# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1354806681 0
# Node ID 110247f9d47e7cc271598f973e1e7277c9b70954

added

diff -r 000000000000 -r 110247f9d47e CpsG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/CpsG.thy	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,1997 @@
+theory CpsG
+imports PrioG 
+begin
+
+lemma not_thread_holdents:
+  fixes th s
+  assumes vt: "vt 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 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_test)
+        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_test 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_test 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 assms thread_exit ih stp not_in vt eq_e have vtp: "vt (P thread cs#s)" by auto
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      hence "holdents (e # s) th  = holdents s th "
+        apply (unfold cntCS_def holdents_test 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 assms thread_V eq_e ih stp not_in vt have vtv: "vt (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: wq_def s_holding_def)
+      from not_in eq_e eq_wq
+      have "\<not> next_th s thread cs th"
+        apply (auto simp:next_th_def)
+      proof -
+        assume ne: "rest \<noteq> []"
+          and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
+        have "?t \<in> set rest"
+        proof(rule someI2)
+          from 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_test 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_test)
+        by (simp add:depend_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+
+
+lemma next_th_neq: 
+  assumes vt: "vt 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 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 \<equiv>
+            {(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 \<equiv> {th'. (Th th', Th th) \<in> child s}"
+
+lemma children_def2:
+  "children s th \<equiv> {th'. \<exists> cs. (Th th', Cs cs) \<in> depend s \<and> (Cs cs, Th th) \<in> depend s}"
+unfolding child_def children_def by simp
+
+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 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 (wq s) 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 (wq s) 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 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 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 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 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 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 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 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 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 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_pre:
+  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 no_dependents:
+  assumes "th' \<noteq> th"
+  shows "th \<notin> dependents s th'"
+proof
+  assume h: "th \<in> dependents s th'"
+  from step_back_step [OF vt_s[unfolded s_def]]
+  have "step s' (Set th prio)" .
+  hence "th \<in> runing s'" by (cases, simp)
+  hence rd_th: "th \<in> readys s'" 
+    by (simp add:readys_def runing_def)
+  from h have "(Th th, Th th') \<in> (depend s')\<^sup>+"
+    by (unfold s_dependents_def, unfold eq_depend, unfold eq_dep, auto)
+  from tranclD[OF this]
+  obtain z where "(Th th, z) \<in> depend s'" by auto
+  with rd_th show "False"
+    apply (case_tac z, auto simp:readys_def s_waiting_def s_depend_def s_waiting_def cs_waiting_def)
+    by (fold wq_def, blast)
+qed
+
+(* Result improved *)
+lemma eq_cp:
+ fixes th' 
+  assumes neq_th: "th' \<noteq> th"
+  shows "cp s th' = cp s' th'"
+proof(rule eq_cp_pre [OF neq_th])
+  from no_dependents[OF neq_th] 
+  show "th \<notin> dependents s th'" .
+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_pre[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_pre)
+              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_pre[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_pre)
+              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 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[folded wq_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 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 wq_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 wq_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: wq_def 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 wq_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 wq_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 wq_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 wq_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 wq_def[symmetric])
+        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 s"
+
+locale step_P_cps_ne =step_P_cps +
+  assumes ne: "wq s' cs \<noteq> []"
+
+locale step_P_cps_e =step_P_cps +
+  assumes ee: "wq s' cs = []"
+
+context step_P_cps_e
+begin
+
+lemma depend_s: "depend s = depend s' \<union> {(Cs cs, Th th)}"
+proof -
+  from ee and  step_depend_p[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 cs) \<in> depend s'" by simp
+      with ee show False
+        by (auto simp:s_depend_def cs_waiting_def)
+    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 cs) \<in> depend s'" by simp
+      with ee show False 
+        by (auto simp:s_depend_def cs_waiting_def)
+    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'"
+      apply (auto simp:child_def)
+      proof -
+        fix th'
+        assume "(Th th', Cs cs) \<in> depend s'"
+        with ee have "False"
+          by (auto simp:s_depend_def cs_waiting_def)
+        thus "\<exists>cs. (Th th', Cs cs) \<in> depend s' \<and> (Cs cs, Th th) \<in> depend s'" by auto 
+      qed
+    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'"
+      apply (auto simp:child_def)
+      proof -
+        fix th'
+        assume "(Th th', Cs cs) \<in> depend s'"
+        with ee have "False"
+          by (auto simp:s_depend_def cs_waiting_def)
+        thus "\<exists>cs. (Th th', Cs cs) \<in> depend s' \<and> (Cs cs, Th th) \<in> depend s'" by auto 
+      qed
+    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 auto
+    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
+
+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 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_test)
+      } 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 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 wq_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 000000000000 -r 110247f9d47e ExtGG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ExtGG.thy	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,1046 @@
+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 s"
+  and threads_s: "th \<in> threads s"
+  and highest: "preced th s = 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 (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 (t@s)" 
+  shows "vt s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt (t @ s) \<Longrightarrow> vt s"
+      and vt_et: "vt ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_gen
+begin
+
+thm extend_highest_gen_axioms_def
+
+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 (t@s); step (t@s) e; 
+                    extend_highest_gen s th prio tm t; 
+                    extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_gen_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt ((e # t') @ s)"
+      and et: "extend_highest_gen s th prio tm (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_gen s th prio tm (e # t')" .
+    next
+      from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+                 preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "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 (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 (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 (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 (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 (wq (t@s)) (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_pre:
+  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
+
+lemmas pv_blocked = pv_blocked_pre[folded detached_eq [OF vt_t]]
+
+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
+    then have not_runing: "th' \<notin> runing (t @ s)" 
+      apply(rule extend_highest_gen.pv_blocked) 
+      using Cons(1) in_thread neq_th' not_holding
+      apply(simp_all add: detached_eq)
+      done
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt (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
+*)
+
+lemmas runing_precond_pre_dtc = runing_precond_pre[folded detached_eq[OF vt_t] detached_eq[OF vt_s]]
+
+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_pre[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 (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 vt_e show "detached (moment (i + k) t @ s) th'"
+            apply(subst detached_eq)
+            apply(auto intro: vt_e evt_cons)
+            done
+        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_eqpv:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_gen.pv_blocked 
+  show ?thesis 
+    using  red_moment [of j] h2 neq_th' h1
+    apply(auto)
+    by (metis extend_highest_gen.pv_blocked_pre)
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and dtc: "detached (moment i t @ s) th'"
+  and le_ij: "i \<le> j"
+  shows "detached (moment j t @ s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from vt_moment[OF vt_t, of "i+length s"] moment_prefix[of i t s]
+  have vt_i: "vt (moment i t @ s)" by auto
+  from vt_moment[OF vt_t, of "j+length s"] moment_prefix[of j t s]
+  have vt_j: "vt  (moment j t @ s)" by auto
+  from moment_blocked_eqpv [OF neq_th' th'_in detached_elim [OF vt_i dtc] le_ij, 
+  folded detached_eq[OF vt_j]]
+  show ?thesis .
+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 (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 (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_eqpv [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 runing_preced_inversion:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "cp (t@s) th' = preced th s"
+proof -
+  from runing' have "cp (t@s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    by (unfold runing_def, auto)
+  also have "\<dots> = preced th s"
+  proof -
+    from max_cp_readys_threads[OF vt_t]
+    have "\<dots> =  Max (cp (t @ s) ` threads (t @ s))" .
+    also have "\<dots> = preced th s"
+    proof -
+      from max_kept
+      and max_cp_eq [OF vt_t]
+      show ?thesis by auto
+    qed
+    finally show ?thesis .
+  qed
+  finally show ?thesis .
+qed
+
+lemma runing_inversion_3:
+  assumes runing': "th' \<in> runing (t@s)"
+  and neq_th: "th' \<noteq> th"
+  shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
+proof -
+  from runing_inversion_2 [OF runing'] 
+    and neq_th 
+    and runing_preced_inversion[OF runing']
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_4:
+  assumes runing': "th' \<in> runing (t@s)"
+  and neq_th: "th' \<noteq> th"
+  shows "th' \<in> threads s"
+  and    "\<not>detached s th'"
+  and    "cp (t@s) th' = preced th s"
+using runing_inversion_3 [OF runing'] 
+  and neq_th 
+  and runing_preced_inversion[OF runing']
+apply(auto simp add: detached_eq[OF vt_s])
+done
+
+
+
+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 000000000000 -r 110247f9d47e Moment.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Moment.thy	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,783 @@
+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
+
+lemma moment_prefix: "(moment i t @ s) = moment (i + length s) (t @ s)"
+proof -
+  from moment_plus_split [of i "length s" "t@s"]
+  have " moment (i + length s) (t @ s) = moment i (restm (length s) (t @ s)) @ moment (length s) (t @ s)"
+    by auto
+  with app_moment_restm[of t s]
+  have "moment (i + length s) (t @ s) = moment i t @ moment (length s) (t @ s)" by simp
+  with moment_app show ?thesis by auto
+qed
+
+end
\ No newline at end of file
diff -r 000000000000 -r 110247f9d47e Precedence_ord.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Precedence_ord.thy	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,34 @@
+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 000000000000 -r 110247f9d47e PrioG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/PrioG.thy	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,2864 @@
+theory PrioG
+imports PrioGDef 
+begin
+
+lemma runing_ready: 
+  shows "runing s \<subseteq> readys s"
+  unfolding runing_def readys_def
+  by auto 
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+  unfolding readys_def
+  by auto
+
+lemma wq_v_neq:
+   "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 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 (wq_fun (schs s) cs)"
+      and h3: "thread \<in> runing s"
+    show "False" 
+    proof -
+      from h3 have "\<And> cs. thread \<in>  set (wq_fun (schs s) cs) \<Longrightarrow> 
+                             thread = hd ((wq_fun (schs s) cs))" 
+        by (simp add:runing_def readys_def s_waiting_def wq_def)
+      from this [OF h2] have "thread = hd (wq_fun (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 (e#s) \<Longrightarrow> vt s"
+  by(ind_cases "vt (e#s)", simp)
+
+lemma step_back_step: "vt (e#s) \<Longrightarrow> step s e"
+  by(ind_cases "vt (e#s)", simp)
+
+lemma block_pre: 
+  fixes thread cs s
+  assumes vt_e: "vt (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 (wq_fun (schs s) cs)"
+        and h2: "q # qs = wq_fun (schs s) cs"
+        and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+        and vt: "vt (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 ((P thread cs)#s)\<rbrakk> \<Longrightarrow> 
+  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (depend s)^+"
+apply (ind_cases "vt ((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 (a#s)"
+
+lemma abs2:
+  assumes vt: "vt (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 (V th cs # s)"
+      and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+      and eq_wq: "wq_fun (schs s) cs = thread # qs"
+    show "False"
+    proof -
+      from wq_distinct[OF 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 s\<rbrakk> \<Longrightarrow> vt (moment t s)"
+proof(induct s, simp)
+  fix a s t
+  assume h: "\<And>t.\<lbrakk>vt s\<rbrakk> \<Longrightarrow> vt (moment t s)"
+    and vt_a: "vt (a # s)"
+  show "vt (moment t (a # s))"
+  proof(cases "t \<ge> 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
+    hence le_t1: "t \<le> length s" by simp
+    from vt_a have "vt s"
+      by (erule_tac evt_cons, simp)
+    from h [OF this] have "vt (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 (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
+*)
+
+lemma waiting_unique_pre:
+  fixes cs1 cs2 s thread
+  assumes vt: "vt 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 (e#moment t2 s)"
+      proof -
+        from vt_moment [OF vt]
+        have "vt (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 ((P thread cs2)# moment t2 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t2 s)" by auto
+        with nn1 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def wq_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume lt12: "t2 < t1"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt  (e#moment t1 s)"
+      proof -
+        from vt_moment [OF vt]
+        have "vt (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 ((P thread cs1)# moment t1 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t1 s)" by auto
+        with nn2 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def wq_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume eqt12: "t1 = t2"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt (e#moment t1 s)"
+      proof -
+        from vt_moment [OF vt]
+        have "vt (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 (e#moment t2 s)" by simp 
+          from abs2 [OF this True eq_th h2 h1]
+          show ?thesis .
+        next
+          case False
+          have vt_e: "vt (e#moment t2 s)"
+          proof -
+            from vt_moment [OF vt] eqt12
+            have "vt (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:
+  fixes s cs1 cs2
+  assumes "vt s"
+  and "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+using waiting_unique_pre assms
+unfolding wq_def s_waiting_def
+by auto
+
+(* not used *)
+lemma held_unique:
+  fixes s::"state"
+  assumes "holding s th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+using assms
+unfolding s_holding_def
+by auto
+
+
+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 (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 (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 (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
+           \<rbrakk>
+          \<Longrightarrow> (next_th s th cs t \<and> cs = c)"
+proof -
+  fix t c 
+  assume vt: "vt (V th cs # s)"
+    and nw: "\<not> waiting (wq (V th cs # s)) t c"
+    and wt: "waiting (wq s) t c"
+  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: "wq_fun (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: "wq_fun (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: "wq_fun (schs s) cs = a # list"
+      from step_back_step[OF vt]
+      show "a = th"
+      proof(cases)
+        assume "holding s th cs"
+        with eq_wq show ?thesis
+          by (unfold s_holding_def wq_def, auto)
+      qed
+    qed
+    with True show ?thesis by simp
+  qed
+qed
+
+lemma step_v_not_wait[consumes 3]:
+  "\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
+  by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
+
+lemma step_v_release:
+  "\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
+proof -
+  assume vt: "vt (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]: 
+        "wq_fun (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
+      and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
+            \<in> set (SOME q. distinct q \<and> set q = set list)"
+      have "set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from 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 (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
+  apply (unfold cs_holding_def next_th_def wq_def,
+         auto simp:Let_def)
+proof -
+  fix rest
+  assume vt: "vt (V th cs # s)"
+    and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest"
+    and nrest: "rest \<noteq> []"
+    and ni: "hd (SOME q. distinct q \<and> set q = set rest)
+            \<notin> set (SOME q. distinct q \<and> set q = set rest)"
+  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 (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> 
+  c = cs \<and> t = th"
+  apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
+  proof -
+    fix a list
+    assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
+    from step_back_step [OF vt] show "a = th"
+    proof(cases)
+      assume "holding s th cs" with eq_wq
+      show ?thesis
+        by (unfold s_holding_def wq_def, auto)
+    qed
+  next
+    fix a list
+    assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
+    from step_back_step [OF vt] show "a = th"
+    proof(cases)
+      assume "holding s th cs" with eq_wq
+      show ?thesis
+        by (unfold s_holding_def wq_def, auto)
+    qed
+  qed
+
+lemma step_v_waiting_mono:
+  "\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
+proof -
+  fix t c
+  let ?s' = "(V th cs # s)"
+  assume vt: "vt ?s'" 
+    and wt: "waiting (wq ?s') t c"
+  show "waiting (wq s) t c"
+  proof(cases "c = cs")
+    case False
+    assume neq_cs: "c \<noteq> cs"
+    hence "waiting (wq ?s') t c = waiting (wq s) t c"
+      by (unfold cs_waiting_def wq_def, auto simp:Let_def)
+    with wt show ?thesis by simp
+  next
+    case True
+    with wt show ?thesis
+      apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
+    proof -
+      fix a list
+      assume not_in: "t \<notin> set list"
+        and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
+        and eq_wq: "wq_fun (schs s) cs = a # list"
+      have "set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from 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: "wq_fun (schs s) cs = t # list"
+      hence "t \<in> set list"
+        apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
+      proof -
+        assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
+        moreover have "\<dots> = set list" 
+        proof(rule someI2)
+          from 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:
+fixes th::thread
+assumes vt:
+  "vt (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 (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(simp only: s_depend_def wq_def)
+  apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
+  apply(case_tac "csa = cs", auto)
+  apply(fold wq_def)
+  apply(drule_tac step_back_step)
+  apply(ind_cases " step s (P (hd (wq s cs)) cs)")
+  apply(auto simp:s_depend_def wq_def cs_holding_def)
+  done
+
+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 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 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 (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)" unfolding s_holding_def wq_def by auto
+        then obtain rest where
+          eq_wq: "wq s cs = th#rest"
+          by (cases "wq s cs", auto)
+        show ?thesis
+        proof(cases "rest = []")
+          case False
+          let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+          from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
+            by (unfold next_th_def, auto)
+          let ?E = "(?A - ?B - ?C)"
+          have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
+          proof
+            assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
+            hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+            from tranclD [OF this]
+            obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
+            hence th_d: "(Th ?th', x) \<in> ?A" by simp
+            from 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 wq_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 wq_def, auto)
+              proof -
+                assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq_fun (schs s) cs = th # rest"
+                have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+                proof(rule someI2)
+                  from wq_distinct[OF vt, of cs] and eq_wq
+                  show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+                next
+                  fix x assume "distinct x \<and> set x = set rest"
+                  with False show "x \<noteq> []" by auto
+                qed
+                hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
+                                  set (SOME q. distinct q \<and> set q = set rest)" by auto
+                moreover have "\<dots> = set rest" 
+                proof(rule someI2)
+                  from wq_distinct[OF vt, of cs] and eq_wq
+                  show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+                next
+                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+                qed
+                moreover note hd_in
+                ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
+              next
+                assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+                have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+                proof(rule someI2)
+                  from 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 (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 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 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 (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 (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 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 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 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)
+        done
+    next
+      case (V th' cs')
+      show ?thesis
+      proof(cases "cs' = cs")
+        case False
+        with h
+        show ?thesis
+          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+          by (drule_tac ih, simp)
+      next
+        case True
+        from h
+        show ?thesis
+        proof(unfold V wq_def)
+          assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+          show "th \<in> threads (V th' cs' # s)"
+          proof(cases "cs = cs'")
+            case False
+            hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
+            with th_in have " th \<in> set (wq s cs)" 
+              by (fold wq_def, simp)
+            from ih [OF this] show ?thesis by simp
+          next
+            case True
+            show ?thesis
+            proof(cases "wq_fun (schs s) cs'")
+              case Nil
+              with h V show ?thesis
+                apply (auto simp:wq_def Let_def split:if_splits)
+                by (fold wq_def, drule_tac ih, simp)
+            next
+              case (Cons a rest)
+              assume eq_wq: "wq_fun (schs s) cs' = a # rest"
+              with h V show ?thesis
+                apply (auto simp:Let_def wq_def split:if_splits)
+              proof -
+                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
+                proof(rule someI2)
+                  from 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 (wq_fun (schs s) cs')" by auto
+                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+              next
+                assume th_in: "th \<in> set (wq_fun (schs s) cs)"
+                from ih[OF this[folded wq_def]]
+                show "th \<in> threads s" .
+              qed
+            qed
+          qed
+        qed
+      qed
+    next
+      case (P th' cs')
+      from h stp
+      show ?thesis
+        apply (unfold P wq_def)
+        apply (auto simp:Let_def split:if_splits, fold wq_def)
+        apply (auto intro:ih)
+        apply(ind_cases "step s (P th' cs')")
+        by (unfold runing_def readys_def, auto)
+    next
+      case (Set thread prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    qed
+  next
+    case vt_nil
+    thus ?case by (auto simp:wq_def)
+  qed
+qed
+
+lemma range_in: "\<lbrakk>vt 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 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 assms show ?thesis
+    apply (auto simp:readys_def)
+    apply(simp add:s_waiting_def[folded wq_def])
+    apply (erule_tac x = csa in allE)
+    apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
+    apply (case_tac "csa = cs", simp)
+    apply (erule_tac x = cs in allE)
+    apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
+    apply(auto simp add: wq_def)
+    apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
+    proof -
+       assume th_nin: "th \<notin> set rest"
+        and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+        and eq_wq: "wq_fun (schs s) cs = thread # rest"
+      have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+      proof(rule someI2)
+        from wq_distinct[OF vt, of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
+        show "distinct rest \<and> set rest = set rest" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+      qed
+      with th_nin th_in show False by auto
+    qed
+qed
+
+lemma chain_building:
+  assumes vt: "vt 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 wq_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 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 wq_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 wq_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def wq_def cs_holding_def, simp)
+
+lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
+  by (unfold s_holding_def cs_holding_def, auto)
+
+lemma unique_depend: "\<lbrakk>vt 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 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 wq_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 wq_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 (P th cs#s)"
+  and "wq s cs = []"
+  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from assms show ?thesis
+  unfolding  holdents_test step_depend_p[OF vt] by (auto)
+qed
+
+lemma step_holdents_p_eq:
+  fixes th cs s
+  assumes vt: "vt (P th cs#s)"
+  and "wq s cs \<noteq> []"
+  shows "holdents (P th cs#s) th = holdents s th"
+proof -
+  from assms show ?thesis
+  unfolding  holdents_test step_depend_p[OF vt] by auto
+qed
+
+
+lemma finite_holding:
+  fixes s th cs
+  assumes vt: "vt 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_test, auto intro:finite_subset)
+qed
+
+lemma cntCS_v_dec: 
+  fixes s thread cs
+  assumes vtv: "vt (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_test s_depend_def, simp)
+    by (unfold cs_holding_def s_holding_def wq_def, auto)
+  moreover have cs_not_in: 
+    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
+    apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
+    apply (unfold holdents_test, 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 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 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_test
+          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_test
+        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 (simp add:Let_def)
+          apply (subst s_waiting_def, simp)
+          done
+        with eq_cnp eq_cnv eq_cncs ih
+        have ?thesis by simp
+      } moreover {
+        assume eq_th: "th = thread"
+        with ih is_runing have " cntP s th = cntV s th + cntCS s th" 
+          by (simp add:runing_def)
+        moreover from eq_th eq_e have "th \<notin> threads (e#s)"
+          by simp
+        moreover note eq_cnp eq_cnv eq_cncs
+        ultimately have ?thesis by auto
+      } ultimately show ?thesis by blast
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+        and is_runing: "thread \<in> runing s"
+        and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+      from thread_P vt stp ih  have vtp: "vt (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 "wq_fun (schs s) cs \<noteq> []", auto)
+            apply (erule_tac x = cs in allE, auto)
+            by (case_tac "(wq_fun (schs s) cs)", auto)
+          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+            apply (simp add:cntCS_def holdents_test)
+            by (unfold  step_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_test, 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_test)
+                  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 wq_def)
+                moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
+                ultimately have "th = hd (wq (e#s) cs)" by blast
+                with eq_wq have "th = hd (wq s cs @ [th])" by simp
+                hence "th = hd (wq s cs)" using False by auto
+                with False eq_wq 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_test 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 assms vt stp ih thread_V have vtv: "vt (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: wq_def s_holding_def)
+      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+      have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+      proof(rule someI2)
+        from 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:wq_def s_waiting_def)
+                qed
+              } moreover {
+                assume neq_cs: "cs1 \<noteq> cs"
+                  have "\<not> waiting (e # s) thread cs1" 
+                  proof -
+                    from wq_v_neq [OF neq_cs[symmetric]]
+                    have "wq (V thread cs # s) cs1 = wq s cs1" .
+                    moreover have "\<not> waiting s thread cs1" 
+                    proof -
+                      from runing_ready and is_runing
+                      have "thread \<in> readys s" by auto
+                      thus ?thesis by (simp add:readys_def)
+                    qed
+                    ultimately show ?thesis 
+                      by (auto simp:wq_def s_waiting_def eq_e)
+                  qed
+              } ultimately show "\<not> waiting (e # s) thread cs1" by blast
+            qed
+            ultimately show ?thesis by (simp add:readys_def)
+          qed
+          moreover note eq_th ih
+          ultimately have ?thesis by auto
+        } moreover {
+          assume neq_th: "th \<noteq> thread"
+          from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" 
+            by (simp add:cntP_def count_def)
+          from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" 
+            by (simp add:cntV_def count_def)
+          have ?thesis
+          proof(cases "th \<in> set rest")
+            case False
+            have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+              apply (insert step_back_vt[OF vtv])
+              by (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_test 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 simp add: wq_def)
+                moreover 
+                from eq_wq and th_in and neq_hd
+                have "\<not> (th \<in> readys (e # s))"
+                  apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
+                  by (rule_tac x = cs in exI, auto simp:eq_set)
+                ultimately show ?thesis by auto
+              qed
+              moreover have "cntCS (e#s) th = cntCS s th" 
+              proof -
+                from eq_wq and  th_in and neq_hd
+                have "(holdents (e # s) th) = (holdents s th)"
+                  apply (unfold eq_e step_depend_v[OF vtv], 
+                         auto simp:next_th_def eq_set s_depend_def holdents_test 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: "wq_fun (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: "wq_fun (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                  and neq_cs: "csa \<noteq> cs"
+                  and t_in': "?t \<in>  set (wq_fun (schs s) csa)"
+                show "?t = hd (wq_fun (schs s) csa)"
+                proof -
+                  { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
+                    from 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 simp add: wq_def)
+                qed
+                moreover have "th \<in> threads s"
+                proof -
+                  from th_in eq_wq
+                  have "th \<in> set (wq s cs)" by simp
+                  from 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_test 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_test
+          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_test s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma not_thread_cncs:
+  fixes th s
+  assumes vt: "vt 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 s"
+      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
+      and stp: "step s e"
+      and not_in: "th \<notin> threads (e # s)"
+    from stp show ?case
+    proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+        and not_in': "thread \<notin> threads s"
+      have "cntCS (e # s) th = cntCS s th"
+        apply (unfold eq_e cntCS_def holdents_test)
+        by (simp add: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_test)
+        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 assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      hence "cntCS (e # s) th  = cntCS s th "
+        apply (unfold cntCS_def holdents_test eq_e)
+        by (unfold step_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 assms thread_V vt stp ih have vtv: "vt (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: wq_def s_holding_def)
+      from not_in eq_e eq_wq
+      have "\<not> next_th s thread cs th"
+        apply (auto simp:next_th_def)
+      proof -
+        assume ne: "rest \<noteq> []"
+          and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
+        have "?t \<in> set rest"
+        proof(rule someI2)
+          from 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_test 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_test)
+        by (simp add:depend_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (unfold cntCS_def, 
+        auto simp:count_def holdents_test 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 wq_def)
+
+lemma dm_depend_threads:
+  fixes th s
+  assumes vt: "vt 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 (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+apply(simp add: Let_def cpreced_initial)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+
+
+lemma runing_unique:
+  fixes th1 th2 s
+  assumes vt: "vt 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)^+)" 
+        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)
+      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)^+)" 
+        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)
+      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 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 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 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_test)
+  qed
+  ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
+    by (unfold cntCS_def holdents_test 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 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 s"
+  shows "finite (threads s)"
+using vt
+by (induct) (auto elim: step.cases)
+
+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 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 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 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 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 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 eq_holding: "holding (wq s) th cs = holding s th cs"
+  apply (unfold s_holding_def cs_holding_def wq_def, simp)
+  done
+
+lemma f_image_eq:
+  assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
+  shows "f ` A = g ` A"
+proof
+  show "f ` A \<subseteq> g ` A"
+    by(rule image_subsetI, auto intro:h)
+next
+  show "g ` A \<subseteq> f ` A"
+   by (rule image_subsetI, auto intro:h[symmetric])
+qed
+
+
+definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
+  where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
+
+
+lemma detached_test:
+  shows "detached s th = (Th th \<notin> Field (depend s))"
+apply(simp add: detached_def Field_def)
+apply(simp add: s_depend_def)
+apply(simp add: s_holding_abv s_waiting_abv)
+apply(simp add: Domain_iff Range_iff)
+apply(simp add: wq_def)
+apply(auto)
+done
+
+lemma detached_intro:
+  fixes s th
+  assumes vt: "vt s"
+  and eq_pv: "cntP s th = cntV s th"
+  shows "detached s th"
+proof -
+ from cnp_cnv_cncs[OF vt]
+  have eq_cnt: "cntP s th =
+    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+  hence cncs_zero: "cntCS s th = 0"
+    by (auto simp:eq_pv split:if_splits)
+  with eq_cnt
+  have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
+  thus ?thesis
+  proof
+    assume "th \<notin> threads s"
+    with range_in[OF vt] dm_depend_threads[OF vt]
+    show ?thesis
+      by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
+  next
+    assume "th \<in> readys s"
+    moreover have "Th th \<notin> Range (depend s)"
+    proof -
+      from card_0_eq [OF finite_holding [OF vt]] and cncs_zero
+      have "holdents s th = {}"
+        by (simp add:cntCS_def)
+      thus ?thesis
+        apply(auto simp:holdents_test)
+        apply(case_tac a)
+        apply(auto simp:holdents_test s_depend_def)
+        done
+    qed
+    ultimately show ?thesis
+      by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def readys_def)
+  qed
+qed
+
+lemma detached_elim:
+  fixes s th
+  assumes vt: "vt s"
+  and dtc: "detached s th"
+  shows "cntP s th = cntV s th"
+proof -
+  from cnp_cnv_cncs[OF vt]
+  have eq_pv: " cntP s th =
+    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+  have cncs_z: "cntCS s th = 0"
+  proof -
+    from dtc have "holdents s th = {}"
+      unfolding detached_def holdents_test s_depend_def
+      by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
+    thus ?thesis by (auto simp:cntCS_def)
+  qed
+  show ?thesis
+  proof(cases "th \<in> threads s")
+    case True
+    with dtc 
+    have "th \<in> readys s"
+      by (unfold readys_def detached_def Field_def Domain_def Range_def, 
+           auto simp:eq_waiting s_depend_def)
+    with cncs_z and eq_pv show ?thesis by simp
+  next
+    case False
+    with cncs_z and eq_pv show ?thesis by simp
+  qed
+qed
+
+lemma detached_eq:
+  fixes s th
+  assumes vt: "vt s"
+  shows "(detached s th) = (cntP s th = cntV s th)"
+  by (insert vt, auto intro:detached_intro detached_elim)
+
+end
\ No newline at end of file
diff -r 000000000000 -r 110247f9d47e PrioGDef.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/PrioGDef.thy	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,483 @@
+(*<*)
+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 \<equiv> 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 th, Cs cs) | th cs. waiting wq th cs} \<union> {(Cs cs, Th th) | cs th. holding wq th cs}"
+  -- {* 
+  \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 = 
+    wq_fun :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
+    cprec_fun :: "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 :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
+  where "cpreced wq s = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
+
+(*<*)
+lemma 
+  cpreced_def2:
+  "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependents wq th})"
+  unfolding cpreced_def image_def
+  apply(rule eq_reflection)
+  apply(rule_tac f="Max" in arg_cong)
+  by (auto)
+(*>*)
+
+abbreviation
+  "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
+
+abbreviation 
+  "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
+ 
+abbreviation
+  "release qs \<equiv> case qs of
+             [] => [] 
+          |  (_#qs) => (SOME q. distinct q \<and> set q = set qs)"
+
+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 [] = (| wq_fun = \<lambda> cs. [],  cprec_fun = (\<lambda>_. Prc 0 0) |)" |
+
+  -- {*
+  \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 (NOT USED). 
+  \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 (Create th prio # s) = 
+       (let wq = wq_fun (schs s) in
+          (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"
+|  "schs (Exit th # s) = 
+       (let wq = wq_fun (schs s) in
+          (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"
+|  "schs (Set th prio # s) = 
+       (let wq = wq_fun (schs s) in
+          (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"
+|  "schs (P th cs # s) = 
+       (let wq = wq_fun (schs s) in
+        let new_wq = wq(cs := (wq cs @ [th])) in
+          (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"
+|  "schs (V th cs # s) = 
+       (let wq = wq_fun (schs s) in
+        let new_wq = wq(cs := release (wq cs)) in
+          (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"
+
+lemma cpreced_initial: 
+  "cpreced (\<lambda> cs. []) [] = (\<lambda>_. (Prc 0 0))"
+apply(simp add: cpreced_def)
+apply(simp add: cs_dependents_def cs_depend_def cs_waiting_def cs_holding_def)
+apply(simp add: preced_def)
+done
+
+lemma sch_old_def:
+  "schs (e#s) = (let ps = schs s in 
+                  let pwq = wq_fun ps in 
+                  let nwq = case e of
+                             P th cs \<Rightarrow>  pwq(cs:=(pwq cs @ [th])) |
+                             V th cs \<Rightarrow> let nq = case (pwq cs) of
+                                                      [] \<Rightarrow> [] | 
+                                                      (_#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
+                                            in pwq(cs:=nq)                 |
+                              _ \<Rightarrow> pwq
+                  in let ncp = cpreced nwq (e#s) in 
+                     \<lparr>wq_fun = nwq, cprec_fun = ncp\<rparr>
+                 )"
+apply(cases e)
+apply(simp_all)
+done
+
+
+text {* 
+  \noindent
+  The following @{text "wq"} is a shorthand for @{text "wq_fun"}. 
+  *}
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
+  where "wq s = wq_fun (schs s)"
+
+text {* \noindent 
+  The following @{text "cp"} is a shorthand for @{text "cprec_fun"}. 
+  *}
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+  where "cp s \<equiv> cprec_fun (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_abv: 
+  "holding (s::state) \<equiv> holding (wq_fun (schs s))"
+  s_waiting_abv: 
+  "waiting (s::state) \<equiv> waiting (wq_fun (schs s))"
+  s_depend_abv: 
+  "depend (s::state) \<equiv> depend (wq_fun (schs s))"
+  s_dependents_abv: 
+  "dependents (s::state) \<equiv> dependents (wq_fun (schs s))"
+
+
+text {* 
+  The following lemma can be proved easily:
+  *}
+lemma
+  s_holding_def: 
+  "holding (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th = hd (wq_fun (schs s) cs))"
+  by (auto simp:s_holding_abv wq_def cs_holding_def)
+
+lemma s_waiting_def: 
+  "waiting (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th \<noteq> hd (wq_fun (schs s) cs))"
+  by (auto simp:s_waiting_abv wq_def cs_waiting_def)
+
+lemma s_depend_def: 
+  "depend (s::state) =
+    {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \<union> {(Cs cs, Th th) | cs th. holding (wq s) th cs}"
+  by (auto simp:s_depend_abv wq_def cs_depend_def)
+
+lemma
+  s_dependents_def: 
+  "dependents (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
+  by (auto simp:s_dependents_abv wq_def cs_dependents_def)
+
+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 \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waiting s th 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 \<equiv> {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 \<equiv> {cs . holding s th cs}"
+
+lemma holdents_test: 
+  "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
+unfolding holdents_def
+unfolding s_depend_def
+unfolding s_holding_abv
+unfolding wq_def
+by (simp)
+
+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> bool"
+  where
+  -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
+  vt_nil[intro]: "vt []" |
+  -- {* 
+  \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 s; step s e\<rbrakk> \<Longrightarrow> vt (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 000000000000 -r 110247f9d47e README
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/README	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,14 @@
+Theories:
+=========
+
+ Precedence_ord.thy      A theory of precedences.
+ Moment.thy              The notion of moment.
+ PrioGDef.thy            The formal definition of the PIP-model.
+ PrioG.thy               Basic properties of the PIP-model.
+ ExtGG.thy               The correctness proof of the PIP-model.
+ CpsG.thy                Properties interesting for an implementation.
+
+The repository can be checked using Isabelle 2011-1.
+
+  isabelle make session
+
diff -r 000000000000 -r 110247f9d47e ROOT.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ROOT.ML	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,2 @@
+use_thy "CpsG"; 
+use_thy "ExtGG";
diff -r 000000000000 -r 110247f9d47e Slides/ROOT1.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/ROOT1.ML	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,7 @@
+(*show_question_marks := false;*)
+
+no_document use_thy "../CpsG"; 
+no_document use_thy "../ExtGG"; 
+no_document use_thy "~~/src/HOL/Library/LaTeXsugar";
+quick_and_dirty := true;
+use_thy "Slides1"
\ No newline at end of file
diff -r 000000000000 -r 110247f9d47e Slides/Slides1.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/Slides1.thy	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,669 @@
+(*<*)
+theory Slides1
+imports "../CpsG" "../ExtGG" "~~/src/HOL/Library/LaTeXsugar"
+begin
+
+notation (latex output)
+  set ("_") and
+  Cons  ("_::/_" [66,65] 65) 
+
+ML {*
+  open Printer;
+  show_question_marks_default := false;
+  *}
+
+notation (latex output)
+  Cons ("_::_" [78,77] 73) and
+  vt ("valid'_state") and
+  runing ("running") and
+  birthtime ("last'_set") and
+  If  ("(\<^raw:\textrm{>if\<^raw:}> (_)/ \<^raw:\textrm{>then\<^raw:}> (_)/ \<^raw:\textrm{>else\<^raw:}> (_))" 10) and
+  Prc ("'(_, _')") and
+  holding ("holds") and
+  waiting ("waits") and
+  Th ("T") and
+  Cs ("C") and
+  readys ("ready") and
+  depend ("RAG") and 
+  preced ("prec") and
+  cpreced ("cprec") and
+  dependents ("dependants") and
+  cp ("cprec") and
+  holdents ("resources") and
+  original_priority ("priority") and
+  DUMMY  ("\<^raw:\mbox{$\_\!\_$}>")
+
+(*>*)
+
+
+
+text_raw {*
+  \renewcommand{\slidecaption}{Nanjing, P.R. China, 1 August 2012}
+  \newcommand{\bl}[1]{#1}                        
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}
+  \frametitle{%
+  \begin{tabular}{@ {}c@ {}}
+  \\[-3mm]
+  \Large Priority Inheritance Protocol \\[-3mm] 
+  \Large Proved Correct \\[0mm]
+  \end{tabular}}
+  
+  \begin{center}
+  \small Xingyuan Zhang \\
+  \small \mbox{PLA University of Science and Technology} \\
+  \small \mbox{Nanjing, China}
+  \end{center}
+
+  \begin{center}
+  \small joint work with \\
+  Christian Urban \\
+  Kings College, University of London, U.K.\\
+  Chunhan Wu \\
+  My Ph.D. student now working for Christian\\
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\large Prioirty Inheritance Protocol (PIP)}
+  \large
+
+  \begin{itemize} 
+  \item Widely used in Real-Time OSs \pause
+  \item One solution of \textcolor{red}{`Priority Inversion'} \pause
+  \item A flawed manual correctness proof (1990)\pause
+        \begin{itemize} \large
+          \item {Notations with no precise definition}
+          \item {Resorts to intuitions}
+        \end{itemize} \pause
+  \item Formal treatments using model-checking \pause
+        \begin{itemize} \large
+          \item {Applicable to small size system models}
+          \item { Unhelpful for human understanding } 
+        \end{itemize} \pause
+  \item Verification of PCP in PVS (2000)\pause
+        \begin{itemize} \large
+           \item {A related protocol}
+           \item {Priority Ceiling Protocol}
+        \end{itemize}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Our Motivation}
+  \large
+
+  \begin{itemize}
+  \item Undergraduate OS course in our university \pause
+        \begin{itemize}
+           \item {\large Experiments using instrutional OSs }
+           \item {\large PINTOS (Stanford) is chosen }
+           \item {\large Core project: Implementing PIP in it}
+        \end{itemize} \pause
+  \item Understanding is crucial for the implemention \pause
+  \item Existing literature of little help \pause
+  \item Some mention the complication
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\mbox{Some excerpts}}
+  
+  \begin{quote}
+  ``Priority inheritance is neither ef$\!$ficient nor reliable. 
+  Implementations are either incomplete (and unreliable) 
+  or surprisingly complex and intrusive.''
+  \end{quote}\medskip
+
+  \pause
+
+  \begin{quote}
+  ``I observed in the kernel code (to my disgust), the Linux 
+  PIP implementation is a nightmare: extremely heavy weight, 
+  involving maintenance of a full wait-for graph, and requiring 
+  updates for a range of events, including priority changes and 
+  interruptions of wait operations.''
+  \end{quote}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Our Aims}
+  \large
+
+  \begin{itemize}
+  \item Formal specification at appropriate abstract level,
+        convenient for:
+  \begin{itemize} \large
+    \item Constructing interactive proofs
+    \item Clarifying the underlying ideas
+  \end{itemize} \pause
+  \item Theorems usable to guide implementation, critical point:
+    \begin{itemize} \large
+      \item Understanding the relationship with real OS code \pause
+      \item Not yet formalized
+    \end{itemize}
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Real-Time OSes}
+  \large
+
+  \begin{itemize}
+  \item Purpose: gurantee the most urgent task to be processed in time
+  \item Processes have priorities\\
+  \item Resources can be locked and unlocked
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Problem}
+  \Large
+
+  \begin{center}
+  \begin{tabular}{l}
+  \alert{H}igh-priority process\\[4mm]
+  \onslide<2->{\alert{M}edium-priority process}\\[4mm]
+  \alert{L}ow-priority process\\[4mm]
+  \end{tabular}
+  \end{center}
+
+  \onslide<3->{
+  \begin{itemize}
+  \item \alert{Priority Inversion} @{text "\<equiv>"} \alert{H $<$ L}
+  \item<4> avoid indefinite priority inversion
+  \end{itemize}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Priority Inversion}
+
+  \begin{center}
+  \includegraphics[scale=0.4]{PriorityInversion.png}
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Mars Pathfinder Mission 1997}
+  \Large
+
+  \begin{center}
+  \includegraphics[scale=0.2]{marspath1.png}
+  \includegraphics[scale=0.22]{marspath3.png}
+  \includegraphics[scale=0.4]{marsrover.png}
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Solution}
+  \Large
+
+  \alert{Priority Inheritance Protocol (PIP):}
+
+  \begin{center}
+  \begin{tabular}{l}
+  \alert{H}igh-priority process\\[4mm]
+  \textcolor{gray}{Medium-priority process}\\[4mm]
+  \alert{L}ow-priority process\\[21mm]
+  {\normalsize (temporarily raise its priority)}
+  \end{tabular}
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{A Correctness ``Proof'' in 1990}
+  \Large
+
+  \begin{itemize}
+  \item a paper$^\star$ 
+  in 1990 ``proved'' the correctness of an algorithm for PIP\\[5mm]
+  \end{itemize}
+
+  \normalsize
+  \begin{quote}
+  \ldots{}after the thread (whose priority has been raised) completes its 
+  critical section and releases the lock, it ``returns to its original 
+  priority level''.
+  \end{quote}\bigskip
+
+  \small
+  $^\star$ in IEEE Transactions on Computers
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{}
+  \Large
+
+  \begin{center}
+  \begin{tabular}{l}
+  \alert{H}igh-priority process 1\\[2mm]
+  \alert{H}igh-priority process 2\\[8mm]
+  \alert{L}ow-priority process
+  \end{tabular}
+  \end{center}
+
+  \onslide<2->{
+  \begin{itemize}
+  \item Solution: \\Return to highest \alert{remaining} priority
+  \end{itemize}}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Event Abstraction}
+
+  \begin{itemize}\large
+     \item Use Inductive Approach of L. Paulson \pause
+     \item System is event-driven \pause
+     \item A \alert{state} is a list of events 
+  \end{itemize}
+
+  \pause
+
+  \begin{center}
+  \includegraphics[scale=0.4]{EventAbstract.png}
+  \end{center}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Events}
+  \Large
+
+  \begin{center}
+  \begin{tabular}{l}
+  Create \textcolor{gray}{thread priority}\\
+  Exit \textcolor{gray}{thread}\\
+  Set \textcolor{gray}{thread priority}\\
+  Lock \textcolor{gray}{thread cs}\\
+  Unlock \textcolor{gray}{thread cs}\\
+  \end{tabular}
+  \end{center}\medskip
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Precedences}
+  \large
+
+  \begin{center}
+  \begin{tabular}{l}
+  @{thm preced_def[where thread="th"]} 
+  \end{tabular}
+  \end{center}
+
+  
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{RAGs}
+
+\begin{center}
+  \newcommand{\fnt}{\fontsize{7}{8}\selectfont}
+  \begin{tikzpicture}[scale=1]
+  %%\draw[step=2mm] (-3,2) grid (1,-1);
+
+  \node (A) at (0,0) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>0"}};
+  \node (B) at (2,0) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>1"}};
+  \node (C) at (4,0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>1"}};
+  \node (D) at (4,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>2"}};
+  \node (E) at (6,-0.7) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>2"}};
+  \node (E1) at (6, 0.2) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>3"}};
+  \node (F) at (8,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>3"}};
+
+  \draw [<-,line width=0.6mm] (A) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding}  (B);
+  \draw [->,line width=0.6mm] (C) to node [pos=0.4,sloped,above=-0.5mm] {\fnt{}waiting}  (B);
+  \draw [->,line width=0.6mm] (D) to node [pos=0.4,sloped,below=-0.5mm] {\fnt{}waiting}  (B);
+  \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,below=-0.5mm] {\fnt{}holding}  (E);
+  \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding}  (E1);
+  \draw [->,line width=0.6mm] (F) to node [pos=0.45,sloped,below=-0.5mm] {\fnt{}waiting}  (E);
+  \end{tikzpicture}
+  \end{center}\bigskip
+
+  \begin{center}
+  \begin{minipage}{0.8\linewidth}
+  \raggedleft
+  @{thm cs_depend_def}
+  \end{minipage}
+  \end{center}\pause
+
+  
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Good Next Events}
+  %%\large
+
+  \begin{center}
+  @{thm[mode=Rule] thread_create[where thread=th]}\bigskip
+
+  @{thm[mode=Rule] thread_exit[where thread=th]}\bigskip
+
+  @{thm[mode=Rule] thread_set[where thread=th]}
+  \end{center}
+
+  
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Good Next Events}
+  %%\large
+
+  \begin{center}
+  @{thm[mode=Rule] thread_P[where thread=th]}\bigskip
+  
+  @{thm[mode=Rule] thread_V[where thread=th]}\bigskip
+  \end{center}
+
+  
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+(*<*)
+context extend_highest_gen
+begin
+(*>*)
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{\mbox{\large Theorem: ``No indefinite priority inversion''}}
+
+  \pause
+
+  Theorem $^\star$: If th is the thread with the highest precedence in state 
+  @{text "s"}: \pause
+  \begin{center}
+  \textcolor{red}{@{thm highest})}
+  \end{center}
+  \pause
+  and @{text "th"} is blocked by a thread @{text "th'"} in 
+  a future state @{text "s'"} (with @{text "s' = t@s"}): \pause
+  \begin{center}
+  \textcolor{red}{@{text "th' \<in> running (t@s)"} and @{text "th' \<noteq> th"}} \pause
+  \end{center}
+  \fbox{ \hspace{1em} \pause
+  \begin{minipage}{0.95\textwidth}
+  \begin{itemize}
+  \item @{text "th'"} did not hold or wait for a resource in s:
+  \begin{center}
+  \textcolor{red}{@{text "\<not>detached s th'"}}  
+  \end{center} \pause
+  \item @{text "th'"} is running with the precedence of @{text "th"}:
+  \begin{center}
+  \textcolor{red}{@{text "cp (t@s) th' = preced th s"}} 
+  \end{center} 
+  \end{itemize}
+  \end{minipage}}
+  \pause
+  \small
+  $^\star$ modulo some further assumptions\bigskip\pause
+  It does not matter which process gets a released lock.
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Implementation}
+  
+  s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
+
+  When @{text "e"} = \alert{Create th prio}, \alert{Exit th} 
+  
+  \begin{itemize}
+  \item @{text "RAG s' = RAG s"}
+  \item No precedence needs to be recomputed
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Implementation}
+
+  s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
+  
+
+  When @{text "e"} = \alert{Set th prio}
+  
+  \begin{itemize}
+  \item @{text "RAG s' = RAG s"}
+  \item No precedence needs to be recomputed
+  \end{itemize}
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Implementation}
+
+  s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
+
+  When @{text "e"} = \alert{Unlock th cs} where there is a thread to take over
+  
+  \begin{itemize}
+  \item @{text "RAG s' = RAG s - {(C cs, T th), (T th', C cs)} \<union> {(C cs, T th')}"}
+  \item we have to recalculate the precedence of the direct descendants
+  \end{itemize}\bigskip
+
+  \pause
+
+  When @{text "e"} =  \alert{Unlock th cs} where no thread takes over
+  
+  \begin{itemize}
+  \item @{text "RAG s' = RAG s - {(C cs, T th)}"}
+  \item no recalculation of precedences
+  \end{itemize}
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[t]
+  \frametitle{Implementation}
+
+  s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
+
+  When @{text "e"} = \alert{Lock th cs} where cs is not locked
+  
+  \begin{itemize}
+  \item @{text "RAG s' = RAG s \<union> {(C cs, T th')}"}
+  \item no recalculation of precedences
+  \end{itemize}\bigskip
+
+  \pause
+
+  When @{text "e"} = \alert{Lock th cs} where cs is locked
+  
+  \begin{itemize}
+  \item @{text "RAG s' = RAG s - {(T th, C cs)}"}
+  \item we have to recalculate the precedence of the descendants
+  \end{itemize}
+
+
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Conclusion}
+  
+  \begin{itemize} \large
+  \item Aims fulfilled \medskip \pause
+  \item Alternative way \pause
+       \begin{itemize} 
+           \item using Isabelle/HOL in OS code development \medskip
+           \item through the Inductive Approach
+       \end{itemize} \pause
+  \item Future research \pause
+       \begin{itemize}
+           \item scheduler in RT-Linux\medskip
+           \item multiprocessor case\medskip
+           \item other ``nails'' ? (networks, \ldots) \medskip \pause
+           \item Refinement to real code and relation between implementations
+        \end{itemize}
+  \end{itemize}
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+text_raw {*
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \mode<presentation>{
+  \begin{frame}[c]
+  \frametitle{Questions?}
+  
+  \begin{itemize} \large
+  \item Thank you for listening!
+  \end{itemize}
+  
+  \end{frame}}
+  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
+*}
+
+
+(*<*)
+end
+end
+(*>*)
\ No newline at end of file
diff -r 000000000000 -r 110247f9d47e Slides/document/beamerthemeplaincu.sty
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/document/beamerthemeplaincu.sty	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,126 @@
+\ProvidesPackage{beamerthemeplaincu}[2003/11/07 ver 0.93]
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+
+% Copyright 2003 by Till Tantau <tantau@cs.tu-berlin.de>.
+%
+% This program can be redistributed and/or modified under the terms
+% of the LaTeX Project Public License Distributed from CTAN
+% archives in directory macros/latex/base/lppl.txt.
+
+\newcommand{\slidecaption}{}
+
+\mode<presentation>
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% comic fonts fonts
+\DeclareFontFamily{T1}{comic}{}%
+\DeclareFontShape{T1}{comic}{m}{n}{<->s*[.9]comic8t}{}%
+\DeclareFontShape{T1}{comic}{m}{it}{<->s*[.9]comic8t}{}%
+\DeclareFontShape{T1}{comic}{m}{sc}{<->s*[.9]comic8t}{}%
+\DeclareFontShape{T1}{comic}{b}{n}{<->s*[.9]comicbd8t}{}%
+\DeclareFontShape{T1}{comic}{b}{it}{<->s*[.9]comicbd8t}{}%
+\DeclareFontShape{T1}{comic}{m}{sl}{<->ssub * comic/m/it}{}%
+\DeclareFontShape{T1}{comic}{b}{sc}{<->sub * comic/m/sc}{}%
+\DeclareFontShape{T1}{comic}{b}{sl}{<->ssub * comic/b/it}{}%
+\DeclareFontShape{T1}{comic}{bx}{n}{<->ssub * comic/b/n}{}%
+\DeclareFontShape{T1}{comic}{bx}{it}{<->ssub * comic/b/it}{}%
+\DeclareFontShape{T1}{comic}{bx}{sc}{<->sub * comic/m/sc}{}%
+\DeclareFontShape{T1}{comic}{bx}{sl}{<->ssub * comic/b/it}{}%
+%
+\renewcommand{\rmdefault}{comic}%
+\renewcommand{\sfdefault}{comic}%
+\renewcommand{\mathfamilydefault}{cmr}% mathfont should be still the old one
+%
+\DeclareMathVersion{bold}% mathfont needs to be bold
+\DeclareSymbolFont{operators}{OT1}{cmr}{b}{n}%
+\SetSymbolFont{operators}{bold}{OT1}{cmr}{b}{n}%
+\DeclareSymbolFont{letters}{OML}{cmm}{b}{it}%
+\SetSymbolFont{letters}{bold}{OML}{cmm}{b}{it}%
+\DeclareSymbolFont{symbols}{OMS}{cmsy}{b}{n}%
+\SetSymbolFont{symbols}{bold}{OMS}{cmsy}{b}{n}%
+\DeclareSymbolFont{largesymbols}{OMX}{cmex}{b}{n}%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%                              % Title page  
+%\usetitlepagetemplate{
+%  \vbox{}
+%  \vfill
+%  \begin{centering}
+%    \Large\structure{\textrm{\textit{{\inserttitle}}}}
+%    \vskip1em\par
+%    \normalsize\insertauthor\vskip1em\par
+%    {\scriptsize\insertinstitute\par}\par\vskip1em
+%    \insertdate\par\vskip1.5em
+%    \inserttitlegraphic
+%  \end{centering}
+%  \vfill
+%  }
+
+                                % Part page  
+%\usepartpagetemplate{
+%  \begin{centering}
+%    \Large\structure{\textrm{\textit{\partname~\@Roman\c@part}}}
+%    \vskip1em\par
+%    \textrm{\textit{\insertpart}}\par
+%  \end{centering}
+%  }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Frametitles
+\setbeamerfont{frametitle}{size={\huge}}
+\setbeamerfont{frametitle}{family={\usefont{T1}{ptm}{b}{n}}}
+\setbeamercolor{frametitle}{fg=gray,bg=white}
+
+\setbeamertemplate{frametitle}{%
+\vskip 2mm  % distance from the top margin
+\hskip -3mm % distance from left margin
+\vbox{%
+\begin{minipage}{1.05\textwidth}%
+\centering%
+\insertframetitle%
+\end{minipage}}%
+}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Foot
+%
+\setbeamertemplate{navigation symbols}{} 
+\usefoottemplate{%
+\vbox{%
+  \tinyline{%
+    \tiny\hfill\textcolor{gray!50}{\slidecaption{} --
+      p.~\insertframenumber/\inserttotalframenumber}}}%
+}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\beamertemplateballitem
+\setlength\leftmargini{2mm}
+\setlength\leftmarginii{0.6cm}
+\setlength\leftmarginiii{1.5cm}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% blocks
+%\definecolor{cream}{rgb}{1,1,.65}
+\definecolor{cream}{rgb}{1,1,.8}
+\setbeamerfont{block title}{size=\normalsize}
+\setbeamercolor{block title}{fg=black,bg=cream}
+\setbeamercolor{block body}{fg=black,bg=cream}
+
+\setbeamertemplate{blocks}[rounded][shadow=true]
+
+\setbeamercolor{boxcolor}{fg=black,bg=cream}
+
+\mode
+<all>
+
+
+
+
+
+
diff -r 000000000000 -r 110247f9d47e Slides/document/mathpartir.sty
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/document/mathpartir.sty	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,446 @@
+%  Mathpartir --- Math Paragraph for Typesetting Inference Rules
+%
+%  Copyright (C) 2001, 2002, 2003, 2004, 2005 Didier Rémy
+%
+%  Author         : Didier Remy 
+%  Version        : 1.2.0
+%  Bug Reports    : to author
+%  Web Site       : http://pauillac.inria.fr/~remy/latex/
+% 
+%  Mathpartir is free software; you can redistribute it and/or modify
+%  it under the terms of the GNU General Public License as published by
+%  the Free Software Foundation; either version 2, or (at your option)
+%  any later version.
+%  
+%  Mathpartir is distributed in the hope that it will be useful,
+%  but WITHOUT ANY WARRANTY; without even the implied warranty of
+%  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+%  GNU General Public License for more details 
+%  (http://pauillac.inria.fr/~remy/license/GPL).
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%  File mathpartir.sty (LaTeX macros)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{mathpartir}
+    [2005/12/20 version 1.2.0 Math Paragraph for Typesetting Inference Rules]
+
+%%
+
+%% Identification
+%% Preliminary declarations
+
+\RequirePackage {keyval}
+
+%% Options
+%% More declarations
+
+%% PART I: Typesetting maths in paragraphe mode
+
+%% \newdimen \mpr@tmpdim
+%% Dimens are a precious ressource. Uses seems to be local.
+\let \mpr@tmpdim \@tempdima
+
+% To ensure hevea \hva compatibility, \hva should expands to nothing 
+% in mathpar or in inferrule
+\let \mpr@hva \empty
+
+%% normal paragraph parametters, should rather be taken dynamically
+\def \mpr@savepar {%
+  \edef \MathparNormalpar
+     {\noexpand \lineskiplimit \the\lineskiplimit
+      \noexpand \lineskip \the\lineskip}%
+  }
+
+\def \mpr@rulelineskip {\lineskiplimit=0.3em\lineskip=0.2em plus 0.1em}
+\def \mpr@lesslineskip {\lineskiplimit=0.6em\lineskip=0.5em plus 0.2em}
+\def \mpr@lineskip  {\lineskiplimit=1.2em\lineskip=1.2em plus 0.2em}
+\let \MathparLineskip \mpr@lineskip
+\def \mpr@paroptions {\MathparLineskip}
+\let \mpr@prebindings \relax
+
+\newskip \mpr@andskip \mpr@andskip 2em plus 0.5fil minus 0.5em
+
+\def \mpr@goodbreakand
+   {\hskip -\mpr@andskip  \penalty -1000\hskip \mpr@andskip}
+\def \mpr@and {\hskip \mpr@andskip}
+\def \mpr@andcr {\penalty 50\mpr@and}
+\def \mpr@cr {\penalty -10000\mpr@and}
+\def \mpr@eqno #1{\mpr@andcr #1\hskip 0em plus -1fil \penalty 10}
+
+\def \mpr@bindings {%
+  \let \and \mpr@andcr
+  \let \par \mpr@andcr
+  \let \\\mpr@cr
+  \let \eqno \mpr@eqno
+  \let \hva \mpr@hva
+  } 
+\let \MathparBindings \mpr@bindings
+
+% \@ifundefined {ignorespacesafterend}
+%    {\def \ignorespacesafterend {\aftergroup \ignorespaces}
+
+\newenvironment{mathpar}[1][]
+  {$$\mpr@savepar \parskip 0em \hsize \linewidth \centering
+     \vbox \bgroup \mpr@prebindings \mpr@paroptions #1\ifmmode $\else
+     \noindent $\displaystyle\fi
+     \MathparBindings}
+  {\unskip \ifmmode $\fi\egroup $$\ignorespacesafterend}
+
+\newenvironment{mathparpagebreakable}[1][]
+  {\begingroup 
+   \par
+   \mpr@savepar \parskip 0em \hsize \linewidth \centering
+      \mpr@prebindings \mpr@paroptions #1%
+      \vskip \abovedisplayskip \vskip -\lineskip%
+     \ifmmode  \else  $\displaystyle\fi
+     \MathparBindings
+  }
+  {\unskip
+   \ifmmode $\fi \par\endgroup
+   \vskip \belowdisplayskip
+   \noindent
+  \ignorespacesafterend}
+
+% \def \math@mathpar #1{\setbox0 \hbox {$\displaystyle #1$}\ifnum
+%     \wd0 < \hsize  $$\box0$$\else \bmathpar #1\emathpar \fi}
+
+%%% HOV BOXES
+
+\def \mathvbox@ #1{\hbox \bgroup \mpr@normallineskip 
+  \vbox \bgroup \tabskip 0em \let \\ \cr
+  \halign \bgroup \hfil $##$\hfil\cr #1\crcr \egroup \egroup
+  \egroup}
+
+\def \mathhvbox@ #1{\setbox0 \hbox {\let \\\qquad $#1$}\ifnum \wd0 < \hsize
+      \box0\else \mathvbox {#1}\fi}
+
+
+%% Part II -- operations on lists
+
+\newtoks \mpr@lista
+\newtoks \mpr@listb
+
+\long \def\mpr@cons #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
+{#2}\edef #2{\the \mpr@lista \the \mpr@listb}}
+
+\long \def\mpr@snoc #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
+{#2}\edef #2{\the \mpr@listb\the\mpr@lista}}
+
+\long \def \mpr@concat#1=#2\mpr@to#3{\mpr@lista \expandafter {#2}\mpr@listb
+\expandafter {#3}\edef #1{\the \mpr@listb\the\mpr@lista}}
+
+\def \mpr@head #1\mpr@to #2{\expandafter \mpr@head@ #1\mpr@head@ #1#2}
+\long \def \mpr@head@ #1#2\mpr@head@ #3#4{\def #4{#1}\def#3{#2}}
+
+\def \mpr@flatten #1\mpr@to #2{\expandafter \mpr@flatten@ #1\mpr@flatten@ #1#2}
+\long \def \mpr@flatten@ \\#1\\#2\mpr@flatten@ #3#4{\def #4{#1}\def #3{\\#2}}
+
+\def \mpr@makelist #1\mpr@to #2{\def \mpr@all {#1}%
+   \mpr@lista {\\}\mpr@listb \expandafter {\mpr@all}\edef \mpr@all {\the
+   \mpr@lista \the \mpr@listb \the \mpr@lista}\let #2\empty 
+   \def \mpr@stripof ##1##2\mpr@stripend{\def \mpr@stripped{##2}}\loop
+     \mpr@flatten \mpr@all \mpr@to \mpr@one
+     \expandafter \mpr@snoc \mpr@one \mpr@to #2\expandafter \mpr@stripof
+     \mpr@all \mpr@stripend  
+     \ifx \mpr@stripped \empty \let \mpr@isempty 0\else \let \mpr@isempty 1\fi
+     \ifx 1\mpr@isempty
+   \repeat
+}
+
+\def \mpr@rev #1\mpr@to #2{\let \mpr@tmp \empty
+   \def \\##1{\mpr@cons ##1\mpr@to \mpr@tmp}#1\let #2\mpr@tmp}
+
+%% Part III -- Type inference rules
+
+\newif \if@premisse
+\newbox \mpr@hlist
+\newbox \mpr@vlist
+\newif \ifmpr@center \mpr@centertrue
+\def \mpr@htovlist {%
+   \setbox \mpr@hlist
+      \hbox {\strut
+             \ifmpr@center \hskip -0.5\wd\mpr@hlist\fi
+             \unhbox \mpr@hlist}%
+   \setbox \mpr@vlist
+      \vbox {\if@premisse  \box \mpr@hlist \unvbox \mpr@vlist
+             \else \unvbox \mpr@vlist \box \mpr@hlist
+             \fi}%
+}
+% OLD version
+% \def \mpr@htovlist {%
+%    \setbox \mpr@hlist
+%       \hbox {\strut \hskip -0.5\wd\mpr@hlist \unhbox \mpr@hlist}%
+%    \setbox \mpr@vlist
+%       \vbox {\if@premisse  \box \mpr@hlist \unvbox \mpr@vlist
+%              \else \unvbox \mpr@vlist \box \mpr@hlist
+%              \fi}%
+% }
+
+\def \mpr@item #1{$\displaystyle #1$}
+\def \mpr@sep{2em}
+\def \mpr@blank { }
+\def \mpr@hovbox #1#2{\hbox
+  \bgroup
+  \ifx #1T\@premissetrue
+  \else \ifx #1B\@premissefalse
+  \else
+     \PackageError{mathpartir}
+       {Premisse orientation should either be T or B}
+       {Fatal error in Package}%
+  \fi \fi
+  \def \@test {#2}\ifx \@test \mpr@blank\else
+  \setbox \mpr@hlist \hbox {}%
+  \setbox \mpr@vlist \vbox {}%
+  \if@premisse \let \snoc \mpr@cons \else \let \snoc \mpr@snoc \fi
+  \let \@hvlist \empty \let \@rev \empty
+  \mpr@tmpdim 0em
+  \expandafter \mpr@makelist #2\mpr@to \mpr@flat
+  \if@premisse \mpr@rev \mpr@flat \mpr@to \@rev \else \let \@rev \mpr@flat \fi
+  \def \\##1{%
+     \def \@test {##1}\ifx \@test \empty
+        \mpr@htovlist
+        \mpr@tmpdim 0em %%% last bug fix not extensively checked
+     \else
+      \setbox0 \hbox{\mpr@item {##1}}\relax
+      \advance \mpr@tmpdim by \wd0
+      %\mpr@tmpdim 1.02\mpr@tmpdim
+      \ifnum \mpr@tmpdim < \hsize
+         \ifnum \wd\mpr@hlist > 0
+           \if@premisse
+             \setbox \mpr@hlist 
+                \hbox {\unhbox0 \hskip \mpr@sep \unhbox \mpr@hlist}%
+           \else
+             \setbox \mpr@hlist
+                \hbox {\unhbox \mpr@hlist  \hskip \mpr@sep \unhbox0}%
+           \fi
+         \else 
+         \setbox \mpr@hlist \hbox {\unhbox0}%
+         \fi
+      \else
+         \ifnum \wd \mpr@hlist > 0
+            \mpr@htovlist 
+            \mpr@tmpdim \wd0
+         \fi
+         \setbox \mpr@hlist \hbox {\unhbox0}%
+      \fi
+      \advance \mpr@tmpdim by \mpr@sep
+   \fi
+   }%
+   \@rev
+   \mpr@htovlist
+   \ifmpr@center \hskip \wd\mpr@vlist\fi \box \mpr@vlist
+   \fi
+   \egroup
+}
+
+%%% INFERENCE RULES
+
+\@ifundefined{@@over}{%
+    \let\@@over\over % fallback if amsmath is not loaded
+    \let\@@overwithdelims\overwithdelims
+    \let\@@atop\atop \let\@@atopwithdelims\atopwithdelims
+    \let\@@above\above \let\@@abovewithdelims\abovewithdelims
+  }{}
+
+%% The default
+
+\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
+    $\displaystyle {#1\mpr@over #2}$}}
+\def \mpr@@nofraction #1#2{\hbox {\advance \hsize by -0.5em
+    $\displaystyle {#1\@@atop #2}$}}
+
+\let \mpr@fraction \mpr@@fraction
+
+%% A generic solution to arrow
+
+\def \mpr@make@fraction #1#2#3#4#5{\hbox {%
+     \def \mpr@tail{#1}%
+     \def \mpr@body{#2}%
+     \def \mpr@head{#3}%
+     \setbox1=\hbox{$#4$}\setbox2=\hbox{$#5$}%
+     \setbox3=\hbox{$\mkern -3mu\mpr@body\mkern -3mu$}%
+     \setbox3=\hbox{$\mkern -3mu \mpr@body\mkern -3mu$}%
+     \dimen0=\dp1\advance\dimen0 by \ht3\relax\dp1\dimen0\relax
+     \dimen0=\ht2\advance\dimen0 by \dp3\relax\ht2\dimen0\relax
+     \setbox0=\hbox {$\box1 \@@atop \box2$}%
+     \dimen0=\wd0\box0
+     \box0 \hskip -\dimen0\relax
+     \hbox to \dimen0 {$%
+       \mathrel{\mpr@tail}\joinrel
+       \xleaders\hbox{\copy3}\hfil\joinrel\mathrel{\mpr@head}%
+     $}}}
+
+%% Old stuff should be removed in next version
+\def \mpr@@nothing #1#2
+    {$\lower 0.01pt \mpr@@nofraction {#1}{#2}$}
+\def \mpr@@reduce #1#2{\hbox
+    {$\lower 0.01pt \mpr@@fraction {#1}{#2}\mkern -15mu\rightarrow$}}
+\def \mpr@@rewrite #1#2#3{\hbox
+    {$\lower 0.01pt \mpr@@fraction {#2}{#3}\mkern -8mu#1$}}
+\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
+
+\def \mpr@empty {}
+\def \mpr@inferrule
+  {\bgroup
+     \ifnum \linewidth<\hsize \hsize \linewidth\fi
+     \mpr@rulelineskip
+     \let \and \qquad
+     \let \hva \mpr@hva
+     \let \@rulename \mpr@empty
+     \let \@rule@options \mpr@empty
+     \let \mpr@over \@@over
+     \mpr@inferrule@}
+\newcommand {\mpr@inferrule@}[3][]
+  {\everymath={\displaystyle}%       
+   \def \@test {#2}\ifx \empty \@test
+      \setbox0 \hbox {$\vcenter {\mpr@hovbox{B}{#3}}$}%
+   \else 
+   \def \@test {#3}\ifx \empty \@test
+      \setbox0 \hbox {$\vcenter {\mpr@hovbox{T}{#2}}$}%
+   \else
+   \setbox0 \mpr@fraction {\mpr@hovbox{T}{#2}}{\mpr@hovbox{B}{#3}}%
+   \fi \fi
+   \def \@test {#1}\ifx \@test\empty \box0
+   \else \vbox 
+%%% Suggestion de Francois pour les etiquettes longues
+%%%   {\hbox to \wd0 {\RefTirName {#1}\hfil}\box0}\fi
+      {\hbox {\RefTirName {#1}}\box0}\fi
+   \egroup}
+
+\def \mpr@vdotfil #1{\vbox to #1{\leaders \hbox{$\cdot$} \vfil}}
+
+% They are two forms
+% \inferrule [label]{[premisses}{conclusions}
+% or
+% \inferrule* [options]{[premisses}{conclusions}
+%
+% Premisses and conclusions are lists of elements separated by \\
+% Each \\ produces a break, attempting horizontal breaks if possible, 
+% and  vertical breaks if needed. 
+% 
+% An empty element obtained by \\\\ produces a vertical break in all cases. 
+%
+% The former rule is aligned on the fraction bar. 
+% The optional label appears on top of the rule
+% The second form to be used in a derivation tree is aligned on the last
+% line of its conclusion
+% 
+% The second form can be parameterized, using the key=val interface. The
+% folloiwng keys are recognized:
+%       
+%  width                set the width of the rule to val
+%  narrower             set the width of the rule to val\hsize
+%  before               execute val at the beginning/left
+%  lab                  put a label [Val] on top of the rule
+%  lskip                add negative skip on the right
+%  left                 put a left label [Val]
+%  Left                 put a left label [Val],  ignoring its width 
+%  right                put a right label [Val]
+%  Right                put a right label [Val], ignoring its width
+%  leftskip             skip negative space on the left-hand side
+%  rightskip            skip negative space on the right-hand side
+%  vdots                lift the rule by val and fill vertical space with dots
+%  after                execute val at the end/right
+%  
+%  Note that most options must come in this order to avoid strange
+%  typesetting (in particular  leftskip must preceed left and Left and
+%  rightskip must follow Right or right; vdots must come last 
+%  or be only followed by rightskip. 
+%  
+
+%% Keys that make sence in all kinds of rules
+\def \mprset #1{\setkeys{mprset}{#1}}
+\define@key {mprset}{andskip}[]{\mpr@andskip=#1}
+\define@key {mprset}{lineskip}[]{\lineskip=#1}
+\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
+\define@key {mprset}{center}[]{\mpr@centertrue}
+\define@key {mprset}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
+\define@key {mprset}{atop}[]{\let \mpr@fraction \mpr@@nofraction}
+\define@key {mprset}{myfraction}[]{\let \mpr@fraction #1}
+\define@key {mprset}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
+\define@key {mprset}{sep}{\def\mpr@sep{#1}}
+
+\newbox \mpr@right
+\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
+\define@key {mpr}{center}[]{\mpr@centertrue}
+\define@key {mpr}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
+\define@key {mpr}{myfraction}[]{\let \mpr@fraction #1}
+\define@key {mpr}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
+\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
+     \advance \hsize by -\wd0\box0}
+\define@key {mpr}{width}{\hsize #1}
+\define@key {mpr}{sep}{\def\mpr@sep{#1}}
+\define@key {mpr}{before}{#1}
+\define@key {mpr}{lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
+\define@key {mpr}{Lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
+\define@key {mpr}{narrower}{\hsize #1\hsize}
+\define@key {mpr}{leftskip}{\hskip -#1}
+\define@key {mpr}{reduce}[]{\let \mpr@fraction \mpr@@reduce}
+\define@key {mpr}{rightskip}
+  {\setbox \mpr@right \hbox {\unhbox \mpr@right \hskip -#1}}
+\define@key {mpr}{LEFT}{\setbox0 \hbox {$#1$}\relax
+     \advance \hsize by -\wd0\box0}
+\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
+     \advance \hsize by -\wd0\box0}
+\define@key {mpr}{Left}{\llap{$\TirName {#1}\;$}}
+\define@key {mpr}{right}
+  {\setbox0 \hbox {$\;\TirName {#1}$}\relax \advance \hsize by -\wd0
+   \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
+\define@key {mpr}{RIGHT}
+  {\setbox0 \hbox {$#1$}\relax \advance \hsize by -\wd0
+   \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
+\define@key {mpr}{Right}
+  {\setbox \mpr@right \hbox {\unhbox \mpr@right \rlap {$\;\TirName {#1}$}}}
+\define@key {mpr}{vdots}{\def \mpr@vdots {\@@atop \mpr@vdotfil{#1}}}
+\define@key {mpr}{after}{\edef \mpr@after {\mpr@after #1}}
+
+\newcommand \mpr@inferstar@ [3][]{\setbox0
+  \hbox {\let \mpr@rulename \mpr@empty \let \mpr@vdots \relax
+         \setbox \mpr@right \hbox{}%
+         $\setkeys{mpr}{#1}%
+          \ifx \mpr@rulename \mpr@empty \mpr@inferrule {#2}{#3}\else
+          \mpr@inferrule [{\mpr@rulename}]{#2}{#3}\fi
+          \box \mpr@right \mpr@vdots$}
+  \setbox1 \hbox {\strut}
+  \@tempdima \dp0 \advance \@tempdima by -\dp1
+  \raise \@tempdima \box0}
+
+\def \mpr@infer {\@ifnextchar *{\mpr@inferstar}{\mpr@inferrule}}
+\newcommand \mpr@err@skipargs[3][]{}
+\def \mpr@inferstar*{\ifmmode 
+    \let \@do \mpr@inferstar@
+  \else 
+    \let \@do \mpr@err@skipargs
+    \PackageError {mathpartir}
+      {\string\inferrule* can only be used in math mode}{}%
+  \fi \@do}
+
+
+%%% Exports
+
+% Envirnonment mathpar
+
+\let \inferrule \mpr@infer
+
+% make a short name \infer is not already defined
+\@ifundefined {infer}{\let \infer \mpr@infer}{}
+
+\def \TirNameStyle #1{\small \textsc{#1}}
+\def \tir@name #1{\hbox {\small \TirNameStyle{#1}}}
+\let \TirName \tir@name
+\let \DefTirName \TirName
+\let \RefTirName \TirName
+
+%%% Other Exports
+
+% \let \listcons \mpr@cons
+% \let \listsnoc \mpr@snoc
+% \let \listhead \mpr@head
+% \let \listmake \mpr@makelist
+
+
+
+
+\endinput
diff -r 000000000000 -r 110247f9d47e Slides/document/root.beamer.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/document/root.beamer.tex	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,12 @@
+\documentclass[14pt,t]{beamer}
+%%%\usepackage{pstricks}
+
+\input{root.tex}
+
+%%% Local Variables:  
+%%% mode: latex
+%%% TeX-master: t
+%%% TeX-command-default: "Slides"
+%%% TeX-view-style: (("." "kghostview --landscape --scale 0.45 --geometry 605x505 %f"))
+%%% End: 
+
diff -r 000000000000 -r 110247f9d47e Slides/document/root.notes.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/document/root.notes.tex	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,18 @@
+\documentclass[11pt]{article}
+%%\usepackage{pstricks}
+\usepackage{dina4}
+\usepackage{beamerarticle}
+\usepackage{times}
+\usepackage{hyperref}
+\usepackage{pgf}
+\usepackage{amssymb}
+\setjobnamebeamerversion{root.beamer}
+\input{root.tex}
+
+%%% Local Variables:  
+%%% mode: latex
+%%% TeX-master: t
+%%% TeX-command-default: "Slides"
+%%% TeX-view-style: (("." "kghostview --landscape --scale 0.45 --geometry 605x505 %f"))
+%%% End: 
+
diff -r 000000000000 -r 110247f9d47e Slides/document/root.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/document/root.tex	Thu Dec 06 15:11:21 2012 +0000
@@ -0,0 +1,147 @@
+\usepackage{beamerthemeplaincu}
+%%\usepackage{ulem}
+\usepackage[T1]{fontenc}
+\usepackage{proof}
+\usepackage[latin1]{inputenc}
+\usepackage{isabelle}
+\usepackage{isabellesym}
+\usepackage{mathpartir}
+\usepackage[absolute, overlay]{textpos}
+\usepackage{proof}
+\usepackage{ifthen}
+\usepackage{animate}
+\usepackage{tikz}
+\usepackage{pgf}
+\usetikzlibrary{arrows}
+\usetikzlibrary{automata}
+\usetikzlibrary{shapes}
+\usetikzlibrary{shadows}
+\usetikzlibrary{calc}
+
+% Isabelle configuration
+%%\urlstyle{rm}
+\isabellestyle{rm}
+\renewcommand{\isastyle}{\rm}%
+\renewcommand{\isastyleminor}{\rm}%
+\renewcommand{\isastylescript}{\footnotesize\rm\slshape}%
+\renewcommand{\isatagproof}{}
+\renewcommand{\endisatagproof}{}
+\renewcommand{\isamarkupcmt}[1]{#1}
+
+% Isabelle characters
+\renewcommand{\isacharunderscore}{\_}
+\renewcommand{\isacharbar}{\isamath{\mid}}
+\renewcommand{\isasymiota}{}
+\renewcommand{\isacharbraceleft}{\{}
+\renewcommand{\isacharbraceright}{\}}
+\renewcommand{\isacharless}{$\langle$}
+\renewcommand{\isachargreater}{$\rangle$}
+\renewcommand{\isasymsharp}{\isamath{\#}}
+\renewcommand{\isasymdots}{\isamath{...}}
+\renewcommand{\isasymbullet}{\act}
+
+% mathpatir
+\mprset{sep=1em}
+
+% general math stuff
+\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
+\newcommand{\dnn}{\stackrel{\mbox{\Large def}}{=}}
+\renewcommand{\isasymequiv}{$\dn$}
+\renewcommand{\emptyset}{\varnothing}% nice round empty set
+\renewcommand{\Gamma}{\varGamma} 
+\DeclareRobustCommand{\flqq}{\mbox{\guillemotleft}}
+\DeclareRobustCommand{\frqq}{\mbox{\guillemotright}}
+\newcommand{\smath}[1]{\textcolor{blue}{\ensuremath{#1}}}
+\newcommand{\fresh}{\mathrel{\#}}
+\newcommand{\act}{{\raisebox{-0.5mm}{\Large$\boldsymbol{\cdot}$}}}% swapping action
+\newcommand{\swap}[2]{(#1\,#2)}% swapping operation
+
+% beamer stuff 
+\renewcommand{\slidecaption}{Salvador, 26.~August 2008}
+
+
+% colours for Isar Code (in article mode everything is black and white)
+\mode<presentation>{
+\definecolor{isacol:brown}{rgb}{.823,.411,.117}
+\definecolor{isacol:lightblue}{rgb}{.274,.509,.705}
+\definecolor{isacol:green}{rgb}{0,.51,0.14}
+\definecolor{isacol:red}{rgb}{.803,0,0}
+\definecolor{isacol:blue}{rgb}{0,0,.803}
+\definecolor{isacol:darkred}{rgb}{.545,0,0}
+\definecolor{isacol:black}{rgb}{0,0,0}}
+\mode<article>{
+\definecolor{isacol:brown}{rgb}{0,0,0}
+\definecolor{isacol:lightblue}{rgb}{0,0,0}
+\definecolor{isacol:green}{rgb}{0,0,0}
+\definecolor{isacol:red}{rgb}{0,0,0}
+\definecolor{isacol:blue}{rgb}{0,0,0}
+\definecolor{isacol:darkred}{rgb}{0,0,0}
+\definecolor{isacol:black}{rgb}{0,0,0}
+}
+
+
+\newcommand{\strong}[1]{{\bfseries {#1}}}
+\newcommand{\bluecmd}[1]{{\color{isacol:lightblue}{\strong{#1}}}}
+\newcommand{\browncmd}[1]{{\color{isacol:brown}{\strong{#1}}}}
+\newcommand{\redcmd}[1]{{\color{isacol:red}{\strong{#1}}}}
+
+\renewcommand{\isakeyword}[1]{%
+\ifthenelse{\equal{#1}{show}}{\browncmd{#1}}{%
+\ifthenelse{\equal{#1}{case}}{\browncmd{#1}}{%
+\ifthenelse{\equal{#1}{assume}}{\browncmd{#1}}{%
+\ifthenelse{\equal{#1}{obtain}}{\browncmd{#1}}{%
+\ifthenelse{\equal{#1}{fix}}{\browncmd{#1}}{%
+\ifthenelse{\equal{#1}{oops}}{\redcmd{#1}}{%
+\ifthenelse{\equal{#1}{thm}}{\redcmd{#1}}{%
+{\bluecmd{#1}}}}}}}}}}%
+
+% inner syntax colour
+\chardef\isachardoublequoteopen=`\"%
+\chardef\isachardoublequoteclose=`\"%
+\chardef\isacharbackquoteopen=`\`%
+\chardef\isacharbackquoteclose=`\`%
+\newenvironment{innersingle}%
+{\isacharbackquoteopen\color{isacol:green}}%
+{\color{isacol:black}\isacharbackquoteclose}
+\newenvironment{innerdouble}%
+{\isachardoublequoteopen\color{isacol:green}}%
+{\color{isacol:black}\isachardoublequoteclose}
+
+%% misc
+\newcommand{\gb}[1]{\textcolor{isacol:green}{#1}}
+\newcommand{\rb}[1]{\textcolor{red}{#1}}
+
+%% animations
+\newcounter{growcnt}
+\newcommand{\grow}[2]
+{\begin{tikzpicture}[baseline=(n.base)]%
+    \node[scale=(0.1 *#1 + 0.001),inner sep=0pt] (n) {#2};
+  \end{tikzpicture}%
+}
+
+%% isatabbing
+%\renewcommand{\isamarkupcmt}[1]%
+%{\ifthenelse{\equal{TABSET}{#1}}{\=}%
+% {\ifthenelse{\equal{TAB}{#1}}{\>}%
+%  {\ifthenelse{\equal{NEWLINE}{#1}}{\\}%
+%   {\ifthenelse{\equal{DOTS}{#1}}{\ldots}{\isastylecmt--- {#1}}}%
+%  }%
+% }%
+%}%
+
+
+\newenvironment{isatabbing}%
+{\renewcommand{\isanewline}{\\}\begin{tabbing}}%
+{\end{tabbing}}
+
+\begin{document}
+\input{session}
+\end{document}
+
+%%% Local Variables:  
+%%% mode: latex
+%%% TeX-master: t
+%%% TeX-command-default: "Slides"
+%%% TeX-view-style: (("." "kghostview --landscape --scale 0.45 --geometry 605x505 %f"))
+%%% End: 
+