theory PrioGimports PrioGDef beginlemma runing_ready: "runing s \<subseteq> readys s" by (auto simp only:runing_def readys_def)lemma wq_v_neq: "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'" by (auto simp:wq_def Let_def cp_def split:list.splits)lemma wq_distinct: "vt step s \<Longrightarrow> distinct (wq s cs)"proof(erule_tac vt.induct, simp add:wq_def) fix s e assume h1: "step s e" and h2: "distinct (wq s cs)" thus "distinct (wq (e # s) cs)" proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits) fix thread s assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+" and h2: "thread \<in> set (waiting_queue (schs s) cs)" and h3: "thread \<in> runing s" show "False" proof - from h3 have "\<And> cs. thread \<in> set (waiting_queue (schs s) cs) \<Longrightarrow> thread = hd ((waiting_queue (schs s) cs))" by (simp add:runing_def readys_def s_waiting_def wq_def) from this [OF h2] have "thread = hd (waiting_queue (schs s) cs)" . with h2 have "(Cs cs, Th thread) \<in> (depend s)" by (simp add:s_depend_def s_holding_def wq_def cs_holding_def) with h1 show False by auto qed next fix thread s a list assume dst: "distinct list" show "distinct (SOME q. distinct q \<and> set q = set list)" proof(rule someI2) from dst show "distinct list \<and> set list = set list" by auto next fix q assume "distinct q \<and> set q = set list" thus "distinct q" by auto qed qedqedlemma step_back_vt: "vt ccs (e#s) \<Longrightarrow> vt ccs s" by(ind_cases "vt ccs (e#s)", simp)lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e" by(ind_cases "vt ccs (e#s)", simp)lemma block_pre: fixes thread cs s assumes vt_e: "vt step (e#s)" and s_ni: "thread \<notin> set (wq s cs)" and s_i: "thread \<in> set (wq (e#s) cs)" shows "e = P thread cs"proof - show ?thesis proof(cases e) case (P th cs) with assms show ?thesis by (auto simp:wq_def Let_def split:if_splits) next case (Create th prio) with assms show ?thesis by (auto simp:wq_def Let_def split:if_splits) next case (Exit th) with assms show ?thesis by (auto simp:wq_def Let_def split:if_splits) next case (Set th prio) with assms show ?thesis by (auto simp:wq_def Let_def split:if_splits) next case (V th cs) with assms show ?thesis apply (auto simp:wq_def Let_def split:if_splits) proof - fix q qs assume h1: "thread \<notin> set (waiting_queue (schs s) cs)" and h2: "q # qs = waiting_queue (schs s) cs" and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)" and vt: "vt step (V th cs # s)" from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp moreover have "thread \<in> set qs" proof - have "set (SOME q. distinct q \<and> set q = set qs) = set qs" proof(rule someI2) from wq_distinct [OF step_back_vt[OF vt], of cs] and h2[symmetric, folded wq_def] show "distinct qs \<and> set qs = set qs" by auto next fix x assume "distinct x \<and> set x = set qs" thus "set x = set qs" by auto qed with h3 show ?thesis by simp qed ultimately show "False" by auto qed qedqedlemma p_pre: "\<lbrakk>vt step ((P thread cs)#s)\<rbrakk> \<Longrightarrow> thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (depend s)^+"apply (ind_cases "vt step ((P thread cs)#s)")apply (ind_cases "step s (P thread cs)")by autolemma abs1: fixes e es assumes ein: "e \<in> set es" and neq: "hd es \<noteq> hd (es @ [x])" shows "False"proof - from ein have "es \<noteq> []" by auto then obtain e ess where "es = e # ess" by (cases es, auto) with neq show ?thesis by autoqedlemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]" by (cases es, auto)inductive_cases evt_cons: "vt cs (a#s)"lemma abs2: assumes vt: "vt step (e#s)" and inq: "thread \<in> set (wq s cs)" and nh: "thread = hd (wq s cs)" and qt: "thread \<noteq> hd (wq (e#s) cs)" and inq': "thread \<in> set (wq (e#s) cs)" shows "False"proof - from assms show "False" apply (cases e) apply ((simp split:if_splits add:Let_def wq_def)[1])+ apply (insert abs1, fast)[1] apply (auto simp:wq_def simp:Let_def split:if_splits list.splits) proof - fix th qs assume vt: "vt step (V th cs # s)" and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)" and eq_wq: "waiting_queue (schs s) cs = thread # qs" show "False" proof - from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp moreover have "thread \<in> set qs" proof - have "set (SOME q. distinct q \<and> set q = set qs) = set qs" proof(rule someI2) from wq_distinct [OF step_back_vt[OF vt], of cs] and eq_wq [folded wq_def] show "distinct qs \<and> set qs = set qs" by auto next fix x assume "distinct x \<and> set x = set qs" thus "set x = set qs" by auto qed with th_in show ?thesis by auto qed ultimately show ?thesis by auto qed qedqedlemma vt_moment: "\<And> t. \<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"proof(induct s, simp) fix a s t assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)" and vt_a: "vt cs (a # s)" and le_t: "t \<le> length (a # s)" show "vt cs (moment t (a # s))" proof(cases "t = length (a#s)") case True from True have "moment t (a#s) = a#s" by simp with vt_a show ?thesis by simp next case False with le_t have le_t1: "t \<le> length s" by simp from vt_a have "vt cs s" by (erule_tac evt_cons, simp) from h [OF this le_t1] have "vt cs (moment t s)" . moreover have "moment t (a#s) = moment t s" proof - from moment_app [OF le_t1, of "[a]"] show ?thesis by simp qed ultimately show ?thesis by auto qedqed(* Wrong: lemma \<lbrakk>thread \<in> set (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"*)lemma waiting_unique_pre: fixes cs1 cs2 s thread assumes vt: "vt step s" and h11: "thread \<in> set (wq s cs1)" and h12: "thread \<noteq> hd (wq s cs1)" assumes h21: "thread \<in> set (wq s cs2)" and h22: "thread \<noteq> hd (wq s cs2)" and neq12: "cs1 \<noteq> cs2" shows "False"proof - let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)" from h11 and h12 have q1: "?Q cs1 s" by simp from h21 and h22 have q2: "?Q cs2 s" by simp have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def) have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def) from p_split [of "?Q cs1", OF q1 nq1] obtain t1 where lt1: "t1 < length s" and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and> thread \<noteq> hd (wq (moment t1 s) cs1))" and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and> thread \<noteq> hd (wq (moment i' s) cs1))" by auto from p_split [of "?Q cs2", OF q2 nq2] obtain t2 where lt2: "t2 < length s" and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and> thread \<noteq> hd (wq (moment t2 s) cs2))" and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and> thread \<noteq> hd (wq (moment i' s) cs2))" by auto show ?thesis proof - { assume lt12: "t1 < t2" let ?t3 = "Suc t2" from lt2 have le_t3: "?t3 \<le> length s" by auto from moment_plus [OF this] obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto have "t2 < ?t3" by simp from nn2 [rule_format, OF this] and eq_m have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto have vt_e: "vt step (e#moment t2 s)" proof - from vt_moment [OF vt le_t3] have "vt step (moment ?t3 s)" . with eq_m show ?thesis by simp qed have ?thesis proof(cases "thread \<in> set (wq (moment t2 s) cs2)") case True from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)" by auto from abs2 [OF vt_e True eq_th h2 h1] show ?thesis by auto next case False from block_pre [OF vt_e False h1] have "e = P thread cs2" . with vt_e have "vt step ((P thread cs2)# moment t2 s)" by simp from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp with runing_ready have "thread \<in> readys (moment t2 s)" by auto with nn1 [rule_format, OF lt12] show ?thesis by (simp add:readys_def s_waiting_def, auto) qed } moreover { assume lt12: "t2 < t1" let ?t3 = "Suc t1" from lt1 have le_t3: "?t3 \<le> length s" by auto from moment_plus [OF this] obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto have lt_t3: "t1 < ?t3" by simp from nn1 [rule_format, OF this] and eq_m have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto have vt_e: "vt step (e#moment t1 s)" proof - from vt_moment [OF vt le_t3] have "vt step (moment ?t3 s)" . with eq_m show ?thesis by simp qed have ?thesis proof(cases "thread \<in> set (wq (moment t1 s) cs1)") case True from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)" by auto from abs2 [OF vt_e True eq_th h2 h1] show ?thesis by auto next case False from block_pre [OF vt_e False h1] have "e = P thread cs1" . with vt_e have "vt step ((P thread cs1)# moment t1 s)" by simp from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp with runing_ready have "thread \<in> readys (moment t1 s)" by auto with nn2 [rule_format, OF lt12] show ?thesis by (simp add:readys_def s_waiting_def, auto) qed } moreover { assume eqt12: "t1 = t2" let ?t3 = "Suc t1" from lt1 have le_t3: "?t3 \<le> length s" by auto from moment_plus [OF this] obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto have lt_t3: "t1 < ?t3" by simp from nn1 [rule_format, OF this] and eq_m have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto have vt_e: "vt step (e#moment t1 s)" proof - from vt_moment [OF vt le_t3] have "vt step (moment ?t3 s)" . with eq_m show ?thesis by simp qed have ?thesis proof(cases "thread \<in> set (wq (moment t1 s) cs1)") case True from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)" by auto from abs2 [OF vt_e True eq_th h2 h1] show ?thesis by auto next case False from block_pre [OF vt_e False h1] have eq_e1: "e = P thread cs1" . have lt_t3: "t1 < ?t3" by simp with eqt12 have "t2 < ?t3" by simp from nn2 [rule_format, OF this] and eq_m and eqt12 have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto show ?thesis proof(cases "thread \<in> set (wq (moment t2 s) cs2)") case True from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)" by auto from vt_e and eqt12 have "vt step (e#moment t2 s)" by simp from abs2 [OF this True eq_th h2 h1] show ?thesis . next case False have vt_e: "vt step (e#moment t2 s)" proof - from vt_moment [OF vt le_t3] eqt12 have "vt step (moment (Suc t2) s)" by auto with eq_m eqt12 show ?thesis by simp qed from block_pre [OF vt_e False h1] have "e = P thread cs2" . with eq_e1 neq12 show ?thesis by auto qed qed } ultimately show ?thesis by arith qedqedlemma waiting_unique: fixes s cs1 cs2 assumes "vt step s" and "waiting s th cs1" and "waiting s th cs2" shows "cs1 = cs2"proof - from waiting_unique_pre and prems show ?thesis by (auto simp add:s_waiting_def)qedlemma held_unique: assumes "vt step s" and "holding s th1 cs" and "holding s th2 cs" shows "th1 = th2"proof - from prems show ?thesis unfolding s_holding_def by autoqedlemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s" apply (induct s, auto) by (case_tac a, auto split:if_splits)lemma birthtime_unique: "\<lbrakk>birthtime th1 s = birthtime th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk> \<Longrightarrow> th1 = th2" apply (induct s, auto) by (case_tac a, auto split:if_splits dest:birthtime_lt)lemma preced_unique : assumes pcd_eq: "preced th1 s = preced th2 s" and th_in1: "th1 \<in> threads s" and th_in2: " th2 \<in> threads s" shows "th1 = th2"proof - from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def) from birthtime_unique [OF this th_in1 th_in2] show ?thesis .qedlemma preced_linorder: assumes neq_12: "th1 \<noteq> th2" and th_in1: "th1 \<in> threads s" and th_in2: " th2 \<in> threads s" shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"proof - from preced_unique [OF _ th_in1 th_in2] and neq_12 have "preced th1 s \<noteq> preced th2 s" by auto thus ?thesis by autoqedlemma unique_minus: fixes x y z r assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c" and xy: "(x, y) \<in> r" and xz: "(x, z) \<in> r^+" and neq: "y \<noteq> z" shows "(y, z) \<in> r^+"proof - from xz and neq show ?thesis proof(induct) case (base ya) have "(x, ya) \<in> r" by fact from unique [OF xy this] have "y = ya" . with base show ?case by auto next case (step ya z) show ?case proof(cases "y = ya") case True from step True show ?thesis by simp next case False from step False show ?thesis by auto qed qedqedlemma unique_base: fixes r x y z assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c" and xy: "(x, y) \<in> r" and xz: "(x, z) \<in> r^+" and neq_yz: "y \<noteq> z" shows "(y, z) \<in> r^+"proof - from xz neq_yz show ?thesis proof(induct) case (base ya) from xy unique base show ?case by auto next case (step ya z) show ?case proof(cases "y = ya") case True from True step show ?thesis by auto next case False from False step have "(y, ya) \<in> r\<^sup>+" by auto with step show ?thesis by auto qed qedqedlemma unique_chain: fixes r x y z assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c" and xy: "(x, y) \<in> r^+" and xz: "(x, z) \<in> r^+" and neq_yz: "y \<noteq> z" shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"proof - from xy xz neq_yz show ?thesis proof(induct) case (base y) have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto from unique_base [OF _ h1 h2 h3] and unique show ?case by auto next case (step y za) show ?case proof(cases "y = z") case True from True step show ?thesis by auto next case False from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto thus ?thesis proof assume "(z, y) \<in> r\<^sup>+" with step have "(z, za) \<in> r\<^sup>+" by auto thus ?thesis by auto next assume h: "(y, z) \<in> r\<^sup>+" from step have yza: "(y, za) \<in> r" by simp from step have "za \<noteq> z" by simp from unique_minus [OF _ yza h this] and unique have "(za, z) \<in> r\<^sup>+" by auto thus ?thesis by auto qed qed qedqedlemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"apply (unfold s_depend_def s_waiting_def wq_def)by (simp add:Let_def)lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"apply (unfold s_depend_def s_waiting_def wq_def)by (simp add:Let_def)lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"apply (unfold s_depend_def s_waiting_def wq_def)by (simp add:Let_def)lemma step_v_hold_inv[elim_format]: "\<And>c t. \<lbrakk>vt step (V th cs # s); \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> next_th s th cs t \<and> c = cs"proof - fix c t assume vt: "vt step (V th cs # s)" and nhd: "\<not> holding (wq s) t c" and hd: "holding (wq (V th cs # s)) t c" show "next_th s th cs t \<and> c = cs" proof(cases "c = cs") case False with nhd hd show ?thesis by (unfold cs_holding_def wq_def, auto simp:Let_def) next case True with step_back_step [OF vt] have "step s (V th c)" by simp hence "next_th s th cs t" proof(cases) assume "holding s th c" with nhd hd show ?thesis apply (unfold s_holding_def cs_holding_def wq_def next_th_def, auto simp:Let_def split:list.splits if_splits) proof - assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])" moreover have "\<dots> = set []" proof(rule someI2) show "distinct [] \<and> [] = []" by auto next fix x assume "distinct x \<and> x = []" thus "set x = set []" by auto qed ultimately show False by auto next assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])" moreover have "\<dots> = set []" proof(rule someI2) show "distinct [] \<and> [] = []" by auto next fix x assume "distinct x \<and> x = []" thus "set x = set []" by auto qed ultimately show False by auto qed qed with True show ?thesis by auto qedqedlemma step_v_wait_inv[elim_format]: "\<And>t c. \<lbrakk>vt step (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c \<rbrakk> \<Longrightarrow> (next_th s th cs t \<and> cs = c)"proof - fix t c assume vt: "vt step (V th cs # s)" and nw: "\<not> waiting (wq (V th cs # s)) t c" and wt: "waiting (wq s) t c" show "next_th s th cs t \<and> cs = c" proof(cases "cs = c") case False with nw wt show ?thesis by (auto simp:cs_waiting_def wq_def Let_def) next case True from nw[folded True] wt[folded True] have "next_th s th cs t" apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits) proof - fix a list assume t_in: "t \<in> set list" and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)" and eq_wq: "waiting_queue (schs s) cs = a # list" have " set (SOME q. distinct q \<and> set q = set list) = set list" proof(rule someI2) from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def] show "distinct list \<and> set list = set list" by auto next show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list" by auto qed with t_ni and t_in show "a = th" by auto next fix a list assume t_in: "t \<in> set list" and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)" and eq_wq: "waiting_queue (schs s) cs = a # list" have " set (SOME q. distinct q \<and> set q = set list) = set list" proof(rule someI2) from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def] show "distinct list \<and> set list = set list" by auto next show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list" by auto qed with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto next fix a list assume eq_wq: "waiting_queue (schs s) cs = a # list" from step_back_step[OF vt] show "a = th" proof(cases) assume "holding s th cs" with eq_wq show ?thesis by (unfold s_holding_def wq_def, auto) qed qed with True show ?thesis by simp qedqedlemma step_v_not_wait[consumes 3]: "\<lbrakk>vt step (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False" by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)lemma step_v_release: "\<lbrakk>vt step (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"proof - assume vt: "vt step (V th cs # s)" and hd: "holding (wq (V th cs # s)) th cs" from step_back_step [OF vt] and hd show "False" proof(cases) assume "holding (wq (V th cs # s)) th cs" and "holding s th cs" thus ?thesis apply (unfold s_holding_def wq_def cs_holding_def) apply (auto simp:Let_def split:list.splits) proof - fix list assume eq_wq[folded wq_def]: "waiting_queue (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list" and hd_in: "hd (SOME q. distinct q \<and> set q = set list) \<in> set (SOME q. distinct q \<and> set q = set list)" have "set (SOME q. distinct q \<and> set q = set list) = set list" proof(rule someI2) from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq show "distinct list \<and> set list = set list" by auto next show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list" by auto qed moreover have "distinct (hd (SOME q. distinct q \<and> set q = set list) # list)" proof - from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq show ?thesis by auto qed moreover note eq_wq and hd_in ultimately show "False" by auto qed qedqedlemma step_v_get_hold: "\<And>th'. \<lbrakk>vt step (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False" apply (unfold cs_holding_def next_th_def wq_def, auto simp:Let_def)proof - fix rest assume vt: "vt step (V th cs # s)" and eq_wq[folded wq_def]: " waiting_queue (schs s) cs = th # rest" and nrest: "rest \<noteq> []" and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set (SOME q. distinct q \<and> set q = set rest)" have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []" proof(rule someI2) from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next fix x assume "distinct x \<and> set x = set rest" hence "set x = set rest" by auto with nrest show "x \<noteq> []" by (case_tac x, auto) qed with ni show "False" by autoqedlemma step_v_release_inv[elim_format]:"\<And>c t. \<lbrakk>vt step (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> c = cs \<and> t = th" apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits) proof - fix a list assume vt: "vt step (V th cs # s)" and eq_wq: "waiting_queue (schs s) cs = a # list" from step_back_step [OF vt] show "a = th" proof(cases) assume "holding s th cs" with eq_wq show ?thesis by (unfold s_holding_def wq_def, auto) qed next fix a list assume vt: "vt step (V th cs # s)" and eq_wq: "waiting_queue (schs s) cs = a # list" from step_back_step [OF vt] show "a = th" proof(cases) assume "holding s th cs" with eq_wq show ?thesis by (unfold s_holding_def wq_def, auto) qed qedlemma step_v_waiting_mono: "\<And>t c. \<lbrakk>vt step (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"proof - fix t c let ?s' = "(V th cs # s)" assume vt: "vt step ?s'" and wt: "waiting (wq ?s') t c" show "waiting (wq s) t c" proof(cases "c = cs") case False assume neq_cs: "c \<noteq> cs" hence "waiting (wq ?s') t c = waiting (wq s) t c" by (unfold cs_waiting_def wq_def, auto simp:Let_def) with wt show ?thesis by simp next case True with wt show ?thesis apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits) proof - fix a list assume not_in: "t \<notin> set list" and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)" and eq_wq: "waiting_queue (schs s) cs = a # list" have "set (SOME q. distinct q \<and> set q = set list) = set list" proof(rule someI2) from wq_distinct [OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def] show "distinct list \<and> set list = set list" by auto next fix x assume "distinct x \<and> set x = set list" thus "set x = set list" by auto qed with not_in is_in show "t = a" by auto next fix list assume is_waiting: "waiting (wq (V th cs # s)) t cs" and eq_wq: "waiting_queue (schs s) cs = t # list" hence "t \<in> set list" apply (unfold wq_def, auto simp:Let_def cs_waiting_def) proof - assume " t \<in> set (SOME q. distinct q \<and> set q = set list)" moreover have "\<dots> = set list" proof(rule someI2) from wq_distinct [OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def] show "distinct list \<and> set list = set list" by auto next fix x assume "distinct x \<and> set x = set list" thus "set x = set list" by auto qed ultimately show "t \<in> set list" by simp qed with eq_wq and wq_distinct [OF step_back_vt[OF vt], of cs, unfolded wq_def] show False by auto qed qedqedlemma step_depend_v:fixes th::threadassumes vt: "vt step (V th cs#s)"shows " depend (V th cs # s) = depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union> {(Cs cs, Th th') |th'. next_th s th cs th'}" apply (insert vt, unfold s_depend_def) apply (auto split:if_splits list.splits simp:Let_def) apply (auto elim: step_v_waiting_mono step_v_hold_inv step_v_release step_v_wait_inv step_v_get_hold step_v_release_inv) apply (erule_tac step_v_not_wait, auto) donelemma step_depend_p: "vt step (P th cs#s) \<Longrightarrow> depend (P th cs # s) = (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)} else depend s \<union> {(Th th, Cs cs)})" apply(simp only: s_depend_def wq_def) apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def) apply(case_tac "csa = cs", auto) apply(fold wq_def) apply(drule_tac step_back_step) apply(ind_cases " step s (P (hd (wq s cs)) cs)") apply(auto simp:s_depend_def wq_def cs_holding_def) donelemma simple_A: fixes A assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "A = {} \<or> (\<exists> a. A = {a})"proof(cases "A = {}") case True thus ?thesis by simpnext case False then obtain a where "a \<in> A" by auto with h have "A = {a}" by auto thus ?thesis by simpqedlemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs" by (unfold s_depend_def, auto)lemma acyclic_depend: fixes s assumes vt: "vt step s" shows "acyclic (depend s)"proof - from vt show ?thesis proof(induct) case (vt_cons s e) assume ih: "acyclic (depend s)" and stp: "step s e" and vt: "vt step s" show ?case proof(cases e) case (Create th prio) with ih show ?thesis by (simp add:depend_create_unchanged) next case (Exit th) with ih show ?thesis by (simp add:depend_exit_unchanged) next case (V th cs) from V vt stp have vtt: "vt step (V th cs#s)" by auto from step_depend_v [OF this] have eq_de: "depend (e # s) = depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union> {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V) from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset) from step_back_step [OF vtt] have "step s (V th cs)" . thus ?thesis proof(cases) assume "holding s th cs" hence th_in: "th \<in> set (wq s cs)" and eq_hd: "th = hd (wq s cs)" by (unfold s_holding_def, auto) then obtain rest where eq_wq: "wq s cs = th#rest" by (cases "wq s cs", auto) show ?thesis proof(cases "rest = []") case False let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)" from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}" by (unfold next_th_def, auto) let ?E = "(?A - ?B - ?C)" have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*" proof assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*" hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl) from tranclD [OF this] obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast hence th_d: "(Th ?th', x) \<in> ?A" by simp from depend_target_th [OF this] obtain cs' where eq_x: "x = Cs cs'" by auto with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp hence wt_th': "waiting s ?th' cs'" unfolding s_depend_def s_waiting_def cs_waiting_def by simp hence "cs' = cs" proof(rule waiting_unique [OF vt]) from eq_wq wq_distinct[OF vt, of cs] show "waiting s ?th' cs" apply (unfold s_waiting_def, auto) proof - assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest" and eq_wq: "wq s cs = th # rest" have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []" proof(rule someI2) from wq_distinct[OF vt, of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next fix x assume "distinct x \<and> set x = set rest" with False show "x \<noteq> []" by auto qed hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> set (SOME q. distinct q \<and> set q = set rest)" by auto moreover have "\<dots> = set rest" proof(rule someI2) from wq_distinct[OF vt, of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed moreover note hd_in ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto next assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest" and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest" have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []" proof(rule someI2) from wq_distinct[OF vt, of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next fix x assume "distinct x \<and> set x = set rest" with False show "x \<noteq> []" by auto qed hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> set (SOME q. distinct q \<and> set q = set rest)" by auto moreover have "\<dots> = set rest" proof(rule someI2) from wq_distinct[OF vt, of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed moreover note hd_in ultimately show False by auto qed qed with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp with False show "False" by (auto simp: next_th_def eq_wq) qed with acyclic_insert[symmetric] and ac and eq_de eq_D show ?thesis by auto next case True with eq_wq have eq_D: "?D = {}" by (unfold next_th_def, auto) with eq_de ac show ?thesis by auto qed qed next case (P th cs) from P vt stp have vtt: "vt step (P th cs#s)" by auto from step_depend_p [OF this] P have "depend (e # s) = (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R") by simp moreover have "acyclic ?R" proof(cases "wq s cs = []") case True hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*" proof assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*" hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl) from tranclD2 [OF this] obtain x where "(x, Cs cs) \<in> depend s" by auto with True show False by (auto simp:s_depend_def cs_waiting_def) qed with acyclic_insert ih eq_r show ?thesis by auto next case False hence eq_r: "?R = depend s \<union> {(Th th, Cs cs)}" by simp have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*" proof assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*" hence "(Cs cs, Th th) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl) moreover from step_back_step [OF vtt] have "step s (P th cs)" . ultimately show False proof - show " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False" by (ind_cases "step s (P th cs)", simp) qed qed with acyclic_insert ih eq_r show ?thesis by auto qed ultimately show ?thesis by simp next case (Set thread prio) with ih thm depend_set_unchanged show ?thesis by (simp add:depend_set_unchanged) qed next case vt_nil show "acyclic (depend ([]::state))" by (auto simp: s_depend_def cs_waiting_def cs_holding_def wq_def acyclic_def) qedqedlemma finite_depend: fixes s assumes vt: "vt step s" shows "finite (depend s)"proof - from vt show ?thesis proof(induct) case (vt_cons s e) assume ih: "finite (depend s)" and stp: "step s e" and vt: "vt step s" show ?case proof(cases e) case (Create th prio) with ih show ?thesis by (simp add:depend_create_unchanged) next case (Exit th) with ih show ?thesis by (simp add:depend_exit_unchanged) next case (V th cs) from V vt stp have vtt: "vt step (V th cs#s)" by auto from step_depend_v [OF this] have eq_de: "depend (e # s) = depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union> {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V) moreover from ih have ac: "finite (?A - ?B - ?C)" by simp moreover have "finite ?D" proof - have "?D = {} \<or> (\<exists> a. ?D = {a})" by (unfold next_th_def, auto) thus ?thesis proof assume h: "?D = {}" show ?thesis by (unfold h, simp) next assume "\<exists> a. ?D = {a}" thus ?thesis by auto qed qed ultimately show ?thesis by simp next case (P th cs) from P vt stp have vtt: "vt step (P th cs#s)" by auto from step_depend_p [OF this] P have "depend (e # s) = (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R") by simp moreover have "finite ?R" proof(cases "wq s cs = []") case True hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp with True and ih show ?thesis by auto next case False hence "?R = depend s \<union> {(Th th, Cs cs)}" by simp with False and ih show ?thesis by auto qed ultimately show ?thesis by auto next case (Set thread prio) with ih show ?thesis by (simp add:depend_set_unchanged) qed next case vt_nil show "finite (depend ([]::state))" by (auto simp: s_depend_def cs_waiting_def cs_holding_def wq_def acyclic_def) qedqedtext {* Several useful lemmas *}lemma wf_dep_converse: fixes s assumes vt: "vt step s" shows "wf ((depend s)^-1)"proof(rule finite_acyclic_wf_converse) from finite_depend [OF vt] show "finite (depend s)" .next from acyclic_depend[OF vt] show "acyclic (depend s)" .qedlemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"by (induct l, auto)lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s" by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)lemma wq_threads: fixes s cs assumes vt: "vt step s" and h: "th \<in> set (wq s cs)" shows "th \<in> threads s"proof - from vt and h show ?thesis proof(induct arbitrary: th cs) case (vt_cons s e) assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s" and stp: "step s e" and vt: "vt step s" and h: "th \<in> set (wq (e # s) cs)" show ?case proof(cases e) case (Create th' prio) with ih h show ?thesis by (auto simp:wq_def Let_def) next case (Exit th') with stp ih h show ?thesis apply (auto simp:wq_def Let_def) apply (ind_cases "step s (Exit th')") apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def s_depend_def s_holding_def cs_holding_def) by (fold wq_def, auto) next case (V th' cs') show ?thesis proof(cases "cs' = cs") case False with h show ?thesis apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def) by (drule_tac ih, simp) next case True from h show ?thesis proof(unfold V wq_def) assume th_in: "th \<in> set (waiting_queue (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l") show "th \<in> threads (V th' cs' # s)" proof(cases "cs = cs'") case False hence "?l = waiting_queue (schs s) cs" by (simp add:Let_def) with th_in have " th \<in> set (wq s cs)" by (fold wq_def, simp) from ih [OF this] show ?thesis by simp next case True show ?thesis proof(cases "waiting_queue (schs s) cs'") case Nil with h V show ?thesis apply (auto simp:wq_def Let_def split:if_splits) by (fold wq_def, drule_tac ih, simp) next case (Cons a rest) assume eq_wq: "waiting_queue (schs s) cs' = a # rest" with h V show ?thesis apply (auto simp:Let_def wq_def split:if_splits) proof - assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)" have "set (SOME q. distinct q \<and> set q = set rest) = set rest" proof(rule someI2) from wq_distinct[OF vt, of cs'] and eq_wq[folded wq_def] show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed with eq_wq th_in have "th \<in> set (waiting_queue (schs s) cs')" by auto from ih[OF this[folded wq_def]] show "th \<in> threads s" . next assume th_in: "th \<in> set (waiting_queue (schs s) cs)" from ih[OF this[folded wq_def]] show "th \<in> threads s" . qed qed qed qed qed next case (P th' cs') from h stp show ?thesis apply (unfold P wq_def) apply (auto simp:Let_def split:if_splits, fold wq_def) apply (auto intro:ih) apply(ind_cases "step s (P th' cs')") by (unfold runing_def readys_def, auto) next case (Set thread prio) with ih h show ?thesis by (auto simp:wq_def Let_def) qed next case vt_nil thus ?case by (auto simp:wq_def) qedqedlemma range_in: "\<lbrakk>vt step s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s" apply(unfold s_depend_def cs_waiting_def cs_holding_def) by (auto intro:wq_threads)lemma readys_v_eq: fixes th thread cs rest assumes vt: "vt step s" and neq_th: "th \<noteq> thread" and eq_wq: "wq s cs = thread#rest" and not_in: "th \<notin> set rest" shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"proof - from prems show ?thesis apply (auto simp:readys_def) apply (case_tac "cs = csa", simp add:s_waiting_def) apply (erule_tac x = csa in allE) apply (simp add:s_waiting_def wq_def Let_def split:if_splits) apply (case_tac "csa = cs", simp) apply (erule_tac x = cs in allE) apply (auto simp:s_waiting_def wq_def Let_def split:list.splits) proof - assume th_nin: "th \<notin> set rest" and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)" and eq_wq: "waiting_queue (schs s) cs = thread # rest" have "set (SOME q. distinct q \<and> set q = set rest) = set rest" proof(rule someI2) from wq_distinct[OF vt, of cs] and eq_wq[folded wq_def] show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed with th_nin th_in show False by auto qedqedlemma chain_building: assumes vt: "vt step s" shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"proof - from wf_dep_converse [OF vt] have h: "wf ((depend s)\<inverse>)" . show ?thesis proof(induct rule:wf_induct [OF h]) fix x assume ih [rule_format]: "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow> y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)" show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)" proof assume x_d: "x \<in> Domain (depend s)" show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+" proof(cases x) case (Th th) from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def) with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast hence "Cs cs \<in> Domain (depend s)" by auto from ih [OF x_in_r this] obtain th' where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto have "(x, Th th') \<in> (depend s)\<^sup>+" using Th x_in cs_in by auto with th'_ready show ?thesis by auto next case (Cs cs) from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (depend s)^-1" by (auto simp:s_depend_def) show ?thesis proof(cases "th' \<in> readys s") case True from True and th'_d show ?thesis by auto next case False from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto with False have "Th th' \<in> Domain (depend s)" by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def) from ih [OF th'_d this] obtain th'' where th''_r: "th'' \<in> readys s" and th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto from th'_d and th''_in have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto with th''_r show ?thesis by auto qed qed qed qedqedlemma th_chain_to_ready: fixes s th assumes vt: "vt step s" and th_in: "th \<in> threads s" shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"proof(cases "th \<in> readys s") case True thus ?thesis by autonext case False from False and th_in have "Th th \<in> Domain (depend s)" by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def) from chain_building [rule_format, OF vt this] show ?thesis by autoqedlemma waiting_eq: "waiting s th cs = waiting (wq s) th cs" by (unfold s_waiting_def cs_waiting_def, auto)lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs" by (unfold s_holding_def cs_holding_def, simp)lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2" by (unfold s_holding_def cs_holding_def, auto)lemma unique_depend: "\<lbrakk>vt step s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2" apply(unfold s_depend_def, auto, fold waiting_eq holding_eq) by(auto elim:waiting_unique holding_unique)lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"by (induct rule:trancl_induct, auto)lemma dchain_unique: assumes vt: "vt step s" and th1_d: "(n, Th th1) \<in> (depend s)^+" and th1_r: "th1 \<in> readys s" and th2_d: "(n, Th th2) \<in> (depend s)^+" and th2_r: "th2 \<in> readys s" shows "th1 = th2"proof - { assume neq: "th1 \<noteq> th2" hence "Th th1 \<noteq> Th th2" by simp from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt] have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto hence "False" proof assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+" from trancl_split [OF this] obtain n where dd: "(Th th1, n) \<in> depend s" by auto then obtain cs where eq_n: "n = Cs cs" by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in) from dd eq_n have "th1 \<notin> readys s" by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def) with th1_r show ?thesis by auto next assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+" from trancl_split [OF this] obtain n where dd: "(Th th2, n) \<in> depend s" by auto then obtain cs where eq_n: "n = Cs cs" by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in) from dd eq_n have "th2 \<notin> readys s" by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def) with th2_r show ?thesis by auto qed } thus ?thesis by autoqedlemma step_holdents_p_add: fixes th cs s assumes vt: "vt step (P th cs#s)" and "wq s cs = []" shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"proof - from prems show ?thesis unfolding holdents_def step_depend_p[OF vt] by autoqedlemma step_holdents_p_eq: fixes th cs s assumes vt: "vt step (P th cs#s)" and "wq s cs \<noteq> []" shows "holdents (P th cs#s) th = holdents s th"proof - from prems show ?thesis unfolding holdents_def step_depend_p[OF vt] by autoqedlemma finite_holding: fixes s th cs assumes vt: "vt step s" shows "finite (holdents s th)"proof - let ?F = "\<lambda> (x, y). the_cs x" from finite_depend [OF vt] have "finite (depend s)" . hence "finite (?F `(depend s))" by simp moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>" proof - { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto fix x assume "(Cs x, Th th) \<in> depend s" hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h) moreover have "?F (Cs x, Th th) = x" by simp ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp } thus ?thesis by auto qed ultimately show ?thesis by (unfold holdents_def, auto intro:finite_subset)qedlemma cntCS_v_dec: fixes s thread cs assumes vtv: "vt step (V thread cs#s)" shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"proof - from step_back_step[OF vtv] have cs_in: "cs \<in> holdents s thread" apply (cases, unfold holdents_def s_depend_def, simp) by (unfold cs_holding_def s_holding_def, auto) moreover have cs_not_in: "(holdents (V thread cs#s) thread) = holdents s thread - {cs}" apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs]) apply (unfold holdents_def, unfold step_depend_v[OF vtv], auto simp:next_th_def) proof - fix rest assume dst: "distinct (rest::thread list)" and ne: "rest \<noteq> []" and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest" moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest" proof(rule someI2) from dst show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set (SOME q. distinct q \<and> set q = set rest)" by simp moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []" proof(rule someI2) from dst show "distinct rest \<and> set rest = set rest" by auto next fix x assume " distinct x \<and> set x = set rest" with ne show "x \<noteq> []" by auto qed ultimately show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s" by auto next fix rest assume dst: "distinct (rest::thread list)" and ne: "rest \<noteq> []" and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest" moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest" proof(rule someI2) from dst show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set (SOME q. distinct q \<and> set q = set rest)" by simp moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []" proof(rule someI2) from dst show "distinct rest \<and> set rest = set rest" by auto next fix x assume " distinct x \<and> set x = set rest" with ne show "x \<noteq> []" by auto qed ultimately show "False" by auto qed ultimately have "holdents s thread = insert cs (holdents (V thread cs#s) thread)" by auto moreover have "card \<dots> = Suc (card ((holdents (V thread cs#s) thread) - {cs}))" proof(rule card_insert) from finite_holding [OF vtv] show " finite (holdents (V thread cs # s) thread)" . qed moreover from cs_not_in have "cs \<notin> (holdents (V thread cs#s) thread)" by auto ultimately show ?thesis by (simp add:cntCS_def)qed lemma cnp_cnv_cncs: fixes s th assumes vt: "vt step s" shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"proof - from vt show ?thesis proof(induct arbitrary:th) case (vt_cons s e) assume vt: "vt step s" and ih: "\<And>th. cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)" and stp: "step s e" from stp show ?case proof(cases) case (thread_create thread prio) assume eq_e: "e = Create thread prio" and not_in: "thread \<notin> threads s" show ?thesis proof - { fix cs assume "thread \<in> set (wq s cs)" from wq_threads [OF vt this] have "thread \<in> threads s" . with not_in have "False" by simp } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}" by (auto simp:readys_def threads.simps s_waiting_def wq_def cs_waiting_def Let_def) from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def) from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def) have eq_cncs: "cntCS (e#s) th = cntCS s th" unfolding cntCS_def holdents_def by (simp add:depend_create_unchanged eq_e) { assume "th \<noteq> thread" with eq_readys eq_e have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = (th \<in> readys (s) \<or> th \<notin> threads (s))" by (simp add:threads.simps) with eq_cnp eq_cnv eq_cncs ih not_in have ?thesis by simp } moreover { assume eq_th: "th = thread" with not_in ih have " cntP s th = cntV s th + cntCS s th" by simp moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp moreover note eq_cnp eq_cnv eq_cncs ultimately have ?thesis by auto } ultimately show ?thesis by blast qed next case (thread_exit thread) assume eq_e: "e = Exit thread" and is_runing: "thread \<in> runing s" and no_hold: "holdents s thread = {}" from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def) from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def) have eq_cncs: "cntCS (e#s) th = cntCS s th" unfolding cntCS_def holdents_def by (simp add:depend_exit_unchanged eq_e) { assume "th \<noteq> thread" with eq_e have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = (th \<in> readys (s) \<or> th \<notin> threads (s))" apply (simp add:threads.simps readys_def) apply (subst s_waiting_def) apply (subst (1 2) wq_def) apply (simp add:Let_def) apply (subst s_waiting_def, simp) by (fold wq_def, simp) with eq_cnp eq_cnv eq_cncs ih have ?thesis by simp } moreover { assume eq_th: "th = thread" with ih is_runing have " cntP s th = cntV s th + cntCS s th" by (simp add:runing_def) moreover from eq_th eq_e have "th \<notin> threads (e#s)" by simp moreover note eq_cnp eq_cnv eq_cncs ultimately have ?thesis by auto } ultimately show ?thesis by blast next case (thread_P thread cs) assume eq_e: "e = P thread cs" and is_runing: "thread \<in> runing s" and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+" from prems have vtp: "vt step (P thread cs#s)" by auto show ?thesis proof - { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast assume neq_th: "th \<noteq> thread" with eq_e have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))" apply (simp add:readys_def s_waiting_def wq_def Let_def) apply (rule_tac hh, clarify) apply (intro iffI allI, clarify) apply (erule_tac x = csa in allE, auto) apply (subgoal_tac "waiting_queue (schs s) cs \<noteq> []", auto) apply (erule_tac x = cs in allE, auto) by (case_tac "(waiting_queue (schs s) cs)", auto) moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th" apply (simp add:cntCS_def holdents_def) by (unfold step_depend_p [OF vtp], auto) moreover from eq_e neq_th have "cntP (e # s) th = cntP s th" by (simp add:cntP_def count_def) moreover from eq_e neq_th have "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def) moreover from eq_e neq_th have "threads (e#s) = threads s" by simp moreover note ih [of th] ultimately have ?thesis by simp } moreover { assume eq_th: "th = thread" have ?thesis proof - from eq_e eq_th have eq_cnp: "cntP (e # s) th = 1 + (cntP s th)" by (simp add:cntP_def count_def) from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def) show ?thesis proof (cases "wq s cs = []") case True with is_runing have "th \<in> readys (e#s)" apply (unfold eq_e wq_def, unfold readys_def s_depend_def) apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def) by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def) moreover have "cntCS (e # s) th = 1 + cntCS s th" proof - have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> depend s} = Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)") proof - have "?L = insert cs ?R" by auto moreover have "card \<dots> = Suc (card (?R - {cs}))" proof(rule card_insert) from finite_holding [OF vt, of thread] show " finite {cs. (Cs cs, Th thread) \<in> depend s}" by (unfold holdents_def, simp) qed moreover have "?R - {cs} = ?R" proof - have "cs \<notin> ?R" proof assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend s}" with no_dep show False by auto qed thus ?thesis by auto qed ultimately show ?thesis by auto qed thus ?thesis apply (unfold eq_e eq_th cntCS_def) apply (simp add: holdents_def) by (unfold step_depend_p [OF vtp], auto simp:True) qed moreover from is_runing have "th \<in> readys s" by (simp add:runing_def eq_th) moreover note eq_cnp eq_cnv ih [of th] ultimately show ?thesis by auto next case False have eq_wq: "wq (e#s) cs = wq s cs @ [th]" by (unfold eq_th eq_e wq_def, auto simp:Let_def) have "th \<notin> readys (e#s)" proof assume "th \<in> readys (e#s)" hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def) from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" . hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)" by (simp add:s_waiting_def) moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto ultimately have "th = hd (wq (e#s) cs)" by blast with eq_wq have "th = hd (wq s cs @ [th])" by simp hence "th = hd (wq s cs)" using False by auto with False eq_wq wq_distinct [OF vtp, of cs] show False by (fold eq_e, auto) qed moreover from is_runing have "th \<in> threads (e#s)" by (unfold eq_e, auto simp:runing_def readys_def eq_th) moreover have "cntCS (e # s) th = cntCS s th" apply (unfold cntCS_def holdents_def eq_e step_depend_p[OF vtp]) by (auto simp:False) moreover note eq_cnp eq_cnv ih[of th] moreover from is_runing have "th \<in> readys s" by (simp add:runing_def eq_th) ultimately show ?thesis by auto qed qed } ultimately show ?thesis by blast qed next case (thread_V thread cs) from prems have vtv: "vt step (V thread cs # s)" by auto assume eq_e: "e = V thread cs" and is_runing: "thread \<in> runing s" and hold: "holding s thread cs" from hold obtain rest where eq_wq: "wq s cs = thread # rest" by (case_tac "wq s cs", auto simp:s_holding_def) have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e) have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest" proof(rule someI2) from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed show ?thesis proof - { assume eq_th: "th = thread" from eq_th have eq_cnp: "cntP (e # s) th = cntP s th" by (unfold eq_e, simp add:cntP_def count_def) moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th" by (unfold eq_e, simp add:cntV_def count_def) moreover from cntCS_v_dec [OF vtv] have "cntCS (e # s) thread + 1 = cntCS s thread" by (simp add:eq_e) moreover from is_runing have rd_before: "thread \<in> readys s" by (unfold runing_def, simp) moreover have "thread \<in> readys (e # s)" proof - from is_runing have "thread \<in> threads (e#s)" by (unfold eq_e, auto simp:runing_def readys_def) moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1" proof fix cs1 { assume eq_cs: "cs1 = cs" have "\<not> waiting (e # s) thread cs1" proof - from eq_wq have "thread \<notin> set (wq (e#s) cs1)" apply(unfold eq_e wq_def eq_cs s_holding_def) apply (auto simp:Let_def) proof - assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)" with eq_set have "thread \<in> set rest" by simp with wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq show False by auto qed thus ?thesis by (simp add:s_waiting_def) qed } moreover { assume neq_cs: "cs1 \<noteq> cs" have "\<not> waiting (e # s) thread cs1" proof - from wq_v_neq [OF neq_cs[symmetric]] have "wq (V thread cs # s) cs1 = wq s cs1" . moreover have "\<not> waiting s thread cs1" proof - from runing_ready and is_runing have "thread \<in> readys s" by auto thus ?thesis by (simp add:readys_def) qed ultimately show ?thesis by (auto simp:s_waiting_def eq_e) qed } ultimately show "\<not> waiting (e # s) thread cs1" by blast qed ultimately show ?thesis by (simp add:readys_def) qed moreover note eq_th ih ultimately have ?thesis by auto } moreover { assume neq_th: "th \<noteq> thread" from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" by (simp add:cntP_def count_def) from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" by (simp add:cntV_def count_def) have ?thesis proof(cases "th \<in> set rest") case False have "(th \<in> readys (e # s)) = (th \<in> readys s)" apply (insert step_back_vt[OF vtv]) by (unfold eq_e, rule readys_v_eq [OF _ neq_th eq_wq False], auto) moreover have "cntCS (e#s) th = cntCS s th" apply (insert neq_th, unfold eq_e cntCS_def holdents_def step_depend_v[OF vtv], auto) proof - have "{csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} = {cs. (Cs cs, Th th) \<in> depend s}" proof - from False eq_wq have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> depend s" apply (unfold next_th_def, auto) proof - assume ne: "rest \<noteq> []" and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest" and eq_wq: "wq s cs = thread # rest" from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set (SOME q. distinct q \<and> set q = set rest) " by simp moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []" proof(rule someI2) from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next fix x assume "distinct x \<and> set x = set rest" with ne show "x \<noteq> []" by auto qed ultimately show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s" by auto qed thus ?thesis by auto qed thus "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} = card {cs. (Cs cs, Th th) \<in> depend s}" by simp qed moreover note ih eq_cnp eq_cnv eq_threads ultimately show ?thesis by auto next case True assume th_in: "th \<in> set rest" show ?thesis proof(cases "next_th s thread cs th") case False with eq_wq and th_in have neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest") by (auto simp:next_th_def) have "(th \<in> readys (e # s)) = (th \<in> readys s)" proof - from eq_wq and th_in have "\<not> th \<in> readys s" apply (auto simp:readys_def s_waiting_def) apply (rule_tac x = cs in exI, auto) by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto) moreover from eq_wq and th_in and neq_hd have "\<not> (th \<in> readys (e # s))" apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits) by (rule_tac x = cs in exI, auto simp:eq_set) ultimately show ?thesis by auto qed moreover have "cntCS (e#s) th = cntCS s th" proof - from eq_wq and th_in and neq_hd have "(holdents (e # s) th) = (holdents s th)" apply (unfold eq_e step_depend_v[OF vtv], auto simp:next_th_def eq_set s_depend_def holdents_def wq_def Let_def cs_holding_def) by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto simp:wq_def) thus ?thesis by (simp add:cntCS_def) qed moreover note ih eq_cnp eq_cnv eq_threads ultimately show ?thesis by auto next case True let ?rest = " (SOME q. distinct q \<and> set q = set rest)" let ?t = "hd ?rest" from True eq_wq th_in neq_th have "th \<in> readys (e # s)" apply (auto simp:eq_e readys_def s_waiting_def wq_def Let_def next_th_def) proof - assume eq_wq: "waiting_queue (schs s) cs = thread # rest" and t_in: "?t \<in> set rest" show "?t \<in> threads s" proof(rule wq_threads[OF step_back_vt[OF vtv]]) from eq_wq and t_in show "?t \<in> set (wq s cs)" by (auto simp:wq_def) qed next fix csa assume eq_wq: "waiting_queue (schs s) cs = thread # rest" and t_in: "?t \<in> set rest" and neq_cs: "csa \<noteq> cs" and t_in': "?t \<in> set (waiting_queue (schs s) csa)" show "?t = hd (waiting_queue (schs s) csa)" proof - { assume neq_hd': "?t \<noteq> hd (waiting_queue (schs s) csa)" from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq[folded wq_def] and t_in eq_wq have "?t \<noteq> thread" by auto with eq_wq and t_in have w1: "waiting s ?t cs" by (auto simp:s_waiting_def wq_def) from t_in' neq_hd' have w2: "waiting s ?t csa" by (auto simp:s_waiting_def wq_def) from waiting_unique[OF step_back_vt[OF vtv] w1 w2] and neq_cs have "False" by auto } thus ?thesis by auto qed qed moreover have "cntP s th = cntV s th + cntCS s th + 1" proof - have "th \<notin> readys s" proof - from True eq_wq neq_th th_in show ?thesis apply (unfold readys_def s_waiting_def, auto) by (rule_tac x = cs in exI, auto) qed moreover have "th \<in> threads s" proof - from th_in eq_wq have "th \<in> set (wq s cs)" by simp from wq_threads [OF step_back_vt[OF vtv] this] show ?thesis . qed ultimately show ?thesis using ih by auto qed moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th" apply (unfold cntCS_def holdents_def eq_e step_depend_v[OF vtv], auto) proof - show "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs} = Suc (card {cs. (Cs cs, Th th) \<in> depend s})" (is "card ?A = Suc (card ?B)") proof - have "?A = insert cs ?B" by auto hence "card ?A = card (insert cs ?B)" by simp also have "\<dots> = Suc (card ?B)" proof(rule card_insert_disjoint) have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` depend s)" apply (auto simp:image_def) by (rule_tac x = "(Cs x, Th th)" in bexI, auto) with finite_depend[OF step_back_vt[OF vtv]] show "finite {cs. (Cs cs, Th th) \<in> depend s}" by (auto intro:finite_subset) next show "cs \<notin> {cs. (Cs cs, Th th) \<in> depend s}" proof assume "cs \<in> {cs. (Cs cs, Th th) \<in> depend s}" hence "(Cs cs, Th th) \<in> depend s" by simp with True neq_th eq_wq show False by (auto simp:next_th_def s_depend_def cs_holding_def) qed qed finally show ?thesis . qed qed moreover note eq_cnp eq_cnv ultimately show ?thesis by simp qed qed } ultimately show ?thesis by blast qed next case (thread_set thread prio) assume eq_e: "e = Set thread prio" and is_runing: "thread \<in> runing s" show ?thesis proof - from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def) from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def) have eq_cncs: "cntCS (e#s) th = cntCS s th" unfolding cntCS_def holdents_def by (simp add:depend_set_unchanged eq_e) from eq_e have eq_readys: "readys (e#s) = readys s" by (simp add:readys_def cs_waiting_def s_waiting_def wq_def, auto simp:Let_def) { assume "th \<noteq> thread" with eq_readys eq_e have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = (th \<in> readys (s) \<or> th \<notin> threads (s))" by (simp add:threads.simps) with eq_cnp eq_cnv eq_cncs ih is_runing have ?thesis by simp } moreover { assume eq_th: "th = thread" with is_runing ih have " cntP s th = cntV s th + cntCS s th" by (unfold runing_def, auto) moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)" by (simp add:runing_def) moreover note eq_cnp eq_cnv eq_cncs ultimately have ?thesis by auto } ultimately show ?thesis by blast qed qed next case vt_nil show ?case by (unfold cntP_def cntV_def cntCS_def, auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def) qedqedlemma not_thread_cncs: fixes th s assumes vt: "vt step s" and not_in: "th \<notin> threads s" shows "cntCS s th = 0"proof - from vt not_in show ?thesis proof(induct arbitrary:th) case (vt_cons s e th) assume vt: "vt step s" and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0" and stp: "step s e" and not_in: "th \<notin> threads (e # s)" from stp show ?case proof(cases) case (thread_create thread prio) assume eq_e: "e = Create thread prio" and not_in': "thread \<notin> threads s" have "cntCS (e # s) th = cntCS s th" apply (unfold eq_e cntCS_def holdents_def) by (simp add:depend_create_unchanged) moreover have "th \<notin> threads s" proof - from not_in eq_e show ?thesis by simp qed moreover note ih ultimately show ?thesis by auto next case (thread_exit thread) assume eq_e: "e = Exit thread" and nh: "holdents s thread = {}" have eq_cns: "cntCS (e # s) th = cntCS s th" apply (unfold eq_e cntCS_def holdents_def) by (simp add:depend_exit_unchanged) show ?thesis proof(cases "th = thread") case True have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True) with eq_cns show ?thesis by simp next case False with not_in and eq_e have "th \<notin> threads s" by simp from ih[OF this] and eq_cns show ?thesis by simp qed next case (thread_P thread cs) assume eq_e: "e = P thread cs" and is_runing: "thread \<in> runing s" from prems have vtp: "vt step (P thread cs#s)" by auto have neq_th: "th \<noteq> thread" proof - from not_in eq_e have "th \<notin> threads s" by simp moreover from is_runing have "thread \<in> threads s" by (simp add:runing_def readys_def) ultimately show ?thesis by auto qed hence "cntCS (e # s) th = cntCS s th " apply (unfold cntCS_def holdents_def eq_e) by (unfold step_depend_p[OF vtp], auto) moreover have "cntCS s th = 0" proof(rule ih) from not_in eq_e show "th \<notin> threads s" by simp qed ultimately show ?thesis by simp next case (thread_V thread cs) assume eq_e: "e = V thread cs" and is_runing: "thread \<in> runing s" and hold: "holding s thread cs" have neq_th: "th \<noteq> thread" proof - from not_in eq_e have "th \<notin> threads s" by simp moreover from is_runing have "thread \<in> threads s" by (simp add:runing_def readys_def) ultimately show ?thesis by auto qed from prems have vtv: "vt step (V thread cs#s)" by auto from hold obtain rest where eq_wq: "wq s cs = thread # rest" by (case_tac "wq s cs", auto simp:s_holding_def) from not_in eq_e eq_wq have "\<not> next_th s thread cs th" apply (auto simp:next_th_def) proof - assume ne: "rest \<noteq> []" and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s") have "?t \<in> set rest" proof(rule someI2) from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next fix x assume "distinct x \<and> set x = set rest" with ne show "hd x \<in> set rest" by (cases x, auto) qed with eq_wq have "?t \<in> set (wq s cs)" by simp from wq_threads[OF step_back_vt[OF vtv], OF this] and ni show False by auto qed moreover note neq_th eq_wq ultimately have "cntCS (e # s) th = cntCS s th" by (unfold eq_e cntCS_def holdents_def step_depend_v[OF vtv], auto) moreover have "cntCS s th = 0" proof(rule ih) from not_in eq_e show "th \<notin> threads s" by simp qed ultimately show ?thesis by simp next case (thread_set thread prio) print_facts assume eq_e: "e = Set thread prio" and is_runing: "thread \<in> runing s" from not_in and eq_e have "th \<notin> threads s" by auto from ih [OF this] and eq_e show ?thesis apply (unfold eq_e cntCS_def holdents_def) by (simp add:depend_set_unchanged) qed next case vt_nil show ?case by (unfold cntCS_def, auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def) qedqedlemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs" by (auto simp:s_waiting_def cs_waiting_def)lemma dm_depend_threads: fixes th s assumes vt: "vt step s" and in_dom: "(Th th) \<in> Domain (depend s)" shows "th \<in> threads s"proof - from in_dom obtain n where "(Th th, n) \<in> depend s" by auto moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto ultimately have "(Th th, Cs cs) \<in> depend s" by simp hence "th \<in> set (wq s cs)" by (unfold s_depend_def, auto simp:cs_waiting_def) from wq_threads [OF vt this] show ?thesis .qedlemma cp_eq_cpreced: "cp s th = cpreced s (wq s) th"proof(unfold cp_def wq_def, induct s) case (Cons e s') show ?case by (auto simp:Let_def)next case Nil show ?case by (auto simp:Let_def)qedlemma runing_unique: fixes th1 th2 s assumes vt: "vt step s" and runing_1: "th1 \<in> runing s" and runing_2: "th2 \<in> runing s" shows "th1 = th2"proof - from runing_1 and runing_2 have "cp s th1 = cp s th2" by (unfold runing_def, simp) hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) = Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))" (is "Max (?f ` ?A) = Max (?f ` ?B)") by (unfold cp_eq_cpreced cpreced_def) obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)" proof - have h1: "finite (?f ` ?A)" proof - have "finite ?A" proof - have "finite (dependents (wq s) th1)" proof- have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}" proof - let ?F = "\<lambda> (x, y). the_th x" have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)" apply (auto simp:image_def) by (rule_tac x = "(Th x, Th th1)" in bexI, auto) moreover have "finite \<dots>" proof - from finite_depend[OF vt] have "finite (depend s)" . hence "finite ((depend (wq s))\<^sup>+)" apply (unfold finite_trancl) by (auto simp: s_depend_def cs_depend_def wq_def) thus ?thesis by auto qed ultimately show ?thesis by (auto intro:finite_subset) qed thus ?thesis by (simp add:cs_dependents_def) qed thus ?thesis by simp qed thus ?thesis by auto qed moreover have h2: "(?f ` ?A) \<noteq> {}" proof - have "?A \<noteq> {}" by simp thus ?thesis by simp qed from Max_in [OF h1 h2] have "Max (?f ` ?A) \<in> (?f ` ?A)" . thus ?thesis by (auto intro:that) qed obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)" proof - have h1: "finite (?f ` ?B)" proof - have "finite ?B" proof - have "finite (dependents (wq s) th2)" proof- have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}" proof - let ?F = "\<lambda> (x, y). the_th x" have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)" apply (auto simp:image_def) by (rule_tac x = "(Th x, Th th2)" in bexI, auto) moreover have "finite \<dots>" proof - from finite_depend[OF vt] have "finite (depend s)" . hence "finite ((depend (wq s))\<^sup>+)" apply (unfold finite_trancl) by (auto simp: s_depend_def cs_depend_def wq_def) thus ?thesis by auto qed ultimately show ?thesis by (auto intro:finite_subset) qed thus ?thesis by (simp add:cs_dependents_def) qed thus ?thesis by simp qed thus ?thesis by auto qed moreover have h2: "(?f ` ?B) \<noteq> {}" proof - have "?B \<noteq> {}" by simp thus ?thesis by simp qed from Max_in [OF h1 h2] have "Max (?f ` ?B) \<in> (?f ` ?B)" . thus ?thesis by (auto intro:that) qed from eq_f_th1 eq_f_th2 eq_max have eq_preced: "preced th1' s = preced th2' s" by auto hence eq_th12: "th1' = th2'" proof (rule preced_unique) from th1_in have "th1' = th1 \<or> (th1' \<in> dependents (wq s) th1)" by simp thus "th1' \<in> threads s" proof assume "th1' \<in> dependents (wq s) th1" hence "(Th th1') \<in> Domain ((depend s)^+)" apply (unfold cs_dependents_def cs_depend_def s_depend_def) by (auto simp:Domain_def) hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain) from dm_depend_threads[OF vt this] show ?thesis . next assume "th1' = th1" with runing_1 show ?thesis by (unfold runing_def readys_def, auto) qed next from th2_in have "th2' = th2 \<or> (th2' \<in> dependents (wq s) th2)" by simp thus "th2' \<in> threads s" proof assume "th2' \<in> dependents (wq s) th2" hence "(Th th2') \<in> Domain ((depend s)^+)" apply (unfold cs_dependents_def cs_depend_def s_depend_def) by (auto simp:Domain_def) hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain) from dm_depend_threads[OF vt this] show ?thesis . next assume "th2' = th2" with runing_2 show ?thesis by (unfold runing_def readys_def, auto) qed qed from th1_in have "th1' = th1 \<or> th1' \<in> dependents (wq s) th1" by simp thus ?thesis proof assume eq_th': "th1' = th1" from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp thus ?thesis proof assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp next assume "th2' \<in> dependents (wq s) th2" with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp hence "(Th th1, Th th2) \<in> (depend s)^+" by (unfold cs_dependents_def s_depend_def cs_depend_def, simp) hence "Th th1 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"] by auto hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain) then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def) from depend_target_th [OF this] obtain cs' where "n = Cs cs'" by auto with d have "(Th th1, Cs cs') \<in> depend s" by simp with runing_1 have "False" apply (unfold runing_def readys_def s_depend_def) by (auto simp:eq_waiting) thus ?thesis by simp qed next assume th1'_in: "th1' \<in> dependents (wq s) th1" from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp thus ?thesis proof assume "th2' = th2" with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp hence "(Th th2, Th th1) \<in> (depend s)^+" by (unfold cs_dependents_def s_depend_def cs_depend_def, simp) hence "Th th2 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"] by auto hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain) then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def) from depend_target_th [OF this] obtain cs' where "n = Cs cs'" by auto with d have "(Th th2, Cs cs') \<in> depend s" by simp with runing_2 have "False" apply (unfold runing_def readys_def s_depend_def) by (auto simp:eq_waiting) thus ?thesis by simp next assume "th2' \<in> dependents (wq s) th2" with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp hence h1: "(Th th1', Th th2) \<in> (depend s)^+" by (unfold cs_dependents_def s_depend_def cs_depend_def, simp) from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+" by (unfold cs_dependents_def s_depend_def cs_depend_def, simp) show ?thesis proof(rule dchain_unique[OF vt h1 _ h2, symmetric]) from runing_1 show "th1 \<in> readys s" by (simp add:runing_def) from runing_2 show "th2 \<in> readys s" by (simp add:runing_def) qed qed qedqedlemma create_pre: assumes stp: "step s e" and not_in: "th \<notin> threads s" and is_in: "th \<in> threads (e#s)" obtains prio where "e = Create th prio"proof - from assms show ?thesis proof(cases) case (thread_create thread prio) with is_in not_in have "e = Create th prio" by simp from that[OF this] show ?thesis . next case (thread_exit thread) with assms show ?thesis by (auto intro!:that) next case (thread_P thread) with assms show ?thesis by (auto intro!:that) next case (thread_V thread) with assms show ?thesis by (auto intro!:that) next case (thread_set thread) with assms show ?thesis by (auto intro!:that) qedqedlemma length_down_to_in: assumes le_ij: "i \<le> j" and le_js: "j \<le> length s" shows "length (down_to j i s) = j - i"proof - have "length (down_to j i s) = length (from_to i j (rev s))" by (unfold down_to_def, auto) also have "\<dots> = j - i" proof(rule length_from_to_in[OF le_ij]) from le_js show "j \<le> length (rev s)" by simp qed finally show ?thesis .qedlemma moment_head: assumes le_it: "Suc i \<le> length t" obtains e where "moment (Suc i) t = e#moment i t"proof - have "i \<le> Suc i" by simp from length_down_to_in [OF this le_it] have "length (down_to (Suc i) i t) = 1" by auto then obtain e where "down_to (Suc i) i t = [e]" apply (cases "(down_to (Suc i) i t)") by auto moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t" by (rule down_to_conc[symmetric], auto) ultimately have eq_me: "moment (Suc i) t = e#(moment i t)" by (auto simp:down_to_moment) from that [OF this] show ?thesis .qedlemma cnp_cnv_eq: fixes th s assumes "vt step s" and "th \<notin> threads s" shows "cntP s th = cntV s th"proof - from assms show ?thesis proof(induct) case (vt_cons s e) have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact have not_in: "th \<notin> threads (e # s)" by fact have "step s e" by fact thus ?case proof(cases) case (thread_create thread prio) assume eq_e: "e = Create thread prio" hence "thread \<in> threads (e#s)" by simp with not_in and eq_e have "th \<notin> threads s" by auto from ih [OF this] show ?thesis using eq_e by (auto simp:cntP_def cntV_def count_def) next case (thread_exit thread) assume eq_e: "e = Exit thread" and not_holding: "holdents s thread = {}" have vt_s: "vt step s" by fact from finite_holding[OF vt_s] have "finite (holdents s thread)" . with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto) moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def) moreover note cnp_cnv_cncs[OF vt_s, of thread] ultimately have eq_thread: "cntP s thread = cntV s thread" by auto show ?thesis proof(cases "th = thread") case True with eq_thread eq_e show ?thesis by (auto simp:cntP_def cntV_def count_def) next case False with not_in and eq_e have "th \<notin> threads s" by simp from ih[OF this] and eq_e show ?thesis by (auto simp:cntP_def cntV_def count_def) qed next case (thread_P thread cs) assume eq_e: "e = P thread cs" have "thread \<in> runing s" by fact with not_in eq_e have neq_th: "thread \<noteq> th" by (auto simp:runing_def readys_def) from not_in eq_e have "th \<notin> threads s" by simp from ih[OF this] and neq_th and eq_e show ?thesis by (auto simp:cntP_def cntV_def count_def) next case (thread_V thread cs) assume eq_e: "e = V thread cs" have "thread \<in> runing s" by fact with not_in eq_e have neq_th: "thread \<noteq> th" by (auto simp:runing_def readys_def) from not_in eq_e have "th \<notin> threads s" by simp from ih[OF this] and neq_th and eq_e show ?thesis by (auto simp:cntP_def cntV_def count_def) next case (thread_set thread prio) assume eq_e: "e = Set thread prio" and "thread \<in> runing s" hence "thread \<in> threads (e#s)" by (simp add:runing_def readys_def) with not_in and eq_e have "th \<notin> threads s" by auto from ih [OF this] show ?thesis using eq_e by (auto simp:cntP_def cntV_def count_def) qed next case vt_nil show ?case by (auto simp:cntP_def cntV_def count_def) qedqedlemma eq_depend: "depend (wq s) = depend s"by (unfold cs_depend_def s_depend_def, auto)lemma count_eq_dependents: assumes vt: "vt step s" and eq_pv: "cntP s th = cntV s th" shows "dependents (wq s) th = {}"proof - from cnp_cnv_cncs[OF vt] and eq_pv have "cntCS s th = 0" by (auto split:if_splits) moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}" proof - from finite_holding[OF vt, of th] show ?thesis by (simp add:holdents_def) qed ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}" by (unfold cntCS_def holdents_def cs_dependents_def, auto) show ?thesis proof(unfold cs_dependents_def) { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto hence "False" proof(cases) assume "(Th th', Th th) \<in> depend (wq s)" thus "False" by (auto simp:cs_depend_def) next fix c assume "(c, Th th) \<in> depend (wq s)" with h and eq_depend show "False" by (cases c, auto simp:cs_depend_def) qed } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto qedqedlemma dependents_threads: fixes s th assumes vt: "vt step s" shows "dependents (wq s) th \<subseteq> threads s"proof { fix th th' assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}" have "Th th \<in> Domain (depend s)" proof - from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def) with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp thus ?thesis using eq_depend by simp qed from dm_depend_threads[OF vt this] have "th \<in> threads s" . } note hh = this fix th1 assume "th1 \<in> dependents (wq s) th" hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}" by (unfold cs_dependents_def, simp) from hh [OF this] show "th1 \<in> threads s" .qedlemma finite_threads: assumes vt: "vt step s" shows "finite (threads s)"proof - from vt show ?thesis proof(induct) case (vt_cons s e) assume vt: "vt step s" and step: "step s e" and ih: "finite (threads s)" from step show ?case proof(cases) case (thread_create thread prio) assume eq_e: "e = Create thread prio" with ih show ?thesis by (unfold eq_e, auto) next case (thread_exit thread) assume eq_e: "e = Exit thread" with ih show ?thesis by (unfold eq_e, auto) next case (thread_P thread cs) assume eq_e: "e = P thread cs" with ih show ?thesis by (unfold eq_e, auto) next case (thread_V thread cs) assume eq_e: "e = V thread cs" with ih show ?thesis by (unfold eq_e, auto) next case (thread_set thread prio) from vt_cons thread_set show ?thesis by simp qed next case vt_nil show ?case by (auto) qedqedlemma Max_f_mono: assumes seq: "A \<subseteq> B" and np: "A \<noteq> {}" and fnt: "finite B" shows "Max (f ` A) \<le> Max (f ` B)"proof(rule Max_mono) from seq show "f ` A \<subseteq> f ` B" by autonext from np show "f ` A \<noteq> {}" by autonext from fnt and seq show "finite (f ` B)" by autoqedlemma cp_le: assumes vt: "vt step s" and th_in: "th \<in> threads s" shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def) show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+})) \<le> Max ((\<lambda>th. preced th s) ` threads s)" (is "Max (?f ` ?A) \<le> Max (?f ` ?B)") proof(rule Max_f_mono) show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp next from finite_threads [OF vt] show "finite (threads s)" . next from th_in show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s" apply (auto simp:Domain_def) apply (rule_tac dm_depend_threads[OF vt]) apply (unfold trancl_domain [of "depend s", symmetric]) by (unfold cs_depend_def s_depend_def, auto simp:Domain_def) qedqedlemma le_cp: assumes vt: "vt step s" shows "preced th s \<le> cp s th"proof(unfold cp_eq_cpreced preced_def cpreced_def, simp) show "Prc (original_priority th s) (birthtime th s) \<le> Max (insert (Prc (original_priority th s) (birthtime th s)) ((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))" (is "?l \<le> Max (insert ?l ?A)") proof(cases "?A = {}") case False have "finite ?A" (is "finite (?f ` ?B)") proof - have "finite ?B" proof- have "finite {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}" proof - let ?F = "\<lambda> (x, y). the_th x" have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)" apply (auto simp:image_def) by (rule_tac x = "(Th x, Th th)" in bexI, auto) moreover have "finite \<dots>" proof - from finite_depend[OF vt] have "finite (depend s)" . hence "finite ((depend (wq s))\<^sup>+)" apply (unfold finite_trancl) by (auto simp: s_depend_def cs_depend_def wq_def) thus ?thesis by auto qed ultimately show ?thesis by (auto intro:finite_subset) qed thus ?thesis by (simp add:cs_dependents_def) qed thus ?thesis by simp qed from Max_insert [OF this False, of ?l] show ?thesis by auto next case True thus ?thesis by auto qedqedlemma max_cp_eq: assumes vt: "vt step s" shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)" (is "?l = ?r")proof(cases "threads s = {}") case True thus ?thesis by autonext case False have "?l \<in> ((cp s) ` threads s)" proof(rule Max_in) from finite_threads[OF vt] show "finite (cp s ` threads s)" by auto next from False show "cp s ` threads s \<noteq> {}" by auto qed then obtain th where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto have "\<dots> \<le> ?r" by (rule cp_le[OF vt th_in]) moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th") proof - have "?r \<in> (?f ` ?A)" proof(rule Max_in) from finite_threads[OF vt] show " finite ((\<lambda>th. preced th s) ` threads s)" by auto next from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto qed then obtain th' where th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto from le_cp [OF vt, of th'] eq_r have "?r \<le> cp s th'" by auto moreover have "\<dots> \<le> cp s th" proof(fold eq_l) show " cp s th' \<le> Max (cp s ` threads s)" proof(rule Max_ge) from th_in' show "cp s th' \<in> cp s ` threads s" by auto next from finite_threads[OF vt] show "finite (cp s ` threads s)" by auto qed qed ultimately show ?thesis by auto qed ultimately show ?thesis using eq_l by autoqedlemma max_cp_readys_threads_pre: assumes vt: "vt step s" and np: "threads s \<noteq> {}" shows "Max (cp s ` readys s) = Max (cp s ` threads s)"proof(unfold max_cp_eq[OF vt]) show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)" proof - let ?p = "Max ((\<lambda>th. preced th s) ` threads s)" let ?f = "(\<lambda>th. preced th s)" have "?p \<in> ((\<lambda>th. preced th s) ` threads s)" proof(rule Max_in) from finite_threads[OF vt] show "finite (?f ` threads s)" by simp next from np show "?f ` threads s \<noteq> {}" by simp qed then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s" by (auto simp:Image_def) from th_chain_to_ready [OF vt tm_in] have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" . thus ?thesis proof assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ " then obtain th' where th'_in: "th' \<in> readys s" and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto have "cp s th' = ?f tm" proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI) from dependents_threads[OF vt] finite_threads[OF vt] show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))" by (auto intro:finite_subset) next fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')" from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" . moreover have "p \<le> \<dots>" proof(rule Max_ge) from finite_threads[OF vt] show "finite ((\<lambda>th. preced th s) ` threads s)" by simp next from p_in and th'_in and dependents_threads[OF vt, of th'] show "p \<in> (\<lambda>th. preced th s) ` threads s" by (auto simp:readys_def) qed ultimately show "p \<le> preced tm s" by auto next show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')" proof - from tm_chain have "tm \<in> dependents (wq s) th'" by (unfold cs_dependents_def s_depend_def cs_depend_def, auto) thus ?thesis by auto qed qed with tm_max have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp show ?thesis proof (fold h, rule Max_eqI) fix q assume "q \<in> cp s ` readys s" then obtain th1 where th1_in: "th1 \<in> readys s" and eq_q: "q = cp s th1" by auto show "q \<le> cp s th'" apply (unfold h eq_q) apply (unfold cp_eq_cpreced cpreced_def) apply (rule Max_mono) proof - from dependents_threads [OF vt, of th1] th1_in show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq> (\<lambda>th. preced th s) ` threads s" by (auto simp:readys_def) next show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp next from finite_threads[OF vt] show " finite ((\<lambda>th. preced th s) ` threads s)" by simp qed next from finite_threads[OF vt] show "finite (cp s ` readys s)" by (auto simp:readys_def) next from th'_in show "cp s th' \<in> cp s ` readys s" by simp qed next assume tm_ready: "tm \<in> readys s" show ?thesis proof(fold tm_max) have cp_eq_p: "cp s tm = preced tm s" proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI) fix y assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)" show "y \<le> preced tm s" proof - { fix y' assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)" have "y' \<le> preced tm s" proof(unfold tm_max, rule Max_ge) from hy' dependents_threads[OF vt, of tm] show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto next from finite_threads[OF vt] show "finite ((\<lambda>th. preced th s) ` threads s)" by simp qed } with hy show ?thesis by auto qed next from dependents_threads[OF vt, of tm] finite_threads[OF vt] show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))" by (auto intro:finite_subset) next show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)" by simp qed moreover have "Max (cp s ` readys s) = cp s tm" proof(rule Max_eqI) from tm_ready show "cp s tm \<in> cp s ` readys s" by simp next from finite_threads[OF vt] show "finite (cp s ` readys s)" by (auto simp:readys_def) next fix y assume "y \<in> cp s ` readys s" then obtain th1 where th1_readys: "th1 \<in> readys s" and h: "y = cp s th1" by auto show "y \<le> cp s tm" apply(unfold cp_eq_p h) apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono) proof - from finite_threads[OF vt] show "finite ((\<lambda>th. preced th s) ` threads s)" by simp next show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp next from dependents_threads[OF vt, of th1] th1_readys show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq> (\<lambda>th. preced th s) ` threads s" by (auto simp:readys_def) qed qed ultimately show " Max (cp s ` readys s) = preced tm s" by simp qed qed qedqedlemma max_cp_readys_threads: assumes vt: "vt step s" shows "Max (cp s ` readys s) = Max (cp s ` threads s)"proof(cases "threads s = {}") case True thus ?thesis by (auto simp:readys_def)next case False show ?thesis by (rule max_cp_readys_threads_pre[OF vt False])qedlemma readys_threads: shows "readys s \<subseteq> threads s"proof fix th assume "th \<in> readys s" thus "th \<in> threads s" by (unfold readys_def, auto)qedlemma eq_holding: "holding (wq s) th cs = holding s th cs" apply (unfold s_holding_def cs_holding_def, simp) donelemma f_image_eq: assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a" shows "f ` A = g ` A"proof show "f ` A \<subseteq> g ` A" by(rule image_subsetI, auto intro:h)next show "g ` A \<subseteq> f ` A" by(rule image_subsetI, auto intro:h[symmetric])qedend