theory ExtSimports Priobeginlocale 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_setbeginlemma 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) qedqedlemma 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)qedlemma highest': "cp s th = Max (cp s ` threads s)"proof - from highest_cp_preced max_cp_eq[OF vt_s, symmetric] show ?thesis by simpqedlemma 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)qedendlocale 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 qedqedcontext extend_highest_setbeginlemma 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 qedqedlemma 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 qedqedlemma 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 simpnext 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) qednext 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 qedqedlemma 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 .qedlemma 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 autoqedlemma 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) qedqedlemma 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 qedqed(*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 autoqedlemma 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 qednext case 0 from assms show ?case by autoqedlemma 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 autoqedlemma 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 simpnext 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 autoqedlemma 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 autoqedlemma live: "runing (t@s) \<noteq> {}"proof(cases "th \<in> runing (t@s)") case True thus ?thesis by autonext 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 autoqedendend