diff -r 2c56b20032a7 -r 0679a84b11ad prio/PrioG.thy --- a/prio/PrioG.thy Mon Dec 03 08:16:58 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,2864 +0,0 @@ -theory PrioG -imports PrioGDef -begin - -lemma runing_ready: - shows "runing s \ readys s" - unfolding runing_def readys_def - by auto - -lemma readys_threads: - shows "readys s \ threads s" - unfolding readys_def - by auto - -lemma wq_v_neq: - "cs \ cs' \ 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 s \ 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) \ (depend s)\<^sup>+" - and h2: "thread \ set (wq_fun (schs s) cs)" - and h3: "thread \ runing s" - show "False" - proof - - from h3 have "\ cs. thread \ set (wq_fun (schs s) cs) \ - thread = hd ((wq_fun (schs s) cs))" - by (simp add:runing_def readys_def s_waiting_def wq_def) - from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" . - with h2 - have "(Cs cs, Th thread) \ (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 \ set q = set list)" - proof(rule someI2) - from dst show "distinct list \ set list = set list" by auto - next - fix q assume "distinct q \ set q = set list" - thus "distinct q" by auto - qed - qed -qed - -lemma step_back_vt: "vt (e#s) \ vt s" - by(ind_cases "vt (e#s)", simp) - -lemma step_back_step: "vt (e#s) \ step s e" - by(ind_cases "vt (e#s)", simp) - -lemma block_pre: - fixes thread cs s - assumes vt_e: "vt (e#s)" - and s_ni: "thread \ set (wq s cs)" - and s_i: "thread \ 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 \ set (wq_fun (schs s) cs)" - and h2: "q # qs = wq_fun (schs s) cs" - and h3: "thread \ set (SOME q. distinct q \ set q = set qs)" - and vt: "vt (V th cs # s)" - from h1 and h2[symmetric] have "thread \ set (q # qs)" by simp - moreover have "thread \ set qs" - proof - - have "set (SOME q. distinct q \ 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 \ set qs = set qs" by auto - next - fix x assume "distinct x \ 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 - qed -qed - -lemma p_pre: "\vt ((P thread cs)#s)\ \ - thread \ runing s \ (Cs cs, Th thread) \ (depend s)^+" -apply (ind_cases "vt ((P thread cs)#s)") -apply (ind_cases "step s (P thread cs)") -by auto - -lemma abs1: - fixes e es - assumes ein: "e \ set es" - and neq: "hd es \ hd (es @ [x])" - shows "False" -proof - - from ein have "es \ []" by auto - then obtain e ess where "es = e # ess" by (cases es, auto) - with neq show ?thesis by auto -qed - -lemma q_head: "Q (hd es) \ hd es = hd [th\es . Q th]" - by (cases es, auto) - -inductive_cases evt_cons: "vt (a#s)" - -lemma abs2: - assumes vt: "vt (e#s)" - and inq: "thread \ set (wq s cs)" - and nh: "thread = hd (wq s cs)" - and qt: "thread \ hd (wq (e#s) cs)" - and inq': "thread \ 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 (V th cs # s)" - and th_in: "thread \ set (SOME q. distinct q \ set q = set qs)" - and eq_wq: "wq_fun (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 \ set qs" - proof - - have "set (SOME q. distinct q \ 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 \ set qs = set qs" by auto - next - fix x assume "distinct x \ 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 - qed -qed - -lemma vt_moment: "\ t. \vt s\ \ vt (moment t s)" -proof(induct s, simp) - fix a s t - assume h: "\t.\vt s\ \ vt (moment t s)" - and vt_a: "vt (a # s)" - show "vt (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 - hence le_t1: "t \ length s" by simp - from vt_a have "vt s" - by (erule_tac evt_cons, simp) - from h [OF this] have "vt (moment t s)" . - moreover have "moment t (a#s) = moment t s" - proof - - from moment_app [OF le_t1, of "[a]"] - show ?thesis by simp - qed - ultimately show ?thesis by auto - qed -qed - -(* Wrong: - lemma \thread \ set (wq_fun cs1 s); thread \ set (wq_fun cs2 s)\ \ cs1 = cs2" -*) - -lemma waiting_unique_pre: - fixes cs1 cs2 s thread - assumes vt: "vt s" - and h11: "thread \ set (wq s cs1)" - and h12: "thread \ hd (wq s cs1)" - assumes h21: "thread \ set (wq s cs2)" - and h22: "thread \ hd (wq s cs2)" - and neq12: "cs1 \ cs2" - shows "False" -proof - - let "?Q cs s" = "thread \ set (wq s cs) \ thread \ 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: "\ ?Q cs1 []" by (simp add:wq_def) - have nq2: "\ ?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: "\(thread \ set (wq (moment t1 s) cs1) \ - thread \ hd (wq (moment t1 s) cs1))" - and nn1: "(\i'>t1. thread \ set (wq (moment i' s) cs1) \ - thread \ 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: "\(thread \ set (wq (moment t2 s) cs2) \ - thread \ hd (wq (moment t2 s) cs2))" - and nn2: "(\i'>t2. thread \ set (wq (moment i' s) cs2) \ - thread \ 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 \ 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 \ set (wq (e#moment t2 s) cs2)" and - h2: "thread \ hd (wq (e#moment t2 s) cs2)" by auto - have vt_e: "vt (e#moment t2 s)" - proof - - from vt_moment [OF vt] - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - have ?thesis - proof(cases "thread \ 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 ((P thread cs2)# moment t2 s)" by simp - from p_pre [OF this] have "thread \ runing (moment t2 s)" by simp - with runing_ready have "thread \ readys (moment t2 s)" by auto - with nn1 [rule_format, OF lt12] - show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto) - qed - } moreover { - assume lt12: "t2 < t1" - let ?t3 = "Suc t1" - from lt1 have le_t3: "?t3 \ 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 \ set (wq (e#moment t1 s) cs1)" and - h2: "thread \ hd (wq (e#moment t1 s) cs1)" by auto - have vt_e: "vt (e#moment t1 s)" - proof - - from vt_moment [OF vt] - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - have ?thesis - proof(cases "thread \ 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 ((P thread cs1)# moment t1 s)" by simp - from p_pre [OF this] have "thread \ runing (moment t1 s)" by simp - with runing_ready have "thread \ readys (moment t1 s)" by auto - with nn2 [rule_format, OF lt12] - show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto) - qed - } moreover { - assume eqt12: "t1 = t2" - let ?t3 = "Suc t1" - from lt1 have le_t3: "?t3 \ 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 \ set (wq (e#moment t1 s) cs1)" and - h2: "thread \ hd (wq (e#moment t1 s) cs1)" by auto - have vt_e: "vt (e#moment t1 s)" - proof - - from vt_moment [OF vt] - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - have ?thesis - proof(cases "thread \ 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 \ set (wq (e#moment t2 s) cs2)" and - h2: "thread \ hd (wq (e#moment t2 s) cs2)" by auto - show ?thesis - proof(cases "thread \ 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 (e#moment t2 s)" by simp - from abs2 [OF this True eq_th h2 h1] - show ?thesis . - next - case False - have vt_e: "vt (e#moment t2 s)" - proof - - from vt_moment [OF vt] eqt12 - have "vt (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 - qed -qed - -lemma waiting_unique: - fixes s cs1 cs2 - assumes "vt s" - and "waiting s th cs1" - and "waiting s th cs2" - shows "cs1 = cs2" -using waiting_unique_pre assms -unfolding wq_def s_waiting_def -by auto - -(* not used *) -lemma held_unique: - fixes s::"state" - assumes "holding s th1 cs" - and "holding s th2 cs" - shows "th1 = th2" -using assms -unfolding s_holding_def -by auto - - -lemma birthtime_lt: "th \ threads s \ birthtime th s < length s" - apply (induct s, auto) - by (case_tac a, auto split:if_splits) - -lemma birthtime_unique: - "\birthtime th1 s = birthtime th2 s; th1 \ threads s; th2 \ threads s\ - \ 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 \ threads s" - and th_in2: " th2 \ threads s" - shows "th1 = th2" -proof - - from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def) - from birthtime_unique [OF this th_in1 th_in2] - show ?thesis . -qed - -lemma preced_linorder: - assumes neq_12: "th1 \ th2" - and th_in1: "th1 \ threads s" - and th_in2: " th2 \ threads s" - shows "preced th1 s < preced th2 s \ preced th1 s > preced th2 s" -proof - - from preced_unique [OF _ th_in1 th_in2] and neq_12 - have "preced th1 s \ preced th2 s" by auto - thus ?thesis by auto -qed - -lemma unique_minus: - fixes x y z r - assumes unique: "\ a b c. \(a, b) \ r; (a, c) \ r\ \ b = c" - and xy: "(x, y) \ r" - and xz: "(x, z) \ r^+" - and neq: "y \ z" - shows "(y, z) \ r^+" -proof - - from xz and neq show ?thesis - proof(induct) - case (base ya) - have "(x, ya) \ r" by fact - from unique [OF xy this] have "y = ya" . - with base show ?case by auto - next - case (step ya z) - show ?case - proof(cases "y = ya") - case True - from step True show ?thesis by simp - next - case False - from step False - show ?thesis by auto - qed - qed -qed - -lemma unique_base: - fixes r x y z - assumes unique: "\ a b c. \(a, b) \ r; (a, c) \ r\ \ b = c" - and xy: "(x, y) \ r" - and xz: "(x, z) \ r^+" - and neq_yz: "y \ z" - shows "(y, z) \ 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) \ r\<^sup>+" by auto - with step show ?thesis by auto - qed - qed -qed - -lemma unique_chain: - fixes r x y z - assumes unique: "\ a b c. \(a, b) \ r; (a, c) \ r\ \ b = c" - and xy: "(x, y) \ r^+" - and xz: "(x, z) \ r^+" - and neq_yz: "y \ z" - shows "(y, z) \ r^+ \ (z, y) \ r^+" -proof - - from xy xz neq_yz show ?thesis - proof(induct) - case (base y) - have h1: "(x, y) \ r" and h2: "(x, z) \ r\<^sup>+" and h3: "y \ 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) \ r\<^sup>+ \ (z, y) \ r\<^sup>+" by auto - thus ?thesis - proof - assume "(z, y) \ r\<^sup>+" - with step have "(z, za) \ r\<^sup>+" by auto - thus ?thesis by auto - next - assume h: "(y, z) \ r\<^sup>+" - from step have yza: "(y, za) \ r" by simp - from step have "za \ z" by simp - from unique_minus [OF _ yza h this] and unique - have "(za, z) \ r\<^sup>+" by auto - thus ?thesis by auto - qed - qed - qed -qed - -lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s" -apply (unfold s_depend_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s" -apply (unfold s_depend_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s" -apply (unfold s_depend_def s_waiting_def wq_def) -by (simp add:Let_def) - - - -lemma step_v_hold_inv[elim_format]: - "\c t. \vt (V th cs # s); - \ holding (wq s) t c; holding (wq (V th cs # s)) t c\ \ next_th s th cs t \ c = cs" -proof - - fix c t - assume vt: "vt (V th cs # s)" - and nhd: "\ holding (wq s) t c" - and hd: "holding (wq (V th cs # s)) t c" - show "next_th s th cs t \ 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 \ q = []) \ set (SOME q. distinct q \ q = [])" - moreover have "\ = set []" - proof(rule someI2) - show "distinct [] \ [] = []" by auto - next - fix x assume "distinct x \ x = []" - thus "set x = set []" by auto - qed - ultimately show False by auto - next - assume " hd (SOME q. distinct q \ q = []) \ set (SOME q. distinct q \ q = [])" - moreover have "\ = set []" - proof(rule someI2) - show "distinct [] \ [] = []" by auto - next - fix x assume "distinct x \ x = []" - thus "set x = set []" by auto - qed - ultimately show False by auto - qed - qed - with True show ?thesis by auto - qed -qed - -lemma step_v_wait_inv[elim_format]: - "\t c. \vt (V th cs # s); \ waiting (wq (V th cs # s)) t c; waiting (wq s) t c - \ - \ (next_th s th cs t \ cs = c)" -proof - - fix t c - assume vt: "vt (V th cs # s)" - and nw: "\ waiting (wq (V th cs # s)) t c" - and wt: "waiting (wq s) t c" - show "next_th s th cs t \ 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 \ set list" - and t_ni: "t \ set (SOME q. distinct q \ set q = set list)" - and eq_wq: "wq_fun (schs s) cs = a # list" - have " set (SOME q. distinct q \ 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 \ set list = set list" by auto - next - show "\x. distinct x \ set x = set list \ 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 \ set list" - and t_ni: "t \ set (SOME q. distinct q \ set q = set list)" - and eq_wq: "wq_fun (schs s) cs = a # list" - have " set (SOME q. distinct q \ 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 \ set list = set list" by auto - next - show "\x. distinct x \ set x = set list \ set x = set list" - by auto - qed - with t_ni and t_in show "t = hd (SOME q. distinct q \ set q = set list)" by auto - next - fix a list - assume eq_wq: "wq_fun (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 - qed -qed - -lemma step_v_not_wait[consumes 3]: - "\vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\ \ False" - by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def) - -lemma step_v_release: - "\vt (V th cs # s); holding (wq (V th cs # s)) th cs\ \ False" -proof - - assume vt: "vt (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]: - "wq_fun (schs s) cs = hd (SOME q. distinct q \ set q = set list) # list" - and hd_in: "hd (SOME q. distinct q \ set q = set list) - \ set (SOME q. distinct q \ set q = set list)" - have "set (SOME q. distinct q \ 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 \ set list = set list" by auto - next - show "\x. distinct x \ set x = set list \ set x = set list" - by auto - qed - moreover have "distinct (hd (SOME q. distinct q \ 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 - qed -qed - -lemma step_v_get_hold: - "\th'. \vt (V th cs # s); \ holding (wq (V th cs # s)) th' cs; next_th s th cs th'\ \ False" - apply (unfold cs_holding_def next_th_def wq_def, - auto simp:Let_def) -proof - - fix rest - assume vt: "vt (V th cs # s)" - and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest" - and nrest: "rest \ []" - and ni: "hd (SOME q. distinct q \ set q = set rest) - \ set (SOME q. distinct q \ set q = set rest)" - have "(SOME q. distinct q \ set q = set rest) \ []" - proof(rule someI2) - from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq - show "distinct rest \ set rest = set rest" by auto - next - fix x assume "distinct x \ set x = set rest" - hence "set x = set rest" by auto - with nrest - show "x \ []" by (case_tac x, auto) - qed - with ni show "False" by auto -qed - -lemma step_v_release_inv[elim_format]: -"\c t. \vt (V th cs # s); \ holding (wq (V th cs # s)) t c; holding (wq s) t c\ \ - c = cs \ 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 (V th cs # s)" and eq_wq: "wq_fun (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 (V th cs # s)" and eq_wq: "wq_fun (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 - -lemma step_v_waiting_mono: - "\t c. \vt (V th cs # s); waiting (wq (V th cs # s)) t c\ \ waiting (wq s) t c" -proof - - fix t c - let ?s' = "(V th cs # s)" - assume vt: "vt ?s'" - and wt: "waiting (wq ?s') t c" - show "waiting (wq s) t c" - proof(cases "c = cs") - case False - assume neq_cs: "c \ 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 \ set list" - and is_in: "t \ set (SOME q. distinct q \ set q = set list)" - and eq_wq: "wq_fun (schs s) cs = a # list" - have "set (SOME q. distinct q \ 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 \ set list = set list" by auto - next - fix x assume "distinct x \ 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: "wq_fun (schs s) cs = t # list" - hence "t \ set list" - apply (unfold wq_def, auto simp:Let_def cs_waiting_def) - proof - - assume " t \ set (SOME q. distinct q \ set q = set list)" - moreover have "\ = 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 \ set list = set list" by auto - next - fix x assume "distinct x \ set x = set list" - thus "set x = set list" by auto - qed - ultimately show "t \ 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 - qed -qed - -lemma step_depend_v: -fixes th::thread -assumes vt: - "vt (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'} \ - {(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) - done - -lemma step_depend_p: - "vt (P th cs#s) \ - depend (P th cs # s) = (if (wq s cs = []) then depend s \ {(Cs cs, Th th)} - else depend s \ {(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) - done - -lemma simple_A: - fixes A - assumes h: "\ x y. \x \ A; y \ A\ \ x = y" - shows "A = {} \ (\ a. A = {a})" -proof(cases "A = {}") - case True thus ?thesis by simp -next - case False then obtain a where "a \ A" by auto - with h have "A = {a}" by auto - thus ?thesis by simp -qed - -lemma depend_target_th: "(Th th, x) \ depend (s::state) \ \ cs. x = Cs cs" - by (unfold s_depend_def, auto) - -lemma acyclic_depend: - fixes s - assumes vt: "vt 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 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 (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'} \ - {(Cs cs, Th th') |th'. next_th s th cs th'}" - (is "?L = (?A - ?B - ?C) \ ?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 \ set (wq s cs)" and - eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by 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 \ 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) \ ?E\<^sup>*" - proof - assume "(Th ?th', Cs cs) \ ?E\<^sup>*" - hence " (Th ?th', Cs cs) \ ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl) - from tranclD [OF this] - obtain x where th'_e: "(Th ?th', x) \ ?E" by blast - hence th_d: "(Th ?th', x) \ ?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') \ ?A" by simp - hence wt_th': "waiting s ?th' cs'" - unfolding s_depend_def s_waiting_def cs_waiting_def wq_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 wq_def, auto) - proof - - assume hd_in: "hd (SOME q. distinct q \ set q = set rest) \ set rest" - and eq_wq: "wq_fun (schs s) cs = th # rest" - have "(SOME q. distinct q \ set q = set rest) \ []" - proof(rule someI2) - from wq_distinct[OF vt, of cs] and eq_wq - show "distinct rest \ set rest = set rest" unfolding wq_def by auto - next - fix x assume "distinct x \ set x = set rest" - with False show "x \ []" by auto - qed - hence "hd (SOME q. distinct q \ set q = set rest) \ - set (SOME q. distinct q \ set q = set rest)" by auto - moreover have "\ = set rest" - proof(rule someI2) - from wq_distinct[OF vt, of cs] and eq_wq - show "distinct rest \ set rest = set rest" unfolding wq_def by auto - next - show "\x. distinct x \ set x = set rest \ set x = set rest" by auto - qed - moreover note hd_in - ultimately show "hd (SOME q. distinct q \ set q = set rest) = th" by auto - next - assume hd_in: "hd (SOME q. distinct q \ set q = set rest) \ set rest" - and eq_wq: "wq s cs = hd (SOME q. distinct q \ set q = set rest) # rest" - have "(SOME q. distinct q \ set q = set rest) \ []" - proof(rule someI2) - from wq_distinct[OF vt, of cs] and eq_wq - show "distinct rest \ set rest = set rest" by auto - next - fix x assume "distinct x \ set x = set rest" - with False show "x \ []" by auto - qed - hence "hd (SOME q. distinct q \ set q = set rest) \ - set (SOME q. distinct q \ set q = set rest)" by auto - moreover have "\ = set rest" - proof(rule someI2) - from wq_distinct[OF vt, of cs] and eq_wq - show "distinct rest \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ 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) \ ?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 (P th cs#s)" by auto - from step_depend_p [OF this] P - have "depend (e # s) = - (if wq s cs = [] then depend s \ {(Cs cs, Th th)} else - depend s \ {(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 \ {(Cs cs, Th th)}" by simp - have "(Th th, Cs cs) \ (depend s)\<^sup>*" - proof - assume "(Th th, Cs cs) \ (depend s)\<^sup>*" - hence "(Th th, Cs cs) \ (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl) - from tranclD2 [OF this] - obtain x where "(x, Cs cs) \ 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 \ {(Th th, Cs cs)}" by simp - have "(Cs cs, Th th) \ (depend s)\<^sup>*" - proof - assume "(Cs cs, Th th) \ (depend s)\<^sup>*" - hence "(Cs cs, Th th) \ (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 " \(Cs cs, Th th) \ (depend s)\<^sup>+; step s (P th cs)\ \ False" - by (ind_cases "step s (P th cs)", simp) - qed - qed - with acyclic_insert ih eq_r show ?thesis by auto - qed - ultimately show ?thesis by simp - next - case (Set thread prio) - with ih - thm depend_set_unchanged - show ?thesis by (simp add:depend_set_unchanged) - qed - next - case vt_nil - show "acyclic (depend ([]::state))" - by (auto simp: s_depend_def cs_waiting_def - cs_holding_def wq_def acyclic_def) - qed -qed - -lemma finite_depend: - fixes s - assumes vt: "vt 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 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 (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'} \ - {(Cs cs, Th th') |th'. next_th s th cs th'} -" - (is "?L = (?A - ?B - ?C) \ ?D") by (simp add:V) - moreover from ih have ac: "finite (?A - ?B - ?C)" by simp - moreover have "finite ?D" - proof - - have "?D = {} \ (\ a. ?D = {a})" - by (unfold next_th_def, auto) - thus ?thesis - proof - assume h: "?D = {}" - show ?thesis by (unfold h, simp) - next - assume "\ 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 (P th cs#s)" by auto - from step_depend_p [OF this] P - have "depend (e # s) = - (if wq s cs = [] then depend s \ {(Cs cs, Th th)} else - depend s \ {(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 \ {(Cs cs, Th th)}" by simp - with True and ih show ?thesis by auto - next - case False - hence "?R = depend s \ {(Th th, Cs cs)}" by simp - with False and ih show ?thesis by auto - qed - ultimately show ?thesis by auto - next - case (Set thread prio) - with ih - show ?thesis by (simp add:depend_set_unchanged) - qed - next - case vt_nil - show "finite (depend ([]::state))" - by (auto simp: s_depend_def cs_waiting_def - cs_holding_def wq_def acyclic_def) - qed -qed - -text {* Several useful lemmas *} - -lemma wf_dep_converse: - fixes s - assumes vt: "vt s" - shows "wf ((depend s)^-1)" -proof(rule finite_acyclic_wf_converse) - from finite_depend [OF vt] - show "finite (depend s)" . -next - from acyclic_depend[OF vt] - show "acyclic (depend s)" . -qed - -lemma hd_np_in: "x \ set l \ hd l \ set l" -by (induct l, auto) - -lemma th_chasing: "(Th th, Cs cs) \ depend (s::state) \ \ th'. (Cs cs, Th th') \ 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 s" - and h: "th \ set (wq s cs)" - shows "th \ threads s" -proof - - from vt and h show ?thesis - proof(induct arbitrary: th cs) - case (vt_cons s e) - assume ih: "\th cs. th \ set (wq s cs) \ th \ threads s" - and stp: "step s e" - and vt: "vt s" - and h: "th \ 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) - done - 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 \ set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \ set ?l") - show "th \ threads (V th' cs' # s)" - proof(cases "cs = cs'") - case False - hence "?l = wq_fun (schs s) cs" by (simp add:Let_def) - with th_in have " th \ set (wq s cs)" - by (fold wq_def, simp) - from ih [OF this] show ?thesis by simp - next - case True - show ?thesis - proof(cases "wq_fun (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: "wq_fun (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 \ set (SOME q. distinct q \ set q = set rest)" - have "set (SOME q. distinct q \ 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 \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ set x = set rest" - by auto - qed - with eq_wq th_in have "th \ set (wq_fun (schs s) cs')" by auto - from ih[OF this[folded wq_def]] show "th \ threads s" . - next - assume th_in: "th \ set (wq_fun (schs s) cs)" - from ih[OF this[folded wq_def]] - show "th \ 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) - qed -qed - -lemma range_in: "\vt s; (Th th) \ Range (depend (s::state))\ \ th \ 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 s" - and neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and not_in: "th \ set rest" - shows "(th \ readys (V thread cs#s)) = (th \ readys s)" -proof - - from assms show ?thesis - apply (auto simp:readys_def) - apply(simp add:s_waiting_def[folded wq_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 add: s_waiting_def[folded wq_def] Let_def split: list.splits) - apply(auto simp add: wq_def) - apply (auto simp:s_waiting_def wq_def Let_def split:list.splits) - proof - - assume th_nin: "th \ set rest" - and th_in: "th \ set (SOME q. distinct q \ set q = set rest)" - and eq_wq: "wq_fun (schs s) cs = thread # rest" - have "set (SOME q. distinct q \ set q = set rest) = set rest" - proof(rule someI2) - from wq_distinct[OF vt, of cs, unfolded wq_def] and eq_wq[unfolded wq_def] - show "distinct rest \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ set x = set rest" by auto - qed - with th_nin th_in show False by auto - qed -qed - -lemma chain_building: - assumes vt: "vt s" - shows "node \ Domain (depend s) \ (\ th'. th' \ readys s \ (node, Th th') \ (depend s)^+)" -proof - - from wf_dep_converse [OF vt] - have h: "wf ((depend s)\)" . - show ?thesis - proof(induct rule:wf_induct [OF h]) - fix x - assume ih [rule_format]: - "\y. (y, x) \ (depend s)\ \ - y \ Domain (depend s) \ (\th'. th' \ readys s \ (y, Th th') \ (depend s)\<^sup>+)" - show "x \ Domain (depend s) \ (\th'. th' \ readys s \ (x, Th th') \ (depend s)\<^sup>+)" - proof - assume x_d: "x \ Domain (depend s)" - show "\th'. th' \ readys s \ (x, Th th') \ (depend s)\<^sup>+" - proof(cases x) - case (Th th) - from x_d Th obtain cs where x_in: "(Th th, Cs cs) \ depend s" by (auto simp:s_depend_def) - with Th have x_in_r: "(Cs cs, x) \ (depend s)^-1" by simp - from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \ depend s" by blast - hence "Cs cs \ Domain (depend s)" by auto - from ih [OF x_in_r this] obtain th' - where th'_ready: " th' \ readys s" and cs_in: "(Cs cs, Th th') \ (depend s)\<^sup>+" by auto - have "(x, Th th') \ (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) \ (depend s)^-1" by (auto simp:s_depend_def) - show ?thesis - proof(cases "th' \ 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' \ threads s" by auto - with False have "Th th' \ Domain (depend s)" - by (auto simp:readys_def wq_def s_waiting_def s_depend_def cs_waiting_def Domain_def) - from ih [OF th'_d this] - obtain th'' where - th''_r: "th'' \ readys s" and - th''_in: "(Th th', Th th'') \ (depend s)\<^sup>+" by auto - from th'_d and th''_in - have "(x, Th th'') \ (depend s)\<^sup>+" by auto - with th''_r show ?thesis by auto - qed - qed - qed - qed -qed - -lemma th_chain_to_ready: - fixes s th - assumes vt: "vt s" - and th_in: "th \ threads s" - shows "th \ readys s \ (\ th'. th' \ readys s \ (Th th, Th th') \ (depend s)^+)" -proof(cases "th \ readys s") - case True - thus ?thesis by auto -next - case False - from False and th_in have "Th th \ Domain (depend s)" - by (auto simp:readys_def s_waiting_def s_depend_def wq_def cs_waiting_def Domain_def) - from chain_building [rule_format, OF vt this] - show ?thesis by auto -qed - -lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs" - by (unfold s_waiting_def cs_waiting_def wq_def, auto) - -lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs" - by (unfold s_holding_def wq_def cs_holding_def, simp) - -lemma holding_unique: "\holding (s::state) th1 cs; holding s th2 cs\ \ th1 = th2" - by (unfold s_holding_def cs_holding_def, auto) - -lemma unique_depend: "\vt s; (n, n1) \ depend s; (n, n2) \ depend s\ \ 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) \ r^+ \ \ c. (a, c) \ r" -by (induct rule:trancl_induct, auto) - -lemma dchain_unique: - assumes vt: "vt s" - and th1_d: "(n, Th th1) \ (depend s)^+" - and th1_r: "th1 \ readys s" - and th2_d: "(n, Th th2) \ (depend s)^+" - and th2_r: "th2 \ readys s" - shows "th1 = th2" -proof - - { assume neq: "th1 \ th2" - hence "Th th1 \ Th th2" by simp - from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt] - have "(Th th1, Th th2) \ (depend s)\<^sup>+ \ (Th th2, Th th1) \ (depend s)\<^sup>+" by auto - hence "False" - proof - assume "(Th th1, Th th2) \ (depend s)\<^sup>+" - from trancl_split [OF this] - obtain n where dd: "(Th th1, n) \ 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 \ readys s" - by (auto simp:readys_def s_depend_def wq_def s_waiting_def cs_waiting_def) - with th1_r show ?thesis by auto - next - assume "(Th th2, Th th1) \ (depend s)\<^sup>+" - from trancl_split [OF this] - obtain n where dd: "(Th th2, n) \ 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 \ readys s" - by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def) - with th2_r show ?thesis by auto - qed - } thus ?thesis by auto -qed - - -lemma step_holdents_p_add: - fixes th cs s - assumes vt: "vt (P th cs#s)" - and "wq s cs = []" - shows "holdents (P th cs#s) th = holdents s th \ {cs}" -proof - - from assms show ?thesis - unfolding holdents_test step_depend_p[OF vt] by (auto) -qed - -lemma step_holdents_p_eq: - fixes th cs s - assumes vt: "vt (P th cs#s)" - and "wq s cs \ []" - shows "holdents (P th cs#s) th = holdents s th" -proof - - from assms show ?thesis - unfolding holdents_test step_depend_p[OF vt] by auto -qed - - -lemma finite_holding: - fixes s th cs - assumes vt: "vt s" - shows "finite (holdents s th)" -proof - - let ?F = "\ (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) \ depend s} \ \" - proof - - { have h: "\ a A f. a \ A \ f a \ f ` A" by auto - fix x assume "(Cs x, Th th) \ depend s" - hence "?F (Cs x, Th th) \ ?F `(depend s)" by (rule h) - moreover have "?F (Cs x, Th th) = x" by simp - ultimately have "x \ (\(x, y). the_cs x) ` depend s" by simp - } thus ?thesis by auto - qed - ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset) -qed - -lemma cntCS_v_dec: - fixes s thread cs - assumes vtv: "vt (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 \ holdents s thread" - apply (cases, unfold holdents_test s_depend_def, simp) - by (unfold cs_holding_def s_holding_def wq_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_test, unfold step_depend_v[OF vtv], - auto simp:next_th_def) - proof - - fix rest - assume dst: "distinct (rest::thread list)" - and ne: "rest \ []" - and hd_ni: "hd (SOME q. distinct q \ set q = set rest) \ set rest" - moreover have "set (SOME q. distinct q \ set q = set rest) = set rest" - proof(rule someI2) - from dst show "distinct rest \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ set x = set rest" by auto - qed - ultimately have "hd (SOME q. distinct q \ set q = set rest) \ - set (SOME q. distinct q \ set q = set rest)" by simp - moreover have "(SOME q. distinct q \ set q = set rest) \ []" - proof(rule someI2) - from dst show "distinct rest \ set rest = set rest" by auto - next - fix x assume " distinct x \ set x = set rest" with ne - show "x \ []" by auto - qed - ultimately - show "(Cs cs, Th (hd (SOME q. distinct q \ set q = set rest))) \ depend s" - by auto - next - fix rest - assume dst: "distinct (rest::thread list)" - and ne: "rest \ []" - and hd_ni: "hd (SOME q. distinct q \ set q = set rest) \ set rest" - moreover have "set (SOME q. distinct q \ set q = set rest) = set rest" - proof(rule someI2) - from dst show "distinct rest \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ set x = set rest" by auto - qed - ultimately have "hd (SOME q. distinct q \ set q = set rest) \ - set (SOME q. distinct q \ set q = set rest)" by simp - moreover have "(SOME q. distinct q \ set q = set rest) \ []" - proof(rule someI2) - from dst show "distinct rest \ set rest = set rest" by auto - next - fix x assume " distinct x \ set x = set rest" with ne - show "x \ []" 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 \ = - 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 \ (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 s" - shows "cntP s th = cntV s th + (if (th \ readys s \ th \ 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 s" - and ih: "\th. cntP s th = cntV s th + - (if (th \ readys s \ th \ 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 \ threads s" - show ?thesis - proof - - { fix cs - assume "thread \ set (wq s cs)" - from wq_threads [OF vt this] have "thread \ threads s" . - with not_in have "False" by simp - } with eq_e have eq_readys: "readys (e#s) = readys s \ {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_test - by (simp add:depend_create_unchanged eq_e) - { assume "th \ thread" - with eq_readys eq_e - have "(th \ readys (e # s) \ th \ threads (e # s)) = - (th \ readys (s) \ th \ 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 \ 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 \ 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_test - by (simp add:depend_exit_unchanged eq_e) - { assume "th \ thread" - with eq_e - have "(th \ readys (e # s) \ th \ threads (e # s)) = - (th \ readys (s) \ th \ threads (s))" - apply (simp add:threads.simps readys_def) - apply (subst s_waiting_def) - apply (simp add:Let_def) - apply (subst s_waiting_def, simp) - done - 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 \ 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 \ runing s" - and no_dep: "(Cs cs, Th thread) \ (depend s)\<^sup>+" - from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto - show ?thesis - proof - - { have hh: "\ A B C. (B = C) \ (A \ B) = (A \ C)" by blast - assume neq_th: "th \ thread" - with eq_e - have eq_readys: "(th \ readys (e#s)) = (th \ 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 "wq_fun (schs s) cs \ []", auto) - apply (erule_tac x = cs in allE, auto) - by (case_tac "(wq_fun (schs s) cs)", auto) - moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th" - apply (simp add:cntCS_def holdents_test) - 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 \ 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 \ (Cs csa, Th thread) \ depend s} = - Suc (card {cs. (Cs cs, Th thread) \ depend s})" (is "card ?L = Suc (card ?R)") - proof - - have "?L = insert cs ?R" by auto - moreover have "card \ = Suc (card (?R - {cs}))" - proof(rule card_insert) - from finite_holding [OF vt, of thread] - show " finite {cs. (Cs cs, Th thread) \ depend s}" - by (unfold holdents_test, simp) - qed - moreover have "?R - {cs} = ?R" - proof - - have "cs \ ?R" - proof - assume "cs \ {cs. (Cs cs, Th thread) \ 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_test) - by (unfold step_depend_p [OF vtp], auto simp:True) - qed - moreover from is_runing have "th \ 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 \ readys (e#s)" - proof - assume "th \ readys (e#s)" - hence "\cs. \ waiting (e # s) th cs" by (simp add:readys_def) - from this[rule_format, of cs] have " \ waiting (e # s) th cs" . - hence "th \ set (wq (e#s) cs) \ th = hd (wq (e#s) cs)" - by (simp add:s_waiting_def wq_def) - moreover from eq_wq have "th \ 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 \ 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_test 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 \ 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 assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto - assume eq_e: "e = V thread cs" - and is_runing: "thread \ 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: wq_def s_holding_def) - have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e) - have eq_set: "set (SOME q. distinct q \ 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 \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ 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 \ readys s" - by (unfold runing_def, simp) - moreover have "thread \ readys (e # s)" - proof - - from is_runing - have "thread \ threads (e#s)" - by (unfold eq_e, auto simp:runing_def readys_def) - moreover have "\ cs1. \ waiting (e#s) thread cs1" - proof - fix cs1 - { assume eq_cs: "cs1 = cs" - have "\ waiting (e # s) thread cs1" - proof - - from eq_wq - have "thread \ set (wq (e#s) cs1)" - apply(unfold eq_e wq_def eq_cs s_holding_def) - apply (auto simp:Let_def) - proof - - assume "thread \ set (SOME q. distinct q \ set q = set rest)" - with eq_set have "thread \ 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:wq_def s_waiting_def) - qed - } moreover { - assume neq_cs: "cs1 \ cs" - have "\ 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 "\ waiting s thread cs1" - proof - - from runing_ready and is_runing - have "thread \ readys s" by auto - thus ?thesis by (simp add:readys_def) - qed - ultimately show ?thesis - by (auto simp:wq_def s_waiting_def eq_e) - qed - } ultimately show "\ 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 \ 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 \ set rest") - case False - have "(th \ readys (e # s)) = (th \ 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_test step_depend_v[OF vtv], auto) - proof - - have "{csa. (Cs csa, Th th) \ depend s \ csa = cs \ next_th s thread cs th} = - {cs. (Cs cs, Th th) \ depend s}" - proof - - from False eq_wq - have " next_th s thread cs th \ (Cs cs, Th th) \ depend s" - apply (unfold next_th_def, auto) - proof - - assume ne: "rest \ []" - and ni: "hd (SOME q. distinct q \ set q = set rest) \ set rest" - and eq_wq: "wq s cs = thread # rest" - from eq_set ni have "hd (SOME q. distinct q \ set q = set rest) \ - set (SOME q. distinct q \ set q = set rest) - " by simp - moreover have "(SOME q. distinct q \ set q = set rest) \ []" - proof(rule someI2) - from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq - show "distinct rest \ set rest = set rest" by auto - next - fix x assume "distinct x \ set x = set rest" - with ne show "x \ []" by auto - qed - ultimately show - "(Cs cs, Th (hd (SOME q. distinct q \ set q = set rest))) \ depend s" - by auto - qed - thus ?thesis by auto - qed - thus "card {csa. (Cs csa, Th th) \ depend s \ csa = cs \ next_th s thread cs th} = - card {cs. (Cs cs, Th th) \ 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 \ set rest" - show ?thesis - proof(cases "next_th s thread cs th") - case False - with eq_wq and th_in have - neq_hd: "th \ hd (SOME q. distinct q \ set q = set rest)" (is "th \ hd ?rest") - by (auto simp:next_th_def) - have "(th \ readys (e # s)) = (th \ readys s)" - proof - - from eq_wq and th_in - have "\ th \ 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 simp add: wq_def) - moreover - from eq_wq and th_in and neq_hd - have "\ (th \ 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_test 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 \ set q = set rest)" - let ?t = "hd ?rest" - from True eq_wq th_in neq_th - have "th \ readys (e # s)" - apply (auto simp:eq_e readys_def s_waiting_def wq_def - Let_def next_th_def) - proof - - assume eq_wq: "wq_fun (schs s) cs = thread # rest" - and t_in: "?t \ set rest" - show "?t \ threads s" - proof(rule wq_threads[OF step_back_vt[OF vtv]]) - from eq_wq and t_in - show "?t \ set (wq s cs)" by (auto simp:wq_def) - qed - next - fix csa - assume eq_wq: "wq_fun (schs s) cs = thread # rest" - and t_in: "?t \ set rest" - and neq_cs: "csa \ cs" - and t_in': "?t \ set (wq_fun (schs s) csa)" - show "?t = hd (wq_fun (schs s) csa)" - proof - - { assume neq_hd': "?t \ hd (wq_fun (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 \ 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 \ 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 simp add: wq_def) - qed - moreover have "th \ threads s" - proof - - from th_in eq_wq - have "th \ 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_test eq_e step_depend_v[OF vtv], auto) - proof - - show "card {csa. (Cs csa, Th th) \ depend s \ csa = cs} = - Suc (card {cs. (Cs cs, Th th) \ 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 "\ = Suc (card ?B)" - proof(rule card_insert_disjoint) - have "?B \ ((\ (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) \ depend s}" by (auto intro:finite_subset) - next - show "cs \ {cs. (Cs cs, Th th) \ depend s}" - proof - assume "cs \ {cs. (Cs cs, Th th) \ depend s}" - hence "(Cs cs, Th th) \ 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 \ 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_test - 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 \ thread" - with eq_readys eq_e - have "(th \ readys (e # s) \ th \ threads (e # s)) = - (th \ readys (s) \ th \ 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 \ 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_test s_depend_def wq_def cs_holding_def) - qed -qed - -lemma not_thread_cncs: - fixes th s - assumes vt: "vt s" - and not_in: "th \ 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 s" - and ih: "\th. th \ threads s \ cntCS s th = 0" - and stp: "step s e" - and not_in: "th \ 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 \ threads s" - have "cntCS (e # s) th = cntCS s th" - apply (unfold eq_e cntCS_def holdents_test) - by (simp add:depend_create_unchanged) - moreover have "th \ 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_test) - 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 \ 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 \ runing s" - from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto - have neq_th: "th \ thread" - proof - - from not_in eq_e have "th \ threads s" by simp - moreover from is_runing have "thread \ 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_test 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 \ 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 \ runing s" - and hold: "holding s thread cs" - have neq_th: "th \ thread" - proof - - from not_in eq_e have "th \ threads s" by simp - moreover from is_runing have "thread \ threads s" - by (simp add:runing_def readys_def) - ultimately show ?thesis by auto - qed - from assms thread_V vt stp ih have vtv: "vt (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: wq_def s_holding_def) - from not_in eq_e eq_wq - have "\ next_th s thread cs th" - apply (auto simp:next_th_def) - proof - - assume ne: "rest \ []" - and ni: "hd (SOME q. distinct q \ set q = set rest) \ threads s" (is "?t \ threads s") - have "?t \ set rest" - proof(rule someI2) - from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq - show "distinct rest \ set rest = set rest" by auto - next - fix x assume "distinct x \ set x = set rest" with ne - show "hd x \ set rest" by (cases x, auto) - qed - with eq_wq have "?t \ 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_test step_depend_v[OF vtv], auto) - moreover have "cntCS s th = 0" - proof(rule ih) - from not_in eq_e show "th \ 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 \ runing s" - from not_in and eq_e have "th \ threads s" by auto - from ih [OF this] and eq_e - show ?thesis - apply (unfold eq_e cntCS_def holdents_test) - by (simp add:depend_set_unchanged) - qed - next - case vt_nil - show ?case - by (unfold cntCS_def, - auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def) - qed -qed - -lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs" - by (auto simp:s_waiting_def cs_waiting_def wq_def) - -lemma dm_depend_threads: - fixes th s - assumes vt: "vt s" - and in_dom: "(Th th) \ Domain (depend s)" - shows "th \ threads s" -proof - - from in_dom obtain n where "(Th th, n) \ 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) \ depend s" by simp - hence "th \ set (wq s cs)" - by (unfold s_depend_def, auto simp:cs_waiting_def) - from wq_threads [OF vt this] show ?thesis . -qed - -lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th" -unfolding cp_def wq_def -apply(induct s rule: schs.induct) -apply(simp add: Let_def cpreced_initial) -apply(simp add: Let_def) -apply(simp add: Let_def) -apply(simp add: Let_def) -apply(subst (2) schs.simps) -apply(simp add: Let_def) -apply(subst (2) schs.simps) -apply(simp add: Let_def) -done - - -lemma runing_unique: - fixes th1 th2 s - assumes vt: "vt s" - and runing_1: "th1 \ runing s" - and runing_2: "th2 \ 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 ((\th. preced th s) ` ({th1} \ dependents (wq s) th1)) = - Max ((\th. preced th s) ` ({th2} \ dependents (wq s) th2))" - (is "Max (?f ` ?A) = Max (?f ` ?B)") - by (unfold cp_eq_cpreced cpreced_def) - obtain th1' where th1_in: "th1' \ ?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) \ (depend (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th1) \ (depend (wq s))\<^sup>+} \ ?F ` ((depend (wq s))\<^sup>+)" - apply (auto simp:image_def) - by (rule_tac x = "(Th x, Th th1)" in bexI, auto) - moreover have "finite \" - 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) \ {}" - proof - - have "?A \ {}" by simp - thus ?thesis by simp - qed - from Max_in [OF h1 h2] - have "Max (?f ` ?A) \ (?f ` ?A)" . - thus ?thesis by (auto intro:that) - qed - obtain th2' where th2_in: "th2' \ ?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) \ (depend (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th2) \ (depend (wq s))\<^sup>+} \ ?F ` ((depend (wq s))\<^sup>+)" - apply (auto simp:image_def) - by (rule_tac x = "(Th x, Th th2)" in bexI, auto) - moreover have "finite \" - 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) \ {}" - proof - - have "?B \ {}" by simp - thus ?thesis by simp - qed - from Max_in [OF h1 h2] - have "Max (?f ` ?B) \ (?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 \ (th1' \ dependents (wq s) th1)" by simp - thus "th1' \ threads s" - proof - assume "th1' \ dependents (wq s) th1" - hence "(Th th1') \ Domain ((depend s)^+)" - apply (unfold cs_dependents_def cs_depend_def s_depend_def) - by (auto simp:Domain_def) - hence "(Th th1') \ 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 \ (th2' \ dependents (wq s) th2)" by simp - thus "th2' \ threads s" - proof - assume "th2' \ dependents (wq s) th2" - hence "(Th th2') \ Domain ((depend s)^+)" - apply (unfold cs_dependents_def cs_depend_def s_depend_def) - by (auto simp:Domain_def) - hence "(Th th2') \ 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 \ th1' \ dependents (wq s) th1" by simp - thus ?thesis - proof - assume eq_th': "th1' = th1" - from th2_in have "th2' = th2 \ th2' \ dependents (wq s) th2" by simp - thus ?thesis - proof - assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp - next - assume "th2' \ dependents (wq s) th2" - with eq_th12 eq_th' have "th1 \ dependents (wq s) th2" by simp - hence "(Th th1, Th th2) \ (depend s)^+" - by (unfold cs_dependents_def s_depend_def cs_depend_def, simp) - hence "Th th1 \ Domain ((depend s)^+)" - apply (unfold cs_dependents_def cs_depend_def s_depend_def) - by (auto simp:Domain_def) - hence "Th th1 \ Domain (depend s)" by (simp add:trancl_domain) - then obtain n where d: "(Th th1, n) \ 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') \ 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' \ dependents (wq s) th1" - from th2_in have "th2' = th2 \ th2' \ dependents (wq s) th2" by simp - thus ?thesis - proof - assume "th2' = th2" - with th1'_in eq_th12 have "th2 \ dependents (wq s) th1" by simp - hence "(Th th2, Th th1) \ (depend s)^+" - by (unfold cs_dependents_def s_depend_def cs_depend_def, simp) - hence "Th th2 \ Domain ((depend s)^+)" - apply (unfold cs_dependents_def cs_depend_def s_depend_def) - by (auto simp:Domain_def) - hence "Th th2 \ Domain (depend s)" by (simp add:trancl_domain) - then obtain n where d: "(Th th2, n) \ 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') \ 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' \ dependents (wq s) th2" - with eq_th12 have "th1' \ dependents (wq s) th2" by simp - hence h1: "(Th th1', Th th2) \ (depend s)^+" - by (unfold cs_dependents_def s_depend_def cs_depend_def, simp) - from th1'_in have h2: "(Th th1', Th th1) \ (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 \ readys s" by (simp add:runing_def) - from runing_2 show "th2 \ readys s" by (simp add:runing_def) - qed - qed - qed -qed - -lemma create_pre: - assumes stp: "step s e" - and not_in: "th \ threads s" - and is_in: "th \ 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) - qed -qed - -lemma length_down_to_in: - assumes le_ij: "i \ j" - and le_js: "j \ 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 "\ = j - i" - proof(rule length_from_to_in[OF le_ij]) - from le_js show "j \ length (rev s)" by simp - qed - finally show ?thesis . -qed - - -lemma moment_head: - assumes le_it: "Suc i \ length t" - obtains e where "moment (Suc i) t = e#moment i t" -proof - - have "i \ Suc i" by simp - from length_down_to_in [OF this le_it] - have "length (down_to (Suc i) i t) = 1" by auto - then obtain e where "down_to (Suc i) i t = [e]" - apply (cases "(down_to (Suc i) i t)") by auto - moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t" - by (rule down_to_conc[symmetric], auto) - ultimately have eq_me: "moment (Suc i) t = e#(moment i t)" - by (auto simp:down_to_moment) - from that [OF this] show ?thesis . -qed - -lemma cnp_cnv_eq: - fixes th s - assumes "vt s" - and "th \ threads s" - shows "cntP s th = cntV s th" -proof - - from assms show ?thesis - proof(induct) - case (vt_cons s e) - have ih: "th \ threads s \ cntP s th = cntV s th" by fact - have not_in: "th \ 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 \ threads (e#s)" by simp - with not_in and eq_e have "th \ 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 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 \ 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 \ 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 \ runing s" by fact - with not_in eq_e have neq_th: "thread \ th" - by (auto simp:runing_def readys_def) - from not_in eq_e have "th \ 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 \ runing s" by fact - with not_in eq_e have neq_th: "thread \ th" - by (auto simp:runing_def readys_def) - from not_in eq_e have "th \ 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 \ runing s" - hence "thread \ threads (e#s)" - by (simp add:runing_def readys_def) - with not_in and eq_e have "th \ threads s" by auto - from ih [OF this] show ?thesis using eq_e - by (auto simp:cntP_def cntV_def count_def) - qed - next - case vt_nil - show ?case by (auto simp:cntP_def cntV_def count_def) - qed -qed - -lemma eq_depend: - "depend (wq s) = depend s" -by (unfold cs_depend_def s_depend_def, auto) - -lemma count_eq_dependents: - assumes vt: "vt 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) \ depend s}" - proof - - from finite_holding[OF vt, of th] show ?thesis - by (simp add:holdents_test) - qed - ultimately have h: "{cs. (Cs cs, Th th) \ depend s} = {}" - by (unfold cntCS_def holdents_test cs_dependents_def, auto) - show ?thesis - proof(unfold cs_dependents_def) - { assume "{th'. (Th th', Th th) \ (depend (wq s))\<^sup>+} \ {}" - then obtain th' where "(Th th', Th th) \ (depend (wq s))\<^sup>+" by auto - hence "False" - proof(cases) - assume "(Th th', Th th) \ depend (wq s)" - thus "False" by (auto simp:cs_depend_def) - next - fix c - assume "(c, Th th) \ 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) \ (depend (wq s))\<^sup>+} = {}" by auto - qed -qed - -lemma dependents_threads: - fixes s th - assumes vt: "vt s" - shows "dependents (wq s) th \ threads s" -proof - { fix th th' - assume h: "th \ {th'a. (Th th'a, Th th') \ (depend (wq s))\<^sup>+}" - have "Th th \ Domain (depend s)" - proof - - from h obtain th' where "(Th th, Th th') \ (depend (wq s))\<^sup>+" by auto - hence "(Th th) \ Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def) - with trancl_domain have "(Th th) \ Domain (depend (wq s))" by simp - thus ?thesis using eq_depend by simp - qed - from dm_depend_threads[OF vt this] - have "th \ threads s" . - } note hh = this - fix th1 - assume "th1 \ dependents (wq s) th" - hence "th1 \ {th'a. (Th th'a, Th th) \ (depend (wq s))\<^sup>+}" - by (unfold cs_dependents_def, simp) - from hh [OF this] show "th1 \ threads s" . -qed - -lemma finite_threads: - assumes vt: "vt s" - shows "finite (threads s)" -using vt -by (induct) (auto elim: step.cases) - -lemma Max_f_mono: - assumes seq: "A \ B" - and np: "A \ {}" - and fnt: "finite B" - shows "Max (f ` A) \ Max (f ` B)" -proof(rule Max_mono) - from seq show "f ` A \ f ` B" by auto -next - from np show "f ` A \ {}" by auto -next - from fnt and seq show "finite (f ` B)" by auto -qed - -lemma cp_le: - assumes vt: "vt s" - and th_in: "th \ threads s" - shows "cp s th \ Max ((\ th. (preced th s)) ` threads s)" -proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def) - show "Max ((\th. preced th s) ` ({th} \ {th'. (Th th', Th th) \ (depend (wq s))\<^sup>+})) - \ Max ((\th. preced th s) ` threads s)" - (is "Max (?f ` ?A) \ Max (?f ` ?B)") - proof(rule Max_f_mono) - show "{th} \ {th'. (Th th', Th th) \ (depend (wq s))\<^sup>+} \ {}" by simp - next - from finite_threads [OF vt] - show "finite (threads s)" . - next - from th_in - show "{th} \ {th'. (Th th', Th th) \ (depend (wq s))\<^sup>+} \ threads s" - apply (auto simp:Domain_def) - apply (rule_tac dm_depend_threads[OF vt]) - apply (unfold trancl_domain [of "depend s", symmetric]) - by (unfold cs_depend_def s_depend_def, auto simp:Domain_def) - qed -qed - -lemma le_cp: - assumes vt: "vt s" - shows "preced th s \ cp s th" -proof(unfold cp_eq_cpreced preced_def cpreced_def, simp) - show "Prc (original_priority th s) (birthtime th s) - \ Max (insert (Prc (original_priority th s) (birthtime th s)) - ((\th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))" - (is "?l \ 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) \ (depend (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th) \ (depend (wq s))\<^sup>+} \ ?F ` ((depend (wq s))\<^sup>+)" - apply (auto simp:image_def) - by (rule_tac x = "(Th x, Th th)" in bexI, auto) - moreover have "finite \" - proof - - from finite_depend[OF vt] have "finite (depend s)" . - hence "finite ((depend (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_depend_def cs_depend_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependents_def) - qed - thus ?thesis by simp - qed - from Max_insert [OF this False, of ?l] show ?thesis by auto - next - case True - thus ?thesis by auto - qed -qed - -lemma max_cp_eq: - assumes vt: "vt s" - shows "Max ((cp s) ` threads s) = Max ((\ th. (preced th s)) ` threads s)" - (is "?l = ?r") -proof(cases "threads s = {}") - case True - thus ?thesis by auto -next - case False - have "?l \ ((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 \ {}" by auto - qed - then obtain th - where th_in: "th \ threads s" and eq_l: "?l = cp s th" by auto - have "\ \ ?r" by (rule cp_le[OF vt th_in]) - moreover have "?r \ cp s th" (is "Max (?f ` ?A) \ cp s th") - proof - - have "?r \ (?f ` ?A)" - proof(rule Max_in) - from finite_threads[OF vt] - show " finite ((\th. preced th s) ` threads s)" by auto - next - from False show " (\th. preced th s) ` threads s \ {}" by auto - qed - then obtain th' where - th_in': "th' \ ?A " and eq_r: "?r = ?f th'" by auto - from le_cp [OF vt, of th'] eq_r - have "?r \ cp s th'" by auto - moreover have "\ \ cp s th" - proof(fold eq_l) - show " cp s th' \ Max (cp s ` threads s)" - proof(rule Max_ge) - from th_in' show "cp s th' \ cp s ` threads s" - by auto - next - from finite_threads[OF vt] - show "finite (cp s ` threads s)" by auto - qed - qed - ultimately show ?thesis by auto - qed - ultimately show ?thesis using eq_l by auto -qed - -lemma max_cp_readys_threads_pre: - assumes vt: "vt s" - and np: "threads s \ {}" - 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 ((\th. preced th s) ` threads s)" - proof - - let ?p = "Max ((\th. preced th s) ` threads s)" - let ?f = "(\th. preced th s)" - have "?p \ ((\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 \ {}" by simp - qed - then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \ threads s" - by (auto simp:Image_def) - from th_chain_to_ready [OF vt tm_in] - have "tm \ readys s \ (\th'. th' \ readys s \ (Th tm, Th th') \ (depend s)\<^sup>+)" . - thus ?thesis - proof - assume "\th'. th' \ readys s \ (Th tm, Th th') \ (depend s)\<^sup>+ " - then obtain th' where th'_in: "th' \ readys s" - and tm_chain:"(Th tm, Th th') \ (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 ((\th. preced th s) ` ({th'} \ dependents (wq s) th'))" - by (auto intro:finite_subset) - next - fix p assume p_in: "p \ (\th. preced th s) ` ({th'} \ dependents (wq s) th')" - from tm_max have " preced tm s = Max ((\th. preced th s) ` threads s)" . - moreover have "p \ \" - proof(rule Max_ge) - from finite_threads[OF vt] - show "finite ((\th. preced th s) ` threads s)" by simp - next - from p_in and th'_in and dependents_threads[OF vt, of th'] - show "p \ (\th. preced th s) ` threads s" - by (auto simp:readys_def) - qed - ultimately show "p \ preced tm s" by auto - next - show "preced tm s \ (\th. preced th s) ` ({th'} \ dependents (wq s) th')" - proof - - from tm_chain - have "tm \ 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 ((\th. preced th s) ` threads s)" by simp - show ?thesis - proof (fold h, rule Max_eqI) - fix q - assume "q \ cp s ` readys s" - then obtain th1 where th1_in: "th1 \ readys s" - and eq_q: "q = cp s th1" by auto - show "q \ 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 "(\th. preced th s) ` ({th1} \ dependents (wq s) th1) \ - (\th. preced th s) ` threads s" - by (auto simp:readys_def) - next - show "(\th. preced th s) ` ({th1} \ dependents (wq s) th1) \ {}" by simp - next - from finite_threads[OF vt] - show " finite ((\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' \ cp s ` readys s" by simp - qed - next - assume tm_ready: "tm \ 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 \ (\th. preced th s) ` ({tm} \ dependents (wq s) tm)" - show "y \ preced tm s" - proof - - { fix y' - assume hy' : "y' \ ((\th. preced th s) ` dependents (wq s) tm)" - have "y' \ preced tm s" - proof(unfold tm_max, rule Max_ge) - from hy' dependents_threads[OF vt, of tm] - show "y' \ (\th. preced th s) ` threads s" by auto - next - from finite_threads[OF vt] - show "finite ((\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 ((\th. preced th s) ` ({tm} \ dependents (wq s) tm))" - by (auto intro:finite_subset) - next - show "preced tm s \ (\th. preced th s) ` ({tm} \ 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 \ 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 \ cp s ` readys s" - then obtain th1 where th1_readys: "th1 \ readys s" - and h: "y = cp s th1" by auto - show "y \ 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 ((\th. preced th s) ` threads s)" by simp - next - show "(\th. preced th s) ` ({th1} \ dependents (wq s) th1) \ {}" - by simp - next - from dependents_threads[OF vt, of th1] th1_readys - show "(\th. preced th s) ` ({th1} \ dependents (wq s) th1) - \ (\th. preced th s) ` threads s" - by (auto simp:readys_def) - qed - qed - ultimately show " Max (cp s ` readys s) = preced tm s" by simp - qed - qed - qed -qed - -lemma max_cp_readys_threads: - assumes vt: "vt s" - shows "Max (cp s ` readys s) = Max (cp s ` threads s)" -proof(cases "threads s = {}") - case True - thus ?thesis - by (auto simp:readys_def) -next - case False - show ?thesis by (rule max_cp_readys_threads_pre[OF vt False]) -qed - - -lemma eq_holding: "holding (wq s) th cs = holding s th cs" - apply (unfold s_holding_def cs_holding_def wq_def, simp) - done - -lemma f_image_eq: - assumes h: "\ a. a \ A \ f a = g a" - shows "f ` A = g ` A" -proof - show "f ` A \ g ` A" - by(rule image_subsetI, auto intro:h) -next - show "g ` A \ f ` A" - by (rule image_subsetI, auto intro:h[symmetric]) -qed - - -definition detached :: "state \ thread \ bool" - where "detached s th \ (\(\ cs. holding s th cs)) \ (\(\cs. waiting s th cs))" - - -lemma detached_test: - shows "detached s th = (Th th \ Field (depend s))" -apply(simp add: detached_def Field_def) -apply(simp add: s_depend_def) -apply(simp add: s_holding_abv s_waiting_abv) -apply(simp add: Domain_iff Range_iff) -apply(simp add: wq_def) -apply(auto) -done - -lemma detached_intro: - fixes s th - assumes vt: "vt s" - and eq_pv: "cntP s th = cntV s th" - shows "detached s th" -proof - - from cnp_cnv_cncs[OF vt] - have eq_cnt: "cntP s th = - cntV s th + (if th \ readys s \ th \ threads s then cntCS s th else cntCS s th + 1)" . - hence cncs_zero: "cntCS s th = 0" - by (auto simp:eq_pv split:if_splits) - with eq_cnt - have "th \ readys s \ th \ threads s" by (auto simp:eq_pv) - thus ?thesis - proof - assume "th \ threads s" - with range_in[OF vt] dm_depend_threads[OF vt] - show ?thesis - by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff) - next - assume "th \ readys s" - moreover have "Th th \ Range (depend s)" - proof - - from card_0_eq [OF finite_holding [OF vt]] and cncs_zero - have "holdents s th = {}" - by (simp add:cntCS_def) - thus ?thesis - apply(auto simp:holdents_test) - apply(case_tac a) - apply(auto simp:holdents_test s_depend_def) - done - qed - ultimately show ?thesis - by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def readys_def) - qed -qed - -lemma detached_elim: - fixes s th - assumes vt: "vt s" - and dtc: "detached s th" - shows "cntP s th = cntV s th" -proof - - from cnp_cnv_cncs[OF vt] - have eq_pv: " cntP s th = - cntV s th + (if th \ readys s \ th \ threads s then cntCS s th else cntCS s th + 1)" . - have cncs_z: "cntCS s th = 0" - proof - - from dtc have "holdents s th = {}" - unfolding detached_def holdents_test s_depend_def - by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff) - thus ?thesis by (auto simp:cntCS_def) - qed - show ?thesis - proof(cases "th \ threads s") - case True - with dtc - have "th \ readys s" - by (unfold readys_def detached_def Field_def Domain_def Range_def, - auto simp:eq_waiting s_depend_def) - with cncs_z and eq_pv show ?thesis by simp - next - case False - with cncs_z and eq_pv show ?thesis by simp - qed -qed - -lemma detached_eq: - fixes s th - assumes vt: "vt s" - shows "(detached s th) = (cntP s th = cntV s th)" - by (insert vt, auto intro:detached_intro detached_elim) - -end \ No newline at end of file