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
-
-