diff -r 12e9aa68d5db -r 4190df6f4488 prio/ExtS.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtS.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1019 @@
+theory ExtS
+imports Prio
+begin
+
+locale highest_set =
+  fixes s' th prio fixes s 
+  defines s_def : "s \<equiv> (Set th prio#s')"
+  assumes vt_s: "vt step s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+
+context highest_set
+begin
+
+
+lemma vt_s': "vt step s'"
+  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_set: "step s' (Set th prio)"
+  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_set_elim: 
+  "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+  by (insert step_set, ind_cases "step s' (Set th prio)", auto)
+
+
+lemma threads_s: "th \<in> threads s"
+  by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
+
+lemma same_depend: "depend s = depend s'"
+  by (insert depend_set_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+  "dependents (wq s) th = dependents (wq s') th"
+  apply (unfold cs_dependents_def)
+  by (unfold eq_depend same_depend, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+  from highest and max_cp_eq[OF vt_s]
+  have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+  have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+  proof -
+    from threads_s and dependents_threads[OF vt_s, of th]
+    show ?thesis by auto
+  qed
+  show ?thesis
+  proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+    show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+  next
+    fix y 
+    assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+    then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+      and eq_y: "y = preced th1 s" by auto
+    show "y \<le> preced th s"
+    proof(unfold is_max, rule Max_ge)
+      from finite_threads[OF vt_s] 
+      show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+    next
+      from sbs th1_in and eq_y 
+      show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+    qed
+  next
+    from sbs and finite_threads[OF vt_s]
+    show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+      by (auto intro:finite_subset)
+  qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+  have "\<forall>cs. \<not> waiting s th cs"
+    apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
+    apply (unfold s_depend_def, unfold runing_def readys_def)
+    apply (auto, fold s_def)
+    apply (erule_tac x = cs in allE, auto simp:waiting_eq)
+  proof -
+    fix cs
+    assume h: 
+      "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
+          {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
+            (is "?L = ?R")
+    and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
+    from wt have "(Th th, Cs cs) \<in> ?L" by auto
+    with h have "(Th th, Cs cs) \<in> ?R" by simp
+    hence "waiting (wq s') th cs" by auto with nwt
+    show False by auto
+  qed    
+  with threads_s show ?thesis 
+    by (unfold readys_def, auto)
+qed
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+  show ?thesis by simp
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+  have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+  proof -
+    have " Max (cp s ` readys s) = cp s th"
+    proof(rule Max_eqI)
+      from finite_threads[OF vt_s] readys_threads finite_subset
+      have "finite (readys s)" by blast
+      thus "finite (cp s ` readys s)" by auto
+    next
+      from is_ready show "cp s th \<in> cp s ` readys s" by auto
+    next
+      fix y
+      assume "y \<in> cp s ` readys s"
+      then obtain th1 where 
+        eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
+      show  "y \<le> cp s th" 
+      proof -
+        have "y \<le> Max (cp s ` threads s)"
+        proof(rule Max_ge)
+          from eq_y and th1_in
+          show "y \<in> cp s ` threads s"
+            by (auto simp:readys_def)
+        next
+          from finite_threads[OF vt_s]
+          show "finite (cp s ` threads s)" by auto
+        qed
+        with highest' show ?thesis by auto
+      qed
+    qed
+    with highest' show ?thesis by auto
+  qed
+  thus ?thesis
+    by (unfold runing_def, insert highest' is_ready, auto)
+qed
+
+end
+
+locale extend_highest_set = highest_set + 
+  fixes t 
+  assumes vt_t: "vt step (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt cs (t@s)" 
+  shows "vt cs s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+      and vt_et: "vt cs ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt cs (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt cs (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+context extend_highest_set
+begin
+
+lemma red_moment:
+  "extend_highest_set s' th prio (moment i t)"
+  apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
+  by (unfold highest_set_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e; 
+                    extend_highest_set s' th prio t; 
+                    extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_set_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt step ((e # t') @ s)"
+      and et: "extend_highest_set s' th prio (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt step (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_set s' th prio (e # t')" .
+    next
+      from et show ext': "extend_highest_set s' th prio t'"
+          by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+        preced th (t@s) = preced th s" (is "?Q t")
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+      by auto
+  next
+    case (Cons e t)
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      assume eq_e: " e = Create thread prio"
+      show ?thesis
+      proof -
+        from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          assume "thread \<notin> threads (t @ s)"
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    next
+      case (Exit thread)
+      assume eq_e: "e = Exit thread"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.exit_diff [OF this] and eq_e
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold eq_e, auto simp:preced_def)
+    next
+      case (P thread cs)
+      assume eq_e: "e = P thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (V thread cs)
+      assume eq_e: "e = V thread cs"
+      with Cons
+      show ?thesis 
+        by (auto simp:eq_e preced_def)
+    next
+      case (Set thread prio')
+      assume eq_e: " e = Set thread prio'"
+      show ?thesis
+      proof -
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.set_diff_low[OF this] and eq_e
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold eq_e, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:eq_e)
+      qed
+    qed
+  qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+    by simp
+next
+  case (Cons e t)
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    assume eq_e: " e = Create thread prio'"
+    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+    hence neq_thread: "thread \<noteq> th"
+    proof(cases)
+      assume "thread \<notin> threads (t @ s)"
+      moreover have "th \<in> threads (t@s)"
+      proof -
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
+      qed
+      ultimately show ?thesis by auto
+    qed
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this]
+    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" 
+      by (auto simp:s_def)
+    from stp
+    have thread_ts: "thread \<notin> threads (t @ s)"
+      by (cases, auto)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+        by (unfold eq_e, simp)
+      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+      proof(rule Max_insert)
+        from Cons have "vt step (t @ s)" by auto
+        from finite_threads[OF this]
+        show "finite (?f ` (threads (t@s)))" by simp
+      next
+        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+      qed
+      moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+      proof -
+        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = 
+          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+        proof -
+          { fix th' 
+            assume "th' \<in> ?B"
+            with thread_ts eq_e
+            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+          } thus ?thesis 
+            apply (auto simp:Image_def)
+          proof -
+            fix th'
+            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> 
+              preced th' (e # t @ s) = preced th' (t @ s)"
+              and h1: "th' \<in> threads (t @ s)"
+            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+            proof -
+              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+              ultimately show ?thesis by simp
+            qed
+          qed
+        qed
+        with Cons show ?thesis by auto
+      qed
+      moreover have "?f thread < ?t"
+      proof -
+        from Cons have " extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.create_low[OF this] and eq_e
+        have "prio' \<le> prio" by auto
+        thus ?thesis
+        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+    qed
+    ultimately show ?thesis by (auto simp:max_def)
+  qed
+next
+    case (Exit thread)
+    assume eq_e: "e = Exit thread"
+    from Cons have vt_e: "vt step (e#(t @ s))" by auto
+    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+    from stp have thread_ts: "thread \<in> threads (t @ s)"
+      by(cases, unfold runing_def readys_def, auto)
+    from Cons have "extend_highest_set s' th prio (e # t)" by auto
+    from extend_highest_set.exit_diff[OF this] and eq_e
+    have neq_thread: "thread \<noteq> th" by auto
+    from Cons have "extend_highest_set s' th prio t" by auto
+    from extend_highest_set.th_kept[OF this, folded s_def]
+    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "threads (t@s) = insert thread ?A"
+        by (insert stp thread_ts, unfold eq_e, auto)
+      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        from finite_threads [OF vt_e]
+        show "finite (?f ` ?A)" by simp
+      next
+        from Cons have "extend_highest_set s' th prio (e # t)" by auto
+        from extend_highest_set.th_kept[OF this]
+        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)
+      qed
+      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+      moreover have "Max (?f ` (threads (t@s))) = ?t"
+      proof -
+        from Cons show ?thesis
+          by (unfold eq_e, auto simp:preced_def)
+      qed
+      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+      moreover have "?f thread < ?t"
+      proof(unfold eq_e, simp add:preced_def, fold preced_def)
+        show "preced thread (t @ s) < ?t"
+        proof -
+          have "preced thread (t @ s) \<le> ?t" 
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" 
+              (is "?t = Max (?g ` ?B)") by simp
+            moreover have "?g thread \<le> \<dots>"
+            proof(rule Max_ge)
+              have "vt step (t@s)" by fact
+              from finite_threads [OF this]
+              show "finite (?g ` ?B)" by simp
+            next
+              from thread_ts
+              show "?g thread \<in> (?g ` ?B)" by auto
+            qed
+            ultimately show ?thesis by auto
+          qed
+          moreover have "preced thread (t @ s) \<noteq> ?t"
+          proof
+            assume "preced thread (t @ s) = preced th s"
+            with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+            from preced_unique [OF this] have "thread = th"
+            proof
+              from h' show "th \<in> threads (t @ s)" by simp
+            next
+              from thread_ts show "thread \<in> threads (t @ s)" .
+            qed(simp)
+            with neq_thread show "False" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      qed
+      ultimately show ?thesis 
+        by (auto simp:max_def split:if_splits)
+    qed
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def)
+  next
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      let ?B = "threads (t@s)"
+      from Cons have "extend_highest_set s' th prio (e # t)" by auto
+      from extend_highest_set.set_diff_low[OF this] and Set
+      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+      also have "\<dots> = ?t"
+      proof(rule Max_eqI)
+        fix y
+        assume y_in: "y \<in> ?f ` ?B"
+        then obtain th1 where 
+          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+        show "y \<le> ?t"
+        proof(cases "th1 = thread")
+          case True
+          with neq_thread le_p eq_y s_def Set
+          show ?thesis
+            by (auto simp:preced_def precedence_le_def)
+        next
+          case False
+          with Set eq_y
+          have "y  = preced th1 (t@s)"
+            by (simp add:preced_def)
+          moreover have "\<dots> \<le> ?t"
+          proof -
+            from Cons
+            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+              by auto
+            moreover have "preced th1 (t@s) \<le> \<dots>"
+            proof(rule Max_ge)
+              from th1_in 
+              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+                by simp
+            next
+              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+              proof -
+                from Cons have "vt step (t @ s)" by auto
+                from finite_threads[OF this] show ?thesis by auto
+              qed
+            qed
+            ultimately show ?thesis by auto
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from Cons and finite_threads
+        show "finite (?f ` ?B)" by auto
+      next
+        from Cons have "extend_highest_set s' th prio t" by auto
+        from extend_highest_set.th_kept [OF this, folded s_def]
+        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+        show "?t \<in> (?f ` ?B)" 
+        proof -
+          from neq_thread Set h
+          have "?t = ?f th" by (auto simp:preced_def)
+          with h show ?thesis by auto
+        qed
+      qed
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" 
+  (is "?L = ?R")
+proof -
+  have "?L = cpreced (t@s) (wq (t@s)) th" 
+    by (unfold cp_eq_cpreced, simp)
+  also have "\<dots> = ?R"
+  proof(unfold cpreced_def)
+    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+    proof(cases "?A = {}")
+      case False
+      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+      proof(rule Max_insert)
+        show "finite (?f ` ?A)"
+        proof -
+          from dependents_threads[OF vt_t]
+          have "?A \<subseteq> threads (t@s)" .
+          moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+          ultimately show ?thesis 
+            by (auto simp:finite_subset)
+        qed
+      next
+        from False show "(?f ` ?A) \<noteq> {}" by simp
+      qed
+      moreover have "\<dots> = Max (?f ` ?B)"
+      proof -
+        from max_preced have "?f th = Max (?f ` ?B)" .
+        moreover have "Max (?f ` ?A) \<le> \<dots>" 
+        proof(rule Max_mono)
+          from False show "(?f ` ?A) \<noteq> {}" by simp
+        next
+          show "?f ` ?A \<subseteq> ?f ` ?B" 
+          proof -
+            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+            thus ?thesis by auto
+          qed
+        next
+          from finite_threads[OF vt_t] 
+          show "finite (?f ` ?B)" by simp
+        qed
+        ultimately show ?thesis
+          by (auto simp:max_def)
+      qed
+      ultimately show ?thesis by auto
+    next
+      case True
+      with max_preced show ?thesis by auto
+    qed
+  qed
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+proof -
+  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+  proof(rule Max_ge)
+    from finite_threads [OF vt_s]
+    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+  next
+    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+      by simp
+  qed
+  moreover have "preced th' s \<noteq> preced th s"
+  proof
+    assume "preced th' s = preced th s"
+    from preced_unique[OF this th'_in] neq_th' is_ready
+    show "False" by  (auto simp:readys_def)
+  qed
+  ultimately show ?thesis using highest_preced_thread
+    by auto
+qed
+
+lemma pv_blocked:
+  fixes th'
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume "th' \<in> runing (t@s)"
+  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" 
+    by (auto simp:runing_def)
+  with max_cp_readys_threads [OF vt_t]
+  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+    by auto
+  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+    by simp
+  finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+  show False
+  proof -
+    have "dependents (wq (t @ s)) th' = {}" 
+      by (rule count_eq_dependents [OF vt_t eq_pv])
+    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+    proof
+      assume "preced th' (t @ s) = preced th (t @ s)"
+      hence "th' = th"
+      proof(rule preced_unique)
+        from th_kept show "th \<in> threads (t @ s)" by simp
+      next
+        from th'_in show "th' \<in> threads (t @ s)" by simp
+      qed
+      with assms show False by simp
+    qed
+    ultimately show ?thesis
+      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+  qed
+qed
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case (Cons e t)
+    from Cons
+    have in_thread: "th' \<in> threads (t @ s)"
+      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    have "extend_highest_set s' th prio t" by fact
+    from extend_highest_set.pv_blocked 
+    [OF this, folded s_def, OF in_thread neq_th' not_holding]
+    have not_runing: "th' \<notin> runing (t @ s)" .
+    show ?case
+    proof(cases e)
+      case (V thread cs)
+      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+      show ?thesis
+      proof -
+        from Cons and V have "step (t@s) (V thread cs)" by auto
+        hence neq_th': "thread \<noteq> th'"
+        proof(cases)
+          assume "thread \<in> runing (t@s)"
+          moreover have "th' \<notin> runing (t@s)" by fact
+          ultimately show ?thesis by auto
+        qed
+        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" 
+          by (unfold V, simp add:cntP_def cntV_def count_def)
+        moreover from in_thread
+        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (P thread cs)
+      from Cons and P have "step (t@s) (P thread cs)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t@s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Create thread prio')
+      from Cons and Create have "step (t@s) (Create thread prio')" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<notin> threads (t @ s)"
+        moreover have "th' \<in> threads (t@s)" by fact
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Create 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Create 
+      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+      ultimately show ?thesis by auto
+    next
+      case (Exit thread)
+      from Cons and Exit have "step (t@s) (Exit thread)" by auto
+      hence neq_th': "thread \<noteq> th'"
+      proof(cases)
+        assume "thread \<in> runing (t @ s)"
+        moreover note not_runing
+        ultimately show ?thesis by auto
+      qed
+      with Cons and Exit 
+      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        by (auto simp:cntP_def cntV_def count_def)
+      moreover from Cons and Exit and neq_th' 
+      have in_thread': "th' \<in> threads ((e # t) @ s)"
+        by auto
+      ultimately show ?thesis by auto
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+  next
+    case Nil
+    with assms
+    show ?case by auto
+  qed
+qed
+
+(*
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<notin> runing (t@s)"
+proof -
+  from runing_precond_pre[OF th'_in eq_pv neq_th']
+  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+  from pv_blocked[OF h1 neq_th' h2] 
+  show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+  proof
+    assume eq_pv: "cntP s th' = cntV s th'"
+    from runing_precond_pre[OF th'_in eq_pv neq_th']
+    have h1: "th' \<in> threads (t @ s)"  
+      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+    with is_runing show "False" by simp
+  qed
+  moreover from cnp_cnv_cncs[OF vt_s, of th'] 
+  have "cntV s th' \<le> cntP s th'" by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+  case (Suc k)
+  show ?case
+  proof -
+    { assume True: "Suc (i+k) \<le> length t"
+      from moment_head [OF this] 
+      obtain e where
+        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+        by blast
+      from red_moment[of "Suc(i+k)"]
+      and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
+      hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+        by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+          highest_set_def s_def, auto)
+      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"
+      proof(unfold s_def)
+        show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
+        proof(rule extend_highest_set.pv_blocked)
+          from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
+            by (simp add:s_def)
+        next
+          from neq_th' show "th' \<noteq> th" .
+        next
+          from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
+        next
+          from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
+            cntV (moment (i + k) t @ Set th prio # s') th'"
+            by (auto simp:s_def)
+        qed
+      qed
+      from step_back_step[OF vt_e]
+      have "step ((moment (i + k) t)@s) e" .
+      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+        th' \<in> threads (e#(moment (i + k) t)@s)
+        "
+      proof(cases)
+        case (thread_create thread prio)
+        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_exit thread)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_P thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_V thread cs)
+        moreover have "thread \<noteq> th'"
+        proof -
+          have "thread \<in> runing (moment (i + k) t @ s)" by fact
+          moreover note not_runing'
+          ultimately show ?thesis by auto
+        qed
+        moreover note Suc 
+        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+      next
+        case (thread_set thread prio')
+        with Suc show ?thesis
+          by (auto simp:cntP_def cntV_def count_def)
+      qed
+      with eq_me have ?thesis using eq_me by auto 
+    } note h = this
+    show ?thesis
+    proof(cases "Suc (i+k) \<le> length t")
+      case True
+      from h [OF this] show ?thesis .
+    next
+      case False
+      with moment_ge
+      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+      with Suc show ?thesis by auto
+    qed
+  qed
+next
+  case 0
+  from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+    and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  with extend_highest_set.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+  case True
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+  case False
+  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+  let ?q = "moment 0 t"
+  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+  from p_split_gen [of ?Q, OF this not_thread]
+  obtain i where lt_its: "i < length t"
+    and le_i: "0 \<le> i"
+    and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+    and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+  from lt_its have "Suc i \<le> length t" by auto
+  from moment_head[OF this] obtain e where 
+   eq_me: "moment (Suc i) t = e # moment i t" by blast
+  from red_moment[of "Suc i"] and eq_me
+  have "extend_highest_set s' th prio (e # moment i t)" by simp
+  hence vt_e: "vt step (e#(moment i t)@s)"
+    by (unfold extend_highest_set_def extend_highest_set_axioms_def 
+      highest_set_def s_def, auto)
+  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+  from post[rule_format, of "Suc i"] and eq_me 
+  have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+  from create_pre[OF stp_i pre this] 
+  obtain prio where eq_e: "e = Create th' prio" .
+  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+  proof(rule cnp_cnv_eq)
+    from step_back_vt [OF vt_e] 
+    show "vt step (moment i t @ s)" .
+  next
+    from eq_e and stp_i 
+    have "step (moment i t @ s) (Create th' prio)" by simp
+    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+  qed
+  with eq_e
+  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" 
+    by (simp add:cntP_def cntV_def count_def)
+  with eq_me[symmetric]
+  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+    by simp
+  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+  with eq_me [symmetric]
+  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+  and moment_ge
+  have "th' \<notin> runing (t @ s)" by auto
+  with runing'
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+  case True thus ?thesis by auto
+next
+  case False
+  then have not_ready: "th \<notin> readys (t@s)"
+    apply (unfold runing_def, 
+            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+    by auto
+  from th_kept have "th \<in> threads (t@s)" by auto
+  from th_chain_to_ready[OF vt_t this] and not_ready
+  obtain th' where th'_in: "th' \<in> readys (t@s)"
+    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+  have "th' \<in> runing (t@s)"
+  proof -
+    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+    proof -
+      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = 
+               preced th (t@s)"
+      proof(rule Max_eqI)
+        fix y
+        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+        then obtain th1 where
+          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"
+          and eq_y: "y = preced th1 (t@s)" by auto
+        show "y \<le> preced th (t @ s)"
+        proof -
+          from max_preced
+          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+          moreover have "y \<le> \<dots>"
+          proof(rule Max_ge)
+            from h1
+            have "th1 \<in> threads (t@s)"
+            proof
+              assume "th1 = th'"
+              with th'_in show ?thesis by (simp add:readys_def)
+            next
+              assume "th1 \<in> dependents (wq (t @ s)) th'"
+              with dependents_threads [OF vt_t]
+              show "th1 \<in> threads (t @ s)" by auto
+            qed
+            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+          next
+            from finite_threads[OF vt_t]
+            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+          qed
+          ultimately show ?thesis by auto
+        qed
+      next
+        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+          by (auto intro:finite_subset)
+      next
+        from dp
+        have "th \<in> dependents (wq (t @ s)) th'" 
+          by (unfold cs_dependents_def, auto simp:eq_depend)
+        thus "preced th (t @ s) \<in> 
+                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+          by auto
+      qed
+      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+      proof -
+        from max_preced and max_cp_eq[OF vt_t, symmetric]
+        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+        with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+      qed
+      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+    qed
+    with th'_in show ?thesis by (auto simp:runing_def)
+  qed
+  thus ?thesis by auto
+qed
+
+end
+
+end
+