-
-lemma preced_unique : 
-  assumes pcd_eq: "preced th1 s = preced th2 s"
-  and th_in1: "th1 \<in> threads s"
-  and th_in2: " th2 \<in> threads s"
-  shows "th1 = th2"
-proof -
-  from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
-  from last_set_unique [OF this th_in1 th_in2]
-  show ?thesis .
-qed
-                      
-lemma preced_linorder: 
-  assumes neq_12: "th1 \<noteq> th2"
-  and th_in1: "th1 \<in> threads s"
-  and th_in2: " th2 \<in> threads s"
-  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
-  from preced_unique [OF _ th_in1 th_in2] and neq_12 
-  have "preced th1 s \<noteq> preced th2 s" by auto
-  thus ?thesis by auto
-qed
-
-lemma in_RAG_E:
-  assumes "(n1, n2) \<in> RAG (s::state)"
-  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
-      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
-  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
-  by auto
-
-lemma count_rec1 [simp]: 
-  assumes "Q e"
-  shows "count Q (e#es) = Suc (count Q es)"
-  using assms
-  by (unfold count_def, auto)
-
-lemma count_rec2 [simp]: 
-  assumes "\<not>Q e"
-  shows "count Q (e#es) = (count Q es)"
-  using assms
-  by (unfold count_def, auto)
-
-lemma count_rec3 [simp]: 
-  shows "count Q [] =  0"
-  by (unfold count_def, auto)
-
-lemma cntP_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"
-proof(cases e)
-  case (P th' pty)
-  show ?thesis
-  by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)", 
-        insert assms P, auto simp:cntP_def)
-qed (insert assms, auto simp:cntP_def)
-  
-lemma cntV_diff_inv:
-  assumes "cntV (e#s) th \<noteq> cntV s th"
-  shows "isV e \<and> actor e = th"
-proof(cases e)
-  case (V th' pty)
-  show ?thesis
-  by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)", 
-        insert assms V, auto simp:cntV_def)
-qed (insert assms, auto simp:cntV_def)
-
-lemma eq_dependants: "dependants (wq s) = dependants s"
-  by (simp add: s_dependants_abv wq_def)
-
-lemma inj_the_preced: 
-  "inj_on (the_preced s) (threads s)"
-  by (metis inj_onI preced_unique the_preced_def)
-
-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
-
-(* ccc *)
-
-section {* Locales used to investigate the execution of PIP *}
-
-text {* 
-  The following locale @{text valid_trace} is used to constrain the 
-  trace to be valid. All properties hold for valid traces are 
-  derived under this locale. 
-*}
-locale valid_trace = 
-  fixes s
-  assumes vt : "vt s"
-
-text {* 
-  The following locale @{text valid_trace_e} describes 
-  the valid extension of a valid trace. The event @{text "e"}
-  represents an event in the system, which corresponds 
-  to a one step operation of the PIP protocol. 
-  It is required that @{text "e"} is an event eligible to happen
-  under state @{text "s"}, which is already required to be valid
-  by the parent locale @{text "valid_trace"}.
-
-  This locale is used to investigate one step execution of PIP, 
-  properties concerning the effects of @{text "e"}'s execution, 
-  for example, how the values of observation functions are changed, 
-  or how desirable properties are kept invariant, are derived
-  under this locale. The state before execution is @{text "s"}, while
-  the state after execution is @{text "e#s"}. Therefore, the lemmas 
-  derived usually relate observations on @{text "e#s"} to those 
-  on @{text "s"}.
-*}
-
-locale valid_trace_e = valid_trace +
-  fixes e
-  assumes vt_e: "vt (e#s)"
-begin
-
-text {*
-  The following lemma shows that @{text "e"} must be a 
-  eligible event (or a valid step) to be taken under
-  the state represented by @{text "s"}.
-*}
-lemma pip_e: "PIP s e"
-  using vt_e by (cases, simp)  
-
-end
-
-text {*
-  Because @{term "e#s"} is also a valid trace, properties 
-  derived for valid trace @{term s} also hold on @{term "e#s"}.
-*}
-sublocale valid_trace_e < vat_es!: valid_trace "e#s" 
-  using vt_e
-  by (unfold_locales, simp)
-
-text {*
-  For each specific event (or operation), there is a sublocale
-  further constraining that the event @{text e} to be that 
-  particular event. 
-
-  For example, the following 
-  locale @{text "valid_trace_create"} is the sublocale for 
-  event @{term "Create"}:
-*}
-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"
-
-text {*
-  locale @{text "valid_trace_p"} is divided further into two 
-  sublocales, namely, @{text "valid_trace_p_h"} 
-  and @{text "valid_trace_p_w"}.
-*}
-
-text {*
-  The following two sublocales @{text "valid_trace_p_h"}
-  and @{text "valid_trace_p_w"} represent two complementary 
-  cases under @{text "valid_trace_p"}, where
-  @{text "valid_trace_p_h"} further constraints that
-  @{text "wq s cs = []"}, which means the waiting queue of 
-  the requested resource @{text "cs"} is empty, in which
-  case,  the requesting thread @{text "th"} 
-  will take hold of @{text "cs"}. 
-
-  Opposite to @{text "valid_trace_p_h"},
-  @{text "valid_trace_p_w"} constraints that
-  @{text "wq s cs \<noteq> []"}, which means the waiting queue of 
-  the requested resource @{text "cs"} is nonempty, in which
-  case,  the requesting thread @{text "th"} will be blocked
-  on @{text "cs"}: 
-
-  Peculiar properties will be derived under respective 
-  locales.
-*}
-
-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
-
-text {*
-  The following @{text "holder"} designates
-  the holder of @{text "cs"} before the @{text "P"}-operation.
-*}
-definition "holder = hd (wq s cs)"
-
-text {*
-  The following @{text "waiters"} designates
-  the list of threads waiting for @{text "cs"} 
-  before the @{text "P"}-operation.
-*}
-definition "waiters = tl (wq s cs)"
-end
-
-text {* 
-  @{text "valid_trace_v"} is set for the @{term V}-operation.
-*}
-locale valid_trace_v = valid_trace_e + 
-  fixes th cs
-  assumes is_v: "e = V th cs"
-begin
-  -- {* The following @{text "rest"} is the tail of 
-        waiting queue of the resource @{text "cs"}
-        to be released by this @{text "V"}-operation.
-     *}
-  definition "rest = tl (wq s cs)"
-
-  text {*
-    The following @{text "wq'"} is the waiting
-    queue of @{term "cs"}
-    after the @{text "V"}-operation, which
-    is simply a reordering of @{term "rest"}. 
-
-    The effect of this reordering needs to be 
-    understood by two cases:
-    \begin{enumerate}
-    \item When @{text "rest = []"},
-    the reordering gives rise to an empty list as well, 
-    which means there is no thread holding or waiting 
-    for resource @{term "cs"}, therefore, it is free.
-
-    \item When @{text "rest \<noteq> []"}, the effect of 
-    this reordering is to arbitrarily 
-    switch one thread in @{term "rest"} to the 
-    head, which, by definition take over the hold
-    of @{term "cs"} and is designated by @{text "taker"}
-    in the following sublocale @{text "valid_trace_v_n"}.
-  *}
-  definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
-
-  text {* 
-  The following @{text "rest'"} is the tail of the 
-  waiting queue after the @{text "V"}-operation. 
-  It plays only auxiliary role to ease reasoning. 
-  *}
-  definition "rest' = tl wq'"
-
-end
-
-text {* 
-  In the following, @{text "valid_trace_v"} is also 
-  divided into two 
-  sublocales: when @{text "rest"} is empty (represented
-  by @{text "valid_trace_v_e"}), which means, there is no thread waiting 
-  for @{text "cs"}, therefore, after the @{text "V"}-operation, 
-  it will become free; otherwise (represented 
-  by @{text "valid_trace_v_n"}), one thread 
-  will be picked from those in @{text "rest"} to take 
-  over @{text "cs"}.
-*}
-
-locale valid_trace_v_e = valid_trace_v +
-  assumes rest_nil: "rest = []"
-
-locale valid_trace_v_n = valid_trace_v +
-  assumes rest_nnl: "rest \<noteq> []"
-begin
-
-text {* 
-  The following @{text "taker"} is the thread to 
-  take over @{text "cs"}. 
-*}
-  definition "taker = hd wq'"
-
-end
-
-
-locale valid_trace_set = valid_trace_e + 
-  fixes th prio
-  assumes is_set: "e = Set th prio"
-
-context valid_trace
-begin
-
-text {*
-  Induction rule introduced to easy the 
-  derivation of properties for valid trace @{term "s"}.
-  One more premises, namely @{term "valid_trace_e s e"}
-  is added, so that an interpretation of 
-  @{text "valid_trace_e"} can be instantiated 
-  so that all properties derived so far becomes 
-  available in the proof of induction step.
-
-  You will see its use in the proofs that follows.
-*}
-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
-
-text {*
-  The following lemma says that if @{text "s"} is a valid state, so 
-  is its any postfix. Where @{term "monent t s"} is the postfix of 
-  @{term "s"} with length @{term "t"}.
-*}
-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
-end
-
-text {*
-  The following locale @{text "valid_moment"} is to inherit the properties 
-  derived on any valid state to the prefix of it, with length @{text "i"}.
-*}
-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)
-
-locale valid_moment_e = valid_moment +
-  assumes less_i: "i < length s"
-begin
-  definition "next_e  = hd (moment (Suc i) s)"
-
-  lemma trace_e: 
-    "moment (Suc i) s = next_e#moment i s"
-   proof -
-    from less_i have "Suc i \<le> length s" by auto
-    from moment_plus[OF this, folded next_e_def]
-    show ?thesis .
-   qed
-
-end
-
-sublocale valid_moment_e < vat_moment_e!: valid_trace_e "moment i s" "next_e"
-  using vt_moment[of "Suc i", unfolded trace_e]
-  by (unfold_locales, simp)
-
-section {* Distinctiveness of waiting queues *}
-
-context valid_trace_create
-begin
-
-lemma wq_kept [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_kept [simp]:
-  shows "wq (e#s) cs' = wq s cs'"
-    using assms unfolding is_exit wq_def
-  by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
-  assumes "distinct (wq s cs')"
-  shows "distinct (wq (e#s) cs')"
-  using assms by simp
-end
-
-context valid_trace_p 
-begin
-
-lemma wq_neq_simp [simp]:
-  assumes "cs' \<noteq> cs"
-  shows "wq (e#s) cs' = wq s cs'"
-    using assms unfolding is_p wq_def
-  by (auto simp:Let_def)
-
-lemma runing_th_s:
-  shows "th \<in> runing s"
-proof -
-  from pip_e[unfolded is_p]
-  show ?thesis by (cases, simp)
-qed
-
-lemma th_not_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 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_kept [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  finite_threads:
-  shows "finite (threads s)"
-  using vt by (induct) (auto elim: step.cases)
-
-lemma finite_readys [simp]: "finite (readys s)"
-  using finite_threads readys_threads rev_finite_subset by blast
-
-lemma wq_distinct: "distinct (wq s cs)"
-proof(induct rule:ind)
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case 
-  proof(cases e)
-    case (Create th prio)
-    interpret vt_create: valid_trace_create s e th prio 
-      using Create by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept) 
-  next
-    case (Exit th)
-    interpret vt_exit: valid_trace_exit s e th  
-        using Exit by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept) 
-  next
-    case (P th cs)
-    interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) 
-  next
-    case (V th cs)
-    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) 
-  next
-    case (Set th prio)
-    interpret vt_set: valid_trace_set s e th prio
-        using Set by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) 
-  qed
-qed (unfold wq_def Let_def, simp)
-
-end
-
-section {* Waiting queues and threads *}
-
-context valid_trace_e
-begin
-
-lemma wq_out_inv: 
-  assumes s_in: "thread \<in> set (wq s cs)"
-  and s_hd: "thread = hd (wq s cs)"
-  and s_i: "thread \<noteq> hd (wq (e#s) cs)"
-  shows "e = V thread cs"
-proof(cases e)
--- {* There are only two non-trivial cases: *}
-  case (V th cs1)
-  show ?thesis
-  proof(cases "cs1 = cs")
-    case True
-    have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
-    thus ?thesis
-    proof(cases)
-      case (thread_V)
-      moreover have "th = thread" using thread_V(2) s_hd
-          by (unfold s_holding_def wq_def, simp)
-      ultimately show ?thesis using V True by simp
-    qed
-  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
-next
-  case (P th cs1)
-  show ?thesis
-  proof(cases "cs1 = cs")
-    case True
-    with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
-      by (auto simp:wq_def Let_def split:if_splits)
-    with s_i s_hd s_in have False
-      by (metis empty_iff hd_append2 list.set(1) wq_def) 
-    thus ?thesis by simp
-  qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
-qed (insert assms, auto simp:wq_def Let_def split:if_splits)
-
-lemma wq_in_inv: 
-  assumes s_ni: "thread \<notin> set (wq s cs)"
-  and s_i: "thread \<in> set (wq (e#s) cs)"
-  shows "e = P thread cs"
-proof(cases e)
-  -- {* This is the only non-trivial case: *}
-  case (V th cs1)
-  have False
-  proof(cases "cs1 = cs")
-    case True
-    show ?thesis
-    proof(cases "(wq s cs1)")
-      case (Cons w_hd w_tl)
-      have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
-      proof -
-        have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
-          using  Cons V by (auto simp:wq_def Let_def True split:if_splits)
-        moreover have "set ... \<subseteq> set (wq s cs)"
-        proof(rule someI2)
-          show "distinct w_tl \<and> set w_tl = set w_tl"
-            by (metis distinct.simps(2) local.Cons wq_distinct)
-        qed (insert Cons True, auto)
-        ultimately show ?thesis by simp
-      qed
-      with assms show ?thesis by auto
-    qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
-  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
-  thus ?thesis by auto
-qed (insert assms, auto simp:wq_def Let_def split:if_splits)
-
-end
-
-lemma (in valid_trace_create)
-  th_not_in_threads: "th \<notin> threads s"
-proof -
-  from pip_e[unfolded is_create]
-  show ?thesis by (cases, simp)
-qed
-
-lemma (in valid_trace_create)
-  threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
-  by (unfold is_create, simp)
-
-lemma (in valid_trace_exit)
-  threads_es [simp]: "threads (e#s) = threads s - {th}"
-  by (unfold is_exit, simp)
-
-lemma (in valid_trace_p)
-  threads_es [simp]: "threads (e#s) = threads s"
-  by (unfold is_p, simp)
-
-lemma (in valid_trace_v)
-  threads_es [simp]: "threads (e#s) = threads s"
-  by (unfold is_v, simp)
-
-lemma (in valid_trace_v)
-  th_not_in_rest[simp]: "th \<notin> set rest"
-proof
-  assume otherwise: "th \<in> set rest"
-  have "distinct (wq s cs)" by (simp add: wq_distinct)
-  from this[unfolded wq_s_cs] and otherwise
-  show False by auto
-qed
-
-lemma (in valid_trace_v) distinct_rest: "distinct rest"
-  by (simp add: distinct_tl rest_def wq_distinct)
-
-lemma (in valid_trace_v)
-  set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
-proof(unfold wq_es_cs wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest) 
-next
-  fix x
-  assume "distinct x \<and> set x = set rest"
-  thus "set x = set (wq s cs) - {th}" 
-      by (unfold wq_s_cs, simp)
-qed
-
-lemma (in valid_trace_exit)
-  th_not_in_wq: "th \<notin> set (wq s cs)"
-proof -
-  from pip_e[unfolded is_exit]
-  show ?thesis
-  by (cases, unfold holdents_def s_holding_def, fold wq_def, 
-             auto elim!:runing_wqE)
-qed
-
-lemma (in valid_trace) wq_threads: 
-  assumes "th \<in> set (wq s cs)"
-  shows "th \<in> threads s"
-  using assms
-proof(induct rule:ind)
-  case (Nil)
-  thus ?case by (auto simp:wq_def)
-next
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case
-  proof(cases e)
-    case (Create th' prio')
-    interpret vt: valid_trace_create s e th' prio'
-      using Create by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems by auto
-  next
-    case (Exit th')
-    interpret vt: valid_trace_exit s e th'
-      using Exit by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto 
-  next
-    case (P th' cs')
-    interpret vt: valid_trace_p s e th' cs'
-      using P by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems readys_threads 
-        runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv 
-        by fastforce 
-  next
-    case (V th' cs')
-    interpret vt: valid_trace_v s e th' cs'
-      using V by (unfold_locales, simp)
-    show ?thesis using Cons
-      using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
-  next
-    case (Set th' prio)
-    interpret vt: valid_trace_set s e th' prio
-      using Set by (unfold_locales, simp)
-    show ?thesis using Cons.hyps(2) Cons.prems vt.is_set 
-        by (auto simp:wq_def Let_def)
-  qed
-qed 
-
-section {* RAG and threads *}
-
-context valid_trace
-begin
-
-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
-
-section {* The change of @{term RAG} *}
-
-text {*
-  The following three lemmas show that @{text "RAG"} does not change
-  by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
-  events, respectively.
-*}
-
-lemma (in valid_trace_set) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s"
-   by (unfold is_set s_RAG_def s_waiting_def wq_def, simp add:Let_def)
-
-lemma (in valid_trace_create) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s"
- by (unfold is_create s_RAG_def s_waiting_def wq_def, simp add:Let_def)
-
-lemma (in valid_trace_exit) RAG_unchanged[simp]: "(RAG (e # s)) = RAG s"
-  by (unfold is_exit s_RAG_def s_waiting_def wq_def, simp add:Let_def)
-
-context valid_trace_v
-begin
-
-lemma holding_cs_eq_th:
-  assumes "holding s t cs"
-  shows "t = th"
-proof -
-  from pip_e[unfolded is_v]
-  show ?thesis
-  proof(cases)
-    case (thread_V)
-    from held_unique[OF this(2) assms]
-    show ?thesis by simp
-  qed
-qed
-
-lemma distinct_wq': "distinct wq'"
-  by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
-  
-lemma set_wq': "set wq' = set rest"
-  by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def)
-    
-lemma th'_in_inv:
-  assumes "th' \<in> set wq'"
-  shows "th' \<in> set rest"
-  using assms set_wq' by simp
-
-lemma runing_th_s:
-  shows "th \<in> runing s"
-proof -
-  from pip_e[unfolded is_v]
-  show ?thesis by (cases, simp)
-qed
-
-lemma neq_t_th: 
-  assumes "waiting (e#s) t c"
-  shows "t \<noteq> th"
-proof
-  assume otherwise: "t = th"
-  show False
-  proof(cases "c = cs")
-    case True
-    have "t \<in> set wq'" 
-     using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
-     by simp 
-    from th'_in_inv[OF this] have "t \<in> set rest" .
-    with wq_s_cs[folded otherwise] wq_distinct[of cs]
-    show ?thesis by simp
-  next
-    case False
-    have "wq (e#s) c = wq s c" using False
-        by (unfold is_v, simp)
-    hence "waiting s t c" using assms 
-        by (simp add: cs_waiting_def waiting_eq)
-    hence "t \<notin> readys s" by (unfold readys_def, auto)
-    hence "t \<notin> runing s" using runing_ready by auto 
-    with runing_th_s[folded otherwise] show ?thesis by auto 
-  qed
-qed
-
-lemma waiting_esI1:
-  assumes "waiting s t c"
-      and "c \<noteq> cs" 
-  shows "waiting (e#s) t c" 
-proof -
-  have "wq (e#s) c = wq s c" 
-    using assms(2) is_v by auto
-  with assms(1) show ?thesis 
-    using cs_waiting_def waiting_eq by auto 
-qed
-
-lemma holding_esI2:
-  assumes "c \<noteq> cs" 
-  and "holding s t c"
-  shows "holding (e#s) t c"
-proof -
-  from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
-  from assms(2)[unfolded s_holding_def, folded wq_def, 
-                folded this, unfolded wq_def, folded s_holding_def]
-  show ?thesis .
-qed
-
-lemma holding_esI1:
-  assumes "holding s t c"
-  and "t \<noteq> th"
-  shows "holding (e#s) t c"
-proof -
-  have "c \<noteq> cs" using assms using holding_cs_eq_th by blast 
-  from holding_esI2[OF this assms(1)]
-  show ?thesis .
-qed
-
-end
-
-context valid_trace_v_n
-begin
-
-lemma neq_wq': "wq' \<noteq> []" 
-proof (unfold wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest) 
-next
-  fix x
-  assume " distinct x \<and> set x = set rest" 
-  thus "x \<noteq> []" using rest_nnl by auto
-qed 
-
-lemma eq_wq': "wq' = taker # rest'"
-  by (simp add: neq_wq' rest'_def taker_def)
-
-lemma next_th_taker: 
-  shows "next_th s th cs taker"
-  using rest_nnl taker_def wq'_def wq_s_cs 
-  by (auto simp:next_th_def)
-
-lemma taker_unique: 
-  assumes "next_th s th cs taker'"
-  shows "taker' = taker"
-proof -
-  from assms
-  obtain rest' where 
-    h: "wq s cs = th # rest'" 
-       "taker' = hd (SOME q. distinct q \<and> set q = set rest')"
-          by (unfold next_th_def, auto)
-  with wq_s_cs have "rest' = rest" by auto
-  thus ?thesis using h(2) taker_def wq'_def by auto 
-qed
-
-lemma waiting_set_eq:
-  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
-  by (smt all_not_in_conv bot.extremum insertI1 insert_subset 
-      mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
-
-lemma holding_set_eq:
-  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {(Cs cs, Th taker)}"
-  using next_th_taker taker_def waiting_set_eq 
-  by fastforce
-   
-lemma holding_taker:
-  shows "holding (e#s) taker cs"
-    by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, 
-        auto simp:neq_wq' taker_def)
-
-lemma waiting_esI2:
-  assumes "waiting s t cs"
-      and "t \<noteq> taker"
-  shows "waiting (e#s) t cs" 
-proof -
-  have "t \<in> set wq'" 
-  proof(unfold wq'_def, rule someI2)
-    show "distinct rest \<and> set rest = set rest"
-          by (simp add: distinct_rest)
-  next
-    fix x
-    assume "distinct x \<and> set x = set rest"
-    moreover have "t \<in> set rest"
-        using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto 
-    ultimately show "t \<in> set x" by simp
-  qed
-  moreover have "t \<noteq> hd wq'"
-    using assms(2) taker_def by auto 
-  ultimately show ?thesis
-    by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
-qed
-
-lemma waiting_esE:
-  assumes "waiting (e#s) t c" 
-  obtains "c \<noteq> cs" "waiting s t c"
-     |    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
-proof(cases "c = cs")
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
-  from that(1)[OF False this] show ?thesis .
-next
-  case True
-  from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
-  have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
-  hence "t \<noteq> taker" by (simp add: taker_def) 
-  moreover hence "t \<noteq> th" using assms neq_t_th by blast 
-  moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv) 
-  ultimately have "waiting s t cs"
-    by (metis cs_waiting_def list.distinct(2) list.sel(1) 
-                list.set_sel(2) rest_def waiting_eq wq_s_cs)  
-  show ?thesis using that(2)
-  using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto   
-qed
-
-lemma holding_esI1:
-  assumes "c = cs"
-  and "t = taker"
-  shows "holding (e#s) t c"
-  by (unfold assms, simp add: holding_taker)
-
-lemma holding_esE:
-  assumes "holding (e#s) t c" 
-  obtains "c = cs" "t = taker"
-      | "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
-  case True
-  from assms[unfolded True, unfolded s_holding_def, 
-             folded wq_def, unfolded wq_es_cs]
-  have "t = taker" by (simp add: taker_def) 
-  from that(1)[OF True this] show ?thesis .
-next
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  from assms[unfolded s_holding_def, folded wq_def, 
-             unfolded this, unfolded wq_def, folded s_holding_def]
-  have "holding s t c"  .
-  from that(2)[OF False this] show ?thesis .
-qed
-
-end 
-
-
-context valid_trace_v_e
-begin
-
-lemma nil_wq': "wq' = []" 
-proof (unfold wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest) 
-next
-  fix x
-  assume " distinct x \<and> set x = set rest" 
-  thus "x = []" using rest_nil by auto
-qed 
-
-lemma no_taker: 
-  assumes "next_th s th cs taker"
-  shows "False"
-proof -
-  from assms[unfolded next_th_def]
-  obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
-    by auto
-  thus ?thesis using rest_def rest_nil by auto 
-qed
-
-lemma waiting_set_eq:
-  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
-  using no_taker by auto
-
-lemma holding_set_eq:
-  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
-  using no_taker by auto
-   
-lemma no_holding:
-  assumes "holding (e#s) taker cs"
-  shows False
-proof -
-  from wq_es_cs[unfolded nil_wq']
-  have " wq (e # s) cs = []" .
-  from assms[unfolded s_holding_def, folded wq_def, unfolded this]
-  show ?thesis by auto
-qed
-
-lemma no_waiting:
-  assumes "waiting (e#s) t cs"
-  shows False
-proof -
-  from wq_es_cs[unfolded nil_wq']
-  have " wq (e # s) cs = []" .
-  from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
-  show ?thesis by auto
-qed
-
-lemma waiting_esI2:
-  assumes "waiting s t c"
-  shows "waiting (e#s) t c"
-proof -
-  have "c \<noteq> cs" using assms
-    using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto 
-  from waiting_esI1[OF assms this]
-  show ?thesis .
-qed
-
-lemma waiting_esE:
-  assumes "waiting (e#s) t c" 
-  obtains "c \<noteq> cs" "waiting s t c"
-proof(cases "c = cs")
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
-  from that(1)[OF False this] show ?thesis .
-next
-  case True
-  from no_waiting[OF assms[unfolded True]]
-  show ?thesis by auto
-qed
-
-lemma holding_esE:
-  assumes "holding (e#s) t c" 
-  obtains "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
-  case True
-  from no_holding[OF assms[unfolded True]] 
-  show ?thesis by auto
-next
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  from assms[unfolded s_holding_def, folded wq_def, 
-             unfolded this, unfolded wq_def, folded s_holding_def]
-  have "holding s t c"  .
-  from that[OF False this] show ?thesis .
-qed
-
-end 
-
-  
-context valid_trace_v
-begin
-
-lemma RAG_es:
-  "RAG (e # s) =
-   RAG s - {(Cs cs, Th th)} -
-     {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-     {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
-proof(rule rel_eqI)
-  fix n1 n2
-  assume "(n1, n2) \<in> ?L"
-  thus "(n1, n2) \<in> ?R"
-  proof(cases rule:in_RAG_E)
-    case (waiting th' cs')
-    show ?thesis
-    proof(cases "rest = []")
-      case False
-      interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-      from waiting(3)
-      show ?thesis
-      proof(cases rule:h_n.waiting_esE)
-        case 1
-        with waiting(1,2)
-        show ?thesis
-        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      next
-        case 2
-        with waiting(1,2)
-        show ?thesis
-         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      qed
-    next
-      case True
-      interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-      from waiting(3)
-      show ?thesis
-      proof(cases rule:h_e.waiting_esE)
-        case 1
-        with waiting(1,2)
-        show ?thesis
-        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      qed
-    qed
-  next
-    case (holding th' cs')
-    show ?thesis
-    proof(cases "rest = []")
-      case False
-      interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-      from holding(3)
-      show ?thesis
-      proof(cases rule:h_n.holding_esE)
-        case 1
-        with holding(1,2)
-        show ?thesis
-        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      next
-        case 2
-        with holding(1,2)
-        show ?thesis
-         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold holding_eq, auto)
-      qed
-    next
-      case True
-      interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-      from holding(3)
-      show ?thesis
-      proof(cases rule:h_e.holding_esE)
-        case 1
-        with holding(1,2)
-        show ?thesis
-        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
-             fold holding_eq, auto)
-      qed
-    qed
-  qed
-next
-  fix n1 n2
-  assume h: "(n1, n2) \<in> ?R"
-  show "(n1, n2) \<in> ?L"
-  proof(cases "rest = []")
-    case False
-    interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-    from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
-    have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
-                            \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or> 
-          (n2 = Th h_n.taker \<and> n1 = Cs cs)" 
-      by auto
-   thus ?thesis
-   proof
-      assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
-      with h_n.holding_taker
-      show ?thesis 
-        by (unfold s_RAG_def, fold holding_eq, auto)
-   next
-    assume h: "(n1, n2) \<in> RAG s \<and>
-        (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
-    hence "(n1, n2) \<in> RAG s" by simp
-    thus ?thesis
-    proof(cases rule:in_RAG_E)
-      case (waiting th' cs')
-      from h and this(1,2)
-      have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
-      hence "waiting (e#s) th' cs'" 
-      proof
-        assume "cs' \<noteq> cs"
-        from waiting_esI1[OF waiting(3) this] 
-        show ?thesis .
-      next
-        assume neq_th': "th' \<noteq> h_n.taker"
-        show ?thesis
-        proof(cases "cs' = cs")
-          case False
-          from waiting_esI1[OF waiting(3) this] 
-          show ?thesis .
-        next
-          case True
-          from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
-          show ?thesis .
-        qed
-      qed
-      thus ?thesis using waiting(1,2)
-        by (unfold s_RAG_def, fold waiting_eq, auto)
-    next
-      case (holding th' cs')
-      from h this(1,2)
-      have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
-      hence "holding (e#s) th' cs'"
-      proof
-        assume "cs' \<noteq> cs"
-        from holding_esI2[OF this holding(3)] 
-        show ?thesis .
-      next
-        assume "th' \<noteq> th"
-        from holding_esI1[OF holding(3) this]
-        show ?thesis .
-      qed
-      thus ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-   qed
- next
-   case True
-   interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-   from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
-   have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)" 
-      by auto
-   from h_s(1)
-   show ?thesis
-   proof(cases rule:in_RAG_E)
-    case (waiting th' cs')
-    from h_e.waiting_esI2[OF this(3)]
-    show ?thesis using waiting(1,2)
-      by (unfold s_RAG_def, fold waiting_eq, auto)
-   next
-    case (holding th' cs')
-    with h_s(2)
-    have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
-    thus ?thesis
-    proof
-      assume neq_cs: "cs' \<noteq> cs"
-      from holding_esI2[OF this holding(3)]
-      show ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    next
-      assume "th' \<noteq> th"
-      from holding_esI1[OF holding(3) this]
-      show ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-   qed
- qed
-qed
-
-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_p
-begin
-
-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 
-
-lemma (in valid_trace_p) th_not_waiting: "\<not> waiting s th c"
-proof -
-  have "th \<in> readys s"
-    using runing_ready runing_th_s by blast 
-  thus ?thesis
-    by (unfold readys_def, auto)
-qed
-
-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_w
-begin
-
-lemma wq_s_cs: "wq s cs = holder#waiters"
-    by (simp add: holder_def waiters_def wne)
-    
-lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
-  by (simp add: wq_es_cs wq_s_cs)
-
-lemma waiting_es_th_cs: "waiting (e#s) th cs"
-  using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
-
-lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
-   by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
-
-lemma holding_esE:
-  assumes "holding (e#s) th' cs'"
-  obtains "holding s th' cs'"
-  using assms 
-proof(cases "cs' = cs")
-  case False
-  hence "wq (e#s) cs' = wq s cs'" by simp
-  with assms show ?thesis
-    using cs_holding_def holding_eq that by auto 
-next
-  case True
-  with assms show ?thesis
-  by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that 
-        wq_es_cs' wq_s_cs) 
-qed
-
-lemma waiting_esE:
-  assumes "waiting (e#s) th' cs'"
-  obtains "th' \<noteq> th" "waiting s th' cs'"
-     |  "th' = th" "cs' = cs"
-proof(cases "waiting s th' cs'")
-  case True
-  have "th' \<noteq> th"
-  proof
-    assume otherwise: "th' = th"
-    from True[unfolded this]
-    show False by (simp add: th_not_waiting)
-  qed
-  from that(1)[OF this True] show ?thesis .
-next
-  case False
-  hence "th' = th \<and> cs' = cs"
-      by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) 
-        set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
-  with that(2) show ?thesis by metis
-qed
-
-lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
-proof(rule rel_eqI)
-  fix n1 n2
-  assume "(n1, n2) \<in> ?L"
-  thus "(n1, n2) \<in> ?R" 
-  proof(cases rule:in_RAG_E)
-    case (waiting th' cs')
-    from this(3)
-    show ?thesis
-    proof(cases rule:waiting_esE)
-      case 1
-      thus ?thesis using waiting(1,2)
-        by (unfold s_RAG_def, fold waiting_eq, auto)
-    next
-      case 2
-      thus ?thesis using waiting(1,2) by auto
-    qed
-  next
-    case (holding th' cs')
-    from this(3)
-    show ?thesis
-    proof(cases rule:holding_esE)
-      case 1
-      with holding(1,2)
-      show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-  qed
-next
-  fix n1 n2
-  assume "(n1, n2) \<in> ?R"
-  hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
-  thus "(n1, n2) \<in> ?L"
-  proof
-    assume "(n1, n2) \<in> RAG s"
-    thus ?thesis
-    proof(cases rule:in_RAG_E)
-      case (waiting th' cs')
-      from waiting_kept[OF this(3)]
-      show ?thesis using waiting(1,2)
-         by (unfold s_RAG_def, fold waiting_eq, auto)
-    next
-      case (holding th' cs')
-      from holding_kept[OF this(3)]
-      show ?thesis using holding(1,2)
-         by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-  next
-    assume "n1 = Th th \<and> n2 = Cs cs"
-    thus ?thesis using RAG_edge by auto
-  qed
-qed
-
-end
-
-context valid_trace_p
-begin
-
-lemma RAG_es: "RAG (e # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
-                                                  else RAG s \<union> {(Th th, Cs cs)})"
-proof(cases "wq s cs = []")
-  case True
-  interpret vt_p: valid_trace_p_h using True
-    by (unfold_locales, simp)
-  show ?thesis by (simp add: vt_p.RAG_es vt_p.we) 
-next
-  case False
-  interpret vt_p: valid_trace_p_w using False
-    by (unfold_locales, simp)
-  show ?thesis by (simp add: vt_p.RAG_es vt_p.wne) 
-qed
-
-end
-
-section {* Finiteness of RAG *}
-
-context valid_trace
-begin
-
-lemma finite_RAG:
-  shows "finite (RAG s)"
-proof(induct rule:ind)
-  case Nil
-  show ?case 
-  by (auto simp: s_RAG_def cs_waiting_def 
-                   cs_holding_def wq_def acyclic_def)
-next
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case
-  proof(cases e)
-    case (Create th prio)
-    interpret vt: valid_trace_create s e th prio using Create
-      by (unfold_locales, simp)
-    show ?thesis using Cons by simp
-  next
-    case (Exit th)
-    interpret vt: valid_trace_exit s e th using Exit
-      by (unfold_locales, simp)
-    show ?thesis using Cons by simp
-  next
-    case (P th cs)
-    interpret vt: valid_trace_p s e th cs using P
-      by (unfold_locales, simp)
-    show ?thesis using Cons using vt.RAG_es by auto 
-  next
-    case (V th cs)
-    interpret vt: valid_trace_v s e th cs using V
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.finite_RAG_kept) 
-  next
-    case (Set th prio)
-    interpret vt: valid_trace_set s e th prio using Set
-      by (unfold_locales, simp)
-    show ?thesis using Cons by simp
-  qed
-qed
-end
-
-section {* RAG is acyclic *}
-
-text {* (* ddd *)
-  The nature of the work is like this: since it starts from a very simple and basic 
-  model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
-  For instance, the fact 
-  that one thread can not be blocked by two critical resources at the same time
-  is obvious, because only running threads can make new requests, if one is waiting for 
-  a critical resource and get blocked, it can not make another resource request and get 
-  blocked the second time (because it is not running). 
-
-  To derive this fact, one needs to prove by contraction and 
-  reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
-  named @{text "p_split"}, which is about status changing along the time axis. It says if 
-  a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
-  but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} 
-  in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history 
-  of events leading to it), such that @{text "Q"} switched 
-  from being @{text "False"} to @{text "True"} and kept being @{text "True"}
-  till the last moment of @{text "s"}.
-
-  Suppose a thread @{text "th"} is blocked
-  on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, 
-  since no thread is blocked at the very beginning, by applying 
-  @{text "p_split"} to these two blocking facts, there exist 
-  two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that 
-  @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} 
-  and kept on blocked on them respectively ever since.
- 
-  Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
-  However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
-  in blocked state at moment @{text "t2"} and could not
-  make any request and get blocked the second time: Contradiction.
-*}
-
-
-context valid_trace
-begin
-
-lemma waiting_unique_pre: (* ddd *)
-  assumes h11: "thread \<in> set (wq s cs1)"
-  and h12: "thread \<noteq> hd (wq s cs1)"
-  assumes h21: "thread \<in> set (wq s cs2)"
-  and h22: "thread \<noteq> hd (wq s cs2)"
-  and neq12: "cs1 \<noteq> cs2"
-  shows "False"
-proof -
-  let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
-  from h11 and h12 have q1: "?Q cs1 s" by simp
-  from h21 and h22 have q2: "?Q cs2 s" by simp
-  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
-  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
-  from p_split [of "?Q cs1", OF q1 nq1]
-  obtain t1 where lt1: "t1 < length s"
-    and np1: "\<not> ?Q cs1 (moment t1 s)"
-    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
-  from p_split [of "?Q cs2", OF q2 nq2]
-  obtain t2 where lt2: "t2 < length s"
-    and np2: "\<not> ?Q cs2 (moment t2 s)"
-    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
-  { fix s cs
-    assume q: "?Q cs s"
-    have "thread \<notin> runing s"
-    proof
-      assume "thread \<in> runing s"
-      hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and> 
-                 thread \<noteq> hd (wq_fun (schs s) cs))"
-        by (unfold runing_def s_waiting_def readys_def, auto)
-      from this[rule_format, of cs] q 
-      show False by (simp add: wq_def) 
-    qed
-  } note q_not_runing = this
-  { fix t1 t2 cs1 cs2
-    assume  lt1: "t1 < length s"
-    and np1: "\<not> ?Q cs1 (moment t1 s)"
-    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
-    and lt2: "t2 < length s"
-    and np2: "\<not> ?Q cs2 (moment t2 s)"
-    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
-    and lt12: "t1 < t2"
-    let ?t3 = "Suc t2" 
-    interpret ve2: valid_moment_e _ t2 using lt2
-     by (unfold_locales, simp)
-    let ?e = ve2.next_e
-    have "t2 < ?t3" by simp
-    from nn2 [rule_format, OF this] and ve2.trace_e
-    have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" and
-         h2: "thread \<noteq> hd (wq (?e#moment t2 s) cs2)" by auto
-    have ?thesis
-    proof -
-      have "thread \<in> runing (moment t2 s)"
-      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-        case True
-        have "?e = V thread cs2"
-        proof -
-          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
-              using True and np2  by auto 
-          thus ?thesis
-            using True h2 ve2.vat_moment_e.wq_out_inv by blast 
-        qed
-        thus ?thesis
-          using step.cases ve2.vat_moment_e.pip_e by auto 
-      next
-        case False
-        hence "?e = P thread cs2"
-          using h1 ve2.vat_moment_e.wq_in_inv by blast 
-        thus ?thesis
-          using step.cases ve2.vat_moment_e.pip_e by auto 
-      qed
-      moreover have "thread \<notin> runing (moment t2 s)"
-        by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
-      ultimately show ?thesis by simp
-    qed
-  } note lt_case = this
-  show ?thesis
-  proof -
-    { assume "t1 < t2"
-      from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
-      have ?thesis .
-    } moreover {
-      assume "t2 < t1"
-      from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
-      have ?thesis .
-    } moreover { 
-      assume eq_12: "t1 = t2"
-      let ?t3 = "Suc t2"
-      interpret ve2: valid_moment_e _ t2 using lt2
-        by (unfold_locales, simp)
-      let ?e = ve2.next_e
-      have "t2 < ?t3" by simp
-      from nn2 [rule_format, OF this] and ve2.trace_e
-      have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" by auto
-      have lt_2: "t2 < ?t3" by simp
-      from nn2 [rule_format, OF this] and ve2.trace_e
-      have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" and
-           h2: "thread \<noteq> hd (wq (?e#moment t2 s) cs2)" by auto
-      from nn1[rule_format, OF lt_2[folded eq_12], unfolded ve2.trace_e[folded eq_12]] 
-           eq_12[symmetric]
-      have g1: "thread \<in> set (wq (?e#moment t1 s) cs1)" and
-           g2: "thread \<noteq> hd (wq (?e#moment t1 s) cs1)" by auto
-      have "?e = V thread cs2 \<or> ?e = P thread cs2"
-        using h1 h2 np2 ve2.vat_moment_e.wq_in_inv 
-              ve2.vat_moment_e.wq_out_inv by blast
-      moreover have "?e = V thread cs1 \<or> ?e = P thread cs1"
-        using eq_12 g1 g2 np1 ve2.vat_moment_e.wq_in_inv 
-              ve2.vat_moment_e.wq_out_inv by blast
-      ultimately have ?thesis using neq12 by auto
-    } ultimately show ?thesis using nat_neq_iff by blast 
-  qed
-qed
-
-text {*
-  This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
-*}
-
-lemma waiting_unique:
-  assumes "waiting s th cs1"
-  and "waiting s th cs2"
-  shows "cs1 = cs2"
-  using waiting_unique_pre assms
-  unfolding wq_def s_waiting_def
-  by auto
-
-end
-
-lemma (in valid_trace_v)
-  preced_es [simp]: "preced th (e#s) = preced th s"
-  by (unfold is_v preced_def, simp)
-
-lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
-proof
-  fix th'
-  show "the_preced (V th cs # s) th' = the_preced s th'"
-    by (unfold the_preced_def preced_def, simp)
-qed
-
-
-lemma (in valid_trace_v)
-  the_preced_es: "the_preced (e#s) = the_preced s"
-  by (unfold is_v preced_def, simp)
-
-context valid_trace_p
-begin
-
-lemma not_holding_s_th_cs: "\<not> holding s th cs"
-proof
-  assume otherwise: "holding s th cs"
-  from pip_e[unfolded is_p]
-  show False
-  proof(cases)
-    case (thread_P)
-    moreover have "(Cs cs, Th th) \<in> RAG s"
-      using otherwise cs_holding_def 
-            holding_eq th_not_in_wq by auto
-    ultimately show ?thesis by auto
-  qed
-qed
-
-end
-
-
-lemma (in valid_trace_v_n) finite_waiting_set:
-  "finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
-    by (simp add: waiting_set_eq)
-
-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_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 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 
-  next
-    case (Exit th)
-    interpret vt: valid_trace_exit s e th using Exit
-      by (unfold_locales, simp)
-    show ?thesis using Cons by simp
-  next
-    case (P th cs)
-    interpret vt: valid_trace_p s e th cs using P
-      by (unfold_locales, simp)
-    show ?thesis
-    proof(cases "wq s cs = []")
-      case True
-      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
-      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 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 (Set th prio)
-    interpret vt: valid_trace_set s e th prio using Set
-      by (unfold_locales, simp)
-    show ?thesis using Cons by simp 
-  qed
-qed
-
-end
-
-section {* RAG is single-valued *}
-
-context valid_trace
-begin
-
-lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
-  apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
-  by(auto elim:waiting_unique held_unique)
-
-lemma sgv_RAG: "single_valued (RAG s)"
-  using unique_RAG by (auto simp:single_valued_def)
-
-end
-
-section {* RAG is well-founded *}
-
-context valid_trace
-begin
-
-lemma wf_RAG: "wf (RAG s)"
-proof(rule finite_acyclic_wf)
-  from finite_RAG show "finite (RAG s)" .
-next
-  from acyclic_RAG show "acyclic (RAG s)" .
-qed
-
-lemma wf_RAG_converse: 
-  shows "wf ((RAG s)^-1)"
-proof(rule finite_acyclic_wf_converse)
-  from finite_RAG 
-  show "finite (RAG s)" .
-next
-  from acyclic_RAG
-  show "acyclic (RAG s)" .
-qed
-
-end
-
-section {* RAG forms a forest (or tree) *}
-
-context valid_trace
-begin
-
-lemma rtree_RAG: "rtree (RAG s)"
-  using sgv_RAG acyclic_RAG
-  by (unfold rtree_def rtree_axioms_def sgv_def, auto)
-
-end
-
-sublocale valid_trace < rtree_RAG: rtree "RAG s"
-  using rtree_RAG .
-
-sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
-proof -
-  show "fsubtree (RAG s)"
-  proof(intro_locales)
-    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
-  next
-    show "fsubtree_axioms (RAG s)"
-    proof(unfold fsubtree_axioms_def)
-      from wf_RAG show "wf (RAG s)" .
-    qed
-  qed
-qed
-
-
-section {* Derived properties for parts of RAG *}
-
-context valid_trace
-begin
-
-lemma acyclic_tRAG: "acyclic (tRAG s)"
-proof(unfold tRAG_def, rule acyclic_compose)
-  show "acyclic (RAG s)" using acyclic_RAG .
-next
-  show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-next
-  show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-qed
-
-lemma sgv_wRAG: "single_valued (wRAG s)"
-  using waiting_unique
-  by (unfold single_valued_def wRAG_def, auto)
-
-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)
-
-end
-
-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 < 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 tRAG_nodeE:
-  assumes "(n1, n2) \<in> tRAG s"
-  obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
-  using assms
-  by (auto simp: tRAG_def wRAG_def hRAG_def)
-
-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 subtree_nodeE:
-  assumes "n \<in> subtree (tRAG s) (Th th)"
-  obtains th1 where "n = Th th1"
-proof -
-  show ?thesis
-  proof(rule subtreeE[OF assms])
-    assume "n = Th th"
-    from that[OF this] show ?thesis .
-  next
-    assume "Th th \<in> ancestors (tRAG s) n"
-    hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
-    hence "\<exists> th1. n = Th th1"
-    proof(induct)
-      case (base y)
-      from tRAG_nodeE[OF this] show ?case by metis
-    next
-      case (step y z)
-      thus ?case by auto
-    qed
-    with that show ?thesis by auto
-  qed
-qed
-
-lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
-proof -
-  have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*" 
-    by (rule rtrancl_mono, auto simp:RAG_split)
-  also have "... \<subseteq> ((RAG s)^*)^*"
-    by (rule rtrancl_mono, auto)
-  also have "... = (RAG s)^*" by simp
-  finally show ?thesis by (unfold tRAG_def, simp)
-qed
-
-lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
-proof -
-  { fix a
-    assume "a \<in> subtree (tRAG s) x"
-    hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
-    with tRAG_star_RAG
-    have "(a, x) \<in> (RAG s)^*" by auto
-    hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
-  } thus ?thesis by auto
-qed
-
-lemma tRAG_trancl_eq:
-   "{th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
-    {th'. (Th th', Th th)  \<in> (RAG s)^+}"
-   (is "?L = ?R")
-proof -
-  { fix th'
-    assume "th' \<in> ?L"
-    hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
-    from tranclD[OF this]
-    obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
-    from tRAG_subtree_RAG and this(2)
-    have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG) 
-    moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto 
-    ultimately have "th' \<in> ?R"  by auto 
-  } moreover 
-  { fix th'
-    assume "th' \<in> ?R"
-    hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
-    from plus_rpath[OF this]
-    obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
-    hence "(Th th', Th th) \<in> (tRAG s)^+"
-    proof(induct xs arbitrary:th' th rule:length_induct)
-      case (1 xs th' th)
-      then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
-      show ?case
-      proof(cases "xs1")
-        case Nil
-        from 1(2)[unfolded Cons1 Nil]
-        have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
-        hence "(Th th', x1) \<in> (RAG s)" 
-          by (cases, auto)
-        then obtain cs where "x1 = Cs cs" 
-              by (unfold s_RAG_def, auto)
-        from rpath_nnl_lastE[OF rp[unfolded this]]
-        show ?thesis by auto
-      next
-        case (Cons x2 xs2)
-        from 1(2)[unfolded Cons1[unfolded this]]
-        have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
-        from rpath_edges_on[OF this]
-        have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
-        have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
-            by (simp add: edges_on_unfold)
-        with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
-        then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
-        have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
-            by (simp add: edges_on_unfold)
-        from this eds
-        have rg2: "(x1, x2) \<in> RAG s" by auto
-        from this[unfolded eq_x1] 
-        obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
-        from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
-        have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
-        from rp have "rpath (RAG s) x2 xs2 (Th th)"
-           by  (elim rpath_ConsE, simp)
-        from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
-        show ?thesis
-        proof(cases "xs2 = []")
-          case True
-          from rpath_nilE[OF rp'[unfolded this]]
-          have "th1 = th" by auto
-          from rt1[unfolded this] show ?thesis by auto
-        next
-          case False
-          from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
-          have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
-          with rt1 show ?thesis by auto
-        qed
-      qed
-    qed
-    hence "th' \<in> ?L" by auto
-  } ultimately show ?thesis by blast
-qed
-
-lemma tRAG_trancl_eq_Th:
-   "{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
-    {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}"
-    using tRAG_trancl_eq by auto
-
-
-lemma tRAG_Field:
-  "Field (tRAG s) \<subseteq> Field (RAG s)"
-  by (unfold tRAG_alt_def Field_def, auto)
-
-lemma tRAG_mono:
-  assumes "RAG s' \<subseteq> RAG s"
-  shows "tRAG s' \<subseteq> tRAG s"
-  using assms 
-  by (unfold tRAG_alt_def, auto)
-
-lemma tRAG_subtree_eq: 
-   "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th'  \<in> (subtree (RAG s) (Th th))}"
-   (is "?L = ?R")
-proof -
-  { fix n 
-    assume h: "n \<in> ?L"
-    hence "n \<in> ?R"
-    by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) 
-  } moreover {
-    fix n
-    assume "n \<in> ?R"
-    then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
-      by (auto simp:subtree_def)
-    from rtranclD[OF this(2)]
-    have "n \<in> ?L"
-    proof
-      assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
-      with h have "n \<in> {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}" by auto
-      thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
-    qed (insert h, auto simp:subtree_def)
-  } ultimately show ?thesis by auto
-qed
-
-lemma threads_set_eq: 
-   "the_thread ` (subtree (tRAG s) (Th th)) = 
-                  {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
-   by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
-
-context valid_trace
-begin
-
-lemma RAG_tRAG_transfer:
-  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 -
-  { 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(1) have cs_in: "(Cs cs', Th th2) \<in> RAG s" by auto
-    from h(3) and assms(1) 
-    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(2) cs_in[unfolded this]
-        show ?thesis using 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(1), auto)
-      qed
-      ultimately show ?thesis by auto
-    next
-      assume eq_n: "(n1, n2) = (Th th, Th th'')"
-      from assms(1, 2) have "(Cs cs, Th th'') \<in> RAG s'" by auto
-      moreover have "(Th th, Cs cs) \<in> RAG s'" using assms(1) by auto
-      ultimately show ?thesis 
-        by (unfold eq_n tRAG_alt_def, auto)
-    qed
-  } ultimately show ?thesis by auto
-qed
-
-lemma subtree_tRAG_thread:
-  assumes "th \<in> threads s"
-  shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
-proof -
-  have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
-    by (unfold tRAG_subtree_eq, simp)
-  also have "... \<subseteq> ?R"
-  proof
-    fix x
-    assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
-    then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
-    from this(2)
-    show "x \<in> ?R"
-    proof(cases rule:subtreeE)
-      case 1
-      thus ?thesis by (simp add: assms h(1)) 
-    next
-      case 2
-      thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) 
-    qed
-  qed
-  finally show ?thesis .
-qed
-
-lemma 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
-
-end
-
-section {* Chain to readys *}
-
-context valid_trace
-begin
-
-lemma chain_building:
-  assumes "node \<in> Domain (RAG s)"
-  obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"
-proof -
-  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
-    ultimately show ?thesis by (auto simp:readys_def)
-  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
-  The following is just an instance of @{text "chain_building"}.
-*}                    
-lemma th_chain_to_ready:
-  assumes th_in: "th \<in> threads s"
-  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
-proof(cases "th \<in> readys s")
-  case True
-  thus ?thesis by auto
-next
-  case False
-  from False and th_in have "Th th \<in> Domain (RAG s)" 
-    by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
-  from chain_building [rule_format, OF this]
-  show ?thesis by auto
-qed
-
-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 runing_unique:
-  assumes runing_1: "th1 \<in> runing s"
-  and runing_2: "th2 \<in> runing s"
-  shows "th1 = th2"
-proof -
-  from runing_1 and runing_2 have "cp s th1 = cp s th2"
-    unfolding runing_def 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
-  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
-  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
-  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 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
-    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
-    ultimately have "xs2 = xs1" by simp
-    from rpath_dest_eq[OF rp1 rp2[unfolded this]]
-    show ?thesis by simp
-  qed
-qed
-
-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
-
-end
-
-
-section {* Relating @{term cp} and @{term the_preced} and @{term preced} *}
-
-context valid_trace
-begin
-
-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 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
-
-
-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
-
-section {* Relating @{term cntP}, @{term cntV}, @{term cntCS} and @{term pvD} *}
-
-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 -
-  { 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
-       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
-  
-lemma (in valid_trace) finite_holdents: "finite (holdents s th)"
-  by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)
-
-context valid_trace_p 
-begin
-
-lemma ready_th_s: "th \<in> readys s"
-  using runing_th_s
-  by (unfold runing_def, auto)
-
-lemma live_th_s: "th \<in> threads s"
-  using readys_threads ready_th_s by auto
-
-lemma live_th_es: "th \<in> threads (e#s)"
-  using live_th_s 
-  by (unfold is_p, simp)
-
-lemma waiting_neq_th: 
-  assumes "waiting s t c"
-  shows "t \<noteq> th"
-  using assms using th_not_waiting by blast 
-
-end
-
-context valid_trace_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
-      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 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 
-        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 
-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
-begin
-
-lemma th_not_waiting: 
-  "\<not> waiting s th c"
-proof -
-  have "th \<in> readys s"
-    using runing_ready runing_th_s by blast 
-  thus ?thesis
-    by (unfold readys_def, auto)
-qed
-
-lemma waiting_neq_th: 
-  assumes "waiting s t c"
-  shows "t \<noteq> th"
-  using assms using th_not_waiting by blast 
-
-end
-
-context valid_trace_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
-      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
-        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 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_kept]
-  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_kept]
-  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_kept]
-  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_kept]
-  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_kept, 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_kept, 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_kept]
-    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_kept]
-         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_kept]
-  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_kept, 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_kept, 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_kept]
-    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_kept]
-         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_kept, 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_kept, 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_kept]
-    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_kept]
-         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 (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
-
-end
-
-section {* Corollaries of @{thm valid_trace.cnp_cnv_cncs} *}
-
-context valid_trace
-begin
-
-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"
-  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 eq_pv_children:
-  assumes eq_pv: "cntP s th = cntV 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)
-    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 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)
-
-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
-
-lemma count_eq_tRAG_plus:
-  assumes "cntP s th = cntV s th"
-  shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
-  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"
-  shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
-  using count_eq_RAG_plus[OF assms] by auto
-
-lemma count_eq_tRAG_plus_Th:
-  assumes "cntP s th = cntV s th"
-  shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
-   using count_eq_tRAG_plus[OF assms] by auto
-
-end
-
-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
-
-section {* Recursive definition of @{term "cp"} *}
-
-lemma cp_alt_def1: 
-  "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
-proof -
-  have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
-       ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
-       by auto
-  thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
-qed
-
-lemma cp_gen_def_cond: 
-  assumes "x = Th th"
-  shows "cp s th = cp_gen s (Th th)"
-by (unfold cp_alt_def1 cp_gen_def, simp)
-
-lemma cp_gen_over_set:
-  assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
-  shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
-proof(rule f_image_eq)
-  fix a
-  assume "a \<in> A"
-  from assms[rule_format, OF this]
-  obtain th where eq_a: "a = Th th" by auto
-  show "cp_gen s a = (cp s \<circ> the_thread) a"
-    by  (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
-qed
-
-
-context valid_trace
-begin
-(* ddd *)
-lemma cp_gen_rec:
-  assumes "x = Th th"
-  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
-proof(cases "children (tRAG s) x = {}")
-  case True
-  show ?thesis
-    by (unfold True cp_gen_def subtree_children, simp add:assms)
-next
-  case False
-  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
-  note fsbttRAGs.finite_subtree[simp]
-  have [simp]: "finite (children (tRAG s) x)"
-     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 
-            rule children_subtree)
-  { fix r x
-    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
-  } note this[simp]
-  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
-  proof -
-    from False obtain q where "q \<in> children (tRAG s) x" by blast
-    moreover have "subtree (tRAG s) q \<noteq> {}" by simp
-    ultimately show ?thesis by blast
-  qed
-  have h: "Max ((the_preced s \<circ> the_thread) `
-                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
-        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
-                     (is "?L = ?R")
-  proof -
-    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
-    let "Max (_ \<union> (?h ` ?B))" = ?R
-    let ?L1 = "?f ` \<Union>(?g ` ?B)"
-    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
-    proof -
-      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
-      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
-      finally have "Max ?L1 = Max ..." by simp
-      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
-        by (subst Max_UNION, simp+)
-      also have "... = Max (cp_gen s ` children (tRAG s) x)"
-          by (unfold image_comp cp_gen_alt_def, simp)
-      finally show ?thesis .
-    qed
-    show ?thesis
-    proof -
-      have "?L = Max (?f ` ?A \<union> ?L1)" by simp
-      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
-            by (subst Max_Un, simp+)
-      also have "... = max (?f x) (Max (?h ` ?B))"
-        by (unfold eq_Max_L1, simp)
-      also have "... =?R"
-        by (rule max_Max_eq, (simp)+, unfold assms, simp)
-      finally show ?thesis .
-    qed
-  qed  thus ?thesis 
-          by (fold h subtree_children, unfold cp_gen_def, simp) 
-qed
-
-lemma cp_rec:
-  "cp s th = Max ({the_preced s th} \<union> 
-                     (cp s o the_thread) ` children (tRAG s) (Th th))"
-proof -
-  have "Th th = Th th" by simp
-  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
-  show ?thesis 
-  proof -
-    have "cp_gen s ` children (tRAG s) (Th th) = 
-                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
-    proof(rule cp_gen_over_set)
-      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
-        by (unfold tRAG_alt_def, auto simp:children_def)
-    qed
-    thus ?thesis by (subst (1) h(1), unfold h(2), simp)
-  qed
-qed
-end
-
-section {* Other properties useful in Implementation.thy or Correctness.thy *}
-
-context valid_trace_e 
-begin
-
-lemma actor_inv: 
-  assumes "\<not> isCreate e"
-  shows "actor e \<in> runing s"
-  using pip_e assms 
-  by (induct, auto)
-end
-
-context valid_trace
-begin
-
-lemma readys_root:
-  assumes "th \<in> readys s"
-  shows "root (RAG s) (Th th)"
-proof -
-  { fix x
-    assume "x \<in> ancestors (RAG s) (Th th)"
-    hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
-    from tranclD[OF this]
-    obtain z where "(Th th, z) \<in> RAG s" by auto
-    with assms(1) have False
-         apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
-         by (fold wq_def, blast)
-  } thus ?thesis by (unfold root_def, auto)
-qed
-
-lemma readys_in_no_subtree:
-  assumes "th \<in> readys s"
-  and "th' \<noteq> th"
-  shows "Th th \<notin> subtree (RAG s) (Th th')" 
-proof
-   assume "Th th \<in> subtree (RAG s) (Th th')"
-   thus False
-   proof(cases rule:subtreeE)
-      case 1
-      with assms show ?thesis by auto
-   next
-      case 2
-      with readys_root[OF assms(1)]
-      show ?thesis by (auto simp:root_def)
-   qed
-qed
-
-lemma not_in_thread_isolated:
-  assumes "th \<notin> threads s"
-  shows "(Th th) \<notin> Field (RAG s)"
-proof
-  assume "(Th th) \<in> Field (RAG s)"
-  with dm_RAG_threads and rg_RAG_threads assms
-  show False by (unfold Field_def, blast)
-qed
-
-lemma next_th_holding:
-  assumes nxt: "next_th s th cs th'"
-  shows "holding (wq s) th cs"
-proof -
-  from nxt[unfolded next_th_def]
-  obtain rest where h: "wq s cs = th # rest"
-                       "rest \<noteq> []" 
-                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
-  thus ?thesis
-    by (unfold cs_holding_def, auto)
-qed
-
-lemma next_th_waiting:
-  assumes nxt: "next_th s th cs th'"
-  shows "waiting (wq s) th' cs"
-proof -
-  from nxt[unfolded next_th_def]
-  obtain rest where h: "wq s cs = th # rest"
-                       "rest \<noteq> []" 
-                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
-  from wq_distinct[of cs, unfolded h]
-  have dst: "distinct (th # rest)" .
-  have in_rest: "th' \<in> set rest"
-  proof(unfold h, rule someI2)
-    show "distinct rest \<and> set rest = set rest" using dst by auto
-  next
-    fix x assume "distinct x \<and> set x = set rest"
-    with h(2)
-    show "hd x \<in> set (rest)" by (cases x, auto)
-  qed
-  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
-  moreover have "th' \<noteq> hd (wq s cs)"
-    by (unfold h(1), insert in_rest dst, auto)
-  ultimately show ?thesis by (auto simp:cs_waiting_def)
-qed
-
-lemma next_th_RAG:
-  assumes nxt: "next_th (s::event list) th cs th'"
-  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
-  using vt assms next_th_holding next_th_waiting
-  by (unfold s_RAG_def, simp)
-
-end 
-
-context valid_trace_p
-begin
-
-find_theorems readys th
-
-end
-
-end
diff -r 5d8ec128518b -r e3cf792db636 CpsG_1.thy
--- a/CpsG_1.thy	Tue Jun 14 13:56:51 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,4403 +0,0 @@
-theory CpsG
-imports PIPDefs 
-begin
-
-lemma Max_f_mono:
-  assumes seq: "A \<subseteq> B"
-  and np: "A \<noteq> {}"
-  and fnt: "finite B"
-  shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
-  from seq show "f ` A \<subseteq> f ` B" by auto
-next
-  from np show "f ` A \<noteq> {}" by auto
-next
-  from fnt and seq show "finite (f ` B)" by auto
-qed
-
-(* I am going to use this file as a start point to retrofiting 
-   PIPBasics.thy, which is originally called CpsG.ghy *)
-
-locale valid_trace = 
-  fixes s
-  assumes vt : "vt s"
-
-locale valid_trace_e = valid_trace +
-  fixes e
-  assumes vt_e: "vt (e#s)"
-begin
-
-lemma pip_e: "PIP s e"
-  using vt_e by (cases, simp)  
-
-end
-
-locale valid_trace_create = valid_trace_e + 
-  fixes th prio
-  assumes is_create: "e = Create th prio"
-
-locale valid_trace_exit = valid_trace_e + 
-  fixes th
-  assumes is_exit: "e = Exit th"
-
-locale valid_trace_p = valid_trace_e + 
-  fixes th cs
-  assumes is_p: "e = P th cs"
-
-locale valid_trace_v = valid_trace_e + 
-  fixes th cs
-  assumes is_v: "e = V th cs"
-begin
-  definition "rest = tl (wq s cs)"
-  definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
-end
-
-locale valid_trace_v_n = valid_trace_v +
-  assumes rest_nnl: "rest \<noteq> []"
-
-locale valid_trace_v_e = valid_trace_v +
-  assumes rest_nil: "rest = []"
-
-locale valid_trace_set= valid_trace_e + 
-  fixes th prio
-  assumes is_set: "e = Set th prio"
-
-context valid_trace
-begin
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
-  assumes "PP []"
-     and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
-                   PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
-     shows "PP s"
-proof(induct rule:vt.induct[OF vt, case_names Init Step])
-  case Init
-  from assms(1) show ?case .
-next
-  case (Step s e)
-  show ?case
-  proof(rule assms(2))
-    show "valid_trace_e s e" using Step by (unfold_locales, auto)
-  next
-    show "PP s" using Step by simp
-  next
-    show "PIP s e" using Step by simp
-  qed
-qed
-
-end
-
-
-lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
-  by  (unfold s_waiting_def cs_waiting_def wq_def, auto)
-
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
-  by (unfold s_holding_def wq_def cs_holding_def, simp)
-
-lemma runing_ready: 
-  shows "runing s \<subseteq> readys s"
-  unfolding runing_def readys_def
-  by auto 
-
-lemma readys_threads:
-  shows "readys s \<subseteq> threads s"
-  unfolding readys_def
-  by auto
-
-lemma wq_v_neq [simp]:
-   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
-  by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-lemma runing_head:
-  assumes "th \<in> runing s"
-  and "th \<in> set (wq_fun (schs s) cs)"
-  shows "th = hd (wq_fun (schs s) cs)"
-  using assms
-  by (simp add:runing_def readys_def s_waiting_def wq_def)
-
-context valid_trace
-begin
-
-lemma runing_wqE:
-  assumes "th \<in> runing s"
-  and "th \<in> set (wq s cs)"
-  obtains rest where "wq s cs = th#rest"
-proof -
-  from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
-    by (meson list.set_cases)
-  have "th' = th"
-  proof(rule ccontr)
-    assume "th' \<noteq> th"
-    hence "th \<noteq> hd (wq s cs)" using eq_wq by auto 
-    with assms(2)
-    have "waiting s th cs" 
-      by (unfold s_waiting_def, fold wq_def, auto)
-    with assms show False 
-      by (unfold runing_def readys_def, auto)
-  qed
-  with eq_wq that show ?thesis by metis
-qed
-
-end
-
-context valid_trace_p
-begin
-
-lemma wq_neq_simp [simp]:
-  assumes "cs' \<noteq> cs"
-  shows "wq (e#s) cs' = wq s cs'"
-    using assms unfolding is_p wq_def
-  by (auto simp:Let_def)
-
-lemma runing_th_s:
-  shows "th \<in> runing s"
-proof -
-  from pip_e[unfolded is_p]
-  show ?thesis by (cases, simp)
-qed
-
-lemma th_not_waiting: 
-  "\<not> waiting s th c"
-proof -
-  have "th \<in> readys s"
-    using runing_ready runing_th_s by blast 
-  thus ?thesis
-    by (unfold readys_def, auto)
-qed
-
-lemma waiting_neq_th: 
-  assumes "waiting s t c"
-  shows "t \<noteq> th"
-  using assms using th_not_waiting by blast 
-
-lemma th_not_in_wq: 
-  shows "th \<notin> set (wq s cs)"
-proof
-  assume otherwise: "th \<in> set (wq s cs)"
-  from runing_wqE[OF runing_th_s this]
-  obtain rest where eq_wq: "wq s cs = th#rest" by blast
-  with otherwise
-  have "holding s th cs"
-    by (unfold s_holding_def, fold wq_def, simp)
-  hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
-    by (unfold s_RAG_def, fold holding_eq, auto)
-  from pip_e[unfolded is_p]
-  show False
-  proof(cases)
-    case (thread_P)
-    with cs_th_RAG show ?thesis by auto
-  qed
-qed
-
-lemma wq_es_cs: 
-  "wq (e#s) cs =  wq s cs @ [th]"
-  by (unfold is_p wq_def, auto simp:Let_def)
-
-lemma wq_distinct_kept:
-  assumes "distinct (wq s cs')"
-  shows "distinct (wq (e#s) cs')"
-proof(cases "cs' = cs")
-  case True
-  show ?thesis using True assms th_not_in_wq
-    by (unfold True wq_es_cs, auto)
-qed (insert assms, simp)
-
-end
-
-
-context valid_trace_v
-begin
-
-lemma wq_neq_simp [simp]:
-  assumes "cs' \<noteq> cs"
-  shows "wq (e#s) cs' = wq s cs'"
-    using assms unfolding is_v wq_def
-  by (auto simp:Let_def)
-
-lemma runing_th_s:
-  shows "th \<in> runing s"
-proof -
-  from pip_e[unfolded is_v]
-  show ?thesis by (cases, simp)
-qed
-
-lemma th_not_waiting: 
-  "\<not> waiting s th c"
-proof -
-  have "th \<in> readys s"
-    using runing_ready runing_th_s by blast 
-  thus ?thesis
-    by (unfold readys_def, auto)
-qed
-
-lemma waiting_neq_th: 
-  assumes "waiting s t c"
-  shows "t \<noteq> th"
-  using assms using th_not_waiting by blast 
-
-lemma wq_s_cs:
-  "wq s cs = th#rest"
-proof -
-  from pip_e[unfolded is_v]
-  show ?thesis
-  proof(cases)
-    case (thread_V)
-    from this(2) show ?thesis
-      by (unfold rest_def s_holding_def, fold wq_def,
-                 metis empty_iff list.collapse list.set(1))
-  qed
-qed
-
-lemma wq_es_cs:
-  "wq (e#s) cs = wq'"
- using wq_s_cs[unfolded wq_def]
- by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) 
-
-lemma wq_distinct_kept:
-  assumes "distinct (wq s cs')"
-  shows "distinct (wq (e#s) cs')"
-proof(cases "cs' = cs")
-  case True
-  show ?thesis
-  proof(unfold True wq_es_cs wq'_def, rule someI2)
-    show "distinct rest \<and> set rest = set rest"
-        using assms[unfolded True wq_s_cs] by auto
-  qed simp
-qed (insert assms, simp)
-
-end
-
-context valid_trace
-begin
-
-lemma actor_inv: 
-  assumes "PIP s e"
-  and "\<not> isCreate e"
-  shows "actor e \<in> runing s"
-  using assms
-  by (induct, auto)
-
-lemma isP_E:
-  assumes "isP e"
-  obtains cs where "e = P (actor e) cs"
-  using assms by (cases e, auto)
-
-lemma isV_E:
-  assumes "isV e"
-  obtains cs where "e = V (actor e) cs"
-  using assms by (cases e, auto) 
-
-lemma wq_distinct: "distinct (wq s cs)"
-proof(induct rule:ind)
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case 
-  proof(cases e)
-    case (V th cs)
-    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) 
-  qed
-qed (unfold wq_def Let_def, simp)
-
-end
-
-context valid_trace_e
-begin
-
-text {*
-  The following lemma shows that only the @{text "P"}
-  operation can add new thread into waiting queues. 
-  Such kind of lemmas are very obvious, but need to be checked formally.
-  This is a kind of confirmation that our modelling is correct.
-*}
-
-lemma wq_in_inv: 
-  assumes s_ni: "thread \<notin> set (wq s cs)"
-  and s_i: "thread \<in> set (wq (e#s) cs)"
-  shows "e = P thread cs"
-proof(cases e)
-  -- {* This is the only non-trivial case: *}
-  case (V th cs1)
-  have False
-  proof(cases "cs1 = cs")
-    case True
-    show ?thesis
-    proof(cases "(wq s cs1)")
-      case (Cons w_hd w_tl)
-      have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
-      proof -
-        have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
-          using  Cons V by (auto simp:wq_def Let_def True split:if_splits)
-        moreover have "set ... \<subseteq> set (wq s cs)"
-        proof(rule someI2)
-          show "distinct w_tl \<and> set w_tl = set w_tl"
-            by (metis distinct.simps(2) local.Cons wq_distinct)
-        qed (insert Cons True, auto)
-        ultimately show ?thesis by simp
-      qed
-      with assms show ?thesis by auto
-    qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
-  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
-  thus ?thesis by auto
-qed (insert assms, auto simp:wq_def Let_def split:if_splits)
-
-lemma wq_out_inv: 
-  assumes s_in: "thread \<in> set (wq s cs)"
-  and s_hd: "thread = hd (wq s cs)"
-  and s_i: "thread \<noteq> hd (wq (e#s) cs)"
-  shows "e = V thread cs"
-proof(cases e)
--- {* There are only two non-trivial cases: *}
-  case (V th cs1)
-  show ?thesis
-  proof(cases "cs1 = cs")
-    case True
-    have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
-    thus ?thesis
-    proof(cases)
-      case (thread_V)
-      moreover have "th = thread" using thread_V(2) s_hd
-          by (unfold s_holding_def wq_def, simp)
-      ultimately show ?thesis using V True by simp
-    qed
-  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
-next
-  case (P th cs1)
-  show ?thesis
-  proof(cases "cs1 = cs")
-    case True
-    with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
-      by (auto simp:wq_def Let_def split:if_splits)
-    with s_i s_hd s_in have False
-      by (metis empty_iff hd_append2 list.set(1) wq_def) 
-    thus ?thesis by simp
-  qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
-qed (insert assms, auto simp:wq_def Let_def split:if_splits)
-
-end
-
-
-
-context valid_trace
-begin
-
-
-text {* (* ddd *)
-  The nature of the work is like this: since it starts from a very simple and basic 
-  model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
-  For instance, the fact 
-  that one thread can not be blocked by two critical resources at the same time
-  is obvious, because only running threads can make new requests, if one is waiting for 
-  a critical resource and get blocked, it can not make another resource request and get 
-  blocked the second time (because it is not running). 
-
-  To derive this fact, one needs to prove by contraction and 
-  reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
-  named @{text "p_split"}, which is about status changing along the time axis. It says if 
-  a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
-  but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} 
-  in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history 
-  of events leading to it), such that @{text "Q"} switched 
-  from being @{text "False"} to @{text "True"} and kept being @{text "True"}
-  till the last moment of @{text "s"}.
-
-  Suppose a thread @{text "th"} is blocked
-  on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, 
-  since no thread is blocked at the very beginning, by applying 
-  @{text "p_split"} to these two blocking facts, there exist 
-  two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that 
-  @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} 
-  and kept on blocked on them respectively ever since.
- 
-  Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
-  However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
-  in blocked state at moment @{text "t2"} and could not
-  make any request and get blocked the second time: Contradiction.
-*}
-
-lemma waiting_unique_pre: (* ddd *)
-  assumes h11: "thread \<in> set (wq s cs1)"
-  and h12: "thread \<noteq> hd (wq s cs1)"
-  assumes h21: "thread \<in> set (wq s cs2)"
-  and h22: "thread \<noteq> hd (wq s cs2)"
-  and neq12: "cs1 \<noteq> cs2"
-  shows "False"
-proof -
-  let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
-  from h11 and h12 have q1: "?Q cs1 s" by simp
-  from h21 and h22 have q2: "?Q cs2 s" by simp
-  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
-  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
-  from p_split [of "?Q cs1", OF q1 nq1]
-  obtain t1 where lt1: "t1 < length s"
-    and np1: "\<not> ?Q cs1 (moment t1 s)"
-    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
-  from p_split [of "?Q cs2", OF q2 nq2]
-  obtain t2 where lt2: "t2 < length s"
-    and np2: "\<not> ?Q cs2 (moment t2 s)"
-    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
-  { fix s cs
-    assume q: "?Q cs s"
-    have "thread \<notin> runing s"
-    proof
-      assume "thread \<in> runing s"
-      hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and> 
-                 thread \<noteq> hd (wq_fun (schs s) cs))"
-        by (unfold runing_def s_waiting_def readys_def, auto)
-      from this[rule_format, of cs] q 
-      show False by (simp add: wq_def) 
-    qed
-  } note q_not_runing = this
-  { fix t1 t2 cs1 cs2
-    assume  lt1: "t1 < length s"
-    and np1: "\<not> ?Q cs1 (moment t1 s)"
-    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
-    and lt2: "t2 < length s"
-    and np2: "\<not> ?Q cs2 (moment t2 s)"
-    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
-    and lt12: "t1 < t2"
-    let ?t3 = "Suc t2"
-    from lt2 have le_t3: "?t3 \<le> length s" by auto
-    from moment_plus [OF this] 
-    obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
-    have "t2 < ?t3" by simp
-    from nn2 [rule_format, OF this] and eq_m
-    have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-         h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-    have "vt (e#moment t2 s)"
-    proof -
-      from vt_moment 
-      have "vt (moment ?t3 s)" .
-      with eq_m show ?thesis by simp
-    qed
-    then interpret vt_e: valid_trace_e "moment t2 s" "e"
-        by (unfold_locales, auto, cases, simp)
-    have ?thesis
-    proof -
-      have "thread \<in> runing (moment t2 s)"
-      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-        case True
-        have "e = V thread cs2"
-        proof -
-          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
-              using True and np2  by auto 
-          from vt_e.wq_out_inv[OF True this h2]
-          show ?thesis .
-        qed
-        thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto
-      next
-        case False
-        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
-        with vt_e.actor_inv[OF vt_e.pip_e]
-        show ?thesis by auto
-      qed
-      moreover have "thread \<notin> runing (moment t2 s)"
-        by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
-      ultimately show ?thesis by simp
-    qed
-  } note lt_case = this
-  show ?thesis
-  proof -
-    { assume "t1 < t2"
-      from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
-      have ?thesis .
-    } moreover {
-      assume "t2 < t1"
-      from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
-      have ?thesis .
-    } moreover {
-      assume eq_12: "t1 = t2"
-      let ?t3 = "Suc t2"
-      from lt2 have le_t3: "?t3 \<le> length s" by auto
-      from moment_plus [OF this] 
-      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
-      have lt_2: "t2 < ?t3" by simp
-      from nn2 [rule_format, OF this] and eq_m
-      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-           h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-      from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12]
-      have g1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
-           g2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
-      have "vt (e#moment t2 s)"
-      proof -
-        from vt_moment 
-        have "vt (moment ?t3 s)" .
-        with eq_m show ?thesis by simp
-      qed
-      then interpret vt_e: valid_trace_e "moment t2 s" "e"
-          by (unfold_locales, auto, cases, simp)
-      have "e = V thread cs2 \<or> e = P thread cs2"
-      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-        case True
-        have "e = V thread cs2"
-        proof -
-          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
-              using True and np2  by auto 
-          from vt_e.wq_out_inv[OF True this h2]
-          show ?thesis .
-        qed
-        thus ?thesis by auto
-      next
-        case False
-        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
-        thus ?thesis by auto
-      qed
-      moreover have "e = V thread cs1 \<or> e = P thread cs1"
-      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
-        case True
-        have eq_th: "thread = hd (wq (moment t1 s) cs1)" 
-              using True and np1  by auto 
-        from vt_e.wq_out_inv[folded eq_12, OF True this g2]
-        have "e = V thread cs1" .
-        thus ?thesis by auto
-      next
-        case False
-        have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
-        thus ?thesis by auto
-      qed
-      ultimately have ?thesis using neq12 by auto
-    } ultimately show ?thesis using nat_neq_iff by blast 
-  qed
-qed
-
-text {*
-  This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
-*}
-
-lemma waiting_unique:
-  assumes "waiting s th cs1"
-  and "waiting s th cs2"
-  shows "cs1 = cs2"
-  using waiting_unique_pre assms
-  unfolding wq_def s_waiting_def
-  by auto
-
-end
-
-(* not used *)
-text {*
-  Every thread can only be blocked on one critical resource, 
-  symmetrically, every critical resource can only be held by one thread. 
-  This fact is much more easier according to our definition. 
-*}
-lemma held_unique:
-  assumes "holding (s::event list) th1 cs"
-  and "holding s th2 cs"
-  shows "th1 = th2"
- by (insert assms, unfold s_holding_def, auto)
-
-
-lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
-  apply (induct s, auto)
-  by (case_tac a, auto split:if_splits)
-
-lemma last_set_unique: 
-  "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
-          \<Longrightarrow> th1 = th2"
-  apply (induct s, auto)
-  by (case_tac a, auto split:if_splits dest:last_set_lt)
-
-lemma preced_unique : 
-  assumes pcd_eq: "preced th1 s = preced th2 s"
-  and th_in1: "th1 \<in> threads s"
-  and th_in2: " th2 \<in> threads s"
-  shows "th1 = th2"
-proof -
-  from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
-  from last_set_unique [OF this th_in1 th_in2]
-  show ?thesis .
-qed
-                      
-lemma preced_linorder: 
-  assumes neq_12: "th1 \<noteq> th2"
-  and th_in1: "th1 \<in> threads s"
-  and th_in2: " th2 \<in> threads s"
-  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
-  from preced_unique [OF _ th_in1 th_in2] and neq_12 
-  have "preced th1 s \<noteq> preced th2 s" by auto
-  thus ?thesis by auto
-qed
-
-(* An aux lemma used later *) 
-lemma unique_minus:
-  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-  and xy: "(x, y) \<in> r"
-  and xz: "(x, z) \<in> r^+"
-  and neq: "y \<noteq> z"
-  shows "(y, z) \<in> r^+"
-proof -
- from xz and neq show ?thesis
- proof(induct)
-   case (base ya)
-   have "(x, ya) \<in> r" by fact
-   from unique [OF xy this] have "y = ya" .
-   with base show ?case by auto
- next
-   case (step ya z)
-   show ?case
-   proof(cases "y = ya")
-     case True
-     from step True show ?thesis by simp
-   next
-     case False
-     from step False
-     show ?thesis by auto
-   qed
- qed
-qed
-
-lemma unique_base:
-  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-  and xy: "(x, y) \<in> r"
-  and xz: "(x, z) \<in> r^+"
-  and neq_yz: "y \<noteq> z"
-  shows "(y, z) \<in> r^+"
-proof -
-  from xz neq_yz show ?thesis
-  proof(induct)
-    case (base ya)
-    from xy unique base show ?case by auto
-  next
-    case (step ya z)
-    show ?case
-    proof(cases "y = ya")
-      case True
-      from True step show ?thesis by auto
-    next
-      case False
-      from False step 
-      have "(y, ya) \<in> r\<^sup>+" by auto
-      with step show ?thesis by auto
-    qed
-  qed
-qed
-
-lemma unique_chain:
-  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-  and xy: "(x, y) \<in> r^+"
-  and xz: "(x, z) \<in> r^+"
-  and neq_yz: "y \<noteq> z"
-  shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
-proof -
-  from xy xz neq_yz show ?thesis
-  proof(induct)
-    case (base y)
-    have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
-    from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
-  next
-    case (step y za)
-    show ?case
-    proof(cases "y = z")
-      case True
-      from True step show ?thesis by auto
-    next
-      case False
-      from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
-      thus ?thesis
-      proof
-        assume "(z, y) \<in> r\<^sup>+"
-        with step have "(z, za) \<in> r\<^sup>+" by auto
-        thus ?thesis by auto
-      next
-        assume h: "(y, z) \<in> r\<^sup>+"
-        from step have yza: "(y, za) \<in> r" by simp
-        from step have "za \<noteq> z" by simp
-        from unique_minus [OF _ yza h this] and unique
-        have "(za, z) \<in> r\<^sup>+" by auto
-        thus ?thesis by auto
-      qed
-    qed
-  qed
-qed
-
-text {*
-  The following three lemmas show that @{text "RAG"} does not change
-  by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
-  events, respectively.
-*}
-
-lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-
-context valid_trace_v
-begin
-
-
-lemma distinct_rest: "distinct rest"
-  by (simp add: distinct_tl rest_def wq_distinct)
-
-definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
-
-lemma runing_th_s:
-  shows "th \<in> runing s"
-proof -
-  from pip_e[unfolded is_v]
-  show ?thesis by (cases, simp)
-qed
-
-lemma holding_cs_eq_th:
-  assumes "holding s t cs"
-  shows "t = th"
-proof -
-  from pip_e[unfolded is_v]
-  show ?thesis
-  proof(cases)
-    case (thread_V)
-    from held_unique[OF this(2) assms]
-    show ?thesis by simp
-  qed
-qed
-
-lemma th_not_waiting: 
-  "\<not> waiting s th c"
-proof -
-  have "th \<in> readys s"
-    using runing_ready runing_th_s by blast 
-  thus ?thesis
-    by (unfold readys_def, auto)
-qed
-
-lemma waiting_neq_th: 
-  assumes "waiting s t c"
-  shows "t \<noteq> th"
-  using assms using th_not_waiting by blast 
-
-lemma wq_s_cs:
-  "wq s cs = th#rest"
-proof -
-  from pip_e[unfolded is_v]
-  show ?thesis
-  proof(cases)
-    case (thread_V)
-    from this(2) show ?thesis
-      by (unfold rest_def s_holding_def, fold wq_def,
-                 metis empty_iff list.collapse list.set(1))
-  qed
-qed
-
-lemma wq_es_cs:
-  "wq (e#s) cs = wq'"
- using wq_s_cs[unfolded wq_def]
- by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) 
-
-lemma distinct_wq': "distinct wq'"
-  by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
-  
-lemma th'_in_inv:
-  assumes "th' \<in> set wq'"
-  shows "th' \<in> set rest"
-  using assms
-  by (metis (mono_tags, lifting) distinct.simps(2) 
-        rest_def some_eq_ex wq'_def wq_distinct wq_s_cs) 
-
-lemma neq_t_th: 
-  assumes "waiting (e#s) t c"
-  shows "t \<noteq> th"
-proof
-  assume otherwise: "t = th"
-  show False
-  proof(cases "c = cs")
-    case True
-    have "t \<in> set wq'" 
-     using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
-     by simp 
-    from th'_in_inv[OF this] have "t \<in> set rest" .
-    with wq_s_cs[folded otherwise] wq_distinct[of cs]
-    show ?thesis by simp
-  next
-    case False
-    have "wq (e#s) c = wq s c" using False
-        by (unfold is_v, simp)
-    hence "waiting s t c" using assms 
-        by (simp add: cs_waiting_def waiting_eq)
-    hence "t \<notin> readys s" by (unfold readys_def, auto)
-    hence "t \<notin> runing s" using runing_ready by auto 
-    with runing_th_s[folded otherwise] show ?thesis by auto
-  qed
-qed
-
-lemma waiting_esI1:
-  assumes "waiting s t c"
-      and "c \<noteq> cs" 
-  shows "waiting (e#s) t c" 
-proof -
-  have "wq (e#s) c = wq s c" 
-    using assms(2) is_v by auto
-  with assms(1) show ?thesis 
-    using cs_waiting_def waiting_eq by auto 
-qed
-
-lemma holding_esI2:
-  assumes "c \<noteq> cs" 
-  and "holding s t c"
-  shows "holding (e#s) t c"
-proof -
-  from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
-  from assms(2)[unfolded s_holding_def, folded wq_def, 
-                folded this, unfolded wq_def, folded s_holding_def]
-  show ?thesis .
-qed
-
-lemma holding_esI1:
-  assumes "holding s t c"
-  and "t \<noteq> th"
-  shows "holding (e#s) t c"
-proof -
-  have "c \<noteq> cs" using assms using holding_cs_eq_th by blast 
-  from holding_esI2[OF this assms(1)]
-  show ?thesis .
-qed
-
-end
-
-context valid_trace_v_n
-begin
-
-lemma neq_wq': "wq' \<noteq> []" 
-proof (unfold wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest) 
-next
-  fix x
-  assume " distinct x \<and> set x = set rest" 
-  thus "x \<noteq> []" using rest_nnl by auto
-qed 
-
-definition "taker = hd wq'"
-
-definition "rest' = tl wq'"
-
-lemma eq_wq': "wq' = taker # rest'"
-  by (simp add: neq_wq' rest'_def taker_def)
-
-lemma next_th_taker: 
-  shows "next_th s th cs taker"
-  using rest_nnl taker_def wq'_def wq_s_cs 
-  by (auto simp:next_th_def)
-
-lemma taker_unique: 
-  assumes "next_th s th cs taker'"
-  shows "taker' = taker"
-proof -
-  from assms
-  obtain rest' where 
-    h: "wq s cs = th # rest'" 
-       "taker' = hd (SOME q. distinct q \<and> set q = set rest')"
-          by (unfold next_th_def, auto)
-  with wq_s_cs have "rest' = rest" by auto
-  thus ?thesis using h(2) taker_def wq'_def by auto 
-qed
-
-lemma waiting_set_eq:
-  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
-  by (smt all_not_in_conv bot.extremum insertI1 insert_subset 
-      mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
-
-lemma holding_set_eq:
-  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {(Cs cs, Th taker)}"
-  using next_th_taker taker_def waiting_set_eq 
-  by fastforce
-   
-lemma holding_taker:
-  shows "holding (e#s) taker cs"
-    by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, 
-        auto simp:neq_wq' taker_def)
-
-lemma waiting_esI2:
-  assumes "waiting s t cs"
-      and "t \<noteq> taker"
-  shows "waiting (e#s) t cs" 
-proof -
-  have "t \<in> set wq'" 
-  proof(unfold wq'_def, rule someI2)
-    show "distinct rest \<and> set rest = set rest"
-          by (simp add: distinct_rest)
-  next
-    fix x
-    assume "distinct x \<and> set x = set rest"
-    moreover have "t \<in> set rest"
-        using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto 
-    ultimately show "t \<in> set x" by simp
-  qed
-  moreover have "t \<noteq> hd wq'"
-    using assms(2) taker_def by auto 
-  ultimately show ?thesis
-    by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
-qed
-
-lemma waiting_esE:
-  assumes "waiting (e#s) t c" 
-  obtains "c \<noteq> cs" "waiting s t c"
-     |    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
-proof(cases "c = cs")
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
-  from that(1)[OF False this] show ?thesis .
-next
-  case True
-  from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
-  have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
-  hence "t \<noteq> taker" by (simp add: taker_def) 
-  moreover hence "t \<noteq> th" using assms neq_t_th by blast 
-  moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv) 
-  ultimately have "waiting s t cs"
-    by (metis cs_waiting_def list.distinct(2) list.sel(1) 
-                list.set_sel(2) rest_def waiting_eq wq_s_cs)  
-  show ?thesis using that(2)
-  using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto   
-qed
-
-lemma holding_esI1:
-  assumes "c = cs"
-  and "t = taker"
-  shows "holding (e#s) t c"
-  by (unfold assms, simp add: holding_taker)
-
-lemma holding_esE:
-  assumes "holding (e#s) t c" 
-  obtains "c = cs" "t = taker"
-      | "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
-  case True
-  from assms[unfolded True, unfolded s_holding_def, 
-             folded wq_def, unfolded wq_es_cs]
-  have "t = taker" by (simp add: taker_def) 
-  from that(1)[OF True this] show ?thesis .
-next
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  from assms[unfolded s_holding_def, folded wq_def, 
-             unfolded this, unfolded wq_def, folded s_holding_def]
-  have "holding s t c"  .
-  from that(2)[OF False this] show ?thesis .
-qed
-
-end 
-
-
-context valid_trace_v_n
-begin
-
-lemma nil_wq': "wq' = []" 
-proof (unfold wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest) 
-next
-  fix x
-  assume " distinct x \<and> set x = set rest" 
-  thus "x = []" using rest_nil by auto
-qed 
-
-lemma no_taker: 
-  assumes "next_th s th cs taker"
-  shows "False"
-proof -
-  from assms[unfolded next_th_def]
-  obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
-    by auto
-  thus ?thesis using rest_def rest_nil by auto 
-qed
-
-lemma waiting_set_eq:
-  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
-  using no_taker by auto
-
-lemma holding_set_eq:
-  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
-  using no_taker by auto
-   
-lemma no_holding:
-  assumes "holding (e#s) taker cs"
-  shows False
-proof -
-  from wq_es_cs[unfolded nil_wq']
-  have " wq (e # s) cs = []" .
-  from assms[unfolded s_holding_def, folded wq_def, unfolded this]
-  show ?thesis by auto
-qed
-
-lemma no_waiting:
-  assumes "waiting (e#s) t cs"
-  shows False
-proof -
-  from wq_es_cs[unfolded nil_wq']
-  have " wq (e # s) cs = []" .
-  from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
-  show ?thesis by auto
-qed
-
-lemma waiting_esI2:
-  assumes "waiting s t c"
-  shows "waiting (e#s) t c"
-proof -
-  have "c \<noteq> cs" using assms
-    using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto 
-  from waiting_esI1[OF assms this]
-  show ?thesis .
-qed
-
-lemma waiting_esE:
-  assumes "waiting (e#s) t c" 
-  obtains "c \<noteq> cs" "waiting s t c"
-proof(cases "c = cs")
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
-  from that(1)[OF False this] show ?thesis .
-next
-  case True
-  from no_waiting[OF assms[unfolded True]]
-  show ?thesis by auto
-qed
-
-lemma holding_esE:
-  assumes "holding (e#s) t c" 
-  obtains "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
-  case True
-  from no_holding[OF assms[unfolded True]] 
-  show ?thesis by auto
-next
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  from assms[unfolded s_holding_def, folded wq_def, 
-             unfolded this, unfolded wq_def, folded s_holding_def]
-  have "holding s t c"  .
-  from that[OF False this] show ?thesis .
-qed
-
-end (* ccc *)
-
-lemma rel_eqI:
-  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
-  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
-  shows "A = B"
-  using assms by auto
-
-lemma in_RAG_E:
-  assumes "(n1, n2) \<in> RAG (s::state)"
-  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
-      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
-  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
-  by auto
-  
-context valid_trace_v
-begin
-
-lemma RAG_es:
-  "RAG (e # s) =
-   RAG s - {(Cs cs, Th th)} -
-     {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-     {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
-proof(rule rel_eqI)
-  fix n1 n2
-  assume "(n1, n2) \<in> ?L"
-  thus "(n1, n2) \<in> ?R"
-  proof(cases rule:in_RAG_E)
-    case (waiting th' cs')
-    show ?thesis
-    proof(cases "rest = []")
-      case False
-      interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-      from waiting(3)
-      show ?thesis
-      proof(cases rule:h_n.waiting_esE)
-        case 1
-        with waiting(1,2)
-        show ?thesis
-        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      next
-        case 2
-        with waiting(1,2)
-        show ?thesis
-         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      qed
-    next
-      case True
-      interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-      from waiting(3)
-      show ?thesis
-      proof(cases rule:h_e.waiting_esE)
-        case 1
-        with waiting(1,2)
-        show ?thesis
-        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      qed
-    qed
-  next
-    case (holding th' cs')
-    show ?thesis
-    proof(cases "rest = []")
-      case False
-      interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-      from holding(3)
-      show ?thesis
-      proof(cases rule:h_n.holding_esE)
-        case 1
-        with holding(1,2)
-        show ?thesis
-        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      next
-        case 2
-        with holding(1,2)
-        show ?thesis
-         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold holding_eq, auto)
-      qed
-    next
-      case True
-      interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-      from holding(3)
-      show ?thesis
-      proof(cases rule:h_e.holding_esE)
-        case 1
-        with holding(1,2)
-        show ?thesis
-        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
-             fold holding_eq, auto)
-      qed
-    qed
-  qed
-next
-  fix n1 n2
-  assume h: "(n1, n2) \<in> ?R"
-  show "(n1, n2) \<in> ?L"
-  proof(cases "rest = []")
-    case False
-    interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-    from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
-    have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
-                            \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or> 
-          (n2 = Th h_n.taker \<and> n1 = Cs cs)" 
-      by auto
-   thus ?thesis
-   proof
-      assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
-      with h_n.holding_taker
-      show ?thesis 
-        by (unfold s_RAG_def, fold holding_eq, auto)
-   next
-    assume h: "(n1, n2) \<in> RAG s \<and>
-        (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
-    hence "(n1, n2) \<in> RAG s" by simp
-    thus ?thesis
-    proof(cases rule:in_RAG_E)
-      case (waiting th' cs')
-      from h and this(1,2)
-      have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
-      hence "waiting (e#s) th' cs'" 
-      proof
-        assume "cs' \<noteq> cs"
-        from waiting_esI1[OF waiting(3) this] 
-        show ?thesis .
-      next
-        assume neq_th': "th' \<noteq> h_n.taker"
-        show ?thesis
-        proof(cases "cs' = cs")
-          case False
-          from waiting_esI1[OF waiting(3) this] 
-          show ?thesis .
-        next
-          case True
-          from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
-          show ?thesis .
-        qed
-      qed
-      thus ?thesis using waiting(1,2)
-        by (unfold s_RAG_def, fold waiting_eq, auto)
-    next
-      case (holding th' cs')
-      from h this(1,2)
-      have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
-      hence "holding (e#s) th' cs'"
-      proof
-        assume "cs' \<noteq> cs"
-        from holding_esI2[OF this holding(3)] 
-        show ?thesis .
-      next
-        assume "th' \<noteq> th"
-        from holding_esI1[OF holding(3) this]
-        show ?thesis .
-      qed
-      thus ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-   qed
- next
-   case True
-   interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-   from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
-   have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)" 
-      by auto
-   from h_s(1)
-   show ?thesis
-   proof(cases rule:in_RAG_E)
-    case (waiting th' cs')
-    from h_e.waiting_esI2[OF this(3)]
-    show ?thesis using waiting(1,2)
-      by (unfold s_RAG_def, fold waiting_eq, auto)
-   next
-    case (holding th' cs')
-    with h_s(2)
-    have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
-    thus ?thesis
-    proof
-      assume neq_cs: "cs' \<noteq> cs"
-      from holding_esI2[OF this holding(3)]
-      show ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    next
-      assume "th' \<noteq> th"
-      from holding_esI1[OF holding(3) this]
-      show ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-   qed
- qed
-qed
-
-end
-
-
-
-context valid_trace
-begin
-
-lemma finite_threads:
-  shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
-unfolding cp_def wq_def
-apply(induct s rule: schs.induct)
-apply(simp add: Let_def cpreced_initial)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-done
-
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
-  by (unfold s_RAG_def, auto)
-
-lemma wq_threads: 
-  assumes h: "th \<in> set (wq s cs)"
-  shows "th \<in> threads s"
-
-
-lemma wq_threads: 
-  assumes h: "th \<in> set (wq s cs)"
-  shows "th \<in> threads s"
-proof -
- from vt and h show ?thesis
-  proof(induct arbitrary: th cs)
-    case (vt_cons s e)
-    interpret vt_s: valid_trace s
-      using vt_cons(1) by (unfold_locales, auto)
-    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
-      and stp: "step s e"
-      and vt: "vt s"
-      and h: "th \<in> set (wq (e # s) cs)"
-    show ?case
-    proof(cases e)
-      case (Create th' prio)
-      with ih h show ?thesis
-        by (auto simp:wq_def Let_def)
-    next
-      case (Exit th')
-      with stp ih h show ?thesis
-        apply (auto simp:wq_def Let_def)
-        apply (ind_cases "step s (Exit th')")
-        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
-               s_RAG_def s_holding_def cs_holding_def)
-        done
-    next
-      case (V th' cs')
-      show ?thesis
-      proof(cases "cs' = cs")
-        case False
-        with h
-        show ?thesis
-          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
-          by (drule_tac ih, simp)
-      next
-        case True
-        from h
-        show ?thesis
-        proof(unfold V wq_def)
-          assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
-          show "th \<in> threads (V th' cs' # s)"
-          proof(cases "cs = cs'")
-            case False
-            hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
-            with th_in have " th \<in> set (wq s cs)" 
-              by (fold wq_def, simp)
-            from ih [OF this] show ?thesis by simp
-          next
-            case True
-            show ?thesis
-            proof(cases "wq_fun (schs s) cs'")
-              case Nil
-              with h V show ?thesis
-                apply (auto simp:wq_def Let_def split:if_splits)
-                by (fold wq_def, drule_tac ih, simp)
-            next
-              case (Cons a rest)
-              assume eq_wq: "wq_fun (schs s) cs' = a # rest"
-              with h V show ?thesis
-                apply (auto simp:Let_def wq_def split:if_splits)
-              proof -
-                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
-                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
-                proof(rule someI2)
-                  from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
-                  show "distinct rest \<and> set rest = set rest" by auto
-                next
-                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
-                    by auto
-                qed
-                with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
-                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
-              next
-                assume th_in: "th \<in> set (wq_fun (schs s) cs)"
-                from ih[OF this[folded wq_def]]
-                show "th \<in> threads s" .
-              qed
-            qed
-          qed
-        qed
-      qed
-    next
-      case (P th' cs')
-      from h stp
-      show ?thesis
-        apply (unfold P wq_def)
-        apply (auto simp:Let_def split:if_splits, fold wq_def)
-        apply (auto intro:ih)
-        apply(ind_cases "step s (P th' cs')")
-        by (unfold runing_def readys_def, auto)
-    next
-      case (Set thread prio)
-      with ih h show ?thesis
-        by (auto simp:wq_def Let_def)
-    qed
-  next
-    case vt_nil
-    thus ?case by (auto simp:wq_def)
-  qed
-qed
-
-lemma dm_RAG_threads:
-  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
-  shows "th \<in> threads s"
-proof -
-  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
-  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
-  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
-  hence "th \<in> set (wq s cs)"
-    by (unfold s_RAG_def, auto simp:cs_waiting_def)
-  from wq_threads [OF this] show ?thesis .
-qed
-
-
-lemma cp_le:
-  assumes th_in: "th \<in> threads s"
-  shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
-  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
-         \<le> Max ((\<lambda>th. preced th s) ` threads s)"
-    (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
-  proof(rule Max_f_mono)
-    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
-  next
-    from finite_threads
-    show "finite (threads s)" .
-  next
-    from th_in
-    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
-      apply (auto simp:Domain_def)
-      apply (rule_tac dm_RAG_threads)
-      apply (unfold trancl_domain [of "RAG s", symmetric])
-      by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
-  qed
-qed
-
-lemma le_cp:
-  shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
-  show "Prc (priority th s) (last_set th s)
-    \<le> Max (insert (Prc (priority th s) (last_set th s))
-            ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
-    (is "?l \<le> Max (insert ?l ?A)")
-  proof(cases "?A = {}")
-    case False
-    have "finite ?A" (is "finite (?f ` ?B)")
-    proof -
-      have "finite ?B" 
-      proof-
-        have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
-        proof -
-          let ?F = "\<lambda> (x, y). the_th x"
-          have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
-            apply (auto simp:image_def)
-            by (rule_tac x = "(Th x, Th th)" in bexI, auto)
-          moreover have "finite \<dots>"
-          proof -
-            from finite_RAG have "finite (RAG s)" .
-            hence "finite ((RAG (wq s))\<^sup>+)"
-              apply (unfold finite_trancl)
-              by (auto simp: s_RAG_def cs_RAG_def wq_def)
-            thus ?thesis by auto
-          qed
-          ultimately show ?thesis by (auto intro:finite_subset)
-        qed
-        thus ?thesis by (simp add:cs_dependants_def)
-      qed
-      thus ?thesis by simp
-    qed
-    from Max_insert [OF this False, of ?l] show ?thesis by auto
-  next
-    case True
-    thus ?thesis by auto
-  qed
-qed
-
-lemma max_cp_eq: 
-  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
-  (is "?l = ?r")
-proof(cases "threads s = {}")
-  case True
-  thus ?thesis by auto
-next
-  case False
-  have "?l \<in> ((cp s) ` threads s)"
-  proof(rule Max_in)
-    from finite_threads
-    show "finite (cp s ` threads s)" by auto
-  next
-    from False show "cp s ` threads s \<noteq> {}" by auto
-  qed
-  then obtain th 
-    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
-  have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
-  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
-  proof -
-    have "?r \<in> (?f ` ?A)"
-    proof(rule Max_in)
-      from finite_threads
-      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
-    next
-      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
-    qed
-    then obtain th' where 
-      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
-    from le_cp [of th']  eq_r
-    have "?r \<le> cp s th'" by auto
-    moreover have "\<dots> \<le> cp s th"
-    proof(fold eq_l)
-      show " cp s th' \<le> Max (cp s ` threads s)"
-      proof(rule Max_ge)
-        from th_in' show "cp s th' \<in> cp s ` threads s"
-          by auto
-      next
-        from finite_threads
-        show "finite (cp s ` threads s)" by auto
-      qed
-    qed
-    ultimately show ?thesis by auto
-  qed
-  ultimately show ?thesis using eq_l by auto
-qed
-
-lemma max_cp_eq_the_preced:
-  shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
-  using max_cp_eq using the_preced_def by presburger 
-
-end
-
-lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
-  by (unfold preced_def, simp)
-
-lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
-proof
-  fix th'
-  show "the_preced (V th cs # s) th' = the_preced s th'"
-    by (unfold the_preced_def preced_def, simp)
-qed
-
-lemma step_RAG_v: 
-assumes vt:
-  "vt (V th cs#s)"
-shows "
-  RAG (V th cs # s) =
-  RAG s - {(Cs cs, Th th)} -
-  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-  {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
-proof -
-  interpret vt_v: valid_trace_v s "V th cs"
-    using assms step_back_vt by (unfold_locales, auto) 
-  show ?thesis using vt_v.RAG_es .
-qed
-
-
-
-
-
-text {* (* ddd *) 
-  The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
-  with the happening of @{text "V"}-events:
-*}
-lemma step_RAG_v:
-assumes vt:
-  "vt (V th cs#s)"
-shows "
-  RAG (V th cs # s) =
-  RAG s - {(Cs cs, Th th)} -
-  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-  {(Cs cs, Th th') |th'.  next_th s th cs th'}"
-  apply (insert vt, unfold s_RAG_def) 
-  apply (auto split:if_splits list.splits simp:Let_def)
-  apply (auto elim: step_v_waiting_mono step_v_hold_inv
-              step_v_release step_v_wait_inv
-              step_v_get_hold step_v_release_inv)
-  apply (erule_tac step_v_not_wait, auto)
-  done
-
-text {* 
-  The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
-  with the happening of @{text "P"}-events:
-*}
-lemma step_RAG_p:
-  "vt (P th cs#s) \<Longrightarrow>
-  RAG (P th cs # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
-                                             else RAG s \<union> {(Th th, Cs cs)})"
-  apply(simp only: s_RAG_def wq_def)
-  apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
-  apply(case_tac "csa = cs", auto)
-  apply(fold wq_def)
-  apply(drule_tac step_back_step)
-  apply(ind_cases " step s (P (hd (wq s cs)) cs)")
-  apply(simp add:s_RAG_def wq_def cs_holding_def)
-  apply(auto)
-  done
-
-
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
-  by (unfold s_RAG_def, auto)
-
-context valid_trace
-begin
-
-text {*
-  The following lemma shows that @{text "RAG"} is acyclic.
-  The overall structure is by induction on the formation of @{text "vt s"}
-  and then case analysis on event @{text "e"}, where the non-trivial cases 
-  for those for @{text "V"} and @{text "P"} events.
-*}
-lemma acyclic_RAG:
-  shows "acyclic (RAG s)"
-using vt
-proof(induct)
-  case (vt_cons s e)
-  interpret vt_s: valid_trace s using vt_cons(1)
-    by (unfold_locales, simp)
-  assume ih: "acyclic (RAG s)"
-    and stp: "step s e"
-    and vt: "vt s"
-  show ?case
-  proof(cases e)
-    case (Create th prio)
-    with ih
-    show ?thesis by (simp add:RAG_create_unchanged)
-  next
-    case (Exit th)
-    with ih show ?thesis by (simp add:RAG_exit_unchanged)
-  next
-    case (V th cs)
-    from V vt stp have vtt: "vt (V th cs#s)" by auto
-    from step_RAG_v [OF this]
-    have eq_de: 
-      "RAG (e # s) = 
-      RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-      {(Cs cs, Th th') |th'. next_th s th cs th'}"
-      (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
-    from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
-    from step_back_step [OF vtt]
-    have "step s (V th cs)" .
-    thus ?thesis
-    proof(cases)
-      assume "holding s th cs"
-      hence th_in: "th \<in> set (wq s cs)" and
-        eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
-      then obtain rest where
-        eq_wq: "wq s cs = th#rest"
-        by (cases "wq s cs", auto)
-      show ?thesis
-      proof(cases "rest = []")
-        case False
-        let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
-        from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
-          by (unfold next_th_def, auto)
-        let ?E = "(?A - ?B - ?C)"
-        have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
-        proof
-          assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
-          hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-          from tranclD [OF this]
-          obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
-          hence th_d: "(Th ?th', x) \<in> ?A" by simp
-          from RAG_target_th [OF this]
-          obtain cs' where eq_x: "x = Cs cs'" by auto
-          with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
-          hence wt_th': "waiting s ?th' cs'"
-            unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
-          hence "cs' = cs"
-          proof(rule vt_s.waiting_unique)
-            from eq_wq vt_s.wq_distinct[of cs]
-            show "waiting s ?th' cs" 
-              apply (unfold s_waiting_def wq_def, auto)
-            proof -
-              assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-                and eq_wq: "wq_fun (schs s) cs = th # rest"
-              have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-              proof(rule someI2)
-                from vt_s.wq_distinct[of cs] and eq_wq
-                show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
-              next
-                fix x assume "distinct x \<and> set x = set rest"
-                with False show "x \<noteq> []" by auto
-              qed
-              hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
-                set (SOME q. distinct q \<and> set q = set rest)" by auto
-              moreover have "\<dots> = set rest" 
-              proof(rule someI2)
-                from vt_s.wq_distinct[of cs] and eq_wq
-                show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
-              next
-                show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-              qed
-              moreover note hd_in
-              ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
-            next
-              assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-                and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
-              have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-              proof(rule someI2)
-                from vt_s.wq_distinct[of cs] and eq_wq
-                show "distinct rest \<and> set rest = set rest" by auto
-              next
-                fix x assume "distinct x \<and> set x = set rest"
-                with False show "x \<noteq> []" by auto
-              qed
-              hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
-                set (SOME q. distinct q \<and> set q = set rest)" by auto
-              moreover have "\<dots> = set rest" 
-              proof(rule someI2)
-                from vt_s.wq_distinct[of cs] and eq_wq
-                show "distinct rest \<and> set rest = set rest" by auto
-              next
-                show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-              qed
-              moreover note hd_in
-              ultimately show False by auto
-            qed
-          qed
-          with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
-          with False
-          show "False" by (auto simp: next_th_def eq_wq)
-        qed
-        with acyclic_insert[symmetric] and ac
-          and eq_de eq_D show ?thesis by auto
-      next
-        case True
-        with eq_wq
-        have eq_D: "?D = {}"
-          by (unfold next_th_def, auto)
-        with eq_de ac
-        show ?thesis by auto
-      qed 
-    qed
-  next
-    case (P th cs)
-    from P vt stp have vtt: "vt (P th cs#s)" by auto
-    from step_RAG_p [OF this] P
-    have "RAG (e # s) = 
-      (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else 
-      RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
-      by simp
-    moreover have "acyclic ?R"
-    proof(cases "wq s cs = []")
-      case True
-      hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
-      have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
-      proof
-        assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
-        hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-        from tranclD2 [OF this]
-        obtain x where "(x, Cs cs) \<in> RAG s" by auto
-        with True show False by (auto simp:s_RAG_def cs_waiting_def)
-      qed
-      with acyclic_insert ih eq_r show ?thesis by auto
-    next
-      case False
-      hence eq_r: "?R =  RAG s \<union> {(Th th, Cs cs)}" by simp
-      have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
-      proof
-        assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
-        hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-        moreover from step_back_step [OF vtt] have "step s (P th cs)" .
-        ultimately show False
-        proof -
-          show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
-            by (ind_cases "step s (P th cs)", simp)
-        qed
-      qed
-      with acyclic_insert ih eq_r show ?thesis by auto
-      qed
-      ultimately show ?thesis by simp
-    next
-      case (Set thread prio)
-      with ih
-      thm RAG_set_unchanged
-      show ?thesis by (simp add:RAG_set_unchanged)
-    qed
-  next
-    case vt_nil
-    show "acyclic (RAG ([]::state))"
-      by (auto simp: s_RAG_def cs_waiting_def 
-        cs_holding_def wq_def acyclic_def)
-qed
-
-
-lemma finite_RAG:
-  shows "finite (RAG s)"
-proof -
-  from vt show ?thesis
-  proof(induct)
-    case (vt_cons s e)
-    interpret vt_s: valid_trace s using vt_cons(1)
-      by (unfold_locales, simp)
-    assume ih: "finite (RAG s)"
-      and stp: "step s e"
-      and vt: "vt s"
-    show ?case
-    proof(cases e)
-      case (Create th prio)
-      with ih
-      show ?thesis by (simp add:RAG_create_unchanged)
-    next
-      case (Exit th)
-      with ih show ?thesis by (simp add:RAG_exit_unchanged)
-    next
-      case (V th cs)
-      from V vt stp have vtt: "vt (V th cs#s)" by auto
-      from step_RAG_v [OF this]
-      have eq_de: "RAG (e # s) = 
-                   RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-                      {(Cs cs, Th th') |th'. next_th s th cs th'}
-"
-        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
-      moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
-      moreover have "finite ?D"
-      proof -
-        have "?D = {} \<or> (\<exists> a. ?D = {a})" 
-          by (unfold next_th_def, auto)
-        thus ?thesis
-        proof
-          assume h: "?D = {}"
-          show ?thesis by (unfold h, simp)
-        next
-          assume "\<exists> a. ?D = {a}"
-          thus ?thesis
-            by (metis finite.simps)
-        qed
-      qed
-      ultimately show ?thesis by simp
-    next
-      case (P th cs)
-      from P vt stp have vtt: "vt (P th cs#s)" by auto
-      from step_RAG_p [OF this] P
-      have "RAG (e # s) = 
-              (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else 
-                                    RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
-        by simp
-      moreover have "finite ?R"
-      proof(cases "wq s cs = []")
-        case True
-        hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
-        with True and ih show ?thesis by auto
-      next
-        case False
-        hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
-        with False and ih show ?thesis by auto
-      qed
-      ultimately show ?thesis by auto
-    next
-      case (Set thread prio)
-      with ih
-      show ?thesis by (simp add:RAG_set_unchanged)
-    qed
-  next
-    case vt_nil
-    show "finite (RAG ([]::state))"
-      by (auto simp: s_RAG_def cs_waiting_def 
-                   cs_holding_def wq_def acyclic_def)
-  qed
-qed
-
-text {* Several useful lemmas *}
-
-lemma wf_dep_converse: 
-  shows "wf ((RAG s)^-1)"
-proof(rule finite_acyclic_wf_converse)
-  from finite_RAG 
-  show "finite (RAG s)" .
-next
-  from acyclic_RAG
-  show "acyclic (RAG s)" .
-qed
-
-end
-
-lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
-  by (induct l, auto)
-
-lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
-  by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-
-context valid_trace
-begin
-
-lemma wq_threads: 
-  assumes h: "th \<in> set (wq s cs)"
-  shows "th \<in> threads s"
-proof -
- from vt and h show ?thesis
-  proof(induct arbitrary: th cs)
-    case (vt_cons s e)
-    interpret vt_s: valid_trace s
-      using vt_cons(1) by (unfold_locales, auto)
-    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
-      and stp: "step s e"
-      and vt: "vt s"
-      and h: "th \<in> set (wq (e # s) cs)"
-    show ?case
-    proof(cases e)
-      case (Create th' prio)
-      with ih h show ?thesis
-        by (auto simp:wq_def Let_def)
-    next
-      case (Exit th')
-      with stp ih h show ?thesis
-        apply (auto simp:wq_def Let_def)
-        apply (ind_cases "step s (Exit th')")
-        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
-               s_RAG_def s_holding_def cs_holding_def)
-        done
-    next
-      case (V th' cs')
-      show ?thesis
-      proof(cases "cs' = cs")
-        case False
-        with h
-        show ?thesis
-          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
-          by (drule_tac ih, simp)
-      next
-        case True
-        from h
-        show ?thesis
-        proof(unfold V wq_def)
-          assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
-          show "th \<in> threads (V th' cs' # s)"
-          proof(cases "cs = cs'")
-            case False
-            hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
-            with th_in have " th \<in> set (wq s cs)" 
-              by (fold wq_def, simp)
-            from ih [OF this] show ?thesis by simp
-          next
-            case True
-            show ?thesis
-            proof(cases "wq_fun (schs s) cs'")
-              case Nil
-              with h V show ?thesis
-                apply (auto simp:wq_def Let_def split:if_splits)
-                by (fold wq_def, drule_tac ih, simp)
-            next
-              case (Cons a rest)
-              assume eq_wq: "wq_fun (schs s) cs' = a # rest"
-              with h V show ?thesis
-                apply (auto simp:Let_def wq_def split:if_splits)
-              proof -
-                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
-                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
-                proof(rule someI2)
-                  from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
-                  show "distinct rest \<and> set rest = set rest" by auto
-                next
-                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
-                    by auto
-                qed
-                with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
-                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
-              next
-                assume th_in: "th \<in> set (wq_fun (schs s) cs)"
-                from ih[OF this[folded wq_def]]
-                show "th \<in> threads s" .
-              qed
-            qed
-          qed
-        qed
-      qed
-    next
-      case (P th' cs')
-      from h stp
-      show ?thesis
-        apply (unfold P wq_def)
-        apply (auto simp:Let_def split:if_splits, fold wq_def)
-        apply (auto intro:ih)
-        apply(ind_cases "step s (P th' cs')")
-        by (unfold runing_def readys_def, auto)
-    next
-      case (Set thread prio)
-      with ih h show ?thesis
-        by (auto simp:wq_def Let_def)
-    qed
-  next
-    case vt_nil
-    thus ?case by (auto simp:wq_def)
-  qed
-qed
-
-lemma range_in: "\<lbrakk>(Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
-  apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
-  by (auto intro:wq_threads)
-
-lemma readys_v_eq:
-  assumes neq_th: "th \<noteq> thread"
-  and eq_wq: "wq s cs = thread#rest"
-  and not_in: "th \<notin>  set rest"
-  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
-  from assms show ?thesis
-    apply (auto simp:readys_def)
-    apply(simp add:s_waiting_def[folded wq_def])
-    apply (erule_tac x = csa in allE)
-    apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
-    apply (case_tac "csa = cs", simp)
-    apply (erule_tac x = cs in allE)
-    apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
-    apply(auto simp add: wq_def)
-    apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
-    proof -
-       assume th_nin: "th \<notin> set rest"
-        and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
-        and eq_wq: "wq_fun (schs s) cs = thread # rest"
-      have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-      proof(rule someI2)
-        from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
-        show "distinct rest \<and> set rest = set rest" by auto
-      next
-        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-      qed
-      with th_nin th_in show False by auto
-    qed
-qed
-
-text {* \noindent
-  The following lemmas shows that: starting from any node in @{text "RAG"}, 
-  by chasing out-going edges, it is always possible to reach a node representing a ready
-  thread. In this lemma, it is the @{text "th'"}.
-*}
-
-lemma chain_building:
-  shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
-proof -
-  from wf_dep_converse
-  have h: "wf ((RAG s)\<inverse>)" .
-  show ?thesis
-  proof(induct rule:wf_induct [OF h])
-    fix x
-    assume ih [rule_format]: 
-      "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow> 
-           y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
-    show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
-    proof
-      assume x_d: "x \<in> Domain (RAG s)"
-      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
-      proof(cases x)
-        case (Th th)
-        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
-        with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
-        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
-        hence "Cs cs \<in> Domain (RAG s)" by auto
-        from ih [OF x_in_r this] obtain th'
-          where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
-        have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
-        with th'_ready show ?thesis by auto
-      next
-        case (Cs cs)
-        from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
-        show ?thesis
-        proof(cases "th' \<in> readys s")
-          case True
-          from True and th'_d show ?thesis by auto
-        next
-          case False
-          from th'_d and range_in  have "th' \<in> threads s" by auto
-          with False have "Th th' \<in> Domain (RAG s)" 
-            by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
-          from ih [OF th'_d this]
-          obtain th'' where 
-            th''_r: "th'' \<in> readys s" and 
-            th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
-          from th'_d and th''_in 
-          have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
-          with th''_r show ?thesis by auto
-        qed
-      qed
-    qed
-  qed
-qed
-
-text {* \noindent
-  The following is just an instance of @{text "chain_building"}.
-*}
-lemma th_chain_to_ready:
-  assumes th_in: "th \<in> threads s"
-  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
-proof(cases "th \<in> readys s")
-  case True
-  thus ?thesis by auto
-next
-  case False
-  from False and th_in have "Th th \<in> Domain (RAG s)" 
-    by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
-  from chain_building [rule_format, OF this]
-  show ?thesis by auto
-qed
-
-end
-
-
-
-lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
-  by (unfold s_holding_def cs_holding_def, auto)
-
-context valid_trace
-begin
-
-lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
-  apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
-  by(auto elim:waiting_unique holding_unique)
-
-end
-
-
-lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
-by (induct rule:trancl_induct, auto)
-
-context valid_trace
-begin
-
-lemma dchain_unique:
-  assumes th1_d: "(n, Th th1) \<in> (RAG s)^+"
-  and th1_r: "th1 \<in> readys s"
-  and th2_d: "(n, Th th2) \<in> (RAG s)^+"
-  and th2_r: "th2 \<in> readys s"
-  shows "th1 = th2"
-proof -
-  { assume neq: "th1 \<noteq> th2"
-    hence "Th th1 \<noteq> Th th2" by simp
-    from unique_chain [OF _ th1_d th2_d this] and unique_RAG 
-    have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
-    hence "False"
-    proof
-      assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
-      from trancl_split [OF this]
-      obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
-      then obtain cs where eq_n: "n = Cs cs"
-        by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-      from dd eq_n have "th1 \<notin> readys s"
-        by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
-      with th1_r show ?thesis by auto
-    next
-      assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
-      from trancl_split [OF this]
-      obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
-      then obtain cs where eq_n: "n = Cs cs"
-        by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-      from dd eq_n have "th2 \<notin> readys s"
-        by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
-      with th2_r show ?thesis by auto
-    qed
-  } thus ?thesis by auto
-qed
-
-end
-             
-
-lemma step_holdents_p_add:
-  assumes vt: "vt (P th cs#s)"
-  and "wq s cs = []"
-  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
-proof -
-  from assms show ?thesis
-  unfolding  holdents_test step_RAG_p[OF vt] by (auto)
-qed
-
-lemma step_holdents_p_eq:
-  assumes vt: "vt (P th cs#s)"
-  and "wq s cs \<noteq> []"
-  shows "holdents (P th cs#s) th = holdents s th"
-proof -
-  from assms show ?thesis
-  unfolding  holdents_test step_RAG_p[OF vt] by auto
-qed
-
-
-lemma (in valid_trace) finite_holding :
-  shows "finite (holdents s th)"
-proof -
-  let ?F = "\<lambda> (x, y). the_cs x"
-  from finite_RAG 
-  have "finite (RAG s)" .
-  hence "finite (?F `(RAG s))" by simp
-  moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>" 
-  proof -
-    { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
-      fix x assume "(Cs x, Th th) \<in> RAG s"
-      hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
-      moreover have "?F (Cs x, Th th) = x" by simp
-      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp 
-    } thus ?thesis by auto
-  qed
-  ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
-qed
-
-lemma cntCS_v_dec: 
-  assumes vtv: "vt (V thread cs#s)"
-  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
-proof -
-  from vtv interpret vt_s: valid_trace s
-    by (cases, unfold_locales, simp)
-  from vtv interpret vt_v: valid_trace "V thread cs#s"
-     by (unfold_locales, simp)
-  from step_back_step[OF vtv]
-  have cs_in: "cs \<in> holdents s thread" 
-    apply (cases, unfold holdents_test s_RAG_def, simp)
-    by (unfold cs_holding_def s_holding_def wq_def, auto)
-  moreover have cs_not_in: 
-    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
-    apply (insert vt_s.wq_distinct[of cs])
-    apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
-            auto simp:next_th_def)
-  proof -
-    fix rest
-    assume dst: "distinct (rest::thread list)"
-      and ne: "rest \<noteq> []"
-    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-    proof(rule someI2)
-      from dst show "distinct rest \<and> set rest = set rest" by auto
-    next
-      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-    qed
-    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
-                     set (SOME q. distinct q \<and> set q = set rest)" by simp
-    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-    proof(rule someI2)
-      from dst show "distinct rest \<and> set rest = set rest" by auto
-    next
-      fix x assume " distinct x \<and> set x = set rest" with ne
-      show "x \<noteq> []" by auto
-    qed
-    ultimately 
-    show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
-      by auto
-  next
-    fix rest
-    assume dst: "distinct (rest::thread list)"
-      and ne: "rest \<noteq> []"
-    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-    proof(rule someI2)
-      from dst show "distinct rest \<and> set rest = set rest" by auto
-    next
-      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-    qed
-    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
-                     set (SOME q. distinct q \<and> set q = set rest)" by simp
-    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-    proof(rule someI2)
-      from dst show "distinct rest \<and> set rest = set rest" by auto
-    next
-      fix x assume " distinct x \<and> set x = set rest" with ne
-      show "x \<noteq> []" by auto
-    qed
-    ultimately show "False" by auto 
-  qed
-  ultimately 
-  have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
-    by auto
-  moreover have "card \<dots> = 
-                    Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
-  proof(rule card_insert)
-    from vt_v.finite_holding
-    show " finite (holdents (V thread cs # s) thread)" .
-  qed
-  moreover from cs_not_in 
-  have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
-  ultimately show ?thesis by (simp add:cntCS_def)
-qed 
-
-lemma count_rec1 [simp]: 
-  assumes "Q e"
-  shows "count Q (e#es) = Suc (count Q es)"
-  using assms
-  by (unfold count_def, auto)
-
-lemma count_rec2 [simp]: 
-  assumes "\<not>Q e"
-  shows "count Q (e#es) = (count Q es)"
-  using assms
-  by (unfold count_def, auto)
-
-lemma count_rec3 [simp]: 
-  shows "count Q [] =  0"
-  by (unfold count_def, auto)
-
-lemma cntP_diff_inv:
-  assumes "cntP (e#s) th \<noteq> cntP s th"
-  shows "isP e \<and> actor e = th"
-proof(cases e)
-  case (P th' pty)
-  show ?thesis
-  by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)", 
-        insert assms P, auto simp:cntP_def)
-qed (insert assms, auto simp:cntP_def)
-  
-lemma cntV_diff_inv:
-  assumes "cntV (e#s) th \<noteq> cntV s th"
-  shows "isV e \<and> actor e = th"
-proof(cases e)
-  case (V th' pty)
-  show ?thesis
-  by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)", 
-        insert assms V, auto simp:cntV_def)
-qed (insert assms, auto simp:cntV_def)
-
-context valid_trace
-begin
-
-text {* (* ddd *) \noindent
-  The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} 
-  of one particular thread. t
-*} 
-
-lemma cnp_cnv_cncs:
-  shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) 
-                                       then cntCS s th else cntCS s th + 1)"
-proof -
-  from vt show ?thesis
-  proof(induct arbitrary:th)
-    case (vt_cons s e)
-    interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
-    assume vt: "vt s"
-    and ih: "\<And>th. cntP s th  = cntV s th +
-               (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
-    and stp: "step s e"
-    from stp show ?case
-    proof(cases)
-      case (thread_create thread prio)
-      assume eq_e: "e = Create thread prio"
-        and not_in: "thread \<notin> threads s"
-      show ?thesis
-      proof -
-        { fix cs 
-          assume "thread \<in> set (wq s cs)"
-          from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
-          with not_in have "False" by simp
-        } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
-          by (auto simp:readys_def threads.simps s_waiting_def 
-            wq_def cs_waiting_def Let_def)
-        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-        have eq_cncs: "cntCS (e#s) th = cntCS s th"
-          unfolding cntCS_def holdents_test
-          by (simp add:RAG_create_unchanged eq_e)
-        { assume "th \<noteq> thread"
-          with eq_readys eq_e
-          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
-                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
-            by (simp add:threads.simps)
-          with eq_cnp eq_cnv eq_cncs ih not_in
-          have ?thesis by simp
-        } moreover {
-          assume eq_th: "th = thread"
-          with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp
-          moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
-          moreover note eq_cnp eq_cnv eq_cncs
-          ultimately have ?thesis by auto
-        } ultimately show ?thesis by blast
-      qed
-    next
-      case (thread_exit thread)
-      assume eq_e: "e = Exit thread" 
-      and is_runing: "thread \<in> runing s"
-      and no_hold: "holdents s thread = {}"
-      from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-      from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-      have eq_cncs: "cntCS (e#s) th = cntCS s th"
-        unfolding cntCS_def holdents_test
-        by (simp add:RAG_exit_unchanged eq_e)
-      { assume "th \<noteq> thread"
-        with eq_e
-        have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
-          (th \<in> readys (s) \<or> th \<notin> threads (s))" 
-          apply (simp add:threads.simps readys_def)
-          apply (subst s_waiting_def)
-          apply (simp add:Let_def)
-          apply (subst s_waiting_def, simp)
-          done
-        with eq_cnp eq_cnv eq_cncs ih
-        have ?thesis by simp
-      } moreover {
-        assume eq_th: "th = thread"
-        with ih is_runing have " cntP s th = cntV s th + cntCS s th" 
-          by (simp add:runing_def)
-        moreover from eq_th eq_e have "th \<notin> threads (e#s)"
-          by simp
-        moreover note eq_cnp eq_cnv eq_cncs
-        ultimately have ?thesis by auto
-      } ultimately show ?thesis by blast
-    next
-      case (thread_P thread cs)
-      assume eq_e: "e = P thread cs"
-        and is_runing: "thread \<in> runing s"
-        and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
-      from thread_P vt stp ih  have vtp: "vt (P thread cs#s)" by auto
-      then interpret vt_p: valid_trace "(P thread cs#s)"
-        by (unfold_locales, simp)
-      show ?thesis 
-      proof -
-        { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
-          assume neq_th: "th \<noteq> thread"
-          with eq_e
-          have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
-            apply (simp add:readys_def s_waiting_def wq_def Let_def)
-            apply (rule_tac hh)
-             apply (intro iffI allI, clarify)
-            apply (erule_tac x = csa in allE, auto)
-            apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
-            apply (erule_tac x = cs in allE, auto)
-            by (case_tac "(wq_fun (schs s) cs)", auto)
-          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
-            apply (simp add:cntCS_def holdents_test)
-            by (unfold  step_RAG_p [OF vtp], auto)
-          moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
-            by (simp add:cntP_def count_def)
-          moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
-            by (simp add:cntV_def count_def)
-          moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
-          moreover note ih [of th] 
-          ultimately have ?thesis by simp
-        } moreover {
-          assume eq_th: "th = thread"
-          have ?thesis
-          proof -
-            from eq_e eq_th have eq_cnp: "cntP (e # s) th  = 1 + (cntP s th)" 
-              by (simp add:cntP_def count_def)
-            from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
-              by (simp add:cntV_def count_def)
-            show ?thesis
-            proof (cases "wq s cs = []")
-              case True
-              with is_runing
-              have "th \<in> readys (e#s)"
-                apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
-                apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
-                by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
-              moreover have "cntCS (e # s) th = 1 + cntCS s th"
-              proof -
-                have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
-                  Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
-                proof -
-                  have "?L = insert cs ?R" by auto
-                  moreover have "card \<dots> = Suc (card (?R - {cs}))" 
-                  proof(rule card_insert)
-                    from vt_s.finite_holding [of thread]
-                    show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
-                      by (unfold holdents_test, simp)
-                  qed
-                  moreover have "?R - {cs} = ?R"
-                  proof -
-                    have "cs \<notin> ?R"
-                    proof
-                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
-                      with no_dep show False by auto
-                    qed
-                    thus ?thesis by auto
-                  qed
-                  ultimately show ?thesis by auto
-                qed
-                thus ?thesis
-                  apply (unfold eq_e eq_th cntCS_def)
-                  apply (simp add: holdents_test)
-                  by (unfold step_RAG_p [OF vtp], auto simp:True)
-              qed
-              moreover from is_runing have "th \<in> readys s"
-                by (simp add:runing_def eq_th)
-              moreover note eq_cnp eq_cnv ih [of th]
-              ultimately show ?thesis by auto
-            next
-              case False
-              have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
-                    by (unfold eq_th eq_e wq_def, auto simp:Let_def)
-              have "th \<notin> readys (e#s)"
-              proof
-                assume "th \<in> readys (e#s)"
-                hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
-                from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
-                hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)" 
-                  by (simp add:s_waiting_def wq_def)
-                moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
-                ultimately have "th = hd (wq (e#s) cs)" by blast
-                with eq_wq have "th = hd (wq s cs @ [th])" by simp
-                hence "th = hd (wq s cs)" using False by auto
-                with False eq_wq vt_p.wq_distinct [of cs]
-                show False by (fold eq_e, auto)
-              qed
-              moreover from is_runing have "th \<in> threads (e#s)" 
-                by (unfold eq_e, auto simp:runing_def readys_def eq_th)
-              moreover have "cntCS (e # s) th = cntCS s th"
-                apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
-                by (auto simp:False)
-              moreover note eq_cnp eq_cnv ih[of th]
-              moreover from is_runing have "th \<in> readys s"
-                by (simp add:runing_def eq_th)
-              ultimately show ?thesis by auto
-            qed
-          qed
-        } ultimately show ?thesis by blast
-      qed
-    next
-      case (thread_V thread cs)
-      from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
-      then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
-      assume eq_e: "e = V thread cs"
-        and is_runing: "thread \<in> runing s"
-        and hold: "holding s thread cs"
-      from hold obtain rest 
-        where eq_wq: "wq s cs = thread # rest"
-        by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
-      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
-      have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-      proof(rule someI2)
-        from vt_v.wq_distinct[of cs] and eq_wq
-        show "distinct rest \<and> set rest = set rest"
-          by (metis distinct.simps(2) vt_s.wq_distinct)
-      next
-        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
-          by auto
-      qed
-      show ?thesis
-      proof -
-        { assume eq_th: "th = thread"
-          from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
-            by (unfold eq_e, simp add:cntP_def count_def)
-          moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
-            by (unfold eq_e, simp add:cntV_def count_def)
-          moreover from cntCS_v_dec [OF vtv] 
-          have "cntCS (e # s) thread + 1 = cntCS s thread"
-            by (simp add:eq_e)
-          moreover from is_runing have rd_before: "thread \<in> readys s"
-            by (unfold runing_def, simp)
-          moreover have "thread \<in> readys (e # s)"
-          proof -
-            from is_runing
-            have "thread \<in> threads (e#s)" 
-              by (unfold eq_e, auto simp:runing_def readys_def)
-            moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
-            proof
-              fix cs1
-              { assume eq_cs: "cs1 = cs" 
-                have "\<not> waiting (e # s) thread cs1"
-                proof -
-                  from eq_wq
-                  have "thread \<notin> set (wq (e#s) cs1)"
-                    apply(unfold eq_e wq_def eq_cs s_holding_def)
-                    apply (auto simp:Let_def)
-                  proof -
-                    assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
-                    with eq_set have "thread \<in> set rest" by simp
-                    with vt_v.wq_distinct[of cs]
-                    and eq_wq show False
-                        by (metis distinct.simps(2) vt_s.wq_distinct)
-                  qed
-                  thus ?thesis by (simp add:wq_def s_waiting_def)
-                qed
-              } moreover {
-                assume neq_cs: "cs1 \<noteq> cs"
-                  have "\<not> waiting (e # s) thread cs1" 
-                  proof -
-                    from wq_v_neq [OF neq_cs[symmetric]]
-                    have "wq (V thread cs # s) cs1 = wq s cs1" .
-                    moreover have "\<not> waiting s thread cs1" 
-                    proof -
-                      from runing_ready and is_runing
-                      have "thread \<in> readys s" by auto
-                      thus ?thesis by (simp add:readys_def)
-                    qed
-                    ultimately show ?thesis 
-                      by (auto simp:wq_def s_waiting_def eq_e)
-                  qed
-              } ultimately show "\<not> waiting (e # s) thread cs1" by blast
-            qed
-            ultimately show ?thesis by (simp add:readys_def)
-          qed
-          moreover note eq_th ih
-          ultimately have ?thesis by auto
-        } moreover {
-          assume neq_th: "th \<noteq> thread"
-          from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" 
-            by (simp add:cntP_def count_def)
-          from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" 
-            by (simp add:cntV_def count_def)
-          have ?thesis
-          proof(cases "th \<in> set rest")
-            case False
-            have "(th \<in> readys (e # s)) = (th \<in> readys s)"
-              apply (insert step_back_vt[OF vtv])
-              by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
-            moreover have "cntCS (e#s) th = cntCS s th"
-              apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
-              proof -
-                have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
-                      {cs. (Cs cs, Th th) \<in> RAG s}"
-                proof -
-                  from False eq_wq
-                  have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
-                    apply (unfold next_th_def, auto)
-                  proof -
-                    assume ne: "rest \<noteq> []"
-                      and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-                      and eq_wq: "wq s cs = thread # rest"
-                    from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
-                                  set (SOME q. distinct q \<and> set q = set rest)
-                                  " by simp
-                    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-                    proof(rule someI2)
-                      from vt_s.wq_distinct[ of cs] and eq_wq
-                      show "distinct rest \<and> set rest = set rest" by auto
-                    next
-                      fix x assume "distinct x \<and> set x = set rest"
-                      with ne show "x \<noteq> []" by auto
-                    qed
-                    ultimately show 
-                      "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
-                      by auto
-                  qed    
-                  thus ?thesis by auto
-                qed
-                thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
-                             card {cs. (Cs cs, Th th) \<in> RAG s}" by simp 
-              qed
-            moreover note ih eq_cnp eq_cnv eq_threads
-            ultimately show ?thesis by auto
-          next
-            case True
-            assume th_in: "th \<in> set rest"
-            show ?thesis
-            proof(cases "next_th s thread cs th")
-              case False
-              with eq_wq and th_in have 
-                neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
-                by (auto simp:next_th_def)
-              have "(th \<in> readys (e # s)) = (th \<in> readys s)"
-              proof -
-                from eq_wq and th_in
-                have "\<not> th \<in> readys s"
-                  apply (auto simp:readys_def s_waiting_def)
-                  apply (rule_tac x = cs in exI, auto)
-                  by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
-                moreover 
-                from eq_wq and th_in and neq_hd
-                have "\<not> (th \<in> readys (e # s))"
-                  apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
-                  by (rule_tac x = cs in exI, auto simp:eq_set)
-                ultimately show ?thesis by auto
-              qed
-              moreover have "cntCS (e#s) th = cntCS s th" 
-              proof -
-                from eq_wq and  th_in and neq_hd
-                have "(holdents (e # s) th) = (holdents s th)"
-                  apply (unfold eq_e step_RAG_v[OF vtv], 
-                         auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
-                                   Let_def cs_holding_def)
-                  by (insert vt_s.wq_distinct[of cs], auto simp:wq_def)
-                thus ?thesis by (simp add:cntCS_def)
-              qed
-              moreover note ih eq_cnp eq_cnv eq_threads
-              ultimately show ?thesis by auto
-            next
-              case True
-              let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
-              let ?t = "hd ?rest"
-              from True eq_wq th_in neq_th
-              have "th \<in> readys (e # s)"
-                apply (auto simp:eq_e readys_def s_waiting_def wq_def
-                        Let_def next_th_def)
-              proof -
-                assume eq_wq: "wq_fun (schs s) cs = thread # rest"
-                  and t_in: "?t \<in> set rest"
-                show "?t \<in> threads s"
-                proof(rule vt_s.wq_threads)
-                  from eq_wq and t_in
-                  show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
-                qed
-              next
-                fix csa
-                assume eq_wq: "wq_fun (schs s) cs = thread # rest"
-                  and t_in: "?t \<in> set rest"
-                  and neq_cs: "csa \<noteq> cs"
-                  and t_in': "?t \<in>  set (wq_fun (schs s) csa)"
-                show "?t = hd (wq_fun (schs s) csa)"
-                proof -
-                  { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
-                    from vt_s.wq_distinct[of cs] and 
-                    eq_wq[folded wq_def] and t_in eq_wq
-                    have "?t \<noteq> thread" by auto
-                    with eq_wq and t_in
-                    have w1: "waiting s ?t cs"
-                      by (auto simp:s_waiting_def wq_def)
-                    from t_in' neq_hd'
-                    have w2: "waiting s ?t csa"
-                      by (auto simp:s_waiting_def wq_def)
-                    from vt_s.waiting_unique[OF w1 w2]
-                    and neq_cs have "False" by auto
-                  } thus ?thesis by auto
-                qed
-              qed
-              moreover have "cntP s th = cntV s th + cntCS s th + 1"
-              proof -
-                have "th \<notin> readys s" 
-                proof -
-                  from True eq_wq neq_th th_in
-                  show ?thesis
-                    apply (unfold readys_def s_waiting_def, auto)
-                    by (rule_tac x = cs in exI, auto simp add: wq_def)
-                qed
-                moreover have "th \<in> threads s"
-                proof -
-                  from th_in eq_wq
-                  have "th \<in> set (wq s cs)" by simp
-                  from vt_s.wq_threads [OF this] 
-                  show ?thesis .
-                qed
-                ultimately show ?thesis using ih by auto
-              qed
-              moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
-                apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
-              proof -
-                show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
-                               Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
-                  (is "card ?A = Suc (card ?B)")
-                proof -
-                  have "?A = insert cs ?B" by auto
-                  hence "card ?A = card (insert cs ?B)" by simp
-                  also have "\<dots> = Suc (card ?B)"
-                  proof(rule card_insert_disjoint)
-                    have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)" 
-                      apply (auto simp:image_def)
-                      by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
-                    with vt_s.finite_RAG
-                    show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
-                  next
-                    show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
-                    proof
-                      assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
-                      hence "(Cs cs, Th th) \<in> RAG s" by simp
-                      with True neq_th eq_wq show False
-                        by (auto simp:next_th_def s_RAG_def cs_holding_def)
-                    qed
-                  qed
-                  finally show ?thesis .
-                qed
-              qed
-              moreover note eq_cnp eq_cnv
-              ultimately show ?thesis by simp
-            qed
-          qed
-        } ultimately show ?thesis by blast
-      qed
-    next
-      case (thread_set thread prio)
-      assume eq_e: "e = Set thread prio"
-        and is_runing: "thread \<in> runing s"
-      show ?thesis
-      proof -
-        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-        have eq_cncs: "cntCS (e#s) th = cntCS s th"
-          unfolding cntCS_def holdents_test
-          by (simp add:RAG_set_unchanged eq_e)
-        from eq_e have eq_readys: "readys (e#s) = readys s" 
-          by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
-                  auto simp:Let_def)
-        { assume "th \<noteq> thread"
-          with eq_readys eq_e
-          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
-                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
-            by (simp add:threads.simps)
-          with eq_cnp eq_cnv eq_cncs ih is_runing
-          have ?thesis by simp
-        } moreover {
-          assume eq_th: "th = thread"
-          with is_runing ih have " cntP s th  = cntV s th + cntCS s th" 
-            by (unfold runing_def, auto)
-          moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
-            by (simp add:runing_def)
-          moreover note eq_cnp eq_cnv eq_cncs
-          ultimately have ?thesis by auto
-        } ultimately show ?thesis by blast
-      qed   
-    qed
-  next
-    case vt_nil
-    show ?case 
-      by (unfold cntP_def cntV_def cntCS_def, 
-        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
-  qed
-qed
-
-lemma not_thread_cncs:
-  assumes not_in: "th \<notin> threads s" 
-  shows "cntCS s th = 0"
-proof -
-  from vt not_in show ?thesis
-  proof(induct arbitrary:th)
-    case (vt_cons s e th)
-    interpret vt_s: valid_trace s using vt_cons(1)
-       by (unfold_locales, simp)
-    assume vt: "vt s"
-      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
-      and stp: "step s e"
-      and not_in: "th \<notin> threads (e # s)"
-    from stp show ?case
-    proof(cases)
-      case (thread_create thread prio)
-      assume eq_e: "e = Create thread prio"
-        and not_in': "thread \<notin> threads s"
-      have "cntCS (e # s) th = cntCS s th"
-        apply (unfold eq_e cntCS_def holdents_test)
-        by (simp add:RAG_create_unchanged)
-      moreover have "th \<notin> threads s" 
-      proof -
-        from not_in eq_e show ?thesis by simp
-      qed
-      moreover note ih ultimately show ?thesis by auto
-    next
-      case (thread_exit thread)
-      assume eq_e: "e = Exit thread"
-      and nh: "holdents s thread = {}"
-      have eq_cns: "cntCS (e # s) th = cntCS s th"
-        apply (unfold eq_e cntCS_def holdents_test)
-        by (simp add:RAG_exit_unchanged)
-      show ?thesis
-      proof(cases "th = thread")
-        case True
-        have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
-        with eq_cns show ?thesis by simp
-      next
-        case False
-        with not_in and eq_e
-        have "th \<notin> threads s" by simp
-        from ih[OF this] and eq_cns show ?thesis by simp
-      qed
-    next
-      case (thread_P thread cs)
-      assume eq_e: "e = P thread cs"
-      and is_runing: "thread \<in> runing s"
-      from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
-      have neq_th: "th \<noteq> thread" 
-      proof -
-        from not_in eq_e have "th \<notin> threads s" by simp
-        moreover from is_runing have "thread \<in> threads s"
-          by (simp add:runing_def readys_def)
-        ultimately show ?thesis by auto
-      qed
-      hence "cntCS (e # s) th  = cntCS s th "
-        apply (unfold cntCS_def holdents_test eq_e)
-        by (unfold step_RAG_p[OF vtp], auto)
-      moreover have "cntCS s th = 0"
-      proof(rule ih)
-        from not_in eq_e show "th \<notin> threads s" by simp
-      qed
-      ultimately show ?thesis by simp
-    next
-      case (thread_V thread cs)
-      assume eq_e: "e = V thread cs"
-        and is_runing: "thread \<in> runing s"
-        and hold: "holding s thread cs"
-      have neq_th: "th \<noteq> thread" 
-      proof -
-        from not_in eq_e have "th \<notin> threads s" by simp
-        moreover from is_runing have "thread \<in> threads s"
-          by (simp add:runing_def readys_def)
-        ultimately show ?thesis by auto
-      qed
-      from assms thread_V vt stp ih 
-      have vtv: "vt (V thread cs#s)" by auto
-      then interpret vt_v: valid_trace "(V thread cs#s)"
-        by (unfold_locales, simp)
-      from hold obtain rest 
-        where eq_wq: "wq s cs = thread # rest"
-        by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
-      from not_in eq_e eq_wq
-      have "\<not> next_th s thread cs th"
-        apply (auto simp:next_th_def)
-      proof -
-        assume ne: "rest \<noteq> []"
-          and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
-        have "?t \<in> set rest"
-        proof(rule someI2)
-          from vt_v.wq_distinct[of cs] and eq_wq
-          show "distinct rest \<and> set rest = set rest"
-            by (metis distinct.simps(2) vt_s.wq_distinct) 
-        next
-          fix x assume "distinct x \<and> set x = set rest" with ne
-          show "hd x \<in> set rest" by (cases x, auto)
-        qed
-        with eq_wq have "?t \<in> set (wq s cs)" by simp
-        from vt_s.wq_threads[OF this] and ni
-        show False
-          using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)` 
-            ni vt_s.wq_threads by blast 
-      qed
-      moreover note neq_th eq_wq
-      ultimately have "cntCS (e # s) th  = cntCS s th"
-        by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
-      moreover have "cntCS s th = 0"
-      proof(rule ih)
-        from not_in eq_e show "th \<notin> threads s" by simp
-      qed
-      ultimately show ?thesis by simp
-    next
-      case (thread_set thread prio)
-      print_facts
-      assume eq_e: "e = Set thread prio"
-        and is_runing: "thread \<in> runing s"
-      from not_in and eq_e have "th \<notin> threads s" by auto
-      from ih [OF this] and eq_e
-      show ?thesis 
-        apply (unfold eq_e cntCS_def holdents_test)
-        by (simp add:RAG_set_unchanged)
-    qed
-    next
-      case vt_nil
-      show ?case
-      by (unfold cntCS_def, 
-        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
-  qed
-qed
-
-end
-
-
-context valid_trace
-begin
-
-lemma dm_RAG_threads:
-  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
-  shows "th \<in> threads s"
-proof -
-  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
-  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
-  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
-  hence "th \<in> set (wq s cs)"
-    by (unfold s_RAG_def, auto simp:cs_waiting_def)
-  from wq_threads [OF this] show ?thesis .
-qed
-
-end
-
-lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
-unfolding cp_def wq_def
-apply(induct s rule: schs.induct)
-thm cpreced_initial
-apply(simp add: Let_def cpreced_initial)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-done
-
-context valid_trace
-begin
-
-lemma runing_unique:
-  assumes runing_1: "th1 \<in> runing s"
-  and runing_2: "th2 \<in> runing s"
-  shows "th1 = th2"
-proof -
-  from runing_1 and runing_2 have "cp s th1 = cp s th2"
-    unfolding runing_def
-    apply(simp)
-    done
-  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
-                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
-    (is "Max (?f ` ?A) = Max (?f ` ?B)")
-    unfolding cp_eq_cpreced 
-    unfolding cpreced_def .
-  obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
-  proof -
-    have h1: "finite (?f ` ?A)"
-    proof -
-      have "finite ?A" 
-      proof -
-        have "finite (dependants (wq s) th1)"
-        proof-
-          have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
-          proof -
-            let ?F = "\<lambda> (x, y). the_th x"
-            have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
-              apply (auto simp:image_def)
-              by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
-            moreover have "finite \<dots>"
-            proof -
-              from finite_RAG have "finite (RAG s)" .
-              hence "finite ((RAG (wq s))\<^sup>+)"
-                apply (unfold finite_trancl)
-                by (auto simp: s_RAG_def cs_RAG_def wq_def)
-              thus ?thesis by auto
-            qed
-            ultimately show ?thesis by (auto intro:finite_subset)
-          qed
-          thus ?thesis by (simp add:cs_dependants_def)
-        qed
-        thus ?thesis by simp
-      qed
-      thus ?thesis by auto
-    qed
-    moreover have h2: "(?f ` ?A) \<noteq> {}"
-    proof -
-      have "?A \<noteq> {}" by simp
-      thus ?thesis by simp
-    qed
-    from Max_in [OF h1 h2]
-    have "Max (?f ` ?A) \<in> (?f ` ?A)" .
-    thus ?thesis 
-      thm cpreced_def
-      unfolding cpreced_def[symmetric] 
-      unfolding cp_eq_cpreced[symmetric] 
-      unfolding cpreced_def 
-      using that[intro] by (auto)
-  qed
-  obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
-  proof -
-    have h1: "finite (?f ` ?B)"
-    proof -
-      have "finite ?B" 
-      proof -
-        have "finite (dependants (wq s) th2)"
-        proof-
-          have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
-          proof -
-            let ?F = "\<lambda> (x, y). the_th x"
-            have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
-              apply (auto simp:image_def)
-              by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
-            moreover have "finite \<dots>"
-            proof -
-              from finite_RAG have "finite (RAG s)" .
-              hence "finite ((RAG (wq s))\<^sup>+)"
-                apply (unfold finite_trancl)
-                by (auto simp: s_RAG_def cs_RAG_def wq_def)
-              thus ?thesis by auto
-            qed
-            ultimately show ?thesis by (auto intro:finite_subset)
-          qed
-          thus ?thesis by (simp add:cs_dependants_def)
-        qed
-        thus ?thesis by simp
-      qed
-      thus ?thesis by auto
-    qed
-    moreover have h2: "(?f ` ?B) \<noteq> {}"
-    proof -
-      have "?B \<noteq> {}" by simp
-      thus ?thesis by simp
-    qed
-    from Max_in [OF h1 h2]
-    have "Max (?f ` ?B) \<in> (?f ` ?B)" .
-    thus ?thesis by (auto intro:that)
-  qed
-  from eq_f_th1 eq_f_th2 eq_max 
-  have eq_preced: "preced th1' s = preced th2' s" by auto
-  hence eq_th12: "th1' = th2'"
-  proof (rule preced_unique)
-    from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
-    thus "th1' \<in> threads s"
-    proof
-      assume "th1' \<in> dependants (wq s) th1"
-      hence "(Th th1') \<in> Domain ((RAG s)^+)"
-        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
-        by (auto simp:Domain_def)
-      hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
-      from dm_RAG_threads[OF this] show ?thesis .
-    next
-      assume "th1' = th1"
-      with runing_1 show ?thesis
-        by (unfold runing_def readys_def, auto)
-    qed
-  next
-    from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
-    thus "th2' \<in> threads s"
-    proof
-      assume "th2' \<in> dependants (wq s) th2"
-      hence "(Th th2') \<in> Domain ((RAG s)^+)"
-        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
-        by (auto simp:Domain_def)
-      hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
-      from dm_RAG_threads[OF this] show ?thesis .
-    next
-      assume "th2' = th2"
-      with runing_2 show ?thesis
-        by (unfold runing_def readys_def, auto)
-    qed
-  qed
-  from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
-  thus ?thesis
-  proof
-    assume eq_th': "th1' = th1"
-    from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
-    thus ?thesis
-    proof
-      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
-    next
-      assume "th2' \<in> dependants (wq s) th2"
-      with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
-      hence "(Th th1, Th th2) \<in> (RAG s)^+"
-        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
-      hence "Th th1 \<in> Domain ((RAG s)^+)" 
-        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
-        by (auto simp:Domain_def)
-      hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
-      then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
-      from RAG_target_th [OF this]
-      obtain cs' where "n = Cs cs'" by auto
-      with d have "(Th th1, Cs cs') \<in> RAG s" by simp
-      with runing_1 have "False"
-        apply (unfold runing_def readys_def s_RAG_def)
-        by (auto simp:waiting_eq)
-      thus ?thesis by simp
-    qed
-  next
-    assume th1'_in: "th1' \<in> dependants (wq s) th1"
-    from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
-    thus ?thesis 
-    proof
-      assume "th2' = th2"
-      with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
-      hence "(Th th2, Th th1) \<in> (RAG s)^+"
-        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
-      hence "Th th2 \<in> Domain ((RAG s)^+)" 
-        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
-        by (auto simp:Domain_def)
-      hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
-      then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
-      from RAG_target_th [OF this]
-      obtain cs' where "n = Cs cs'" by auto
-      with d have "(Th th2, Cs cs') \<in> RAG s" by simp
-      with runing_2 have "False"
-        apply (unfold runing_def readys_def s_RAG_def)
-        by (auto simp:waiting_eq)
-      thus ?thesis by simp
-    next
-      assume "th2' \<in> dependants (wq s) th2"
-      with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
-      hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
-        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
-      from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
-        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
-      show ?thesis
-      proof(rule dchain_unique[OF h1 _ h2, symmetric])
-        from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
-        from runing_2 show "th2 \<in> readys s" by (simp add:runing_def) 
-      qed
-    qed
-  qed
-qed
-
-
-lemma "card (runing s) \<le> 1"
-apply(subgoal_tac "finite (runing s)")
-prefer 2
-apply (metis finite_nat_set_iff_bounded lessI runing_unique)
-apply(rule ccontr)
-apply(simp)
-apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
-apply(subst (asm) card_le_Suc_iff)
-apply(simp)
-apply(auto)[1]
-apply (metis insertCI runing_unique)
-apply(auto) 
-done
-
-end
-
-
-lemma create_pre:
-  assumes stp: "step s e"
-  and not_in: "th \<notin> threads s"
-  and is_in: "th \<in> threads (e#s)"
-  obtains prio where "e = Create th prio"
-proof -
-  from assms  
-  show ?thesis
-  proof(cases)
-    case (thread_create thread prio)
-    with is_in not_in have "e = Create th prio" by simp
-    from that[OF this] show ?thesis .
-  next
-    case (thread_exit thread)
-    with assms show ?thesis by (auto intro!:that)
-  next
-    case (thread_P thread)
-    with assms show ?thesis by (auto intro!:that)
-  next
-    case (thread_V thread)
-    with assms show ?thesis by (auto intro!:that)
-  next 
-    case (thread_set thread)
-    with assms show ?thesis by (auto intro!:that)
-  qed
-qed
-
-context valid_trace
-begin
-
-lemma cnp_cnv_eq:
-  assumes "th \<notin> threads s"
-  shows "cntP s th = cntV s th"
-  using assms
-  using cnp_cnv_cncs not_thread_cncs by auto
-
-end
-
-
-lemma eq_RAG: 
-  "RAG (wq s) = RAG s"
-by (unfold cs_RAG_def s_RAG_def, auto)
-
-context valid_trace
-begin
-
-lemma count_eq_dependants:
-  assumes eq_pv: "cntP s th = cntV s th"
-  shows "dependants (wq s) th = {}"
-proof -
-  from cnp_cnv_cncs and eq_pv
-  have "cntCS s th = 0" 
-    by (auto split:if_splits)
-  moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
-  proof -
-    from finite_holding[of th] show ?thesis
-      by (simp add:holdents_test)
-  qed
-  ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
-    by (unfold cntCS_def holdents_test cs_dependants_def, auto)
-  show ?thesis
-  proof(unfold cs_dependants_def)
-    { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
-      then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
-      hence "False"
-      proof(cases)
-        assume "(Th th', Th th) \<in> RAG (wq s)"
-        thus "False" by (auto simp:cs_RAG_def)
-      next
-        fix c
-        assume "(c, Th th) \<in> RAG (wq s)"
-        with h and eq_RAG show "False"
-          by (cases c, auto simp:cs_RAG_def)
-      qed
-    } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
-  qed
-qed
-
-lemma dependants_threads:
-  shows "dependants (wq s) th \<subseteq> threads s"
-proof
-  { fix th th'
-    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
-    have "Th th \<in> Domain (RAG s)"
-    proof -
-      from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
-      hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
-      with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
-      thus ?thesis using eq_RAG by simp
-    qed
-    from dm_RAG_threads[OF this]
-    have "th \<in> threads s" .
-  } note hh = this
-  fix th1 
-  assume "th1 \<in> dependants (wq s) th"
-  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
-    by (unfold cs_dependants_def, simp)
-  from hh [OF this] show "th1 \<in> threads s" .
-qed
-
-lemma finite_threads:
-  shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-end
-
-lemma Max_f_mono:
-  assumes seq: "A \<subseteq> B"
-  and np: "A \<noteq> {}"
-  and fnt: "finite B"
-  shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
-  from seq show "f ` A \<subseteq> f ` B" by auto
-next
-  from np show "f ` A \<noteq> {}" by auto
-next
-  from fnt and seq show "finite (f ` B)" by auto
-qed
-
-context valid_trace
-begin
-
-lemma cp_le:
-  assumes th_in: "th \<in> threads s"
-  shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
-  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
-         \<le> Max ((\<lambda>th. preced th s) ` threads s)"
-    (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
-  proof(rule Max_f_mono)
-    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
-  next
-    from finite_threads
-    show "finite (threads s)" .
-  next
-    from th_in
-    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
-      apply (auto simp:Domain_def)
-      apply (rule_tac dm_RAG_threads)
-      apply (unfold trancl_domain [of "RAG s", symmetric])
-      by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
-  qed
-qed
-
-lemma le_cp:
-  shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
-  show "Prc (priority th s) (last_set th s)
-    \<le> Max (insert (Prc (priority th s) (last_set th s))
-            ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
-    (is "?l \<le> Max (insert ?l ?A)")
-  proof(cases "?A = {}")
-    case False
-    have "finite ?A" (is "finite (?f ` ?B)")
-    proof -
-      have "finite ?B" 
-      proof-
-        have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
-        proof -
-          let ?F = "\<lambda> (x, y). the_th x"
-          have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
-            apply (auto simp:image_def)
-            by (rule_tac x = "(Th x, Th th)" in bexI, auto)
-          moreover have "finite \<dots>"
-          proof -
-            from finite_RAG have "finite (RAG s)" .
-            hence "finite ((RAG (wq s))\<^sup>+)"
-              apply (unfold finite_trancl)
-              by (auto simp: s_RAG_def cs_RAG_def wq_def)
-            thus ?thesis by auto
-          qed
-          ultimately show ?thesis by (auto intro:finite_subset)
-        qed
-        thus ?thesis by (simp add:cs_dependants_def)
-      qed
-      thus ?thesis by simp
-    qed
-    from Max_insert [OF this False, of ?l] show ?thesis by auto
-  next
-    case True
-    thus ?thesis by auto
-  qed
-qed
-
-lemma max_cp_eq: 
-  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
-  (is "?l = ?r")
-proof(cases "threads s = {}")
-  case True
-  thus ?thesis by auto
-next
-  case False
-  have "?l \<in> ((cp s) ` threads s)"
-  proof(rule Max_in)
-    from finite_threads
-    show "finite (cp s ` threads s)" by auto
-  next
-    from False show "cp s ` threads s \<noteq> {}" by auto
-  qed
-  then obtain th 
-    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
-  have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
-  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
-  proof -
-    have "?r \<in> (?f ` ?A)"
-    proof(rule Max_in)
-      from finite_threads
-      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
-    next
-      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
-    qed
-    then obtain th' where 
-      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
-    from le_cp [of th']  eq_r
-    have "?r \<le> cp s th'" by auto
-    moreover have "\<dots> \<le> cp s th"
-    proof(fold eq_l)
-      show " cp s th' \<le> Max (cp s ` threads s)"
-      proof(rule Max_ge)
-        from th_in' show "cp s th' \<in> cp s ` threads s"
-          by auto
-      next
-        from finite_threads
-        show "finite (cp s ` threads s)" by auto
-      qed
-    qed
-    ultimately show ?thesis by auto
-  qed
-  ultimately show ?thesis using eq_l by auto
-qed
-
-lemma max_cp_readys_threads_pre:
-  assumes np: "threads s \<noteq> {}"
-  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(unfold max_cp_eq)
-  show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
-  proof -
-    let ?p = "Max ((\<lambda>th. preced th s) ` threads s)" 
-    let ?f = "(\<lambda>th. preced th s)"
-    have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
-    proof(rule Max_in)
-      from finite_threads show "finite (?f ` threads s)" by simp
-    next
-      from np show "?f ` threads s \<noteq> {}" by simp
-    qed
-    then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
-      by (auto simp:Image_def)
-    from th_chain_to_ready [OF tm_in]
-    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
-    thus ?thesis
-    proof
-      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
-      then obtain th' where th'_in: "th' \<in> readys s" 
-        and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
-      have "cp s th' = ?f tm"
-      proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
-        from dependants_threads finite_threads
-        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))" 
-          by (auto intro:finite_subset)
-      next
-        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
-        from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
-        moreover have "p \<le> \<dots>"
-        proof(rule Max_ge)
-          from finite_threads
-          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-        next
-          from p_in and th'_in and dependants_threads[of th']
-          show "p \<in> (\<lambda>th. preced th s) ` threads s"
-            by (auto simp:readys_def)
-        qed
-        ultimately show "p \<le> preced tm s" by auto
-      next
-        show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
-        proof -
-          from tm_chain
-          have "tm \<in> dependants (wq s) th'"
-            by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
-          thus ?thesis by auto
-        qed
-      qed
-      with tm_max
-      have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
-      show ?thesis
-      proof (fold h, rule Max_eqI)
-        fix q 
-        assume "q \<in> cp s ` readys s"
-        then obtain th1 where th1_in: "th1 \<in> readys s"
-          and eq_q: "q = cp s th1" by auto
-        show "q \<le> cp s th'"
-          apply (unfold h eq_q)
-          apply (unfold cp_eq_cpreced cpreced_def)
-          apply (rule Max_mono)
-        proof -
-          from dependants_threads [of th1] th1_in
-          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq> 
-                 (\<lambda>th. preced th s) ` threads s"
-            by (auto simp:readys_def)
-        next
-          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
-        next
-          from finite_threads 
-          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
-        qed
-      next
-        from finite_threads
-        show "finite (cp s ` readys s)" by (auto simp:readys_def)
-      next
-        from th'_in
-        show "cp s th' \<in> cp s ` readys s" by simp
-      qed
-    next
-      assume tm_ready: "tm \<in> readys s"
-      show ?thesis
-      proof(fold tm_max)
-        have cp_eq_p: "cp s tm = preced tm s"
-        proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
-          fix y 
-          assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
-          show "y \<le> preced tm s"
-          proof -
-            { fix y'
-              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
-              have "y' \<le> preced tm s"
-              proof(unfold tm_max, rule Max_ge)
-                from hy' dependants_threads[of tm]
-                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
-              next
-                from finite_threads
-                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-              qed
-            } with hy show ?thesis by auto
-          qed
-        next
-          from dependants_threads[of tm] finite_threads
-          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
-            by (auto intro:finite_subset)
-        next
-          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
-            by simp
-        qed 
-        moreover have "Max (cp s ` readys s) = cp s tm"
-        proof(rule Max_eqI)
-          from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
-        next
-          from finite_threads
-          show "finite (cp s ` readys s)" by (auto simp:readys_def)
-        next
-          fix y assume "y \<in> cp s ` readys s"
-          then obtain th1 where th1_readys: "th1 \<in> readys s"
-            and h: "y = cp s th1" by auto
-          show "y \<le> cp s tm"
-            apply(unfold cp_eq_p h)
-            apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
-          proof -
-            from finite_threads
-            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-          next
-            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
-              by simp
-          next
-            from dependants_threads[of th1] th1_readys
-            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) 
-                    \<subseteq> (\<lambda>th. preced th s) ` threads s"
-              by (auto simp:readys_def)
-          qed
-        qed
-        ultimately show " Max (cp s ` readys s) = preced tm s" by simp
-      qed 
-    qed
-  qed
-qed
-
-text {* (* ccc *) \noindent
-  Since the current precedence of the threads in ready queue will always be boosted,
-  there must be one inside it has the maximum precedence of the whole system. 
-*}
-lemma max_cp_readys_threads:
-  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(cases "threads s = {}")
-  case True
-  thus ?thesis 
-    by (auto simp:readys_def)
-next
-  case False
-  show ?thesis by (rule max_cp_readys_threads_pre[OF False])
-qed
-
-end
-
-lemma eq_holding: "holding (wq s) th cs = holding s th cs"
-  apply (unfold s_holding_def cs_holding_def wq_def, simp)
-  done
-
-lemma f_image_eq:
-  assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
-  shows "f ` A = g ` A"
-proof
-  show "f ` A \<subseteq> g ` A"
-    by(rule image_subsetI, auto intro:h)
-next
-  show "g ` A \<subseteq> f ` A"
-   by (rule image_subsetI, auto intro:h[symmetric])
-qed
-
-
-definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
-  where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
-
-lemma detached_test:
-  shows "detached s th = (Th th \<notin> Field (RAG s))"
-apply(simp add: detached_def Field_def)
-apply(simp add: s_RAG_def)
-apply(simp add: s_holding_abv s_waiting_abv)
-apply(simp add: Domain_iff Range_iff)
-apply(simp add: wq_def)
-apply(auto)
-done
-
-context valid_trace
-begin
-
-lemma detached_intro:
-  assumes eq_pv: "cntP s th = cntV s th"
-  shows "detached s th"
-proof -
- from cnp_cnv_cncs
-  have eq_cnt: "cntP s th =
-    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
-  hence cncs_zero: "cntCS s th = 0"
-    by (auto simp:eq_pv split:if_splits)
-  with eq_cnt
-  have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
-  thus ?thesis
-  proof
-    assume "th \<notin> threads s"
-    with range_in dm_RAG_threads
-    show ?thesis
-      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
-  next
-    assume "th \<in> readys s"
-    moreover have "Th th \<notin> Range (RAG s)"
-    proof -
-      from card_0_eq [OF finite_holding] and cncs_zero
-      have "holdents s th = {}"
-        by (simp add:cntCS_def)
-      thus ?thesis
-        apply(auto simp:holdents_test)
-        apply(case_tac a)
-        apply(auto simp:holdents_test s_RAG_def)
-        done
-    qed
-    ultimately show ?thesis
-      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
-  qed
-qed
-
-lemma detached_elim:
-  assumes dtc: "detached s th"
-  shows "cntP s th = cntV s th"
-proof -
-  from cnp_cnv_cncs
-  have eq_pv: " cntP s th =
-    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
-  have cncs_z: "cntCS s th = 0"
-  proof -
-    from dtc have "holdents s th = {}"
-      unfolding detached_def holdents_test s_RAG_def
-      by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
-    thus ?thesis by (auto simp:cntCS_def)
-  qed
-  show ?thesis
-  proof(cases "th \<in> threads s")
-    case True
-    with dtc 
-    have "th \<in> readys s"
-      by (unfold readys_def detached_def Field_def Domain_def Range_def, 
-           auto simp:waiting_eq s_RAG_def)
-    with cncs_z and eq_pv show ?thesis by simp
-  next
-    case False
-    with cncs_z and eq_pv show ?thesis by simp
-  qed
-qed
-
-lemma detached_eq:
-  shows "(detached s th) = (cntP s th = cntV s th)"
-  by (insert vt, auto intro:detached_intro detached_elim)
-
-end
-
-text {* 
-  The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived
-  from the concise and miniature model of PIP given in PrioGDef.thy.
-*}
-
-lemma eq_dependants: "dependants (wq s) = dependants s"
-  by (simp add: s_dependants_abv wq_def)
-
-lemma next_th_unique: 
-  assumes nt1: "next_th s th cs th1"
-  and nt2: "next_th s th cs th2"
-  shows "th1 = th2"
-using assms by (unfold next_th_def, auto)
-
-lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
-  apply (induct s, simp)
-proof -
-  fix a s
-  assume ih: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
-    and eq_as: "a # s \<noteq> []"
-  show "last_set th (a # s) < length (a # s)"
-  proof(cases "s \<noteq> []")
-    case False
-    from False show ?thesis
-      by (cases a, auto simp:last_set.simps)
-  next
-    case True
-    from ih [OF True] show ?thesis
-      by (cases a, auto simp:last_set.simps)
-  qed
-qed
-
-lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
-  by (induct s, auto simp:threads.simps)
-
-lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
-  apply (drule_tac th_in_ne)
-  by (unfold preced_def, auto intro: birth_time_lt)
-
-lemma inj_the_preced: 
-  "inj_on (the_preced s) (threads s)"
-  by (metis inj_onI preced_unique the_preced_def)
-
-lemma tRAG_alt_def: 
-  "tRAG s = {(Th th1, Th th2) | th1 th2. 
-                  \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
- by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
-
-lemma tRAG_Field:
-  "Field (tRAG s) \<subseteq> Field (RAG s)"
-  by (unfold tRAG_alt_def Field_def, auto)
-
-lemma tRAG_ancestorsE:
-  assumes "x \<in> ancestors (tRAG s) u"
-  obtains th where "x = Th th"
-proof -
-  from assms have "(u, x) \<in> (tRAG s)^+" 
-      by (unfold ancestors_def, auto)
-  from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
-  then obtain th where "x = Th th"
-    by (unfold tRAG_alt_def, auto)
-  from that[OF this] show ?thesis .
-qed
-
-lemma tRAG_mono:
-  assumes "RAG s' \<subseteq> RAG s"
-  shows "tRAG s' \<subseteq> tRAG s"
-  using assms 
-  by (unfold tRAG_alt_def, auto)
-
-lemma holding_next_thI:
-  assumes "holding s th cs"
-  and "length (wq s cs) > 1"
-  obtains th' where "next_th s th cs th'"
-proof -
-  from assms(1)[folded eq_holding, unfolded cs_holding_def]
-  have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
-  then obtain rest where h1: "wq s cs = th#rest" 
-    by (cases "wq s cs", auto)
-  with assms(2) have h2: "rest \<noteq> []" by auto
-  let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
-  have "next_th s th cs ?th'" using  h1(1) h2 
-    by (unfold next_th_def, auto)
-  from that[OF this] show ?thesis .
-qed
-
-lemma RAG_tRAG_transfer:
-  assumes "vt s'"
-  assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
-  and "(Cs cs, Th th'') \<in> RAG s'"
-  shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
-proof -
-  interpret vt_s': valid_trace "s'" using assms(1)
-    by (unfold_locales, simp)
-  interpret rtree: rtree "RAG s'"
-  proof
-  show "single_valued (RAG s')"
-  apply (intro_locales)
-    by (unfold single_valued_def, 
-        auto intro:vt_s'.unique_RAG)
-
-  show "acyclic (RAG s')"
-     by (rule vt_s'.acyclic_RAG)
-  qed
-  { fix n1 n2
-    assume "(n1, n2) \<in> ?L"
-    from this[unfolded tRAG_alt_def]
-    obtain th1 th2 cs' where 
-      h: "n1 = Th th1" "n2 = Th th2" 
-         "(Th th1, Cs cs') \<in> RAG s"
-         "(Cs cs', Th th2) \<in> RAG s" by auto
-    from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
-    from h(3) and assms(2) 
-    have "(Th th1, Cs cs') = (Th th, Cs cs) \<or> 
-          (Th th1, Cs cs') \<in> RAG s'" by auto
-    hence "(n1, n2) \<in> ?R"
-    proof
-      assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
-      hence eq_th1: "th1 = th" by simp
-      moreover have "th2 = th''"
-      proof -
-        from h1 have "cs' = cs" by simp
-        from assms(3) cs_in[unfolded this] rtree.sgv
-        show ?thesis
-          by (unfold single_valued_def, auto)
-      qed
-      ultimately show ?thesis using h(1,2) by auto
-    next
-      assume "(Th th1, Cs cs') \<in> RAG s'"
-      with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
-        by (unfold tRAG_alt_def, auto)
-      from this[folded h(1, 2)] show ?thesis by auto
-    qed
-  } moreover {
-    fix n1 n2
-    assume "(n1, n2) \<in> ?R"
-    hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
-    hence "(n1, n2) \<in> ?L" 
-    proof
-      assume "(n1, n2) \<in> tRAG s'"
-      moreover have "... \<subseteq> ?L"
-      proof(rule tRAG_mono)
-        show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
-      qed
-      ultimately show ?thesis by auto
-    next
-      assume eq_n: "(n1, n2) = (Th th, Th th'')"
-      from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
-      moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
-      ultimately show ?thesis 
-        by (unfold eq_n tRAG_alt_def, auto)
-    qed
-  } ultimately show ?thesis by auto
-qed
-
-context valid_trace
-begin
-
-lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
-
-end
-
-lemma cp_alt_def:
-  "cp s th =  
-           Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
-proof -
-  have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
-        Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})" 
-          (is "Max (_ ` ?L) = Max (_ ` ?R)")
-  proof -
-    have "?L = ?R" 
-    by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
-    thus ?thesis by simp
-  qed
-  thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
-qed
-
-lemma cp_gen_alt_def:
-  "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
-    by (auto simp:cp_gen_def)
-
-lemma tRAG_nodeE:
-  assumes "(n1, n2) \<in> tRAG s"
-  obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
-  using assms
-  by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
-
-lemma subtree_nodeE:
-  assumes "n \<in> subtree (tRAG s) (Th th)"
-  obtains th1 where "n = Th th1"
-proof -
-  show ?thesis
-  proof(rule subtreeE[OF assms])
-    assume "n = Th th"
-    from that[OF this] show ?thesis .
-  next
-    assume "Th th \<in> ancestors (tRAG s) n"
-    hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
-    hence "\<exists> th1. n = Th th1"
-    proof(induct)
-      case (base y)
-      from tRAG_nodeE[OF this] show ?case by metis
-    next
-      case (step y z)
-      thus ?case by auto
-    qed
-    with that show ?thesis by auto
-  qed
-qed
-
-lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
-proof -
-  have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*" 
-    by (rule rtrancl_mono, auto simp:RAG_split)
-  also have "... \<subseteq> ((RAG s)^*)^*"
-    by (rule rtrancl_mono, auto)
-  also have "... = (RAG s)^*" by simp
-  finally show ?thesis by (unfold tRAG_def, simp)
-qed
-
-lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
-proof -
-  { fix a
-    assume "a \<in> subtree (tRAG s) x"
-    hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
-    with tRAG_star_RAG[of s]
-    have "(a, x) \<in> (RAG s)^*" by auto
-    hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
-  } thus ?thesis by auto
-qed
-
-lemma tRAG_trancl_eq:
-   "{th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
-    {th'. (Th th', Th th)  \<in> (RAG s)^+}"
-   (is "?L = ?R")
-proof -
-  { fix th'
-    assume "th' \<in> ?L"
-    hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
-    from tranclD[OF this]
-    obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
-    from tRAG_subtree_RAG[of s] and this(2)
-    have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG) 
-    moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto 
-    ultimately have "th' \<in> ?R"  by auto 
-  } moreover 
-  { fix th'
-    assume "th' \<in> ?R"
-    hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
-    from plus_rpath[OF this]
-    obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
-    hence "(Th th', Th th) \<in> (tRAG s)^+"
-    proof(induct xs arbitrary:th' th rule:length_induct)
-      case (1 xs th' th)
-      then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
-      show ?case
-      proof(cases "xs1")
-        case Nil
-        from 1(2)[unfolded Cons1 Nil]
-        have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
-        hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
-        then obtain cs where "x1 = Cs cs" 
-              by (unfold s_RAG_def, auto)
-        from rpath_nnl_lastE[OF rp[unfolded this]]
-        show ?thesis by auto
-      next
-        case (Cons x2 xs2)
-        from 1(2)[unfolded Cons1[unfolded this]]
-        have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
-        from rpath_edges_on[OF this]
-        have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
-        have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
-            by (simp add: edges_on_unfold)
-        with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
-        then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
-        have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
-            by (simp add: edges_on_unfold)
-        from this eds
-        have rg2: "(x1, x2) \<in> RAG s" by auto
-        from this[unfolded eq_x1] 
-        obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
-        from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
-        have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
-        from rp have "rpath (RAG s) x2 xs2 (Th th)"
-           by  (elim rpath_ConsE, simp)
-        from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
-        show ?thesis
-        proof(cases "xs2 = []")
-          case True
-          from rpath_nilE[OF rp'[unfolded this]]
-          have "th1 = th" by auto
-          from rt1[unfolded this] show ?thesis by auto
-        next
-          case False
-          from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
-          have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
-          with rt1 show ?thesis by auto
-        qed
-      qed
-    qed
-    hence "th' \<in> ?L" by auto
-  } ultimately show ?thesis by blast
-qed
-
-lemma tRAG_trancl_eq_Th:
-   "{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
-    {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}"
-    using tRAG_trancl_eq by auto
-
-lemma dependants_alt_def:
-  "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
-  by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
-  
-context valid_trace
-begin
-
-lemma count_eq_tRAG_plus:
-  assumes "cntP s th = cntV s th"
-  shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
-  using assms count_eq_dependants dependants_alt_def eq_dependants by auto 
-
-lemma count_eq_RAG_plus:
-  assumes "cntP s th = cntV s th"
-  shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
-  using assms count_eq_dependants cs_dependants_def eq_RAG by auto
-
-lemma count_eq_RAG_plus_Th:
-  assumes "cntP s th = cntV s th"
-  shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
-  using count_eq_RAG_plus[OF assms] by auto
-
-lemma count_eq_tRAG_plus_Th:
-  assumes "cntP s th = cntV s th"
-  shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
-   using count_eq_tRAG_plus[OF assms] by auto
-
-end
-
-lemma tRAG_subtree_eq: 
-   "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th'  \<in> (subtree (RAG s) (Th th))}"
-   (is "?L = ?R")
-proof -
-  { fix n 
-    assume h: "n \<in> ?L"
-    hence "n \<in> ?R"
-    by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) 
-  } moreover {
-    fix n
-    assume "n \<in> ?R"
-    then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
-      by (auto simp:subtree_def)
-    from rtranclD[OF this(2)]
-    have "n \<in> ?L"
-    proof
-      assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
-      with h have "n \<in> {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}" by auto
-      thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
-    qed (insert h, auto simp:subtree_def)
-  } ultimately show ?thesis by auto
-qed
-
-lemma threads_set_eq: 
-   "the_thread ` (subtree (tRAG s) (Th th)) = 
-                  {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
-   by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
-
-lemma cp_alt_def1: 
-  "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
-proof -
-  have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
-       ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
-       by auto
-  thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
-qed
-
-lemma cp_gen_def_cond: 
-  assumes "x = Th th"
-  shows "cp s th = cp_gen s (Th th)"
-by (unfold cp_alt_def1 cp_gen_def, simp)
-
-lemma cp_gen_over_set:
-  assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
-  shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
-proof(rule f_image_eq)
-  fix a
-  assume "a \<in> A"
-  from assms[rule_format, OF this]
-  obtain th where eq_a: "a = Th th" by auto
-  show "cp_gen s a = (cp s \<circ> the_thread) a"
-    by  (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
-qed
-
-
-context valid_trace
-begin
-
-lemma RAG_threads:
-  assumes "(Th th) \<in> Field (RAG s)"
-  shows "th \<in> threads s"
-  using assms
-  by (metis Field_def UnE dm_RAG_threads range_in vt)
-
-lemma subtree_tRAG_thread:
-  assumes "th \<in> threads s"
-  shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
-proof -
-  have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
-    by (unfold tRAG_subtree_eq, simp)
-  also have "... \<subseteq> ?R"
-  proof
-    fix x
-    assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
-    then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
-    from this(2)
-    show "x \<in> ?R"
-    proof(cases rule:subtreeE)
-      case 1
-      thus ?thesis by (simp add: assms h(1)) 
-    next
-      case 2
-      thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) 
-    qed
-  qed
-  finally show ?thesis .
-qed
-
-lemma readys_root:
-  assumes "th \<in> readys s"
-  shows "root (RAG s) (Th th)"
-proof -
-  { fix x
-    assume "x \<in> ancestors (RAG s) (Th th)"
-    hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
-    from tranclD[OF this]
-    obtain z where "(Th th, z) \<in> RAG s" by auto
-    with assms(1) have False
-         apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
-         by (fold wq_def, blast)
-  } thus ?thesis by (unfold root_def, auto)
-qed
-
-lemma readys_in_no_subtree:
-  assumes "th \<in> readys s"
-  and "th' \<noteq> th"
-  shows "Th th \<notin> subtree (RAG s) (Th th')" 
-proof
-   assume "Th th \<in> subtree (RAG s) (Th th')"
-   thus False
-   proof(cases rule:subtreeE)
-      case 1
-      with assms show ?thesis by auto
-   next
-      case 2
-      with readys_root[OF assms(1)]
-      show ?thesis by (auto simp:root_def)
-   qed
-qed
-
-lemma not_in_thread_isolated:
-  assumes "th \<notin> threads s"
-  shows "(Th th) \<notin> Field (RAG s)"
-proof
-  assume "(Th th) \<in> Field (RAG s)"
-  with dm_RAG_threads and range_in assms
-  show False by (unfold Field_def, blast)
-qed
-
-lemma wf_RAG: "wf (RAG s)"
-proof(rule finite_acyclic_wf)
-  from finite_RAG show "finite (RAG s)" .
-next
-  from acyclic_RAG show "acyclic (RAG s)" .
-qed
-
-lemma sgv_wRAG: "single_valued (wRAG s)"
-  using waiting_unique
-  by (unfold single_valued_def wRAG_def, auto)
-
-lemma sgv_hRAG: "single_valued (hRAG s)"
-  using holding_unique 
-  by (unfold single_valued_def hRAG_def, auto)
-
-lemma sgv_tRAG: "single_valued (tRAG s)"
-  by (unfold tRAG_def, rule single_valued_relcomp, 
-              insert sgv_wRAG sgv_hRAG, auto)
-
-lemma acyclic_tRAG: "acyclic (tRAG s)"
-proof(unfold tRAG_def, rule acyclic_compose)
-  show "acyclic (RAG s)" using acyclic_RAG .
-next
-  show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-next
-  show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-qed
-
-lemma sgv_RAG: "single_valued (RAG s)"
-  using unique_RAG by (auto simp:single_valued_def)
-
-lemma rtree_RAG: "rtree (RAG s)"
-  using sgv_RAG acyclic_RAG
-  by (unfold rtree_def rtree_axioms_def sgv_def, auto)
-
-end
-
-sublocale valid_trace < rtree_RAG: rtree "RAG s"
-proof
-  show "single_valued (RAG s)"
-  apply (intro_locales)
-    by (unfold single_valued_def, 
-        auto intro:unique_RAG)
-
-  show "acyclic (RAG s)"
-     by (rule acyclic_RAG)
-qed
-
-sublocale valid_trace < rtree_s: rtree "tRAG s"
-proof(unfold_locales)
-  from sgv_tRAG show "single_valued (tRAG s)" .
-next
-  from acyclic_tRAG show "acyclic (tRAG s)" .
-qed
-
-sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
-proof -
-  show "fsubtree (RAG s)"
-  proof(intro_locales)
-    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
-  next
-    show "fsubtree_axioms (RAG s)"
-    proof(unfold fsubtree_axioms_def)
-      from wf_RAG show "wf (RAG s)" .
-    qed
-  qed
-qed
-
-sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
-proof -
-  have "fsubtree (tRAG s)"
-  proof -
-    have "fbranch (tRAG s)"
-    proof(unfold tRAG_def, rule fbranch_compose)
-        show "fbranch (wRAG s)"
-        proof(rule finite_fbranchI)
-           from finite_RAG show "finite (wRAG s)"
-           by (unfold RAG_split, auto)
-        qed
-    next
-        show "fbranch (hRAG s)"
-        proof(rule finite_fbranchI)
-           from finite_RAG 
-           show "finite (hRAG s)" by (unfold RAG_split, auto)
-        qed
-    qed
-    moreover have "wf (tRAG s)"
-    proof(rule wf_subset)
-      show "wf (RAG s O RAG s)" using wf_RAG
-        by (fold wf_comp_self, simp)
-    next
-      show "tRAG s \<subseteq> (RAG s O RAG s)"
-        by (unfold tRAG_alt_def, auto)
-    qed
-    ultimately show ?thesis
-      by (unfold fsubtree_def fsubtree_axioms_def,auto)
-  qed
-  from this[folded tRAG_def] show "fsubtree (tRAG s)" .
-qed
-
-lemma Max_UNION: 
-  assumes "finite A"
-  and "A \<noteq> {}"
-  and "\<forall> M \<in> f ` A. finite M"
-  and "\<forall> M \<in> f ` A. M \<noteq> {}"
-  shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
-  using assms[simp]
-proof -
-  have "?L = Max (\<Union>(f ` A))"
-    by (fold Union_image_eq, simp)
-  also have "... = ?R"
-    by (subst Max_Union, simp+)
-  finally show ?thesis .
-qed
-
-lemma max_Max_eq:
-  assumes "finite A"
-    and "A \<noteq> {}"
-    and "x = y"
-  shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
-proof -
-  have "?R = Max (insert y A)" by simp
-  also from assms have "... = ?L"
-      by (subst Max.insert, simp+)
-  finally show ?thesis by simp
-qed
-
-context valid_trace
-begin
-
-(* ddd *)
-lemma cp_gen_rec:
-  assumes "x = Th th"
-  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
-proof(cases "children (tRAG s) x = {}")
-  case True
-  show ?thesis
-    by (unfold True cp_gen_def subtree_children, simp add:assms)
-next
-  case False
-  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
-  note fsbttRAGs.finite_subtree[simp]
-  have [simp]: "finite (children (tRAG s) x)"
-     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 
-            rule children_subtree)
-  { fix r x
-    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
-  } note this[simp]
-  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
-  proof -
-    from False obtain q where "q \<in> children (tRAG s) x" by blast
-    moreover have "subtree (tRAG s) q \<noteq> {}" by simp
-    ultimately show ?thesis by blast
-  qed
-  have h: "Max ((the_preced s \<circ> the_thread) `
-                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
-        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
-                     (is "?L = ?R")
-  proof -
-    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
-    let "Max (_ \<union> (?h ` ?B))" = ?R
-    let ?L1 = "?f ` \<Union>(?g ` ?B)"
-    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
-    proof -
-      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
-      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
-      finally have "Max ?L1 = Max ..." by simp
-      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
-        by (subst Max_UNION, simp+)
-      also have "... = Max (cp_gen s ` children (tRAG s) x)"
-          by (unfold image_comp cp_gen_alt_def, simp)
-      finally show ?thesis .
-    qed
-    show ?thesis
-    proof -
-      have "?L = Max (?f ` ?A \<union> ?L1)" by simp
-      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
-            by (subst Max_Un, simp+)
-      also have "... = max (?f x) (Max (?h ` ?B))"
-        by (unfold eq_Max_L1, simp)
-      also have "... =?R"
-        by (rule max_Max_eq, (simp)+, unfold assms, simp)
-      finally show ?thesis .
-    qed
-  qed  thus ?thesis 
-          by (fold h subtree_children, unfold cp_gen_def, simp) 
-qed
-
-lemma cp_rec:
-  "cp s th = Max ({the_preced s th} \<union> 
-                     (cp s o the_thread) ` children (tRAG s) (Th th))"
-proof -
-  have "Th th = Th th" by simp
-  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
-  show ?thesis 
-  proof -
-    have "cp_gen s ` children (tRAG s) (Th th) = 
-                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
-    proof(rule cp_gen_over_set)
-      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
-        by (unfold tRAG_alt_def, auto simp:children_def)
-    qed
-    thus ?thesis by (subst (1) h(1), unfold h(2), simp)
-  qed
-qed
-
-end
-
-(* keep *)
-lemma next_th_holding:
-  assumes vt: "vt s"
-  and nxt: "next_th s th cs th'"
-  shows "holding (wq s) th cs"
-proof -
-  from nxt[unfolded next_th_def]
-  obtain rest where h: "wq s cs = th # rest"
-                       "rest \<noteq> []" 
-                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
-  thus ?thesis
-    by (unfold cs_holding_def, auto)
-qed
-
-context valid_trace
-begin
-
-lemma next_th_waiting:
-  assumes nxt: "next_th s th cs th'"
-  shows "waiting (wq s) th' cs"
-proof -
-  from nxt[unfolded next_th_def]
-  obtain rest where h: "wq s cs = th # rest"
-                       "rest \<noteq> []" 
-                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
-  from wq_distinct[of cs, unfolded h]
-  have dst: "distinct (th # rest)" .
-  have in_rest: "th' \<in> set rest"
-  proof(unfold h, rule someI2)
-    show "distinct rest \<and> set rest = set rest" using dst by auto
-  next
-    fix x assume "distinct x \<and> set x = set rest"
-    with h(2)
-    show "hd x \<in> set (rest)" by (cases x, auto)
-  qed
-  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
-  moreover have "th' \<noteq> hd (wq s cs)"
-    by (unfold h(1), insert in_rest dst, auto)
-  ultimately show ?thesis by (auto simp:cs_waiting_def)
-qed
-
-lemma next_th_RAG:
-  assumes nxt: "next_th (s::event list) th cs th'"
-  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
-  using vt assms next_th_holding next_th_waiting
-  by (unfold s_RAG_def, simp)
-
-end
-
--- {* A useless definition *}
-definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
-where "cps s = {(th, cp s th) | th . th \<in> threads s}"
-
-lemma "wq (V th cs # s) cs1 = ttt"
-  apply (unfold wq_def, auto simp:Let_def)
-
-end
-
diff -r 5d8ec128518b -r e3cf792db636 CpsG_2.thy
--- a/CpsG_2.thy	Tue Jun 14 13:56:51 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,3557 +0,0 @@
-theory CpsG
-imports PIPDefs
-begin
-
-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 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"
-
-locale valid_trace_e = valid_trace +
-  fixes e
-  assumes vt_e: "vt (e#s)"
-begin
-
-lemma pip_e: "PIP s e"
-  using vt_e by (cases, simp)  
-
-end
-
-locale valid_trace_create = valid_trace_e + 
-  fixes th prio
-  assumes is_create: "e = Create th prio"
-
-locale valid_trace_exit = valid_trace_e + 
-  fixes th
-  assumes is_exit: "e = Exit th"
-
-locale valid_trace_p = valid_trace_e + 
-  fixes th cs
-  assumes is_p: "e = P th cs"
-
-locale valid_trace_v = valid_trace_e + 
-  fixes th cs
-  assumes is_v: "e = V th cs"
-begin
-  definition "rest = tl (wq s cs)"
-  definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
-end
-
-locale valid_trace_v_n = valid_trace_v +
-  assumes rest_nnl: "rest \<noteq> []"
-
-locale valid_trace_v_e = valid_trace_v +
-  assumes rest_nil: "rest = []"
-
-locale valid_trace_set= valid_trace_e + 
-  fixes th prio
-  assumes is_set: "e = Set th prio"
-
-context valid_trace
-begin
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
-  assumes "PP []"
-     and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
-                   PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
-     shows "PP s"
-proof(induct rule:vt.induct[OF vt, case_names Init Step])
-  case Init
-  from assms(1) show ?case .
-next
-  case (Step s e)
-  show ?case
-  proof(rule assms(2))
-    show "valid_trace_e s e" using Step by (unfold_locales, auto)
-  next
-    show "PP s" using Step by simp
-  next
-    show "PIP s e" using Step by simp
-  qed
-qed
-
-lemma  vt_moment: "\<And> t. vt (moment t s)"
-proof(induct rule:ind)
-  case Nil
-  thus ?case by (simp add:vt_nil)
-next
-  case (Cons s e t)
-  show ?case
-  proof(cases "t \<ge> length (e#s)")
-    case True
-    from True have "moment t (e#s) = e#s" by simp
-    thus ?thesis using Cons
-      by (simp add:valid_trace_def valid_trace_e_def, auto)
-  next
-    case False
-    from Cons have "vt (moment t s)" by simp
-    moreover have "moment t (e#s) = moment t s"
-    proof -
-      from False have "t \<le> length s" by simp
-      from moment_app [OF this, of "[e]"] 
-      show ?thesis by simp
-    qed
-    ultimately show ?thesis by simp
-  qed
-qed
-
-lemma finite_threads:
-  shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-end
-
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
-  by (unfold s_RAG_def, auto)
-
-locale valid_moment = valid_trace + 
-  fixes i :: nat
-
-sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
-  by (unfold_locales, insert vt_moment, auto)
-
-lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
-  by  (unfold s_waiting_def cs_waiting_def wq_def, auto)
-
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
-  by (unfold s_holding_def wq_def cs_holding_def, simp)
-
-lemma runing_ready: 
-  shows "runing s \<subseteq> readys s"
-  unfolding runing_def readys_def
-  by auto 
-
-lemma readys_threads:
-  shows "readys s \<subseteq> threads s"
-  unfolding readys_def
-  by auto
-
-lemma wq_v_neq [simp]:
-   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
-  by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-lemma runing_head:
-  assumes "th \<in> runing s"
-  and "th \<in> set (wq_fun (schs s) cs)"
-  shows "th = hd (wq_fun (schs s) cs)"
-  using assms
-  by (simp add:runing_def readys_def s_waiting_def wq_def)
-
-context valid_trace
-begin
-
-lemma runing_wqE:
-  assumes "th \<in> runing s"
-  and "th \<in> set (wq s cs)"
-  obtains rest where "wq s cs = th#rest"
-proof -
-  from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
-    by (meson list.set_cases)
-  have "th' = th"
-  proof(rule ccontr)
-    assume "th' \<noteq> th"
-    hence "th \<noteq> hd (wq s cs)" using eq_wq by auto 
-    with assms(2)
-    have "waiting s th cs" 
-      by (unfold s_waiting_def, fold wq_def, auto)
-    with assms show False 
-      by (unfold runing_def readys_def, auto)
-  qed
-  with eq_wq that show ?thesis by metis
-qed
-
-end
-
-context valid_trace_create
-begin
-
-lemma wq_neq_simp [simp]:
-  shows "wq (e#s) cs' = wq s cs'"
-    using assms unfolding is_create wq_def
-  by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
-  assumes "distinct (wq s cs')"
-  shows "distinct (wq (e#s) cs')"
-  using assms by simp
-end
-
-context valid_trace_exit
-begin
-
-lemma wq_neq_simp [simp]:
-  shows "wq (e#s) cs' = wq s cs'"
-    using assms unfolding is_exit wq_def
-  by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
-  assumes "distinct (wq s cs')"
-  shows "distinct (wq (e#s) cs')"
-  using assms by simp
-end
-
-context valid_trace_p
-begin
-
-lemma wq_neq_simp [simp]:
-  assumes "cs' \<noteq> cs"
-  shows "wq (e#s) cs' = wq s cs'"
-    using assms unfolding is_p wq_def
-  by (auto simp:Let_def)
-
-lemma runing_th_s:
-  shows "th \<in> runing s"
-proof -
-  from pip_e[unfolded is_p]
-  show ?thesis by (cases, simp)
-qed
-
-lemma 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"
-  shows "actor e \<in> runing s"
-  using assms
-  by (induct, auto)
-
-lemma isP_E:
-  assumes "isP e"
-  obtains cs where "e = P (actor e) cs"
-  using assms by (cases e, auto)
-
-lemma isV_E:
-  assumes "isV e"
-  obtains cs where "e = V (actor e) cs"
-  using assms by (cases e, auto) 
-
-lemma wq_distinct: "distinct (wq s cs)"
-proof(induct rule:ind)
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case 
-  proof(cases e)
-    case (Create th prio)
-    interpret vt_create: valid_trace_create s e th prio 
-      using Create by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept) 
-  next
-    case (Exit th)
-    interpret vt_exit: valid_trace_exit s e th  
-        using Exit by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept) 
-  next
-    case (P th cs)
-    interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) 
-  next
-    case (V th cs)
-    interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) 
-  next
-    case (Set th prio)
-    interpret vt_set: valid_trace_set s e th prio
-        using Set by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) 
-  qed
-qed (unfold wq_def Let_def, simp)
-
-end
-
-context valid_trace_e
-begin
-
-text {*
-  The following lemma shows that only the @{text "P"}
-  operation can add new thread into waiting queues. 
-  Such kind of lemmas are very obvious, but need to be checked formally.
-  This is a kind of confirmation that our modelling is correct.
-*}
-
-lemma wq_in_inv: 
-  assumes s_ni: "thread \<notin> set (wq s cs)"
-  and s_i: "thread \<in> set (wq (e#s) cs)"
-  shows "e = P thread cs"
-proof(cases e)
-  -- {* This is the only non-trivial case: *}
-  case (V th cs1)
-  have False
-  proof(cases "cs1 = cs")
-    case True
-    show ?thesis
-    proof(cases "(wq s cs1)")
-      case (Cons w_hd w_tl)
-      have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
-      proof -
-        have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
-          using  Cons V by (auto simp:wq_def Let_def True split:if_splits)
-        moreover have "set ... \<subseteq> set (wq s cs)"
-        proof(rule someI2)
-          show "distinct w_tl \<and> set w_tl = set w_tl"
-            by (metis distinct.simps(2) local.Cons wq_distinct)
-        qed (insert Cons True, auto)
-        ultimately show ?thesis by simp
-      qed
-      with assms show ?thesis by auto
-    qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
-  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
-  thus ?thesis by auto
-qed (insert assms, auto simp:wq_def Let_def split:if_splits)
-
-lemma wq_out_inv: 
-  assumes s_in: "thread \<in> set (wq s cs)"
-  and s_hd: "thread = hd (wq s cs)"
-  and s_i: "thread \<noteq> hd (wq (e#s) cs)"
-  shows "e = V thread cs"
-proof(cases e)
--- {* There are only two non-trivial cases: *}
-  case (V th cs1)
-  show ?thesis
-  proof(cases "cs1 = cs")
-    case True
-    have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
-    thus ?thesis
-    proof(cases)
-      case (thread_V)
-      moreover have "th = thread" using thread_V(2) s_hd
-          by (unfold s_holding_def wq_def, simp)
-      ultimately show ?thesis using V True by simp
-    qed
-  qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
-next
-  case (P th cs1)
-  show ?thesis
-  proof(cases "cs1 = cs")
-    case True
-    with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
-      by (auto simp:wq_def Let_def split:if_splits)
-    with s_i s_hd s_in have False
-      by (metis empty_iff hd_append2 list.set(1) wq_def) 
-    thus ?thesis by simp
-  qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
-qed (insert assms, auto simp:wq_def Let_def split:if_splits)
-
-end
-
-
-context valid_trace
-begin
-
-
-text {* (* ddd *)
-  The nature of the work is like this: since it starts from a very simple and basic 
-  model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
-  For instance, the fact 
-  that one thread can not be blocked by two critical resources at the same time
-  is obvious, because only running threads can make new requests, if one is waiting for 
-  a critical resource and get blocked, it can not make another resource request and get 
-  blocked the second time (because it is not running). 
-
-  To derive this fact, one needs to prove by contraction and 
-  reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
-  named @{text "p_split"}, which is about status changing along the time axis. It says if 
-  a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
-  but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} 
-  in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history 
-  of events leading to it), such that @{text "Q"} switched 
-  from being @{text "False"} to @{text "True"} and kept being @{text "True"}
-  till the last moment of @{text "s"}.
-
-  Suppose a thread @{text "th"} is blocked
-  on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, 
-  since no thread is blocked at the very beginning, by applying 
-  @{text "p_split"} to these two blocking facts, there exist 
-  two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that 
-  @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} 
-  and kept on blocked on them respectively ever since.
- 
-  Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
-  However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
-  in blocked state at moment @{text "t2"} and could not
-  make any request and get blocked the second time: Contradiction.
-*}
-
-lemma waiting_unique_pre: (* ddd *)
-  assumes h11: "thread \<in> set (wq s cs1)"
-  and h12: "thread \<noteq> hd (wq s cs1)"
-  assumes h21: "thread \<in> set (wq s cs2)"
-  and h22: "thread \<noteq> hd (wq s cs2)"
-  and neq12: "cs1 \<noteq> cs2"
-  shows "False"
-proof -
-  let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
-  from h11 and h12 have q1: "?Q cs1 s" by simp
-  from h21 and h22 have q2: "?Q cs2 s" by simp
-  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
-  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
-  from p_split [of "?Q cs1", OF q1 nq1]
-  obtain t1 where lt1: "t1 < length s"
-    and np1: "\<not> ?Q cs1 (moment t1 s)"
-    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
-  from p_split [of "?Q cs2", OF q2 nq2]
-  obtain t2 where lt2: "t2 < length s"
-    and np2: "\<not> ?Q cs2 (moment t2 s)"
-    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
-  { fix s cs
-    assume q: "?Q cs s"
-    have "thread \<notin> runing s"
-    proof
-      assume "thread \<in> runing s"
-      hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and> 
-                 thread \<noteq> hd (wq_fun (schs s) cs))"
-        by (unfold runing_def s_waiting_def readys_def, auto)
-      from this[rule_format, of cs] q 
-      show False by (simp add: wq_def) 
-    qed
-  } note q_not_runing = this
-  { fix t1 t2 cs1 cs2
-    assume  lt1: "t1 < length s"
-    and np1: "\<not> ?Q cs1 (moment t1 s)"
-    and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
-    and lt2: "t2 < length s"
-    and np2: "\<not> ?Q cs2 (moment t2 s)"
-    and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
-    and lt12: "t1 < t2"
-    let ?t3 = "Suc t2"
-    from lt2 have le_t3: "?t3 \<le> length s" by auto
-    from moment_plus [OF this] 
-    obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
-    have "t2 < ?t3" by simp
-    from nn2 [rule_format, OF this] and eq_m
-    have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-         h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-    have "vt (e#moment t2 s)"
-    proof -
-      from vt_moment 
-      have "vt (moment ?t3 s)" .
-      with eq_m show ?thesis by simp
-    qed
-    then interpret vt_e: valid_trace_e "moment t2 s" "e"
-        by (unfold_locales, auto, cases, simp)
-    have ?thesis
-    proof -
-      have "thread \<in> runing (moment t2 s)"
-      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-        case True
-        have "e = V thread cs2"
-        proof -
-          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
-              using True and np2  by auto 
-          from vt_e.wq_out_inv[OF True this h2]
-          show ?thesis .
-        qed
-        thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto
-      next
-        case False
-        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
-        with vt_e.actor_inv[OF vt_e.pip_e]
-        show ?thesis by auto
-      qed
-      moreover have "thread \<notin> runing (moment t2 s)"
-        by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
-      ultimately show ?thesis by simp
-    qed
-  } note lt_case = this
-  show ?thesis
-  proof -
-    { assume "t1 < t2"
-      from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
-      have ?thesis .
-    } moreover {
-      assume "t2 < t1"
-      from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
-      have ?thesis .
-    } moreover {
-      assume eq_12: "t1 = t2"
-      let ?t3 = "Suc t2"
-      from lt2 have le_t3: "?t3 \<le> length s" by auto
-      from moment_plus [OF this] 
-      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
-      have lt_2: "t2 < ?t3" by simp
-      from nn2 [rule_format, OF this] and eq_m
-      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-           h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-      from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12]
-      have g1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
-           g2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
-      have "vt (e#moment t2 s)"
-      proof -
-        from vt_moment 
-        have "vt (moment ?t3 s)" .
-        with eq_m show ?thesis by simp
-      qed
-      then interpret vt_e: valid_trace_e "moment t2 s" "e"
-          by (unfold_locales, auto, cases, simp)
-      have "e = V thread cs2 \<or> e = P thread cs2"
-      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-        case True
-        have "e = V thread cs2"
-        proof -
-          have eq_th: "thread = hd (wq (moment t2 s) cs2)" 
-              using True and np2  by auto 
-          from vt_e.wq_out_inv[OF True this h2]
-          show ?thesis .
-        qed
-        thus ?thesis by auto
-      next
-        case False
-        have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
-        thus ?thesis by auto
-      qed
-      moreover have "e = V thread cs1 \<or> e = P thread cs1"
-      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
-        case True
-        have eq_th: "thread = hd (wq (moment t1 s) cs1)" 
-              using True and np1  by auto 
-        from vt_e.wq_out_inv[folded eq_12, OF True this g2]
-        have "e = V thread cs1" .
-        thus ?thesis by auto
-      next
-        case False
-        have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
-        thus ?thesis by auto
-      qed
-      ultimately have ?thesis using neq12 by auto
-    } ultimately show ?thesis using nat_neq_iff by blast 
-  qed
-qed
-
-text {*
-  This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
-*}
-
-lemma waiting_unique:
-  assumes "waiting s th cs1"
-  and "waiting s th cs2"
-  shows "cs1 = cs2"
-  using waiting_unique_pre assms
-  unfolding wq_def s_waiting_def
-  by auto
-
-end
-
-(* not used *)
-text {*
-  Every thread can only be blocked on one critical resource, 
-  symmetrically, every critical resource can only be held by one thread. 
-  This fact is much more easier according to our definition. 
-*}
-lemma held_unique:
-  assumes "holding (s::event list) th1 cs"
-  and "holding s th2 cs"
-  shows "th1 = th2"
- by (insert assms, unfold s_holding_def, auto)
-
-lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
-  apply (induct s, auto)
-  by (case_tac a, auto split:if_splits)
-
-lemma last_set_unique: 
-  "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
-          \<Longrightarrow> th1 = th2"
-  apply (induct s, auto)
-  by (case_tac a, auto split:if_splits dest:last_set_lt)
-
-lemma preced_unique : 
-  assumes pcd_eq: "preced th1 s = preced th2 s"
-  and th_in1: "th1 \<in> threads s"
-  and th_in2: " th2 \<in> threads s"
-  shows "th1 = th2"
-proof -
-  from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
-  from last_set_unique [OF this th_in1 th_in2]
-  show ?thesis .
-qed
-                      
-lemma preced_linorder: 
-  assumes neq_12: "th1 \<noteq> th2"
-  and th_in1: "th1 \<in> threads s"
-  and th_in2: " th2 \<in> threads s"
-  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
-  from preced_unique [OF _ th_in1 th_in2] and neq_12 
-  have "preced th1 s \<noteq> preced th2 s" by auto
-  thus ?thesis by auto
-qed
-
-text {*
-  The following three lemmas show that @{text "RAG"} does not change
-  by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
-  events, respectively.
-*}
-
-lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma (in valid_trace_set)
-   RAG_unchanged: "(RAG (e # s)) = RAG s"
-   by (unfold is_set RAG_set_unchanged, simp)
-
-lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma (in valid_trace_create)
-   RAG_unchanged: "(RAG (e # s)) = RAG s"
-   by (unfold is_create RAG_create_unchanged, simp)
-
-lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma (in valid_trace_exit)
-   RAG_unchanged: "(RAG (e # s)) = RAG s"
-   by (unfold is_exit RAG_exit_unchanged, simp)
-
-context valid_trace_v
-begin
-
-lemma distinct_rest: "distinct rest"
-  by (simp add: distinct_tl rest_def wq_distinct)
-
-lemma holding_cs_eq_th:
-  assumes "holding s t cs"
-  shows "t = th"
-proof -
-  from pip_e[unfolded is_v]
-  show ?thesis
-  proof(cases)
-    case (thread_V)
-    from held_unique[OF this(2) assms]
-    show ?thesis by simp
-  qed
-qed
-
-lemma distinct_wq': "distinct wq'"
-  by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
-  
-lemma 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
-    have "t \<in> set wq'" 
-     using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
-     by simp 
-    from th'_in_inv[OF this] have "t \<in> set rest" .
-    with wq_s_cs[folded otherwise] wq_distinct[of cs]
-    show ?thesis by simp
-  next
-    case False
-    have "wq (e#s) c = wq s c" using False
-        by (unfold is_v, simp)
-    hence "waiting s t c" using assms 
-        by (simp add: cs_waiting_def waiting_eq)
-    hence "t \<notin> readys s" by (unfold readys_def, auto)
-    hence "t \<notin> runing s" using runing_ready by auto 
-    with runing_th_s[folded otherwise] show ?thesis by auto
-  qed
-qed
-
-lemma waiting_esI1:
-  assumes "waiting s t c"
-      and "c \<noteq> cs" 
-  shows "waiting (e#s) t c" 
-proof -
-  have "wq (e#s) c = wq s c" 
-    using assms(2) is_v by auto
-  with assms(1) show ?thesis 
-    using cs_waiting_def waiting_eq by auto 
-qed
-
-lemma holding_esI2:
-  assumes "c \<noteq> cs" 
-  and "holding s t c"
-  shows "holding (e#s) t c"
-proof -
-  from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
-  from assms(2)[unfolded s_holding_def, folded wq_def, 
-                folded this, unfolded wq_def, folded s_holding_def]
-  show ?thesis .
-qed
-
-lemma holding_esI1:
-  assumes "holding s t c"
-  and "t \<noteq> th"
-  shows "holding (e#s) t c"
-proof -
-  have "c \<noteq> cs" using assms using holding_cs_eq_th by blast 
-  from holding_esI2[OF this assms(1)]
-  show ?thesis .
-qed
-
-end
-
-context valid_trace_v_n
-begin
-
-lemma neq_wq': "wq' \<noteq> []" 
-proof (unfold wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest) 
-next
-  fix x
-  assume " distinct x \<and> set x = set rest" 
-  thus "x \<noteq> []" using rest_nnl by auto
-qed 
-
-definition "taker = hd wq'"
-
-definition "rest' = tl wq'"
-
-lemma eq_wq': "wq' = taker # rest'"
-  by (simp add: neq_wq' rest'_def taker_def)
-
-lemma next_th_taker: 
-  shows "next_th s th cs taker"
-  using rest_nnl taker_def wq'_def wq_s_cs 
-  by (auto simp:next_th_def)
-
-lemma taker_unique: 
-  assumes "next_th s th cs taker'"
-  shows "taker' = taker"
-proof -
-  from assms
-  obtain rest' where 
-    h: "wq s cs = th # rest'" 
-       "taker' = hd (SOME q. distinct q \<and> set q = set rest')"
-          by (unfold next_th_def, auto)
-  with wq_s_cs have "rest' = rest" by auto
-  thus ?thesis using h(2) taker_def wq'_def by auto 
-qed
-
-lemma waiting_set_eq:
-  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
-  by (smt all_not_in_conv bot.extremum insertI1 insert_subset 
-      mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
-
-lemma holding_set_eq:
-  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {(Cs cs, Th taker)}"
-  using next_th_taker taker_def waiting_set_eq 
-  by fastforce
-   
-lemma holding_taker:
-  shows "holding (e#s) taker cs"
-    by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, 
-        auto simp:neq_wq' taker_def)
-
-lemma waiting_esI2:
-  assumes "waiting s t cs"
-      and "t \<noteq> taker"
-  shows "waiting (e#s) t cs" 
-proof -
-  have "t \<in> set wq'" 
-  proof(unfold wq'_def, rule someI2)
-    show "distinct rest \<and> set rest = set rest"
-          by (simp add: distinct_rest)
-  next
-    fix x
-    assume "distinct x \<and> set x = set rest"
-    moreover have "t \<in> set rest"
-        using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto 
-    ultimately show "t \<in> set x" by simp
-  qed
-  moreover have "t \<noteq> hd wq'"
-    using assms(2) taker_def by auto 
-  ultimately show ?thesis
-    by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
-qed
-
-lemma waiting_esE:
-  assumes "waiting (e#s) t c" 
-  obtains "c \<noteq> cs" "waiting s t c"
-     |    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
-proof(cases "c = cs")
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
-  from that(1)[OF False this] show ?thesis .
-next
-  case True
-  from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
-  have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
-  hence "t \<noteq> taker" by (simp add: taker_def) 
-  moreover hence "t \<noteq> th" using assms neq_t_th by blast 
-  moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv) 
-  ultimately have "waiting s t cs"
-    by (metis cs_waiting_def list.distinct(2) list.sel(1) 
-                list.set_sel(2) rest_def waiting_eq wq_s_cs)  
-  show ?thesis using that(2)
-  using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto   
-qed
-
-lemma holding_esI1:
-  assumes "c = cs"
-  and "t = taker"
-  shows "holding (e#s) t c"
-  by (unfold assms, simp add: holding_taker)
-
-lemma holding_esE:
-  assumes "holding (e#s) t c" 
-  obtains "c = cs" "t = taker"
-      | "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
-  case True
-  from assms[unfolded True, unfolded s_holding_def, 
-             folded wq_def, unfolded wq_es_cs]
-  have "t = taker" by (simp add: taker_def) 
-  from that(1)[OF True this] show ?thesis .
-next
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  from assms[unfolded s_holding_def, folded wq_def, 
-             unfolded this, unfolded wq_def, folded s_holding_def]
-  have "holding s t c"  .
-  from that(2)[OF False this] show ?thesis .
-qed
-
-end 
-
-
-context valid_trace_v_e
-begin
-
-lemma nil_wq': "wq' = []" 
-proof (unfold wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest) 
-next
-  fix x
-  assume " distinct x \<and> set x = set rest" 
-  thus "x = []" using rest_nil by auto
-qed 
-
-lemma no_taker: 
-  assumes "next_th s th cs taker"
-  shows "False"
-proof -
-  from assms[unfolded next_th_def]
-  obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
-    by auto
-  thus ?thesis using rest_def rest_nil by auto 
-qed
-
-lemma waiting_set_eq:
-  "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
-  using no_taker by auto
-
-lemma holding_set_eq:
-  "{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
-  using no_taker by auto
-   
-lemma no_holding:
-  assumes "holding (e#s) taker cs"
-  shows False
-proof -
-  from wq_es_cs[unfolded nil_wq']
-  have " wq (e # s) cs = []" .
-  from assms[unfolded s_holding_def, folded wq_def, unfolded this]
-  show ?thesis by auto
-qed
-
-lemma no_waiting:
-  assumes "waiting (e#s) t cs"
-  shows False
-proof -
-  from wq_es_cs[unfolded nil_wq']
-  have " wq (e # s) cs = []" .
-  from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
-  show ?thesis by auto
-qed
-
-lemma waiting_esI2:
-  assumes "waiting s t c"
-  shows "waiting (e#s) t c"
-proof -
-  have "c \<noteq> cs" using assms
-    using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto 
-  from waiting_esI1[OF assms this]
-  show ?thesis .
-qed
-
-lemma waiting_esE:
-  assumes "waiting (e#s) t c" 
-  obtains "c \<noteq> cs" "waiting s t c"
-proof(cases "c = cs")
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto 
-  from that(1)[OF False this] show ?thesis .
-next
-  case True
-  from no_waiting[OF assms[unfolded True]]
-  show ?thesis by auto
-qed
-
-lemma holding_esE:
-  assumes "holding (e#s) t c" 
-  obtains "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
-  case True
-  from no_holding[OF assms[unfolded True]] 
-  show ?thesis by auto
-next
-  case False
-  hence "wq (e#s) c = wq s c" using is_v by auto
-  from assms[unfolded s_holding_def, folded wq_def, 
-             unfolded this, unfolded wq_def, folded s_holding_def]
-  have "holding s t c"  .
-  from that[OF False this] show ?thesis .
-qed
-
-end 
-
-lemma rel_eqI:
-  assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
-  and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
-  shows "A = B"
-  using assms by auto
-
-lemma in_RAG_E:
-  assumes "(n1, n2) \<in> RAG (s::state)"
-  obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
-      | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
-  using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
-  by auto
-  
-context valid_trace_v
-begin
-
-lemma RAG_es:
-  "RAG (e # s) =
-   RAG s - {(Cs cs, Th th)} -
-     {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-     {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
-proof(rule rel_eqI)
-  fix n1 n2
-  assume "(n1, n2) \<in> ?L"
-  thus "(n1, n2) \<in> ?R"
-  proof(cases rule:in_RAG_E)
-    case (waiting th' cs')
-    show ?thesis
-    proof(cases "rest = []")
-      case False
-      interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-      from waiting(3)
-      show ?thesis
-      proof(cases rule:h_n.waiting_esE)
-        case 1
-        with waiting(1,2)
-        show ?thesis
-        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      next
-        case 2
-        with waiting(1,2)
-        show ?thesis
-         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      qed
-    next
-      case True
-      interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-      from waiting(3)
-      show ?thesis
-      proof(cases rule:h_e.waiting_esE)
-        case 1
-        with waiting(1,2)
-        show ?thesis
-        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      qed
-    qed
-  next
-    case (holding th' cs')
-    show ?thesis
-    proof(cases "rest = []")
-      case False
-      interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-      from holding(3)
-      show ?thesis
-      proof(cases rule:h_n.holding_esE)
-        case 1
-        with holding(1,2)
-        show ?thesis
-        by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold waiting_eq, auto)
-      next
-        case 2
-        with holding(1,2)
-        show ?thesis
-         by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, 
-             fold holding_eq, auto)
-      qed
-    next
-      case True
-      interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-      from holding(3)
-      show ?thesis
-      proof(cases rule:h_e.holding_esE)
-        case 1
-        with holding(1,2)
-        show ?thesis
-        by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, 
-             fold holding_eq, auto)
-      qed
-    qed
-  qed
-next
-  fix n1 n2
-  assume h: "(n1, n2) \<in> ?R"
-  show "(n1, n2) \<in> ?L"
-  proof(cases "rest = []")
-    case False
-    interpret h_n: valid_trace_v_n s e th cs
-        by (unfold_locales, insert False, simp)
-    from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
-    have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
-                            \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or> 
-          (n2 = Th h_n.taker \<and> n1 = Cs cs)" 
-      by auto
-   thus ?thesis
-   proof
-      assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
-      with h_n.holding_taker
-      show ?thesis 
-        by (unfold s_RAG_def, fold holding_eq, auto)
-   next
-    assume h: "(n1, n2) \<in> RAG s \<and>
-        (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
-    hence "(n1, n2) \<in> RAG s" by simp
-    thus ?thesis
-    proof(cases rule:in_RAG_E)
-      case (waiting th' cs')
-      from h and this(1,2)
-      have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
-      hence "waiting (e#s) th' cs'" 
-      proof
-        assume "cs' \<noteq> cs"
-        from waiting_esI1[OF waiting(3) this] 
-        show ?thesis .
-      next
-        assume neq_th': "th' \<noteq> h_n.taker"
-        show ?thesis
-        proof(cases "cs' = cs")
-          case False
-          from waiting_esI1[OF waiting(3) this] 
-          show ?thesis .
-        next
-          case True
-          from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
-          show ?thesis .
-        qed
-      qed
-      thus ?thesis using waiting(1,2)
-        by (unfold s_RAG_def, fold waiting_eq, auto)
-    next
-      case (holding th' cs')
-      from h this(1,2)
-      have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
-      hence "holding (e#s) th' cs'"
-      proof
-        assume "cs' \<noteq> cs"
-        from holding_esI2[OF this holding(3)] 
-        show ?thesis .
-      next
-        assume "th' \<noteq> th"
-        from holding_esI1[OF holding(3) this]
-        show ?thesis .
-      qed
-      thus ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-   qed
- next
-   case True
-   interpret h_e: valid_trace_v_e s e th cs
-        by (unfold_locales, insert True, simp)
-   from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
-   have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)" 
-      by auto
-   from h_s(1)
-   show ?thesis
-   proof(cases rule:in_RAG_E)
-    case (waiting th' cs')
-    from h_e.waiting_esI2[OF this(3)]
-    show ?thesis using waiting(1,2)
-      by (unfold s_RAG_def, fold waiting_eq, auto)
-   next
-    case (holding th' cs')
-    with h_s(2)
-    have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
-    thus ?thesis
-    proof
-      assume neq_cs: "cs' \<noteq> cs"
-      from holding_esI2[OF this holding(3)]
-      show ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    next
-      assume "th' \<noteq> th"
-      from holding_esI1[OF holding(3) this]
-      show ?thesis using holding(1,2)
-        by (unfold s_RAG_def, fold holding_eq, auto)
-    qed
-   qed
- qed
-qed
-
-end
-
-lemma step_RAG_v: 
-assumes vt:
-  "vt (V th cs#s)"
-shows "
-  RAG (V th cs # s) =
-  RAG s - {(Cs cs, Th th)} -
-  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-  {(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
-proof -
-  interpret vt_v: valid_trace_v s "V th cs"
-    using assms step_back_vt by (unfold_locales, auto) 
-  show ?thesis using vt_v.RAG_es .
-qed
-
-lemma (in valid_trace_create)
-  th_not_in_threads: "th \<notin> threads s"
-proof -
-  from pip_e[unfolded is_create]
-  show ?thesis by (cases, simp)
-qed
-
-lemma (in valid_trace_create)
-  threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
-  by (unfold is_create, simp)
-
-lemma (in valid_trace_exit)
-  threads_es [simp]: "threads (e#s) = threads s - {th}"
-  by (unfold is_exit, simp)
-
-lemma (in valid_trace_p)
-  threads_es [simp]: "threads (e#s) = threads s"
-  by (unfold is_p, simp)
-
-lemma (in valid_trace_v)
-  threads_es [simp]: "threads (e#s) = threads s"
-  by (unfold is_v, simp)
-
-lemma (in valid_trace_v)
-  th_not_in_rest[simp]: "th \<notin> set rest"
-proof
-  assume otherwise: "th \<in> set rest"
-  have "distinct (wq s cs)" by (simp add: wq_distinct)
-  from this[unfolded wq_s_cs] and otherwise
-  show False by auto
-qed
-
-lemma (in valid_trace_v)
-  set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
-proof(unfold wq_es_cs wq'_def, rule someI2)
-  show "distinct rest \<and> set rest = set rest"
-    by (simp add: distinct_rest)
-next
-  fix x
-  assume "distinct x \<and> set x = set rest"
-  thus "set x = set (wq s cs) - {th}" 
-      by (unfold wq_s_cs, simp)
-qed
-
-lemma (in valid_trace_exit)
-  th_not_in_wq: "th \<notin> set (wq s cs)"
-proof -
-  from pip_e[unfolded is_exit]
-  show ?thesis
-  by (cases, unfold holdents_def s_holding_def, fold wq_def, 
-             auto elim!:runing_wqE)
-qed
-
-lemma (in valid_trace) wq_threads: 
-  assumes "th \<in> set (wq s cs)"
-  shows "th \<in> threads s"
-  using assms
-proof(induct rule:ind)
-  case (Nil)
-  thus ?case by (auto simp:wq_def)
-next
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case
-  proof(cases e)
-    case (Create th' prio')
-    interpret vt: valid_trace_create s e th' prio'
-      using Create by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems by auto
-  next
-    case (Exit th')
-    interpret vt: valid_trace_exit s e th'
-      using Exit by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto 
-  next
-    case (P th' cs')
-    interpret vt: valid_trace_p s e th' cs'
-      using P by (unfold_locales, simp)
-    show ?thesis
-      using Cons.hyps(2) Cons.prems readys_threads 
-        runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv 
-        by fastforce 
-  next
-    case (V th' cs')
-    interpret vt: valid_trace_v s e th' cs'
-      using V by (unfold_locales, simp)
-    show ?thesis using Cons
-      using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
-  next
-    case (Set th' prio)
-    interpret vt: valid_trace_set s e th' prio
-      using Set by (unfold_locales, simp)
-    show ?thesis using Cons.hyps(2) Cons.prems vt.is_set 
-        by (auto simp:wq_def Let_def)
-  qed
-qed 
-
-context valid_trace
-begin
-
-lemma  dm_RAG_threads:
-  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
-  shows "th \<in> threads s"
-proof -
-  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
-  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
-  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
-  hence "th \<in> set (wq s cs)"
-    by (unfold s_RAG_def, auto simp:cs_waiting_def)
-  from wq_threads [OF this] show ?thesis .
-qed
-
-lemma 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)
-
-end
-
-
-
-
-lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
-  by (unfold preced_def, simp)
-
-lemma (in valid_trace_v)
-  preced_es: "preced th (e#s) = preced th s"
-  by (unfold is_v preced_def, simp)
-
-lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
-proof
-  fix th'
-  show "the_preced (V th cs # s) th' = the_preced s th'"
-    by (unfold the_preced_def preced_def, simp)
-qed
-
-lemma (in valid_trace_v)
-  the_preced_es: "the_preced (e#s) = the_preced s"
-  by (unfold is_v preced_def, simp)
-
-context valid_trace_p
-begin
-
-lemma not_holding_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)
-  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)
-    interpret vt: valid_trace_create s e th prio using Create
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
-  next
-    case (Exit th)
-    interpret vt: valid_trace_exit s e th using Exit
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged)
-  next
-    case (P th cs)
-    interpret vt: valid_trace_p s e th cs using P
-      by (unfold_locales, simp)
-    show ?thesis using Cons using vt.RAG_es' by auto 
-  next
-    case (V th cs)
-    interpret vt: valid_trace_v s e th cs using V
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.finite_RAG_kept) 
-  next
-    case (Set th prio)
-    interpret vt: valid_trace_set s e th prio using Set
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
-  qed
-qed
-
-lemma acyclic_RAG:
-  shows "acyclic (RAG s)"
-proof(induct rule:ind)
-  case Nil
-  show ?case 
-  by (auto simp: s_RAG_def cs_waiting_def 
-                   cs_holding_def wq_def acyclic_def)
-next
-  case (Cons s e)
-  interpret vt_e: valid_trace_e s e using Cons by simp
-  show ?case
-  proof(cases e)
-    case (Create th prio)
-    interpret vt: valid_trace_create s e th prio using Create
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged) 
-  next
-    case (Exit th)
-    interpret vt: valid_trace_exit s e th using Exit
-      by (unfold_locales, simp)
-    show ?thesis using Cons by (simp add: vt.RAG_unchanged)
-  next
-    case (P th cs)
-    interpret vt: valid_trace_p s e th cs using P
-      by (unfold_locales, simp)
-    show ?thesis
-    proof(cases "wq s cs = []")
-      case True
-      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
-      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 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 (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 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 < 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
-
-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: 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 max_cp_eq_the_preced:
-  shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
-  using max_cp_eq using the_preced_def by presburger 
-
-lemma wf_RAG_converse: 
-  shows "wf ((RAG s)^-1)"
-proof(rule finite_acyclic_wf_converse)
-  from finite_RAG 
-  show "finite (RAG s)" .
-next
-  from acyclic_RAG
-  show "acyclic (RAG s)" .
-qed
-
-lemma chain_building:
-  assumes "node \<in> Domain (RAG s)"
-  obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"
-proof -
-  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
-    ultimately show ?thesis by (auto simp:readys_def)
-  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
-  The following is just an instance of @{text "chain_building"}.
-*}
-lemma th_chain_to_ready:
-  assumes th_in: "th \<in> threads s"
-  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
-proof(cases "th \<in> readys s")
-  case True
-  thus ?thesis by auto
-next
-  case False
-  from False and th_in have "Th th \<in> Domain (RAG s)" 
-    by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
-  from chain_building [rule_format, OF this]
-  show ?thesis by auto
-qed
-
-end
-
-lemma count_rec1 [simp]: 
-  assumes "Q e"
-  shows "count Q (e#es) = Suc (count Q es)"
-  using assms
-  by (unfold count_def, auto)
-
-lemma count_rec2 [simp]: 
-  assumes "\<not>Q e"
-  shows "count Q (e#es) = (count Q es)"
-  using assms
-  by (unfold count_def, auto)
-
-lemma count_rec3 [simp]: 
-  shows "count Q [] =  0"
-  by (unfold count_def, auto)
-
-lemma cntP_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"
-proof(cases e)
-  case (P th' pty)
-  show ?thesis
-  by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)", 
-        insert assms P, auto simp:cntP_def)
-qed (insert assms, auto simp:cntP_def)
-  
-lemma cntV_diff_inv:
-  assumes "cntV (e#s) th \<noteq> cntV s th"
-  shows "isV e \<and> actor e = th"
-proof(cases e)
-  case (V th' pty)
-  show ?thesis
-  by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)", 
-        insert assms V, auto simp:cntV_def)
-qed (insert assms, auto simp:cntV_def)
-
-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)
-
-fun the_cs :: "node \<Rightarrow> cs" where
-  "the_cs (Cs cs) = cs"
-
-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
-
-lemma finite_holdents: "finite (holdents s th)"
-  by (unfold holdents_alt_def, insert 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 -
-  { 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
-       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
-      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 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 
-        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:
-  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
-      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
-        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 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 (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"
-  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)
-
-end
-
-
-
-end
-
diff -r 5d8ec128518b -r e3cf792db636 Moment.thy
--- a/Moment.thy	Tue Jun 14 13:56:51 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,105 +0,0 @@
-theory Moment
-imports Main
-begin
-
-definition moment :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "moment n s = rev (take n (rev s))"
-
-value "moment 3 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9::int]"
-value "moment 2 [5, 4, 3, 2, 1, 0::int]"
-
-lemma moment_app [simp]:
-  assumes ile: "i \<le> length s"
-  shows "moment i (s' @ s) = moment i s"
-using assms unfolding moment_def by simp
-
-lemma moment_eq [simp]: "moment (length s) (s' @ s) = s"
-  unfolding moment_def by simp
-
-lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"
-  by (unfold moment_def, simp)
-
-lemma moment_zero [simp]: "moment 0 s = []"
-  by (simp add:moment_def)
-
-lemma least_idx:
-  assumes "Q (i::nat)"
-  obtains j where "j \<le> i" "Q j" "\<forall> k < j. \<not> Q k"
-  using assms
-  by (metis ex_least_nat_le le0 not_less0) 
-
-lemma duration_idx:
-  assumes "\<not> Q (i::nat)"
-  and "Q j"
-  and "i \<le> j"
-  obtains k where "i \<le> k" "k < j" "\<not> Q k" "\<forall> i'. k < i' \<and> i' \<le> j \<longrightarrow> Q i'" 
-proof -
-  let ?Q = "\<lambda> t. t \<le> j \<and> \<not> Q (j - t)"
-  have "?Q (j - i)" using assms by (simp add: assms(1)) 
-  from least_idx [of ?Q, OF this]
-  obtain l
-  where h: "l \<le> j - i" "\<not> Q (j - l)" "\<forall>k<l. \<not> (k \<le> j \<and> \<not> Q (j - k))"
-    by metis
-  let ?k = "j - l"
-  have "i \<le> ?k" using assms(3) h(1) by linarith 
-  moreover have "?k < j" by (metis assms(2) diff_le_self h(2) le_neq_implies_less) 
-  moreover have "\<not> Q ?k" by (simp add: h(2)) 
-  moreover have "\<forall> i'. ?k < i' \<and> i' \<le> j \<longrightarrow> Q i'"
-      by (metis diff_diff_cancel diff_le_self diff_less_mono2 h(3) 
-              less_imp_diff_less not_less) 
-  ultimately show ?thesis using that by metis
-qed
-
-lemma p_split_gen: 
-  assumes "Q s"
-  and "\<not> Q (moment k s)"
-  shows "(\<exists> i. i < length s \<and> k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
-proof(cases "k \<le> length s")
-  case True
-  let ?Q = "\<lambda> t. Q (moment t s)"
-  have "?Q (length s)" using assms(1) by simp
-  from duration_idx[of ?Q, OF assms(2) this True]
-  obtain i where h: "k \<le> i" "i < length s" "\<not> Q (moment i s)"
-    "\<forall>i'. i < i' \<and> i' \<le> length s \<longrightarrow> Q (moment i' s)" by metis
-  moreover have "(\<forall> i' > i. Q (moment i' s))" using h(4) assms(1) not_less
-    by fastforce
-  ultimately show ?thesis by metis
-qed (insert assms, auto)
-
-lemma p_split: 
-  assumes qs: "Q s"
-  and nq: "\<not> Q []"
-  shows "(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
-proof -
-  from nq have "\<not> Q (moment 0 s)" by simp
-  from p_split_gen [of Q s 0, OF qs this]
-  show ?thesis by auto
-qed
-
-lemma moment_Suc_tl:
-  assumes "Suc i \<le> length s"
-  shows "tl (moment (Suc i) s) = moment i s"
-  using assms 
-  by (simp add:moment_def rev_take, 
-      metis Suc_diff_le diff_Suc_Suc drop_Suc tl_drop)
-
-lemma moment_Suc_hd:
-  assumes "Suc i \<le> length s"
-  shows "hd (moment (Suc i) s) = s!(length s - Suc i)"
-  by (simp add:moment_def rev_take, 
-      subst hd_drop_conv_nth, insert assms, auto)
-  
-lemma moment_plus:
-  assumes "Suc i \<le> length s"
-  shows "(moment (Suc i) s) = (hd (moment (Suc i) s)) # (moment i s)"
-proof -
-  have "(moment (Suc i) s) \<noteq> []" using assms 
-    by (simp add:moment_def rev_take)
-  hence "(moment (Suc i) s) = (hd (moment (Suc i) s)) #  tl (moment (Suc i) s)"
-    by auto
-  with moment_Suc_tl[OF assms]
-  show ?thesis by metis
-qed
-
-end
-
diff -r 5d8ec128518b -r e3cf792db636 Moment_1.thy
--- a/Moment_1.thy	Tue Jun 14 13:56:51 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,896 +0,0 @@
-theory Moment
-imports Main
-begin
-
-fun firstn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
-  "firstn 0 s = []" |
-  "firstn (Suc n) [] = []" |
-  "firstn (Suc n) (e#s) = e#(firstn n s)"
-
-lemma upto_map_plus: "map (op + k) [i..j] = [i+k..j+k]"
-proof(induct "[i..j]" arbitrary:i j rule:length_induct)
-  case (1 i j)
-  thus ?case
-  proof(cases "i \<le> j")
-    case True
-    hence le_k: "i + k \<le> j + k" by simp
-    show ?thesis (is "?L = ?R")
-    proof -
-      have "?L  = (k + i) # map (op + k) [i + 1..j]"
-         by (simp add: upto_rec1[OF True])
-      moreover have "?R = (i + k) # [i + k + 1..j + k]"
-        by (simp add: upto_rec1[OF le_k])
-      moreover have "map (op + k) [i + 1..j] = [i + k + 1..j + k]"
-      proof -
-        have h: "i + k + 1 = (i + 1) + k" by simp
-        show ?thesis
-        proof(unfold h, rule 1[rule_format])
-          show "length [i + 1..j] < length [i..j]"
-            using upto_rec1[OF True] by simp
-        qed simp
-      qed
-      ultimately show ?thesis by simp
-    qed
-  qed auto
-qed
-
-lemma firstn_alt_def:
-  "firstn n s = map (\<lambda> i. s!(nat i)) [0..(int (min (length s) n)) - 1]"
-proof(induct n arbitrary:s)
-  case (0 s)
-  thus ?case by auto
-next
-  case (Suc n s)
-  thus ?case (is "?L = ?R")
-  proof(cases s)
-    case Nil
-    thus ?thesis by simp
-  next
-    case (Cons e es)
-    with Suc 
-    have "?L =  e # map (\<lambda>i. es ! nat i) [0..int (min (length es) n) - 1]"
-      by simp
-    also have "... = map (\<lambda>i. (e # es) ! nat i) [0..int (min (length es) n)]"
-      (is "?L1 = ?R1")
-    proof -
-      have "?R1 =   e # map (\<lambda>i. (e # es) ! nat i) 
-                            [1..int (min (length es) n)]" 
-      proof -
-        have "[0..int (min (length es) n)] = 0#[1..int (min (length es) n)]"
-          by (simp add: upto.simps)
-        thus ?thesis by simp
-      qed
-      also have "... = ?L1" (is "_#?L2 = _#?R2")
-      proof -
-        have "?L2 = ?R2"
-        proof -
-          have "map (\<lambda>i. (e # es) ! nat i) [1..int (min (length es) n)] =  
-                map ((\<lambda>i. (e # es) ! nat i) \<circ> op + 1) [0..int (min (length es) n) - 1]" 
-          proof -
-            have "[1..int (min (length es) n)] = 
-                             map (op + 1) [0..int (min (length es) n) - 1]"
-                     by (unfold upto_map_plus, simp)
-            thus ?thesis by simp
-          qed
-          also have "... = map (\<lambda>i. es ! nat i) [0..int (min (length es) n) - 1]"
-          proof(rule map_cong)
-            fix x
-            assume "x \<in> set [0..int (min (length es) n) - 1]"
-            thus "((\<lambda>i. (e # es) ! nat i) \<circ> op + 1) x = es ! nat x"
-              by (metis atLeastLessThan_iff atLeastLessThan_upto 
-                    comp_apply local.Cons nat_0_le nat_int nth_Cons_Suc of_nat_Suc)
-          qed auto
-          finally show ?thesis .
-        qed
-        thus ?thesis by simp
-      qed
-      finally show ?thesis by simp
-    qed
-    also have "... = ?R"
-      by (unfold Cons, simp)
-    finally show ?thesis .
-  qed
-qed
-
-fun restn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "restn n s = rev (firstn (length s - n) (rev s))"
-
-definition moment :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "moment n s = rev (firstn n (rev s))"
-
-definition restm :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "restm n s = rev (restn n (rev s))"
-
-definition from_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-  where "from_to i j s = firstn (j - i) (restn i s)"
-
-definition down_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "down_to j i s = rev (from_to i j (rev s))"
-
-value "down_to 6 2 [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]"
-value "from_to 2 6 [0, 1, 2, 3, 4, 5, 6, 7]"
-
-lemma length_eq_elim_l: "\<lbrakk>length xs = length ys; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"
-  by auto
-
-lemma length_eq_elim_r: "\<lbrakk>length us = length vs; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"
-  by simp
-
-lemma firstn_nil [simp]: "firstn n [] = []"
-  by (cases n, simp+)
-
-
-value "from_to 0 2 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] @ 
-       from_to 2 5 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"
-
-lemma firstn_le: "\<And> n s'. n \<le> length s \<Longrightarrow> firstn n (s@s') = firstn n s"
-proof (induct s, simp)
-  fix a s n s'
-  assume ih: "\<And>n s'. n \<le> length s \<Longrightarrow> firstn n (s @ s') = firstn n s"
-  and le_n: " n \<le> length (a # s)"
-  show "firstn n ((a # s) @ s') = firstn n (a # s)"
-  proof(cases n, simp)
-    fix k
-    assume eq_n: "n = Suc k"
-    with le_n have "k \<le> length s" by auto
-    from ih [OF this] and eq_n
-    show "firstn n ((a # s) @ s') = firstn n (a # s)" by auto
-  qed
-qed
-
-lemma firstn_ge [simp]: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"
-proof(induct s, simp)
-  fix a s n
-  assume ih: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"
-    and le: "length (a # s) \<le> n"
-  show "firstn n (a # s) = a # s"
-  proof(cases n)
-    assume eq_n: "n = 0" with le show ?thesis by simp
-  next
-    fix k
-    assume eq_n: "n = Suc k"
-    with le have le_k: "length s \<le> k" by simp
-    from ih [OF this] have "firstn k s = s" .
-    from eq_n and this
-    show ?thesis by simp
-  qed
-qed
-
-lemma firstn_eq [simp]: "firstn (length s) s = s"
-  by simp
-
-lemma firstn_restn_s: "(firstn n (s::'a list)) @ (restn n s) = s"
-proof(induct n arbitrary:s, simp)
-  fix n s
-  assume ih: "\<And>t. firstn n (t::'a list) @ restn n t = t"
-  show "firstn (Suc n) (s::'a list) @ restn (Suc n) s = s"
-  proof(cases s, simp)
-    fix x xs
-    assume eq_s: "s = x#xs"
-    show "firstn (Suc n) s @ restn (Suc n) s = s"
-    proof -
-      have "firstn (Suc n) s @ restn (Suc n) s =  x # (firstn n xs @ restn n xs)"
-      proof -
-        from eq_s have "firstn (Suc n) s =  x # firstn n xs" by simp
-        moreover have "restn (Suc n) s = restn n xs"
-        proof -
-          from eq_s have "restn (Suc n) s = rev (firstn (length xs - n) (rev xs @ [x]))" by simp
-          also have "\<dots> = restn n xs"
-          proof -
-            have "(firstn (length xs - n) (rev xs @ [x])) = (firstn (length xs - n) (rev xs))"
-              by(rule firstn_le, simp)
-            hence "rev (firstn (length xs - n) (rev xs @ [x])) = 
-              rev (firstn (length xs - n) (rev xs))" by simp
-            also have "\<dots> = rev (firstn (length (rev xs) - n) (rev xs))" by simp
-            finally show ?thesis by simp
-          qed
-          finally show ?thesis by simp
-        qed
-        ultimately show ?thesis by simp
-      qed with ih eq_s show ?thesis by simp
-    qed
-  qed
-qed
-
-lemma moment_restm_s: "(restm n s)@(moment n s) = s"
-proof -
-  have " rev  ((firstn n (rev s)) @ (restn n (rev s))) = s" (is "rev ?x = s")
-  proof -
-    have "?x = rev s" by (simp only:firstn_restn_s)
-    thus ?thesis by auto
-  qed
-  thus ?thesis 
-    by (auto simp:restm_def moment_def)
-qed
-
-declare restn.simps [simp del] firstn.simps[simp del]
-
-lemma length_firstn_ge: "length s \<le> n \<Longrightarrow> length (firstn n s) = length s"
-proof(induct n arbitrary:s, simp add:firstn.simps)
-  case (Suc k)
-  assume ih: "\<And> s. length (s::'a list) \<le> k \<Longrightarrow> length (firstn k s) = length s"
-  and le: "length s \<le> Suc k"
-  show ?case
-  proof(cases s)
-    case Nil
-    from Nil show ?thesis by simp
-  next
-    case (Cons x xs)
-    from le and Cons have "length xs \<le> k" by simp
-    from ih [OF this] have "length (firstn k xs) = length xs" .
-    moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))" 
-      by (simp add:firstn.simps)
-    moreover note Cons
-    ultimately show ?thesis by simp
-  qed
-qed
-
-lemma length_firstn_le: "n \<le> length s \<Longrightarrow> length (firstn n s) = n"
-proof(induct n arbitrary:s, simp add:firstn.simps)
-  case (Suc k)
-  assume ih: "\<And>s. k \<le> length (s::'a list) \<Longrightarrow> length (firstn k s) = k"
-    and le: "Suc k \<le> length s"
-  show ?case
-  proof(cases s)
-    case Nil
-    from Nil and le show ?thesis by auto
-  next
-    case (Cons x xs)
-    from le and Cons have "k \<le> length xs" by simp
-    from ih [OF this] have "length (firstn k xs) = k" .
-    moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))" 
-      by (simp add:firstn.simps)
-    ultimately show ?thesis by simp
-  qed
-qed
-
-lemma app_firstn_restn: 
-  fixes s1 s2
-  shows "s1 = firstn (length s1) (s1 @ s2) \<and> s2 = restn (length s1) (s1 @ s2)"
-proof(rule length_eq_elim_l)
-  have "length s1 \<le> length (s1 @ s2)" by simp
-  from length_firstn_le [OF this]
-  show "length s1 = length (firstn (length s1) (s1 @ s2))" by simp
-next
-  from firstn_restn_s 
-  show "s1 @ s2 = firstn (length s1) (s1 @ s2) @ restn (length s1) (s1 @ s2)"
-    by metis
-qed
-
-
-lemma length_moment_le:
-  fixes k s
-  assumes le_k: "k \<le> length s"
-  shows "length (moment k s) = k"
-proof -
-  have "length (rev (firstn k (rev s))) = k"
-  proof -
-    have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
-    also have "\<dots> = k" 
-    proof(rule length_firstn_le)
-      from le_k show "k \<le> length (rev s)" by simp
-    qed
-    finally show ?thesis .
-  qed
-  thus ?thesis by (simp add:moment_def)
-qed
-
-lemma app_moment_restm: 
-  fixes s1 s2
-  shows "s1 = restm (length s2) (s1 @ s2) \<and> s2 = moment (length s2) (s1 @ s2)"
-proof(rule length_eq_elim_r)
-  have "length s2 \<le> length (s1 @ s2)" by simp
-  from length_moment_le [OF this]
-  show "length s2 = length (moment (length s2) (s1 @ s2))" by simp
-next
-  from moment_restm_s 
-  show "s1 @ s2 = restm (length s2) (s1 @ s2) @ moment (length s2) (s1 @ s2)"
-    by metis
-qed
-
-lemma length_moment_ge:
-  fixes k s
-  assumes le_k: "length s \<le> k"
-  shows "length (moment k s) = (length s)"
-proof -
-  have "length (rev (firstn k (rev s))) = length s"
-  proof -
-    have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
-    also have "\<dots> = length s" 
-    proof -
-      have "\<dots> = length (rev s)"
-      proof(rule length_firstn_ge)
-        from le_k show "length (rev s) \<le> k" by simp
-      qed
-      also have "\<dots> = length s" by simp
-      finally show ?thesis .
-    qed
-    finally show ?thesis .
-  qed
-  thus ?thesis by (simp add:moment_def)
-qed
-
-lemma length_firstn: "(length (firstn n s) = length s) \<or> (length (firstn n s) = n)"
-proof(cases "n \<le> length s")
-  case True
-  from length_firstn_le [OF True] show ?thesis by auto
-next
-  case False
-  from False have "length s \<le> n" by simp
-  from firstn_ge [OF this] show ?thesis by auto
-qed
-
-lemma firstn_conc: 
-  fixes m n
-  assumes le_mn: "m \<le> n"
-  shows "firstn m s = firstn m (firstn n  s)"
-proof(cases "m \<le> length s")
-  case True
-  have "s = (firstn n s) @ (restn n s)" by (simp add:firstn_restn_s)
-  hence "firstn m s = firstn m \<dots>" by simp
-  also have "\<dots> = firstn m (firstn n s)" 
-  proof -
-    from length_firstn [of n s]
-    have "m \<le> length (firstn n s)"
-    proof
-      assume "length (firstn n s) = length s" with True show ?thesis by simp
-    next
-      assume "length (firstn n s) = n " with le_mn show ?thesis by simp
-    qed
-    from firstn_le [OF this, of "restn n s"]
-    show ?thesis .
-  qed
-  finally show ?thesis by simp
-next
-  case False
-  from False and le_mn have "length s \<le> n"  by simp
-  from firstn_ge [OF this] show ?thesis by simp
-qed
-
-lemma restn_conc: 
-  fixes i j k s
-  assumes eq_k: "j + i = k"
-  shows "restn k s = restn j (restn i s)"
-proof -
-  have "(firstn (length s - k) (rev s)) =
-        (firstn (length (rev (firstn (length s - i) (rev s))) - j) 
-                            (rev (rev (firstn (length s - i) (rev s)))))"
-  proof  -
-    have "(firstn (length s - k) (rev s)) =
-            (firstn (length (rev (firstn (length s - i) (rev s))) - j) 
-                                           (firstn (length s - i) (rev s)))"
-    proof -
-      have " (length (rev (firstn (length s - i) (rev s))) - j) = length s - k"
-      proof -
-        have "(length (rev (firstn (length s - i) (rev s))) - j) = (length s - i) - j"
-        proof -
-          have "(length (rev (firstn (length s - i) (rev s))) - j) = 
-                                         length ((firstn (length s - i) (rev s))) - j"
-            by simp
-          also have "\<dots> = length ((firstn (length (rev s) - i) (rev s))) - j" by simp
-          also have "\<dots> = (length (rev s) - i) - j" 
-          proof -
-            have "length ((firstn (length (rev s) - i) (rev s))) = (length (rev s) - i)"
-              by (rule length_firstn_le, simp)
-            thus ?thesis by simp
-          qed
-          also have "\<dots> = (length s - i) - j" by simp
-          finally show ?thesis .
-        qed
-        with eq_k show ?thesis by auto
-      qed
-      moreover have "(firstn (length s - k) (rev s)) =
-                             (firstn (length s - k) (firstn (length s - i) (rev s)))"
-      proof(rule firstn_conc)
-        from eq_k show "length s - k \<le> length s - i" by simp
-      qed
-      ultimately show ?thesis by simp
-    qed
-    thus ?thesis by simp
-  qed
-  thus ?thesis by (simp only:restn.simps)
-qed
-
-(*
-value "down_to 2 0 [5, 4, 3, 2, 1, 0]"
-value "moment 2 [5, 4, 3, 2, 1, 0]"
-*)
-
-lemma from_to_firstn: "from_to 0 k s = firstn k s"
-by (simp add:from_to_def restn.simps)
-
-lemma moment_app [simp]:
-  assumes 
-  ile: "i \<le> length s"
-  shows "moment i (s'@s) = moment i s"
-proof -
-  have "moment i (s'@s) = rev (firstn i (rev (s'@s)))" by (simp add:moment_def)
-  moreover have "firstn i (rev (s'@s)) = firstn i (rev s @ rev s')" by simp
-  moreover have "\<dots> = firstn i (rev s)"
-  proof(rule firstn_le)
-    have "length (rev s) = length s" by simp
-    with ile show "i \<le> length (rev s)" by simp
-  qed
-  ultimately show ?thesis by (simp add:moment_def)
-qed
-
-lemma moment_eq [simp]: "moment (length s) (s'@s) = s"
-proof -
-  have "length s \<le> length s" by simp
-  from moment_app [OF this, of s'] 
-  have " moment (length s) (s' @ s) = moment (length s) s" .
-  moreover have "\<dots> = s" by (simp add:moment_def)
-  ultimately show ?thesis by simp
-qed
-
-lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"
-  by (unfold moment_def, simp)
-
-lemma moment_zero [simp]: "moment 0 s = []"
-  by (simp add:moment_def firstn.simps)
-
-lemma p_split_gen: 
-  "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk> \<Longrightarrow>
-  (\<exists> i. i < length s \<and> k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
-proof (induct s, simp)
-  fix a s
-  assume ih: "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk>
-           \<Longrightarrow> \<exists>i<length s. k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall>i'>i. Q (moment i' s))"
-    and nq: "\<not> Q (moment k (a # s))" and qa: "Q (a # s)"
-  have le_k: "k \<le> length s"
-  proof -
-    { assume "length s < k"
-      hence "length (a#s) \<le> k" by simp
-      from moment_ge [OF this] and nq and qa
-      have "False" by auto
-    } thus ?thesis by arith
-  qed
-  have nq_k: "\<not> Q (moment k s)"
-  proof -
-    have "moment k (a#s) = moment k s"
-    proof -
-      from moment_app [OF le_k, of "[a]"] show ?thesis by simp
-    qed
-    with nq show ?thesis by simp
-  qed
-  show "\<exists>i<length (a # s). k \<le> i \<and> \<not> Q (moment i (a # s)) \<and> (\<forall>i'>i. Q (moment i' (a # s)))"
-  proof -
-    { assume "Q s"
-      from ih [OF this nq_k]
-      obtain i where lti: "i < length s" 
-        and nq: "\<not> Q (moment i s)" 
-        and rst: "\<forall>i'>i. Q (moment i' s)" 
-        and lki: "k \<le> i" by auto
-      have ?thesis 
-      proof -
-        from lti have "i < length (a # s)" by auto
-        moreover have " \<not> Q (moment i (a # s))"
-        proof -
-          from lti have "i \<le> (length s)" by simp
-          from moment_app [OF this, of "[a]"]
-          have "moment i (a # s) = moment i s" by simp
-          with nq show ?thesis by auto
-        qed
-        moreover have " (\<forall>i'>i. Q (moment i' (a # s)))"
-        proof -
-          {
-            fix i'
-            assume lti': "i < i'"
-            have "Q (moment i' (a # s))"
-            proof(cases "length (a#s) \<le> i'")
-              case True
-              from True have "moment i' (a#s) = a#s" by simp
-              with qa show ?thesis by simp
-            next
-              case False
-              from False have "i' \<le> length s" by simp
-              from moment_app [OF this, of "[a]"]
-              have "moment i' (a#s) = moment i' s" by simp
-              with rst lti' show ?thesis by auto
-            qed
-          } thus ?thesis by auto
-        qed
-        moreover note lki
-        ultimately show ?thesis by auto
-      qed
-    } moreover {
-      assume ns: "\<not> Q s"
-      have ?thesis
-      proof -
-        let ?i = "length s"
-        have "\<not> Q (moment ?i (a#s))"
-        proof -
-          have "?i \<le> length s" by simp
-          from moment_app [OF this, of "[a]"]
-          have "moment ?i (a#s) = moment ?i s" by simp
-          moreover have "\<dots> = s" by simp
-          ultimately show ?thesis using ns by auto
-        qed
-        moreover have "\<forall> i' > ?i. Q (moment i' (a#s))" 
-        proof -
-          { fix i'
-            assume "i' > ?i"
-            hence "length (a#s) \<le> i'" by simp
-            from moment_ge [OF this] 
-            have " moment i' (a # s) = a # s" .
-            with qa have "Q (moment i' (a#s))" by simp
-          } thus ?thesis by auto
-        qed
-        moreover have "?i < length (a#s)" by simp
-        moreover note le_k
-        ultimately show ?thesis by auto
-      qed
-    } ultimately show ?thesis by auto
-  qed
-qed
-
-lemma p_split: 
-  "\<And> s Q. \<lbrakk>Q s; \<not> Q []\<rbrakk> \<Longrightarrow> 
-       (\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
-proof -
-  fix s Q
-  assume qs: "Q s" and nq: "\<not> Q []"
-  from nq have "\<not> Q (moment 0 s)" by simp
-  from p_split_gen [of Q s 0, OF qs this]
-  show "(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
-    by auto
-qed
-
-lemma moment_plus: 
-  "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = (hd (moment (Suc i) s)) # (moment i s)"
-proof(induct s, simp+)
-  fix a s
-  assume ih: "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = hd (moment (Suc i) s) # moment i s"
-    and le_i: "i \<le> length s"
-  show "moment (Suc i) (a # s) = hd (moment (Suc i) (a # s)) # moment i (a # s)"
-  proof(cases "i= length s")
-    case True
-    hence "Suc i = length (a#s)" by simp
-    with moment_eq have "moment (Suc i) (a#s) = a#s" by auto
-    moreover have "moment i (a#s) = s"
-    proof -
-      from moment_app [OF le_i, of "[a]"]
-      and True show ?thesis by simp
-    qed
-    ultimately show ?thesis by auto
-  next
-    case False
-    from False and le_i have lti: "i < length s" by arith
-    hence les_i: "Suc i \<le> length s" by arith
-    show ?thesis 
-    proof -
-      from moment_app [OF les_i, of "[a]"]
-      have "moment (Suc i) (a # s) = moment (Suc i) s" by simp
-      moreover have "moment i (a#s) = moment i s" 
-      proof -
-        from lti have "i \<le> length s" by simp
-        from moment_app [OF this, of "[a]"] show ?thesis by simp
-      qed
-      moreover note ih [OF les_i]
-      ultimately show ?thesis by auto
-    qed
-  qed
-qed
-
-lemma from_to_conc:
-  fixes i j k s
-  assumes le_ij: "i \<le> j"
-  and le_jk: "j \<le> k"
-  shows "from_to i j s @ from_to j k s = from_to i k s"
-proof -
-  let ?ris = "restn i s"
-  have "firstn (j - i) (restn i s) @ firstn (k - j) (restn j s) =
-         firstn (k - i) (restn i s)" (is "?x @ ?y = ?z")
-  proof -
-    let "firstn (k-j) ?u" = "?y"
-    let ?rst = " restn (k - j) (restn (j - i) ?ris)"
-    let ?rst' = "restn (k - i) ?ris"
-    have "?u = restn (j-i) ?ris"
-    proof(rule restn_conc)
-      from le_ij show "j - i + i = j" by simp
-    qed
-    hence "?x @ ?y = ?x @ firstn (k-j) (restn (j-i) ?ris)" by simp
-    moreover have "firstn (k - j) (restn (j - i) (restn i s)) @ ?rst = 
-                      restn (j-i) ?ris" by (simp add:firstn_restn_s)
-    ultimately have "?x @ ?y @ ?rst = ?x @ (restn (j-i) ?ris)" by simp
-    also have "\<dots> = ?ris" by (simp add:firstn_restn_s)
-    finally have "?x @ ?y @ ?rst = ?ris" .
-    moreover have "?z @ ?rst = ?ris"
-    proof -
-      have "?z @ ?rst' = ?ris" by (simp add:firstn_restn_s)
-      moreover have "?rst' = ?rst"
-      proof(rule restn_conc)
-        from le_ij le_jk show "k - j + (j - i) = k - i" by auto
-      qed
-      ultimately show ?thesis by simp
-    qed
-    ultimately have "?x @ ?y @ ?rst = ?z @ ?rst" by simp
-    thus ?thesis by auto    
-  qed
-  thus ?thesis by (simp only:from_to_def)
-qed
-
-lemma down_to_conc:
-  fixes i j k s
-  assumes le_ij: "i \<le> j"
-  and le_jk: "j \<le> k"
-  shows "down_to k j s @ down_to j i s = down_to k i s"
-proof -
-  have "rev (from_to j k (rev s)) @ rev (from_to i j (rev s)) = rev (from_to i k (rev s))"
-    (is "?L = ?R")
-  proof -
-    have "?L = rev (from_to i j (rev s) @ from_to j k (rev s))" by simp
-    also have "\<dots> = ?R" (is "rev ?x = rev ?y")
-    proof -
-      have "?x = ?y" by (rule from_to_conc[OF le_ij le_jk])
-      thus ?thesis by simp
-    qed
-    finally show ?thesis .
-  qed
-  thus ?thesis by (simp add:down_to_def)
-qed
-
-lemma restn_ge:
-  fixes s k
-  assumes le_k: "length s \<le> k"
-  shows "restn k s = []"
-proof -
-  from firstn_restn_s [of k s, symmetric] have "s = (firstn k s) @ (restn k s)" .
-  hence "length s = length \<dots>" by simp
-  also have "\<dots> = length (firstn k s) + length (restn k s)" by simp
-  finally have "length s = ..." by simp
-  moreover from length_firstn_ge and le_k 
-  have "length (firstn k s) = length s" by simp
-  ultimately have "length (restn k s) = 0" by auto
-  thus ?thesis by auto
-qed
-
-lemma from_to_ge: "length s \<le> k \<Longrightarrow> from_to k j s = []"
-proof(simp only:from_to_def)
-  assume "length s \<le> k"
-  from restn_ge [OF this] 
-  show "firstn (j - k) (restn k s) = []" by simp
-qed
-
-(*
-value "from_to 2 5 [0, 1, 2, 3, 4]"
-value "restn 2  [0, 1, 2, 3, 4]"
-*)
-
-lemma from_to_restn: 
-  fixes k j s
-  assumes le_j: "length s \<le> j"
-  shows "from_to k j s = restn k s"
-proof -
-  have "from_to 0 k s @ from_to k j s = from_to 0 j s"
-  proof(cases "k \<le> j")
-    case True
-    from from_to_conc True show ?thesis by auto
-  next
-    case False
-    from False le_j have lek: "length s \<le>  k" by auto
-    from from_to_ge [OF this] have "from_to k j s = []" .
-    hence "from_to 0 k s @ from_to k j s = from_to 0 k s" by simp
-    also have "\<dots> = s"
-    proof -
-      from from_to_firstn [of k s]
-      have "\<dots> = firstn k s" .
-      also have "\<dots> = s" by (rule firstn_ge [OF lek])
-      finally show ?thesis .
-    qed
-    finally have "from_to 0 k s @ from_to k j s = s" .
-    moreover have "from_to 0 j s = s"
-    proof -
-      have "from_to 0 j s = firstn j s" by (simp add:from_to_firstn)
-      also have "\<dots> = s"
-      proof(rule firstn_ge)
-        from le_j show "length s \<le> j " by simp
-      qed
-      finally show ?thesis .
-    qed
-    ultimately show ?thesis by auto
-  qed
-  also have "\<dots> = s" 
-  proof -
-    from from_to_firstn have "\<dots> = firstn j s" .
-    also have "\<dots> = s"
-    proof(rule firstn_ge)
-      from le_j show "length s \<le> j" by simp
-    qed
-    finally show ?thesis .
-  qed
-  finally have "from_to 0 k s @ from_to k j s = s" .
-  moreover have "from_to 0 k s @ restn k s = s"
-  proof -
-    from from_to_firstn [of k s]
-    have "from_to 0 k s = firstn k s" .
-    thus ?thesis by (simp add:firstn_restn_s)
-  qed
-  ultimately have "from_to 0 k s @ from_to k j s  = 
-                    from_to 0 k s @ restn k s" by simp
-  thus ?thesis by auto
-qed
-
-lemma down_to_moment: "down_to k 0 s = moment k s"
-proof -
-  have "rev (from_to 0 k (rev s)) = rev (firstn k (rev s))" 
-    using from_to_firstn by metis
-  thus ?thesis by (simp add:down_to_def moment_def)
-qed
-
-lemma down_to_restm:
-  assumes le_s: "length s \<le> j"
-  shows "down_to j k s = restm k s"
-proof -
-  have "rev (from_to k j (rev s)) = rev (restn k (rev s))" (is "?L = ?R")
-  proof -
-    from le_s have "length (rev s) \<le> j" by simp
-    from from_to_restn [OF this, of k] show ?thesis by simp
-  qed
-  thus ?thesis by (simp add:down_to_def restm_def)
-qed
-
-lemma moment_split: "moment (m+i) s = down_to (m+i) i s @down_to i 0 s"
-proof -
-  have "moment (m + i) s = down_to (m+i) 0 s" using down_to_moment by metis
-  also have "\<dots> = (down_to (m+i) i s) @ (down_to i 0 s)" 
-    by(rule down_to_conc[symmetric], auto)
-  finally show ?thesis .
-qed
-
-lemma length_restn: "length (restn i s) = length s - i"
-proof(cases "i \<le> length s")
-  case True
-  from length_firstn_le [OF this] have "length (firstn i s) = i" .
-  moreover have "length s = length (firstn i s) + length (restn i s)"
-  proof -
-    have "s = firstn i s @ restn i s" using firstn_restn_s by metis
-    hence "length s = length \<dots>" by simp
-    thus ?thesis by simp
-  qed
-  ultimately show ?thesis by simp
-next 
-  case False
-  hence "length s \<le> i" by simp
-  from restn_ge [OF this] have "restn i s = []" .
-  with False show ?thesis by simp
-qed
-
-lemma length_from_to_in:
-  fixes i j s
-  assumes le_ij: "i \<le> j"
-  and le_j: "j \<le> length s"
-  shows "length (from_to i j s) = j - i"
-proof -
-  have "from_to 0 j s = from_to 0 i s @ from_to i j s"
-    by (rule from_to_conc[symmetric, OF _ le_ij], simp)
-  moreover have "length (from_to 0 j s) = j"
-  proof -
-    have "from_to 0 j s = firstn j s" using from_to_firstn by metis
-    moreover have "length \<dots> = j" by (rule length_firstn_le [OF le_j])
-    ultimately show ?thesis by simp
-  qed
-  moreover have "length (from_to 0 i s) = i"
-  proof -
-    have "from_to 0 i s = firstn i s" using from_to_firstn by metis
-    moreover have "length \<dots> = i" 
-    proof (rule length_firstn_le)
-      from le_ij le_j show "i \<le> length s" by simp
-    qed
-    ultimately show ?thesis by simp
-  qed
-  ultimately show ?thesis by auto
-qed
-
-lemma firstn_restn_from_to: "from_to i (m + i) s = firstn m (restn i s)"
-proof(cases "m+i \<le> length s")
-  case True
-  have "restn i s = from_to i (m+i) s @ from_to (m+i) (length s) s"
-  proof -
-    have "restn i s = from_to i (length s) s"
-      by(rule from_to_restn[symmetric], simp)
-    also have "\<dots> = from_to i (m+i) s @ from_to (m+i) (length s) s"
-      by(rule from_to_conc[symmetric, OF _ True], simp)
-    finally show ?thesis .
-  qed
-  hence "firstn m (restn i s) = firstn m \<dots>" by simp
-  moreover have "\<dots> = firstn (length (from_to i (m+i) s)) 
-                    (from_to i (m+i) s @ from_to (m+i) (length s) s)"
-  proof -
-    have "length (from_to i (m+i) s) = m"
-    proof -
-      have "length (from_to i (m+i) s) = (m+i) - i"
-        by(rule length_from_to_in [OF _ True], simp)
-      thus ?thesis by simp
-    qed
-    thus ?thesis by simp
-  qed
-  ultimately show ?thesis using app_firstn_restn by metis
-next
-  case False
-  hence "length s \<le> m + i" by simp
-  from from_to_restn [OF this]
-  have "from_to i (m + i) s = restn i s" .
-  moreover have "firstn m (restn i s) = restn i s" 
-  proof(rule firstn_ge)
-    show "length (restn i s) \<le> m"
-    proof -
-      have "length (restn i s) = length s - i" using length_restn by metis
-      with False show ?thesis by simp
-    qed
-  qed
-  ultimately show ?thesis by simp
-qed
-
-lemma down_to_moment_restm:
-  fixes m i s
-  shows "down_to (m + i) i s = moment m (restm i s)"
-  by (simp add:firstn_restn_from_to down_to_def moment_def restm_def)
-
-lemma moment_plus_split:
-  fixes m i s
-  shows "moment (m + i) s = moment m (restm i s) @ moment i s"
-proof -
-  from moment_split [of m i s]
-  have "moment (m + i) s = down_to (m + i) i s @ down_to i 0 s" .
-  also have "\<dots> = down_to (m+i) i s @ moment i s" using down_to_moment by simp
-  also from down_to_moment_restm have "\<dots> = moment m (restm i s) @ moment i s"
-    by simp
-  finally show ?thesis .
-qed
-
-lemma length_restm: "length (restm i s) = length s - i"
-proof -
-  have "length (rev (restn i (rev s))) = length s - i" (is "?L = ?R")
-  proof -
-    have "?L = length (restn i (rev s))" by simp
-    also have "\<dots>  = length (rev s) - i" using length_restn by metis
-    also have "\<dots> = ?R" by simp
-    finally show ?thesis .
-  qed
-  thus ?thesis by (simp add:restm_def)
-qed
-
-lemma moment_prefix: "(moment i t @ s) = moment (i + length s) (t @ s)"
-proof -
-  from moment_plus_split [of i "length s" "t@s"]
-  have " moment (i + length s) (t @ s) = moment i (restm (length s) (t @ s)) @ moment (length s) (t @ s)"
-    by auto
-  with app_moment_restm[of t s]
-  have "moment (i + length s) (t @ s) = moment i t @ moment (length s) (t @ s)" by simp
-  with moment_app show ?thesis by auto
-qed
-
-lemma length_down_to_in: 
-  assumes le_ij: "i \<le> j"
-    and le_js: "j \<le> length s"
-  shows "length (down_to j i s) = j - i"
-proof -
-  have "length (down_to j i s) = length (from_to i j (rev s))"
-    by (unfold down_to_def, auto)
-  also have "\<dots> = j - i"
-  proof(rule length_from_to_in[OF le_ij])
-    from le_js show "j \<le> length (rev s)" by simp
-  qed
-  finally show ?thesis .
-qed
-
-
-lemma moment_head: 
-  assumes le_it: "Suc i \<le> length t"
-  obtains e where "moment (Suc i) t = e#moment i t"
-proof -
-  have "i \<le> Suc i" by simp
-  from length_down_to_in [OF this le_it]
-  have "length (down_to (Suc i) i t) = 1" by auto
-  then obtain e where "down_to (Suc i) i t = [e]"
-    apply (cases "(down_to (Suc i) i t)") by auto
-  moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
-    by (rule down_to_conc[symmetric], auto)
-  ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
-    by (auto simp:down_to_moment)
-  from that [OF this] show ?thesis .
-qed
-
-end
diff -r 5d8ec128518b -r e3cf792db636 PIPBasics.thy
--- a/PIPBasics.thy	Tue Jun 14 13:56:51 2016 +0100
+++ b/PIPBasics.thy	Tue Jun 14 15:06:16 2016 +0100
@@ -511,10 +511,12 @@
   } ultimately show ?thesis by blast
 qed
 
+(*
 lemma image_eq:
   assumes "A = B"
   shows "f ` A = f ` B"
   using assms by auto
+*)
 
 lemma tRAG_trancl_eq_Th:
    "{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
@@ -907,40 +909,6 @@
   qed
 qed
 
-text {*
-  The following lemma says that if @{text "s"} is a valid state, so 
-  is its any postfix. Where @{term "monent t s"} is the postfix of 
-  @{term "s"} with length @{term "t"}.
-*}
-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
-
-text {*
-  The following two lemmas are fundamental, because they assure
-  that the numbers of both living and ready threads are finite.
-*}
 
 lemma finite_threads:
   shows "finite (threads s)"
@@ -951,34 +919,6 @@
 
 end
 
-text {*
-  The following locale @{text "valid_moment"} is to inherit the properties 
-  derived on any valid state to the prefix of it, with length @{text "i"}.
-*}
-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)
-
-locale valid_moment_e = valid_moment +
-  assumes less_i: "i < length s"
-begin
-  definition "next_e  = hd (moment (Suc i) s)"
-
-  lemma trace_e: 
-    "moment (Suc i) s = next_e#moment i s"
-   proof -
-    from less_i have "Suc i \<le> length s" by auto
-    from moment_plus[OF this, folded next_e_def]
-    show ?thesis .
-   qed
-
-end
-
-sublocale valid_moment_e < vat_moment_e: valid_trace_e "moment i s" "next_e"
-  using vt_moment[of "Suc i", unfolded trace_e]
-  by (unfold_locales, simp)
 
 section {* Waiting queues are distinct *}
 
@@ -1385,13 +1325,16 @@
 
 end
 
+context valid_trace
+begin
+
 text {*
   The is the main lemma of this section, which is derived
   by induction, case analysis on event @{text e} and 
   assembling the @{text "wq_threads_kept"}-results of 
   all possible cases of @{text "e"}.
 *}
-lemma (in valid_trace) wq_threads: 
+lemma wq_threads: 
   assumes "th \<in> set (wq s cs)"
   shows "th \<in> threads s"
   using assms
@@ -1432,8 +1375,6 @@
 
 subsection {* RAG and threads *}
 
-context valid_trace
-begin
 
 text {*
   As corollaries of @{thm wq_threads}, it is shown in this subsection
@@ -1869,7 +1810,7 @@
   "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")
+     {(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"
@@ -3208,14 +3149,15 @@
   qed
 qed
 
-lemma card_running: "card (running s) \<le> 1"
+lemma card_running: 
+  shows "card (running s) \<le> 1"
 proof(cases "running s = {}")
   case True
   thus ?thesis by auto
 next
   case False
-  then obtain th where [simp]: "th \<in> running s" by auto
-  from running_unique[OF this]
+  then obtain th where "th \<in> running s" by auto
+  with running_unique
   have "running s = {th}" by auto
   thus ?thesis by auto
 qed
diff -r 5d8ec128518b -r e3cf792db636 PIPDefs.thy
--- a/PIPDefs.thy	Tue Jun 14 13:56:51 2016 +0100
+++ b/PIPDefs.thy	Tue Jun 14 15:06:16 2016 +0100
@@ -1,6 +1,6 @@
 (*<*)
 theory PIPDefs
-imports Precedence_ord Moment RTree Max
+imports Precedence_ord RTree Max
 begin
 (*>*)