# HG changeset patch
# User urbanc
# Date 1328475612 0
# Node ID a3b4eed091d22e36aca5c78c35d7d0ccf9d01d51
# Parent  e5bfdd2d1ac827275e73b4d0eb87e0672ec9f23a
moved unused theories to Attic

diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Attic/Ext.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Attic/Ext.thy	Sun Feb 05 21:00:12 2012 +0000
@@ -0,0 +1,1057 @@
+theory Ext
+imports Prio
+begin
+
+locale highest_create =
+  fixes s' th prio fixes s 
+  defines s_def : "s \<equiv> (Create th prio#s')"
+  assumes vt_s: "vt step s"
+  and highest: "cp s th = Max ((cp s)`threads s)"
+
+context highest_create
+begin
+
+lemma threads_s: "threads s = threads s' \<union> {th}"
+  by (unfold s_def, simp)
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_create: "step s' (Create th prio)"
+  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_create_elim: 
+  "\<lbrakk>\<And>max_prio. \<lbrakk>prio \<le> max_prio; th \<notin> threads s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+  by (insert step_create, ind_cases "step s' (Create th prio)", auto)
+
+lemma eq_cp_s: 
+  assumes th'_in: "th' \<in> threads s'"
+  shows "cp s th' = cp s' th'"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def s_def 
+    eq_depend depend_create_unchanged)
+  show "Max ((\<lambda>tha. preced tha (Create th prio # s')) `
+         ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+})) =
+        Max ((\<lambda>th. preced th s') ` ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+}))"
+    (is "Max (?f ` ?A) = Max (?g ` ?A)")
+  proof -
+    have "?f ` ?A = ?g ` ?A"
+    proof(rule f_image_eq)
+      fix a
+      assume a_in: "a \<in> ?A"
+      thus "?f a = ?g a" 
+      proof -
+        from a_in
+        have "a = th' \<or> (Th a, Th th') \<in> (depend s')\<^sup>+" by auto 
+        hence "a \<noteq> th"
+        proof
+          assume "a = th'"
+          moreover have "th' \<noteq> th"
+          proof(rule step_create_elim)
+            assume th_not_in: "th \<notin> threads s'" with th'_in
+            show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        next
+          assume "(Th a, Th th') \<in> (depend s')\<^sup>+"
+          hence "Th a \<in> Domain \<dots>"
+            by (auto simp:Domain_def)
+          hence "Th a \<in> Domain (depend s')"
+            by (simp add:trancl_domain)
+          from dm_depend_threads[OF vt_s' this]
+          have h: "a \<in> threads s'" .
+          show ?thesis
+          proof(rule step_create_elim)
+            assume "th \<notin> threads s'" with h
+            show ?thesis by auto
+          qed
+        qed
+        thus ?thesis 
+          by (unfold preced_def, auto)
+      qed
+    qed
+    thus ?thesis by auto
+  qed
+qed
+
+lemma same_depend: "depend s = depend s'"
+  by (insert depend_create_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+  "dependents (wq s) th = dependents (wq s') th"
+  apply (unfold cs_dependents_def)
+  by (unfold eq_depend same_depend, simp)
+
+lemma nil_dependents_s': "dependents (wq s') th = {}"
+proof -
+  { assume ne: "dependents (wq s') th \<noteq> {}"
+    then obtain th' where "th' \<in>  dependents (wq s') th"
+      by (unfold cs_dependents_def, auto)
+    hence "(Th th', Th th) \<in> (depend (wq s'))^+"
+      by (unfold cs_dependents_def, auto)
+    hence "(Th th', Th th) \<in> (depend s')^+"
+      by (simp add:eq_depend)
+    hence "Th th \<in> Range ((depend s')^+)" by (auto simp:Range_def Domain_def)
+    hence "Th th \<in> Range (depend s')" by (simp add:trancl_range)
+    from range_in [OF vt_s' this]
+    have h: "th \<in> threads s'" .
+    have "False"
+    proof(rule step_create_elim)
+      assume "th \<notin> threads s'" with h show ?thesis by auto
+    qed
+  } thus ?thesis by auto
+qed
+
+lemma nil_dependents: "dependents (wq s) th = {}"
+proof -
+  have "wq s' = wq s"
+    by (unfold wq_def s_def, auto simp:Let_def)
+  with nil_dependents_s' show ?thesis by auto
+qed
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+  by (unfold cp_eq_cpreced cpreced_def nil_dependents, auto)
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+  { assume "th \<notin> readys s"
+    with threads_s obtain cs where 
+      "waiting s th cs"
+      by (unfold readys_def, auto)
+    hence "(Th th, Cs cs) \<in> depend s"
+      by (unfold s_depend_def, unfold eq_waiting, simp)
+    hence "Th th \<in> Domain (depend s')"
+      by (unfold same_depend, auto simp:Domain_def)
+    from dm_depend_threads [OF vt_s' this] 
+    have h: "th \<in> threads s'" .
+    have "False"
+    proof (rule_tac step_create_elim)
+      assume "th \<notin> threads s'" with h show ?thesis by simp
+    qed
+  } thus ?thesis by auto
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+  proof -
+    have " Max (cp s ` readys s) = cp s th"
+    proof(rule Max_eqI)
+      from finite_threads[OF vt_s] readys_threads finite_subset
+      have "finite (readys s)" by blast
+      thus "finite (cp s ` readys s)" by auto
+    next
+      from is_ready show "cp s th \<in> cp s ` readys s" by auto
+    next
+      fix y
+      assume h: "y \<in> cp s ` readys s"
+      have "y \<le> Max (cp s ` readys s)"
+      proof(rule Max_ge [OF _ h])
+        from finite_threads[OF vt_s] readys_threads finite_subset
+        have "finite (readys s)" by blast
+        thus "finite (cp s ` readys s)" by auto
+      qed
+      moreover have "\<dots> \<le> Max (cp s ` threads s)"
+      proof(rule Max_mono)
+        from readys_threads 
+        show "cp s ` readys s \<subseteq> cp s ` threads s" by auto
+      next
+        from is_ready show "cp s ` readys s \<noteq> {}" by auto
+      next
+        from finite_threads [OF vt_s]
+        show "finite (cp s ` threads s)" by auto
+      qed
+      moreover note highest
+      ultimately show "y \<le> cp s th" by auto
+    qed
+    with highest show ?thesis by auto
+  qed
+  thus ?thesis
+    by (unfold runing_def, insert highest is_ready, auto)
+qed
+
+end
+
+locale extend_highest = highest_create + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest
+begin
+
+lemma red_moment:
+  "extend_highest s' th prio (moment i t)"
+  apply (insert extend_highest_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_def extend_highest_axioms_def, clarsimp)
+  by (unfold highest_create_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest s' th prio t; 
+                    extend_highest s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest s' th prio (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest s' th prio t'"
+          by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest s' th prio (e # t')" .
+    next
+      from et show ext': "extend_highest s' th prio t'"
+          by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest s' th prio (e # t)" by auto
+      from extend_highest.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest s' th prio (e # t)" by auto
+        from extend_highest.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest s' th prio t" by auto
+        from extend_highest.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest s' th prio t" by auto
+    from extend_highest.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have " extend_highest s' th prio (e # t)" by auto
+        from extend_highest.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest s' th prio (e # t)" by auto
+    from extend_highest.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest s' th prio t" by auto
+    from extend_highest.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest s' th prio (e # t)" by auto
+        from extend_highest.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest s' th prio (e # t)" by auto
+      from extend_highest.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest s' th prio t" by auto
+        from extend_highest.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' is_ready
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    have "extend_highest s' th prio t" by fact
+    from extend_highest.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest s' th prio (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_def extend_highest_axioms_def 
+          highest_create_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ Create th prio # s')"
+        proof(rule extend_highest.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ Create th prio # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest s' th prio (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ Create th prio # s') th' =
+            cntV (moment (i + k) t @ Create th prio # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest s' th prio (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_def extend_highest_axioms_def 
+      highest_create_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Attic/ExtGG_1.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Attic/ExtGG_1.thy	Sun Feb 05 21:00:12 2012 +0000
@@ -0,0 +1,973 @@
+theory ExtGG
+imports PrioG
+begin
+
+lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+  apply (induct s, simp)
+proof -
+  fix a s
+  assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+    and eq_as: "a # s \<noteq> []"
+  show "birthtime th (a # s) < length (a # s)"
+  proof(cases "s \<noteq> []")
+    case False
+    from False show ?thesis
+      by (cases a, auto simp:birthtime.simps)
+  next
+    case True
+    from ih [OF True] show ?thesis
+      by (cases a, auto simp:birthtime.simps)
+  qed
+qed
+
+lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
+  by (induct s, auto simp:threads.simps)
+
+lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
+  apply (drule_tac th_in_ne)
+  by (unfold preced_def, auto intro: birth_time_lt)
+
+locale highest_gen =
+  fixes s' th s e' prio tm
+  defines s_def : "s \<equiv> (e'#s')"
+  assumes vt_s: "vt step s"
+  and threads_s: "th \<in> threads s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+  and nh: "preced th s' \<noteq> Max ((cp s)`threads s')"
+  and preced_th: "preced th s = Prc prio tm"
+
+context highest_gen
+begin
+
+lemma lt_tm: "tm < length s"
+  by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+  from highest and max_cp_eq[OF vt_s]
+  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+  proof -
+    from threads_s and dependents_threads[OF vt_s, of th]
+    show ?thesis by auto
+  qed
+  show ?thesis
+  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+  next
+    fix y 
+    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+      and eq_y: "y = preced th1 s" by auto
+    show "y \<le> preced th s"
+    proof(unfold is_max, rule Max_ge)
+      from finite_threads[OF vt_s] 
+      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+    next
+      from sbs th1_in and eq_y 
+      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+    qed
+  next
+    from sbs and finite_threads[OF vt_s]
+    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+      by (auto intro:finite_subset)
+  qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+  show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_gen
+begin
+
+lemma red_moment:
+  "extend_highest_gen s' th e' prio tm (moment i t)"
+  apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+  by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest_gen s' th e' prio tm t; 
+                    extend_highest_gen s' th e' prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_gen_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_gen s' th e' prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest_gen s' th e' prio tm (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_gen s' th e' prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_gen s' th e' prio tm (e # t')" .
+    next
+      from et show ext': "extend_highest_gen s' th e' prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+      from extend_highest_gen.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+        from extend_highest_gen.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+        from extend_highest_gen.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+    from extend_highest_gen.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+        from extend_highest_gen.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold preced_th, unfold eq_e, insert lt_tm, 
+          auto simp:preced_def s_def precedence_less_def preced_th)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+    from extend_highest_gen.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+    from extend_highest_gen.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+        from extend_highest_gen.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+      from extend_highest_gen.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            apply (subst preced_th, insert lt_tm)
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+        from extend_highest_gen.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' threads_s
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+    from extend_highest_gen.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest_gen s' th e' prio tm (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_gen_def extend_highest_gen_axioms_def 
+                          highest_gen_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ e' # s')"
+        proof(rule extend_highest_gen.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ e' # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest_gen s' th e' prio tm (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ e' # s') th' = cntV (moment (i + k) t @ e' # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_gen.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest_gen s' th e' prio tm (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_gen_def extend_highest_gen_axioms_def 
+      highest_gen_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+
+
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Attic/ExtS.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Attic/ExtS.thy	Sun Feb 05 21:00:12 2012 +0000
@@ -0,0 +1,1019 @@
+theory ExtS
+imports Prio
+begin
+
+locale highest_set =
+  fixes s' th prio fixes s 
+  defines s_def : "s \<equiv> (Set th prio#s')"
+  assumes vt_s: "vt step s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+
+context highest_set
+begin
+
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_set: "step s' (Set th prio)"
+  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_set_elim: 
+  "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+  by (insert step_set, ind_cases "step s' (Set th prio)", auto)
+
+
+lemma threads_s: "th \<in> threads s"
+  by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
+
+lemma same_depend: "depend s = depend s'"
+  by (insert depend_set_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+  "dependents (wq s) th = dependents (wq s') th"
+  apply (unfold cs_dependents_def)
+  by (unfold eq_depend same_depend, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+  from highest and max_cp_eq[OF vt_s]
+  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+  proof -
+    from threads_s and dependents_threads[OF vt_s, of th]
+    show ?thesis by auto
+  qed
+  show ?thesis
+  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+  next
+    fix y 
+    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+      and eq_y: "y = preced th1 s" by auto
+    show "y \<le> preced th s"
+    proof(unfold is_max, rule Max_ge)
+      from finite_threads[OF vt_s] 
+      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+    next
+      from sbs th1_in and eq_y 
+      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+    qed
+  next
+    from sbs and finite_threads[OF vt_s]
+    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+      by (auto intro:finite_subset)
+  qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+  have "\<forall>cs. \<not> waiting s th cs"
+    apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
+    apply (unfold s_depend_def, unfold runing_def readys_def)
+    apply (auto, fold s_def)
+    apply (erule_tac x = cs in allE, auto simp:waiting_eq)
+  proof -
+    fix cs
+    assume h: 
+      "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
+          {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
+            (is "?L = ?R")
+    and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
+    from wt have "(Th th, Cs cs) \<in> ?L" by auto
+    with h have "(Th th, Cs cs) \<in> ?R" by simp
+    hence "waiting (wq s') th cs" by auto with nwt
+    show False by auto
+  qed    
+  with threads_s show ?thesis 
+    by (unfold readys_def, auto)
+qed
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+  show ?thesis by simp
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+  proof -
+    have " Max (cp s ` readys s) = cp s th"
+    proof(rule Max_eqI)
+      from finite_threads[OF vt_s] readys_threads finite_subset
+      have "finite (readys s)" by blast
+      thus "finite (cp s ` readys s)" by auto
+    next
+      from is_ready show "cp s th \<in> cp s ` readys s" by auto
+    next
+      fix y
+      assume "y \<in> cp s ` readys s"
+      then obtain th1 where 
+        eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
+      show  "y \<le> cp s th" 
+      proof -
+        have "y \<le> Max (cp s ` threads s)"
+        proof(rule Max_ge)
+          from eq_y and th1_in
+          show "y \<in> cp s ` threads s"
+            by (auto simp:readys_def)
+        next
+          from finite_threads[OF vt_s]
+          show "finite (cp s ` threads s)" by auto
+        qed
+        with highest' show ?thesis by auto
+      qed
+    qed
+    with highest' show ?thesis by auto
+  qed
+  thus ?thesis
+    by (unfold runing_def, insert highest' is_ready, auto)
+qed
+
+end
+
+locale extend_highest_set = highest_set + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_set
+begin
+
+lemma red_moment:
+  "extend_highest_set s' th prio (moment i t)"
+  apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
+  by (unfold highest_set_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest_set s' th prio t; 
+                    extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_set_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest_set s' th prio (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_set s' th prio (e # t')" .
+    next
+      from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have " extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest_set s' th prio (e # t)" by auto
+    from extend_highest_set.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' is_ready
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    have "extend_highest_set s' th prio t" by fact
+    from extend_highest_set.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+          highest_set_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
+        proof(rule extend_highest_set.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
+            cntV (moment (i + k) t @ Set th prio # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_set.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest_set s' th prio (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+      highest_set_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Attic/ExtSG.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Attic/ExtSG.thy	Sun Feb 05 21:00:12 2012 +0000
@@ -0,0 +1,1019 @@
+theory ExtSG
+imports PrioG
+begin
+
+locale highest_set =
+  fixes s' th prio fixes s 
+  defines s_def : "s \<equiv> (Set th prio#s')"
+  assumes vt_s: "vt step s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+
+context highest_set
+begin
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_set: "step s' (Set th prio)"
+  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_set_elim: 
+  "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+  by (insert step_set, ind_cases "step s' (Set th prio)", auto)
+
+
+lemma threads_s: "th \<in> threads s"
+  by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
+
+lemma same_depend: "depend s = depend s'"
+  by (insert depend_set_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+  "dependents (wq s) th = dependents (wq s') th"
+  apply (unfold cs_dependents_def)
+  by (unfold eq_depend same_depend, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+  from highest and max_cp_eq[OF vt_s]
+  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+  proof -
+    from threads_s and dependents_threads[OF vt_s, of th]
+    show ?thesis by auto
+  qed
+  show ?thesis
+  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+  next
+    fix y 
+    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+      and eq_y: "y = preced th1 s" by auto
+    show "y \<le> preced th s"
+    proof(unfold is_max, rule Max_ge)
+      from finite_threads[OF vt_s] 
+      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+    next
+      from sbs th1_in and eq_y 
+      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+    qed
+  next
+    from sbs and finite_threads[OF vt_s]
+    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+      by (auto intro:finite_subset)
+  qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+  have "\<forall>cs. \<not> waiting s th cs"
+    apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
+    apply (unfold s_depend_def, unfold runing_def readys_def)
+    apply (auto, fold s_def)
+    apply (erule_tac x = cs in allE, auto simp:waiting_eq)
+  proof -
+    fix cs
+    assume h: 
+      "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
+          {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
+            (is "?L = ?R")
+    and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
+    from wt have "(Th th, Cs cs) \<in> ?L" by auto
+    with h have "(Th th, Cs cs) \<in> ?R" by simp
+    hence "waiting (wq s') th cs" by auto with nwt
+    show False by auto
+  qed    
+  with threads_s show ?thesis 
+    by (unfold readys_def, auto)
+qed
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+  show ?thesis by simp
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+  proof -
+    have " Max (cp s ` readys s) = cp s th"
+    proof(rule Max_eqI)
+      from finite_threads[OF vt_s] readys_threads finite_subset
+      have "finite (readys s)" by blast
+      thus "finite (cp s ` readys s)" by auto
+    next
+      from is_ready show "cp s th \<in> cp s ` readys s" by auto
+    next
+      fix y
+      assume "y \<in> cp s ` readys s"
+      then obtain th1 where 
+        eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
+      show  "y \<le> cp s th" 
+      proof -
+        have "y \<le> Max (cp s ` threads s)"
+        proof(rule Max_ge)
+          from eq_y and th1_in
+          show "y \<in> cp s ` threads s"
+            by (auto simp:readys_def)
+        next
+          from finite_threads[OF vt_s]
+          show "finite (cp s ` threads s)" by auto
+        qed
+        with highest' show ?thesis by auto
+      qed
+    qed
+    with highest' show ?thesis by auto
+  qed
+  thus ?thesis
+    by (unfold runing_def, insert highest' is_ready, auto)
+qed
+
+end
+
+locale extend_highest_set = highest_set + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_set
+begin
+
+lemma red_moment:
+  "extend_highest_set s' th prio (moment i t)"
+  apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
+  by (unfold highest_set_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest_set s' th prio t; 
+                    extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_set_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest_set s' th prio (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_set s' th prio (e # t')" .
+    next
+      from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have " extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest_set s' th prio (e # t)" by auto
+    from extend_highest_set.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' is_ready
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    have "extend_highest_set s' th prio t" by fact
+    from extend_highest_set.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+                          highest_set_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
+        proof(rule extend_highest_set.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
+            cntV (moment (i + k) t @ Set th prio # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_set.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest_set s' th prio (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+      highest_set_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+
+
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Attic/Happen_within.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Attic/Happen_within.thy	Sun Feb 05 21:00:12 2012 +0000
@@ -0,0 +1,126 @@
+theory Happen_within
+imports Main Moment
+begin
+
+(* 
+  lemma 
+  fixes P :: "('a list) \<Rightarrow> bool"
+  and Q :: "('a list) \<Rightarrow> bool"
+  and k :: nat
+  and f :: "('a list) \<Rightarrow> nat"
+  assumes "\<And> s t. \<lbrakk>P s; \<not> Q s; P (t@s); k < length t\<rbrakk> \<Longrightarrow> f (t@s) < f s"
+  shows "\<And> s t. \<lbrakk> P s;  P(t @ s); f(s) * k < length t\<rbrakk> \<Longrightarrow> Q (t@s)"
+  sorry
+*)
+
+text {* 
+  The following two notions are introduced to improve the situation.
+  *}
+
+definition all_future :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> ('a list) \<Rightarrow> bool"
+where "all_future G R s = (\<forall> t. G (t@s) \<longrightarrow> R t)"
+
+definition happen_within :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> ('a list) \<Rightarrow> bool"
+where "happen_within G R k s = all_future G (\<lambda> t. k < length t \<longrightarrow> 
+                                  (\<exists> i \<le> k. R (moment i t @ s) \<and> G (moment i t @ s))) s"
+
+lemma happen_within_intro:
+  fixes P :: "('a list) \<Rightarrow> bool"
+  and Q :: "('a list) \<Rightarrow> bool"
+  and k :: nat
+  and f :: "('a list) \<Rightarrow> nat"
+  assumes 
+  lt_k: "0 < k"
+  and step: "\<And> s. \<lbrakk>P s; \<not> Q s\<rbrakk> \<Longrightarrow> happen_within P (\<lambda> s'. f s' < f s) k s"
+  shows "\<And> s. P s \<Longrightarrow> happen_within P Q ((f s + 1) * k) s"
+proof -
+  fix s
+  assume "P s"
+  thus "happen_within P Q ((f s + 1) * k) s"
+  proof(induct n == "f s + 1" arbitrary:s rule:nat_less_induct)
+    fix s
+    assume ih [rule_format]: "\<forall>m<f s + 1. \<forall>x. m = f x + 1 \<longrightarrow> P x 
+                                 \<longrightarrow> happen_within P Q ((f x + 1) * k) x"
+      and ps: "P s"
+    show "happen_within P Q ((f s + 1) * k) s"
+    proof(cases "Q s")
+      case True
+      show ?thesis 
+      proof -
+        { fix t
+          from True and ps have "0 \<le> ((f s + 1)*k) \<and> Q (moment 0 t @ s) \<and> P (moment 0 t @ s)" by auto
+          hence "\<exists>i\<le>(f s + 1) * k. Q (moment i t @ s) \<and> P (moment i t @ s)" by auto
+        } thus ?thesis by (auto simp: happen_within_def all_future_def)
+      qed
+    next
+      case False
+      from step [OF ps False] have kk: "happen_within P (\<lambda>s'. f s' < f s) k s" .
+      show ?thesis
+      proof -
+        { fix t
+          assume pts: "P (t @ s)" and ltk: "(f s + 1) * k < length t"
+          from ltk have lt_k_lt: "k < length t" by auto
+          with kk pts obtain i 
+            where le_ik: "i \<le> k" 
+            and lt_f: "f (moment i t @ s) < f s" 
+            and p_m: "P (moment i t @ s)"
+            by (auto simp:happen_within_def all_future_def)
+          from ih [of "f (moment i t @ s) + 1" "(moment i t @ s)", OF _ _ p_m] and lt_f
+          have hw: "happen_within P Q ((f (moment i t @ s) + 1) * k) (moment i t @ s)" by auto
+          have "(\<exists>j\<le>(f s + 1) * k. Q (moment j t @ s) \<and>  P (moment j t @ s))" (is "\<exists> j. ?T j")
+          proof -
+            let ?t = "restm i t"
+            have eq_t: "t = ?t @ moment i t" by (simp add:moment_restm_s) 
+            have h1: "P (restm i t @ moment i t @ s)" 
+            proof -
+              from pts and eq_t have "P ((restm i t @ moment i t) @ s)" by simp
+              thus ?thesis by simp
+            qed
+            moreover have h2: "(f (moment i t @ s) + 1) * k < length (restm i t)"
+            proof -
+              have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
+              from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
+              from h [OF this, of k]
+              have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
+              moreover from le_ik have "\<dots> \<le> ((f s) * k + k - i)" by simp
+              moreover from le_ik lt_k_lt and ltk have "(f s) * k + k - i < length t - i" by simp
+              moreover have "length (restm i t) = length t - i" using length_restm by metis
+              ultimately show ?thesis by simp
+            qed
+            from hw [unfolded happen_within_def all_future_def, rule_format, OF h1 h2]
+            obtain m where le_m: "m \<le> (f (moment i t @ s) + 1) * k"
+              and q_m: "Q (moment m ?t @ moment i t @ s)" 
+              and p_m: "P (moment m ?t @ moment i t @ s)" by auto
+            have eq_mm: "moment m ?t @ moment i t @ s = (moment (m+i) t)@s"
+            proof -
+              have "moment m (restm i t) @ moment i t = moment (m + i) t"
+                using moment_plus_split by metis
+              thus ?thesis by simp
+            qed
+            let ?j = "m + i"
+            have "?T ?j"
+            proof -
+              have "m + i \<le> (f s + 1) * k"
+              proof -
+                have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
+                from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
+                from h [OF this, of k]
+                have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
+                with le_m have "m \<le> f s * k" by simp
+                hence "m + i \<le> f s * k + i" by simp
+                with le_ik show ?thesis by simp
+              qed
+              moreover from eq_mm q_m have " Q (moment (m + i) t @ s)" by metis
+              moreover from eq_mm p_m have " P (moment (m + i) t @ s)" by metis
+              ultimately show ?thesis by blast
+            qed
+            thus ?thesis by blast
+          qed
+        } thus ?thesis by  (simp add:happen_within_def all_future_def firstn.simps)
+      qed
+    qed
+  qed
+qed
+
+end
+
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Attic/Lsp.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Attic/Lsp.thy	Sun Feb 05 21:00:12 2012 +0000
@@ -0,0 +1,323 @@
+theory Lsp
+imports Main
+begin
+
+fun lsp :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list)"
+where 
+   "lsp f [] = ([], [], [])" |
+   "lsp f [x] = ([], [x], [])" |
+   "lsp f (x#xs) = (case (lsp f xs) of
+                     (l, [], r) \<Rightarrow> ([], [x], []) |
+                     (l, y#ys, r) \<Rightarrow> if f x \<ge> f y then ([], [x], xs) else (x#l, y#ys, r))"
+
+inductive lsp_p :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list) \<Rightarrow> bool"
+for f :: "('a \<Rightarrow> ('b::linorder))"
+where
+  lsp_nil [intro]: "lsp_p f [] ([], [], [])" |
+  lsp_single [intro]: "lsp_p f [x] ([], [x], [])" |
+  lsp_cons_1 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x \<ge> f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) ([], [x], xs)" |
+  lsp_cons_2 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) (x#l, [m], r)"
+
+lemma lsp_p_lsp_1: "lsp_p f x y \<Longrightarrow> y = lsp f x"
+proof (induct rule:lsp_p.induct)
+  case (lsp_cons_1 xs  l m r x)
+  assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
+    and le_mx: "f m \<le> f x"
+  show ?case (is "?L = ?R")
+  proof(cases xs, simp)
+    case (Cons v vs)
+    show ?thesis
+      apply (simp add:Cons)
+      apply (fold Cons)
+      by (simp add:lsp_xs le_mx)
+  qed
+next
+  case (lsp_cons_2 xs l m r x)
+  assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
+    and lt_xm: "f x < f m"
+  show ?case (is "?L = ?R")
+  proof(cases xs)
+    case (Cons v vs)
+    show ?thesis
+      apply (simp add:Cons)
+      apply (fold Cons)
+      apply (simp add:lsp_xs)
+      by (insert lt_xm, auto)
+  next
+    case Nil
+    from prems show ?thesis by simp
+  qed
+qed auto
+
+lemma lsp_mid_nil: "lsp f xs = (a, [], c) \<Longrightarrow> xs = []"
+  apply (induct xs arbitrary:a c, auto)
+  apply (case_tac xs, auto)
+  by (case_tac "(lsp f (ab # list))", auto split:if_splits list.splits)
+
+
+lemma lsp_mid_length: "lsp f x = (u, v, w) \<Longrightarrow> length v \<le> 1"
+proof(induct x arbitrary:u v w, simp)
+  case (Cons x xs)
+  assume ih: "\<And> u v w. lsp f xs = (u, v, w) \<Longrightarrow> length v \<le> 1"
+  and h: "lsp f (x # xs) = (u, v, w)"
+  show "length v \<le> 1" using h
+  proof(cases xs, simp add:h)
+    case (Cons z zs)
+    assume eq_xs: "xs = z # zs"
+    show ?thesis
+    proof(cases "lsp f xs")
+      fix l m r
+      assume eq_lsp: "lsp f xs = (l, m, r)"
+      show ?thesis
+      proof(cases m)
+        case Nil
+        from Nil and eq_lsp have "lsp f xs = (l, [], r)" by simp
+        from lsp_mid_nil [OF this] have "xs = []" .
+        with h show ?thesis by auto
+      next
+        case (Cons y ys)
+        assume eq_m: "m = y # ys"
+        from ih [OF eq_lsp] have eq_xs_1: "length m \<le> 1" .
+        show ?thesis
+        proof(cases "f x \<ge> f y")
+          case True
+          from eq_xs eq_xs_1 True h eq_lsp show ?thesis 
+            by (auto split:list.splits if_splits)
+        next
+          case False
+          from eq_xs eq_xs_1 False h eq_lsp show ?thesis 
+             by (auto split:list.splits if_splits)
+        qed
+      qed
+    qed
+  next
+    assume "[] = u \<and> [x] = v \<and> [] = w"
+    hence "v = [x]" by simp
+    thus "length v \<le> Suc 0" by simp
+  qed
+qed
+
+lemma lsp_p_lsp_2: "lsp_p f x (lsp f x)"
+proof(induct x, auto)
+  case (Cons x xs)
+  assume ih: "lsp_p f xs (lsp f xs)"
+  show ?case
+  proof(cases xs)
+    case Nil
+    thus ?thesis by auto
+  next
+    case (Cons v vs)
+    show ?thesis
+    proof(cases "xs")
+      case Nil
+      thus ?thesis by auto
+    next
+      case (Cons v vs)
+      assume eq_xs: "xs = v # vs"
+      show ?thesis
+      proof(cases "lsp f xs")
+        fix l m r
+        assume eq_lsp_xs: "lsp f xs = (l, m, r)"
+        show ?thesis
+        proof(cases m)
+          case Nil
+          from eq_lsp_xs and Nil have "lsp f xs = (l, [], r)" by simp
+          from lsp_mid_nil [OF this] have eq_xs: "xs = []" .
+          hence "lsp f (x#xs) = ([], [x], [])" by simp
+          with eq_xs show ?thesis by auto
+        next
+          case (Cons y ys)
+          assume eq_m: "m = y # ys"
+          show ?thesis
+          proof(cases "f x \<ge> f y")
+            case True
+            from eq_xs eq_lsp_xs Cons True
+            have eq_lsp: "lsp f (x#xs) = ([], [x], v # vs)" by simp
+            show ?thesis
+            proof (simp add:eq_lsp)
+              show "lsp_p f (x # xs) ([], [x], v # vs)"
+              proof(fold eq_xs, rule lsp_cons_1 [OF _])
+                from eq_xs show "xs \<noteq> []" by simp
+              next
+                from lsp_mid_length [OF eq_lsp_xs] and Cons
+                have "m = [y]" by simp
+                with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
+                with ih show "lsp_p f xs (l, [y], r)" by simp
+              next
+                from True show "f y \<le> f x" by simp
+              qed
+            qed
+          next
+            case False
+            from eq_xs eq_lsp_xs Cons False
+            have eq_lsp: "lsp f (x#xs) = (x # l, y # ys, r) " by simp
+            show ?thesis
+            proof (simp add:eq_lsp)
+              from lsp_mid_length [OF eq_lsp_xs] and eq_m
+              have "ys = []" by simp
+              moreover have "lsp_p f (x # xs) (x # l, [y], r)"
+              proof(rule lsp_cons_2)
+                from eq_xs show "xs \<noteq> []" by simp
+              next
+                from lsp_mid_length [OF eq_lsp_xs] and Cons
+                have "m = [y]" by simp
+                with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
+                with ih show "lsp_p f xs (l, [y], r)" by simp
+              next
+                from False show "f x < f y" by simp
+              qed
+              ultimately show "lsp_p f (x # xs) (x # l, y # ys, r)" by simp
+            qed
+          qed
+        qed
+      qed
+    qed
+  qed
+qed
+
+lemma lsp_induct:
+  fixes f x1 x2 P
+  assumes h: "lsp f x1 = x2"
+  and p1: "P [] ([], [], [])"
+  and p2: "\<And>x. P [x] ([], [x], [])"
+  and p3: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f m \<le> f x\<rbrakk> \<Longrightarrow> P (x # xs) ([], [x], xs)"
+  and p4: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> P (x # xs) (x # l, [m], r)"
+  shows "P x1 x2"
+proof(rule lsp_p.induct)
+  from lsp_p_lsp_2 and h
+  show "lsp_p f x1 x2" by metis
+next
+  from p1 show "P [] ([], [], [])" by metis
+next
+  from p2 show "\<And>x. P [x] ([], [x], [])" by metis
+next
+  fix xs l m r x 
+  assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f m \<le> f x"
+  show "P (x # xs) ([], [x], xs)" 
+  proof(rule p3 [OF h1 _ h3 h4])
+    from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
+  qed
+next
+  fix xs l m r x 
+  assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f x < f m"
+  show "P (x # xs) (x # l, [m], r)"
+  proof(rule p4 [OF h1 _ h3 h4])
+    from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
+  qed
+qed
+
+lemma lsp_set_eq: 
+  fixes f x u v w
+  assumes h: "lsp f x = (u, v, w)"
+  shows "x = u@v@w"
+proof -
+  have "\<And> f x r. lsp f x = r \<Longrightarrow> \<forall> u v w. (r = (u, v, w) \<longrightarrow> x = u@v@w)" 
+    by (erule lsp_induct, simp+)
+  from this [rule_format, OF h] show ?thesis by simp
+qed
+
+lemma lsp_set: 
+  assumes h: "(u, v, w) = lsp f x"
+  shows "set (u@v@w) = set x"
+proof -
+  from lsp_set_eq [OF h[symmetric]] 
+  show ?thesis by simp
+qed
+
+lemma max_insert_gt:
+  fixes S fx
+  assumes h: "fx < Max S"
+  and np: "S \<noteq> {}"
+  and fn: "finite S" 
+  shows "Max S = Max (insert fx S)"
+proof -
+  from Max_insert [OF fn np]
+  have "Max (insert fx S) = max fx (Max S)" .
+  moreover have "\<dots> = Max S"
+  proof(cases "fx \<le> Max S")
+    case False
+    with h
+    show ?thesis by (simp add:max_def)
+  next
+    case True
+    thus ?thesis by (simp add:max_def)
+  qed
+  ultimately show ?thesis by simp
+qed
+
+lemma max_insert_le: 
+  fixes S fx
+  assumes h: "Max S \<le> fx"
+  and fn: "finite S"
+  shows "fx = Max (insert fx S)"
+proof(cases "S = {}")
+  case True
+  thus ?thesis by simp
+next
+  case False
+  from Max_insert [OF fn False]
+  have "Max (insert fx S) = max fx (Max S)" .
+  moreover have "\<dots> = fx"
+  proof(cases "fx \<le> Max S")
+    case False
+    thus ?thesis by (simp add:max_def)
+  next
+    case True
+    have hh: "\<And> x y. \<lbrakk> x \<le> (y::('a::linorder)); y \<le> x\<rbrakk> \<Longrightarrow> x = y" by auto
+    from hh [OF True h]
+    have "fx = Max S" .
+    thus ?thesis by simp
+  qed
+  ultimately show ?thesis by simp
+qed
+  
+lemma lsp_max: 
+  fixes f x u m w
+  assumes h: "lsp f x = (u, [m], w)"
+  shows "f m = Max (f ` (set x))"
+proof -
+  { fix y
+    have "lsp f x = y \<Longrightarrow> \<forall> u m w. y = (u, [m], w) \<longrightarrow> f m = Max (f ` (set x))"
+    proof(erule lsp_induct, simp)
+      { fix x u m w
+        assume "(([]::'a list), ([x]::'a list), ([]::'a list)) = (u, [m], w)"
+        hence "f m = Max (f ` set [x])"  by simp
+      } thus "\<And>x. \<forall>u m w. ([], [x], []) = (u, [m], w) \<longrightarrow> f m = Max (f ` set [x])" by simp
+    next
+      fix xs l m r x
+      assume h1: "xs \<noteq> []"
+        and h2: " lsp f xs = (l, [m], r)"
+        and h3: "\<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
+        and h4: "f m \<le> f x"
+      show " \<forall>u m w. ([], [x], xs) = (u, [m], w) \<longrightarrow> f m = Max (f ` set (x # xs))"
+      proof -
+        have "f x = Max (f ` set (x # xs))"
+        proof -
+          from h2 h3 have "f m = Max (f ` set xs)" by simp
+          with h4 show ?thesis
+            apply auto
+            by (rule_tac max_insert_le, auto)
+        qed
+        thus ?thesis by simp
+      qed
+    next
+      fix xs l m r x
+      assume h1: "xs \<noteq> []"
+        and h2: " lsp f xs = (l, [m], r)"
+        and h3: " \<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
+        and h4: "f x < f m"
+      show "\<forall>u ma w. (x # l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set (x # xs))"
+      proof -
+        from h2 h3 have "f m = Max (f ` set xs)" by simp
+        with h4
+        have "f m =  Max (f ` set (x # xs))"
+          apply auto
+          apply (rule_tac max_insert_gt, simp+)
+          by (insert h1, simp+)
+        thus ?thesis by auto
+      qed
+    qed
+  } with h show ?thesis by metis
+qed
+
+end
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Attic/Prio.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Attic/Prio.thy	Sun Feb 05 21:00:12 2012 +0000
@@ -0,0 +1,2813 @@
+theory Prio
+imports Precedence_ord Moment Lsp Happen_within
+begin
+
+type_synonym thread = nat
+type_synonym priority = nat
+type_synonym cs = nat
+
+datatype event = 
+  Create thread priority |
+  Exit thread |
+  P thread cs |
+  V thread cs |
+  Set thread priority
+
+datatype node = 
+   Th "thread" |
+   Cs "cs"
+
+type_synonym state = "event list"
+
+fun threads :: "state \<Rightarrow> thread set"
+where 
+  "threads [] = {}" |
+  "threads (Create thread prio#s) = {thread} \<union> threads s" |
+  "threads (Exit thread # s) = (threads s) - {thread}" |
+  "threads (e#s) = threads s"
+
+fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> nat"
+where
+  "original_priority thread [] = 0" |
+  "original_priority thread (Create thread' prio#s) = 
+     (if thread' = thread then prio else original_priority thread s)" |
+  "original_priority thread (Set thread' prio#s) = 
+     (if thread' = thread then prio else original_priority thread s)" |
+  "original_priority thread (e#s) = original_priority thread s"
+
+fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
+where
+  "birthtime thread [] = 0" |
+  "birthtime thread ((Create thread' prio)#s) = (if (thread = thread') then length s 
+                                                  else birthtime thread s)" |
+  "birthtime thread ((Set thread' prio)#s) = (if (thread = thread') then length s 
+                                                  else birthtime thread s)" |
+  "birthtime thread (e#s) = birthtime thread s"
+
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+  where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
+
+consts holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+       waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+       depend :: "'b \<Rightarrow> (node \<times> node) set"
+       dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+defs (overloaded) cs_holding_def: "holding wq thread cs == (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
+                  cs_waiting_def: "waiting wq thread cs == (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
+                  cs_depend_def: "depend (wq::cs \<Rightarrow> thread list) == {(Th t, Cs c) | t c. waiting wq t c} \<union> 
+                                               {(Cs c, Th t) | c t. holding wq t c}"
+                  cs_dependents_def: "dependents (wq::cs \<Rightarrow> thread list) th == {th' . (Th th', Th th) \<in> (depend wq)^+}"
+
+record schedule_state = 
+    waiting_queue :: "cs \<Rightarrow> thread list"
+    cur_preced :: "thread \<Rightarrow> precedence"
+
+
+definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
+where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
+
+fun schs :: "state \<Rightarrow> schedule_state"
+where
+   "schs [] = \<lparr>waiting_queue = \<lambda> cs. [], 
+               cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
+   "schs (e#s) = (let ps = schs s in
+                  let pwq = waiting_queue ps in
+                  let pcp = cur_preced ps in
+                  let nwq = case e of
+                             P thread cs \<Rightarrow>  pwq(cs:=(pwq cs @ [thread])) |
+                             V thread cs \<Rightarrow> let nq = case (pwq cs) of
+                                                      [] \<Rightarrow> [] | 
+                                                      (th#pq) \<Rightarrow> case (lsp pcp pq) of
+                                                                   (l, [], r) \<Rightarrow> []
+                                                                 | (l, m#ms, r) \<Rightarrow> m#(l@ms@r)
+                                            in pwq(cs:=nq)                 |
+                              _ \<Rightarrow> pwq
+                  in let ncp = cpreced (e#s) nwq in 
+                     \<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
+                 )"
+
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
+where "wq s == waiting_queue (schs s)"
+
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+where "cp s = cur_preced (schs s)"
+
+defs (overloaded) s_holding_def: "holding (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
+                  s_waiting_def: "waiting (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
+                  s_depend_def: "depend (s::state) == {(Th t, Cs c) | t c. waiting (wq s) t c} \<union> 
+                                               {(Cs c, Th t) | c t. holding (wq s) t c}"
+                  s_dependents_def: "dependents (s::state) th == {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
+
+definition readys :: "state \<Rightarrow> thread set"
+where
+  "readys s = 
+     {thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
+
+definition runing :: "state \<Rightarrow> thread set"
+where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
+
+definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+  where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
+
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+where
+  thread_create: "\<lbrakk>prio \<le> max_prio; thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
+  thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
+  thread_P: "\<lbrakk>thread \<in> runing s;  (Cs cs, Th thread)  \<notin> (depend s)^+\<rbrakk> \<Longrightarrow> step s (P thread cs)" |
+  thread_V: "\<lbrakk>thread \<in> runing s;  holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
+  thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
+ for cs
+where
+  vt_nil[intro]: "vt cs []" |
+  vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
+
+lemma runing_ready: "runing s \<subseteq> readys s"
+  by (auto simp only:runing_def readys_def)
+
+lemma wq_v_eq_nil: 
+  fixes s cs thread rest
+  assumes eq_wq: "wq s cs = thread # rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [], r)"
+  shows "wq (V thread cs#s) cs = []"
+proof -
+  from prems show ?thesis
+    by (auto simp:wq_def Let_def cp_def split:list.splits)
+qed
+
+lemma wq_v_eq: 
+  fixes s cs thread rest
+  assumes eq_wq: "wq s cs = thread # rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  shows "wq (V thread cs#s) cs = th'#l@r"
+proof -
+  from prems show ?thesis
+    by (auto simp:wq_def Let_def cp_def split:list.splits)
+qed
+
+lemma wq_v_neq:
+   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
+  by (auto simp:wq_def Let_def cp_def split:list.splits)
+
+lemma wq_distinct: "vt step s \<Longrightarrow> distinct (wq s cs)"
+proof(erule_tac vt.induct, simp add:wq_def)
+  fix s e
+  assume h1: "step s e"
+  and h2: "distinct (wq s cs)"
+  thus "distinct (wq (e # s) cs)"
+  proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
+    fix thread s
+    assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+      and h2: "thread \<in> set (waiting_queue (schs s) cs)"
+      and h3: "thread \<in> runing s"
+    show "False" 
+    proof -
+      from h3 have "\<And> cs. thread \<in>  set (waiting_queue (schs s) cs) \<Longrightarrow> 
+                             thread = hd ((waiting_queue (schs s) cs))" 
+        by (simp add:runing_def readys_def s_waiting_def wq_def)
+      from this [OF h2] have "thread = hd (waiting_queue (schs s) cs)" .
+      with h2
+      have "(Cs cs, Th thread) \<in> (depend s)"
+        by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
+      with h1 show False by auto
+    qed
+  next
+    fix thread s a list
+    assume h1: "thread \<in> runing s" 
+      and h2: "holding s thread cs"
+      and h3: "waiting_queue (schs s) cs = a # list"
+      and h4: "a \<notin> set list"
+      and h5: "distinct list"
+    thus "distinct
+           ((\<lambda>(l, a, r). case a of [] \<Rightarrow> [] | m # ms \<Rightarrow> m # l @ ms @ r)
+             (lsp (cur_preced (schs s)) list))"
+    apply (cases "(lsp (cur_preced (schs s)) list)", simp)
+    apply (case_tac b, simp)
+    by (drule_tac lsp_set_eq, simp)
+  qed
+qed
+
+lemma block_pre: 
+  fixes thread cs s
+  assumes s_ni: "thread \<notin>  set (wq s cs)"
+  and s_i: "thread \<in> set (wq (e#s) cs)"
+  shows "e = P thread cs"
+proof -
+  have ee: "\<And> x y. \<lbrakk>x = y\<rbrakk> \<Longrightarrow> set x = set y"
+    by auto
+  from s_ni s_i show ?thesis
+  proof (cases e, auto split:if_splits simp add:Let_def wq_def)
+    fix uu uub uuc uud uue
+    assume h: "(uuc, thread # uu, uub) = lsp (cur_preced (schs s)) uud"
+      and h1 [symmetric]: "uue # uud = waiting_queue (schs s) cs"
+      and h2: "thread \<notin> set (waiting_queue (schs s) cs)"
+    from lsp_set [OF h] have "set (uuc @ (thread # uu) @ uub) = set uud" .
+    hence "thread \<in> set uud" by auto
+    with h1 have "thread \<in> set (waiting_queue (schs s) cs)" by auto
+    with h2 show False by auto
+  next
+    fix uu uua uub uuc uud uue
+    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+      and h2: "uue # uud = waiting_queue (schs s) cs"
+      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+      and h4: "thread \<in> set uuc"
+    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+    with h4 have "thread \<in> set uud" by auto
+    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
+      apply (drule_tac ee) by auto
+    with h1 show "False" by fast
+  next
+    fix uu uua uub uuc uud uue
+    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+      and h2: "uue # uud = waiting_queue (schs s) cs"
+      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+      and h4: "thread \<in> set uu"
+    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+    with h4 have "thread \<in> set uud" by auto
+    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
+      apply (drule_tac ee) by auto
+    with h1 show "False" by fast
+  next
+    fix uu uua uub uuc uud uue
+    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+      and h2: "uue # uud = waiting_queue (schs s) cs"
+      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+      and h4: "thread \<in> set uub"
+    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+    with h4 have "thread \<in> set uud" by auto
+    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
+      apply (drule_tac ee) by auto
+    with h1 show "False" by fast
+  qed
+qed
+
+lemma p_pre: "\<lbrakk>vt step ((P thread cs)#s)\<rbrakk> \<Longrightarrow> 
+  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (depend s)^+"
+apply (ind_cases "vt step ((P thread cs)#s)")
+apply (ind_cases "step s (P thread cs)")
+by auto
+
+lemma abs1:
+  fixes e es
+  assumes ein: "e \<in> set es"
+  and neq: "hd es \<noteq> hd (es @ [x])"
+  shows "False"
+proof -
+  from ein have "es \<noteq> []" by auto
+  then obtain e ess where "es = e # ess" by (cases es, auto)
+  with neq show ?thesis by auto
+qed
+
+lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
+  by (cases es, auto)
+
+inductive_cases evt_cons: "vt cs (a#s)"
+
+lemma abs2:
+  assumes vt: "vt step (e#s)"
+  and inq: "thread \<in> set (wq s cs)"
+  and nh: "thread = hd (wq s cs)"
+  and qt: "thread \<noteq> hd (wq (e#s) cs)"
+  and inq': "thread \<in> set (wq (e#s) cs)"
+  shows "False"
+proof -
+  have ee: "\<And> uuc thread uu uub s list. (uuc, thread # uu, uub) = lsp (cur_preced (schs s)) list \<Longrightarrow> 
+                 lsp (cur_preced (schs s)) list = (uuc, thread # uu, uub) 
+               " by simp
+  from prems show "False"
+    apply (cases e)
+    apply ((simp split:if_splits add:Let_def wq_def)[1])+
+    apply (insert abs1, fast)[1] 
+    apply ((simp split:if_splits add:Let_def)[1])+
+    apply (simp split:if_splits list.splits add:Let_def wq_def) 
+    apply (auto dest!:ee)
+    apply (drule_tac lsp_set_eq, simp)
+    apply (subgoal_tac "distinct (waiting_queue (schs s) cs)", simp, fold wq_def)
+    apply (rule_tac wq_distinct, auto)
+    apply (erule_tac evt_cons, auto)
+    apply (drule_tac lsp_set_eq, simp)
+    apply (subgoal_tac "distinct (wq s cs)", simp)
+    apply (rule_tac wq_distinct, auto)
+    apply (erule_tac evt_cons, auto)
+    apply (drule_tac lsp_set_eq, simp)
+    apply (subgoal_tac "distinct (wq s cs)", simp)
+    apply (rule_tac wq_distinct, auto)
+    apply (erule_tac evt_cons, auto)
+    apply (auto simp:wq_def Let_def split:if_splits prod.splits)
+    done
+qed
+
+lemma vt_moment: "\<And> t. \<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
+proof(induct s, simp)
+  fix a s t
+  assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
+    and vt_a: "vt cs (a # s)"
+    and le_t: "t \<le> length (a # s)"
+  show "vt cs (moment t (a # s))"
+  proof(cases "t = length (a#s)")
+    case True
+    from True have "moment t (a#s) = a#s" by simp
+    with vt_a show ?thesis by simp
+  next
+    case False
+    with le_t have le_t1: "t \<le> length s" by simp
+    from vt_a have "vt cs s"
+      by (erule_tac evt_cons, simp)
+    from h [OF this le_t1] have "vt cs (moment t s)" .
+    moreover have "moment t (a#s) = moment t s"
+    proof -
+      from moment_app [OF le_t1, of "[a]"] 
+      show ?thesis by simp
+    qed
+    ultimately show ?thesis by auto
+  qed
+qed
+
+(* Wrong:
+    lemma \<lbrakk>thread \<in> set (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
+*)
+
+lemma waiting_unique_pre:
+  fixes cs1 cs2 s thread
+  assumes vt: "vt step s"
+  and h11: "thread \<in> set (wq s cs1)"
+  and h12: "thread \<noteq> hd (wq s cs1)"
+  assumes h21: "thread \<in> set (wq s cs2)"
+  and h22: "thread \<noteq> hd (wq s cs2)"
+  and neq12: "cs1 \<noteq> cs2"
+  shows "False"
+proof -
+  let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  from h11 and h12 have q1: "?Q cs1 s" by simp
+  from h21 and h22 have q2: "?Q cs2 s" by simp
+  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+  from p_split [of "?Q cs1", OF q1 nq1]
+  obtain t1 where lt1: "t1 < length s"
+    and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
+        thread \<noteq> hd (wq (moment t1 s) cs1))"
+    and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
+             thread \<noteq> hd (wq (moment i' s) cs1))" by auto
+  from p_split [of "?Q cs2", OF q2 nq2]
+  obtain t2 where lt2: "t2 < length s"
+    and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
+        thread \<noteq> hd (wq (moment t2 s) cs2))"
+    and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
+             thread \<noteq> hd (wq (moment i' s) cs2))" by auto
+  show ?thesis
+  proof -
+    { 
+      assume lt12: "t1 < t2"
+      let ?t3 = "Suc t2"
+      from lt2 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+      have "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+        h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+      have vt_e: "vt step (e#moment t2 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF False h1]
+        have "e = P thread cs2" .
+        with vt_e have "vt step ((P thread cs2)# moment t2 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t2 s)" by auto
+        with nn1 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume lt12: "t2 < t1"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt step (e#moment t1 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF False h1]
+        have "e = P thread cs1" .
+        with vt_e have "vt step ((P thread cs1)# moment t1 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t1 s)" by auto
+        with nn2 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume eqt12: "t1 = t2"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt step (e#moment t1 s)"
+      proof -
+        from vt_moment [OF vt le_t3]
+        have "vt step (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+          by auto
+        from abs2 [OF vt_e True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from block_pre [OF False h1]
+        have eq_e1: "e = P thread cs1" .
+        have lt_t3: "t1 < ?t3" by simp
+        with eqt12 have "t2 < ?t3" by simp
+        from nn2 [rule_format, OF this] and eq_m and eqt12
+        have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+          h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+        show ?thesis
+        proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+          case True
+          from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+            by auto
+          from vt_e and eqt12 have "vt step (e#moment t2 s)" by simp 
+          from abs2 [OF this True eq_th h2 h1]
+          show ?thesis .
+        next
+          case False
+          from block_pre [OF False h1]
+          have "e = P thread cs2" .
+          with eq_e1 neq12 show ?thesis by auto
+        qed
+      qed
+    } ultimately show ?thesis by arith
+  qed
+qed
+
+lemma waiting_unique:
+  assumes "vt step s"
+  and "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+proof -
+  from waiting_unique_pre and prems
+  show ?thesis
+    by (auto simp add:s_waiting_def)
+qed
+
+lemma holded_unique:
+  assumes "vt step s"
+  and "holding s th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+proof -
+  from prems show ?thesis
+    unfolding s_holding_def
+    by auto
+qed
+
+lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits)
+
+lemma birthtime_unique: 
+  "\<lbrakk>birthtime th1 s = birthtime th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
+          \<Longrightarrow> th1 = th2"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits dest:birthtime_lt)
+
+lemma preced_unique : 
+  assumes pcd_eq: "preced th1 s = preced th2 s"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "th1 = th2"
+proof -
+  from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
+  from birthtime_unique [OF this th_in1 th_in2]
+  show ?thesis .
+qed
+
+lemma preced_linorder: 
+  assumes neq_12: "th1 \<noteq> th2"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
+proof -
+  from preced_unique [OF _ th_in1 th_in2] and neq_12 
+  have "preced th1 s \<noteq> preced th2 s" by auto
+  thus ?thesis by auto
+qed
+
+lemma unique_minus:
+  fixes x y z r
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r"
+  and xz: "(x, z) \<in> r^+"
+  and neq: "y \<noteq> z"
+  shows "(y, z) \<in> r^+"
+proof -
+ from xz and neq show ?thesis
+ proof(induct)
+   case (base ya)
+   have "(x, ya) \<in> r" by fact
+   from unique [OF xy this] have "y = ya" .
+   with base show ?case by auto
+ next
+   case (step ya z)
+   show ?case
+   proof(cases "y = ya")
+     case True
+     from step True show ?thesis by simp
+   next
+     case False
+     from step False
+     show ?thesis by auto
+   qed
+ qed
+qed
+
+lemma unique_base:
+  fixes r x y z
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r"
+  and xz: "(x, z) \<in> r^+"
+  and neq_yz: "y \<noteq> z"
+  shows "(y, z) \<in> r^+"
+proof -
+  from xz neq_yz show ?thesis
+  proof(induct)
+    case (base ya)
+    from xy unique base show ?case by auto
+  next
+    case (step ya z)
+    show ?case
+    proof(cases "y = ya")
+      case True
+      from True step show ?thesis by auto
+    next
+      case False
+      from False step 
+      have "(y, ya) \<in> r\<^sup>+" by auto
+      with step show ?thesis by auto
+    qed
+  qed
+qed
+
+lemma unique_chain:
+  fixes r x y z
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r^+"
+  and xz: "(x, z) \<in> r^+"
+  and neq_yz: "y \<noteq> z"
+  shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
+proof -
+  from xy xz neq_yz show ?thesis
+  proof(induct)
+    case (base y)
+    have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
+    from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
+  next
+    case (step y za)
+    show ?case
+    proof(cases "y = z")
+      case True
+      from True step show ?thesis by auto
+    next
+      case False
+      from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
+      thus ?thesis
+      proof
+        assume "(z, y) \<in> r\<^sup>+"
+        with step have "(z, za) \<in> r\<^sup>+" by auto
+        thus ?thesis by auto
+      next
+        assume h: "(y, z) \<in> r\<^sup>+"
+        from step have yza: "(y, za) \<in> r" by simp
+        from step have "za \<noteq> z" by simp
+        from unique_minus [OF _ yza h this] and unique
+        have "(za, z) \<in> r\<^sup>+" by auto
+        thus ?thesis by auto
+      qed
+    qed
+  qed
+qed
+
+lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+definition head_of :: "('a \<Rightarrow> 'b::linorder) \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where "head_of f A = {a . a \<in> A \<and> f a = Max (f ` A)}"
+
+definition wq_head :: "state \<Rightarrow> cs \<Rightarrow> thread set"
+  where "wq_head s cs = head_of (cp s) (set (wq s cs))"
+
+lemma f_nil_simp: "\<lbrakk>f cs = []\<rbrakk> \<Longrightarrow> f(cs:=[]) = f"
+proof
+  fix x
+  assume h:"f cs = []"
+  show "(f(cs := [])) x = f x"
+  proof(cases "cs = x")
+    case True
+    with h show ?thesis by simp
+  next
+    case False
+    with h show ?thesis by simp
+  qed
+qed
+
+lemma step_back_vt: "vt ccs (e#s) \<Longrightarrow> vt ccs s"
+  by(ind_cases "vt ccs (e#s)", simp)
+
+lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e"
+  by(ind_cases "vt ccs (e#s)", simp)
+
+lemma holding_nil:
+    "\<lbrakk>wq s cs = []; holding (wq s) th cs\<rbrakk> \<Longrightarrow> False"
+  by (unfold cs_holding_def, auto)
+
+lemma waiting_kept_1: "
+       \<lbrakk>vt step (V th cs#s); wq s cs = a # list; waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c;
+        lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
+       \<Longrightarrow> waiting (wq s) t c"
+  apply (drule_tac step_back_vt, drule_tac wq_distinct[of _ cs])
+  apply(drule_tac lsp_set_eq)
+  by (unfold cs_waiting_def, auto split:if_splits)
+ 
+lemma waiting_kept_2: 
+  "\<And>a list t c aa ca.
+       \<lbrakk>wq s cs = a # list; waiting ((wq s)(cs := [])) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
+       \<Longrightarrow> waiting (wq s) t c"
+  apply(drule_tac lsp_set_eq)
+  by (unfold cs_waiting_def, auto split:if_splits)
+  
+
+lemma holding_nil_simp: "\<lbrakk>holding ((wq s)(cs := [])) t c\<rbrakk> \<Longrightarrow> holding (wq s) t c"
+  by(unfold cs_holding_def, auto)
+
+lemma step_wq_elim: "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; a \<noteq> th\<rbrakk> \<Longrightarrow> False"
+  apply(drule_tac step_back_step)
+  apply(ind_cases "step s (V th cs)")
+  by(unfold s_holding_def, auto)
+
+lemma holding_cs_neq_simp: "c \<noteq> cs \<Longrightarrow> holding ((wq s)(cs := u)) t c = holding (wq s) t c"
+  by (unfold cs_holding_def, auto)
+
+lemma holding_th_neq_elim:
+  "\<And>a list c t aa ca ab lista.
+       \<lbrakk>\<not> holding (wq s) t c; holding ((wq s)(cs := ab # aa @ lista @ ca)) t c;
+         ab \<noteq> t\<rbrakk>
+       \<Longrightarrow> False"
+  by (unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_nil_abs:
+  "\<not> holding ((wq s)(cs := [])) th cs"
+  by (unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_abs: "\<lbrakk>holding ((wq s)(cs := ab # aa @ lista @ c)) th cs; ab \<noteq> th\<rbrakk>
+       \<Longrightarrow> False"
+    by (unfold cs_holding_def, auto split:if_splits)
+
+lemma waiting_abs: "\<not> waiting ((wq s)(cs := t # l @ r)) t cs"
+    by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_1: 
+  "\<lbrakk>\<not> waiting ((wq s)(cs := [])) t c; waiting (wq s) t c; c \<noteq> cs\<rbrakk>
+       \<Longrightarrow> False"
+    by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_2: "
+       \<lbrakk>\<not> waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c; waiting (wq s) t c;
+        c \<noteq> cs\<rbrakk>
+       \<Longrightarrow> False"
+  by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_3:
+     "\<lbrakk>wq s cs = a # list; \<not> waiting ((wq s)(cs := [])) t c;
+        waiting (wq s) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
+       \<Longrightarrow> False"
+  apply(drule_tac lsp_mid_nil, simp)
+  by(unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_simp: "c \<noteq> cs \<Longrightarrow> waiting ((wq s)(cs:=z)) t c = waiting (wq s) t c"
+   by(unfold cs_waiting_def, auto split:if_splits)
+
+lemma holding_cs_eq:
+  "\<lbrakk>\<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> c = cs"
+   by(unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_cs_eq_1:
+  "\<lbrakk>\<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c\<rbrakk>
+       \<Longrightarrow> c = cs"
+   by(unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_th_eq: 
+       "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; \<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c;
+        lsp (cp s) list = (aa, [], ca)\<rbrakk>
+       \<Longrightarrow> t = th"
+  apply(drule_tac lsp_mid_nil, simp)
+  apply(unfold cs_holding_def, auto split:if_splits)
+  apply(drule_tac step_back_step)
+  apply(ind_cases "step s (V th cs)")
+  by (unfold s_holding_def, auto split:if_splits)
+
+lemma holding_th_eq_1:
+  "\<lbrakk>vt step (V th cs#s); 
+     wq s cs = a # list; \<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c;
+        lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
+       \<Longrightarrow> t = th"
+  apply(drule_tac step_back_step)
+  apply(ind_cases "step s (V th cs)")
+  apply(unfold s_holding_def cs_holding_def)
+  by (auto split:if_splits)
+
+lemma holding_th_eq_2: "\<lbrakk>holding ((wq s)(cs := ac # x)) th cs\<rbrakk>
+       \<Longrightarrow> ac = th"
+  by (unfold cs_holding_def, auto)
+
+lemma holding_th_eq_3: "
+       \<lbrakk>\<not> holding (wq s) t c;
+        holding ((wq s)(cs := ac # x)) t c\<rbrakk>
+       \<Longrightarrow> ac = t"
+  by (unfold cs_holding_def, auto)
+
+lemma holding_wq_eq: "holding ((wq s)(cs := th' # l @ r)) th' cs"
+   by (unfold cs_holding_def, auto)
+
+lemma waiting_th_eq: "
+       \<lbrakk>waiting (wq s) t c; wq s cs = a # list;
+        lsp (cp s) list = (aa, ac # lista, ba); \<not> waiting ((wq s)(cs := ac # aa @ lista @ ba)) t c\<rbrakk>
+       \<Longrightarrow> ac = t"
+  apply(drule_tac lsp_set_eq, simp)
+  by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma step_depend_v:
+  "vt step (V th cs#s) \<Longrightarrow>
+  depend (V th cs # s) =
+  depend s - {(Cs cs, Th th)} -
+  {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+  {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+  apply (unfold s_depend_def wq_def, 
+         auto split:list.splits simp:Let_def f_nil_simp holding_wq_eq, fold wq_def cp_def)
+  apply (auto split:list.splits prod.splits  
+               simp:Let_def f_nil_simp holding_nil_simp holding_cs_neq_simp holding_nil_abs
+                    waiting_abs waiting_simp holding_wq_eq
+               elim:holding_nil waiting_kept_1 waiting_kept_2 step_wq_elim holding_th_neq_elim 
+               holding_abs waiting_abs_1 waiting_abs_3 holding_cs_eq holding_cs_eq_1
+               holding_th_eq holding_th_eq_1 holding_th_eq_2 holding_th_eq_3 waiting_th_eq
+               dest:lsp_mid_length)
+  done
+
+lemma step_depend_p:
+  "vt step (P th cs#s) \<Longrightarrow>
+  depend (P th cs # s) =  (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
+                                             else depend s \<union> {(Th th, Cs cs)})"
+  apply(unfold s_depend_def wq_def)
+  apply (auto split:list.splits prod.splits simp:Let_def cs_waiting_def cs_holding_def)
+  apply(case_tac "c = cs", auto)
+  apply(fold wq_def)
+  apply(drule_tac step_back_step)
+  by (ind_cases " step s (P (hd (wq s cs)) cs)", 
+    auto simp:s_depend_def wq_def cs_holding_def)
+
+lemma simple_A:
+  fixes A
+  assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
+  shows "A = {} \<or> (\<exists> a. A = {a})"
+proof(cases "A = {}")
+  case True thus ?thesis by simp
+next
+  case False then obtain a where "a \<in> A" by auto
+  with h have "A = {a}" by auto
+  thus ?thesis by simp
+qed
+
+lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_depend_def, auto)
+
+lemma acyclic_depend: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "acyclic (depend s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume ih: "acyclic (depend s)"
+      and stp: "step s e"
+      and vt: "vt step s"
+    show ?case
+    proof(cases e)
+      case (Create th prio)
+      with ih
+      show ?thesis by (simp add:depend_create_unchanged)
+    next
+      case (Exit th)
+      with ih show ?thesis by (simp add:depend_exit_unchanged)
+    next
+      case (V th cs)
+      from V vt stp have vtt: "vt step (V th cs#s)" by auto
+      from step_depend_v [OF this]
+      have eq_de: "depend (e # s) = 
+        depend s - {(Cs cs, Th th)} -
+        {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+        {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+      from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+      have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
+      thus ?thesis
+      proof(cases "wq s cs")
+        case Nil
+        hence "?D = {}" by simp
+        with ac and eq_de show ?thesis by simp
+      next
+        case (Cons tth rest)
+        from stp and V have "step s (V th cs)" by simp
+        hence eq_wq: "wq s cs = th # rest"
+        proof -
+          show "step s (V th cs) \<Longrightarrow> wq s cs = th # rest"
+            apply(ind_cases "step s (V th cs)")
+            by(insert Cons, unfold s_holding_def, simp)
+        qed
+        show ?thesis
+        proof(cases "lsp (cp s) rest")
+          fix l b r
+          assume eq_lsp: "lsp (cp s) rest = (l, b, r) "
+          show ?thesis
+          proof(cases "b")
+            case Nil
+            with eq_lsp and eq_wq have "?D = {}" by simp
+            with ac and eq_de show ?thesis by simp
+          next
+            case (Cons th' m)
+            with eq_lsp 
+            have eq_lsp: "lsp (cp s) rest = (l, [th'], r)" 
+              apply simp
+              by (drule_tac lsp_mid_length, simp)
+            from eq_wq and eq_lsp
+            have eq_D: "?D = {(Cs cs, Th th')}" by auto
+            from eq_wq and eq_lsp
+            have eq_C: "?C = {(Th th', Cs cs)}" by auto
+            let ?E = "(?A - ?B - ?C)"
+            have "(Th th', Cs cs) \<notin> ?E\<^sup>*"
+            proof
+              assume "(Th th', Cs cs) \<in> ?E\<^sup>*"
+              hence " (Th th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+              from tranclD [OF this]
+              obtain x where th'_e: "(Th th', x) \<in> ?E" by blast
+              hence th_d: "(Th th', x) \<in> ?A" by simp
+              from depend_target_th [OF this]
+              obtain cs' where eq_x: "x = Cs cs'" by auto
+              with th_d have "(Th th', Cs cs') \<in> ?A" by simp
+              hence wt_th': "waiting s th' cs'"
+                unfolding s_depend_def s_waiting_def cs_waiting_def by simp
+              hence "cs' = cs"
+              proof(rule waiting_unique [OF vt])
+                from eq_wq eq_lsp wq_distinct[OF vt, of cs]
+                show "waiting s th' cs" by(unfold s_waiting_def, auto dest:lsp_set_eq)
+              qed
+              with th'_e eq_x have "(Th th', Cs cs) \<in> ?E" by simp
+              with eq_C show "False" by simp
+            qed
+            with acyclic_insert[symmetric] and ac and eq_D
+            and eq_de show ?thesis by simp
+          qed 
+        qed
+      qed
+    next
+      case (P th cs)
+      from P vt stp have vtt: "vt step (P th cs#s)" by auto
+      from step_depend_p [OF this] P
+      have "depend (e # s) = 
+              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
+                                    depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+        by simp
+      moreover have "acyclic ?R"
+      proof(cases "wq s cs = []")
+        case True
+        hence eq_r: "?R =  depend s \<union> {(Cs cs, Th th)}" by simp
+        have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
+        proof
+          assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
+          hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+          from tranclD2 [OF this]
+          obtain x where "(x, Cs cs) \<in> depend s" by auto
+          with True show False by (auto simp:s_depend_def cs_waiting_def)
+        qed
+        with acyclic_insert ih eq_r show ?thesis by auto
+      next
+        case False
+        hence eq_r: "?R =  depend s \<union> {(Th th, Cs cs)}" by simp
+        have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
+        proof
+          assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
+          hence "(Cs cs, Th th) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+          moreover from step_back_step [OF vtt] have "step s (P th cs)" .
+          ultimately show False
+          proof -
+            show " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+              by (ind_cases "step s (P th cs)", simp)
+          qed
+        qed
+        with acyclic_insert ih eq_r show ?thesis by auto
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (Set thread prio)
+      with ih
+      thm depend_set_unchanged
+      show ?thesis by (simp add:depend_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "acyclic (depend ([]::state))"
+      by (auto simp: s_depend_def cs_waiting_def 
+                      cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+lemma finite_depend: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "finite (depend s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume ih: "finite (depend s)"
+      and stp: "step s e"
+      and vt: "vt step s"
+    show ?case
+    proof(cases e)
+      case (Create th prio)
+      with ih
+      show ?thesis by (simp add:depend_create_unchanged)
+    next
+      case (Exit th)
+      with ih show ?thesis by (simp add:depend_exit_unchanged)
+    next
+      case (V th cs)
+      from V vt stp have vtt: "vt step (V th cs#s)" by auto
+      from step_depend_v [OF this]
+      have eq_de: "depend (e # s) = 
+        depend s - {(Cs cs, Th th)} -
+        {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+        {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+      moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
+      moreover have "finite ?D"
+      proof -
+        have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
+        thus ?thesis
+        proof
+          assume h: "?D = {}"
+          show ?thesis by (unfold h, simp)
+        next
+          assume "\<exists> a. ?D = {a}"
+          thus ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (P th cs)
+      from P vt stp have vtt: "vt step (P th cs#s)" by auto
+      from step_depend_p [OF this] P
+      have "depend (e # s) = 
+              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
+                                    depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+        by simp
+      moreover have "finite ?R"
+      proof(cases "wq s cs = []")
+        case True
+        hence eq_r: "?R =  depend s \<union> {(Cs cs, Th th)}" by simp
+        with True and ih show ?thesis by auto
+      next
+        case False
+        hence "?R = depend s \<union> {(Th th, Cs cs)}" by simp
+        with False and ih show ?thesis by auto
+      qed
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio)
+      with ih
+      show ?thesis by (simp add:depend_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "finite (depend ([]::state))"
+      by (auto simp: s_depend_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+text {* Several useful lemmas *}
+
+thm wf_trancl
+thm finite_acyclic_wf
+thm finite_acyclic_wf_converse
+thm wf_induct
+
+
+lemma wf_dep_converse: 
+  fixes s
+  assumes vt: "vt step s"
+  shows "wf ((depend s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+  from finite_depend [OF vt]
+  show "finite (depend s)" .
+next
+  from acyclic_depend[OF vt]
+  show "acyclic (depend s)" .
+qed
+
+lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
+by (induct l, auto)
+
+lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
+  by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+
+lemma wq_threads: 
+  fixes s cs
+  assumes vt: "vt step s"
+  and h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+proof -
+ from vt and h show ?thesis
+  proof(induct arbitrary: th cs)
+    case (vt_cons s e)
+    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+      and stp: "step s e"
+      and vt: "vt step s"
+      and h: "th \<in> set (wq (e # s) cs)"
+    show ?case
+    proof(cases e)
+      case (Create th' prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    next
+      case (Exit th')
+      with stp ih h show ?thesis
+        apply (auto simp:wq_def Let_def)
+        apply (ind_cases "step s (Exit th')")
+        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
+               s_depend_def s_holding_def cs_holding_def)
+        by (fold wq_def, auto)
+    next
+      case (V th' cs')
+      show ?thesis
+      proof(cases "cs' = cs")
+        case False
+        with h
+        show ?thesis
+          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+          by (drule_tac ih, simp)
+      next
+        case True
+        from h
+        show ?thesis
+        proof(unfold V wq_def)
+          assume th_in: "th \<in> set (waiting_queue (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+          show "th \<in> threads (V th' cs' # s)"
+          proof(cases "cs = cs'")
+            case False
+            hence "?l = waiting_queue (schs s) cs" by (simp add:Let_def)
+            with th_in have " th \<in> set (wq s cs)" 
+              by (fold wq_def, simp)
+            from ih [OF this] show ?thesis by simp
+          next
+            case True
+            show ?thesis
+            proof(cases "waiting_queue (schs s) cs'")
+              case Nil
+              with h V show ?thesis
+                apply (auto simp:wq_def Let_def split:if_splits)
+                by (fold wq_def, drule_tac ih, simp)
+            next
+              case (Cons a rest)
+              assume eq_wq: "waiting_queue (schs s) cs' = a # rest"
+              with h V show ?thesis
+              proof(cases "(lsp (cur_preced (schs s)) rest)", unfold V)
+                fix l m r
+                assume eq_lsp: "lsp (cur_preced (schs s)) rest = (l, m, r)"
+                  and eq_wq: "waiting_queue (schs s) cs' = a # rest"
+                  and th_in_set: "th \<in> set (wq (V th' cs' # s) cs)"
+                show ?thesis
+                proof(cases "m")
+                  case Nil
+                  with eq_lsp have "rest = []" using lsp_mid_nil by auto
+                  with eq_wq have "waiting_queue (schs s) cs' = [a]" by simp
+                  with h[unfolded V wq_def] True 
+                  show ?thesis
+                    by (simp add:Let_def)
+                next
+                  case (Cons b rb)
+                  with lsp_mid_length[OF eq_lsp] have eq_m: "m = [b]" by auto
+                  with eq_lsp have "lsp (cur_preced (schs s)) rest = (l, [b], r)" by simp
+                  with h[unfolded V wq_def] True lsp_set_eq [OF this] eq_wq
+                  show ?thesis
+                    apply (auto simp:Let_def, fold wq_def)
+                    by (rule_tac ih [of _ cs'], auto)+
+                qed
+              qed
+            qed
+          qed
+        qed
+      qed
+    next
+      case (P th' cs')
+      from h stp
+      show ?thesis
+        apply (unfold P wq_def)
+        apply (auto simp:Let_def split:if_splits, fold wq_def)
+        apply (auto intro:ih)
+        apply(ind_cases "step s (P th' cs')")
+        by (unfold runing_def readys_def, auto)
+    next
+      case (Set thread prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    qed
+  next
+    case vt_nil
+    thus ?case by (auto simp:wq_def)
+  qed
+qed
+
+lemma range_in: "\<lbrakk>vt step s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+  apply(unfold s_depend_def cs_waiting_def cs_holding_def)
+  by (auto intro:wq_threads)
+
+lemma readys_v_eq:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and not_in: "th \<notin>  set rest"
+  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
+proof -
+  from prems show ?thesis
+    apply (auto simp:readys_def)
+    apply (case_tac "cs = csa", simp add:s_waiting_def)
+    apply (erule_tac x = csa in allE)
+    apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
+    apply (case_tac "csa = cs", simp)
+    apply (erule_tac x = cs in allE)
+    by (auto simp:s_waiting_def wq_def Let_def split:list.splits prod.splits 
+            dest:lsp_set_eq)
+qed
+
+lemma readys_v_eq_1:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  and neq_th': "th \<noteq> th'"
+  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
+proof -
+  from prems show ?thesis
+    apply (auto simp:readys_def)
+    apply (case_tac "cs = csa", simp add:s_waiting_def)
+    apply (erule_tac x = cs in allE)
+    apply (simp add:s_waiting_def wq_def Let_def split:prod.splits list.splits)
+    apply (drule_tac lsp_mid_nil,simp, clarify, fold cp_def, clarsimp)
+    apply (frule_tac lsp_set_eq, simp)
+    apply (erule_tac x = csa in allE)
+    apply (subst (asm) (2) s_waiting_def, unfold wq_def)
+    apply (auto simp:Let_def split:list.splits prod.splits if_splits 
+            dest:lsp_set_eq)
+    apply (unfold s_waiting_def)
+    apply (fold wq_def, clarsimp)
+    apply (clarsimp)+
+    apply (case_tac "csa = cs", simp)
+    apply (erule_tac x = cs in allE, simp)
+    apply (unfold wq_def)
+    by (auto simp:Let_def split:list.splits prod.splits if_splits 
+            dest:lsp_set_eq)
+qed
+
+lemma readys_v_eq_2:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  and neq_th': "th = th'"
+  and vt: "vt step s"
+  shows "(th \<in> readys (V thread cs#s))"
+proof -
+  from prems show ?thesis
+    apply (auto simp:readys_def)
+    apply (rule_tac wq_threads [of s _ cs], auto dest:lsp_set_eq)
+    apply (unfold s_waiting_def wq_def)
+    apply (auto simp:Let_def split:list.splits prod.splits if_splits 
+            dest:lsp_set_eq lsp_mid_nil lsp_mid_length)
+    apply (fold cp_def, simp+, clarsimp)
+    apply (frule_tac lsp_set_eq, simp)
+    apply (fold wq_def)
+    apply (subgoal_tac "csa = cs", simp)
+    apply (rule_tac waiting_unique [of s th'], simp)
+    by (auto simp:s_waiting_def)
+qed
+
+lemma chain_building:
+  assumes vt: "vt step s"
+  shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
+proof -
+  from wf_dep_converse [OF vt]
+  have h: "wf ((depend s)\<inverse>)" .
+  show ?thesis
+  proof(induct rule:wf_induct [OF h])
+    fix x
+    assume ih [rule_format]: 
+      "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow> 
+           y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
+    show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
+    proof
+      assume x_d: "x \<in> Domain (depend s)"
+      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
+      proof(cases x)
+        case (Th th)
+        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
+        with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
+        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
+        hence "Cs cs \<in> Domain (depend s)" by auto
+        from ih [OF x_in_r this] obtain th'
+          where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto
+        have "(x, Th th') \<in> (depend s)\<^sup>+" using Th x_in cs_in by auto
+        with th'_ready show ?thesis by auto
+      next
+        case (Cs cs)
+        from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (depend s)^-1" by (auto simp:s_depend_def)
+        show ?thesis
+        proof(cases "th' \<in> readys s")
+          case True
+          from True and th'_d show ?thesis by auto
+        next
+          case False
+          from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto
+          with False have "Th th' \<in> Domain (depend s)" 
+            by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
+          from ih [OF th'_d this]
+          obtain th'' where 
+            th''_r: "th'' \<in> readys s" and 
+            th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto
+          from th'_d and th''_in 
+          have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
+          with th''_r show ?thesis by auto
+        qed
+      qed
+    qed
+  qed
+qed
+
+lemma th_chain_to_ready:
+  fixes s th
+  assumes vt: "vt step s"
+  and th_in: "th \<in> threads s"
+  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"
+proof(cases "th \<in> readys s")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  from False and th_in have "Th th \<in> Domain (depend s)" 
+    by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
+  from chain_building [rule_format, OF vt this]
+  show ?thesis by auto
+qed
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+  by  (unfold s_waiting_def cs_waiting_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def cs_holding_def, simp)
+
+lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
+  by (unfold s_holding_def cs_holding_def, auto)
+
+lemma unique_depend: "\<lbrakk>vt step s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
+  apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
+  by(auto elim:waiting_unique holding_unique)
+
+lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
+by (induct rule:trancl_induct, auto)
+
+lemma dchain_unique:
+  assumes vt: "vt step s"
+  and th1_d: "(n, Th th1) \<in> (depend s)^+"
+  and th1_r: "th1 \<in> readys s"
+  and th2_d: "(n, Th th2) \<in> (depend s)^+"
+  and th2_r: "th2 \<in> readys s"
+  shows "th1 = th2"
+proof -
+  { assume neq: "th1 \<noteq> th2"
+    hence "Th th1 \<noteq> Th th2" by simp
+    from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]
+    have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
+    hence "False"
+    proof
+      assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th1, n) \<in> depend s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+      from dd eq_n have "th1 \<notin> readys s"
+        by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
+      with th1_r show ?thesis by auto
+    next
+      assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
+      from trancl_split [OF this]
+      obtain n where dd: "(Th th2, n) \<in> depend s" by auto
+      then obtain cs where eq_n: "n = Cs cs"
+        by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+      from dd eq_n have "th2 \<notin> readys s"
+        by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
+      with th2_r show ?thesis by auto
+    qed
+  } thus ?thesis by auto
+qed
+             
+definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
+where "count Q l = length (filter Q l)"
+
+definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
+
+definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+
+
+lemma step_holdents_p_add:
+  fixes th cs s
+  assumes vt: "vt step (P th cs#s)"
+  and "wq s cs = []"
+  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_p[OF vt] by auto
+qed
+
+lemma step_holdents_p_eq:
+  fixes th cs s
+  assumes vt: "vt step (P th cs#s)"
+  and "wq s cs \<noteq> []"
+  shows "holdents (P th cs#s) th = holdents s th"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_p[OF vt] by auto
+qed
+
+lemma step_holdents_v_minus:
+  fixes th cs s
+  assumes vt: "vt step (V th cs#s)"
+  shows "holdents (V th cs#s) th = holdents s th - {cs}"
+proof -
+  { fix rest l r
+    assume eq_wq: "wq s cs = th # rest" 
+      and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
+    have "False" 
+    proof -
+      from lsp_set_eq [OF eq_lsp] have " rest = l @ [th] @ r" .
+      with eq_wq have "wq s cs = th#\<dots>" by simp
+      with wq_distinct [OF step_back_vt[OF vt], of cs]
+      show ?thesis by auto
+    qed
+  } thus ?thesis unfolding holdents_def step_depend_v[OF vt] by auto
+qed
+
+lemma step_holdents_v_add:
+  fixes th th' cs s rest l r
+  assumes vt: "vt step (V th' cs#s)"
+  and neq_th: "th \<noteq> th'" 
+  and eq_wq: "wq s cs = th' # rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
+  shows "holdents (V th' cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_v[OF vt] by auto
+qed
+
+lemma step_holdents_v_eq:
+  fixes th th' cs s rest l r th''
+  assumes vt: "vt step (V th' cs#s)"
+  and neq_th: "th \<noteq> th'" 
+  and eq_wq: "wq s cs = th' # rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th''], r)"
+  and neq_th': "th \<noteq> th''"
+  shows "holdents (V th' cs#s) th = holdents s th"
+proof -
+  from prems show ?thesis
+  unfolding  holdents_def step_depend_v[OF vt] by auto
+qed
+
+definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntCS s th = card (holdents s th)"
+
+lemma cntCS_v_eq:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and not_in: "th \<notin>  set rest"
+  and vtv: "vt step (V thread cs#s)"
+  shows "cntCS (V thread cs#s) th = cntCS s th"
+proof -
+  from prems show ?thesis
+    apply (unfold cntCS_def holdents_def step_depend_v)
+    apply auto
+    apply (subgoal_tac "\<not>  (\<exists>l r. lsp (cp s) rest = (l, [th], r))", auto)
+    by (drule_tac lsp_set_eq, auto)
+qed
+
+lemma cntCS_v_eq_1:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  and neq_th': "th \<noteq> th'"
+  and vtv: "vt step (V thread cs#s)"
+  shows "cntCS (V thread cs#s) th = cntCS s th"
+proof -
+  from prems show ?thesis
+    apply (unfold cntCS_def holdents_def step_depend_v)
+    by auto
+qed
+
+fun the_cs :: "node \<Rightarrow> cs"
+where "the_cs (Cs cs) = cs"
+
+lemma cntCS_v_eq_2:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+  and neq_th': "th = th'"
+  and vtv: "vt step (V thread cs#s)"
+  shows "cntCS (V thread cs#s) th = 1 + cntCS s th"
+proof -
+  have "card {csa. csa = cs \<or> (Cs csa, Th th') \<in> depend s} = 
+                     Suc (card {cs. (Cs cs, Th th') \<in> depend s})" 
+    (is "card ?A = Suc (card ?B)")
+  proof -
+    have h: "?A = insert cs ?B" by auto
+    moreover have h1: "?B = ?B - {cs}"
+    proof -
+      { assume "(Cs cs, Th th') \<in> depend s"
+        moreover have "(Th th', Cs cs) \<in> depend s"
+        proof -
+          from wq_distinct [OF step_back_vt[OF vtv], of cs]
+          eq_wq lsp_set_eq [OF eq_lsp] show ?thesis
+            apply (auto simp:s_depend_def)
+            by (unfold cs_waiting_def, auto)
+        qed
+        moreover note acyclic_depend [OF step_back_vt[OF vtv]]
+        ultimately have "False"
+          apply (auto simp:acyclic_def)
+          apply (erule_tac x="Cs cs" in allE)
+          apply (subgoal_tac "(Cs cs, Cs cs) \<in> (depend s)\<^sup>+", simp)
+          by (rule_tac trancl_into_trancl [where b = "Th th'"], auto)
+      } thus ?thesis by auto
+    qed
+    moreover have "card (insert cs ?B) = Suc (card (?B - {cs}))"
+    proof(rule card_insert)
+      from finite_depend [OF step_back_vt [OF vtv]]
+      have fnt: "finite (depend s)" .
+      show " finite {cs. (Cs cs, Th th') \<in> depend s}" (is "finite ?B")
+      proof -
+        have "?B \<subseteq>  (\<lambda> (a, b). the_cs a) ` (depend s)"
+          apply (auto simp:image_def)
+          by (rule_tac x = "(Cs x, Th th')" in bexI, auto)
+        with fnt show ?thesis by (auto intro:finite_subset)
+      qed
+    qed
+    ultimately show ?thesis by simp
+  qed
+  with prems show ?thesis
+    apply (unfold cntCS_def holdents_def step_depend_v[OF vtv])
+    by auto
+qed
+
+lemma finite_holding:
+  fixes s th cs
+  assumes vt: "vt step s"
+  shows "finite (holdents s th)"
+proof -
+  let ?F = "\<lambda> (x, y). the_cs x"
+  from finite_depend [OF vt]
+  have "finite (depend s)" .
+  hence "finite (?F `(depend s))" by simp
+  moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>" 
+  proof -
+    { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
+      fix x assume "(Cs x, Th th) \<in> depend s"
+      hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
+      moreover have "?F (Cs x, Th th) = x" by simp
+      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp 
+    } thus ?thesis by auto
+  qed
+  ultimately show ?thesis by (unfold holdents_def, auto intro:finite_subset)
+qed
+
+inductive_cases case_step_v: "step s (V thread cs)"
+
+lemma cntCS_v_dec: 
+  fixes s thread cs
+  assumes vtv: "vt step (V thread cs#s)"
+  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
+proof -
+  have cs_in: "cs \<in> holdents s thread" using step_back_step[OF vtv]
+    apply (erule_tac case_step_v)
+    apply (unfold holdents_def s_depend_def, simp)
+    by (unfold cs_holding_def s_holding_def, auto)
+  moreover have cs_not_in: 
+    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
+    apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
+    by (unfold holdents_def, unfold step_depend_v[OF vtv], 
+            auto dest:lsp_set_eq)
+  ultimately 
+  have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
+    by auto
+  moreover have "card \<dots> = 
+                    Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
+  proof(rule card_insert)
+    from finite_holding [OF vtv]
+    show " finite (holdents (V thread cs # s) thread)" .
+  qed
+  moreover from cs_not_in 
+  have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
+  ultimately show ?thesis by (simp add:cntCS_def)
+qed 
+
+lemma cnp_cnv_cncs:
+  fixes s th
+  assumes vt: "vt step s"
+  shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) 
+                                       then cntCS s th else cntCS s th + 1)"
+proof -
+  from vt show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e)
+    assume vt: "vt step s"
+    and ih: "\<And>th. cntP s th  = cntV s th +
+               (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
+    and stp: "step s e"
+    from stp show ?case
+    proof(cases)
+      case (thread_create prio max_prio thread)
+      assume eq_e: "e = Create thread prio"
+        and not_in: "thread \<notin> threads s"
+      show ?thesis
+      proof -
+        { fix cs 
+          assume "thread \<in> set (wq s cs)"
+          from wq_threads [OF vt this] have "thread \<in> threads s" .
+          with not_in have "False" by simp
+        } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
+          by (auto simp:readys_def threads.simps s_waiting_def 
+            wq_def cs_waiting_def Let_def)
+        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+        have eq_cncs: "cntCS (e#s) th = cntCS s th"
+          unfolding cntCS_def holdents_def
+          by (simp add:depend_create_unchanged eq_e)
+        { assume "th \<noteq> thread"
+          with eq_readys eq_e
+          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+            by (simp add:threads.simps)
+          with eq_cnp eq_cnv eq_cncs ih not_in
+          have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp
+          moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
+          moreover note eq_cnp eq_cnv eq_cncs
+          ultimately have ?thesis by auto
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread" 
+      and is_runing: "thread \<in> runing s"
+      and no_hold: "holdents s thread = {}"
+      from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+      from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+      have eq_cncs: "cntCS (e#s) th = cntCS s th"
+        unfolding cntCS_def holdents_def
+        by (simp add:depend_exit_unchanged eq_e)
+      { assume "th \<noteq> thread"
+        with eq_e
+        have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+          (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+          apply (simp add:threads.simps readys_def)
+          apply (subst s_waiting_def)
+          apply (subst (1 2) wq_def)
+          apply (simp add:Let_def)
+          apply (subst s_waiting_def, simp)
+          by (fold wq_def, simp)
+        with eq_cnp eq_cnv eq_cncs ih
+        have ?thesis by simp
+      } moreover {
+        assume eq_th: "th = thread"
+        with ih is_runing have " cntP s th = cntV s th + cntCS s th" 
+          by (simp add:runing_def)
+        moreover from eq_th eq_e have "th \<notin> threads (e#s)"
+          by simp
+        moreover note eq_cnp eq_cnv eq_cncs
+        ultimately have ?thesis by auto
+      } ultimately show ?thesis by blast
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+        and is_runing: "thread \<in> runing s"
+        and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+      from prems have vtp: "vt step (P thread cs#s)" by auto
+      show ?thesis 
+      proof -
+        { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
+          assume neq_th: "th \<noteq> thread"
+          with eq_e
+          have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
+            apply (simp add:readys_def s_waiting_def wq_def Let_def)
+            apply (rule_tac hh, clarify)
+            apply (intro iffI allI, clarify)
+            apply (erule_tac x = csa in allE, auto)
+            apply (subgoal_tac "waiting_queue (schs s) cs \<noteq> []", auto)
+            apply (erule_tac x = cs in allE, auto)
+            by (case_tac "(waiting_queue (schs s) cs)", auto)
+          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+            apply (simp add:cntCS_def holdents_def)
+            by (unfold  step_depend_p [OF vtp], auto)
+          moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
+            by (simp add:cntP_def count_def)
+          moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
+            by (simp add:cntV_def count_def)
+          moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
+          moreover note ih [of th] 
+          ultimately have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          have ?thesis
+          proof -
+            from eq_e eq_th have eq_cnp: "cntP (e # s) th  = 1 + (cntP s th)" 
+              by (simp add:cntP_def count_def)
+            from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
+              by (simp add:cntV_def count_def)
+            show ?thesis
+            proof (cases "wq s cs = []")
+              case True
+              with is_runing
+              have "th \<in> readys (e#s)"
+                apply (unfold eq_e wq_def, unfold readys_def s_depend_def)
+                apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
+                by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
+              moreover have "cntCS (e # s) th = 1 + cntCS s th"
+              proof -
+                have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> depend s} =
+                  Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")
+                proof -
+                  have "?L = insert cs ?R" by auto
+                  moreover have "card \<dots> = Suc (card (?R - {cs}))" 
+                  proof(rule card_insert)
+                    from finite_holding [OF vt, of thread]
+                    show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
+                      by (unfold holdents_def, simp)
+                  qed
+                  moreover have "?R - {cs} = ?R"
+                  proof -
+                    have "cs \<notin> ?R"
+                    proof
+                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend s}"
+                      with no_dep show False by auto
+                    qed
+                    thus ?thesis by auto
+                  qed
+                  ultimately show ?thesis by auto
+                qed
+                thus ?thesis
+                  apply (unfold eq_e eq_th cntCS_def)
+                  apply (simp add: holdents_def)
+                  by (unfold step_depend_p [OF vtp], auto simp:True)
+              qed
+              moreover from is_runing have "th \<in> readys s"
+                by (simp add:runing_def eq_th)
+              moreover note eq_cnp eq_cnv ih [of th]
+              ultimately show ?thesis by auto
+            next
+              case False
+              have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
+                    by (unfold eq_th eq_e wq_def, auto simp:Let_def)
+              have "th \<notin> readys (e#s)"
+              proof
+                assume "th \<in> readys (e#s)"
+                hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
+                from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
+                hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)" 
+                  by (simp add:s_waiting_def)
+                moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
+                ultimately have "th = hd (wq (e#s) cs)" by blast
+                with eq_wq have "th = hd (wq s cs @ [th])" by simp
+                hence "th = hd (wq s cs)" using False by auto
+                with False eq_wq wq_distinct [OF vtp, of cs]
+                show False by (fold eq_e, auto)
+              qed
+              moreover from is_runing have "th \<in> threads (e#s)" 
+                by (unfold eq_e, auto simp:runing_def readys_def eq_th)
+              moreover have "cntCS (e # s) th = cntCS s th"
+                apply (unfold cntCS_def holdents_def eq_e step_depend_p[OF vtp])
+                by (auto simp:False)
+              moreover note eq_cnp eq_cnv ih[of th]
+              moreover from is_runing have "th \<in> readys s"
+                by (simp add:runing_def eq_th)
+              ultimately show ?thesis by auto
+            qed
+          qed
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_V thread cs)
+      from prems have vtv: "vt step (V thread cs # s)" by auto
+      assume eq_e: "e = V thread cs"
+        and is_runing: "thread \<in> runing s"
+        and hold: "holding s thread cs"
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp:s_holding_def)
+      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+      show ?thesis
+      proof -
+        { assume eq_th: "th = thread"
+          from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
+            by (unfold eq_e, simp add:cntP_def count_def)
+          moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
+            by (unfold eq_e, simp add:cntV_def count_def)
+          moreover from cntCS_v_dec [OF vtv] 
+          have "cntCS (e # s) thread + 1 = cntCS s thread"
+            by (simp add:eq_e)
+          moreover from is_runing have rd_before: "thread \<in> readys s"
+            by (unfold runing_def, simp)
+          moreover have "thread \<in> readys (e # s)"
+          proof -
+            from is_runing
+            have "thread \<in> threads (e#s)" 
+              by (unfold eq_e, auto simp:runing_def readys_def)
+            moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
+            proof
+              fix cs1
+              { assume eq_cs: "cs1 = cs" 
+                have "\<not> waiting (e # s) thread cs1"
+                proof -
+                  have "thread \<notin> set (wq (e#s) cs1)"
+                  proof(cases "lsp (cp s) rest")
+                    fix l m r 
+                    assume h: "lsp (cp s) rest = (l, m, r)"
+                    show ?thesis
+                    proof(cases "m")
+                      case Nil
+                      from wq_v_eq_nil [OF eq_wq] h Nil eq_e
+                      have " wq (e # s) cs = []" by auto
+                      thus ?thesis using eq_cs by auto
+                    next
+                      case (Cons th' l')
+                      from lsp_mid_length [OF h] and Cons h
+                      have eqh: "lsp (cp s) rest = (l, [th'], r)" by auto
+                      from wq_v_eq [OF eq_wq this]
+                      have "wq (V thread cs # s) cs = th' # l @ r" .
+                      moreover from lsp_set_eq [OF eqh]
+                      have "set rest = set \<dots>" by auto
+                      moreover have "thread \<notin> set rest"
+                      proof -
+                        from wq_distinct [OF step_back_vt[OF vtv], of cs]
+                        and eq_wq show ?thesis by auto
+                      qed
+                      moreover note eq_e eq_cs
+                      ultimately show ?thesis by simp
+                    qed
+                  qed
+                  thus ?thesis by (simp add:s_waiting_def)
+                qed
+              } moreover {
+                assume neq_cs: "cs1 \<noteq> cs"
+                  have "\<not> waiting (e # s) thread cs1" 
+                  proof -
+                    from wq_v_neq [OF neq_cs[symmetric]]
+                    have "wq (V thread cs # s) cs1 = wq s cs1" .
+                    moreover have "\<not> waiting s thread cs1" 
+                    proof -
+                      from runing_ready and is_runing
+                      have "thread \<in> readys s" by auto
+                      thus ?thesis by (simp add:readys_def)
+                    qed
+                    ultimately show ?thesis 
+                      by (auto simp:s_waiting_def eq_e)
+                  qed
+              } ultimately show "\<not> waiting (e # s) thread cs1" by blast
+            qed
+            ultimately show ?thesis by (simp add:readys_def)
+          qed
+          moreover note eq_th ih
+          ultimately have ?thesis by auto
+        } moreover {
+          assume neq_th: "th \<noteq> thread"
+          from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" 
+            by (simp add:cntP_def count_def)
+          from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" 
+            by (simp add:cntV_def count_def)
+          have ?thesis
+          proof(cases "th \<in> set rest")
+            case False
+            have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+              by(unfold eq_e, rule readys_v_eq [OF neq_th eq_wq False])
+            moreover have "cntCS (e#s) th = cntCS s th"
+              by(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq False vtv]) 
+            moreover note ih eq_cnp eq_cnv eq_threads
+            ultimately show ?thesis by auto
+          next
+            case True
+            obtain l m r where eq_lsp: "lsp (cp s) rest = (l, m, r)" 
+              by (cases "lsp (cp s) rest", auto)
+            with True have "m \<noteq> []" by  (auto dest:lsp_mid_nil)
+            with eq_lsp obtain th' where eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+              by (case_tac m, auto dest:lsp_mid_length)
+            show ?thesis
+            proof(cases "th = th'")
+              case False
+              have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+                by (unfold eq_e, rule readys_v_eq_1 [OF neq_th eq_wq eq_lsp False])
+              moreover have "cntCS (e#s) th = cntCS s th" 
+                by (unfold eq_e, rule cntCS_v_eq_1[OF neq_th eq_wq eq_lsp False vtv])
+              moreover note ih eq_cnp eq_cnv eq_threads
+              ultimately show ?thesis by auto
+            next
+              case True
+              have "th \<in> readys (e # s)"
+                by (unfold eq_e, rule readys_v_eq_2 [OF neq_th eq_wq eq_lsp True vt])
+              moreover have "cntP s th = cntV s th + cntCS s th + 1"
+              proof -
+                have "th \<notin> readys s" 
+                proof -
+                  from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
+                  show ?thesis
+                    apply (unfold readys_def s_waiting_def, auto)
+                    by (rule_tac x = cs in exI, auto)
+                qed
+                moreover have "th \<in> threads s"
+                proof -
+                  from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
+                  have "th \<in> set (wq s cs)" by simp
+                  from wq_threads [OF step_back_vt[OF vtv] this] 
+                  show ?thesis .
+                qed
+                ultimately show ?thesis using ih by auto
+              qed
+              moreover have "cntCS (e # s) th = 1 + cntCS s th"
+                by (unfold eq_e, rule cntCS_v_eq_2 [OF neq_th eq_wq eq_lsp True vtv])
+              moreover note eq_cnp eq_cnv
+              ultimately show ?thesis by simp
+            qed
+          qed
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_set thread prio)
+      assume eq_e: "e = Set thread prio"
+        and is_runing: "thread \<in> runing s"
+      show ?thesis
+      proof -
+        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+        have eq_cncs: "cntCS (e#s) th = cntCS s th"
+          unfolding cntCS_def holdents_def
+          by (simp add:depend_set_unchanged eq_e)
+        from eq_e have eq_readys: "readys (e#s) = readys s" 
+          by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
+                  auto simp:Let_def)
+        { assume "th \<noteq> thread"
+          with eq_readys eq_e
+          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+            by (simp add:threads.simps)
+          with eq_cnp eq_cnv eq_cncs ih is_runing
+          have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          with is_runing ih have " cntP s th  = cntV s th + cntCS s th" 
+            by (unfold runing_def, auto)
+          moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
+            by (simp add:runing_def)
+          moreover note eq_cnp eq_cnv eq_cncs
+          ultimately have ?thesis by auto
+        } ultimately show ?thesis by blast
+      qed   
+    qed
+  next
+    case vt_nil
+    show ?case 
+      by (unfold cntP_def cntV_def cntCS_def, 
+        auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma not_thread_cncs:
+  fixes th s
+  assumes vt: "vt step s"
+  and not_in: "th \<notin> threads s" 
+  shows "cntCS s th = 0"
+proof -
+  from vt not_in show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e th)
+    assume vt: "vt step s"
+      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
+      and stp: "step s e"
+      and not_in: "th \<notin> threads (e # s)"
+    from stp show ?case
+    proof(cases)
+      case (thread_create prio max_prio thread)
+      assume eq_e: "e = Create thread prio"
+        and not_in': "thread \<notin> threads s"
+      have "cntCS (e # s) th = cntCS s th"
+        apply (unfold eq_e cntCS_def holdents_def)
+        by (simp add:depend_create_unchanged)
+      moreover have "th \<notin> threads s" 
+      proof -
+        from not_in eq_e show ?thesis by simp
+      qed
+      moreover note ih ultimately show ?thesis by auto
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+      and nh: "holdents s thread = {}"
+      have eq_cns: "cntCS (e # s) th = cntCS s th"
+        apply (unfold eq_e cntCS_def holdents_def)
+        by (simp add:depend_exit_unchanged)
+      show ?thesis
+      proof(cases "th = thread")
+        case True
+        have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
+        with eq_cns show ?thesis by simp
+      next
+        case False
+        with not_in and eq_e
+        have "th \<notin> threads s" by simp
+        from ih[OF this] and eq_cns show ?thesis by simp
+      qed
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      and is_runing: "thread \<in> runing s"
+      from prems have vtp: "vt step (P thread cs#s)" by auto
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      hence "cntCS (e # s) th  = cntCS s th "
+        apply (unfold cntCS_def holdents_def eq_e)
+        by (unfold step_depend_p[OF vtp], auto)
+      moreover have "cntCS s th = 0"
+      proof(rule ih)
+        from not_in eq_e show "th \<notin> threads s" by simp
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+        and is_runing: "thread \<in> runing s"
+        and hold: "holding s thread cs"
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      from prems have vtv: "vt step (V thread cs#s)" by auto
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp:s_holding_def)
+      have "cntCS (e # s) th  = cntCS s th"
+      proof(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq _ vtv])
+        show "th \<notin> set rest" 
+        proof
+          assume "th \<in> set rest"
+          with eq_wq have "th \<in> set (wq s cs)" by simp
+          from wq_threads [OF vt this] eq_e not_in 
+          show False by simp
+        qed
+      qed
+      moreover have "cntCS s th = 0"
+      proof(rule ih)
+        from not_in eq_e show "th \<notin> threads s" by simp
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (thread_set thread prio)
+      print_facts
+      assume eq_e: "e = Set thread prio"
+        and is_runing: "thread \<in> runing s"
+      from not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] and eq_e
+      show ?thesis 
+        apply (unfold eq_e cntCS_def holdents_def)
+        by (simp add:depend_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (unfold cntCS_def, 
+        auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
+  by (auto simp:s_waiting_def cs_waiting_def)
+
+lemma dm_depend_threads:
+  fixes th s
+  assumes vt: "vt step s"
+  and in_dom: "(Th th) \<in> Domain (depend s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
+  moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
+  ultimately have "(Th th, Cs cs) \<in> depend s" by simp
+  hence "th \<in> set (wq s cs)"
+    by (unfold s_depend_def, auto simp:cs_waiting_def)
+  from wq_threads [OF vt this] show ?thesis .
+qed
+
+lemma cp_eq_cpreced: "cp s th = cpreced s (wq s) th"
+proof(unfold cp_def wq_def, induct s)
+  case (Cons e s')
+  show ?case
+    by (auto simp:Let_def)
+next
+  case Nil
+  show ?case by (auto simp:Let_def)
+qed
+
+fun the_th :: "node \<Rightarrow> thread"
+  where "the_th (Th th) = th"
+
+lemma runing_unique:
+  fixes th1 th2 s
+  assumes vt: "vt step s"
+  and runing_1: "th1 \<in> runing s"
+  and runing_2: "th2 \<in> runing s"
+  shows "th1 = th2"
+proof -
+  from runing_1 and runing_2 have "cp s th1 = cp s th2"
+    by (unfold runing_def, simp)
+  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
+                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
+    (is "Max (?f ` ?A) = Max (?f ` ?B)")
+    by (unfold cp_eq_cpreced cpreced_def)
+  obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
+  proof -
+    have h1: "finite (?f ` ?A)"
+    proof -
+      have "finite ?A" 
+      proof -
+        have "finite (dependents (wq s) th1)"
+        proof-
+          have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+              apply (auto simp:image_def)
+              by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
+            moreover have "finite \<dots>"
+            proof -
+              from finite_depend[OF vt] have "finite (depend s)" .
+              hence "finite ((depend (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_depend_def cs_depend_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependents_def)
+        qed
+        thus ?thesis by simp
+      qed
+      thus ?thesis by auto
+    qed
+    moreover have h2: "(?f ` ?A) \<noteq> {}"
+    proof -
+      have "?A \<noteq> {}" by simp
+      thus ?thesis by simp
+    qed
+    from Max_in [OF h1 h2]
+    have "Max (?f ` ?A) \<in> (?f ` ?A)" .
+    thus ?thesis by (auto intro:that)
+  qed
+  obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
+  proof -
+    have h1: "finite (?f ` ?B)"
+    proof -
+      have "finite ?B" 
+      proof -
+        have "finite (dependents (wq s) th2)"
+        proof-
+          have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
+          proof -
+            let ?F = "\<lambda> (x, y). the_th x"
+            have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+              apply (auto simp:image_def)
+              by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
+            moreover have "finite \<dots>"
+            proof -
+              from finite_depend[OF vt] have "finite (depend s)" .
+              hence "finite ((depend (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_depend_def cs_depend_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependents_def)
+        qed
+        thus ?thesis by simp
+      qed
+      thus ?thesis by auto
+    qed
+    moreover have h2: "(?f ` ?B) \<noteq> {}"
+    proof -
+      have "?B \<noteq> {}" by simp
+      thus ?thesis by simp
+    qed
+    from Max_in [OF h1 h2]
+    have "Max (?f ` ?B) \<in> (?f ` ?B)" .
+    thus ?thesis by (auto intro:that)
+  qed
+  from eq_f_th1 eq_f_th2 eq_max 
+  have eq_preced: "preced th1' s = preced th2' s" by auto
+  hence eq_th12: "th1' = th2'"
+  proof (rule preced_unique)
+    from th1_in have "th1' = th1 \<or> (th1' \<in> dependents (wq s) th1)" by simp
+    thus "th1' \<in> threads s"
+    proof
+      assume "th1' \<in> dependents (wq s) th1"
+      hence "(Th th1') \<in> Domain ((depend s)^+)"
+        apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+        by (auto simp:Domain_def)
+      hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
+      from dm_depend_threads[OF vt this] show ?thesis .
+    next
+      assume "th1' = th1"
+      with runing_1 show ?thesis
+        by (unfold runing_def readys_def, auto)
+    qed
+  next
+    from th2_in have "th2' = th2 \<or> (th2' \<in> dependents (wq s) th2)" by simp
+    thus "th2' \<in> threads s"
+    proof
+      assume "th2' \<in> dependents (wq s) th2"
+      hence "(Th th2') \<in> Domain ((depend s)^+)"
+        apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+        by (auto simp:Domain_def)
+      hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
+      from dm_depend_threads[OF vt this] show ?thesis .
+    next
+      assume "th2' = th2"
+      with runing_2 show ?thesis
+        by (unfold runing_def readys_def, auto)
+    qed
+  qed
+  from th1_in have "th1' = th1 \<or> th1' \<in> dependents (wq s) th1" by simp
+  thus ?thesis
+  proof
+    assume eq_th': "th1' = th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
+    thus ?thesis
+    proof
+      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+    next
+      assume "th2' \<in> dependents (wq s) th2"
+      with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
+      hence "(Th th1, Th th2) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      hence "Th th1 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
+        by auto
+      hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
+      from depend_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th1, Cs cs') \<in> depend s" by simp
+      with runing_1 have "False"
+        apply (unfold runing_def readys_def s_depend_def)
+        by (auto simp:eq_waiting)
+      thus ?thesis by simp
+    qed
+  next
+    assume th1'_in: "th1' \<in> dependents (wq s) th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
+    thus ?thesis 
+    proof
+      assume "th2' = th2"
+      with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
+      hence "(Th th2, Th th1) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      hence "Th th2 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
+        by auto
+      hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
+      from depend_target_th [OF this]
+      obtain cs' where "n = Cs cs'" by auto
+      with d have "(Th th2, Cs cs') \<in> depend s" by simp
+      with runing_2 have "False"
+        apply (unfold runing_def readys_def s_depend_def)
+        by (auto simp:eq_waiting)
+      thus ?thesis by simp
+    next
+      assume "th2' \<in> dependents (wq s) th2"
+      with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
+      hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
+        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+      show ?thesis
+      proof(rule dchain_unique[OF vt h1 _ h2, symmetric])
+        from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
+        from runing_2 show "th2 \<in> readys s" by (simp add:runing_def) 
+      qed
+    qed
+  qed
+qed
+
+lemma create_pre:
+  assumes stp: "step s e"
+  and not_in: "th \<notin> threads s"
+  and is_in: "th \<in> threads (e#s)"
+  obtains prio where "e = Create th prio"
+proof -
+  from assms  
+  show ?thesis
+  proof(cases)
+    case (thread_create prio max_prio thread)
+    with is_in not_in have "e = Create th prio" by simp
+    from that[OF this] show ?thesis .
+  next
+    case (thread_exit thread)
+    with assms show ?thesis by (auto intro!:that)
+  next
+    case (thread_P thread)
+    with assms show ?thesis by (auto intro!:that)
+  next
+    case (thread_V thread)
+    with assms show ?thesis by (auto intro!:that)
+  next 
+    case (thread_set thread)
+    with assms show ?thesis by (auto intro!:that)
+  qed
+qed
+
+lemma length_down_to_in: 
+  assumes le_ij: "i \<le> j"
+    and le_js: "j \<le> length s"
+  shows "length (down_to j i s) = j - i"
+proof -
+  have "length (down_to j i s) = length (from_to i j (rev s))"
+    by (unfold down_to_def, auto)
+  also have "\<dots> = j - i"
+  proof(rule length_from_to_in[OF le_ij])
+    from le_js show "j \<le> length (rev s)" by simp
+  qed
+  finally show ?thesis .
+qed
+
+
+lemma moment_head: 
+  assumes le_it: "Suc i \<le> length t"
+  obtains e where "moment (Suc i) t = e#moment i t"
+proof -
+  have "i \<le> Suc i" by simp
+  from length_down_to_in [OF this le_it]
+  have "length (down_to (Suc i) i t) = 1" by auto
+  then obtain e where "down_to (Suc i) i t = [e]"
+    apply (cases "(down_to (Suc i) i t)") by auto
+  moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
+    by (rule down_to_conc[symmetric], auto)
+  ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
+    by (auto simp:down_to_moment)
+  from that [OF this] show ?thesis .
+qed
+
+lemma cnp_cnv_eq:
+  fixes th s
+  assumes "vt step s"
+  and "th \<notin> threads s"
+  shows "cntP s th = cntV s th"
+proof -
+  from assms show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
+    have not_in: "th \<notin> threads (e # s)" by fact
+    have "step s e" by fact
+    thus ?case proof(cases)
+      case (thread_create prio max_prio thread)
+      assume eq_e: "e = Create thread prio"
+      hence "thread \<in> threads (e#s)" by simp
+      with not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] show ?thesis using eq_e
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+        and not_holding: "holdents s thread = {}"
+      have vt_s: "vt step s" by fact
+      from finite_holding[OF vt_s] have "finite (holdents s thread)" .
+      with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
+      moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
+      moreover note cnp_cnv_cncs[OF vt_s, of thread]
+      ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
+      show ?thesis
+      proof(cases "th = thread")
+        case True
+        with eq_thread eq_e show ?thesis 
+          by (auto simp:cntP_def cntV_def count_def)
+      next
+        case False
+        with not_in and eq_e have "th \<notin> threads s" by simp
+        from ih[OF this] and eq_e show ?thesis 
+           by (auto simp:cntP_def cntV_def count_def)
+      qed
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      have "thread \<in> runing s" by fact
+      with not_in eq_e have neq_th: "thread \<noteq> th" 
+        by (auto simp:runing_def readys_def)
+      from not_in eq_e have "th \<notin> threads s" by simp
+      from ih[OF this] and neq_th and eq_e show ?thesis
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+      have "thread \<in> runing s" by fact
+      with not_in eq_e have neq_th: "thread \<noteq> th" 
+        by (auto simp:runing_def readys_def)
+      from not_in eq_e have "th \<notin> threads s" by simp
+      from ih[OF this] and neq_th and eq_e show ?thesis
+        by (auto simp:cntP_def cntV_def count_def)
+    next
+      case (thread_set thread prio)
+      assume eq_e: "e = Set thread prio"
+        and "thread \<in> runing s"
+      hence "thread \<in> threads (e#s)" 
+        by (simp add:runing_def readys_def)
+      with not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] show ?thesis using eq_e
+        by (auto simp:cntP_def cntV_def count_def)  
+    qed
+  next
+    case vt_nil
+    show ?case by (auto simp:cntP_def cntV_def count_def)
+  qed
+qed
+
+lemma eq_depend: 
+  "depend (wq s) = depend s"
+by (unfold cs_depend_def s_depend_def, auto)
+
+lemma count_eq_dependents:
+  assumes vt: "vt step s"
+  and eq_pv: "cntP s th = cntV s th"
+  shows "dependents (wq s) th = {}"
+proof -
+  from cnp_cnv_cncs[OF vt] and eq_pv
+  have "cntCS s th = 0" 
+    by (auto split:if_splits)
+  moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
+  proof -
+    from finite_holding[OF vt, of th] show ?thesis
+      by (simp add:holdents_def)
+  qed
+  ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
+    by (unfold cntCS_def holdents_def cs_dependents_def, auto)
+  show ?thesis
+  proof(unfold cs_dependents_def)
+    { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
+      then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
+      hence "False"
+      proof(cases)
+        assume "(Th th', Th th) \<in> depend (wq s)"
+        thus "False" by (auto simp:cs_depend_def)
+      next
+        fix c
+        assume "(c, Th th) \<in> depend (wq s)"
+        with h and eq_depend show "False"
+          by (cases c, auto simp:cs_depend_def)
+      qed
+    } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
+  qed
+qed
+
+lemma dependents_threads:
+  fixes s th
+  assumes vt: "vt step s"
+  shows "dependents (wq s) th \<subseteq> threads s"
+proof
+  { fix th th'
+    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
+    have "Th th \<in> Domain (depend s)"
+    proof -
+      from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
+      hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
+      with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
+      thus ?thesis using eq_depend by simp
+    qed
+    from dm_depend_threads[OF vt this]
+    have "th \<in> threads s" .
+  } note hh = this
+  fix th1 
+  assume "th1 \<in> dependents (wq s) th"
+  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
+    by (unfold cs_dependents_def, simp)
+  from hh [OF this] show "th1 \<in> threads s" .
+qed
+
+lemma finite_threads:
+  assumes vt: "vt step s"
+  shows "finite (threads s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    assume vt: "vt step s"
+    and step: "step s e"
+    and ih: "finite (threads s)"
+    from step
+    show ?case
+    proof(cases)
+      case (thread_create prio max_prio thread)
+      assume eq_e: "e = Create thread prio"
+      with ih
+      show ?thesis by (unfold eq_e, auto)
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+      with ih show ?thesis 
+        by (unfold eq_e, auto)
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      with ih show ?thesis by (unfold eq_e, auto)
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+      with ih show ?thesis by (unfold eq_e, auto)
+    next 
+      case (thread_set thread prio)
+      from vt_cons thread_set show ?thesis by simp
+    qed
+  next
+    case vt_nil
+    show ?case by (auto)
+  qed
+qed
+
+lemma Max_f_mono:
+  assumes seq: "A \<subseteq> B"
+  and np: "A \<noteq> {}"
+  and fnt: "finite B"
+  shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+  from seq show "f ` A \<subseteq> f ` B" by auto
+next
+  from np show "f ` A \<noteq> {}" by auto
+next
+  from fnt and seq show "finite (f ` B)" by auto
+qed
+
+lemma cp_le:
+  assumes vt: "vt step s"
+  and th_in: "th \<in> threads s"
+  shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def)
+  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))
+         \<le> Max ((\<lambda>th. preced th s) ` threads s)"
+    (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
+  proof(rule Max_f_mono)
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
+  next
+    from finite_threads [OF vt]
+    show "finite (threads s)" .
+  next
+    from th_in
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
+      apply (auto simp:Domain_def)
+      apply (rule_tac dm_depend_threads[OF vt])
+      apply (unfold trancl_domain [of "depend s", symmetric])
+      by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
+  qed
+qed
+
+lemma le_cp:
+  assumes vt: "vt step s"
+  shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+  show "Prc (original_priority th s) (birthtime th s)
+    \<le> Max (insert (Prc (original_priority th s) (birthtime th s))
+            ((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))"
+    (is "?l \<le> Max (insert ?l ?A)")
+  proof(cases "?A = {}")
+    case False
+    have "finite ?A" (is "finite (?f ` ?B)")
+    proof -
+      have "finite ?B" 
+      proof-
+        have "finite {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}"
+        proof -
+          let ?F = "\<lambda> (x, y). the_th x"
+          have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+            apply (auto simp:image_def)
+            by (rule_tac x = "(Th x, Th th)" in bexI, auto)
+          moreover have "finite \<dots>"
+          proof -
+            from finite_depend[OF vt] have "finite (depend s)" .
+            hence "finite ((depend (wq s))\<^sup>+)"
+              apply (unfold finite_trancl)
+              by (auto simp: s_depend_def cs_depend_def wq_def)
+            thus ?thesis by auto
+          qed
+          ultimately show ?thesis by (auto intro:finite_subset)
+        qed
+        thus ?thesis by (simp add:cs_dependents_def)
+      qed
+      thus ?thesis by simp
+    qed
+    from Max_insert [OF this False, of ?l] show ?thesis by auto
+  next
+    case True
+    thus ?thesis by auto
+  qed
+qed
+
+lemma max_cp_eq: 
+  assumes vt: "vt step s"
+  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
+  (is "?l = ?r")
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  have "?l \<in> ((cp s) ` threads s)"
+  proof(rule Max_in)
+    from finite_threads[OF vt] 
+    show "finite (cp s ` threads s)" by auto
+  next
+    from False show "cp s ` threads s \<noteq> {}" by auto
+  qed
+  then obtain th 
+    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
+  have "\<dots> \<le> ?r" by (rule cp_le[OF vt th_in])
+  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
+  proof -
+    have "?r \<in> (?f ` ?A)"
+    proof(rule Max_in)
+      from finite_threads[OF vt]
+      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
+    next
+      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
+    qed
+    then obtain th' where 
+      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
+    from le_cp [OF vt, of th']  eq_r
+    have "?r \<le> cp s th'" by auto
+    moreover have "\<dots> \<le> cp s th"
+    proof(fold eq_l)
+      show " cp s th' \<le> Max (cp s ` threads s)"
+      proof(rule Max_ge)
+        from th_in' show "cp s th' \<in> cp s ` threads s"
+          by auto
+      next
+        from finite_threads[OF vt]
+        show "finite (cp s ` threads s)" by auto
+      qed
+    qed
+    ultimately show ?thesis by auto
+  qed
+  ultimately show ?thesis using eq_l by auto
+qed
+
+lemma max_cp_readys_threads_pre:
+  assumes vt: "vt step s"
+  and np: "threads s \<noteq> {}"
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq[OF vt])
+  show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
+  proof -
+    let ?p = "Max ((\<lambda>th. preced th s) ` threads s)" 
+    let ?f = "(\<lambda>th. preced th s)"
+    have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
+    proof(rule Max_in)
+      from finite_threads[OF vt] show "finite (?f ` threads s)" by simp
+    next
+      from np show "?f ` threads s \<noteq> {}" by simp
+    qed
+    then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
+      by (auto simp:Image_def)
+    from th_chain_to_ready [OF vt tm_in]
+    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
+    thus ?thesis
+    proof
+      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
+      then obtain th' where th'_in: "th' \<in> readys s" 
+        and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
+      have "cp s th' = ?f tm"
+      proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
+        from dependents_threads[OF vt] finite_threads[OF vt]
+        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))" 
+          by (auto intro:finite_subset)
+      next
+        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
+        from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
+        moreover have "p \<le> \<dots>"
+        proof(rule Max_ge)
+          from finite_threads[OF vt]
+          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        next
+          from p_in and th'_in and dependents_threads[OF vt, of th']
+          show "p \<in> (\<lambda>th. preced th s) ` threads s"
+            by (auto simp:readys_def)
+        qed
+        ultimately show "p \<le> preced tm s" by auto
+      next
+        show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
+        proof -
+          from tm_chain
+          have "tm \<in> dependents (wq s) th'"
+            by (unfold cs_dependents_def s_depend_def cs_depend_def, auto)
+          thus ?thesis by auto
+        qed
+      qed
+      with tm_max
+      have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+      show ?thesis
+      proof (fold h, rule Max_eqI)
+        fix q 
+        assume "q \<in> cp s ` readys s"
+        then obtain th1 where th1_in: "th1 \<in> readys s"
+          and eq_q: "q = cp s th1" by auto
+        show "q \<le> cp s th'"
+          apply (unfold h eq_q)
+          apply (unfold cp_eq_cpreced cpreced_def)
+          apply (rule Max_mono)
+        proof -
+          from dependents_threads [OF vt, of th1] th1_in
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq> 
+                 (\<lambda>th. preced th s) ` threads s"
+            by (auto simp:readys_def)
+        next
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp
+        next
+          from finite_threads[OF vt] 
+          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        qed
+      next
+        from finite_threads[OF vt]
+        show "finite (cp s ` readys s)" by (auto simp:readys_def)
+      next
+        from th'_in
+        show "cp s th' \<in> cp s ` readys s" by simp
+      qed
+    next
+      assume tm_ready: "tm \<in> readys s"
+      show ?thesis
+      proof(fold tm_max)
+        have cp_eq_p: "cp s tm = preced tm s"
+        proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+          fix y 
+          assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
+          show "y \<le> preced tm s"
+          proof -
+            { fix y'
+              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
+              have "y' \<le> preced tm s"
+              proof(unfold tm_max, rule Max_ge)
+                from hy' dependents_threads[OF vt, of tm]
+                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+              next
+                from finite_threads[OF vt] 
+                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+              qed
+            } with hy show ?thesis by auto
+          qed
+        next
+          from dependents_threads[OF vt, of tm] finite_threads[OF vt]
+          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
+            by (auto intro:finite_subset)
+        next
+          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
+            by simp
+        qed 
+        moreover have "Max (cp s ` readys s) = cp s tm"
+        proof(rule Max_eqI)
+          from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
+        next
+          from finite_threads[OF vt]
+          show "finite (cp s ` readys s)" by (auto simp:readys_def)
+        next
+          fix y assume "y \<in> cp s ` readys s"
+          then obtain th1 where th1_readys: "th1 \<in> readys s"
+            and h: "y = cp s th1" by auto
+          show "y \<le> cp s tm"
+            apply(unfold cp_eq_p h)
+            apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
+          proof -
+            from finite_threads[OF vt]
+            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+          next
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
+              by simp
+          next
+            from dependents_threads[OF vt, of th1] th1_readys
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) 
+                    \<subseteq> (\<lambda>th. preced th s) ` threads s"
+              by (auto simp:readys_def)
+          qed
+        qed
+        ultimately show " Max (cp s ` readys s) = preced tm s" by simp
+      qed 
+    qed
+  qed
+qed
+
+lemma max_cp_readys_threads:
+  assumes vt: "vt step s"
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis 
+    by (auto simp:readys_def)
+next
+  case False
+  show ?thesis by (rule max_cp_readys_threads_pre[OF vt False])
+qed
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+proof
+  fix th
+  assume "th \<in> readys s"
+  thus "th \<in> threads s"
+    by (unfold readys_def, auto)
+qed
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+  apply (unfold s_holding_def cs_holding_def, simp)
+  done
+
+lemma f_image_eq:
+  assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
+  shows "f ` A = g ` A"
+proof
+  show "f ` A \<subseteq> g ` A"
+    by(rule image_subsetI, auto intro:h)
+next
+  show "g ` A \<subseteq> f ` A"
+   by(rule image_subsetI, auto intro:h[symmetric])
+qed
+
+end
\ No newline at end of file
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Ext.thy
--- a/prio/Ext.thy	Sun Feb 05 14:29:08 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1057 +0,0 @@
-theory Ext
-imports Prio
-begin
-
-locale highest_create =
-  fixes s' th prio fixes s 
-  defines s_def : "s \<equiv> (Create th prio#s')"
-  assumes vt_s: "vt step s"
-  and highest: "cp s th = Max ((cp s)`threads s)"
-
-context highest_create
-begin
-
-lemma threads_s: "threads s = threads s' \<union> {th}"
-  by (unfold s_def, simp)
-
-lemma vt_s': "vt step s'"
-  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
-
-lemma step_create: "step s' (Create th prio)"
-  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
-
-lemma step_create_elim: 
-  "\<lbrakk>\<And>max_prio. \<lbrakk>prio \<le> max_prio; th \<notin> threads s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
-  by (insert step_create, ind_cases "step s' (Create th prio)", auto)
-
-lemma eq_cp_s: 
-  assumes th'_in: "th' \<in> threads s'"
-  shows "cp s th' = cp s' th'"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def s_def 
-    eq_depend depend_create_unchanged)
-  show "Max ((\<lambda>tha. preced tha (Create th prio # s')) `
-         ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+})) =
-        Max ((\<lambda>th. preced th s') ` ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+}))"
-    (is "Max (?f ` ?A) = Max (?g ` ?A)")
-  proof -
-    have "?f ` ?A = ?g ` ?A"
-    proof(rule f_image_eq)
-      fix a
-      assume a_in: "a \<in> ?A"
-      thus "?f a = ?g a" 
-      proof -
-        from a_in
-        have "a = th' \<or> (Th a, Th th') \<in> (depend s')\<^sup>+" by auto 
-        hence "a \<noteq> th"
-        proof
-          assume "a = th'"
-          moreover have "th' \<noteq> th"
-          proof(rule step_create_elim)
-            assume th_not_in: "th \<notin> threads s'" with th'_in
-            show ?thesis by auto
-          qed
-          ultimately show ?thesis by auto
-        next
-          assume "(Th a, Th th') \<in> (depend s')\<^sup>+"
-          hence "Th a \<in> Domain \<dots>"
-            by (auto simp:Domain_def)
-          hence "Th a \<in> Domain (depend s')"
-            by (simp add:trancl_domain)
-          from dm_depend_threads[OF vt_s' this]
-          have h: "a \<in> threads s'" .
-          show ?thesis
-          proof(rule step_create_elim)
-            assume "th \<notin> threads s'" with h
-            show ?thesis by auto
-          qed
-        qed
-        thus ?thesis 
-          by (unfold preced_def, auto)
-      qed
-    qed
-    thus ?thesis by auto
-  qed
-qed
-
-lemma same_depend: "depend s = depend s'"
-  by (insert depend_create_unchanged, unfold s_def, simp)
-
-lemma same_dependents:
-  "dependents (wq s) th = dependents (wq s') th"
-  apply (unfold cs_dependents_def)
-  by (unfold eq_depend same_depend, simp)
-
-lemma nil_dependents_s': "dependents (wq s') th = {}"
-proof -
-  { assume ne: "dependents (wq s') th \<noteq> {}"
-    then obtain th' where "th' \<in>  dependents (wq s') th"
-      by (unfold cs_dependents_def, auto)
-    hence "(Th th', Th th) \<in> (depend (wq s'))^+"
-      by (unfold cs_dependents_def, auto)
-    hence "(Th th', Th th) \<in> (depend s')^+"
-      by (simp add:eq_depend)
-    hence "Th th \<in> Range ((depend s')^+)" by (auto simp:Range_def Domain_def)
-    hence "Th th \<in> Range (depend s')" by (simp add:trancl_range)
-    from range_in [OF vt_s' this]
-    have h: "th \<in> threads s'" .
-    have "False"
-    proof(rule step_create_elim)
-      assume "th \<notin> threads s'" with h show ?thesis by auto
-    qed
-  } thus ?thesis by auto
-qed
-
-lemma nil_dependents: "dependents (wq s) th = {}"
-proof -
-  have "wq s' = wq s"
-    by (unfold wq_def s_def, auto simp:Let_def)
-  with nil_dependents_s' show ?thesis by auto
-qed
-
-lemma eq_cp_s_th: "cp s th = preced th s"
-  by (unfold cp_eq_cpreced cpreced_def nil_dependents, auto)
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
-  by (fold max_cp_eq[OF vt_s], unfold highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
-  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma is_ready: "th \<in> readys s"
-proof -
-  { assume "th \<notin> readys s"
-    with threads_s obtain cs where 
-      "waiting s th cs"
-      by (unfold readys_def, auto)
-    hence "(Th th, Cs cs) \<in> depend s"
-      by (unfold s_depend_def, unfold eq_waiting, simp)
-    hence "Th th \<in> Domain (depend s')"
-      by (unfold same_depend, auto simp:Domain_def)
-    from dm_depend_threads [OF vt_s' this] 
-    have h: "th \<in> threads s'" .
-    have "False"
-    proof (rule_tac step_create_elim)
-      assume "th \<notin> threads s'" with h show ?thesis by simp
-    qed
-  } thus ?thesis by auto
-qed
-
-lemma is_runing: "th \<in> runing s"
-proof -
-  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
-  proof -
-    have " Max (cp s ` readys s) = cp s th"
-    proof(rule Max_eqI)
-      from finite_threads[OF vt_s] readys_threads finite_subset
-      have "finite (readys s)" by blast
-      thus "finite (cp s ` readys s)" by auto
-    next
-      from is_ready show "cp s th \<in> cp s ` readys s" by auto
-    next
-      fix y
-      assume h: "y \<in> cp s ` readys s"
-      have "y \<le> Max (cp s ` readys s)"
-      proof(rule Max_ge [OF _ h])
-        from finite_threads[OF vt_s] readys_threads finite_subset
-        have "finite (readys s)" by blast
-        thus "finite (cp s ` readys s)" by auto
-      qed
-      moreover have "\<dots> \<le> Max (cp s ` threads s)"
-      proof(rule Max_mono)
-        from readys_threads 
-        show "cp s ` readys s \<subseteq> cp s ` threads s" by auto
-      next
-        from is_ready show "cp s ` readys s \<noteq> {}" by auto
-      next
-        from finite_threads [OF vt_s]
-        show "finite (cp s ` threads s)" by auto
-      qed
-      moreover note highest
-      ultimately show "y \<le> cp s th" by auto
-    qed
-    with highest show ?thesis by auto
-  qed
-  thus ?thesis
-    by (unfold runing_def, insert highest is_ready, auto)
-qed
-
-end
-
-locale extend_highest = highest_create + 
-  fixes t 
-  assumes vt_t: "vt step (t@s)"
-  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
-  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
-  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app: 
-  assumes vt_ts: "vt cs (t@s)" 
-  shows "vt cs s"
-proof -
-  from vt_ts show ?thesis
-  proof(induct t)
-    case Nil
-    from Nil show ?case by auto
-  next
-    case (Cons e t)
-    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
-      and vt_et: "vt cs ((e # t) @ s)"
-    show ?case
-    proof(rule ih)
-      show "vt cs (t @ s)"
-      proof(rule step_back_vt)
-        from vt_et show "vt cs (e # t @ s)" by simp
-      qed
-    qed
-  qed
-qed
-
-context extend_highest
-begin
-
-lemma red_moment:
-  "extend_highest s' th prio (moment i t)"
-  apply (insert extend_highest_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
-  apply (unfold extend_highest_def extend_highest_axioms_def, clarsimp)
-  by (unfold highest_create_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
-  assumes 
-    h0: "R []"
-  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
-                    extend_highest s' th prio t; 
-                    extend_highest s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
-  shows "R t"
-proof -
-  from vt_t extend_highest_axioms show ?thesis
-  proof(induct t)
-    from h0 show "R []" .
-  next
-    case (Cons e t')
-    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
-      and vt_e: "vt step ((e # t') @ s)"
-      and et: "extend_highest s' th prio (e # t')"
-    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
-    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
-    show ?case
-    proof(rule h2 [OF vt_ts stp _ _ _ ])
-      show "R t'"
-      proof(rule ih)
-        from et show ext': "extend_highest s' th prio t'"
-          by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
-      next
-        from vt_ts show "vt step (t' @ s)" .
-      qed
-    next
-      from et show "extend_highest s' th prio (e # t')" .
-    next
-      from et show ext': "extend_highest s' th prio t'"
-          by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
-    qed
-  qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and> 
-        preced th (t@s) = preced th s" (is "?Q t")
-proof -
-  show ?thesis
-  proof(induct rule:ind)
-    case Nil
-    from threads_s
-    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
-      by auto
-  next
-    case (Cons e t)
-    show ?case
-    proof(cases e)
-      case (Create thread prio)
-      assume eq_e: " e = Create thread prio"
-      show ?thesis
-      proof -
-        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
-        hence "th \<noteq> thread"
-        proof(cases)
-          assume "thread \<notin> threads (t @ s)"
-          with Cons show ?thesis by auto
-        qed
-        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
-          by (unfold eq_e, auto simp:preced_def)
-        moreover note Cons
-        ultimately show ?thesis
-          by (auto simp:eq_e)
-      qed
-    next
-      case (Exit thread)
-      assume eq_e: "e = Exit thread"
-      from Cons have "extend_highest s' th prio (e # t)" by auto
-      from extend_highest.exit_diff [OF this] and eq_e
-      have neq_th: "thread \<noteq> th" by auto
-      with Cons
-      show ?thesis
-        by (unfold eq_e, auto simp:preced_def)
-    next
-      case (P thread cs)
-      assume eq_e: "e = P thread cs"
-      with Cons
-      show ?thesis 
-        by (auto simp:eq_e preced_def)
-    next
-      case (V thread cs)
-      assume eq_e: "e = V thread cs"
-      with Cons
-      show ?thesis 
-        by (auto simp:eq_e preced_def)
-    next
-      case (Set thread prio')
-      assume eq_e: " e = Set thread prio'"
-      show ?thesis
-      proof -
-        from Cons have "extend_highest s' th prio (e # t)" by auto
-        from extend_highest.set_diff_low[OF this] and eq_e
-        have "th \<noteq> thread" by auto
-        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
-          by (unfold eq_e, auto simp:preced_def)
-        moreover note Cons
-        ultimately show ?thesis
-          by (auto simp:eq_e)
-      qed
-    qed
-  qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
-  case Nil
-  from highest_preced_thread
-  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
-    by simp
-next
-  case (Cons e t)
-  show ?case
-  proof(cases e)
-    case (Create thread prio')
-    assume eq_e: " e = Create thread prio'"
-    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
-    hence neq_thread: "thread \<noteq> th"
-    proof(cases)
-      assume "thread \<notin> threads (t @ s)"
-      moreover have "th \<in> threads (t@s)"
-      proof -
-        from Cons have "extend_highest s' th prio t" by auto
-        from extend_highest.th_kept[OF this] show ?thesis by (simp add:s_def)
-      qed
-      ultimately show ?thesis by auto
-    qed
-    from Cons have "extend_highest s' th prio t" by auto
-    from extend_highest.th_kept[OF this]
-    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
-      by (auto simp:s_def)
-    from stp
-    have thread_ts: "thread \<notin> threads (t @ s)"
-      by (cases, auto)
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
-        by (unfold eq_e, simp)
-      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
-      proof(rule Max_insert)
-        from Cons have "vt step (t @ s)" by auto
-        from finite_threads[OF this]
-        show "finite (?f ` (threads (t@s)))" by simp
-      next
-        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
-      qed
-      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
-      proof -
-        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
-          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
-        proof -
-          { fix th' 
-            assume "th' \<in> ?B"
-            with thread_ts eq_e
-            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
-          } thus ?thesis 
-            apply (auto simp:Image_def)
-          proof -
-            fix th'
-            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
-              preced th' (e # t @ s) = preced th' (t @ s)"
-              and h1: "th' \<in> threads (t @ s)"
-            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
-            proof -
-              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
-              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
-              ultimately show ?thesis by simp
-            qed
-          qed
-        qed
-        with Cons show ?thesis by auto
-      qed
-      moreover have "?f thread < ?t"
-      proof -
-        from Cons have " extend_highest s' th prio (e # t)" by auto
-        from extend_highest.create_low[OF this] and eq_e
-        have "prio' \<le> prio" by auto
-        thus ?thesis
-        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
-    qed
-    ultimately show ?thesis by (auto simp:max_def)
-  qed
-next
-    case (Exit thread)
-    assume eq_e: "e = Exit thread"
-    from Cons have vt_e: "vt step (e#(t @ s))" by auto
-    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
-    from stp have thread_ts: "thread \<in> threads (t @ s)"
-      by(cases, unfold runing_def readys_def, auto)
-    from Cons have "extend_highest s' th prio (e # t)" by auto
-    from extend_highest.exit_diff[OF this] and eq_e
-    have neq_thread: "thread \<noteq> th" by auto
-    from Cons have "extend_highest s' th prio t" by auto
-    from extend_highest.th_kept[OF this, folded s_def]
-    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      have "threads (t@s) = insert thread ?A"
-        by (insert stp thread_ts, unfold eq_e, auto)
-      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
-      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
-      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
-      proof(rule Max_insert)
-        from finite_threads [OF vt_e]
-        show "finite (?f ` ?A)" by simp
-      next
-        from Cons have "extend_highest s' th prio (e # t)" by auto
-        from extend_highest.th_kept[OF this]
-        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
-      qed
-      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
-      moreover have "Max (?f ` (threads (t@s))) = ?t"
-      proof -
-        from Cons show ?thesis
-          by (unfold eq_e, auto simp:preced_def)
-      qed
-      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
-      moreover have "?f thread < ?t"
-      proof(unfold eq_e, simp add:preced_def, fold preced_def)
-        show "preced thread (t @ s) < ?t"
-        proof -
-          have "preced thread (t @ s) \<le> ?t" 
-          proof -
-            from Cons
-            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
-              (is "?t = Max (?g ` ?B)") by simp
-            moreover have "?g thread \<le> \<dots>"
-            proof(rule Max_ge)
-              have "vt step (t@s)" by fact
-              from finite_threads [OF this]
-              show "finite (?g ` ?B)" by simp
-            next
-              from thread_ts
-              show "?g thread \<in> (?g ` ?B)" by auto
-            qed
-            ultimately show ?thesis by auto
-          qed
-          moreover have "preced thread (t @ s) \<noteq> ?t"
-          proof
-            assume "preced thread (t @ s) = preced th s"
-            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
-            from preced_unique [OF this] have "thread = th"
-            proof
-              from h' show "th \<in> threads (t @ s)" by simp
-            next
-              from thread_ts show "thread \<in> threads (t @ s)" .
-            qed(simp)
-            with neq_thread show "False" by simp
-          qed
-          ultimately show ?thesis by auto
-        qed
-      qed
-      ultimately show ?thesis 
-        by (auto simp:max_def split:if_splits)
-    qed
-  next
-    case (P thread cs)
-    with Cons
-    show ?thesis by (auto simp:preced_def)
-  next
-    case (V thread cs)
-    with Cons
-    show ?thesis by (auto simp:preced_def)
-  next
-    case (Set thread prio')
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      let ?B = "threads (t@s)"
-      from Cons have "extend_highest s' th prio (e # t)" by auto
-      from extend_highest.set_diff_low[OF this] and Set
-      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
-      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
-      also have "\<dots> = ?t"
-      proof(rule Max_eqI)
-        fix y
-        assume y_in: "y \<in> ?f ` ?B"
-        then obtain th1 where 
-          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
-        show "y \<le> ?t"
-        proof(cases "th1 = thread")
-          case True
-          with neq_thread le_p eq_y s_def Set
-          show ?thesis
-            by (auto simp:preced_def precedence_le_def)
-        next
-          case False
-          with Set eq_y
-          have "y  = preced th1 (t@s)"
-            by (simp add:preced_def)
-          moreover have "\<dots> \<le> ?t"
-          proof -
-            from Cons
-            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
-              by auto
-            moreover have "preced th1 (t@s) \<le> \<dots>"
-            proof(rule Max_ge)
-              from th1_in 
-              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
-                by simp
-            next
-              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
-              proof -
-                from Cons have "vt step (t @ s)" by auto
-                from finite_threads[OF this] show ?thesis by auto
-              qed
-            qed
-            ultimately show ?thesis by auto
-          qed
-          ultimately show ?thesis by auto
-        qed
-      next
-        from Cons and finite_threads
-        show "finite (?f ` ?B)" by auto
-      next
-        from Cons have "extend_highest s' th prio t" by auto
-        from extend_highest.th_kept [OF this, folded s_def]
-        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
-        show "?t \<in> (?f ` ?B)" 
-        proof -
-          from neq_thread Set h
-          have "?t = ?f th" by (auto simp:preced_def)
-          with h show ?thesis by auto
-        qed
-      qed
-      finally show ?thesis .
-    qed
-  qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
-  by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
-  (is "?L = ?R")
-proof -
-  have "?L = cpreced (t@s) (wq (t@s)) th" 
-    by (unfold cp_eq_cpreced, simp)
-  also have "\<dots> = ?R"
-  proof(unfold cpreced_def)
-    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
-          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
-      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
-    proof(cases "?A = {}")
-      case False
-      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
-      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
-      proof(rule Max_insert)
-        show "finite (?f ` ?A)"
-        proof -
-          from dependents_threads[OF vt_t]
-          have "?A \<subseteq> threads (t@s)" .
-          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
-          ultimately show ?thesis 
-            by (auto simp:finite_subset)
-        qed
-      next
-        from False show "(?f ` ?A) \<noteq> {}" by simp
-      qed
-      moreover have "\<dots> = Max (?f ` ?B)"
-      proof -
-        from max_preced have "?f th = Max (?f ` ?B)" .
-        moreover have "Max (?f ` ?A) \<le> \<dots>" 
-        proof(rule Max_mono)
-          from False show "(?f ` ?A) \<noteq> {}" by simp
-        next
-          show "?f ` ?A \<subseteq> ?f ` ?B" 
-          proof -
-            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
-            thus ?thesis by auto
-          qed
-        next
-          from finite_threads[OF vt_t] 
-          show "finite (?f ` ?B)" by simp
-        qed
-        ultimately show ?thesis
-          by (auto simp:max_def)
-      qed
-      ultimately show ?thesis by auto
-    next
-      case True
-      with max_preced show ?thesis by auto
-    qed
-  qed
-  finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
-  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
-  by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less':
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and neq_th': "th' \<noteq> th"
-  shows "preced th' s < preced th s"
-proof -
-  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
-  proof(rule Max_ge)
-    from finite_threads [OF vt_s]
-    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
-  next
-    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
-      by simp
-  qed
-  moreover have "preced th' s \<noteq> preced th s"
-  proof
-    assume "preced th' s = preced th s"
-    from preced_unique[OF this th'_in] neq_th' is_ready
-    show "False" by  (auto simp:readys_def)
-  qed
-  ultimately show ?thesis using highest_preced_thread
-    by auto
-qed
-
-lemma pv_blocked:
-  fixes th'
-  assumes th'_in: "th' \<in> threads (t@s)"
-  and neq_th': "th' \<noteq> th"
-  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
-  shows "th' \<notin> runing (t@s)"
-proof
-  assume "th' \<in> runing (t@s)"
-  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
-    by (auto simp:runing_def)
-  with max_cp_readys_threads [OF vt_t]
-  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
-    by auto
-  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
-  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
-  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
-    by simp
-  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
-  show False
-  proof -
-    have "dependents (wq (t @ s)) th' = {}" 
-      by (rule count_eq_dependents [OF vt_t eq_pv])
-    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
-    proof
-      assume "preced th' (t @ s) = preced th (t @ s)"
-      hence "th' = th"
-      proof(rule preced_unique)
-        from th_kept show "th \<in> threads (t @ s)" by simp
-      next
-        from th'_in show "th' \<in> threads (t @ s)" by simp
-      qed
-      with assms show False by simp
-    qed
-    ultimately show ?thesis
-      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
-  qed
-qed
-
-lemma runing_precond_pre:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and eq_pv: "cntP s th' = cntV s th'"
-  and neq_th': "th' \<noteq> th"
-  shows "th' \<in> threads (t@s) \<and>
-         cntP (t@s) th' = cntV (t@s) th'"
-proof -
-  show ?thesis
-  proof(induct rule:ind)
-    case (Cons e t)
-    from Cons
-    have in_thread: "th' \<in> threads (t @ s)"
-      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-    have "extend_highest s' th prio t" by fact
-    from extend_highest.pv_blocked 
-    [OF this, folded s_def, OF in_thread neq_th' not_holding]
-    have not_runing: "th' \<notin> runing (t @ s)" .
-    show ?case
-    proof(cases e)
-      case (V thread cs)
-      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
-
-      show ?thesis
-      proof -
-        from Cons and V have "step (t@s) (V thread cs)" by auto
-        hence neq_th': "thread \<noteq> th'"
-        proof(cases)
-          assume "thread \<in> runing (t@s)"
-          moreover have "th' \<notin> runing (t@s)" by fact
-          ultimately show ?thesis by auto
-        qed
-        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
-          by (unfold V, simp add:cntP_def cntV_def count_def)
-        moreover from in_thread
-        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
-        ultimately show ?thesis by auto
-      qed
-    next
-      case (P thread cs)
-      from Cons and P have "step (t@s) (P thread cs)" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<in> runing (t@s)"
-        moreover note not_runing
-        ultimately show ?thesis by auto
-      qed
-      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
-        by auto
-      ultimately show ?thesis by auto
-    next
-      case (Create thread prio')
-      from Cons and Create have "step (t@s) (Create thread prio')" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<notin> threads (t @ s)"
-        moreover have "th' \<in> threads (t@s)" by fact
-        ultimately show ?thesis by auto
-      qed
-      with Cons and Create 
-      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and Create 
-      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
-      ultimately show ?thesis by auto
-    next
-      case (Exit thread)
-      from Cons and Exit have "step (t@s) (Exit thread)" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<in> runing (t @ s)"
-        moreover note not_runing
-        ultimately show ?thesis by auto
-      qed
-      with Cons and Exit 
-      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and Exit and neq_th' 
-      have in_thread': "th' \<in> threads ((e # t) @ s)"
-        by auto
-      ultimately show ?thesis by auto
-    next
-      case (Set thread prio')
-      with Cons
-      show ?thesis 
-        by (auto simp:cntP_def cntV_def count_def)
-    qed
-  next
-    case Nil
-    with assms
-    show ?case by auto
-  qed
-qed
-
-(*
-lemma runing_precond:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and eq_pv: "cntP s th' = cntV s th'"
-  and neq_th': "th' \<noteq> th"
-  shows "th' \<notin> runing (t@s)"
-proof -
-  from runing_precond_pre[OF th'_in eq_pv neq_th']
-  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-  from pv_blocked[OF h1 neq_th' h2] 
-  show ?thesis .
-qed
-*)
-
-lemma runing_precond:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and neq_th': "th' \<noteq> th"
-  and is_runing: "th' \<in> runing (t@s)"
-  shows "cntP s th' > cntV s th'"
-proof -
-  have "cntP s th' \<noteq> cntV s th'"
-  proof
-    assume eq_pv: "cntP s th' = cntV s th'"
-    from runing_precond_pre[OF th'_in eq_pv neq_th']
-    have h1: "th' \<in> threads (t @ s)"  
-      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
-    with is_runing show "False" by simp
-  qed
-  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
-  have "cntV s th' \<le> cntP s th'" by auto
-  ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
-  assumes neq_th': "th' \<noteq> th"
-  and th'_in: "th' \<in> threads ((moment i t)@s)"
-  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
-  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
-         th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
-  case (Suc k)
-  show ?case
-  proof -
-    { assume True: "Suc (i+k) \<le> length t"
-      from moment_head [OF this] 
-      obtain e where
-        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
-        by blast
-      from red_moment[of "Suc(i+k)"]
-      and eq_me have "extend_highest s' th prio (e # moment (i + k) t)" by simp
-      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
-        by (unfold extend_highest_def extend_highest_axioms_def 
-          highest_create_def s_def, auto)
-      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
-      proof(unfold s_def)
-        show "th' \<notin> runing (moment (i + k) t @ Create th prio # s')"
-        proof(rule extend_highest.pv_blocked)
-          from Suc show "th' \<in> threads (moment (i + k) t @ Create th prio # s')"
-            by (simp add:s_def)
-        next
-          from neq_th' show "th' \<noteq> th" .
-        next
-          from red_moment show "extend_highest s' th prio (moment (i + k) t)" .
-        next
-          from Suc show "cntP (moment (i + k) t @ Create th prio # s') th' =
-            cntV (moment (i + k) t @ Create th prio # s') th'"
-            by (auto simp:s_def)
-        qed
-      qed
-      from step_back_step[OF vt_e]
-      have "step ((moment (i + k) t)@s) e" .
-      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
-        th' \<in> threads (e#(moment (i + k) t)@s)
-        "
-      proof(cases)
-        case (thread_create thread prio)
-        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_exit thread)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_P thread cs)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_V thread cs)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_set thread prio')
-        with Suc show ?thesis
-          by (auto simp:cntP_def cntV_def count_def)
-      qed
-      with eq_me have ?thesis using eq_me by auto 
-    } note h = this
-    show ?thesis
-    proof(cases "Suc (i+k) \<le> length t")
-      case True
-      from h [OF this] show ?thesis .
-    next
-      case False
-      with moment_ge
-      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
-      with Suc show ?thesis by auto
-    qed
-  qed
-next
-  case 0
-  from assms show ?case by auto
-qed
-
-lemma moment_blocked:
-  assumes neq_th': "th' \<noteq> th"
-  and th'_in: "th' \<in> threads ((moment i t)@s)"
-  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
-  and le_ij: "i \<le> j"
-  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
-         th' \<in> threads ((moment j t)@s) \<and>
-         th' \<notin> runing ((moment j t)@s)"
-proof -
-  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
-  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
-    and h2: "th' \<in> threads ((moment j t)@s)" by auto
-  with extend_highest.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
-  show ?thesis by auto
-qed
-
-lemma runing_inversion_1:
-  assumes neq_th': "th' \<noteq> th"
-  and runing': "th' \<in> runing (t@s)"
-  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
-  case True
-  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
-  case False
-  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
-  let ?q = "moment 0 t"
-  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
-  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
-  from p_split_gen [of ?Q, OF this not_thread]
-  obtain i where lt_its: "i < length t"
-    and le_i: "0 \<le> i"
-    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
-    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
-  from lt_its have "Suc i \<le> length t" by auto
-  from moment_head[OF this] obtain e where 
-   eq_me: "moment (Suc i) t = e # moment i t" by blast
-  from red_moment[of "Suc i"] and eq_me
-  have "extend_highest s' th prio (e # moment i t)" by simp
-  hence vt_e: "vt step (e#(moment i t)@s)"
-    by (unfold extend_highest_def extend_highest_axioms_def 
-      highest_create_def s_def, auto)
-  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
-  from post[rule_format, of "Suc i"] and eq_me 
-  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
-  from create_pre[OF stp_i pre this] 
-  obtain prio where eq_e: "e = Create th' prio" .
-  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
-  proof(rule cnp_cnv_eq)
-    from step_back_vt [OF vt_e] 
-    show "vt step (moment i t @ s)" .
-  next
-    from eq_e and stp_i 
-    have "step (moment i t @ s) (Create th' prio)" by simp
-    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
-  qed
-  with eq_e
-  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
-    by (simp add:cntP_def cntV_def count_def)
-  with eq_me[symmetric]
-  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
-    by simp
-  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
-  with eq_me [symmetric]
-  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
-  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
-  and moment_ge
-  have "th' \<notin> runing (t @ s)" by auto
-  with runing'
-  show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
-  assumes runing': "th' \<in> runing (t@s)"
-  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
-  from runing_inversion_1[OF _ runing']
-  show ?thesis by auto
-qed
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
-  case True thus ?thesis by auto
-next
-  case False
-  then have not_ready: "th \<notin> readys (t@s)"
-    apply (unfold runing_def, 
-            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
-    by auto
-  from th_kept have "th \<in> threads (t@s)" by auto
-  from th_chain_to_ready[OF vt_t this] and not_ready
-  obtain th' where th'_in: "th' \<in> readys (t@s)"
-    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
-  have "th' \<in> runing (t@s)"
-  proof -
-    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
-    proof -
-      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
-               preced th (t@s)"
-      proof(rule Max_eqI)
-        fix y
-        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
-        then obtain th1 where
-          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
-          and eq_y: "y = preced th1 (t@s)" by auto
-        show "y \<le> preced th (t @ s)"
-        proof -
-          from max_preced
-          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
-          moreover have "y \<le> \<dots>"
-          proof(rule Max_ge)
-            from h1
-            have "th1 \<in> threads (t@s)"
-            proof
-              assume "th1 = th'"
-              with th'_in show ?thesis by (simp add:readys_def)
-            next
-              assume "th1 \<in> dependents (wq (t @ s)) th'"
-              with dependents_threads [OF vt_t]
-              show "th1 \<in> threads (t @ s)" by auto
-            qed
-            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
-          next
-            from finite_threads[OF vt_t]
-            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
-          qed
-          ultimately show ?thesis by auto
-        qed
-      next
-        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
-        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
-          by (auto intro:finite_subset)
-      next
-        from dp
-        have "th \<in> dependents (wq (t @ s)) th'" 
-          by (unfold cs_dependents_def, auto simp:eq_depend)
-        thus "preced th (t @ s) \<in> 
-                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
-          by auto
-      qed
-      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
-      proof -
-        from max_preced and max_cp_eq[OF vt_t, symmetric]
-        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
-        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
-      qed
-      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
-    qed
-    with th'_in show ?thesis by (auto simp:runing_def)
-  qed
-  thus ?thesis by auto
-qed
-
-end
-
-end
-
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/ExtGG_1.thy
--- a/prio/ExtGG_1.thy	Sun Feb 05 14:29:08 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,973 +0,0 @@
-theory ExtGG
-imports PrioG
-begin
-
-lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
-  apply (induct s, simp)
-proof -
-  fix a s
-  assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
-    and eq_as: "a # s \<noteq> []"
-  show "birthtime th (a # s) < length (a # s)"
-  proof(cases "s \<noteq> []")
-    case False
-    from False show ?thesis
-      by (cases a, auto simp:birthtime.simps)
-  next
-    case True
-    from ih [OF True] show ?thesis
-      by (cases a, auto simp:birthtime.simps)
-  qed
-qed
-
-lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
-  by (induct s, auto simp:threads.simps)
-
-lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
-  apply (drule_tac th_in_ne)
-  by (unfold preced_def, auto intro: birth_time_lt)
-
-locale highest_gen =
-  fixes s' th s e' prio tm
-  defines s_def : "s \<equiv> (e'#s')"
-  assumes vt_s: "vt step s"
-  and threads_s: "th \<in> threads s"
-  and highest: "preced th s = Max ((cp s)`threads s)"
-  and nh: "preced th s' \<noteq> Max ((cp s)`threads s')"
-  and preced_th: "preced th s = Prc prio tm"
-
-context highest_gen
-begin
-
-lemma lt_tm: "tm < length s"
-  by (insert preced_tm_lt[OF threads_s preced_th], simp)
-
-lemma vt_s': "vt step s'"
-  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
-
-lemma eq_cp_s_th: "cp s th = preced th s"
-proof -
-  from highest and max_cp_eq[OF vt_s]
-  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
-  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
-  proof -
-    from threads_s and dependents_threads[OF vt_s, of th]
-    show ?thesis by auto
-  qed
-  show ?thesis
-  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
-    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
-  next
-    fix y 
-    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
-    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
-      and eq_y: "y = preced th1 s" by auto
-    show "y \<le> preced th s"
-    proof(unfold is_max, rule Max_ge)
-      from finite_threads[OF vt_s] 
-      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-    next
-      from sbs th1_in and eq_y 
-      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
-    qed
-  next
-    from sbs and finite_threads[OF vt_s]
-    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
-      by (auto intro:finite_subset)
-  qed
-qed
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
-  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
-  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
-proof -
-  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
-  show ?thesis by simp
-qed
-
-end
-
-locale extend_highest_gen = highest_gen + 
-  fixes t 
-  assumes vt_t: "vt step (t@s)"
-  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
-  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
-  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app: 
-  assumes vt_ts: "vt cs (t@s)" 
-  shows "vt cs s"
-proof -
-  from vt_ts show ?thesis
-  proof(induct t)
-    case Nil
-    from Nil show ?case by auto
-  next
-    case (Cons e t)
-    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
-      and vt_et: "vt cs ((e # t) @ s)"
-    show ?case
-    proof(rule ih)
-      show "vt cs (t @ s)"
-      proof(rule step_back_vt)
-        from vt_et show "vt cs (e # t @ s)" by simp
-      qed
-    qed
-  qed
-qed
-
-context extend_highest_gen
-begin
-
-lemma red_moment:
-  "extend_highest_gen s' th e' prio tm (moment i t)"
-  apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
-  apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
-  by (unfold highest_gen_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
-  assumes 
-    h0: "R []"
-  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
-                    extend_highest_gen s' th e' prio tm t; 
-                    extend_highest_gen s' th e' prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
-  shows "R t"
-proof -
-  from vt_t extend_highest_gen_axioms show ?thesis
-  proof(induct t)
-    from h0 show "R []" .
-  next
-    case (Cons e t')
-    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_gen s' th e' prio tm t'\<rbrakk> \<Longrightarrow> R t'"
-      and vt_e: "vt step ((e # t') @ s)"
-      and et: "extend_highest_gen s' th e' prio tm (e # t')"
-    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
-    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
-    show ?case
-    proof(rule h2 [OF vt_ts stp _ _ _ ])
-      show "R t'"
-      proof(rule ih)
-        from et show ext': "extend_highest_gen s' th e' prio tm t'"
-          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
-      next
-        from vt_ts show "vt step (t' @ s)" .
-      qed
-    next
-      from et show "extend_highest_gen s' th e' prio tm (e # t')" .
-    next
-      from et show ext': "extend_highest_gen s' th e' prio tm t'"
-          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
-    qed
-  qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and> 
-        preced th (t@s) = preced th s" (is "?Q t")
-proof -
-  show ?thesis
-  proof(induct rule:ind)
-    case Nil
-    from threads_s
-    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
-      by auto
-  next
-    case (Cons e t)
-    show ?case
-    proof(cases e)
-      case (Create thread prio)
-      assume eq_e: " e = Create thread prio"
-      show ?thesis
-      proof -
-        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
-        hence "th \<noteq> thread"
-        proof(cases)
-          assume "thread \<notin> threads (t @ s)"
-          with Cons show ?thesis by auto
-        qed
-        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
-          by (unfold eq_e, auto simp:preced_def)
-        moreover note Cons
-        ultimately show ?thesis
-          by (auto simp:eq_e)
-      qed
-    next
-      case (Exit thread)
-      assume eq_e: "e = Exit thread"
-      from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
-      from extend_highest_gen.exit_diff [OF this] and eq_e
-      have neq_th: "thread \<noteq> th" by auto
-      with Cons
-      show ?thesis
-        by (unfold eq_e, auto simp:preced_def)
-    next
-      case (P thread cs)
-      assume eq_e: "e = P thread cs"
-      with Cons
-      show ?thesis 
-        by (auto simp:eq_e preced_def)
-    next
-      case (V thread cs)
-      assume eq_e: "e = V thread cs"
-      with Cons
-      show ?thesis 
-        by (auto simp:eq_e preced_def)
-    next
-      case (Set thread prio')
-      assume eq_e: " e = Set thread prio'"
-      show ?thesis
-      proof -
-        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
-        from extend_highest_gen.set_diff_low[OF this] and eq_e
-        have "th \<noteq> thread" by auto
-        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
-          by (unfold eq_e, auto simp:preced_def)
-        moreover note Cons
-        ultimately show ?thesis
-          by (auto simp:eq_e)
-      qed
-    qed
-  qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
-  case Nil
-  from highest_preced_thread
-  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
-    by simp
-next
-  case (Cons e t)
-  show ?case
-  proof(cases e)
-    case (Create thread prio')
-    assume eq_e: " e = Create thread prio'"
-    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
-    hence neq_thread: "thread \<noteq> th"
-    proof(cases)
-      assume "thread \<notin> threads (t @ s)"
-      moreover have "th \<in> threads (t@s)"
-      proof -
-        from Cons have "extend_highest_gen s' th e' prio tm t" by auto
-        from extend_highest_gen.th_kept[OF this] show ?thesis by (simp add:s_def)
-      qed
-      ultimately show ?thesis by auto
-    qed
-    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
-    from extend_highest_gen.th_kept[OF this]
-    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
-      by (auto simp:s_def)
-    from stp
-    have thread_ts: "thread \<notin> threads (t @ s)"
-      by (cases, auto)
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
-        by (unfold eq_e, simp)
-      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
-      proof(rule Max_insert)
-        from Cons have "vt step (t @ s)" by auto
-        from finite_threads[OF this]
-        show "finite (?f ` (threads (t@s)))" by simp
-      next
-        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
-      qed
-      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
-      proof -
-        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
-          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
-        proof -
-          { fix th' 
-            assume "th' \<in> ?B"
-            with thread_ts eq_e
-            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
-          } thus ?thesis 
-            apply (auto simp:Image_def)
-          proof -
-            fix th'
-            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
-              preced th' (e # t @ s) = preced th' (t @ s)"
-              and h1: "th' \<in> threads (t @ s)"
-            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
-            proof -
-              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
-              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
-              ultimately show ?thesis by simp
-            qed
-          qed
-        qed
-        with Cons show ?thesis by auto
-      qed
-      moreover have "?f thread < ?t"
-      proof -
-        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
-        from extend_highest_gen.create_low[OF this] and eq_e
-        have "prio' \<le> prio" by auto
-        thus ?thesis
-        by (unfold preced_th, unfold eq_e, insert lt_tm, 
-          auto simp:preced_def s_def precedence_less_def preced_th)
-    qed
-    ultimately show ?thesis by (auto simp:max_def)
-  qed
-next
-    case (Exit thread)
-    assume eq_e: "e = Exit thread"
-    from Cons have vt_e: "vt step (e#(t @ s))" by auto
-    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
-    from stp have thread_ts: "thread \<in> threads (t @ s)"
-      by(cases, unfold runing_def readys_def, auto)
-    from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
-    from extend_highest_gen.exit_diff[OF this] and eq_e
-    have neq_thread: "thread \<noteq> th" by auto
-    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
-    from extend_highest_gen.th_kept[OF this, folded s_def]
-    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      have "threads (t@s) = insert thread ?A"
-        by (insert stp thread_ts, unfold eq_e, auto)
-      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
-      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
-      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
-      proof(rule Max_insert)
-        from finite_threads [OF vt_e]
-        show "finite (?f ` ?A)" by simp
-      next
-        from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
-        from extend_highest_gen.th_kept[OF this]
-        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
-      qed
-      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
-      moreover have "Max (?f ` (threads (t@s))) = ?t"
-      proof -
-        from Cons show ?thesis
-          by (unfold eq_e, auto simp:preced_def)
-      qed
-      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
-      moreover have "?f thread < ?t"
-      proof(unfold eq_e, simp add:preced_def, fold preced_def)
-        show "preced thread (t @ s) < ?t"
-        proof -
-          have "preced thread (t @ s) \<le> ?t" 
-          proof -
-            from Cons
-            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
-              (is "?t = Max (?g ` ?B)") by simp
-            moreover have "?g thread \<le> \<dots>"
-            proof(rule Max_ge)
-              have "vt step (t@s)" by fact
-              from finite_threads [OF this]
-              show "finite (?g ` ?B)" by simp
-            next
-              from thread_ts
-              show "?g thread \<in> (?g ` ?B)" by auto
-            qed
-            ultimately show ?thesis by auto
-          qed
-          moreover have "preced thread (t @ s) \<noteq> ?t"
-          proof
-            assume "preced thread (t @ s) = preced th s"
-            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
-            from preced_unique [OF this] have "thread = th"
-            proof
-              from h' show "th \<in> threads (t @ s)" by simp
-            next
-              from thread_ts show "thread \<in> threads (t @ s)" .
-            qed(simp)
-            with neq_thread show "False" by simp
-          qed
-          ultimately show ?thesis by auto
-        qed
-      qed
-      ultimately show ?thesis 
-        by (auto simp:max_def split:if_splits)
-    qed
-  next
-    case (P thread cs)
-    with Cons
-    show ?thesis by (auto simp:preced_def)
-  next
-    case (V thread cs)
-    with Cons
-    show ?thesis by (auto simp:preced_def)
-  next
-    case (Set thread prio')
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      let ?B = "threads (t@s)"
-      from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
-      from extend_highest_gen.set_diff_low[OF this] and Set
-      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
-      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
-      also have "\<dots> = ?t"
-      proof(rule Max_eqI)
-        fix y
-        assume y_in: "y \<in> ?f ` ?B"
-        then obtain th1 where 
-          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
-        show "y \<le> ?t"
-        proof(cases "th1 = thread")
-          case True
-          with neq_thread le_p eq_y s_def Set
-          show ?thesis
-            apply (subst preced_th, insert lt_tm)
-            by (auto simp:preced_def precedence_le_def)
-        next
-          case False
-          with Set eq_y
-          have "y  = preced th1 (t@s)"
-            by (simp add:preced_def)
-          moreover have "\<dots> \<le> ?t"
-          proof -
-            from Cons
-            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
-              by auto
-            moreover have "preced th1 (t@s) \<le> \<dots>"
-            proof(rule Max_ge)
-              from th1_in 
-              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
-                by simp
-            next
-              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
-              proof -
-                from Cons have "vt step (t @ s)" by auto
-                from finite_threads[OF this] show ?thesis by auto
-              qed
-            qed
-            ultimately show ?thesis by auto
-          qed
-          ultimately show ?thesis by auto
-        qed
-      next
-        from Cons and finite_threads
-        show "finite (?f ` ?B)" by auto
-      next
-        from Cons have "extend_highest_gen s' th e' prio tm t" by auto
-        from extend_highest_gen.th_kept [OF this, folded s_def]
-        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
-        show "?t \<in> (?f ` ?B)" 
-        proof -
-          from neq_thread Set h
-          have "?t = ?f th" by (auto simp:preced_def)
-          with h show ?thesis by auto
-        qed
-      qed
-      finally show ?thesis .
-    qed
-  qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
-  by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
-  (is "?L = ?R")
-proof -
-  have "?L = cpreced (t@s) (wq (t@s)) th" 
-    by (unfold cp_eq_cpreced, simp)
-  also have "\<dots> = ?R"
-  proof(unfold cpreced_def)
-    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
-          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
-      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
-    proof(cases "?A = {}")
-      case False
-      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
-      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
-      proof(rule Max_insert)
-        show "finite (?f ` ?A)"
-        proof -
-          from dependents_threads[OF vt_t]
-          have "?A \<subseteq> threads (t@s)" .
-          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
-          ultimately show ?thesis 
-            by (auto simp:finite_subset)
-        qed
-      next
-        from False show "(?f ` ?A) \<noteq> {}" by simp
-      qed
-      moreover have "\<dots> = Max (?f ` ?B)"
-      proof -
-        from max_preced have "?f th = Max (?f ` ?B)" .
-        moreover have "Max (?f ` ?A) \<le> \<dots>" 
-        proof(rule Max_mono)
-          from False show "(?f ` ?A) \<noteq> {}" by simp
-        next
-          show "?f ` ?A \<subseteq> ?f ` ?B" 
-          proof -
-            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
-            thus ?thesis by auto
-          qed
-        next
-          from finite_threads[OF vt_t] 
-          show "finite (?f ` ?B)" by simp
-        qed
-        ultimately show ?thesis
-          by (auto simp:max_def)
-      qed
-      ultimately show ?thesis by auto
-    next
-      case True
-      with max_preced show ?thesis by auto
-    qed
-  qed
-  finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
-  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
-  by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less':
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and neq_th': "th' \<noteq> th"
-  shows "preced th' s < preced th s"
-proof -
-  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
-  proof(rule Max_ge)
-    from finite_threads [OF vt_s]
-    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
-  next
-    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
-      by simp
-  qed
-  moreover have "preced th' s \<noteq> preced th s"
-  proof
-    assume "preced th' s = preced th s"
-    from preced_unique[OF this th'_in] neq_th' threads_s
-    show "False" by  (auto simp:readys_def)
-  qed
-  ultimately show ?thesis using highest_preced_thread
-    by auto
-qed
-
-lemma pv_blocked:
-  fixes th'
-  assumes th'_in: "th' \<in> threads (t@s)"
-  and neq_th': "th' \<noteq> th"
-  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
-  shows "th' \<notin> runing (t@s)"
-proof
-  assume "th' \<in> runing (t@s)"
-  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
-    by (auto simp:runing_def)
-  with max_cp_readys_threads [OF vt_t]
-  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
-    by auto
-  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
-  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
-  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
-    by simp
-  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
-  show False
-  proof -
-    have "dependents (wq (t @ s)) th' = {}" 
-      by (rule count_eq_dependents [OF vt_t eq_pv])
-    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
-    proof
-      assume "preced th' (t @ s) = preced th (t @ s)"
-      hence "th' = th"
-      proof(rule preced_unique)
-        from th_kept show "th \<in> threads (t @ s)" by simp
-      next
-        from th'_in show "th' \<in> threads (t @ s)" by simp
-      qed
-      with assms show False by simp
-    qed
-    ultimately show ?thesis
-      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
-  qed
-qed
-
-lemma runing_precond_pre:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and eq_pv: "cntP s th' = cntV s th'"
-  and neq_th': "th' \<noteq> th"
-  shows "th' \<in> threads (t@s) \<and>
-         cntP (t@s) th' = cntV (t@s) th'"
-proof -
-  show ?thesis
-  proof(induct rule:ind)
-    case (Cons e t)
-    from Cons
-    have in_thread: "th' \<in> threads (t @ s)"
-      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-    from Cons have "extend_highest_gen s' th e' prio tm t" by auto
-    from extend_highest_gen.pv_blocked 
-    [OF this, folded s_def, OF in_thread neq_th' not_holding]
-    have not_runing: "th' \<notin> runing (t @ s)" .
-    show ?case
-    proof(cases e)
-      case (V thread cs)
-      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
-
-      show ?thesis
-      proof -
-        from Cons and V have "step (t@s) (V thread cs)" by auto
-        hence neq_th': "thread \<noteq> th'"
-        proof(cases)
-          assume "thread \<in> runing (t@s)"
-          moreover have "th' \<notin> runing (t@s)" by fact
-          ultimately show ?thesis by auto
-        qed
-        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
-          by (unfold V, simp add:cntP_def cntV_def count_def)
-        moreover from in_thread
-        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
-        ultimately show ?thesis by auto
-      qed
-    next
-      case (P thread cs)
-      from Cons and P have "step (t@s) (P thread cs)" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<in> runing (t@s)"
-        moreover note not_runing
-        ultimately show ?thesis by auto
-      qed
-      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
-        by auto
-      ultimately show ?thesis by auto
-    next
-      case (Create thread prio')
-      from Cons and Create have "step (t@s) (Create thread prio')" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<notin> threads (t @ s)"
-        moreover have "th' \<in> threads (t@s)" by fact
-        ultimately show ?thesis by auto
-      qed
-      with Cons and Create 
-      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and Create 
-      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
-      ultimately show ?thesis by auto
-    next
-      case (Exit thread)
-      from Cons and Exit have "step (t@s) (Exit thread)" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<in> runing (t @ s)"
-        moreover note not_runing
-        ultimately show ?thesis by auto
-      qed
-      with Cons and Exit 
-      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and Exit and neq_th' 
-      have in_thread': "th' \<in> threads ((e # t) @ s)"
-        by auto
-      ultimately show ?thesis by auto
-    next
-      case (Set thread prio')
-      with Cons
-      show ?thesis 
-        by (auto simp:cntP_def cntV_def count_def)
-    qed
-  next
-    case Nil
-    with assms
-    show ?case by auto
-  qed
-qed
-
-(*
-lemma runing_precond:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and eq_pv: "cntP s th' = cntV s th'"
-  and neq_th': "th' \<noteq> th"
-  shows "th' \<notin> runing (t@s)"
-proof -
-  from runing_precond_pre[OF th'_in eq_pv neq_th']
-  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-  from pv_blocked[OF h1 neq_th' h2] 
-  show ?thesis .
-qed
-*)
-
-lemma runing_precond:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and neq_th': "th' \<noteq> th"
-  and is_runing: "th' \<in> runing (t@s)"
-  shows "cntP s th' > cntV s th'"
-proof -
-  have "cntP s th' \<noteq> cntV s th'"
-  proof
-    assume eq_pv: "cntP s th' = cntV s th'"
-    from runing_precond_pre[OF th'_in eq_pv neq_th']
-    have h1: "th' \<in> threads (t @ s)"  
-      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
-    with is_runing show "False" by simp
-  qed
-  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
-  have "cntV s th' \<le> cntP s th'" by auto
-  ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
-  assumes neq_th': "th' \<noteq> th"
-  and th'_in: "th' \<in> threads ((moment i t)@s)"
-  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
-  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
-         th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
-  case (Suc k)
-  show ?case
-  proof -
-    { assume True: "Suc (i+k) \<le> length t"
-      from moment_head [OF this] 
-      obtain e where
-        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
-        by blast
-      from red_moment[of "Suc(i+k)"]
-      and eq_me have "extend_highest_gen s' th e' prio tm (e # moment (i + k) t)" by simp
-      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
-        by (unfold extend_highest_gen_def extend_highest_gen_axioms_def 
-                          highest_gen_def s_def, auto)
-      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
-      proof(unfold s_def)
-        show "th' \<notin> runing (moment (i + k) t @ e' # s')"
-        proof(rule extend_highest_gen.pv_blocked)
-          from Suc show "th' \<in> threads (moment (i + k) t @ e' # s')"
-            by (simp add:s_def)
-        next
-          from neq_th' show "th' \<noteq> th" .
-        next
-          from red_moment show "extend_highest_gen s' th e' prio tm (moment (i + k) t)" .
-        next
-          from Suc show "cntP (moment (i + k) t @ e' # s') th' = cntV (moment (i + k) t @ e' # s') th'"
-            by (auto simp:s_def)
-        qed
-      qed
-      from step_back_step[OF vt_e]
-      have "step ((moment (i + k) t)@s) e" .
-      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
-        th' \<in> threads (e#(moment (i + k) t)@s)
-        "
-      proof(cases)
-        case (thread_create thread prio)
-        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_exit thread)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_P thread cs)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_V thread cs)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_set thread prio')
-        with Suc show ?thesis
-          by (auto simp:cntP_def cntV_def count_def)
-      qed
-      with eq_me have ?thesis using eq_me by auto 
-    } note h = this
-    show ?thesis
-    proof(cases "Suc (i+k) \<le> length t")
-      case True
-      from h [OF this] show ?thesis .
-    next
-      case False
-      with moment_ge
-      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
-      with Suc show ?thesis by auto
-    qed
-  qed
-next
-  case 0
-  from assms show ?case by auto
-qed
-
-lemma moment_blocked:
-  assumes neq_th': "th' \<noteq> th"
-  and th'_in: "th' \<in> threads ((moment i t)@s)"
-  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
-  and le_ij: "i \<le> j"
-  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
-         th' \<in> threads ((moment j t)@s) \<and>
-         th' \<notin> runing ((moment j t)@s)"
-proof -
-  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
-  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
-    and h2: "th' \<in> threads ((moment j t)@s)" by auto
-  with extend_highest_gen.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
-  show ?thesis by auto
-qed
-
-lemma runing_inversion_1:
-  assumes neq_th': "th' \<noteq> th"
-  and runing': "th' \<in> runing (t@s)"
-  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
-  case True
-  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
-  case False
-  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
-  let ?q = "moment 0 t"
-  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
-  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
-  from p_split_gen [of ?Q, OF this not_thread]
-  obtain i where lt_its: "i < length t"
-    and le_i: "0 \<le> i"
-    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
-    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
-  from lt_its have "Suc i \<le> length t" by auto
-  from moment_head[OF this] obtain e where 
-   eq_me: "moment (Suc i) t = e # moment i t" by blast
-  from red_moment[of "Suc i"] and eq_me
-  have "extend_highest_gen s' th e' prio tm (e # moment i t)" by simp
-  hence vt_e: "vt step (e#(moment i t)@s)"
-    by (unfold extend_highest_gen_def extend_highest_gen_axioms_def 
-      highest_gen_def s_def, auto)
-  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
-  from post[rule_format, of "Suc i"] and eq_me 
-  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
-  from create_pre[OF stp_i pre this] 
-  obtain prio where eq_e: "e = Create th' prio" .
-  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
-  proof(rule cnp_cnv_eq)
-    from step_back_vt [OF vt_e] 
-    show "vt step (moment i t @ s)" .
-  next
-    from eq_e and stp_i 
-    have "step (moment i t @ s) (Create th' prio)" by simp
-    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
-  qed
-  with eq_e
-  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
-    by (simp add:cntP_def cntV_def count_def)
-  with eq_me[symmetric]
-  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
-    by simp
-  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
-  with eq_me [symmetric]
-  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
-  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
-  and moment_ge
-  have "th' \<notin> runing (t @ s)" by auto
-  with runing'
-  show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
-  assumes runing': "th' \<in> runing (t@s)"
-  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
-  from runing_inversion_1[OF _ runing']
-  show ?thesis by auto
-qed
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
-  case True thus ?thesis by auto
-next
-  case False
-  then have not_ready: "th \<notin> readys (t@s)"
-    apply (unfold runing_def, 
-            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
-    by auto
-  from th_kept have "th \<in> threads (t@s)" by auto
-  from th_chain_to_ready[OF vt_t this] and not_ready
-  obtain th' where th'_in: "th' \<in> readys (t@s)"
-    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
-  have "th' \<in> runing (t@s)"
-  proof -
-    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
-    proof -
-      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
-               preced th (t@s)"
-      proof(rule Max_eqI)
-        fix y
-        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
-        then obtain th1 where
-          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
-          and eq_y: "y = preced th1 (t@s)" by auto
-        show "y \<le> preced th (t @ s)"
-        proof -
-          from max_preced
-          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
-          moreover have "y \<le> \<dots>"
-          proof(rule Max_ge)
-            from h1
-            have "th1 \<in> threads (t@s)"
-            proof
-              assume "th1 = th'"
-              with th'_in show ?thesis by (simp add:readys_def)
-            next
-              assume "th1 \<in> dependents (wq (t @ s)) th'"
-              with dependents_threads [OF vt_t]
-              show "th1 \<in> threads (t @ s)" by auto
-            qed
-            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
-          next
-            from finite_threads[OF vt_t]
-            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
-          qed
-          ultimately show ?thesis by auto
-        qed
-      next
-        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
-        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
-          by (auto intro:finite_subset)
-      next
-        from dp
-        have "th \<in> dependents (wq (t @ s)) th'" 
-          by (unfold cs_dependents_def, auto simp:eq_depend)
-        thus "preced th (t @ s) \<in> 
-                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
-          by auto
-      qed
-      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
-      proof -
-        from max_preced and max_cp_eq[OF vt_t, symmetric]
-        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
-        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
-      qed
-      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
-    qed
-    with th'_in show ?thesis by (auto simp:runing_def)
-  qed
-  thus ?thesis by auto
-qed
-
-end
-
-end
-
-
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/ExtS.thy
--- a/prio/ExtS.thy	Sun Feb 05 14:29:08 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1019 +0,0 @@
-theory ExtS
-imports Prio
-begin
-
-locale highest_set =
-  fixes s' th prio fixes s 
-  defines s_def : "s \<equiv> (Set th prio#s')"
-  assumes vt_s: "vt step s"
-  and highest: "preced th s = Max ((cp s)`threads s)"
-
-context highest_set
-begin
-
-
-lemma vt_s': "vt step s'"
-  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
-
-lemma step_set: "step s' (Set th prio)"
-  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
-
-lemma step_set_elim: 
-  "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
-  by (insert step_set, ind_cases "step s' (Set th prio)", auto)
-
-
-lemma threads_s: "th \<in> threads s"
-  by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
-
-lemma same_depend: "depend s = depend s'"
-  by (insert depend_set_unchanged, unfold s_def, simp)
-
-lemma same_dependents:
-  "dependents (wq s) th = dependents (wq s') th"
-  apply (unfold cs_dependents_def)
-  by (unfold eq_depend same_depend, simp)
-
-lemma eq_cp_s_th: "cp s th = preced th s"
-proof -
-  from highest and max_cp_eq[OF vt_s]
-  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
-  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
-  proof -
-    from threads_s and dependents_threads[OF vt_s, of th]
-    show ?thesis by auto
-  qed
-  show ?thesis
-  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
-    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
-  next
-    fix y 
-    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
-    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
-      and eq_y: "y = preced th1 s" by auto
-    show "y \<le> preced th s"
-    proof(unfold is_max, rule Max_ge)
-      from finite_threads[OF vt_s] 
-      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-    next
-      from sbs th1_in and eq_y 
-      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
-    qed
-  next
-    from sbs and finite_threads[OF vt_s]
-    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
-      by (auto intro:finite_subset)
-  qed
-qed
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
-  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
-  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma is_ready: "th \<in> readys s"
-proof -
-  have "\<forall>cs. \<not> waiting s th cs"
-    apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
-    apply (unfold s_depend_def, unfold runing_def readys_def)
-    apply (auto, fold s_def)
-    apply (erule_tac x = cs in allE, auto simp:waiting_eq)
-  proof -
-    fix cs
-    assume h: 
-      "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
-          {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
-            (is "?L = ?R")
-    and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
-    from wt have "(Th th, Cs cs) \<in> ?L" by auto
-    with h have "(Th th, Cs cs) \<in> ?R" by simp
-    hence "waiting (wq s') th cs" by auto with nwt
-    show False by auto
-  qed    
-  with threads_s show ?thesis 
-    by (unfold readys_def, auto)
-qed
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
-proof -
-  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
-  show ?thesis by simp
-qed
-
-lemma is_runing: "th \<in> runing s"
-proof -
-  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
-  proof -
-    have " Max (cp s ` readys s) = cp s th"
-    proof(rule Max_eqI)
-      from finite_threads[OF vt_s] readys_threads finite_subset
-      have "finite (readys s)" by blast
-      thus "finite (cp s ` readys s)" by auto
-    next
-      from is_ready show "cp s th \<in> cp s ` readys s" by auto
-    next
-      fix y
-      assume "y \<in> cp s ` readys s"
-      then obtain th1 where 
-        eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
-      show  "y \<le> cp s th" 
-      proof -
-        have "y \<le> Max (cp s ` threads s)"
-        proof(rule Max_ge)
-          from eq_y and th1_in
-          show "y \<in> cp s ` threads s"
-            by (auto simp:readys_def)
-        next
-          from finite_threads[OF vt_s]
-          show "finite (cp s ` threads s)" by auto
-        qed
-        with highest' show ?thesis by auto
-      qed
-    qed
-    with highest' show ?thesis by auto
-  qed
-  thus ?thesis
-    by (unfold runing_def, insert highest' is_ready, auto)
-qed
-
-end
-
-locale extend_highest_set = highest_set + 
-  fixes t 
-  assumes vt_t: "vt step (t@s)"
-  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
-  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
-  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app: 
-  assumes vt_ts: "vt cs (t@s)" 
-  shows "vt cs s"
-proof -
-  from vt_ts show ?thesis
-  proof(induct t)
-    case Nil
-    from Nil show ?case by auto
-  next
-    case (Cons e t)
-    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
-      and vt_et: "vt cs ((e # t) @ s)"
-    show ?case
-    proof(rule ih)
-      show "vt cs (t @ s)"
-      proof(rule step_back_vt)
-        from vt_et show "vt cs (e # t @ s)" by simp
-      qed
-    qed
-  qed
-qed
-
-context extend_highest_set
-begin
-
-lemma red_moment:
-  "extend_highest_set s' th prio (moment i t)"
-  apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
-  apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
-  by (unfold highest_set_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
-  assumes 
-    h0: "R []"
-  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
-                    extend_highest_set s' th prio t; 
-                    extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
-  shows "R t"
-proof -
-  from vt_t extend_highest_set_axioms show ?thesis
-  proof(induct t)
-    from h0 show "R []" .
-  next
-    case (Cons e t')
-    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
-      and vt_e: "vt step ((e # t') @ s)"
-      and et: "extend_highest_set s' th prio (e # t')"
-    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
-    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
-    show ?case
-    proof(rule h2 [OF vt_ts stp _ _ _ ])
-      show "R t'"
-      proof(rule ih)
-        from et show ext': "extend_highest_set s' th prio t'"
-          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
-      next
-        from vt_ts show "vt step (t' @ s)" .
-      qed
-    next
-      from et show "extend_highest_set s' th prio (e # t')" .
-    next
-      from et show ext': "extend_highest_set s' th prio t'"
-          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
-    qed
-  qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and> 
-        preced th (t@s) = preced th s" (is "?Q t")
-proof -
-  show ?thesis
-  proof(induct rule:ind)
-    case Nil
-    from threads_s
-    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
-      by auto
-  next
-    case (Cons e t)
-    show ?case
-    proof(cases e)
-      case (Create thread prio)
-      assume eq_e: " e = Create thread prio"
-      show ?thesis
-      proof -
-        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
-        hence "th \<noteq> thread"
-        proof(cases)
-          assume "thread \<notin> threads (t @ s)"
-          with Cons show ?thesis by auto
-        qed
-        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
-          by (unfold eq_e, auto simp:preced_def)
-        moreover note Cons
-        ultimately show ?thesis
-          by (auto simp:eq_e)
-      qed
-    next
-      case (Exit thread)
-      assume eq_e: "e = Exit thread"
-      from Cons have "extend_highest_set s' th prio (e # t)" by auto
-      from extend_highest_set.exit_diff [OF this] and eq_e
-      have neq_th: "thread \<noteq> th" by auto
-      with Cons
-      show ?thesis
-        by (unfold eq_e, auto simp:preced_def)
-    next
-      case (P thread cs)
-      assume eq_e: "e = P thread cs"
-      with Cons
-      show ?thesis 
-        by (auto simp:eq_e preced_def)
-    next
-      case (V thread cs)
-      assume eq_e: "e = V thread cs"
-      with Cons
-      show ?thesis 
-        by (auto simp:eq_e preced_def)
-    next
-      case (Set thread prio')
-      assume eq_e: " e = Set thread prio'"
-      show ?thesis
-      proof -
-        from Cons have "extend_highest_set s' th prio (e # t)" by auto
-        from extend_highest_set.set_diff_low[OF this] and eq_e
-        have "th \<noteq> thread" by auto
-        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
-          by (unfold eq_e, auto simp:preced_def)
-        moreover note Cons
-        ultimately show ?thesis
-          by (auto simp:eq_e)
-      qed
-    qed
-  qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
-  case Nil
-  from highest_preced_thread
-  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
-    by simp
-next
-  case (Cons e t)
-  show ?case
-  proof(cases e)
-    case (Create thread prio')
-    assume eq_e: " e = Create thread prio'"
-    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
-    hence neq_thread: "thread \<noteq> th"
-    proof(cases)
-      assume "thread \<notin> threads (t @ s)"
-      moreover have "th \<in> threads (t@s)"
-      proof -
-        from Cons have "extend_highest_set s' th prio t" by auto
-        from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
-      qed
-      ultimately show ?thesis by auto
-    qed
-    from Cons have "extend_highest_set s' th prio t" by auto
-    from extend_highest_set.th_kept[OF this]
-    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
-      by (auto simp:s_def)
-    from stp
-    have thread_ts: "thread \<notin> threads (t @ s)"
-      by (cases, auto)
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
-        by (unfold eq_e, simp)
-      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
-      proof(rule Max_insert)
-        from Cons have "vt step (t @ s)" by auto
-        from finite_threads[OF this]
-        show "finite (?f ` (threads (t@s)))" by simp
-      next
-        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
-      qed
-      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
-      proof -
-        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
-          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
-        proof -
-          { fix th' 
-            assume "th' \<in> ?B"
-            with thread_ts eq_e
-            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
-          } thus ?thesis 
-            apply (auto simp:Image_def)
-          proof -
-            fix th'
-            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
-              preced th' (e # t @ s) = preced th' (t @ s)"
-              and h1: "th' \<in> threads (t @ s)"
-            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
-            proof -
-              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
-              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
-              ultimately show ?thesis by simp
-            qed
-          qed
-        qed
-        with Cons show ?thesis by auto
-      qed
-      moreover have "?f thread < ?t"
-      proof -
-        from Cons have " extend_highest_set s' th prio (e # t)" by auto
-        from extend_highest_set.create_low[OF this] and eq_e
-        have "prio' \<le> prio" by auto
-        thus ?thesis
-        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
-    qed
-    ultimately show ?thesis by (auto simp:max_def)
-  qed
-next
-    case (Exit thread)
-    assume eq_e: "e = Exit thread"
-    from Cons have vt_e: "vt step (e#(t @ s))" by auto
-    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
-    from stp have thread_ts: "thread \<in> threads (t @ s)"
-      by(cases, unfold runing_def readys_def, auto)
-    from Cons have "extend_highest_set s' th prio (e # t)" by auto
-    from extend_highest_set.exit_diff[OF this] and eq_e
-    have neq_thread: "thread \<noteq> th" by auto
-    from Cons have "extend_highest_set s' th prio t" by auto
-    from extend_highest_set.th_kept[OF this, folded s_def]
-    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      have "threads (t@s) = insert thread ?A"
-        by (insert stp thread_ts, unfold eq_e, auto)
-      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
-      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
-      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
-      proof(rule Max_insert)
-        from finite_threads [OF vt_e]
-        show "finite (?f ` ?A)" by simp
-      next
-        from Cons have "extend_highest_set s' th prio (e # t)" by auto
-        from extend_highest_set.th_kept[OF this]
-        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
-      qed
-      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
-      moreover have "Max (?f ` (threads (t@s))) = ?t"
-      proof -
-        from Cons show ?thesis
-          by (unfold eq_e, auto simp:preced_def)
-      qed
-      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
-      moreover have "?f thread < ?t"
-      proof(unfold eq_e, simp add:preced_def, fold preced_def)
-        show "preced thread (t @ s) < ?t"
-        proof -
-          have "preced thread (t @ s) \<le> ?t" 
-          proof -
-            from Cons
-            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
-              (is "?t = Max (?g ` ?B)") by simp
-            moreover have "?g thread \<le> \<dots>"
-            proof(rule Max_ge)
-              have "vt step (t@s)" by fact
-              from finite_threads [OF this]
-              show "finite (?g ` ?B)" by simp
-            next
-              from thread_ts
-              show "?g thread \<in> (?g ` ?B)" by auto
-            qed
-            ultimately show ?thesis by auto
-          qed
-          moreover have "preced thread (t @ s) \<noteq> ?t"
-          proof
-            assume "preced thread (t @ s) = preced th s"
-            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
-            from preced_unique [OF this] have "thread = th"
-            proof
-              from h' show "th \<in> threads (t @ s)" by simp
-            next
-              from thread_ts show "thread \<in> threads (t @ s)" .
-            qed(simp)
-            with neq_thread show "False" by simp
-          qed
-          ultimately show ?thesis by auto
-        qed
-      qed
-      ultimately show ?thesis 
-        by (auto simp:max_def split:if_splits)
-    qed
-  next
-    case (P thread cs)
-    with Cons
-    show ?thesis by (auto simp:preced_def)
-  next
-    case (V thread cs)
-    with Cons
-    show ?thesis by (auto simp:preced_def)
-  next
-    case (Set thread prio')
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      let ?B = "threads (t@s)"
-      from Cons have "extend_highest_set s' th prio (e # t)" by auto
-      from extend_highest_set.set_diff_low[OF this] and Set
-      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
-      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
-      also have "\<dots> = ?t"
-      proof(rule Max_eqI)
-        fix y
-        assume y_in: "y \<in> ?f ` ?B"
-        then obtain th1 where 
-          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
-        show "y \<le> ?t"
-        proof(cases "th1 = thread")
-          case True
-          with neq_thread le_p eq_y s_def Set
-          show ?thesis
-            by (auto simp:preced_def precedence_le_def)
-        next
-          case False
-          with Set eq_y
-          have "y  = preced th1 (t@s)"
-            by (simp add:preced_def)
-          moreover have "\<dots> \<le> ?t"
-          proof -
-            from Cons
-            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
-              by auto
-            moreover have "preced th1 (t@s) \<le> \<dots>"
-            proof(rule Max_ge)
-              from th1_in 
-              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
-                by simp
-            next
-              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
-              proof -
-                from Cons have "vt step (t @ s)" by auto
-                from finite_threads[OF this] show ?thesis by auto
-              qed
-            qed
-            ultimately show ?thesis by auto
-          qed
-          ultimately show ?thesis by auto
-        qed
-      next
-        from Cons and finite_threads
-        show "finite (?f ` ?B)" by auto
-      next
-        from Cons have "extend_highest_set s' th prio t" by auto
-        from extend_highest_set.th_kept [OF this, folded s_def]
-        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
-        show "?t \<in> (?f ` ?B)" 
-        proof -
-          from neq_thread Set h
-          have "?t = ?f th" by (auto simp:preced_def)
-          with h show ?thesis by auto
-        qed
-      qed
-      finally show ?thesis .
-    qed
-  qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
-  by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
-  (is "?L = ?R")
-proof -
-  have "?L = cpreced (t@s) (wq (t@s)) th" 
-    by (unfold cp_eq_cpreced, simp)
-  also have "\<dots> = ?R"
-  proof(unfold cpreced_def)
-    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
-          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
-      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
-    proof(cases "?A = {}")
-      case False
-      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
-      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
-      proof(rule Max_insert)
-        show "finite (?f ` ?A)"
-        proof -
-          from dependents_threads[OF vt_t]
-          have "?A \<subseteq> threads (t@s)" .
-          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
-          ultimately show ?thesis 
-            by (auto simp:finite_subset)
-        qed
-      next
-        from False show "(?f ` ?A) \<noteq> {}" by simp
-      qed
-      moreover have "\<dots> = Max (?f ` ?B)"
-      proof -
-        from max_preced have "?f th = Max (?f ` ?B)" .
-        moreover have "Max (?f ` ?A) \<le> \<dots>" 
-        proof(rule Max_mono)
-          from False show "(?f ` ?A) \<noteq> {}" by simp
-        next
-          show "?f ` ?A \<subseteq> ?f ` ?B" 
-          proof -
-            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
-            thus ?thesis by auto
-          qed
-        next
-          from finite_threads[OF vt_t] 
-          show "finite (?f ` ?B)" by simp
-        qed
-        ultimately show ?thesis
-          by (auto simp:max_def)
-      qed
-      ultimately show ?thesis by auto
-    next
-      case True
-      with max_preced show ?thesis by auto
-    qed
-  qed
-  finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
-  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
-  by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less':
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and neq_th': "th' \<noteq> th"
-  shows "preced th' s < preced th s"
-proof -
-  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
-  proof(rule Max_ge)
-    from finite_threads [OF vt_s]
-    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
-  next
-    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
-      by simp
-  qed
-  moreover have "preced th' s \<noteq> preced th s"
-  proof
-    assume "preced th' s = preced th s"
-    from preced_unique[OF this th'_in] neq_th' is_ready
-    show "False" by  (auto simp:readys_def)
-  qed
-  ultimately show ?thesis using highest_preced_thread
-    by auto
-qed
-
-lemma pv_blocked:
-  fixes th'
-  assumes th'_in: "th' \<in> threads (t@s)"
-  and neq_th': "th' \<noteq> th"
-  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
-  shows "th' \<notin> runing (t@s)"
-proof
-  assume "th' \<in> runing (t@s)"
-  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
-    by (auto simp:runing_def)
-  with max_cp_readys_threads [OF vt_t]
-  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
-    by auto
-  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
-  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
-  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
-    by simp
-  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
-  show False
-  proof -
-    have "dependents (wq (t @ s)) th' = {}" 
-      by (rule count_eq_dependents [OF vt_t eq_pv])
-    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
-    proof
-      assume "preced th' (t @ s) = preced th (t @ s)"
-      hence "th' = th"
-      proof(rule preced_unique)
-        from th_kept show "th \<in> threads (t @ s)" by simp
-      next
-        from th'_in show "th' \<in> threads (t @ s)" by simp
-      qed
-      with assms show False by simp
-    qed
-    ultimately show ?thesis
-      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
-  qed
-qed
-
-lemma runing_precond_pre:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and eq_pv: "cntP s th' = cntV s th'"
-  and neq_th': "th' \<noteq> th"
-  shows "th' \<in> threads (t@s) \<and>
-         cntP (t@s) th' = cntV (t@s) th'"
-proof -
-  show ?thesis
-  proof(induct rule:ind)
-    case (Cons e t)
-    from Cons
-    have in_thread: "th' \<in> threads (t @ s)"
-      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-    have "extend_highest_set s' th prio t" by fact
-    from extend_highest_set.pv_blocked 
-    [OF this, folded s_def, OF in_thread neq_th' not_holding]
-    have not_runing: "th' \<notin> runing (t @ s)" .
-    show ?case
-    proof(cases e)
-      case (V thread cs)
-      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
-
-      show ?thesis
-      proof -
-        from Cons and V have "step (t@s) (V thread cs)" by auto
-        hence neq_th': "thread \<noteq> th'"
-        proof(cases)
-          assume "thread \<in> runing (t@s)"
-          moreover have "th' \<notin> runing (t@s)" by fact
-          ultimately show ?thesis by auto
-        qed
-        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
-          by (unfold V, simp add:cntP_def cntV_def count_def)
-        moreover from in_thread
-        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
-        ultimately show ?thesis by auto
-      qed
-    next
-      case (P thread cs)
-      from Cons and P have "step (t@s) (P thread cs)" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<in> runing (t@s)"
-        moreover note not_runing
-        ultimately show ?thesis by auto
-      qed
-      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
-        by auto
-      ultimately show ?thesis by auto
-    next
-      case (Create thread prio')
-      from Cons and Create have "step (t@s) (Create thread prio')" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<notin> threads (t @ s)"
-        moreover have "th' \<in> threads (t@s)" by fact
-        ultimately show ?thesis by auto
-      qed
-      with Cons and Create 
-      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and Create 
-      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
-      ultimately show ?thesis by auto
-    next
-      case (Exit thread)
-      from Cons and Exit have "step (t@s) (Exit thread)" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<in> runing (t @ s)"
-        moreover note not_runing
-        ultimately show ?thesis by auto
-      qed
-      with Cons and Exit 
-      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and Exit and neq_th' 
-      have in_thread': "th' \<in> threads ((e # t) @ s)"
-        by auto
-      ultimately show ?thesis by auto
-    next
-      case (Set thread prio')
-      with Cons
-      show ?thesis 
-        by (auto simp:cntP_def cntV_def count_def)
-    qed
-  next
-    case Nil
-    with assms
-    show ?case by auto
-  qed
-qed
-
-(*
-lemma runing_precond:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and eq_pv: "cntP s th' = cntV s th'"
-  and neq_th': "th' \<noteq> th"
-  shows "th' \<notin> runing (t@s)"
-proof -
-  from runing_precond_pre[OF th'_in eq_pv neq_th']
-  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-  from pv_blocked[OF h1 neq_th' h2] 
-  show ?thesis .
-qed
-*)
-
-lemma runing_precond:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and neq_th': "th' \<noteq> th"
-  and is_runing: "th' \<in> runing (t@s)"
-  shows "cntP s th' > cntV s th'"
-proof -
-  have "cntP s th' \<noteq> cntV s th'"
-  proof
-    assume eq_pv: "cntP s th' = cntV s th'"
-    from runing_precond_pre[OF th'_in eq_pv neq_th']
-    have h1: "th' \<in> threads (t @ s)"  
-      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
-    with is_runing show "False" by simp
-  qed
-  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
-  have "cntV s th' \<le> cntP s th'" by auto
-  ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
-  assumes neq_th': "th' \<noteq> th"
-  and th'_in: "th' \<in> threads ((moment i t)@s)"
-  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
-  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
-         th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
-  case (Suc k)
-  show ?case
-  proof -
-    { assume True: "Suc (i+k) \<le> length t"
-      from moment_head [OF this] 
-      obtain e where
-        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
-        by blast
-      from red_moment[of "Suc(i+k)"]
-      and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
-      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
-        by (unfold extend_highest_set_def extend_highest_set_axioms_def 
-          highest_set_def s_def, auto)
-      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
-      proof(unfold s_def)
-        show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
-        proof(rule extend_highest_set.pv_blocked)
-          from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
-            by (simp add:s_def)
-        next
-          from neq_th' show "th' \<noteq> th" .
-        next
-          from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
-        next
-          from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
-            cntV (moment (i + k) t @ Set th prio # s') th'"
-            by (auto simp:s_def)
-        qed
-      qed
-      from step_back_step[OF vt_e]
-      have "step ((moment (i + k) t)@s) e" .
-      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
-        th' \<in> threads (e#(moment (i + k) t)@s)
-        "
-      proof(cases)
-        case (thread_create thread prio)
-        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_exit thread)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_P thread cs)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_V thread cs)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_set thread prio')
-        with Suc show ?thesis
-          by (auto simp:cntP_def cntV_def count_def)
-      qed
-      with eq_me have ?thesis using eq_me by auto 
-    } note h = this
-    show ?thesis
-    proof(cases "Suc (i+k) \<le> length t")
-      case True
-      from h [OF this] show ?thesis .
-    next
-      case False
-      with moment_ge
-      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
-      with Suc show ?thesis by auto
-    qed
-  qed
-next
-  case 0
-  from assms show ?case by auto
-qed
-
-lemma moment_blocked:
-  assumes neq_th': "th' \<noteq> th"
-  and th'_in: "th' \<in> threads ((moment i t)@s)"
-  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
-  and le_ij: "i \<le> j"
-  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
-         th' \<in> threads ((moment j t)@s) \<and>
-         th' \<notin> runing ((moment j t)@s)"
-proof -
-  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
-  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
-    and h2: "th' \<in> threads ((moment j t)@s)" by auto
-  with extend_highest_set.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
-  show ?thesis by auto
-qed
-
-lemma runing_inversion_1:
-  assumes neq_th': "th' \<noteq> th"
-  and runing': "th' \<in> runing (t@s)"
-  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
-  case True
-  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
-  case False
-  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
-  let ?q = "moment 0 t"
-  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
-  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
-  from p_split_gen [of ?Q, OF this not_thread]
-  obtain i where lt_its: "i < length t"
-    and le_i: "0 \<le> i"
-    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
-    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
-  from lt_its have "Suc i \<le> length t" by auto
-  from moment_head[OF this] obtain e where 
-   eq_me: "moment (Suc i) t = e # moment i t" by blast
-  from red_moment[of "Suc i"] and eq_me
-  have "extend_highest_set s' th prio (e # moment i t)" by simp
-  hence vt_e: "vt step (e#(moment i t)@s)"
-    by (unfold extend_highest_set_def extend_highest_set_axioms_def 
-      highest_set_def s_def, auto)
-  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
-  from post[rule_format, of "Suc i"] and eq_me 
-  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
-  from create_pre[OF stp_i pre this] 
-  obtain prio where eq_e: "e = Create th' prio" .
-  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
-  proof(rule cnp_cnv_eq)
-    from step_back_vt [OF vt_e] 
-    show "vt step (moment i t @ s)" .
-  next
-    from eq_e and stp_i 
-    have "step (moment i t @ s) (Create th' prio)" by simp
-    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
-  qed
-  with eq_e
-  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
-    by (simp add:cntP_def cntV_def count_def)
-  with eq_me[symmetric]
-  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
-    by simp
-  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
-  with eq_me [symmetric]
-  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
-  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
-  and moment_ge
-  have "th' \<notin> runing (t @ s)" by auto
-  with runing'
-  show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
-  assumes runing': "th' \<in> runing (t@s)"
-  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
-  from runing_inversion_1[OF _ runing']
-  show ?thesis by auto
-qed
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
-  case True thus ?thesis by auto
-next
-  case False
-  then have not_ready: "th \<notin> readys (t@s)"
-    apply (unfold runing_def, 
-            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
-    by auto
-  from th_kept have "th \<in> threads (t@s)" by auto
-  from th_chain_to_ready[OF vt_t this] and not_ready
-  obtain th' where th'_in: "th' \<in> readys (t@s)"
-    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
-  have "th' \<in> runing (t@s)"
-  proof -
-    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
-    proof -
-      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
-               preced th (t@s)"
-      proof(rule Max_eqI)
-        fix y
-        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
-        then obtain th1 where
-          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
-          and eq_y: "y = preced th1 (t@s)" by auto
-        show "y \<le> preced th (t @ s)"
-        proof -
-          from max_preced
-          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
-          moreover have "y \<le> \<dots>"
-          proof(rule Max_ge)
-            from h1
-            have "th1 \<in> threads (t@s)"
-            proof
-              assume "th1 = th'"
-              with th'_in show ?thesis by (simp add:readys_def)
-            next
-              assume "th1 \<in> dependents (wq (t @ s)) th'"
-              with dependents_threads [OF vt_t]
-              show "th1 \<in> threads (t @ s)" by auto
-            qed
-            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
-          next
-            from finite_threads[OF vt_t]
-            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
-          qed
-          ultimately show ?thesis by auto
-        qed
-      next
-        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
-        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
-          by (auto intro:finite_subset)
-      next
-        from dp
-        have "th \<in> dependents (wq (t @ s)) th'" 
-          by (unfold cs_dependents_def, auto simp:eq_depend)
-        thus "preced th (t @ s) \<in> 
-                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
-          by auto
-      qed
-      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
-      proof -
-        from max_preced and max_cp_eq[OF vt_t, symmetric]
-        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
-        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
-      qed
-      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
-    qed
-    with th'_in show ?thesis by (auto simp:runing_def)
-  qed
-  thus ?thesis by auto
-qed
-
-end
-
-end
-
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/ExtSG.thy
--- a/prio/ExtSG.thy	Sun Feb 05 14:29:08 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1019 +0,0 @@
-theory ExtSG
-imports PrioG
-begin
-
-locale highest_set =
-  fixes s' th prio fixes s 
-  defines s_def : "s \<equiv> (Set th prio#s')"
-  assumes vt_s: "vt step s"
-  and highest: "preced th s = Max ((cp s)`threads s)"
-
-context highest_set
-begin
-
-lemma vt_s': "vt step s'"
-  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
-
-lemma step_set: "step s' (Set th prio)"
-  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
-
-lemma step_set_elim: 
-  "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
-  by (insert step_set, ind_cases "step s' (Set th prio)", auto)
-
-
-lemma threads_s: "th \<in> threads s"
-  by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
-
-lemma same_depend: "depend s = depend s'"
-  by (insert depend_set_unchanged, unfold s_def, simp)
-
-lemma same_dependents:
-  "dependents (wq s) th = dependents (wq s') th"
-  apply (unfold cs_dependents_def)
-  by (unfold eq_depend same_depend, simp)
-
-lemma eq_cp_s_th: "cp s th = preced th s"
-proof -
-  from highest and max_cp_eq[OF vt_s]
-  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
-  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
-  proof -
-    from threads_s and dependents_threads[OF vt_s, of th]
-    show ?thesis by auto
-  qed
-  show ?thesis
-  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
-    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
-  next
-    fix y 
-    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
-    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
-      and eq_y: "y = preced th1 s" by auto
-    show "y \<le> preced th s"
-    proof(unfold is_max, rule Max_ge)
-      from finite_threads[OF vt_s] 
-      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-    next
-      from sbs th1_in and eq_y 
-      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
-    qed
-  next
-    from sbs and finite_threads[OF vt_s]
-    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
-      by (auto intro:finite_subset)
-  qed
-qed
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
-  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
-  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma is_ready: "th \<in> readys s"
-proof -
-  have "\<forall>cs. \<not> waiting s th cs"
-    apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
-    apply (unfold s_depend_def, unfold runing_def readys_def)
-    apply (auto, fold s_def)
-    apply (erule_tac x = cs in allE, auto simp:waiting_eq)
-  proof -
-    fix cs
-    assume h: 
-      "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
-          {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
-            (is "?L = ?R")
-    and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
-    from wt have "(Th th, Cs cs) \<in> ?L" by auto
-    with h have "(Th th, Cs cs) \<in> ?R" by simp
-    hence "waiting (wq s') th cs" by auto with nwt
-    show False by auto
-  qed    
-  with threads_s show ?thesis 
-    by (unfold readys_def, auto)
-qed
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
-proof -
-  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
-  show ?thesis by simp
-qed
-
-lemma is_runing: "th \<in> runing s"
-proof -
-  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
-  proof -
-    have " Max (cp s ` readys s) = cp s th"
-    proof(rule Max_eqI)
-      from finite_threads[OF vt_s] readys_threads finite_subset
-      have "finite (readys s)" by blast
-      thus "finite (cp s ` readys s)" by auto
-    next
-      from is_ready show "cp s th \<in> cp s ` readys s" by auto
-    next
-      fix y
-      assume "y \<in> cp s ` readys s"
-      then obtain th1 where 
-        eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
-      show  "y \<le> cp s th" 
-      proof -
-        have "y \<le> Max (cp s ` threads s)"
-        proof(rule Max_ge)
-          from eq_y and th1_in
-          show "y \<in> cp s ` threads s"
-            by (auto simp:readys_def)
-        next
-          from finite_threads[OF vt_s]
-          show "finite (cp s ` threads s)" by auto
-        qed
-        with highest' show ?thesis by auto
-      qed
-    qed
-    with highest' show ?thesis by auto
-  qed
-  thus ?thesis
-    by (unfold runing_def, insert highest' is_ready, auto)
-qed
-
-end
-
-locale extend_highest_set = highest_set + 
-  fixes t 
-  assumes vt_t: "vt step (t@s)"
-  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
-  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
-  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app: 
-  assumes vt_ts: "vt cs (t@s)" 
-  shows "vt cs s"
-proof -
-  from vt_ts show ?thesis
-  proof(induct t)
-    case Nil
-    from Nil show ?case by auto
-  next
-    case (Cons e t)
-    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
-      and vt_et: "vt cs ((e # t) @ s)"
-    show ?case
-    proof(rule ih)
-      show "vt cs (t @ s)"
-      proof(rule step_back_vt)
-        from vt_et show "vt cs (e # t @ s)" by simp
-      qed
-    qed
-  qed
-qed
-
-context extend_highest_set
-begin
-
-lemma red_moment:
-  "extend_highest_set s' th prio (moment i t)"
-  apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
-  apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
-  by (unfold highest_set_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
-  assumes 
-    h0: "R []"
-  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
-                    extend_highest_set s' th prio t; 
-                    extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
-  shows "R t"
-proof -
-  from vt_t extend_highest_set_axioms show ?thesis
-  proof(induct t)
-    from h0 show "R []" .
-  next
-    case (Cons e t')
-    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
-      and vt_e: "vt step ((e # t') @ s)"
-      and et: "extend_highest_set s' th prio (e # t')"
-    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
-    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
-    show ?case
-    proof(rule h2 [OF vt_ts stp _ _ _ ])
-      show "R t'"
-      proof(rule ih)
-        from et show ext': "extend_highest_set s' th prio t'"
-          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
-      next
-        from vt_ts show "vt step (t' @ s)" .
-      qed
-    next
-      from et show "extend_highest_set s' th prio (e # t')" .
-    next
-      from et show ext': "extend_highest_set s' th prio t'"
-          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
-    qed
-  qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and> 
-        preced th (t@s) = preced th s" (is "?Q t")
-proof -
-  show ?thesis
-  proof(induct rule:ind)
-    case Nil
-    from threads_s
-    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
-      by auto
-  next
-    case (Cons e t)
-    show ?case
-    proof(cases e)
-      case (Create thread prio)
-      assume eq_e: " e = Create thread prio"
-      show ?thesis
-      proof -
-        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
-        hence "th \<noteq> thread"
-        proof(cases)
-          assume "thread \<notin> threads (t @ s)"
-          with Cons show ?thesis by auto
-        qed
-        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
-          by (unfold eq_e, auto simp:preced_def)
-        moreover note Cons
-        ultimately show ?thesis
-          by (auto simp:eq_e)
-      qed
-    next
-      case (Exit thread)
-      assume eq_e: "e = Exit thread"
-      from Cons have "extend_highest_set s' th prio (e # t)" by auto
-      from extend_highest_set.exit_diff [OF this] and eq_e
-      have neq_th: "thread \<noteq> th" by auto
-      with Cons
-      show ?thesis
-        by (unfold eq_e, auto simp:preced_def)
-    next
-      case (P thread cs)
-      assume eq_e: "e = P thread cs"
-      with Cons
-      show ?thesis 
-        by (auto simp:eq_e preced_def)
-    next
-      case (V thread cs)
-      assume eq_e: "e = V thread cs"
-      with Cons
-      show ?thesis 
-        by (auto simp:eq_e preced_def)
-    next
-      case (Set thread prio')
-      assume eq_e: " e = Set thread prio'"
-      show ?thesis
-      proof -
-        from Cons have "extend_highest_set s' th prio (e # t)" by auto
-        from extend_highest_set.set_diff_low[OF this] and eq_e
-        have "th \<noteq> thread" by auto
-        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
-          by (unfold eq_e, auto simp:preced_def)
-        moreover note Cons
-        ultimately show ?thesis
-          by (auto simp:eq_e)
-      qed
-    qed
-  qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
-  case Nil
-  from highest_preced_thread
-  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
-    by simp
-next
-  case (Cons e t)
-  show ?case
-  proof(cases e)
-    case (Create thread prio')
-    assume eq_e: " e = Create thread prio'"
-    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
-    hence neq_thread: "thread \<noteq> th"
-    proof(cases)
-      assume "thread \<notin> threads (t @ s)"
-      moreover have "th \<in> threads (t@s)"
-      proof -
-        from Cons have "extend_highest_set s' th prio t" by auto
-        from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
-      qed
-      ultimately show ?thesis by auto
-    qed
-    from Cons have "extend_highest_set s' th prio t" by auto
-    from extend_highest_set.th_kept[OF this]
-    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
-      by (auto simp:s_def)
-    from stp
-    have thread_ts: "thread \<notin> threads (t @ s)"
-      by (cases, auto)
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
-        by (unfold eq_e, simp)
-      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
-      proof(rule Max_insert)
-        from Cons have "vt step (t @ s)" by auto
-        from finite_threads[OF this]
-        show "finite (?f ` (threads (t@s)))" by simp
-      next
-        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
-      qed
-      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
-      proof -
-        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
-          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
-        proof -
-          { fix th' 
-            assume "th' \<in> ?B"
-            with thread_ts eq_e
-            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
-          } thus ?thesis 
-            apply (auto simp:Image_def)
-          proof -
-            fix th'
-            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
-              preced th' (e # t @ s) = preced th' (t @ s)"
-              and h1: "th' \<in> threads (t @ s)"
-            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
-            proof -
-              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
-              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
-              ultimately show ?thesis by simp
-            qed
-          qed
-        qed
-        with Cons show ?thesis by auto
-      qed
-      moreover have "?f thread < ?t"
-      proof -
-        from Cons have " extend_highest_set s' th prio (e # t)" by auto
-        from extend_highest_set.create_low[OF this] and eq_e
-        have "prio' \<le> prio" by auto
-        thus ?thesis
-        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
-    qed
-    ultimately show ?thesis by (auto simp:max_def)
-  qed
-next
-    case (Exit thread)
-    assume eq_e: "e = Exit thread"
-    from Cons have vt_e: "vt step (e#(t @ s))" by auto
-    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
-    from stp have thread_ts: "thread \<in> threads (t @ s)"
-      by(cases, unfold runing_def readys_def, auto)
-    from Cons have "extend_highest_set s' th prio (e # t)" by auto
-    from extend_highest_set.exit_diff[OF this] and eq_e
-    have neq_thread: "thread \<noteq> th" by auto
-    from Cons have "extend_highest_set s' th prio t" by auto
-    from extend_highest_set.th_kept[OF this, folded s_def]
-    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      have "threads (t@s) = insert thread ?A"
-        by (insert stp thread_ts, unfold eq_e, auto)
-      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
-      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
-      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
-      proof(rule Max_insert)
-        from finite_threads [OF vt_e]
-        show "finite (?f ` ?A)" by simp
-      next
-        from Cons have "extend_highest_set s' th prio (e # t)" by auto
-        from extend_highest_set.th_kept[OF this]
-        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
-      qed
-      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
-      moreover have "Max (?f ` (threads (t@s))) = ?t"
-      proof -
-        from Cons show ?thesis
-          by (unfold eq_e, auto simp:preced_def)
-      qed
-      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
-      moreover have "?f thread < ?t"
-      proof(unfold eq_e, simp add:preced_def, fold preced_def)
-        show "preced thread (t @ s) < ?t"
-        proof -
-          have "preced thread (t @ s) \<le> ?t" 
-          proof -
-            from Cons
-            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
-              (is "?t = Max (?g ` ?B)") by simp
-            moreover have "?g thread \<le> \<dots>"
-            proof(rule Max_ge)
-              have "vt step (t@s)" by fact
-              from finite_threads [OF this]
-              show "finite (?g ` ?B)" by simp
-            next
-              from thread_ts
-              show "?g thread \<in> (?g ` ?B)" by auto
-            qed
-            ultimately show ?thesis by auto
-          qed
-          moreover have "preced thread (t @ s) \<noteq> ?t"
-          proof
-            assume "preced thread (t @ s) = preced th s"
-            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
-            from preced_unique [OF this] have "thread = th"
-            proof
-              from h' show "th \<in> threads (t @ s)" by simp
-            next
-              from thread_ts show "thread \<in> threads (t @ s)" .
-            qed(simp)
-            with neq_thread show "False" by simp
-          qed
-          ultimately show ?thesis by auto
-        qed
-      qed
-      ultimately show ?thesis 
-        by (auto simp:max_def split:if_splits)
-    qed
-  next
-    case (P thread cs)
-    with Cons
-    show ?thesis by (auto simp:preced_def)
-  next
-    case (V thread cs)
-    with Cons
-    show ?thesis by (auto simp:preced_def)
-  next
-    case (Set thread prio')
-    show ?thesis (is "Max (?f ` ?A) = ?t")
-    proof -
-      let ?B = "threads (t@s)"
-      from Cons have "extend_highest_set s' th prio (e # t)" by auto
-      from extend_highest_set.set_diff_low[OF this] and Set
-      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
-      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
-      also have "\<dots> = ?t"
-      proof(rule Max_eqI)
-        fix y
-        assume y_in: "y \<in> ?f ` ?B"
-        then obtain th1 where 
-          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
-        show "y \<le> ?t"
-        proof(cases "th1 = thread")
-          case True
-          with neq_thread le_p eq_y s_def Set
-          show ?thesis
-            by (auto simp:preced_def precedence_le_def)
-        next
-          case False
-          with Set eq_y
-          have "y  = preced th1 (t@s)"
-            by (simp add:preced_def)
-          moreover have "\<dots> \<le> ?t"
-          proof -
-            from Cons
-            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
-              by auto
-            moreover have "preced th1 (t@s) \<le> \<dots>"
-            proof(rule Max_ge)
-              from th1_in 
-              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
-                by simp
-            next
-              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
-              proof -
-                from Cons have "vt step (t @ s)" by auto
-                from finite_threads[OF this] show ?thesis by auto
-              qed
-            qed
-            ultimately show ?thesis by auto
-          qed
-          ultimately show ?thesis by auto
-        qed
-      next
-        from Cons and finite_threads
-        show "finite (?f ` ?B)" by auto
-      next
-        from Cons have "extend_highest_set s' th prio t" by auto
-        from extend_highest_set.th_kept [OF this, folded s_def]
-        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
-        show "?t \<in> (?f ` ?B)" 
-        proof -
-          from neq_thread Set h
-          have "?t = ?f th" by (auto simp:preced_def)
-          with h show ?thesis by auto
-        qed
-      qed
-      finally show ?thesis .
-    qed
-  qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
-  by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
-  (is "?L = ?R")
-proof -
-  have "?L = cpreced (t@s) (wq (t@s)) th" 
-    by (unfold cp_eq_cpreced, simp)
-  also have "\<dots> = ?R"
-  proof(unfold cpreced_def)
-    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
-          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
-      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
-    proof(cases "?A = {}")
-      case False
-      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
-      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
-      proof(rule Max_insert)
-        show "finite (?f ` ?A)"
-        proof -
-          from dependents_threads[OF vt_t]
-          have "?A \<subseteq> threads (t@s)" .
-          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
-          ultimately show ?thesis 
-            by (auto simp:finite_subset)
-        qed
-      next
-        from False show "(?f ` ?A) \<noteq> {}" by simp
-      qed
-      moreover have "\<dots> = Max (?f ` ?B)"
-      proof -
-        from max_preced have "?f th = Max (?f ` ?B)" .
-        moreover have "Max (?f ` ?A) \<le> \<dots>" 
-        proof(rule Max_mono)
-          from False show "(?f ` ?A) \<noteq> {}" by simp
-        next
-          show "?f ` ?A \<subseteq> ?f ` ?B" 
-          proof -
-            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
-            thus ?thesis by auto
-          qed
-        next
-          from finite_threads[OF vt_t] 
-          show "finite (?f ` ?B)" by simp
-        qed
-        ultimately show ?thesis
-          by (auto simp:max_def)
-      qed
-      ultimately show ?thesis by auto
-    next
-      case True
-      with max_preced show ?thesis by auto
-    qed
-  qed
-  finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
-  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
-  by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less':
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and neq_th': "th' \<noteq> th"
-  shows "preced th' s < preced th s"
-proof -
-  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
-  proof(rule Max_ge)
-    from finite_threads [OF vt_s]
-    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
-  next
-    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
-      by simp
-  qed
-  moreover have "preced th' s \<noteq> preced th s"
-  proof
-    assume "preced th' s = preced th s"
-    from preced_unique[OF this th'_in] neq_th' is_ready
-    show "False" by  (auto simp:readys_def)
-  qed
-  ultimately show ?thesis using highest_preced_thread
-    by auto
-qed
-
-lemma pv_blocked:
-  fixes th'
-  assumes th'_in: "th' \<in> threads (t@s)"
-  and neq_th': "th' \<noteq> th"
-  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
-  shows "th' \<notin> runing (t@s)"
-proof
-  assume "th' \<in> runing (t@s)"
-  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
-    by (auto simp:runing_def)
-  with max_cp_readys_threads [OF vt_t]
-  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
-    by auto
-  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
-  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
-  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
-    by simp
-  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
-  show False
-  proof -
-    have "dependents (wq (t @ s)) th' = {}" 
-      by (rule count_eq_dependents [OF vt_t eq_pv])
-    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
-    proof
-      assume "preced th' (t @ s) = preced th (t @ s)"
-      hence "th' = th"
-      proof(rule preced_unique)
-        from th_kept show "th \<in> threads (t @ s)" by simp
-      next
-        from th'_in show "th' \<in> threads (t @ s)" by simp
-      qed
-      with assms show False by simp
-    qed
-    ultimately show ?thesis
-      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
-  qed
-qed
-
-lemma runing_precond_pre:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and eq_pv: "cntP s th' = cntV s th'"
-  and neq_th': "th' \<noteq> th"
-  shows "th' \<in> threads (t@s) \<and>
-         cntP (t@s) th' = cntV (t@s) th'"
-proof -
-  show ?thesis
-  proof(induct rule:ind)
-    case (Cons e t)
-    from Cons
-    have in_thread: "th' \<in> threads (t @ s)"
-      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-    have "extend_highest_set s' th prio t" by fact
-    from extend_highest_set.pv_blocked 
-    [OF this, folded s_def, OF in_thread neq_th' not_holding]
-    have not_runing: "th' \<notin> runing (t @ s)" .
-    show ?case
-    proof(cases e)
-      case (V thread cs)
-      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
-
-      show ?thesis
-      proof -
-        from Cons and V have "step (t@s) (V thread cs)" by auto
-        hence neq_th': "thread \<noteq> th'"
-        proof(cases)
-          assume "thread \<in> runing (t@s)"
-          moreover have "th' \<notin> runing (t@s)" by fact
-          ultimately show ?thesis by auto
-        qed
-        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
-          by (unfold V, simp add:cntP_def cntV_def count_def)
-        moreover from in_thread
-        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
-        ultimately show ?thesis by auto
-      qed
-    next
-      case (P thread cs)
-      from Cons and P have "step (t@s) (P thread cs)" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<in> runing (t@s)"
-        moreover note not_runing
-        ultimately show ?thesis by auto
-      qed
-      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
-        by auto
-      ultimately show ?thesis by auto
-    next
-      case (Create thread prio')
-      from Cons and Create have "step (t@s) (Create thread prio')" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<notin> threads (t @ s)"
-        moreover have "th' \<in> threads (t@s)" by fact
-        ultimately show ?thesis by auto
-      qed
-      with Cons and Create 
-      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and Create 
-      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
-      ultimately show ?thesis by auto
-    next
-      case (Exit thread)
-      from Cons and Exit have "step (t@s) (Exit thread)" by auto
-      hence neq_th': "thread \<noteq> th'"
-      proof(cases)
-        assume "thread \<in> runing (t @ s)"
-        moreover note not_runing
-        ultimately show ?thesis by auto
-      qed
-      with Cons and Exit 
-      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
-        by (auto simp:cntP_def cntV_def count_def)
-      moreover from Cons and Exit and neq_th' 
-      have in_thread': "th' \<in> threads ((e # t) @ s)"
-        by auto
-      ultimately show ?thesis by auto
-    next
-      case (Set thread prio')
-      with Cons
-      show ?thesis 
-        by (auto simp:cntP_def cntV_def count_def)
-    qed
-  next
-    case Nil
-    with assms
-    show ?case by auto
-  qed
-qed
-
-(*
-lemma runing_precond:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and eq_pv: "cntP s th' = cntV s th'"
-  and neq_th': "th' \<noteq> th"
-  shows "th' \<notin> runing (t@s)"
-proof -
-  from runing_precond_pre[OF th'_in eq_pv neq_th']
-  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-  from pv_blocked[OF h1 neq_th' h2] 
-  show ?thesis .
-qed
-*)
-
-lemma runing_precond:
-  fixes th'
-  assumes th'_in: "th' \<in> threads s"
-  and neq_th': "th' \<noteq> th"
-  and is_runing: "th' \<in> runing (t@s)"
-  shows "cntP s th' > cntV s th'"
-proof -
-  have "cntP s th' \<noteq> cntV s th'"
-  proof
-    assume eq_pv: "cntP s th' = cntV s th'"
-    from runing_precond_pre[OF th'_in eq_pv neq_th']
-    have h1: "th' \<in> threads (t @ s)"  
-      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
-    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
-    with is_runing show "False" by simp
-  qed
-  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
-  have "cntV s th' \<le> cntP s th'" by auto
-  ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
-  assumes neq_th': "th' \<noteq> th"
-  and th'_in: "th' \<in> threads ((moment i t)@s)"
-  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
-  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
-         th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
-  case (Suc k)
-  show ?case
-  proof -
-    { assume True: "Suc (i+k) \<le> length t"
-      from moment_head [OF this] 
-      obtain e where
-        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
-        by blast
-      from red_moment[of "Suc(i+k)"]
-      and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
-      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
-        by (unfold extend_highest_set_def extend_highest_set_axioms_def 
-                          highest_set_def s_def, auto)
-      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
-      proof(unfold s_def)
-        show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
-        proof(rule extend_highest_set.pv_blocked)
-          from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
-            by (simp add:s_def)
-        next
-          from neq_th' show "th' \<noteq> th" .
-        next
-          from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
-        next
-          from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
-            cntV (moment (i + k) t @ Set th prio # s') th'"
-            by (auto simp:s_def)
-        qed
-      qed
-      from step_back_step[OF vt_e]
-      have "step ((moment (i + k) t)@s) e" .
-      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
-        th' \<in> threads (e#(moment (i + k) t)@s)
-        "
-      proof(cases)
-        case (thread_create thread prio)
-        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_exit thread)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_P thread cs)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_V thread cs)
-        moreover have "thread \<noteq> th'"
-        proof -
-          have "thread \<in> runing (moment (i + k) t @ s)" by fact
-          moreover note not_runing'
-          ultimately show ?thesis by auto
-        qed
-        moreover note Suc 
-        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
-      next
-        case (thread_set thread prio')
-        with Suc show ?thesis
-          by (auto simp:cntP_def cntV_def count_def)
-      qed
-      with eq_me have ?thesis using eq_me by auto 
-    } note h = this
-    show ?thesis
-    proof(cases "Suc (i+k) \<le> length t")
-      case True
-      from h [OF this] show ?thesis .
-    next
-      case False
-      with moment_ge
-      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
-      with Suc show ?thesis by auto
-    qed
-  qed
-next
-  case 0
-  from assms show ?case by auto
-qed
-
-lemma moment_blocked:
-  assumes neq_th': "th' \<noteq> th"
-  and th'_in: "th' \<in> threads ((moment i t)@s)"
-  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
-  and le_ij: "i \<le> j"
-  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
-         th' \<in> threads ((moment j t)@s) \<and>
-         th' \<notin> runing ((moment j t)@s)"
-proof -
-  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
-  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
-    and h2: "th' \<in> threads ((moment j t)@s)" by auto
-  with extend_highest_set.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
-  show ?thesis by auto
-qed
-
-lemma runing_inversion_1:
-  assumes neq_th': "th' \<noteq> th"
-  and runing': "th' \<in> runing (t@s)"
-  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
-  case True
-  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
-  case False
-  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
-  let ?q = "moment 0 t"
-  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
-  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
-  from p_split_gen [of ?Q, OF this not_thread]
-  obtain i where lt_its: "i < length t"
-    and le_i: "0 \<le> i"
-    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
-    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
-  from lt_its have "Suc i \<le> length t" by auto
-  from moment_head[OF this] obtain e where 
-   eq_me: "moment (Suc i) t = e # moment i t" by blast
-  from red_moment[of "Suc i"] and eq_me
-  have "extend_highest_set s' th prio (e # moment i t)" by simp
-  hence vt_e: "vt step (e#(moment i t)@s)"
-    by (unfold extend_highest_set_def extend_highest_set_axioms_def 
-      highest_set_def s_def, auto)
-  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
-  from post[rule_format, of "Suc i"] and eq_me 
-  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
-  from create_pre[OF stp_i pre this] 
-  obtain prio where eq_e: "e = Create th' prio" .
-  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
-  proof(rule cnp_cnv_eq)
-    from step_back_vt [OF vt_e] 
-    show "vt step (moment i t @ s)" .
-  next
-    from eq_e and stp_i 
-    have "step (moment i t @ s) (Create th' prio)" by simp
-    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
-  qed
-  with eq_e
-  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
-    by (simp add:cntP_def cntV_def count_def)
-  with eq_me[symmetric]
-  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
-    by simp
-  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
-  with eq_me [symmetric]
-  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
-  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
-  and moment_ge
-  have "th' \<notin> runing (t @ s)" by auto
-  with runing'
-  show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
-  assumes runing': "th' \<in> runing (t@s)"
-  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
-  from runing_inversion_1[OF _ runing']
-  show ?thesis by auto
-qed
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
-  case True thus ?thesis by auto
-next
-  case False
-  then have not_ready: "th \<notin> readys (t@s)"
-    apply (unfold runing_def, 
-            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
-    by auto
-  from th_kept have "th \<in> threads (t@s)" by auto
-  from th_chain_to_ready[OF vt_t this] and not_ready
-  obtain th' where th'_in: "th' \<in> readys (t@s)"
-    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
-  have "th' \<in> runing (t@s)"
-  proof -
-    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
-    proof -
-      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
-               preced th (t@s)"
-      proof(rule Max_eqI)
-        fix y
-        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
-        then obtain th1 where
-          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
-          and eq_y: "y = preced th1 (t@s)" by auto
-        show "y \<le> preced th (t @ s)"
-        proof -
-          from max_preced
-          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
-          moreover have "y \<le> \<dots>"
-          proof(rule Max_ge)
-            from h1
-            have "th1 \<in> threads (t@s)"
-            proof
-              assume "th1 = th'"
-              with th'_in show ?thesis by (simp add:readys_def)
-            next
-              assume "th1 \<in> dependents (wq (t @ s)) th'"
-              with dependents_threads [OF vt_t]
-              show "th1 \<in> threads (t @ s)" by auto
-            qed
-            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
-          next
-            from finite_threads[OF vt_t]
-            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
-          qed
-          ultimately show ?thesis by auto
-        qed
-      next
-        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
-        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
-          by (auto intro:finite_subset)
-      next
-        from dp
-        have "th \<in> dependents (wq (t @ s)) th'" 
-          by (unfold cs_dependents_def, auto simp:eq_depend)
-        thus "preced th (t @ s) \<in> 
-                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
-          by auto
-      qed
-      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
-      proof -
-        from max_preced and max_cp_eq[OF vt_t, symmetric]
-        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
-        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
-      qed
-      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
-    qed
-    with th'_in show ?thesis by (auto simp:runing_def)
-  qed
-  thus ?thesis by auto
-qed
-
-end
-
-end
-
-
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Happen_within.thy
--- a/prio/Happen_within.thy	Sun Feb 05 14:29:08 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,126 +0,0 @@
-theory Happen_within
-imports Main Moment
-begin
-
-(* 
-  lemma 
-  fixes P :: "('a list) \<Rightarrow> bool"
-  and Q :: "('a list) \<Rightarrow> bool"
-  and k :: nat
-  and f :: "('a list) \<Rightarrow> nat"
-  assumes "\<And> s t. \<lbrakk>P s; \<not> Q s; P (t@s); k < length t\<rbrakk> \<Longrightarrow> f (t@s) < f s"
-  shows "\<And> s t. \<lbrakk> P s;  P(t @ s); f(s) * k < length t\<rbrakk> \<Longrightarrow> Q (t@s)"
-  sorry
-*)
-
-text {* 
-  The following two notions are introduced to improve the situation.
-  *}
-
-definition all_future :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> ('a list) \<Rightarrow> bool"
-where "all_future G R s = (\<forall> t. G (t@s) \<longrightarrow> R t)"
-
-definition happen_within :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> ('a list) \<Rightarrow> bool"
-where "happen_within G R k s = all_future G (\<lambda> t. k < length t \<longrightarrow> 
-                                  (\<exists> i \<le> k. R (moment i t @ s) \<and> G (moment i t @ s))) s"
-
-lemma happen_within_intro:
-  fixes P :: "('a list) \<Rightarrow> bool"
-  and Q :: "('a list) \<Rightarrow> bool"
-  and k :: nat
-  and f :: "('a list) \<Rightarrow> nat"
-  assumes 
-  lt_k: "0 < k"
-  and step: "\<And> s. \<lbrakk>P s; \<not> Q s\<rbrakk> \<Longrightarrow> happen_within P (\<lambda> s'. f s' < f s) k s"
-  shows "\<And> s. P s \<Longrightarrow> happen_within P Q ((f s + 1) * k) s"
-proof -
-  fix s
-  assume "P s"
-  thus "happen_within P Q ((f s + 1) * k) s"
-  proof(induct n == "f s + 1" arbitrary:s rule:nat_less_induct)
-    fix s
-    assume ih [rule_format]: "\<forall>m<f s + 1. \<forall>x. m = f x + 1 \<longrightarrow> P x 
-                                 \<longrightarrow> happen_within P Q ((f x + 1) * k) x"
-      and ps: "P s"
-    show "happen_within P Q ((f s + 1) * k) s"
-    proof(cases "Q s")
-      case True
-      show ?thesis 
-      proof -
-        { fix t
-          from True and ps have "0 \<le> ((f s + 1)*k) \<and> Q (moment 0 t @ s) \<and> P (moment 0 t @ s)" by auto
-          hence "\<exists>i\<le>(f s + 1) * k. Q (moment i t @ s) \<and> P (moment i t @ s)" by auto
-        } thus ?thesis by (auto simp: happen_within_def all_future_def)
-      qed
-    next
-      case False
-      from step [OF ps False] have kk: "happen_within P (\<lambda>s'. f s' < f s) k s" .
-      show ?thesis
-      proof -
-        { fix t
-          assume pts: "P (t @ s)" and ltk: "(f s + 1) * k < length t"
-          from ltk have lt_k_lt: "k < length t" by auto
-          with kk pts obtain i 
-            where le_ik: "i \<le> k" 
-            and lt_f: "f (moment i t @ s) < f s" 
-            and p_m: "P (moment i t @ s)"
-            by (auto simp:happen_within_def all_future_def)
-          from ih [of "f (moment i t @ s) + 1" "(moment i t @ s)", OF _ _ p_m] and lt_f
-          have hw: "happen_within P Q ((f (moment i t @ s) + 1) * k) (moment i t @ s)" by auto
-          have "(\<exists>j\<le>(f s + 1) * k. Q (moment j t @ s) \<and>  P (moment j t @ s))" (is "\<exists> j. ?T j")
-          proof -
-            let ?t = "restm i t"
-            have eq_t: "t = ?t @ moment i t" by (simp add:moment_restm_s) 
-            have h1: "P (restm i t @ moment i t @ s)" 
-            proof -
-              from pts and eq_t have "P ((restm i t @ moment i t) @ s)" by simp
-              thus ?thesis by simp
-            qed
-            moreover have h2: "(f (moment i t @ s) + 1) * k < length (restm i t)"
-            proof -
-              have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
-              from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
-              from h [OF this, of k]
-              have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
-              moreover from le_ik have "\<dots> \<le> ((f s) * k + k - i)" by simp
-              moreover from le_ik lt_k_lt and ltk have "(f s) * k + k - i < length t - i" by simp
-              moreover have "length (restm i t) = length t - i" using length_restm by metis
-              ultimately show ?thesis by simp
-            qed
-            from hw [unfolded happen_within_def all_future_def, rule_format, OF h1 h2]
-            obtain m where le_m: "m \<le> (f (moment i t @ s) + 1) * k"
-              and q_m: "Q (moment m ?t @ moment i t @ s)" 
-              and p_m: "P (moment m ?t @ moment i t @ s)" by auto
-            have eq_mm: "moment m ?t @ moment i t @ s = (moment (m+i) t)@s"
-            proof -
-              have "moment m (restm i t) @ moment i t = moment (m + i) t"
-                using moment_plus_split by metis
-              thus ?thesis by simp
-            qed
-            let ?j = "m + i"
-            have "?T ?j"
-            proof -
-              have "m + i \<le> (f s + 1) * k"
-              proof -
-                have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
-                from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
-                from h [OF this, of k]
-                have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
-                with le_m have "m \<le> f s * k" by simp
-                hence "m + i \<le> f s * k + i" by simp
-                with le_ik show ?thesis by simp
-              qed
-              moreover from eq_mm q_m have " Q (moment (m + i) t @ s)" by metis
-              moreover from eq_mm p_m have " P (moment (m + i) t @ s)" by metis
-              ultimately show ?thesis by blast
-            qed
-            thus ?thesis by blast
-          qed
-        } thus ?thesis by  (simp add:happen_within_def all_future_def firstn.simps)
-      qed
-    qed
-  qed
-qed
-
-end
-
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/IsaMakefile
--- a/prio/IsaMakefile	Sun Feb 05 14:29:08 2012 +0000
+++ b/prio/IsaMakefile	Sun Feb 05 21:00:12 2012 +0000
@@ -2,7 +2,7 @@
 ## targets
 
 default: itp
-all: session paper
+all: session itp
 
 ## global settings
 
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Lsp.thy
--- a/prio/Lsp.thy	Sun Feb 05 14:29:08 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,323 +0,0 @@
-theory Lsp
-imports Main
-begin
-
-fun lsp :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list)"
-where 
-   "lsp f [] = ([], [], [])" |
-   "lsp f [x] = ([], [x], [])" |
-   "lsp f (x#xs) = (case (lsp f xs) of
-                     (l, [], r) \<Rightarrow> ([], [x], []) |
-                     (l, y#ys, r) \<Rightarrow> if f x \<ge> f y then ([], [x], xs) else (x#l, y#ys, r))"
-
-inductive lsp_p :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list) \<Rightarrow> bool"
-for f :: "('a \<Rightarrow> ('b::linorder))"
-where
-  lsp_nil [intro]: "lsp_p f [] ([], [], [])" |
-  lsp_single [intro]: "lsp_p f [x] ([], [x], [])" |
-  lsp_cons_1 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x \<ge> f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) ([], [x], xs)" |
-  lsp_cons_2 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) (x#l, [m], r)"
-
-lemma lsp_p_lsp_1: "lsp_p f x y \<Longrightarrow> y = lsp f x"
-proof (induct rule:lsp_p.induct)
-  case (lsp_cons_1 xs  l m r x)
-  assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
-    and le_mx: "f m \<le> f x"
-  show ?case (is "?L = ?R")
-  proof(cases xs, simp)
-    case (Cons v vs)
-    show ?thesis
-      apply (simp add:Cons)
-      apply (fold Cons)
-      by (simp add:lsp_xs le_mx)
-  qed
-next
-  case (lsp_cons_2 xs l m r x)
-  assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
-    and lt_xm: "f x < f m"
-  show ?case (is "?L = ?R")
-  proof(cases xs)
-    case (Cons v vs)
-    show ?thesis
-      apply (simp add:Cons)
-      apply (fold Cons)
-      apply (simp add:lsp_xs)
-      by (insert lt_xm, auto)
-  next
-    case Nil
-    from prems show ?thesis by simp
-  qed
-qed auto
-
-lemma lsp_mid_nil: "lsp f xs = (a, [], c) \<Longrightarrow> xs = []"
-  apply (induct xs arbitrary:a c, auto)
-  apply (case_tac xs, auto)
-  by (case_tac "(lsp f (ab # list))", auto split:if_splits list.splits)
-
-
-lemma lsp_mid_length: "lsp f x = (u, v, w) \<Longrightarrow> length v \<le> 1"
-proof(induct x arbitrary:u v w, simp)
-  case (Cons x xs)
-  assume ih: "\<And> u v w. lsp f xs = (u, v, w) \<Longrightarrow> length v \<le> 1"
-  and h: "lsp f (x # xs) = (u, v, w)"
-  show "length v \<le> 1" using h
-  proof(cases xs, simp add:h)
-    case (Cons z zs)
-    assume eq_xs: "xs = z # zs"
-    show ?thesis
-    proof(cases "lsp f xs")
-      fix l m r
-      assume eq_lsp: "lsp f xs = (l, m, r)"
-      show ?thesis
-      proof(cases m)
-        case Nil
-        from Nil and eq_lsp have "lsp f xs = (l, [], r)" by simp
-        from lsp_mid_nil [OF this] have "xs = []" .
-        with h show ?thesis by auto
-      next
-        case (Cons y ys)
-        assume eq_m: "m = y # ys"
-        from ih [OF eq_lsp] have eq_xs_1: "length m \<le> 1" .
-        show ?thesis
-        proof(cases "f x \<ge> f y")
-          case True
-          from eq_xs eq_xs_1 True h eq_lsp show ?thesis 
-            by (auto split:list.splits if_splits)
-        next
-          case False
-          from eq_xs eq_xs_1 False h eq_lsp show ?thesis 
-             by (auto split:list.splits if_splits)
-        qed
-      qed
-    qed
-  next
-    assume "[] = u \<and> [x] = v \<and> [] = w"
-    hence "v = [x]" by simp
-    thus "length v \<le> Suc 0" by simp
-  qed
-qed
-
-lemma lsp_p_lsp_2: "lsp_p f x (lsp f x)"
-proof(induct x, auto)
-  case (Cons x xs)
-  assume ih: "lsp_p f xs (lsp f xs)"
-  show ?case
-  proof(cases xs)
-    case Nil
-    thus ?thesis by auto
-  next
-    case (Cons v vs)
-    show ?thesis
-    proof(cases "xs")
-      case Nil
-      thus ?thesis by auto
-    next
-      case (Cons v vs)
-      assume eq_xs: "xs = v # vs"
-      show ?thesis
-      proof(cases "lsp f xs")
-        fix l m r
-        assume eq_lsp_xs: "lsp f xs = (l, m, r)"
-        show ?thesis
-        proof(cases m)
-          case Nil
-          from eq_lsp_xs and Nil have "lsp f xs = (l, [], r)" by simp
-          from lsp_mid_nil [OF this] have eq_xs: "xs = []" .
-          hence "lsp f (x#xs) = ([], [x], [])" by simp
-          with eq_xs show ?thesis by auto
-        next
-          case (Cons y ys)
-          assume eq_m: "m = y # ys"
-          show ?thesis
-          proof(cases "f x \<ge> f y")
-            case True
-            from eq_xs eq_lsp_xs Cons True
-            have eq_lsp: "lsp f (x#xs) = ([], [x], v # vs)" by simp
-            show ?thesis
-            proof (simp add:eq_lsp)
-              show "lsp_p f (x # xs) ([], [x], v # vs)"
-              proof(fold eq_xs, rule lsp_cons_1 [OF _])
-                from eq_xs show "xs \<noteq> []" by simp
-              next
-                from lsp_mid_length [OF eq_lsp_xs] and Cons
-                have "m = [y]" by simp
-                with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
-                with ih show "lsp_p f xs (l, [y], r)" by simp
-              next
-                from True show "f y \<le> f x" by simp
-              qed
-            qed
-          next
-            case False
-            from eq_xs eq_lsp_xs Cons False
-            have eq_lsp: "lsp f (x#xs) = (x # l, y # ys, r) " by simp
-            show ?thesis
-            proof (simp add:eq_lsp)
-              from lsp_mid_length [OF eq_lsp_xs] and eq_m
-              have "ys = []" by simp
-              moreover have "lsp_p f (x # xs) (x # l, [y], r)"
-              proof(rule lsp_cons_2)
-                from eq_xs show "xs \<noteq> []" by simp
-              next
-                from lsp_mid_length [OF eq_lsp_xs] and Cons
-                have "m = [y]" by simp
-                with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
-                with ih show "lsp_p f xs (l, [y], r)" by simp
-              next
-                from False show "f x < f y" by simp
-              qed
-              ultimately show "lsp_p f (x # xs) (x # l, y # ys, r)" by simp
-            qed
-          qed
-        qed
-      qed
-    qed
-  qed
-qed
-
-lemma lsp_induct:
-  fixes f x1 x2 P
-  assumes h: "lsp f x1 = x2"
-  and p1: "P [] ([], [], [])"
-  and p2: "\<And>x. P [x] ([], [x], [])"
-  and p3: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f m \<le> f x\<rbrakk> \<Longrightarrow> P (x # xs) ([], [x], xs)"
-  and p4: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> P (x # xs) (x # l, [m], r)"
-  shows "P x1 x2"
-proof(rule lsp_p.induct)
-  from lsp_p_lsp_2 and h
-  show "lsp_p f x1 x2" by metis
-next
-  from p1 show "P [] ([], [], [])" by metis
-next
-  from p2 show "\<And>x. P [x] ([], [x], [])" by metis
-next
-  fix xs l m r x 
-  assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f m \<le> f x"
-  show "P (x # xs) ([], [x], xs)" 
-  proof(rule p3 [OF h1 _ h3 h4])
-    from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
-  qed
-next
-  fix xs l m r x 
-  assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f x < f m"
-  show "P (x # xs) (x # l, [m], r)"
-  proof(rule p4 [OF h1 _ h3 h4])
-    from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
-  qed
-qed
-
-lemma lsp_set_eq: 
-  fixes f x u v w
-  assumes h: "lsp f x = (u, v, w)"
-  shows "x = u@v@w"
-proof -
-  have "\<And> f x r. lsp f x = r \<Longrightarrow> \<forall> u v w. (r = (u, v, w) \<longrightarrow> x = u@v@w)" 
-    by (erule lsp_induct, simp+)
-  from this [rule_format, OF h] show ?thesis by simp
-qed
-
-lemma lsp_set: 
-  assumes h: "(u, v, w) = lsp f x"
-  shows "set (u@v@w) = set x"
-proof -
-  from lsp_set_eq [OF h[symmetric]] 
-  show ?thesis by simp
-qed
-
-lemma max_insert_gt:
-  fixes S fx
-  assumes h: "fx < Max S"
-  and np: "S \<noteq> {}"
-  and fn: "finite S" 
-  shows "Max S = Max (insert fx S)"
-proof -
-  from Max_insert [OF fn np]
-  have "Max (insert fx S) = max fx (Max S)" .
-  moreover have "\<dots> = Max S"
-  proof(cases "fx \<le> Max S")
-    case False
-    with h
-    show ?thesis by (simp add:max_def)
-  next
-    case True
-    thus ?thesis by (simp add:max_def)
-  qed
-  ultimately show ?thesis by simp
-qed
-
-lemma max_insert_le: 
-  fixes S fx
-  assumes h: "Max S \<le> fx"
-  and fn: "finite S"
-  shows "fx = Max (insert fx S)"
-proof(cases "S = {}")
-  case True
-  thus ?thesis by simp
-next
-  case False
-  from Max_insert [OF fn False]
-  have "Max (insert fx S) = max fx (Max S)" .
-  moreover have "\<dots> = fx"
-  proof(cases "fx \<le> Max S")
-    case False
-    thus ?thesis by (simp add:max_def)
-  next
-    case True
-    have hh: "\<And> x y. \<lbrakk> x \<le> (y::('a::linorder)); y \<le> x\<rbrakk> \<Longrightarrow> x = y" by auto
-    from hh [OF True h]
-    have "fx = Max S" .
-    thus ?thesis by simp
-  qed
-  ultimately show ?thesis by simp
-qed
-  
-lemma lsp_max: 
-  fixes f x u m w
-  assumes h: "lsp f x = (u, [m], w)"
-  shows "f m = Max (f ` (set x))"
-proof -
-  { fix y
-    have "lsp f x = y \<Longrightarrow> \<forall> u m w. y = (u, [m], w) \<longrightarrow> f m = Max (f ` (set x))"
-    proof(erule lsp_induct, simp)
-      { fix x u m w
-        assume "(([]::'a list), ([x]::'a list), ([]::'a list)) = (u, [m], w)"
-        hence "f m = Max (f ` set [x])"  by simp
-      } thus "\<And>x. \<forall>u m w. ([], [x], []) = (u, [m], w) \<longrightarrow> f m = Max (f ` set [x])" by simp
-    next
-      fix xs l m r x
-      assume h1: "xs \<noteq> []"
-        and h2: " lsp f xs = (l, [m], r)"
-        and h3: "\<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
-        and h4: "f m \<le> f x"
-      show " \<forall>u m w. ([], [x], xs) = (u, [m], w) \<longrightarrow> f m = Max (f ` set (x # xs))"
-      proof -
-        have "f x = Max (f ` set (x # xs))"
-        proof -
-          from h2 h3 have "f m = Max (f ` set xs)" by simp
-          with h4 show ?thesis
-            apply auto
-            by (rule_tac max_insert_le, auto)
-        qed
-        thus ?thesis by simp
-      qed
-    next
-      fix xs l m r x
-      assume h1: "xs \<noteq> []"
-        and h2: " lsp f xs = (l, [m], r)"
-        and h3: " \<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
-        and h4: "f x < f m"
-      show "\<forall>u ma w. (x # l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set (x # xs))"
-      proof -
-        from h2 h3 have "f m = Max (f ` set xs)" by simp
-        with h4
-        have "f m =  Max (f ` set (x # xs))"
-          apply auto
-          apply (rule_tac max_insert_gt, simp+)
-          by (insert h1, simp+)
-        thus ?thesis by auto
-      qed
-    qed
-  } with h show ?thesis by metis
-qed
-
-end
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/Prio.thy
--- a/prio/Prio.thy	Sun Feb 05 14:29:08 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2813 +0,0 @@
-theory Prio
-imports Precedence_ord Moment Lsp Happen_within
-begin
-
-type_synonym thread = nat
-type_synonym priority = nat
-type_synonym cs = nat
-
-datatype event = 
-  Create thread priority |
-  Exit thread |
-  P thread cs |
-  V thread cs |
-  Set thread priority
-
-datatype node = 
-   Th "thread" |
-   Cs "cs"
-
-type_synonym state = "event list"
-
-fun threads :: "state \<Rightarrow> thread set"
-where 
-  "threads [] = {}" |
-  "threads (Create thread prio#s) = {thread} \<union> threads s" |
-  "threads (Exit thread # s) = (threads s) - {thread}" |
-  "threads (e#s) = threads s"
-
-fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> nat"
-where
-  "original_priority thread [] = 0" |
-  "original_priority thread (Create thread' prio#s) = 
-     (if thread' = thread then prio else original_priority thread s)" |
-  "original_priority thread (Set thread' prio#s) = 
-     (if thread' = thread then prio else original_priority thread s)" |
-  "original_priority thread (e#s) = original_priority thread s"
-
-fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
-where
-  "birthtime thread [] = 0" |
-  "birthtime thread ((Create thread' prio)#s) = (if (thread = thread') then length s 
-                                                  else birthtime thread s)" |
-  "birthtime thread ((Set thread' prio)#s) = (if (thread = thread') then length s 
-                                                  else birthtime thread s)" |
-  "birthtime thread (e#s) = birthtime thread s"
-
-definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
-  where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
-
-consts holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
-       waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
-       depend :: "'b \<Rightarrow> (node \<times> node) set"
-       dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
-
-defs (overloaded) cs_holding_def: "holding wq thread cs == (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
-                  cs_waiting_def: "waiting wq thread cs == (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
-                  cs_depend_def: "depend (wq::cs \<Rightarrow> thread list) == {(Th t, Cs c) | t c. waiting wq t c} \<union> 
-                                               {(Cs c, Th t) | c t. holding wq t c}"
-                  cs_dependents_def: "dependents (wq::cs \<Rightarrow> thread list) th == {th' . (Th th', Th th) \<in> (depend wq)^+}"
-
-record schedule_state = 
-    waiting_queue :: "cs \<Rightarrow> thread list"
-    cur_preced :: "thread \<Rightarrow> precedence"
-
-
-definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
-where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
-
-fun schs :: "state \<Rightarrow> schedule_state"
-where
-   "schs [] = \<lparr>waiting_queue = \<lambda> cs. [], 
-               cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
-   "schs (e#s) = (let ps = schs s in
-                  let pwq = waiting_queue ps in
-                  let pcp = cur_preced ps in
-                  let nwq = case e of
-                             P thread cs \<Rightarrow>  pwq(cs:=(pwq cs @ [thread])) |
-                             V thread cs \<Rightarrow> let nq = case (pwq cs) of
-                                                      [] \<Rightarrow> [] | 
-                                                      (th#pq) \<Rightarrow> case (lsp pcp pq) of
-                                                                   (l, [], r) \<Rightarrow> []
-                                                                 | (l, m#ms, r) \<Rightarrow> m#(l@ms@r)
-                                            in pwq(cs:=nq)                 |
-                              _ \<Rightarrow> pwq
-                  in let ncp = cpreced (e#s) nwq in 
-                     \<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
-                 )"
-
-definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
-where "wq s == waiting_queue (schs s)"
-
-definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
-where "cp s = cur_preced (schs s)"
-
-defs (overloaded) s_holding_def: "holding (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
-                  s_waiting_def: "waiting (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
-                  s_depend_def: "depend (s::state) == {(Th t, Cs c) | t c. waiting (wq s) t c} \<union> 
-                                               {(Cs c, Th t) | c t. holding (wq s) t c}"
-                  s_dependents_def: "dependents (s::state) th == {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
-
-definition readys :: "state \<Rightarrow> thread set"
-where
-  "readys s = 
-     {thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
-
-definition runing :: "state \<Rightarrow> thread set"
-where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
-
-definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
-  where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
-
-inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
-where
-  thread_create: "\<lbrakk>prio \<le> max_prio; thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
-  thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
-  thread_P: "\<lbrakk>thread \<in> runing s;  (Cs cs, Th thread)  \<notin> (depend s)^+\<rbrakk> \<Longrightarrow> step s (P thread cs)" |
-  thread_V: "\<lbrakk>thread \<in> runing s;  holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
-  thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
-
-inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
- for cs
-where
-  vt_nil[intro]: "vt cs []" |
-  vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
-
-lemma runing_ready: "runing s \<subseteq> readys s"
-  by (auto simp only:runing_def readys_def)
-
-lemma wq_v_eq_nil: 
-  fixes s cs thread rest
-  assumes eq_wq: "wq s cs = thread # rest"
-  and eq_lsp: "lsp (cp s) rest = (l, [], r)"
-  shows "wq (V thread cs#s) cs = []"
-proof -
-  from prems show ?thesis
-    by (auto simp:wq_def Let_def cp_def split:list.splits)
-qed
-
-lemma wq_v_eq: 
-  fixes s cs thread rest
-  assumes eq_wq: "wq s cs = thread # rest"
-  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
-  shows "wq (V thread cs#s) cs = th'#l@r"
-proof -
-  from prems show ?thesis
-    by (auto simp:wq_def Let_def cp_def split:list.splits)
-qed
-
-lemma wq_v_neq:
-   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
-  by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-lemma wq_distinct: "vt step s \<Longrightarrow> distinct (wq s cs)"
-proof(erule_tac vt.induct, simp add:wq_def)
-  fix s e
-  assume h1: "step s e"
-  and h2: "distinct (wq s cs)"
-  thus "distinct (wq (e # s) cs)"
-  proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
-    fix thread s
-    assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
-      and h2: "thread \<in> set (waiting_queue (schs s) cs)"
-      and h3: "thread \<in> runing s"
-    show "False" 
-    proof -
-      from h3 have "\<And> cs. thread \<in>  set (waiting_queue (schs s) cs) \<Longrightarrow> 
-                             thread = hd ((waiting_queue (schs s) cs))" 
-        by (simp add:runing_def readys_def s_waiting_def wq_def)
-      from this [OF h2] have "thread = hd (waiting_queue (schs s) cs)" .
-      with h2
-      have "(Cs cs, Th thread) \<in> (depend s)"
-        by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
-      with h1 show False by auto
-    qed
-  next
-    fix thread s a list
-    assume h1: "thread \<in> runing s" 
-      and h2: "holding s thread cs"
-      and h3: "waiting_queue (schs s) cs = a # list"
-      and h4: "a \<notin> set list"
-      and h5: "distinct list"
-    thus "distinct
-           ((\<lambda>(l, a, r). case a of [] \<Rightarrow> [] | m # ms \<Rightarrow> m # l @ ms @ r)
-             (lsp (cur_preced (schs s)) list))"
-    apply (cases "(lsp (cur_preced (schs s)) list)", simp)
-    apply (case_tac b, simp)
-    by (drule_tac lsp_set_eq, simp)
-  qed
-qed
-
-lemma block_pre: 
-  fixes thread cs s
-  assumes s_ni: "thread \<notin>  set (wq s cs)"
-  and s_i: "thread \<in> set (wq (e#s) cs)"
-  shows "e = P thread cs"
-proof -
-  have ee: "\<And> x y. \<lbrakk>x = y\<rbrakk> \<Longrightarrow> set x = set y"
-    by auto
-  from s_ni s_i show ?thesis
-  proof (cases e, auto split:if_splits simp add:Let_def wq_def)
-    fix uu uub uuc uud uue
-    assume h: "(uuc, thread # uu, uub) = lsp (cur_preced (schs s)) uud"
-      and h1 [symmetric]: "uue # uud = waiting_queue (schs s) cs"
-      and h2: "thread \<notin> set (waiting_queue (schs s) cs)"
-    from lsp_set [OF h] have "set (uuc @ (thread # uu) @ uub) = set uud" .
-    hence "thread \<in> set uud" by auto
-    with h1 have "thread \<in> set (waiting_queue (schs s) cs)" by auto
-    with h2 show False by auto
-  next
-    fix uu uua uub uuc uud uue
-    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
-      and h2: "uue # uud = waiting_queue (schs s) cs"
-      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
-      and h4: "thread \<in> set uuc"
-    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
-    with h4 have "thread \<in> set uud" by auto
-    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
-      apply (drule_tac ee) by auto
-    with h1 show "False" by fast
-  next
-    fix uu uua uub uuc uud uue
-    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
-      and h2: "uue # uud = waiting_queue (schs s) cs"
-      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
-      and h4: "thread \<in> set uu"
-    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
-    with h4 have "thread \<in> set uud" by auto
-    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
-      apply (drule_tac ee) by auto
-    with h1 show "False" by fast
-  next
-    fix uu uua uub uuc uud uue
-    assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
-      and h2: "uue # uud = waiting_queue (schs s) cs"
-      and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
-      and h4: "thread \<in> set uub"
-    from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
-    with h4 have "thread \<in> set uud" by auto
-    with h2 have "thread \<in> set (waiting_queue (schs s) cs)" 
-      apply (drule_tac ee) by auto
-    with h1 show "False" by fast
-  qed
-qed
-
-lemma p_pre: "\<lbrakk>vt step ((P thread cs)#s)\<rbrakk> \<Longrightarrow> 
-  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (depend s)^+"
-apply (ind_cases "vt step ((P thread cs)#s)")
-apply (ind_cases "step s (P thread cs)")
-by auto
-
-lemma abs1:
-  fixes e es
-  assumes ein: "e \<in> set es"
-  and neq: "hd es \<noteq> hd (es @ [x])"
-  shows "False"
-proof -
-  from ein have "es \<noteq> []" by auto
-  then obtain e ess where "es = e # ess" by (cases es, auto)
-  with neq show ?thesis by auto
-qed
-
-lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
-  by (cases es, auto)
-
-inductive_cases evt_cons: "vt cs (a#s)"
-
-lemma abs2:
-  assumes vt: "vt step (e#s)"
-  and inq: "thread \<in> set (wq s cs)"
-  and nh: "thread = hd (wq s cs)"
-  and qt: "thread \<noteq> hd (wq (e#s) cs)"
-  and inq': "thread \<in> set (wq (e#s) cs)"
-  shows "False"
-proof -
-  have ee: "\<And> uuc thread uu uub s list. (uuc, thread # uu, uub) = lsp (cur_preced (schs s)) list \<Longrightarrow> 
-                 lsp (cur_preced (schs s)) list = (uuc, thread # uu, uub) 
-               " by simp
-  from prems show "False"
-    apply (cases e)
-    apply ((simp split:if_splits add:Let_def wq_def)[1])+
-    apply (insert abs1, fast)[1] 
-    apply ((simp split:if_splits add:Let_def)[1])+
-    apply (simp split:if_splits list.splits add:Let_def wq_def) 
-    apply (auto dest!:ee)
-    apply (drule_tac lsp_set_eq, simp)
-    apply (subgoal_tac "distinct (waiting_queue (schs s) cs)", simp, fold wq_def)
-    apply (rule_tac wq_distinct, auto)
-    apply (erule_tac evt_cons, auto)
-    apply (drule_tac lsp_set_eq, simp)
-    apply (subgoal_tac "distinct (wq s cs)", simp)
-    apply (rule_tac wq_distinct, auto)
-    apply (erule_tac evt_cons, auto)
-    apply (drule_tac lsp_set_eq, simp)
-    apply (subgoal_tac "distinct (wq s cs)", simp)
-    apply (rule_tac wq_distinct, auto)
-    apply (erule_tac evt_cons, auto)
-    apply (auto simp:wq_def Let_def split:if_splits prod.splits)
-    done
-qed
-
-lemma vt_moment: "\<And> t. \<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
-proof(induct s, simp)
-  fix a s t
-  assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
-    and vt_a: "vt cs (a # s)"
-    and le_t: "t \<le> length (a # s)"
-  show "vt cs (moment t (a # s))"
-  proof(cases "t = length (a#s)")
-    case True
-    from True have "moment t (a#s) = a#s" by simp
-    with vt_a show ?thesis by simp
-  next
-    case False
-    with le_t have le_t1: "t \<le> length s" by simp
-    from vt_a have "vt cs s"
-      by (erule_tac evt_cons, simp)
-    from h [OF this le_t1] have "vt cs (moment t s)" .
-    moreover have "moment t (a#s) = moment t s"
-    proof -
-      from moment_app [OF le_t1, of "[a]"] 
-      show ?thesis by simp
-    qed
-    ultimately show ?thesis by auto
-  qed
-qed
-
-(* Wrong:
-    lemma \<lbrakk>thread \<in> set (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
-*)
-
-lemma waiting_unique_pre:
-  fixes cs1 cs2 s thread
-  assumes vt: "vt step s"
-  and h11: "thread \<in> set (wq s cs1)"
-  and h12: "thread \<noteq> hd (wq s cs1)"
-  assumes h21: "thread \<in> set (wq s cs2)"
-  and h22: "thread \<noteq> hd (wq s cs2)"
-  and neq12: "cs1 \<noteq> cs2"
-  shows "False"
-proof -
-  let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
-  from h11 and h12 have q1: "?Q cs1 s" by simp
-  from h21 and h22 have q2: "?Q cs2 s" by simp
-  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
-  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
-  from p_split [of "?Q cs1", OF q1 nq1]
-  obtain t1 where lt1: "t1 < length s"
-    and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
-        thread \<noteq> hd (wq (moment t1 s) cs1))"
-    and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
-             thread \<noteq> hd (wq (moment i' s) cs1))" by auto
-  from p_split [of "?Q cs2", OF q2 nq2]
-  obtain t2 where lt2: "t2 < length s"
-    and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
-        thread \<noteq> hd (wq (moment t2 s) cs2))"
-    and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
-             thread \<noteq> hd (wq (moment i' s) cs2))" by auto
-  show ?thesis
-  proof -
-    { 
-      assume lt12: "t1 < t2"
-      let ?t3 = "Suc t2"
-      from lt2 have le_t3: "?t3 \<le> length s" by auto
-      from moment_plus [OF this] 
-      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
-      have "t2 < ?t3" by simp
-      from nn2 [rule_format, OF this] and eq_m
-      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-        h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-      have vt_e: "vt step (e#moment t2 s)"
-      proof -
-        from vt_moment [OF vt le_t3]
-        have "vt step (moment ?t3 s)" .
-        with eq_m show ?thesis by simp
-      qed
-      have ?thesis
-      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-        case True
-        from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
-          by auto
-        from abs2 [OF vt_e True eq_th h2 h1]
-        show ?thesis by auto
-      next
-        case False
-        from block_pre [OF False h1]
-        have "e = P thread cs2" .
-        with vt_e have "vt step ((P thread cs2)# moment t2 s)" by simp
-        from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
-        with runing_ready have "thread \<in> readys (moment t2 s)" by auto
-        with nn1 [rule_format, OF lt12]
-        show ?thesis  by (simp add:readys_def s_waiting_def, auto)
-      qed
-    } moreover {
-      assume lt12: "t2 < t1"
-      let ?t3 = "Suc t1"
-      from lt1 have le_t3: "?t3 \<le> length s" by auto
-      from moment_plus [OF this] 
-      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
-      have lt_t3: "t1 < ?t3" by simp
-      from nn1 [rule_format, OF this] and eq_m
-      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
-        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
-      have vt_e: "vt step (e#moment t1 s)"
-      proof -
-        from vt_moment [OF vt le_t3]
-        have "vt step (moment ?t3 s)" .
-        with eq_m show ?thesis by simp
-      qed
-      have ?thesis
-      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
-        case True
-        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
-          by auto
-        from abs2 [OF vt_e True eq_th h2 h1]
-        show ?thesis by auto
-      next
-        case False
-        from block_pre [OF False h1]
-        have "e = P thread cs1" .
-        with vt_e have "vt step ((P thread cs1)# moment t1 s)" by simp
-        from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
-        with runing_ready have "thread \<in> readys (moment t1 s)" by auto
-        with nn2 [rule_format, OF lt12]
-        show ?thesis  by (simp add:readys_def s_waiting_def, auto)
-      qed
-    } moreover {
-      assume eqt12: "t1 = t2"
-      let ?t3 = "Suc t1"
-      from lt1 have le_t3: "?t3 \<le> length s" by auto
-      from moment_plus [OF this] 
-      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
-      have lt_t3: "t1 < ?t3" by simp
-      from nn1 [rule_format, OF this] and eq_m
-      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
-        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
-      have vt_e: "vt step (e#moment t1 s)"
-      proof -
-        from vt_moment [OF vt le_t3]
-        have "vt step (moment ?t3 s)" .
-        with eq_m show ?thesis by simp
-      qed
-      have ?thesis
-      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
-        case True
-        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
-          by auto
-        from abs2 [OF vt_e True eq_th h2 h1]
-        show ?thesis by auto
-      next
-        case False
-        from block_pre [OF False h1]
-        have eq_e1: "e = P thread cs1" .
-        have lt_t3: "t1 < ?t3" by simp
-        with eqt12 have "t2 < ?t3" by simp
-        from nn2 [rule_format, OF this] and eq_m and eqt12
-        have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-          h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-        show ?thesis
-        proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-          case True
-          from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
-            by auto
-          from vt_e and eqt12 have "vt step (e#moment t2 s)" by simp 
-          from abs2 [OF this True eq_th h2 h1]
-          show ?thesis .
-        next
-          case False
-          from block_pre [OF False h1]
-          have "e = P thread cs2" .
-          with eq_e1 neq12 show ?thesis by auto
-        qed
-      qed
-    } ultimately show ?thesis by arith
-  qed
-qed
-
-lemma waiting_unique:
-  assumes "vt step s"
-  and "waiting s th cs1"
-  and "waiting s th cs2"
-  shows "cs1 = cs2"
-proof -
-  from waiting_unique_pre and prems
-  show ?thesis
-    by (auto simp add:s_waiting_def)
-qed
-
-lemma holded_unique:
-  assumes "vt step s"
-  and "holding s th1 cs"
-  and "holding s th2 cs"
-  shows "th1 = th2"
-proof -
-  from prems show ?thesis
-    unfolding s_holding_def
-    by auto
-qed
-
-lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
-  apply (induct s, auto)
-  by (case_tac a, auto split:if_splits)
-
-lemma birthtime_unique: 
-  "\<lbrakk>birthtime th1 s = birthtime th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
-          \<Longrightarrow> th1 = th2"
-  apply (induct s, auto)
-  by (case_tac a, auto split:if_splits dest:birthtime_lt)
-
-lemma preced_unique : 
-  assumes pcd_eq: "preced th1 s = preced th2 s"
-  and th_in1: "th1 \<in> threads s"
-  and th_in2: " th2 \<in> threads s"
-  shows "th1 = th2"
-proof -
-  from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
-  from birthtime_unique [OF this th_in1 th_in2]
-  show ?thesis .
-qed
-
-lemma preced_linorder: 
-  assumes neq_12: "th1 \<noteq> th2"
-  and th_in1: "th1 \<in> threads s"
-  and th_in2: " th2 \<in> threads s"
-  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
-  from preced_unique [OF _ th_in1 th_in2] and neq_12 
-  have "preced th1 s \<noteq> preced th2 s" by auto
-  thus ?thesis by auto
-qed
-
-lemma unique_minus:
-  fixes x y z r
-  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-  and xy: "(x, y) \<in> r"
-  and xz: "(x, z) \<in> r^+"
-  and neq: "y \<noteq> z"
-  shows "(y, z) \<in> r^+"
-proof -
- from xz and neq show ?thesis
- proof(induct)
-   case (base ya)
-   have "(x, ya) \<in> r" by fact
-   from unique [OF xy this] have "y = ya" .
-   with base show ?case by auto
- next
-   case (step ya z)
-   show ?case
-   proof(cases "y = ya")
-     case True
-     from step True show ?thesis by simp
-   next
-     case False
-     from step False
-     show ?thesis by auto
-   qed
- qed
-qed
-
-lemma unique_base:
-  fixes r x y z
-  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-  and xy: "(x, y) \<in> r"
-  and xz: "(x, z) \<in> r^+"
-  and neq_yz: "y \<noteq> z"
-  shows "(y, z) \<in> r^+"
-proof -
-  from xz neq_yz show ?thesis
-  proof(induct)
-    case (base ya)
-    from xy unique base show ?case by auto
-  next
-    case (step ya z)
-    show ?case
-    proof(cases "y = ya")
-      case True
-      from True step show ?thesis by auto
-    next
-      case False
-      from False step 
-      have "(y, ya) \<in> r\<^sup>+" by auto
-      with step show ?thesis by auto
-    qed
-  qed
-qed
-
-lemma unique_chain:
-  fixes r x y z
-  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-  and xy: "(x, y) \<in> r^+"
-  and xz: "(x, z) \<in> r^+"
-  and neq_yz: "y \<noteq> z"
-  shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
-proof -
-  from xy xz neq_yz show ?thesis
-  proof(induct)
-    case (base y)
-    have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
-    from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
-  next
-    case (step y za)
-    show ?case
-    proof(cases "y = z")
-      case True
-      from True step show ?thesis by auto
-    next
-      case False
-      from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
-      thus ?thesis
-      proof
-        assume "(z, y) \<in> r\<^sup>+"
-        with step have "(z, za) \<in> r\<^sup>+" by auto
-        thus ?thesis by auto
-      next
-        assume h: "(y, z) \<in> r\<^sup>+"
-        from step have yza: "(y, za) \<in> r" by simp
-        from step have "za \<noteq> z" by simp
-        from unique_minus [OF _ yza h this] and unique
-        have "(za, z) \<in> r\<^sup>+" by auto
-        thus ?thesis by auto
-      qed
-    qed
-  qed
-qed
-
-lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-definition head_of :: "('a \<Rightarrow> 'b::linorder) \<Rightarrow> 'a set \<Rightarrow> 'a set"
-  where "head_of f A = {a . a \<in> A \<and> f a = Max (f ` A)}"
-
-definition wq_head :: "state \<Rightarrow> cs \<Rightarrow> thread set"
-  where "wq_head s cs = head_of (cp s) (set (wq s cs))"
-
-lemma f_nil_simp: "\<lbrakk>f cs = []\<rbrakk> \<Longrightarrow> f(cs:=[]) = f"
-proof
-  fix x
-  assume h:"f cs = []"
-  show "(f(cs := [])) x = f x"
-  proof(cases "cs = x")
-    case True
-    with h show ?thesis by simp
-  next
-    case False
-    with h show ?thesis by simp
-  qed
-qed
-
-lemma step_back_vt: "vt ccs (e#s) \<Longrightarrow> vt ccs s"
-  by(ind_cases "vt ccs (e#s)", simp)
-
-lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e"
-  by(ind_cases "vt ccs (e#s)", simp)
-
-lemma holding_nil:
-    "\<lbrakk>wq s cs = []; holding (wq s) th cs\<rbrakk> \<Longrightarrow> False"
-  by (unfold cs_holding_def, auto)
-
-lemma waiting_kept_1: "
-       \<lbrakk>vt step (V th cs#s); wq s cs = a # list; waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c;
-        lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
-       \<Longrightarrow> waiting (wq s) t c"
-  apply (drule_tac step_back_vt, drule_tac wq_distinct[of _ cs])
-  apply(drule_tac lsp_set_eq)
-  by (unfold cs_waiting_def, auto split:if_splits)
- 
-lemma waiting_kept_2: 
-  "\<And>a list t c aa ca.
-       \<lbrakk>wq s cs = a # list; waiting ((wq s)(cs := [])) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
-       \<Longrightarrow> waiting (wq s) t c"
-  apply(drule_tac lsp_set_eq)
-  by (unfold cs_waiting_def, auto split:if_splits)
-  
-
-lemma holding_nil_simp: "\<lbrakk>holding ((wq s)(cs := [])) t c\<rbrakk> \<Longrightarrow> holding (wq s) t c"
-  by(unfold cs_holding_def, auto)
-
-lemma step_wq_elim: "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; a \<noteq> th\<rbrakk> \<Longrightarrow> False"
-  apply(drule_tac step_back_step)
-  apply(ind_cases "step s (V th cs)")
-  by(unfold s_holding_def, auto)
-
-lemma holding_cs_neq_simp: "c \<noteq> cs \<Longrightarrow> holding ((wq s)(cs := u)) t c = holding (wq s) t c"
-  by (unfold cs_holding_def, auto)
-
-lemma holding_th_neq_elim:
-  "\<And>a list c t aa ca ab lista.
-       \<lbrakk>\<not> holding (wq s) t c; holding ((wq s)(cs := ab # aa @ lista @ ca)) t c;
-         ab \<noteq> t\<rbrakk>
-       \<Longrightarrow> False"
-  by (unfold cs_holding_def, auto split:if_splits)
-
-lemma holding_nil_abs:
-  "\<not> holding ((wq s)(cs := [])) th cs"
-  by (unfold cs_holding_def, auto split:if_splits)
-
-lemma holding_abs: "\<lbrakk>holding ((wq s)(cs := ab # aa @ lista @ c)) th cs; ab \<noteq> th\<rbrakk>
-       \<Longrightarrow> False"
-    by (unfold cs_holding_def, auto split:if_splits)
-
-lemma waiting_abs: "\<not> waiting ((wq s)(cs := t # l @ r)) t cs"
-    by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_abs_1: 
-  "\<lbrakk>\<not> waiting ((wq s)(cs := [])) t c; waiting (wq s) t c; c \<noteq> cs\<rbrakk>
-       \<Longrightarrow> False"
-    by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_abs_2: "
-       \<lbrakk>\<not> waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c; waiting (wq s) t c;
-        c \<noteq> cs\<rbrakk>
-       \<Longrightarrow> False"
-  by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_abs_3:
-     "\<lbrakk>wq s cs = a # list; \<not> waiting ((wq s)(cs := [])) t c;
-        waiting (wq s) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
-       \<Longrightarrow> False"
-  apply(drule_tac lsp_mid_nil, simp)
-  by(unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_simp: "c \<noteq> cs \<Longrightarrow> waiting ((wq s)(cs:=z)) t c = waiting (wq s) t c"
-   by(unfold cs_waiting_def, auto split:if_splits)
-
-lemma holding_cs_eq:
-  "\<lbrakk>\<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> c = cs"
-   by(unfold cs_holding_def, auto split:if_splits)
-
-lemma holding_cs_eq_1:
-  "\<lbrakk>\<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c\<rbrakk>
-       \<Longrightarrow> c = cs"
-   by(unfold cs_holding_def, auto split:if_splits)
-
-lemma holding_th_eq: 
-       "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; \<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c;
-        lsp (cp s) list = (aa, [], ca)\<rbrakk>
-       \<Longrightarrow> t = th"
-  apply(drule_tac lsp_mid_nil, simp)
-  apply(unfold cs_holding_def, auto split:if_splits)
-  apply(drule_tac step_back_step)
-  apply(ind_cases "step s (V th cs)")
-  by (unfold s_holding_def, auto split:if_splits)
-
-lemma holding_th_eq_1:
-  "\<lbrakk>vt step (V th cs#s); 
-     wq s cs = a # list; \<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c;
-        lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
-       \<Longrightarrow> t = th"
-  apply(drule_tac step_back_step)
-  apply(ind_cases "step s (V th cs)")
-  apply(unfold s_holding_def cs_holding_def)
-  by (auto split:if_splits)
-
-lemma holding_th_eq_2: "\<lbrakk>holding ((wq s)(cs := ac # x)) th cs\<rbrakk>
-       \<Longrightarrow> ac = th"
-  by (unfold cs_holding_def, auto)
-
-lemma holding_th_eq_3: "
-       \<lbrakk>\<not> holding (wq s) t c;
-        holding ((wq s)(cs := ac # x)) t c\<rbrakk>
-       \<Longrightarrow> ac = t"
-  by (unfold cs_holding_def, auto)
-
-lemma holding_wq_eq: "holding ((wq s)(cs := th' # l @ r)) th' cs"
-   by (unfold cs_holding_def, auto)
-
-lemma waiting_th_eq: "
-       \<lbrakk>waiting (wq s) t c; wq s cs = a # list;
-        lsp (cp s) list = (aa, ac # lista, ba); \<not> waiting ((wq s)(cs := ac # aa @ lista @ ba)) t c\<rbrakk>
-       \<Longrightarrow> ac = t"
-  apply(drule_tac lsp_set_eq, simp)
-  by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma step_depend_v:
-  "vt step (V th cs#s) \<Longrightarrow>
-  depend (V th cs # s) =
-  depend s - {(Cs cs, Th th)} -
-  {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
-  {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
-  apply (unfold s_depend_def wq_def, 
-         auto split:list.splits simp:Let_def f_nil_simp holding_wq_eq, fold wq_def cp_def)
-  apply (auto split:list.splits prod.splits  
-               simp:Let_def f_nil_simp holding_nil_simp holding_cs_neq_simp holding_nil_abs
-                    waiting_abs waiting_simp holding_wq_eq
-               elim:holding_nil waiting_kept_1 waiting_kept_2 step_wq_elim holding_th_neq_elim 
-               holding_abs waiting_abs_1 waiting_abs_3 holding_cs_eq holding_cs_eq_1
-               holding_th_eq holding_th_eq_1 holding_th_eq_2 holding_th_eq_3 waiting_th_eq
-               dest:lsp_mid_length)
-  done
-
-lemma step_depend_p:
-  "vt step (P th cs#s) \<Longrightarrow>
-  depend (P th cs # s) =  (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
-                                             else depend s \<union> {(Th th, Cs cs)})"
-  apply(unfold s_depend_def wq_def)
-  apply (auto split:list.splits prod.splits simp:Let_def cs_waiting_def cs_holding_def)
-  apply(case_tac "c = cs", auto)
-  apply(fold wq_def)
-  apply(drule_tac step_back_step)
-  by (ind_cases " step s (P (hd (wq s cs)) cs)", 
-    auto simp:s_depend_def wq_def cs_holding_def)
-
-lemma simple_A:
-  fixes A
-  assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
-  shows "A = {} \<or> (\<exists> a. A = {a})"
-proof(cases "A = {}")
-  case True thus ?thesis by simp
-next
-  case False then obtain a where "a \<in> A" by auto
-  with h have "A = {a}" by auto
-  thus ?thesis by simp
-qed
-
-lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
-  by (unfold s_depend_def, auto)
-
-lemma acyclic_depend: 
-  fixes s
-  assumes vt: "vt step s"
-  shows "acyclic (depend s)"
-proof -
-  from vt show ?thesis
-  proof(induct)
-    case (vt_cons s e)
-    assume ih: "acyclic (depend s)"
-      and stp: "step s e"
-      and vt: "vt step s"
-    show ?case
-    proof(cases e)
-      case (Create th prio)
-      with ih
-      show ?thesis by (simp add:depend_create_unchanged)
-    next
-      case (Exit th)
-      with ih show ?thesis by (simp add:depend_exit_unchanged)
-    next
-      case (V th cs)
-      from V vt stp have vtt: "vt step (V th cs#s)" by auto
-      from step_depend_v [OF this]
-      have eq_de: "depend (e # s) = 
-        depend s - {(Cs cs, Th th)} -
-        {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
-        {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
-        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
-      from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
-      have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
-      thus ?thesis
-      proof(cases "wq s cs")
-        case Nil
-        hence "?D = {}" by simp
-        with ac and eq_de show ?thesis by simp
-      next
-        case (Cons tth rest)
-        from stp and V have "step s (V th cs)" by simp
-        hence eq_wq: "wq s cs = th # rest"
-        proof -
-          show "step s (V th cs) \<Longrightarrow> wq s cs = th # rest"
-            apply(ind_cases "step s (V th cs)")
-            by(insert Cons, unfold s_holding_def, simp)
-        qed
-        show ?thesis
-        proof(cases "lsp (cp s) rest")
-          fix l b r
-          assume eq_lsp: "lsp (cp s) rest = (l, b, r) "
-          show ?thesis
-          proof(cases "b")
-            case Nil
-            with eq_lsp and eq_wq have "?D = {}" by simp
-            with ac and eq_de show ?thesis by simp
-          next
-            case (Cons th' m)
-            with eq_lsp 
-            have eq_lsp: "lsp (cp s) rest = (l, [th'], r)" 
-              apply simp
-              by (drule_tac lsp_mid_length, simp)
-            from eq_wq and eq_lsp
-            have eq_D: "?D = {(Cs cs, Th th')}" by auto
-            from eq_wq and eq_lsp
-            have eq_C: "?C = {(Th th', Cs cs)}" by auto
-            let ?E = "(?A - ?B - ?C)"
-            have "(Th th', Cs cs) \<notin> ?E\<^sup>*"
-            proof
-              assume "(Th th', Cs cs) \<in> ?E\<^sup>*"
-              hence " (Th th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-              from tranclD [OF this]
-              obtain x where th'_e: "(Th th', x) \<in> ?E" by blast
-              hence th_d: "(Th th', x) \<in> ?A" by simp
-              from depend_target_th [OF this]
-              obtain cs' where eq_x: "x = Cs cs'" by auto
-              with th_d have "(Th th', Cs cs') \<in> ?A" by simp
-              hence wt_th': "waiting s th' cs'"
-                unfolding s_depend_def s_waiting_def cs_waiting_def by simp
-              hence "cs' = cs"
-              proof(rule waiting_unique [OF vt])
-                from eq_wq eq_lsp wq_distinct[OF vt, of cs]
-                show "waiting s th' cs" by(unfold s_waiting_def, auto dest:lsp_set_eq)
-              qed
-              with th'_e eq_x have "(Th th', Cs cs) \<in> ?E" by simp
-              with eq_C show "False" by simp
-            qed
-            with acyclic_insert[symmetric] and ac and eq_D
-            and eq_de show ?thesis by simp
-          qed 
-        qed
-      qed
-    next
-      case (P th cs)
-      from P vt stp have vtt: "vt step (P th cs#s)" by auto
-      from step_depend_p [OF this] P
-      have "depend (e # s) = 
-              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
-                                    depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
-        by simp
-      moreover have "acyclic ?R"
-      proof(cases "wq s cs = []")
-        case True
-        hence eq_r: "?R =  depend s \<union> {(Cs cs, Th th)}" by simp
-        have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
-        proof
-          assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
-          hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-          from tranclD2 [OF this]
-          obtain x where "(x, Cs cs) \<in> depend s" by auto
-          with True show False by (auto simp:s_depend_def cs_waiting_def)
-        qed
-        with acyclic_insert ih eq_r show ?thesis by auto
-      next
-        case False
-        hence eq_r: "?R =  depend s \<union> {(Th th, Cs cs)}" by simp
-        have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
-        proof
-          assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
-          hence "(Cs cs, Th th) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-          moreover from step_back_step [OF vtt] have "step s (P th cs)" .
-          ultimately show False
-          proof -
-            show " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
-              by (ind_cases "step s (P th cs)", simp)
-          qed
-        qed
-        with acyclic_insert ih eq_r show ?thesis by auto
-      qed
-      ultimately show ?thesis by simp
-    next
-      case (Set thread prio)
-      with ih
-      thm depend_set_unchanged
-      show ?thesis by (simp add:depend_set_unchanged)
-    qed
-  next
-    case vt_nil
-    show "acyclic (depend ([]::state))"
-      by (auto simp: s_depend_def cs_waiting_def 
-                      cs_holding_def wq_def acyclic_def)
-  qed
-qed
-
-lemma finite_depend: 
-  fixes s
-  assumes vt: "vt step s"
-  shows "finite (depend s)"
-proof -
-  from vt show ?thesis
-  proof(induct)
-    case (vt_cons s e)
-    assume ih: "finite (depend s)"
-      and stp: "step s e"
-      and vt: "vt step s"
-    show ?case
-    proof(cases e)
-      case (Create th prio)
-      with ih
-      show ?thesis by (simp add:depend_create_unchanged)
-    next
-      case (Exit th)
-      with ih show ?thesis by (simp add:depend_exit_unchanged)
-    next
-      case (V th cs)
-      from V vt stp have vtt: "vt step (V th cs#s)" by auto
-      from step_depend_v [OF this]
-      have eq_de: "depend (e # s) = 
-        depend s - {(Cs cs, Th th)} -
-        {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
-        {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
-        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
-      moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
-      moreover have "finite ?D"
-      proof -
-        have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
-        thus ?thesis
-        proof
-          assume h: "?D = {}"
-          show ?thesis by (unfold h, simp)
-        next
-          assume "\<exists> a. ?D = {a}"
-          thus ?thesis by auto
-        qed
-      qed
-      ultimately show ?thesis by simp
-    next
-      case (P th cs)
-      from P vt stp have vtt: "vt step (P th cs#s)" by auto
-      from step_depend_p [OF this] P
-      have "depend (e # s) = 
-              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
-                                    depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
-        by simp
-      moreover have "finite ?R"
-      proof(cases "wq s cs = []")
-        case True
-        hence eq_r: "?R =  depend s \<union> {(Cs cs, Th th)}" by simp
-        with True and ih show ?thesis by auto
-      next
-        case False
-        hence "?R = depend s \<union> {(Th th, Cs cs)}" by simp
-        with False and ih show ?thesis by auto
-      qed
-      ultimately show ?thesis by auto
-    next
-      case (Set thread prio)
-      with ih
-      show ?thesis by (simp add:depend_set_unchanged)
-    qed
-  next
-    case vt_nil
-    show "finite (depend ([]::state))"
-      by (auto simp: s_depend_def cs_waiting_def 
-                   cs_holding_def wq_def acyclic_def)
-  qed
-qed
-
-text {* Several useful lemmas *}
-
-thm wf_trancl
-thm finite_acyclic_wf
-thm finite_acyclic_wf_converse
-thm wf_induct
-
-
-lemma wf_dep_converse: 
-  fixes s
-  assumes vt: "vt step s"
-  shows "wf ((depend s)^-1)"
-proof(rule finite_acyclic_wf_converse)
-  from finite_depend [OF vt]
-  show "finite (depend s)" .
-next
-  from acyclic_depend[OF vt]
-  show "acyclic (depend s)" .
-qed
-
-lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
-by (induct l, auto)
-
-lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
-  by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-
-lemma wq_threads: 
-  fixes s cs
-  assumes vt: "vt step s"
-  and h: "th \<in> set (wq s cs)"
-  shows "th \<in> threads s"
-proof -
- from vt and h show ?thesis
-  proof(induct arbitrary: th cs)
-    case (vt_cons s e)
-    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
-      and stp: "step s e"
-      and vt: "vt step s"
-      and h: "th \<in> set (wq (e # s) cs)"
-    show ?case
-    proof(cases e)
-      case (Create th' prio)
-      with ih h show ?thesis
-        by (auto simp:wq_def Let_def)
-    next
-      case (Exit th')
-      with stp ih h show ?thesis
-        apply (auto simp:wq_def Let_def)
-        apply (ind_cases "step s (Exit th')")
-        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
-               s_depend_def s_holding_def cs_holding_def)
-        by (fold wq_def, auto)
-    next
-      case (V th' cs')
-      show ?thesis
-      proof(cases "cs' = cs")
-        case False
-        with h
-        show ?thesis
-          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
-          by (drule_tac ih, simp)
-      next
-        case True
-        from h
-        show ?thesis
-        proof(unfold V wq_def)
-          assume th_in: "th \<in> set (waiting_queue (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
-          show "th \<in> threads (V th' cs' # s)"
-          proof(cases "cs = cs'")
-            case False
-            hence "?l = waiting_queue (schs s) cs" by (simp add:Let_def)
-            with th_in have " th \<in> set (wq s cs)" 
-              by (fold wq_def, simp)
-            from ih [OF this] show ?thesis by simp
-          next
-            case True
-            show ?thesis
-            proof(cases "waiting_queue (schs s) cs'")
-              case Nil
-              with h V show ?thesis
-                apply (auto simp:wq_def Let_def split:if_splits)
-                by (fold wq_def, drule_tac ih, simp)
-            next
-              case (Cons a rest)
-              assume eq_wq: "waiting_queue (schs s) cs' = a # rest"
-              with h V show ?thesis
-              proof(cases "(lsp (cur_preced (schs s)) rest)", unfold V)
-                fix l m r
-                assume eq_lsp: "lsp (cur_preced (schs s)) rest = (l, m, r)"
-                  and eq_wq: "waiting_queue (schs s) cs' = a # rest"
-                  and th_in_set: "th \<in> set (wq (V th' cs' # s) cs)"
-                show ?thesis
-                proof(cases "m")
-                  case Nil
-                  with eq_lsp have "rest = []" using lsp_mid_nil by auto
-                  with eq_wq have "waiting_queue (schs s) cs' = [a]" by simp
-                  with h[unfolded V wq_def] True 
-                  show ?thesis
-                    by (simp add:Let_def)
-                next
-                  case (Cons b rb)
-                  with lsp_mid_length[OF eq_lsp] have eq_m: "m = [b]" by auto
-                  with eq_lsp have "lsp (cur_preced (schs s)) rest = (l, [b], r)" by simp
-                  with h[unfolded V wq_def] True lsp_set_eq [OF this] eq_wq
-                  show ?thesis
-                    apply (auto simp:Let_def, fold wq_def)
-                    by (rule_tac ih [of _ cs'], auto)+
-                qed
-              qed
-            qed
-          qed
-        qed
-      qed
-    next
-      case (P th' cs')
-      from h stp
-      show ?thesis
-        apply (unfold P wq_def)
-        apply (auto simp:Let_def split:if_splits, fold wq_def)
-        apply (auto intro:ih)
-        apply(ind_cases "step s (P th' cs')")
-        by (unfold runing_def readys_def, auto)
-    next
-      case (Set thread prio)
-      with ih h show ?thesis
-        by (auto simp:wq_def Let_def)
-    qed
-  next
-    case vt_nil
-    thus ?case by (auto simp:wq_def)
-  qed
-qed
-
-lemma range_in: "\<lbrakk>vt step s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
-  apply(unfold s_depend_def cs_waiting_def cs_holding_def)
-  by (auto intro:wq_threads)
-
-lemma readys_v_eq:
-  fixes th thread cs rest
-  assumes neq_th: "th \<noteq> thread"
-  and eq_wq: "wq s cs = thread#rest"
-  and not_in: "th \<notin>  set rest"
-  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
-  from prems show ?thesis
-    apply (auto simp:readys_def)
-    apply (case_tac "cs = csa", simp add:s_waiting_def)
-    apply (erule_tac x = csa in allE)
-    apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
-    apply (case_tac "csa = cs", simp)
-    apply (erule_tac x = cs in allE)
-    by (auto simp:s_waiting_def wq_def Let_def split:list.splits prod.splits 
-            dest:lsp_set_eq)
-qed
-
-lemma readys_v_eq_1:
-  fixes th thread cs rest
-  assumes neq_th: "th \<noteq> thread"
-  and eq_wq: "wq s cs = thread#rest"
-  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
-  and neq_th': "th \<noteq> th'"
-  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
-  from prems show ?thesis
-    apply (auto simp:readys_def)
-    apply (case_tac "cs = csa", simp add:s_waiting_def)
-    apply (erule_tac x = cs in allE)
-    apply (simp add:s_waiting_def wq_def Let_def split:prod.splits list.splits)
-    apply (drule_tac lsp_mid_nil,simp, clarify, fold cp_def, clarsimp)
-    apply (frule_tac lsp_set_eq, simp)
-    apply (erule_tac x = csa in allE)
-    apply (subst (asm) (2) s_waiting_def, unfold wq_def)
-    apply (auto simp:Let_def split:list.splits prod.splits if_splits 
-            dest:lsp_set_eq)
-    apply (unfold s_waiting_def)
-    apply (fold wq_def, clarsimp)
-    apply (clarsimp)+
-    apply (case_tac "csa = cs", simp)
-    apply (erule_tac x = cs in allE, simp)
-    apply (unfold wq_def)
-    by (auto simp:Let_def split:list.splits prod.splits if_splits 
-            dest:lsp_set_eq)
-qed
-
-lemma readys_v_eq_2:
-  fixes th thread cs rest
-  assumes neq_th: "th \<noteq> thread"
-  and eq_wq: "wq s cs = thread#rest"
-  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
-  and neq_th': "th = th'"
-  and vt: "vt step s"
-  shows "(th \<in> readys (V thread cs#s))"
-proof -
-  from prems show ?thesis
-    apply (auto simp:readys_def)
-    apply (rule_tac wq_threads [of s _ cs], auto dest:lsp_set_eq)
-    apply (unfold s_waiting_def wq_def)
-    apply (auto simp:Let_def split:list.splits prod.splits if_splits 
-            dest:lsp_set_eq lsp_mid_nil lsp_mid_length)
-    apply (fold cp_def, simp+, clarsimp)
-    apply (frule_tac lsp_set_eq, simp)
-    apply (fold wq_def)
-    apply (subgoal_tac "csa = cs", simp)
-    apply (rule_tac waiting_unique [of s th'], simp)
-    by (auto simp:s_waiting_def)
-qed
-
-lemma chain_building:
-  assumes vt: "vt step s"
-  shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
-proof -
-  from wf_dep_converse [OF vt]
-  have h: "wf ((depend s)\<inverse>)" .
-  show ?thesis
-  proof(induct rule:wf_induct [OF h])
-    fix x
-    assume ih [rule_format]: 
-      "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow> 
-           y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
-    show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
-    proof
-      assume x_d: "x \<in> Domain (depend s)"
-      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
-      proof(cases x)
-        case (Th th)
-        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
-        with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
-        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
-        hence "Cs cs \<in> Domain (depend s)" by auto
-        from ih [OF x_in_r this] obtain th'
-          where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto
-        have "(x, Th th') \<in> (depend s)\<^sup>+" using Th x_in cs_in by auto
-        with th'_ready show ?thesis by auto
-      next
-        case (Cs cs)
-        from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (depend s)^-1" by (auto simp:s_depend_def)
-        show ?thesis
-        proof(cases "th' \<in> readys s")
-          case True
-          from True and th'_d show ?thesis by auto
-        next
-          case False
-          from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto
-          with False have "Th th' \<in> Domain (depend s)" 
-            by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
-          from ih [OF th'_d this]
-          obtain th'' where 
-            th''_r: "th'' \<in> readys s" and 
-            th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto
-          from th'_d and th''_in 
-          have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
-          with th''_r show ?thesis by auto
-        qed
-      qed
-    qed
-  qed
-qed
-
-lemma th_chain_to_ready:
-  fixes s th
-  assumes vt: "vt step s"
-  and th_in: "th \<in> threads s"
-  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"
-proof(cases "th \<in> readys s")
-  case True
-  thus ?thesis by auto
-next
-  case False
-  from False and th_in have "Th th \<in> Domain (depend s)" 
-    by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
-  from chain_building [rule_format, OF vt this]
-  show ?thesis by auto
-qed
-
-lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
-  by  (unfold s_waiting_def cs_waiting_def, auto)
-
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
-  by (unfold s_holding_def cs_holding_def, simp)
-
-lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
-  by (unfold s_holding_def cs_holding_def, auto)
-
-lemma unique_depend: "\<lbrakk>vt step s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
-  apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
-  by(auto elim:waiting_unique holding_unique)
-
-lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
-by (induct rule:trancl_induct, auto)
-
-lemma dchain_unique:
-  assumes vt: "vt step s"
-  and th1_d: "(n, Th th1) \<in> (depend s)^+"
-  and th1_r: "th1 \<in> readys s"
-  and th2_d: "(n, Th th2) \<in> (depend s)^+"
-  and th2_r: "th2 \<in> readys s"
-  shows "th1 = th2"
-proof -
-  { assume neq: "th1 \<noteq> th2"
-    hence "Th th1 \<noteq> Th th2" by simp
-    from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]
-    have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
-    hence "False"
-    proof
-      assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
-      from trancl_split [OF this]
-      obtain n where dd: "(Th th1, n) \<in> depend s" by auto
-      then obtain cs where eq_n: "n = Cs cs"
-        by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-      from dd eq_n have "th1 \<notin> readys s"
-        by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
-      with th1_r show ?thesis by auto
-    next
-      assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
-      from trancl_split [OF this]
-      obtain n where dd: "(Th th2, n) \<in> depend s" by auto
-      then obtain cs where eq_n: "n = Cs cs"
-        by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-      from dd eq_n have "th2 \<notin> readys s"
-        by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
-      with th2_r show ?thesis by auto
-    qed
-  } thus ?thesis by auto
-qed
-             
-definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
-where "count Q l = length (filter Q l)"
-
-definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
-where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
-
-definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
-where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
-
-
-lemma step_holdents_p_add:
-  fixes th cs s
-  assumes vt: "vt step (P th cs#s)"
-  and "wq s cs = []"
-  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
-proof -
-  from prems show ?thesis
-  unfolding  holdents_def step_depend_p[OF vt] by auto
-qed
-
-lemma step_holdents_p_eq:
-  fixes th cs s
-  assumes vt: "vt step (P th cs#s)"
-  and "wq s cs \<noteq> []"
-  shows "holdents (P th cs#s) th = holdents s th"
-proof -
-  from prems show ?thesis
-  unfolding  holdents_def step_depend_p[OF vt] by auto
-qed
-
-lemma step_holdents_v_minus:
-  fixes th cs s
-  assumes vt: "vt step (V th cs#s)"
-  shows "holdents (V th cs#s) th = holdents s th - {cs}"
-proof -
-  { fix rest l r
-    assume eq_wq: "wq s cs = th # rest" 
-      and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
-    have "False" 
-    proof -
-      from lsp_set_eq [OF eq_lsp] have " rest = l @ [th] @ r" .
-      with eq_wq have "wq s cs = th#\<dots>" by simp
-      with wq_distinct [OF step_back_vt[OF vt], of cs]
-      show ?thesis by auto
-    qed
-  } thus ?thesis unfolding holdents_def step_depend_v[OF vt] by auto
-qed
-
-lemma step_holdents_v_add:
-  fixes th th' cs s rest l r
-  assumes vt: "vt step (V th' cs#s)"
-  and neq_th: "th \<noteq> th'" 
-  and eq_wq: "wq s cs = th' # rest"
-  and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
-  shows "holdents (V th' cs#s) th = holdents s th \<union> {cs}"
-proof -
-  from prems show ?thesis
-  unfolding  holdents_def step_depend_v[OF vt] by auto
-qed
-
-lemma step_holdents_v_eq:
-  fixes th th' cs s rest l r th''
-  assumes vt: "vt step (V th' cs#s)"
-  and neq_th: "th \<noteq> th'" 
-  and eq_wq: "wq s cs = th' # rest"
-  and eq_lsp: "lsp (cp s) rest = (l, [th''], r)"
-  and neq_th': "th \<noteq> th''"
-  shows "holdents (V th' cs#s) th = holdents s th"
-proof -
-  from prems show ?thesis
-  unfolding  holdents_def step_depend_v[OF vt] by auto
-qed
-
-definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
-where "cntCS s th = card (holdents s th)"
-
-lemma cntCS_v_eq:
-  fixes th thread cs rest
-  assumes neq_th: "th \<noteq> thread"
-  and eq_wq: "wq s cs = thread#rest"
-  and not_in: "th \<notin>  set rest"
-  and vtv: "vt step (V thread cs#s)"
-  shows "cntCS (V thread cs#s) th = cntCS s th"
-proof -
-  from prems show ?thesis
-    apply (unfold cntCS_def holdents_def step_depend_v)
-    apply auto
-    apply (subgoal_tac "\<not>  (\<exists>l r. lsp (cp s) rest = (l, [th], r))", auto)
-    by (drule_tac lsp_set_eq, auto)
-qed
-
-lemma cntCS_v_eq_1:
-  fixes th thread cs rest
-  assumes neq_th: "th \<noteq> thread"
-  and eq_wq: "wq s cs = thread#rest"
-  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
-  and neq_th': "th \<noteq> th'"
-  and vtv: "vt step (V thread cs#s)"
-  shows "cntCS (V thread cs#s) th = cntCS s th"
-proof -
-  from prems show ?thesis
-    apply (unfold cntCS_def holdents_def step_depend_v)
-    by auto
-qed
-
-fun the_cs :: "node \<Rightarrow> cs"
-where "the_cs (Cs cs) = cs"
-
-lemma cntCS_v_eq_2:
-  fixes th thread cs rest
-  assumes neq_th: "th \<noteq> thread"
-  and eq_wq: "wq s cs = thread#rest"
-  and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
-  and neq_th': "th = th'"
-  and vtv: "vt step (V thread cs#s)"
-  shows "cntCS (V thread cs#s) th = 1 + cntCS s th"
-proof -
-  have "card {csa. csa = cs \<or> (Cs csa, Th th') \<in> depend s} = 
-                     Suc (card {cs. (Cs cs, Th th') \<in> depend s})" 
-    (is "card ?A = Suc (card ?B)")
-  proof -
-    have h: "?A = insert cs ?B" by auto
-    moreover have h1: "?B = ?B - {cs}"
-    proof -
-      { assume "(Cs cs, Th th') \<in> depend s"
-        moreover have "(Th th', Cs cs) \<in> depend s"
-        proof -
-          from wq_distinct [OF step_back_vt[OF vtv], of cs]
-          eq_wq lsp_set_eq [OF eq_lsp] show ?thesis
-            apply (auto simp:s_depend_def)
-            by (unfold cs_waiting_def, auto)
-        qed
-        moreover note acyclic_depend [OF step_back_vt[OF vtv]]
-        ultimately have "False"
-          apply (auto simp:acyclic_def)
-          apply (erule_tac x="Cs cs" in allE)
-          apply (subgoal_tac "(Cs cs, Cs cs) \<in> (depend s)\<^sup>+", simp)
-          by (rule_tac trancl_into_trancl [where b = "Th th'"], auto)
-      } thus ?thesis by auto
-    qed
-    moreover have "card (insert cs ?B) = Suc (card (?B - {cs}))"
-    proof(rule card_insert)
-      from finite_depend [OF step_back_vt [OF vtv]]
-      have fnt: "finite (depend s)" .
-      show " finite {cs. (Cs cs, Th th') \<in> depend s}" (is "finite ?B")
-      proof -
-        have "?B \<subseteq>  (\<lambda> (a, b). the_cs a) ` (depend s)"
-          apply (auto simp:image_def)
-          by (rule_tac x = "(Cs x, Th th')" in bexI, auto)
-        with fnt show ?thesis by (auto intro:finite_subset)
-      qed
-    qed
-    ultimately show ?thesis by simp
-  qed
-  with prems show ?thesis
-    apply (unfold cntCS_def holdents_def step_depend_v[OF vtv])
-    by auto
-qed
-
-lemma finite_holding:
-  fixes s th cs
-  assumes vt: "vt step s"
-  shows "finite (holdents s th)"
-proof -
-  let ?F = "\<lambda> (x, y). the_cs x"
-  from finite_depend [OF vt]
-  have "finite (depend s)" .
-  hence "finite (?F `(depend s))" by simp
-  moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>" 
-  proof -
-    { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
-      fix x assume "(Cs x, Th th) \<in> depend s"
-      hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
-      moreover have "?F (Cs x, Th th) = x" by simp
-      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp 
-    } thus ?thesis by auto
-  qed
-  ultimately show ?thesis by (unfold holdents_def, auto intro:finite_subset)
-qed
-
-inductive_cases case_step_v: "step s (V thread cs)"
-
-lemma cntCS_v_dec: 
-  fixes s thread cs
-  assumes vtv: "vt step (V thread cs#s)"
-  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
-proof -
-  have cs_in: "cs \<in> holdents s thread" using step_back_step[OF vtv]
-    apply (erule_tac case_step_v)
-    apply (unfold holdents_def s_depend_def, simp)
-    by (unfold cs_holding_def s_holding_def, auto)
-  moreover have cs_not_in: 
-    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
-    apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
-    by (unfold holdents_def, unfold step_depend_v[OF vtv], 
-            auto dest:lsp_set_eq)
-  ultimately 
-  have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
-    by auto
-  moreover have "card \<dots> = 
-                    Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
-  proof(rule card_insert)
-    from finite_holding [OF vtv]
-    show " finite (holdents (V thread cs # s) thread)" .
-  qed
-  moreover from cs_not_in 
-  have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
-  ultimately show ?thesis by (simp add:cntCS_def)
-qed 
-
-lemma cnp_cnv_cncs:
-  fixes s th
-  assumes vt: "vt step s"
-  shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) 
-                                       then cntCS s th else cntCS s th + 1)"
-proof -
-  from vt show ?thesis
-  proof(induct arbitrary:th)
-    case (vt_cons s e)
-    assume vt: "vt step s"
-    and ih: "\<And>th. cntP s th  = cntV s th +
-               (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
-    and stp: "step s e"
-    from stp show ?case
-    proof(cases)
-      case (thread_create prio max_prio thread)
-      assume eq_e: "e = Create thread prio"
-        and not_in: "thread \<notin> threads s"
-      show ?thesis
-      proof -
-        { fix cs 
-          assume "thread \<in> set (wq s cs)"
-          from wq_threads [OF vt this] have "thread \<in> threads s" .
-          with not_in have "False" by simp
-        } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
-          by (auto simp:readys_def threads.simps s_waiting_def 
-            wq_def cs_waiting_def Let_def)
-        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-        have eq_cncs: "cntCS (e#s) th = cntCS s th"
-          unfolding cntCS_def holdents_def
-          by (simp add:depend_create_unchanged eq_e)
-        { assume "th \<noteq> thread"
-          with eq_readys eq_e
-          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
-                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
-            by (simp add:threads.simps)
-          with eq_cnp eq_cnv eq_cncs ih not_in
-          have ?thesis by simp
-        } moreover {
-          assume eq_th: "th = thread"
-          with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp
-          moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
-          moreover note eq_cnp eq_cnv eq_cncs
-          ultimately have ?thesis by auto
-        } ultimately show ?thesis by blast
-      qed
-    next
-      case (thread_exit thread)
-      assume eq_e: "e = Exit thread" 
-      and is_runing: "thread \<in> runing s"
-      and no_hold: "holdents s thread = {}"
-      from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-      from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-      have eq_cncs: "cntCS (e#s) th = cntCS s th"
-        unfolding cntCS_def holdents_def
-        by (simp add:depend_exit_unchanged eq_e)
-      { assume "th \<noteq> thread"
-        with eq_e
-        have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
-          (th \<in> readys (s) \<or> th \<notin> threads (s))" 
-          apply (simp add:threads.simps readys_def)
-          apply (subst s_waiting_def)
-          apply (subst (1 2) wq_def)
-          apply (simp add:Let_def)
-          apply (subst s_waiting_def, simp)
-          by (fold wq_def, simp)
-        with eq_cnp eq_cnv eq_cncs ih
-        have ?thesis by simp
-      } moreover {
-        assume eq_th: "th = thread"
-        with ih is_runing have " cntP s th = cntV s th + cntCS s th" 
-          by (simp add:runing_def)
-        moreover from eq_th eq_e have "th \<notin> threads (e#s)"
-          by simp
-        moreover note eq_cnp eq_cnv eq_cncs
-        ultimately have ?thesis by auto
-      } ultimately show ?thesis by blast
-    next
-      case (thread_P thread cs)
-      assume eq_e: "e = P thread cs"
-        and is_runing: "thread \<in> runing s"
-        and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
-      from prems have vtp: "vt step (P thread cs#s)" by auto
-      show ?thesis 
-      proof -
-        { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
-          assume neq_th: "th \<noteq> thread"
-          with eq_e
-          have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
-            apply (simp add:readys_def s_waiting_def wq_def Let_def)
-            apply (rule_tac hh, clarify)
-            apply (intro iffI allI, clarify)
-            apply (erule_tac x = csa in allE, auto)
-            apply (subgoal_tac "waiting_queue (schs s) cs \<noteq> []", auto)
-            apply (erule_tac x = cs in allE, auto)
-            by (case_tac "(waiting_queue (schs s) cs)", auto)
-          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
-            apply (simp add:cntCS_def holdents_def)
-            by (unfold  step_depend_p [OF vtp], auto)
-          moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
-            by (simp add:cntP_def count_def)
-          moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
-            by (simp add:cntV_def count_def)
-          moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
-          moreover note ih [of th] 
-          ultimately have ?thesis by simp
-        } moreover {
-          assume eq_th: "th = thread"
-          have ?thesis
-          proof -
-            from eq_e eq_th have eq_cnp: "cntP (e # s) th  = 1 + (cntP s th)" 
-              by (simp add:cntP_def count_def)
-            from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
-              by (simp add:cntV_def count_def)
-            show ?thesis
-            proof (cases "wq s cs = []")
-              case True
-              with is_runing
-              have "th \<in> readys (e#s)"
-                apply (unfold eq_e wq_def, unfold readys_def s_depend_def)
-                apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
-                by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
-              moreover have "cntCS (e # s) th = 1 + cntCS s th"
-              proof -
-                have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> depend s} =
-                  Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")
-                proof -
-                  have "?L = insert cs ?R" by auto
-                  moreover have "card \<dots> = Suc (card (?R - {cs}))" 
-                  proof(rule card_insert)
-                    from finite_holding [OF vt, of thread]
-                    show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
-                      by (unfold holdents_def, simp)
-                  qed
-                  moreover have "?R - {cs} = ?R"
-                  proof -
-                    have "cs \<notin> ?R"
-                    proof
-                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend s}"
-                      with no_dep show False by auto
-                    qed
-                    thus ?thesis by auto
-                  qed
-                  ultimately show ?thesis by auto
-                qed
-                thus ?thesis
-                  apply (unfold eq_e eq_th cntCS_def)
-                  apply (simp add: holdents_def)
-                  by (unfold step_depend_p [OF vtp], auto simp:True)
-              qed
-              moreover from is_runing have "th \<in> readys s"
-                by (simp add:runing_def eq_th)
-              moreover note eq_cnp eq_cnv ih [of th]
-              ultimately show ?thesis by auto
-            next
-              case False
-              have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
-                    by (unfold eq_th eq_e wq_def, auto simp:Let_def)
-              have "th \<notin> readys (e#s)"
-              proof
-                assume "th \<in> readys (e#s)"
-                hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
-                from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
-                hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)" 
-                  by (simp add:s_waiting_def)
-                moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
-                ultimately have "th = hd (wq (e#s) cs)" by blast
-                with eq_wq have "th = hd (wq s cs @ [th])" by simp
-                hence "th = hd (wq s cs)" using False by auto
-                with False eq_wq wq_distinct [OF vtp, of cs]
-                show False by (fold eq_e, auto)
-              qed
-              moreover from is_runing have "th \<in> threads (e#s)" 
-                by (unfold eq_e, auto simp:runing_def readys_def eq_th)
-              moreover have "cntCS (e # s) th = cntCS s th"
-                apply (unfold cntCS_def holdents_def eq_e step_depend_p[OF vtp])
-                by (auto simp:False)
-              moreover note eq_cnp eq_cnv ih[of th]
-              moreover from is_runing have "th \<in> readys s"
-                by (simp add:runing_def eq_th)
-              ultimately show ?thesis by auto
-            qed
-          qed
-        } ultimately show ?thesis by blast
-      qed
-    next
-      case (thread_V thread cs)
-      from prems have vtv: "vt step (V thread cs # s)" by auto
-      assume eq_e: "e = V thread cs"
-        and is_runing: "thread \<in> runing s"
-        and hold: "holding s thread cs"
-      from hold obtain rest 
-        where eq_wq: "wq s cs = thread # rest"
-        by (case_tac "wq s cs", auto simp:s_holding_def)
-      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
-      show ?thesis
-      proof -
-        { assume eq_th: "th = thread"
-          from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
-            by (unfold eq_e, simp add:cntP_def count_def)
-          moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
-            by (unfold eq_e, simp add:cntV_def count_def)
-          moreover from cntCS_v_dec [OF vtv] 
-          have "cntCS (e # s) thread + 1 = cntCS s thread"
-            by (simp add:eq_e)
-          moreover from is_runing have rd_before: "thread \<in> readys s"
-            by (unfold runing_def, simp)
-          moreover have "thread \<in> readys (e # s)"
-          proof -
-            from is_runing
-            have "thread \<in> threads (e#s)" 
-              by (unfold eq_e, auto simp:runing_def readys_def)
-            moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
-            proof
-              fix cs1
-              { assume eq_cs: "cs1 = cs" 
-                have "\<not> waiting (e # s) thread cs1"
-                proof -
-                  have "thread \<notin> set (wq (e#s) cs1)"
-                  proof(cases "lsp (cp s) rest")
-                    fix l m r 
-                    assume h: "lsp (cp s) rest = (l, m, r)"
-                    show ?thesis
-                    proof(cases "m")
-                      case Nil
-                      from wq_v_eq_nil [OF eq_wq] h Nil eq_e
-                      have " wq (e # s) cs = []" by auto
-                      thus ?thesis using eq_cs by auto
-                    next
-                      case (Cons th' l')
-                      from lsp_mid_length [OF h] and Cons h
-                      have eqh: "lsp (cp s) rest = (l, [th'], r)" by auto
-                      from wq_v_eq [OF eq_wq this]
-                      have "wq (V thread cs # s) cs = th' # l @ r" .
-                      moreover from lsp_set_eq [OF eqh]
-                      have "set rest = set \<dots>" by auto
-                      moreover have "thread \<notin> set rest"
-                      proof -
-                        from wq_distinct [OF step_back_vt[OF vtv], of cs]
-                        and eq_wq show ?thesis by auto
-                      qed
-                      moreover note eq_e eq_cs
-                      ultimately show ?thesis by simp
-                    qed
-                  qed
-                  thus ?thesis by (simp add:s_waiting_def)
-                qed
-              } moreover {
-                assume neq_cs: "cs1 \<noteq> cs"
-                  have "\<not> waiting (e # s) thread cs1" 
-                  proof -
-                    from wq_v_neq [OF neq_cs[symmetric]]
-                    have "wq (V thread cs # s) cs1 = wq s cs1" .
-                    moreover have "\<not> waiting s thread cs1" 
-                    proof -
-                      from runing_ready and is_runing
-                      have "thread \<in> readys s" by auto
-                      thus ?thesis by (simp add:readys_def)
-                    qed
-                    ultimately show ?thesis 
-                      by (auto simp:s_waiting_def eq_e)
-                  qed
-              } ultimately show "\<not> waiting (e # s) thread cs1" by blast
-            qed
-            ultimately show ?thesis by (simp add:readys_def)
-          qed
-          moreover note eq_th ih
-          ultimately have ?thesis by auto
-        } moreover {
-          assume neq_th: "th \<noteq> thread"
-          from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" 
-            by (simp add:cntP_def count_def)
-          from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" 
-            by (simp add:cntV_def count_def)
-          have ?thesis
-          proof(cases "th \<in> set rest")
-            case False
-            have "(th \<in> readys (e # s)) = (th \<in> readys s)"
-              by(unfold eq_e, rule readys_v_eq [OF neq_th eq_wq False])
-            moreover have "cntCS (e#s) th = cntCS s th"
-              by(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq False vtv]) 
-            moreover note ih eq_cnp eq_cnv eq_threads
-            ultimately show ?thesis by auto
-          next
-            case True
-            obtain l m r where eq_lsp: "lsp (cp s) rest = (l, m, r)" 
-              by (cases "lsp (cp s) rest", auto)
-            with True have "m \<noteq> []" by  (auto dest:lsp_mid_nil)
-            with eq_lsp obtain th' where eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
-              by (case_tac m, auto dest:lsp_mid_length)
-            show ?thesis
-            proof(cases "th = th'")
-              case False
-              have "(th \<in> readys (e # s)) = (th \<in> readys s)"
-                by (unfold eq_e, rule readys_v_eq_1 [OF neq_th eq_wq eq_lsp False])
-              moreover have "cntCS (e#s) th = cntCS s th" 
-                by (unfold eq_e, rule cntCS_v_eq_1[OF neq_th eq_wq eq_lsp False vtv])
-              moreover note ih eq_cnp eq_cnv eq_threads
-              ultimately show ?thesis by auto
-            next
-              case True
-              have "th \<in> readys (e # s)"
-                by (unfold eq_e, rule readys_v_eq_2 [OF neq_th eq_wq eq_lsp True vt])
-              moreover have "cntP s th = cntV s th + cntCS s th + 1"
-              proof -
-                have "th \<notin> readys s" 
-                proof -
-                  from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
-                  show ?thesis
-                    apply (unfold readys_def s_waiting_def, auto)
-                    by (rule_tac x = cs in exI, auto)
-                qed
-                moreover have "th \<in> threads s"
-                proof -
-                  from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
-                  have "th \<in> set (wq s cs)" by simp
-                  from wq_threads [OF step_back_vt[OF vtv] this] 
-                  show ?thesis .
-                qed
-                ultimately show ?thesis using ih by auto
-              qed
-              moreover have "cntCS (e # s) th = 1 + cntCS s th"
-                by (unfold eq_e, rule cntCS_v_eq_2 [OF neq_th eq_wq eq_lsp True vtv])
-              moreover note eq_cnp eq_cnv
-              ultimately show ?thesis by simp
-            qed
-          qed
-        } ultimately show ?thesis by blast
-      qed
-    next
-      case (thread_set thread prio)
-      assume eq_e: "e = Set thread prio"
-        and is_runing: "thread \<in> runing s"
-      show ?thesis
-      proof -
-        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-        have eq_cncs: "cntCS (e#s) th = cntCS s th"
-          unfolding cntCS_def holdents_def
-          by (simp add:depend_set_unchanged eq_e)
-        from eq_e have eq_readys: "readys (e#s) = readys s" 
-          by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
-                  auto simp:Let_def)
-        { assume "th \<noteq> thread"
-          with eq_readys eq_e
-          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
-                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
-            by (simp add:threads.simps)
-          with eq_cnp eq_cnv eq_cncs ih is_runing
-          have ?thesis by simp
-        } moreover {
-          assume eq_th: "th = thread"
-          with is_runing ih have " cntP s th  = cntV s th + cntCS s th" 
-            by (unfold runing_def, auto)
-          moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
-            by (simp add:runing_def)
-          moreover note eq_cnp eq_cnv eq_cncs
-          ultimately have ?thesis by auto
-        } ultimately show ?thesis by blast
-      qed   
-    qed
-  next
-    case vt_nil
-    show ?case 
-      by (unfold cntP_def cntV_def cntCS_def, 
-        auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
-  qed
-qed
-
-lemma not_thread_cncs:
-  fixes th s
-  assumes vt: "vt step s"
-  and not_in: "th \<notin> threads s" 
-  shows "cntCS s th = 0"
-proof -
-  from vt not_in show ?thesis
-  proof(induct arbitrary:th)
-    case (vt_cons s e th)
-    assume vt: "vt step s"
-      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
-      and stp: "step s e"
-      and not_in: "th \<notin> threads (e # s)"
-    from stp show ?case
-    proof(cases)
-      case (thread_create prio max_prio thread)
-      assume eq_e: "e = Create thread prio"
-        and not_in': "thread \<notin> threads s"
-      have "cntCS (e # s) th = cntCS s th"
-        apply (unfold eq_e cntCS_def holdents_def)
-        by (simp add:depend_create_unchanged)
-      moreover have "th \<notin> threads s" 
-      proof -
-        from not_in eq_e show ?thesis by simp
-      qed
-      moreover note ih ultimately show ?thesis by auto
-    next
-      case (thread_exit thread)
-      assume eq_e: "e = Exit thread"
-      and nh: "holdents s thread = {}"
-      have eq_cns: "cntCS (e # s) th = cntCS s th"
-        apply (unfold eq_e cntCS_def holdents_def)
-        by (simp add:depend_exit_unchanged)
-      show ?thesis
-      proof(cases "th = thread")
-        case True
-        have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
-        with eq_cns show ?thesis by simp
-      next
-        case False
-        with not_in and eq_e
-        have "th \<notin> threads s" by simp
-        from ih[OF this] and eq_cns show ?thesis by simp
-      qed
-    next
-      case (thread_P thread cs)
-      assume eq_e: "e = P thread cs"
-      and is_runing: "thread \<in> runing s"
-      from prems have vtp: "vt step (P thread cs#s)" by auto
-      have neq_th: "th \<noteq> thread" 
-      proof -
-        from not_in eq_e have "th \<notin> threads s" by simp
-        moreover from is_runing have "thread \<in> threads s"
-          by (simp add:runing_def readys_def)
-        ultimately show ?thesis by auto
-      qed
-      hence "cntCS (e # s) th  = cntCS s th "
-        apply (unfold cntCS_def holdents_def eq_e)
-        by (unfold step_depend_p[OF vtp], auto)
-      moreover have "cntCS s th = 0"
-      proof(rule ih)
-        from not_in eq_e show "th \<notin> threads s" by simp
-      qed
-      ultimately show ?thesis by simp
-    next
-      case (thread_V thread cs)
-      assume eq_e: "e = V thread cs"
-        and is_runing: "thread \<in> runing s"
-        and hold: "holding s thread cs"
-      have neq_th: "th \<noteq> thread" 
-      proof -
-        from not_in eq_e have "th \<notin> threads s" by simp
-        moreover from is_runing have "thread \<in> threads s"
-          by (simp add:runing_def readys_def)
-        ultimately show ?thesis by auto
-      qed
-      from prems have vtv: "vt step (V thread cs#s)" by auto
-      from hold obtain rest 
-        where eq_wq: "wq s cs = thread # rest"
-        by (case_tac "wq s cs", auto simp:s_holding_def)
-      have "cntCS (e # s) th  = cntCS s th"
-      proof(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq _ vtv])
-        show "th \<notin> set rest" 
-        proof
-          assume "th \<in> set rest"
-          with eq_wq have "th \<in> set (wq s cs)" by simp
-          from wq_threads [OF vt this] eq_e not_in 
-          show False by simp
-        qed
-      qed
-      moreover have "cntCS s th = 0"
-      proof(rule ih)
-        from not_in eq_e show "th \<notin> threads s" by simp
-      qed
-      ultimately show ?thesis by simp
-    next
-      case (thread_set thread prio)
-      print_facts
-      assume eq_e: "e = Set thread prio"
-        and is_runing: "thread \<in> runing s"
-      from not_in and eq_e have "th \<notin> threads s" by auto
-      from ih [OF this] and eq_e
-      show ?thesis 
-        apply (unfold eq_e cntCS_def holdents_def)
-        by (simp add:depend_set_unchanged)
-    qed
-    next
-      case vt_nil
-      show ?case
-      by (unfold cntCS_def, 
-        auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
-  qed
-qed
-
-lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
-  by (auto simp:s_waiting_def cs_waiting_def)
-
-lemma dm_depend_threads:
-  fixes th s
-  assumes vt: "vt step s"
-  and in_dom: "(Th th) \<in> Domain (depend s)"
-  shows "th \<in> threads s"
-proof -
-  from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
-  moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
-  ultimately have "(Th th, Cs cs) \<in> depend s" by simp
-  hence "th \<in> set (wq s cs)"
-    by (unfold s_depend_def, auto simp:cs_waiting_def)
-  from wq_threads [OF vt this] show ?thesis .
-qed
-
-lemma cp_eq_cpreced: "cp s th = cpreced s (wq s) th"
-proof(unfold cp_def wq_def, induct s)
-  case (Cons e s')
-  show ?case
-    by (auto simp:Let_def)
-next
-  case Nil
-  show ?case by (auto simp:Let_def)
-qed
-
-fun the_th :: "node \<Rightarrow> thread"
-  where "the_th (Th th) = th"
-
-lemma runing_unique:
-  fixes th1 th2 s
-  assumes vt: "vt step s"
-  and runing_1: "th1 \<in> runing s"
-  and runing_2: "th2 \<in> runing s"
-  shows "th1 = th2"
-proof -
-  from runing_1 and runing_2 have "cp s th1 = cp s th2"
-    by (unfold runing_def, simp)
-  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
-                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
-    (is "Max (?f ` ?A) = Max (?f ` ?B)")
-    by (unfold cp_eq_cpreced cpreced_def)
-  obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
-  proof -
-    have h1: "finite (?f ` ?A)"
-    proof -
-      have "finite ?A" 
-      proof -
-        have "finite (dependents (wq s) th1)"
-        proof-
-          have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
-          proof -
-            let ?F = "\<lambda> (x, y). the_th x"
-            have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
-              apply (auto simp:image_def)
-              by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
-            moreover have "finite \<dots>"
-            proof -
-              from finite_depend[OF vt] have "finite (depend s)" .
-              hence "finite ((depend (wq s))\<^sup>+)"
-                apply (unfold finite_trancl)
-                by (auto simp: s_depend_def cs_depend_def wq_def)
-              thus ?thesis by auto
-            qed
-            ultimately show ?thesis by (auto intro:finite_subset)
-          qed
-          thus ?thesis by (simp add:cs_dependents_def)
-        qed
-        thus ?thesis by simp
-      qed
-      thus ?thesis by auto
-    qed
-    moreover have h2: "(?f ` ?A) \<noteq> {}"
-    proof -
-      have "?A \<noteq> {}" by simp
-      thus ?thesis by simp
-    qed
-    from Max_in [OF h1 h2]
-    have "Max (?f ` ?A) \<in> (?f ` ?A)" .
-    thus ?thesis by (auto intro:that)
-  qed
-  obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
-  proof -
-    have h1: "finite (?f ` ?B)"
-    proof -
-      have "finite ?B" 
-      proof -
-        have "finite (dependents (wq s) th2)"
-        proof-
-          have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
-          proof -
-            let ?F = "\<lambda> (x, y). the_th x"
-            have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
-              apply (auto simp:image_def)
-              by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
-            moreover have "finite \<dots>"
-            proof -
-              from finite_depend[OF vt] have "finite (depend s)" .
-              hence "finite ((depend (wq s))\<^sup>+)"
-                apply (unfold finite_trancl)
-                by (auto simp: s_depend_def cs_depend_def wq_def)
-              thus ?thesis by auto
-            qed
-            ultimately show ?thesis by (auto intro:finite_subset)
-          qed
-          thus ?thesis by (simp add:cs_dependents_def)
-        qed
-        thus ?thesis by simp
-      qed
-      thus ?thesis by auto
-    qed
-    moreover have h2: "(?f ` ?B) \<noteq> {}"
-    proof -
-      have "?B \<noteq> {}" by simp
-      thus ?thesis by simp
-    qed
-    from Max_in [OF h1 h2]
-    have "Max (?f ` ?B) \<in> (?f ` ?B)" .
-    thus ?thesis by (auto intro:that)
-  qed
-  from eq_f_th1 eq_f_th2 eq_max 
-  have eq_preced: "preced th1' s = preced th2' s" by auto
-  hence eq_th12: "th1' = th2'"
-  proof (rule preced_unique)
-    from th1_in have "th1' = th1 \<or> (th1' \<in> dependents (wq s) th1)" by simp
-    thus "th1' \<in> threads s"
-    proof
-      assume "th1' \<in> dependents (wq s) th1"
-      hence "(Th th1') \<in> Domain ((depend s)^+)"
-        apply (unfold cs_dependents_def cs_depend_def s_depend_def)
-        by (auto simp:Domain_def)
-      hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
-      from dm_depend_threads[OF vt this] show ?thesis .
-    next
-      assume "th1' = th1"
-      with runing_1 show ?thesis
-        by (unfold runing_def readys_def, auto)
-    qed
-  next
-    from th2_in have "th2' = th2 \<or> (th2' \<in> dependents (wq s) th2)" by simp
-    thus "th2' \<in> threads s"
-    proof
-      assume "th2' \<in> dependents (wq s) th2"
-      hence "(Th th2') \<in> Domain ((depend s)^+)"
-        apply (unfold cs_dependents_def cs_depend_def s_depend_def)
-        by (auto simp:Domain_def)
-      hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
-      from dm_depend_threads[OF vt this] show ?thesis .
-    next
-      assume "th2' = th2"
-      with runing_2 show ?thesis
-        by (unfold runing_def readys_def, auto)
-    qed
-  qed
-  from th1_in have "th1' = th1 \<or> th1' \<in> dependents (wq s) th1" by simp
-  thus ?thesis
-  proof
-    assume eq_th': "th1' = th1"
-    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
-    thus ?thesis
-    proof
-      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
-    next
-      assume "th2' \<in> dependents (wq s) th2"
-      with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
-      hence "(Th th1, Th th2) \<in> (depend s)^+"
-        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
-      hence "Th th1 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
-        by auto
-      hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
-      then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
-      from depend_target_th [OF this]
-      obtain cs' where "n = Cs cs'" by auto
-      with d have "(Th th1, Cs cs') \<in> depend s" by simp
-      with runing_1 have "False"
-        apply (unfold runing_def readys_def s_depend_def)
-        by (auto simp:eq_waiting)
-      thus ?thesis by simp
-    qed
-  next
-    assume th1'_in: "th1' \<in> dependents (wq s) th1"
-    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
-    thus ?thesis 
-    proof
-      assume "th2' = th2"
-      with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
-      hence "(Th th2, Th th1) \<in> (depend s)^+"
-        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
-      hence "Th th2 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
-        by auto
-      hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
-      then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
-      from depend_target_th [OF this]
-      obtain cs' where "n = Cs cs'" by auto
-      with d have "(Th th2, Cs cs') \<in> depend s" by simp
-      with runing_2 have "False"
-        apply (unfold runing_def readys_def s_depend_def)
-        by (auto simp:eq_waiting)
-      thus ?thesis by simp
-    next
-      assume "th2' \<in> dependents (wq s) th2"
-      with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
-      hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
-        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
-      from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
-        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
-      show ?thesis
-      proof(rule dchain_unique[OF vt h1 _ h2, symmetric])
-        from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
-        from runing_2 show "th2 \<in> readys s" by (simp add:runing_def) 
-      qed
-    qed
-  qed
-qed
-
-lemma create_pre:
-  assumes stp: "step s e"
-  and not_in: "th \<notin> threads s"
-  and is_in: "th \<in> threads (e#s)"
-  obtains prio where "e = Create th prio"
-proof -
-  from assms  
-  show ?thesis
-  proof(cases)
-    case (thread_create prio max_prio thread)
-    with is_in not_in have "e = Create th prio" by simp
-    from that[OF this] show ?thesis .
-  next
-    case (thread_exit thread)
-    with assms show ?thesis by (auto intro!:that)
-  next
-    case (thread_P thread)
-    with assms show ?thesis by (auto intro!:that)
-  next
-    case (thread_V thread)
-    with assms show ?thesis by (auto intro!:that)
-  next 
-    case (thread_set thread)
-    with assms show ?thesis by (auto intro!:that)
-  qed
-qed
-
-lemma length_down_to_in: 
-  assumes le_ij: "i \<le> j"
-    and le_js: "j \<le> length s"
-  shows "length (down_to j i s) = j - i"
-proof -
-  have "length (down_to j i s) = length (from_to i j (rev s))"
-    by (unfold down_to_def, auto)
-  also have "\<dots> = j - i"
-  proof(rule length_from_to_in[OF le_ij])
-    from le_js show "j \<le> length (rev s)" by simp
-  qed
-  finally show ?thesis .
-qed
-
-
-lemma moment_head: 
-  assumes le_it: "Suc i \<le> length t"
-  obtains e where "moment (Suc i) t = e#moment i t"
-proof -
-  have "i \<le> Suc i" by simp
-  from length_down_to_in [OF this le_it]
-  have "length (down_to (Suc i) i t) = 1" by auto
-  then obtain e where "down_to (Suc i) i t = [e]"
-    apply (cases "(down_to (Suc i) i t)") by auto
-  moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
-    by (rule down_to_conc[symmetric], auto)
-  ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
-    by (auto simp:down_to_moment)
-  from that [OF this] show ?thesis .
-qed
-
-lemma cnp_cnv_eq:
-  fixes th s
-  assumes "vt step s"
-  and "th \<notin> threads s"
-  shows "cntP s th = cntV s th"
-proof -
-  from assms show ?thesis
-  proof(induct)
-    case (vt_cons s e)
-    have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
-    have not_in: "th \<notin> threads (e # s)" by fact
-    have "step s e" by fact
-    thus ?case proof(cases)
-      case (thread_create prio max_prio thread)
-      assume eq_e: "e = Create thread prio"
-      hence "thread \<in> threads (e#s)" by simp
-      with not_in and eq_e have "th \<notin> threads s" by auto
-      from ih [OF this] show ?thesis using eq_e
-        by (auto simp:cntP_def cntV_def count_def)
-    next
-      case (thread_exit thread)
-      assume eq_e: "e = Exit thread"
-        and not_holding: "holdents s thread = {}"
-      have vt_s: "vt step s" by fact
-      from finite_holding[OF vt_s] have "finite (holdents s thread)" .
-      with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
-      moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
-      moreover note cnp_cnv_cncs[OF vt_s, of thread]
-      ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
-      show ?thesis
-      proof(cases "th = thread")
-        case True
-        with eq_thread eq_e show ?thesis 
-          by (auto simp:cntP_def cntV_def count_def)
-      next
-        case False
-        with not_in and eq_e have "th \<notin> threads s" by simp
-        from ih[OF this] and eq_e show ?thesis 
-           by (auto simp:cntP_def cntV_def count_def)
-      qed
-    next
-      case (thread_P thread cs)
-      assume eq_e: "e = P thread cs"
-      have "thread \<in> runing s" by fact
-      with not_in eq_e have neq_th: "thread \<noteq> th" 
-        by (auto simp:runing_def readys_def)
-      from not_in eq_e have "th \<notin> threads s" by simp
-      from ih[OF this] and neq_th and eq_e show ?thesis
-        by (auto simp:cntP_def cntV_def count_def)
-    next
-      case (thread_V thread cs)
-      assume eq_e: "e = V thread cs"
-      have "thread \<in> runing s" by fact
-      with not_in eq_e have neq_th: "thread \<noteq> th" 
-        by (auto simp:runing_def readys_def)
-      from not_in eq_e have "th \<notin> threads s" by simp
-      from ih[OF this] and neq_th and eq_e show ?thesis
-        by (auto simp:cntP_def cntV_def count_def)
-    next
-      case (thread_set thread prio)
-      assume eq_e: "e = Set thread prio"
-        and "thread \<in> runing s"
-      hence "thread \<in> threads (e#s)" 
-        by (simp add:runing_def readys_def)
-      with not_in and eq_e have "th \<notin> threads s" by auto
-      from ih [OF this] show ?thesis using eq_e
-        by (auto simp:cntP_def cntV_def count_def)  
-    qed
-  next
-    case vt_nil
-    show ?case by (auto simp:cntP_def cntV_def count_def)
-  qed
-qed
-
-lemma eq_depend: 
-  "depend (wq s) = depend s"
-by (unfold cs_depend_def s_depend_def, auto)
-
-lemma count_eq_dependents:
-  assumes vt: "vt step s"
-  and eq_pv: "cntP s th = cntV s th"
-  shows "dependents (wq s) th = {}"
-proof -
-  from cnp_cnv_cncs[OF vt] and eq_pv
-  have "cntCS s th = 0" 
-    by (auto split:if_splits)
-  moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
-  proof -
-    from finite_holding[OF vt, of th] show ?thesis
-      by (simp add:holdents_def)
-  qed
-  ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
-    by (unfold cntCS_def holdents_def cs_dependents_def, auto)
-  show ?thesis
-  proof(unfold cs_dependents_def)
-    { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
-      then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
-      hence "False"
-      proof(cases)
-        assume "(Th th', Th th) \<in> depend (wq s)"
-        thus "False" by (auto simp:cs_depend_def)
-      next
-        fix c
-        assume "(c, Th th) \<in> depend (wq s)"
-        with h and eq_depend show "False"
-          by (cases c, auto simp:cs_depend_def)
-      qed
-    } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
-  qed
-qed
-
-lemma dependents_threads:
-  fixes s th
-  assumes vt: "vt step s"
-  shows "dependents (wq s) th \<subseteq> threads s"
-proof
-  { fix th th'
-    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
-    have "Th th \<in> Domain (depend s)"
-    proof -
-      from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
-      hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
-      with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
-      thus ?thesis using eq_depend by simp
-    qed
-    from dm_depend_threads[OF vt this]
-    have "th \<in> threads s" .
-  } note hh = this
-  fix th1 
-  assume "th1 \<in> dependents (wq s) th"
-  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
-    by (unfold cs_dependents_def, simp)
-  from hh [OF this] show "th1 \<in> threads s" .
-qed
-
-lemma finite_threads:
-  assumes vt: "vt step s"
-  shows "finite (threads s)"
-proof -
-  from vt show ?thesis
-  proof(induct)
-    case (vt_cons s e)
-    assume vt: "vt step s"
-    and step: "step s e"
-    and ih: "finite (threads s)"
-    from step
-    show ?case
-    proof(cases)
-      case (thread_create prio max_prio thread)
-      assume eq_e: "e = Create thread prio"
-      with ih
-      show ?thesis by (unfold eq_e, auto)
-    next
-      case (thread_exit thread)
-      assume eq_e: "e = Exit thread"
-      with ih show ?thesis 
-        by (unfold eq_e, auto)
-    next
-      case (thread_P thread cs)
-      assume eq_e: "e = P thread cs"
-      with ih show ?thesis by (unfold eq_e, auto)
-    next
-      case (thread_V thread cs)
-      assume eq_e: "e = V thread cs"
-      with ih show ?thesis by (unfold eq_e, auto)
-    next 
-      case (thread_set thread prio)
-      from vt_cons thread_set show ?thesis by simp
-    qed
-  next
-    case vt_nil
-    show ?case by (auto)
-  qed
-qed
-
-lemma Max_f_mono:
-  assumes seq: "A \<subseteq> B"
-  and np: "A \<noteq> {}"
-  and fnt: "finite B"
-  shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
-  from seq show "f ` A \<subseteq> f ` B" by auto
-next
-  from np show "f ` A \<noteq> {}" by auto
-next
-  from fnt and seq show "finite (f ` B)" by auto
-qed
-
-lemma cp_le:
-  assumes vt: "vt step s"
-  and th_in: "th \<in> threads s"
-  shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def)
-  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))
-         \<le> Max ((\<lambda>th. preced th s) ` threads s)"
-    (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
-  proof(rule Max_f_mono)
-    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
-  next
-    from finite_threads [OF vt]
-    show "finite (threads s)" .
-  next
-    from th_in
-    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
-      apply (auto simp:Domain_def)
-      apply (rule_tac dm_depend_threads[OF vt])
-      apply (unfold trancl_domain [of "depend s", symmetric])
-      by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
-  qed
-qed
-
-lemma le_cp:
-  assumes vt: "vt step s"
-  shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
-  show "Prc (original_priority th s) (birthtime th s)
-    \<le> Max (insert (Prc (original_priority th s) (birthtime th s))
-            ((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))"
-    (is "?l \<le> Max (insert ?l ?A)")
-  proof(cases "?A = {}")
-    case False
-    have "finite ?A" (is "finite (?f ` ?B)")
-    proof -
-      have "finite ?B" 
-      proof-
-        have "finite {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}"
-        proof -
-          let ?F = "\<lambda> (x, y). the_th x"
-          have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
-            apply (auto simp:image_def)
-            by (rule_tac x = "(Th x, Th th)" in bexI, auto)
-          moreover have "finite \<dots>"
-          proof -
-            from finite_depend[OF vt] have "finite (depend s)" .
-            hence "finite ((depend (wq s))\<^sup>+)"
-              apply (unfold finite_trancl)
-              by (auto simp: s_depend_def cs_depend_def wq_def)
-            thus ?thesis by auto
-          qed
-          ultimately show ?thesis by (auto intro:finite_subset)
-        qed
-        thus ?thesis by (simp add:cs_dependents_def)
-      qed
-      thus ?thesis by simp
-    qed
-    from Max_insert [OF this False, of ?l] show ?thesis by auto
-  next
-    case True
-    thus ?thesis by auto
-  qed
-qed
-
-lemma max_cp_eq: 
-  assumes vt: "vt step s"
-  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
-  (is "?l = ?r")
-proof(cases "threads s = {}")
-  case True
-  thus ?thesis by auto
-next
-  case False
-  have "?l \<in> ((cp s) ` threads s)"
-  proof(rule Max_in)
-    from finite_threads[OF vt] 
-    show "finite (cp s ` threads s)" by auto
-  next
-    from False show "cp s ` threads s \<noteq> {}" by auto
-  qed
-  then obtain th 
-    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
-  have "\<dots> \<le> ?r" by (rule cp_le[OF vt th_in])
-  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
-  proof -
-    have "?r \<in> (?f ` ?A)"
-    proof(rule Max_in)
-      from finite_threads[OF vt]
-      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
-    next
-      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
-    qed
-    then obtain th' where 
-      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
-    from le_cp [OF vt, of th']  eq_r
-    have "?r \<le> cp s th'" by auto
-    moreover have "\<dots> \<le> cp s th"
-    proof(fold eq_l)
-      show " cp s th' \<le> Max (cp s ` threads s)"
-      proof(rule Max_ge)
-        from th_in' show "cp s th' \<in> cp s ` threads s"
-          by auto
-      next
-        from finite_threads[OF vt]
-        show "finite (cp s ` threads s)" by auto
-      qed
-    qed
-    ultimately show ?thesis by auto
-  qed
-  ultimately show ?thesis using eq_l by auto
-qed
-
-lemma max_cp_readys_threads_pre:
-  assumes vt: "vt step s"
-  and np: "threads s \<noteq> {}"
-  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(unfold max_cp_eq[OF vt])
-  show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
-  proof -
-    let ?p = "Max ((\<lambda>th. preced th s) ` threads s)" 
-    let ?f = "(\<lambda>th. preced th s)"
-    have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
-    proof(rule Max_in)
-      from finite_threads[OF vt] show "finite (?f ` threads s)" by simp
-    next
-      from np show "?f ` threads s \<noteq> {}" by simp
-    qed
-    then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
-      by (auto simp:Image_def)
-    from th_chain_to_ready [OF vt tm_in]
-    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
-    thus ?thesis
-    proof
-      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
-      then obtain th' where th'_in: "th' \<in> readys s" 
-        and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
-      have "cp s th' = ?f tm"
-      proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
-        from dependents_threads[OF vt] finite_threads[OF vt]
-        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))" 
-          by (auto intro:finite_subset)
-      next
-        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
-        from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
-        moreover have "p \<le> \<dots>"
-        proof(rule Max_ge)
-          from finite_threads[OF vt]
-          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-        next
-          from p_in and th'_in and dependents_threads[OF vt, of th']
-          show "p \<in> (\<lambda>th. preced th s) ` threads s"
-            by (auto simp:readys_def)
-        qed
-        ultimately show "p \<le> preced tm s" by auto
-      next
-        show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
-        proof -
-          from tm_chain
-          have "tm \<in> dependents (wq s) th'"
-            by (unfold cs_dependents_def s_depend_def cs_depend_def, auto)
-          thus ?thesis by auto
-        qed
-      qed
-      with tm_max
-      have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
-      show ?thesis
-      proof (fold h, rule Max_eqI)
-        fix q 
-        assume "q \<in> cp s ` readys s"
-        then obtain th1 where th1_in: "th1 \<in> readys s"
-          and eq_q: "q = cp s th1" by auto
-        show "q \<le> cp s th'"
-          apply (unfold h eq_q)
-          apply (unfold cp_eq_cpreced cpreced_def)
-          apply (rule Max_mono)
-        proof -
-          from dependents_threads [OF vt, of th1] th1_in
-          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq> 
-                 (\<lambda>th. preced th s) ` threads s"
-            by (auto simp:readys_def)
-        next
-          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp
-        next
-          from finite_threads[OF vt] 
-          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
-        qed
-      next
-        from finite_threads[OF vt]
-        show "finite (cp s ` readys s)" by (auto simp:readys_def)
-      next
-        from th'_in
-        show "cp s th' \<in> cp s ` readys s" by simp
-      qed
-    next
-      assume tm_ready: "tm \<in> readys s"
-      show ?thesis
-      proof(fold tm_max)
-        have cp_eq_p: "cp s tm = preced tm s"
-        proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
-          fix y 
-          assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
-          show "y \<le> preced tm s"
-          proof -
-            { fix y'
-              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
-              have "y' \<le> preced tm s"
-              proof(unfold tm_max, rule Max_ge)
-                from hy' dependents_threads[OF vt, of tm]
-                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
-              next
-                from finite_threads[OF vt] 
-                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-              qed
-            } with hy show ?thesis by auto
-          qed
-        next
-          from dependents_threads[OF vt, of tm] finite_threads[OF vt]
-          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
-            by (auto intro:finite_subset)
-        next
-          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
-            by simp
-        qed 
-        moreover have "Max (cp s ` readys s) = cp s tm"
-        proof(rule Max_eqI)
-          from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
-        next
-          from finite_threads[OF vt]
-          show "finite (cp s ` readys s)" by (auto simp:readys_def)
-        next
-          fix y assume "y \<in> cp s ` readys s"
-          then obtain th1 where th1_readys: "th1 \<in> readys s"
-            and h: "y = cp s th1" by auto
-          show "y \<le> cp s tm"
-            apply(unfold cp_eq_p h)
-            apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
-          proof -
-            from finite_threads[OF vt]
-            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-          next
-            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
-              by simp
-          next
-            from dependents_threads[OF vt, of th1] th1_readys
-            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) 
-                    \<subseteq> (\<lambda>th. preced th s) ` threads s"
-              by (auto simp:readys_def)
-          qed
-        qed
-        ultimately show " Max (cp s ` readys s) = preced tm s" by simp
-      qed 
-    qed
-  qed
-qed
-
-lemma max_cp_readys_threads:
-  assumes vt: "vt step s"
-  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(cases "threads s = {}")
-  case True
-  thus ?thesis 
-    by (auto simp:readys_def)
-next
-  case False
-  show ?thesis by (rule max_cp_readys_threads_pre[OF vt False])
-qed
-
-lemma readys_threads:
-  shows "readys s \<subseteq> threads s"
-proof
-  fix th
-  assume "th \<in> readys s"
-  thus "th \<in> threads s"
-    by (unfold readys_def, auto)
-qed
-
-lemma eq_holding: "holding (wq s) th cs = holding s th cs"
-  apply (unfold s_holding_def cs_holding_def, simp)
-  done
-
-lemma f_image_eq:
-  assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
-  shows "f ` A = g ` A"
-proof
-  show "f ` A \<subseteq> g ` A"
-    by(rule image_subsetI, auto intro:h)
-next
-  show "g ` A \<subseteq> f ` A"
-   by(rule image_subsetI, auto intro:h[symmetric])
-qed
-
-end
\ No newline at end of file
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/README
--- a/prio/README	Sun Feb 05 14:29:08 2012 +0000
+++ b/prio/README	Sun Feb 05 21:00:12 2012 +0000
@@ -1,6 +1,14 @@
-Precedence_ord.thy      A theory for precedence.
-Moment.thy              A theory for the notion of moment.
-PrioGDef.thy            The formal definition of the model.
-PrioG.thy               Basic properties of the formal model.
-ExtGG.thy               Formal correctness proof of the formal model.
-CpsG.thy                Properties used to guide implementation.
\ No newline at end of file
+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 e5bfdd2d1ac8 -r a3b4eed091d2 prio/README.txt
--- a/prio/README.txt	Sun Feb 05 14:29:08 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2 +0,0 @@
-Overview of files:
-
diff -r e5bfdd2d1ac8 -r a3b4eed091d2 prio/paper.pdf
Binary file prio/paper.pdf has changed