diff -r 2c56b20032a7 -r 0679a84b11ad prio/Attic/Prio.thy --- a/prio/Attic/Prio.thy Mon Dec 03 08:16:58 2012 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,2813 +0,0 @@ -theory Prio -imports Precedence_ord Moment Lsp Happen_within -begin - -type_synonym thread = nat -type_synonym priority = nat -type_synonym cs = nat - -datatype event = - Create thread priority | - Exit thread | - P thread cs | - V thread cs | - Set thread priority - -datatype node = - Th "thread" | - Cs "cs" - -type_synonym state = "event list" - -fun threads :: "state \ thread set" -where - "threads [] = {}" | - "threads (Create thread prio#s) = {thread} \ threads s" | - "threads (Exit thread # s) = (threads s) - {thread}" | - "threads (e#s) = threads s" - -fun original_priority :: "thread \ state \ nat" -where - "original_priority thread [] = 0" | - "original_priority thread (Create thread' prio#s) = - (if thread' = thread then prio else original_priority thread s)" | - "original_priority thread (Set thread' prio#s) = - (if thread' = thread then prio else original_priority thread s)" | - "original_priority thread (e#s) = original_priority thread s" - -fun birthtime :: "thread \ state \ nat" -where - "birthtime thread [] = 0" | - "birthtime thread ((Create thread' prio)#s) = (if (thread = thread') then length s - else birthtime thread s)" | - "birthtime thread ((Set thread' prio)#s) = (if (thread = thread') then length s - else birthtime thread s)" | - "birthtime thread (e#s) = birthtime thread s" - -definition preced :: "thread \ state \ precedence" - where "preced thread s = Prc (original_priority thread s) (birthtime thread s)" - -consts holding :: "'b \ thread \ cs \ bool" - waiting :: "'b \ thread \ cs \ bool" - depend :: "'b \ (node \ node) set" - dependents :: "'b \ thread \ thread set" - -defs (overloaded) cs_holding_def: "holding wq thread cs == (thread \ set (wq cs) \ thread = hd (wq cs))" - cs_waiting_def: "waiting wq thread cs == (thread \ set (wq cs) \ thread \ hd (wq cs))" - cs_depend_def: "depend (wq::cs \ thread list) == {(Th t, Cs c) | t c. waiting wq t c} \ - {(Cs c, Th t) | c t. holding wq t c}" - cs_dependents_def: "dependents (wq::cs \ thread list) th == {th' . (Th th', Th th) \ (depend wq)^+}" - -record schedule_state = - waiting_queue :: "cs \ thread list" - cur_preced :: "thread \ precedence" - - -definition cpreced :: "state \ (cs \ thread list) \ thread \ precedence" -where "cpreced s wq = (\ th. Max ((\ th. preced th s) ` ({th} \ dependents wq th)))" - -fun schs :: "state \ schedule_state" -where - "schs [] = \waiting_queue = \ cs. [], - cur_preced = cpreced [] (\ cs. [])\" | - "schs (e#s) = (let ps = schs s in - let pwq = waiting_queue ps in - let pcp = cur_preced ps in - let nwq = case e of - P thread cs \ pwq(cs:=(pwq cs @ [thread])) | - V thread cs \ let nq = case (pwq cs) of - [] \ [] | - (th#pq) \ case (lsp pcp pq) of - (l, [], r) \ [] - | (l, m#ms, r) \ m#(l@ms@r) - in pwq(cs:=nq) | - _ \ pwq - in let ncp = cpreced (e#s) nwq in - \waiting_queue = nwq, cur_preced = ncp\ - )" - -definition wq :: "state \ cs \ thread list" -where "wq s == waiting_queue (schs s)" - -definition cp :: "state \ thread \ precedence" -where "cp s = cur_preced (schs s)" - -defs (overloaded) s_holding_def: "holding (s::state) thread cs == (thread \ set (wq s cs) \ thread = hd (wq s cs))" - s_waiting_def: "waiting (s::state) thread cs == (thread \ set (wq s cs) \ thread \ hd (wq s cs))" - s_depend_def: "depend (s::state) == {(Th t, Cs c) | t c. waiting (wq s) t c} \ - {(Cs c, Th t) | c t. holding (wq s) t c}" - s_dependents_def: "dependents (s::state) th == {th' . (Th th', Th th) \ (depend (wq s))^+}" - -definition readys :: "state \ thread set" -where - "readys s = - {thread . thread \ threads s \ (\ cs. \ waiting s thread cs)}" - -definition runing :: "state \ thread set" -where "runing s = {th . th \ readys s \ cp s th = Max ((cp s) ` (readys s))}" - -definition holdents :: "state \ thread \ cs set" - where "holdents s th = {cs . (Cs cs, Th th) \ depend s}" - -inductive step :: "state \ event \ bool" -where - thread_create: "\prio \ max_prio; thread \ threads s\ \ step s (Create thread prio)" | - thread_exit: "\thread \ runing s; holdents s thread = {}\ \ step s (Exit thread)" | - thread_P: "\thread \ runing s; (Cs cs, Th thread) \ (depend s)^+\ \ step s (P thread cs)" | - thread_V: "\thread \ runing s; holding s thread cs\ \ step s (V thread cs)" | - thread_set: "\thread \ runing s\ \ step s (Set thread prio)" - -inductive vt :: "(state \ event \ bool) \ state \ bool" - for cs -where - vt_nil[intro]: "vt cs []" | - vt_cons[intro]: "\vt cs s; cs s e\ \ vt cs (e#s)" - -lemma runing_ready: "runing s \ readys s" - by (auto simp only:runing_def readys_def) - -lemma wq_v_eq_nil: - fixes s cs thread rest - assumes eq_wq: "wq s cs = thread # rest" - and eq_lsp: "lsp (cp s) rest = (l, [], r)" - shows "wq (V thread cs#s) cs = []" -proof - - from prems show ?thesis - by (auto simp:wq_def Let_def cp_def split:list.splits) -qed - -lemma wq_v_eq: - fixes s cs thread rest - assumes eq_wq: "wq s cs = thread # rest" - and eq_lsp: "lsp (cp s) rest = (l, [th'], r)" - shows "wq (V thread cs#s) cs = th'#l@r" -proof - - from prems show ?thesis - by (auto simp:wq_def Let_def cp_def split:list.splits) -qed - -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 step 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 (waiting_queue (schs s) cs)" - and h3: "thread \ runing s" - show "False" - proof - - from h3 have "\ cs. thread \ set (waiting_queue (schs s) cs) \ - 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) \ (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 h1: "thread \ runing s" - and h2: "holding s thread cs" - and h3: "waiting_queue (schs s) cs = a # list" - and h4: "a \ set list" - and h5: "distinct list" - thus "distinct - ((\(l, a, r). case a of [] \ [] | m # ms \ m # l @ ms @ r) - (lsp (cur_preced (schs s)) list))" - apply (cases "(lsp (cur_preced (schs s)) list)", simp) - apply (case_tac b, simp) - by (drule_tac lsp_set_eq, simp) - qed -qed - -lemma block_pre: - fixes thread cs s - assumes s_ni: "thread \ set (wq s cs)" - and s_i: "thread \ set (wq (e#s) cs)" - shows "e = P thread cs" -proof - - have ee: "\ x y. \x = y\ \ set x = set y" - by auto - from s_ni s_i show ?thesis - proof (cases e, auto split:if_splits simp add:Let_def wq_def) - fix uu uub uuc uud uue - assume h: "(uuc, thread # uu, uub) = lsp (cur_preced (schs s)) uud" - and h1 [symmetric]: "uue # uud = waiting_queue (schs s) cs" - and h2: "thread \ set (waiting_queue (schs s) cs)" - from lsp_set [OF h] have "set (uuc @ (thread # uu) @ uub) = set uud" . - hence "thread \ set uud" by auto - with h1 have "thread \ set (waiting_queue (schs s) cs)" by auto - with h2 show False by auto - next - fix uu uua uub uuc uud uue - assume h1: "thread \ set (waiting_queue (schs s) cs)" - and h2: "uue # uud = waiting_queue (schs s) cs" - and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud" - and h4: "thread \ set uuc" - from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" . - with h4 have "thread \ set uud" by auto - with h2 have "thread \ set (waiting_queue (schs s) cs)" - apply (drule_tac ee) by auto - with h1 show "False" by fast - next - fix uu uua uub uuc uud uue - assume h1: "thread \ set (waiting_queue (schs s) cs)" - and h2: "uue # uud = waiting_queue (schs s) cs" - and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud" - and h4: "thread \ set uu" - from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" . - with h4 have "thread \ set uud" by auto - with h2 have "thread \ set (waiting_queue (schs s) cs)" - apply (drule_tac ee) by auto - with h1 show "False" by fast - next - fix uu uua uub uuc uud uue - assume h1: "thread \ set (waiting_queue (schs s) cs)" - and h2: "uue # uud = waiting_queue (schs s) cs" - and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud" - and h4: "thread \ set uub" - from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" . - with h4 have "thread \ set uud" by auto - with h2 have "thread \ set (waiting_queue (schs s) cs)" - apply (drule_tac ee) by auto - with h1 show "False" by fast - qed -qed - -lemma p_pre: "\vt step ((P thread cs)#s)\ \ - thread \ runing s \ (Cs cs, Th thread) \ (depend s)^+" -apply (ind_cases "vt step ((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 cs (a#s)" - -lemma abs2: - assumes vt: "vt step (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 - - have ee: "\ uuc thread uu uub s list. (uuc, thread # uu, uub) = lsp (cur_preced (schs s)) list \ - lsp (cur_preced (schs s)) list = (uuc, thread # uu, uub) - " by simp - from prems show "False" - apply (cases e) - apply ((simp split:if_splits add:Let_def wq_def)[1])+ - apply (insert abs1, fast)[1] - apply ((simp split:if_splits add:Let_def)[1])+ - apply (simp split:if_splits list.splits add:Let_def wq_def) - apply (auto dest!:ee) - apply (drule_tac lsp_set_eq, simp) - apply (subgoal_tac "distinct (waiting_queue (schs s) cs)", simp, fold wq_def) - apply (rule_tac wq_distinct, auto) - apply (erule_tac evt_cons, auto) - apply (drule_tac lsp_set_eq, simp) - apply (subgoal_tac "distinct (wq s cs)", simp) - apply (rule_tac wq_distinct, auto) - apply (erule_tac evt_cons, auto) - apply (drule_tac lsp_set_eq, simp) - apply (subgoal_tac "distinct (wq s cs)", simp) - apply (rule_tac wq_distinct, auto) - apply (erule_tac evt_cons, auto) - apply (auto simp:wq_def Let_def split:if_splits prod.splits) - done -qed - -lemma vt_moment: "\ t. \vt cs s; t \ length s\ \ vt cs (moment t s)" -proof(induct s, simp) - fix a s t - assume h: "\t.\vt cs s; t \ length s\ \ vt cs (moment t s)" - and vt_a: "vt cs (a # s)" - and le_t: "t \ 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 \ 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 - qed -qed - -(* Wrong: - lemma \thread \ set (waiting_queue cs1 s); thread \ set (waiting_queue cs2 s)\ \ cs1 = cs2" -*) - -lemma waiting_unique_pre: - fixes cs1 cs2 s thread - assumes vt: "vt step 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 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 \ 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 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 \ 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 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 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 \ 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 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 \ 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 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 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 \ 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 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 step (e#moment t2 s)" by simp - from abs2 [OF this True eq_th h2 h1] - show ?thesis . - next - case False - from block_pre [OF 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: - 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) -qed - -lemma holded_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 auto -qed - -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) - -definition head_of :: "('a \ 'b::linorder) \ 'a set \ 'a set" - where "head_of f A = {a . a \ A \ f a = Max (f ` A)}" - -definition wq_head :: "state \ cs \ thread set" - where "wq_head s cs = head_of (cp s) (set (wq s cs))" - -lemma f_nil_simp: "\f cs = []\ \ f(cs:=[]) = f" -proof - fix x - assume h:"f cs = []" - show "(f(cs := [])) x = f x" - proof(cases "cs = x") - case True - with h show ?thesis by simp - next - case False - with h show ?thesis by simp - qed -qed - -lemma step_back_vt: "vt ccs (e#s) \ vt ccs s" - by(ind_cases "vt ccs (e#s)", simp) - -lemma step_back_step: "vt ccs (e#s) \ ccs s e" - by(ind_cases "vt ccs (e#s)", simp) - -lemma holding_nil: - "\wq s cs = []; holding (wq s) th cs\ \ False" - by (unfold cs_holding_def, auto) - -lemma waiting_kept_1: " - \vt step (V th cs#s); wq s cs = a # list; waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c; - lsp (cp s) list = (aa, ab # lista, ca)\ - \ waiting (wq s) t c" - apply (drule_tac step_back_vt, drule_tac wq_distinct[of _ cs]) - apply(drule_tac lsp_set_eq) - by (unfold cs_waiting_def, auto split:if_splits) - -lemma waiting_kept_2: - "\a list t c aa ca. - \wq s cs = a # list; waiting ((wq s)(cs := [])) t c; lsp (cp s) list = (aa, [], ca)\ - \ waiting (wq s) t c" - apply(drule_tac lsp_set_eq) - by (unfold cs_waiting_def, auto split:if_splits) - - -lemma holding_nil_simp: "\holding ((wq s)(cs := [])) t c\ \ holding (wq s) t c" - by(unfold cs_holding_def, auto) - -lemma step_wq_elim: "\vt step (V th cs#s); wq s cs = a # list; a \ th\ \ False" - apply(drule_tac step_back_step) - apply(ind_cases "step s (V th cs)") - by(unfold s_holding_def, auto) - -lemma holding_cs_neq_simp: "c \ cs \ holding ((wq s)(cs := u)) t c = holding (wq s) t c" - by (unfold cs_holding_def, auto) - -lemma holding_th_neq_elim: - "\a list c t aa ca ab lista. - \\ holding (wq s) t c; holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; - ab \ t\ - \ False" - by (unfold cs_holding_def, auto split:if_splits) - -lemma holding_nil_abs: - "\ holding ((wq s)(cs := [])) th cs" - by (unfold cs_holding_def, auto split:if_splits) - -lemma holding_abs: "\holding ((wq s)(cs := ab # aa @ lista @ c)) th cs; ab \ th\ - \ False" - by (unfold cs_holding_def, auto split:if_splits) - -lemma waiting_abs: "\ waiting ((wq s)(cs := t # l @ r)) t cs" - by (unfold cs_waiting_def, auto split:if_splits) - -lemma waiting_abs_1: - "\\ waiting ((wq s)(cs := [])) t c; waiting (wq s) t c; c \ cs\ - \ False" - by (unfold cs_waiting_def, auto split:if_splits) - -lemma waiting_abs_2: " - \\ waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c; waiting (wq s) t c; - c \ cs\ - \ False" - by (unfold cs_waiting_def, auto split:if_splits) - -lemma waiting_abs_3: - "\wq s cs = a # list; \ waiting ((wq s)(cs := [])) t c; - waiting (wq s) t c; lsp (cp s) list = (aa, [], ca)\ - \ False" - apply(drule_tac lsp_mid_nil, simp) - by(unfold cs_waiting_def, auto split:if_splits) - -lemma waiting_simp: "c \ cs \ waiting ((wq s)(cs:=z)) t c = waiting (wq s) t c" - by(unfold cs_waiting_def, auto split:if_splits) - -lemma holding_cs_eq: - "\\ holding ((wq s)(cs := [])) t c; holding (wq s) t c\ \ c = cs" - by(unfold cs_holding_def, auto split:if_splits) - -lemma holding_cs_eq_1: - "\\ holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c\ - \ c = cs" - by(unfold cs_holding_def, auto split:if_splits) - -lemma holding_th_eq: - "\vt step (V th cs#s); wq s cs = a # list; \ holding ((wq s)(cs := [])) t c; holding (wq s) t c; - lsp (cp s) list = (aa, [], ca)\ - \ t = th" - apply(drule_tac lsp_mid_nil, simp) - apply(unfold cs_holding_def, auto split:if_splits) - apply(drule_tac step_back_step) - apply(ind_cases "step s (V th cs)") - by (unfold s_holding_def, auto split:if_splits) - -lemma holding_th_eq_1: - "\vt step (V th cs#s); - wq s cs = a # list; \ holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c; - lsp (cp s) list = (aa, ab # lista, ca)\ - \ t = th" - apply(drule_tac step_back_step) - apply(ind_cases "step s (V th cs)") - apply(unfold s_holding_def cs_holding_def) - by (auto split:if_splits) - -lemma holding_th_eq_2: "\holding ((wq s)(cs := ac # x)) th cs\ - \ ac = th" - by (unfold cs_holding_def, auto) - -lemma holding_th_eq_3: " - \\ holding (wq s) t c; - holding ((wq s)(cs := ac # x)) t c\ - \ ac = t" - by (unfold cs_holding_def, auto) - -lemma holding_wq_eq: "holding ((wq s)(cs := th' # l @ r)) th' cs" - by (unfold cs_holding_def, auto) - -lemma waiting_th_eq: " - \waiting (wq s) t c; wq s cs = a # list; - lsp (cp s) list = (aa, ac # lista, ba); \ waiting ((wq s)(cs := ac # aa @ lista @ ba)) t c\ - \ ac = t" - apply(drule_tac lsp_set_eq, simp) - by (unfold cs_waiting_def, auto split:if_splits) - -lemma step_depend_v: - "vt step (V th cs#s) \ - depend (V th cs # s) = - depend s - {(Cs cs, Th th)} - - {(Th th', Cs cs) |th'. \rest. wq s cs = th # rest \ (\l r. lsp (cp s) rest = (l, [th'], r))} \ - {(Cs cs, Th th') |th'. \rest. wq s cs = th # rest \ (\l r. lsp (cp s) rest = (l, [th'], r))}" - apply (unfold s_depend_def wq_def, - auto split:list.splits simp:Let_def f_nil_simp holding_wq_eq, fold wq_def cp_def) - apply (auto split:list.splits prod.splits - simp:Let_def f_nil_simp holding_nil_simp holding_cs_neq_simp holding_nil_abs - waiting_abs waiting_simp holding_wq_eq - elim:holding_nil waiting_kept_1 waiting_kept_2 step_wq_elim holding_th_neq_elim - holding_abs waiting_abs_1 waiting_abs_3 holding_cs_eq holding_cs_eq_1 - holding_th_eq holding_th_eq_1 holding_th_eq_2 holding_th_eq_3 waiting_th_eq - dest:lsp_mid_length) - done - -lemma step_depend_p: - "vt step (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(unfold s_depend_def wq_def) - apply (auto split:list.splits prod.splits simp:Let_def cs_waiting_def cs_holding_def) - apply(case_tac "c = cs", auto) - apply(fold wq_def) - apply(drule_tac step_back_step) - by (ind_cases " step s (P (hd (wq s cs)) cs)", - auto simp:s_depend_def wq_def cs_holding_def) - -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 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'. \rest. wq s cs = th # rest \ (\l r. lsp (cp s) rest = (l, [th'], r))} \ - {(Cs cs, Th th') |th'. \rest. wq s cs = th # rest \ (\l r. lsp (cp s) rest = (l, [th'], r))}" - (is "?L = (?A - ?B - ?C) \ ?D") by (simp add:V) - from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset) - have "?D = {} \ (\ a. ?D = {a})" by (rule simple_A, auto) - thus ?thesis - proof(cases "wq s cs") - case Nil - hence "?D = {}" by simp - with ac and eq_de show ?thesis by simp - next - case (Cons tth rest) - from stp and V have "step s (V th cs)" by simp - hence eq_wq: "wq s cs = th # rest" - proof - - show "step s (V th cs) \ wq s cs = th # rest" - apply(ind_cases "step s (V th cs)") - by(insert Cons, unfold s_holding_def, simp) - qed - show ?thesis - proof(cases "lsp (cp s) rest") - fix l b r - assume eq_lsp: "lsp (cp s) rest = (l, b, r) " - show ?thesis - proof(cases "b") - case Nil - with eq_lsp and eq_wq have "?D = {}" by simp - with ac and eq_de show ?thesis by simp - next - case (Cons th' m) - with eq_lsp - have eq_lsp: "lsp (cp s) rest = (l, [th'], r)" - apply simp - by (drule_tac lsp_mid_length, simp) - from eq_wq and eq_lsp - have eq_D: "?D = {(Cs cs, Th th')}" by auto - from eq_wq and eq_lsp - have eq_C: "?C = {(Th th', Cs cs)}" by 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 by simp - hence "cs' = cs" - proof(rule waiting_unique [OF vt]) - from eq_wq eq_lsp wq_distinct[OF vt, of cs] - show "waiting s th' cs" by(unfold s_waiting_def, auto dest:lsp_set_eq) - qed - with th'_e eq_x have "(Th th', Cs cs) \ ?E" by simp - with eq_C show "False" by simp - qed - with acyclic_insert[symmetric] and ac and eq_D - and eq_de show ?thesis by simp - qed - 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 \ {(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 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'. \rest. wq s cs = th # rest \ (\l r. lsp (cp s) rest = (l, [th'], r))} \ - {(Cs cs, Th th') |th'. \rest. wq s cs = th # rest \ (\l r. lsp (cp s) rest = (l, [th'], r))}" - (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 (rule simple_A, 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 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 \ {(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 *} - -thm wf_trancl -thm finite_acyclic_wf -thm finite_acyclic_wf_converse -thm wf_induct - - -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)" . -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 step 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 step 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) - 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 \ set (waiting_queue (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 = waiting_queue (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 "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 - proof(cases "(lsp (cur_preced (schs s)) rest)", unfold V) - fix l m r - assume eq_lsp: "lsp (cur_preced (schs s)) rest = (l, m, r)" - and eq_wq: "waiting_queue (schs s) cs' = a # rest" - and th_in_set: "th \ set (wq (V th' cs' # s) cs)" - show ?thesis - proof(cases "m") - case Nil - with eq_lsp have "rest = []" using lsp_mid_nil by auto - with eq_wq have "waiting_queue (schs s) cs' = [a]" by simp - with h[unfolded V wq_def] True - show ?thesis - by (simp add:Let_def) - next - case (Cons b rb) - with lsp_mid_length[OF eq_lsp] have eq_m: "m = [b]" by auto - with eq_lsp have "lsp (cur_preced (schs s)) rest = (l, [b], r)" by simp - with h[unfolded V wq_def] True lsp_set_eq [OF this] eq_wq - show ?thesis - apply (auto simp:Let_def, fold wq_def) - by (rule_tac ih [of _ cs'], auto)+ - qed - 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 step 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 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 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) - by (auto simp:s_waiting_def wq_def Let_def split:list.splits prod.splits - dest:lsp_set_eq) -qed - -lemma readys_v_eq_1: - fixes th thread cs rest - assumes neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and eq_lsp: "lsp (cp s) rest = (l, [th'], r)" - and neq_th': "th \ th'" - shows "(th \ readys (V thread cs#s)) = (th \ 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 = cs in allE) - apply (simp add:s_waiting_def wq_def Let_def split:prod.splits list.splits) - apply (drule_tac lsp_mid_nil,simp, clarify, fold cp_def, clarsimp) - apply (frule_tac lsp_set_eq, simp) - apply (erule_tac x = csa in allE) - apply (subst (asm) (2) s_waiting_def, unfold wq_def) - apply (auto simp:Let_def split:list.splits prod.splits if_splits - dest:lsp_set_eq) - apply (unfold s_waiting_def) - apply (fold wq_def, clarsimp) - apply (clarsimp)+ - apply (case_tac "csa = cs", simp) - apply (erule_tac x = cs in allE, simp) - apply (unfold wq_def) - by (auto simp:Let_def split:list.splits prod.splits if_splits - dest:lsp_set_eq) -qed - -lemma readys_v_eq_2: - fixes th thread cs rest - assumes neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and eq_lsp: "lsp (cp s) rest = (l, [th'], r)" - and neq_th': "th = th'" - and vt: "vt step s" - shows "(th \ readys (V thread cs#s))" -proof - - from prems show ?thesis - apply (auto simp:readys_def) - apply (rule_tac wq_threads [of s _ cs], auto dest:lsp_set_eq) - apply (unfold s_waiting_def wq_def) - apply (auto simp:Let_def split:list.splits prod.splits if_splits - dest:lsp_set_eq lsp_mid_nil lsp_mid_length) - apply (fold cp_def, simp+, clarsimp) - apply (frule_tac lsp_set_eq, simp) - apply (fold wq_def) - apply (subgoal_tac "csa = cs", simp) - apply (rule_tac waiting_unique [of s th'], simp) - by (auto simp:s_waiting_def) -qed - -lemma chain_building: - assumes vt: "vt step 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 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 step 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 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, 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: "\holding (s::state) th1 cs; holding s th2 cs\ \ th1 = th2" - by (unfold s_holding_def cs_holding_def, auto) - -lemma unique_depend: "\vt step 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 step 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 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 s_depend_def s_waiting_def cs_waiting_def) - with th2_r show ?thesis by auto - qed - } thus ?thesis by auto -qed - -definition count :: "('a \ bool) \ 'a list \ nat" -where "count Q l = length (filter Q l)" - -definition cntP :: "state \ thread \ nat" -where "cntP s th = count (\ e. \ cs. e = P th cs) s" - -definition cntV :: "state \ thread \ nat" -where "cntV s th = count (\ e. \ cs. e = V th cs) s" - - -lemma 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 \ {cs}" -proof - - from prems show ?thesis - unfolding holdents_def step_depend_p[OF vt] by auto -qed - -lemma step_holdents_p_eq: - 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" -proof - - from prems show ?thesis - unfolding holdents_def step_depend_p[OF vt] by auto -qed - -lemma step_holdents_v_minus: - fixes th cs s - assumes vt: "vt step (V th cs#s)" - shows "holdents (V th cs#s) th = holdents s th - {cs}" -proof - - { fix rest l r - assume eq_wq: "wq s cs = th # rest" - and eq_lsp: "lsp (cp s) rest = (l, [th], r)" - have "False" - proof - - from lsp_set_eq [OF eq_lsp] have " rest = l @ [th] @ r" . - with eq_wq have "wq s cs = th#\" by simp - with wq_distinct [OF step_back_vt[OF vt], of cs] - show ?thesis by auto - qed - } thus ?thesis unfolding holdents_def step_depend_v[OF vt] by auto -qed - -lemma step_holdents_v_add: - fixes th th' cs s rest l r - assumes vt: "vt step (V th' cs#s)" - and neq_th: "th \ th'" - and eq_wq: "wq s cs = th' # rest" - and eq_lsp: "lsp (cp s) rest = (l, [th], r)" - shows "holdents (V th' cs#s) th = holdents s th \ {cs}" -proof - - from prems show ?thesis - unfolding holdents_def step_depend_v[OF vt] by auto -qed - -lemma step_holdents_v_eq: - fixes th th' cs s rest l r th'' - assumes vt: "vt step (V th' cs#s)" - and neq_th: "th \ th'" - and eq_wq: "wq s cs = th' # rest" - and eq_lsp: "lsp (cp s) rest = (l, [th''], r)" - and neq_th': "th \ th''" - shows "holdents (V th' cs#s) th = holdents s th" -proof - - from prems show ?thesis - unfolding holdents_def step_depend_v[OF vt] by auto -qed - -definition cntCS :: "state \ thread \ nat" -where "cntCS s th = card (holdents s th)" - -lemma cntCS_v_eq: - fixes th thread cs rest - assumes neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and not_in: "th \ set rest" - and vtv: "vt step (V thread cs#s)" - shows "cntCS (V thread cs#s) th = cntCS s th" -proof - - from prems show ?thesis - apply (unfold cntCS_def holdents_def step_depend_v) - apply auto - apply (subgoal_tac "\ (\l r. lsp (cp s) rest = (l, [th], r))", auto) - by (drule_tac lsp_set_eq, auto) -qed - -lemma cntCS_v_eq_1: - fixes th thread cs rest - assumes neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and eq_lsp: "lsp (cp s) rest = (l, [th'], r)" - and neq_th': "th \ th'" - and vtv: "vt step (V thread cs#s)" - shows "cntCS (V thread cs#s) th = cntCS s th" -proof - - from prems show ?thesis - apply (unfold cntCS_def holdents_def step_depend_v) - by auto -qed - -fun the_cs :: "node \ cs" -where "the_cs (Cs cs) = cs" - -lemma cntCS_v_eq_2: - fixes th thread cs rest - assumes neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and eq_lsp: "lsp (cp s) rest = (l, [th'], r)" - and neq_th': "th = th'" - and vtv: "vt step (V thread cs#s)" - shows "cntCS (V thread cs#s) th = 1 + cntCS s th" -proof - - have "card {csa. csa = cs \ (Cs csa, Th th') \ depend s} = - Suc (card {cs. (Cs cs, Th th') \ depend s})" - (is "card ?A = Suc (card ?B)") - proof - - have h: "?A = insert cs ?B" by auto - moreover have h1: "?B = ?B - {cs}" - proof - - { assume "(Cs cs, Th th') \ depend s" - moreover have "(Th th', Cs cs) \ depend s" - proof - - from wq_distinct [OF step_back_vt[OF vtv], of cs] - eq_wq lsp_set_eq [OF eq_lsp] show ?thesis - apply (auto simp:s_depend_def) - by (unfold cs_waiting_def, auto) - qed - moreover note acyclic_depend [OF step_back_vt[OF vtv]] - ultimately have "False" - apply (auto simp:acyclic_def) - apply (erule_tac x="Cs cs" in allE) - apply (subgoal_tac "(Cs cs, Cs cs) \ (depend s)\<^sup>+", simp) - by (rule_tac trancl_into_trancl [where b = "Th th'"], auto) - } thus ?thesis by auto - qed - moreover have "card (insert cs ?B) = Suc (card (?B - {cs}))" - proof(rule card_insert) - from finite_depend [OF step_back_vt [OF vtv]] - have fnt: "finite (depend s)" . - show " finite {cs. (Cs cs, Th th') \ depend s}" (is "finite ?B") - proof - - have "?B \ (\ (a, b). the_cs a) ` (depend s)" - apply (auto simp:image_def) - by (rule_tac x = "(Cs x, Th th')" in bexI, auto) - with fnt show ?thesis by (auto intro:finite_subset) - qed - qed - ultimately show ?thesis by simp - qed - with prems show ?thesis - apply (unfold cntCS_def holdents_def step_depend_v[OF vtv]) - by auto -qed - -lemma finite_holding: - fixes s th cs - assumes vt: "vt step 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_def, auto intro:finite_subset) -qed - -inductive_cases case_step_v: "step s (V thread cs)" - -lemma 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 - - have cs_in: "cs \ holdents s thread" using step_back_step[OF vtv] - apply (erule_tac case_step_v) - apply (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]) - by (unfold holdents_def, unfold step_depend_v[OF vtv], - auto dest:lsp_set_eq) - 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 step 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 step 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 prio max_prio thread) - 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_def - 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_def - 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 (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 \ 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 prems have vtp: "vt step (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 "waiting_queue (schs s) cs \ []", 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 \ 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_def, 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_def) - 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) - 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_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 \ 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 \ 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) - 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 - - have "thread \ set (wq (e#s) cs1)" - proof(cases "lsp (cp s) rest") - fix l m r - assume h: "lsp (cp s) rest = (l, m, r)" - show ?thesis - proof(cases "m") - case Nil - from wq_v_eq_nil [OF eq_wq] h Nil eq_e - have " wq (e # s) cs = []" by auto - thus ?thesis using eq_cs by auto - next - case (Cons th' l') - from lsp_mid_length [OF h] and Cons h - have eqh: "lsp (cp s) rest = (l, [th'], r)" by auto - from wq_v_eq [OF eq_wq this] - have "wq (V thread cs # s) cs = th' # l @ r" . - moreover from lsp_set_eq [OF eqh] - have "set rest = set \" by auto - moreover have "thread \ set rest" - proof - - from wq_distinct [OF step_back_vt[OF vtv], of cs] - and eq_wq show ?thesis by auto - qed - moreover note eq_e eq_cs - ultimately show ?thesis by simp - qed - qed - thus ?thesis by (simp add: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: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)" - by(unfold eq_e, rule readys_v_eq [OF neq_th eq_wq False]) - moreover have "cntCS (e#s) th = cntCS s th" - by(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq False vtv]) - moreover note ih eq_cnp eq_cnv eq_threads - ultimately show ?thesis by auto - next - case True - obtain l m r where eq_lsp: "lsp (cp s) rest = (l, m, r)" - by (cases "lsp (cp s) rest", auto) - with True have "m \ []" by (auto dest:lsp_mid_nil) - with eq_lsp obtain th' where eq_lsp: "lsp (cp s) rest = (l, [th'], r)" - by (case_tac m, auto dest:lsp_mid_length) - show ?thesis - proof(cases "th = th'") - case False - have "(th \ readys (e # s)) = (th \ readys s)" - by (unfold eq_e, rule readys_v_eq_1 [OF neq_th eq_wq eq_lsp False]) - moreover have "cntCS (e#s) th = cntCS s th" - by (unfold eq_e, rule cntCS_v_eq_1[OF neq_th eq_wq eq_lsp False vtv]) - moreover note ih eq_cnp eq_cnv eq_threads - ultimately show ?thesis by auto - next - case True - have "th \ readys (e # s)" - by (unfold eq_e, rule readys_v_eq_2 [OF neq_th eq_wq eq_lsp True vt]) - moreover have "cntP s th = cntV s th + cntCS s th + 1" - proof - - have "th \ readys s" - proof - - from True eq_wq lsp_set_eq [OF eq_lsp] neq_th - show ?thesis - apply (unfold readys_def s_waiting_def, auto) - by (rule_tac x = cs in exI, auto) - qed - moreover have "th \ threads s" - proof - - from True eq_wq lsp_set_eq [OF eq_lsp] neq_th - 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 have "cntCS (e # s) th = 1 + cntCS s th" - by (unfold eq_e, rule cntCS_v_eq_2 [OF neq_th eq_wq eq_lsp True vtv]) - 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_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 \ 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_def s_depend_def wq_def cs_holding_def) - qed -qed - -lemma not_thread_cncs: - fixes th s - assumes vt: "vt step 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 step 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 prio max_prio thread) - 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_def) - 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_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 \ 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 prems have vtp: "vt step (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_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 \ 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 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) - have "cntCS (e # s) th = cntCS s th" - proof(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq _ vtv]) - show "th \ set rest" - proof - assume "th \ set rest" - with eq_wq have "th \ set (wq s cs)" by simp - from wq_threads [OF vt this] eq_e not_in - show False by simp - qed - qed - 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_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) - qed -qed - -lemma 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) \ 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 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) -qed - -fun the_th :: "node \ thread" - where "the_th (Th th) = th" - -lemma runing_unique: - fixes th1 th2 s - assumes vt: "vt step 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)^+)" using Domain_def [of "(depend s)^+"] - by auto - 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)^+)" using Domain_def [of "(depend s)^+"] - by auto - 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 prio max_prio thread) - 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 step 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 prio max_prio thread) - 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 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 \ 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 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) \ 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) \ 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) \ (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 step 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 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 prio max_prio thread) - 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) - qed -qed - -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 step 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 step 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 step 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 step 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 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]) -qed - -lemma readys_threads: - shows "readys s \ threads s" -proof - fix th - assume "th \ readys s" - thus "th \ threads s" - by (unfold readys_def, auto) -qed - -lemma eq_holding: "holding (wq s) th cs = holding s th cs" - apply (unfold s_holding_def cs_holding_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 - -end \ No newline at end of file