# HG changeset patch
# User urbanc
# Date 1327364409 0
# Node ID 4190df6f4488e18f25ec69631b9f89348f41b965
# Parent  12e9aa68d5db341fba30390df150d7dc2654e969
initial version of the PIP formalisation

diff -r 12e9aa68d5db -r 4190df6f4488 Matcher.thy
--- a/Matcher.thy	Mon Dec 26 08:21:00 2011 +0000
+++ b/Matcher.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -171,6 +171,17 @@
 
 section {* Examples *}
 
+definition 
+  "CHRA \<equiv> CHAR (CHR ''a'')"
+
+definition 
+  "ALT1 \<equiv> ALT CHRA EMPTY"
+
+definition 
+  "SEQ3 \<equiv> SEQ (SEQ ALT1 ALT1) ALT1"
+
+value "matcher SEQ3 ''aaa''"
+
 value "matcher NULL []"
 value "matcher (CHAR (CHR ''a'')) [CHR ''a'']"
 value "matcher (CHAR a) [a,a]"
diff -r 12e9aa68d5db -r 4190df6f4488 prio/CpsG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/CpsG.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1826 @@
+theory CpsG
+imports PrioG 
+begin
+
+lemma not_thread_holdents:
+  fixes th s
+  assumes vt: "vt step s"
+  and not_in: "th \<notin> threads s" 
+  shows "holdents s th = {}"
+proof -
+  from vt not_in show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e th)
+    assume vt: "vt step s"
+      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> holdents s th = {}"
+      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 "holdents (e # s) th = holdents s th"
+        apply (unfold eq_e holdents_def)
+        by (simp add:depend_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 = {}"
+      show ?thesis
+      proof(cases "th = thread")
+        case True
+        with nh eq_e
+        show ?thesis 
+          by (auto simp:holdents_def depend_exit_unchanged)
+      next
+        case False
+        with not_in and eq_e
+        have "th \<notin> threads s" by simp
+        from ih[OF this] False eq_e show ?thesis 
+          by (auto simp:holdents_def depend_exit_unchanged)
+      qed
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      and is_runing: "thread \<in> runing s"
+      from prems have vtp: "vt step (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 "holdents (e # s) th  = holdents s th "
+        apply (unfold cntCS_def holdents_def eq_e)
+        by (unfold step_depend_p[OF vtp], auto)
+      moreover have "holdents s th = {}"
+      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 prems have vtv: "vt step (V thread cs#s)" by auto
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp: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 wq_distinct[OF step_back_vt[OF vtv], 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 "hd x \<in> set rest" by (cases x, auto)
+        qed
+        with eq_wq have "?t \<in> set (wq s cs)" by simp
+        from wq_threads[OF step_back_vt[OF vtv], OF this] and ni
+        show False by auto
+      qed
+      moreover note neq_th eq_wq
+      ultimately have "holdents (e # s) th  = holdents s th"
+        by (unfold eq_e cntCS_def holdents_def step_depend_v[OF vtv], auto)
+      moreover have "holdents s th = {}"
+      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_def)
+        by (simp add:depend_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+
+
+lemma next_th_neq: 
+  assumes vt: "vt step s"
+  and nt: "next_th s th cs th'"
+  shows "th' \<noteq> th"
+proof -
+  from nt show ?thesis
+    apply (auto simp:next_th_def)
+  proof -
+    fix rest
+    assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+      and ne: "rest \<noteq> []"
+    have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest" 
+    proof(rule someI2)
+      from wq_distinct[OF vt, of cs] eq_wq
+      show "distinct rest \<and> set rest = set rest" by auto
+    next
+      fix x
+      assume "distinct x \<and> set x = set rest"
+      hence eq_set: "set x = set rest" by auto
+      with ne have "x \<noteq> []" by auto
+      hence "hd x \<in> set x" by auto
+      with eq_set show "hd x \<in> set rest" by auto
+    qed
+    with wq_distinct[OF vt, of cs] eq_wq show False by auto
+  qed
+qed
+
+lemma next_th_unique: 
+  assumes nt1: "next_th s th cs th1"
+  and nt2: "next_th s th cs th2"
+  shows "th1 = th2"
+proof -
+  from assms show ?thesis
+    by (unfold next_th_def, auto)
+qed
+
+lemma pp_sub: "(r^+)^+ \<subseteq> r^+"
+  by auto
+
+lemma wf_depend:
+  assumes vt: "vt step s"
+  shows "wf (depend s)"
+proof(rule finite_acyclic_wf)
+  from finite_depend[OF vt] show "finite (depend s)" .
+next
+  from acyclic_depend[OF vt] show "acyclic (depend s)" .
+qed
+
+lemma Max_Union:
+  assumes fc: "finite C"
+  and ne: "C \<noteq> {}"
+  and fa: "\<And> A. A \<in> C \<Longrightarrow> finite A \<and> A \<noteq> {}"
+  shows "Max (\<Union> C) = Max (Max ` C)"
+proof -
+  from fc ne fa show ?thesis
+  proof(induct)
+    case (insert x F)
+    assume ih: "\<lbrakk>F \<noteq> {}; \<And>A. A \<in> F \<Longrightarrow> finite A \<and> A \<noteq> {}\<rbrakk> \<Longrightarrow> Max (\<Union>F) = Max (Max ` F)"
+    and h: "\<And> A. A \<in> insert x F \<Longrightarrow> finite A \<and> A \<noteq> {}"
+    show ?case (is "?L = ?R")
+    proof(cases "F = {}")
+      case False
+      from Union_insert have "?L = Max (x \<union> (\<Union> F))" by simp
+      also have "\<dots> = max (Max x) (Max(\<Union> F))"
+      proof(rule Max_Un)
+        from h[of x] show "finite x" by auto
+      next
+        from h[of x] show "x \<noteq> {}" by auto
+      next
+        show "finite (\<Union>F)"
+        proof(rule finite_Union)
+          show "finite F" by fact
+        next
+          from h show "\<And>M. M \<in> F \<Longrightarrow> finite M" by auto
+        qed
+      next
+        from False and h show "\<Union>F \<noteq> {}" by auto
+      qed
+      also have "\<dots> = ?R"
+      proof -
+        have "?R = Max (Max ` ({x} \<union> F))" by simp
+        also have "\<dots> = Max ({Max x} \<union> (Max ` F))" by simp
+        also have "\<dots> = max (Max x) (Max (\<Union>F))"
+        proof -
+          have "Max ({Max x} \<union> Max ` F) = max (Max {Max x}) (Max (Max ` F))"
+          proof(rule Max_Un)
+            show "finite {Max x}" by simp
+          next
+            show "{Max x} \<noteq> {}" by simp
+          next
+            from insert show "finite (Max ` F)" by auto
+          next
+            from False show "Max ` F \<noteq> {}" by auto
+          qed
+          moreover have "Max {Max x} = Max x" by simp
+          moreover have "Max (\<Union>F) = Max (Max ` F)"
+          proof(rule ih)
+            show "F \<noteq> {}" by fact
+          next
+            from h show "\<And>A. A \<in> F \<Longrightarrow> finite A \<and> A \<noteq> {}"
+              by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+        finally show ?thesis by simp
+      qed
+      finally show ?thesis by simp
+    next
+      case True
+      thus ?thesis by auto
+    qed
+  next
+    case empty
+    assume "{} \<noteq> {}" show ?case by auto
+  qed
+qed
+
+definition child :: "state \<Rightarrow> (node \<times> node) set"
+  where "child s =
+            {(Th th', Th th) | th th'. \<exists> cs. (Th th', Cs cs) \<in> depend s \<and> (Cs cs, Th th) \<in> depend s}"
+
+definition children :: "state \<Rightarrow> thread \<Rightarrow> thread set"
+  where "children s th = {th'. (Th th', Th th) \<in> child s}"
+
+
+lemma children_dependents: "children s th \<subseteq> dependents (wq s) th"
+  by (unfold children_def child_def cs_dependents_def, auto simp:eq_depend)
+
+lemma child_unique:
+  assumes vt: "vt step s"
+  and ch1: "(Th th, Th th1) \<in> child s"
+  and ch2: "(Th th, Th th2) \<in> child s"
+  shows "th1 = th2"
+proof -
+  from ch1 ch2 show ?thesis
+  proof(unfold child_def, clarsimp)
+    fix cs csa
+    assume h1: "(Th th, Cs cs) \<in> depend s"
+      and h2: "(Cs cs, Th th1) \<in> depend s"
+      and h3: "(Th th, Cs csa) \<in> depend s"
+      and h4: "(Cs csa, Th th2) \<in> depend s"
+    from unique_depend[OF vt h1 h3] have "cs = csa" by simp
+    with h4 have "(Cs cs, Th th2) \<in> depend s" by simp
+    from unique_depend[OF vt h2 this]
+    show "th1 = th2" by simp
+  qed 
+qed
+
+
+lemma cp_eq_cpreced_f: "cp s = cpreced s (wq s)"
+proof -
+  from fun_eq_iff 
+  have h:"\<And>f g. (\<forall> x. f x = g x) \<Longrightarrow> f = g" by auto
+  show ?thesis
+  proof(rule h)
+    from cp_eq_cpreced show "\<forall>x. cp s x = cpreced s (wq s) x" by auto
+  qed
+qed
+
+lemma depend_children:
+  assumes h: "(Th th1, Th th2) \<in> (depend s)^+"
+  shows "th1 \<in> children s th2 \<or> (\<exists> th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)^+)"
+proof -
+  from h show ?thesis
+  proof(induct rule: tranclE)
+    fix c th2
+    assume h1: "(Th th1, c) \<in> (depend s)\<^sup>+"
+    and h2: "(c, Th th2) \<in> depend s"
+    from h2 obtain cs where eq_c: "c = Cs cs"
+      by (case_tac c, auto simp:s_depend_def)
+    show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)\<^sup>+)"
+    proof(rule tranclE[OF h1])
+      fix ca
+      assume h3: "(Th th1, ca) \<in> (depend s)\<^sup>+"
+        and h4: "(ca, c) \<in> depend s"
+      show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)\<^sup>+)"
+      proof -
+        from eq_c and h4 obtain th3 where eq_ca: "ca = Th th3"
+          by (case_tac ca, auto simp:s_depend_def)
+        from eq_ca h4 h2 eq_c
+        have "th3 \<in> children s th2" by (auto simp:children_def child_def)
+        moreover from h3 eq_ca have "(Th th1, Th th3) \<in> (depend s)\<^sup>+" by simp
+        ultimately show ?thesis by auto
+      qed
+    next
+      assume "(Th th1, c) \<in> depend s"
+      with h2 eq_c
+      have "th1 \<in> children s th2"
+        by (auto simp:children_def child_def)
+      thus ?thesis by auto
+    qed
+  next
+    assume "(Th th1, Th th2) \<in> depend s"
+    thus ?thesis
+      by (auto simp:s_depend_def)
+  qed
+qed
+
+lemma sub_child: "child s \<subseteq> (depend s)^+"
+  by (unfold child_def, auto)
+
+lemma wf_child: 
+  assumes vt: "vt step s"
+  shows "wf (child s)"
+proof(rule wf_subset)
+  from wf_trancl[OF wf_depend[OF vt]]
+  show "wf ((depend s)\<^sup>+)" .
+next
+  from sub_child show "child s \<subseteq> (depend s)\<^sup>+" .
+qed
+
+lemma depend_child_pre:
+  assumes vt: "vt step s"
+  shows
+  "(Th th, n) \<in> (depend s)^+ \<longrightarrow> (\<forall> th'. n = (Th th') \<longrightarrow> (Th th, Th th') \<in> (child s)^+)" (is "?P n")
+proof -
+  from wf_trancl[OF wf_depend[OF vt]]
+  have wf: "wf ((depend s)^+)" .
+  show ?thesis
+  proof(rule wf_induct[OF wf, of ?P], clarsimp)
+    fix th'
+    assume ih[rule_format]: "\<forall>y. (y, Th th') \<in> (depend s)\<^sup>+ \<longrightarrow>
+               (Th th, y) \<in> (depend s)\<^sup>+ \<longrightarrow> (\<forall>th'. y = Th th' \<longrightarrow> (Th th, Th th') \<in> (child s)\<^sup>+)"
+    and h: "(Th th, Th th') \<in> (depend s)\<^sup>+"
+    show "(Th th, Th th') \<in> (child s)\<^sup>+"
+    proof -
+      from depend_children[OF h]
+      have "th \<in> children s th' \<or> (\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (depend s)\<^sup>+)" .
+      thus ?thesis
+      proof
+        assume "th \<in> children s th'"
+        thus "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def)
+      next
+        assume "\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (depend s)\<^sup>+"
+        then obtain th3 where th3_in: "th3 \<in> children s th'" 
+          and th_dp: "(Th th, Th th3) \<in> (depend s)\<^sup>+" by auto
+        from th3_in have "(Th th3, Th th') \<in> (depend s)^+" by (auto simp:children_def child_def)
+        from ih[OF this th_dp, of th3] have "(Th th, Th th3) \<in> (child s)\<^sup>+" by simp
+        with th3_in show "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def)
+      qed
+    qed
+  qed
+qed
+
+lemma depend_child: "\<lbrakk>vt step s; (Th th, Th th') \<in> (depend s)^+\<rbrakk> \<Longrightarrow> (Th th, Th th') \<in> (child s)^+"
+  by (insert depend_child_pre, auto)
+
+lemma child_depend_p:
+  assumes "(n1, n2) \<in> (child s)^+"
+  shows "(n1, n2) \<in> (depend s)^+"
+proof -
+  from assms show ?thesis
+  proof(induct)
+    case (base y)
+    with sub_child show ?case by auto
+  next
+    case (step y z)
+    assume "(y, z) \<in> child s"
+    with sub_child have "(y, z) \<in> (depend s)^+" by auto
+    moreover have "(n1, y) \<in> (depend s)^+" by fact
+    ultimately show ?case by auto
+  qed
+qed
+
+lemma child_depend_eq: 
+  assumes vt: "vt step s"
+  shows 
+  "((Th th1, Th th2) \<in> (child s)^+) = 
+   ((Th th1, Th th2) \<in> (depend s)^+)"
+  by (auto intro: depend_child[OF vt] child_depend_p)
+
+lemma children_no_dep:
+  fixes s th th1 th2 th3
+  assumes vt: "vt step s"
+  and ch1: "(Th th1, Th th) \<in> child s"
+  and ch2: "(Th th2, Th th) \<in> child s"
+  and ch3: "(Th th1, Th th2) \<in> (depend s)^+"
+  shows "False"
+proof -
+  from depend_child[OF vt ch3]
+  have "(Th th1, Th th2) \<in> (child s)\<^sup>+" .
+  thus ?thesis
+  proof(rule converse_tranclE)
+    thm tranclD
+    assume "(Th th1, Th th2) \<in> child s"
+    from child_unique[OF vt ch1 this] have "th = th2" by simp
+    with ch2 have "(Th th2, Th th2) \<in> child s" by simp
+    with wf_child[OF vt] show ?thesis by auto
+  next
+    fix c
+    assume h1: "(Th th1, c) \<in> child s"
+      and h2: "(c, Th th2) \<in> (child s)\<^sup>+"
+    from h1 obtain th3 where eq_c: "c = Th th3" by (unfold child_def, auto)
+    with h1 have "(Th th1, Th th3) \<in> child s" by simp
+    from child_unique[OF vt ch1 this] have eq_th3: "th3 = th" by simp
+    with eq_c and h2 have "(Th th, Th th2) \<in> (child s)\<^sup>+" by simp
+    with ch2 have "(Th th, Th th) \<in> (child s)\<^sup>+" by auto
+    moreover have "wf ((child s)\<^sup>+)"
+    proof(rule wf_trancl)
+      from wf_child[OF vt] show "wf (child s)" .
+    qed
+    ultimately show False by auto
+  qed
+qed
+
+lemma unique_depend_p:
+  assumes vt: "vt step s"
+  and dp1: "(n, n1) \<in> (depend s)^+"
+  and dp2: "(n, n2) \<in> (depend s)^+"
+  and neq: "n1 \<noteq> n2"
+  shows "(n1, n2) \<in> (depend s)^+ \<or> (n2, n1) \<in> (depend s)^+"
+proof(rule unique_chain [OF _ dp1 dp2 neq])
+  from unique_depend[OF vt]
+  show "\<And>a b c. \<lbrakk>(a, b) \<in> depend s; (a, c) \<in> depend s\<rbrakk> \<Longrightarrow> b = c" by auto
+qed
+
+lemma dependents_child_unique:
+  fixes s th th1 th2 th3
+  assumes vt: "vt step s"
+  and ch1: "(Th th1, Th th) \<in> child s"
+  and ch2: "(Th th2, Th th) \<in> child s"
+  and dp1: "th3 \<in> dependents s th1"
+  and dp2: "th3 \<in> dependents s th2"
+shows "th1 = th2"
+proof -
+  { assume neq: "th1 \<noteq> th2"
+    from dp1 have dp1: "(Th th3, Th th1) \<in> (depend s)^+" 
+      by (simp add:s_dependents_def eq_depend)
+    from dp2 have dp2: "(Th th3, Th th2) \<in> (depend s)^+" 
+      by (simp add:s_dependents_def eq_depend)
+    from unique_depend_p[OF vt dp1 dp2] and neq
+    have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
+    hence False
+    proof
+      assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+ "
+      from children_no_dep[OF vt ch1 ch2 this] show ?thesis .
+    next
+      assume " (Th th2, Th th1) \<in> (depend s)\<^sup>+"
+      from children_no_dep[OF vt ch2 ch1 this] show ?thesis .
+    qed
+  } thus ?thesis by auto
+qed
+
+lemma cp_rec:
+  fixes s th
+  assumes vt: "vt step s"
+  shows "cp s th = Max ({preced th s} \<union> (cp s ` children s th))"
+proof(unfold cp_eq_cpreced_f cpreced_def)
+  let ?f = "(\<lambda>th. preced th s)"
+  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) =
+        Max ({preced th s} \<union> (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th)"
+  proof(cases " children s th = {}")
+    case False
+    have "(\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th = 
+          {Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) | th' . th' \<in> children s th}"
+      (is "?L = ?R")
+      by auto
+    also have "\<dots> = 
+      Max ` {((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) | th' . th' \<in> children s th}"
+      (is "_ = Max ` ?C")
+      by auto
+    finally have "Max ?L = Max (Max ` ?C)" by auto
+    also have "\<dots> = Max (\<Union> ?C)"
+    proof(rule Max_Union[symmetric])
+      from children_dependents[of s th] finite_threads[OF vt] and dependents_threads[OF vt, of th]
+      show "finite {(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+        by (auto simp:finite_subset)
+    next
+      from False
+      show "{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th} \<noteq> {}"
+        by simp
+    next
+      show "\<And>A. A \<in> {(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th} \<Longrightarrow>
+        finite A \<and> A \<noteq> {}"
+        apply (auto simp:finite_subset)
+      proof -
+        fix th'
+        from finite_threads[OF vt] and dependents_threads[OF vt, of th']
+        show "finite ((\<lambda>th. preced th s) ` dependents (wq s) th')" by (auto simp:finite_subset)
+      qed
+    qed
+    also have "\<dots> = Max ((\<lambda>th. preced th s) ` dependents (wq s) th)"
+      (is "Max ?A = Max ?B")
+    proof -
+      have "?A = ?B"
+      proof
+        show "\<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}
+                    \<subseteq> (\<lambda>th. preced th s) ` dependents (wq s) th"
+        proof
+          fix x 
+          assume "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+          then obtain th' where 
+             th'_in: "th' \<in> children s th"
+            and x_in: "x \<in> ?f ` ({th'} \<union> dependents (wq s) th')" by auto
+          hence "x = ?f th' \<or> x \<in> (?f ` dependents (wq s) th')" by auto
+          thus "x \<in> ?f ` dependents (wq s) th"
+          proof
+            assume "x = preced th' s"
+            with th'_in and children_dependents
+            show "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th" by auto
+          next
+            assume "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th'"
+            moreover note th'_in
+            ultimately show " x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th"
+              by (unfold cs_dependents_def children_def child_def, auto simp:eq_depend)
+          qed
+        qed
+      next
+        show "?f ` dependents (wq s) th
+           \<subseteq> \<Union>{?f ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+        proof
+          fix x 
+          assume x_in: "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th"
+          then obtain th' where
+            eq_x: "x = ?f th'" and dp: "(Th th', Th th) \<in> (depend s)^+" 
+            by (auto simp:cs_dependents_def eq_depend)
+          from depend_children[OF dp]
+          have "th' \<in> children s th \<or> (\<exists>th3. th3 \<in> children s th \<and> (Th th', Th th3) \<in> (depend s)\<^sup>+)" .
+          thus "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+          proof
+            assume "th' \<in> children s th"
+            with eq_x
+            show "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+              by auto
+          next
+            assume "\<exists>th3. th3 \<in> children s th \<and> (Th th', Th th3) \<in> (depend s)\<^sup>+"
+            then obtain th3 where th3_in: "th3 \<in> children s th"
+              and dp3: "(Th th', Th th3) \<in> (depend s)\<^sup>+" by auto
+            show "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+            proof -
+              from dp3
+              have "th' \<in> dependents (wq s) th3"
+                by (auto simp:cs_dependents_def eq_depend)
+              with eq_x th3_in show ?thesis by auto
+            qed
+          qed          
+        qed
+      qed
+      thus ?thesis by simp
+    qed
+    finally have "Max ((\<lambda>th. preced th s) ` dependents (wq s) th) = Max (?L)" 
+      (is "?X = ?Y") by auto
+    moreover have "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) = 
+                   max (?f th) ?X" 
+    proof -
+      have "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) = 
+            Max ({?f th} \<union> ?f ` (dependents (wq s) th))" by simp
+      also have "\<dots> = max (Max {?f th}) (Max (?f ` (dependents (wq s) th)))"
+      proof(rule Max_Un, auto)
+        from finite_threads[OF vt] and dependents_threads[OF vt, of th]
+        show "finite ((\<lambda>th. preced th s) ` dependents (wq s) th)" by (auto simp:finite_subset)
+      next
+        assume "dependents (wq s) th = {}"
+        with False and children_dependents show False by auto
+      qed
+      also have "\<dots> = max (?f th) ?X" by simp
+      finally show ?thesis .
+    qed
+    moreover have "Max ({preced th s} \<union> 
+                     (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th) = 
+                   max (?f th) ?Y"
+    proof -
+      have "Max ({preced th s} \<union> 
+                     (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th) = 
+            max (Max {preced th s}) ?Y"
+      proof(rule Max_Un, auto)
+        from finite_threads[OF vt] dependents_threads[OF vt, of th] children_dependents [of s th]
+        show "finite ((\<lambda>th. Max (insert (preced th s) ((\<lambda>th. preced th s) ` dependents (wq s) th))) ` 
+                       children s th)" 
+          by (auto simp:finite_subset)
+      next
+        assume "children s th = {}"
+        with False show False by auto
+      qed
+      thus ?thesis by simp
+    qed
+    ultimately show ?thesis by auto
+  next
+    case True
+    moreover have "dependents (wq s) th = {}"
+    proof -
+      { fix th'
+        assume "th' \<in> dependents (wq s) th"
+        hence " (Th th', Th th) \<in> (depend s)\<^sup>+" by (simp add:cs_dependents_def eq_depend)
+        from depend_children[OF this] and True
+        have "False" by auto
+      } thus ?thesis by auto
+    qed
+    ultimately show ?thesis by auto
+  qed
+qed
+
+definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+
+locale step_set_cps =
+  fixes s' th prio s 
+  defines s_def : "s \<equiv> (Set th prio#s')"
+  assumes vt_s: "vt step s"
+
+context step_set_cps 
+begin
+
+lemma eq_preced:
+  fixes th'
+  assumes "th' \<noteq> th"
+  shows "preced th' s = preced th' s'"
+proof -
+  from assms show ?thesis 
+    by (unfold s_def, auto simp:preced_def)
+qed
+
+lemma eq_dep: "depend s = depend s'"
+  by (unfold s_def depend_set_unchanged, auto)
+
+lemma eq_cp:
+  fixes th' 
+  assumes neq_th: "th' \<noteq> th"
+  and nd: "th \<notin> dependents s th'"
+  shows "cp s th' = cp s' th'"
+  apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+  have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
+    by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
+  moreover {
+    fix th1 
+    assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
+    hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
+    hence "preced th1 s = preced th1 s'"
+    proof
+      assume "th1 = th'"
+      with eq_preced[OF neq_th]
+      show "preced th1 s = preced th1 s'" by simp
+    next
+      assume "th1 \<in> dependents (wq s') th'"
+      with nd and eq_dp have "th1 \<noteq> th"
+        by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
+      from eq_preced[OF this] show "preced th1 s = preced th1 s'" by simp
+    qed
+  } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) = 
+                     ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" 
+    by (auto simp:image_def)
+  thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+        Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+lemma eq_up:
+  fixes th' th''
+  assumes dp1: "th \<in> dependents s th'"
+  and dp2: "th' \<in> dependents s th''"
+  and eq_cps: "cp s th' = cp s' th'"
+  shows "cp s th'' = cp s' th''"
+proof -
+  from dp2
+  have "(Th th', Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
+  from depend_child[OF vt_s this[unfolded eq_depend]]
+  have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" .
+  moreover { fix n th''
+    have "\<lbrakk>(Th th', n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
+                   (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
+    proof(erule trancl_induct, auto)
+      fix y th''
+      assume y_ch: "(y, Th th'') \<in> child s"
+        and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
+        and ch': "(Th th', y) \<in> (child s)\<^sup>+"
+      from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
+      with ih have eq_cpy:"cp s thy = cp s' thy" by blast
+      from dp1 have "(Th th, Th th') \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
+      moreover from child_depend_p[OF ch'] and eq_y
+      have "(Th th', Th thy) \<in> (depend s)^+" by simp
+      ultimately have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by auto
+      show "cp s th'' = cp s' th''"
+        apply (subst cp_rec[OF vt_s])
+      proof -
+        have "preced th'' s = preced th'' s'"
+        proof(rule eq_preced)
+          show "th'' \<noteq> th"
+          proof
+            assume "th'' = th"
+            with dp_thy y_ch[unfolded eq_y] 
+            have "(Th th, Th th) \<in> (depend s)^+"
+              by (auto simp:child_def)
+            with wf_trancl[OF wf_depend[OF vt_s]] 
+            show False by auto
+          qed
+        qed
+        moreover { 
+          fix th1
+          assume th1_in: "th1 \<in> children s th''"
+          have "cp s th1 = cp s' th1"
+          proof(cases "th1 = thy")
+            case True
+            with eq_cpy show ?thesis by simp
+          next
+            case False
+            have neq_th1: "th1 \<noteq> th"
+            proof
+              assume eq_th1: "th1 = th"
+              with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+              from children_no_dep[OF vt_s _ _ this] and 
+              th1_in y_ch eq_y show False by (auto simp:children_def)
+            qed
+            have "th \<notin> dependents s th1"
+            proof
+              assume h:"th \<in> dependents s th1"
+              from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
+              from dependents_child_unique[OF vt_s _ _ h this]
+              th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
+              with False show False by auto
+            qed
+            from eq_cp[OF neq_th1 this]
+            show ?thesis .
+          qed
+        }
+        ultimately have "{preced th'' s} \<union> (cp s ` children s th'') = 
+          {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+        moreover have "children s th'' = children s' th''"
+          by (unfold children_def child_def s_def depend_set_unchanged, simp)
+        ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+          by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+      qed
+    next
+      fix th''
+      assume dp': "(Th th', Th th'') \<in> child s"
+      show "cp s th'' = cp s' th''"
+        apply (subst cp_rec[OF vt_s])
+      proof -
+        have "preced th'' s = preced th'' s'"
+        proof(rule eq_preced)
+          show "th'' \<noteq> th"
+          proof
+            assume "th'' = th"
+            with dp1 dp'
+            have "(Th th, Th th) \<in> (depend s)^+"
+              by (auto simp:child_def s_dependents_def eq_depend)
+            with wf_trancl[OF wf_depend[OF vt_s]] 
+            show False by auto
+          qed
+        qed
+        moreover { 
+          fix th1
+          assume th1_in: "th1 \<in> children s th''"
+          have "cp s th1 = cp s' th1"
+          proof(cases "th1 = th'")
+            case True
+            with eq_cps show ?thesis by simp
+          next
+            case False
+            have neq_th1: "th1 \<noteq> th"
+            proof
+              assume eq_th1: "th1 = th"
+              with dp1 have "(Th th1, Th th') \<in> (depend s)^+" 
+                by (auto simp:s_dependents_def eq_depend)
+              from children_no_dep[OF vt_s _ _ this]
+              th1_in dp'
+              show False by (auto simp:children_def)
+            qed
+            thus ?thesis
+            proof(rule eq_cp)
+              show "th \<notin> dependents s th1"
+              proof
+                assume "th \<in> dependents s th1"
+                from dependents_child_unique[OF vt_s _ _ this dp1]
+                th1_in dp' have "th1 = th'"
+                  by (auto simp:children_def)
+                with False show False by auto
+              qed
+            qed
+          qed
+        }
+        ultimately have "{preced th'' s} \<union> (cp s ` children s th'') = 
+          {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+        moreover have "children s th'' = children s' th''"
+          by (unfold children_def child_def s_def depend_set_unchanged, simp)
+        ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+          by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+      qed     
+    qed
+  }
+  ultimately show ?thesis by auto
+qed
+
+lemma eq_up_self:
+  fixes th' th''
+  assumes dp: "th \<in> dependents s th''"
+  and eq_cps: "cp s th = cp s' th"
+  shows "cp s th'' = cp s' th''"
+proof -
+  from dp
+  have "(Th th, Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
+  from depend_child[OF vt_s this[unfolded eq_depend]]
+  have ch_th': "(Th th, Th th'') \<in> (child s)\<^sup>+" .
+  moreover { fix n th''
+    have "\<lbrakk>(Th th, n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
+                   (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
+    proof(erule trancl_induct, auto)
+      fix y th''
+      assume y_ch: "(y, Th th'') \<in> child s"
+        and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
+        and ch': "(Th th, y) \<in> (child s)\<^sup>+"
+      from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
+      with ih have eq_cpy:"cp s thy = cp s' thy" by blast
+      from child_depend_p[OF ch'] and eq_y
+      have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by simp
+      show "cp s th'' = cp s' th''"
+        apply (subst cp_rec[OF vt_s])
+      proof -
+        have "preced th'' s = preced th'' s'"
+        proof(rule eq_preced)
+          show "th'' \<noteq> th"
+          proof
+            assume "th'' = th"
+            with dp_thy y_ch[unfolded eq_y] 
+            have "(Th th, Th th) \<in> (depend s)^+"
+              by (auto simp:child_def)
+            with wf_trancl[OF wf_depend[OF vt_s]] 
+            show False by auto
+          qed
+        qed
+        moreover { 
+          fix th1
+          assume th1_in: "th1 \<in> children s th''"
+          have "cp s th1 = cp s' th1"
+          proof(cases "th1 = thy")
+            case True
+            with eq_cpy show ?thesis by simp
+          next
+            case False
+            have neq_th1: "th1 \<noteq> th"
+            proof
+              assume eq_th1: "th1 = th"
+              with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+              from children_no_dep[OF vt_s _ _ this] and 
+              th1_in y_ch eq_y show False by (auto simp:children_def)
+            qed
+            have "th \<notin> dependents s th1"
+            proof
+              assume h:"th \<in> dependents s th1"
+              from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
+              from dependents_child_unique[OF vt_s _ _ h this]
+              th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
+              with False show False by auto
+            qed
+            from eq_cp[OF neq_th1 this]
+            show ?thesis .
+          qed
+        }
+        ultimately have "{preced th'' s} \<union> (cp s ` children s th'') = 
+          {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+        moreover have "children s th'' = children s' th''"
+          by (unfold children_def child_def s_def depend_set_unchanged, simp)
+        ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+          by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+      qed
+    next
+      fix th''
+      assume dp': "(Th th, Th th'') \<in> child s"
+      show "cp s th'' = cp s' th''"
+        apply (subst cp_rec[OF vt_s])
+      proof -
+        have "preced th'' s = preced th'' s'"
+        proof(rule eq_preced)
+          show "th'' \<noteq> th"
+          proof
+            assume "th'' = th"
+            with dp dp'
+            have "(Th th, Th th) \<in> (depend s)^+"
+              by (auto simp:child_def s_dependents_def eq_depend)
+            with wf_trancl[OF wf_depend[OF vt_s]] 
+            show False by auto
+          qed
+        qed
+        moreover { 
+          fix th1
+          assume th1_in: "th1 \<in> children s th''"
+          have "cp s th1 = cp s' th1"
+          proof(cases "th1 = th")
+            case True
+            with eq_cps show ?thesis by simp
+          next
+            case False
+            assume neq_th1: "th1 \<noteq> th"
+            thus ?thesis
+            proof(rule eq_cp)
+              show "th \<notin> dependents s th1"
+              proof
+                assume "th \<in> dependents s th1"
+                hence "(Th th, Th th1) \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
+                from children_no_dep[OF vt_s _ _ this]
+                and th1_in dp' show False
+                  by (auto simp:children_def)
+              qed
+            qed
+          qed
+        }
+        ultimately have "{preced th'' s} \<union> (cp s ` children s th'') = 
+          {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+        moreover have "children s th'' = children s' th''"
+          by (unfold children_def child_def s_def depend_set_unchanged, simp)
+        ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+          by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+      qed     
+    qed
+  }
+  ultimately show ?thesis by auto
+qed
+end
+
+lemma next_waiting:
+  assumes vt: "vt step s"
+  and nxt: "next_th s th cs th'"
+  shows "waiting s th' cs"
+proof -
+  from assms show ?thesis
+    apply (auto simp:next_th_def s_waiting_def)
+  proof -
+    fix rest
+    assume ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+      and eq_wq: "wq s cs = th # rest"
+      and ne: "rest \<noteq> []"
+    have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
+    proof(rule someI2)
+      from wq_distinct[OF vt, of cs] 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
+    with 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 wq_distinct[OF vt, of cs] eq_wq
+      show "distinct rest \<and> set rest = set rest" by auto
+    next
+      from ne show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> x \<noteq> []" by auto
+    qed
+    ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
+  next
+    fix rest
+    assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+      and ne: "rest \<noteq> []"
+    have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+    proof(rule someI2)
+      from wq_distinct[OF vt, of cs] eq_wq
+      show "distinct rest \<and> set rest = set rest" by auto
+    next
+      from ne show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> 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 "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
+    proof(rule someI2)
+      from wq_distinct[OF vt, of cs] 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
+    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest" by simp
+    with eq_wq and wq_distinct[OF vt, of cs]
+    show False by auto
+  qed
+qed
+
+locale step_v_cps =
+  fixes s' th cs s 
+  defines s_def : "s \<equiv> (V th cs#s')"
+  assumes vt_s: "vt step s"
+
+locale step_v_cps_nt = step_v_cps +
+  fixes th'
+  assumes nt: "next_th s' th cs th'"
+
+context step_v_cps_nt
+begin
+
+lemma depend_s:
+  "depend s = (depend s' - {(Cs cs, Th th)} - {(Th th', Cs cs)}) \<union>
+                                         {(Cs cs, Th th')}"
+proof -
+  from step_depend_v[OF vt_s[unfolded s_def], folded s_def]
+    and nt show ?thesis  by (auto intro:next_th_unique)
+qed
+
+lemma dependents_kept:
+  fixes th''
+  assumes neq1: "th'' \<noteq> th"
+  and neq2: "th'' \<noteq> th'"
+  shows "dependents (wq s) th'' = dependents (wq s') th''"
+proof(auto)
+  fix x
+  assume "x \<in> dependents (wq s) th''"
+  hence dp: "(Th x, Th th'') \<in> (depend s)^+"
+    by (auto simp:cs_dependents_def eq_depend)
+  { fix n
+    have "(n, Th th'') \<in> (depend s)^+ \<Longrightarrow>  (n, Th th'') \<in> (depend s')^+"
+    proof(induct rule:converse_trancl_induct)
+      fix y 
+      assume "(y, Th th'') \<in> depend s"
+      with depend_s neq1 neq2
+      have "(y, Th th'') \<in> depend s'" by auto
+      thus "(y, Th th'') \<in> (depend s')\<^sup>+" by auto
+    next
+      fix y z 
+      assume yz: "(y, z) \<in> depend s"
+        and ztp: "(z, Th th'') \<in> (depend s)\<^sup>+"
+        and ztp': "(z, Th th'') \<in> (depend s')\<^sup>+"
+      have "y \<noteq> Cs cs \<and> y \<noteq> Th th'"
+      proof
+        show "y \<noteq> Cs cs"
+        proof
+          assume eq_y: "y = Cs cs"
+          with yz have dp_yz: "(Cs cs, z) \<in> depend s" by simp
+          from depend_s
+          have cst': "(Cs cs, Th th') \<in> depend s" by simp
+          from unique_depend[OF vt_s this dp_yz] 
+          have eq_z: "z = Th th'" by simp
+          with ztp have "(Th th', Th th'') \<in> (depend s)^+" by simp
+          from converse_tranclE[OF this]
+          obtain cs' where dp'': "(Th th', Cs cs') \<in> depend s"
+            by (auto simp:s_depend_def)
+          with depend_s have dp': "(Th th', Cs cs') \<in> depend s'" by auto
+          from dp'' eq_y yz eq_z have "(Cs cs, Cs cs') \<in> (depend s)^+" by auto
+          moreover have "cs' = cs"
+          proof -
+            from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
+            have "(Th th', Cs cs) \<in> depend s'"
+              by (auto simp:s_waiting_def s_depend_def cs_waiting_def)
+            from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this dp']
+            show ?thesis by simp
+          qed
+          ultimately have "(Cs cs, Cs cs) \<in> (depend s)^+" by simp
+          moreover note wf_trancl[OF wf_depend[OF vt_s]]
+          ultimately show False by auto
+        qed
+      next
+        show "y \<noteq> Th th'"
+        proof
+          assume eq_y: "y = Th th'"
+          with yz have dps: "(Th th', z) \<in> depend s" by simp
+          with depend_s have dps': "(Th th', z) \<in> depend s'" by auto
+          have "z = Cs cs"
+          proof -
+            from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
+            have "(Th th', Cs cs) \<in> depend s'"
+              by (auto simp:s_waiting_def s_depend_def cs_waiting_def)
+            from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] dps' this]
+            show ?thesis .
+          qed
+          with dps depend_s show False by auto
+        qed
+      qed
+      with depend_s yz have "(y, z) \<in> depend s'" by auto
+      with ztp'
+      show "(y, Th th'') \<in> (depend s')\<^sup>+" by auto
+    qed    
+  }
+  from this[OF dp]
+  show "x \<in> dependents (wq s') th''" 
+    by (auto simp:cs_dependents_def eq_depend)
+next
+  fix x
+  assume "x \<in> dependents (wq s') th''"
+  hence dp: "(Th x, Th th'') \<in> (depend s')^+"
+    by (auto simp:cs_dependents_def eq_depend)
+  { fix n
+    have "(n, Th th'') \<in> (depend s')^+ \<Longrightarrow>  (n, Th th'') \<in> (depend s)^+"
+    proof(induct rule:converse_trancl_induct)
+      fix y 
+      assume "(y, Th th'') \<in> depend s'"
+      with depend_s neq1 neq2
+      have "(y, Th th'') \<in> depend s" by auto
+      thus "(y, Th th'') \<in> (depend s)\<^sup>+" by auto
+    next
+      fix y z 
+      assume yz: "(y, z) \<in> depend s'"
+        and ztp: "(z, Th th'') \<in> (depend s')\<^sup>+"
+        and ztp': "(z, Th th'') \<in> (depend s)\<^sup>+"
+      have "y \<noteq> Cs cs \<and> y \<noteq> Th th'"
+      proof
+        show "y \<noteq> Cs cs"
+        proof
+          assume eq_y: "y = Cs cs"
+          with yz have dp_yz: "(Cs cs, z) \<in> depend s'" by simp
+          from this have eq_z: "z = Th th"
+          proof -
+            from step_back_step[OF vt_s[unfolded s_def]]
+            have "(Cs cs, Th th) \<in> depend s'"
+              by(cases, auto simp: s_depend_def cs_holding_def s_holding_def)
+            from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this dp_yz]
+            show ?thesis by simp
+          qed
+          from converse_tranclE[OF ztp]
+          obtain u where "(z, u) \<in> depend s'" by auto
+          moreover 
+          from step_back_step[OF vt_s[unfolded s_def]]
+          have "th \<in> readys s'" by (cases, simp add:runing_def)
+          moreover note eq_z
+          ultimately show False 
+            by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
+        qed
+      next
+        show "y \<noteq> Th th'"
+        proof
+          assume eq_y: "y = Th th'"
+          with yz have dps: "(Th th', z) \<in> depend s'" by simp
+          have "z = Cs cs"
+          proof -
+            from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
+            have "(Th th', Cs cs) \<in> depend s'"
+              by (auto simp:s_waiting_def s_depend_def cs_waiting_def)
+            from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] dps this]
+            show ?thesis .
+          qed
+          with ztp have cs_i: "(Cs cs, Th th'') \<in>  (depend s')\<^sup>+" by simp
+          from step_back_step[OF vt_s[unfolded s_def]]
+          have cs_th: "(Cs cs, Th th) \<in> depend s'"
+            by(cases, auto simp: s_depend_def cs_holding_def s_holding_def)
+          have "(Cs cs, Th th'') \<notin>  depend s'"
+          proof
+            assume "(Cs cs, Th th'') \<in> depend s'"
+            from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this cs_th]
+            and neq1 show "False" by simp
+          qed
+          with converse_tranclE[OF cs_i]
+          obtain u where cu: "(Cs cs, u) \<in> depend s'"  
+            and u_t: "(u, Th th'') \<in> (depend s')\<^sup>+" by auto
+          have "u = Th th"
+          proof -
+            from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] cu cs_th]
+            show ?thesis .
+          qed
+          with u_t have "(Th th, Th th'') \<in> (depend s')\<^sup>+" by simp
+          from converse_tranclE[OF this]
+          obtain v where "(Th th, v) \<in> (depend s')" by auto
+          moreover from step_back_step[OF vt_s[unfolded s_def]]
+          have "th \<in> readys s'" by (cases, simp add:runing_def)
+          ultimately show False 
+            by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
+        qed
+      qed
+      with depend_s yz have "(y, z) \<in> depend s" by auto
+      with ztp'
+      show "(y, Th th'') \<in> (depend s)\<^sup>+" by auto
+    qed    
+  }
+  from this[OF dp]
+  show "x \<in> dependents (wq s) th''"
+    by (auto simp:cs_dependents_def eq_depend)
+qed
+
+lemma cp_kept:
+  fixes th''
+  assumes neq1: "th'' \<noteq> th"
+  and neq2: "th'' \<noteq> th'"
+  shows "cp s th'' = cp s' th''"
+proof -
+  from dependents_kept[OF neq1 neq2]
+  have "dependents (wq s) th'' = dependents (wq s') th''" .
+  moreover {
+    fix th1
+    assume "th1 \<in> dependents (wq s) th''"
+    have "preced th1 s = preced th1 s'" 
+      by (unfold s_def, auto simp:preced_def)
+  }
+  moreover have "preced th'' s = preced th'' s'" 
+    by (unfold s_def, auto simp:preced_def)
+  ultimately have "((\<lambda>th. preced th s) ` ({th''} \<union> dependents (wq s) th'')) = 
+    ((\<lambda>th. preced th s') ` ({th''} \<union> dependents (wq s') th''))"
+    by (auto simp:image_def)
+  thus ?thesis
+    by (unfold cp_eq_cpreced cpreced_def, simp)
+qed
+
+end
+
+locale step_v_cps_nnt = step_v_cps +
+  assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
+
+context step_v_cps_nnt
+begin
+
+lemma nw_cs: "(Th th1, Cs cs) \<notin> depend s'"
+proof
+  assume "(Th th1, Cs cs) \<in> depend s'"
+  thus "False"
+    apply (auto simp:s_depend_def cs_waiting_def)
+  proof -
+    assume h1: "th1 \<in> set (wq s' cs)"
+      and h2: "th1 \<noteq> hd (wq s' cs)"
+    from step_back_step[OF vt_s[unfolded s_def]]
+    show "False"
+    proof(cases)
+      assume "holding s' th cs" 
+      then obtain rest where
+        eq_wq: "wq s' cs = th#rest"
+        apply (unfold s_holding_def)
+        by (case_tac "(wq s' cs)", auto)
+      with h1 h2 have ne: "rest \<noteq> []" by auto
+      with eq_wq
+      have "next_th s' th cs (hd (SOME q. distinct q \<and> set q = set rest))"
+        by(unfold next_th_def, rule_tac x = "rest" in exI, auto)
+      with nnt show ?thesis by auto
+    qed
+  qed
+qed
+
+lemma depend_s: "depend s = depend s' - {(Cs cs, Th th)}"
+proof -
+  from nnt and  step_depend_v[OF vt_s[unfolded s_def], folded s_def]
+  show ?thesis by auto
+qed
+
+lemma child_kept_left:
+  assumes 
+  "(n1, n2) \<in> (child s')^+"
+  shows "(n1, n2) \<in> (child s)^+"
+proof -
+  from assms show ?thesis 
+  proof(induct rule: converse_trancl_induct)
+    case (base y)
+    from base obtain th1 cs1 th2
+      where h1: "(Th th1, Cs cs1) \<in> depend s'"
+      and h2: "(Cs cs1, Th th2) \<in> depend s'"
+      and eq_y: "y = Th th1" and eq_n2: "n2 = Th th2"  by (auto simp:child_def)
+    have "cs1 \<noteq> cs"
+    proof
+      assume eq_cs: "cs1 = cs"
+      with h1 have "(Th th1, Cs cs1) \<in> depend s'" by simp
+      with nw_cs eq_cs show False by auto
+    qed
+    with h1 h2 depend_s have 
+      h1': "(Th th1, Cs cs1) \<in> depend s" and
+      h2': "(Cs cs1, Th th2) \<in> depend s" by auto
+    hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
+    with eq_y eq_n2 have "(y, n2) \<in> child s" by simp
+    thus ?case by auto
+  next
+    case (step y z)
+    have "(y, z) \<in> child s'" by fact
+    then obtain th1 cs1 th2
+      where h1: "(Th th1, Cs cs1) \<in> depend s'"
+      and h2: "(Cs cs1, Th th2) \<in> depend s'"
+      and eq_y: "y = Th th1" and eq_z: "z = Th th2"  by (auto simp:child_def)
+    have "cs1 \<noteq> cs"
+    proof
+      assume eq_cs: "cs1 = cs"
+      with h1 have "(Th th1, Cs cs1) \<in> depend s'" by simp
+      with nw_cs eq_cs show False by auto
+    qed
+    with h1 h2 depend_s have 
+      h1': "(Th th1, Cs cs1) \<in> depend s" and
+      h2': "(Cs cs1, Th th2) \<in> depend s" by auto
+    hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
+    with eq_y eq_z have "(y, z) \<in> child s" by simp
+    moreover have "(z, n2) \<in> (child s)^+" by fact
+    ultimately show ?case by auto
+  qed
+qed
+
+lemma  child_kept_right:
+  assumes
+  "(n1, n2) \<in> (child s)^+"
+  shows "(n1, n2) \<in> (child s')^+"
+proof -
+  from assms show ?thesis
+  proof(induct)
+    case (base y)
+    from base and depend_s 
+    have "(n1, y) \<in> child s'"
+      by (auto simp:child_def)
+    thus ?case by auto
+  next
+    case (step y z)
+    have "(y, z) \<in> child s" by fact
+    with depend_s have "(y, z) \<in> child s'"
+      by (auto simp:child_def)
+    moreover have "(n1, y) \<in> (child s')\<^sup>+" by fact
+    ultimately show ?case by auto
+  qed
+qed
+
+lemma eq_child: "(child s)^+ = (child s')^+"
+  by (insert child_kept_left child_kept_right, auto)
+
+lemma eq_cp:
+  fixes th' 
+  shows "cp s th' = cp s' th'"
+  apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+  have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
+    apply (unfold cs_dependents_def, unfold eq_depend)
+  proof -
+    from eq_child
+    have "\<And>th. {th'. (Th th', Th th) \<in> (child s)\<^sup>+} = {th'. (Th th', Th th) \<in> (child s')\<^sup>+}"
+      by simp
+    with child_depend_eq[OF vt_s] child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+    show "\<And>th. {th'. (Th th', Th th) \<in> (depend s)\<^sup>+} = {th'. (Th th', Th th) \<in> (depend s')\<^sup>+}"
+      by simp
+  qed
+  moreover {
+    fix th1 
+    assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
+    hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
+    hence "preced th1 s = preced th1 s'"
+    proof
+      assume "th1 = th'"
+      show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
+    next
+      assume "th1 \<in> dependents (wq s') th'"
+      show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
+    qed
+  } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) = 
+                     ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" 
+    by (auto simp:image_def)
+  thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+        Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+end
+
+locale step_P_cps =
+  fixes s' th cs s 
+  defines s_def : "s \<equiv> (P th cs#s')"
+  assumes vt_s: "vt step s"
+
+locale step_P_cps_ne =step_P_cps +
+  assumes ne: "wq s' cs \<noteq> []"
+
+context step_P_cps_ne
+begin
+
+lemma depend_s: "depend s = depend s' \<union> {(Th th, Cs cs)}"
+proof -
+  from step_depend_p[OF vt_s[unfolded s_def]] and ne
+  show ?thesis by (simp add:s_def)
+qed
+
+lemma eq_child_left:
+  assumes nd: "(Th th, Th th') \<notin> (child s)^+"
+  shows "(n1, Th th') \<in> (child s)^+ \<Longrightarrow> (n1, Th th') \<in> (child s')^+"
+proof(induct rule:converse_trancl_induct)
+  case (base y)
+  from base obtain th1 cs1
+    where h1: "(Th th1, Cs cs1) \<in> depend s"
+    and h2: "(Cs cs1, Th th') \<in> depend s"
+    and eq_y: "y = Th th1"   by (auto simp:child_def)
+  have "th1 \<noteq> th"
+  proof
+    assume "th1 = th"
+    with base eq_y have "(Th th, Th th') \<in> child s" by simp
+    with nd show False by auto
+  qed
+  with h1 h2 depend_s 
+  have h1': "(Th th1, Cs cs1) \<in> depend s'" and 
+       h2': "(Cs cs1, Th th') \<in> depend s'" by auto
+  with eq_y show ?case by (auto simp:child_def)
+next
+  case (step y z)
+  have yz: "(y, z) \<in> child s" by fact
+  then obtain th1 cs1 th2
+    where h1: "(Th th1, Cs cs1) \<in> depend s"
+    and h2: "(Cs cs1, Th th2) \<in> depend s"
+    and eq_y: "y = Th th1" and eq_z: "z = Th th2"  by (auto simp:child_def)
+  have "th1 \<noteq> th"
+  proof
+    assume "th1 = th"
+    with yz eq_y have "(Th th, z) \<in> child s" by simp
+    moreover have "(z, Th th') \<in> (child s)^+" by fact
+    ultimately have "(Th th, Th th') \<in> (child s)^+" by auto
+    with nd show False by auto
+  qed
+  with h1 h2 depend_s have h1': "(Th th1, Cs cs1) \<in> depend s'"
+                       and h2': "(Cs cs1, Th th2) \<in> depend s'" by auto
+  with eq_y eq_z have "(y, z) \<in> child s'" by (auto simp:child_def)
+  moreover have "(z, Th th') \<in> (child s')^+" by fact
+  ultimately show ?case by auto
+qed
+
+lemma eq_child_right:
+  shows "(n1, Th th') \<in> (child s')^+ \<Longrightarrow> (n1, Th th') \<in> (child s)^+"
+proof(induct rule:converse_trancl_induct)
+  case (base y)
+  with depend_s show ?case by (auto simp:child_def)
+next
+  case (step y z)
+  have "(y, z) \<in> child s'" by fact
+  with depend_s have "(y, z) \<in> child s" by (auto simp:child_def)
+  moreover have "(z, Th th') \<in> (child s)^+" by fact
+  ultimately show ?case by auto
+qed
+
+lemma eq_child:
+  assumes nd: "(Th th, Th th') \<notin> (child s)^+"
+  shows "((n1, Th th') \<in> (child s)^+) = ((n1, Th th') \<in> (child s')^+)"
+  by (insert eq_child_left[OF nd] eq_child_right, auto)
+
+lemma eq_cp:
+  fixes th' 
+  assumes nd: "th \<notin> dependents s th'"
+  shows "cp s th' = cp s' th'"
+  apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+  have nd': "(Th th, Th th') \<notin> (child s)^+"
+  proof
+    assume "(Th th, Th th') \<in> (child s)\<^sup>+"
+    with child_depend_eq[OF vt_s]
+    have "(Th th, Th th') \<in> (depend s)\<^sup>+" by simp
+    with nd show False 
+      by (simp add:s_dependents_def eq_depend)
+  qed
+  have eq_dp: "dependents (wq s) th' = dependents (wq s') th'"
+  proof(auto)
+    fix x assume " x \<in> dependents (wq s) th'"
+    thus "x \<in> dependents (wq s') th'"
+      apply (auto simp:cs_dependents_def eq_depend)
+    proof -
+      assume "(Th x, Th th') \<in> (depend s)\<^sup>+"
+      with  child_depend_eq[OF vt_s] have "(Th x, Th th') \<in> (child s)\<^sup>+" by simp
+      with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s')^+" by simp
+      with child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+      show "(Th x, Th th') \<in> (depend s')\<^sup>+" by simp
+    qed
+  next
+    fix x assume "x \<in> dependents (wq s') th'"
+    thus "x \<in> dependents (wq s) th'"
+      apply (auto simp:cs_dependents_def eq_depend)
+    proof -
+      assume "(Th x, Th th') \<in> (depend s')\<^sup>+"
+      with child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]] 
+      have "(Th x, Th th') \<in> (child s')\<^sup>+" by simp
+      with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s)^+" by simp
+      with  child_depend_eq[OF vt_s]
+      show "(Th x, Th th') \<in> (depend s)\<^sup>+" by simp
+    qed
+  qed
+  moreover {
+    fix th1 have "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
+  } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) = 
+                     ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" 
+    by (auto simp:image_def)
+  thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+        Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+lemma eq_up:
+  fixes th' th''
+  assumes dp1: "th \<in> dependents s th'"
+  and dp2: "th' \<in> dependents s th''"
+  and eq_cps: "cp s th' = cp s' th'"
+  shows "cp s th'' = cp s' th''"
+proof -
+  from dp2
+  have "(Th th', Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
+  from depend_child[OF vt_s this[unfolded eq_depend]]
+  have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" .
+  moreover {
+    fix n th''
+    have "\<lbrakk>(Th th', n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
+                   (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
+    proof(erule trancl_induct, auto)
+      fix y th''
+      assume y_ch: "(y, Th th'') \<in> child s"
+        and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
+        and ch': "(Th th', y) \<in> (child s)\<^sup>+"
+      from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
+      with ih have eq_cpy:"cp s thy = cp s' thy" by blast
+      from dp1 have "(Th th, Th th') \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
+      moreover from child_depend_p[OF ch'] and eq_y
+      have "(Th th', Th thy) \<in> (depend s)^+" by simp
+      ultimately have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by auto
+      show "cp s th'' = cp s' th''"
+        apply (subst cp_rec[OF vt_s])
+      proof -
+        have "preced th'' s = preced th'' s'"
+          by (simp add:s_def preced_def)
+        moreover { 
+          fix th1
+          assume th1_in: "th1 \<in> children s th''"
+          have "cp s th1 = cp s' th1"
+          proof(cases "th1 = thy")
+            case True
+            with eq_cpy show ?thesis by simp
+          next
+            case False
+            have neq_th1: "th1 \<noteq> th"
+            proof
+              assume eq_th1: "th1 = th"
+              with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+              from children_no_dep[OF vt_s _ _ this] and 
+              th1_in y_ch eq_y show False by (auto simp:children_def)
+            qed
+            have "th \<notin> dependents s th1"
+            proof
+              assume h:"th \<in> dependents s th1"
+              from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
+              from dependents_child_unique[OF vt_s _ _ h this]
+              th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
+              with False show False by auto
+            qed
+            from eq_cp[OF this]
+            show ?thesis .
+          qed
+        }
+        ultimately have "{preced th'' s} \<union> (cp s ` children s th'') = 
+          {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+        moreover have "children s th'' = children s' th''"
+          apply (unfold children_def child_def s_def depend_set_unchanged, simp)
+          apply (fold s_def, auto simp:depend_s)
+          proof -
+            assume "(Cs cs, Th th'') \<in> depend s'"
+            with depend_s have cs_th': "(Cs cs, Th th'') \<in> depend s" by auto
+            from dp1 have "(Th th, Th th') \<in> (depend s)^+"
+              by (auto simp:s_dependents_def eq_depend)
+            from converse_tranclE[OF this]
+            obtain cs1 where h1: "(Th th, Cs cs1) \<in> depend s"
+              and h2: "(Cs cs1 , Th th') \<in> (depend s)\<^sup>+"
+              by (auto simp:s_depend_def)
+            have eq_cs: "cs1 = cs" 
+            proof -
+              from depend_s have "(Th th, Cs cs) \<in> depend s" by simp
+              from unique_depend[OF vt_s this h1]
+              show ?thesis by simp
+            qed
+            have False
+            proof(rule converse_tranclE[OF h2])
+              assume "(Cs cs1, Th th') \<in> depend s"
+              with eq_cs have "(Cs cs, Th th') \<in> depend s" by simp
+              from unique_depend[OF vt_s this cs_th']
+              have "th' = th''" by simp
+              with ch' y_ch have "(Th th'', Th th'') \<in> (child s)^+" by auto
+              with wf_trancl[OF wf_child[OF vt_s]] 
+              show False by auto
+            next
+              fix y
+              assume "(Cs cs1, y) \<in> depend s"
+                and ytd: " (y, Th th') \<in> (depend s)\<^sup>+"
+              with eq_cs have csy: "(Cs cs, y) \<in> depend s" by simp
+              from unique_depend[OF vt_s this cs_th']
+              have "y = Th th''" .
+              with ytd have "(Th th'', Th th') \<in> (depend s)^+" by simp
+              from depend_child[OF vt_s this]
+              have "(Th th'', Th th') \<in> (child s)\<^sup>+" .
+              moreover from ch' y_ch have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto
+              ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto 
+              with wf_trancl[OF wf_child[OF vt_s]] 
+              show False by auto
+            qed
+            thus "\<exists>cs. (Th th, Cs cs) \<in> depend s' \<and> (Cs cs, Th th'') \<in> depend s'" by auto
+          qed
+          ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+          by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+      qed
+    next
+      fix th''
+      assume dp': "(Th th', Th th'') \<in> child s"
+      show "cp s th'' = cp s' th''"
+        apply (subst cp_rec[OF vt_s])
+      proof -
+        have "preced th'' s = preced th'' s'"
+          by (simp add:s_def preced_def)
+        moreover { 
+          fix th1
+          assume th1_in: "th1 \<in> children s th''"
+          have "cp s th1 = cp s' th1"
+          proof(cases "th1 = th'")
+            case True
+            with eq_cps show ?thesis by simp
+          next
+            case False
+            have neq_th1: "th1 \<noteq> th"
+            proof
+              assume eq_th1: "th1 = th"
+              with dp1 have "(Th th1, Th th') \<in> (depend s)^+" 
+                by (auto simp:s_dependents_def eq_depend)
+              from children_no_dep[OF vt_s _ _ this]
+              th1_in dp'
+              show False by (auto simp:children_def)
+            qed
+            show ?thesis
+            proof(rule eq_cp)
+              show "th \<notin> dependents s th1"
+              proof
+                assume "th \<in> dependents s th1"
+                from dependents_child_unique[OF vt_s _ _ this dp1]
+                th1_in dp' have "th1 = th'"
+                  by (auto simp:children_def)
+                with False show False by auto
+              qed
+            qed
+          qed
+        }
+        ultimately have "{preced th'' s} \<union> (cp s ` children s th'') = 
+          {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+        moreover have "children s th'' = children s' th''"
+          apply (unfold children_def child_def s_def depend_set_unchanged, simp)
+          apply (fold s_def, auto simp:depend_s)
+          proof -
+            assume "(Cs cs, Th th'') \<in> depend s'"
+            with depend_s have cs_th': "(Cs cs, Th th'') \<in> depend s" by auto
+            from dp1 have "(Th th, Th th') \<in> (depend s)^+"
+              by (auto simp:s_dependents_def eq_depend)
+            from converse_tranclE[OF this]
+            obtain cs1 where h1: "(Th th, Cs cs1) \<in> depend s"
+              and h2: "(Cs cs1 , Th th') \<in> (depend s)\<^sup>+"
+              by (auto simp:s_depend_def)
+            have eq_cs: "cs1 = cs" 
+            proof -
+              from depend_s have "(Th th, Cs cs) \<in> depend s" by simp
+              from unique_depend[OF vt_s this h1]
+              show ?thesis by simp
+            qed
+            have False
+            proof(rule converse_tranclE[OF h2])
+              assume "(Cs cs1, Th th') \<in> depend s"
+              with eq_cs have "(Cs cs, Th th') \<in> depend s" by simp
+              from unique_depend[OF vt_s this cs_th']
+              have "th' = th''" by simp
+              with dp' have "(Th th'', Th th'') \<in> (child s)^+" by auto
+              with wf_trancl[OF wf_child[OF vt_s]] 
+              show False by auto
+            next
+              fix y
+              assume "(Cs cs1, y) \<in> depend s"
+                and ytd: " (y, Th th') \<in> (depend s)\<^sup>+"
+              with eq_cs have csy: "(Cs cs, y) \<in> depend s" by simp
+              from unique_depend[OF vt_s this cs_th']
+              have "y = Th th''" .
+              with ytd have "(Th th'', Th th') \<in> (depend s)^+" by simp
+              from depend_child[OF vt_s this]
+              have "(Th th'', Th th') \<in> (child s)\<^sup>+" .
+              moreover from dp' have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto
+              ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto 
+              with wf_trancl[OF wf_child[OF vt_s]] 
+              show False by auto
+            qed
+            thus "\<exists>cs. (Th th, Cs cs) \<in> depend s' \<and> (Cs cs, Th th'') \<in> depend s'" by auto
+          qed
+        ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+          by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+      qed     
+    qed
+  }
+  ultimately show ?thesis by auto
+qed
+
+end
+
+locale step_create_cps =
+  fixes s' th prio s 
+  defines s_def : "s \<equiv> (Create th prio#s')"
+  assumes vt_s: "vt step s"
+
+context step_create_cps
+begin
+
+lemma eq_dep: "depend s = depend s'"
+  by (unfold s_def depend_create_unchanged, auto)
+
+lemma eq_cp:
+  fixes th' 
+  assumes neq_th: "th' \<noteq> th"
+  shows "cp s th' = cp s' th'"
+  apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+  have nd: "th \<notin> dependents s th'"
+  proof
+    assume "th \<in> dependents s th'"
+    hence "(Th th, Th th') \<in> (depend s)^+" by (simp add:s_dependents_def eq_depend)
+    with eq_dep have "(Th th, Th th') \<in> (depend s')^+" by simp
+    from converse_tranclE[OF this]
+    obtain y where "(Th th, y) \<in> depend s'" by auto
+    with dm_depend_threads[OF step_back_vt[OF vt_s[unfolded s_def]]]
+    have in_th: "th \<in> threads s'" by auto
+    from step_back_step[OF vt_s[unfolded s_def]]
+    show False
+    proof(cases)
+      assume "th \<notin> threads s'" 
+      with in_th show ?thesis by simp
+    qed
+  qed
+  have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
+    by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
+  moreover {
+    fix th1 
+    assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
+    hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
+    hence "preced th1 s = preced th1 s'"
+    proof
+      assume "th1 = th'"
+      with neq_th
+      show "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
+    next
+      assume "th1 \<in> dependents (wq s') th'"
+      with nd and eq_dp have "th1 \<noteq> th"
+        by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
+      thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
+    qed
+  } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) = 
+                     ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" 
+    by (auto simp:image_def)
+  thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+        Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+lemma nil_dependents: "dependents s th = {}"
+proof -
+  from step_back_step[OF vt_s[unfolded s_def]]
+  show ?thesis
+  proof(cases)
+    assume "th \<notin> threads s'"
+    from not_thread_holdents[OF step_back_vt[OF vt_s[unfolded s_def]] this]
+    have hdn: " holdents s' th = {}" .
+    have "dependents s' th = {}"
+    proof -
+      { assume "dependents s' th \<noteq> {}"
+        then obtain th' where dp: "(Th th', Th th) \<in> (depend s')^+"
+          by (auto simp:s_dependents_def eq_depend)
+        from tranclE[OF this] obtain cs' where 
+          "(Cs cs', Th th) \<in> depend s'" by (auto simp:s_depend_def)
+        with hdn
+        have False by (auto simp:holdents_def)
+      } thus ?thesis by auto
+    qed
+    thus ?thesis 
+      by (unfold s_def s_dependents_def eq_depend depend_create_unchanged, simp)
+  qed
+qed
+
+lemma eq_cp_th: "cp s th = preced th s"
+  apply (unfold cp_eq_cpreced cpreced_def)
+  by (insert nil_dependents, unfold s_dependents_def cs_dependents_def, auto)
+
+end
+
+
+locale step_exit_cps =
+  fixes s' th prio s 
+  defines s_def : "s \<equiv> (Exit th#s')"
+  assumes vt_s: "vt step s"
+
+context step_exit_cps
+begin
+
+lemma eq_dep: "depend s = depend s'"
+  by (unfold s_def depend_exit_unchanged, auto)
+
+lemma eq_cp:
+  fixes th' 
+  assumes neq_th: "th' \<noteq> th"
+  shows "cp s th' = cp s' th'"
+  apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+  have nd: "th \<notin> dependents s th'"
+  proof
+    assume "th \<in> dependents s th'"
+    hence "(Th th, Th th') \<in> (depend s)^+" by (simp add:s_dependents_def eq_depend)
+    with eq_dep have "(Th th, Th th') \<in> (depend s')^+" by simp
+    from converse_tranclE[OF this]
+    obtain cs' where bk: "(Th th, Cs cs') \<in> depend s'"
+      by (auto simp:s_depend_def)
+    from step_back_step[OF vt_s[unfolded s_def]]
+    show False
+    proof(cases)
+      assume "th \<in> runing s'"
+      with bk show ?thesis
+        apply (unfold runing_def readys_def s_waiting_def s_depend_def)
+        by (auto simp:cs_waiting_def)
+    qed
+  qed
+  have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
+    by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
+  moreover {
+    fix th1 
+    assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
+    hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
+    hence "preced th1 s = preced th1 s'"
+    proof
+      assume "th1 = th'"
+      with neq_th
+      show "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
+    next
+      assume "th1 \<in> dependents (wq s') th'"
+      with nd and eq_dp have "th1 \<noteq> th"
+        by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
+      thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
+    qed
+  } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) = 
+                     ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" 
+    by (auto simp:image_def)
+  thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+        Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+end
+end
+
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Ext.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Ext.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1057 @@
+theory Ext
+imports Prio
+begin
+
+locale highest_create =
+  fixes s' th prio fixes s 
+  defines s_def : "s \<equiv> (Create th prio#s')"
+  assumes vt_s: "vt step s"
+  and highest: "cp s th = Max ((cp s)`threads s)"
+
+context highest_create
+begin
+
+lemma threads_s: "threads s = threads s' \<union> {th}"
+  by (unfold s_def, simp)
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_create: "step s' (Create th prio)"
+  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_create_elim: 
+  "\<lbrakk>\<And>max_prio. \<lbrakk>prio \<le> max_prio; th \<notin> threads s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+  by (insert step_create, ind_cases "step s' (Create th prio)", auto)
+
+lemma eq_cp_s: 
+  assumes th'_in: "th' \<in> threads s'"
+  shows "cp s th' = cp s' th'"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def s_def 
+    eq_depend depend_create_unchanged)
+  show "Max ((\<lambda>tha. preced tha (Create th prio # s')) `
+         ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+})) =
+        Max ((\<lambda>th. preced th s') ` ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+}))"
+    (is "Max (?f ` ?A) = Max (?g ` ?A)")
+  proof -
+    have "?f ` ?A = ?g ` ?A"
+    proof(rule f_image_eq)
+      fix a
+      assume a_in: "a \<in> ?A"
+      thus "?f a = ?g a" 
+      proof -
+        from a_in
+        have "a = th' \<or> (Th a, Th th') \<in> (depend s')\<^sup>+" by auto 
+        hence "a \<noteq> th"
+        proof
+          assume "a = th'"
+          moreover have "th' \<noteq> th"
+          proof(rule step_create_elim)
+            assume th_not_in: "th \<notin> threads s'" with th'_in
+            show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        next
+          assume "(Th a, Th th') \<in> (depend s')\<^sup>+"
+          hence "Th a \<in> Domain \<dots>"
+            by (auto simp:Domain_def)
+          hence "Th a \<in> Domain (depend s')"
+            by (simp add:trancl_domain)
+          from dm_depend_threads[OF vt_s' this]
+          have h: "a \<in> threads s'" .
+          show ?thesis
+          proof(rule step_create_elim)
+            assume "th \<notin> threads s'" with h
+            show ?thesis by auto
+          qed
+        qed
+        thus ?thesis 
+          by (unfold preced_def, auto)
+      qed
+    qed
+    thus ?thesis by auto
+  qed
+qed
+
+lemma same_depend: "depend s = depend s'"
+  by (insert depend_create_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+  "dependents (wq s) th = dependents (wq s') th"
+  apply (unfold cs_dependents_def)
+  by (unfold eq_depend same_depend, simp)
+
+lemma nil_dependents_s': "dependents (wq s') th = {}"
+proof -
+  { assume ne: "dependents (wq s') th \<noteq> {}"
+    then obtain th' where "th' \<in>  dependents (wq s') th"
+      by (unfold cs_dependents_def, auto)
+    hence "(Th th', Th th) \<in> (depend (wq s'))^+"
+      by (unfold cs_dependents_def, auto)
+    hence "(Th th', Th th) \<in> (depend s')^+"
+      by (simp add:eq_depend)
+    hence "Th th \<in> Range ((depend s')^+)" by (auto simp:Range_def Domain_def)
+    hence "Th th \<in> Range (depend s')" by (simp add:trancl_range)
+    from range_in [OF vt_s' this]
+    have h: "th \<in> threads s'" .
+    have "False"
+    proof(rule step_create_elim)
+      assume "th \<notin> threads s'" with h show ?thesis by auto
+    qed
+  } thus ?thesis by auto
+qed
+
+lemma nil_dependents: "dependents (wq s) th = {}"
+proof -
+  have "wq s' = wq s"
+    by (unfold wq_def s_def, auto simp:Let_def)
+  with nil_dependents_s' show ?thesis by auto
+qed
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+  by (unfold cp_eq_cpreced cpreced_def nil_dependents, auto)
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+  { assume "th \<notin> readys s"
+    with threads_s obtain cs where 
+      "waiting s th cs"
+      by (unfold readys_def, auto)
+    hence "(Th th, Cs cs) \<in> depend s"
+      by (unfold s_depend_def, unfold eq_waiting, simp)
+    hence "Th th \<in> Domain (depend s')"
+      by (unfold same_depend, auto simp:Domain_def)
+    from dm_depend_threads [OF vt_s' this] 
+    have h: "th \<in> threads s'" .
+    have "False"
+    proof (rule_tac step_create_elim)
+      assume "th \<notin> threads s'" with h show ?thesis by simp
+    qed
+  } thus ?thesis by auto
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+  proof -
+    have " Max (cp s ` readys s) = cp s th"
+    proof(rule Max_eqI)
+      from finite_threads[OF vt_s] readys_threads finite_subset
+      have "finite (readys s)" by blast
+      thus "finite (cp s ` readys s)" by auto
+    next
+      from is_ready show "cp s th \<in> cp s ` readys s" by auto
+    next
+      fix y
+      assume h: "y \<in> cp s ` readys s"
+      have "y \<le> Max (cp s ` readys s)"
+      proof(rule Max_ge [OF _ h])
+        from finite_threads[OF vt_s] readys_threads finite_subset
+        have "finite (readys s)" by blast
+        thus "finite (cp s ` readys s)" by auto
+      qed
+      moreover have "\<dots> \<le> Max (cp s ` threads s)"
+      proof(rule Max_mono)
+        from readys_threads 
+        show "cp s ` readys s \<subseteq> cp s ` threads s" by auto
+      next
+        from is_ready show "cp s ` readys s \<noteq> {}" by auto
+      next
+        from finite_threads [OF vt_s]
+        show "finite (cp s ` threads s)" by auto
+      qed
+      moreover note highest
+      ultimately show "y \<le> cp s th" by auto
+    qed
+    with highest show ?thesis by auto
+  qed
+  thus ?thesis
+    by (unfold runing_def, insert highest is_ready, auto)
+qed
+
+end
+
+locale extend_highest = highest_create + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest
+begin
+
+lemma red_moment:
+  "extend_highest s' th prio (moment i t)"
+  apply (insert extend_highest_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_def extend_highest_axioms_def, clarsimp)
+  by (unfold highest_create_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest s' th prio t; 
+                    extend_highest s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest s' th prio (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest s' th prio t'"
+          by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest s' th prio (e # t')" .
+    next
+      from et show ext': "extend_highest s' th prio t'"
+          by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest s' th prio (e # t)" by auto
+      from extend_highest.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest s' th prio (e # t)" by auto
+        from extend_highest.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest s' th prio t" by auto
+        from extend_highest.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest s' th prio t" by auto
+    from extend_highest.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have " extend_highest s' th prio (e # t)" by auto
+        from extend_highest.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest s' th prio (e # t)" by auto
+    from extend_highest.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest s' th prio t" by auto
+    from extend_highest.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest s' th prio (e # t)" by auto
+        from extend_highest.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest s' th prio (e # t)" by auto
+      from extend_highest.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest s' th prio t" by auto
+        from extend_highest.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' is_ready
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    have "extend_highest s' th prio t" by fact
+    from extend_highest.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest s' th prio (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_def extend_highest_axioms_def 
+          highest_create_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ Create th prio # s')"
+        proof(rule extend_highest.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ Create th prio # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest s' th prio (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ Create th prio # s') th' =
+            cntV (moment (i + k) t @ Create th prio # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest s' th prio (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_def extend_highest_axioms_def 
+      highest_create_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+
diff -r 12e9aa68d5db -r 4190df6f4488 prio/ExtGG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtGG.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,970 @@
+theory ExtGG
+imports PrioG
+begin
+
+lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+  apply (induct s, simp)
+proof -
+  fix a s
+  assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+    and eq_as: "a # s \<noteq> []"
+  show "birthtime th (a # s) < length (a # s)"
+  proof(cases "s \<noteq> []")
+    case False
+    from False show ?thesis
+      by (cases a, auto simp:birthtime.simps)
+  next
+    case True
+    from ih [OF True] show ?thesis
+      by (cases a, auto simp:birthtime.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)
+
+locale highest_gen =
+  fixes s th prio tm
+  assumes vt_s: "vt step s"
+  and threads_s: "th \<in> threads s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+  and nh: "preced th s' \<noteq> Max ((cp s)`threads s')"
+  and preced_th: "preced th s = Prc prio tm"
+
+context highest_gen
+begin
+
+lemma lt_tm: "tm < length s"
+  by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+  from highest and max_cp_eq[OF vt_s]
+  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+  proof -
+    from threads_s and dependents_threads[OF vt_s, of th]
+    show ?thesis by auto
+  qed
+  show ?thesis
+  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+  next
+    fix y 
+    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+      and eq_y: "y = preced th1 s" by auto
+    show "y \<le> preced th s"
+    proof(unfold is_max, rule Max_ge)
+      from finite_threads[OF vt_s] 
+      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+    next
+      from sbs th1_in and eq_y 
+      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+    qed
+  next
+    from sbs and finite_threads[OF vt_s]
+    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+      by (auto intro:finite_subset)
+  qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+  show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_gen
+begin
+
+thm extend_highest_gen.axioms
+
+lemma red_moment:
+  "extend_highest_gen s th prio tm (moment i t)"
+  apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+  by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest_gen s th prio tm t; 
+                    extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_gen_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest_gen s th prio tm (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_gen s th prio tm (e # t')" .
+    next
+      from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+      from extend_highest_gen.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+        from extend_highest_gen.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest_gen s th prio tm t" by auto
+        from extend_highest_gen.th_kept[OF this] show ?thesis by (simp)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest_gen s th prio tm t" by auto
+    from extend_highest_gen.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+        from extend_highest_gen.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold preced_th, unfold eq_e, insert lt_tm, 
+          auto simp:preced_def precedence_less_def preced_th)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+    from extend_highest_gen.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest_gen s th prio tm t" by auto
+    from extend_highest_gen.th_kept[OF this]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+        from extend_highest_gen.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  auto
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+      from extend_highest_gen.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y Set
+          show ?thesis
+            apply (subst preced_th, insert lt_tm)
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest_gen s th prio tm t" by auto
+        from extend_highest_gen.th_kept [OF this]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' threads_s
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from Cons have "extend_highest_gen s th prio tm t" by auto
+    from extend_highest_gen.pv_blocked 
+    [OF this, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest_gen s th prio tm (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_gen_def extend_highest_gen_axioms_def 
+                          highest_gen_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof -
+        show "th' \<notin> runing (moment (i + k) t @ s)"
+        proof(rule extend_highest_gen.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ s)"
+            by simp
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest_gen s th prio tm (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ s) th' = cntV (moment (i + k) t @ s) th'"
+            by (auto)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_gen.pv_blocked [OF  red_moment [of j], OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest_gen s th prio tm (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_gen_def extend_highest_gen_axioms_def 
+      highest_gen_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+end
+
+
diff -r 12e9aa68d5db -r 4190df6f4488 prio/ExtGG_1.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtGG_1.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,973 @@
+theory ExtGG
+imports PrioG
+begin
+
+lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+  apply (induct s, simp)
+proof -
+  fix a s
+  assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+    and eq_as: "a # s \<noteq> []"
+  show "birthtime th (a # s) < length (a # s)"
+  proof(cases "s \<noteq> []")
+    case False
+    from False show ?thesis
+      by (cases a, auto simp:birthtime.simps)
+  next
+    case True
+    from ih [OF True] show ?thesis
+      by (cases a, auto simp:birthtime.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)
+
+locale highest_gen =
+  fixes s' th s e' prio tm
+  defines s_def : "s \<equiv> (e'#s')"
+  assumes vt_s: "vt step s"
+  and threads_s: "th \<in> threads s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+  and nh: "preced th s' \<noteq> Max ((cp s)`threads s')"
+  and preced_th: "preced th s = Prc prio tm"
+
+context highest_gen
+begin
+
+lemma lt_tm: "tm < length s"
+  by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+  from highest and max_cp_eq[OF vt_s]
+  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+  proof -
+    from threads_s and dependents_threads[OF vt_s, of th]
+    show ?thesis by auto
+  qed
+  show ?thesis
+  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+  next
+    fix y 
+    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+      and eq_y: "y = preced th1 s" by auto
+    show "y \<le> preced th s"
+    proof(unfold is_max, rule Max_ge)
+      from finite_threads[OF vt_s] 
+      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+    next
+      from sbs th1_in and eq_y 
+      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+    qed
+  next
+    from sbs and finite_threads[OF vt_s]
+    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+      by (auto intro:finite_subset)
+  qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+  show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_gen
+begin
+
+lemma red_moment:
+  "extend_highest_gen s' th e' prio tm (moment i t)"
+  apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+  by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest_gen s' th e' prio tm t; 
+                    extend_highest_gen s' th e' prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_gen_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_gen s' th e' prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest_gen s' th e' prio tm (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_gen s' th e' prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_gen s' th e' prio tm (e # t')" .
+    next
+      from et show ext': "extend_highest_gen s' th e' prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+      from extend_highest_gen.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+        from extend_highest_gen.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+        from extend_highest_gen.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+    from extend_highest_gen.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+        from extend_highest_gen.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold preced_th, unfold eq_e, insert lt_tm, 
+          auto simp:preced_def s_def precedence_less_def preced_th)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+    from extend_highest_gen.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+    from extend_highest_gen.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+        from extend_highest_gen.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+      from extend_highest_gen.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            apply (subst preced_th, insert lt_tm)
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+        from extend_highest_gen.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' threads_s
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+    from extend_highest_gen.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest_gen s' th e' prio tm (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_gen_def extend_highest_gen_axioms_def 
+                          highest_gen_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ e' # s')"
+        proof(rule extend_highest_gen.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ e' # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest_gen s' th e' prio tm (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ e' # s') th' = cntV (moment (i + k) t @ e' # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_gen.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest_gen s' th e' prio tm (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_gen_def extend_highest_gen_axioms_def 
+      highest_gen_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+
+
diff -r 12e9aa68d5db -r 4190df6f4488 prio/ExtS.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtS.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1019 @@
+theory ExtS
+imports Prio
+begin
+
+locale highest_set =
+  fixes s' th prio fixes s 
+  defines s_def : "s \<equiv> (Set th prio#s')"
+  assumes vt_s: "vt step s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+
+context highest_set
+begin
+
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_set: "step s' (Set th prio)"
+  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_set_elim: 
+  "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+  by (insert step_set, ind_cases "step s' (Set th prio)", auto)
+
+
+lemma threads_s: "th \<in> threads s"
+  by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
+
+lemma same_depend: "depend s = depend s'"
+  by (insert depend_set_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+  "dependents (wq s) th = dependents (wq s') th"
+  apply (unfold cs_dependents_def)
+  by (unfold eq_depend same_depend, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+  from highest and max_cp_eq[OF vt_s]
+  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+  proof -
+    from threads_s and dependents_threads[OF vt_s, of th]
+    show ?thesis by auto
+  qed
+  show ?thesis
+  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+  next
+    fix y 
+    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+      and eq_y: "y = preced th1 s" by auto
+    show "y \<le> preced th s"
+    proof(unfold is_max, rule Max_ge)
+      from finite_threads[OF vt_s] 
+      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+    next
+      from sbs th1_in and eq_y 
+      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+    qed
+  next
+    from sbs and finite_threads[OF vt_s]
+    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+      by (auto intro:finite_subset)
+  qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+  have "\<forall>cs. \<not> waiting s th cs"
+    apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
+    apply (unfold s_depend_def, unfold runing_def readys_def)
+    apply (auto, fold s_def)
+    apply (erule_tac x = cs in allE, auto simp:waiting_eq)
+  proof -
+    fix cs
+    assume h: 
+      "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
+          {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
+            (is "?L = ?R")
+    and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
+    from wt have "(Th th, Cs cs) \<in> ?L" by auto
+    with h have "(Th th, Cs cs) \<in> ?R" by simp
+    hence "waiting (wq s') th cs" by auto with nwt
+    show False by auto
+  qed    
+  with threads_s show ?thesis 
+    by (unfold readys_def, auto)
+qed
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+  show ?thesis by simp
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+  proof -
+    have " Max (cp s ` readys s) = cp s th"
+    proof(rule Max_eqI)
+      from finite_threads[OF vt_s] readys_threads finite_subset
+      have "finite (readys s)" by blast
+      thus "finite (cp s ` readys s)" by auto
+    next
+      from is_ready show "cp s th \<in> cp s ` readys s" by auto
+    next
+      fix y
+      assume "y \<in> cp s ` readys s"
+      then obtain th1 where 
+        eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
+      show  "y \<le> cp s th" 
+      proof -
+        have "y \<le> Max (cp s ` threads s)"
+        proof(rule Max_ge)
+          from eq_y and th1_in
+          show "y \<in> cp s ` threads s"
+            by (auto simp:readys_def)
+        next
+          from finite_threads[OF vt_s]
+          show "finite (cp s ` threads s)" by auto
+        qed
+        with highest' show ?thesis by auto
+      qed
+    qed
+    with highest' show ?thesis by auto
+  qed
+  thus ?thesis
+    by (unfold runing_def, insert highest' is_ready, auto)
+qed
+
+end
+
+locale extend_highest_set = highest_set + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_set
+begin
+
+lemma red_moment:
+  "extend_highest_set s' th prio (moment i t)"
+  apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
+  by (unfold highest_set_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest_set s' th prio t; 
+                    extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_set_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest_set s' th prio (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_set s' th prio (e # t')" .
+    next
+      from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have " extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest_set s' th prio (e # t)" by auto
+    from extend_highest_set.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' is_ready
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    have "extend_highest_set s' th prio t" by fact
+    from extend_highest_set.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+          highest_set_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
+        proof(rule extend_highest_set.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
+            cntV (moment (i + k) t @ Set th prio # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_set.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest_set s' th prio (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+      highest_set_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+
diff -r 12e9aa68d5db -r 4190df6f4488 prio/ExtSG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtSG.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1019 @@
+theory ExtSG
+imports PrioG
+begin
+
+locale highest_set =
+  fixes s' th prio fixes s 
+  defines s_def : "s \<equiv> (Set th prio#s')"
+  assumes vt_s: "vt step s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+
+context highest_set
+begin
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_set: "step s' (Set th prio)"
+  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_set_elim: 
+  "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+  by (insert step_set, ind_cases "step s' (Set th prio)", auto)
+
+
+lemma threads_s: "th \<in> threads s"
+  by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
+
+lemma same_depend: "depend s = depend s'"
+  by (insert depend_set_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+  "dependents (wq s) th = dependents (wq s') th"
+  apply (unfold cs_dependents_def)
+  by (unfold eq_depend same_depend, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+  from highest and max_cp_eq[OF vt_s]
+  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+  proof -
+    from threads_s and dependents_threads[OF vt_s, of th]
+    show ?thesis by auto
+  qed
+  show ?thesis
+  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+  next
+    fix y 
+    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+      and eq_y: "y = preced th1 s" by auto
+    show "y \<le> preced th s"
+    proof(unfold is_max, rule Max_ge)
+      from finite_threads[OF vt_s] 
+      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+    next
+      from sbs th1_in and eq_y 
+      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+    qed
+  next
+    from sbs and finite_threads[OF vt_s]
+    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+      by (auto intro:finite_subset)
+  qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+  have "\<forall>cs. \<not> waiting s th cs"
+    apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
+    apply (unfold s_depend_def, unfold runing_def readys_def)
+    apply (auto, fold s_def)
+    apply (erule_tac x = cs in allE, auto simp:waiting_eq)
+  proof -
+    fix cs
+    assume h: 
+      "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
+          {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
+            (is "?L = ?R")
+    and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
+    from wt have "(Th th, Cs cs) \<in> ?L" by auto
+    with h have "(Th th, Cs cs) \<in> ?R" by simp
+    hence "waiting (wq s') th cs" by auto with nwt
+    show False by auto
+  qed    
+  with threads_s show ?thesis 
+    by (unfold readys_def, auto)
+qed
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+  show ?thesis by simp
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+  proof -
+    have " Max (cp s ` readys s) = cp s th"
+    proof(rule Max_eqI)
+      from finite_threads[OF vt_s] readys_threads finite_subset
+      have "finite (readys s)" by blast
+      thus "finite (cp s ` readys s)" by auto
+    next
+      from is_ready show "cp s th \<in> cp s ` readys s" by auto
+    next
+      fix y
+      assume "y \<in> cp s ` readys s"
+      then obtain th1 where 
+        eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
+      show  "y \<le> cp s th" 
+      proof -
+        have "y \<le> Max (cp s ` threads s)"
+        proof(rule Max_ge)
+          from eq_y and th1_in
+          show "y \<in> cp s ` threads s"
+            by (auto simp:readys_def)
+        next
+          from finite_threads[OF vt_s]
+          show "finite (cp s ` threads s)" by auto
+        qed
+        with highest' show ?thesis by auto
+      qed
+    qed
+    with highest' show ?thesis by auto
+  qed
+  thus ?thesis
+    by (unfold runing_def, insert highest' is_ready, auto)
+qed
+
+end
+
+locale extend_highest_set = highest_set + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_set
+begin
+
+lemma red_moment:
+  "extend_highest_set s' th prio (moment i t)"
+  apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
+  by (unfold highest_set_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest_set s' th prio t; 
+                    extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_set_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest_set s' th prio (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_set s' th prio (e # t')" .
+    next
+      from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have " extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest_set s' th prio (e # t)" by auto
+    from extend_highest_set.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' is_ready
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    have "extend_highest_set s' th prio t" by fact
+    from extend_highest_set.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+                          highest_set_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
+        proof(rule extend_highest_set.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
+            cntV (moment (i + k) t @ Set th prio # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_set.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest_set s' th prio (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+      highest_set_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+
+
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Happen_within.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Happen_within.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,126 @@
+theory Happen_within
+imports Main Moment
+begin
+
+(* 
+  lemma 
+  fixes P :: "('a list) \<Rightarrow> bool"
+  and Q :: "('a list) \<Rightarrow> bool"
+  and k :: nat
+  and f :: "('a list) \<Rightarrow> nat"
+  assumes "\<And> s t. \<lbrakk>P s; \<not> Q s; P (t@s); k < length t\<rbrakk> \<Longrightarrow> f (t@s) < f s"
+  shows "\<And> s t. \<lbrakk> P s;  P(t @ s); f(s) * k < length t\<rbrakk> \<Longrightarrow> Q (t@s)"
+  sorry
+*)
+
+text {* 
+  The following two notions are introduced to improve the situation.
+  *}
+
+definition all_future :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> ('a list) \<Rightarrow> bool"
+where "all_future G R s = (\<forall> t. G (t@s) \<longrightarrow> R t)"
+
+definition happen_within :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> ('a list) \<Rightarrow> bool"
+where "happen_within G R k s = all_future G (\<lambda> t. k < length t \<longrightarrow> 
+                                  (\<exists> i \<le> k. R (moment i t @ s) \<and> G (moment i t @ s))) s"
+
+lemma happen_within_intro:
+  fixes P :: "('a list) \<Rightarrow> bool"
+  and Q :: "('a list) \<Rightarrow> bool"
+  and k :: nat
+  and f :: "('a list) \<Rightarrow> nat"
+  assumes 
+  lt_k: "0 < k"
+  and step: "\<And> s. \<lbrakk>P s; \<not> Q s\<rbrakk> \<Longrightarrow> happen_within P (\<lambda> s'. f s' < f s) k s"
+  shows "\<And> s. P s \<Longrightarrow> happen_within P Q ((f s + 1) * k) s"
+proof -
+  fix s
+  assume "P s"
+  thus "happen_within P Q ((f s + 1) * k) s"
+  proof(induct n == "f s + 1" arbitrary:s rule:nat_less_induct)
+    fix s
+    assume ih [rule_format]: "\<forall>m<f s + 1. \<forall>x. m = f x + 1 \<longrightarrow> P x 
+                                 \<longrightarrow> happen_within P Q ((f x + 1) * k) x"
+      and ps: "P s"
+    show "happen_within P Q ((f s + 1) * k) s"
+    proof(cases "Q s")
+      case True
+      show ?thesis 
+      proof -
+        { fix t
+          from True and ps have "0 \<le> ((f s + 1)*k) \<and> Q (moment 0 t @ s) \<and> P (moment 0 t @ s)" by auto
+          hence "\<exists>i\<le>(f s + 1) * k. Q (moment i t @ s) \<and> P (moment i t @ s)" by auto
+        } thus ?thesis by (auto simp: happen_within_def all_future_def)
+      qed
+    next
+      case False
+      from step [OF ps False] have kk: "happen_within P (\<lambda>s'. f s' < f s) k s" .
+      show ?thesis
+      proof -
+        { fix t
+          assume pts: "P (t @ s)" and ltk: "(f s + 1) * k < length t"
+          from ltk have lt_k_lt: "k < length t" by auto
+          with kk pts obtain i 
+            where le_ik: "i \<le> k" 
+            and lt_f: "f (moment i t @ s) < f s" 
+            and p_m: "P (moment i t @ s)"
+            by (auto simp:happen_within_def all_future_def)
+          from ih [of "f (moment i t @ s) + 1" "(moment i t @ s)", OF _ _ p_m] and lt_f
+          have hw: "happen_within P Q ((f (moment i t @ s) + 1) * k) (moment i t @ s)" by auto
+          have "(\<exists>j\<le>(f s + 1) * k. Q (moment j t @ s) \<and>  P (moment j t @ s))" (is "\<exists> j. ?T j")
+          proof -
+            let ?t = "restm i t"
+            have eq_t: "t = ?t @ moment i t" by (simp add:moment_restm_s) 
+            have h1: "P (restm i t @ moment i t @ s)" 
+            proof -
+              from pts and eq_t have "P ((restm i t @ moment i t) @ s)" by simp
+              thus ?thesis by simp
+            qed
+            moreover have h2: "(f (moment i t @ s) + 1) * k < length (restm i t)"
+            proof -
+              have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
+              from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
+              from h [OF this, of k]
+              have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
+              moreover from le_ik have "\<dots> \<le> ((f s) * k + k - i)" by simp
+              moreover from le_ik lt_k_lt and ltk have "(f s) * k + k - i < length t - i" by simp
+              moreover have "length (restm i t) = length t - i" using length_restm by metis
+              ultimately show ?thesis by simp
+            qed
+            from hw [unfolded happen_within_def all_future_def, rule_format, OF h1 h2]
+            obtain m where le_m: "m \<le> (f (moment i t @ s) + 1) * k"
+              and q_m: "Q (moment m ?t @ moment i t @ s)" 
+              and p_m: "P (moment m ?t @ moment i t @ s)" by auto
+            have eq_mm: "moment m ?t @ moment i t @ s = (moment (m+i) t)@s"
+            proof -
+              have "moment m (restm i t) @ moment i t = moment (m + i) t"
+                using moment_plus_split by metis
+              thus ?thesis by simp
+            qed
+            let ?j = "m + i"
+            have "?T ?j"
+            proof -
+              have "m + i \<le> (f s + 1) * k"
+              proof -
+                have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
+                from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
+                from h [OF this, of k]
+                have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
+                with le_m have "m \<le> f s * k" by simp
+                hence "m + i \<le> f s * k + i" by simp
+                with le_ik show ?thesis by simp
+              qed
+              moreover from eq_mm q_m have " Q (moment (m + i) t @ s)" by metis
+              moreover from eq_mm p_m have " P (moment (m + i) t @ s)" by metis
+              ultimately show ?thesis by blast
+            qed
+            thus ?thesis by blast
+          qed
+        } thus ?thesis by  (simp add:happen_within_def all_future_def firstn.simps)
+      qed
+    qed
+  qed
+qed
+
+end
+
diff -r 12e9aa68d5db -r 4190df6f4488 prio/IsaMakefile
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/IsaMakefile	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,29 @@
+
+## targets
+
+default: paper
+all: session paper
+
+## global settings
+
+SRC = $(ISABELLE_HOME)/src
+OUT = $(ISABELLE_OUTPUT)
+LOG = $(OUT)/log
+
+
+USEDIR = $(ISABELLE_TOOL) usedir -v true -t true 
+
+
+## Slides
+
+session: ./ROOT.ML ./*.thy
+	@$(USEDIR) -b -D generated -f ROOT.ML HOL Prio
+	
+paper: Paper/ROOT.ML \
+       Paper/*.thy 
+	@$(USEDIR) -D generated -f ROOT.ML Prio Paper
+	rm -f Paper/generated/*.aux # otherwise latex will fall over  
+	cd Paper/generated ; $(ISABELLE_TOOL) latex -o pdf root.tex 
+	cd Paper/generated ; bibtex root
+	cd Paper/generated ; $(ISABELLE_TOOL) latex -o pdf root.tex
+	cp Paper/generated/root.pdf paper.pdf 
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Lsp.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Lsp.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,323 @@
+theory Lsp
+imports Main
+begin
+
+fun lsp :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list)"
+where 
+   "lsp f [] = ([], [], [])" |
+   "lsp f [x] = ([], [x], [])" |
+   "lsp f (x#xs) = (case (lsp f xs) of
+                     (l, [], r) \<Rightarrow> ([], [x], []) |
+                     (l, y#ys, r) \<Rightarrow> if f x \<ge> f y then ([], [x], xs) else (x#l, y#ys, r))"
+
+inductive lsp_p :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list) \<Rightarrow> bool"
+for f :: "('a \<Rightarrow> ('b::linorder))"
+where
+  lsp_nil [intro]: "lsp_p f [] ([], [], [])" |
+  lsp_single [intro]: "lsp_p f [x] ([], [x], [])" |
+  lsp_cons_1 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x \<ge> f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) ([], [x], xs)" |
+  lsp_cons_2 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) (x#l, [m], r)"
+
+lemma lsp_p_lsp_1: "lsp_p f x y \<Longrightarrow> y = lsp f x"
+proof (induct rule:lsp_p.induct)
+  case (lsp_cons_1 xs  l m r x)
+  assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
+    and le_mx: "f m \<le> f x"
+  show ?case (is "?L = ?R")
+  proof(cases xs, simp)
+    case (Cons v vs)
+    show ?thesis
+      apply (simp add:Cons)
+      apply (fold Cons)
+      by (simp add:lsp_xs le_mx)
+  qed
+next
+  case (lsp_cons_2 xs l m r x)
+  assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
+    and lt_xm: "f x < f m"
+  show ?case (is "?L = ?R")
+  proof(cases xs)
+    case (Cons v vs)
+    show ?thesis
+      apply (simp add:Cons)
+      apply (fold Cons)
+      apply (simp add:lsp_xs)
+      by (insert lt_xm, auto)
+  next
+    case Nil
+    from prems show ?thesis by simp
+  qed
+qed auto
+
+lemma lsp_mid_nil: "lsp f xs = (a, [], c) \<Longrightarrow> xs = []"
+  apply (induct xs arbitrary:a c, auto)
+  apply (case_tac xs, auto)
+  by (case_tac "(lsp f (ab # list))", auto split:if_splits list.splits)
+
+
+lemma lsp_mid_length: "lsp f x = (u, v, w) \<Longrightarrow> length v \<le> 1"
+proof(induct x arbitrary:u v w, simp)
+  case (Cons x xs)
+  assume ih: "\<And> u v w. lsp f xs = (u, v, w) \<Longrightarrow> length v \<le> 1"
+  and h: "lsp f (x # xs) = (u, v, w)"
+  show "length v \<le> 1" using h
+  proof(cases xs, simp add:h)
+    case (Cons z zs)
+    assume eq_xs: "xs = z # zs"
+    show ?thesis
+    proof(cases "lsp f xs")
+      fix l m r
+      assume eq_lsp: "lsp f xs = (l, m, r)"
+      show ?thesis
+      proof(cases m)
+        case Nil
+        from Nil and eq_lsp have "lsp f xs = (l, [], r)" by simp
+        from lsp_mid_nil [OF this] have "xs = []" .
+        with h show ?thesis by auto
+      next
+        case (Cons y ys)
+        assume eq_m: "m = y # ys"
+        from ih [OF eq_lsp] have eq_xs_1: "length m \<le> 1" .
+        show ?thesis
+        proof(cases "f x \<ge> f y")
+          case True
+          from eq_xs eq_xs_1 True h eq_lsp show ?thesis 
+            by (auto split:list.splits if_splits)
+        next
+          case False
+          from eq_xs eq_xs_1 False h eq_lsp show ?thesis 
+             by (auto split:list.splits if_splits)
+        qed
+      qed
+    qed
+  next
+    assume "[] = u \<and> [x] = v \<and> [] = w"
+    hence "v = [x]" by simp
+    thus "length v \<le> Suc 0" by simp
+  qed
+qed
+
+lemma lsp_p_lsp_2: "lsp_p f x (lsp f x)"
+proof(induct x, auto)
+  case (Cons x xs)
+  assume ih: "lsp_p f xs (lsp f xs)"
+  show ?case
+  proof(cases xs)
+    case Nil
+    thus ?thesis by auto
+  next
+    case (Cons v vs)
+    show ?thesis
+    proof(cases "xs")
+      case Nil
+      thus ?thesis by auto
+    next
+      case (Cons v vs)
+      assume eq_xs: "xs = v # vs"
+      show ?thesis
+      proof(cases "lsp f xs")
+        fix l m r
+        assume eq_lsp_xs: "lsp f xs = (l, m, r)"
+        show ?thesis
+        proof(cases m)
+          case Nil
+          from eq_lsp_xs and Nil have "lsp f xs = (l, [], r)" by simp
+          from lsp_mid_nil [OF this] have eq_xs: "xs = []" .
+          hence "lsp f (x#xs) = ([], [x], [])" by simp
+          with eq_xs show ?thesis by auto
+        next
+          case (Cons y ys)
+          assume eq_m: "m = y # ys"
+          show ?thesis
+          proof(cases "f x \<ge> f y")
+            case True
+            from eq_xs eq_lsp_xs Cons True
+            have eq_lsp: "lsp f (x#xs) = ([], [x], v # vs)" by simp
+            show ?thesis
+            proof (simp add:eq_lsp)
+              show "lsp_p f (x # xs) ([], [x], v # vs)"
+              proof(fold eq_xs, rule lsp_cons_1 [OF _])
+                from eq_xs show "xs \<noteq> []" by simp
+              next
+                from lsp_mid_length [OF eq_lsp_xs] and Cons
+                have "m = [y]" by simp
+                with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
+                with ih show "lsp_p f xs (l, [y], r)" by simp
+              next
+                from True show "f y \<le> f x" by simp
+              qed
+            qed
+          next
+            case False
+            from eq_xs eq_lsp_xs Cons False
+            have eq_lsp: "lsp f (x#xs) = (x # l, y # ys, r) " by simp
+            show ?thesis
+            proof (simp add:eq_lsp)
+              from lsp_mid_length [OF eq_lsp_xs] and eq_m
+              have "ys = []" by simp
+              moreover have "lsp_p f (x # xs) (x # l, [y], r)"
+              proof(rule lsp_cons_2)
+                from eq_xs show "xs \<noteq> []" by simp
+              next
+                from lsp_mid_length [OF eq_lsp_xs] and Cons
+                have "m = [y]" by simp
+                with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
+                with ih show "lsp_p f xs (l, [y], r)" by simp
+              next
+                from False show "f x < f y" by simp
+              qed
+              ultimately show "lsp_p f (x # xs) (x # l, y # ys, r)" by simp
+            qed
+          qed
+        qed
+      qed
+    qed
+  qed
+qed
+
+lemma lsp_induct:
+  fixes f x1 x2 P
+  assumes h: "lsp f x1 = x2"
+  and p1: "P [] ([], [], [])"
+  and p2: "\<And>x. P [x] ([], [x], [])"
+  and p3: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f m \<le> f x\<rbrakk> \<Longrightarrow> P (x # xs) ([], [x], xs)"
+  and p4: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> P (x # xs) (x # l, [m], r)"
+  shows "P x1 x2"
+proof(rule lsp_p.induct)
+  from lsp_p_lsp_2 and h
+  show "lsp_p f x1 x2" by metis
+next
+  from p1 show "P [] ([], [], [])" by metis
+next
+  from p2 show "\<And>x. P [x] ([], [x], [])" by metis
+next
+  fix xs l m r x 
+  assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f m \<le> f x"
+  show "P (x # xs) ([], [x], xs)" 
+  proof(rule p3 [OF h1 _ h3 h4])
+    from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
+  qed
+next
+  fix xs l m r x 
+  assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f x < f m"
+  show "P (x # xs) (x # l, [m], r)"
+  proof(rule p4 [OF h1 _ h3 h4])
+    from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
+  qed
+qed
+
+lemma lsp_set_eq: 
+  fixes f x u v w
+  assumes h: "lsp f x = (u, v, w)"
+  shows "x = u@v@w"
+proof -
+  have "\<And> f x r. lsp f x = r \<Longrightarrow> \<forall> u v w. (r = (u, v, w) \<longrightarrow> x = u@v@w)" 
+    by (erule lsp_induct, simp+)
+  from this [rule_format, OF h] show ?thesis by simp
+qed
+
+lemma lsp_set: 
+  assumes h: "(u, v, w) = lsp f x"
+  shows "set (u@v@w) = set x"
+proof -
+  from lsp_set_eq [OF h[symmetric]] 
+  show ?thesis by simp
+qed
+
+lemma max_insert_gt:
+  fixes S fx
+  assumes h: "fx < Max S"
+  and np: "S \<noteq> {}"
+  and fn: "finite S" 
+  shows "Max S = Max (insert fx S)"
+proof -
+  from Max_insert [OF fn np]
+  have "Max (insert fx S) = max fx (Max S)" .
+  moreover have "\<dots> = Max S"
+  proof(cases "fx \<le> Max S")
+    case False
+    with h
+    show ?thesis by (simp add:max_def)
+  next
+    case True
+    thus ?thesis by (simp add:max_def)
+  qed
+  ultimately show ?thesis by simp
+qed
+
+lemma max_insert_le: 
+  fixes S fx
+  assumes h: "Max S \<le> fx"
+  and fn: "finite S"
+  shows "fx = Max (insert fx S)"
+proof(cases "S = {}")
+  case True
+  thus ?thesis by simp
+next
+  case False
+  from Max_insert [OF fn False]
+  have "Max (insert fx S) = max fx (Max S)" .
+  moreover have "\<dots> = fx"
+  proof(cases "fx \<le> Max S")
+    case False
+    thus ?thesis by (simp add:max_def)
+  next
+    case True
+    have hh: "\<And> x y. \<lbrakk> x \<le> (y::('a::linorder)); y \<le> x\<rbrakk> \<Longrightarrow> x = y" by auto
+    from hh [OF True h]
+    have "fx = Max S" .
+    thus ?thesis by simp
+  qed
+  ultimately show ?thesis by simp
+qed
+  
+lemma lsp_max: 
+  fixes f x u m w
+  assumes h: "lsp f x = (u, [m], w)"
+  shows "f m = Max (f ` (set x))"
+proof -
+  { fix y
+    have "lsp f x = y \<Longrightarrow> \<forall> u m w. y = (u, [m], w) \<longrightarrow> f m = Max (f ` (set x))"
+    proof(erule lsp_induct, simp)
+      { fix x u m w
+        assume "(([]::'a list), ([x]::'a list), ([]::'a list)) = (u, [m], w)"
+        hence "f m = Max (f ` set [x])"  by simp
+      } thus "\<And>x. \<forall>u m w. ([], [x], []) = (u, [m], w) \<longrightarrow> f m = Max (f ` set [x])" by simp
+    next
+      fix xs l m r x
+      assume h1: "xs \<noteq> []"
+        and h2: " lsp f xs = (l, [m], r)"
+        and h3: "\<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
+        and h4: "f m \<le> f x"
+      show " \<forall>u m w. ([], [x], xs) = (u, [m], w) \<longrightarrow> f m = Max (f ` set (x # xs))"
+      proof -
+        have "f x = Max (f ` set (x # xs))"
+        proof -
+          from h2 h3 have "f m = Max (f ` set xs)" by simp
+          with h4 show ?thesis
+            apply auto
+            by (rule_tac max_insert_le, auto)
+        qed
+        thus ?thesis by simp
+      qed
+    next
+      fix xs l m r x
+      assume h1: "xs \<noteq> []"
+        and h2: " lsp f xs = (l, [m], r)"
+        and h3: " \<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
+        and h4: "f x < f m"
+      show "\<forall>u ma w. (x # l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set (x # xs))"
+      proof -
+        from h2 h3 have "f m = Max (f ` set xs)" by simp
+        with h4
+        have "f m =  Max (f ` set (x # xs))"
+          apply auto
+          apply (rule_tac max_insert_gt, simp+)
+          by (insert h1, simp+)
+        thus ?thesis by auto
+      qed
+    qed
+  } with h show ?thesis by metis
+qed
+
+end
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Moment.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Moment.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,773 @@
+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)"
+
+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
+
+end
\ No newline at end of file
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Paper/Paper.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/Paper.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,168 @@
+(*<*)
+theory Paper
+imports CpsG ExtGG
+begin
+(*>*)
+
+section {* Introduction *}
+
+text {*
+
+  Priority inversion referrers to the phenomena where tasks with higher 
+  priority are blocked by ones with lower priority. If priority inversion 
+  is not controlled, there will be no guarantee the urgent tasks will be 
+  processed in time. As reported in \cite{Reeves-Glenn-1998}, 
+  priority inversion used to cause software system resets and data lose in 
+  JPL's Mars pathfinder project. Therefore, the avoiding, detecting and controlling 
+  of priority inversion is a key issue to attain predictability in priority 
+  based real-time systems. 
+
+  The priority inversion phenomenon was first published in \cite{Lampson:Redell:cacm:1980}. 
+  The two protocols widely used to eliminate priority inversion, namely 
+  PI (Priority Inheritance) and PCE (Priority Ceiling Emulation), were proposed 
+  in \cite{journals/tc/ShaRL90}. PCE is less convenient to use because it requires 
+  static analysis of programs. Therefore, PI is more commonly used in 
+  practice\cite{locke-july02}. However, as pointed out in the literature, 
+  the analysis of priority inheritance protocol is quite subtle\cite{yodaiken-july02}. 
+  A formal analysis will certainly be helpful for us to understand and correctly 
+  implement PI. All existing formal analysis of PI
+  \cite{conf/fase/JahierHR09,WellingsBSB07,Faria08} are based on the model checking 
+  technology. Because of the state explosion problem, model check 
+  is much like an exhaustive testing of finite models with limited size. 
+  The results obtained can not be safely generalized to models with arbitrarily 
+  large size. Worse still, since model checking is fully automatic, it give little 
+  insight on why the formal model is correct. It is therefore 
+  definitely desirable to analyze PI using theorem proving, which gives 
+  more general results as well as deeper insight. And this is the purpose 
+  of this paper which gives a formal analysis of PI in the interactive 
+  theorem prover Isabelle using Higher Order Logic (HOL). The formalization 
+  focuses on on two issues:
+
+  \begin{enumerate}
+  \item The correctness of the protocol model itself. A series of desirable properties is 
+    derived until we are fully convinced that the formal model of PI does 
+    eliminate priority inversion. And a better understanding of PI is so obtained 
+    in due course. For example, we find through formalization that the choice of 
+    next thread to take hold when a 
+    resource is released is irrelevant for the very basic property of PI to hold. 
+    A point never mentioned in literature. 
+  \item The correctness of the implementation. A series of properties is derived the meaning 
+    of which can be used as guidelines on how PI can be implemented efficiently and correctly. 
+  \end{enumerate} 
+
+  The rest of the paper is organized as follows: Section \ref{overview} gives an overview 
+  of PI. Section \ref{model} introduces the formal model of PI. Section \ref{general} 
+  discusses a series of basic properties of PI. Section \ref{extension} shows formally 
+  how priority inversion is controlled by PI. Section \ref{implement} gives properties 
+  which can be used for guidelines of implementation. Section \ref{related} discusses 
+  related works. Section \ref{conclusion} concludes the whole paper.
+*}
+
+section {* An overview of priority inversion and priority inheritance \label{overview} *}
+
+text {*
+
+  Priority inversion refers to the phenomenon when a thread with high priority is blocked 
+  by a thread with low priority. Priority happens when the high priority thread requests 
+  for some critical resource already taken by the low priority thread. Since the high 
+  priority thread has to wait for the low priority thread to complete, it is said to be 
+  blocked by the low priority thread. Priority inversion might prevent high priority 
+  thread from fulfill its task in time if the duration of priority inversion is indefinite 
+  and unpredictable. Indefinite priority inversion happens when indefinite number 
+  of threads with medium priorities is activated during the period when the high 
+  priority thread is blocked by the low priority thread. Although these medium 
+  priority threads can not preempt the high priority thread directly, they are able 
+  to preempt the low priority threads and cause it to stay in critical section for 
+  an indefinite long duration. In this way, the high priority thread may be blocked indefinitely. 
+  
+  Priority inheritance is one protocol proposed to avoid indefinite priority inversion. 
+  The basic idea is to let the high priority thread donate its priority to the low priority 
+  thread holding the critical resource, so that it will not be preempted by medium priority 
+  threads. The thread with highest priority will not be blocked unless it is requesting 
+  some critical resource already taken by other threads. Viewed from a different angle, 
+  any thread which is able to block the highest priority threads must already hold some 
+  critical resource. Further more, it must have hold some critical resource at the 
+  moment the highest priority is created, otherwise, it may never get change to run and 
+  get hold. Since the number of such resource holding lower priority threads is finite, 
+  if every one of them finishes with its own critical section in a definite duration, 
+  the duration the highest priority thread is blocked is definite as well. The key to 
+  guarantee lower priority threads to finish in definite is to donate them the highest 
+  priority. In such cases, the lower priority threads is said to have inherited the 
+  highest priority. And this explains the name of the protocol: 
+  {\em Priority Inheritance} and how Priority Inheritance prevents indefinite delay.
+
+  The objectives of this paper are:
+  \begin{enumerate}
+  \item Build the above mentioned idea into formal model and prove a series of properties 
+    until we are convinced that the formal model does fulfill the original idea. 
+  \item Show how formally derived properties can be used as guidelines for correct 
+    and efficient implementation.
+  \end{enumerate}
+  The proof is totally formal in the sense that every detail is reduced to the 
+  very first principles of Higher Order Logic. The nature of interactive theorem 
+  proving is for the human user to persuade computer program to accept its arguments. 
+  A clear and simple understanding of the problem at hand is both a prerequisite and a 
+  byproduct of such an effort, because everything has finally be reduced to the very 
+  first principle to be checked mechanically. The former intuitive explanation of 
+  Priority Inheritance is just such a byproduct. 
+  *}
+
+section {* Formal model of Priority Inheritance \label{model} *}
+text {*
+  \input{../../generated/PrioGDef}
+*}
+
+section {* General properties of Priority Inheritance \label{general} *}
+
+section {* Key properties \label{extension} *}
+
+section {* Properties to guide implementation \label{implement} *}
+
+section {* Related works \label{related} *}
+
+text {*
+  \begin{enumerate}
+  \item {\em Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java}
+    \cite{WellingsBSB07} models and verifies the combination of Priority Inheritance (PI) and 
+    Priority Ceiling Emulation (PCE) protocols in the setting of Java virtual machine 
+    using extended Timed Automata(TA) formalism of the UPPAAL tool. Although a detailed 
+    formal model of combined PI and PCE is given, the number of properties is quite 
+    small and the focus is put on the harmonious working of PI and PCE. Most key features of PI 
+    (as well as PCE) are not shown. Because of the limitation of the model checking technique
+    used there, properties are shown only for a small number of scenarios. Therefore, 
+    the verification does not show the correctness of the formal model itself in a 
+    convincing way.  
+  \item {\em Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC}
+    \cite{Faria08}. A formal model of PI is given in TLA+. Only 3 properties are shown 
+    for PI using model checking. The limitation of model checking is intrinsic to the work.
+  \item {\em Synchronous modeling and validation of priority inheritance schedulers}
+    \cite{conf/fase/JahierHR09}. Gives a formal model
+    of PI and PCE in AADL (Architecture Analysis \& Design Language) and checked 
+    several properties using model checking. The number of properties shown there is 
+    less than here and the scale is also limited by the model checking technique. 
+  \item {\em The Priority Ceiling Protocol: Formalization and Analysis Using PVS}
+    \cite{dutertre99b}. Formalized another protocol for Priority Inversion in the 
+    interactive theorem proving system PVS.
+\end{enumerate}
+
+
+  There are several works on inversion avoidance:
+  \begin{enumerate}
+  \item {\em Solving the group priority inversion problem in a timed asynchronous system}
+    \cite{Wang:2002:SGP}. The notion of Group Priority Inversion is introduced. The main 
+    strategy is still inversion avoidance. The method is by reordering requests 
+    in the setting of Client-Server.
+  \item {\em A Formalization of Priority Inversion} \cite{journals/rts/BabaogluMS93}. 
+    Formalized the notion of Priority 
+    Inversion and proposes methods to avoid it. 
+  \end{enumerate}
+
+  {\em Examples of inaccurate specification of the protocol ???}.
+
+*}
+
+section {* Conclusions \label{conclusion} *}
+
+(*<*)
+end
+(*>*)
\ No newline at end of file
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Paper/PrioGDef.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/PrioGDef.tex	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,488 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{PrioGDef}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\begin{isamarkuptext}%
+In this section, the formal model of Priority Inheritance is presented. First, the identifiers of {\em threads},
+  {\em priority} and {\em critical resources } (abbreviated as \isa{cs}) are all represented as natural numbers,
+  i.e. standard Isabelle/HOL type \isa{nat}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ thread\ {\isaliteral{3D}{\isacharequal}}\ nat\ %
+\isamarkupcmt{Type for thread identifiers.%
+}
+\isanewline
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ priority\ {\isaliteral{3D}{\isacharequal}}\ nat\ \ %
+\isamarkupcmt{Type for priorities.%
+}
+\isanewline
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ cs\ {\isaliteral{3D}{\isacharequal}}\ nat\ %
+\isamarkupcmt{Type for critical sections (or critical resources).%
+}
+%
+\begin{isamarkuptext}%
+Priority Inheritance protocol is modeled as an event driven system, where every event represents an 
+  system call. Event format is given by the following type definition:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ event\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ Create\ thread\ priority\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Thread \isa{thread} is created with priority \isa{priority}.%
+}
+\isanewline
+\ \ Exit\ thread\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Thread \isa{thread} finishing its execution.%
+}
+\isanewline
+\ \ P\ thread\ cs\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Thread \isa{thread} requesting critical resource \isa{cs}.%
+}
+\isanewline
+\ \ V\ thread\ cs\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Thread \isa{thread}  releasing critical resource \isa{cs}.%
+}
+\isanewline
+\ \ Set\ thread\ priority\ %
+\isamarkupcmt{Thread \isa{thread} resets its priority to \isa{priority}.%
+}
+%
+\begin{isamarkuptext}%
+Resource Allocation Graph (RAG for short) is used extensively in the analysis of Priority Inheritance. 
+  The following type \isa{node} is used to model nodes in RAG.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ node\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ Th\ {\isaliteral{22}{\isachardoublequoteopen}}thread{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Node for thread.%
+}
+\isanewline
+\ \ \ Cs\ {\isaliteral{22}{\isachardoublequoteopen}}cs{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\isamarkupcmt{Node for critical resource.%
+}
+%
+\begin{isamarkuptext}%
+The protocol is analyzed using Paulson's inductive protocol verification method, where 
+  the state of the system is modelled as the list of events happened so far with the latest 
+  event at the head. Therefore, the state of the system is represented by the following
+  type \isa{state}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ state\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}event\ list{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following \isa{threads} is used to calculate the set of live threads (\isa{threads\ s})
+  in state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ threads\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ \isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{At the start of the system, the set of threads is empty.%
+}
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}Create\ thread\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}thread{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ threads\ s{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{New thread is added to the \isa{threads}.%
+}
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}Exit\ thread\ {\isaliteral{23}{\isacharhash}}\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}threads\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{7B}{\isacharbraceleft}}thread{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Finished thread is removed.%
+}
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ threads\ s{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\isamarkupcmt{other kind of events does not affect the value of \isa{threads}.%
+}
+%
+\begin{isamarkuptext}%
+Functions such as \isa{threads}, which extract information out of system states, are called
+  {\em observing functions}. A series of observing functions will be defined in the sequel in order to 
+  model the protocol. 
+  Observing function \isa{original{\isaliteral{5F}{\isacharunderscore}}priority} calculates 
+  the {\em original priority} of thread \isa{th} in state \isa{s}, expressed as
+  : \isa{original{\isaliteral{5F}{\isacharunderscore}}priority\ th\ s}. The {\em original priority} is the priority 
+  assigned to a thread when it is created or when it is reset by system call \isa{Set\ thread\ priority}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ original{\isaliteral{5F}{\isacharunderscore}}priority\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ priority{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{\isa{{\isadigit{0}}} is assigned to threads which have never been created.%
+}
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}Create\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ thread{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ thread\ then\ prio\ else\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}Set\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ thread{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ thread\ then\ prio\ else\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{birthtime\ th\ s} is the time when thread \isa{th} is created, observed from state \isa{s}.
+  The time in the system is measured by the number of events happened so far since the very beginning.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ birthtime\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Create\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{3D}{\isacharequal}}\ thread{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ then\ length\ s\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Set\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{3D}{\isacharequal}}\ thread{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ then\ length\ s\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ birthtime\ thread\ s{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of 
+  a thread is the combination of its {\em original priority} and {\em birth time}. The intention is
+  to discriminate threads with the same priority by giving threads with the earlier assigned priority
+  higher precedence in scheduling. This explains the following definition:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ preced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}preced\ thread\ s\ {\isaliteral{3D}{\isacharequal}}\ Prc\ {\isaliteral{28}{\isacharparenleft}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+A number of important notions are defined here:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{consts}\isamarkupfalse%
+\ \isanewline
+\ \ holding\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isanewline
+\ \ \ \ \ \ \ waiting\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \ \ \ depend\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}node\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ node{\isaliteral{29}{\isacharparenright}}\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \ \ \ dependents\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The definition of the following several functions, it is supposed that
+  the waiting queue of every critical resource is given by a waiting queue 
+  function \isa{wq}, which servers as arguments of these functions.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{defs}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{overloaded}{\isaliteral{29}{\isacharparenright}}\ \isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+  We define that the thread which is at the head of waiting queue of resource \isa{cs}
+  is holding the resource. This definition is slightly different from tradition where
+  all threads in the waiting queue are considered as waiting for the resource.
+  This notion is reflected in the definition of \isa{holding\ wq\ th\ cs} as follows:
+  \end{minipage}%
+}
+\isanewline
+\ \ cs{\isaliteral{5F}{\isacharunderscore}}holding{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}holding\ wq\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+  In accordance with the definition of \isa{holding\ wq\ th\ cs}, 
+  a thread \isa{th} is considered waiting for \isa{cs} if 
+  it is in the {\em waiting queue} of critical resource \isa{cs}, but not at the head.
+  This is reflected in the definition of \isa{waiting\ wq\ th\ cs} as follows:
+  \end{minipage}%
+}
+\isanewline
+\ \ cs{\isaliteral{5F}{\isacharunderscore}}waiting{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}waiting\ wq\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+  \isa{depend\ wq} represents the Resource Allocation Graph of the system under the waiting 
+  queue function \isa{wq}.
+  \end{minipage}%
+}
+\isanewline
+\ \ cs{\isaliteral{5F}{\isacharunderscore}}depend{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}depend\ {\isaliteral{28}{\isacharparenleft}}wq{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Th\ t{\isaliteral{2C}{\isacharcomma}}\ Cs\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ t\ c{\isaliteral{2E}{\isachardot}}\ waiting\ wq\ t\ c{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Cs\ c{\isaliteral{2C}{\isacharcomma}}\ Th\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ c\ t{\isaliteral{2E}{\isachardot}}\ holding\ wq\ t\ c{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+  \isa{dependents\ wq\ th} represents the set of threads which are depending on
+  thread \isa{th} in Resource Allocation Graph \isa{depend\ wq}:
+  \end{minipage}%
+}
+\isanewline
+\ \ cs{\isaliteral{5F}{\isacharunderscore}}dependents{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dependents\ {\isaliteral{28}{\isacharparenleft}}wq{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ th\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{27}{\isacharprime}}\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ wq{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The data structure used by the operating system for scheduling is referred to as 
+  {\em schedule state}. It is represented as a record consisting of 
+  a function assigning waiting queue to resources and a function assigning precedence to 
+  threads:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{record}\isamarkupfalse%
+\ schedule{\isaliteral{5F}{\isacharunderscore}}state\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ \ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\isamarkupcmt{The function assigning waiting queue.%
+}
+\isanewline
+\ \ \ \ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\isamarkupcmt{The function assigning precedence.%
+}
+%
+\begin{isamarkuptext}%
+\isa{cpreced\ s\ th} gives the {\em current precedence} of thread \isa{th} under
+  state \isa{s}. The definition of \isa{cpreced} reflects the basic idea of 
+  Priority Inheritance that the {\em current precedence} of a thread is the precedence 
+  inherited from the maximum of all its dependents, i.e. the threads which are waiting 
+  directly or indirectly waiting for some resources from it. If no such thread exits, 
+  \isa{th}'s {\em current precedence} equals its original precedence, i.e. 
+  \isa{preced\ th\ s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cpreced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cpreced\ s\ wq\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ th{\isaliteral{2E}{\isachardot}}\ Max\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ th{\isaliteral{2E}{\isachardot}}\ preced\ th\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ dependents\ wq\ th{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following function \isa{schs} is used to calculate the schedule state \isa{schs\ s}.
+  It is the key function to model Priority Inheritance:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ schs\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ schedule{\isaliteral{5F}{\isacharunderscore}}state{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ \ {\isaliteral{22}{\isachardoublequoteopen}}schs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{2C}{\isacharcomma}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3D}{\isacharequal}}\ cpreced\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+  \begin{enumerate}
+  \item \isa{ps} is the schedule state of last moment.
+  \item \isa{pwq} is the waiting queue function of last moment.
+  \item \isa{pcp} is the precedence function of last moment. 
+  \item \isa{nwq} is the new waiting queue function. It is calculated using a \isa{case} statement:
+  \begin{enumerate}
+      \item If the happening event is \isa{P\ thread\ cs}, \isa{thread} is added to 
+            the end of \isa{cs}'s waiting queue.
+      \item If the happening event is \isa{V\ thread\ cs} and \isa{s} is a legal state,
+            \isa{th{\isaliteral{27}{\isacharprime}}} must equal to \isa{thread}, 
+            because \isa{thread} is the one currently holding \isa{cs}. 
+            The case \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} may never be executed in a legal state.
+            the \isa{{\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ qs{\isaliteral{29}{\isacharparenright}}} is used to choose arbitrarily one 
+            thread in waiting to take over the released resource \isa{cs}. In our representation,
+            this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
+      \item For other happening event, the schedule state just does not change.
+  \end{enumerate}
+  \item \isa{ncp} is new precedence function, it is calculated from the newly updated waiting queue 
+        function. The dependency of precedence function on waiting queue function is the reason to 
+        put them in the same record so that they can evolve together.
+  \end{enumerate}
+  \end{minipage}%
+}
+\isanewline
+\ \ \ {\isaliteral{22}{\isachardoublequoteopen}}schs\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}let\ ps\ {\isaliteral{3D}{\isacharequal}}\ schs\ s\ in\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ pwq\ {\isaliteral{3D}{\isacharequal}}\ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ ps\ in\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ pcp\ {\isaliteral{3D}{\isacharequal}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ ps\ in\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ nwq\ {\isaliteral{3D}{\isacharequal}}\ case\ e\ of\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P\ thread\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ \ pwq{\isaliteral{28}{\isacharparenleft}}cs{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{28}{\isacharparenleft}}pwq\ cs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}thread{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ V\ thread\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ let\ nq\ {\isaliteral{3D}{\isacharequal}}\ case\ {\isaliteral{28}{\isacharparenleft}}pwq\ cs{\isaliteral{29}{\isacharparenright}}\ of\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7C}{\isacharbar}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}th{\isaliteral{27}{\isacharprime}}{\isaliteral{23}{\isacharhash}}qs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ qs{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ pwq{\isaliteral{28}{\isacharparenleft}}cs{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}nq{\isaliteral{29}{\isacharparenright}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ pwq\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ let\ ncp\ {\isaliteral{3D}{\isacharequal}}\ cpreced\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ nwq\ in\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3D}{\isacharequal}}\ nwq{\isaliteral{2C}{\isacharcomma}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3D}{\isacharequal}}\ ncp{\isaliteral{5C3C72706172723E}{\isasymrparr}}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{wq} is a shorthand for \isa{waiting{\isaliteral{5F}{\isacharunderscore}}queue}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ wq\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{22}{\isachardoublequoteclose}}\ \isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}wq\ s\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{28}{\isacharparenleft}}schs\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{cp} is a shorthand for \isa{cur{\isaliteral{5F}{\isacharunderscore}}preced}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cp\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cp\ s\ {\isaliteral{3D}{\isacharequal}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{28}{\isacharparenleft}}schs\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+Functions \isa{holding}, \isa{waiting}, \isa{depend} and \isa{dependents} still have the 
+  same meaning, but redefined so that they no longer depend on the fictitious {\em waiting queue function}
+  \isa{wq}, but on system state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{defs}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{overloaded}{\isaliteral{29}{\isacharparenright}}\ \isanewline
+\ \ s{\isaliteral{5F}{\isacharunderscore}}holding{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}holding\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ s{\isaliteral{5F}{\isacharunderscore}}waiting{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}waiting\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ s{\isaliteral{5F}{\isacharunderscore}}depend{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}depend\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Th\ t{\isaliteral{2C}{\isacharcomma}}\ Cs\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ t\ c{\isaliteral{2E}{\isachardot}}\ waiting\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}\ t\ c{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Cs\ c{\isaliteral{2C}{\isacharcomma}}\ Th\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ c\ t{\isaliteral{2E}{\isachardot}}\ holding\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}\ t\ c{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ s{\isaliteral{5F}{\isacharunderscore}}dependents{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dependents\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ th\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{27}{\isacharprime}}\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following function \isa{readys} calculates the set of ready threads. A thread is {\em ready} 
+  for running if it is a live thread and it is not waiting for any critical resource.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ readys\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}readys\ s\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}thread\ {\isaliteral{2E}{\isachardot}}\ thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ threads\ s\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ waiting\ s\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following function \isa{runing} calculates the set of running thread, which is the ready 
+  thread with the highest precedence.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ runing\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}runing\ s\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th\ {\isaliteral{2E}{\isachardot}}\ th\ {\isaliteral{5C3C696E3E}{\isasymin}}\ readys\ s\ {\isaliteral{5C3C616E643E}{\isasymand}}\ cp\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ Max\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}cp\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{28}{\isacharparenleft}}readys\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following function \isa{holdents\ s\ th} returns the set of resources held by thread 
+  \isa{th} in state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ holdents\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}holdents\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}cs\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ depend\ s{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{cntCS\ s\ th} returns the number of resources held by thread \isa{th} in
+  state \isa{s}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cntCS\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntCS\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ card\ {\isaliteral{28}{\isacharparenleft}}holdents\ s\ th{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The fact that event \isa{e} is eligible to happen next in state \isa{s} 
+  is expressed as \isa{step\ s\ e}. The predicate \isa{step} is inductively defined as 
+  follows:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{inductive}\isamarkupfalse%
+\ step\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ event\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ %
+\isamarkupcmt{A thread can be created if it is not a live thread:%
+}
+\isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}create{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ threads\ s{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Create\ thread\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{A thread can exit if it no longer hold any resource:%
+}
+\isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}exit{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ holdents\ s\ thread\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Exit\ thread{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{A thread can request for an critical resource \isa{cs}, if it is running and 
+  the request does not form a loop in the current RAG. The latter condition 
+  is set up to avoid deadlock. The condition also reflects our assumption all threads are 
+  carefully programmed so that deadlock can not happen.%
+}
+\isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}P{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ \ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{2C}{\isacharcomma}}\ Th\ thread{\isaliteral{29}{\isacharparenright}}\ \ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}P\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{A thread can release a critical resource \isa{cs} if it is running and holding that resource.%
+}
+\isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}V{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ \ holding\ s\ thread\ cs{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}V\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{A thread can adjust its own priority as long as it is current running.%
+}
+\ \ \isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}set{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Set\ thread\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+With predicate \isa{step}, the fact that \isa{s} is a legal state in 
+  Priority Inheritance protocol can be expressed as: \isa{vt\ step\ s}, where
+  the predicate \isa{vt} can be defined as the following:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{inductive}\isamarkupfalse%
+\ vt\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ event\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \isakeyword{for}\ cs\ %
+\isamarkupcmt{\isa{cs} is an argument representing any step predicate.%
+}
+\isanewline
+\isakeyword{where}\isanewline
+\ \ %
+\isamarkupcmt{Empty list \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} is a legal state in any protocol:%
+}
+\isanewline
+\ \ vt{\isaliteral{5F}{\isacharunderscore}}nil{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}vt\ cs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{If \isa{s} a legal state, and event \isa{e} is eligible to happen
+        in state \isa{s}, then \isa{e{\isaliteral{23}{\isacharhash}}{\isaliteral{23}{\isacharhash}}s} is a legal state as well:%
+}
+\isanewline
+\ \ vt{\isaliteral{5F}{\isacharunderscore}}cons{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}vt\ cs\ s{\isaliteral{3B}{\isacharsemicolon}}\ cs\ s\ e{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ vt\ cs\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+It is easy to see that the definition of \isa{vt} is generic. It can be applied to 
+  any step predicate to get the set of legal states.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The following two functions \isa{the{\isaliteral{5F}{\isacharunderscore}}cs} and \isa{the{\isaliteral{5F}{\isacharunderscore}}th} are used to extract
+  critical resource and thread respectively out of RAG nodes.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ the{\isaliteral{5F}{\isacharunderscore}}cs\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}node\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}the{\isaliteral{5F}{\isacharunderscore}}cs\ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ cs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\isacommand{fun}\isamarkupfalse%
+\ the{\isaliteral{5F}{\isacharunderscore}}th\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}node\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}the{\isaliteral{5F}{\isacharunderscore}}th\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ th{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following predicate \isa{next{\isaliteral{5F}{\isacharunderscore}}th} describe the next thread to 
+  take over when a critical resource is released. In \isa{next{\isaliteral{5F}{\isacharunderscore}}th\ s\ th\ cs\ t}, 
+  \isa{th} is the thread to release, \isa{t} is the one to take over.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ next{\isaliteral{5F}{\isacharunderscore}}th{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}next{\isaliteral{5F}{\isacharunderscore}}th\ s\ th\ cs\ t\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}\ rest{\isaliteral{2E}{\isachardot}}\ wq\ s\ cs\ {\isaliteral{3D}{\isacharequal}}\ th{\isaliteral{23}{\isacharhash}}rest\ {\isaliteral{5C3C616E643E}{\isasymand}}\ rest\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ rest{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The function \isa{count\ Q\ l} is used to count the occurrence of situation \isa{Q}
+  in list \isa{l}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ count\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}count\ Q\ l\ {\isaliteral{3D}{\isacharequal}}\ length\ {\isaliteral{28}{\isacharparenleft}}filter\ Q\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{cntP\ s} returns the number of operation \isa{P} happened 
+  before reaching state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cntP\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntP\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ count\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ e{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}\ cs{\isaliteral{2E}{\isachardot}}\ e\ {\isaliteral{3D}{\isacharequal}}\ P\ th\ cs{\isaliteral{29}{\isacharparenright}}\ s{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{cntV\ s} returns the number of operation \isa{V} happened 
+  before reaching state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cntV\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntV\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ count\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ e{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}\ cs{\isaliteral{2E}{\isachardot}}\ e\ {\isaliteral{3D}{\isacharequal}}\ V\ th\ cs{\isaliteral{29}{\isacharparenright}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+\isanewline
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Paper/ROOT.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/ROOT.ML	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1 @@
+use_thy "Paper";
\ No newline at end of file
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Paper/document/llncs.cls
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/document/llncs.cls	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1189 @@
+% LLNCS DOCUMENT CLASS -- version 2.13 (28-Jan-2002)
+% Springer Verlag LaTeX2e support for Lecture Notes in Computer Science
+%
+%%
+%% \CharacterTable
+%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%%   Digits        \0\1\2\3\4\5\6\7\8\9
+%%   Exclamation   \!     Double quote  \"     Hash (number) \#
+%%   Dollar        \$     Percent       \%     Ampersand     \&
+%%   Acute accent  \'     Left paren    \(     Right paren   \)
+%%   Asterisk      \*     Plus          \+     Comma         \,
+%%   Minus         \-     Point         \.     Solidus       \/
+%%   Colon         \:     Semicolon     \;     Less than     \<
+%%   Equals        \=     Greater than  \>     Question mark \?
+%%   Commercial at \@     Left bracket  \[     Backslash     \\
+%%   Right bracket \]     Circumflex    \^     Underscore    \_
+%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
+%%   Right brace   \}     Tilde         \~}
+%%
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesClass{llncs}[2002/01/28 v2.13
+^^J LaTeX document class for Lecture Notes in Computer Science]
+% Options
+\let\if@envcntreset\iffalse
+\DeclareOption{envcountreset}{\let\if@envcntreset\iftrue}
+\DeclareOption{citeauthoryear}{\let\citeauthoryear=Y}
+\DeclareOption{oribibl}{\let\oribibl=Y}
+\let\if@custvec\iftrue
+\DeclareOption{orivec}{\let\if@custvec\iffalse}
+\let\if@envcntsame\iffalse
+\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue}
+\let\if@envcntsect\iffalse
+\DeclareOption{envcountsect}{\let\if@envcntsect\iftrue}
+\let\if@runhead\iffalse
+\DeclareOption{runningheads}{\let\if@runhead\iftrue}
+
+\let\if@openbib\iffalse
+\DeclareOption{openbib}{\let\if@openbib\iftrue}
+
+% languages
+\let\switcht@@therlang\relax
+\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}}
+\def\ds@francais{\def\switcht@@therlang{\switcht@francais}}
+
+\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
+
+\ProcessOptions
+
+\LoadClass[twoside]{article}
+\RequirePackage{multicol} % needed for the list of participants, index
+
+\setlength{\textwidth}{12.2cm}
+\setlength{\textheight}{19.3cm}
+\renewcommand\@pnumwidth{2em}
+\renewcommand\@tocrmarg{3.5em}
+%
+\def\@dottedtocline#1#2#3#4#5{%
+  \ifnum #1>\c@tocdepth \else
+    \vskip \z@ \@plus.2\p@
+    {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by 0pt plus 2cm
+               \parfillskip -\rightskip \pretolerance=10000
+     \parindent #2\relax\@afterindenttrue
+     \interlinepenalty\@M
+     \leavevmode
+     \@tempdima #3\relax
+     \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
+     {#4}\nobreak
+     \leaders\hbox{$\m@th
+        \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
+        mu$}\hfill
+     \nobreak
+     \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}%
+     \par}%
+  \fi}
+%
+\def\switcht@albion{%
+\def\abstractname{Abstract.}
+\def\ackname{Acknowledgement.}
+\def\andname{and}
+\def\lastandname{\unskip, and}
+\def\appendixname{Appendix}
+\def\chaptername{Chapter}
+\def\claimname{Claim}
+\def\conjecturename{Conjecture}
+\def\contentsname{Table of Contents}
+\def\corollaryname{Corollary}
+\def\definitionname{Definition}
+\def\examplename{Example}
+\def\exercisename{Exercise}
+\def\figurename{Fig.}
+\def\keywordname{{\bf Key words:}}
+\def\indexname{Index}
+\def\lemmaname{Lemma}
+\def\contriblistname{List of Contributors}
+\def\listfigurename{List of Figures}
+\def\listtablename{List of Tables}
+\def\mailname{{\it Correspondence to\/}:}
+\def\noteaddname{Note added in proof}
+\def\notename{Note}
+\def\partname{Part}
+\def\problemname{Problem}
+\def\proofname{Proof}
+\def\propertyname{Property}
+\def\propositionname{Proposition}
+\def\questionname{Question}
+\def\remarkname{Remark}
+\def\seename{see}
+\def\solutionname{Solution}
+\def\subclassname{{\it Subject Classifications\/}:}
+\def\tablename{Table}
+\def\theoremname{Theorem}}
+\switcht@albion
+% Names of theorem like environments are already defined
+% but must be translated if another language is chosen
+%
+% French section
+\def\switcht@francais{%\typeout{On parle francais.}%
+ \def\abstractname{R\'esum\'e.}%
+ \def\ackname{Remerciements.}%
+ \def\andname{et}%
+ \def\lastandname{ et}%
+ \def\appendixname{Appendice}
+ \def\chaptername{Chapitre}%
+ \def\claimname{Pr\'etention}%
+ \def\conjecturename{Hypoth\`ese}%
+ \def\contentsname{Table des mati\`eres}%
+ \def\corollaryname{Corollaire}%
+ \def\definitionname{D\'efinition}%
+ \def\examplename{Exemple}%
+ \def\exercisename{Exercice}%
+ \def\figurename{Fig.}%
+ \def\keywordname{{\bf Mots-cl\'e:}}
+ \def\indexname{Index}
+ \def\lemmaname{Lemme}%
+ \def\contriblistname{Liste des contributeurs}
+ \def\listfigurename{Liste des figures}%
+ \def\listtablename{Liste des tables}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}%
+ \def\notename{Remarque}%
+ \def\partname{Partie}%
+ \def\problemname{Probl\`eme}%
+ \def\proofname{Preuve}%
+ \def\propertyname{Caract\'eristique}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Question}%
+ \def\remarkname{Remarque}%
+ \def\seename{voir}
+ \def\solutionname{Solution}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tableau}%
+ \def\theoremname{Th\'eor\`eme}%
+}
+%
+% German section
+\def\switcht@deutsch{%\typeout{Man spricht deutsch.}%
+ \def\abstractname{Zusammenfassung.}%
+ \def\ackname{Danksagung.}%
+ \def\andname{und}%
+ \def\lastandname{ und}%
+ \def\appendixname{Anhang}%
+ \def\chaptername{Kapitel}%
+ \def\claimname{Behauptung}%
+ \def\conjecturename{Hypothese}%
+ \def\contentsname{Inhaltsverzeichnis}%
+ \def\corollaryname{Korollar}%
+%\def\definitionname{Definition}%
+ \def\examplename{Beispiel}%
+ \def\exercisename{\"Ubung}%
+ \def\figurename{Abb.}%
+ \def\keywordname{{\bf Schl\"usselw\"orter:}}
+ \def\indexname{Index}
+%\def\lemmaname{Lemma}%
+ \def\contriblistname{Mitarbeiter}
+ \def\listfigurename{Abbildungsverzeichnis}%
+ \def\listtablename{Tabellenverzeichnis}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Nachtrag}%
+ \def\notename{Anmerkung}%
+ \def\partname{Teil}%
+%\def\problemname{Problem}%
+ \def\proofname{Beweis}%
+ \def\propertyname{Eigenschaft}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Frage}%
+ \def\remarkname{Anmerkung}%
+ \def\seename{siehe}
+ \def\solutionname{L\"osung}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tabelle}%
+%\def\theoremname{Theorem}%
+}
+
+% Ragged bottom for the actual page
+\def\thisbottomragged{\def\@textbottom{\vskip\z@ plus.0001fil
+\global\let\@textbottom\relax}}
+
+\renewcommand\small{%
+   \@setfontsize\small\@ixpt{11}%
+   \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@
+   \abovedisplayshortskip \z@ \@plus2\p@
+   \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@
+   \def\@listi{\leftmargin\leftmargini
+               \parsep 0\p@ \@plus1\p@ \@minus\p@
+               \topsep 8\p@ \@plus2\p@ \@minus4\p@
+               \itemsep0\p@}%
+   \belowdisplayskip \abovedisplayskip
+}
+
+\frenchspacing
+\widowpenalty=10000
+\clubpenalty=10000
+
+\setlength\oddsidemargin   {63\p@}
+\setlength\evensidemargin  {63\p@}
+\setlength\marginparwidth  {90\p@}
+
+\setlength\headsep   {16\p@}
+
+\setlength\footnotesep{7.7\p@}
+\setlength\textfloatsep{8mm\@plus 2\p@ \@minus 4\p@}
+\setlength\intextsep   {8mm\@plus 2\p@ \@minus 2\p@}
+
+\setcounter{secnumdepth}{2}
+
+\newcounter {chapter}
+\renewcommand\thechapter      {\@arabic\c@chapter}
+
+\newif\if@mainmatter \@mainmattertrue
+\newcommand\frontmatter{\cleardoublepage
+            \@mainmatterfalse\pagenumbering{Roman}}
+\newcommand\mainmatter{\cleardoublepage
+       \@mainmattertrue\pagenumbering{arabic}}
+\newcommand\backmatter{\if@openright\cleardoublepage\else\clearpage\fi
+      \@mainmatterfalse}
+
+\renewcommand\part{\cleardoublepage
+                 \thispagestyle{empty}%
+                 \if@twocolumn
+                     \onecolumn
+                     \@tempswatrue
+                   \else
+                     \@tempswafalse
+                 \fi
+                 \null\vfil
+                 \secdef\@part\@spart}
+
+\def\@part[#1]#2{%
+    \ifnum \c@secnumdepth >-2\relax
+      \refstepcounter{part}%
+      \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
+    \else
+      \addcontentsline{toc}{part}{#1}%
+    \fi
+    \markboth{}{}%
+    {\centering
+     \interlinepenalty \@M
+     \normalfont
+     \ifnum \c@secnumdepth >-2\relax
+       \huge\bfseries \partname~\thepart
+       \par
+       \vskip 20\p@
+     \fi
+     \Huge \bfseries #2\par}%
+    \@endpart}
+\def\@spart#1{%
+    {\centering
+     \interlinepenalty \@M
+     \normalfont
+     \Huge \bfseries #1\par}%
+    \@endpart}
+\def\@endpart{\vfil\newpage
+              \if@twoside
+                \null
+                \thispagestyle{empty}%
+                \newpage
+              \fi
+              \if@tempswa
+                \twocolumn
+              \fi}
+
+\newcommand\chapter{\clearpage
+                    \thispagestyle{empty}%
+                    \global\@topnum\z@
+                    \@afterindentfalse
+                    \secdef\@chapter\@schapter}
+\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
+                       \if@mainmatter
+                         \refstepcounter{chapter}%
+                         \typeout{\@chapapp\space\thechapter.}%
+                         \addcontentsline{toc}{chapter}%
+                                  {\protect\numberline{\thechapter}#1}%
+                       \else
+                         \addcontentsline{toc}{chapter}{#1}%
+                       \fi
+                    \else
+                      \addcontentsline{toc}{chapter}{#1}%
+                    \fi
+                    \chaptermark{#1}%
+                    \addtocontents{lof}{\protect\addvspace{10\p@}}%
+                    \addtocontents{lot}{\protect\addvspace{10\p@}}%
+                    \if@twocolumn
+                      \@topnewpage[\@makechapterhead{#2}]%
+                    \else
+                      \@makechapterhead{#2}%
+                      \@afterheading
+                    \fi}
+\def\@makechapterhead#1{%
+% \vspace*{50\p@}%
+  {\centering
+    \ifnum \c@secnumdepth >\m@ne
+      \if@mainmatter
+        \large\bfseries \@chapapp{} \thechapter
+        \par\nobreak
+        \vskip 20\p@
+      \fi
+    \fi
+    \interlinepenalty\@M
+    \Large \bfseries #1\par\nobreak
+    \vskip 40\p@
+  }}
+\def\@schapter#1{\if@twocolumn
+                   \@topnewpage[\@makeschapterhead{#1}]%
+                 \else
+                   \@makeschapterhead{#1}%
+                   \@afterheading
+                 \fi}
+\def\@makeschapterhead#1{%
+% \vspace*{50\p@}%
+  {\centering
+    \normalfont
+    \interlinepenalty\@M
+    \Large \bfseries  #1\par\nobreak
+    \vskip 40\p@
+  }}
+
+\renewcommand\section{\@startsection{section}{1}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {12\p@ \@plus 4\p@ \@minus 4\p@}%
+                       {\normalfont\large\bfseries\boldmath
+                        \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {8\p@ \@plus 4\p@ \@minus 4\p@}%
+                       {\normalfont\normalsize\bfseries\boldmath
+                        \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {-0.5em \@plus -0.22em \@minus -0.1em}%
+                       {\normalfont\normalsize\bfseries\boldmath}}
+\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
+                       {-12\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {-0.5em \@plus -0.22em \@minus -0.1em}%
+                       {\normalfont\normalsize\itshape}}
+\renewcommand\subparagraph[1]{\typeout{LLNCS warning: You should not use
+                  \string\subparagraph\space with this class}\vskip0.5cm
+You should not use \verb|\subparagraph| with this class.\vskip0.5cm}
+
+\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
+\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
+\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
+\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
+\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
+\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
+\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
+\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
+\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
+\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
+\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}
+
+\let\footnotesize\small
+
+\if@custvec
+\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}}
+{\mbox{\boldmath$\textstyle#1$}}
+{\mbox{\boldmath$\scriptstyle#1$}}
+{\mbox{\boldmath$\scriptscriptstyle#1$}}}
+\fi
+
+\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}}
+\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil
+\penalty50\hskip1em\null\nobreak\hfil\squareforqed
+\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
+
+\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+\gets\cr\to\cr}}}}}
+\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+<\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
+>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.8pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.3pt}<\cr}}}}}
+\def\bbbr{{\rm I\!R}} %reelle Zahlen
+\def\bbbm{{\rm I\!M}}
+\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
+\def\bbbf{{\rm I\!F}}
+\def\bbbh{{\rm I\!H}}
+\def\bbbk{{\rm I\!K}}
+\def\bbbp{{\rm I\!P}}
+\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
+{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
+\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
+\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbs{{\mathchoice
+{\setbox0=\hbox{$\displaystyle     \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle        \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle      \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
+\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
+{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
+
+\let\ts\,
+
+\setlength\leftmargini  {17\p@}
+\setlength\leftmargin    {\leftmargini}
+\setlength\leftmarginii  {\leftmargini}
+\setlength\leftmarginiii {\leftmargini}
+\setlength\leftmarginiv  {\leftmargini}
+\setlength  \labelsep  {.5em}
+\setlength  \labelwidth{\leftmargini}
+\addtolength\labelwidth{-\labelsep}
+
+\def\@listI{\leftmargin\leftmargini
+            \parsep 0\p@ \@plus1\p@ \@minus\p@
+            \topsep 8\p@ \@plus2\p@ \@minus4\p@
+            \itemsep0\p@}
+\let\@listi\@listI
+\@listi
+\def\@listii {\leftmargin\leftmarginii
+              \labelwidth\leftmarginii
+              \advance\labelwidth-\labelsep
+              \topsep    0\p@ \@plus2\p@ \@minus\p@}
+\def\@listiii{\leftmargin\leftmarginiii
+              \labelwidth\leftmarginiii
+              \advance\labelwidth-\labelsep
+              \topsep    0\p@ \@plus\p@\@minus\p@
+              \parsep    \z@
+              \partopsep \p@ \@plus\z@ \@minus\p@}
+
+\renewcommand\labelitemi{\normalfont\bfseries --}
+\renewcommand\labelitemii{$\m@th\bullet$}
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
+                                                    {{\contentsname}}}
+ \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
+ \def\lastand{\ifnum\value{auco}=2\relax
+                 \unskip{} \andname\
+              \else
+                 \unskip \lastandname\
+              \fi}%
+ \def\and{\stepcounter{@auth}\relax
+          \ifnum\value{@auth}=\value{auco}%
+             \lastand
+          \else
+             \unskip,
+          \fi}%
+ \@starttoc{toc}\if@restonecol\twocolumn\fi}
+
+\def\l@part#1#2{\addpenalty{\@secpenalty}%
+   \addvspace{2em plus\p@}%  % space above part line
+   \begingroup
+     \parindent \z@
+     \rightskip \z@ plus 5em
+     \hrule\vskip5pt
+     \large               % same size as for a contribution heading
+     \bfseries\boldmath   % set line in boldface
+     \leavevmode          % TeX command to enter horizontal mode.
+     #1\par
+     \vskip5pt
+     \hrule
+     \vskip1pt
+     \nobreak             % Never break after part entry
+   \endgroup}
+
+\def\@dotsep{2}
+
+\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
+{chapter.\thechapter}\fi}
+
+\def\addnumcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
+                     {\thechapter}#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmarkwop#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}
+
+\def\@adcmk[#1]{\ifcase #1 \or
+\def\@gtempa{\addnumcontentsmark}%
+  \or    \def\@gtempa{\addcontentsmark}%
+  \or    \def\@gtempa{\addcontentsmarkwop}%
+  \fi\@gtempa{toc}{chapter}}
+\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
+
+\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
+ \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ {\large\bfseries\boldmath#1}\ifx0#2\hfil\null
+ \else
+      \nobreak
+      \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+      \@dotsep mu$}\hfill
+      \nobreak\hbox to\@pnumwidth{\hss #2}%
+ \fi\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@title#1#2{\addpenalty{-\@highpenalty}
+ \addvspace{8pt plus 1pt}
+ \@tempdima \z@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ #1\nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hss #2}\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@author#1#2{\addpenalty{\@highpenalty}
+ \@tempdima=\z@ %15\p@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
+ \textit{#1}\par
+ \penalty\@highpenalty \endgroup}
+
+%\setcounter{tocdepth}{0}
+\newdimen\tocchpnum
+\newdimen\tocsecnum
+\newdimen\tocsectotal
+\newdimen\tocsubsecnum
+\newdimen\tocsubsectotal
+\newdimen\tocsubsubsecnum
+\newdimen\tocsubsubsectotal
+\newdimen\tocparanum
+\newdimen\tocparatotal
+\newdimen\tocsubparanum
+\tocchpnum=\z@            % no chapter numbers
+\tocsecnum=15\p@          % section 88. plus 2.222pt
+\tocsubsecnum=23\p@       % subsection 88.8 plus 2.222pt
+\tocsubsubsecnum=27\p@    % subsubsection 88.8.8 plus 1.444pt
+\tocparanum=35\p@         % paragraph 88.8.8.8 plus 1.666pt
+\tocsubparanum=43\p@      % subparagraph 88.8.8.8.8 plus 1.888pt
+\def\calctocindent{%
+\tocsectotal=\tocchpnum
+\advance\tocsectotal by\tocsecnum
+\tocsubsectotal=\tocsectotal
+\advance\tocsubsectotal by\tocsubsecnum
+\tocsubsubsectotal=\tocsubsectotal
+\advance\tocsubsubsectotal by\tocsubsubsecnum
+\tocparatotal=\tocsubsubsectotal
+\advance\tocparatotal by\tocparanum}
+\calctocindent
+
+\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
+\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
+\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
+\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
+\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
+
+\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
+ \@starttoc{lof}\if@restonecol\twocolumn\fi}
+\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
+
+\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
+ \@starttoc{lot}\if@restonecol\twocolumn\fi}
+\let\l@table\l@figure
+
+\renewcommand\listoffigures{%
+    \section*{\listfigurename
+      \@mkboth{\listfigurename}{\listfigurename}}%
+    \@starttoc{lof}%
+    }
+
+\renewcommand\listoftables{%
+    \section*{\listtablename
+      \@mkboth{\listtablename}{\listtablename}}%
+    \@starttoc{lot}%
+    }
+
+\ifx\oribibl\undefined
+\ifx\citeauthoryear\undefined
+\renewenvironment{thebibliography}[1]
+     {\section*{\refname}
+      \def\@biblabel##1{##1.}
+      \small
+      \list{\@biblabel{\@arabic\c@enumiv}}%
+           {\settowidth\labelwidth{\@biblabel{#1}}%
+            \leftmargin\labelwidth
+            \advance\leftmargin\labelsep
+            \if@openbib
+              \advance\leftmargin\bibindent
+              \itemindent -\bibindent
+              \listparindent \itemindent
+              \parsep \z@
+            \fi
+            \usecounter{enumiv}%
+            \let\p@enumiv\@empty
+            \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+      \if@openbib
+        \renewcommand\newblock{\par}%
+      \else
+        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+      \fi
+      \sloppy\clubpenalty4000\widowpenalty4000%
+      \sfcode`\.=\@m}
+     {\def\@noitemerr
+       {\@latex@warning{Empty `thebibliography' environment}}%
+      \endlist}
+\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
+     {\let\protect\noexpand\immediate
+     \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+\newcount\@tempcntc
+\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
+  \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
+    {\@ifundefined
+       {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
+        ?}\@warning
+       {Citation `\@citeb' on page \thepage \space undefined}}%
+    {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
+     \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
+       \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
+     \else
+      \advance\@tempcntb\@ne
+      \ifnum\@tempcntb=\@tempcntc
+      \else\advance\@tempcntb\m@ne\@citeo
+      \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
+\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
+               \@citea\def\@citea{,\,\hskip\z@skip}%
+               \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
+               {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
+                \def\@citea{--}\fi
+      \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
+\else
+\renewenvironment{thebibliography}[1]
+     {\section*{\refname}
+      \small
+      \list{}%
+           {\settowidth\labelwidth{}%
+            \leftmargin\parindent
+            \itemindent=-\parindent
+            \labelsep=\z@
+            \if@openbib
+              \advance\leftmargin\bibindent
+              \itemindent -\bibindent
+              \listparindent \itemindent
+              \parsep \z@
+            \fi
+            \usecounter{enumiv}%
+            \let\p@enumiv\@empty
+            \renewcommand\theenumiv{}}%
+      \if@openbib
+        \renewcommand\newblock{\par}%
+      \else
+        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+      \fi
+      \sloppy\clubpenalty4000\widowpenalty4000%
+      \sfcode`\.=\@m}
+     {\def\@noitemerr
+       {\@latex@warning{Empty `thebibliography' environment}}%
+      \endlist}
+      \def\@cite#1{#1}%
+      \def\@lbibitem[#1]#2{\item[]\if@filesw
+        {\def\protect##1{\string ##1\space}\immediate
+      \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+   \fi
+\else
+\@cons\@openbib@code{\noexpand\small}
+\fi
+
+\def\idxquad{\hskip 10\p@}% space that divides entry from number
+
+\def\@idxitem{\par\hangindent 10\p@}
+
+\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
+                \noindent\hangindent\wd0\box0}% index entry
+
+\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
+                \noindent\hangindent\wd0\box0}% order index entry
+
+\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
+
+\renewenvironment{theindex}
+               {\@mkboth{\indexname}{\indexname}%
+                \thispagestyle{empty}\parindent\z@
+                \parskip\z@ \@plus .3\p@\relax
+                \let\item\par
+                \def\,{\relax\ifmmode\mskip\thinmuskip
+                             \else\hskip0.2em\ignorespaces\fi}%
+                \normalfont\small
+                \begin{multicols}{2}[\@makeschapterhead{\indexname}]%
+                }
+                {\end{multicols}}
+
+\renewcommand\footnoterule{%
+  \kern-3\p@
+  \hrule\@width 2truecm
+  \kern2.6\p@}
+  \newdimen\fnindent
+  \fnindent1em
+\long\def\@makefntext#1{%
+    \parindent \fnindent%
+    \leftskip \fnindent%
+    \noindent
+    \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
+
+\long\def\@makecaption#1#2{%
+  \vskip\abovecaptionskip
+  \sbox\@tempboxa{{\bfseries #1.} #2}%
+  \ifdim \wd\@tempboxa >\hsize
+    {\bfseries #1.} #2\par
+  \else
+    \global \@minipagefalse
+    \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
+  \fi
+  \vskip\belowcaptionskip}
+
+\def\fps@figure{htbp}
+\def\fnum@figure{\figurename\thinspace\thefigure}
+\def \@floatboxreset {%
+        \reset@font
+        \small
+        \@setnobreak
+        \@setminipage
+}
+\def\fps@table{htbp}
+\def\fnum@table{\tablename~\thetable}
+\renewenvironment{table}
+               {\setlength\abovecaptionskip{0\p@}%
+                \setlength\belowcaptionskip{10\p@}%
+                \@float{table}}
+               {\end@float}
+\renewenvironment{table*}
+               {\setlength\abovecaptionskip{0\p@}%
+                \setlength\belowcaptionskip{10\p@}%
+                \@dblfloat{table}}
+               {\end@dblfloat}
+
+\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
+  ext@#1\endcsname}{#1}{\protect\numberline{\csname
+  the#1\endcsname}{\ignorespaces #2}}\begingroup
+    \@parboxrestore
+    \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+  \endgroup}
+
+% LaTeX does not provide a command to enter the authors institute
+% addresses. The \institute command is defined here.
+
+\newcounter{@inst}
+\newcounter{@auth}
+\newcounter{auco}
+\newdimen\instindent
+\newbox\authrun
+\newtoks\authorrunning
+\newtoks\tocauthor
+\newbox\titrun
+\newtoks\titlerunning
+\newtoks\toctitle
+
+\def\clearheadinfo{\gdef\@author{No Author Given}%
+                   \gdef\@title{No Title Given}%
+                   \gdef\@subtitle{}%
+                   \gdef\@institute{No Institute Given}%
+                   \gdef\@thanks{}%
+                   \global\titlerunning={}\global\authorrunning={}%
+                   \global\toctitle={}\global\tocauthor={}}
+
+\def\institute#1{\gdef\@institute{#1}}
+
+\def\institutename{\par
+ \begingroup
+ \parskip=\z@
+ \parindent=\z@
+ \setcounter{@inst}{1}%
+ \def\and{\par\stepcounter{@inst}%
+ \noindent$^{\the@inst}$\enspace\ignorespaces}%
+ \setbox0=\vbox{\def\thanks##1{}\@institute}%
+ \ifnum\c@@inst=1\relax
+   \gdef\fnnstart{0}%
+ \else
+   \xdef\fnnstart{\c@@inst}%
+   \setcounter{@inst}{1}%
+   \noindent$^{\the@inst}$\enspace
+ \fi
+ \ignorespaces
+ \@institute\par
+ \endgroup}
+
+\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
+   {\star\star\star}\or \dagger\or \ddagger\or
+   \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
+   \or \ddagger\ddagger \else\@ctrerr\fi}}
+
+\def\inst#1{\unskip$^{#1}$}
+\def\fnmsep{\unskip$^,$}
+\def\email#1{{\tt#1}}
+\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
+\@ifpackageloaded{babel}{%
+\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
+\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
+\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
+}{\switcht@@therlang}%
+}
+\def\homedir{\~{ }}
+
+\def\subtitle#1{\gdef\@subtitle{#1}}
+\clearheadinfo
+
+\renewcommand\maketitle{\newpage
+  \refstepcounter{chapter}%
+  \stepcounter{section}%
+  \setcounter{section}{0}%
+  \setcounter{subsection}{0}%
+  \setcounter{figure}{0}
+  \setcounter{table}{0}
+  \setcounter{equation}{0}
+  \setcounter{footnote}{0}%
+  \begingroup
+    \parindent=\z@
+    \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
+    \if@twocolumn
+      \ifnum \col@number=\@ne
+        \@maketitle
+      \else
+        \twocolumn[\@maketitle]%
+      \fi
+    \else
+      \newpage
+      \global\@topnum\z@   % Prevents figures from going at top of page.
+      \@maketitle
+    \fi
+    \thispagestyle{empty}\@thanks
+%
+    \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
+    \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
+    \instindent=\hsize
+    \advance\instindent by-\headlineindent
+%    \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
+%       \addcontentsline{toc}{title}{\the\toctitle}\fi
+    \if@runhead
+       \if!\the\titlerunning!\else
+         \edef\@title{\the\titlerunning}%
+       \fi
+       \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
+       \ifdim\wd\titrun>\instindent
+          \typeout{Title too long for running head. Please supply}%
+          \typeout{a shorter form with \string\titlerunning\space prior to
+                   \string\maketitle}%
+          \global\setbox\titrun=\hbox{\small\rm
+          Title Suppressed Due to Excessive Length}%
+       \fi
+       \xdef\@title{\copy\titrun}%
+    \fi
+%
+    \if!\the\tocauthor!\relax
+      {\def\and{\noexpand\protect\noexpand\and}%
+      \protected@xdef\toc@uthor{\@author}}%
+    \else
+      \def\\{\noexpand\protect\noexpand\newline}%
+      \protected@xdef\scratch{\the\tocauthor}%
+      \protected@xdef\toc@uthor{\scratch}%
+    \fi
+%    \addcontentsline{toc}{author}{\toc@uthor}%
+    \if@runhead
+       \if!\the\authorrunning!
+         \value{@inst}=\value{@auth}%
+         \setcounter{@auth}{1}%
+       \else
+         \edef\@author{\the\authorrunning}%
+       \fi
+       \global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}%
+       \ifdim\wd\authrun>\instindent
+          \typeout{Names of authors too long for running head. Please supply}%
+          \typeout{a shorter form with \string\authorrunning\space prior to
+                   \string\maketitle}%
+          \global\setbox\authrun=\hbox{\small\rm
+          Authors Suppressed Due to Excessive Length}%
+       \fi
+       \xdef\@author{\copy\authrun}%
+       \markboth{\@author}{\@title}%
+     \fi
+  \endgroup
+  \setcounter{footnote}{\fnnstart}%
+  \clearheadinfo}
+%
+\def\@maketitle{\newpage
+ \markboth{}{}%
+ \def\lastand{\ifnum\value{@inst}=2\relax
+                 \unskip{} \andname\
+              \else
+                 \unskip \lastandname\
+              \fi}%
+ \def\and{\stepcounter{@auth}\relax
+          \ifnum\value{@auth}=\value{@inst}%
+             \lastand
+          \else
+             \unskip,
+          \fi}%
+ \begin{center}%
+ \let\newline\\
+ {\Large \bfseries\boldmath
+  \pretolerance=10000
+  \@title \par}\vskip .8cm
+\if!\@subtitle!\else {\large \bfseries\boldmath
+  \vskip -.65cm
+  \pretolerance=10000
+  \@subtitle \par}\vskip .8cm\fi
+ \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
+ \def\thanks##1{}\@author}%
+ \global\value{@inst}=\value{@auth}%
+ \global\value{auco}=\value{@auth}%
+ \setcounter{@auth}{1}%
+{\lineskip .5em
+\noindent\ignorespaces
+\@author\vskip.35cm}
+ {\small\institutename}
+ \end{center}%
+ }
+
+% definition of the "\spnewtheorem" command.
+%
+% Usage:
+%
+%     \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
+% or  \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
+% or  \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
+%
+% New is "cap_font" and "body_font". It stands for
+% fontdefinition of the caption and the text itself.
+%
+% "\spnewtheorem*" gives a theorem without number.
+%
+% A defined spnewthoerem environment is used as described
+% by Lamport.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\@thmcountersep{}
+\def\@thmcounterend{.}
+
+\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
+
+% definition of \spnewtheorem with number
+
+\def\@spnthm#1#2{%
+  \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
+\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
+
+\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
+   {\@definecounter{#1}\@addtoreset{#1}{#3}%
+   \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
+     \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
+   \expandafter\xdef\csname #1name\endcsname{#2}%
+   \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
+                              \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+   {\@definecounter{#1}%
+   \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
+   \expandafter\xdef\csname #1name\endcsname{#2}%
+   \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
+                               \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spothm#1[#2]#3#4#5{%
+  \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
+  {\expandafter\@ifdefinable\csname #1\endcsname
+  {\global\@namedef{the#1}{\@nameuse{the#2}}%
+  \expandafter\xdef\csname #1name\endcsname{#3}%
+  \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
+  \global\@namedef{end#1}{\@endtheorem}}}}
+
+\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\refstepcounter{#1}%
+\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
+
+\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
+                    \ignorespaces}
+
+\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
+       the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
+
+\def\@spbegintheorem#1#2#3#4{\trivlist
+                 \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
+
+\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
+      \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
+
+% definition of \spnewtheorem* without number
+
+\def\@sthm#1#2{\@Ynthm{#1}{#2}}
+
+\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+   {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
+    \expandafter\xdef\csname #1name\endcsname{#2}%
+    \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
+
+\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
+
+\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
+       {#4}{#2}{#3}\ignorespaces}
+
+\def\@Begintheorem#1#2#3{#3\trivlist
+                           \item[\hskip\labelsep{#2#1\@thmcounterend}]}
+
+\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
+      \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
+
+\if@envcntsect
+   \def\@thmcountersep{.}
+   \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
+\else
+   \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
+   \if@envcntreset
+      \@addtoreset{theorem}{section}
+   \else
+      \@addtoreset{theorem}{chapter}
+   \fi
+\fi
+
+%definition of divers theorem environments
+\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
+\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
+\if@envcntsame % alle Umgebungen wie Theorem.
+   \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
+\else % alle Umgebungen mit eigenem Zaehler
+   \if@envcntsect % mit section numeriert
+      \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
+   \else % nicht mit section numeriert
+      \if@envcntreset
+         \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+                                   \@addtoreset{#1}{section}}
+      \else
+         \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+                                   \@addtoreset{#1}{chapter}}%
+      \fi
+   \fi
+\fi
+\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
+\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
+\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
+\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
+\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
+\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
+\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
+\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
+\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
+\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
+\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
+\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
+\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
+\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
+
+\def\@takefromreset#1#2{%
+    \def\@tempa{#1}%
+    \let\@tempd\@elt
+    \def\@elt##1{%
+        \def\@tempb{##1}%
+        \ifx\@tempa\@tempb\else
+            \@addtoreset{##1}{#2}%
+        \fi}%
+    \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
+    \expandafter\def\csname cl@#2\endcsname{}%
+    \@tempc
+    \let\@elt\@tempd}
+
+\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+      \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
+                  \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
+      \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
+      }
+
+\renewenvironment{abstract}{%
+      \list{}{\advance\topsep by0.35cm\relax\small
+      \leftmargin=1cm
+      \labelwidth=\z@
+      \listparindent=\z@
+      \itemindent\listparindent
+      \rightmargin\leftmargin}\item[\hskip\labelsep
+                                    \bfseries\abstractname]}
+    {\endlist}
+
+\newdimen\headlineindent             % dimension for space between
+\headlineindent=1.166cm              % number and text of headings.
+
+\def\ps@headings{\let\@mkboth\@gobbletwo
+   \let\@oddfoot\@empty\let\@evenfoot\@empty
+   \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+                  \leftmark\hfil}
+   \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
+                 \llap{\thepage}}
+   \def\chaptermark##1{}%
+   \def\sectionmark##1{}%
+   \def\subsectionmark##1{}}
+
+\def\ps@titlepage{\let\@mkboth\@gobbletwo
+   \let\@oddfoot\@empty\let\@evenfoot\@empty
+   \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+                  \hfil}
+   \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
+                 \llap{\thepage}}
+   \def\chaptermark##1{}%
+   \def\sectionmark##1{}%
+   \def\subsectionmark##1{}}
+
+\if@runhead\ps@headings\else
+\ps@empty\fi
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\endinput
+%end of file llncs.cls
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Paper/document/root.bib
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/document/root.bib	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,152 @@
+@Article{Lampson:Redell:cacm:1980,
+  author =	"B. Lampson and D. Redell",
+  title =	"{Experience with processes and monitors in Mesa}",
+  journal =	"Communications of the ACM",
+  volume =	"23",
+  number =	"2",
+  pages =	"105--117",
+  month =	feb,
+  year = 	"1980",
+  keywords =	"Mesa, processes, monitors",
+}
+
+@Article{journals/tc/ShaRL90,
+  title =	"Priority Inheritance Protocols: An Approach to
+		 Real-Time Synchronization",
+  author =	"S. Liu  and R. Rajkumar and J. P. Lehoczky",
+  journal =	"IEEE Trans. Computers",
+  year = 	"1990",
+  number =	"9",
+  volume =	"39",
+  bibdate =	"2011-10-27",
+  bibsource =	"DBLP,
+		 http://dblp.uni-trier.de/db/journals/tc/tc39.html#ShaRL90",
+  pages =	"1175--1185",
+  URL =  	"http://doi.ieeecomputersociety.org/10.1109/12.57058",
+}
+
+@MISC{yodaiken-july02,
+author = {V. Yodaiken},
+title = {Against Priority Inheritance},
+month = July,
+year = {2002},
+howpublished={\url{http://www.linuxfordevices.com/files/misc/yodaiken-july02.pdf}},
+}
+
+@MISC{locke-july02,
+author = {D. Locke},
+title = {Priority Inheritance: The Real Story},
+month = July,
+year = {2002},
+howpublished={\url{http://www.math.unipd.it/~tullio/SCD/2007/Materiale/Locke.pdf}},
+}
+
+@MISC{Faria08,
+author = {J. M. S. Faria},
+title = {Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC},
+year = {2008},
+howpublished={\url{http://repositorio-aberto.up.pt/bitstream/10216/11466/2/Texto%20integral.pdf}},
+}
+
+
+http://repositorio-aberto.up.pt/bitstream/10216/11466/2/Texto%20integral.pdf
+
+@Article{Reeves-Glenn-1998,
+  title =	"Re: What Really Happened on Mars?",
+  author =	"G. Reeves",
+  journal =	"Risks-Forum Digest",
+  year = 	"1998",
+  month = "January",
+  number =	"58",
+  volume =	"19",
+}
+
+@TechReport{dutertre99b,
+  title =	"The {Priority Ceiling Protocol}: Formalization and
+		 Analysis Using {PVS}",
+  author =	"B. Dutertre",
+  month =	Oct,
+  year = 	"1999",
+  institution =  "System Design Laboratory, SRI International",
+  address =	"Menlo Park, CA",
+  note = 	"Available at
+		 \url{http://www.sdl.sri.com/dsa/publis/prio-ceiling.html}",
+}
+
+@InProceedings{conf/fase/JahierHR09,
+  title =	"Synchronous Modeling and Validation of Priority
+		 Inheritance Schedulers",
+  author =	"E. Jahier and B. Halbwachs and P.
+		 Raymond",
+  bibdate =	"2009-04-01",
+  bibsource =	"DBLP,
+		 http://dblp.uni-trier.de/db/conf/fase/fase2009.html#JahierHR09",
+  booktitle =	"FASE",
+  booktitle =	"Fundamental Approaches to Software Engineering, 12th
+		 International Conference, {FASE} 2009, Held as Part of
+		 the Joint European Conferences on Theory and Practice
+		 of Software, {ETAPS} 2009, York, {UK}, March 22-29,
+		 2009. Proceedings",
+  publisher =	"Springer",
+  year = 	"2009",
+  volume =	"5503",
+  editor =	"Marsha Chechik and Martin Wirsing",
+  ISBN = 	"978-3-642-00592-3",
+  pages =	"140--154",
+  series =	"Lecture Notes in Computer Science",
+  URL =  	"http://dx.doi.org/10.1007/978-3-642-00593-0",
+}
+
+@InProceedings{WellingsBSB07,
+  title =	"Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java",
+  author =	"A. J. Wellings and A. Burns and O. M. Santos and B. M. Brosgol",
+  publisher =	"IEEE Computer Society",
+  year = 	"2007",
+  booktitle =	"Proceedings of the 10th IEEE International Symposium on Object 
+                and Component-Oriented Real-Time Distributed Computing",
+  pages =	"115--123",
+}
+
+@Article{Wang:2002:SGP,
+  author =	"Y. Wang and E. Anceaume and F. Brasileiro and F.
+		 Greve and M. Hurfin",
+  title =	"Solving the group priority inversion problem in a
+		 timed asynchronous system",
+  journal =	"IEEE Transactions on Computers",
+  volume =	"51",
+  number =	"8",
+  pages =	"900--915",
+  month =	aug,
+  year = 	"2002",
+  CODEN =	"ITCOB4",
+  doi =  	"http://dx.doi.org/10.1109/TC.2002.1024738",
+  ISSN = 	"0018-9340 (print), 1557-9956 (electronic)",
+  issn-l =	"0018-9340",
+  bibdate =	"Tue Jul 5 09:41:56 MDT 2011",
+  bibsource =	"http://www.computer.org/tc/;
+		 http://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib",
+  URL =  	"http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1024738",
+  acknowledgement = "Nelson H. F. Beebe, University of Utah, Department
+		 of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake
+		 City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1
+		 801 581 4148, e-mail: \path|beebe@math.utah.edu|,
+		 \path|beebe@acm.org|, \path|beebe@computer.org|
+		 (Internet), URL:
+		 \path|http://www.math.utah.edu/~beebe/|",
+  fjournal =	"IEEE Transactions on Computers",
+  doi-url =	"http://dx.doi.org/10.1109/TC.2002.1024738",
+}
+
+@Article{journals/rts/BabaogluMS93,
+  title =	"A Formalization of Priority Inversion",
+  author =	"{\"O} Babaoglu and K. Marzullo and F. B. Schneider",
+  journal =	"Real-Time Systems",
+  year = 	"1993",
+  number =	"4",
+  volume =	"5",
+  bibdate =	"2011-06-03",
+  bibsource =	"DBLP,
+		 http://dblp.uni-trier.de/db/journals/rts/rts5.html#BabaogluMS93",
+  pages =	"285--303",
+  URL =  	"http://dx.doi.org/10.1007/BF01088832",
+}
\ No newline at end of file
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Paper/document/root.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/document/root.tex	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,74 @@
+\documentclass[runningheads]{llncs}
+\usepackage{isabelle}
+\usepackage{isabellesym}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{tikz}
+\usepackage{pgf}
+%\usetikzlibrary{arrows,automata,decorations,fit,calc}
+%\usetikzlibrary{shapes,shapes.arrows,snakes,positioning}
+%\usepgflibrary{shapes.misc} % LATEX and plain TEX and pure pgf
+%\usetikzlibrary{matrix}
+\usepackage{pdfsetup}
+\usepackage{ot1patch}
+\usepackage{times}
+%%\usepackage{proof}
+%%\usepackage{mathabx}
+\usepackage{stmaryrd}
+\usepackage{url}
+
+\titlerunning{Myhill-Nerode using Regular Expressions}
+
+
+\urlstyle{rm}
+\isabellestyle{it}
+\renewcommand{\isastyleminor}{\it}%
+\renewcommand{\isastyle}{\normalsize\it}%
+
+
+\def\dn{\,\stackrel{\mbox{\scriptsize def}}{=}\,}
+\renewcommand{\isasymequiv}{$\dn$}
+\renewcommand{\isasymemptyset}{$\varnothing$}
+\renewcommand{\isacharunderscore}{\mbox{$\_\!\_$}}
+
+\newcommand{\isasymcalL}{\ensuremath{\cal{L}}}
+\newcommand{\isasymbigplus}{\ensuremath{\bigplus}}
+
+\newcommand{\bigplus}{\mbox{\Large\bf$+$}}
+\begin{document}
+
+\title{A Formalisation of Priority Inheritance Protocol \\ 
+       for Correct and Efficient Implementation}
+\author{Xingyuan Zhang\inst{1} \and Christian Urban\inst{2} \and Chunhan Wu\inst{1}}
+\institute{PLA University of Science and Technology, China \and 
+           King's College, University of London, U.K.}
+\maketitle
+
+%\mbox{}\\[-10mm]
+\begin{abstract}
+Despite the wide use of Priority Inheritance Protocol in real time operating
+system, it's correctness has never been formally proved and mechanically checked. 
+All existing verification are based on model checking technology. Full automatic
+verification gives little help to understand why the protocol is correct. 
+And results such obtained only apply to models of limited size. 
+This paper presents a formal verification based on theorem proving. 
+Machine checked formal proof does help to get deeper understanding. We found 
+the fact which is not mentioned in the literature, that the choice of next 
+thread to take over when an critical resource is release does not affect the correctness
+of the protocol. The paper also shows how formal proof can help to construct 
+correct and efficient implementation. 
+\end{abstract}
+
+
+\input{session}
+
+%%\mbox{}\\[-10mm]
+\bibliographystyle{plain}
+\bibliography{root}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Paper/tt.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/tt.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,94 @@
+
+There are several works on inversion avoidance:
+\begin{enumerate}
+\item {\em Solving the group priority inversion problem in a timed asynchronous system}. 
+The notion of \<exclamdown>\<degree>Group Priority Inversion\<exclamdown>\<plusminus> is introduced. The main strategy is still inversion avoidance. 
+The method is by reordering requests in the setting of Client-Server.
+\item {\em Examples of inaccurate specification of the protocol}.
+\end{enumerate}
+
+
+
+
+
+
+section{* Related works *}
+
+text {*
+1.	<<Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java>> models and 
+verifies the combination of Priority Inheritance (PI) and Priority Ceiling Emulation (PCE) protocols in 
+the setting of Java virtual machine using extended Timed Automata(TA) formalism of the UPPAAL tool. 
+Although a detailed formal model of combined PI and PCE is given, the number of properties is quite 
+small and the focus is put on the harmonious working of PI and PCE. Most key features of PI 
+(as well as PCE) are not shown. Because of the limitation of the model checking technique
+ used there, properties are shown only for a small number of scenarios. Therefore, the verification 
+does not show the correctness of the formal model itself in a convincing way.  
+2.	<< Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC>>. A formal model 
+of PI is given in TLA+. Only 3 properties are shown for PI using model checking. The limitation of 
+model checking is intrinsic to the work.
+3.	<<Synchronous modeling and validation of priority inheritance schedulers>>. Gives a formal model 
+of PI and PCE in AADL (Architecture Analysis & Design Language) and checked several properties 
+using model checking. The number of properties shown there is less than here and the scale 
+is also limited by the model checking technique. 
+
+
+There are several works on inversion avoidance:
+1.	<<Solving the group priority inversion problem in a timed asynchronous system>>. 
+The notion of \<exclamdown>\<degree>Group Priority Inversion\<exclamdown>\<plusminus> is introduced. The main strategy is still inversion avoidance. 
+The method is by reordering requests in the setting of Client-Server.
+
+
+<<Examples of inaccurate specification of the protocol>>.
+
+*}
+
+text {*
+
+\section{An overview of priority inversion and priority inheritance}
+
+Priority inversion refers to the phenomenon when a thread with high priority is blocked 
+by a thread with low priority. Priority happens when the high priority thread requests 
+for some critical resource already taken by the low priority thread. Since the high 
+priority thread has to wait for the low priority thread to complete, it is said to be 
+blocked by the low priority thread. Priority inversion might prevent high priority 
+thread from fulfill its task in time if the duration of priority inversion is indefinite 
+and unpredictable. Indefinite priority inversion happens when indefinite number 
+of threads with medium priorities is activated during the period when the high 
+priority thread is blocked by the low priority thread. Although these medium 
+priority threads can not preempt the high priority thread directly, they are able 
+to preempt the low priority threads and cause it to stay in critical section for 
+an indefinite long duration. In this way, the high priority thread may be blocked indefinitely. 
+
+Priority inheritance is one protocol proposed to avoid indefinite priority inversion. 
+The basic idea is to let the high priority thread donate its priority to the low priority 
+thread holding the critical resource, so that it will not be preempted by medium priority 
+threads. The thread with highest priority will not be blocked unless it is requesting 
+some critical resource already taken by other threads. Viewed from a different angle, 
+any thread which is able to block the highest priority threads must already hold some 
+critical resource. Further more, it must have hold some critical resource at the 
+moment the highest priority is created, otherwise, it may never get change to run and 
+get hold. Since the number of such resource holding lower priority threads is finite, 
+if every one of them finishes with its own critical section in a definite duration, 
+the duration the highest priority thread is blocked is definite as well. The key to 
+guarantee lower priority threads to finish in definite is to donate them the highest 
+priority. In such cases, the lower priority threads is said to have inherited the 
+highest priority. And this explains the name of the protocol: 
+{\em Priority Inheritance} and how Priority Inheritance prevents indefinite delay.
+
+The objectives of this paper are:
+\begin{enumerate}
+\item Build the above mentioned idea into formal model and prove a series of properties 
+until we are convinced that the formal model does fulfill the original idea. 
+\item Show how formally derived properties can be used as guidelines for correct 
+and efficient implementation.
+\end{enumerate}.
+The proof is totally formal in the sense that every detail is reduced to the 
+very first principles of Higher Order Logic. The nature of interactive theorem 
+proving is for the human user to persuade computer program to accept its arguments. 
+A clear and simple understanding of the problem at hand is both a prerequisite and a 
+byproduct of such an effort, because everything has finally be reduced to the very 
+first principle to be checked mechanically. The former intuitive explanation of 
+Priority Inheritance is just such a byproduct. 
+*}
+
+*)
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Precedence_ord.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Precedence_ord.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,38 @@
+(*  Title:      HOL/Library/Product_ord.thy
+    Author:     Norbert Voelker
+*)
+
+header {* Order on product types *}
+
+theory Precedence_ord
+imports Main
+begin
+
+datatype precedence = Prc nat nat
+
+instantiation precedence :: order
+begin
+
+definition
+  precedence_le_def: "x \<le> y \<longleftrightarrow> (case (x, y) of
+                                   (Prc fx sx, Prc fy sy) \<Rightarrow> 
+                                 fx < fy \<or> (fx \<le> fy \<and> sy \<le> sx))"
+
+definition
+  precedence_less_def: "x < y \<longleftrightarrow> (case (x, y) of
+                               (Prc fx sx, Prc fy sy) \<Rightarrow> 
+                                     fx < fy \<or> (fx \<le> fy \<and> sy < sx))"
+
+instance
+proof
+qed (auto simp: precedence_le_def precedence_less_def 
+              intro: order_trans split:precedence.splits)
+end
+
+instance precedence :: preorder ..
+
+instance precedence :: linorder proof
+qed (auto simp: precedence_le_def precedence_less_def 
+              intro: order_trans split:precedence.splits)
+
+end
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Prio.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Prio.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,2813 @@
+theory Prio
+imports Precedence_ord Moment Lsp Happen_within
+begin
+
+type_synonym thread = nat
+type_synonym priority = nat
+type_synonym cs = nat
+
+datatype event = 
+  Create thread priority |
+  Exit thread |
+  P thread cs |
+  V thread cs |
+  Set thread priority
+
+datatype node = 
+   Th "thread" |
+   Cs "cs"
+
+type_synonym state = "event list"
+
+fun threads :: "state \<Rightarrow> thread set"
+where 
+  "threads [] = {}" |
+  "threads (Create thread prio#s) = {thread} \<union> threads s" |
+  "threads (Exit thread # s) = (threads s) - {thread}" |
+  "threads (e#s) = threads s"
+
+fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> nat"
+where
+  "original_priority thread [] = 0" |
+  "original_priority thread (Create thread' prio#s) = 
+     (if thread' = thread then prio else original_priority thread s)" |
+  "original_priority thread (Set thread' prio#s) = 
+     (if thread' = thread then prio else original_priority thread s)" |
+  "original_priority thread (e#s) = original_priority thread s"
+
+fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
+where
+  "birthtime thread [] = 0" |
+  "birthtime thread ((Create thread' prio)#s) = (if (thread = thread') then length s 
+                                                  else birthtime thread s)" |
+  "birthtime thread ((Set thread' prio)#s) = (if (thread = thread') then length s 
+                                                  else birthtime thread s)" |
+  "birthtime thread (e#s) = birthtime thread s"
+
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+  where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
+
+consts holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+       waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+       depend :: "'b \<Rightarrow> (node \<times> node) set"
+       dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+defs (overloaded) cs_holding_def: "holding wq thread cs == (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
+                  cs_waiting_def: "waiting wq thread cs == (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
+                  cs_depend_def: "depend (wq::cs \<Rightarrow> thread list) == {(Th t, Cs c) | t c. waiting wq t c} \<union> 
+                                               {(Cs c, Th t) | c t. holding wq t c}"
+                  cs_dependents_def: "dependents (wq::cs \<Rightarrow> thread list) th == {th' . (Th th', Th th) \<in> (depend wq)^+}"
+
+record schedule_state = 
+    waiting_queue :: "cs \<Rightarrow> thread list"
+    cur_preced :: "thread \<Rightarrow> precedence"
+
+
+definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
+where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
+
+fun schs :: "state \<Rightarrow> schedule_state"
+where
+   "schs [] = \<lparr>waiting_queue = \<lambda> cs. [], 
+               cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
+   "schs (e#s) = (let ps = schs s in
+                  let pwq = waiting_queue ps in
+                  let pcp = cur_preced ps in
+                  let nwq = case e of
+                             P thread cs \<Rightarrow>  pwq(cs:=(pwq cs @ [thread])) |
+                             V thread cs \<Rightarrow> let nq = case (pwq cs) of
+                                                      [] \<Rightarrow> [] | 
+                                                      (th#pq) \<Rightarrow> case (lsp pcp pq) of
+                                                                   (l, [], r) \<Rightarrow> []
+                                                                 | (l, m#ms, r) \<Rightarrow> m#(l@ms@r)
+                                            in pwq(cs:=nq)                 |
+                              _ \<Rightarrow> pwq
+                  in let ncp = cpreced (e#s) nwq in 
+                     \<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
+                 )"
+
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
+where "wq s == waiting_queue (schs s)"
+
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+where "cp s = cur_preced (schs s)"
+
+defs (overloaded) s_holding_def: "holding (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
+                  s_waiting_def: "waiting (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
+                  s_depend_def: "depend (s::state) == {(Th t, Cs c) | t c. waiting (wq s) t c} \<union> 
+                                               {(Cs c, Th t) | c t. holding (wq s) t c}"
+                  s_dependents_def: "dependents (s::state) th == {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
+
+definition readys :: "state \<Rightarrow> thread set"
+where
+  "readys s = 
+     {thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
+
+definition runing :: "state \<Rightarrow> thread set"
+where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
+
+definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+  where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
+
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+where
+  thread_create: "\<lbrakk>prio \<le> max_prio; thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
+  thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
+  thread_P: "\<lbrakk>thread \<in> runing s;  (Cs cs, Th thread)  \<notin> (depend s)^+\<rbrakk> \<Longrightarrow> step s (P thread cs)" |
+  thread_V: "\<lbrakk>thread \<in> runing s;  holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
+  thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
+ for cs
+where
+  vt_nil[intro]: "vt cs []" |
+  vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
+
+lemma runing_ready: "runing s \<subseteq> readys s"
+  by (auto simp only:runing_def readys_def)
+
+lemma wq_v_eq_nil: 
+  fixes s cs thread rest
+  assumes eq_wq: "wq s cs = thread # rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [], r)"
+  shows "wq (V thread cs#s) cs = []"
+proof -
+  from prems show ?thesis
+    by (auto simp:wq_def Let_def cp_def split:list.splits)
+qed
+
+lemma wq_v_eq: 
+  fixes s cs thread rest
+  assumes eq_wq: "wq s cs = thread # rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  shows "wq (V thread cs#s) cs = th'#l@r"
+proof -
+  from prems show ?thesis
+    by (auto simp:wq_def Let_def cp_def split:list.splits)
+qed
+
+lemma wq_v_neq:
+   "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 wq_distinct: "vt step s \<Longrightarrow> distinct (wq s cs)"
+proof(erule_tac vt.induct, simp add:wq_def)
+  fix s e
+  assume h1: "step s e"
+  and h2: "distinct (wq s cs)"
+  thus "distinct (wq (e # s) cs)"
+  proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
+    fix thread s
+    assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+      and h2: "thread \<in> set (waiting_queue (schs s) cs)"
+      and h3: "thread \<in> runing s"
+    show "False" 
+    proof -
+      from h3 have "\<And> cs. thread \<in>  set (waiting_queue (schs s) cs) \<Longrightarrow> 
+                             thread = hd ((waiting_queue (schs s) cs))" 
+        by (simp add:runing_def readys_def s_waiting_def wq_def)
+      from this [OF h2] have "thread = hd (waiting_queue (schs s) cs)" .
+      with h2
+      have "(Cs cs, Th thread) \<in> (depend s)"
+        by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
+      with h1 show False by auto
+    qed
+  next
+    fix thread s a list
+    assume h1: "thread \<in> runing s" 
+      and h2: "holding s thread cs"
+      and h3: "waiting_queue (schs s) cs = a # list"
+      and h4: "a \<notin> set list"
+      and h5: "distinct list"
+    thus "distinct
+           ((\<lambda>(l, a, r). case a of [] \<Rightarrow> [] | m # ms \<Rightarrow> m # l @ ms @ r)
+             (lsp (cur_preced (schs s)) list))"
+    apply (cases "(lsp (cur_preced (schs s)) list)", simp)
+    apply (case_tac b, simp)
+    by (drule_tac lsp_set_eq, simp)
+  qed
+qed
+
+lemma block_pre: 
+  fixes thread cs s
+  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 -
+  have ee: "\<And> x y. \<lbrakk>x = y\<rbrakk> \<Longrightarrow> set x = set y"
+    by auto
+  from s_ni s_i show ?thesis
+  proof (cases e, auto split:if_splits simp add:Let_def wq_def)
+    fix uu uub uuc uud uue
+    assume h: "(uuc, thread # uu, uub) = lsp (cur_preced (schs s)) uud"
+      and h1 [symmetric]: "uue # uud = waiting_queue (schs s) cs"
+      and h2: "thread \<notin> set (waiting_queue (schs s) cs)"
+    from lsp_set [OF h] have "set (uuc @ (thread # uu) @ uub) = set uud" .
+    hence "thread \<in> set uud" by auto
+    with h1 have "thread \<in> set (waiting_queue (schs s) cs)" by auto
+    with h2 show False by auto
+  next
+    fix uu uua uub uuc uud uue
+    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+      and h2: "uue # uud = waiting_queue (schs s) cs"
+      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+      and h4: "thread \<in> set uuc"
+    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+    with h4 have "thread \<in> set uud" by auto
+    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
+      apply (drule_tac ee) by auto
+    with h1 show "False" by fast
+  next
+    fix uu uua uub uuc uud uue
+    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+      and h2: "uue # uud = waiting_queue (schs s) cs"
+      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+      and h4: "thread \<in> set uu"
+    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+    with h4 have "thread \<in> set uud" by auto
+    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
+      apply (drule_tac ee) by auto
+    with h1 show "False" by fast
+  next
+    fix uu uua uub uuc uud uue
+    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+      and h2: "uue # uud = waiting_queue (schs s) cs"
+      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+      and h4: "thread \<in> set uub"
+    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+    with h4 have "thread \<in> set uud" by auto
+    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
+      apply (drule_tac ee) by auto
+    with h1 show "False" by fast
+  qed
+qed
+
+lemma p_pre: "\<lbrakk>vt step ((P thread cs)#s)\<rbrakk> \<Longrightarrow> 
+  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (depend s)^+"
+apply (ind_cases "vt step ((P thread cs)#s)")
+apply (ind_cases "step s (P thread cs)")
+by auto
+
+lemma abs1:
+  fixes e es
+  assumes ein: "e \<in> set es"
+  and neq: "hd es \<noteq> hd (es @ [x])"
+  shows "False"
+proof -
+  from ein have "es \<noteq> []" by auto
+  then obtain e ess where "es = e # ess" by (cases es, auto)
+  with neq show ?thesis by auto
+qed
+
+lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
+  by (cases es, auto)
+
+inductive_cases evt_cons: "vt cs (a#s)"
+
+lemma abs2:
+  assumes vt: "vt step (e#s)"
+  and inq: "thread \<in> set (wq s cs)"
+  and nh: "thread = hd (wq s cs)"
+  and qt: "thread \<noteq> hd (wq (e#s) cs)"
+  and inq': "thread \<in> set (wq (e#s) cs)"
+  shows "False"
+proof -
+  have ee: "\<And> uuc thread uu uub s list. (uuc, thread # uu, uub) = lsp (cur_preced (schs s)) list \<Longrightarrow> 
+                 lsp (cur_preced (schs s)) list = (uuc, thread # uu, uub) 
+               " by simp
+  from prems show "False"
+    apply (cases e)
+    apply ((simp split:if_splits add:Let_def wq_def)[1])+
+    apply (insert abs1, fast)[1] 
+    apply ((simp split:if_splits add:Let_def)[1])+
+    apply (simp split:if_splits list.splits add:Let_def wq_def) 
+    apply (auto dest!:ee)
+    apply (drule_tac lsp_set_eq, simp)
+    apply (subgoal_tac "distinct (waiting_queue (schs s) cs)", simp, fold wq_def)
+    apply (rule_tac wq_distinct, auto)
+    apply (erule_tac evt_cons, auto)
+    apply (drule_tac lsp_set_eq, simp)
+    apply (subgoal_tac "distinct (wq s cs)", simp)
+    apply (rule_tac wq_distinct, auto)
+    apply (erule_tac evt_cons, auto)
+    apply (drule_tac lsp_set_eq, simp)
+    apply (subgoal_tac "distinct (wq s cs)", simp)
+    apply (rule_tac wq_distinct, auto)
+    apply (erule_tac evt_cons, auto)
+    apply (auto simp:wq_def Let_def split:if_splits prod.splits)
+    done
+qed
+
+lemma vt_moment: "\<And> t. \<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
+proof(induct s, simp)
+  fix a s t
+  assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
+    and vt_a: "vt cs (a # s)"
+    and le_t: "t \<le> length (a # s)"
+  show "vt cs (moment t (a # s))"
+  proof(cases "t = length (a#s)")
+    case True
+    from True have "moment t (a#s) = a#s" by simp
+    with vt_a show ?thesis by simp
+  next
+    case False
+    with le_t have le_t1: "t \<le> length s" by simp
+    from vt_a have "vt cs s"
+      by (erule_tac evt_cons, simp)
+    from h [OF this le_t1] have "vt cs (moment t s)" .
+    moreover have "moment t (a#s) = moment t s"
+    proof -
+      from moment_app [OF le_t1, of "[a]"] 
+      show ?thesis by simp
+    qed
+    ultimately show ?thesis by auto
+  qed
+qed
+
+(* Wrong:
+    lemma \<lbrakk>thread \<in> set (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
+*)
+
+lemma waiting_unique_pre:
+  fixes cs1 cs2 s thread
+  assumes vt: "vt step s"
+  and 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 cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  from h11 and h12 have q1: "?Q cs1 s" by simp
+  from h21 and h22 have q2: "?Q cs2 s" by simp
+  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+  from p_split [of "?Q cs1", OF q1 nq1]
+  obtain t1 where lt1: "t1 < length s"
+    and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
+        thread \<noteq> hd (wq (moment t1 s) cs1))"
+    and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
+             thread \<noteq> hd (wq (moment i' s) cs1))" by auto
+  from p_split [of "?Q cs2", OF q2 nq2]
+  obtain t2 where lt2: "t2 < length s"
+    and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
+        thread \<noteq> hd (wq (moment t2 s) cs2))"
+    and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
+             thread \<noteq> hd (wq (moment i' s) cs2))" by auto
+  show ?thesis
+  proof -
+    { 
+      assume 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: "vt step (e#moment t2 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF False h1]
+        have "e = P thread cs2" .
+        with vt_e have "vt step ((P thread cs2)# moment t2 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t2 s)" by auto
+        with nn1 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume lt12: "t2 < t1"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt step (e#moment t1 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF False h1]
+        have "e = P thread cs1" .
+        with vt_e have "vt step ((P thread cs1)# moment t1 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t1 s)" by auto
+        with nn2 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume eqt12: "t1 = t2"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt step (e#moment t1 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF False h1]
+        have eq_e1: "e = P thread cs1" .
+        have lt_t3: "t1 < ?t3" by simp
+        with eqt12 have "t2 < ?t3" by simp
+        from nn2 [rule_format, OF this] and eq_m and eqt12
+        have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+          h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+        show ?thesis
+        proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+          case True
+          from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+            by auto
+          from vt_e and eqt12 have "vt step (e#moment t2 s)" by simp 
+          from abs2 [OF this True eq_th h2 h1]
+          show ?thesis .
+        next
+          case False
+          from block_pre [OF False h1]
+          have "e = P thread cs2" .
+          with eq_e1 neq12 show ?thesis by auto
+        qed
+      qed
+    } ultimately show ?thesis by arith
+  qed
+qed
+
+lemma waiting_unique:
+  assumes "vt step s"
+  and "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+proof -
+  from waiting_unique_pre and prems
+  show ?thesis
+    by (auto simp add:s_waiting_def)
+qed
+
+lemma holded_unique:
+  assumes "vt step s"
+  and "holding s th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+proof -
+  from prems show ?thesis
+    unfolding s_holding_def
+    by auto
+qed
+
+lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits)
+
+lemma birthtime_unique: 
+  "\<lbrakk>birthtime th1 s = birthtime 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:birthtime_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 "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
+  from birthtime_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 unique_minus:
+  fixes x y z r
+  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:
+  fixes r x y z
+  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:
+  fixes r x y z
+  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
+
+lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+definition head_of :: "('a \<Rightarrow> 'b::linorder) \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where "head_of f A = {a . a \<in> A \<and> f a = Max (f ` A)}"
+
+definition wq_head :: "state \<Rightarrow> cs \<Rightarrow> thread set"
+  where "wq_head s cs = head_of (cp s) (set (wq s cs))"
+
+lemma f_nil_simp: "\<lbrakk>f cs = []\<rbrakk> \<Longrightarrow> f(cs:=[]) = f"
+proof
+  fix x
+  assume h:"f cs = []"
+  show "(f(cs := [])) x = f x"
+  proof(cases "cs = x")
+    case True
+    with h show ?thesis by simp
+  next
+    case False
+    with h show ?thesis by simp
+  qed
+qed
+
+lemma step_back_vt: "vt ccs (e#s) \<Longrightarrow> vt ccs s"
+  by(ind_cases "vt ccs (e#s)", simp)
+
+lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e"
+  by(ind_cases "vt ccs (e#s)", simp)
+
+lemma holding_nil:
+    "\<lbrakk>wq s cs = []; holding (wq s) th cs\<rbrakk> \<Longrightarrow> False"
+  by (unfold cs_holding_def, auto)
+
+lemma waiting_kept_1: "
+       \<lbrakk>vt step (V th cs#s); wq s cs = a # list; waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c;
+        lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
+       \<Longrightarrow> waiting (wq s) t c"
+  apply (drule_tac step_back_vt, drule_tac wq_distinct[of _ cs])
+  apply(drule_tac lsp_set_eq)
+  by (unfold cs_waiting_def, auto split:if_splits)
+ 
+lemma waiting_kept_2: 
+  "\<And>a list t c aa ca.
+       \<lbrakk>wq s cs = a # list; waiting ((wq s)(cs := [])) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
+       \<Longrightarrow> waiting (wq s) t c"
+  apply(drule_tac lsp_set_eq)
+  by (unfold cs_waiting_def, auto split:if_splits)
+  
+
+lemma holding_nil_simp: "\<lbrakk>holding ((wq s)(cs := [])) t c\<rbrakk> \<Longrightarrow> holding (wq s) t c"
+  by(unfold cs_holding_def, auto)
+
+lemma step_wq_elim: "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; a \<noteq> th\<rbrakk> \<Longrightarrow> False"
+  apply(drule_tac step_back_step)
+  apply(ind_cases "step s (V th cs)")
+  by(unfold s_holding_def, auto)
+
+lemma holding_cs_neq_simp: "c \<noteq> cs \<Longrightarrow> holding ((wq s)(cs := u)) t c = holding (wq s) t c"
+  by (unfold cs_holding_def, auto)
+
+lemma holding_th_neq_elim:
+  "\<And>a list c t aa ca ab lista.
+       \<lbrakk>\<not> holding (wq s) t c; holding ((wq s)(cs := ab # aa @ lista @ ca)) t c;
+         ab \<noteq> t\<rbrakk>
+       \<Longrightarrow> False"
+  by (unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_nil_abs:
+  "\<not> holding ((wq s)(cs := [])) th cs"
+  by (unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_abs: "\<lbrakk>holding ((wq s)(cs := ab # aa @ lista @ c)) th cs; ab \<noteq> th\<rbrakk>
+       \<Longrightarrow> False"
+    by (unfold cs_holding_def, auto split:if_splits)
+
+lemma waiting_abs: "\<not> waiting ((wq s)(cs := t # l @ r)) t cs"
+    by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_1: 
+  "\<lbrakk>\<not> waiting ((wq s)(cs := [])) t c; waiting (wq s) t c; c \<noteq> cs\<rbrakk>
+       \<Longrightarrow> False"
+    by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_2: "
+       \<lbrakk>\<not> waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c; waiting (wq s) t c;
+        c \<noteq> cs\<rbrakk>
+       \<Longrightarrow> False"
+  by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_3:
+     "\<lbrakk>wq s cs = a # list; \<not> waiting ((wq s)(cs := [])) t c;
+        waiting (wq s) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
+       \<Longrightarrow> False"
+  apply(drule_tac lsp_mid_nil, simp)
+  by(unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_simp: "c \<noteq> cs \<Longrightarrow> waiting ((wq s)(cs:=z)) t c = waiting (wq s) t c"
+   by(unfold cs_waiting_def, auto split:if_splits)
+
+lemma holding_cs_eq:
+  "\<lbrakk>\<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> c = cs"
+   by(unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_cs_eq_1:
+  "\<lbrakk>\<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c\<rbrakk>
+       \<Longrightarrow> c = cs"
+   by(unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_th_eq: 
+       "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; \<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c;
+        lsp (cp s) list = (aa, [], ca)\<rbrakk>
+       \<Longrightarrow> t = th"
+  apply(drule_tac lsp_mid_nil, simp)
+  apply(unfold cs_holding_def, auto split:if_splits)
+  apply(drule_tac step_back_step)
+  apply(ind_cases "step s (V th cs)")
+  by (unfold s_holding_def, auto split:if_splits)
+
+lemma holding_th_eq_1:
+  "\<lbrakk>vt step (V th cs#s); 
+     wq s cs = a # list; \<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c;
+        lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
+       \<Longrightarrow> t = th"
+  apply(drule_tac step_back_step)
+  apply(ind_cases "step s (V th cs)")
+  apply(unfold s_holding_def cs_holding_def)
+  by (auto split:if_splits)
+
+lemma holding_th_eq_2: "\<lbrakk>holding ((wq s)(cs := ac # x)) th cs\<rbrakk>
+       \<Longrightarrow> ac = th"
+  by (unfold cs_holding_def, auto)
+
+lemma holding_th_eq_3: "
+       \<lbrakk>\<not> holding (wq s) t c;
+        holding ((wq s)(cs := ac # x)) t c\<rbrakk>
+       \<Longrightarrow> ac = t"
+  by (unfold cs_holding_def, auto)
+
+lemma holding_wq_eq: "holding ((wq s)(cs := th' # l @ r)) th' cs"
+   by (unfold cs_holding_def, auto)
+
+lemma waiting_th_eq: "
+       \<lbrakk>waiting (wq s) t c; wq s cs = a # list;
+        lsp (cp s) list = (aa, ac # lista, ba); \<not> waiting ((wq s)(cs := ac # aa @ lista @ ba)) t c\<rbrakk>
+       \<Longrightarrow> ac = t"
+  apply(drule_tac lsp_set_eq, simp)
+  by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma step_depend_v:
+  "vt step (V th cs#s) \<Longrightarrow>
+  depend (V th cs # s) =
+  depend s - {(Cs cs, Th th)} -
+  {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+  {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+  apply (unfold s_depend_def wq_def, 
+         auto split:list.splits simp:Let_def f_nil_simp holding_wq_eq, fold wq_def cp_def)
+  apply (auto split:list.splits prod.splits  
+               simp:Let_def f_nil_simp holding_nil_simp holding_cs_neq_simp holding_nil_abs
+                    waiting_abs waiting_simp holding_wq_eq
+               elim:holding_nil waiting_kept_1 waiting_kept_2 step_wq_elim holding_th_neq_elim 
+               holding_abs waiting_abs_1 waiting_abs_3 holding_cs_eq holding_cs_eq_1
+               holding_th_eq holding_th_eq_1 holding_th_eq_2 holding_th_eq_3 waiting_th_eq
+               dest:lsp_mid_length)
+  done
+
+lemma step_depend_p:
+  "vt step (P th cs#s) \<Longrightarrow>
+  depend (P th cs # s) =  (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
+                                             else depend s \<union> {(Th th, Cs cs)})"
+  apply(unfold s_depend_def wq_def)
+  apply (auto split:list.splits prod.splits simp:Let_def cs_waiting_def cs_holding_def)
+  apply(case_tac "c = cs", auto)
+  apply(fold wq_def)
+  apply(drule_tac step_back_step)
+  by (ind_cases " step s (P (hd (wq s cs)) cs)", 
+    auto simp:s_depend_def wq_def cs_holding_def)
+
+lemma simple_A:
+  fixes A
+  assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
+  shows "A = {} \<or> (\<exists> a. A = {a})"
+proof(cases "A = {}")
+  case True thus ?thesis by simp
+next
+  case False then obtain a where "a \<in> A" by auto
+  with h have "A = {a}" by auto
+  thus ?thesis by simp
+qed
+
+lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_depend_def, auto)
+
+lemma acyclic_depend: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "acyclic (depend s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume ih: "acyclic (depend s)"
+      and stp: "step s e"
+      and vt: "vt step s"
+    show ?case
+    proof(cases e)
+      case (Create th prio)
+      with ih
+      show ?thesis by (simp add:depend_create_unchanged)
+    next
+      case (Exit th)
+      with ih show ?thesis by (simp add:depend_exit_unchanged)
+    next
+      case (V th cs)
+      from V vt stp have vtt: "vt step (V th cs#s)" by auto
+      from step_depend_v [OF this]
+      have eq_de: "depend (e # s) = 
+        depend s - {(Cs cs, Th th)} -
+        {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+        {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+      from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+      have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
+      thus ?thesis
+      proof(cases "wq s cs")
+        case Nil
+        hence "?D = {}" by simp
+        with ac and eq_de show ?thesis by simp
+      next
+        case (Cons tth rest)
+        from stp and V have "step s (V th cs)" by simp
+        hence eq_wq: "wq s cs = th # rest"
+        proof -
+          show "step s (V th cs) \<Longrightarrow> wq s cs = th # rest"
+            apply(ind_cases "step s (V th cs)")
+            by(insert Cons, unfold s_holding_def, simp)
+        qed
+        show ?thesis
+        proof(cases "lsp (cp s) rest")
+          fix l b r
+          assume eq_lsp: "lsp (cp s) rest = (l, b, r) "
+          show ?thesis
+          proof(cases "b")
+            case Nil
+            with eq_lsp and eq_wq have "?D = {}" by simp
+            with ac and eq_de show ?thesis by simp
+          next
+            case (Cons th' m)
+            with eq_lsp 
+            have eq_lsp: "lsp (cp s) rest = (l, [th'], r)" 
+              apply simp
+              by (drule_tac lsp_mid_length, simp)
+            from eq_wq and eq_lsp
+            have eq_D: "?D = {(Cs cs, Th th')}" by auto
+            from eq_wq and eq_lsp
+            have eq_C: "?C = {(Th th', Cs cs)}" by 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 depend_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_depend_def s_waiting_def cs_waiting_def by simp
+              hence "cs' = cs"
+              proof(rule waiting_unique [OF vt])
+                from eq_wq eq_lsp wq_distinct[OF vt, of cs]
+                show "waiting s th' cs" by(unfold s_waiting_def, auto dest:lsp_set_eq)
+              qed
+              with th'_e eq_x have "(Th th', Cs cs) \<in> ?E" by simp
+              with eq_C show "False" by simp
+            qed
+            with acyclic_insert[symmetric] and ac and eq_D
+            and eq_de show ?thesis by simp
+          qed 
+        qed
+      qed
+    next
+      case (P th cs)
+      from P vt stp have vtt: "vt step (P th cs#s)" by auto
+      from step_depend_p [OF this] P
+      have "depend (e # s) = 
+              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
+                                    depend 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 =  depend s \<union> {(Cs cs, Th th)}" by simp
+        have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
+        proof
+          assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
+          hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+          from tranclD2 [OF this]
+          obtain x where "(x, Cs cs) \<in> depend s" by auto
+          with True show False by (auto simp:s_depend_def cs_waiting_def)
+        qed
+        with acyclic_insert ih eq_r show ?thesis by auto
+      next
+        case False
+        hence eq_r: "?R =  depend s \<union> {(Th th, Cs cs)}" by simp
+        have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
+        proof
+          assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
+          hence "(Cs cs, Th th) \<in> (depend 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> (depend 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 depend_set_unchanged
+      show ?thesis by (simp add:depend_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "acyclic (depend ([]::state))"
+      by (auto simp: s_depend_def cs_waiting_def 
+                      cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+lemma finite_depend: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "finite (depend s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume ih: "finite (depend s)"
+      and stp: "step s e"
+      and vt: "vt step s"
+    show ?case
+    proof(cases e)
+      case (Create th prio)
+      with ih
+      show ?thesis by (simp add:depend_create_unchanged)
+    next
+      case (Exit th)
+      with ih show ?thesis by (simp add:depend_exit_unchanged)
+    next
+      case (V th cs)
+      from V vt stp have vtt: "vt step (V th cs#s)" by auto
+      from step_depend_v [OF this]
+      have eq_de: "depend (e # s) = 
+        depend s - {(Cs cs, Th th)} -
+        {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+        {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+        (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 (rule simple_A, auto)
+        thus ?thesis
+        proof
+          assume h: "?D = {}"
+          show ?thesis by (unfold h, simp)
+        next
+          assume "\<exists> a. ?D = {a}"
+          thus ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (P th cs)
+      from P vt stp have vtt: "vt step (P th cs#s)" by auto
+      from step_depend_p [OF this] P
+      have "depend (e # s) = 
+              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
+                                    depend 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 =  depend s \<union> {(Cs cs, Th th)}" by simp
+        with True and ih show ?thesis by auto
+      next
+        case False
+        hence "?R = depend 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:depend_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "finite (depend ([]::state))"
+      by (auto simp: s_depend_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+text {* Several useful lemmas *}
+
+thm wf_trancl
+thm finite_acyclic_wf
+thm finite_acyclic_wf_converse
+thm wf_induct
+
+
+lemma wf_dep_converse: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "wf ((depend s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+  from finite_depend [OF vt]
+  show "finite (depend s)" .
+next
+  from acyclic_depend[OF vt]
+  show "acyclic (depend s)" .
+qed
+
+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> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
+  by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+
+lemma wq_threads: 
+  fixes s cs
+  assumes vt: "vt step s"
+  and 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)
+    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+      and stp: "step s e"
+      and vt: "vt step 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_depend_def s_holding_def cs_holding_def)
+        by (fold wq_def, auto)
+    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 (waiting_queue (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 = waiting_queue (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 "waiting_queue (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: "waiting_queue (schs s) cs' = a # rest"
+              with h V show ?thesis
+              proof(cases "(lsp (cur_preced (schs s)) rest)", unfold V)
+                fix l m r
+                assume eq_lsp: "lsp (cur_preced (schs s)) rest = (l, m, r)"
+                  and eq_wq: "waiting_queue (schs s) cs' = a # rest"
+                  and th_in_set: "th \<in> set (wq (V th' cs' # s) cs)"
+                show ?thesis
+                proof(cases "m")
+                  case Nil
+                  with eq_lsp have "rest = []" using lsp_mid_nil by auto
+                  with eq_wq have "waiting_queue (schs s) cs' = [a]" by simp
+                  with h[unfolded V wq_def] True 
+                  show ?thesis
+                    by (simp add:Let_def)
+                next
+                  case (Cons b rb)
+                  with lsp_mid_length[OF eq_lsp] have eq_m: "m = [b]" by auto
+                  with eq_lsp have "lsp (cur_preced (schs s)) rest = (l, [b], r)" by simp
+                  with h[unfolded V wq_def] True lsp_set_eq [OF this] eq_wq
+                  show ?thesis
+                    apply (auto simp:Let_def, fold wq_def)
+                    by (rule_tac ih [of _ cs'], auto)+
+                qed
+              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>vt step s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+  apply(unfold s_depend_def cs_waiting_def cs_holding_def)
+  by (auto intro:wq_threads)
+
+lemma readys_v_eq:
+  fixes th thread cs rest
+  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 prems show ?thesis
+    apply (auto simp:readys_def)
+    apply (case_tac "cs = csa", simp add:s_waiting_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)
+    by (auto simp:s_waiting_def wq_def Let_def split:list.splits prod.splits 
+            dest:lsp_set_eq)
+qed
+
+lemma readys_v_eq_1:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  and neq_th': "th \<noteq> th'"
+  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
+proof -
+  from prems show ?thesis
+    apply (auto simp:readys_def)
+    apply (case_tac "cs = csa", simp add:s_waiting_def)
+    apply (erule_tac x = cs in allE)
+    apply (simp add:s_waiting_def wq_def Let_def split:prod.splits list.splits)
+    apply (drule_tac lsp_mid_nil,simp, clarify, fold cp_def, clarsimp)
+    apply (frule_tac lsp_set_eq, simp)
+    apply (erule_tac x = csa in allE)
+    apply (subst (asm) (2) s_waiting_def, unfold wq_def)
+    apply (auto simp:Let_def split:list.splits prod.splits if_splits 
+            dest:lsp_set_eq)
+    apply (unfold s_waiting_def)
+    apply (fold wq_def, clarsimp)
+    apply (clarsimp)+
+    apply (case_tac "csa = cs", simp)
+    apply (erule_tac x = cs in allE, simp)
+    apply (unfold wq_def)
+    by (auto simp:Let_def split:list.splits prod.splits if_splits 
+            dest:lsp_set_eq)
+qed
+
+lemma readys_v_eq_2:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  and neq_th': "th = th'"
+  and vt: "vt step s"
+  shows "(th \<in> readys (V thread cs#s))"
+proof -
+  from prems show ?thesis
+    apply (auto simp:readys_def)
+    apply (rule_tac wq_threads [of s _ cs], auto dest:lsp_set_eq)
+    apply (unfold s_waiting_def wq_def)
+    apply (auto simp:Let_def split:list.splits prod.splits if_splits 
+            dest:lsp_set_eq lsp_mid_nil lsp_mid_length)
+    apply (fold cp_def, simp+, clarsimp)
+    apply (frule_tac lsp_set_eq, simp)
+    apply (fold wq_def)
+    apply (subgoal_tac "csa = cs", simp)
+    apply (rule_tac waiting_unique [of s th'], simp)
+    by (auto simp:s_waiting_def)
+qed
+
+lemma chain_building:
+  assumes vt: "vt step s"
+  shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
+proof -
+  from wf_dep_converse [OF vt]
+  have h: "wf ((depend s)\<inverse>)" .
+  show ?thesis
+  proof(induct rule:wf_induct [OF h])
+    fix x
+    assume ih [rule_format]: 
+      "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow> 
+           y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
+    show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
+    proof
+      assume x_d: "x \<in> Domain (depend s)"
+      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
+      proof(cases x)
+        case (Th th)
+        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
+        with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
+        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
+        hence "Cs cs \<in> Domain (depend 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> (depend s)\<^sup>+" by auto
+        have "(x, Th th') \<in> (depend 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> (depend s)^-1" by (auto simp:s_depend_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 [OF vt] have "th' \<in> threads s" by auto
+          with False have "Th th' \<in> Domain (depend s)" 
+            by (auto simp:readys_def s_waiting_def s_depend_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> (depend s)\<^sup>+" by auto
+          from th'_d and th''_in 
+          have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
+          with th''_r show ?thesis by auto
+        qed
+      qed
+    qed
+  qed
+qed
+
+lemma th_chain_to_ready:
+  fixes s th
+  assumes vt: "vt step s"
+  and th_in: "th \<in> threads s"
+  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend 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 (depend s)" 
+    by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
+  from chain_building [rule_format, OF vt this]
+  show ?thesis by auto
+qed
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+  by  (unfold s_waiting_def cs_waiting_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def cs_holding_def, simp)
+
+lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
+  by (unfold s_holding_def cs_holding_def, auto)
+
+lemma unique_depend: "\<lbrakk>vt step s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
+  apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
+  by(auto elim:waiting_unique holding_unique)
+
+lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
+by (induct rule:trancl_induct, auto)
+
+lemma dchain_unique:
+  assumes vt: "vt step s"
+  and th1_d: "(n, Th th1) \<in> (depend s)^+"
+  and th1_r: "th1 \<in> readys s"
+  and th2_d: "(n, Th th2) \<in> (depend 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_depend [OF vt]
+    have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
+    hence "False"
+    proof
+      assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th1, n) \<in> depend s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_depend_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_depend_def s_waiting_def cs_waiting_def)
+      with th1_r show ?thesis by auto
+    next
+      assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th2, n) \<in> depend s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_depend_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 s_depend_def s_waiting_def cs_waiting_def)
+      with th2_r show ?thesis by auto
+    qed
+  } thus ?thesis by auto
+qed
+             
+definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
+where "count Q l = length (filter Q l)"
+
+definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
+
+definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+
+
+lemma step_holdents_p_add:
+  fixes th cs s
+  assumes vt: "vt step (P th cs#s)"
+  and "wq s cs = []"
+  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_p[OF vt] by auto
+qed
+
+lemma step_holdents_p_eq:
+  fixes th cs s
+  assumes vt: "vt step (P th cs#s)"
+  and "wq s cs \<noteq> []"
+  shows "holdents (P th cs#s) th = holdents s th"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_p[OF vt] by auto
+qed
+
+lemma step_holdents_v_minus:
+  fixes th cs s
+  assumes vt: "vt step (V th cs#s)"
+  shows "holdents (V th cs#s) th = holdents s th - {cs}"
+proof -
+  { fix rest l r
+    assume eq_wq: "wq s cs = th # rest" 
+      and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
+    have "False" 
+    proof -
+      from lsp_set_eq [OF eq_lsp] have " rest = l @ [th] @ r" .
+      with eq_wq have "wq s cs = th#\<dots>" by simp
+      with wq_distinct [OF step_back_vt[OF vt], of cs]
+      show ?thesis by auto
+    qed
+  } thus ?thesis unfolding holdents_def step_depend_v[OF vt] by auto
+qed
+
+lemma step_holdents_v_add:
+  fixes th th' cs s rest l r
+  assumes vt: "vt step (V th' cs#s)"
+  and neq_th: "th \<noteq> th'" 
+  and eq_wq: "wq s cs = th' # rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
+  shows "holdents (V th' cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_v[OF vt] by auto
+qed
+
+lemma step_holdents_v_eq:
+  fixes th th' cs s rest l r th''
+  assumes vt: "vt step (V th' cs#s)"
+  and neq_th: "th \<noteq> th'" 
+  and eq_wq: "wq s cs = th' # rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th''], r)"
+  and neq_th': "th \<noteq> th''"
+  shows "holdents (V th' cs#s) th = holdents s th"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_v[OF vt] by auto
+qed
+
+definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntCS s th = card (holdents s th)"
+
+lemma cntCS_v_eq:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and not_in: "th \<notin>  set rest"
+  and vtv: "vt step (V thread cs#s)"
+  shows "cntCS (V thread cs#s) th = cntCS s th"
+proof -
+  from prems show ?thesis
+    apply (unfold cntCS_def holdents_def step_depend_v)
+    apply auto
+    apply (subgoal_tac "\<not>  (\<exists>l r. lsp (cp s) rest = (l, [th], r))", auto)
+    by (drule_tac lsp_set_eq, auto)
+qed
+
+lemma cntCS_v_eq_1:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  and neq_th': "th \<noteq> th'"
+  and vtv: "vt step (V thread cs#s)"
+  shows "cntCS (V thread cs#s) th = cntCS s th"
+proof -
+  from prems show ?thesis
+    apply (unfold cntCS_def holdents_def step_depend_v)
+    by auto
+qed
+
+fun the_cs :: "node \<Rightarrow> cs"
+where "the_cs (Cs cs) = cs"
+
+lemma cntCS_v_eq_2:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  and neq_th': "th = th'"
+  and vtv: "vt step (V thread cs#s)"
+  shows "cntCS (V thread cs#s) th = 1 + cntCS s th"
+proof -
+  have "card {csa. csa = cs \<or> (Cs csa, Th th') \<in> depend s} = 
+                     Suc (card {cs. (Cs cs, Th th') \<in> depend s})" 
+    (is "card ?A = Suc (card ?B)")
+  proof -
+    have h: "?A = insert cs ?B" by auto
+    moreover have h1: "?B = ?B - {cs}"
+    proof -
+      { assume "(Cs cs, Th th') \<in> depend s"
+        moreover have "(Th th', Cs cs) \<in> depend s"
+        proof -
+          from wq_distinct [OF step_back_vt[OF vtv], of cs]
+          eq_wq lsp_set_eq [OF eq_lsp] show ?thesis
+            apply (auto simp:s_depend_def)
+            by (unfold cs_waiting_def, auto)
+        qed
+        moreover note acyclic_depend [OF step_back_vt[OF vtv]]
+        ultimately have "False"
+          apply (auto simp:acyclic_def)
+          apply (erule_tac x="Cs cs" in allE)
+          apply (subgoal_tac "(Cs cs, Cs cs) \<in> (depend s)\<^sup>+", simp)
+          by (rule_tac trancl_into_trancl [where b = "Th th'"], auto)
+      } thus ?thesis by auto
+    qed
+    moreover have "card (insert cs ?B) = Suc (card (?B - {cs}))"
+    proof(rule card_insert)
+      from finite_depend [OF step_back_vt [OF vtv]]
+      have fnt: "finite (depend s)" .
+      show " finite {cs. (Cs cs, Th th') \<in> depend s}" (is "finite ?B")
+      proof -
+        have "?B \<subseteq>  (\<lambda> (a, b). the_cs a) ` (depend s)"
+          apply (auto simp:image_def)
+          by (rule_tac x = "(Cs x, Th th')" in bexI, auto)
+        with fnt show ?thesis by (auto intro:finite_subset)
+      qed
+    qed
+    ultimately show ?thesis by simp
+  qed
+  with prems show ?thesis
+    apply (unfold cntCS_def holdents_def step_depend_v[OF vtv])
+    by auto
+qed
+
+lemma finite_holding:
+  fixes s th cs
+  assumes vt: "vt step s"
+  shows "finite (holdents s th)"
+proof -
+  let ?F = "\<lambda> (x, y). the_cs x"
+  from finite_depend [OF vt]
+  have "finite (depend s)" .
+  hence "finite (?F `(depend s))" by simp
+  moreover have "{cs . (Cs cs, Th th) \<in> depend 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> depend s"
+      hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
+      moreover have "?F (Cs x, Th th) = x" by simp
+      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp 
+    } thus ?thesis by auto
+  qed
+  ultimately show ?thesis by (unfold holdents_def, auto intro:finite_subset)
+qed
+
+inductive_cases case_step_v: "step s (V thread cs)"
+
+lemma cntCS_v_dec: 
+  fixes s thread cs
+  assumes vtv: "vt step (V thread cs#s)"
+  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
+proof -
+  have cs_in: "cs \<in> holdents s thread" using step_back_step[OF vtv]
+    apply (erule_tac case_step_v)
+    apply (unfold holdents_def s_depend_def, simp)
+    by (unfold cs_holding_def s_holding_def, auto)
+  moreover have cs_not_in: 
+    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
+    apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
+    by (unfold holdents_def, unfold step_depend_v[OF vtv], 
+            auto dest:lsp_set_eq)
+  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 finite_holding [OF vtv]
+    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 cnp_cnv_cncs:
+  fixes s th
+  assumes vt: "vt step s"
+  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)
+    assume vt: "vt step 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 prio max_prio thread)
+      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 wq_threads [OF vt 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_def
+          by (simp add:depend_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_def
+        by (simp add:depend_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 (subst (1 2) wq_def)
+          apply (simp add:Let_def)
+          apply (subst s_waiting_def, simp)
+          by (fold wq_def, simp)
+        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> (depend s)\<^sup>+"
+      from prems have vtp: "vt step (P thread cs#s)" by auto
+      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, clarify)
+            apply (intro iffI allI, clarify)
+            apply (erule_tac x = csa in allE, auto)
+            apply (subgoal_tac "waiting_queue (schs s) cs \<noteq> []", auto)
+            apply (erule_tac x = cs in allE, auto)
+            by (case_tac "(waiting_queue (schs s) cs)", auto)
+          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+            apply (simp add:cntCS_def holdents_def)
+            by (unfold  step_depend_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_depend_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> depend s} =
+                  Suc (card {cs. (Cs cs, Th thread) \<in> depend 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 finite_holding [OF vt, of thread]
+                    show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
+                      by (unfold holdents_def, simp)
+                  qed
+                  moreover have "?R - {cs} = ?R"
+                  proof -
+                    have "cs \<notin> ?R"
+                    proof
+                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend 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_def)
+                  by (unfold step_depend_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)
+                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 wq_distinct [OF vtp, 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_def eq_e step_depend_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 prems have vtv: "vt step (V thread cs # s)" by auto
+      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:s_holding_def)
+      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+      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 -
+                  have "thread \<notin> set (wq (e#s) cs1)"
+                  proof(cases "lsp (cp s) rest")
+                    fix l m r 
+                    assume h: "lsp (cp s) rest = (l, m, r)"
+                    show ?thesis
+                    proof(cases "m")
+                      case Nil
+                      from wq_v_eq_nil [OF eq_wq] h Nil eq_e
+                      have " wq (e # s) cs = []" by auto
+                      thus ?thesis using eq_cs by auto
+                    next
+                      case (Cons th' l')
+                      from lsp_mid_length [OF h] and Cons h
+                      have eqh: "lsp (cp s) rest = (l, [th'], r)" by auto
+                      from wq_v_eq [OF eq_wq this]
+                      have "wq (V thread cs # s) cs = th' # l @ r" .
+                      moreover from lsp_set_eq [OF eqh]
+                      have "set rest = set \<dots>" by auto
+                      moreover have "thread \<notin> set rest"
+                      proof -
+                        from wq_distinct [OF step_back_vt[OF vtv], of cs]
+                        and eq_wq show ?thesis by auto
+                      qed
+                      moreover note eq_e eq_cs
+                      ultimately show ?thesis by simp
+                    qed
+                  qed
+                  thus ?thesis by (simp add: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: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)"
+              by(unfold eq_e, rule readys_v_eq [OF neq_th eq_wq False])
+            moreover have "cntCS (e#s) th = cntCS s th"
+              by(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq False vtv]) 
+            moreover note ih eq_cnp eq_cnv eq_threads
+            ultimately show ?thesis by auto
+          next
+            case True
+            obtain l m r where eq_lsp: "lsp (cp s) rest = (l, m, r)" 
+              by (cases "lsp (cp s) rest", auto)
+            with True have "m \<noteq> []" by  (auto dest:lsp_mid_nil)
+            with eq_lsp obtain th' where eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+              by (case_tac m, auto dest:lsp_mid_length)
+            show ?thesis
+            proof(cases "th = th'")
+              case False
+              have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+                by (unfold eq_e, rule readys_v_eq_1 [OF neq_th eq_wq eq_lsp False])
+              moreover have "cntCS (e#s) th = cntCS s th" 
+                by (unfold eq_e, rule cntCS_v_eq_1[OF neq_th eq_wq eq_lsp False vtv])
+              moreover note ih eq_cnp eq_cnv eq_threads
+              ultimately show ?thesis by auto
+            next
+              case True
+              have "th \<in> readys (e # s)"
+                by (unfold eq_e, rule readys_v_eq_2 [OF neq_th eq_wq eq_lsp True vt])
+              moreover have "cntP s th = cntV s th + cntCS s th + 1"
+              proof -
+                have "th \<notin> readys s" 
+                proof -
+                  from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
+                  show ?thesis
+                    apply (unfold readys_def s_waiting_def, auto)
+                    by (rule_tac x = cs in exI, auto)
+                qed
+                moreover have "th \<in> threads s"
+                proof -
+                  from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
+                  have "th \<in> set (wq s cs)" by simp
+                  from wq_threads [OF step_back_vt[OF vtv] this] 
+                  show ?thesis .
+                qed
+                ultimately show ?thesis using ih by auto
+              qed
+              moreover have "cntCS (e # s) th = 1 + cntCS s th"
+                by (unfold eq_e, rule cntCS_v_eq_2 [OF neq_th eq_wq eq_lsp True vtv])
+              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_def
+          by (simp add:depend_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_def s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma not_thread_cncs:
+  fixes th s
+  assumes vt: "vt step s"
+  and 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)
+    assume vt: "vt step 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 prio max_prio thread)
+      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_def)
+        by (simp add:depend_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_def)
+        by (simp add:depend_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 prems have vtp: "vt step (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_def eq_e)
+        by (unfold step_depend_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 prems have vtv: "vt step (V thread cs#s)" by auto
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp:s_holding_def)
+      have "cntCS (e # s) th  = cntCS s th"
+      proof(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq _ vtv])
+        show "th \<notin> set rest" 
+        proof
+          assume "th \<in> set rest"
+          with eq_wq have "th \<in> set (wq s cs)" by simp
+          from wq_threads [OF vt this] eq_e not_in 
+          show False by simp
+        qed
+      qed
+      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_def)
+        by (simp add:depend_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (unfold cntCS_def, 
+        auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
+  by (auto simp:s_waiting_def cs_waiting_def)
+
+lemma dm_depend_threads:
+  fixes th s
+  assumes vt: "vt step s"
+  and in_dom: "(Th th) \<in> Domain (depend s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
+  moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
+  ultimately have "(Th th, Cs cs) \<in> depend s" by simp
+  hence "th \<in> set (wq s cs)"
+    by (unfold s_depend_def, auto simp:cs_waiting_def)
+  from wq_threads [OF vt this] show ?thesis .
+qed
+
+lemma cp_eq_cpreced: "cp s th = cpreced s (wq s) th"
+proof(unfold cp_def wq_def, induct s)
+  case (Cons e s')
+  show ?case
+    by (auto simp:Let_def)
+next
+  case Nil
+  show ?case by (auto simp:Let_def)
+qed
+
+fun the_th :: "node \<Rightarrow> thread"
+  where "the_th (Th th) = th"
+
+lemma runing_unique:
+  fixes th1 th2 s
+  assumes vt: "vt step s"
+  and 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"
+    by (unfold runing_def, simp)
+  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
+                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
+    (is "Max (?f ` ?A) = Max (?f ` ?B)")
+    by (unfold cp_eq_cpreced 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 (dependents (wq s) th1)"
+        proof-
+          have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+              hence "finite ((depend (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_depend_def cs_depend_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependents_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 by (auto intro:that)
+  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 (dependents (wq s) th2)"
+        proof-
+          have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+              hence "finite ((depend (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_depend_def cs_depend_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependents_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> dependents (wq s) th1)" by simp
+    thus "th1' \<in> threads s"
+    proof
+      assume "th1' \<in> dependents (wq s) th1"
+      hence "(Th th1') \<in> Domain ((depend s)^+)"
+        apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+        by (auto simp:Domain_def)
+      hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
+      from dm_depend_threads[OF vt 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> dependents (wq s) th2)" by simp
+    thus "th2' \<in> threads s"
+    proof
+      assume "th2' \<in> dependents (wq s) th2"
+      hence "(Th th2') \<in> Domain ((depend s)^+)"
+        apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+        by (auto simp:Domain_def)
+      hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
+      from dm_depend_threads[OF vt 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> dependents (wq s) th1" by simp
+  thus ?thesis
+  proof
+    assume eq_th': "th1' = th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
+    thus ?thesis
+    proof
+      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+    next
+      assume "th2' \<in> dependents (wq s) th2"
+      with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
+      hence "(Th th1, Th th2) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      hence "Th th1 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
+        by auto
+      hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
+      from depend_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th1, Cs cs') \<in> depend s" by simp
+      with runing_1 have "False"
+        apply (unfold runing_def readys_def s_depend_def)
+        by (auto simp:eq_waiting)
+      thus ?thesis by simp
+    qed
+  next
+    assume th1'_in: "th1' \<in> dependents (wq s) th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
+    thus ?thesis 
+    proof
+      assume "th2' = th2"
+      with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
+      hence "(Th th2, Th th1) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      hence "Th th2 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
+        by auto
+      hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
+      from depend_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th2, Cs cs') \<in> depend s" by simp
+      with runing_2 have "False"
+        apply (unfold runing_def readys_def s_depend_def)
+        by (auto simp:eq_waiting)
+      thus ?thesis by simp
+    next
+      assume "th2' \<in> dependents (wq s) th2"
+      with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
+      hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      show ?thesis
+      proof(rule dchain_unique[OF vt 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 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 prio max_prio thread)
+    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
+
+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
+
+lemma cnp_cnv_eq:
+  fixes th s
+  assumes "vt step s"
+  and "th \<notin> threads s"
+  shows "cntP s th = cntV s th"
+proof -
+  from assms show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
+    have not_in: "th \<notin> threads (e # s)" by fact
+    have "step s e" by fact
+    thus ?case proof(cases)
+      case (thread_create prio max_prio thread)
+      assume eq_e: "e = Create thread prio"
+      hence "thread \<in> threads (e#s)" by simp
+      with not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] show ?thesis using eq_e
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+        and not_holding: "holdents s thread = {}"
+      have vt_s: "vt step s" by fact
+      from finite_holding[OF vt_s] have "finite (holdents s thread)" .
+      with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
+      moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
+      moreover note cnp_cnv_cncs[OF vt_s, of thread]
+      ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
+      show ?thesis
+      proof(cases "th = thread")
+        case True
+        with eq_thread eq_e show ?thesis 
+          by (auto simp:cntP_def cntV_def count_def)
+      next
+        case False
+        with not_in and eq_e have "th \<notin> threads s" by simp
+        from ih[OF this] and eq_e show ?thesis 
+           by (auto simp:cntP_def cntV_def count_def)
+      qed
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      have "thread \<in> runing s" by fact
+      with not_in eq_e have neq_th: "thread \<noteq> th" 
+        by (auto simp:runing_def readys_def)
+      from not_in eq_e have "th \<notin> threads s" by simp
+      from ih[OF this] and neq_th and eq_e show ?thesis
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+      have "thread \<in> runing s" by fact
+      with not_in eq_e have neq_th: "thread \<noteq> th" 
+        by (auto simp:runing_def readys_def)
+      from not_in eq_e have "th \<notin> threads s" by simp
+      from ih[OF this] and neq_th and eq_e show ?thesis
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_set thread prio)
+      assume eq_e: "e = Set thread prio"
+        and "thread \<in> runing s"
+      hence "thread \<in> threads (e#s)" 
+        by (simp add:runing_def readys_def)
+      with not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] show ?thesis using eq_e
+        by (auto simp:cntP_def cntV_def count_def)  
+    qed
+  next
+    case vt_nil
+    show ?case by (auto simp:cntP_def cntV_def count_def)
+  qed
+qed
+
+lemma eq_depend: 
+  "depend (wq s) = depend s"
+by (unfold cs_depend_def s_depend_def, auto)
+
+lemma count_eq_dependents:
+  assumes vt: "vt step s"
+  and eq_pv: "cntP s th = cntV s th"
+  shows "dependents (wq s) th = {}"
+proof -
+  from cnp_cnv_cncs[OF vt] and eq_pv
+  have "cntCS s th = 0" 
+    by (auto split:if_splits)
+  moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
+  proof -
+    from finite_holding[OF vt, of th] show ?thesis
+      by (simp add:holdents_def)
+  qed
+  ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
+    by (unfold cntCS_def holdents_def cs_dependents_def, auto)
+  show ?thesis
+  proof(unfold cs_dependents_def)
+    { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
+      then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
+      hence "False"
+      proof(cases)
+        assume "(Th th', Th th) \<in> depend (wq s)"
+        thus "False" by (auto simp:cs_depend_def)
+      next
+        fix c
+        assume "(c, Th th) \<in> depend (wq s)"
+        with h and eq_depend show "False"
+          by (cases c, auto simp:cs_depend_def)
+      qed
+    } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
+  qed
+qed
+
+lemma dependents_threads:
+  fixes s th
+  assumes vt: "vt step s"
+  shows "dependents (wq s) th \<subseteq> threads s"
+proof
+  { fix th th'
+    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
+    have "Th th \<in> Domain (depend s)"
+    proof -
+      from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
+      hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
+      with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
+      thus ?thesis using eq_depend by simp
+    qed
+    from dm_depend_threads[OF vt this]
+    have "th \<in> threads s" .
+  } note hh = this
+  fix th1 
+  assume "th1 \<in> dependents (wq s) th"
+  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
+    by (unfold cs_dependents_def, simp)
+  from hh [OF this] show "th1 \<in> threads s" .
+qed
+
+lemma finite_threads:
+  assumes vt: "vt step s"
+  shows "finite (threads s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume vt: "vt step s"
+    and step: "step s e"
+    and ih: "finite (threads s)"
+    from step
+    show ?case
+    proof(cases)
+      case (thread_create prio max_prio thread)
+      assume eq_e: "e = Create thread prio"
+      with ih
+      show ?thesis by (unfold eq_e, auto)
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+      with ih show ?thesis 
+        by (unfold eq_e, auto)
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      with ih show ?thesis by (unfold eq_e, auto)
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+      with ih show ?thesis by (unfold eq_e, auto)
+    next 
+      case (thread_set thread prio)
+      from vt_cons thread_set show ?thesis by simp
+    qed
+  next
+    case vt_nil
+    show ?case by (auto)
+  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 cp_le:
+  assumes vt: "vt step s"
+  and 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_dependents_def)
+  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (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> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
+  next
+    from finite_threads [OF vt]
+    show "finite (threads s)" .
+  next
+    from th_in
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
+      apply (auto simp:Domain_def)
+      apply (rule_tac dm_depend_threads[OF vt])
+      apply (unfold trancl_domain [of "depend s", symmetric])
+      by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
+  qed
+qed
+
+lemma le_cp:
+  assumes vt: "vt step s"
+  shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+  show "Prc (original_priority th s) (birthtime th s)
+    \<le> Max (insert (Prc (original_priority th s) (birthtime th s))
+            ((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (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> (depend (wq s))\<^sup>+}"
+        proof -
+          let ?F = "\<lambda> (x, y). the_th x"
+          have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+            hence "finite ((depend (wq s))\<^sup>+)"
+              apply (unfold finite_trancl)
+              by (auto simp: s_depend_def cs_depend_def wq_def)
+            thus ?thesis by auto
+          qed
+          ultimately show ?thesis by (auto intro:finite_subset)
+        qed
+        thus ?thesis by (simp add:cs_dependents_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: 
+  assumes vt: "vt step s"
+  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[OF vt] 
+    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 vt 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[OF vt]
+      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 vt, 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[OF vt]
+        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 vt: "vt step s"
+  and np: "threads s \<noteq> {}"
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq[OF vt])
+  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[OF vt] 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 vt tm_in]
+    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
+    thus ?thesis
+    proof
+      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
+      then obtain th' where th'_in: "th' \<in> readys s" 
+        and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
+      have "cp s th' = ?f tm"
+      proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
+        from dependents_threads[OF vt] finite_threads[OF vt]
+        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))" 
+          by (auto intro:finite_subset)
+      next
+        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (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[OF vt]
+          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        next
+          from p_in and th'_in and dependents_threads[OF vt, 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> dependents (wq s) th')"
+        proof -
+          from tm_chain
+          have "tm \<in> dependents (wq s) th'"
+            by (unfold cs_dependents_def s_depend_def cs_depend_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 dependents_threads [OF vt, of th1] th1_in
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (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> dependents (wq s) th1) \<noteq> {}" by simp
+        next
+          from finite_threads[OF vt] 
+          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        qed
+      next
+        from finite_threads[OF vt]
+        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> dependents (wq s) tm)"
+          show "y \<le> preced tm s"
+          proof -
+            { fix y'
+              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
+              have "y' \<le> preced tm s"
+              proof(unfold tm_max, rule Max_ge)
+                from hy' dependents_threads[OF vt, of tm]
+                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+              next
+                from finite_threads[OF vt] 
+                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+              qed
+            } with hy show ?thesis by auto
+          qed
+        next
+          from dependents_threads[OF vt, of tm] finite_threads[OF vt]
+          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
+            by (auto intro:finite_subset)
+        next
+          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (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[OF vt]
+          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[OF vt]
+            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+          next
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
+              by simp
+          next
+            from dependents_threads[OF vt, of th1] th1_readys
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (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
+
+lemma max_cp_readys_threads:
+  assumes vt: "vt step s"
+  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 vt False])
+qed
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+proof
+  fix th
+  assume "th \<in> readys s"
+  thus "th \<in> threads s"
+    by (unfold readys_def, auto)
+qed
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+  apply (unfold s_holding_def cs_holding_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
+
+end
\ No newline at end of file
diff -r 12e9aa68d5db -r 4190df6f4488 prio/PrioG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/PrioG.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,2805 @@
+theory PrioG
+imports PrioGDef
+begin
+
+lemma runing_ready: "runing s \<subseteq> readys s"
+  by (auto simp only:runing_def readys_def)
+
+lemma wq_v_neq:
+   "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 wq_distinct: "vt step s \<Longrightarrow> distinct (wq s cs)"
+proof(erule_tac vt.induct, simp add:wq_def)
+  fix s e
+  assume h1: "step s e"
+  and h2: "distinct (wq s cs)"
+  thus "distinct (wq (e # s) cs)"
+  proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
+    fix thread s
+    assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+      and h2: "thread \<in> set (waiting_queue (schs s) cs)"
+      and h3: "thread \<in> runing s"
+    show "False" 
+    proof -
+      from h3 have "\<And> cs. thread \<in>  set (waiting_queue (schs s) cs) \<Longrightarrow> 
+                             thread = hd ((waiting_queue (schs s) cs))" 
+        by (simp add:runing_def readys_def s_waiting_def wq_def)
+      from this [OF h2] have "thread = hd (waiting_queue (schs s) cs)" .
+      with h2
+      have "(Cs cs, Th thread) \<in> (depend s)"
+        by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
+      with h1 show False by auto
+    qed
+  next
+    fix thread s a list
+    assume dst: "distinct list"
+    show "distinct (SOME q. distinct q \<and> set q = set list)"
+    proof(rule someI2)
+      from dst show  "distinct list \<and> set list = set list" by auto
+    next
+      fix q assume "distinct q \<and> set q = set list"
+      thus "distinct q" by auto
+    qed
+  qed
+qed
+
+lemma step_back_vt: "vt ccs (e#s) \<Longrightarrow> vt ccs s"
+  by(ind_cases "vt ccs (e#s)", simp)
+
+lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e"
+  by(ind_cases "vt ccs (e#s)", simp)
+
+lemma block_pre: 
+  fixes thread cs s
+  assumes vt_e: "vt step (e#s)"
+  and s_ni: "thread \<notin>  set (wq s cs)"
+  and s_i: "thread \<in> set (wq (e#s) cs)"
+  shows "e = P thread cs"
+proof -
+  show ?thesis
+  proof(cases e)
+    case (P th cs)
+    with assms
+    show ?thesis
+      by (auto simp:wq_def Let_def split:if_splits)
+  next
+    case (Create th prio)
+    with assms show ?thesis
+      by (auto simp:wq_def Let_def split:if_splits)
+  next
+    case (Exit th)
+    with assms show ?thesis
+      by (auto simp:wq_def Let_def split:if_splits)
+  next
+    case (Set th prio)
+    with assms show ?thesis
+      by (auto simp:wq_def Let_def split:if_splits)
+  next
+    case (V th cs)
+    with assms show ?thesis
+      apply (auto simp:wq_def Let_def split:if_splits)
+    proof -
+      fix q qs
+      assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+        and h2: "q # qs = waiting_queue (schs s) cs"
+        and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+        and vt: "vt step (V th cs # s)"
+      from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp
+      moreover have "thread \<in> set qs"
+      proof -
+        have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
+        proof(rule someI2)
+          from wq_distinct [OF step_back_vt[OF vt], of cs]
+          and h2[symmetric, folded wq_def]
+          show "distinct qs \<and> set qs = set qs" by auto
+        next
+          fix x assume "distinct x \<and> set x = set qs"
+          thus "set x = set qs" by auto
+        qed
+        with h3 show ?thesis by simp
+      qed
+      ultimately show "False" by auto
+      qed
+  qed
+qed
+
+lemma p_pre: "\<lbrakk>vt step ((P thread cs)#s)\<rbrakk> \<Longrightarrow> 
+  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (depend s)^+"
+apply (ind_cases "vt step ((P thread cs)#s)")
+apply (ind_cases "step s (P thread cs)")
+by auto
+
+lemma abs1:
+  fixes e es
+  assumes ein: "e \<in> set es"
+  and neq: "hd es \<noteq> hd (es @ [x])"
+  shows "False"
+proof -
+  from ein have "es \<noteq> []" by auto
+  then obtain e ess where "es = e # ess" by (cases es, auto)
+  with neq show ?thesis by auto
+qed
+
+lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
+  by (cases es, auto)
+
+inductive_cases evt_cons: "vt cs (a#s)"
+
+lemma abs2:
+  assumes vt: "vt step (e#s)"
+  and inq: "thread \<in> set (wq s cs)"
+  and nh: "thread = hd (wq s cs)"
+  and qt: "thread \<noteq> hd (wq (e#s) cs)"
+  and inq': "thread \<in> set (wq (e#s) cs)"
+  shows "False"
+proof -
+  from assms show "False"
+    apply (cases e)
+    apply ((simp split:if_splits add:Let_def wq_def)[1])+
+    apply (insert abs1, fast)[1]
+    apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)
+  proof -
+    fix th qs
+    assume vt: "vt step (V th cs # s)"
+      and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+      and eq_wq: "waiting_queue (schs s) cs = thread # qs"
+    show "False"
+    proof -
+      from wq_distinct[OF step_back_vt[OF vt], of cs]
+        and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
+      moreover have "thread \<in> set qs"
+      proof -
+        have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
+        proof(rule someI2)
+          from wq_distinct [OF step_back_vt[OF vt], of cs]
+          and eq_wq [folded wq_def]
+          show "distinct qs \<and> set qs = set qs" by auto
+        next
+          fix x assume "distinct x \<and> set x = set qs"
+          thus "set x = set qs" by auto
+        qed
+        with th_in show ?thesis by auto
+      qed
+      ultimately show ?thesis by auto
+    qed
+  qed
+qed
+
+lemma vt_moment: "\<And> t. \<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
+proof(induct s, simp)
+  fix a s t
+  assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
+    and vt_a: "vt cs (a # s)"
+    and le_t: "t \<le> length (a # s)"
+  show "vt cs (moment t (a # s))"
+  proof(cases "t = length (a#s)")
+    case True
+    from True have "moment t (a#s) = a#s" by simp
+    with vt_a show ?thesis by simp
+  next
+    case False
+    with le_t have le_t1: "t \<le> length s" by simp
+    from vt_a have "vt cs s"
+      by (erule_tac evt_cons, simp)
+    from h [OF this le_t1] have "vt cs (moment t s)" .
+    moreover have "moment t (a#s) = moment t s"
+    proof -
+      from moment_app [OF le_t1, of "[a]"] 
+      show ?thesis by simp
+    qed
+    ultimately show ?thesis by auto
+  qed
+qed
+
+(* Wrong:
+    lemma \<lbrakk>thread \<in> set (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
+*)
+
+lemma waiting_unique_pre:
+  fixes cs1 cs2 s thread
+  assumes vt: "vt step s"
+  and 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 cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  from h11 and h12 have q1: "?Q cs1 s" by simp
+  from h21 and h22 have q2: "?Q cs2 s" by simp
+  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+  from p_split [of "?Q cs1", OF q1 nq1]
+  obtain t1 where lt1: "t1 < length s"
+    and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
+        thread \<noteq> hd (wq (moment t1 s) cs1))"
+    and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
+             thread \<noteq> hd (wq (moment i' s) cs1))" by auto
+  from p_split [of "?Q cs2", OF q2 nq2]
+  obtain t2 where lt2: "t2 < length s"
+    and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
+        thread \<noteq> hd (wq (moment t2 s) cs2))"
+    and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
+             thread \<noteq> hd (wq (moment i' s) cs2))" by auto
+  show ?thesis
+  proof -
+    { 
+      assume 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: "vt step (e#moment t2 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF vt_e False h1]
+        have "e = P thread cs2" .
+        with vt_e have "vt step ((P thread cs2)# moment t2 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t2 s)" by auto
+        with nn1 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume lt12: "t2 < t1"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt step (e#moment t1 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF vt_e False h1]
+        have "e = P thread cs1" .
+        with vt_e have "vt step ((P thread cs1)# moment t1 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t1 s)" by auto
+        with nn2 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume eqt12: "t1 = t2"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt step (e#moment t1 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF vt_e False h1]
+        have eq_e1: "e = P thread cs1" .
+        have lt_t3: "t1 < ?t3" by simp
+        with eqt12 have "t2 < ?t3" by simp
+        from nn2 [rule_format, OF this] and eq_m and eqt12
+        have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+          h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+        show ?thesis
+        proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+          case True
+          from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+            by auto
+          from vt_e and eqt12 have "vt step (e#moment t2 s)" by simp 
+          from abs2 [OF this True eq_th h2 h1]
+          show ?thesis .
+        next
+          case False
+          have vt_e: "vt step (e#moment t2 s)"
+          proof -
+            from vt_moment [OF vt le_t3] eqt12
+            have "vt step (moment (Suc t2) s)" by auto
+            with eq_m eqt12 show ?thesis by simp
+          qed
+          from block_pre [OF vt_e False h1]
+          have "e = P thread cs2" .
+          with eq_e1 neq12 show ?thesis by auto
+        qed
+      qed
+    } ultimately show ?thesis by arith
+  qed
+qed
+
+lemma waiting_unique:
+  assumes "vt step s"
+  and "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+proof -
+  from waiting_unique_pre and prems
+  show ?thesis
+    by (auto simp add:s_waiting_def)
+qed
+
+lemma holded_unique:
+  assumes "vt step s"
+  and "holding s th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+proof -
+  from prems show ?thesis
+    unfolding s_holding_def
+    by auto
+qed
+
+lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits)
+
+lemma birthtime_unique: 
+  "\<lbrakk>birthtime th1 s = birthtime 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:birthtime_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 "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
+  from birthtime_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 unique_minus:
+  fixes x y z r
+  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:
+  fixes r x y z
+  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:
+  fixes r x y z
+  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
+
+lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+
+
+lemma step_v_hold_inv[elim_format]:
+  "\<And>c t. \<lbrakk>vt step (V th cs # s); 
+  \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> next_th s th cs t \<and> c = cs"
+proof -
+  fix c t
+  assume vt: "vt step (V th cs # s)"
+    and nhd: "\<not> holding (wq s) t c"
+    and hd: "holding (wq (V th cs # s)) t c"
+  show "next_th s th cs t \<and> c = cs"
+  proof(cases "c = cs")
+    case False
+    with nhd hd show ?thesis
+      by (unfold cs_holding_def wq_def, auto simp:Let_def)
+  next
+    case True
+    with step_back_step [OF vt] 
+    have "step s (V th c)" by simp
+    hence "next_th s th cs t"
+    proof(cases)
+      assume "holding s th c"
+      with nhd hd show ?thesis
+        apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
+               auto simp:Let_def split:list.splits if_splits)
+        proof -
+          assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
+          moreover have "\<dots> = set []"
+          proof(rule someI2)
+            show "distinct [] \<and> [] = []" by auto
+          next
+            fix x assume "distinct x \<and> x = []"
+            thus "set x = set []" by auto
+          qed
+          ultimately show False by auto
+        next
+          assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
+          moreover have "\<dots> = set []"
+          proof(rule someI2)
+            show "distinct [] \<and> [] = []" by auto
+          next
+            fix x assume "distinct x \<and> x = []"
+            thus "set x = set []" by auto
+          qed
+          ultimately show False by auto
+        qed
+    qed
+    with True show ?thesis by auto
+  qed
+qed
+
+lemma step_v_wait_inv[elim_format]:
+    "\<And>t c. \<lbrakk>vt step (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
+           \<rbrakk>
+          \<Longrightarrow> (next_th s th cs t \<and> cs = c)"
+proof -
+  fix t c 
+  assume vt: "vt step (V th cs # s)"
+    and nw: "\<not> waiting (wq (V th cs # s)) t c"
+    and wt: "waiting (wq s) t c"
+  show "next_th s th cs t \<and> cs = c"
+  proof(cases "cs = c")
+    case False
+    with nw wt show ?thesis
+      by (auto simp:cs_waiting_def wq_def Let_def)
+  next
+    case True
+    from nw[folded True] wt[folded True]
+    have "next_th s th cs t"
+      apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
+    proof -
+      fix a list
+      assume t_in: "t \<in> set list"
+        and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
+        and eq_wq: "waiting_queue (schs s) cs = a # list"
+      have " set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def]
+        show "distinct list \<and> set list = set list" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+          by auto
+      qed
+      with t_ni and t_in show "a = th" by auto
+    next
+      fix a list
+      assume t_in: "t \<in> set list"
+        and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
+        and eq_wq: "waiting_queue (schs s) cs = a # list"
+      have " set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def]
+        show "distinct list \<and> set list = set list" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+          by auto
+      qed
+      with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto
+    next
+      fix a list
+      assume eq_wq: "waiting_queue (schs s) cs = a # list"
+      from step_back_step[OF vt]
+      show "a = th"
+      proof(cases)
+        assume "holding s th cs"
+        with eq_wq show ?thesis
+          by (unfold s_holding_def wq_def, auto)
+      qed
+    qed
+    with True show ?thesis by simp
+  qed
+qed
+
+lemma step_v_not_wait[consumes 3]:
+  "\<lbrakk>vt step (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
+  by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
+
+lemma step_v_release:
+  "\<lbrakk>vt step (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
+proof -
+  assume vt: "vt step (V th cs # s)"
+    and hd: "holding (wq (V th cs # s)) th cs"
+  from step_back_step [OF vt] and hd
+  show "False"
+  proof(cases)
+    assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"
+    thus ?thesis
+      apply (unfold s_holding_def wq_def cs_holding_def)
+      apply (auto simp:Let_def split:list.splits)
+    proof -
+      fix list
+      assume eq_wq[folded wq_def]: 
+        "waiting_queue (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
+      and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
+            \<in> set (SOME q. distinct q \<and> set q = set list)"
+      have "set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
+        show "distinct list \<and> set list = set list" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+          by auto
+      qed
+      moreover have "distinct  (hd (SOME q. distinct q \<and> set q = set list) # list)"
+      proof -
+        from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
+        show ?thesis by auto
+      qed
+      moreover note eq_wq and hd_in
+      ultimately show "False" by auto
+    qed
+  qed
+qed
+
+lemma step_v_get_hold:
+  "\<And>th'. \<lbrakk>vt step (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
+  apply (unfold cs_holding_def next_th_def wq_def,
+         auto simp:Let_def)
+proof -
+  fix rest
+  assume vt: "vt step (V th cs # s)"
+    and eq_wq[folded wq_def]: " waiting_queue (schs s) cs = th # rest"
+    and nrest: "rest \<noteq> []"
+    and ni: "hd (SOME q. distinct q \<and> set q = set rest)
+            \<notin> set (SOME q. distinct q \<and> set q = set rest)"
+  have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+  proof(rule someI2)
+    from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
+    show "distinct rest \<and> set rest = set rest" by auto
+  next
+    fix x assume "distinct x \<and> set x = set rest"
+    hence "set x = set rest" by auto
+    with nrest
+    show "x \<noteq> []" by (case_tac x, auto)
+  qed
+  with ni show "False" by auto
+qed
+
+lemma step_v_release_inv[elim_format]:
+"\<And>c t. \<lbrakk>vt step (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> 
+  c = cs \<and> t = th"
+  apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
+  proof -
+    fix a list
+    assume vt: "vt step (V th cs # s)" and eq_wq: "waiting_queue (schs s) cs = a # list"
+    from step_back_step [OF vt] show "a = th"
+    proof(cases)
+      assume "holding s th cs" with eq_wq
+      show ?thesis
+        by (unfold s_holding_def wq_def, auto)
+    qed
+  next
+    fix a list
+    assume vt: "vt step (V th cs # s)" and eq_wq: "waiting_queue (schs s) cs = a # list"
+    from step_back_step [OF vt] show "a = th"
+    proof(cases)
+      assume "holding s th cs" with eq_wq
+      show ?thesis
+        by (unfold s_holding_def wq_def, auto)
+    qed
+  qed
+
+lemma step_v_waiting_mono:
+  "\<And>t c. \<lbrakk>vt step (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
+proof -
+  fix t c
+  let ?s' = "(V th cs # s)"
+  assume vt: "vt step ?s'" 
+    and wt: "waiting (wq ?s') t c"
+  show "waiting (wq s) t c"
+  proof(cases "c = cs")
+    case False
+    assume neq_cs: "c \<noteq> cs"
+    hence "waiting (wq ?s') t c = waiting (wq s) t c"
+      by (unfold cs_waiting_def wq_def, auto simp:Let_def)
+    with wt show ?thesis by simp
+  next
+    case True
+    with wt show ?thesis
+      apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
+    proof -
+      fix a list
+      assume not_in: "t \<notin> set list"
+        and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
+        and eq_wq: "waiting_queue (schs s) cs = a # list"
+      have "set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from wq_distinct [OF step_back_vt[OF vt], of cs]
+        and eq_wq[folded wq_def]
+        show "distinct list \<and> set list = set list" by auto
+      next
+        fix x assume "distinct x \<and> set x = set list"
+        thus "set x = set list" by auto
+      qed
+      with not_in is_in show "t = a" by auto
+    next
+      fix list
+      assume is_waiting: "waiting (wq (V th cs # s)) t cs"
+      and eq_wq: "waiting_queue (schs s) cs = t # list"
+      hence "t \<in> set list"
+        apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
+      proof -
+        assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
+        moreover have "\<dots> = set list" 
+        proof(rule someI2)
+          from wq_distinct [OF step_back_vt[OF vt], of cs]
+            and eq_wq[folded wq_def]
+          show "distinct list \<and> set list = set list" by auto
+        next
+          fix x assume "distinct x \<and> set x = set list" 
+          thus "set x = set list" by auto
+        qed
+        ultimately show "t \<in> set list" by simp
+      qed
+      with eq_wq and wq_distinct [OF step_back_vt[OF vt], of cs, unfolded wq_def]
+      show False by auto
+    qed
+  qed
+qed
+
+lemma step_depend_v:
+assumes vt:
+  "vt step (V th cs#s)"
+shows "
+  depend (V th cs # s) =
+  depend 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_depend_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
+
+lemma step_depend_p:
+  "vt step (P th cs#s) \<Longrightarrow>
+  depend (P th cs # s) =  (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
+                                             else depend s \<union> {(Th th, Cs cs)})"
+  apply(unfold s_depend_def wq_def)
+  apply (auto split:list.splits prod.splits simp:Let_def cs_waiting_def cs_holding_def)
+  apply(case_tac "c = cs", auto)
+  apply(fold wq_def)
+  apply(drule_tac step_back_step)
+  by (ind_cases " step s (P (hd (wq s cs)) cs)", 
+    auto simp:s_depend_def wq_def cs_holding_def)
+
+lemma simple_A:
+  fixes A
+  assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
+  shows "A = {} \<or> (\<exists> a. A = {a})"
+proof(cases "A = {}")
+  case True thus ?thesis by simp
+next
+  case False then obtain a where "a \<in> A" by auto
+  with h have "A = {a}" by auto
+  thus ?thesis by simp
+qed
+
+lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_depend_def, auto)
+
+lemma acyclic_depend: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "acyclic (depend s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume ih: "acyclic (depend s)"
+      and stp: "step s e"
+      and vt: "vt step s"
+    show ?case
+    proof(cases e)
+      case (Create th prio)
+      with ih
+      show ?thesis by (simp add:depend_create_unchanged)
+    next
+      case (Exit th)
+      with ih show ?thesis by (simp add:depend_exit_unchanged)
+    next
+      case (V th cs)
+      from V vt stp have vtt: "vt step (V th cs#s)" by auto
+      from step_depend_v [OF this]
+      have eq_de: 
+        "depend (e # s) = 
+            depend 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)" by (unfold s_holding_def, 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 depend_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_depend_def s_waiting_def cs_waiting_def by simp
+            hence "cs' = cs"
+            proof(rule waiting_unique [OF vt])
+              from eq_wq wq_distinct[OF vt, of cs]
+              show "waiting s ?th' cs" 
+                apply (unfold s_waiting_def, auto)
+              proof -
+                assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq s cs = th # rest"
+                have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+                proof(rule someI2)
+                  from wq_distinct[OF vt, 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 wq_distinct[OF vt, 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 "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 wq_distinct[OF vt, 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 wq_distinct[OF vt, 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 step (P th cs#s)" by auto
+    from step_depend_p [OF this] P
+    have "depend (e # s) = 
+      (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
+      depend 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 =  depend s \<union> {(Cs cs, Th th)}" by simp
+      have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
+      proof
+        assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
+        hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+        from tranclD2 [OF this]
+        obtain x where "(x, Cs cs) \<in> depend s" by auto
+        with True show False by (auto simp:s_depend_def cs_waiting_def)
+      qed
+      with acyclic_insert ih eq_r show ?thesis by auto
+    next
+      case False
+      hence eq_r: "?R =  depend s \<union> {(Th th, Cs cs)}" by simp
+      have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
+      proof
+        assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
+        hence "(Cs cs, Th th) \<in> (depend 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> (depend 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 depend_set_unchanged
+      show ?thesis by (simp add:depend_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "acyclic (depend ([]::state))"
+      by (auto simp: s_depend_def cs_waiting_def 
+                      cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+lemma finite_depend: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "finite (depend s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume ih: "finite (depend s)"
+      and stp: "step s e"
+      and vt: "vt step s"
+    show ?case
+    proof(cases e)
+      case (Create th prio)
+      with ih
+      show ?thesis by (simp add:depend_create_unchanged)
+    next
+      case (Exit th)
+      with ih show ?thesis by (simp add:depend_exit_unchanged)
+    next
+      case (V th cs)
+      from V vt stp have vtt: "vt step (V th cs#s)" by auto
+      from step_depend_v [OF this]
+      have eq_de: "depend (e # s) = 
+                   depend 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 auto
+        qed
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (P th cs)
+      from P vt stp have vtt: "vt step (P th cs#s)" by auto
+      from step_depend_p [OF this] P
+      have "depend (e # s) = 
+              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
+                                    depend 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 =  depend s \<union> {(Cs cs, Th th)}" by simp
+        with True and ih show ?thesis by auto
+      next
+        case False
+        hence "?R = depend 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:depend_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "finite (depend ([]::state))"
+      by (auto simp: s_depend_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+text {* Several useful lemmas *}
+
+lemma wf_dep_converse: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "wf ((depend s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+  from finite_depend [OF vt]
+  show "finite (depend s)" .
+next
+  from acyclic_depend[OF vt]
+  show "acyclic (depend s)" .
+qed
+
+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> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
+  by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+
+lemma wq_threads: 
+  fixes s cs
+  assumes vt: "vt step s"
+  and 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)
+    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+      and stp: "step s e"
+      and vt: "vt step 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_depend_def s_holding_def cs_holding_def)
+        by (fold wq_def, auto)
+    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 (waiting_queue (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 = waiting_queue (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 "waiting_queue (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: "waiting_queue (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 wq_distinct[OF vt, 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 (waiting_queue (schs s) cs')" by auto
+                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+              next
+                assume th_in: "th \<in> set (waiting_queue (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>vt step s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+  apply(unfold s_depend_def cs_waiting_def cs_holding_def)
+  by (auto intro:wq_threads)
+
+lemma readys_v_eq:
+  fixes th thread cs rest
+  assumes vt: "vt step s"
+  and 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 prems show ?thesis
+    apply (auto simp:readys_def)
+    apply (case_tac "cs = csa", simp add:s_waiting_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: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: "waiting_queue (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 vt, 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 th_nin th_in show False by auto
+    qed
+qed
+
+lemma chain_building:
+  assumes vt: "vt step s"
+  shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
+proof -
+  from wf_dep_converse [OF vt]
+  have h: "wf ((depend s)\<inverse>)" .
+  show ?thesis
+  proof(induct rule:wf_induct [OF h])
+    fix x
+    assume ih [rule_format]: 
+      "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow> 
+           y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
+    show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
+    proof
+      assume x_d: "x \<in> Domain (depend s)"
+      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
+      proof(cases x)
+        case (Th th)
+        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
+        with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
+        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
+        hence "Cs cs \<in> Domain (depend 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> (depend s)\<^sup>+" by auto
+        have "(x, Th th') \<in> (depend 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> (depend s)^-1" by (auto simp:s_depend_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 [OF vt] have "th' \<in> threads s" by auto
+          with False have "Th th' \<in> Domain (depend s)" 
+            by (auto simp:readys_def s_waiting_def s_depend_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> (depend s)\<^sup>+" by auto
+          from th'_d and th''_in 
+          have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
+          with th''_r show ?thesis by auto
+        qed
+      qed
+    qed
+  qed
+qed
+
+lemma th_chain_to_ready:
+  fixes s th
+  assumes vt: "vt step s"
+  and th_in: "th \<in> threads s"
+  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend 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 (depend s)" 
+    by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
+  from chain_building [rule_format, OF vt this]
+  show ?thesis by auto
+qed
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+  by  (unfold s_waiting_def cs_waiting_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def cs_holding_def, simp)
+
+lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
+  by (unfold s_holding_def cs_holding_def, auto)
+
+lemma unique_depend: "\<lbrakk>vt step s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
+  apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
+  by(auto elim:waiting_unique holding_unique)
+
+lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
+by (induct rule:trancl_induct, auto)
+
+lemma dchain_unique:
+  assumes vt: "vt step s"
+  and th1_d: "(n, Th th1) \<in> (depend s)^+"
+  and th1_r: "th1 \<in> readys s"
+  and th2_d: "(n, Th th2) \<in> (depend 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_depend [OF vt]
+    have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
+    hence "False"
+    proof
+      assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th1, n) \<in> depend s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_depend_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_depend_def s_waiting_def cs_waiting_def)
+      with th1_r show ?thesis by auto
+    next
+      assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th2, n) \<in> depend s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_depend_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 s_depend_def s_waiting_def cs_waiting_def)
+      with th2_r show ?thesis by auto
+    qed
+  } thus ?thesis by auto
+qed
+             
+
+lemma step_holdents_p_add:
+  fixes th cs s
+  assumes vt: "vt step (P th cs#s)"
+  and "wq s cs = []"
+  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_p[OF vt] by auto
+qed
+
+lemma step_holdents_p_eq:
+  fixes th cs s
+  assumes vt: "vt step (P th cs#s)"
+  and "wq s cs \<noteq> []"
+  shows "holdents (P th cs#s) th = holdents s th"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_p[OF vt] by auto
+qed
+
+
+lemma finite_holding:
+  fixes s th cs
+  assumes vt: "vt step s"
+  shows "finite (holdents s th)"
+proof -
+  let ?F = "\<lambda> (x, y). the_cs x"
+  from finite_depend [OF vt]
+  have "finite (depend s)" .
+  hence "finite (?F `(depend s))" by simp
+  moreover have "{cs . (Cs cs, Th th) \<in> depend 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> depend s"
+      hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
+      moreover have "?F (Cs x, Th th) = x" by simp
+      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp 
+    } thus ?thesis by auto
+  qed
+  ultimately show ?thesis by (unfold holdents_def, auto intro:finite_subset)
+qed
+
+lemma cntCS_v_dec: 
+  fixes s thread cs
+  assumes vtv: "vt step (V thread cs#s)"
+  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
+proof -
+  from step_back_step[OF vtv]
+  have cs_in: "cs \<in> holdents s thread" 
+    apply (cases, unfold holdents_def s_depend_def, simp)
+    by (unfold cs_holding_def s_holding_def, auto)
+  moreover have cs_not_in: 
+    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
+    apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
+    apply (unfold holdents_def, unfold step_depend_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> depend 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 finite_holding [OF vtv]
+    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 cnp_cnv_cncs:
+  fixes s th
+  assumes vt: "vt step s"
+  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)
+    assume vt: "vt step 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 wq_threads [OF vt 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_def
+          by (simp add:depend_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_def
+        by (simp add:depend_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 (subst (1 2) wq_def)
+          apply (simp add:Let_def)
+          apply (subst s_waiting_def, simp)
+          by (fold wq_def, simp)
+        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> (depend s)\<^sup>+"
+      from prems have vtp: "vt step (P thread cs#s)" by auto
+      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, clarify)
+            apply (intro iffI allI, clarify)
+            apply (erule_tac x = csa in allE, auto)
+            apply (subgoal_tac "waiting_queue (schs s) cs \<noteq> []", auto)
+            apply (erule_tac x = cs in allE, auto)
+            by (case_tac "(waiting_queue (schs s) cs)", auto)
+          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+            apply (simp add:cntCS_def holdents_def)
+            by (unfold  step_depend_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_depend_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> depend s} =
+                  Suc (card {cs. (Cs cs, Th thread) \<in> depend 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 finite_holding [OF vt, of thread]
+                    show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
+                      by (unfold holdents_def, simp)
+                  qed
+                  moreover have "?R - {cs} = ?R"
+                  proof -
+                    have "cs \<notin> ?R"
+                    proof
+                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend 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_def)
+                  by (unfold step_depend_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)
+                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 wq_distinct [OF vtp, 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_def eq_e step_depend_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 prems have vtv: "vt step (V thread cs # s)" by auto
+      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: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 wq_distinct[OF step_back_vt[OF vtv], 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
+      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 wq_distinct[OF step_back_vt[OF vtv], of cs]
+                    and eq_wq show False by auto
+                  qed
+                  thus ?thesis by (simp add: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: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 (unfold eq_e, rule readys_v_eq [OF _ neq_th eq_wq False], auto)
+            moreover have "cntCS (e#s) th = cntCS s th"
+              apply (insert neq_th, unfold eq_e cntCS_def holdents_def step_depend_v[OF vtv], auto)
+              proof -
+                have "{csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
+                      {cs. (Cs cs, Th th) \<in> depend s}"
+                proof -
+                  from False eq_wq
+                  have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> depend 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 wq_distinct[OF step_back_vt[OF vtv], 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> depend s"
+                      by auto
+                  qed    
+                  thus ?thesis by auto
+                qed
+                thus "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
+                             card {cs. (Cs cs, Th th) \<in> depend 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 wq_distinct[OF step_back_vt[OF vtv], of cs], auto)
+                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_depend_v[OF vtv], 
+                         auto simp:next_th_def eq_set s_depend_def holdents_def wq_def
+                                   Let_def cs_holding_def)
+                  by (insert wq_distinct[OF step_back_vt[OF vtv], 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: "waiting_queue (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                show "?t \<in> threads s"
+                proof(rule wq_threads[OF step_back_vt[OF vtv]])
+                  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: "waiting_queue (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                  and neq_cs: "csa \<noteq> cs"
+                  and t_in': "?t \<in>  set (waiting_queue (schs s) csa)"
+                show "?t = hd (waiting_queue (schs s) csa)"
+                proof -
+                  { assume neq_hd': "?t \<noteq> hd (waiting_queue (schs s) csa)"
+                    from wq_distinct[OF step_back_vt[OF vtv], 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 waiting_unique[OF step_back_vt[OF vtv] 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)
+                qed
+                moreover have "th \<in> threads s"
+                proof -
+                  from th_in eq_wq
+                  have "th \<in> set (wq s cs)" by simp
+                  from wq_threads [OF step_back_vt[OF vtv] 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_def eq_e step_depend_v[OF vtv], auto)
+              proof -
+                show "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs} =
+                               Suc (card {cs. (Cs cs, Th th) \<in> depend 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) ` depend s)" 
+                      apply (auto simp:image_def)
+                      by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
+                    with finite_depend[OF step_back_vt[OF vtv]]
+                    show "finite {cs. (Cs cs, Th th) \<in> depend s}" by (auto intro:finite_subset)
+                  next
+                    show "cs \<notin> {cs. (Cs cs, Th th) \<in> depend s}"
+                    proof
+                      assume "cs \<in> {cs. (Cs cs, Th th) \<in> depend s}"
+                      hence "(Cs cs, Th th) \<in> depend s" by simp
+                      with True neq_th eq_wq show False
+                        by (auto simp:next_th_def s_depend_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_def
+          by (simp add:depend_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_def s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma not_thread_cncs:
+  fixes th s
+  assumes vt: "vt step s"
+  and 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)
+    assume vt: "vt step 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_def)
+        by (simp add:depend_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_def)
+        by (simp add:depend_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 prems have vtp: "vt step (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_def eq_e)
+        by (unfold step_depend_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 prems have vtv: "vt step (V thread cs#s)" by auto
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp: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 wq_distinct[OF step_back_vt[OF vtv], 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 "hd x \<in> set rest" by (cases x, auto)
+        qed
+        with eq_wq have "?t \<in> set (wq s cs)" by simp
+        from wq_threads[OF step_back_vt[OF vtv], OF this] and ni
+        show False by auto
+      qed
+      moreover note neq_th eq_wq
+      ultimately have "cntCS (e # s) th  = cntCS s th"
+        by (unfold eq_e cntCS_def holdents_def step_depend_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_def)
+        by (simp add:depend_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (unfold cntCS_def, 
+        auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
+  by (auto simp:s_waiting_def cs_waiting_def)
+
+lemma dm_depend_threads:
+  fixes th s
+  assumes vt: "vt step s"
+  and in_dom: "(Th th) \<in> Domain (depend s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
+  moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
+  ultimately have "(Th th, Cs cs) \<in> depend s" by simp
+  hence "th \<in> set (wq s cs)"
+    by (unfold s_depend_def, auto simp:cs_waiting_def)
+  from wq_threads [OF vt this] show ?thesis .
+qed
+
+lemma cp_eq_cpreced: "cp s th = cpreced s (wq s) th"
+proof(unfold cp_def wq_def, induct s)
+  case (Cons e s')
+  show ?case
+    by (auto simp:Let_def)
+next
+  case Nil
+  show ?case by (auto simp:Let_def)
+qed
+
+
+lemma runing_unique:
+  fixes th1 th2 s
+  assumes vt: "vt step s"
+  and 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"
+    by (unfold runing_def, simp)
+  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
+                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
+    (is "Max (?f ` ?A) = Max (?f ` ?B)")
+    by (unfold cp_eq_cpreced 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 (dependents (wq s) th1)"
+        proof-
+          have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+              hence "finite ((depend (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_depend_def cs_depend_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependents_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 by (auto intro:that)
+  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 (dependents (wq s) th2)"
+        proof-
+          have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+              hence "finite ((depend (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_depend_def cs_depend_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependents_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> dependents (wq s) th1)" by simp
+    thus "th1' \<in> threads s"
+    proof
+      assume "th1' \<in> dependents (wq s) th1"
+      hence "(Th th1') \<in> Domain ((depend s)^+)"
+        apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+        by (auto simp:Domain_def)
+      hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
+      from dm_depend_threads[OF vt 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> dependents (wq s) th2)" by simp
+    thus "th2' \<in> threads s"
+    proof
+      assume "th2' \<in> dependents (wq s) th2"
+      hence "(Th th2') \<in> Domain ((depend s)^+)"
+        apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+        by (auto simp:Domain_def)
+      hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
+      from dm_depend_threads[OF vt 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> dependents (wq s) th1" by simp
+  thus ?thesis
+  proof
+    assume eq_th': "th1' = th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
+    thus ?thesis
+    proof
+      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+    next
+      assume "th2' \<in> dependents (wq s) th2"
+      with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
+      hence "(Th th1, Th th2) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      hence "Th th1 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
+        by auto
+      hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
+      from depend_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th1, Cs cs') \<in> depend s" by simp
+      with runing_1 have "False"
+        apply (unfold runing_def readys_def s_depend_def)
+        by (auto simp:eq_waiting)
+      thus ?thesis by simp
+    qed
+  next
+    assume th1'_in: "th1' \<in> dependents (wq s) th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
+    thus ?thesis 
+    proof
+      assume "th2' = th2"
+      with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
+      hence "(Th th2, Th th1) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      hence "Th th2 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
+        by auto
+      hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
+      from depend_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th2, Cs cs') \<in> depend s" by simp
+      with runing_2 have "False"
+        apply (unfold runing_def readys_def s_depend_def)
+        by (auto simp:eq_waiting)
+      thus ?thesis by simp
+    next
+      assume "th2' \<in> dependents (wq s) th2"
+      with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
+      hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      show ?thesis
+      proof(rule dchain_unique[OF vt 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 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
+
+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
+
+lemma cnp_cnv_eq:
+  fixes th s
+  assumes "vt step s"
+  and "th \<notin> threads s"
+  shows "cntP s th = cntV s th"
+proof -
+  from assms show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
+    have not_in: "th \<notin> threads (e # s)" by fact
+    have "step s e" by fact
+    thus ?case proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+      hence "thread \<in> threads (e#s)" by simp
+      with not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] show ?thesis using eq_e
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+        and not_holding: "holdents s thread = {}"
+      have vt_s: "vt step s" by fact
+      from finite_holding[OF vt_s] have "finite (holdents s thread)" .
+      with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
+      moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
+      moreover note cnp_cnv_cncs[OF vt_s, of thread]
+      ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
+      show ?thesis
+      proof(cases "th = thread")
+        case True
+        with eq_thread eq_e show ?thesis 
+          by (auto simp:cntP_def cntV_def count_def)
+      next
+        case False
+        with not_in and eq_e have "th \<notin> threads s" by simp
+        from ih[OF this] and eq_e show ?thesis 
+           by (auto simp:cntP_def cntV_def count_def)
+      qed
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      have "thread \<in> runing s" by fact
+      with not_in eq_e have neq_th: "thread \<noteq> th" 
+        by (auto simp:runing_def readys_def)
+      from not_in eq_e have "th \<notin> threads s" by simp
+      from ih[OF this] and neq_th and eq_e show ?thesis
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+      have "thread \<in> runing s" by fact
+      with not_in eq_e have neq_th: "thread \<noteq> th" 
+        by (auto simp:runing_def readys_def)
+      from not_in eq_e have "th \<notin> threads s" by simp
+      from ih[OF this] and neq_th and eq_e show ?thesis
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_set thread prio)
+      assume eq_e: "e = Set thread prio"
+        and "thread \<in> runing s"
+      hence "thread \<in> threads (e#s)" 
+        by (simp add:runing_def readys_def)
+      with not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] show ?thesis using eq_e
+        by (auto simp:cntP_def cntV_def count_def)  
+    qed
+  next
+    case vt_nil
+    show ?case by (auto simp:cntP_def cntV_def count_def)
+  qed
+qed
+
+lemma eq_depend: 
+  "depend (wq s) = depend s"
+by (unfold cs_depend_def s_depend_def, auto)
+
+lemma count_eq_dependents:
+  assumes vt: "vt step s"
+  and eq_pv: "cntP s th = cntV s th"
+  shows "dependents (wq s) th = {}"
+proof -
+  from cnp_cnv_cncs[OF vt] and eq_pv
+  have "cntCS s th = 0" 
+    by (auto split:if_splits)
+  moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
+  proof -
+    from finite_holding[OF vt, of th] show ?thesis
+      by (simp add:holdents_def)
+  qed
+  ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
+    by (unfold cntCS_def holdents_def cs_dependents_def, auto)
+  show ?thesis
+  proof(unfold cs_dependents_def)
+    { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
+      then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
+      hence "False"
+      proof(cases)
+        assume "(Th th', Th th) \<in> depend (wq s)"
+        thus "False" by (auto simp:cs_depend_def)
+      next
+        fix c
+        assume "(c, Th th) \<in> depend (wq s)"
+        with h and eq_depend show "False"
+          by (cases c, auto simp:cs_depend_def)
+      qed
+    } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
+  qed
+qed
+
+lemma dependents_threads:
+  fixes s th
+  assumes vt: "vt step s"
+  shows "dependents (wq s) th \<subseteq> threads s"
+proof
+  { fix th th'
+    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
+    have "Th th \<in> Domain (depend s)"
+    proof -
+      from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
+      hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
+      with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
+      thus ?thesis using eq_depend by simp
+    qed
+    from dm_depend_threads[OF vt this]
+    have "th \<in> threads s" .
+  } note hh = this
+  fix th1 
+  assume "th1 \<in> dependents (wq s) th"
+  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
+    by (unfold cs_dependents_def, simp)
+  from hh [OF this] show "th1 \<in> threads s" .
+qed
+
+lemma finite_threads:
+  assumes vt: "vt step s"
+  shows "finite (threads s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume vt: "vt step s"
+    and step: "step s e"
+    and ih: "finite (threads s)"
+    from step
+    show ?case
+    proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+      with ih
+      show ?thesis by (unfold eq_e, auto)
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+      with ih show ?thesis 
+        by (unfold eq_e, auto)
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      with ih show ?thesis by (unfold eq_e, auto)
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+      with ih show ?thesis by (unfold eq_e, auto)
+    next 
+      case (thread_set thread prio)
+      from vt_cons thread_set show ?thesis by simp
+    qed
+  next
+    case vt_nil
+    show ?case by (auto)
+  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 cp_le:
+  assumes vt: "vt step s"
+  and 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_dependents_def)
+  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (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> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
+  next
+    from finite_threads [OF vt]
+    show "finite (threads s)" .
+  next
+    from th_in
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
+      apply (auto simp:Domain_def)
+      apply (rule_tac dm_depend_threads[OF vt])
+      apply (unfold trancl_domain [of "depend s", symmetric])
+      by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
+  qed
+qed
+
+lemma le_cp:
+  assumes vt: "vt step s"
+  shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+  show "Prc (original_priority th s) (birthtime th s)
+    \<le> Max (insert (Prc (original_priority th s) (birthtime th s))
+            ((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (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> (depend (wq s))\<^sup>+}"
+        proof -
+          let ?F = "\<lambda> (x, y). the_th x"
+          have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+            hence "finite ((depend (wq s))\<^sup>+)"
+              apply (unfold finite_trancl)
+              by (auto simp: s_depend_def cs_depend_def wq_def)
+            thus ?thesis by auto
+          qed
+          ultimately show ?thesis by (auto intro:finite_subset)
+        qed
+        thus ?thesis by (simp add:cs_dependents_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: 
+  assumes vt: "vt step s"
+  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[OF vt] 
+    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 vt 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[OF vt]
+      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 vt, 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[OF vt]
+        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 vt: "vt step s"
+  and np: "threads s \<noteq> {}"
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq[OF vt])
+  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[OF vt] 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 vt tm_in]
+    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
+    thus ?thesis
+    proof
+      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
+      then obtain th' where th'_in: "th' \<in> readys s" 
+        and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
+      have "cp s th' = ?f tm"
+      proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
+        from dependents_threads[OF vt] finite_threads[OF vt]
+        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))" 
+          by (auto intro:finite_subset)
+      next
+        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (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[OF vt]
+          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        next
+          from p_in and th'_in and dependents_threads[OF vt, 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> dependents (wq s) th')"
+        proof -
+          from tm_chain
+          have "tm \<in> dependents (wq s) th'"
+            by (unfold cs_dependents_def s_depend_def cs_depend_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 dependents_threads [OF vt, of th1] th1_in
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (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> dependents (wq s) th1) \<noteq> {}" by simp
+        next
+          from finite_threads[OF vt] 
+          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        qed
+      next
+        from finite_threads[OF vt]
+        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> dependents (wq s) tm)"
+          show "y \<le> preced tm s"
+          proof -
+            { fix y'
+              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
+              have "y' \<le> preced tm s"
+              proof(unfold tm_max, rule Max_ge)
+                from hy' dependents_threads[OF vt, of tm]
+                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+              next
+                from finite_threads[OF vt] 
+                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+              qed
+            } with hy show ?thesis by auto
+          qed
+        next
+          from dependents_threads[OF vt, of tm] finite_threads[OF vt]
+          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
+            by (auto intro:finite_subset)
+        next
+          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (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[OF vt]
+          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[OF vt]
+            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+          next
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
+              by simp
+          next
+            from dependents_threads[OF vt, of th1] th1_readys
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (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
+
+lemma max_cp_readys_threads:
+  assumes vt: "vt step s"
+  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 vt False])
+qed
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+proof
+  fix th
+  assume "th \<in> readys s"
+  thus "th \<in> threads s"
+    by (unfold readys_def, auto)
+qed
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+  apply (unfold s_holding_def cs_holding_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
+
+end
\ No newline at end of file
diff -r 12e9aa68d5db -r 4190df6f4488 prio/PrioGDef.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/PrioGDef.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,401 @@
+(*<*)
+theory PrioGDef
+imports Precedence_ord Moment
+begin
+(*>*)
+
+text {*
+  In this section, the formal model of Priority Inheritance is presented. 
+  The model is based on Paulson's inductive protocol verification method, where 
+  the state of the system is modelled as a list of events happened so far with the latest 
+  event put at the head. 
+
+  To define events, the identifiers of {\em threads},
+  {\em priority} and {\em critical resources } (abbreviated as @{text "cs"}) 
+  need to be represented. All three are represetned using standard 
+  Isabelle/HOL type @{typ "nat"}:
+*}
+
+type_synonym thread = nat -- {* Type for thread identifiers. *}
+type_synonym priority = nat  -- {* Type for priorities. *}
+type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
+
+text {*
+  \noindent
+  Every event in the system corresponds to a system call, the formats of which are
+  defined as follows:
+  *}
+
+datatype event = 
+  Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *}
+  Exit thread | -- {* Thread @{text "thread"} finishing its execution. *}
+  P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *}
+  V thread cs | -- {* Thread @{text "thread"}  releasing critical resource @{text "cs"}. *}
+  Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
+
+text {* 
+\noindent
+  Resource Allocation Graph (RAG for short) is used extensively in our formal analysis. 
+  The following type @{text "node"} is used to represent nodes in RAG.
+  *}
+datatype node = 
+   Th "thread" | -- {* Node for thread. *}
+   Cs "cs" -- {* Node for critical resource. *}
+
+text {* 
+  In Paulson's inductive method, the states of system are represented as lists of events,
+  which is defined by the following type @{text "state"}:
+  *}
+type_synonym state = "event list"
+
+text {*
+  \noindent
+  The following function
+  @{text "threads"} is used to calculate the set of live threads (@{text "threads s"})
+  in state @{text "s"}.
+  *}
+fun threads :: "state \<Rightarrow> thread set"
+  where 
+  -- {* At the start of the system, the set of threads is empty: *}
+  "threads [] = {}" | 
+  -- {* New thread is added to the @{text "threads"}: *}
+  "threads (Create thread prio#s) = {thread} \<union> threads s" | 
+  -- {* Finished thread is removed: *}
+  "threads (Exit thread # s) = (threads s) - {thread}" | 
+  -- {* Other kind of events does not affect the value of @{text "threads"}: *}
+  "threads (e#s) = threads s" 
+text {* \noindent
+  Functions such as @{text "threads"}, which extract information out of system states, are called
+  {\em observing functions}. A series of observing functions will be defined in the sequel in order to 
+  model the protocol. 
+  Observing function @{text "original_priority"} calculates 
+  the {\em original priority} of thread @{text "th"} in state @{text "s"}, expressed as
+  : @{text "original_priority th s" }. The {\em original priority} is the priority 
+  assigned to a thread when it is created or when it is reset by system call 
+  @{text "Set thread priority"}.
+*}
+
+fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
+  where
+  -- {* @{text "0"} is assigned to threads which have never been created: *}
+  "original_priority thread [] = 0" |
+  "original_priority thread (Create thread' prio#s) = 
+     (if thread' = thread then prio else original_priority thread s)" |
+  "original_priority thread (Set thread' prio#s) = 
+     (if thread' = thread then prio else original_priority thread s)" |
+  "original_priority thread (e#s) = original_priority thread s"
+
+text {*
+  \noindent
+  In the following,
+  @{text "birthtime th s"} is the time when thread @{text "th"} is created, 
+  observed from state @{text "s"}.
+  The time in the system is measured by the number of events happened so far since the very beginning.
+*}
+fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
+  where
+  "birthtime thread [] = 0" |
+  "birthtime thread ((Create thread' prio)#s) = 
+       (if (thread = thread') then length s else birthtime thread s)" |
+  "birthtime thread ((Set thread' prio)#s) = 
+       (if (thread = thread') then length s else birthtime thread s)" |
+  "birthtime thread (e#s) = birthtime thread s"
+
+text {*
+  \noindent 
+  The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of 
+  a thread is the combination of its {\em original priority} and {\em birth time}. The intention is
+  to discriminate threads with the same priority by giving threads whose priority
+  is assigned earlier higher precedences, becasue such threads are more urgent to finish. 
+  This explains the following definition:
+  *}
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+  where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
+
+
+text {*
+  \noindent
+  A number of important notions are defined here:
+  *}
+
+consts 
+  holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool" 
+  waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+  depend :: "'b \<Rightarrow> (node \<times> node) set"
+  dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+text {*
+  \noindent
+  In the definition of the following several functions, it is supposed that
+  the waiting queue of every critical resource is given by a waiting queue 
+  function @{text "wq"}, which servers as arguments of these functions.
+  *}
+defs (overloaded) 
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  We define that the thread which is at the head of waiting queue of resource @{text "cs"}
+  is holding the resource. This definition is slightly different from tradition where
+  all threads in the waiting queue are considered as waiting for the resource.
+  This notion is reflected in the definition of @{text "holding wq th cs"} as follows:
+  \end{minipage}
+  *}
+  cs_holding_def: 
+  "holding wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  In accordance with the definition of @{text "holding wq th cs"}, 
+  a thread @{text "th"} is considered waiting for @{text "cs"} if 
+  it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head.
+  This is reflected in the definition of @{text "waiting wq th cs"} as follows:
+  \end{minipage}
+  *}
+  cs_waiting_def: 
+  "waiting wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  @{text "depend wq"} represents the Resource Allocation Graph of the system under the waiting 
+  queue function @{text "wq"}.
+  \end{minipage}
+  *}
+  cs_depend_def: 
+  "depend (wq::cs \<Rightarrow> thread list) \<equiv>
+      {(Th t, Cs c) | t c. waiting wq t c} \<union> {(Cs c, Th t) | c t. holding wq t c}"
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  The following @{text "dependents wq th"} represents the set of threads which are depending on
+  thread @{text "th"} in Resource Allocation Graph @{text "depend wq"}:
+  \end{minipage}
+  *}
+  cs_dependents_def: 
+  "dependents (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (depend wq)^+}"
+
+text {*
+  The data structure used by the operating system for scheduling is referred to as 
+  {\em schedule state}. It is represented as a record consisting of 
+  a function assigning waiting queue to resources and a function assigning precedence to 
+  threads:
+  *}
+record schedule_state = 
+    waiting_queue :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
+    cur_preced :: "thread \<Rightarrow> precedence" -- {* The function assigning precedence. *}
+
+text {* \noindent 
+  The following
+  @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under
+  state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of 
+  Priority Inheritance that the {\em current precedence} of a thread is the precedence 
+  inherited from the maximum of all its dependents, i.e. the threads which are waiting 
+  directly or indirectly waiting for some resources from it. If no such thread exits, 
+  @{text "th"}'s {\em current precedence} equals its original precedence, i.e. 
+  @{text "preced th s"}.
+  *}
+definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
+  where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
+
+text {* \noindent
+  The following function @{text "schs"} is used to calculate the schedule state @{text "schs s"}.
+  It is the key function to model Priority Inheritance:
+  *}
+fun schs :: "state \<Rightarrow> schedule_state"
+  where "schs [] = \<lparr>waiting_queue = \<lambda> cs. [],  cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
+  -- {*
+  \begin{minipage}{0.9\textwidth}
+  \begin{enumerate}
+  \item @{text "ps"} is the schedule state of last moment.
+  \item @{text "pwq"} is the waiting queue function of last moment.
+  \item @{text "pcp"} is the precedence function of last moment. 
+  \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement:
+  \begin{enumerate}
+      \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to 
+            the end of @{text "cs"}'s waiting queue.
+      \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state,
+            @{text "th'"} must equal to @{text "thread"}, 
+            because @{text "thread"} is the one currently holding @{text "cs"}. 
+            The case @{text "[] \<Longrightarrow> []"} may never be executed in a legal state.
+            the @{text "(SOME q. distinct q \<and> set q = set qs)"} is used to choose arbitrarily one 
+            thread in waiting to take over the released resource @{text "cs"}. In our representation,
+            this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
+      \item For other happening event, the schedule state just does not change.
+  \end{enumerate}
+  \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue 
+        function. The dependency of precedence function on waiting queue function is the reason to 
+        put them in the same record so that they can evolve together.
+  \end{enumerate}
+  \end{minipage}
+     *}
+   "schs (e#s) = (let ps = schs s in 
+                  let pwq = waiting_queue ps in 
+                  let pcp = cur_preced ps in
+                  let nwq = case e of
+                             P thread cs \<Rightarrow>  pwq(cs:=(pwq cs @ [thread])) |
+                             V thread cs \<Rightarrow> let nq = case (pwq cs) of
+                                                      [] \<Rightarrow> [] | 
+                                                      (th'#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
+                                            in pwq(cs:=nq)                 |
+                              _ \<Rightarrow> pwq
+                  in let ncp = cpreced (e#s) nwq in 
+                     \<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
+                 )"
+
+text {* 
+  \noindent
+  The following @{text "wq"} is a shorthand for @{text "waiting_queue"}. 
+  *}
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
+  where "wq s = waiting_queue (schs s)"
+
+text {* \noindent 
+  The following @{text "cp"} is a shorthand for @{text "cur_preced"}. 
+  *}
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+  where "cp s = cur_preced (schs s)"
+
+text {* \noindent
+  Functions @{text "holding"}, @{text "waiting"}, @{text "depend"} and 
+  @{text "dependents"} still have the 
+  same meaning, but redefined so that they no longer depend on the 
+  fictitious {\em waiting queue function}
+  @{text "wq"}, but on system state @{text "s"}.
+  *}
+defs (overloaded) 
+  s_holding_def: 
+  "holding (s::state) thread cs \<equiv> (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
+  s_waiting_def: 
+  "waiting (s::state) thread cs \<equiv> (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
+  s_depend_def: 
+  "depend (s::state) \<equiv> 
+    {(Th t, Cs c) | t c. waiting (wq s) t c} \<union> {(Cs c, Th t) | c t. holding (wq s) t c}"
+  s_dependents_def: 
+  "dependents (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
+
+text {*
+  The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready} 
+  for running if it is a live thread and it is not waiting for any critical resource.
+  *}
+definition readys :: "state \<Rightarrow> thread set"
+  where "readys s = {thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
+
+text {* \noindent
+  The following function @{text "runing"} calculates the set of running thread, which is the ready 
+  thread with the highest precedence. 
+  *}
+definition runing :: "state \<Rightarrow> thread set"
+  where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
+
+text {* \noindent
+  The following function @{text "holdents s th"} returns the set of resources held by thread 
+  @{text "th"} in state @{text "s"}.
+  *}
+definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+  where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
+
+text {* \noindent
+  @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in
+  state @{text "s"}:
+  *}
+definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
+  where "cntCS s th = card (holdents s th)"
+
+text {* \noindent
+  The fact that event @{text "e"} is eligible to happen next in state @{text "s"} 
+  is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as 
+  follows:
+  *}
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+  where
+  -- {* 
+  A thread can be created if it is not a live thread:
+  *}
+  thread_create: "\<lbrakk>thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
+  -- {*
+  A thread can exit if it no longer hold any resource:
+  *}
+  thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
+  -- {*
+  \begin{minipage}{0.9\textwidth}
+  A thread can request for an critical resource @{text "cs"}, if it is running and 
+  the request does not form a loop in the current RAG. The latter condition 
+  is set up to avoid deadlock. The condition also reflects our assumption all threads are 
+  carefully programmed so that deadlock can not happen:
+  \end{minipage}
+  *}
+  thread_P: "\<lbrakk>thread \<in> runing s;  (Cs cs, Th thread)  \<notin> (depend s)^+\<rbrakk> \<Longrightarrow> 
+                                                                step s (P thread cs)" |
+  -- {*
+  \begin{minipage}{0.9\textwidth}
+  A thread can release a critical resource @{text "cs"} 
+  if it is running and holding that resource:
+  \end{minipage}
+  *}
+  thread_V: "\<lbrakk>thread \<in> runing s;  holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
+  -- {*
+  A thread can adjust its own priority as long as it is current running:
+  *}  
+  thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+text {* \noindent
+  With predicate @{text "step"}, the fact that @{text "s"} is a legal state in 
+  Priority Inheritance protocol can be expressed as: @{text "vt step s"}, where
+  the predicate @{text "vt"} can be defined as the following:
+  *}
+inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
+ for cs -- {* @{text "cs"} is an argument representing any step predicate. *}
+  where
+  -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
+  vt_nil[intro]: "vt cs []" |
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  If @{text "s"} a legal state, and event @{text "e"} is eligible to happen
+  in state @{text "s"}, then @{text "e#s"} is a legal state as well:
+  \end{minipage}
+  *}
+  vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
+
+text {*  \noindent
+  It is easy to see that the definition of @{text "vt"} is generic. It can be applied to 
+  any step predicate to get the set of legal states.
+  *}
+
+text {* \noindent
+  The following two functions @{text "the_cs"} and @{text "the_th"} are used to extract
+  critical resource and thread respectively out of RAG nodes.
+  *}
+fun the_cs :: "node \<Rightarrow> cs"
+  where "the_cs (Cs cs) = cs"
+
+fun the_th :: "node \<Rightarrow> thread"
+  where "the_th (Th th) = th"
+
+text {* \noindent
+  The following predicate @{text "next_th"} describe the next thread to 
+  take over when a critical resource is released. In @{text "next_th s th cs t"}, 
+  @{text "th"} is the thread to release, @{text "t"} is the one to take over.
+  *}
+definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
+  where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and> 
+                                                t = hd (SOME q. distinct q \<and> set q = set rest))"
+
+text {* \noindent
+  The function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
+  in list @{text "l"}:
+  *}
+definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
+  where "count Q l = length (filter Q l)"
+
+text {* \noindent
+  The following @{text "cntP s"} returns the number of operation @{text "P"} happened 
+  before reaching state @{text "s"}.
+  *}
+definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
+  where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
+
+text {* \noindent
+  The following @{text "cntV s"} returns the number of operation @{text "V"} happened 
+  before reaching state @{text "s"}.
+  *}
+definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
+  where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+(*<*)
+end
+(*>*)
+
diff -r 12e9aa68d5db -r 4190df6f4488 prio/ROOT.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ROOT.ML	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,2 @@
+use_thy "CpsG"; 
+use_thy "ExtGG";
diff -r 12e9aa68d5db -r 4190df6f4488 prio/document/llncs.cls
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/document/llncs.cls	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1189 @@
+% LLNCS DOCUMENT CLASS -- version 2.13 (28-Jan-2002)
+% Springer Verlag LaTeX2e support for Lecture Notes in Computer Science
+%
+%%
+%% \CharacterTable
+%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%%   Digits        \0\1\2\3\4\5\6\7\8\9
+%%   Exclamation   \!     Double quote  \"     Hash (number) \#
+%%   Dollar        \$     Percent       \%     Ampersand     \&
+%%   Acute accent  \'     Left paren    \(     Right paren   \)
+%%   Asterisk      \*     Plus          \+     Comma         \,
+%%   Minus         \-     Point         \.     Solidus       \/
+%%   Colon         \:     Semicolon     \;     Less than     \<
+%%   Equals        \=     Greater than  \>     Question mark \?
+%%   Commercial at \@     Left bracket  \[     Backslash     \\
+%%   Right bracket \]     Circumflex    \^     Underscore    \_
+%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
+%%   Right brace   \}     Tilde         \~}
+%%
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesClass{llncs}[2002/01/28 v2.13
+^^J LaTeX document class for Lecture Notes in Computer Science]
+% Options
+\let\if@envcntreset\iffalse
+\DeclareOption{envcountreset}{\let\if@envcntreset\iftrue}
+\DeclareOption{citeauthoryear}{\let\citeauthoryear=Y}
+\DeclareOption{oribibl}{\let\oribibl=Y}
+\let\if@custvec\iftrue
+\DeclareOption{orivec}{\let\if@custvec\iffalse}
+\let\if@envcntsame\iffalse
+\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue}
+\let\if@envcntsect\iffalse
+\DeclareOption{envcountsect}{\let\if@envcntsect\iftrue}
+\let\if@runhead\iffalse
+\DeclareOption{runningheads}{\let\if@runhead\iftrue}
+
+\let\if@openbib\iffalse
+\DeclareOption{openbib}{\let\if@openbib\iftrue}
+
+% languages
+\let\switcht@@therlang\relax
+\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}}
+\def\ds@francais{\def\switcht@@therlang{\switcht@francais}}
+
+\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
+
+\ProcessOptions
+
+\LoadClass[twoside]{article}
+\RequirePackage{multicol} % needed for the list of participants, index
+
+\setlength{\textwidth}{12.2cm}
+\setlength{\textheight}{19.3cm}
+\renewcommand\@pnumwidth{2em}
+\renewcommand\@tocrmarg{3.5em}
+%
+\def\@dottedtocline#1#2#3#4#5{%
+  \ifnum #1>\c@tocdepth \else
+    \vskip \z@ \@plus.2\p@
+    {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by 0pt plus 2cm
+               \parfillskip -\rightskip \pretolerance=10000
+     \parindent #2\relax\@afterindenttrue
+     \interlinepenalty\@M
+     \leavevmode
+     \@tempdima #3\relax
+     \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
+     {#4}\nobreak
+     \leaders\hbox{$\m@th
+        \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
+        mu$}\hfill
+     \nobreak
+     \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}%
+     \par}%
+  \fi}
+%
+\def\switcht@albion{%
+\def\abstractname{Abstract.}
+\def\ackname{Acknowledgement.}
+\def\andname{and}
+\def\lastandname{\unskip, and}
+\def\appendixname{Appendix}
+\def\chaptername{Chapter}
+\def\claimname{Claim}
+\def\conjecturename{Conjecture}
+\def\contentsname{Table of Contents}
+\def\corollaryname{Corollary}
+\def\definitionname{Definition}
+\def\examplename{Example}
+\def\exercisename{Exercise}
+\def\figurename{Fig.}
+\def\keywordname{{\bf Key words:}}
+\def\indexname{Index}
+\def\lemmaname{Lemma}
+\def\contriblistname{List of Contributors}
+\def\listfigurename{List of Figures}
+\def\listtablename{List of Tables}
+\def\mailname{{\it Correspondence to\/}:}
+\def\noteaddname{Note added in proof}
+\def\notename{Note}
+\def\partname{Part}
+\def\problemname{Problem}
+\def\proofname{Proof}
+\def\propertyname{Property}
+\def\propositionname{Proposition}
+\def\questionname{Question}
+\def\remarkname{Remark}
+\def\seename{see}
+\def\solutionname{Solution}
+\def\subclassname{{\it Subject Classifications\/}:}
+\def\tablename{Table}
+\def\theoremname{Theorem}}
+\switcht@albion
+% Names of theorem like environments are already defined
+% but must be translated if another language is chosen
+%
+% French section
+\def\switcht@francais{%\typeout{On parle francais.}%
+ \def\abstractname{R\'esum\'e.}%
+ \def\ackname{Remerciements.}%
+ \def\andname{et}%
+ \def\lastandname{ et}%
+ \def\appendixname{Appendice}
+ \def\chaptername{Chapitre}%
+ \def\claimname{Pr\'etention}%
+ \def\conjecturename{Hypoth\`ese}%
+ \def\contentsname{Table des mati\`eres}%
+ \def\corollaryname{Corollaire}%
+ \def\definitionname{D\'efinition}%
+ \def\examplename{Exemple}%
+ \def\exercisename{Exercice}%
+ \def\figurename{Fig.}%
+ \def\keywordname{{\bf Mots-cl\'e:}}
+ \def\indexname{Index}
+ \def\lemmaname{Lemme}%
+ \def\contriblistname{Liste des contributeurs}
+ \def\listfigurename{Liste des figures}%
+ \def\listtablename{Liste des tables}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}%
+ \def\notename{Remarque}%
+ \def\partname{Partie}%
+ \def\problemname{Probl\`eme}%
+ \def\proofname{Preuve}%
+ \def\propertyname{Caract\'eristique}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Question}%
+ \def\remarkname{Remarque}%
+ \def\seename{voir}
+ \def\solutionname{Solution}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tableau}%
+ \def\theoremname{Th\'eor\`eme}%
+}
+%
+% German section
+\def\switcht@deutsch{%\typeout{Man spricht deutsch.}%
+ \def\abstractname{Zusammenfassung.}%
+ \def\ackname{Danksagung.}%
+ \def\andname{und}%
+ \def\lastandname{ und}%
+ \def\appendixname{Anhang}%
+ \def\chaptername{Kapitel}%
+ \def\claimname{Behauptung}%
+ \def\conjecturename{Hypothese}%
+ \def\contentsname{Inhaltsverzeichnis}%
+ \def\corollaryname{Korollar}%
+%\def\definitionname{Definition}%
+ \def\examplename{Beispiel}%
+ \def\exercisename{\"Ubung}%
+ \def\figurename{Abb.}%
+ \def\keywordname{{\bf Schl\"usselw\"orter:}}
+ \def\indexname{Index}
+%\def\lemmaname{Lemma}%
+ \def\contriblistname{Mitarbeiter}
+ \def\listfigurename{Abbildungsverzeichnis}%
+ \def\listtablename{Tabellenverzeichnis}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Nachtrag}%
+ \def\notename{Anmerkung}%
+ \def\partname{Teil}%
+%\def\problemname{Problem}%
+ \def\proofname{Beweis}%
+ \def\propertyname{Eigenschaft}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Frage}%
+ \def\remarkname{Anmerkung}%
+ \def\seename{siehe}
+ \def\solutionname{L\"osung}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tabelle}%
+%\def\theoremname{Theorem}%
+}
+
+% Ragged bottom for the actual page
+\def\thisbottomragged{\def\@textbottom{\vskip\z@ plus.0001fil
+\global\let\@textbottom\relax}}
+
+\renewcommand\small{%
+   \@setfontsize\small\@ixpt{11}%
+   \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@
+   \abovedisplayshortskip \z@ \@plus2\p@
+   \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@
+   \def\@listi{\leftmargin\leftmargini
+               \parsep 0\p@ \@plus1\p@ \@minus\p@
+               \topsep 8\p@ \@plus2\p@ \@minus4\p@
+               \itemsep0\p@}%
+   \belowdisplayskip \abovedisplayskip
+}
+
+\frenchspacing
+\widowpenalty=10000
+\clubpenalty=10000
+
+\setlength\oddsidemargin   {63\p@}
+\setlength\evensidemargin  {63\p@}
+\setlength\marginparwidth  {90\p@}
+
+\setlength\headsep   {16\p@}
+
+\setlength\footnotesep{7.7\p@}
+\setlength\textfloatsep{8mm\@plus 2\p@ \@minus 4\p@}
+\setlength\intextsep   {8mm\@plus 2\p@ \@minus 2\p@}
+
+\setcounter{secnumdepth}{2}
+
+\newcounter {chapter}
+\renewcommand\thechapter      {\@arabic\c@chapter}
+
+\newif\if@mainmatter \@mainmattertrue
+\newcommand\frontmatter{\cleardoublepage
+            \@mainmatterfalse\pagenumbering{Roman}}
+\newcommand\mainmatter{\cleardoublepage
+       \@mainmattertrue\pagenumbering{arabic}}
+\newcommand\backmatter{\if@openright\cleardoublepage\else\clearpage\fi
+      \@mainmatterfalse}
+
+\renewcommand\part{\cleardoublepage
+                 \thispagestyle{empty}%
+                 \if@twocolumn
+                     \onecolumn
+                     \@tempswatrue
+                   \else
+                     \@tempswafalse
+                 \fi
+                 \null\vfil
+                 \secdef\@part\@spart}
+
+\def\@part[#1]#2{%
+    \ifnum \c@secnumdepth >-2\relax
+      \refstepcounter{part}%
+      \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
+    \else
+      \addcontentsline{toc}{part}{#1}%
+    \fi
+    \markboth{}{}%
+    {\centering
+     \interlinepenalty \@M
+     \normalfont
+     \ifnum \c@secnumdepth >-2\relax
+       \huge\bfseries \partname~\thepart
+       \par
+       \vskip 20\p@
+     \fi
+     \Huge \bfseries #2\par}%
+    \@endpart}
+\def\@spart#1{%
+    {\centering
+     \interlinepenalty \@M
+     \normalfont
+     \Huge \bfseries #1\par}%
+    \@endpart}
+\def\@endpart{\vfil\newpage
+              \if@twoside
+                \null
+                \thispagestyle{empty}%
+                \newpage
+              \fi
+              \if@tempswa
+                \twocolumn
+              \fi}
+
+\newcommand\chapter{\clearpage
+                    \thispagestyle{empty}%
+                    \global\@topnum\z@
+                    \@afterindentfalse
+                    \secdef\@chapter\@schapter}
+\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
+                       \if@mainmatter
+                         \refstepcounter{chapter}%
+                         \typeout{\@chapapp\space\thechapter.}%
+                         \addcontentsline{toc}{chapter}%
+                                  {\protect\numberline{\thechapter}#1}%
+                       \else
+                         \addcontentsline{toc}{chapter}{#1}%
+                       \fi
+                    \else
+                      \addcontentsline{toc}{chapter}{#1}%
+                    \fi
+                    \chaptermark{#1}%
+                    \addtocontents{lof}{\protect\addvspace{10\p@}}%
+                    \addtocontents{lot}{\protect\addvspace{10\p@}}%
+                    \if@twocolumn
+                      \@topnewpage[\@makechapterhead{#2}]%
+                    \else
+                      \@makechapterhead{#2}%
+                      \@afterheading
+                    \fi}
+\def\@makechapterhead#1{%
+% \vspace*{50\p@}%
+  {\centering
+    \ifnum \c@secnumdepth >\m@ne
+      \if@mainmatter
+        \large\bfseries \@chapapp{} \thechapter
+        \par\nobreak
+        \vskip 20\p@
+      \fi
+    \fi
+    \interlinepenalty\@M
+    \Large \bfseries #1\par\nobreak
+    \vskip 40\p@
+  }}
+\def\@schapter#1{\if@twocolumn
+                   \@topnewpage[\@makeschapterhead{#1}]%
+                 \else
+                   \@makeschapterhead{#1}%
+                   \@afterheading
+                 \fi}
+\def\@makeschapterhead#1{%
+% \vspace*{50\p@}%
+  {\centering
+    \normalfont
+    \interlinepenalty\@M
+    \Large \bfseries  #1\par\nobreak
+    \vskip 40\p@
+  }}
+
+\renewcommand\section{\@startsection{section}{1}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {12\p@ \@plus 4\p@ \@minus 4\p@}%
+                       {\normalfont\large\bfseries\boldmath
+                        \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {8\p@ \@plus 4\p@ \@minus 4\p@}%
+                       {\normalfont\normalsize\bfseries\boldmath
+                        \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+                       {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {-0.5em \@plus -0.22em \@minus -0.1em}%
+                       {\normalfont\normalsize\bfseries\boldmath}}
+\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
+                       {-12\p@ \@plus -4\p@ \@minus -4\p@}%
+                       {-0.5em \@plus -0.22em \@minus -0.1em}%
+                       {\normalfont\normalsize\itshape}}
+\renewcommand\subparagraph[1]{\typeout{LLNCS warning: You should not use
+                  \string\subparagraph\space with this class}\vskip0.5cm
+You should not use \verb|\subparagraph| with this class.\vskip0.5cm}
+
+\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
+\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
+\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
+\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
+\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
+\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
+\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
+\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
+\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
+\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
+\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}
+
+\let\footnotesize\small
+
+\if@custvec
+\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}}
+{\mbox{\boldmath$\textstyle#1$}}
+{\mbox{\boldmath$\scriptstyle#1$}}
+{\mbox{\boldmath$\scriptscriptstyle#1$}}}
+\fi
+
+\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}}
+\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil
+\penalty50\hskip1em\null\nobreak\hfil\squareforqed
+\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
+
+\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+\gets\cr\to\cr}}}}}
+\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+<\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
+>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.8pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.3pt}<\cr}}}}}
+\def\bbbr{{\rm I\!R}} %reelle Zahlen
+\def\bbbm{{\rm I\!M}}
+\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
+\def\bbbf{{\rm I\!F}}
+\def\bbbh{{\rm I\!H}}
+\def\bbbk{{\rm I\!K}}
+\def\bbbp{{\rm I\!P}}
+\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
+{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
+\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
+\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbs{{\mathchoice
+{\setbox0=\hbox{$\displaystyle     \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle        \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle      \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
+\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
+{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
+
+\let\ts\,
+
+\setlength\leftmargini  {17\p@}
+\setlength\leftmargin    {\leftmargini}
+\setlength\leftmarginii  {\leftmargini}
+\setlength\leftmarginiii {\leftmargini}
+\setlength\leftmarginiv  {\leftmargini}
+\setlength  \labelsep  {.5em}
+\setlength  \labelwidth{\leftmargini}
+\addtolength\labelwidth{-\labelsep}
+
+\def\@listI{\leftmargin\leftmargini
+            \parsep 0\p@ \@plus1\p@ \@minus\p@
+            \topsep 8\p@ \@plus2\p@ \@minus4\p@
+            \itemsep0\p@}
+\let\@listi\@listI
+\@listi
+\def\@listii {\leftmargin\leftmarginii
+              \labelwidth\leftmarginii
+              \advance\labelwidth-\labelsep
+              \topsep    0\p@ \@plus2\p@ \@minus\p@}
+\def\@listiii{\leftmargin\leftmarginiii
+              \labelwidth\leftmarginiii
+              \advance\labelwidth-\labelsep
+              \topsep    0\p@ \@plus\p@\@minus\p@
+              \parsep    \z@
+              \partopsep \p@ \@plus\z@ \@minus\p@}
+
+\renewcommand\labelitemi{\normalfont\bfseries --}
+\renewcommand\labelitemii{$\m@th\bullet$}
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
+                                                    {{\contentsname}}}
+ \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
+ \def\lastand{\ifnum\value{auco}=2\relax
+                 \unskip{} \andname\
+              \else
+                 \unskip \lastandname\
+              \fi}%
+ \def\and{\stepcounter{@auth}\relax
+          \ifnum\value{@auth}=\value{auco}%
+             \lastand
+          \else
+             \unskip,
+          \fi}%
+ \@starttoc{toc}\if@restonecol\twocolumn\fi}
+
+\def\l@part#1#2{\addpenalty{\@secpenalty}%
+   \addvspace{2em plus\p@}%  % space above part line
+   \begingroup
+     \parindent \z@
+     \rightskip \z@ plus 5em
+     \hrule\vskip5pt
+     \large               % same size as for a contribution heading
+     \bfseries\boldmath   % set line in boldface
+     \leavevmode          % TeX command to enter horizontal mode.
+     #1\par
+     \vskip5pt
+     \hrule
+     \vskip1pt
+     \nobreak             % Never break after part entry
+   \endgroup}
+
+\def\@dotsep{2}
+
+\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
+{chapter.\thechapter}\fi}
+
+\def\addnumcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
+                     {\thechapter}#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmarkwop#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}
+
+\def\@adcmk[#1]{\ifcase #1 \or
+\def\@gtempa{\addnumcontentsmark}%
+  \or    \def\@gtempa{\addcontentsmark}%
+  \or    \def\@gtempa{\addcontentsmarkwop}%
+  \fi\@gtempa{toc}{chapter}}
+\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
+
+\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
+ \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ {\large\bfseries\boldmath#1}\ifx0#2\hfil\null
+ \else
+      \nobreak
+      \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+      \@dotsep mu$}\hfill
+      \nobreak\hbox to\@pnumwidth{\hss #2}%
+ \fi\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@title#1#2{\addpenalty{-\@highpenalty}
+ \addvspace{8pt plus 1pt}
+ \@tempdima \z@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ #1\nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hss #2}\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@author#1#2{\addpenalty{\@highpenalty}
+ \@tempdima=\z@ %15\p@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
+ \textit{#1}\par
+ \penalty\@highpenalty \endgroup}
+
+%\setcounter{tocdepth}{0}
+\newdimen\tocchpnum
+\newdimen\tocsecnum
+\newdimen\tocsectotal
+\newdimen\tocsubsecnum
+\newdimen\tocsubsectotal
+\newdimen\tocsubsubsecnum
+\newdimen\tocsubsubsectotal
+\newdimen\tocparanum
+\newdimen\tocparatotal
+\newdimen\tocsubparanum
+\tocchpnum=\z@            % no chapter numbers
+\tocsecnum=15\p@          % section 88. plus 2.222pt
+\tocsubsecnum=23\p@       % subsection 88.8 plus 2.222pt
+\tocsubsubsecnum=27\p@    % subsubsection 88.8.8 plus 1.444pt
+\tocparanum=35\p@         % paragraph 88.8.8.8 plus 1.666pt
+\tocsubparanum=43\p@      % subparagraph 88.8.8.8.8 plus 1.888pt
+\def\calctocindent{%
+\tocsectotal=\tocchpnum
+\advance\tocsectotal by\tocsecnum
+\tocsubsectotal=\tocsectotal
+\advance\tocsubsectotal by\tocsubsecnum
+\tocsubsubsectotal=\tocsubsectotal
+\advance\tocsubsubsectotal by\tocsubsubsecnum
+\tocparatotal=\tocsubsubsectotal
+\advance\tocparatotal by\tocparanum}
+\calctocindent
+
+\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
+\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
+\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
+\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
+\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
+
+\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
+ \@starttoc{lof}\if@restonecol\twocolumn\fi}
+\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
+
+\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
+ \@starttoc{lot}\if@restonecol\twocolumn\fi}
+\let\l@table\l@figure
+
+\renewcommand\listoffigures{%
+    \section*{\listfigurename
+      \@mkboth{\listfigurename}{\listfigurename}}%
+    \@starttoc{lof}%
+    }
+
+\renewcommand\listoftables{%
+    \section*{\listtablename
+      \@mkboth{\listtablename}{\listtablename}}%
+    \@starttoc{lot}%
+    }
+
+\ifx\oribibl\undefined
+\ifx\citeauthoryear\undefined
+\renewenvironment{thebibliography}[1]
+     {\section*{\refname}
+      \def\@biblabel##1{##1.}
+      \small
+      \list{\@biblabel{\@arabic\c@enumiv}}%
+           {\settowidth\labelwidth{\@biblabel{#1}}%
+            \leftmargin\labelwidth
+            \advance\leftmargin\labelsep
+            \if@openbib
+              \advance\leftmargin\bibindent
+              \itemindent -\bibindent
+              \listparindent \itemindent
+              \parsep \z@
+            \fi
+            \usecounter{enumiv}%
+            \let\p@enumiv\@empty
+            \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+      \if@openbib
+        \renewcommand\newblock{\par}%
+      \else
+        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+      \fi
+      \sloppy\clubpenalty4000\widowpenalty4000%
+      \sfcode`\.=\@m}
+     {\def\@noitemerr
+       {\@latex@warning{Empty `thebibliography' environment}}%
+      \endlist}
+\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
+     {\let\protect\noexpand\immediate
+     \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+\newcount\@tempcntc
+\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
+  \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
+    {\@ifundefined
+       {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
+        ?}\@warning
+       {Citation `\@citeb' on page \thepage \space undefined}}%
+    {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
+     \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
+       \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
+     \else
+      \advance\@tempcntb\@ne
+      \ifnum\@tempcntb=\@tempcntc
+      \else\advance\@tempcntb\m@ne\@citeo
+      \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
+\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
+               \@citea\def\@citea{,\,\hskip\z@skip}%
+               \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
+               {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
+                \def\@citea{--}\fi
+      \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
+\else
+\renewenvironment{thebibliography}[1]
+     {\section*{\refname}
+      \small
+      \list{}%
+           {\settowidth\labelwidth{}%
+            \leftmargin\parindent
+            \itemindent=-\parindent
+            \labelsep=\z@
+            \if@openbib
+              \advance\leftmargin\bibindent
+              \itemindent -\bibindent
+              \listparindent \itemindent
+              \parsep \z@
+            \fi
+            \usecounter{enumiv}%
+            \let\p@enumiv\@empty
+            \renewcommand\theenumiv{}}%
+      \if@openbib
+        \renewcommand\newblock{\par}%
+      \else
+        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+      \fi
+      \sloppy\clubpenalty4000\widowpenalty4000%
+      \sfcode`\.=\@m}
+     {\def\@noitemerr
+       {\@latex@warning{Empty `thebibliography' environment}}%
+      \endlist}
+      \def\@cite#1{#1}%
+      \def\@lbibitem[#1]#2{\item[]\if@filesw
+        {\def\protect##1{\string ##1\space}\immediate
+      \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+   \fi
+\else
+\@cons\@openbib@code{\noexpand\small}
+\fi
+
+\def\idxquad{\hskip 10\p@}% space that divides entry from number
+
+\def\@idxitem{\par\hangindent 10\p@}
+
+\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
+                \noindent\hangindent\wd0\box0}% index entry
+
+\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
+                \noindent\hangindent\wd0\box0}% order index entry
+
+\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
+
+\renewenvironment{theindex}
+               {\@mkboth{\indexname}{\indexname}%
+                \thispagestyle{empty}\parindent\z@
+                \parskip\z@ \@plus .3\p@\relax
+                \let\item\par
+                \def\,{\relax\ifmmode\mskip\thinmuskip
+                             \else\hskip0.2em\ignorespaces\fi}%
+                \normalfont\small
+                \begin{multicols}{2}[\@makeschapterhead{\indexname}]%
+                }
+                {\end{multicols}}
+
+\renewcommand\footnoterule{%
+  \kern-3\p@
+  \hrule\@width 2truecm
+  \kern2.6\p@}
+  \newdimen\fnindent
+  \fnindent1em
+\long\def\@makefntext#1{%
+    \parindent \fnindent%
+    \leftskip \fnindent%
+    \noindent
+    \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
+
+\long\def\@makecaption#1#2{%
+  \vskip\abovecaptionskip
+  \sbox\@tempboxa{{\bfseries #1.} #2}%
+  \ifdim \wd\@tempboxa >\hsize
+    {\bfseries #1.} #2\par
+  \else
+    \global \@minipagefalse
+    \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
+  \fi
+  \vskip\belowcaptionskip}
+
+\def\fps@figure{htbp}
+\def\fnum@figure{\figurename\thinspace\thefigure}
+\def \@floatboxreset {%
+        \reset@font
+        \small
+        \@setnobreak
+        \@setminipage
+}
+\def\fps@table{htbp}
+\def\fnum@table{\tablename~\thetable}
+\renewenvironment{table}
+               {\setlength\abovecaptionskip{0\p@}%
+                \setlength\belowcaptionskip{10\p@}%
+                \@float{table}}
+               {\end@float}
+\renewenvironment{table*}
+               {\setlength\abovecaptionskip{0\p@}%
+                \setlength\belowcaptionskip{10\p@}%
+                \@dblfloat{table}}
+               {\end@dblfloat}
+
+\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
+  ext@#1\endcsname}{#1}{\protect\numberline{\csname
+  the#1\endcsname}{\ignorespaces #2}}\begingroup
+    \@parboxrestore
+    \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+  \endgroup}
+
+% LaTeX does not provide a command to enter the authors institute
+% addresses. The \institute command is defined here.
+
+\newcounter{@inst}
+\newcounter{@auth}
+\newcounter{auco}
+\newdimen\instindent
+\newbox\authrun
+\newtoks\authorrunning
+\newtoks\tocauthor
+\newbox\titrun
+\newtoks\titlerunning
+\newtoks\toctitle
+
+\def\clearheadinfo{\gdef\@author{No Author Given}%
+                   \gdef\@title{No Title Given}%
+                   \gdef\@subtitle{}%
+                   \gdef\@institute{No Institute Given}%
+                   \gdef\@thanks{}%
+                   \global\titlerunning={}\global\authorrunning={}%
+                   \global\toctitle={}\global\tocauthor={}}
+
+\def\institute#1{\gdef\@institute{#1}}
+
+\def\institutename{\par
+ \begingroup
+ \parskip=\z@
+ \parindent=\z@
+ \setcounter{@inst}{1}%
+ \def\and{\par\stepcounter{@inst}%
+ \noindent$^{\the@inst}$\enspace\ignorespaces}%
+ \setbox0=\vbox{\def\thanks##1{}\@institute}%
+ \ifnum\c@@inst=1\relax
+   \gdef\fnnstart{0}%
+ \else
+   \xdef\fnnstart{\c@@inst}%
+   \setcounter{@inst}{1}%
+   \noindent$^{\the@inst}$\enspace
+ \fi
+ \ignorespaces
+ \@institute\par
+ \endgroup}
+
+\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
+   {\star\star\star}\or \dagger\or \ddagger\or
+   \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
+   \or \ddagger\ddagger \else\@ctrerr\fi}}
+
+\def\inst#1{\unskip$^{#1}$}
+\def\fnmsep{\unskip$^,$}
+\def\email#1{{\tt#1}}
+\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
+\@ifpackageloaded{babel}{%
+\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
+\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
+\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
+}{\switcht@@therlang}%
+}
+\def\homedir{\~{ }}
+
+\def\subtitle#1{\gdef\@subtitle{#1}}
+\clearheadinfo
+
+\renewcommand\maketitle{\newpage
+  \refstepcounter{chapter}%
+  \stepcounter{section}%
+  \setcounter{section}{0}%
+  \setcounter{subsection}{0}%
+  \setcounter{figure}{0}
+  \setcounter{table}{0}
+  \setcounter{equation}{0}
+  \setcounter{footnote}{0}%
+  \begingroup
+    \parindent=\z@
+    \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
+    \if@twocolumn
+      \ifnum \col@number=\@ne
+        \@maketitle
+      \else
+        \twocolumn[\@maketitle]%
+      \fi
+    \else
+      \newpage
+      \global\@topnum\z@   % Prevents figures from going at top of page.
+      \@maketitle
+    \fi
+    \thispagestyle{empty}\@thanks
+%
+    \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
+    \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
+    \instindent=\hsize
+    \advance\instindent by-\headlineindent
+%    \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
+%       \addcontentsline{toc}{title}{\the\toctitle}\fi
+    \if@runhead
+       \if!\the\titlerunning!\else
+         \edef\@title{\the\titlerunning}%
+       \fi
+       \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
+       \ifdim\wd\titrun>\instindent
+          \typeout{Title too long for running head. Please supply}%
+          \typeout{a shorter form with \string\titlerunning\space prior to
+                   \string\maketitle}%
+          \global\setbox\titrun=\hbox{\small\rm
+          Title Suppressed Due to Excessive Length}%
+       \fi
+       \xdef\@title{\copy\titrun}%
+    \fi
+%
+    \if!\the\tocauthor!\relax
+      {\def\and{\noexpand\protect\noexpand\and}%
+      \protected@xdef\toc@uthor{\@author}}%
+    \else
+      \def\\{\noexpand\protect\noexpand\newline}%
+      \protected@xdef\scratch{\the\tocauthor}%
+      \protected@xdef\toc@uthor{\scratch}%
+    \fi
+%    \addcontentsline{toc}{author}{\toc@uthor}%
+    \if@runhead
+       \if!\the\authorrunning!
+         \value{@inst}=\value{@auth}%
+         \setcounter{@auth}{1}%
+       \else
+         \edef\@author{\the\authorrunning}%
+       \fi
+       \global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}%
+       \ifdim\wd\authrun>\instindent
+          \typeout{Names of authors too long for running head. Please supply}%
+          \typeout{a shorter form with \string\authorrunning\space prior to
+                   \string\maketitle}%
+          \global\setbox\authrun=\hbox{\small\rm
+          Authors Suppressed Due to Excessive Length}%
+       \fi
+       \xdef\@author{\copy\authrun}%
+       \markboth{\@author}{\@title}%
+     \fi
+  \endgroup
+  \setcounter{footnote}{\fnnstart}%
+  \clearheadinfo}
+%
+\def\@maketitle{\newpage
+ \markboth{}{}%
+ \def\lastand{\ifnum\value{@inst}=2\relax
+                 \unskip{} \andname\
+              \else
+                 \unskip \lastandname\
+              \fi}%
+ \def\and{\stepcounter{@auth}\relax
+          \ifnum\value{@auth}=\value{@inst}%
+             \lastand
+          \else
+             \unskip,
+          \fi}%
+ \begin{center}%
+ \let\newline\\
+ {\Large \bfseries\boldmath
+  \pretolerance=10000
+  \@title \par}\vskip .8cm
+\if!\@subtitle!\else {\large \bfseries\boldmath
+  \vskip -.65cm
+  \pretolerance=10000
+  \@subtitle \par}\vskip .8cm\fi
+ \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
+ \def\thanks##1{}\@author}%
+ \global\value{@inst}=\value{@auth}%
+ \global\value{auco}=\value{@auth}%
+ \setcounter{@auth}{1}%
+{\lineskip .5em
+\noindent\ignorespaces
+\@author\vskip.35cm}
+ {\small\institutename}
+ \end{center}%
+ }
+
+% definition of the "\spnewtheorem" command.
+%
+% Usage:
+%
+%     \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
+% or  \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
+% or  \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
+%
+% New is "cap_font" and "body_font". It stands for
+% fontdefinition of the caption and the text itself.
+%
+% "\spnewtheorem*" gives a theorem without number.
+%
+% A defined spnewthoerem environment is used as described
+% by Lamport.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\@thmcountersep{}
+\def\@thmcounterend{.}
+
+\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
+
+% definition of \spnewtheorem with number
+
+\def\@spnthm#1#2{%
+  \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
+\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
+
+\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
+   {\@definecounter{#1}\@addtoreset{#1}{#3}%
+   \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
+     \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
+   \expandafter\xdef\csname #1name\endcsname{#2}%
+   \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
+                              \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+   {\@definecounter{#1}%
+   \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
+   \expandafter\xdef\csname #1name\endcsname{#2}%
+   \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
+                               \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spothm#1[#2]#3#4#5{%
+  \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
+  {\expandafter\@ifdefinable\csname #1\endcsname
+  {\global\@namedef{the#1}{\@nameuse{the#2}}%
+  \expandafter\xdef\csname #1name\endcsname{#3}%
+  \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
+  \global\@namedef{end#1}{\@endtheorem}}}}
+
+\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\refstepcounter{#1}%
+\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
+
+\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
+                    \ignorespaces}
+
+\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
+       the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
+
+\def\@spbegintheorem#1#2#3#4{\trivlist
+                 \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
+
+\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
+      \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
+
+% definition of \spnewtheorem* without number
+
+\def\@sthm#1#2{\@Ynthm{#1}{#2}}
+
+\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+   {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
+    \expandafter\xdef\csname #1name\endcsname{#2}%
+    \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
+
+\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
+
+\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
+       {#4}{#2}{#3}\ignorespaces}
+
+\def\@Begintheorem#1#2#3{#3\trivlist
+                           \item[\hskip\labelsep{#2#1\@thmcounterend}]}
+
+\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
+      \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
+
+\if@envcntsect
+   \def\@thmcountersep{.}
+   \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
+\else
+   \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
+   \if@envcntreset
+      \@addtoreset{theorem}{section}
+   \else
+      \@addtoreset{theorem}{chapter}
+   \fi
+\fi
+
+%definition of divers theorem environments
+\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
+\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
+\if@envcntsame % alle Umgebungen wie Theorem.
+   \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
+\else % alle Umgebungen mit eigenem Zaehler
+   \if@envcntsect % mit section numeriert
+      \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
+   \else % nicht mit section numeriert
+      \if@envcntreset
+         \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+                                   \@addtoreset{#1}{section}}
+      \else
+         \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+                                   \@addtoreset{#1}{chapter}}%
+      \fi
+   \fi
+\fi
+\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
+\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
+\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
+\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
+\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
+\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
+\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
+\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
+\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
+\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
+\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
+\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
+\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
+\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
+
+\def\@takefromreset#1#2{%
+    \def\@tempa{#1}%
+    \let\@tempd\@elt
+    \def\@elt##1{%
+        \def\@tempb{##1}%
+        \ifx\@tempa\@tempb\else
+            \@addtoreset{##1}{#2}%
+        \fi}%
+    \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
+    \expandafter\def\csname cl@#2\endcsname{}%
+    \@tempc
+    \let\@elt\@tempd}
+
+\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+      \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
+                  \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
+      \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
+      }
+
+\renewenvironment{abstract}{%
+      \list{}{\advance\topsep by0.35cm\relax\small
+      \leftmargin=1cm
+      \labelwidth=\z@
+      \listparindent=\z@
+      \itemindent\listparindent
+      \rightmargin\leftmargin}\item[\hskip\labelsep
+                                    \bfseries\abstractname]}
+    {\endlist}
+
+\newdimen\headlineindent             % dimension for space between
+\headlineindent=1.166cm              % number and text of headings.
+
+\def\ps@headings{\let\@mkboth\@gobbletwo
+   \let\@oddfoot\@empty\let\@evenfoot\@empty
+   \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+                  \leftmark\hfil}
+   \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
+                 \llap{\thepage}}
+   \def\chaptermark##1{}%
+   \def\sectionmark##1{}%
+   \def\subsectionmark##1{}}
+
+\def\ps@titlepage{\let\@mkboth\@gobbletwo
+   \let\@oddfoot\@empty\let\@evenfoot\@empty
+   \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+                  \hfil}
+   \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
+                 \llap{\thepage}}
+   \def\chaptermark##1{}%
+   \def\sectionmark##1{}%
+   \def\subsectionmark##1{}}
+
+\if@runhead\ps@headings\else
+\ps@empty\fi
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\endinput
+%end of file llncs.cls
diff -r 12e9aa68d5db -r 4190df6f4488 prio/document/root.bib
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/document/root.bib	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,111 @@
+@article{OwensReppyTuron09,
+  author = {S.~Owens and J.~Reppy and A.~Turon},
+  title = {{R}egular-{E}xpression {D}erivatives {R}e-{E}xamined},
+  journal = {Journal of Functional Programming},
+  volume = 19,
+  number = {2},
+  year = 2009,
+  pages = {173--190}
+}
+
+
+
+@Unpublished{KraussNipkow11,
+  author = 	 {A.~Kraus and T.~Nipkow},
+  title = 	 {{P}roof {P}earl: {R}egular {E}xpression {E}quivalence and {R}elation {A}lgebra},
+  note = 	 {To appear in Journal of Automated Reasoning},
+  year = 	 {2011}
+}
+
+@Book{Kozen97,
+  author = 	 {D.~Kozen},
+  title = 	 {{A}utomata and {C}omputability},
+  publisher = 	 {Springer Verlag},
+  year = 	 {1997}
+}
+
+
+@incollection{Constable00,
+  author    = {R.~L.~Constable and
+               P.~B.~Jackson and
+               P.~Naumov and
+               J.~C.~Uribe},
+  title     = {{C}onstructively {F}ormalizing {A}utomata {T}heory},
+  booktitle = {Proof, Language, and Interaction},
+  year      = {2000},
+  publisher = {MIT Press},
+  pages     = {213-238}
+}
+
+
+@techreport{Filliatre97,
+  author = {J.-C. Filli\^atre},
+  institution = {LIP - ENS Lyon},
+  number = {97--04},
+  title = {{F}inite {A}utomata {T}heory in {C}oq: 
+           {A} {C}onstructive {P}roof of {K}leene's {T}heorem},
+  type = {Research Report},
+  year = {1997}
+}
+
+@article{OwensSlind08,
+  author    = {S.~Owens and K.~Slind},
+  title     = {{A}dapting {F}unctional {P}rograms to {H}igher {O}rder {L}ogic},
+  journal   = {Higher-Order and Symbolic Computation},
+  volume    = {21},
+  number    = {4},
+  year      = {2008},
+  pages     = {377--409}
+}
+
+@article{Brzozowski64,
+ author = {J.~A.~Brzozowski},
+ title = {{D}erivatives of {R}egular {E}xpressions},
+ journal = {J.~ACM},
+ volume = {11},
+ issue = {4},
+ year = {1964},
+ pages = {481--494},
+ publisher = {ACM}
+} 
+
+@inproceedings{Nipkow98,
+ author={T.~Nipkow},
+ title={{V}erified {L}exical {A}nalysis},
+ booktitle={Proc.~of the 11th International Conference on Theorem Proving in Higher Order Logics},
+ series={LNCS},
+ volume=1479,
+ pages={1--15},
+ year=1998
+}
+
+@inproceedings{BerghoferNipkow00,
+  author={S.~Berghofer and T.~Nipkow},
+  title={{E}xecuting {H}igher {O}rder {L}ogic},
+  booktitle={Proc.~of the International Workshop on Types for Proofs and Programs},
+  year=2002,
+  series={LNCS},
+  volume=2277,
+  pages="24--40"
+}
+
+@book{HopcroftUllman69,
+  author    = {J.~E.~Hopcroft and
+               J.~D.~Ullman},
+  title     = {{F}ormal {L}anguages and {T}heir {R}elation to {A}utomata},
+  publisher = {Addison-Wesley},
+  year      = {1969}
+}
+
+
+@inproceedings{BerghoferReiter09,
+  author    = {S.~Berghofer and
+               M.~Reiter},
+  title     = {{F}ormalizing the {L}ogic-{A}utomaton {C}onnection},
+  booktitle = {Proc.~of the 22nd International
+               Conference on Theorem Proving in Higher Order Logics},
+  year      = {2009},
+  pages     = {147-163},
+  series    = {LNCS},
+  volume    = {5674}
+}
\ No newline at end of file
diff -r 12e9aa68d5db -r 4190df6f4488 prio/document/root.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/document/root.tex	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,73 @@
+\documentclass[runningheads]{llncs}
+\usepackage{isabelle}
+\usepackage{isabellesym}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{tikz}
+\usepackage{pgf}
+\usetikzlibrary{arrows,automata,decorations,fit,calc}
+\usetikzlibrary{shapes,shapes.arrows,snakes,positioning}
+\usepgflibrary{shapes.misc} % LATEX and plain TEX and pure pgf
+\usetikzlibrary{matrix}
+\usepackage{pdfsetup}
+\usepackage{ot1patch}
+\usepackage{times}
+%%\usepackage{proof}
+%%\usepackage{mathabx}
+\usepackage{stmaryrd}
+
+\titlerunning{Myhill-Nerode using Regular Expressions}
+
+
+\urlstyle{rm}
+\isabellestyle{it}
+\renewcommand{\isastyleminor}{\it}%
+\renewcommand{\isastyle}{\normalsize\it}%
+
+
+\def\dn{\,\stackrel{\mbox{\scriptsize def}}{=}\,}
+\renewcommand{\isasymequiv}{$\dn$}
+\renewcommand{\isasymemptyset}{$\varnothing$}
+\renewcommand{\isacharunderscore}{\mbox{$\_\!\_$}}
+
+\newcommand{\isasymcalL}{\ensuremath{\cal{L}}}
+\newcommand{\isasymbigplus}{\ensuremath{\bigplus}}
+
+\newcommand{\bigplus}{\mbox{\Large\bf$+$}}
+\begin{document}
+
+\title{A Formalisation of the Myhill-Nerode Theorem\\ based on Regular
+  Expressions (Proof Pearl)}
+\author{Chunhan Wu\inst{1} \and Xingyuan Zhang\inst{1} \and Christian Urban\inst{2}}
+\institute{PLA University of Science and Technology, China \and TU Munich, Germany}
+\maketitle
+
+%\mbox{}\\[-10mm]
+\begin{abstract} 
+There are numerous textbooks on regular languages. Nearly all of them
+introduce the subject by describing finite automata and only mentioning on the
+side a connection with regular expressions. Unfortunately, automata are difficult
+to formalise in HOL-based theorem provers. The reason is that
+they need to be represented as graphs, matrices or functions, none of which
+are inductive datatypes. Also convenient operations for disjoint unions of
+graphs and functions are not easily formalisiable in HOL. In contrast, regular
+expressions can be defined conveniently as a datatype and a corresponding
+reasoning infrastructure comes for free. We show in this paper that a central
+result from formal language theory---the Myhill-Nerode theorem---can be
+recreated using only regular expressions.
+
+\end{abstract}
+
+
+\input{session}
+
+%%\mbox{}\\[-10mm]
+\bibliographystyle{plain}
+\bibliography{root}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
diff -r 12e9aa68d5db -r 4190df6f4488 prio/paper.pdf
Binary file prio/paper.pdf has changed