# HG changeset patch # User zhangx # Date 1454031967 -28800 # Node ID 0525670d8e6aaf211669e9bae7d525e825de3546 # Parent ed938e2246b9bacb4ce92250a483329605635c5c Removed *.*~, #***#, log, etc. diff -r ed938e2246b9 -r 0525670d8e6a #PIPBasics.thy# --- a/#PIPBasics.thy# Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,3787 +0,0 @@ -theory PIPBasics -imports PIPDefs -begin - -locale valid_trace = - fixes s - assumes vt : "vt s" - -locale valid_trace_e = valid_trace + - fixes e - assumes vt_e: "vt (e#s)" -begin - -lemma pip_e: "PIP s e" - using vt_e by (cases, simp) - -end - -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 runing_head: - assumes "th \ runing s" - and "th \ set (wq_fun (schs s) cs)" - shows "th = hd (wq_fun (schs s) cs)" - using assms - by (simp add:runing_def readys_def s_waiting_def wq_def) - -context valid_trace -begin - -lemma actor_inv: - assumes "PIP s e" - and "\ isCreate e" - shows "actor e \ runing s" - using assms - by (induct, auto) - -lemma ind [consumes 0, case_names Nil Cons, induct type]: - assumes "PP []" - and "(\s e. valid_trace s \ valid_trace (e#s) \ - PP s \ PIP s e \ PP (e # s))" - shows "PP s" -proof(rule vt.induct[OF vt]) - from assms(1) show "PP []" . -next - fix s e - assume h: "vt s" "PP s" "PIP s e" - show "PP (e # s)" - proof(cases rule:assms(2)) - from h(1) show v1: "valid_trace s" by (unfold_locales, simp) - next - from h(1,3) have "vt (e#s)" by auto - thus "valid_trace (e # s)" by (unfold_locales, simp) - qed (insert h, auto) -qed - -lemma wq_distinct: "distinct (wq s cs)" -proof(induct rule:ind) - case (Cons s e) - from Cons(4,3) - show ?case - proof(induct) - case (thread_P th s cs1) - show ?case - proof(cases "cs = cs1") - case True - thus ?thesis (is "distinct ?L") - proof - - have "?L = wq_fun (schs s) cs1 @ [th]" using True - by (simp add:wq_def wf_def Let_def split:list.splits) - moreover have "distinct ..." - proof - - have "th \ set (wq_fun (schs s) cs1)" - proof - assume otherwise: "th \ set (wq_fun (schs s) cs1)" - from runing_head[OF thread_P(1) this] - have "th = hd (wq_fun (schs s) cs1)" . - hence "(Cs cs1, Th th) \ (RAG s)" using otherwise - by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def) - with thread_P(2) show False by auto - qed - moreover have "distinct (wq_fun (schs s) cs1)" - using True thread_P wq_def by auto - ultimately show ?thesis by auto - qed - ultimately show ?thesis by simp - qed - next - case False - with thread_P(3) - show ?thesis - by (auto simp:wq_def wf_def Let_def split:list.splits) - qed - next - case (thread_V th s cs1) - thus ?case - proof(cases "cs = cs1") - case True - show ?thesis (is "distinct ?L") - proof(cases "(wq s cs)") - case Nil - thus ?thesis - by (auto simp:wq_def wf_def Let_def split:list.splits) - next - case (Cons w_hd w_tl) - moreover have "distinct (SOME q. distinct q \ set q = set w_tl)" - proof(rule someI2) - from thread_V(3)[unfolded Cons] - show "distinct w_tl \ set w_tl = set w_tl" by auto - qed auto - ultimately show ?thesis - by (auto simp:wq_def wf_def Let_def True split:list.splits) - qed - next - case False - with thread_V(3) - show ?thesis - by (auto simp:wq_def wf_def Let_def split:list.splits) - qed - qed (insert Cons, auto simp: wq_def Let_def split:list.splits) -qed (unfold wq_def Let_def, simp) - -end - - -context valid_trace_e -begin - -text {* - The following lemma shows that only the @{text "P"} - operation can add new thread into waiting queues. - Such kind of lemmas are very obvious, but need to be checked formally. - This is a kind of confirmation that our modelling is correct. -*} - -lemma block_pre: - assumes s_ni: "thread \ set (wq s cs)" - and s_i: "thread \ set (wq (e#s) cs)" - shows "e = P thread cs" -proof(cases e) - -- {* This is the only non-trivial case: *} - case (V th cs1) - have False - proof(cases "cs1 = cs") - case True - show ?thesis - proof(cases "(wq s cs1)") - case (Cons w_hd w_tl) - have "set (wq (e#s) cs) \ set (wq s cs)" - proof - - have "(wq (e#s) cs) = (SOME q. distinct q \ set q = set w_tl)" - using Cons V by (auto simp:wq_def Let_def True split:if_splits) - moreover have "set ... \ set (wq s cs)" - proof(rule someI2) - show "distinct w_tl \ set w_tl = set w_tl" - by (metis distinct.simps(2) local.Cons wq_distinct) - qed (insert Cons True, auto) - ultimately show ?thesis by simp - qed - with assms show ?thesis by auto - qed (insert assms V True, auto simp:wq_def Let_def split:if_splits) - qed (insert assms V, auto simp:wq_def Let_def split:if_splits) - thus ?thesis by auto -qed (insert assms, auto simp:wq_def Let_def split:if_splits) - -end - -text {* - The following lemmas is also obvious and shallow. It says - that only running thread can request for a critical resource - and that the requested resource must be one which is - not current held by the thread. -*} - -lemma p_pre: "\vt ((P thread cs)#s)\ \ - thread \ runing s \ (Cs cs, Th thread) \ (RAG s)^+" -apply (ind_cases "vt ((P thread cs)#s)") -apply (ind_cases "step s (P thread cs)") -by auto - -lemma abs1: - 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)" - -context valid_trace_e -begin - -lemma abs2: - assumes 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 vt_e 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 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 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 - -end - - -context valid_trace -begin -lemma vt_moment: "\ t. vt (moment t s)" -proof(induct rule:ind) - case Nil - thus ?case by (simp add:vt_nil) -next - case (Cons s e t) - show ?case - proof(cases "t \ length (e#s)") - case True - from True have "moment t (e#s) = e#s" by simp - thus ?thesis using Cons - by (simp add:valid_trace_def) - next - case False - from Cons have "vt (moment t s)" by simp - moreover have "moment t (e#s) = moment t s" - proof - - from False have "t \ length s" by simp - from moment_app [OF this, of "[e]"] - show ?thesis by simp - qed - ultimately show ?thesis by simp - qed -qed -end - -locale valid_moment = valid_trace + - fixes i :: nat - -sublocale valid_moment < vat_moment: valid_trace "(moment i s)" - by (unfold_locales, insert vt_moment, auto) - -context valid_trace -begin - - -text {* (* ddd *) - The nature of the work is like this: since it starts from a very simple and basic - model, even intuitively very `basic` and `obvious` properties need to derived from scratch. - For instance, the fact - that one thread can not be blocked by two critical resources at the same time - is obvious, because only running threads can make new requests, if one is waiting for - a critical resource and get blocked, it can not make another resource request and get - blocked the second time (because it is not running). - - To derive this fact, one needs to prove by contraction and - reason about time (or @{text "moement"}). The reasoning is based on a generic theorem - named @{text "p_split"}, which is about status changing along the time axis. It says if - a condition @{text "Q"} is @{text "True"} at a state @{text "s"}, - but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} - in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history - of events leading to it), such that @{text "Q"} switched - from being @{text "False"} to @{text "True"} and kept being @{text "True"} - till the last moment of @{text "s"}. - - Suppose a thread @{text "th"} is blocked - on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, - since no thread is blocked at the very beginning, by applying - @{text "p_split"} to these two blocking facts, there exist - two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that - @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} - and kept on blocked on them respectively ever since. - - Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}. - However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still - in blocked state at moment @{text "t2"} and could not - make any request and get blocked the second time: Contradiction. -*} - -lemma waiting_unique_pre: (* ccc *) - assumes 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#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - 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 vt_e.abs2 [OF True eq_th h2 h1] - show ?thesis by auto - next - case False - from vt_e.block_pre[OF False h1] - have "e = P thread cs2" . - with vt_e.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#moment t1 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t1 s" e - by (unfold_locales, auto, cases, auto) - 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 vt_e.abs2 True eq_th h2 h1 - show ?thesis by auto - next - case False - from vt_e.block_pre [OF False h1] - have "e = P thread cs1" . - with vt_e.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 - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t1 s" e - by (unfold_locales, auto, cases, auto) - 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 vt_e.abs2 [OF True eq_th h2 h1] - show ?thesis by auto - next - case False - from vt_e.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 (e#moment t2 s)" by simp - then interpret vt_e2: valid_trace_e "moment t2 s" e - by (unfold_locales, auto, cases, auto) - from vt_e2.abs2 [OF True eq_th h2 h1] - show ?thesis . - next - case False - have "vt (e#moment t2 s)" - proof - - from vt_moment eqt12 - have "vt (moment (Suc t2) s)" by auto - with eq_m eqt12 show ?thesis by simp - qed - then interpret vt_e2: valid_trace_e "moment t2 s" e - by (unfold_locales, auto, cases, auto) - from vt_e2.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 - -text {* - This lemma is a simple corrolary of @{text "waiting_unique_pre"}. -*} - -lemma waiting_unique: - assumes "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 - -end - -(* not used *) -text {* - Every thread can only be blocked on one critical resource, - symmetrically, every critical resource can only be held by one thread. - This fact is much more easier according to our definition. -*} -lemma held_unique: - assumes "holding (s::event list) th1 cs" - and "holding s th2 cs" - shows "th1 = th2" - by (insert assms, unfold s_holding_def, auto) - - -lemma last_set_lt: "th \ threads s \ last_set th s < length s" - apply (induct s, auto) - by (case_tac a, auto split:if_splits) - -lemma last_set_unique: - "\last_set th1 s = last_set th2 s; th1 \ threads s; th2 \ threads s\ - \ th1 = th2" - apply (induct s, auto) - by (case_tac a, auto split:if_splits dest:last_set_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 "last_set th1 s = last_set th2 s" by (simp add:preced_def) - from last_set_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 - -(* An aux lemma used later *) -lemma unique_minus: - 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: - 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: - 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 - -text {* - The following three lemmas show that @{text "RAG"} does not change - by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"} - events, respectively. -*} - -lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - - -text {* - The following lemmas are used in the proof of - lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed - by @{text "V"}-events. - However, since our model is very concise, such seemingly obvious lemmas need to be derived from scratch, - starting from the model definitions. -*} -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 - -text {* - The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be - derived from scratch, which confirms the correctness of the definition of @{text "next_th"}. -*} -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" - from vt interpret vt_v: valid_trace_e s "V th cs" - by (cases, unfold_locales, simp) - 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 vt_v.wq_distinct[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 vt_v.wq_distinct[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 vt interpret vt_v: valid_trace_e s "V th cs" - by (cases, unfold_locales, simp+) - 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 vt_v.wq_distinct[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 vt_v.wq_distinct[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)" - from vt interpret vt_v: valid_trace_e s "V th cs" - by (cases, unfold_locales, simp+) - have "(SOME q. distinct q \ set q = set rest) \ []" - proof(rule someI2) - from vt_v.wq_distinct[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" - from vt interpret vt_v: valid_trace_e s "V th cs" - by (cases, unfold_locales, simp+) - 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 vt_v.wq_distinct [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 vt_v.wq_distinct [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 vt_v.wq_distinct [of cs, unfolded wq_def] - show False by auto - qed - qed -qed - -text {* (* ddd *) - The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed - with the happening of @{text "V"}-events: -*} -lemma step_RAG_v: -assumes vt: - "vt (V th cs#s)" -shows " - RAG (V th cs # s) = - RAG 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_RAG_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 - -text {* - The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed - with the happening of @{text "P"}-events: -*} -lemma step_RAG_p: - "vt (P th cs#s) \ - RAG (P th cs # s) = (if (wq s cs = []) then RAG s \ {(Cs cs, Th th)} - else RAG s \ {(Th th, Cs cs)})" - apply(simp only: s_RAG_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(simp add:s_RAG_def wq_def cs_holding_def) - apply(auto) - done - - -lemma RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" - by (unfold s_RAG_def, auto) - -context valid_trace -begin - -text {* - The following lemma shows that @{text "RAG"} is acyclic. - The overall structure is by induction on the formation of @{text "vt s"} - and then case analysis on event @{text "e"}, where the non-trivial cases - for those for @{text "V"} and @{text "P"} events. -*} -lemma acyclic_RAG: - shows "acyclic (RAG s)" -using vt -proof(induct) - case (vt_cons s e) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - assume ih: "acyclic (RAG 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:RAG_create_unchanged) - next - case (Exit th) - with ih show ?thesis by (simp add:RAG_exit_unchanged) - next - case (V th cs) - from V vt stp have vtt: "vt (V th cs#s)" by auto - from step_RAG_v [OF this] - have eq_de: - "RAG (e # s) = - RAG 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 RAG_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_RAG_def s_waiting_def cs_waiting_def wq_def by simp - hence "cs' = cs" - proof(rule vt_s.waiting_unique) - from eq_wq vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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_RAG_p [OF this] P - have "RAG (e # s) = - (if wq s cs = [] then RAG s \ {(Cs cs, Th th)} else - RAG 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 = RAG s \ {(Cs cs, Th th)}" by simp - have "(Th th, Cs cs) \ (RAG s)\<^sup>*" - proof - assume "(Th th, Cs cs) \ (RAG s)\<^sup>*" - hence "(Th th, Cs cs) \ (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl) - from tranclD2 [OF this] - obtain x where "(x, Cs cs) \ RAG s" by auto - with True show False by (auto simp:s_RAG_def cs_waiting_def) - qed - with acyclic_insert ih eq_r show ?thesis by auto - next - case False - hence eq_r: "?R = RAG s \ {(Th th, Cs cs)}" by simp - have "(Cs cs, Th th) \ (RAG s)\<^sup>*" - proof - assume "(Cs cs, Th th) \ (RAG s)\<^sup>*" - hence "(Cs cs, Th th) \ (RAG 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) \ (RAG 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 RAG_set_unchanged - show ?thesis by (simp add:RAG_set_unchanged) - qed - next - case vt_nil - show "acyclic (RAG ([]::state))" - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -qed - - -lemma finite_RAG: - shows "finite (RAG s)" -proof - - from vt show ?thesis - proof(induct) - case (vt_cons s e) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - assume ih: "finite (RAG 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:RAG_create_unchanged) - next - case (Exit th) - with ih show ?thesis by (simp add:RAG_exit_unchanged) - next - case (V th cs) - from V vt stp have vtt: "vt (V th cs#s)" by auto - from step_RAG_v [OF this] - have eq_de: "RAG (e # s) = - RAG 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 (metis finite.simps) - 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_RAG_p [OF this] P - have "RAG (e # s) = - (if wq s cs = [] then RAG s \ {(Cs cs, Th th)} else - RAG 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 = RAG s \ {(Cs cs, Th th)}" by simp - with True and ih show ?thesis by auto - next - case False - hence "?R = RAG 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:RAG_set_unchanged) - qed - next - case vt_nil - show "finite (RAG ([]::state))" - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) - qed -qed - -text {* Several useful lemmas *} - -lemma wf_dep_converse: - shows "wf ((RAG s)^-1)" -proof(rule finite_acyclic_wf_converse) - from finite_RAG - show "finite (RAG s)" . -next - from acyclic_RAG - show "acyclic (RAG s)" . -qed - -end - -lemma hd_np_in: "x \ set l \ hd l \ set l" - by (induct l, auto) - -lemma th_chasing: "(Th th, Cs cs) \ RAG (s::state) \ \ th'. (Cs cs, Th th') \ RAG s" - by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in) - -context valid_trace -begin - -lemma wq_threads: - assumes 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) - interpret vt_s: valid_trace s - using vt_cons(1) by (unfold_locales, auto) - 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_RAG_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 vt_s.wq_distinct[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: "\(Th th) \ Range (RAG (s::state))\ \ th \ threads s" - apply(unfold s_RAG_def cs_waiting_def cs_holding_def) - by (auto intro:wq_threads) - -lemma readys_v_eq: - assumes neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and not_in: "th \ set rest" - shows "(th \ readys (V thread cs#s)) = (th \ readys s)" -proof - - from assms show ?thesis - apply (auto simp:readys_def) - apply(simp add:s_waiting_def[folded wq_def]) - apply (erule_tac x = csa in allE) - apply (simp add:s_waiting_def wq_def Let_def split:if_splits) - apply (case_tac "csa = cs", simp) - apply (erule_tac x = cs in allE) - apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits) - apply(auto simp add: wq_def) - apply (auto simp:s_waiting_def wq_def Let_def split:list.splits) - proof - - assume th_nin: "th \ set rest" - and th_in: "th \ set (SOME q. distinct q \ set q = set rest)" - and eq_wq: "wq_fun (schs s) cs = thread # rest" - have "set (SOME q. distinct q \ set q = set rest) = set rest" - proof(rule someI2) - from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def] - show "distinct rest \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ set x = set rest" by auto - qed - with th_nin th_in show False by auto - qed -qed - -text {* \noindent - The following lemmas shows that: starting from any node in @{text "RAG"}, - by chasing out-going edges, it is always possible to reach a node representing a ready - thread. In this lemma, it is the @{text "th'"}. -*} - -lemma chain_building: - shows "node \ Domain (RAG s) \ (\ th'. th' \ readys s \ (node, Th th') \ (RAG s)^+)" -proof - - from wf_dep_converse - have h: "wf ((RAG s)\)" . - show ?thesis - proof(induct rule:wf_induct [OF h]) - fix x - assume ih [rule_format]: - "\y. (y, x) \ (RAG s)\ \ - y \ Domain (RAG s) \ (\th'. th' \ readys s \ (y, Th th') \ (RAG s)\<^sup>+)" - show "x \ Domain (RAG s) \ (\th'. th' \ readys s \ (x, Th th') \ (RAG s)\<^sup>+)" - proof - assume x_d: "x \ Domain (RAG s)" - show "\th'. th' \ readys s \ (x, Th th') \ (RAG s)\<^sup>+" - proof(cases x) - case (Th th) - from x_d Th obtain cs where x_in: "(Th th, Cs cs) \ RAG s" by (auto simp:s_RAG_def) - with Th have x_in_r: "(Cs cs, x) \ (RAG s)^-1" by simp - from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \ RAG s" by blast - hence "Cs cs \ Domain (RAG 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') \ (RAG s)\<^sup>+" by auto - have "(x, Th th') \ (RAG 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) \ (RAG s)^-1" by (auto simp:s_RAG_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 have "th' \ threads s" by auto - with False have "Th th' \ Domain (RAG s)" - by (auto simp:readys_def wq_def s_waiting_def s_RAG_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'') \ (RAG s)\<^sup>+" by auto - from th'_d and th''_in - have "(x, Th th'') \ (RAG s)\<^sup>+" by auto - with th''_r show ?thesis by auto - qed - qed - qed - qed -qed - -text {* \noindent - The following is just an instance of @{text "chain_building"}. -*} -lemma th_chain_to_ready: - assumes th_in: "th \ threads s" - shows "th \ readys s \ (\ th'. th' \ readys s \ (Th th, Th th') \ (RAG s)^+)" -proof(cases "th \ readys s") - case True - thus ?thesis by auto -next - case False - from False and th_in have "Th th \ Domain (RAG s)" - by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def) - from chain_building [rule_format, OF this] - show ?thesis by auto -qed - -end - -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) - -context valid_trace -begin - -lemma unique_RAG: "\(n, n1) \ RAG s; (n, n2) \ RAG s\ \ n1 = n2" - apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq) - by(auto elim:waiting_unique holding_unique) - -end - - -lemma trancl_split: "(a, b) \ r^+ \ \ c. (a, c) \ r" -by (induct rule:trancl_induct, auto) - -context valid_trace -begin - -lemma dchain_unique: - assumes th1_d: "(n, Th th1) \ (RAG s)^+" - and th1_r: "th1 \ readys s" - and th2_d: "(n, Th th2) \ (RAG 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_RAG - have "(Th th1, Th th2) \ (RAG s)\<^sup>+ \ (Th th2, Th th1) \ (RAG s)\<^sup>+" by auto - hence "False" - proof - assume "(Th th1, Th th2) \ (RAG s)\<^sup>+" - from trancl_split [OF this] - obtain n where dd: "(Th th1, n) \ RAG s" by auto - then obtain cs where eq_n: "n = Cs cs" - by (auto simp:s_RAG_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_RAG_def wq_def s_waiting_def cs_waiting_def) - with th1_r show ?thesis by auto - next - assume "(Th th2, Th th1) \ (RAG s)\<^sup>+" - from trancl_split [OF this] - obtain n where dd: "(Th th2, n) \ RAG s" by auto - then obtain cs where eq_n: "n = Cs cs" - by (auto simp:s_RAG_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_RAG_def s_waiting_def cs_waiting_def) - with th2_r show ?thesis by auto - qed - } thus ?thesis by auto -qed - -end - - -lemma step_holdents_p_add: - 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_RAG_p[OF vt] by (auto) -qed - -lemma step_holdents_p_eq: - 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_RAG_p[OF vt] by auto -qed - - -lemma (in valid_trace) finite_holding : - shows "finite (holdents s th)" -proof - - let ?F = "\ (x, y). the_cs x" - from finite_RAG - have "finite (RAG s)" . - hence "finite (?F `(RAG s))" by simp - moreover have "{cs . (Cs cs, Th th) \ RAG s} \ \" - proof - - { have h: "\ a A f. a \ A \ f a \ f ` A" by auto - fix x assume "(Cs x, Th th) \ RAG s" - hence "?F (Cs x, Th th) \ ?F `(RAG s)" by (rule h) - moreover have "?F (Cs x, Th th) = x" by simp - ultimately have "x \ (\(x, y). the_cs x) ` RAG s" by simp - } thus ?thesis by auto - qed - ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset) -qed - -lemma cntCS_v_dec: - assumes vtv: "vt (V thread cs#s)" - shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread" -proof - - from vtv interpret vt_s: valid_trace s - by (cases, unfold_locales, simp) - from vtv interpret vt_v: valid_trace "V thread cs#s" - by (unfold_locales, simp) - from step_back_step[OF vtv] - have cs_in: "cs \ holdents s thread" - apply (cases, unfold holdents_test s_RAG_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 vt_s.wq_distinct[of cs]) - apply (unfold holdents_test, unfold step_RAG_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))) \ RAG 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 vt_v.finite_holding - 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 count_rec1 [simp]: - assumes "Q e" - shows "count Q (e#es) = Suc (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec2 [simp]: - assumes "\Q e" - shows "count Q (e#es) = (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec3 [simp]: - shows "count Q [] = 0" - by (unfold count_def, auto) - -lemma cntP_diff_inv: - assumes "cntP (e#s) th \ cntP s th" - shows "isP e \ actor e = th" -proof(cases e) - case (P th' pty) - show ?thesis - by (cases "(\e. \cs. e = P th cs) (P th' pty)", - insert assms P, auto simp:cntP_def) -qed (insert assms, auto simp:cntP_def) - -lemma isP_E: - assumes "isP e" - obtains cs where "e = P (actor e) cs" - using assms by (cases e, auto) - -lemma isV_E: - assumes "isV e" - obtains cs where "e = V (actor e) cs" - using assms by (cases e, auto) (* ccc *) - -lemma cntV_diff_inv: - assumes "cntV (e#s) th \ cntV s th" - shows "isV e \ actor e = th" -proof(cases e) - case (V th' pty) - show ?thesis - by (cases "(\e. \cs. e = V th cs) (V th' pty)", - insert assms V, auto simp:cntV_def) -qed (insert assms, auto simp:cntV_def) - -context valid_trace -begin - -text {* (* ddd *) \noindent - The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} - of one particular thread. -*} - -lemma cnp_cnv_cncs: - 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) - interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp) - 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 vt_s.wq_threads [OF 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:RAG_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:RAG_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) \ (RAG s)\<^sup>+" - from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto - then interpret vt_p: valid_trace "(P thread cs#s)" - by (unfold_locales, simp) - 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) - 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_RAG_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_RAG_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) \ RAG s} = - Suc (card {cs. (Cs cs, Th thread) \ RAG 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 vt_s.finite_holding [of thread] - show " finite {cs. (Cs cs, Th thread) \ RAG s}" - by (unfold holdents_test, simp) - qed - moreover have "?R - {cs} = ?R" - proof - - have "cs \ ?R" - proof - assume "cs \ {cs. (Cs cs, Th thread) \ RAG 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_RAG_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 vt_p.wq_distinct [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_RAG_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 - then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp) - 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 vt_v.wq_distinct[of cs] and eq_wq - show "distinct rest \ set rest = set rest" - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 vt_v.wq_distinct[of cs] - and eq_wq show False - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq) - moreover have "cntCS (e#s) th = cntCS s th" - apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto) - proof - - have "{csa. (Cs csa, Th th) \ RAG s \ csa = cs \ next_th s thread cs th} = - {cs. (Cs cs, Th th) \ RAG s}" - proof - - from False eq_wq - have " next_th s thread cs th \ (Cs cs, Th th) \ RAG 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 vt_s.wq_distinct[ 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))) \ RAG s" - by auto - qed - thus ?thesis by auto - qed - thus "card {csa. (Cs csa, Th th) \ RAG s \ csa = cs \ next_th s thread cs th} = - card {cs. (Cs cs, Th th) \ RAG 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 vt_s.wq_distinct[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_RAG_v[OF vtv], - auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def - Let_def cs_holding_def) - by (insert vt_s.wq_distinct[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 vt_s.wq_threads) - 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 vt_s.wq_distinct[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 vt_s.waiting_unique[OF 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 vt_s.wq_threads [OF 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_RAG_v[OF vtv], auto) - proof - - show "card {csa. (Cs csa, Th th) \ RAG s \ csa = cs} = - Suc (card {cs. (Cs cs, Th th) \ RAG 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) ` RAG s)" - apply (auto simp:image_def) - by (rule_tac x = "(Cs x, Th th)" in bexI, auto) - with vt_s.finite_RAG - show "finite {cs. (Cs cs, Th th) \ RAG s}" by (auto intro:finite_subset) - next - show "cs \ {cs. (Cs cs, Th th) \ RAG s}" - proof - assume "cs \ {cs. (Cs cs, Th th) \ RAG s}" - hence "(Cs cs, Th th) \ RAG s" by simp - with True neq_th eq_wq show False - by (auto simp:next_th_def s_RAG_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:RAG_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_RAG_def wq_def cs_holding_def) - qed -qed - -lemma not_thread_cncs: - assumes 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) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - 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:RAG_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:RAG_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_RAG_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 - then interpret vt_v: valid_trace "(V thread cs#s)" - by (unfold_locales, simp) - 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 vt_v.wq_distinct[of cs] and eq_wq - show "distinct rest \ set rest = set rest" - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 vt_s.wq_threads[OF this] and ni - show False - using `hd (SOME q. distinct q \ set q = set rest) \ set (wq s cs)` - ni vt_s.wq_threads by blast - 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_RAG_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:RAG_set_unchanged) - qed - next - case vt_nil - show ?case - by (unfold cntCS_def, - auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def) - qed -qed - -end - -lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs" - by (auto simp:s_waiting_def cs_waiting_def wq_def) - -context valid_trace -begin - -lemma dm_RAG_threads: - assumes in_dom: "(Th th) \ Domain (RAG s)" - shows "th \ threads s" -proof - - from in_dom obtain n where "(Th th, n) \ RAG s" by auto - moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto - ultimately have "(Th th, Cs cs) \ RAG s" by simp - hence "th \ set (wq s cs)" - by (unfold s_RAG_def, auto simp:cs_waiting_def) - from wq_threads [OF this] show ?thesis . -qed - -end - -lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th" -unfolding cp_def wq_def -apply(induct s rule: schs.induct) -thm cpreced_initial -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 - -context valid_trace -begin - -lemma runing_unique: - assumes 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" - unfolding runing_def - apply(simp) - done - hence eq_max: "Max ((\th. preced th s) ` ({th1} \ dependants (wq s) th1)) = - Max ((\th. preced th s) ` ({th2} \ dependants (wq s) th2))" - (is "Max (?f ` ?A) = Max (?f ` ?B)") - unfolding cp_eq_cpreced - unfolding 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 (dependants (wq s) th1)" - proof- - have "finite {th'. (Th th', Th th1) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th1) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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 - thm cpreced_def - unfolding cpreced_def[symmetric] - unfolding cp_eq_cpreced[symmetric] - unfolding cpreced_def - using that[intro] by (auto) - 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 (dependants (wq s) th2)" - proof- - have "finite {th'. (Th th', Th th2) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th2) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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' \ dependants (wq s) th1)" by simp - thus "th1' \ threads s" - proof - assume "th1' \ dependants (wq s) th1" - hence "(Th th1') \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "(Th th1') \ Domain (RAG s)" by (simp add:trancl_domain) - from dm_RAG_threads[OF 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' \ dependants (wq s) th2)" by simp - thus "th2' \ threads s" - proof - assume "th2' \ dependants (wq s) th2" - hence "(Th th2') \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "(Th th2') \ Domain (RAG s)" by (simp add:trancl_domain) - from dm_RAG_threads[OF 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' \ dependants (wq s) th1" by simp - thus ?thesis - proof - assume eq_th': "th1' = th1" - from th2_in have "th2' = th2 \ th2' \ dependants (wq s) th2" by simp - thus ?thesis - proof - assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp - next - assume "th2' \ dependants (wq s) th2" - with eq_th12 eq_th' have "th1 \ dependants (wq s) th2" by simp - hence "(Th th1, Th th2) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - hence "Th th1 \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "Th th1 \ Domain (RAG s)" by (simp add:trancl_domain) - then obtain n where d: "(Th th1, n) \ RAG s" by (auto simp:Domain_def) - from RAG_target_th [OF this] - obtain cs' where "n = Cs cs'" by auto - with d have "(Th th1, Cs cs') \ RAG s" by simp - with runing_1 have "False" - apply (unfold runing_def readys_def s_RAG_def) - by (auto simp:eq_waiting) - thus ?thesis by simp - qed - next - assume th1'_in: "th1' \ dependants (wq s) th1" - from th2_in have "th2' = th2 \ th2' \ dependants (wq s) th2" by simp - thus ?thesis - proof - assume "th2' = th2" - with th1'_in eq_th12 have "th2 \ dependants (wq s) th1" by simp - hence "(Th th2, Th th1) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - hence "Th th2 \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "Th th2 \ Domain (RAG s)" by (simp add:trancl_domain) - then obtain n where d: "(Th th2, n) \ RAG s" by (auto simp:Domain_def) - from RAG_target_th [OF this] - obtain cs' where "n = Cs cs'" by auto - with d have "(Th th2, Cs cs') \ RAG s" by simp - with runing_2 have "False" - apply (unfold runing_def readys_def s_RAG_def) - by (auto simp:eq_waiting) - thus ?thesis by simp - next - assume "th2' \ dependants (wq s) th2" - with eq_th12 have "th1' \ dependants (wq s) th2" by simp - hence h1: "(Th th1', Th th2) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - from th1'_in have h2: "(Th th1', Th th1) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - show ?thesis - proof(rule dchain_unique[OF 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 "card (runing s) \ 1" -apply(subgoal_tac "finite (runing s)") -prefer 2 -apply (metis finite_nat_set_iff_bounded lessI runing_unique) -apply(rule ccontr) -apply(simp) -apply(case_tac "Suc (Suc 0) \ card (runing s)") -apply(subst (asm) card_le_Suc_iff) -apply(simp) -apply(auto)[1] -apply (metis insertCI runing_unique) -apply(auto) -done - -end - - -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 - - -context valid_trace -begin - -lemma cnp_cnv_eq: - assumes "th \ threads s" - shows "cntP s th = cntV s th" - using assms - using cnp_cnv_cncs not_thread_cncs by auto - -end - - -lemma eq_RAG: - "RAG (wq s) = RAG s" -by (unfold cs_RAG_def s_RAG_def, auto) - -context valid_trace -begin - -lemma count_eq_dependants: - assumes eq_pv: "cntP s th = cntV s th" - shows "dependants (wq s) th = {}" -proof - - from cnp_cnv_cncs and eq_pv - have "cntCS s th = 0" - by (auto split:if_splits) - moreover have "finite {cs. (Cs cs, Th th) \ RAG s}" - proof - - from finite_holding[of th] show ?thesis - by (simp add:holdents_test) - qed - ultimately have h: "{cs. (Cs cs, Th th) \ RAG s} = {}" - by (unfold cntCS_def holdents_test cs_dependants_def, auto) - show ?thesis - proof(unfold cs_dependants_def) - { assume "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ {}" - then obtain th' where "(Th th', Th th) \ (RAG (wq s))\<^sup>+" by auto - hence "False" - proof(cases) - assume "(Th th', Th th) \ RAG (wq s)" - thus "False" by (auto simp:cs_RAG_def) - next - fix c - assume "(c, Th th) \ RAG (wq s)" - with h and eq_RAG show "False" - by (cases c, auto simp:cs_RAG_def) - qed - } thus "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} = {}" by auto - qed -qed - -lemma dependants_threads: - shows "dependants (wq s) th \ threads s" -proof - { fix th th' - assume h: "th \ {th'a. (Th th'a, Th th') \ (RAG (wq s))\<^sup>+}" - have "Th th \ Domain (RAG s)" - proof - - from h obtain th' where "(Th th, Th th') \ (RAG (wq s))\<^sup>+" by auto - hence "(Th th) \ Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def) - with trancl_domain have "(Th th) \ Domain (RAG (wq s))" by simp - thus ?thesis using eq_RAG by simp - qed - from dm_RAG_threads[OF this] - have "th \ threads s" . - } note hh = this - fix th1 - assume "th1 \ dependants (wq s) th" - hence "th1 \ {th'a. (Th th'a, Th th) \ (RAG (wq s))\<^sup>+}" - by (unfold cs_dependants_def, simp) - from hh [OF this] show "th1 \ threads s" . -qed - -lemma finite_threads: - shows "finite (threads s)" -using vt by (induct) (auto elim: step.cases) - -end - -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 - -context valid_trace -begin - -lemma cp_le: - assumes th_in: "th \ threads s" - shows "cp s th \ Max ((\ th. (preced th s)) ` threads s)" -proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def) - show "Max ((\th. preced th s) ` ({th} \ {th'. (Th th', Th th) \ (RAG (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) \ (RAG (wq s))\<^sup>+} \ {}" by simp - next - from finite_threads - show "finite (threads s)" . - next - from th_in - show "{th} \ {th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ threads s" - apply (auto simp:Domain_def) - apply (rule_tac dm_RAG_threads) - apply (unfold trancl_domain [of "RAG s", symmetric]) - by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def) - qed -qed - -lemma le_cp: - shows "preced th s \ cp s th" -proof(unfold cp_eq_cpreced preced_def cpreced_def, simp) - show "Prc (priority th s) (last_set th s) - \ Max (insert (Prc (priority th s) (last_set th s)) - ((\th. Prc (priority th s) (last_set th s)) ` dependants (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) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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: - 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 - 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 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 - 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 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 - 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 np: "threads s \ {}" - shows "Max (cp s ` readys s) = Max (cp s ` threads s)" -proof(unfold max_cp_eq) - 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 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 tm_in] - have "tm \ readys s \ (\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+)" . - thus ?thesis - proof - assume "\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+ " - then obtain th' where th'_in: "th' \ readys s" - and tm_chain:"(Th tm, Th th') \ (RAG s)\<^sup>+" by auto - have "cp s th' = ?f tm" - proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI) - from dependants_threads finite_threads - show "finite ((\th. preced th s) ` ({th'} \ dependants (wq s) th'))" - by (auto intro:finite_subset) - next - fix p assume p_in: "p \ (\th. preced th s) ` ({th'} \ dependants (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 - show "finite ((\th. preced th s) ` threads s)" by simp - next - from p_in and th'_in and dependants_threads[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'} \ dependants (wq s) th')" - proof - - from tm_chain - have "tm \ dependants (wq s) th'" - by (unfold cs_dependants_def s_RAG_def cs_RAG_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 dependants_threads [of th1] th1_in - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ - (\th. preced th s) ` threads s" - by (auto simp:readys_def) - next - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" by simp - next - from finite_threads - show " finite ((\th. preced th s) ` threads s)" by simp - qed - next - from finite_threads - 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} \ dependants (wq s) tm)" - show "y \ preced tm s" - proof - - { fix y' - assume hy' : "y' \ ((\th. preced th s) ` dependants (wq s) tm)" - have "y' \ preced tm s" - proof(unfold tm_max, rule Max_ge) - from hy' dependants_threads[of tm] - show "y' \ (\th. preced th s) ` threads s" by auto - next - from finite_threads - show "finite ((\th. preced th s) ` threads s)" by simp - qed - } with hy show ?thesis by auto - qed - next - from dependants_threads[of tm] finite_threads - show "finite ((\th. preced th s) ` ({tm} \ dependants (wq s) tm))" - by (auto intro:finite_subset) - next - show "preced tm s \ (\th. preced th s) ` ({tm} \ dependants (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 - 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 - show "finite ((\th. preced th s) ` threads s)" by simp - next - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" - by simp - next - from dependants_threads[of th1] th1_readys - show "(\th. preced th s) ` ({th1} \ dependants (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 - -text {* (* ccc *) \noindent - Since the current precedence of the threads in ready queue will always be boosted, - there must be one inside it has the maximum precedence of the whole system. -*} -lemma max_cp_readys_threads: - 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 False]) -qed - -end - -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 (RAG s))" -apply(simp add: detached_def Field_def) -apply(simp add: s_RAG_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 - -context valid_trace -begin - -lemma detached_intro: - assumes eq_pv: "cntP s th = cntV s th" - shows "detached s th" -proof - - from cnp_cnv_cncs - 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 dm_RAG_threads - show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff) - next - assume "th \ readys s" - moreover have "Th th \ Range (RAG s)" - proof - - from card_0_eq [OF finite_holding] 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_RAG_def) - done - qed - ultimately show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def) - qed -qed - -lemma detached_elim: - assumes dtc: "detached s th" - shows "cntP s th = cntV s th" -proof - - from cnp_cnv_cncs - 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_RAG_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_RAG_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: - shows "(detached s th) = (cntP s th = cntV s th)" - by (insert vt, auto intro:detached_intro detached_elim) - -end - -text {* - The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived - from the concise and miniature model of PIP given in PrioGDef.thy. -*} - -lemma eq_dependants: "dependants (wq s) = dependants s" - by (simp add: s_dependants_abv wq_def) - -lemma next_th_unique: - assumes nt1: "next_th s th cs th1" - and nt2: "next_th s th cs th2" - shows "th1 = th2" -using assms by (unfold next_th_def, auto) - -lemma birth_time_lt: "s \ [] \ last_set th s < length s" - apply (induct s, simp) -proof - - fix a s - assume ih: "s \ [] \ last_set th s < length s" - and eq_as: "a # s \ []" - show "last_set th (a # s) < length (a # s)" - proof(cases "s \ []") - case False - from False show ?thesis - by (cases a, auto simp:last_set.simps) - next - case True - from ih [OF True] show ?thesis - by (cases a, auto simp:last_set.simps) - qed -qed - -lemma th_in_ne: "th \ threads s \ s \ []" - by (induct s, auto simp:threads.simps) - -lemma preced_tm_lt: "th \ threads s \ preced th s = Prc x y \ y < length s" - apply (drule_tac th_in_ne) - by (unfold preced_def, auto intro: birth_time_lt) - -lemma inj_the_preced: - "inj_on (the_preced s) (threads s)" - by (metis inj_onI preced_unique the_preced_def) - -lemma tRAG_alt_def: - "tRAG s = {(Th th1, Th th2) | th1 th2. - \ cs. (Th th1, Cs cs) \ RAG s \ (Cs cs, Th th2) \ RAG s}" - by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def) - -lemma tRAG_Field: - "Field (tRAG s) \ Field (RAG s)" - by (unfold tRAG_alt_def Field_def, auto) - -lemma tRAG_ancestorsE: - assumes "x \ ancestors (tRAG s) u" - obtains th where "x = Th th" -proof - - from assms have "(u, x) \ (tRAG s)^+" - by (unfold ancestors_def, auto) - from tranclE[OF this] obtain c where "(c, x) \ tRAG s" by auto - then obtain th where "x = Th th" - by (unfold tRAG_alt_def, auto) - from that[OF this] show ?thesis . -qed - -lemma tRAG_mono: - assumes "RAG s' \ RAG s" - shows "tRAG s' \ tRAG s" - using assms - by (unfold tRAG_alt_def, auto) - -lemma holding_next_thI: - assumes "holding s th cs" - and "length (wq s cs) > 1" - obtains th' where "next_th s th cs th'" -proof - - from assms(1)[folded eq_holding, unfolded cs_holding_def] - have " th \ set (wq s cs) \ th = hd (wq s cs)" . - then obtain rest where h1: "wq s cs = th#rest" - by (cases "wq s cs", auto) - with assms(2) have h2: "rest \ []" by auto - let ?th' = "hd (SOME q. distinct q \ set q = set rest)" - have "next_th s th cs ?th'" using h1(1) h2 - by (unfold next_th_def, auto) - from that[OF this] show ?thesis . -qed - -lemma RAG_tRAG_transfer: - assumes "vt s'" - assumes "RAG s = RAG s' \ {(Th th, Cs cs)}" - and "(Cs cs, Th th'') \ RAG s'" - shows "tRAG s = tRAG s' \ {(Th th, Th th'')}" (is "?L = ?R") -proof - - interpret vt_s': valid_trace "s'" using assms(1) - by (unfold_locales, simp) - interpret rtree: rtree "RAG s'" - proof - show "single_valued (RAG s')" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:vt_s'.unique_RAG) - - show "acyclic (RAG s')" - by (rule vt_s'.acyclic_RAG) - qed - { fix n1 n2 - assume "(n1, n2) \ ?L" - from this[unfolded tRAG_alt_def] - obtain th1 th2 cs' where - h: "n1 = Th th1" "n2 = Th th2" - "(Th th1, Cs cs') \ RAG s" - "(Cs cs', Th th2) \ RAG s" by auto - from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \ RAG s'" by auto - from h(3) and assms(2) - have "(Th th1, Cs cs') = (Th th, Cs cs) \ - (Th th1, Cs cs') \ RAG s'" by auto - hence "(n1, n2) \ ?R" - proof - assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)" - hence eq_th1: "th1 = th" by simp - moreover have "th2 = th''" - proof - - from h1 have "cs' = cs" by simp - from assms(3) cs_in[unfolded this] rtree.sgv - show ?thesis - by (unfold single_valued_def, auto) - qed - ultimately show ?thesis using h(1,2) by auto - next - assume "(Th th1, Cs cs') \ RAG s'" - with cs_in have "(Th th1, Th th2) \ tRAG s'" - by (unfold tRAG_alt_def, auto) - from this[folded h(1, 2)] show ?thesis by auto - qed - } moreover { - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \tRAG s' \ (n1, n2) = (Th th, Th th'')" by auto - hence "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ tRAG s'" - moreover have "... \ ?L" - proof(rule tRAG_mono) - show "RAG s' \ RAG s" by (unfold assms(2), auto) - qed - ultimately show ?thesis by auto - next - assume eq_n: "(n1, n2) = (Th th, Th th'')" - from assms(2, 3) have "(Cs cs, Th th'') \ RAG s" by auto - moreover have "(Th th, Cs cs) \ RAG s" using assms(2) by auto - ultimately show ?thesis - by (unfold eq_n tRAG_alt_def, auto) - qed - } ultimately show ?thesis by auto -qed - -context valid_trace -begin - -lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt] - -end - -lemma cp_alt_def: - "cp s th = - Max ((the_preced s) ` {th'. Th th' \ (subtree (RAG s) (Th th))})" -proof - - have "Max (the_preced s ` ({th} \ dependants (wq s) th)) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "Max (_ ` ?L) = Max (_ ` ?R)") - proof - - have "?L = ?R" - by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def) - thus ?thesis by simp - qed - thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp) -qed - -lemma cp_gen_alt_def: - "cp_gen s = (Max \ (\x. (the_preced s \ the_thread) ` subtree (tRAG s) x))" - by (auto simp:cp_gen_def) - -lemma tRAG_nodeE: - assumes "(n1, n2) \ tRAG s" - obtains th1 th2 where "n1 = Th th1" "n2 = Th th2" - using assms - by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def) - -lemma subtree_nodeE: - assumes "n \ subtree (tRAG s) (Th th)" - obtains th1 where "n = Th th1" -proof - - show ?thesis - proof(rule subtreeE[OF assms]) - assume "n = Th th" - from that[OF this] show ?thesis . - next - assume "Th th \ ancestors (tRAG s) n" - hence "(n, Th th) \ (tRAG s)^+" by (auto simp:ancestors_def) - hence "\ th1. n = Th th1" - proof(induct) - case (base y) - from tRAG_nodeE[OF this] show ?case by metis - next - case (step y z) - thus ?case by auto - qed - with that show ?thesis by auto - qed -qed - -lemma tRAG_star_RAG: "(tRAG s)^* \ (RAG s)^*" -proof - - have "(wRAG s O hRAG s)^* \ (RAG s O RAG s)^*" - by (rule rtrancl_mono, auto simp:RAG_split) - also have "... \ ((RAG s)^*)^*" - by (rule rtrancl_mono, auto) - also have "... = (RAG s)^*" by simp - finally show ?thesis by (unfold tRAG_def, simp) -qed - -lemma tRAG_subtree_RAG: "subtree (tRAG s) x \ subtree (RAG s) x" -proof - - { fix a - assume "a \ subtree (tRAG s) x" - hence "(a, x) \ (tRAG s)^*" by (auto simp:subtree_def) - with tRAG_star_RAG[of s] - have "(a, x) \ (RAG s)^*" by auto - hence "a \ subtree (RAG s) x" by (auto simp:subtree_def) - } thus ?thesis by auto -qed - -lemma tRAG_trancl_eq: - "{th'. (Th th', Th th) \ (tRAG s)^+} = - {th'. (Th th', Th th) \ (RAG s)^+}" - (is "?L = ?R") -proof - - { fix th' - assume "th' \ ?L" - hence "(Th th', Th th) \ (tRAG s)^+" by auto - from tranclD[OF this] - obtain z where h: "(Th th', z) \ tRAG s" "(z, Th th) \ (tRAG s)\<^sup>*" by auto - from tRAG_subtree_RAG[of s] and this(2) - have "(z, Th th) \ (RAG s)^*" by (meson subsetCE tRAG_star_RAG) - moreover from h(1) have "(Th th', z) \ (RAG s)^+" using tRAG_alt_def by auto - ultimately have "th' \ ?R" by auto - } moreover - { fix th' - assume "th' \ ?R" - hence "(Th th', Th th) \ (RAG s)^+" by (auto) - from plus_rpath[OF this] - obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \ []" by auto - hence "(Th th', Th th) \ (tRAG s)^+" - proof(induct xs arbitrary:th' th rule:length_induct) - case (1 xs th' th) - then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto) - show ?case - proof(cases "xs1") - case Nil - from 1(2)[unfolded Cons1 Nil] - have rp: "rpath (RAG s) (Th th') [x1] (Th th)" . - hence "(Th th', x1) \ (RAG s)" by (cases, simp) - then obtain cs where "x1 = Cs cs" - by (unfold s_RAG_def, auto) - from rpath_nnl_lastE[OF rp[unfolded this]] - show ?thesis by auto - next - case (Cons x2 xs2) - from 1(2)[unfolded Cons1[unfolded this]] - have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" . - from rpath_edges_on[OF this] - have eds: "edges_on (Th th' # x1 # x2 # xs2) \ RAG s" . - have "(Th th', x1) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - with eds have rg1: "(Th th', x1) \ RAG s" by auto - then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto) - have "(x1, x2) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - from this eds - have rg2: "(x1, x2) \ RAG s" by auto - from this[unfolded eq_x1] - obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto) - from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2] - have rt1: "(Th th', Th th1) \ tRAG s" by (unfold tRAG_alt_def, auto) - from rp have "rpath (RAG s) x2 xs2 (Th th)" - by (elim rpath_ConsE, simp) - from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" . - show ?thesis - proof(cases "xs2 = []") - case True - from rpath_nilE[OF rp'[unfolded this]] - have "th1 = th" by auto - from rt1[unfolded this] show ?thesis by auto - next - case False - from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons] - have "(Th th1, Th th) \ (tRAG s)\<^sup>+" by simp - with rt1 show ?thesis by auto - qed - qed - qed - hence "th' \ ?L" by auto - } ultimately show ?thesis by blast -qed - -lemma tRAG_trancl_eq_Th: - "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = - {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" - using tRAG_trancl_eq by auto - -lemma dependants_alt_def: - "dependants s th = {th'. (Th th', Th th) \ (tRAG s)^+}" - by (metis eq_RAG s_dependants_def tRAG_trancl_eq) - -context valid_trace -begin - -lemma count_eq_tRAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using assms count_eq_dependants dependants_alt_def eq_dependants by auto - -lemma count_eq_RAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (RAG s)^+} = {}" - using assms count_eq_dependants cs_dependants_def eq_RAG by auto - -lemma count_eq_RAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (RAG s)^+} = {}" - using count_eq_RAG_plus[OF assms] by auto - -lemma count_eq_tRAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using count_eq_tRAG_plus[OF assms] by auto - -end - -lemma tRAG_subtree_eq: - "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \ (subtree (RAG s) (Th th))}" - (is "?L = ?R") -proof - - { fix n - assume h: "n \ ?L" - hence "n \ ?R" - by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) - } moreover { - fix n - assume "n \ ?R" - then obtain th' where h: "n = Th th'" "(Th th', Th th) \ (RAG s)^*" - by (auto simp:subtree_def) - from rtranclD[OF this(2)] - have "n \ ?L" - proof - assume "Th th' \ Th th \ (Th th', Th th) \ (RAG s)\<^sup>+" - with h have "n \ {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" by auto - thus ?thesis using subtree_def tRAG_trancl_eq by fastforce - qed (insert h, auto simp:subtree_def) - } ultimately show ?thesis by auto -qed - -lemma threads_set_eq: - "the_thread ` (subtree (tRAG s) (Th th)) = - {th'. Th th' \ (subtree (RAG s) (Th th))}" (is "?L = ?R") - by (auto intro:rev_image_eqI simp:tRAG_subtree_eq) - -lemma cp_alt_def1: - "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))" -proof - - have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) = - ((the_preced s \ the_thread) ` subtree (tRAG s) (Th th))" - by auto - thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto) -qed - -lemma cp_gen_def_cond: - assumes "x = Th th" - shows "cp s th = cp_gen s (Th th)" -by (unfold cp_alt_def1 cp_gen_def, simp) - -lemma cp_gen_over_set: - assumes "\ x \ A. \ th. x = Th th" - shows "cp_gen s ` A = (cp s \ the_thread) ` A" -proof(rule f_image_eq) - fix a - assume "a \ A" - from assms[rule_format, OF this] - obtain th where eq_a: "a = Th th" by auto - show "cp_gen s a = (cp s \ the_thread) a" - by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp) -qed - - -context valid_trace -begin - -lemma RAG_threads: - assumes "(Th th) \ Field (RAG s)" - shows "th \ threads s" - using assms - by (metis Field_def UnE dm_RAG_threads range_in vt) - -lemma subtree_tRAG_thread: - assumes "th \ threads s" - shows "subtree (tRAG s) (Th th) \ Th ` threads s" (is "?L \ ?R") -proof - - have "?L = {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - by (unfold tRAG_subtree_eq, simp) - also have "... \ ?R" - proof - fix x - assume "x \ {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - then obtain th' where h: "x = Th th'" "Th th' \ subtree (RAG s) (Th th)" by auto - from this(2) - show "x \ ?R" - proof(cases rule:subtreeE) - case 1 - thus ?thesis by (simp add: assms h(1)) - next - case 2 - thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) - qed - qed - finally show ?thesis . -qed - -lemma readys_root: - assumes "th \ readys s" - shows "root (RAG s) (Th th)" -proof - - { fix x - assume "x \ ancestors (RAG s) (Th th)" - hence h: "(Th th, x) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - obtain z where "(Th th, z) \ RAG s" by auto - with assms(1) have False - apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def) - by (fold wq_def, blast) - } thus ?thesis by (unfold root_def, auto) -qed - -lemma readys_in_no_subtree: - assumes "th \ readys s" - and "th' \ th" - shows "Th th \ subtree (RAG s) (Th th')" -proof - assume "Th th \ subtree (RAG s) (Th th')" - thus False - proof(cases rule:subtreeE) - case 1 - with assms show ?thesis by auto - next - case 2 - with readys_root[OF assms(1)] - show ?thesis by (auto simp:root_def) - qed -qed - -lemma not_in_thread_isolated: - assumes "th \ threads s" - shows "(Th th) \ Field (RAG s)" -proof - assume "(Th th) \ Field (RAG s)" - with dm_RAG_threads and range_in assms - show False by (unfold Field_def, blast) -qed - -lemma wf_RAG: "wf (RAG s)" -proof(rule finite_acyclic_wf) - from finite_RAG show "finite (RAG s)" . -next - from acyclic_RAG show "acyclic (RAG s)" . -qed - -lemma sgv_wRAG: "single_valued (wRAG s)" - using waiting_unique - by (unfold single_valued_def wRAG_def, auto) - -lemma sgv_hRAG: "single_valued (hRAG s)" - using holding_unique - by (unfold single_valued_def hRAG_def, auto) - -lemma sgv_tRAG: "single_valued (tRAG s)" - by (unfold tRAG_def, rule single_valued_relcomp, - insert sgv_wRAG sgv_hRAG, auto) - -lemma acyclic_tRAG: "acyclic (tRAG s)" -proof(unfold tRAG_def, rule acyclic_compose) - show "acyclic (RAG s)" using acyclic_RAG . -next - show "wRAG s \ RAG s" unfolding RAG_split by auto -next - show "hRAG s \ RAG s" unfolding RAG_split by auto -qed - -lemma sgv_RAG: "single_valued (RAG s)" - using unique_RAG by (auto simp:single_valued_def) - -lemma rtree_RAG: "rtree (RAG s)" - using sgv_RAG acyclic_RAG - by (unfold rtree_def rtree_axioms_def sgv_def, auto) - -end - -sublocale valid_trace < rtree_RAG: rtree "RAG s" -proof - show "single_valued (RAG s)" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:unique_RAG) - - show "acyclic (RAG s)" - by (rule acyclic_RAG) -qed - -sublocale valid_trace < rtree_s: rtree "tRAG s" -proof(unfold_locales) - from sgv_tRAG show "single_valued (tRAG s)" . -next - from acyclic_tRAG show "acyclic (tRAG s)" . -qed - -sublocale valid_trace < fsbtRAGs : fsubtree "RAG s" -proof - - show "fsubtree (RAG s)" - proof(intro_locales) - show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] . - next - show "fsubtree_axioms (RAG s)" - proof(unfold fsubtree_axioms_def) - from wf_RAG show "wf (RAG s)" . - qed - qed -qed - -sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s" -proof - - have "fsubtree (tRAG s)" - proof - - have "fbranch (tRAG s)" - proof(unfold tRAG_def, rule fbranch_compose) - show "fbranch (wRAG s)" - proof(rule finite_fbranchI) - from finite_RAG show "finite (wRAG s)" - by (unfold RAG_split, auto) - qed - next - show "fbranch (hRAG s)" - proof(rule finite_fbranchI) - from finite_RAG - show "finite (hRAG s)" by (unfold RAG_split, auto) - qed - qed - moreover have "wf (tRAG s)" - proof(rule wf_subset) - show "wf (RAG s O RAG s)" using wf_RAG - by (fold wf_comp_self, simp) - next - show "tRAG s \ (RAG s O RAG s)" - by (unfold tRAG_alt_def, auto) - qed - ultimately show ?thesis - by (unfold fsubtree_def fsubtree_axioms_def,auto) - qed - from this[folded tRAG_def] show "fsubtree (tRAG s)" . -qed - -lemma Max_UNION: - assumes "finite A" - and "A \ {}" - and "\ M \ f ` A. finite M" - and "\ M \ f ` A. M \ {}" - shows "Max (\x\ A. f x) = Max (Max ` f ` A)" (is "?L = ?R") - using assms[simp] -proof - - have "?L = Max (\(f ` A))" - by (fold Union_image_eq, simp) - also have "... = ?R" - by (subst Max_Union, simp+) - finally show ?thesis . -qed - -lemma max_Max_eq: - assumes "finite A" - and "A \ {}" - and "x = y" - shows "max x (Max A) = Max ({y} \ A)" (is "?L = ?R") -proof - - have "?R = Max (insert y A)" by simp - also from assms have "... = ?L" - by (subst Max.insert, simp+) - finally show ?thesis by simp -qed - -context valid_trace -begin - -(* ddd *) -lemma cp_gen_rec: - assumes "x = Th th" - shows "cp_gen s x = Max ({the_preced s th} \ (cp_gen s) ` children (tRAG s) x)" -proof(cases "children (tRAG s) x = {}") - case True - show ?thesis - by (unfold True cp_gen_def subtree_children, simp add:assms) -next - case False - hence [simp]: "children (tRAG s) x \ {}" by auto - note fsbttRAGs.finite_subtree[simp] - have [simp]: "finite (children (tRAG s) x)" - by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], - rule children_subtree) - { fix r x - have "subtree r x \ {}" by (auto simp:subtree_def) - } note this[simp] - have [simp]: "\x\children (tRAG s) x. subtree (tRAG s) x \ {}" - proof - - from False obtain q where "q \ children (tRAG s) x" by blast - moreover have "subtree (tRAG s) q \ {}" by simp - ultimately show ?thesis by blast - qed - have h: "Max ((the_preced s \ the_thread) ` - ({x} \ \(subtree (tRAG s) ` children (tRAG s) x))) = - Max ({the_preced s th} \ cp_gen s ` children (tRAG s) x)" - (is "?L = ?R") - proof - - let "Max (?f ` (?A \ \ (?g ` ?B)))" = ?L - let "Max (_ \ (?h ` ?B))" = ?R - let ?L1 = "?f ` \(?g ` ?B)" - have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)" - proof - - have "?L1 = ?f ` (\ x \ ?B.(?g x))" by simp - also have "... = (\ x \ ?B. ?f ` (?g x))" by auto - finally have "Max ?L1 = Max ..." by simp - also have "... = Max (Max ` (\x. ?f ` subtree (tRAG s) x) ` ?B)" - by (subst Max_UNION, simp+) - also have "... = Max (cp_gen s ` children (tRAG s) x)" - by (unfold image_comp cp_gen_alt_def, simp) - finally show ?thesis . - qed - show ?thesis - proof - - have "?L = Max (?f ` ?A \ ?L1)" by simp - also have "... = max (the_preced s (the_thread x)) (Max ?L1)" - by (subst Max_Un, simp+) - also have "... = max (?f x) (Max (?h ` ?B))" - by (unfold eq_Max_L1, simp) - also have "... =?R" - by (rule max_Max_eq, (simp)+, unfold assms, simp) - finally show ?thesis . - qed - qed thus ?thesis - by (fold h subtree_children, unfold cp_gen_def, simp) -qed - -lemma cp_rec: - "cp s th = Max ({the_preced s th} \ - (cp s o the_thread) ` children (tRAG s) (Th th))" -proof - - have "Th th = Th th" by simp - note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this] - show ?thesis - proof - - have "cp_gen s ` children (tRAG s) (Th th) = - (cp s \ the_thread) ` children (tRAG s) (Th th)" - proof(rule cp_gen_over_set) - show " \x\children (tRAG s) (Th th). \th. x = Th th" - by (unfold tRAG_alt_def, auto simp:children_def) - qed - thus ?thesis by (subst (1) h(1), unfold h(2), simp) - qed -qed - -end - -(* keep *) -lemma next_th_holding: - assumes vt: "vt s" - and nxt: "next_th s th cs th'" - shows "holding (wq s) th cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - thus ?thesis - by (unfold cs_holding_def, auto) -qed - -context valid_trace -begin - -lemma next_th_waiting: - assumes nxt: "next_th s th cs th'" - shows "waiting (wq s) th' cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - from wq_distinct[of cs, unfolded h] - have dst: "distinct (th # rest)" . - have in_rest: "th' \ set rest" - proof(unfold h, rule someI2) - show "distinct rest \ set rest = set rest" using dst by auto - next - fix x assume "distinct x \ set x = set rest" - with h(2) - show "hd x \ set (rest)" by (cases x, auto) - qed - hence "th' \ set (wq s cs)" by (unfold h(1), auto) - moreover have "th' \ hd (wq s cs)" - by (unfold h(1), insert in_rest dst, auto) - ultimately show ?thesis by (auto simp:cs_waiting_def) -qed - -lemma next_th_RAG: - assumes nxt: "next_th (s::event list) th cs th'" - shows "{(Cs cs, Th th), (Th th', Cs cs)} \ RAG s" - using vt assms next_th_holding next_th_waiting - by (unfold s_RAG_def, simp) - -end - --- {* A useless definition *} -definition cps:: "state \ (thread \ precedence) set" -where "cps s = {(th, cp s th) | th . th \ threads s}" - - -find_theorems "waiting" holding -context valid_trace -begin - -find_theorems "waiting" holding - -end - -end diff -r ed938e2246b9 -r 0525670d8e6a .hgignore --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/.hgignore Fri Jan 29 09:46:07 2016 +0800 @@ -0,0 +1,6 @@ +syntax: glob +*.*~ +log +*.patch +*.hg +*.*.orig diff -r ed938e2246b9 -r 0525670d8e6a Correctness.thy --- a/Correctness.thy Thu Jan 28 21:14:17 2016 +0800 +++ b/Correctness.thy Fri Jan 29 09:46:07 2016 +0800 @@ -694,7 +694,7 @@ characterisationof the blocking thread @{text "th'"}: *} - + lemma runing_inversion: (* ddd, one of the main lemmas to present *) assumes runing': "th' \ runing (t@s)" and neq_th: "th' \ th" diff -r ed938e2246b9 -r 0525670d8e6a CpsG - 副本.thy~ --- a/CpsG - 副本.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,4410 +0,0 @@ -theory CpsG -imports PIPDefs -begin - -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 - -(* I am going to use this file as a start point to retrofiting - PIPBasics.thy, which is originally called CpsG.ghy *) - -locale valid_trace = - fixes s - assumes vt : "vt s" - -locale valid_trace_e = valid_trace + - fixes e - assumes vt_e: "vt (e#s)" -begin - -lemma pip_e: "PIP s e" - using vt_e by (cases, simp) - -end - -locale valid_trace_create = valid_trace_e + - fixes th prio - assumes is_create: "e = Create th prio" - -locale valid_trace_exit = valid_trace_e + - fixes th - assumes is_exit: "e = Exit th" - -locale valid_trace_p = valid_trace_e + - fixes th cs - assumes is_p: "e = P th cs" - -locale valid_trace_v = valid_trace_e + - fixes th cs - assumes is_v: "e = V th cs" -begin - definition "rest = tl (wq s cs)" - definition "wq' = (SOME q. distinct q \ set q = set rest)" -end - -locale valid_trace_v_n = valid_trace_v + - assumes rest_nnl: "rest \ []" - -locale valid_trace_v_e = valid_trace_v + - assumes rest_nil: "rest = []" - -locale valid_trace_set= valid_trace_e + - fixes th prio - assumes is_set: "e = Set th prio" - -context valid_trace -begin - -lemma ind [consumes 0, case_names Nil Cons, induct type]: - assumes "PP []" - and "(\s e. valid_trace_e s e \ - PP s \ PIP s e \ PP (e # s))" - shows "PP s" -proof(induct rule:vt.induct[OF vt, case_names Init Step]) - case Init - from assms(1) show ?case . -next - case (Step s e) - show ?case - proof(rule assms(2)) - show "valid_trace_e s e" using Step by (unfold_locales, auto) - next - show "PP s" using Step by simp - next - show "PIP s e" using Step by simp - qed -qed - -end - - -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 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 [simp]: - "cs \ cs' \ wq (V thread cs#s) cs' = wq s cs'" - by (auto simp:wq_def Let_def cp_def split:list.splits) - -lemma runing_head: - assumes "th \ runing s" - and "th \ set (wq_fun (schs s) cs)" - shows "th = hd (wq_fun (schs s) cs)" - using assms - by (simp add:runing_def readys_def s_waiting_def wq_def) - -context valid_trace -begin - -lemma runing_wqE: - assumes "th \ runing s" - and "th \ set (wq s cs)" - obtains rest where "wq s cs = th#rest" -proof - - from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest" - by (meson list.set_cases) - have "th' = th" - proof(rule ccontr) - assume "th' \ th" - hence "th \ hd (wq s cs)" using eq_wq by auto - with assms(2) - have "waiting s th cs" - by (unfold s_waiting_def, fold wq_def, auto) - with assms show False - by (unfold runing_def readys_def, auto) - qed - with eq_wq that show ?thesis by metis -qed - -end - -context valid_trace_p -begin - -lemma wq_neq_simp [simp]: - assumes "cs' \ cs" - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_p wq_def - by (auto simp:Let_def) - -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_p] - show ?thesis by (cases, simp) -qed - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - -lemma th_not_in_wq: - shows "th \ set (wq s cs)" -proof - assume otherwise: "th \ set (wq s cs)" - from runing_wqE[OF runing_th_s this] - obtain rest where eq_wq: "wq s cs = th#rest" by blast - with otherwise - have "holding s th cs" - by (unfold s_holding_def, fold wq_def, simp) - hence cs_th_RAG: "(Cs cs, Th th) \ RAG s" - by (unfold s_RAG_def, fold holding_eq, auto) - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - with cs_th_RAG show ?thesis by auto - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq s cs @ [th]" - by (unfold is_p wq_def, auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" -proof(cases "cs' = cs") - case True - show ?thesis using True assms th_not_in_wq - by (unfold True wq_es_cs, auto) -qed (insert assms, simp) - -end - - -context valid_trace_v -begin - -lemma wq_neq_simp [simp]: - assumes "cs' \ cs" - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_v wq_def - by (auto simp:Let_def) - -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_v] - show ?thesis by (cases, simp) -qed - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - -lemma wq_s_cs: - "wq s cs = th#rest" -proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from this(2) show ?thesis - by (unfold rest_def s_holding_def, fold wq_def, - metis empty_iff list.collapse list.set(1)) - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq'" - using wq_s_cs[unfolded wq_def] - by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" -proof(cases "cs' = cs") - case True - show ?thesis - proof(unfold True wq_es_cs wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - using assms[unfolded True wq_s_cs] by auto - qed simp -qed (insert assms, simp) - -end - -context valid_trace -begin - -lemma actor_inv: - assumes "PIP s e" - and "\ isCreate e" - shows "actor e \ runing s" - using assms - by (induct, auto) - -lemma isP_E: - assumes "isP e" - obtains cs where "e = P (actor e) cs" - using assms by (cases e, auto) - -lemma isV_E: - assumes "isV e" - obtains cs where "e = V (actor e) cs" - using assms by (cases e, auto) - -lemma wq_distinct: "distinct (wq s cs)" -proof(induct rule:ind) - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (V th cs) - interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) - qed -qed (unfold wq_def Let_def, simp) - -end - -context valid_trace_e -begin - -text {* - The following lemma shows that only the @{text "P"} - operation can add new thread into waiting queues. - Such kind of lemmas are very obvious, but need to be checked formally. - This is a kind of confirmation that our modelling is correct. -*} - -lemma wq_in_inv: - assumes s_ni: "thread \ set (wq s cs)" - and s_i: "thread \ set (wq (e#s) cs)" - shows "e = P thread cs" -proof(cases e) - -- {* This is the only non-trivial case: *} - case (V th cs1) - have False - proof(cases "cs1 = cs") - case True - show ?thesis - proof(cases "(wq s cs1)") - case (Cons w_hd w_tl) - have "set (wq (e#s) cs) \ set (wq s cs)" - proof - - have "(wq (e#s) cs) = (SOME q. distinct q \ set q = set w_tl)" - using Cons V by (auto simp:wq_def Let_def True split:if_splits) - moreover have "set ... \ set (wq s cs)" - proof(rule someI2) - show "distinct w_tl \ set w_tl = set w_tl" - by (metis distinct.simps(2) local.Cons wq_distinct) - qed (insert Cons True, auto) - ultimately show ?thesis by simp - qed - with assms show ?thesis by auto - qed (insert assms V True, auto simp:wq_def Let_def split:if_splits) - qed (insert assms V, auto simp:wq_def Let_def split:if_splits) - thus ?thesis by auto -qed (insert assms, auto simp:wq_def Let_def split:if_splits) - -lemma wq_out_inv: - assumes s_in: "thread \ set (wq s cs)" - and s_hd: "thread = hd (wq s cs)" - and s_i: "thread \ hd (wq (e#s) cs)" - shows "e = V thread cs" -proof(cases e) --- {* There are only two non-trivial cases: *} - case (V th cs1) - show ?thesis - proof(cases "cs1 = cs") - case True - have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] . - thus ?thesis - proof(cases) - case (thread_V) - moreover have "th = thread" using thread_V(2) s_hd - by (unfold s_holding_def wq_def, simp) - ultimately show ?thesis using V True by simp - qed - qed (insert assms V, auto simp:wq_def Let_def split:if_splits) -next - case (P th cs1) - show ?thesis - proof(cases "cs1 = cs") - case True - with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]" - by (auto simp:wq_def Let_def split:if_splits) - with s_i s_hd s_in have False - by (metis empty_iff hd_append2 list.set(1) wq_def) - thus ?thesis by simp - qed (insert assms P, auto simp:wq_def Let_def split:if_splits) -qed (insert assms, auto simp:wq_def Let_def split:if_splits) - -end - - -context valid_trace -begin - -end - - - -context valid_trace -begin - - -text {* (* ddd *) - The nature of the work is like this: since it starts from a very simple and basic - model, even intuitively very `basic` and `obvious` properties need to derived from scratch. - For instance, the fact - that one thread can not be blocked by two critical resources at the same time - is obvious, because only running threads can make new requests, if one is waiting for - a critical resource and get blocked, it can not make another resource request and get - blocked the second time (because it is not running). - - To derive this fact, one needs to prove by contraction and - reason about time (or @{text "moement"}). The reasoning is based on a generic theorem - named @{text "p_split"}, which is about status changing along the time axis. It says if - a condition @{text "Q"} is @{text "True"} at a state @{text "s"}, - but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} - in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history - of events leading to it), such that @{text "Q"} switched - from being @{text "False"} to @{text "True"} and kept being @{text "True"} - till the last moment of @{text "s"}. - - Suppose a thread @{text "th"} is blocked - on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, - since no thread is blocked at the very beginning, by applying - @{text "p_split"} to these two blocking facts, there exist - two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that - @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} - and kept on blocked on them respectively ever since. - - Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}. - However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still - in blocked state at moment @{text "t2"} and could not - make any request and get blocked the second time: Contradiction. -*} - -lemma waiting_unique_pre: (* ddd *) - assumes 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: "\ ?Q cs1 (moment t1 s)" - and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" by auto - from p_split [of "?Q cs2", OF q2 nq2] - obtain t2 where lt2: "t2 < length s" - and np2: "\ ?Q cs2 (moment t2 s)" - and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" by auto - { fix s cs - assume q: "?Q cs s" - have "thread \ runing s" - proof - assume "thread \ runing s" - hence " \cs. \ (thread \ set (wq_fun (schs s) cs) \ - thread \ hd (wq_fun (schs s) cs))" - by (unfold runing_def s_waiting_def readys_def, auto) - from this[rule_format, of cs] q - show False by (simp add: wq_def) - qed - } note q_not_runing = this - { fix t1 t2 cs1 cs2 - assume lt1: "t1 < length s" - and np1: "\ ?Q cs1 (moment t1 s)" - and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" - and lt2: "t2 < length s" - and np2: "\ ?Q cs2 (moment t2 s)" - and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" - and 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#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - have ?thesis - proof - - have "thread \ runing (moment t2 s)" - proof(cases "thread \ set (wq (moment t2 s) cs2)") - case True - have "e = V thread cs2" - proof - - have eq_th: "thread = hd (wq (moment t2 s) cs2)" - using True and np2 by auto - from vt_e.wq_out_inv[OF True this h2] - show ?thesis . - qed - thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto - next - case False - have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] . - with vt_e.actor_inv[OF vt_e.pip_e] - show ?thesis by auto - qed - moreover have "thread \ runing (moment t2 s)" - by (rule q_not_runing[OF nn1[rule_format, OF lt12]]) - ultimately show ?thesis by simp - qed - } note lt_case = this - show ?thesis - proof - - { assume "t1 < t2" - from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this] - have ?thesis . - } moreover { - assume "t2 < t1" - from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this] - have ?thesis . - } moreover { - assume eq_12: "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 lt_2: "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 - from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12] - have g1: "thread \ set (wq (e#moment t1 s) cs1)" and - g2: "thread \ hd (wq (e#moment t1 s) cs1)" by auto - have "vt (e#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - have "e = V thread cs2 \ e = P thread cs2" - proof(cases "thread \ set (wq (moment t2 s) cs2)") - case True - have "e = V thread cs2" - proof - - have eq_th: "thread = hd (wq (moment t2 s) cs2)" - using True and np2 by auto - from vt_e.wq_out_inv[OF True this h2] - show ?thesis . - qed - thus ?thesis by auto - next - case False - have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] . - thus ?thesis by auto - qed - moreover have "e = V thread cs1 \ e = P thread cs1" - proof(cases "thread \ set (wq (moment t1 s) cs1)") - case True - have eq_th: "thread = hd (wq (moment t1 s) cs1)" - using True and np1 by auto - from vt_e.wq_out_inv[folded eq_12, OF True this g2] - have "e = V thread cs1" . - thus ?thesis by auto - next - case False - have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] . - thus ?thesis by auto - qed - ultimately have ?thesis using neq12 by auto - } ultimately show ?thesis using nat_neq_iff by blast - qed -qed - -text {* - This lemma is a simple corrolary of @{text "waiting_unique_pre"}. -*} - -lemma waiting_unique: - assumes "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 - -end - -(* not used *) -text {* - Every thread can only be blocked on one critical resource, - symmetrically, every critical resource can only be held by one thread. - This fact is much more easier according to our definition. -*} -lemma held_unique: - assumes "holding (s::event list) th1 cs" - and "holding s th2 cs" - shows "th1 = th2" - by (insert assms, unfold s_holding_def, auto) - - -lemma last_set_lt: "th \ threads s \ last_set th s < length s" - apply (induct s, auto) - by (case_tac a, auto split:if_splits) - -lemma last_set_unique: - "\last_set th1 s = last_set th2 s; th1 \ threads s; th2 \ threads s\ - \ th1 = th2" - apply (induct s, auto) - by (case_tac a, auto split:if_splits dest:last_set_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 "last_set th1 s = last_set th2 s" by (simp add:preced_def) - from last_set_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 - -(* An aux lemma used later *) -lemma unique_minus: - 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: - 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: - 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 - -text {* - The following three lemmas show that @{text "RAG"} does not change - by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"} - events, respectively. -*} - -lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - - - -context valid_trace_v -begin - - -lemma distinct_rest: "distinct rest" - by (simp add: distinct_tl rest_def wq_distinct) - -definition "wq' = (SOME q. distinct q \ set q = set rest)" - -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_v] - show ?thesis by (cases, simp) -qed - -lemma holding_cs_eq_th: - assumes "holding s t cs" - shows "t = th" -proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from held_unique[OF this(2) assms] - show ?thesis by simp - qed -qed - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - -lemma wq_s_cs: - "wq s cs = th#rest" -proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from this(2) show ?thesis - by (unfold rest_def s_holding_def, fold wq_def, - metis empty_iff list.collapse list.set(1)) - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq'" - using wq_s_cs[unfolded wq_def] - by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) - -lemma distinct_wq': "distinct wq'" - by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def) - -lemma th'_in_inv: - assumes "th' \ set wq'" - shows "th' \ set rest" - using assms - by (metis (mono_tags, lifting) distinct.simps(2) - rest_def some_eq_ex wq'_def wq_distinct wq_s_cs) - -lemma neq_t_th: - assumes "waiting (e#s) t c" - shows "t \ th" -proof - assume otherwise: "t = th" - show False - proof(cases "c = cs") - case True - have "t \ set wq'" - using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs] - by simp - from th'_in_inv[OF this] have "t \ set rest" . - with wq_s_cs[folded otherwise] wq_distinct[of cs] - show ?thesis by simp - next - case False - have "wq (e#s) c = wq s c" using False - by (unfold is_v, simp) - hence "waiting s t c" using assms - by (simp add: cs_waiting_def waiting_eq) - hence "t \ readys s" by (unfold readys_def, auto) - hence "t \ runing s" using runing_ready by auto - with runing_th_s[folded otherwise] show ?thesis by auto - qed -qed - -lemma waiting_esI1: - assumes "waiting s t c" - and "c \ cs" - shows "waiting (e#s) t c" -proof - - have "wq (e#s) c = wq s c" - using assms(2) is_v by auto - with assms(1) show ?thesis - using cs_waiting_def waiting_eq by auto -qed - -lemma holding_esI2: - assumes "c \ cs" - and "holding s t c" - shows "holding (e#s) t c" -proof - - from assms(1) have "wq (e#s) c = wq s c" using is_v by auto - from assms(2)[unfolded s_holding_def, folded wq_def, - folded this, unfolded wq_def, folded s_holding_def] - show ?thesis . -qed - -lemma holding_esI1: - assumes "holding s t c" - and "t \ th" - shows "holding (e#s) t c" -proof - - have "c \ cs" using assms using holding_cs_eq_th by blast - from holding_esI2[OF this assms(1)] - show ?thesis . -qed - -end - -context valid_trace_v_n -begin - -lemma neq_wq': "wq' \ []" -proof (unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume " distinct x \ set x = set rest" - thus "x \ []" using rest_nnl by auto -qed - -definition "taker = hd wq'" - -definition "rest' = tl wq'" - -lemma eq_wq': "wq' = taker # rest'" - by (simp add: neq_wq' rest'_def taker_def) - -lemma next_th_taker: - shows "next_th s th cs taker" - using rest_nnl taker_def wq'_def wq_s_cs - by (auto simp:next_th_def) - -lemma taker_unique: - assumes "next_th s th cs taker'" - shows "taker' = taker" -proof - - from assms - obtain rest' where - h: "wq s cs = th # rest'" - "taker' = hd (SOME q. distinct q \ set q = set rest')" - by (unfold next_th_def, auto) - with wq_s_cs have "rest' = rest" by auto - thus ?thesis using h(2) taker_def wq'_def by auto -qed - -lemma waiting_set_eq: - "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}" - by (smt all_not_in_conv bot.extremum insertI1 insert_subset - mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique) - -lemma holding_set_eq: - "{(Cs cs, Th th') |th'. next_th s th cs th'} = {(Cs cs, Th taker)}" - using next_th_taker taker_def waiting_set_eq - by fastforce - -lemma holding_taker: - shows "holding (e#s) taker cs" - by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, - auto simp:neq_wq' taker_def) - -lemma waiting_esI2: - assumes "waiting s t cs" - and "t \ taker" - shows "waiting (e#s) t cs" -proof - - have "t \ set wq'" - proof(unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) - next - fix x - assume "distinct x \ set x = set rest" - moreover have "t \ set rest" - using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto - ultimately show "t \ set x" by simp - qed - moreover have "t \ hd wq'" - using assms(2) taker_def by auto - ultimately show ?thesis - by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp) -qed - -lemma waiting_esE: - assumes "waiting (e#s) t c" - obtains "c \ cs" "waiting s t c" - | "c = cs" "t \ taker" "waiting s t cs" "t \ set rest'" -proof(cases "c = cs") - case False - hence "wq (e#s) c = wq s c" using is_v by auto - with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto - from that(1)[OF False this] show ?thesis . -next - case True - from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs] - have "t \ hd wq'" "t \ set wq'" by auto - hence "t \ taker" by (simp add: taker_def) - moreover hence "t \ th" using assms neq_t_th by blast - moreover have "t \ set rest" by (simp add: `t \ set wq'` th'_in_inv) - ultimately have "waiting s t cs" - by (metis cs_waiting_def list.distinct(2) list.sel(1) - list.set_sel(2) rest_def waiting_eq wq_s_cs) - show ?thesis using that(2) - using True `t \ set wq'` `t \ taker` `waiting s t cs` eq_wq' by auto -qed - -lemma holding_esI1: - assumes "c = cs" - and "t = taker" - shows "holding (e#s) t c" - by (unfold assms, simp add: holding_taker) - -lemma holding_esE: - assumes "holding (e#s) t c" - obtains "c = cs" "t = taker" - | "c \ cs" "holding s t c" -proof(cases "c = cs") - case True - from assms[unfolded True, unfolded s_holding_def, - folded wq_def, unfolded wq_es_cs] - have "t = taker" by (simp add: taker_def) - from that(1)[OF True this] show ?thesis . -next - case False - hence "wq (e#s) c = wq s c" using is_v by auto - from assms[unfolded s_holding_def, folded wq_def, - unfolded this, unfolded wq_def, folded s_holding_def] - have "holding s t c" . - from that(2)[OF False this] show ?thesis . -qed - -end - - -context valid_trace_v_n -begin - -lemma nil_wq': "wq' = []" -proof (unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume " distinct x \ set x = set rest" - thus "x = []" using rest_nil by auto -qed - -lemma no_taker: - assumes "next_th s th cs taker" - shows "False" -proof - - from assms[unfolded next_th_def] - obtain rest' where "wq s cs = th # rest'" "rest' \ []" - by auto - thus ?thesis using rest_def rest_nil by auto -qed - -lemma waiting_set_eq: - "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}" - using no_taker by auto - -lemma holding_set_eq: - "{(Cs cs, Th th') |th'. next_th s th cs th'} = {}" - using no_taker by auto - -lemma no_holding: - assumes "holding (e#s) taker cs" - shows False -proof - - from wq_es_cs[unfolded nil_wq'] - have " wq (e # s) cs = []" . - from assms[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis by auto -qed - -lemma no_waiting: - assumes "waiting (e#s) t cs" - shows False -proof - - from wq_es_cs[unfolded nil_wq'] - have " wq (e # s) cs = []" . - from assms[unfolded s_waiting_def, folded wq_def, unfolded this] - show ?thesis by auto -qed - -lemma waiting_esI2: - assumes "waiting s t c" - shows "waiting (e#s) t c" -proof - - have "c \ cs" using assms - using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto - from waiting_esI1[OF assms this] - show ?thesis . -qed - -lemma waiting_esE: - assumes "waiting (e#s) t c" - obtains "c \ cs" "waiting s t c" -proof(cases "c = cs") - case False - hence "wq (e#s) c = wq s c" using is_v by auto - with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto - from that(1)[OF False this] show ?thesis . -next - case True - from no_waiting[OF assms[unfolded True]] - show ?thesis by auto -qed - -lemma holding_esE: - assumes "holding (e#s) t c" - obtains "c \ cs" "holding s t c" -proof(cases "c = cs") - case True - from no_holding[OF assms[unfolded True]] - show ?thesis by auto -next - case False - hence "wq (e#s) c = wq s c" using is_v by auto - from assms[unfolded s_holding_def, folded wq_def, - unfolded this, unfolded wq_def, folded s_holding_def] - have "holding s t c" . - from that[OF False this] show ?thesis . -qed - -end (* ccc *) - -lemma rel_eqI: - assumes "\ x y. (x,y) \ A \ (x,y) \ B" - and "\ x y. (x,y) \ B \ (x, y) \ A" - shows "A = B" - using assms by auto - -lemma in_RAG_E: - assumes "(n1, n2) \ RAG (s::state)" - obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs" - | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs" - using assms[unfolded s_RAG_def, folded waiting_eq holding_eq] - by auto - -context valid_trace_v -begin - -lemma RAG_es: - "RAG (e # s) = - RAG 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 = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - show ?thesis - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from waiting(3) - show ?thesis - proof(cases rule:h_n.waiting_esE) - case 1 - with waiting(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - next - case 2 - with waiting(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from waiting(3) - show ?thesis - proof(cases rule:h_e.waiting_esE) - case 1 - with waiting(1,2) - show ?thesis - by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - qed - qed - next - case (holding th' cs') - show ?thesis - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from holding(3) - show ?thesis - proof(cases rule:h_n.holding_esE) - case 1 - with holding(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - next - case 2 - with holding(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold holding_eq, auto) - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from holding(3) - show ?thesis - proof(cases rule:h_e.holding_esE) - case 1 - with holding(1,2) - show ?thesis - by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, - fold holding_eq, auto) - qed - qed - qed -next - fix n1 n2 - assume h: "(n1, n2) \ ?R" - show "(n1, n2) \ ?L" - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq] - have "((n1, n2) \ RAG s \ (n1 \ Cs cs \ n2 \ Th th) - \ (n1 \ Th h_n.taker \ n2 \ Cs cs)) \ - (n2 = Th h_n.taker \ n1 = Cs cs)" - by auto - thus ?thesis - proof - assume "n2 = Th h_n.taker \ n1 = Cs cs" - with h_n.holding_taker - show ?thesis - by (unfold s_RAG_def, fold holding_eq, auto) - next - assume h: "(n1, n2) \ RAG s \ - (n1 \ Cs cs \ n2 \ Th th) \ (n1 \ Th h_n.taker \ n2 \ Cs cs)" - hence "(n1, n2) \ RAG s" by simp - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from h and this(1,2) - have "th' \ h_n.taker \ cs' \ cs" by auto - hence "waiting (e#s) th' cs'" - proof - assume "cs' \ cs" - from waiting_esI1[OF waiting(3) this] - show ?thesis . - next - assume neq_th': "th' \ h_n.taker" - show ?thesis - proof(cases "cs' = cs") - case False - from waiting_esI1[OF waiting(3) this] - show ?thesis . - next - case True - from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True] - show ?thesis . - qed - qed - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - from h this(1,2) - have "cs' \ cs \ th' \ th" by auto - hence "holding (e#s) th' cs'" - proof - assume "cs' \ cs" - from holding_esI2[OF this holding(3)] - show ?thesis . - next - assume "th' \ th" - from holding_esI1[OF holding(3) this] - show ?thesis . - qed - thus ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq] - have h_s: "(n1, n2) \ RAG s" "(n1, n2) \ (Cs cs, Th th)" - by auto - from h_s(1) - show ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from h_e.waiting_esI2[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - with h_s(2) - have "cs' \ cs \ th' \ th" by auto - thus ?thesis - proof - assume neq_cs: "cs' \ cs" - from holding_esI2[OF this holding(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - next - assume "th' \ th" - from holding_esI1[OF holding(3) this] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed - qed -qed - -end - - - -context valid_trace -begin - -lemma finite_threads: - shows "finite (threads s)" -using vt by (induct) (auto elim: step.cases) - -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 RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" - by (unfold s_RAG_def, auto) - -lemma wq_threads: - assumes h: "th \ set (wq s cs)" - shows "th \ threads s" - - -lemma wq_threads: - assumes 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) - interpret vt_s: valid_trace s - using vt_cons(1) by (unfold_locales, auto) - 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_RAG_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 vt_s.wq_distinct[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 dm_RAG_threads: - assumes in_dom: "(Th th) \ Domain (RAG s)" - shows "th \ threads s" -proof - - from in_dom obtain n where "(Th th, n) \ RAG s" by auto - moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto - ultimately have "(Th th, Cs cs) \ RAG s" by simp - hence "th \ set (wq s cs)" - by (unfold s_RAG_def, auto simp:cs_waiting_def) - from wq_threads [OF this] show ?thesis . -qed - - -lemma cp_le: - assumes th_in: "th \ threads s" - shows "cp s th \ Max ((\ th. (preced th s)) ` threads s)" -proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def) - show "Max ((\th. preced th s) ` ({th} \ {th'. (Th th', Th th) \ (RAG (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) \ (RAG (wq s))\<^sup>+} \ {}" by simp - next - from finite_threads - show "finite (threads s)" . - next - from th_in - show "{th} \ {th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ threads s" - apply (auto simp:Domain_def) - apply (rule_tac dm_RAG_threads) - apply (unfold trancl_domain [of "RAG s", symmetric]) - by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def) - qed -qed - -lemma le_cp: - shows "preced th s \ cp s th" -proof(unfold cp_eq_cpreced preced_def cpreced_def, simp) - show "Prc (priority th s) (last_set th s) - \ Max (insert (Prc (priority th s) (last_set th s)) - ((\th. Prc (priority th s) (last_set th s)) ` dependants (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) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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: - 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 - 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 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 - 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 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 - 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_eq_the_preced: - shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)" - using max_cp_eq using the_preced_def by presburger - -end - -lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s" - by (unfold preced_def, simp) - -lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s" -proof - fix th' - show "the_preced (V th cs # s) th' = the_preced s th'" - by (unfold the_preced_def preced_def, simp) -qed - -lemma step_RAG_v: -assumes vt: - "vt (V th cs#s)" -shows " - RAG (V th cs # s) = - RAG 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 = ?R") -proof - - interpret vt_v: valid_trace_v s "V th cs" - using assms step_back_vt by (unfold_locales, auto) - show ?thesis using vt_v.RAG_es . -qed - - - - - -text {* (* ddd *) - The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed - with the happening of @{text "V"}-events: -*} -lemma step_RAG_v: -assumes vt: - "vt (V th cs#s)" -shows " - RAG (V th cs # s) = - RAG 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_RAG_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 - -text {* - The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed - with the happening of @{text "P"}-events: -*} -lemma step_RAG_p: - "vt (P th cs#s) \ - RAG (P th cs # s) = (if (wq s cs = []) then RAG s \ {(Cs cs, Th th)} - else RAG s \ {(Th th, Cs cs)})" - apply(simp only: s_RAG_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(simp add:s_RAG_def wq_def cs_holding_def) - apply(auto) - done - - -lemma RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" - by (unfold s_RAG_def, auto) - -context valid_trace -begin - -text {* - The following lemma shows that @{text "RAG"} is acyclic. - The overall structure is by induction on the formation of @{text "vt s"} - and then case analysis on event @{text "e"}, where the non-trivial cases - for those for @{text "V"} and @{text "P"} events. -*} -lemma acyclic_RAG: - shows "acyclic (RAG s)" -using vt -proof(induct) - case (vt_cons s e) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - assume ih: "acyclic (RAG 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:RAG_create_unchanged) - next - case (Exit th) - with ih show ?thesis by (simp add:RAG_exit_unchanged) - next - case (V th cs) - from V vt stp have vtt: "vt (V th cs#s)" by auto - from step_RAG_v [OF this] - have eq_de: - "RAG (e # s) = - RAG 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 RAG_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_RAG_def s_waiting_def cs_waiting_def wq_def by simp - hence "cs' = cs" - proof(rule vt_s.waiting_unique) - from eq_wq vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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_RAG_p [OF this] P - have "RAG (e # s) = - (if wq s cs = [] then RAG s \ {(Cs cs, Th th)} else - RAG 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 = RAG s \ {(Cs cs, Th th)}" by simp - have "(Th th, Cs cs) \ (RAG s)\<^sup>*" - proof - assume "(Th th, Cs cs) \ (RAG s)\<^sup>*" - hence "(Th th, Cs cs) \ (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl) - from tranclD2 [OF this] - obtain x where "(x, Cs cs) \ RAG s" by auto - with True show False by (auto simp:s_RAG_def cs_waiting_def) - qed - with acyclic_insert ih eq_r show ?thesis by auto - next - case False - hence eq_r: "?R = RAG s \ {(Th th, Cs cs)}" by simp - have "(Cs cs, Th th) \ (RAG s)\<^sup>*" - proof - assume "(Cs cs, Th th) \ (RAG s)\<^sup>*" - hence "(Cs cs, Th th) \ (RAG 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) \ (RAG 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 RAG_set_unchanged - show ?thesis by (simp add:RAG_set_unchanged) - qed - next - case vt_nil - show "acyclic (RAG ([]::state))" - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -qed - - -lemma finite_RAG: - shows "finite (RAG s)" -proof - - from vt show ?thesis - proof(induct) - case (vt_cons s e) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - assume ih: "finite (RAG 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:RAG_create_unchanged) - next - case (Exit th) - with ih show ?thesis by (simp add:RAG_exit_unchanged) - next - case (V th cs) - from V vt stp have vtt: "vt (V th cs#s)" by auto - from step_RAG_v [OF this] - have eq_de: "RAG (e # s) = - RAG 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 (metis finite.simps) - 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_RAG_p [OF this] P - have "RAG (e # s) = - (if wq s cs = [] then RAG s \ {(Cs cs, Th th)} else - RAG 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 = RAG s \ {(Cs cs, Th th)}" by simp - with True and ih show ?thesis by auto - next - case False - hence "?R = RAG 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:RAG_set_unchanged) - qed - next - case vt_nil - show "finite (RAG ([]::state))" - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) - qed -qed - -text {* Several useful lemmas *} - -lemma wf_dep_converse: - shows "wf ((RAG s)^-1)" -proof(rule finite_acyclic_wf_converse) - from finite_RAG - show "finite (RAG s)" . -next - from acyclic_RAG - show "acyclic (RAG s)" . -qed - -end - -lemma hd_np_in: "x \ set l \ hd l \ set l" - by (induct l, auto) - -lemma th_chasing: "(Th th, Cs cs) \ RAG (s::state) \ \ th'. (Cs cs, Th th') \ RAG s" - by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in) - -context valid_trace -begin - -lemma wq_threads: - assumes 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) - interpret vt_s: valid_trace s - using vt_cons(1) by (unfold_locales, auto) - 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_RAG_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 vt_s.wq_distinct[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: "\(Th th) \ Range (RAG (s::state))\ \ th \ threads s" - apply(unfold s_RAG_def cs_waiting_def cs_holding_def) - by (auto intro:wq_threads) - -lemma readys_v_eq: - assumes neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and not_in: "th \ set rest" - shows "(th \ readys (V thread cs#s)) = (th \ readys s)" -proof - - from assms show ?thesis - apply (auto simp:readys_def) - apply(simp add:s_waiting_def[folded wq_def]) - apply (erule_tac x = csa in allE) - apply (simp add:s_waiting_def wq_def Let_def split:if_splits) - apply (case_tac "csa = cs", simp) - apply (erule_tac x = cs in allE) - apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits) - apply(auto simp add: wq_def) - apply (auto simp:s_waiting_def wq_def Let_def split:list.splits) - proof - - assume th_nin: "th \ set rest" - and th_in: "th \ set (SOME q. distinct q \ set q = set rest)" - and eq_wq: "wq_fun (schs s) cs = thread # rest" - have "set (SOME q. distinct q \ set q = set rest) = set rest" - proof(rule someI2) - from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def] - show "distinct rest \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ set x = set rest" by auto - qed - with th_nin th_in show False by auto - qed -qed - -text {* \noindent - The following lemmas shows that: starting from any node in @{text "RAG"}, - by chasing out-going edges, it is always possible to reach a node representing a ready - thread. In this lemma, it is the @{text "th'"}. -*} - -lemma chain_building: - shows "node \ Domain (RAG s) \ (\ th'. th' \ readys s \ (node, Th th') \ (RAG s)^+)" -proof - - from wf_dep_converse - have h: "wf ((RAG s)\)" . - show ?thesis - proof(induct rule:wf_induct [OF h]) - fix x - assume ih [rule_format]: - "\y. (y, x) \ (RAG s)\ \ - y \ Domain (RAG s) \ (\th'. th' \ readys s \ (y, Th th') \ (RAG s)\<^sup>+)" - show "x \ Domain (RAG s) \ (\th'. th' \ readys s \ (x, Th th') \ (RAG s)\<^sup>+)" - proof - assume x_d: "x \ Domain (RAG s)" - show "\th'. th' \ readys s \ (x, Th th') \ (RAG s)\<^sup>+" - proof(cases x) - case (Th th) - from x_d Th obtain cs where x_in: "(Th th, Cs cs) \ RAG s" by (auto simp:s_RAG_def) - with Th have x_in_r: "(Cs cs, x) \ (RAG s)^-1" by simp - from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \ RAG s" by blast - hence "Cs cs \ Domain (RAG 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') \ (RAG s)\<^sup>+" by auto - have "(x, Th th') \ (RAG 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) \ (RAG s)^-1" by (auto simp:s_RAG_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 have "th' \ threads s" by auto - with False have "Th th' \ Domain (RAG s)" - by (auto simp:readys_def wq_def s_waiting_def s_RAG_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'') \ (RAG s)\<^sup>+" by auto - from th'_d and th''_in - have "(x, Th th'') \ (RAG s)\<^sup>+" by auto - with th''_r show ?thesis by auto - qed - qed - qed - qed -qed - -text {* \noindent - The following is just an instance of @{text "chain_building"}. -*} -lemma th_chain_to_ready: - assumes th_in: "th \ threads s" - shows "th \ readys s \ (\ th'. th' \ readys s \ (Th th, Th th') \ (RAG s)^+)" -proof(cases "th \ readys s") - case True - thus ?thesis by auto -next - case False - from False and th_in have "Th th \ Domain (RAG s)" - by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def) - from chain_building [rule_format, OF this] - show ?thesis by auto -qed - -end - - - -lemma holding_unique: "\holding (s::state) th1 cs; holding s th2 cs\ \ th1 = th2" - by (unfold s_holding_def cs_holding_def, auto) - -context valid_trace -begin - -lemma unique_RAG: "\(n, n1) \ RAG s; (n, n2) \ RAG s\ \ n1 = n2" - apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq) - by(auto elim:waiting_unique holding_unique) - -end - - -lemma trancl_split: "(a, b) \ r^+ \ \ c. (a, c) \ r" -by (induct rule:trancl_induct, auto) - -context valid_trace -begin - -lemma dchain_unique: - assumes th1_d: "(n, Th th1) \ (RAG s)^+" - and th1_r: "th1 \ readys s" - and th2_d: "(n, Th th2) \ (RAG 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_RAG - have "(Th th1, Th th2) \ (RAG s)\<^sup>+ \ (Th th2, Th th1) \ (RAG s)\<^sup>+" by auto - hence "False" - proof - assume "(Th th1, Th th2) \ (RAG s)\<^sup>+" - from trancl_split [OF this] - obtain n where dd: "(Th th1, n) \ RAG s" by auto - then obtain cs where eq_n: "n = Cs cs" - by (auto simp:s_RAG_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_RAG_def wq_def s_waiting_def cs_waiting_def) - with th1_r show ?thesis by auto - next - assume "(Th th2, Th th1) \ (RAG s)\<^sup>+" - from trancl_split [OF this] - obtain n where dd: "(Th th2, n) \ RAG s" by auto - then obtain cs where eq_n: "n = Cs cs" - by (auto simp:s_RAG_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_RAG_def s_waiting_def cs_waiting_def) - with th2_r show ?thesis by auto - qed - } thus ?thesis by auto -qed - -end - - -lemma step_holdents_p_add: - 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_RAG_p[OF vt] by (auto) -qed - -lemma step_holdents_p_eq: - 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_RAG_p[OF vt] by auto -qed - - -lemma (in valid_trace) finite_holding : - shows "finite (holdents s th)" -proof - - let ?F = "\ (x, y). the_cs x" - from finite_RAG - have "finite (RAG s)" . - hence "finite (?F `(RAG s))" by simp - moreover have "{cs . (Cs cs, Th th) \ RAG s} \ \" - proof - - { have h: "\ a A f. a \ A \ f a \ f ` A" by auto - fix x assume "(Cs x, Th th) \ RAG s" - hence "?F (Cs x, Th th) \ ?F `(RAG s)" by (rule h) - moreover have "?F (Cs x, Th th) = x" by simp - ultimately have "x \ (\(x, y). the_cs x) ` RAG s" by simp - } thus ?thesis by auto - qed - ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset) -qed - -lemma cntCS_v_dec: - assumes vtv: "vt (V thread cs#s)" - shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread" -proof - - from vtv interpret vt_s: valid_trace s - by (cases, unfold_locales, simp) - from vtv interpret vt_v: valid_trace "V thread cs#s" - by (unfold_locales, simp) - from step_back_step[OF vtv] - have cs_in: "cs \ holdents s thread" - apply (cases, unfold holdents_test s_RAG_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 vt_s.wq_distinct[of cs]) - apply (unfold holdents_test, unfold step_RAG_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))) \ RAG 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 vt_v.finite_holding - 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 count_rec1 [simp]: - assumes "Q e" - shows "count Q (e#es) = Suc (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec2 [simp]: - assumes "\Q e" - shows "count Q (e#es) = (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec3 [simp]: - shows "count Q [] = 0" - by (unfold count_def, auto) - -lemma cntP_diff_inv: - assumes "cntP (e#s) th \ cntP s th" - shows "isP e \ actor e = th" -proof(cases e) - case (P th' pty) - show ?thesis - by (cases "(\e. \cs. e = P th cs) (P th' pty)", - insert assms P, auto simp:cntP_def) -qed (insert assms, auto simp:cntP_def) - -lemma cntV_diff_inv: - assumes "cntV (e#s) th \ cntV s th" - shows "isV e \ actor e = th" -proof(cases e) - case (V th' pty) - show ?thesis - by (cases "(\e. \cs. e = V th cs) (V th' pty)", - insert assms V, auto simp:cntV_def) -qed (insert assms, auto simp:cntV_def) - -context valid_trace -begin - -text {* (* ddd *) \noindent - The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} - of one particular thread. -*} - -lemma cnp_cnv_cncs: - 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) - interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp) - 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 vt_s.wq_threads [OF 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:RAG_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:RAG_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) \ (RAG s)\<^sup>+" - from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto - then interpret vt_p: valid_trace "(P thread cs#s)" - by (unfold_locales, simp) - 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) - 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_RAG_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_RAG_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) \ RAG s} = - Suc (card {cs. (Cs cs, Th thread) \ RAG 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 vt_s.finite_holding [of thread] - show " finite {cs. (Cs cs, Th thread) \ RAG s}" - by (unfold holdents_test, simp) - qed - moreover have "?R - {cs} = ?R" - proof - - have "cs \ ?R" - proof - assume "cs \ {cs. (Cs cs, Th thread) \ RAG 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_RAG_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 vt_p.wq_distinct [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_RAG_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 - then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp) - 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 vt_v.wq_distinct[of cs] and eq_wq - show "distinct rest \ set rest = set rest" - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 vt_v.wq_distinct[of cs] - and eq_wq show False - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq) - moreover have "cntCS (e#s) th = cntCS s th" - apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto) - proof - - have "{csa. (Cs csa, Th th) \ RAG s \ csa = cs \ next_th s thread cs th} = - {cs. (Cs cs, Th th) \ RAG s}" - proof - - from False eq_wq - have " next_th s thread cs th \ (Cs cs, Th th) \ RAG 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 vt_s.wq_distinct[ 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))) \ RAG s" - by auto - qed - thus ?thesis by auto - qed - thus "card {csa. (Cs csa, Th th) \ RAG s \ csa = cs \ next_th s thread cs th} = - card {cs. (Cs cs, Th th) \ RAG 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 vt_s.wq_distinct[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_RAG_v[OF vtv], - auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def - Let_def cs_holding_def) - by (insert vt_s.wq_distinct[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 vt_s.wq_threads) - 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 vt_s.wq_distinct[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 vt_s.waiting_unique[OF 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 vt_s.wq_threads [OF 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_RAG_v[OF vtv], auto) - proof - - show "card {csa. (Cs csa, Th th) \ RAG s \ csa = cs} = - Suc (card {cs. (Cs cs, Th th) \ RAG 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) ` RAG s)" - apply (auto simp:image_def) - by (rule_tac x = "(Cs x, Th th)" in bexI, auto) - with vt_s.finite_RAG - show "finite {cs. (Cs cs, Th th) \ RAG s}" by (auto intro:finite_subset) - next - show "cs \ {cs. (Cs cs, Th th) \ RAG s}" - proof - assume "cs \ {cs. (Cs cs, Th th) \ RAG s}" - hence "(Cs cs, Th th) \ RAG s" by simp - with True neq_th eq_wq show False - by (auto simp:next_th_def s_RAG_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:RAG_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_RAG_def wq_def cs_holding_def) - qed -qed - -lemma not_thread_cncs: - assumes 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) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - 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:RAG_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:RAG_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_RAG_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 - then interpret vt_v: valid_trace "(V thread cs#s)" - by (unfold_locales, simp) - 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 vt_v.wq_distinct[of cs] and eq_wq - show "distinct rest \ set rest = set rest" - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 vt_s.wq_threads[OF this] and ni - show False - using `hd (SOME q. distinct q \ set q = set rest) \ set (wq s cs)` - ni vt_s.wq_threads by blast - 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_RAG_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:RAG_set_unchanged) - qed - next - case vt_nil - show ?case - by (unfold cntCS_def, - auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def) - qed -qed - -end - - -context valid_trace -begin - -lemma dm_RAG_threads: - assumes in_dom: "(Th th) \ Domain (RAG s)" - shows "th \ threads s" -proof - - from in_dom obtain n where "(Th th, n) \ RAG s" by auto - moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto - ultimately have "(Th th, Cs cs) \ RAG s" by simp - hence "th \ set (wq s cs)" - by (unfold s_RAG_def, auto simp:cs_waiting_def) - from wq_threads [OF this] show ?thesis . -qed - -end - -lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th" -unfolding cp_def wq_def -apply(induct s rule: schs.induct) -thm cpreced_initial -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 - -context valid_trace -begin - -lemma runing_unique: - assumes 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" - unfolding runing_def - apply(simp) - done - hence eq_max: "Max ((\th. preced th s) ` ({th1} \ dependants (wq s) th1)) = - Max ((\th. preced th s) ` ({th2} \ dependants (wq s) th2))" - (is "Max (?f ` ?A) = Max (?f ` ?B)") - unfolding cp_eq_cpreced - unfolding 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 (dependants (wq s) th1)" - proof- - have "finite {th'. (Th th', Th th1) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th1) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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 - thm cpreced_def - unfolding cpreced_def[symmetric] - unfolding cp_eq_cpreced[symmetric] - unfolding cpreced_def - using that[intro] by (auto) - 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 (dependants (wq s) th2)" - proof- - have "finite {th'. (Th th', Th th2) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th2) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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' \ dependants (wq s) th1)" by simp - thus "th1' \ threads s" - proof - assume "th1' \ dependants (wq s) th1" - hence "(Th th1') \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "(Th th1') \ Domain (RAG s)" by (simp add:trancl_domain) - from dm_RAG_threads[OF 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' \ dependants (wq s) th2)" by simp - thus "th2' \ threads s" - proof - assume "th2' \ dependants (wq s) th2" - hence "(Th th2') \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "(Th th2') \ Domain (RAG s)" by (simp add:trancl_domain) - from dm_RAG_threads[OF 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' \ dependants (wq s) th1" by simp - thus ?thesis - proof - assume eq_th': "th1' = th1" - from th2_in have "th2' = th2 \ th2' \ dependants (wq s) th2" by simp - thus ?thesis - proof - assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp - next - assume "th2' \ dependants (wq s) th2" - with eq_th12 eq_th' have "th1 \ dependants (wq s) th2" by simp - hence "(Th th1, Th th2) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - hence "Th th1 \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "Th th1 \ Domain (RAG s)" by (simp add:trancl_domain) - then obtain n where d: "(Th th1, n) \ RAG s" by (auto simp:Domain_def) - from RAG_target_th [OF this] - obtain cs' where "n = Cs cs'" by auto - with d have "(Th th1, Cs cs') \ RAG s" by simp - with runing_1 have "False" - apply (unfold runing_def readys_def s_RAG_def) - by (auto simp:waiting_eq) - thus ?thesis by simp - qed - next - assume th1'_in: "th1' \ dependants (wq s) th1" - from th2_in have "th2' = th2 \ th2' \ dependants (wq s) th2" by simp - thus ?thesis - proof - assume "th2' = th2" - with th1'_in eq_th12 have "th2 \ dependants (wq s) th1" by simp - hence "(Th th2, Th th1) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - hence "Th th2 \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "Th th2 \ Domain (RAG s)" by (simp add:trancl_domain) - then obtain n where d: "(Th th2, n) \ RAG s" by (auto simp:Domain_def) - from RAG_target_th [OF this] - obtain cs' where "n = Cs cs'" by auto - with d have "(Th th2, Cs cs') \ RAG s" by simp - with runing_2 have "False" - apply (unfold runing_def readys_def s_RAG_def) - by (auto simp:waiting_eq) - thus ?thesis by simp - next - assume "th2' \ dependants (wq s) th2" - with eq_th12 have "th1' \ dependants (wq s) th2" by simp - hence h1: "(Th th1', Th th2) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - from th1'_in have h2: "(Th th1', Th th1) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - show ?thesis - proof(rule dchain_unique[OF 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 "card (runing s) \ 1" -apply(subgoal_tac "finite (runing s)") -prefer 2 -apply (metis finite_nat_set_iff_bounded lessI runing_unique) -apply(rule ccontr) -apply(simp) -apply(case_tac "Suc (Suc 0) \ card (runing s)") -apply(subst (asm) card_le_Suc_iff) -apply(simp) -apply(auto)[1] -apply (metis insertCI runing_unique) -apply(auto) -done - -end - - -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 - -context valid_trace -begin - -lemma cnp_cnv_eq: - assumes "th \ threads s" - shows "cntP s th = cntV s th" - using assms - using cnp_cnv_cncs not_thread_cncs by auto - -end - - -lemma eq_RAG: - "RAG (wq s) = RAG s" -by (unfold cs_RAG_def s_RAG_def, auto) - -context valid_trace -begin - -lemma count_eq_dependants: - assumes eq_pv: "cntP s th = cntV s th" - shows "dependants (wq s) th = {}" -proof - - from cnp_cnv_cncs and eq_pv - have "cntCS s th = 0" - by (auto split:if_splits) - moreover have "finite {cs. (Cs cs, Th th) \ RAG s}" - proof - - from finite_holding[of th] show ?thesis - by (simp add:holdents_test) - qed - ultimately have h: "{cs. (Cs cs, Th th) \ RAG s} = {}" - by (unfold cntCS_def holdents_test cs_dependants_def, auto) - show ?thesis - proof(unfold cs_dependants_def) - { assume "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ {}" - then obtain th' where "(Th th', Th th) \ (RAG (wq s))\<^sup>+" by auto - hence "False" - proof(cases) - assume "(Th th', Th th) \ RAG (wq s)" - thus "False" by (auto simp:cs_RAG_def) - next - fix c - assume "(c, Th th) \ RAG (wq s)" - with h and eq_RAG show "False" - by (cases c, auto simp:cs_RAG_def) - qed - } thus "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} = {}" by auto - qed -qed - -lemma dependants_threads: - shows "dependants (wq s) th \ threads s" -proof - { fix th th' - assume h: "th \ {th'a. (Th th'a, Th th') \ (RAG (wq s))\<^sup>+}" - have "Th th \ Domain (RAG s)" - proof - - from h obtain th' where "(Th th, Th th') \ (RAG (wq s))\<^sup>+" by auto - hence "(Th th) \ Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def) - with trancl_domain have "(Th th) \ Domain (RAG (wq s))" by simp - thus ?thesis using eq_RAG by simp - qed - from dm_RAG_threads[OF this] - have "th \ threads s" . - } note hh = this - fix th1 - assume "th1 \ dependants (wq s) th" - hence "th1 \ {th'a. (Th th'a, Th th) \ (RAG (wq s))\<^sup>+}" - by (unfold cs_dependants_def, simp) - from hh [OF this] show "th1 \ threads s" . -qed - -lemma finite_threads: - shows "finite (threads s)" -using vt by (induct) (auto elim: step.cases) - -end - -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 - -context valid_trace -begin - -lemma cp_le: - assumes th_in: "th \ threads s" - shows "cp s th \ Max ((\ th. (preced th s)) ` threads s)" -proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def) - show "Max ((\th. preced th s) ` ({th} \ {th'. (Th th', Th th) \ (RAG (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) \ (RAG (wq s))\<^sup>+} \ {}" by simp - next - from finite_threads - show "finite (threads s)" . - next - from th_in - show "{th} \ {th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ threads s" - apply (auto simp:Domain_def) - apply (rule_tac dm_RAG_threads) - apply (unfold trancl_domain [of "RAG s", symmetric]) - by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def) - qed -qed - -lemma le_cp: - shows "preced th s \ cp s th" -proof(unfold cp_eq_cpreced preced_def cpreced_def, simp) - show "Prc (priority th s) (last_set th s) - \ Max (insert (Prc (priority th s) (last_set th s)) - ((\th. Prc (priority th s) (last_set th s)) ` dependants (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) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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: - 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 - 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 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 - 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 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 - 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 np: "threads s \ {}" - shows "Max (cp s ` readys s) = Max (cp s ` threads s)" -proof(unfold max_cp_eq) - 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 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 tm_in] - have "tm \ readys s \ (\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+)" . - thus ?thesis - proof - assume "\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+ " - then obtain th' where th'_in: "th' \ readys s" - and tm_chain:"(Th tm, Th th') \ (RAG s)\<^sup>+" by auto - have "cp s th' = ?f tm" - proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI) - from dependants_threads finite_threads - show "finite ((\th. preced th s) ` ({th'} \ dependants (wq s) th'))" - by (auto intro:finite_subset) - next - fix p assume p_in: "p \ (\th. preced th s) ` ({th'} \ dependants (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 - show "finite ((\th. preced th s) ` threads s)" by simp - next - from p_in and th'_in and dependants_threads[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'} \ dependants (wq s) th')" - proof - - from tm_chain - have "tm \ dependants (wq s) th'" - by (unfold cs_dependants_def s_RAG_def cs_RAG_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 dependants_threads [of th1] th1_in - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ - (\th. preced th s) ` threads s" - by (auto simp:readys_def) - next - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" by simp - next - from finite_threads - show " finite ((\th. preced th s) ` threads s)" by simp - qed - next - from finite_threads - 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} \ dependants (wq s) tm)" - show "y \ preced tm s" - proof - - { fix y' - assume hy' : "y' \ ((\th. preced th s) ` dependants (wq s) tm)" - have "y' \ preced tm s" - proof(unfold tm_max, rule Max_ge) - from hy' dependants_threads[of tm] - show "y' \ (\th. preced th s) ` threads s" by auto - next - from finite_threads - show "finite ((\th. preced th s) ` threads s)" by simp - qed - } with hy show ?thesis by auto - qed - next - from dependants_threads[of tm] finite_threads - show "finite ((\th. preced th s) ` ({tm} \ dependants (wq s) tm))" - by (auto intro:finite_subset) - next - show "preced tm s \ (\th. preced th s) ` ({tm} \ dependants (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 - 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 - show "finite ((\th. preced th s) ` threads s)" by simp - next - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" - by simp - next - from dependants_threads[of th1] th1_readys - show "(\th. preced th s) ` ({th1} \ dependants (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 - -text {* (* ccc *) \noindent - Since the current precedence of the threads in ready queue will always be boosted, - there must be one inside it has the maximum precedence of the whole system. -*} -lemma max_cp_readys_threads: - 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 False]) -qed - -end - -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 (RAG s))" -apply(simp add: detached_def Field_def) -apply(simp add: s_RAG_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 - -context valid_trace -begin - -lemma detached_intro: - assumes eq_pv: "cntP s th = cntV s th" - shows "detached s th" -proof - - from cnp_cnv_cncs - 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 dm_RAG_threads - show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff) - next - assume "th \ readys s" - moreover have "Th th \ Range (RAG s)" - proof - - from card_0_eq [OF finite_holding] 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_RAG_def) - done - qed - ultimately show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def) - qed -qed - -lemma detached_elim: - assumes dtc: "detached s th" - shows "cntP s th = cntV s th" -proof - - from cnp_cnv_cncs - 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_RAG_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:waiting_eq s_RAG_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: - shows "(detached s th) = (cntP s th = cntV s th)" - by (insert vt, auto intro:detached_intro detached_elim) - -end - -text {* - The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived - from the concise and miniature model of PIP given in PrioGDef.thy. -*} - -lemma eq_dependants: "dependants (wq s) = dependants s" - by (simp add: s_dependants_abv wq_def) - -lemma next_th_unique: - assumes nt1: "next_th s th cs th1" - and nt2: "next_th s th cs th2" - shows "th1 = th2" -using assms by (unfold next_th_def, auto) - -lemma birth_time_lt: "s \ [] \ last_set th s < length s" - apply (induct s, simp) -proof - - fix a s - assume ih: "s \ [] \ last_set th s < length s" - and eq_as: "a # s \ []" - show "last_set th (a # s) < length (a # s)" - proof(cases "s \ []") - case False - from False show ?thesis - by (cases a, auto simp:last_set.simps) - next - case True - from ih [OF True] show ?thesis - by (cases a, auto simp:last_set.simps) - qed -qed - -lemma th_in_ne: "th \ threads s \ s \ []" - by (induct s, auto simp:threads.simps) - -lemma preced_tm_lt: "th \ threads s \ preced th s = Prc x y \ y < length s" - apply (drule_tac th_in_ne) - by (unfold preced_def, auto intro: birth_time_lt) - -lemma inj_the_preced: - "inj_on (the_preced s) (threads s)" - by (metis inj_onI preced_unique the_preced_def) - -lemma tRAG_alt_def: - "tRAG s = {(Th th1, Th th2) | th1 th2. - \ cs. (Th th1, Cs cs) \ RAG s \ (Cs cs, Th th2) \ RAG s}" - by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def) - -lemma tRAG_Field: - "Field (tRAG s) \ Field (RAG s)" - by (unfold tRAG_alt_def Field_def, auto) - -lemma tRAG_ancestorsE: - assumes "x \ ancestors (tRAG s) u" - obtains th where "x = Th th" -proof - - from assms have "(u, x) \ (tRAG s)^+" - by (unfold ancestors_def, auto) - from tranclE[OF this] obtain c where "(c, x) \ tRAG s" by auto - then obtain th where "x = Th th" - by (unfold tRAG_alt_def, auto) - from that[OF this] show ?thesis . -qed - -lemma tRAG_mono: - assumes "RAG s' \ RAG s" - shows "tRAG s' \ tRAG s" - using assms - by (unfold tRAG_alt_def, auto) - -lemma holding_next_thI: - assumes "holding s th cs" - and "length (wq s cs) > 1" - obtains th' where "next_th s th cs th'" -proof - - from assms(1)[folded eq_holding, unfolded cs_holding_def] - have " th \ set (wq s cs) \ th = hd (wq s cs)" . - then obtain rest where h1: "wq s cs = th#rest" - by (cases "wq s cs", auto) - with assms(2) have h2: "rest \ []" by auto - let ?th' = "hd (SOME q. distinct q \ set q = set rest)" - have "next_th s th cs ?th'" using h1(1) h2 - by (unfold next_th_def, auto) - from that[OF this] show ?thesis . -qed - -lemma RAG_tRAG_transfer: - assumes "vt s'" - assumes "RAG s = RAG s' \ {(Th th, Cs cs)}" - and "(Cs cs, Th th'') \ RAG s'" - shows "tRAG s = tRAG s' \ {(Th th, Th th'')}" (is "?L = ?R") -proof - - interpret vt_s': valid_trace "s'" using assms(1) - by (unfold_locales, simp) - interpret rtree: rtree "RAG s'" - proof - show "single_valued (RAG s')" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:vt_s'.unique_RAG) - - show "acyclic (RAG s')" - by (rule vt_s'.acyclic_RAG) - qed - { fix n1 n2 - assume "(n1, n2) \ ?L" - from this[unfolded tRAG_alt_def] - obtain th1 th2 cs' where - h: "n1 = Th th1" "n2 = Th th2" - "(Th th1, Cs cs') \ RAG s" - "(Cs cs', Th th2) \ RAG s" by auto - from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \ RAG s'" by auto - from h(3) and assms(2) - have "(Th th1, Cs cs') = (Th th, Cs cs) \ - (Th th1, Cs cs') \ RAG s'" by auto - hence "(n1, n2) \ ?R" - proof - assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)" - hence eq_th1: "th1 = th" by simp - moreover have "th2 = th''" - proof - - from h1 have "cs' = cs" by simp - from assms(3) cs_in[unfolded this] rtree.sgv - show ?thesis - by (unfold single_valued_def, auto) - qed - ultimately show ?thesis using h(1,2) by auto - next - assume "(Th th1, Cs cs') \ RAG s'" - with cs_in have "(Th th1, Th th2) \ tRAG s'" - by (unfold tRAG_alt_def, auto) - from this[folded h(1, 2)] show ?thesis by auto - qed - } moreover { - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \tRAG s' \ (n1, n2) = (Th th, Th th'')" by auto - hence "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ tRAG s'" - moreover have "... \ ?L" - proof(rule tRAG_mono) - show "RAG s' \ RAG s" by (unfold assms(2), auto) - qed - ultimately show ?thesis by auto - next - assume eq_n: "(n1, n2) = (Th th, Th th'')" - from assms(2, 3) have "(Cs cs, Th th'') \ RAG s" by auto - moreover have "(Th th, Cs cs) \ RAG s" using assms(2) by auto - ultimately show ?thesis - by (unfold eq_n tRAG_alt_def, auto) - qed - } ultimately show ?thesis by auto -qed - -context valid_trace -begin - -lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt] - -end - -lemma cp_alt_def: - "cp s th = - Max ((the_preced s) ` {th'. Th th' \ (subtree (RAG s) (Th th))})" -proof - - have "Max (the_preced s ` ({th} \ dependants (wq s) th)) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "Max (_ ` ?L) = Max (_ ` ?R)") - proof - - have "?L = ?R" - by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def) - thus ?thesis by simp - qed - thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp) -qed - -lemma cp_gen_alt_def: - "cp_gen s = (Max \ (\x. (the_preced s \ the_thread) ` subtree (tRAG s) x))" - by (auto simp:cp_gen_def) - -lemma tRAG_nodeE: - assumes "(n1, n2) \ tRAG s" - obtains th1 th2 where "n1 = Th th1" "n2 = Th th2" - using assms - by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def) - -lemma subtree_nodeE: - assumes "n \ subtree (tRAG s) (Th th)" - obtains th1 where "n = Th th1" -proof - - show ?thesis - proof(rule subtreeE[OF assms]) - assume "n = Th th" - from that[OF this] show ?thesis . - next - assume "Th th \ ancestors (tRAG s) n" - hence "(n, Th th) \ (tRAG s)^+" by (auto simp:ancestors_def) - hence "\ th1. n = Th th1" - proof(induct) - case (base y) - from tRAG_nodeE[OF this] show ?case by metis - next - case (step y z) - thus ?case by auto - qed - with that show ?thesis by auto - qed -qed - -lemma tRAG_star_RAG: "(tRAG s)^* \ (RAG s)^*" -proof - - have "(wRAG s O hRAG s)^* \ (RAG s O RAG s)^*" - by (rule rtrancl_mono, auto simp:RAG_split) - also have "... \ ((RAG s)^*)^*" - by (rule rtrancl_mono, auto) - also have "... = (RAG s)^*" by simp - finally show ?thesis by (unfold tRAG_def, simp) -qed - -lemma tRAG_subtree_RAG: "subtree (tRAG s) x \ subtree (RAG s) x" -proof - - { fix a - assume "a \ subtree (tRAG s) x" - hence "(a, x) \ (tRAG s)^*" by (auto simp:subtree_def) - with tRAG_star_RAG[of s] - have "(a, x) \ (RAG s)^*" by auto - hence "a \ subtree (RAG s) x" by (auto simp:subtree_def) - } thus ?thesis by auto -qed - -lemma tRAG_trancl_eq: - "{th'. (Th th', Th th) \ (tRAG s)^+} = - {th'. (Th th', Th th) \ (RAG s)^+}" - (is "?L = ?R") -proof - - { fix th' - assume "th' \ ?L" - hence "(Th th', Th th) \ (tRAG s)^+" by auto - from tranclD[OF this] - obtain z where h: "(Th th', z) \ tRAG s" "(z, Th th) \ (tRAG s)\<^sup>*" by auto - from tRAG_subtree_RAG[of s] and this(2) - have "(z, Th th) \ (RAG s)^*" by (meson subsetCE tRAG_star_RAG) - moreover from h(1) have "(Th th', z) \ (RAG s)^+" using tRAG_alt_def by auto - ultimately have "th' \ ?R" by auto - } moreover - { fix th' - assume "th' \ ?R" - hence "(Th th', Th th) \ (RAG s)^+" by (auto) - from plus_rpath[OF this] - obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \ []" by auto - hence "(Th th', Th th) \ (tRAG s)^+" - proof(induct xs arbitrary:th' th rule:length_induct) - case (1 xs th' th) - then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto) - show ?case - proof(cases "xs1") - case Nil - from 1(2)[unfolded Cons1 Nil] - have rp: "rpath (RAG s) (Th th') [x1] (Th th)" . - hence "(Th th', x1) \ (RAG s)" by (cases, simp) - then obtain cs where "x1 = Cs cs" - by (unfold s_RAG_def, auto) - from rpath_nnl_lastE[OF rp[unfolded this]] - show ?thesis by auto - next - case (Cons x2 xs2) - from 1(2)[unfolded Cons1[unfolded this]] - have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" . - from rpath_edges_on[OF this] - have eds: "edges_on (Th th' # x1 # x2 # xs2) \ RAG s" . - have "(Th th', x1) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - with eds have rg1: "(Th th', x1) \ RAG s" by auto - then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto) - have "(x1, x2) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - from this eds - have rg2: "(x1, x2) \ RAG s" by auto - from this[unfolded eq_x1] - obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto) - from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2] - have rt1: "(Th th', Th th1) \ tRAG s" by (unfold tRAG_alt_def, auto) - from rp have "rpath (RAG s) x2 xs2 (Th th)" - by (elim rpath_ConsE, simp) - from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" . - show ?thesis - proof(cases "xs2 = []") - case True - from rpath_nilE[OF rp'[unfolded this]] - have "th1 = th" by auto - from rt1[unfolded this] show ?thesis by auto - next - case False - from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons] - have "(Th th1, Th th) \ (tRAG s)\<^sup>+" by simp - with rt1 show ?thesis by auto - qed - qed - qed - hence "th' \ ?L" by auto - } ultimately show ?thesis by blast -qed - -lemma tRAG_trancl_eq_Th: - "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = - {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" - using tRAG_trancl_eq by auto - -lemma dependants_alt_def: - "dependants s th = {th'. (Th th', Th th) \ (tRAG s)^+}" - by (metis eq_RAG s_dependants_def tRAG_trancl_eq) - -context valid_trace -begin - -lemma count_eq_tRAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using assms count_eq_dependants dependants_alt_def eq_dependants by auto - -lemma count_eq_RAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (RAG s)^+} = {}" - using assms count_eq_dependants cs_dependants_def eq_RAG by auto - -lemma count_eq_RAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (RAG s)^+} = {}" - using count_eq_RAG_plus[OF assms] by auto - -lemma count_eq_tRAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using count_eq_tRAG_plus[OF assms] by auto - -end - -lemma tRAG_subtree_eq: - "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \ (subtree (RAG s) (Th th))}" - (is "?L = ?R") -proof - - { fix n - assume h: "n \ ?L" - hence "n \ ?R" - by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) - } moreover { - fix n - assume "n \ ?R" - then obtain th' where h: "n = Th th'" "(Th th', Th th) \ (RAG s)^*" - by (auto simp:subtree_def) - from rtranclD[OF this(2)] - have "n \ ?L" - proof - assume "Th th' \ Th th \ (Th th', Th th) \ (RAG s)\<^sup>+" - with h have "n \ {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" by auto - thus ?thesis using subtree_def tRAG_trancl_eq by fastforce - qed (insert h, auto simp:subtree_def) - } ultimately show ?thesis by auto -qed - -lemma threads_set_eq: - "the_thread ` (subtree (tRAG s) (Th th)) = - {th'. Th th' \ (subtree (RAG s) (Th th))}" (is "?L = ?R") - by (auto intro:rev_image_eqI simp:tRAG_subtree_eq) - -lemma cp_alt_def1: - "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))" -proof - - have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) = - ((the_preced s \ the_thread) ` subtree (tRAG s) (Th th))" - by auto - thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto) -qed - -lemma cp_gen_def_cond: - assumes "x = Th th" - shows "cp s th = cp_gen s (Th th)" -by (unfold cp_alt_def1 cp_gen_def, simp) - -lemma cp_gen_over_set: - assumes "\ x \ A. \ th. x = Th th" - shows "cp_gen s ` A = (cp s \ the_thread) ` A" -proof(rule f_image_eq) - fix a - assume "a \ A" - from assms[rule_format, OF this] - obtain th where eq_a: "a = Th th" by auto - show "cp_gen s a = (cp s \ the_thread) a" - by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp) -qed - - -context valid_trace -begin - -lemma RAG_threads: - assumes "(Th th) \ Field (RAG s)" - shows "th \ threads s" - using assms - by (metis Field_def UnE dm_RAG_threads range_in vt) - -lemma subtree_tRAG_thread: - assumes "th \ threads s" - shows "subtree (tRAG s) (Th th) \ Th ` threads s" (is "?L \ ?R") -proof - - have "?L = {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - by (unfold tRAG_subtree_eq, simp) - also have "... \ ?R" - proof - fix x - assume "x \ {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - then obtain th' where h: "x = Th th'" "Th th' \ subtree (RAG s) (Th th)" by auto - from this(2) - show "x \ ?R" - proof(cases rule:subtreeE) - case 1 - thus ?thesis by (simp add: assms h(1)) - next - case 2 - thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) - qed - qed - finally show ?thesis . -qed - -lemma readys_root: - assumes "th \ readys s" - shows "root (RAG s) (Th th)" -proof - - { fix x - assume "x \ ancestors (RAG s) (Th th)" - hence h: "(Th th, x) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - obtain z where "(Th th, z) \ RAG s" by auto - with assms(1) have False - apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def) - by (fold wq_def, blast) - } thus ?thesis by (unfold root_def, auto) -qed - -lemma readys_in_no_subtree: - assumes "th \ readys s" - and "th' \ th" - shows "Th th \ subtree (RAG s) (Th th')" -proof - assume "Th th \ subtree (RAG s) (Th th')" - thus False - proof(cases rule:subtreeE) - case 1 - with assms show ?thesis by auto - next - case 2 - with readys_root[OF assms(1)] - show ?thesis by (auto simp:root_def) - qed -qed - -lemma not_in_thread_isolated: - assumes "th \ threads s" - shows "(Th th) \ Field (RAG s)" -proof - assume "(Th th) \ Field (RAG s)" - with dm_RAG_threads and range_in assms - show False by (unfold Field_def, blast) -qed - -lemma wf_RAG: "wf (RAG s)" -proof(rule finite_acyclic_wf) - from finite_RAG show "finite (RAG s)" . -next - from acyclic_RAG show "acyclic (RAG s)" . -qed - -lemma sgv_wRAG: "single_valued (wRAG s)" - using waiting_unique - by (unfold single_valued_def wRAG_def, auto) - -lemma sgv_hRAG: "single_valued (hRAG s)" - using holding_unique - by (unfold single_valued_def hRAG_def, auto) - -lemma sgv_tRAG: "single_valued (tRAG s)" - by (unfold tRAG_def, rule single_valued_relcomp, - insert sgv_wRAG sgv_hRAG, auto) - -lemma acyclic_tRAG: "acyclic (tRAG s)" -proof(unfold tRAG_def, rule acyclic_compose) - show "acyclic (RAG s)" using acyclic_RAG . -next - show "wRAG s \ RAG s" unfolding RAG_split by auto -next - show "hRAG s \ RAG s" unfolding RAG_split by auto -qed - -lemma sgv_RAG: "single_valued (RAG s)" - using unique_RAG by (auto simp:single_valued_def) - -lemma rtree_RAG: "rtree (RAG s)" - using sgv_RAG acyclic_RAG - by (unfold rtree_def rtree_axioms_def sgv_def, auto) - -end - -sublocale valid_trace < rtree_RAG: rtree "RAG s" -proof - show "single_valued (RAG s)" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:unique_RAG) - - show "acyclic (RAG s)" - by (rule acyclic_RAG) -qed - -sublocale valid_trace < rtree_s: rtree "tRAG s" -proof(unfold_locales) - from sgv_tRAG show "single_valued (tRAG s)" . -next - from acyclic_tRAG show "acyclic (tRAG s)" . -qed - -sublocale valid_trace < fsbtRAGs : fsubtree "RAG s" -proof - - show "fsubtree (RAG s)" - proof(intro_locales) - show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] . - next - show "fsubtree_axioms (RAG s)" - proof(unfold fsubtree_axioms_def) - from wf_RAG show "wf (RAG s)" . - qed - qed -qed - -sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s" -proof - - have "fsubtree (tRAG s)" - proof - - have "fbranch (tRAG s)" - proof(unfold tRAG_def, rule fbranch_compose) - show "fbranch (wRAG s)" - proof(rule finite_fbranchI) - from finite_RAG show "finite (wRAG s)" - by (unfold RAG_split, auto) - qed - next - show "fbranch (hRAG s)" - proof(rule finite_fbranchI) - from finite_RAG - show "finite (hRAG s)" by (unfold RAG_split, auto) - qed - qed - moreover have "wf (tRAG s)" - proof(rule wf_subset) - show "wf (RAG s O RAG s)" using wf_RAG - by (fold wf_comp_self, simp) - next - show "tRAG s \ (RAG s O RAG s)" - by (unfold tRAG_alt_def, auto) - qed - ultimately show ?thesis - by (unfold fsubtree_def fsubtree_axioms_def,auto) - qed - from this[folded tRAG_def] show "fsubtree (tRAG s)" . -qed - -lemma Max_UNION: - assumes "finite A" - and "A \ {}" - and "\ M \ f ` A. finite M" - and "\ M \ f ` A. M \ {}" - shows "Max (\x\ A. f x) = Max (Max ` f ` A)" (is "?L = ?R") - using assms[simp] -proof - - have "?L = Max (\(f ` A))" - by (fold Union_image_eq, simp) - also have "... = ?R" - by (subst Max_Union, simp+) - finally show ?thesis . -qed - -lemma max_Max_eq: - assumes "finite A" - and "A \ {}" - and "x = y" - shows "max x (Max A) = Max ({y} \ A)" (is "?L = ?R") -proof - - have "?R = Max (insert y A)" by simp - also from assms have "... = ?L" - by (subst Max.insert, simp+) - finally show ?thesis by simp -qed - -context valid_trace -begin - -(* ddd *) -lemma cp_gen_rec: - assumes "x = Th th" - shows "cp_gen s x = Max ({the_preced s th} \ (cp_gen s) ` children (tRAG s) x)" -proof(cases "children (tRAG s) x = {}") - case True - show ?thesis - by (unfold True cp_gen_def subtree_children, simp add:assms) -next - case False - hence [simp]: "children (tRAG s) x \ {}" by auto - note fsbttRAGs.finite_subtree[simp] - have [simp]: "finite (children (tRAG s) x)" - by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], - rule children_subtree) - { fix r x - have "subtree r x \ {}" by (auto simp:subtree_def) - } note this[simp] - have [simp]: "\x\children (tRAG s) x. subtree (tRAG s) x \ {}" - proof - - from False obtain q where "q \ children (tRAG s) x" by blast - moreover have "subtree (tRAG s) q \ {}" by simp - ultimately show ?thesis by blast - qed - have h: "Max ((the_preced s \ the_thread) ` - ({x} \ \(subtree (tRAG s) ` children (tRAG s) x))) = - Max ({the_preced s th} \ cp_gen s ` children (tRAG s) x)" - (is "?L = ?R") - proof - - let "Max (?f ` (?A \ \ (?g ` ?B)))" = ?L - let "Max (_ \ (?h ` ?B))" = ?R - let ?L1 = "?f ` \(?g ` ?B)" - have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)" - proof - - have "?L1 = ?f ` (\ x \ ?B.(?g x))" by simp - also have "... = (\ x \ ?B. ?f ` (?g x))" by auto - finally have "Max ?L1 = Max ..." by simp - also have "... = Max (Max ` (\x. ?f ` subtree (tRAG s) x) ` ?B)" - by (subst Max_UNION, simp+) - also have "... = Max (cp_gen s ` children (tRAG s) x)" - by (unfold image_comp cp_gen_alt_def, simp) - finally show ?thesis . - qed - show ?thesis - proof - - have "?L = Max (?f ` ?A \ ?L1)" by simp - also have "... = max (the_preced s (the_thread x)) (Max ?L1)" - by (subst Max_Un, simp+) - also have "... = max (?f x) (Max (?h ` ?B))" - by (unfold eq_Max_L1, simp) - also have "... =?R" - by (rule max_Max_eq, (simp)+, unfold assms, simp) - finally show ?thesis . - qed - qed thus ?thesis - by (fold h subtree_children, unfold cp_gen_def, simp) -qed - -lemma cp_rec: - "cp s th = Max ({the_preced s th} \ - (cp s o the_thread) ` children (tRAG s) (Th th))" -proof - - have "Th th = Th th" by simp - note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this] - show ?thesis - proof - - have "cp_gen s ` children (tRAG s) (Th th) = - (cp s \ the_thread) ` children (tRAG s) (Th th)" - proof(rule cp_gen_over_set) - show " \x\children (tRAG s) (Th th). \th. x = Th th" - by (unfold tRAG_alt_def, auto simp:children_def) - qed - thus ?thesis by (subst (1) h(1), unfold h(2), simp) - qed -qed - -end - -(* keep *) -lemma next_th_holding: - assumes vt: "vt s" - and nxt: "next_th s th cs th'" - shows "holding (wq s) th cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - thus ?thesis - by (unfold cs_holding_def, auto) -qed - -context valid_trace -begin - -lemma next_th_waiting: - assumes nxt: "next_th s th cs th'" - shows "waiting (wq s) th' cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - from wq_distinct[of cs, unfolded h] - have dst: "distinct (th # rest)" . - have in_rest: "th' \ set rest" - proof(unfold h, rule someI2) - show "distinct rest \ set rest = set rest" using dst by auto - next - fix x assume "distinct x \ set x = set rest" - with h(2) - show "hd x \ set (rest)" by (cases x, auto) - qed - hence "th' \ set (wq s cs)" by (unfold h(1), auto) - moreover have "th' \ hd (wq s cs)" - by (unfold h(1), insert in_rest dst, auto) - ultimately show ?thesis by (auto simp:cs_waiting_def) -qed - -lemma next_th_RAG: - assumes nxt: "next_th (s::event list) th cs th'" - shows "{(Cs cs, Th th), (Th th', Cs cs)} \ RAG s" - using vt assms next_th_holding next_th_waiting - by (unfold s_RAG_def, simp) - -end - --- {* A useless definition *} -definition cps:: "state \ (thread \ precedence) set" -where "cps s = {(th, cp s th) | th . th \ threads s}" - -lemma "wq (V th cs # s) cs1 = ttt" - apply (unfold wq_def, auto simp:Let_def) - -end - diff -r ed938e2246b9 -r 0525670d8e6a CpsG.thy~ --- a/CpsG.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,4656 +0,0 @@ -theory CpsG -imports PIPDefs -begin - -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 - -lemma Max_fg_mono: - assumes "finite A" - and "\ a \ A. f a \ g a" - shows "Max (f ` A) \ Max (g ` A)" -proof(cases "A = {}") - case True - thus ?thesis by auto -next - case False - show ?thesis - proof(rule Max.boundedI) - from assms show "finite (f ` A)" by auto - next - from False show "f ` A \ {}" by auto - next - fix fa - assume "fa \ f ` A" - then obtain a where h_fa: "a \ A" "fa = f a" by auto - show "fa \ Max (g ` A)" - proof(rule Max_ge_iff[THEN iffD2]) - from assms show "finite (g ` A)" by auto - next - from False show "g ` A \ {}" by auto - next - from h_fa have "g a \ g ` A" by auto - moreover have "fa \ g a" using h_fa assms(2) by auto - ultimately show "\a\g ` A. fa \ a" by auto - qed - 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 Max_UNION: - assumes "finite A" - and "A \ {}" - and "\ M \ f ` A. finite M" - and "\ M \ f ` A. M \ {}" - shows "Max (\x\ A. f x) = Max (Max ` f ` A)" (is "?L = ?R") - using assms[simp] -proof - - have "?L = Max (\(f ` A))" - by (fold Union_image_eq, simp) - also have "... = ?R" - by (subst Max_Union, simp+) - finally show ?thesis . -qed - -lemma max_Max_eq: - assumes "finite A" - and "A \ {}" - and "x = y" - shows "max x (Max A) = Max ({y} \ A)" (is "?L = ?R") -proof - - have "?R = Max (insert y A)" by simp - also from assms have "... = ?L" - by (subst Max.insert, simp+) - finally show ?thesis by simp -qed - -lemma birth_time_lt: - assumes "s \ []" - shows "last_set th s < length s" - using assms -proof(induct s) - case (Cons a s) - show ?case - proof(cases "s \ []") - case False - thus ?thesis - by (cases a, auto) - next - case True - show ?thesis using Cons(1)[OF True] - by (cases a, auto) - qed -qed simp - -lemma th_in_ne: "th \ threads s \ s \ []" - by (induct s, auto) - -lemma preced_tm_lt: "th \ threads s \ preced th s = Prc x y \ y < length s" - by (drule_tac th_in_ne, unfold preced_def, auto intro: birth_time_lt) - -lemma eq_RAG: - "RAG (wq s) = RAG s" - by (unfold cs_RAG_def s_RAG_def, auto) - -lemma waiting_holding: - assumes "waiting (s::state) th cs" - obtains th' where "holding s th' cs" -proof - - from assms[unfolded s_waiting_def, folded wq_def] - obtain th' where "th' \ set (wq s cs)" "th' = hd (wq s cs)" - by (metis empty_iff hd_in_set list.set(1)) - hence "holding s th' cs" - by (unfold s_holding_def, fold wq_def, auto) - from that[OF 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 cp_alt_def: - "cp s th = - Max ((the_preced s) ` {th'. Th th' \ (subtree (RAG s) (Th th))})" -proof - - have "Max (the_preced s ` ({th} \ dependants (wq s) th)) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "Max (_ ` ?L) = Max (_ ` ?R)") - proof - - have "?L = ?R" - by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def) - thus ?thesis by simp - qed - thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp) -qed - -(* ccc *) - - -locale valid_trace = - fixes s - assumes vt : "vt s" - -locale valid_trace_e = valid_trace + - fixes e - assumes vt_e: "vt (e#s)" -begin - -lemma pip_e: "PIP s e" - using vt_e by (cases, simp) - -end - -locale valid_trace_create = valid_trace_e + - fixes th prio - assumes is_create: "e = Create th prio" - -locale valid_trace_exit = valid_trace_e + - fixes th - assumes is_exit: "e = Exit th" - -locale valid_trace_p = valid_trace_e + - fixes th cs - assumes is_p: "e = P th cs" - -locale valid_trace_v = valid_trace_e + - fixes th cs - assumes is_v: "e = V th cs" -begin - definition "rest = tl (wq s cs)" - definition "wq' = (SOME q. distinct q \ set q = set rest)" -end - -locale valid_trace_v_n = valid_trace_v + - assumes rest_nnl: "rest \ []" - -locale valid_trace_v_e = valid_trace_v + - assumes rest_nil: "rest = []" - -locale valid_trace_set= valid_trace_e + - fixes th prio - assumes is_set: "e = Set th prio" - -context valid_trace -begin - -lemma ind [consumes 0, case_names Nil Cons, induct type]: - assumes "PP []" - and "(\s e. valid_trace_e s e \ - PP s \ PIP s e \ PP (e # s))" - shows "PP s" -proof(induct rule:vt.induct[OF vt, case_names Init Step]) - case Init - from assms(1) show ?case . -next - case (Step s e) - show ?case - proof(rule assms(2)) - show "valid_trace_e s e" using Step by (unfold_locales, auto) - next - show "PP s" using Step by simp - next - show "PIP s e" using Step by simp - qed -qed - -lemma vt_moment: "\ t. vt (moment t s)" -proof(induct rule:ind) - case Nil - thus ?case by (simp add:vt_nil) -next - case (Cons s e t) - show ?case - proof(cases "t \ length (e#s)") - case True - from True have "moment t (e#s) = e#s" by simp - thus ?thesis using Cons - by (simp add:valid_trace_def valid_trace_e_def, auto) - next - case False - from Cons have "vt (moment t s)" by simp - moreover have "moment t (e#s) = moment t s" - proof - - from False have "t \ length s" by simp - from moment_app [OF this, of "[e]"] - show ?thesis by simp - qed - ultimately show ?thesis by simp - qed -qed - -lemma finite_threads: - shows "finite (threads s)" -using vt by (induct) (auto elim: step.cases) - -end - -lemma RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" - by (unfold s_RAG_def, auto) - -locale valid_moment = valid_trace + - fixes i :: nat - -sublocale valid_moment < vat_moment: valid_trace "(moment i s)" - by (unfold_locales, insert vt_moment, auto) - -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 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 [simp]: - "cs \ cs' \ wq (V thread cs#s) cs' = wq s cs'" - by (auto simp:wq_def Let_def cp_def split:list.splits) - -lemma runing_head: - assumes "th \ runing s" - and "th \ set (wq_fun (schs s) cs)" - shows "th = hd (wq_fun (schs s) cs)" - using assms - by (simp add:runing_def readys_def s_waiting_def wq_def) - -context valid_trace -begin - -lemma runing_wqE: - assumes "th \ runing s" - and "th \ set (wq s cs)" - obtains rest where "wq s cs = th#rest" -proof - - from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest" - by (meson list.set_cases) - have "th' = th" - proof(rule ccontr) - assume "th' \ th" - hence "th \ hd (wq s cs)" using eq_wq by auto - with assms(2) - have "waiting s th cs" - by (unfold s_waiting_def, fold wq_def, auto) - with assms show False - by (unfold runing_def readys_def, auto) - qed - with eq_wq that show ?thesis by metis -qed - -end - -context valid_trace_create -begin - -lemma wq_neq_simp [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_create wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace_exit -begin - -lemma wq_neq_simp [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_exit wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace_p -begin - -lemma wq_neq_simp [simp]: - assumes "cs' \ cs" - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_p wq_def - by (auto simp:Let_def) - -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_p] - show ?thesis by (cases, simp) -qed - -lemma ready_th_s: "th \ readys s" - using runing_th_s - by (unfold runing_def, auto) - -lemma live_th_s: "th \ threads s" - using readys_threads ready_th_s by auto - -lemma live_th_es: "th \ threads (e#s)" - using live_th_s - by (unfold is_p, simp) - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - -lemma th_not_in_wq: - shows "th \ set (wq s cs)" -proof - assume otherwise: "th \ set (wq s cs)" - from runing_wqE[OF runing_th_s this] - obtain rest where eq_wq: "wq s cs = th#rest" by blast - with otherwise - have "holding s th cs" - by (unfold s_holding_def, fold wq_def, simp) - hence cs_th_RAG: "(Cs cs, Th th) \ RAG s" - by (unfold s_RAG_def, fold holding_eq, auto) - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - with cs_th_RAG show ?thesis by auto - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq s cs @ [th]" - by (unfold is_p wq_def, auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" -proof(cases "cs' = cs") - case True - show ?thesis using True assms th_not_in_wq - by (unfold True wq_es_cs, auto) -qed (insert assms, simp) - -end - -context valid_trace_v -begin - -lemma wq_neq_simp [simp]: - assumes "cs' \ cs" - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_v wq_def - by (auto simp:Let_def) - -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_v] - show ?thesis by (cases, simp) -qed - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - -lemma wq_s_cs: - "wq s cs = th#rest" -proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from this(2) show ?thesis - by (unfold rest_def s_holding_def, fold wq_def, - metis empty_iff list.collapse list.set(1)) - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq'" - using wq_s_cs[unfolded wq_def] - by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" -proof(cases "cs' = cs") - case True - show ?thesis - proof(unfold True wq_es_cs wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - using assms[unfolded True wq_s_cs] by auto - qed simp -qed (insert assms, simp) - -end - -context valid_trace_set -begin - -lemma wq_neq_simp [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_set wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace -begin - -lemma actor_inv: - assumes "PIP s e" - and "\ isCreate e" - shows "actor e \ runing s" - using assms - by (induct, auto) - -lemma isP_E: - assumes "isP e" - obtains cs where "e = P (actor e) cs" - using assms by (cases e, auto) - -lemma isV_E: - assumes "isV e" - obtains cs where "e = V (actor e) cs" - using assms by (cases e, auto) - -lemma wq_distinct: "distinct (wq s cs)" -proof(induct rule:ind) - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt_create: valid_trace_create s e th prio - using Create by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept) - next - case (Exit th) - interpret vt_exit: valid_trace_exit s e th - using Exit by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept) - next - case (P th cs) - interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) - next - case (V th cs) - interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) - next - case (Set th prio) - interpret vt_set: valid_trace_set s e th prio - using Set by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) - qed -qed (unfold wq_def Let_def, simp) - -end - -context valid_trace_e -begin - -text {* - The following lemma shows that only the @{text "P"} - operation can add new thread into waiting queues. - Such kind of lemmas are very obvious, but need to be checked formally. - This is a kind of confirmation that our modelling is correct. -*} - -lemma wq_in_inv: - assumes s_ni: "thread \ set (wq s cs)" - and s_i: "thread \ set (wq (e#s) cs)" - shows "e = P thread cs" -proof(cases e) - -- {* This is the only non-trivial case: *} - case (V th cs1) - have False - proof(cases "cs1 = cs") - case True - show ?thesis - proof(cases "(wq s cs1)") - case (Cons w_hd w_tl) - have "set (wq (e#s) cs) \ set (wq s cs)" - proof - - have "(wq (e#s) cs) = (SOME q. distinct q \ set q = set w_tl)" - using Cons V by (auto simp:wq_def Let_def True split:if_splits) - moreover have "set ... \ set (wq s cs)" - proof(rule someI2) - show "distinct w_tl \ set w_tl = set w_tl" - by (metis distinct.simps(2) local.Cons wq_distinct) - qed (insert Cons True, auto) - ultimately show ?thesis by simp - qed - with assms show ?thesis by auto - qed (insert assms V True, auto simp:wq_def Let_def split:if_splits) - qed (insert assms V, auto simp:wq_def Let_def split:if_splits) - thus ?thesis by auto -qed (insert assms, auto simp:wq_def Let_def split:if_splits) - -lemma wq_out_inv: - assumes s_in: "thread \ set (wq s cs)" - and s_hd: "thread = hd (wq s cs)" - and s_i: "thread \ hd (wq (e#s) cs)" - shows "e = V thread cs" -proof(cases e) --- {* There are only two non-trivial cases: *} - case (V th cs1) - show ?thesis - proof(cases "cs1 = cs") - case True - have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] . - thus ?thesis - proof(cases) - case (thread_V) - moreover have "th = thread" using thread_V(2) s_hd - by (unfold s_holding_def wq_def, simp) - ultimately show ?thesis using V True by simp - qed - qed (insert assms V, auto simp:wq_def Let_def split:if_splits) -next - case (P th cs1) - show ?thesis - proof(cases "cs1 = cs") - case True - with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]" - by (auto simp:wq_def Let_def split:if_splits) - with s_i s_hd s_in have False - by (metis empty_iff hd_append2 list.set(1) wq_def) - thus ?thesis by simp - qed (insert assms P, auto simp:wq_def Let_def split:if_splits) -qed (insert assms, auto simp:wq_def Let_def split:if_splits) - -end - - -context valid_trace -begin - - -text {* (* ddd *) - The nature of the work is like this: since it starts from a very simple and basic - model, even intuitively very `basic` and `obvious` properties need to derived from scratch. - For instance, the fact - that one thread can not be blocked by two critical resources at the same time - is obvious, because only running threads can make new requests, if one is waiting for - a critical resource and get blocked, it can not make another resource request and get - blocked the second time (because it is not running). - - To derive this fact, one needs to prove by contraction and - reason about time (or @{text "moement"}). The reasoning is based on a generic theorem - named @{text "p_split"}, which is about status changing along the time axis. It says if - a condition @{text "Q"} is @{text "True"} at a state @{text "s"}, - but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} - in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history - of events leading to it), such that @{text "Q"} switched - from being @{text "False"} to @{text "True"} and kept being @{text "True"} - till the last moment of @{text "s"}. - - Suppose a thread @{text "th"} is blocked - on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, - since no thread is blocked at the very beginning, by applying - @{text "p_split"} to these two blocking facts, there exist - two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that - @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} - and kept on blocked on them respectively ever since. - - Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}. - However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still - in blocked state at moment @{text "t2"} and could not - make any request and get blocked the second time: Contradiction. -*} - -lemma waiting_unique_pre: (* ddd *) - assumes 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: "\ ?Q cs1 (moment t1 s)" - and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" by auto - from p_split [of "?Q cs2", OF q2 nq2] - obtain t2 where lt2: "t2 < length s" - and np2: "\ ?Q cs2 (moment t2 s)" - and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" by auto - { fix s cs - assume q: "?Q cs s" - have "thread \ runing s" - proof - assume "thread \ runing s" - hence " \cs. \ (thread \ set (wq_fun (schs s) cs) \ - thread \ hd (wq_fun (schs s) cs))" - by (unfold runing_def s_waiting_def readys_def, auto) - from this[rule_format, of cs] q - show False by (simp add: wq_def) - qed - } note q_not_runing = this - { fix t1 t2 cs1 cs2 - assume lt1: "t1 < length s" - and np1: "\ ?Q cs1 (moment t1 s)" - and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" - and lt2: "t2 < length s" - and np2: "\ ?Q cs2 (moment t2 s)" - and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" - and 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#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - have ?thesis - proof - - have "thread \ runing (moment t2 s)" - proof(cases "thread \ set (wq (moment t2 s) cs2)") - case True - have "e = V thread cs2" - proof - - have eq_th: "thread = hd (wq (moment t2 s) cs2)" - using True and np2 by auto - from vt_e.wq_out_inv[OF True this h2] - show ?thesis . - qed - thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto - next - case False - have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] . - with vt_e.actor_inv[OF vt_e.pip_e] - show ?thesis by auto - qed - moreover have "thread \ runing (moment t2 s)" - by (rule q_not_runing[OF nn1[rule_format, OF lt12]]) - ultimately show ?thesis by simp - qed - } note lt_case = this - show ?thesis - proof - - { assume "t1 < t2" - from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this] - have ?thesis . - } moreover { - assume "t2 < t1" - from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this] - have ?thesis . - } moreover { - assume eq_12: "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 lt_2: "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 - from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12] - have g1: "thread \ set (wq (e#moment t1 s) cs1)" and - g2: "thread \ hd (wq (e#moment t1 s) cs1)" by auto - have "vt (e#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - have "e = V thread cs2 \ e = P thread cs2" - proof(cases "thread \ set (wq (moment t2 s) cs2)") - case True - have "e = V thread cs2" - proof - - have eq_th: "thread = hd (wq (moment t2 s) cs2)" - using True and np2 by auto - from vt_e.wq_out_inv[OF True this h2] - show ?thesis . - qed - thus ?thesis by auto - next - case False - have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] . - thus ?thesis by auto - qed - moreover have "e = V thread cs1 \ e = P thread cs1" - proof(cases "thread \ set (wq (moment t1 s) cs1)") - case True - have eq_th: "thread = hd (wq (moment t1 s) cs1)" - using True and np1 by auto - from vt_e.wq_out_inv[folded eq_12, OF True this g2] - have "e = V thread cs1" . - thus ?thesis by auto - next - case False - have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] . - thus ?thesis by auto - qed - ultimately have ?thesis using neq12 by auto - } ultimately show ?thesis using nat_neq_iff by blast - qed -qed - -text {* - This lemma is a simple corrolary of @{text "waiting_unique_pre"}. -*} - -lemma waiting_unique: - assumes "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 - -end - -(* not used *) -text {* - Every thread can only be blocked on one critical resource, - symmetrically, every critical resource can only be held by one thread. - This fact is much more easier according to our definition. -*} -lemma held_unique: - assumes "holding (s::event list) th1 cs" - and "holding s th2 cs" - shows "th1 = th2" - by (insert assms, unfold s_holding_def, auto) - -lemma last_set_lt: "th \ threads s \ last_set th s < length s" - apply (induct s, auto) - by (case_tac a, auto split:if_splits) - -lemma last_set_unique: - "\last_set th1 s = last_set th2 s; th1 \ threads s; th2 \ threads s\ - \ th1 = th2" - apply (induct s, auto) - by (case_tac a, auto split:if_splits dest:last_set_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 "last_set th1 s = last_set th2 s" by (simp add:preced_def) - from last_set_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 - -text {* - The following three lemmas show that @{text "RAG"} does not change - by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"} - events, respectively. -*} - -lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma (in valid_trace_set) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_set RAG_set_unchanged, simp) - -lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma (in valid_trace_create) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_create RAG_create_unchanged, simp) - -lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma (in valid_trace_exit) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_exit RAG_exit_unchanged, simp) - -context valid_trace_v -begin - -lemma distinct_rest: "distinct rest" - by (simp add: distinct_tl rest_def wq_distinct) - -lemma holding_cs_eq_th: - assumes "holding s t cs" - shows "t = th" -proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from held_unique[OF this(2) assms] - show ?thesis by simp - qed -qed - -lemma distinct_wq': "distinct wq'" - by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def) - -lemma set_wq': "set wq' = set rest" - by (metis (mono_tags, lifting) distinct_rest rest_def - some_eq_ex wq'_def) - -lemma th'_in_inv: - assumes "th' \ set wq'" - shows "th' \ set rest" - using assms set_wq' by simp - -lemma neq_t_th: - assumes "waiting (e#s) t c" - shows "t \ th" -proof - assume otherwise: "t = th" - show False - proof(cases "c = cs") - case True - have "t \ set wq'" - using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs] - by simp - from th'_in_inv[OF this] have "t \ set rest" . - with wq_s_cs[folded otherwise] wq_distinct[of cs] - show ?thesis by simp - next - case False - have "wq (e#s) c = wq s c" using False - by (unfold is_v, simp) - hence "waiting s t c" using assms - by (simp add: cs_waiting_def waiting_eq) - hence "t \ readys s" by (unfold readys_def, auto) - hence "t \ runing s" using runing_ready by auto - with runing_th_s[folded otherwise] show ?thesis by auto - qed -qed - -lemma waiting_esI1: - assumes "waiting s t c" - and "c \ cs" - shows "waiting (e#s) t c" -proof - - have "wq (e#s) c = wq s c" - using assms(2) is_v by auto - with assms(1) show ?thesis - using cs_waiting_def waiting_eq by auto -qed - -lemma holding_esI2: - assumes "c \ cs" - and "holding s t c" - shows "holding (e#s) t c" -proof - - from assms(1) have "wq (e#s) c = wq s c" using is_v by auto - from assms(2)[unfolded s_holding_def, folded wq_def, - folded this, unfolded wq_def, folded s_holding_def] - show ?thesis . -qed - -lemma holding_esI1: - assumes "holding s t c" - and "t \ th" - shows "holding (e#s) t c" -proof - - have "c \ cs" using assms using holding_cs_eq_th by blast - from holding_esI2[OF this assms(1)] - show ?thesis . -qed - -end - -context valid_trace_v_n -begin - -lemma neq_wq': "wq' \ []" -proof (unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume " distinct x \ set x = set rest" - thus "x \ []" using rest_nnl by auto -qed - -definition "taker = hd wq'" - -definition "rest' = tl wq'" - -lemma eq_wq': "wq' = taker # rest'" - by (simp add: neq_wq' rest'_def taker_def) - -lemma next_th_taker: - shows "next_th s th cs taker" - using rest_nnl taker_def wq'_def wq_s_cs - by (auto simp:next_th_def) - -lemma taker_unique: - assumes "next_th s th cs taker'" - shows "taker' = taker" -proof - - from assms - obtain rest' where - h: "wq s cs = th # rest'" - "taker' = hd (SOME q. distinct q \ set q = set rest')" - by (unfold next_th_def, auto) - with wq_s_cs have "rest' = rest" by auto - thus ?thesis using h(2) taker_def wq'_def by auto -qed - -lemma waiting_set_eq: - "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}" - by (smt all_not_in_conv bot.extremum insertI1 insert_subset - mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique) - -lemma holding_set_eq: - "{(Cs cs, Th th') |th'. next_th s th cs th'} = {(Cs cs, Th taker)}" - using next_th_taker taker_def waiting_set_eq - by fastforce - -lemma holding_taker: - shows "holding (e#s) taker cs" - by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, - auto simp:neq_wq' taker_def) - -lemma waiting_esI2: - assumes "waiting s t cs" - and "t \ taker" - shows "waiting (e#s) t cs" -proof - - have "t \ set wq'" - proof(unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) - next - fix x - assume "distinct x \ set x = set rest" - moreover have "t \ set rest" - using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto - ultimately show "t \ set x" by simp - qed - moreover have "t \ hd wq'" - using assms(2) taker_def by auto - ultimately show ?thesis - by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp) -qed - -lemma waiting_esE: - assumes "waiting (e#s) t c" - obtains "c \ cs" "waiting s t c" - | "c = cs" "t \ taker" "waiting s t cs" "t \ set rest'" -proof(cases "c = cs") - case False - hence "wq (e#s) c = wq s c" using is_v by auto - with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto - from that(1)[OF False this] show ?thesis . -next - case True - from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs] - have "t \ hd wq'" "t \ set wq'" by auto - hence "t \ taker" by (simp add: taker_def) - moreover hence "t \ th" using assms neq_t_th by blast - moreover have "t \ set rest" by (simp add: `t \ set wq'` th'_in_inv) - ultimately have "waiting s t cs" - by (metis cs_waiting_def list.distinct(2) list.sel(1) - list.set_sel(2) rest_def waiting_eq wq_s_cs) - show ?thesis using that(2) - using True `t \ set wq'` `t \ taker` `waiting s t cs` eq_wq' by auto -qed - -lemma holding_esI1: - assumes "c = cs" - and "t = taker" - shows "holding (e#s) t c" - by (unfold assms, simp add: holding_taker) - -lemma holding_esE: - assumes "holding (e#s) t c" - obtains "c = cs" "t = taker" - | "c \ cs" "holding s t c" -proof(cases "c = cs") - case True - from assms[unfolded True, unfolded s_holding_def, - folded wq_def, unfolded wq_es_cs] - have "t = taker" by (simp add: taker_def) - from that(1)[OF True this] show ?thesis . -next - case False - hence "wq (e#s) c = wq s c" using is_v by auto - from assms[unfolded s_holding_def, folded wq_def, - unfolded this, unfolded wq_def, folded s_holding_def] - have "holding s t c" . - from that(2)[OF False this] show ?thesis . -qed - -end - - -context valid_trace_v_e -begin - -lemma nil_wq': "wq' = []" -proof (unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume " distinct x \ set x = set rest" - thus "x = []" using rest_nil by auto -qed - -lemma no_taker: - assumes "next_th s th cs taker" - shows "False" -proof - - from assms[unfolded next_th_def] - obtain rest' where "wq s cs = th # rest'" "rest' \ []" - by auto - thus ?thesis using rest_def rest_nil by auto -qed - -lemma waiting_set_eq: - "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}" - using no_taker by auto - -lemma holding_set_eq: - "{(Cs cs, Th th') |th'. next_th s th cs th'} = {}" - using no_taker by auto - -lemma no_holding: - assumes "holding (e#s) taker cs" - shows False -proof - - from wq_es_cs[unfolded nil_wq'] - have " wq (e # s) cs = []" . - from assms[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis by auto -qed - -lemma no_waiting: - assumes "waiting (e#s) t cs" - shows False -proof - - from wq_es_cs[unfolded nil_wq'] - have " wq (e # s) cs = []" . - from assms[unfolded s_waiting_def, folded wq_def, unfolded this] - show ?thesis by auto -qed - -lemma waiting_esI2: - assumes "waiting s t c" - shows "waiting (e#s) t c" -proof - - have "c \ cs" using assms - using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto - from waiting_esI1[OF assms this] - show ?thesis . -qed - -lemma waiting_esE: - assumes "waiting (e#s) t c" - obtains "c \ cs" "waiting s t c" -proof(cases "c = cs") - case False - hence "wq (e#s) c = wq s c" using is_v by auto - with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto - from that(1)[OF False this] show ?thesis . -next - case True - from no_waiting[OF assms[unfolded True]] - show ?thesis by auto -qed - -lemma holding_esE: - assumes "holding (e#s) t c" - obtains "c \ cs" "holding s t c" -proof(cases "c = cs") - case True - from no_holding[OF assms[unfolded True]] - show ?thesis by auto -next - case False - hence "wq (e#s) c = wq s c" using is_v by auto - from assms[unfolded s_holding_def, folded wq_def, - unfolded this, unfolded wq_def, folded s_holding_def] - have "holding s t c" . - from that[OF False this] show ?thesis . -qed - -end - -lemma rel_eqI: - assumes "\ x y. (x,y) \ A \ (x,y) \ B" - and "\ x y. (x,y) \ B \ (x, y) \ A" - shows "A = B" - using assms by auto - -lemma in_RAG_E: - assumes "(n1, n2) \ RAG (s::state)" - obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs" - | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs" - using assms[unfolded s_RAG_def, folded waiting_eq holding_eq] - by auto - -context valid_trace_v -begin - -lemma RAG_es: - "RAG (e # s) = - RAG 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 = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - show ?thesis - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from waiting(3) - show ?thesis - proof(cases rule:h_n.waiting_esE) - case 1 - with waiting(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - next - case 2 - with waiting(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from waiting(3) - show ?thesis - proof(cases rule:h_e.waiting_esE) - case 1 - with waiting(1,2) - show ?thesis - by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - qed - qed - next - case (holding th' cs') - show ?thesis - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from holding(3) - show ?thesis - proof(cases rule:h_n.holding_esE) - case 1 - with holding(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - next - case 2 - with holding(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold holding_eq, auto) - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from holding(3) - show ?thesis - proof(cases rule:h_e.holding_esE) - case 1 - with holding(1,2) - show ?thesis - by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, - fold holding_eq, auto) - qed - qed - qed -next - fix n1 n2 - assume h: "(n1, n2) \ ?R" - show "(n1, n2) \ ?L" - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq] - have "((n1, n2) \ RAG s \ (n1 \ Cs cs \ n2 \ Th th) - \ (n1 \ Th h_n.taker \ n2 \ Cs cs)) \ - (n2 = Th h_n.taker \ n1 = Cs cs)" - by auto - thus ?thesis - proof - assume "n2 = Th h_n.taker \ n1 = Cs cs" - with h_n.holding_taker - show ?thesis - by (unfold s_RAG_def, fold holding_eq, auto) - next - assume h: "(n1, n2) \ RAG s \ - (n1 \ Cs cs \ n2 \ Th th) \ (n1 \ Th h_n.taker \ n2 \ Cs cs)" - hence "(n1, n2) \ RAG s" by simp - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from h and this(1,2) - have "th' \ h_n.taker \ cs' \ cs" by auto - hence "waiting (e#s) th' cs'" - proof - assume "cs' \ cs" - from waiting_esI1[OF waiting(3) this] - show ?thesis . - next - assume neq_th': "th' \ h_n.taker" - show ?thesis - proof(cases "cs' = cs") - case False - from waiting_esI1[OF waiting(3) this] - show ?thesis . - next - case True - from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True] - show ?thesis . - qed - qed - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - from h this(1,2) - have "cs' \ cs \ th' \ th" by auto - hence "holding (e#s) th' cs'" - proof - assume "cs' \ cs" - from holding_esI2[OF this holding(3)] - show ?thesis . - next - assume "th' \ th" - from holding_esI1[OF holding(3) this] - show ?thesis . - qed - thus ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq] - have h_s: "(n1, n2) \ RAG s" "(n1, n2) \ (Cs cs, Th th)" - by auto - from h_s(1) - show ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from h_e.waiting_esI2[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - with h_s(2) - have "cs' \ cs \ th' \ th" by auto - thus ?thesis - proof - assume neq_cs: "cs' \ cs" - from holding_esI2[OF this holding(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - next - assume "th' \ th" - from holding_esI1[OF holding(3) this] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed - qed -qed - -end - -lemma step_RAG_v: -assumes vt: - "vt (V th cs#s)" -shows " - RAG (V th cs # s) = - RAG 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 = ?R") -proof - - interpret vt_v: valid_trace_v s "V th cs" - using assms step_back_vt by (unfold_locales, auto) - show ?thesis using vt_v.RAG_es . -qed - -lemma (in valid_trace_create) - th_not_in_threads: "th \ threads s" -proof - - from pip_e[unfolded is_create] - show ?thesis by (cases, simp) -qed - -lemma (in valid_trace_create) - threads_es [simp]: "threads (e#s) = threads s \ {th}" - by (unfold is_create, simp) - -lemma (in valid_trace_exit) - threads_es [simp]: "threads (e#s) = threads s - {th}" - by (unfold is_exit, simp) - -lemma (in valid_trace_p) - threads_es [simp]: "threads (e#s) = threads s" - by (unfold is_p, simp) - -lemma (in valid_trace_v) - threads_es [simp]: "threads (e#s) = threads s" - by (unfold is_v, simp) - -lemma (in valid_trace_v) - th_not_in_rest[simp]: "th \ set rest" -proof - assume otherwise: "th \ set rest" - have "distinct (wq s cs)" by (simp add: wq_distinct) - from this[unfolded wq_s_cs] and otherwise - show False by auto -qed - -lemma (in valid_trace_v) - set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}" -proof(unfold wq_es_cs wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume "distinct x \ set x = set rest" - thus "set x = set (wq s cs) - {th}" - by (unfold wq_s_cs, simp) -qed - -lemma (in valid_trace_exit) - th_not_in_wq: "th \ set (wq s cs)" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold holdents_def s_holding_def, fold wq_def, - auto elim!:runing_wqE) -qed - -lemma (in valid_trace) wq_threads: - assumes "th \ set (wq s cs)" - shows "th \ threads s" - using assms -proof(induct rule:ind) - case (Nil) - thus ?case by (auto simp:wq_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th' prio') - interpret vt: valid_trace_create s e th' prio' - using Create by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems by auto - next - case (Exit th') - interpret vt: valid_trace_exit s e th' - using Exit by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto - next - case (P th' cs') - interpret vt: valid_trace_p s e th' cs' - using P by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems readys_threads - runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv - by fastforce - next - case (V th' cs') - interpret vt: valid_trace_v s e th' cs' - using V by (unfold_locales, simp) - show ?thesis using Cons - using vt.is_v vt.threads_es vt_e.wq_in_inv by blast - next - case (Set th' prio) - interpret vt: valid_trace_set s e th' prio - using Set by (unfold_locales, simp) - show ?thesis using Cons.hyps(2) Cons.prems vt.is_set - by (auto simp:wq_def Let_def) - qed -qed - -context valid_trace -begin - -lemma dm_RAG_threads: - assumes in_dom: "(Th th) \ Domain (RAG s)" - shows "th \ threads s" -proof - - from in_dom obtain n where "(Th th, n) \ RAG s" by auto - moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto - ultimately have "(Th th, Cs cs) \ RAG s" by simp - hence "th \ set (wq s cs)" - by (unfold s_RAG_def, auto simp:cs_waiting_def) - from wq_threads [OF this] show ?thesis . -qed - -lemma rg_RAG_threads: - assumes "(Th th) \ Range (RAG s)" - shows "th \ threads s" - using assms - by (unfold s_RAG_def cs_waiting_def cs_holding_def, - auto intro:wq_threads) - -lemma RAG_threads: - assumes "(Th th) \ Field (RAG s)" - shows "th \ threads s" - using assms - by (metis Field_def UnE dm_RAG_threads rg_RAG_threads) - -end - -lemma (in valid_trace_v) - preced_es [simp]: "preced th (e#s) = preced th s" - by (unfold is_v preced_def, simp) - -lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s" -proof - fix th' - show "the_preced (V th cs # s) th' = the_preced s th'" - by (unfold the_preced_def preced_def, simp) -qed - -lemma (in valid_trace_v) - the_preced_es: "the_preced (e#s) = the_preced s" - by (unfold is_v preced_def, simp) - -context valid_trace_p -begin - -lemma not_holding_s_th_cs: "\ holding s th cs" -proof - assume otherwise: "holding s th cs" - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - moreover have "(Cs cs, Th th) \ RAG s" - using otherwise cs_holding_def - holding_eq th_not_in_wq by auto - ultimately show ?thesis by auto - qed -qed - -lemma waiting_kept: - assumes "waiting s th' cs'" - shows "waiting (e#s) th' cs'" - using assms - by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2) - rotate1.simps(2) self_append_conv2 set_rotate1 - th_not_in_wq waiting_eq wq_es_cs wq_neq_simp) - -lemma holding_kept: - assumes "holding s th' cs'" - shows "holding (e#s) th' cs'" -proof(cases "cs' = cs") - case False - hence "wq (e#s) cs' = wq s cs'" by simp - with assms show ?thesis using cs_holding_def holding_eq by auto -next - case True - from assms[unfolded s_holding_def, folded wq_def] - obtain rest where eq_wq: "wq s cs' = th'#rest" - by (metis empty_iff list.collapse list.set(1)) - hence "wq (e#s) cs' = th'#(rest@[th])" - by (simp add: True wq_es_cs) - thus ?thesis - by (simp add: cs_holding_def holding_eq) -qed - -end - -locale valid_trace_p_h = valid_trace_p + - assumes we: "wq s cs = []" - -locale valid_trace_p_w = valid_trace_p + - assumes wne: "wq s cs \ []" -begin - -definition "holder = hd (wq s cs)" -definition "waiters = tl (wq s cs)" -definition "waiters' = waiters @ [th]" - -lemma wq_s_cs: "wq s cs = holder#waiters" - by (simp add: holder_def waiters_def wne) - -lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]" - by (simp add: wq_es_cs wq_s_cs) - -lemma waiting_es_th_cs: "waiting (e#s) th cs" - using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto - -lemma RAG_edge: "(Th th, Cs cs) \ RAG (e#s)" - by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto) - -lemma holding_esE: - assumes "holding (e#s) th' cs'" - obtains "holding s th' cs'" - using assms -proof(cases "cs' = cs") - case False - hence "wq (e#s) cs' = wq s cs'" by simp - with assms show ?thesis - using cs_holding_def holding_eq that by auto -next - case True - with assms show ?thesis - by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that - wq_es_cs' wq_s_cs) -qed - -lemma waiting_esE: - assumes "waiting (e#s) th' cs'" - obtains "th' \ th" "waiting s th' cs'" - | "th' = th" "cs' = cs" -proof(cases "waiting s th' cs'") - case True - have "th' \ th" - proof - assume otherwise: "th' = th" - from True[unfolded this] - show False by (simp add: th_not_waiting) - qed - from that(1)[OF this True] show ?thesis . -next - case False - hence "th' = th \ cs' = cs" - by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) - set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp) - with that(2) show ?thesis by metis -qed - -lemma RAG_es: "RAG (e # s) = RAG s \ {(Th th, Cs cs)}" (is "?L = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from this(3) - show ?thesis - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case 2 - thus ?thesis using waiting(1,2) by auto - qed - next - case (holding th' cs') - from this(3) - show ?thesis - proof(cases rule:holding_esE) - case 1 - with holding(1,2) - show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed -next - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \ RAG s \ (n1 = Th th \ n2 = Cs cs)" by auto - thus "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ RAG s" - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from waiting_kept[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - from holding_kept[OF this(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - next - assume "n1 = Th th \ n2 = Cs cs" - thus ?thesis using RAG_edge by auto - qed -qed - -end - -context valid_trace_p_h -begin - -lemma wq_es_cs': "wq (e#s) cs = [th]" - using wq_es_cs[unfolded we] by simp - -lemma holding_es_th_cs: - shows "holding (e#s) th cs" -proof - - from wq_es_cs' - have "th \ set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto - thus ?thesis using cs_holding_def holding_eq by blast -qed - -lemma RAG_edge: "(Cs cs, Th th) \ RAG (e#s)" - by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto) - -lemma waiting_esE: - assumes "waiting (e#s) th' cs'" - obtains "waiting s th' cs'" - using assms - by (metis cs_waiting_def event.distinct(15) is_p list.sel(1) - set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv) - -lemma holding_esE: - assumes "holding (e#s) th' cs'" - obtains "cs' \ cs" "holding s th' cs'" - | "cs' = cs" "th' = th" -proof(cases "cs' = cs") - case True - from held_unique[OF holding_es_th_cs assms[unfolded True]] - have "th' = th" by simp - from that(2)[OF True this] show ?thesis . -next - case False - have "holding s th' cs'" using assms - using False cs_holding_def holding_eq by auto - from that(1)[OF False this] show ?thesis . -qed - -lemma RAG_es: "RAG (e # s) = RAG s \ {(Cs cs, Th th)}" (is "?L = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from this(3) - show ?thesis - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - qed - next - case (holding th' cs') - from this(3) - show ?thesis - proof(cases rule:holding_esE) - case 1 - with holding(1,2) - show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) - next - case 2 - with holding(1,2) show ?thesis by auto - qed - qed -next - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \ RAG s \ (n1 = Cs cs \ n2 = Th th)" by auto - thus "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ RAG s" - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from waiting_kept[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - from holding_kept[OF this(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - next - assume "n1 = Cs cs \ n2 = Th th" - with holding_es_th_cs - show ?thesis - by (unfold s_RAG_def, fold holding_eq, auto) - qed -qed - -end - -context valid_trace_p -begin - -lemma RAG_es': "RAG (e # s) = (if (wq s cs = []) then RAG s \ {(Cs cs, Th th)} - else RAG s \ {(Th th, Cs cs)})" -proof(cases "wq s cs = []") - case True - interpret vt_p: valid_trace_p_h using True - by (unfold_locales, simp) - show ?thesis by (simp add: vt_p.RAG_es vt_p.we) -next - case False - interpret vt_p: valid_trace_p_w using False - by (unfold_locales, simp) - show ?thesis by (simp add: vt_p.RAG_es vt_p.wne) -qed - -end - -lemma (in valid_trace_v_n) finite_waiting_set: - "finite {(Th th', Cs cs) |th'. next_th s th cs th'}" - by (simp add: waiting_set_eq) - -lemma (in valid_trace_v_n) finite_holding_set: - "finite {(Cs cs, Th th') |th'. next_th s th cs th'}" - by (simp add: holding_set_eq) - -lemma (in valid_trace_v_e) finite_waiting_set: - "finite {(Th th', Cs cs) |th'. next_th s th cs th'}" - by (simp add: waiting_set_eq) - -lemma (in valid_trace_v_e) finite_holding_set: - "finite {(Cs cs, Th th') |th'. next_th s th cs th'}" - by (simp add: holding_set_eq) - -context valid_trace_v -begin - -lemma - finite_RAG_kept: - assumes "finite (RAG s)" - shows "finite (RAG (e#s))" -proof(cases "rest = []") - case True - interpret vt: valid_trace_v_e using True - by (unfold_locales, simp) - show ?thesis using assms - by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp) -next - case False - interpret vt: valid_trace_v_n using False - by (unfold_locales, simp) - show ?thesis using assms - by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp) -qed - -end - -context valid_trace_v_e -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof(rule acyclic_subset[OF assms]) - show "RAG (e # s) \ RAG s" - by (unfold RAG_es waiting_set_eq holding_set_eq, auto) -qed - -end - -context valid_trace_v_n -begin - -lemma waiting_taker: "waiting s taker cs" - apply (unfold s_waiting_def, fold wq_def, unfold wq_s_cs taker_def) - using eq_wq' th'_in_inv wq'_def by fastforce - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic ((RAG s - {(Cs cs, Th th)} - {(Th taker, Cs cs)}) \ - {(Cs cs, Th taker)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Th taker, Cs cs) \ ?A^*" - proof - assume otherwise: "(Th taker, Cs cs) \ ?A^*" - hence "(Th taker, Cs cs) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - from tranclD[OF this] - obtain cs' where h: "(Th taker, Cs cs') \ ?A" - "(Th taker, Cs cs') \ RAG s" - by (unfold s_RAG_def, auto) - from this(2) have "waiting s taker cs'" - by (unfold s_RAG_def, fold waiting_eq, auto) - from waiting_unique[OF this waiting_taker] - have "cs' = cs" . - from h(1)[unfolded this] show False by auto - qed - ultimately show ?thesis by auto - qed - thus ?thesis - by (unfold RAG_es waiting_set_eq holding_set_eq, simp) -qed - -end - -context valid_trace_p_h -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic (RAG s \ {(Cs cs, Th th)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Th th, Cs cs) \ ?A^*" - proof - assume otherwise: "(Th th, Cs cs) \ ?A^*" - hence "(Th th, Cs cs) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - from tranclD[OF this] - obtain cs' where h: "(Th th, Cs cs') \ RAG s" - by (unfold s_RAG_def, auto) - hence "waiting s th cs'" - by (unfold s_RAG_def, fold waiting_eq, auto) - with th_not_waiting show False by auto - qed - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold RAG_es, simp) -qed - -end - -context valid_trace_p_w -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic (RAG s \ {(Th th, Cs cs)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Cs cs, Th th) \ ?A^*" - proof - assume otherwise: "(Cs cs, Th th) \ ?A^*" - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - moreover from otherwise have "(Cs cs, Th th) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - ultimately show ?thesis by auto - qed - qed - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold RAG_es, simp) -qed - -end - -context valid_trace -begin - -lemma finite_RAG: - shows "finite (RAG s)" -proof(induct rule:ind) - case Nil - show ?case - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt: valid_trace_create s e th prio using Create - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (Exit th) - interpret vt: valid_trace_exit s e th using Exit - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (P th cs) - interpret vt: valid_trace_p s e th cs using P - by (unfold_locales, simp) - show ?thesis using Cons using vt.RAG_es' by auto - next - case (V th cs) - interpret vt: valid_trace_v s e th cs using V - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.finite_RAG_kept) - next - case (Set th prio) - interpret vt: valid_trace_set s e th prio using Set - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - qed -qed - -lemma acyclic_RAG: - shows "acyclic (RAG s)" -proof(induct rule:ind) - case Nil - show ?case - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt: valid_trace_create s e th prio using Create - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (Exit th) - interpret vt: valid_trace_exit s e th using Exit - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (P th cs) - interpret vt: valid_trace_p s e th cs using P - by (unfold_locales, simp) - show ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt_h: valid_trace_p_h s e th cs - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_h.acylic_RAG_kept) - next - case False - then interpret vt_w: valid_trace_p_w s e th cs - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_w.acylic_RAG_kept) - qed - next - case (V th cs) - interpret vt: valid_trace_v s e th cs using V - by (unfold_locales, simp) - show ?thesis - proof(cases "vt.rest = []") - case True - then interpret vt_e: valid_trace_v_e s e th cs - by (unfold_locales, simp) - show ?thesis by (simp add: Cons.hyps(2) vt_e.acylic_RAG_kept) - next - case False - then interpret vt_n: valid_trace_v_n s e th cs - by (unfold_locales, simp) - show ?thesis by (simp add: Cons.hyps(2) vt_n.acylic_RAG_kept) - qed - next - case (Set th prio) - interpret vt: valid_trace_set s e th prio using Set - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - qed -qed - -lemma wf_RAG: "wf (RAG s)" -proof(rule finite_acyclic_wf) - from finite_RAG show "finite (RAG s)" . -next - from acyclic_RAG show "acyclic (RAG s)" . -qed - -lemma sgv_wRAG: "single_valued (wRAG s)" - using waiting_unique - by (unfold single_valued_def wRAG_def, auto) - -lemma sgv_hRAG: "single_valued (hRAG s)" - using held_unique - by (unfold single_valued_def hRAG_def, auto) - -lemma sgv_tRAG: "single_valued (tRAG s)" - by (unfold tRAG_def, rule single_valued_relcomp, - insert sgv_wRAG sgv_hRAG, auto) - -lemma acyclic_tRAG: "acyclic (tRAG s)" -proof(unfold tRAG_def, rule acyclic_compose) - show "acyclic (RAG s)" using acyclic_RAG . -next - show "wRAG s \ RAG s" unfolding RAG_split by auto -next - show "hRAG s \ RAG s" unfolding RAG_split by auto -qed - -lemma unique_RAG: "\(n, n1) \ RAG s; (n, n2) \ RAG s\ \ n1 = n2" - apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq) - by(auto elim:waiting_unique held_unique) - -lemma sgv_RAG: "single_valued (RAG s)" - using unique_RAG by (auto simp:single_valued_def) - -lemma rtree_RAG: "rtree (RAG s)" - using sgv_RAG acyclic_RAG - by (unfold rtree_def rtree_axioms_def sgv_def, auto) - -end - -sublocale valid_trace < rtree_RAG: rtree "RAG s" -proof - show "single_valued (RAG s)" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:unique_RAG) - - show "acyclic (RAG s)" - by (rule acyclic_RAG) -qed - -sublocale valid_trace < rtree_s: rtree "tRAG s" -proof(unfold_locales) - from sgv_tRAG show "single_valued (tRAG s)" . -next - from acyclic_tRAG show "acyclic (tRAG s)" . -qed - -sublocale valid_trace < fsbtRAGs : fsubtree "RAG s" -proof - - show "fsubtree (RAG s)" - proof(intro_locales) - show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] . - next - show "fsubtree_axioms (RAG s)" - proof(unfold fsubtree_axioms_def) - from wf_RAG show "wf (RAG s)" . - qed - qed -qed - -lemma tRAG_alt_def: - "tRAG s = {(Th th1, Th th2) | th1 th2. - \ cs. (Th th1, Cs cs) \ RAG s \ (Cs cs, Th th2) \ RAG s}" - by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def) - -sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s" -proof - - have "fsubtree (tRAG s)" - proof - - have "fbranch (tRAG s)" - proof(unfold tRAG_def, rule fbranch_compose) - show "fbranch (wRAG s)" - proof(rule finite_fbranchI) - from finite_RAG show "finite (wRAG s)" - by (unfold RAG_split, auto) - qed - next - show "fbranch (hRAG s)" - proof(rule finite_fbranchI) - from finite_RAG - show "finite (hRAG s)" by (unfold RAG_split, auto) - qed - qed - moreover have "wf (tRAG s)" - proof(rule wf_subset) - show "wf (RAG s O RAG s)" using wf_RAG - by (fold wf_comp_self, simp) - next - show "tRAG s \ (RAG s O RAG s)" - by (unfold tRAG_alt_def, auto) - qed - ultimately show ?thesis - by (unfold fsubtree_def fsubtree_axioms_def,auto) - qed - from this[folded tRAG_def] show "fsubtree (tRAG s)" . -qed - - -context valid_trace -begin - -lemma finite_subtree_threads: - "finite {th'. Th th' \ subtree (RAG s) (Th th)}" (is "finite ?A") -proof - - have "?A = the_thread ` {Th th' | th' . Th th' \ subtree (RAG s) (Th th)}" - by (auto, insert image_iff, fastforce) - moreover have "finite {Th th' | th' . Th th' \ subtree (RAG s) (Th th)}" - (is "finite ?B") - proof - - have "?B = (subtree (RAG s) (Th th)) \ {Th th' | th'. True}" - by auto - moreover have "... \ (subtree (RAG s) (Th th))" by auto - moreover have "finite ..." by (simp add: fsbtRAGs.finite_subtree) - ultimately show ?thesis by auto - qed - ultimately show ?thesis by auto -qed - -lemma le_cp: - shows "preced th s \ cp s th" - proof(unfold cp_alt_def, rule Max_ge) - show "finite (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - by (simp add: finite_subtree_threads) - next - show "preced th s \ the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}" - by (simp add: subtree_def the_preced_def) - qed - -lemma cp_le: - assumes th_in: "th \ threads s" - shows "cp s th \ Max (the_preced s ` threads s)" -proof(unfold cp_alt_def, rule Max_f_mono) - show "finite (threads s)" by (simp add: finite_threads) -next - show " {th'. Th th' \ subtree (RAG s) (Th th)} \ {}" - using subtree_def by fastforce -next - show "{th'. Th th' \ subtree (RAG s) (Th th)} \ threads s" - using assms - by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq - node.inject(1) rtranclD subsetI subtree_def trancl_domain) -qed - -lemma max_cp_eq: - shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)" - (is "?L = ?R") -proof - - have "?L \ ?R" - proof(cases "threads s = {}") - case False - show ?thesis - by (rule Max.boundedI, - insert cp_le, - auto simp:finite_threads False) - qed auto - moreover have "?R \ ?L" - by (rule Max_fg_mono, - simp add: finite_threads, - simp add: le_cp the_preced_def) - ultimately show ?thesis by auto -qed - -lemma wf_RAG_converse: - shows "wf ((RAG s)^-1)" -proof(rule finite_acyclic_wf_converse) - from finite_RAG - show "finite (RAG s)" . -next - from acyclic_RAG - show "acyclic (RAG s)" . -qed - -lemma chain_building: - assumes "node \ Domain (RAG s)" - obtains th' where "th' \ readys s" "(node, Th th') \ (RAG s)^+" -proof - - from assms have "node \ Range ((RAG s)^-1)" by auto - from wf_base[OF wf_RAG_converse this] - obtain b where h_b: "(b, node) \ ((RAG s)\)\<^sup>+" "\c. (c, b) \ (RAG s)\" by auto - obtain th' where eq_b: "b = Th th'" - proof(cases b) - case (Cs cs) - from h_b(1)[unfolded trancl_converse] - have "(node, b) \ ((RAG s)\<^sup>+)" by auto - from tranclE[OF this] - obtain n where "(n, b) \ RAG s" by auto - from this[unfolded Cs] - obtain th1 where "waiting s th1 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - from waiting_holding[OF this] - obtain th2 where "holding s th2 cs" . - hence "(Cs cs, Th th2) \ RAG s" - by (unfold s_RAG_def, fold holding_eq, auto) - with h_b(2)[unfolded Cs, rule_format] - have False by auto - thus ?thesis by auto - qed auto - have "th' \ readys s" - proof - - from h_b(2)[unfolded eq_b] - have "\cs. \ waiting s th' cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - moreover have "th' \ threads s" - proof(rule rg_RAG_threads) - from tranclD[OF h_b(1), unfolded eq_b] - obtain z where "(z, Th th') \ (RAG s)" by auto - thus "Th th' \ Range (RAG s)" by auto - qed - ultimately show ?thesis by (auto simp:readys_def) - qed - moreover have "(node, Th th') \ (RAG s)^+" - using h_b(1)[unfolded trancl_converse] eq_b by auto - ultimately show ?thesis using that by metis -qed - -text {* \noindent - The following is just an instance of @{text "chain_building"}. -*} -lemma th_chain_to_ready: - assumes th_in: "th \ threads s" - shows "th \ readys s \ (\ th'. th' \ readys s \ (Th th, Th th') \ (RAG s)^+)" -proof(cases "th \ readys s") - case True - thus ?thesis by auto -next - case False - from False and th_in have "Th th \ Domain (RAG s)" - by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def) - from chain_building [rule_format, OF this] - show ?thesis by auto -qed - -end - -lemma count_rec1 [simp]: - assumes "Q e" - shows "count Q (e#es) = Suc (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec2 [simp]: - assumes "\Q e" - shows "count Q (e#es) = (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec3 [simp]: - shows "count Q [] = 0" - by (unfold count_def, auto) - -lemma cntP_simp1[simp]: - "cntP (P th cs'#s) th = cntP s th + 1" - by (unfold cntP_def, simp) - -lemma cntP_simp2[simp]: - assumes "th' \ th" - shows "cntP (P th cs'#s) th' = cntP s th'" - using assms - by (unfold cntP_def, simp) - -lemma cntP_simp3[simp]: - assumes "\ isP e" - shows "cntP (e#s) th' = cntP s th'" - using assms - by (unfold cntP_def, cases e, simp+) - -lemma cntV_simp1[simp]: - "cntV (V th cs'#s) th = cntV s th + 1" - by (unfold cntV_def, simp) - -lemma cntV_simp2[simp]: - assumes "th' \ th" - shows "cntV (V th cs'#s) th' = cntV s th'" - using assms - by (unfold cntV_def, simp) - -lemma cntV_simp3[simp]: - assumes "\ isV e" - shows "cntV (e#s) th' = cntV s th'" - using assms - by (unfold cntV_def, cases e, simp+) - -lemma cntP_diff_inv: - assumes "cntP (e#s) th \ cntP s th" - shows "isP e \ actor e = th" -proof(cases e) - case (P th' pty) - show ?thesis - by (cases "(\e. \cs. e = P th cs) (P th' pty)", - insert assms P, auto simp:cntP_def) -qed (insert assms, auto simp:cntP_def) - -lemma cntV_diff_inv: - assumes "cntV (e#s) th \ cntV s th" - shows "isV e \ actor e = th" -proof(cases e) - case (V th' pty) - show ?thesis - by (cases "(\e. \cs. e = V th cs) (V th' pty)", - insert assms V, auto simp:cntV_def) -qed (insert assms, auto simp:cntV_def) - -lemma children_RAG_alt_def: - "children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}" - by (unfold s_RAG_def, auto simp:children_def holding_eq) - -lemma holdents_alt_def: - "holdents s th = the_cs ` (children (RAG (s::state)) (Th th))" - by (unfold children_RAG_alt_def holdents_def, simp add: image_image) - -lemma cntCS_alt_def: - "cntCS s th = card (children (RAG s) (Th th))" - apply (unfold children_RAG_alt_def cntCS_def holdents_def) - by (rule card_image[symmetric], auto simp:inj_on_def) - -context valid_trace -begin - -lemma finite_holdents: "finite (holdents s th)" - by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto) - -end - -context valid_trace_p_w -begin - -lemma holding_s_holder: "holding s holder cs" - by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto) - -lemma holding_es_holder: "holding (e#s) holder cs" - by (unfold s_holding_def, fold wq_def, unfold wq_es_cs wq_s_cs, auto) - -lemma holdents_es: - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence h: "holding (e#s) th' cs'" by (auto simp:holdents_def) - have "holding s th' cs'" - proof(cases "cs' = cs") - case True - from held_unique[OF h[unfolded True] holding_es_holder] - have "th' = holder" . - thus ?thesis - by (unfold True holdents_def, insert holding_s_holder, simp) - next - case False - hence "wq (e#s) cs' = wq s cs'" by simp - from h[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis - by (unfold s_holding_def, fold wq_def, auto) - qed - hence "cs' \ ?R" by (auto simp:holdents_def) - } moreover { - fix cs' - assume "cs' \ ?R" - hence h: "holding s th' cs'" by (auto simp:holdents_def) - have "holding (e#s) th' cs'" - proof(cases "cs' = cs") - case True - from held_unique[OF h[unfolded True] holding_s_holder] - have "th' = holder" . - thus ?thesis - by (unfold True holdents_def, insert holding_es_holder, simp) - next - case False - hence "wq s cs' = wq (e#s) cs'" by simp - from h[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis - by (unfold s_holding_def, fold wq_def, auto) - qed - hence "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th[simp]: "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_es, simp) - -lemma th_not_ready_es: - shows "th \ readys (e#s)" - using waiting_es_th_cs - by (unfold readys_def, auto) - -end - -context valid_trace_p_h -begin - -lemma th_not_waiting': - "\ waiting (e#s) th cs'" -proof(cases "cs' = cs") - case True - show ?thesis - by (unfold True s_waiting_def, fold wq_def, unfold wq_es_cs', auto) -next - case False - from th_not_waiting[of cs', unfolded s_waiting_def, folded wq_def] - show ?thesis - by (unfold s_waiting_def, fold wq_def, insert False, simp) -qed - -lemma ready_th_es: - shows "th \ readys (e#s)" - using th_not_waiting' - by (unfold readys_def, insert live_th_es, auto) - -lemma holdents_es_th: - "holdents (e#s) th = (holdents s th) \ {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" - by (unfold holdents_def, auto) - hence "cs' \ ?R" - by (cases rule:holding_esE, auto simp:holdents_def) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s th cs' \ cs' = cs" - by (auto simp:holdents_def) - hence "cs' \ ?L" - proof - assume "holding s th cs'" - from holding_kept[OF this] - show ?thesis by (auto simp:holdents_def) - next - assume "cs' = cs" - thus ?thesis using holding_es_th_cs - by (unfold holdents_def, auto) - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th: "cntCS (e#s) th = cntCS s th + 1" -proof - - have "card (holdents s th \ {cs}) = card (holdents s th) + 1" - proof(subst card_Un_disjoint) - show "holdents s th \ {cs} = {}" - using not_holding_s_th_cs by (auto simp:holdents_def) - qed (auto simp:finite_holdents) - thus ?thesis - by (unfold cntCS_def holdents_es_th, simp) -qed - -lemma no_holder: - "\ holding s th' cs" -proof - assume otherwise: "holding s th' cs" - from this[unfolded s_holding_def, folded wq_def, unfolded we] - show False by auto -qed - -lemma holdents_es_th': - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence h_e: "holding (e#s) th' cs'" by (auto simp:holdents_def) - have "cs' \ cs" - proof - assume "cs' = cs" - from held_unique[OF h_e[unfolded this] holding_es_th_cs] - have "th' = th" . - with assms show False by simp - qed - from h_e[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp[OF this]] - have "th' \ set (wq s cs') \ th' = hd (wq s cs')" . - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, auto) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s th' cs'" by (auto simp:holdents_def) - from holding_kept[OF this] - have "holding (e # s) th' cs'" . - hence "cs' \ ?L" - by (unfold holdents_def, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th'[simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_es_th'[OF assms], simp) - -end - -context valid_trace_p -begin - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - show ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h - by (unfold_locales, simp) - show ?thesis using n_wait wait waiting_kept by auto - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis using n_wait wait waiting_kept by blast - qed - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - show ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h - by (unfold_locales, simp) - show ?thesis using n_wait vt.waiting_esE wait by blast - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis using assms(1) n_wait vt.waiting_esE wait by auto - qed - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: (* ddd *) - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof(cases "th' = th") - case True - note eq_th' = this - show ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h by (unfold_locales, simp) - show ?thesis - using assms eq_th' is_p ready_th_s vt.cntCS_es_th vt.ready_th_es pvD_def by auto - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis - using add.commute add.left_commute assms eq_th' is_p live_th_s - ready_th_s vt.th_not_ready_es pvD_def - apply (auto) - by (fold is_p, simp) - qed -next - case False - note h_False = False - thus ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h by (unfold_locales, simp) - show ?thesis using assms - by (insert True h_False pvD_def, auto split:if_splits,unfold is_p, auto) - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis using assms - by (insert False h_False pvD_def, auto split:if_splits,unfold is_p, auto) - qed -qed - -end - - -context valid_trace_v (* ccc *) -begin - -lemma holding_th_cs_s: - "holding s th cs" - by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto) - -lemma th_ready_s [simp]: "th \ readys s" - using runing_th_s - by (unfold runing_def readys_def, auto) - -lemma th_live_s [simp]: "th \ threads s" - using th_ready_s by (unfold readys_def, auto) - -lemma th_ready_es [simp]: "th \ readys (e#s)" - using runing_th_s neq_t_th - by (unfold is_v runing_def readys_def, auto) - -lemma th_live_es [simp]: "th \ threads (e#s)" - using th_ready_es by (unfold readys_def, auto) - -lemma pvD_th_s[simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es[simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma cntCS_s_th [simp]: "cntCS s th > 0" -proof - - have "cs \ holdents s th" using holding_th_cs_s - by (unfold holdents_def, simp) - moreover have "finite (holdents s th)" using finite_holdents - by simp - ultimately show ?thesis - by (unfold cntCS_def, - auto intro!:card_gt_0_iff[symmetric, THEN iffD1]) -qed - -end - -context valid_trace_v_n -begin - -lemma not_ready_taker_s[simp]: - "taker \ readys s" - using waiting_taker - by (unfold readys_def, auto) - -lemma taker_live_s [simp]: "taker \ threads s" -proof - - have "taker \ set wq'" by (simp add: eq_wq') - from th'_in_inv[OF this] - have "taker \ set rest" . - hence "taker \ set (wq s cs)" by (simp add: wq_s_cs) - thus ?thesis using wq_threads by auto -qed - -lemma taker_live_es [simp]: "taker \ threads (e#s)" - using taker_live_s threads_es by blast - -lemma taker_ready_es [simp]: - shows "taker \ readys (e#s)" -proof - - { fix cs' - assume "waiting (e#s) taker cs'" - hence False - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting_taker waiting_unique by auto - qed simp - } thus ?thesis by (unfold readys_def, auto) -qed - -lemma neq_taker_th: "taker \ th" - using th_not_waiting waiting_taker by blast - -lemma not_holding_taker_s_cs: - shows "\ holding s taker cs" - using holding_cs_eq_th neq_taker_th by auto - -lemma holdents_es_taker: - "holdents (e#s) taker = holdents s taker \ {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) taker cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 2 - thus ?thesis by (auto simp:holdents_def) - qed auto - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s taker cs' \ cs' = cs" by (auto simp:holdents_def) - hence "cs' \ ?L" - proof - assume "holding s taker cs'" - hence "holding (e#s) taker cs'" - using holding_esI2 holding_taker by fastforce - thus ?thesis by (auto simp:holdents_def) - next - assume "cs' = cs" - with holding_taker - show ?thesis by (auto simp:holdents_def) - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_taker [simp]: "cntCS (e#s) taker = cntCS s taker + 1" -proof - - have "card (holdents s taker \ {cs}) = card (holdents s taker) + 1" - proof(subst card_Un_disjoint) - show "holdents s taker \ {cs} = {}" - using not_holding_taker_s_cs by (auto simp:holdents_def) - qed (auto simp:finite_holdents) - thus ?thesis - by (unfold cntCS_def, insert holdents_es_taker, simp) -qed - -lemma pvD_taker_s[simp]: "pvD s taker = 1" - by (unfold pvD_def, simp) - -lemma pvD_taker_es[simp]: "pvD (e#s) taker = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_s[simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es[simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_es_th: - "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 2 - thus ?thesis by (auto simp:holdents_def) - qed (insert neq_taker_th, auto) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "cs' \ cs" "holding s th cs'" by (auto simp:holdents_def) - from holding_esI2[OF this] - have "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1" -proof - - have "card (holdents s th - {cs}) = card (holdents s th) - 1" - proof - - have "cs \ holdents s th" using holding_th_cs_s - by (auto simp:holdents_def) - moreover have "finite (holdents s th)" - by (simp add: finite_holdents) - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold cntCS_def holdents_es_th) -qed - -lemma holdents_kept: - assumes "th' \ taker" - and "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - have "cs' \ ?R" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding (e#s) th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, auto) - next - case True - from h[unfolded this] - have "holding (e#s) th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_taker] - have "th' = taker" . - with assms show ?thesis by auto - qed - } moreover { - fix cs' - assume h: "cs' \ ?R" - have "cs' \ ?L" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding s th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp) - next - case True - from h[unfolded this] - have "holding s th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_th_cs_s] - have "th' = th" . - with assms show ?thesis by auto - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ taker" - and "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_kept[OF assms], simp) - -lemma readys_kept1: - assumes "th' \ taker" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set (th # rest) \ th' \ hd (th # rest)" - using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - moreover have "\ (th' \ set rest \ th' \ hd (taker # rest'))" - using n_wait[unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs set_wq', unfolded eq_wq'] . - ultimately have "th' = taker" by auto - with assms(1) - show ?thesis by simp - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ taker" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set rest \ th' \ hd (taker # rest')" - using wait [unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs set_wq', unfolded eq_wq'] . - moreover have "\ (th' \ set (th # rest) \ th' \ hd (th # rest))" - using n_wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - ultimately have "th' = taker" by auto - with assms(1) - show ?thesis by simp - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ taker" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { assume eq_th': "th' = taker" - have ?thesis - apply (unfold eq_th' pvD_taker_es cntCS_es_taker) - by (insert neq_taker_th assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume eq_th': "th' = th" - have ?thesis - apply (unfold eq_th' pvD_th_es cntCS_es_th) - by (insert assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume h: "th' \ taker" "th' \ th" - have ?thesis using assms - apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp) - by (fold is_v, unfold pvD_def, simp) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_v_e -begin - -lemma holdents_es_th: - "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 1 - thus ?thesis by (auto simp:holdents_def) - qed - } moreover { - fix cs' - assume "cs' \ ?R" - hence "cs' \ cs" "holding s th cs'" by (auto simp:holdents_def) - from holding_esI2[OF this] - have "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1" -proof - - have "card (holdents s th - {cs}) = card (holdents s th) - 1" - proof - - have "cs \ holdents s th" using holding_th_cs_s - by (auto simp:holdents_def) - moreover have "finite (holdents s th)" - by (simp add: finite_holdents) - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold cntCS_def holdents_es_th) -qed - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - have "cs' \ ?R" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding (e#s) th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, auto) - next - case True - from h[unfolded this] - have "holding (e#s) th' cs" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, - unfolded wq_es_cs nil_wq'] - show ?thesis by auto - qed - } moreover { - fix cs' - assume h: "cs' \ ?R" - have "cs' \ ?L" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding s th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp) - next - case True - from h[unfolded this] - have "holding s th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_th_cs_s] - have "th' = th" . - with assms show ?thesis by auto - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_kept[OF assms], simp) - -lemma readys_kept1: - assumes "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(1)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set (th # rest) \ th' \ hd (th # rest)" - using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - hence "th' \ set rest" by auto - with set_wq' have "th' \ set wq'" by metis - with nil_wq' show ?thesis by simp - qed - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set [] \ th' \ hd []" - using wait[unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs nil_wq'] . - thus ?thesis by simp - qed - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis - apply (unfold eq_th' pvD_th_es cntCS_es_th) - by (insert assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume h: "th' \ th" - have ?thesis using assms - apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp) - by (fold is_v, unfold pvD_def, simp) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_v -begin - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof(cases "rest = []") - case True - then interpret vt: valid_trace_v_e by (unfold_locales, simp) - show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast -next - case False - then interpret vt: valid_trace_v_n by (unfold_locales, simp) - show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast -qed - -end - -context valid_trace_create -begin - -lemma th_not_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_create] - show ?thesis by (cases, simp) -qed - -lemma th_not_ready_s [simp]: "th \ readys s" - using th_not_live_s by (unfold readys_def, simp) - -lemma th_live_es [simp]: "th \ threads (e#s)" - by (unfold is_create, simp) - -lemma not_waiting_th_s [simp]: "\ waiting s th cs'" -proof - assume "waiting s th cs'" - from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have "th \ set (wq s cs')" by auto - from wq_threads[OF this] have "th \ threads s" . - with th_not_live_s show False by simp -qed - -lemma not_holding_th_s [simp]: "\ holding s th cs'" -proof - assume "holding s th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] - have "th \ set (wq s cs')" by auto - from wq_threads[OF this] have "th \ threads s" . - with th_not_live_s show False by simp -qed - -lemma not_waiting_th_es [simp]: "\ waiting (e#s) th cs'" -proof - assume "waiting (e # s) th cs'" - from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have "th \ set (wq s cs')" by auto - from wq_threads[OF this] have "th \ threads s" . - with th_not_live_s show False by simp -qed - -lemma not_holding_th_es [simp]: "\ holding (e#s) th cs'" -proof - assume "holding (e # s) th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] - have "th \ set (wq s cs')" by auto - from wq_threads[OF this] have "th \ threads s" . - with th_not_live_s show False by simp -qed - -lemma ready_th_es [simp]: "th \ readys (e#s)" - by (simp add:readys_def) - -lemma holdents_th_s: "holdents s th = {}" - by (unfold holdents_def, auto) - -lemma holdents_th_es: "holdents (e#s) th = {}" - by (unfold holdents_def, auto) - -lemma cntCS_th_s [simp]: "cntCS s th = 0" - by (unfold cntCS_def, simp add:holdents_th_s) - -lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0" - by (unfold cntCS_def, simp add:holdents_th_es) - -lemma pvD_th_s [simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es [simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept[OF assms] - by (unfold cntCS_def, simp) - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have False by auto - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2) by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma pvD_kept [simp]: - assumes "th' \ th" - shows "pvD (e#s) th' = pvD s th'" - using assms - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis using assms - by (unfold eq_th', simp, unfold is_create, simp) - } moreover { - assume h: "th' \ th" - hence ?thesis using assms - by (simp, simp add:is_create) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_exit -begin - -lemma th_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold runing_def readys_def, simp) -qed - -lemma th_ready_s [simp]: "th \ readys s" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold runing_def, simp) -qed - -lemma th_not_live_es [simp]: "th \ threads (e#s)" - by (unfold is_exit, simp) - -lemma not_holding_th_s [simp]: "\ holding s th cs'" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold holdents_def, auto) -qed - -lemma cntCS_th_s [simp]: "cntCS s th = 0" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold cntCS_def, simp) -qed - -lemma not_holding_th_es [simp]: "\ holding (e#s) th cs'" -proof - assume "holding (e # s) th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] - have "holding s th cs'" - by (unfold s_holding_def, fold wq_def, auto) - with not_holding_th_s - show False by simp -qed - -lemma ready_th_es [simp]: "th \ readys (e#s)" - by (simp add:readys_def) - -lemma holdents_th_s: "holdents s th = {}" - by (unfold holdents_def, auto) - -lemma holdents_th_es: "holdents (e#s) th = {}" - by (unfold holdents_def, auto) - -lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0" - by (unfold cntCS_def, simp add:holdents_th_es) - -lemma pvD_th_s [simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es [simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept[OF assms] - by (unfold cntCS_def, simp) - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have False by auto - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2) by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma pvD_kept [simp]: - assumes "th' \ th" - shows "pvD (e#s) th' = pvD s th'" - using assms - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis using assms - by (unfold eq_th', simp, unfold is_exit, simp) - } moreover { - assume h: "th' \ th" - hence ?thesis using assms - by (simp, simp add:is_exit) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_set -begin - -lemma th_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_set] - show ?thesis - by (cases, unfold runing_def readys_def, simp) -qed - -lemma th_ready_s [simp]: "th \ readys s" -proof - - from pip_e[unfolded is_set] - show ?thesis - by (cases, unfold runing_def, simp) -qed - -lemma th_not_live_es [simp]: "th \ threads (e#s)" - by (unfold is_set, simp) - - -lemma holdents_kept: - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept - by (unfold cntCS_def, simp) - -lemma threads_kept[simp]: - "threads (e#s) = threads s" - by (unfold is_set, simp) - -lemma readys_kept1: - assumes "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have False by auto - } moreover have "th' \ threads s" - using assms[unfolded readys_def] by auto - ultimately show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1 readys_kept2 - by metis - -lemma pvD_kept [simp]: - shows "pvD (e#s) th' = pvD s th'" - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" - using assms - by (unfold is_set, simp, fold is_set, simp) - -end - -context valid_trace -begin - -lemma cnp_cnv_cncs: "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" -proof(induct rule:ind) - case Nil - thus ?case - by (unfold cntP_def cntV_def pvD_def cntCS_def holdents_def - s_holding_def, simp) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt_create: valid_trace_create s e th prio - using Create by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_create.cnp_cnv_cncs_kept) - next - case (Exit th) - interpret vt_exit: valid_trace_exit s e th - using Exit by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_exit.cnp_cnv_cncs_kept) - next - case (P th cs) - interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_p.cnp_cnv_cncs_kept) - next - case (V th cs) - interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_v.cnp_cnv_cncs_kept) - next - case (Set th prio) - interpret vt_set: valid_trace_set s e th prio - using Set by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_set.cnp_cnv_cncs_kept) - qed -qed - -lemma not_thread_holdents: - assumes not_in: "th \ threads s" - shows "holdents s th = {}" -proof - - { fix cs - assume "cs \ holdents s th" - hence "holding s th cs" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def] - have "th \ set (wq s cs)" by auto - with wq_threads have "th \ threads s" by auto - with assms - have False by simp - } thus ?thesis by auto -qed - -lemma not_thread_cncs: - assumes not_in: "th \ threads s" - shows "cntCS s th = 0" - using not_thread_holdents[OF assms] - by (simp add:cntCS_def) - -lemma cnp_cnv_eq: - assumes "th \ threads s" - shows "cntP s th = cntV s th" - using assms cnp_cnv_cncs not_thread_cncs pvD_def - by (auto) - -lemma runing_unique: - assumes 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" - unfolding runing_def by auto - from this[unfolded cp_alt_def] - have eq_max: - "Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th1)}) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th2)})" - (is "Max ?L = Max ?R") . - have "Max ?L \ ?L" - proof(rule Max_in) - show "finite ?L" by (simp add: finite_subtree_threads) - next - show "?L \ {}" using subtree_def by fastforce - qed - then obtain th1' where - h_1: "Th th1' \ subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L" - by auto - have "Max ?R \ ?R" - proof(rule Max_in) - show "finite ?R" by (simp add: finite_subtree_threads) - next - show "?R \ {}" using subtree_def by fastforce - qed - then obtain th2' where - h_2: "Th th2' \ subtree (RAG s) (Th th2)" "the_preced s th2' = Max ?R" - by auto - have "th1' = th2'" - proof(rule preced_unique) - from h_1(1) - show "th1' \ threads s" - proof(cases rule:subtreeE) - case 1 - hence "th1' = th1" by simp - with runing_1 show ?thesis by (auto simp:runing_def readys_def) - next - case 2 - from this(2) - have "(Th th1', Th th1) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - have "(Th th1') \ Domain (RAG s)" by auto - from dm_RAG_threads[OF this] show ?thesis . - qed - next - from h_2(1) - show "th2' \ threads s" - proof(cases rule:subtreeE) - case 1 - hence "th2' = th2" by simp - with runing_2 show ?thesis by (auto simp:runing_def readys_def) - next - case 2 - from this(2) - have "(Th th2', Th th2) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - have "(Th th2') \ Domain (RAG s)" by auto - from dm_RAG_threads[OF this] show ?thesis . - qed - next - have "the_preced s th1' = the_preced s th2'" - using eq_max h_1(2) h_2(2) by metis - thus "preced th1' s = preced th2' s" by (simp add:the_preced_def) - qed - from h_1(1)[unfolded this] - have star1: "(Th th2', Th th1) \ (RAG s)^*" by (auto simp:subtree_def) - from h_2(1)[unfolded this] - have star2: "(Th th2', Th th2) \ (RAG s)^*" by (auto simp:subtree_def) - from star_rpath[OF star1] obtain xs1 - where rp1: "rpath (RAG s) (Th th2') xs1 (Th th1)" - by auto - from star_rpath[OF star2] obtain xs2 - where rp2: "rpath (RAG s) (Th th2') xs2 (Th th2)" - by auto - from rp1 rp2 - show ?thesis - proof(cases) - case (less_1 xs') - moreover have "xs' = []" - proof(rule ccontr) - assume otherwise: "xs' \ []" - from rpath_plus[OF less_1(3) this] - have "(Th th1, Th th2) \ (RAG s)\<^sup>+" . - from tranclD[OF this] - obtain cs where "waiting s th1 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - with runing_1 show False - by (unfold runing_def readys_def, auto) - qed - ultimately have "xs2 = xs1" by simp - from rpath_dest_eq[OF rp1 rp2[unfolded this]] - show ?thesis by simp - next - case (less_2 xs') - moreover have "xs' = []" - proof(rule ccontr) - assume otherwise: "xs' \ []" - from rpath_plus[OF less_2(3) this] - have "(Th th2, Th th1) \ (RAG s)\<^sup>+" . - from tranclD[OF this] - obtain cs where "waiting s th2 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - with runing_2 show False - by (unfold runing_def readys_def, auto) - qed - ultimately have "xs2 = xs1" by simp - from rpath_dest_eq[OF rp1 rp2[unfolded this]] - show ?thesis by simp - qed -qed - -lemma card_runing: "card (runing s) \ 1" -proof(cases "runing s = {}") - case True - thus ?thesis by auto -next - case False - then obtain th where [simp]: "th \ runing s" by auto - from runing_unique[OF this] - have "runing s = {th}" by auto - thus ?thesis by auto -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 eq_pv_children: - assumes eq_pv: "cntP s th = cntV s th" - shows "children (RAG s) (Th th) = {}" -proof - - from cnp_cnv_cncs and eq_pv - have "cntCS s th = 0" - by (auto split:if_splits) - from this[unfolded cntCS_def holdents_alt_def] - have card_0: "card (the_cs ` children (RAG s) (Th th)) = 0" . - have "finite (the_cs ` children (RAG s) (Th th))" - by (simp add: fsbtRAGs.finite_children) - from card_0[unfolded card_0_eq[OF this]] - show ?thesis by auto -qed - -lemma eq_pv_holdents: - assumes eq_pv: "cntP s th = cntV s th" - shows "holdents s th = {}" - by (unfold holdents_alt_def eq_pv_children[OF assms], simp) - -lemma eq_pv_subtree: - assumes eq_pv: "cntP s th = cntV s th" - shows "subtree (RAG s) (Th th) = {Th th}" - using eq_pv_children[OF assms] - by (unfold subtree_children, simp) - -end - -lemma cp_gen_alt_def: - "cp_gen s = (Max \ (\x. (the_preced s \ the_thread) ` subtree (tRAG s) x))" - by (auto simp:cp_gen_def) - -lemma tRAG_nodeE: - assumes "(n1, n2) \ tRAG s" - obtains th1 th2 where "n1 = Th th1" "n2 = Th th2" - using assms - by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def) - -lemma subtree_nodeE: - assumes "n \ subtree (tRAG s) (Th th)" - obtains th1 where "n = Th th1" -proof - - show ?thesis - proof(rule subtreeE[OF assms]) - assume "n = Th th" - from that[OF this] show ?thesis . - next - assume "Th th \ ancestors (tRAG s) n" - hence "(n, Th th) \ (tRAG s)^+" by (auto simp:ancestors_def) - hence "\ th1. n = Th th1" - proof(induct) - case (base y) - from tRAG_nodeE[OF this] show ?case by metis - next - case (step y z) - thus ?case by auto - qed - with that show ?thesis by auto - qed -qed - -lemma tRAG_star_RAG: "(tRAG s)^* \ (RAG s)^*" -proof - - have "(wRAG s O hRAG s)^* \ (RAG s O RAG s)^*" - by (rule rtrancl_mono, auto simp:RAG_split) - also have "... \ ((RAG s)^*)^*" - by (rule rtrancl_mono, auto) - also have "... = (RAG s)^*" by simp - finally show ?thesis by (unfold tRAG_def, simp) -qed - -lemma tRAG_subtree_RAG: "subtree (tRAG s) x \ subtree (RAG s) x" -proof - - { fix a - assume "a \ subtree (tRAG s) x" - hence "(a, x) \ (tRAG s)^*" by (auto simp:subtree_def) - with tRAG_star_RAG - have "(a, x) \ (RAG s)^*" by auto - hence "a \ subtree (RAG s) x" by (auto simp:subtree_def) - } thus ?thesis by auto -qed - -lemma tRAG_trancl_eq: - "{th'. (Th th', Th th) \ (tRAG s)^+} = - {th'. (Th th', Th th) \ (RAG s)^+}" - (is "?L = ?R") -proof - - { fix th' - assume "th' \ ?L" - hence "(Th th', Th th) \ (tRAG s)^+" by auto - from tranclD[OF this] - obtain z where h: "(Th th', z) \ tRAG s" "(z, Th th) \ (tRAG s)\<^sup>*" by auto - from tRAG_subtree_RAG and this(2) - have "(z, Th th) \ (RAG s)^*" by (meson subsetCE tRAG_star_RAG) - moreover from h(1) have "(Th th', z) \ (RAG s)^+" using tRAG_alt_def by auto - ultimately have "th' \ ?R" by auto - } moreover - { fix th' - assume "th' \ ?R" - hence "(Th th', Th th) \ (RAG s)^+" by (auto) - from plus_rpath[OF this] - obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \ []" by auto - hence "(Th th', Th th) \ (tRAG s)^+" - proof(induct xs arbitrary:th' th rule:length_induct) - case (1 xs th' th) - then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto) - show ?case - proof(cases "xs1") - case Nil - from 1(2)[unfolded Cons1 Nil] - have rp: "rpath (RAG s) (Th th') [x1] (Th th)" . - hence "(Th th', x1) \ (RAG s)" - by (cases, auto) - then obtain cs where "x1 = Cs cs" - by (unfold s_RAG_def, auto) - from rpath_nnl_lastE[OF rp[unfolded this]] - show ?thesis by auto - next - case (Cons x2 xs2) - from 1(2)[unfolded Cons1[unfolded this]] - have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" . - from rpath_edges_on[OF this] - have eds: "edges_on (Th th' # x1 # x2 # xs2) \ RAG s" . - have "(Th th', x1) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - with eds have rg1: "(Th th', x1) \ RAG s" by auto - then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto) - have "(x1, x2) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - from this eds - have rg2: "(x1, x2) \ RAG s" by auto - from this[unfolded eq_x1] - obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto) - from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2] - have rt1: "(Th th', Th th1) \ tRAG s" by (unfold tRAG_alt_def, auto) - from rp have "rpath (RAG s) x2 xs2 (Th th)" - by (elim rpath_ConsE, simp) - from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" . - show ?thesis - proof(cases "xs2 = []") - case True - from rpath_nilE[OF rp'[unfolded this]] - have "th1 = th" by auto - from rt1[unfolded this] show ?thesis by auto - next - case False - from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons] - have "(Th th1, Th th) \ (tRAG s)\<^sup>+" by simp - with rt1 show ?thesis by auto - qed - qed - qed - hence "th' \ ?L" by auto - } ultimately show ?thesis by blast -qed - -lemma tRAG_trancl_eq_Th: - "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = - {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" - using tRAG_trancl_eq by auto - -lemma dependants_alt_def: - "dependants s th = {th'. (Th th', Th th) \ (tRAG s)^+}" - by (metis eq_RAG s_dependants_def tRAG_trancl_eq) - -lemma dependants_alt_def1: - "dependants (s::state) th = {th'. (Th th', Th th) \ (RAG s)^+}" - using dependants_alt_def tRAG_trancl_eq by auto - -context valid_trace -begin -lemma count_eq_RAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (RAG s)^+} = {}" -proof(rule ccontr) - assume otherwise: "{th'. (Th th', Th th) \ (RAG s)\<^sup>+} \ {}" - then obtain th' where "(Th th', Th th) \ (RAG s)^+" by auto - from tranclD2[OF this] - obtain z where "z \ children (RAG s) (Th th)" - by (auto simp:children_def) - with eq_pv_children[OF assms] - show False by simp -qed - -lemma eq_pv_dependants: - assumes eq_pv: "cntP s th = cntV s th" - shows "dependants s th = {}" -proof - - from count_eq_RAG_plus[OF assms, folded dependants_alt_def1] - show ?thesis . -qed - -end - -lemma eq_dependants: "dependants (wq s) = dependants s" - by (simp add: s_dependants_abv wq_def) - -context valid_trace -begin - -lemma count_eq_tRAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using assms eq_pv_dependants dependants_alt_def eq_dependants by auto - -lemma count_eq_RAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (RAG s)^+} = {}" - using count_eq_RAG_plus[OF assms] by auto - -lemma count_eq_tRAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using count_eq_tRAG_plus[OF assms] by auto -end - -lemma inj_the_preced: - "inj_on (the_preced s) (threads s)" - by (metis inj_onI preced_unique the_preced_def) - -lemma tRAG_Field: - "Field (tRAG s) \ Field (RAG s)" - by (unfold tRAG_alt_def Field_def, auto) - -lemma tRAG_ancestorsE: - assumes "x \ ancestors (tRAG s) u" - obtains th where "x = Th th" -proof - - from assms have "(u, x) \ (tRAG s)^+" - by (unfold ancestors_def, auto) - from tranclE[OF this] obtain c where "(c, x) \ tRAG s" by auto - then obtain th where "x = Th th" - by (unfold tRAG_alt_def, auto) - from that[OF this] show ?thesis . -qed - -lemma tRAG_mono: - assumes "RAG s' \ RAG s" - shows "tRAG s' \ tRAG s" - using assms - by (unfold tRAG_alt_def, auto) - -lemma holding_next_thI: - assumes "holding s th cs" - and "length (wq s cs) > 1" - obtains th' where "next_th s th cs th'" -proof - - from assms(1)[folded holding_eq, unfolded cs_holding_def] - have " th \ set (wq s cs) \ th = hd (wq s cs)" - by (unfold s_holding_def, fold wq_def, auto) - then obtain rest where h1: "wq s cs = th#rest" - by (cases "wq s cs", auto) - with assms(2) have h2: "rest \ []" by auto - let ?th' = "hd (SOME q. distinct q \ set q = set rest)" - have "next_th s th cs ?th'" using h1(1) h2 - by (unfold next_th_def, auto) - from that[OF this] show ?thesis . -qed - -lemma RAG_tRAG_transfer: - assumes "vt s'" - assumes "RAG s = RAG s' \ {(Th th, Cs cs)}" - and "(Cs cs, Th th'') \ RAG s'" - shows "tRAG s = tRAG s' \ {(Th th, Th th'')}" (is "?L = ?R") -proof - - interpret vt_s': valid_trace "s'" using assms(1) - by (unfold_locales, simp) - { fix n1 n2 - assume "(n1, n2) \ ?L" - from this[unfolded tRAG_alt_def] - obtain th1 th2 cs' where - h: "n1 = Th th1" "n2 = Th th2" - "(Th th1, Cs cs') \ RAG s" - "(Cs cs', Th th2) \ RAG s" by auto - from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \ RAG s'" by auto - from h(3) and assms(2) - have "(Th th1, Cs cs') = (Th th, Cs cs) \ - (Th th1, Cs cs') \ RAG s'" by auto - hence "(n1, n2) \ ?R" - proof - assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)" - hence eq_th1: "th1 = th" by simp - moreover have "th2 = th''" - proof - - from h1 have "cs' = cs" by simp - from assms(3) cs_in[unfolded this] - show ?thesis using vt_s'.unique_RAG by auto - qed - ultimately show ?thesis using h(1,2) by auto - next - assume "(Th th1, Cs cs') \ RAG s'" - with cs_in have "(Th th1, Th th2) \ tRAG s'" - by (unfold tRAG_alt_def, auto) - from this[folded h(1, 2)] show ?thesis by auto - qed - } moreover { - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \tRAG s' \ (n1, n2) = (Th th, Th th'')" by auto - hence "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ tRAG s'" - moreover have "... \ ?L" - proof(rule tRAG_mono) - show "RAG s' \ RAG s" by (unfold assms(2), auto) - qed - ultimately show ?thesis by auto - next - assume eq_n: "(n1, n2) = (Th th, Th th'')" - from assms(2, 3) have "(Cs cs, Th th'') \ RAG s" by auto - moreover have "(Th th, Cs cs) \ RAG s" using assms(2) by auto - ultimately show ?thesis - by (unfold eq_n tRAG_alt_def, auto) - qed - } ultimately show ?thesis by auto -qed - -context valid_trace -begin - -lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt] - -end - -lemma tRAG_subtree_eq: - "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \ (subtree (RAG s) (Th th))}" - (is "?L = ?R") -proof - - { fix n - assume h: "n \ ?L" - hence "n \ ?R" - by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) - } moreover { - fix n - assume "n \ ?R" - then obtain th' where h: "n = Th th'" "(Th th', Th th) \ (RAG s)^*" - by (auto simp:subtree_def) - from rtranclD[OF this(2)] - have "n \ ?L" - proof - assume "Th th' \ Th th \ (Th th', Th th) \ (RAG s)\<^sup>+" - with h have "n \ {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" by auto - thus ?thesis using subtree_def tRAG_trancl_eq by fastforce - qed (insert h, auto simp:subtree_def) - } ultimately show ?thesis by auto -qed - -lemma threads_set_eq: - "the_thread ` (subtree (tRAG s) (Th th)) = - {th'. Th th' \ (subtree (RAG s) (Th th))}" (is "?L = ?R") - by (auto intro:rev_image_eqI simp:tRAG_subtree_eq) - -lemma cp_alt_def1: - "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))" -proof - - have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) = - ((the_preced s \ the_thread) ` subtree (tRAG s) (Th th))" - by auto - thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto) -qed - -lemma cp_gen_def_cond: - assumes "x = Th th" - shows "cp s th = cp_gen s (Th th)" -by (unfold cp_alt_def1 cp_gen_def, simp) - -lemma cp_gen_over_set: - assumes "\ x \ A. \ th. x = Th th" - shows "cp_gen s ` A = (cp s \ the_thread) ` A" -proof(rule f_image_eq) - fix a - assume "a \ A" - from assms[rule_format, OF this] - obtain th where eq_a: "a = Th th" by auto - show "cp_gen s a = (cp s \ the_thread) a" - by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp) -qed - -context valid_trace -begin - -lemma subtree_tRAG_thread: - assumes "th \ threads s" - shows "subtree (tRAG s) (Th th) \ Th ` threads s" (is "?L \ ?R") -proof - - have "?L = {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - by (unfold tRAG_subtree_eq, simp) - also have "... \ ?R" - proof - fix x - assume "x \ {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - then obtain th' where h: "x = Th th'" "Th th' \ subtree (RAG s) (Th th)" by auto - from this(2) - show "x \ ?R" - proof(cases rule:subtreeE) - case 1 - thus ?thesis by (simp add: assms h(1)) - next - case 2 - thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) - qed - qed - finally show ?thesis . -qed - -lemma readys_root: - assumes "th \ readys s" - shows "root (RAG s) (Th th)" -proof - - { fix x - assume "x \ ancestors (RAG s) (Th th)" - hence h: "(Th th, x) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - obtain z where "(Th th, z) \ RAG s" by auto - with assms(1) have False - apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def) - by (fold wq_def, blast) - } thus ?thesis by (unfold root_def, auto) -qed - -lemma readys_in_no_subtree: - assumes "th \ readys s" - and "th' \ th" - shows "Th th \ subtree (RAG s) (Th th')" -proof - assume "Th th \ subtree (RAG s) (Th th')" - thus False - proof(cases rule:subtreeE) - case 1 - with assms show ?thesis by auto - next - case 2 - with readys_root[OF assms(1)] - show ?thesis by (auto simp:root_def) - qed -qed - -lemma not_in_thread_isolated: - assumes "th \ threads s" - shows "(Th th) \ Field (RAG s)" -proof - assume "(Th th) \ Field (RAG s)" - with dm_RAG_threads and rg_RAG_threads assms - show False by (unfold Field_def, blast) -qed - -end - -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 (RAG s))" -apply(simp add: detached_def Field_def) -apply(simp add: s_RAG_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 - -context valid_trace -begin - -lemma detached_intro: - assumes eq_pv: "cntP s th = cntV s th" - shows "detached s th" -proof - - from eq_pv cnp_cnv_cncs - have "th \ readys s \ th \ threads s" by (auto simp:pvD_def) - thus ?thesis - proof - assume "th \ threads s" - with rg_RAG_threads dm_RAG_threads - show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv - s_holding_abv wq_def Domain_iff Range_iff) - next - assume "th \ readys s" - moreover have "Th th \ Range (RAG s)" - proof - - from eq_pv_children[OF assms] - have "children (RAG s) (Th th) = {}" . - thus ?thesis - by (unfold children_def, auto) - qed - ultimately show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv - s_holding_abv wq_def readys_def) - qed -qed - -lemma detached_elim: - assumes dtc: "detached s th" - shows "cntP s th = cntV s th" -proof - - have cncs_z: "cntCS s th = 0" - proof - - from dtc have "holdents s th = {}" - unfolding detached_def holdents_test s_RAG_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:waiting_eq s_RAG_def) - with cncs_z show ?thesis using cnp_cnv_cncs by (simp add:pvD_def) - next - case False - with cncs_z and cnp_cnv_cncs show ?thesis by (simp add:pvD_def) - qed -qed - -lemma detached_eq: - shows "(detached s th) = (cntP s th = cntV s th)" - by (insert vt, auto intro:detached_intro detached_elim) - -end - -context valid_trace -begin -(* ddd *) -lemma cp_gen_rec: - assumes "x = Th th" - shows "cp_gen s x = Max ({the_preced s th} \ (cp_gen s) ` children (tRAG s) x)" -proof(cases "children (tRAG s) x = {}") - case True - show ?thesis - by (unfold True cp_gen_def subtree_children, simp add:assms) -next - case False - hence [simp]: "children (tRAG s) x \ {}" by auto - note fsbttRAGs.finite_subtree[simp] - have [simp]: "finite (children (tRAG s) x)" - by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], - rule children_subtree) - { fix r x - have "subtree r x \ {}" by (auto simp:subtree_def) - } note this[simp] - have [simp]: "\x\children (tRAG s) x. subtree (tRAG s) x \ {}" - proof - - from False obtain q where "q \ children (tRAG s) x" by blast - moreover have "subtree (tRAG s) q \ {}" by simp - ultimately show ?thesis by blast - qed - have h: "Max ((the_preced s \ the_thread) ` - ({x} \ \(subtree (tRAG s) ` children (tRAG s) x))) = - Max ({the_preced s th} \ cp_gen s ` children (tRAG s) x)" - (is "?L = ?R") - proof - - let "Max (?f ` (?A \ \ (?g ` ?B)))" = ?L - let "Max (_ \ (?h ` ?B))" = ?R - let ?L1 = "?f ` \(?g ` ?B)" - have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)" - proof - - have "?L1 = ?f ` (\ x \ ?B.(?g x))" by simp - also have "... = (\ x \ ?B. ?f ` (?g x))" by auto - finally have "Max ?L1 = Max ..." by simp - also have "... = Max (Max ` (\x. ?f ` subtree (tRAG s) x) ` ?B)" - by (subst Max_UNION, simp+) - also have "... = Max (cp_gen s ` children (tRAG s) x)" - by (unfold image_comp cp_gen_alt_def, simp) - finally show ?thesis . - qed - show ?thesis - proof - - have "?L = Max (?f ` ?A \ ?L1)" by simp - also have "... = max (the_preced s (the_thread x)) (Max ?L1)" - by (subst Max_Un, simp+) - also have "... = max (?f x) (Max (?h ` ?B))" - by (unfold eq_Max_L1, simp) - also have "... =?R" - by (rule max_Max_eq, (simp)+, unfold assms, simp) - finally show ?thesis . - qed - qed thus ?thesis - by (fold h subtree_children, unfold cp_gen_def, simp) -qed - -lemma cp_rec: - "cp s th = Max ({the_preced s th} \ - (cp s o the_thread) ` children (tRAG s) (Th th))" -proof - - have "Th th = Th th" by simp - note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this] - show ?thesis - proof - - have "cp_gen s ` children (tRAG s) (Th th) = - (cp s \ the_thread) ` children (tRAG s) (Th th)" - proof(rule cp_gen_over_set) - show " \x\children (tRAG s) (Th th). \th. x = Th th" - by (unfold tRAG_alt_def, auto simp:children_def) - qed - thus ?thesis by (subst (1) h(1), unfold h(2), simp) - qed -qed - -lemma next_th_holding: - assumes nxt: "next_th s th cs th'" - shows "holding (wq s) th cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - thus ?thesis - by (unfold cs_holding_def, auto) -qed - -lemma next_th_waiting: - assumes nxt: "next_th s th cs th'" - shows "waiting (wq s) th' cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - from wq_distinct[of cs, unfolded h] - have dst: "distinct (th # rest)" . - have in_rest: "th' \ set rest" - proof(unfold h, rule someI2) - show "distinct rest \ set rest = set rest" using dst by auto - next - fix x assume "distinct x \ set x = set rest" - with h(2) - show "hd x \ set (rest)" by (cases x, auto) - qed - hence "th' \ set (wq s cs)" by (unfold h(1), auto) - moreover have "th' \ hd (wq s cs)" - by (unfold h(1), insert in_rest dst, auto) - ultimately show ?thesis by (auto simp:cs_waiting_def) -qed - -lemma next_th_RAG: - assumes nxt: "next_th (s::event list) th cs th'" - shows "{(Cs cs, Th th), (Th th', Cs cs)} \ RAG s" - using vt assms next_th_holding next_th_waiting - by (unfold s_RAG_def, simp) - -end - -lemma next_th_unique: - assumes nt1: "next_th s th cs th1" - and nt2: "next_th s th cs th2" - shows "th1 = th2" -using assms by (unfold next_th_def, auto) - -context valid_trace -begin - -thm th_chain_to_ready - -find_theorems subtree Th RAG - -lemma "(threads s) = (\ th \ readys s. {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "?L = ?R") -proof - - { fix th1 - assume "th1 \ ?L" - from th_chain_to_ready[OF this] - have "th1 \ readys s \ (\th'a. th'a \ readys s \ (Th th1, Th th'a) \ (RAG s)\<^sup>+)" . - hence "th1 \ ?R" - proof - assume "th1 \ readys s" - thus ?thesis by (auto simp:subtree_def) - next - assume "\th'a. th'a \ readys s \ (Th th1, Th th'a) \ (RAG s)\<^sup>+" - thus ?thesis - qed - } moreover - { fix th' - assume "th' \ ?R" - have "th' \ ?L" sorry - } ultimately show ?thesis by auto -qed - -lemma max_cp_readys_threads_pre: (* ccc *) - assumes np: "threads s \ {}" - shows "Max (cp s ` readys s) = Max (cp s ` threads s)" -proof(unfold max_cp_eq) - show "Max (cp s ` readys s) = Max (the_preced s ` threads s)" - proof - - let ?p = "Max (the_preced s ` threads s)" - let ?f = "the_preced s" - have "?p \ (?f ` threads s)" - proof(rule Max_in) - from finite_threads 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 tm_in] - have "tm \ readys s \ (\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+)" . - thus ?thesis - proof - assume "\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+ " - then obtain th' where th'_in: "th' \ readys s" - and tm_chain:"(Th tm, Th th') \ (RAG s)\<^sup>+" by auto - have "cp s th' = ?f tm" - proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI) - from dependants_threads finite_threads - show "finite ((\th. preced th s) ` ({th'} \ dependants (wq s) th'))" - by (auto intro:finite_subset) - next - fix p assume p_in: "p \ (\th. preced th s) ` ({th'} \ dependants (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 - show "finite ((\th. preced th s) ` threads s)" by simp - next - from p_in and th'_in and dependants_threads[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'} \ dependants (wq s) th')" - proof - - from tm_chain - have "tm \ dependants (wq s) th'" - by (unfold cs_dependants_def s_RAG_def cs_RAG_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 dependants_threads [of th1] th1_in - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ - (\th. preced th s) ` threads s" - by (auto simp:readys_def) - next - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" by simp - next - from finite_threads - show " finite ((\th. preced th s) ` threads s)" by simp - qed - next - from finite_threads - 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} \ dependants (wq s) tm)" - show "y \ preced tm s" - proof - - { fix y' - assume hy' : "y' \ ((\th. preced th s) ` dependants (wq s) tm)" - have "y' \ preced tm s" - proof(unfold tm_max, rule Max_ge) - from hy' dependants_threads[of tm] - show "y' \ (\th. preced th s) ` threads s" by auto - next - from finite_threads - show "finite ((\th. preced th s) ` threads s)" by simp - qed - } with hy show ?thesis by auto - qed - next - from dependants_threads[of tm] finite_threads - show "finite ((\th. preced th s) ` ({tm} \ dependants (wq s) tm))" - by (auto intro:finite_subset) - next - show "preced tm s \ (\th. preced th s) ` ({tm} \ dependants (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 - 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 - show "finite ((\th. preced th s) ` threads s)" by simp - next - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" - by simp - next - from dependants_threads[of th1] th1_readys - show "(\th. preced th s) ` ({th1} \ dependants (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 - -text {* (* ccc *) \noindent - Since the current precedence of the threads in ready queue will always be boosted, - there must be one inside it has the maximum precedence of the whole system. -*} -lemma max_cp_readys_threads: - 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 False]) -qed - -end - -end - diff -r ed938e2246b9 -r 0525670d8e6a ExtGG.thy~ --- a/ExtGG.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,920 +0,0 @@ -section {* - This file contains lemmas used to guide the recalculation of current precedence - after every system call (or system operation) -*} -theory Implementation -imports PIPBasics -begin - -text {* (* ddd *) - One beauty of our modelling is that we follow the definitional extension tradition of HOL. - The benefit of such a concise and miniature model is that large number of intuitively - obvious facts are derived as lemmas, rather than asserted as axioms. -*} - -text {* - However, the lemmas in the forthcoming several locales are no longer - obvious. These lemmas show how the current precedences should be recalculated - after every execution step (in our model, every step is represented by an event, - which in turn, represents a system call, or operation). Each operation is - treated in a separate locale. - - The complication of current precedence recalculation comes - because the changing of RAG needs to be taken into account, - in addition to the changing of precedence. - - The reason RAG changing affects current precedence is that, - according to the definition, current precedence - of a thread is the maximum of the precedences of every threads in its subtree, - where the notion of sub-tree in RAG is defined in RTree.thy. - - Therefore, for each operation, lemmas about the change of precedences - and RAG are derived first, on which lemmas about current precedence - recalculation are based on. -*} - -section {* The @{term Set} operation *} - -text {* (* ddd *) - The following locale @{text "step_set_cps"} investigates the recalculation - after the @{text "Set"} operation. -*} -locale step_set_cps = - fixes s' th prio s - -- {* @{text "s'"} is the system state before the operation *} - -- {* @{text "s"} is the system state after the operation *} - defines s_def : "s \ (Set th prio#s')" - -- {* @{text "s"} is assumed to be a legitimate state, from which - the legitimacy of @{text "s"} can be derived. *} - assumes vt_s: "vt s" - -sublocale step_set_cps < vat_s : valid_trace "s" -proof - from vt_s show "vt s" . -qed - -sublocale step_set_cps < vat_s' : valid_trace "s'" -proof - from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" . -qed - -context step_set_cps -begin - -text {* (* ddd *) - The following two lemmas confirm that @{text "Set"}-operation - only changes the precedence of the initiating thread (or actor) - of the operation (or event). -*} - -lemma eq_preced: - assumes "th' \ th" - shows "preced th' s = preced th' s'" -proof - - from assms show ?thesis - by (unfold s_def, auto simp:preced_def) -qed - -lemma eq_the_preced: - assumes "th' \ th" - shows "the_preced s th' = the_preced s' th'" - using assms - by (unfold the_preced_def, intro eq_preced, simp) - -text {* - The following lemma assures that the resetting of priority does not change the RAG. -*} - -lemma eq_dep: "RAG s = RAG s'" - by (unfold s_def RAG_set_unchanged, auto) - -text {* (* ddd *) - Th following lemma @{text "eq_cp_pre"} says that the priority change of @{text "th"} - only affects those threads, which as @{text "Th th"} in their sub-trees. - - The proof of this lemma is simplified by using the alternative definition - of @{text "cp"}. -*} - -lemma eq_cp_pre: - assumes nd: "Th th \ subtree (RAG s') (Th th')" - shows "cp s th' = cp s' th'" -proof - - -- {* After unfolding using the alternative definition, elements - affecting the @{term "cp"}-value of threads become explicit. - We only need to prove the following: *} - have "Max (the_preced s ` {th'a. Th th'a \ subtree (RAG s) (Th th')}) = - Max (the_preced s' ` {th'a. Th th'a \ subtree (RAG s') (Th th')})" - (is "Max (?f ` ?S1) = Max (?g ` ?S2)") - proof - - -- {* The base sets are equal. *} - have "?S1 = ?S2" using eq_dep by simp - -- {* The function values on the base set are equal as well. *} - moreover have "\ e \ ?S2. ?f e = ?g e" - proof - fix th1 - assume "th1 \ ?S2" - with nd have "th1 \ th" by (auto) - from eq_the_preced[OF this] - show "the_preced s th1 = the_preced s' th1" . - qed - -- {* Therefore, the image of the functions are equal. *} - ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq) - thus ?thesis by simp - qed - thus ?thesis by (simp add:cp_alt_def) -qed - -text {* - The following lemma shows that @{term "th"} is not in the - sub-tree of any other thread. -*} -lemma th_in_no_subtree: - assumes "th' \ th" - shows "Th th \ subtree (RAG s') (Th th')" -proof - - have "th \ readys s'" - proof - - from step_back_step [OF vt_s[unfolded s_def]] - have "step s' (Set th prio)" . - hence "th \ runing s'" by (cases, simp) - thus ?thesis by (simp add:readys_def runing_def) - qed - from vat_s'.readys_in_no_subtree[OF this assms(1)] - show ?thesis by blast -qed - -text {* - By combining @{thm "eq_cp_pre"} and @{thm "th_in_no_subtree"}, - it is obvious that the change of priority only affects the @{text "cp"}-value - of the initiating thread @{text "th"}. -*} -lemma eq_cp: - assumes "th' \ th" - shows "cp s th' = cp s' th'" - by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]]) - -end - -section {* The @{term V} operation *} - -text {* - The following @{text "step_v_cps"} is the locale for @{text "V"}-operation. -*} - -locale step_v_cps = - -- {* @{text "th"} is the initiating thread *} - -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *} - fixes s' th cs s -- {* @{text "s'"} is the state before operation*} - defines s_def : "s \ (V th cs#s')" -- {* @{text "s"} is the state after operation*} - -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *} - assumes vt_s: "vt s" - -sublocale step_v_cps < vat_s : valid_trace "s" -proof - from vt_s show "vt s" . -qed - -sublocale step_v_cps < vat_s' : valid_trace "s'" -proof - from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" . -qed - -context step_v_cps -begin - -lemma ready_th_s': "th \ readys s'" - using step_back_step[OF vt_s[unfolded s_def]] - by (cases, simp add:runing_def) - -lemma ancestors_th: "ancestors (RAG s') (Th th) = {}" -proof - - from vat_s'.readys_root[OF ready_th_s'] - show ?thesis - by (unfold root_def, simp) -qed - -lemma holding_th: "holding s' th cs" -proof - - from vt_s[unfolded s_def] - have " PIP s' (V th cs)" by (cases, simp) - thus ?thesis by (cases, auto) -qed - -lemma edge_of_th: - "(Cs cs, Th th) \ RAG s'" -proof - - from holding_th - show ?thesis - by (unfold s_RAG_def holding_eq, auto) -qed - -lemma ancestors_cs: - "ancestors (RAG s') (Cs cs) = {Th th}" -proof - - have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th) \ {Th th}" - proof(rule vat_s'.rtree_RAG.ancestors_accum) - from vt_s[unfolded s_def] - have " PIP s' (V th cs)" by (cases, simp) - thus "(Cs cs, Th th) \ RAG s'" - proof(cases) - assume "holding s' th cs" - from this[unfolded holding_eq] - show ?thesis by (unfold s_RAG_def, auto) - qed - qed - from this[unfolded ancestors_th] show ?thesis by simp -qed - -lemma preced_kept: "the_preced s = the_preced s'" - by (auto simp: s_def the_preced_def preced_def) - -end - -text {* - The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation, - which represents the case when there is another thread @{text "th'"} - to take over the critical resource released by the initiating thread @{text "th"}. -*} -locale step_v_cps_nt = step_v_cps + - fixes th' - -- {* @{text "th'"} is assumed to take over @{text "cs"} *} - assumes nt: "next_th s' th cs th'" - -context step_v_cps_nt -begin - -text {* - Lemma @{text "RAG_s"} confirms the change of RAG: - two edges removed and one added, as shown by the following diagram. -*} - -(* - RAG before the V-operation - th1 ----| - | - th' ----| - |----> cs -----| - th2 ----| | - | | - th3 ----| | - |------> th - th4 ----| | - | | - th5 ----| | - |----> cs'-----| - th6 ----| - | - th7 ----| - - RAG after the V-operation - th1 ----| - | - |----> cs ----> th' - th2 ----| - | - th3 ----| - - th4 ----| - | - th5 ----| - |----> cs'----> th - th6 ----| - | - th7 ----| -*) - -lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \ RAG s'" - using next_th_RAG[OF nt] . - -lemma ancestors_th': - "ancestors (RAG s') (Th th') = {Th th, Cs cs}" -proof - - have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \ {Cs cs}" - proof(rule vat_s'.rtree_RAG.ancestors_accum) - from sub_RAGs' show "(Th th', Cs cs) \ RAG s'" by auto - qed - thus ?thesis using ancestors_th ancestors_cs by auto -qed - -lemma RAG_s: - "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \ - {(Cs cs, Th th')}" -proof - - from step_RAG_v[OF vt_s[unfolded s_def], folded s_def] - and nt show ?thesis by (auto intro:next_th_unique) -qed - -lemma subtree_kept: (* ddd *) - assumes "th1 \ {th, th'}" - shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R") -proof - - let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})" - let ?RAG'' = "?RAG' \ {(Cs cs, Th th')}" - have "subtree ?RAG' (Th th1) = ?R" - proof(rule subset_del_subtree_outside) - show "Range {(Cs cs, Th th), (Th th', Cs cs)} \ subtree (RAG s') (Th th1) = {}" - proof - - have "(Th th) \ subtree (RAG s') (Th th1)" - proof(rule subtree_refute) - show "Th th1 \ ancestors (RAG s') (Th th)" - by (unfold ancestors_th, simp) - next - from assms show "Th th1 \ Th th" by simp - qed - moreover have "(Cs cs) \ subtree (RAG s') (Th th1)" - proof(rule subtree_refute) - show "Th th1 \ ancestors (RAG s') (Cs cs)" - by (unfold ancestors_cs, insert assms, auto) - qed simp - ultimately have "{Th th, Cs cs} \ subtree (RAG s') (Th th1) = {}" by auto - thus ?thesis by simp - qed - qed - moreover have "subtree ?RAG'' (Th th1) = subtree ?RAG' (Th th1)" - proof(rule subtree_insert_next) - show "Th th' \ subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)" - proof(rule subtree_refute) - show "Th th1 \ ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')" - (is "_ \ ?R") - proof - - have "?R \ ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto) - moreover have "Th th1 \ ..." using ancestors_th' assms by simp - ultimately show ?thesis by auto - qed - next - from assms show "Th th1 \ Th th'" by simp - qed - qed - ultimately show ?thesis by (unfold RAG_s, simp) -qed - -lemma cp_kept: - assumes "th1 \ {th, th'}" - shows "cp s th1 = cp s' th1" - by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp) - -end - -locale step_v_cps_nnt = step_v_cps + - assumes nnt: "\ th'. (\ next_th s' th cs th')" - -context step_v_cps_nnt -begin - -lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}" -proof - - from nnt and step_RAG_v[OF vt_s[unfolded s_def], folded s_def] - show ?thesis by auto -qed - -lemma subtree_kept: - assumes "th1 \ th" - shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" -proof(unfold RAG_s, rule subset_del_subtree_outside) - show "Range {(Cs cs, Th th)} \ subtree (RAG s') (Th th1) = {}" - proof - - have "(Th th) \ subtree (RAG s') (Th th1)" - proof(rule subtree_refute) - show "Th th1 \ ancestors (RAG s') (Th th)" - by (unfold ancestors_th, simp) - next - from assms show "Th th1 \ Th th" by simp - qed - thus ?thesis by auto - qed -qed - -lemma cp_kept_1: - assumes "th1 \ th" - shows "cp s th1 = cp s' th1" - by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp) - -lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}" -proof - - { fix n - have "(Cs cs) \ ancestors (RAG s') n" - proof - assume "Cs cs \ ancestors (RAG s') n" - hence "(n, Cs cs) \ (RAG s')^+" by (auto simp:ancestors_def) - from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \ RAG s'" by auto - then obtain th' where "nn = Th th'" - by (unfold s_RAG_def, auto) - from h[unfolded this] have "(Th th', Cs cs) \ RAG s'" . - from this[unfolded s_RAG_def] - have "waiting (wq s') th' cs" by auto - from this[unfolded cs_waiting_def] - have "1 < length (wq s' cs)" - by (cases "wq s' cs", auto) - from holding_next_thI[OF holding_th this] - obtain th' where "next_th s' th cs th'" by auto - with nnt show False by auto - qed - } note h = this - { fix n - assume "n \ subtree (RAG s') (Cs cs)" - hence "n = (Cs cs)" - by (elim subtreeE, insert h, auto) - } moreover have "(Cs cs) \ subtree (RAG s') (Cs cs)" - by (auto simp:subtree_def) - ultimately show ?thesis by auto -qed - -lemma subtree_th: - "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}" -proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside) - from edge_of_th - show "(Cs cs, Th th) \ edges_in (RAG s') (Th th)" - by (unfold edges_in_def, auto simp:subtree_def) -qed - -lemma cp_kept_2: - shows "cp s th = cp s' th" - by (unfold cp_alt_def subtree_th preced_kept, auto) - -lemma eq_cp: - shows "cp s th' = cp s' th'" - using cp_kept_1 cp_kept_2 - by (cases "th' = th", auto) -end - - -locale step_P_cps = - fixes s' th cs s - defines s_def : "s \ (P th cs#s')" - assumes vt_s: "vt s" - -sublocale step_P_cps < vat_s : valid_trace "s" -proof - from vt_s show "vt s" . -qed - -section {* The @{term P} operation *} - -sublocale step_P_cps < vat_s' : valid_trace "s'" -proof - from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" . -qed - -context step_P_cps -begin - -lemma readys_th: "th \ readys s'" -proof - - from step_back_step [OF vt_s[unfolded s_def]] - have "PIP s' (P th cs)" . - hence "th \ runing s'" by (cases, simp) - thus ?thesis by (simp add:readys_def runing_def) -qed - -lemma root_th: "root (RAG s') (Th th)" - using readys_root[OF readys_th] . - -lemma in_no_others_subtree: - assumes "th' \ th" - shows "Th th \ subtree (RAG s') (Th th')" -proof - assume "Th th \ subtree (RAG s') (Th th')" - thus False - proof(cases rule:subtreeE) - case 1 - with assms show ?thesis by auto - next - case 2 - with root_th show ?thesis by (auto simp:root_def) - qed -qed - -lemma preced_kept: "the_preced s = the_preced s'" - by (auto simp: s_def the_preced_def preced_def) - -end - -locale step_P_cps_ne =step_P_cps + - fixes th' - assumes ne: "wq s' cs \ []" - defines th'_def: "th' \ hd (wq s' cs)" - -locale step_P_cps_e =step_P_cps + - assumes ee: "wq s' cs = []" - -context step_P_cps_e -begin - -lemma RAG_s: "RAG s = RAG s' \ {(Cs cs, Th th)}" -proof - - from ee and step_RAG_p[OF vt_s[unfolded s_def], folded s_def] - show ?thesis by auto -qed - -lemma subtree_kept: - assumes "th' \ th" - shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')" -proof(unfold RAG_s, rule subtree_insert_next) - from in_no_others_subtree[OF assms] - show "Th th \ subtree (RAG s') (Th th')" . -qed - -lemma cp_kept: - assumes "th' \ th" - shows "cp s th' = cp s' th'" -proof - - have "(the_preced s ` {th'a. Th th'a \ subtree (RAG s) (Th th')}) = - (the_preced s' ` {th'a. Th th'a \ subtree (RAG s') (Th th')})" - by (unfold preced_kept subtree_kept[OF assms], simp) - thus ?thesis by (unfold cp_alt_def, simp) -qed - -end - -context step_P_cps_ne -begin - -lemma RAG_s: "RAG s = RAG s' \ {(Th th, Cs cs)}" -proof - - from step_RAG_p[OF vt_s[unfolded s_def]] and ne - show ?thesis by (simp add:s_def) -qed - -lemma cs_held: "(Cs cs, Th th') \ RAG s'" -proof - - have "(Cs cs, Th th') \ hRAG s'" - proof - - from ne - have " holding s' th' cs" - by (unfold th'_def holding_eq cs_holding_def, auto) - thus ?thesis - by (unfold hRAG_def, auto) - qed - thus ?thesis by (unfold RAG_split, auto) -qed - -lemma tRAG_s: - "tRAG s = tRAG s' \ {(Th th, Th th')}" - using RAG_tRAG_transfer[OF RAG_s cs_held] . - -lemma cp_kept: - assumes "Th th'' \ ancestors (tRAG s) (Th th)" - shows "cp s th'' = cp s' th''" -proof - - have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')" - proof - - have "Th th' \ subtree (tRAG s') (Th th'')" - proof - assume "Th th' \ subtree (tRAG s') (Th th'')" - thus False - proof(rule subtreeE) - assume "Th th' = Th th''" - from assms[unfolded tRAG_s ancestors_def, folded this] - show ?thesis by auto - next - assume "Th th'' \ ancestors (tRAG s') (Th th')" - moreover have "... \ ancestors (tRAG s) (Th th')" - proof(rule ancestors_mono) - show "tRAG s' \ tRAG s" by (unfold tRAG_s, auto) - qed - ultimately have "Th th'' \ ancestors (tRAG s) (Th th')" by auto - moreover have "Th th' \ ancestors (tRAG s) (Th th)" - by (unfold tRAG_s, auto simp:ancestors_def) - ultimately have "Th th'' \ ancestors (tRAG s) (Th th)" - by (auto simp:ancestors_def) - with assms show ?thesis by auto - qed - qed - from subtree_insert_next[OF this] - have "subtree (tRAG s' \ {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" . - from this[folded tRAG_s] show ?thesis . - qed - show ?thesis by (unfold cp_alt_def1 h preced_kept, simp) -qed - -lemma cp_gen_update_stop: (* ddd *) - assumes "u \ ancestors (tRAG s) (Th th)" - and "cp_gen s u = cp_gen s' u" - and "y \ ancestors (tRAG s) u" - shows "cp_gen s y = cp_gen s' y" - using assms(3) -proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf]) - case (1 x) - show ?case (is "?L = ?R") - proof - - from tRAG_ancestorsE[OF 1(2)] - obtain th2 where eq_x: "x = Th th2" by blast - from vat_s.cp_gen_rec[OF this] - have "?L = - Max ({the_preced s th2} \ cp_gen s ` RTree.children (tRAG s) x)" . - also have "... = - Max ({the_preced s' th2} \ cp_gen s' ` RTree.children (tRAG s') x)" - - proof - - from preced_kept have "the_preced s th2 = the_preced s' th2" by simp - moreover have "cp_gen s ` RTree.children (tRAG s) x = - cp_gen s' ` RTree.children (tRAG s') x" - proof - - have "RTree.children (tRAG s) x = RTree.children (tRAG s') x" - proof(unfold tRAG_s, rule children_union_kept) - have start: "(Th th, Th th') \ tRAG s" - by (unfold tRAG_s, auto) - note x_u = 1(2) - show "x \ Range {(Th th, Th th')}" - proof - assume "x \ Range {(Th th, Th th')}" - hence eq_x: "x = Th th'" using RangeE by auto - show False - proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start]) - case 1 - from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG - show ?thesis by (auto simp:ancestors_def acyclic_def) - next - case 2 - with x_u[unfolded eq_x] - have "(Th th', Th th') \ (tRAG s)^+" by (auto simp:ancestors_def) - with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def) - qed - qed - qed - moreover have "cp_gen s ` RTree.children (tRAG s) x = - cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A") - proof(rule f_image_eq) - fix a - assume a_in: "a \ ?A" - from 1(2) - show "?f a = ?g a" - proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch]) - case in_ch - show ?thesis - proof(cases "a = u") - case True - from assms(2)[folded this] show ?thesis . - next - case False - have a_not_in: "a \ ancestors (tRAG s) (Th th)" - proof - assume a_in': "a \ ancestors (tRAG s) (Th th)" - have "a = u" - proof(rule vat_s.rtree_s.ancestors_children_unique) - from a_in' a_in show "a \ ancestors (tRAG s) (Th th) \ - RTree.children (tRAG s) x" by auto - next - from assms(1) in_ch show "u \ ancestors (tRAG s) (Th th) \ - RTree.children (tRAG s) x" by auto - qed - with False show False by simp - qed - from a_in obtain th_a where eq_a: "a = Th th_a" - by (unfold RTree.children_def tRAG_alt_def, auto) - from cp_kept[OF a_not_in[unfolded eq_a]] - have "cp s th_a = cp s' th_a" . - from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a] - show ?thesis . - qed - next - case (out_ch z) - hence h: "z \ ancestors (tRAG s) u" "z \ RTree.children (tRAG s) x" by auto - show ?thesis - proof(cases "a = z") - case True - from h(2) have zx_in: "(z, x) \ (tRAG s)" by (auto simp:RTree.children_def) - from 1(1)[rule_format, OF this h(1)] - have eq_cp_gen: "cp_gen s z = cp_gen s' z" . - with True show ?thesis by metis - next - case False - from a_in obtain th_a where eq_a: "a = Th th_a" - by (auto simp:RTree.children_def tRAG_alt_def) - have "a \ ancestors (tRAG s) (Th th)" - proof - assume a_in': "a \ ancestors (tRAG s) (Th th)" - have "a = z" - proof(rule vat_s.rtree_s.ancestors_children_unique) - from assms(1) h(1) have "z \ ancestors (tRAG s) (Th th)" - by (auto simp:ancestors_def) - with h(2) show " z \ ancestors (tRAG s) (Th th) \ - RTree.children (tRAG s) x" by auto - next - from a_in a_in' - show "a \ ancestors (tRAG s) (Th th) \ RTree.children (tRAG s) x" - by auto - qed - with False show False by auto - qed - from cp_kept[OF this[unfolded eq_a]] - have "cp s th_a = cp s' th_a" . - from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a] - show ?thesis . - qed - qed - qed - ultimately show ?thesis by metis - qed - ultimately show ?thesis by simp - qed - also have "... = ?R" - by (fold vat_s'.cp_gen_rec[OF eq_x], simp) - finally show ?thesis . - qed -qed - -lemma cp_up: - assumes "(Th th') \ ancestors (tRAG s) (Th th)" - and "cp s th' = cp s' th'" - and "(Th th'') \ ancestors (tRAG s) (Th th')" - shows "cp s th'' = cp s' th''" -proof - - have "cp_gen s (Th th'') = cp_gen s' (Th th'')" - proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)]) - from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]] - show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis - qed - with cp_gen_def_cond[OF refl[of "Th th''"]] - show ?thesis by metis -qed - -end - -section {* The @{term Create} operation *} - -locale step_create_cps = - fixes s' th prio s - defines s_def : "s \ (Create th prio#s')" - assumes vt_s: "vt s" - -sublocale step_create_cps < vat_s: valid_trace "s" - by (unfold_locales, insert vt_s, simp) - -sublocale step_create_cps < vat_s': valid_trace "s'" - by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp) - -context step_create_cps -begin - -lemma RAG_kept: "RAG s = RAG s'" - by (unfold s_def RAG_create_unchanged, auto) - -lemma tRAG_kept: "tRAG s = tRAG s'" - by (unfold tRAG_alt_def RAG_kept, auto) - -lemma preced_kept: - assumes "th' \ th" - shows "the_preced s th' = the_preced s' th'" - by (unfold s_def the_preced_def preced_def, insert assms, auto) - -lemma th_not_in: "Th th \ Field (tRAG s')" -proof - - from vt_s[unfolded s_def] - have "PIP s' (Create th prio)" by (cases, simp) - hence "th \ threads s'" by(cases, simp) - from vat_s'.not_in_thread_isolated[OF this] - have "Th th \ Field (RAG s')" . - with tRAG_Field show ?thesis by auto -qed - -lemma eq_cp: - assumes neq_th: "th' \ th" - shows "cp s th' = cp s' th'" -proof - - have "(the_preced s \ the_thread) ` subtree (tRAG s) (Th th') = - (the_preced s' \ the_thread) ` subtree (tRAG s') (Th th')" - proof(unfold tRAG_kept, rule f_image_eq) - fix a - assume a_in: "a \ subtree (tRAG s') (Th th')" - then obtain th_a where eq_a: "a = Th th_a" - proof(cases rule:subtreeE) - case 2 - from ancestors_Field[OF 2(2)] - and that show ?thesis by (unfold tRAG_alt_def, auto) - qed auto - have neq_th_a: "th_a \ th" - proof - - have "(Th th) \ subtree (tRAG s') (Th th')" - proof - assume "Th th \ subtree (tRAG s') (Th th')" - thus False - proof(cases rule:subtreeE) - case 2 - from ancestors_Field[OF this(2)] - and th_not_in[unfolded Field_def] - show ?thesis by auto - qed (insert assms, auto) - qed - with a_in[unfolded eq_a] show ?thesis by auto - qed - from preced_kept[OF this] - show "(the_preced s \ the_thread) a = (the_preced s' \ the_thread) a" - by (unfold eq_a, simp) - qed - thus ?thesis by (unfold cp_alt_def1, simp) -qed - -lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}" -proof - - { fix a - assume "a \ RTree.children (tRAG s) (Th th)" - hence "(a, Th th) \ tRAG s" by (auto simp:RTree.children_def) - with th_not_in have False - by (unfold Field_def tRAG_kept, auto) - } thus ?thesis by auto -qed - -lemma eq_cp_th: "cp s th = preced th s" - by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def) - -end - -locale step_exit_cps = - fixes s' th prio s - defines s_def : "s \ Exit th # s'" - assumes vt_s: "vt s" - -sublocale step_exit_cps < vat_s: valid_trace "s" - by (unfold_locales, insert vt_s, simp) - -sublocale step_exit_cps < vat_s': valid_trace "s'" - by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp) - -context step_exit_cps -begin - -lemma preced_kept: - assumes "th' \ th" - shows "the_preced s th' = the_preced s' th'" - by (unfold s_def the_preced_def preced_def, insert assms, auto) - -lemma RAG_kept: "RAG s = RAG s'" - by (unfold s_def RAG_exit_unchanged, auto) - -lemma tRAG_kept: "tRAG s = tRAG s'" - by (unfold tRAG_alt_def RAG_kept, auto) - -lemma th_ready: "th \ readys s'" -proof - - from vt_s[unfolded s_def] - have "PIP s' (Exit th)" by (cases, simp) - hence h: "th \ runing s' \ holdents s' th = {}" by (cases, metis) - thus ?thesis by (unfold runing_def, auto) -qed - -lemma th_holdents: "holdents s' th = {}" -proof - - from vt_s[unfolded s_def] - have "PIP s' (Exit th)" by (cases, simp) - thus ?thesis by (cases, metis) -qed - -lemma th_RAG: "Th th \ Field (RAG s')" -proof - - have "Th th \ Range (RAG s')" - proof - assume "Th th \ Range (RAG s')" - then obtain cs where "holding (wq s') th cs" - by (unfold Range_iff s_RAG_def, auto) - with th_holdents[unfolded holdents_def] - show False by (unfold eq_holding, auto) - qed - moreover have "Th th \ Domain (RAG s')" - proof - assume "Th th \ Domain (RAG s')" - then obtain cs where "waiting (wq s') th cs" - by (unfold Domain_iff s_RAG_def, auto) - with th_ready show False by (unfold readys_def eq_waiting, auto) - qed - ultimately show ?thesis by (auto simp:Field_def) -qed - -lemma th_tRAG: "(Th th) \ Field (tRAG s')" - using th_RAG tRAG_Field[of s'] by auto - -lemma eq_cp: - assumes neq_th: "th' \ th" - shows "cp s th' = cp s' th'" -proof - - have "(the_preced s \ the_thread) ` subtree (tRAG s) (Th th') = - (the_preced s' \ the_thread) ` subtree (tRAG s') (Th th')" - proof(unfold tRAG_kept, rule f_image_eq) - fix a - assume a_in: "a \ subtree (tRAG s') (Th th')" - then obtain th_a where eq_a: "a = Th th_a" - proof(cases rule:subtreeE) - case 2 - from ancestors_Field[OF 2(2)] - and that show ?thesis by (unfold tRAG_alt_def, auto) - qed auto - have neq_th_a: "th_a \ th" - proof - - from vat_s'.readys_in_no_subtree[OF th_ready assms] - have "(Th th) \ subtree (RAG s') (Th th')" . - with tRAG_subtree_RAG[of s' "Th th'"] - have "(Th th) \ subtree (tRAG s') (Th th')" by auto - with a_in[unfolded eq_a] show ?thesis by auto - qed - from preced_kept[OF this] - show "(the_preced s \ the_thread) a = (the_preced s' \ the_thread) a" - by (unfold eq_a, simp) - qed - thus ?thesis by (unfold cp_alt_def1, simp) -qed - -end - -end - diff -r ed938e2246b9 -r 0525670d8e6a Implementation.thy~ --- a/Implementation.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,913 +0,0 @@ -section {* - This file contains lemmas used to guide the recalculation of current precedence - after every system call (or system operation) -*} -theory Implementation -imports PIPBasics -begin - -text {* (* ddd *) - One beauty of our modelling is that we follow the definitional extension tradition of HOL. - The benefit of such a concise and miniature model is that large number of intuitively - obvious facts are derived as lemmas, rather than asserted as axioms. -*} - -text {* - However, the lemmas in the forthcoming several locales are no longer - obvious. These lemmas show how the current precedences should be recalculated - after every execution step (in our model, every step is represented by an event, - which in turn, represents a system call, or operation). Each operation is - treated in a separate locale. - - The complication of current precedence recalculation comes - because the changing of RAG needs to be taken into account, - in addition to the changing of precedence. - The reason RAG changing affects current precedence is that, - according to the definition, current precedence - of a thread is the maximum of the precedences of its dependants, - where the dependants are defined in terms of RAG. - - Therefore, each operation, lemmas concerning the change of the precedences - and RAG are derived first, so that the lemmas about - current precedence recalculation can be based on. -*} - -text {* (* ddd *) - The following locale @{text "step_set_cps"} investigates the recalculation - after the @{text "Set"} operation. -*} -locale step_set_cps = - fixes s' th prio s - -- {* @{text "s'"} is the system state before the operation *} - -- {* @{text "s"} is the system state after the operation *} - defines s_def : "s \ (Set th prio#s')" - -- {* @{text "s"} is assumed to be a legitimate state, from which - the legitimacy of @{text "s"} can be derived. *} - assumes vt_s: "vt s" - -sublocale step_set_cps < vat_s : valid_trace "s" -proof - from vt_s show "vt s" . -qed - -sublocale step_set_cps < vat_s' : valid_trace "s'" -proof - from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" . -qed - -context step_set_cps -begin - -text {* (* ddd *) - The following two lemmas confirm that @{text "Set"}-operating only changes the precedence - of the initiating thread. -*} - -lemma eq_preced: - assumes "th' \ th" - shows "preced th' s = preced th' s'" -proof - - from assms show ?thesis - by (unfold s_def, auto simp:preced_def) -qed - -lemma eq_the_preced: - fixes th' - assumes "th' \ th" - shows "the_preced s th' = the_preced s' th'" - using assms - by (unfold the_preced_def, intro eq_preced, simp) - -text {* - The following lemma assures that the resetting of priority does not change the RAG. -*} - -lemma eq_dep: "RAG s = RAG s'" - by (unfold s_def RAG_set_unchanged, auto) - -text {* (* ddd *) - Th following lemma @{text "eq_cp_pre"} says the priority change of @{text "th"} - only affects those threads, which as @{text "Th th"} in their sub-trees. - - The proof of this lemma is simplified by using the alternative definition of @{text "cp"}. -*} - -lemma eq_cp_pre: - fixes th' - assumes nd: "Th th \ subtree (RAG s') (Th th')" - shows "cp s th' = cp s' th'" -proof - - -- {* After unfolding using the alternative definition, elements - affecting the @{term "cp"}-value of threads become explicit. - We only need to prove the following: *} - have "Max (the_preced s ` {th'a. Th th'a \ subtree (RAG s) (Th th')}) = - Max (the_preced s' ` {th'a. Th th'a \ subtree (RAG s') (Th th')})" - (is "Max (?f ` ?S1) = Max (?g ` ?S2)") - proof - - -- {* The base sets are equal. *} - have "?S1 = ?S2" using eq_dep by simp - -- {* The function values on the base set are equal as well. *} - moreover have "\ e \ ?S2. ?f e = ?g e" - proof - fix th1 - assume "th1 \ ?S2" - with nd have "th1 \ th" by (auto) - from eq_the_preced[OF this] - show "the_preced s th1 = the_preced s' th1" . - qed - -- {* Therefore, the image of the functions are equal. *} - ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq) - thus ?thesis by simp - qed - thus ?thesis by (simp add:cp_alt_def) -qed - -text {* - The following lemma shows that @{term "th"} is not in the - sub-tree of any other thread. -*} -lemma th_in_no_subtree: - assumes "th' \ th" - shows "Th th \ subtree (RAG s') (Th th')" -proof - - have "th \ readys s'" - proof - - from step_back_step [OF vt_s[unfolded s_def]] - have "step s' (Set th prio)" . - hence "th \ runing s'" by (cases, simp) - thus ?thesis by (simp add:readys_def runing_def) - qed - from vat_s'.readys_in_no_subtree[OF this assms(1)] - show ?thesis by blast -qed - -text {* - By combining @{thm "eq_cp_pre"} and @{thm "th_in_no_subtree"}, - it is obvious that the change of priority only affects the @{text "cp"}-value - of the initiating thread @{text "th"}. -*} -lemma eq_cp: - fixes th' - assumes "th' \ th" - shows "cp s th' = cp s' th'" - by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]]) - -end - -text {* - The following @{text "step_v_cps"} is the locale for @{text "V"}-operation. -*} - -locale step_v_cps = - -- {* @{text "th"} is the initiating thread *} - -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *} - fixes s' th cs s -- {* @{text "s'"} is the state before operation*} - defines s_def : "s \ (V th cs#s')" -- {* @{text "s"} is the state after operation*} - -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *} - assumes vt_s: "vt s" - -sublocale step_v_cps < vat_s : valid_trace "s" -proof - from vt_s show "vt s" . -qed - -sublocale step_v_cps < vat_s' : valid_trace "s'" -proof - from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" . -qed - -context step_v_cps -begin - -lemma ready_th_s': "th \ readys s'" - using step_back_step[OF vt_s[unfolded s_def]] - by (cases, simp add:runing_def) - -lemma ancestors_th: "ancestors (RAG s') (Th th) = {}" -proof - - from vat_s'.readys_root[OF ready_th_s'] - show ?thesis - by (unfold root_def, simp) -qed - -lemma holding_th: "holding s' th cs" -proof - - from vt_s[unfolded s_def] - have " PIP s' (V th cs)" by (cases, simp) - thus ?thesis by (cases, auto) -qed - -lemma edge_of_th: - "(Cs cs, Th th) \ RAG s'" -proof - - from holding_th - show ?thesis - by (unfold s_RAG_def holding_eq, auto) -qed - -lemma ancestors_cs: - "ancestors (RAG s') (Cs cs) = {Th th}" -proof - - have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th) \ {Th th}" - proof(rule vat_s'.rtree_RAG.ancestors_accum) - from vt_s[unfolded s_def] - have " PIP s' (V th cs)" by (cases, simp) - thus "(Cs cs, Th th) \ RAG s'" - proof(cases) - assume "holding s' th cs" - from this[unfolded holding_eq] - show ?thesis by (unfold s_RAG_def, auto) - qed - qed - from this[unfolded ancestors_th] show ?thesis by simp -qed - -lemma preced_kept: "the_preced s = the_preced s'" - by (auto simp: s_def the_preced_def preced_def) - -end - -text {* - The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation, - which represents the case when there is another thread @{text "th'"} - to take over the critical resource released by the initiating thread @{text "th"}. -*} -locale step_v_cps_nt = step_v_cps + - fixes th' - -- {* @{text "th'"} is assumed to take over @{text "cs"} *} - assumes nt: "next_th s' th cs th'" - -context step_v_cps_nt -begin - -text {* - Lemma @{text "RAG_s"} confirms the change of RAG: - two edges removed and one added, as shown by the following diagram. -*} - -(* - RAG before the V-operation - th1 ----| - | - th' ----| - |----> cs -----| - th2 ----| | - | | - th3 ----| | - |------> th - th4 ----| | - | | - th5 ----| | - |----> cs'-----| - th6 ----| - | - th7 ----| - - RAG after the V-operation - th1 ----| - | - |----> cs ----> th' - th2 ----| - | - th3 ----| - - th4 ----| - | - th5 ----| - |----> cs'----> th - th6 ----| - | - th7 ----| -*) - -lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \ RAG s'" - using next_th_RAG[OF nt] . - -lemma ancestors_th': - "ancestors (RAG s') (Th th') = {Th th, Cs cs}" -proof - - have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \ {Cs cs}" - proof(rule vat_s'.rtree_RAG.ancestors_accum) - from sub_RAGs' show "(Th th', Cs cs) \ RAG s'" by auto - qed - thus ?thesis using ancestors_th ancestors_cs by auto -qed - -lemma RAG_s: - "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \ - {(Cs cs, Th th')}" -proof - - from step_RAG_v[OF vt_s[unfolded s_def], folded s_def] - and nt show ?thesis by (auto intro:next_th_unique) -qed - -lemma subtree_kept: - assumes "th1 \ {th, th'}" - shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R") -proof - - let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})" - let ?RAG'' = "?RAG' \ {(Cs cs, Th th')}" - have "subtree ?RAG' (Th th1) = ?R" - proof(rule subset_del_subtree_outside) - show "Range {(Cs cs, Th th), (Th th', Cs cs)} \ subtree (RAG s') (Th th1) = {}" - proof - - have "(Th th) \ subtree (RAG s') (Th th1)" - proof(rule subtree_refute) - show "Th th1 \ ancestors (RAG s') (Th th)" - by (unfold ancestors_th, simp) - next - from assms show "Th th1 \ Th th" by simp - qed - moreover have "(Cs cs) \ subtree (RAG s') (Th th1)" - proof(rule subtree_refute) - show "Th th1 \ ancestors (RAG s') (Cs cs)" - by (unfold ancestors_cs, insert assms, auto) - qed simp - ultimately have "{Th th, Cs cs} \ subtree (RAG s') (Th th1) = {}" by auto - thus ?thesis by simp - qed - qed - moreover have "subtree ?RAG'' (Th th1) = subtree ?RAG' (Th th1)" - proof(rule subtree_insert_next) - show "Th th' \ subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)" - proof(rule subtree_refute) - show "Th th1 \ ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')" - (is "_ \ ?R") - proof - - have "?R \ ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto) - moreover have "Th th1 \ ..." using ancestors_th' assms by simp - ultimately show ?thesis by auto - qed - next - from assms show "Th th1 \ Th th'" by simp - qed - qed - ultimately show ?thesis by (unfold RAG_s, simp) -qed - -lemma cp_kept: - assumes "th1 \ {th, th'}" - shows "cp s th1 = cp s' th1" - by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp) - -end - -locale step_v_cps_nnt = step_v_cps + - assumes nnt: "\ th'. (\ next_th s' th cs th')" - -context step_v_cps_nnt -begin - -lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}" -proof - - from nnt and step_RAG_v[OF vt_s[unfolded s_def], folded s_def] - show ?thesis by auto -qed - -lemma subtree_kept: - assumes "th1 \ th" - shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" -proof(unfold RAG_s, rule subset_del_subtree_outside) - show "Range {(Cs cs, Th th)} \ subtree (RAG s') (Th th1) = {}" - proof - - have "(Th th) \ subtree (RAG s') (Th th1)" - proof(rule subtree_refute) - show "Th th1 \ ancestors (RAG s') (Th th)" - by (unfold ancestors_th, simp) - next - from assms show "Th th1 \ Th th" by simp - qed - thus ?thesis by auto - qed -qed - -lemma cp_kept_1: - assumes "th1 \ th" - shows "cp s th1 = cp s' th1" - by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp) - -lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}" -proof - - { fix n - have "(Cs cs) \ ancestors (RAG s') n" - proof - assume "Cs cs \ ancestors (RAG s') n" - hence "(n, Cs cs) \ (RAG s')^+" by (auto simp:ancestors_def) - from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \ RAG s'" by auto - then obtain th' where "nn = Th th'" - by (unfold s_RAG_def, auto) - from h[unfolded this] have "(Th th', Cs cs) \ RAG s'" . - from this[unfolded s_RAG_def] - have "waiting (wq s') th' cs" by auto - from this[unfolded cs_waiting_def] - have "1 < length (wq s' cs)" - by (cases "wq s' cs", auto) - from holding_next_thI[OF holding_th this] - obtain th' where "next_th s' th cs th'" by auto - with nnt show False by auto - qed - } note h = this - { fix n - assume "n \ subtree (RAG s') (Cs cs)" - hence "n = (Cs cs)" - by (elim subtreeE, insert h, auto) - } moreover have "(Cs cs) \ subtree (RAG s') (Cs cs)" - by (auto simp:subtree_def) - ultimately show ?thesis by auto -qed - -lemma subtree_th: - "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}" -proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside) - from edge_of_th - show "(Cs cs, Th th) \ edges_in (RAG s') (Th th)" - by (unfold edges_in_def, auto simp:subtree_def) -qed - -lemma cp_kept_2: - shows "cp s th = cp s' th" - by (unfold cp_alt_def subtree_th preced_kept, auto) - -lemma eq_cp: - fixes th' - shows "cp s th' = cp s' th'" - using cp_kept_1 cp_kept_2 - by (cases "th' = th", auto) -end - - -locale step_P_cps = - fixes s' th cs s - defines s_def : "s \ (P th cs#s')" - assumes vt_s: "vt s" - -sublocale step_P_cps < vat_s : valid_trace "s" -proof - from vt_s show "vt s" . -qed - -sublocale step_P_cps < vat_s' : valid_trace "s'" -proof - from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" . -qed - -context step_P_cps -begin - -lemma readys_th: "th \ readys s'" -proof - - from step_back_step [OF vt_s[unfolded s_def]] - have "PIP s' (P th cs)" . - hence "th \ runing s'" by (cases, simp) - thus ?thesis by (simp add:readys_def runing_def) -qed - -lemma root_th: "root (RAG s') (Th th)" - using readys_root[OF readys_th] . - -lemma in_no_others_subtree: - assumes "th' \ th" - shows "Th th \ subtree (RAG s') (Th th')" -proof - assume "Th th \ subtree (RAG s') (Th th')" - thus False - proof(cases rule:subtreeE) - case 1 - with assms show ?thesis by auto - next - case 2 - with root_th show ?thesis by (auto simp:root_def) - qed -qed - -lemma preced_kept: "the_preced s = the_preced s'" - by (auto simp: s_def the_preced_def preced_def) - -end - -locale step_P_cps_ne =step_P_cps + - fixes th' - assumes ne: "wq s' cs \ []" - defines th'_def: "th' \ hd (wq s' cs)" - -locale step_P_cps_e =step_P_cps + - assumes ee: "wq s' cs = []" - -context step_P_cps_e -begin - -lemma RAG_s: "RAG s = RAG s' \ {(Cs cs, Th th)}" -proof - - from ee and step_RAG_p[OF vt_s[unfolded s_def], folded s_def] - show ?thesis by auto -qed - -lemma subtree_kept: - assumes "th' \ th" - shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')" -proof(unfold RAG_s, rule subtree_insert_next) - from in_no_others_subtree[OF assms] - show "Th th \ subtree (RAG s') (Th th')" . -qed - -lemma cp_kept: - assumes "th' \ th" - shows "cp s th' = cp s' th'" -proof - - have "(the_preced s ` {th'a. Th th'a \ subtree (RAG s) (Th th')}) = - (the_preced s' ` {th'a. Th th'a \ subtree (RAG s') (Th th')})" - by (unfold preced_kept subtree_kept[OF assms], simp) - thus ?thesis by (unfold cp_alt_def, simp) -qed - -end - -context step_P_cps_ne -begin - -lemma RAG_s: "RAG s = RAG s' \ {(Th th, Cs cs)}" -proof - - from step_RAG_p[OF vt_s[unfolded s_def]] and ne - show ?thesis by (simp add:s_def) -qed - -lemma cs_held: "(Cs cs, Th th') \ RAG s'" -proof - - have "(Cs cs, Th th') \ hRAG s'" - proof - - from ne - have " holding s' th' cs" - by (unfold th'_def holding_eq cs_holding_def, auto) - thus ?thesis - by (unfold hRAG_def, auto) - qed - thus ?thesis by (unfold RAG_split, auto) -qed - -lemma tRAG_s: - "tRAG s = tRAG s' \ {(Th th, Th th')}" - using RAG_tRAG_transfer[OF RAG_s cs_held] . - -lemma cp_kept: - assumes "Th th'' \ ancestors (tRAG s) (Th th)" - shows "cp s th'' = cp s' th''" -proof - - have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')" - proof - - have "Th th' \ subtree (tRAG s') (Th th'')" - proof - assume "Th th' \ subtree (tRAG s') (Th th'')" - thus False - proof(rule subtreeE) - assume "Th th' = Th th''" - from assms[unfolded tRAG_s ancestors_def, folded this] - show ?thesis by auto - next - assume "Th th'' \ ancestors (tRAG s') (Th th')" - moreover have "... \ ancestors (tRAG s) (Th th')" - proof(rule ancestors_mono) - show "tRAG s' \ tRAG s" by (unfold tRAG_s, auto) - qed - ultimately have "Th th'' \ ancestors (tRAG s) (Th th')" by auto - moreover have "Th th' \ ancestors (tRAG s) (Th th)" - by (unfold tRAG_s, auto simp:ancestors_def) - ultimately have "Th th'' \ ancestors (tRAG s) (Th th)" - by (auto simp:ancestors_def) - with assms show ?thesis by auto - qed - qed - from subtree_insert_next[OF this] - have "subtree (tRAG s' \ {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" . - from this[folded tRAG_s] show ?thesis . - qed - show ?thesis by (unfold cp_alt_def1 h preced_kept, simp) -qed - -lemma cp_gen_update_stop: (* ddd *) - assumes "u \ ancestors (tRAG s) (Th th)" - and "cp_gen s u = cp_gen s' u" - and "y \ ancestors (tRAG s) u" - shows "cp_gen s y = cp_gen s' y" - using assms(3) -proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf]) - case (1 x) - show ?case (is "?L = ?R") - proof - - from tRAG_ancestorsE[OF 1(2)] - obtain th2 where eq_x: "x = Th th2" by blast - from vat_s.cp_gen_rec[OF this] - have "?L = - Max ({the_preced s th2} \ cp_gen s ` RTree.children (tRAG s) x)" . - also have "... = - Max ({the_preced s' th2} \ cp_gen s' ` RTree.children (tRAG s') x)" - - proof - - from preced_kept have "the_preced s th2 = the_preced s' th2" by simp - moreover have "cp_gen s ` RTree.children (tRAG s) x = - cp_gen s' ` RTree.children (tRAG s') x" - proof - - have "RTree.children (tRAG s) x = RTree.children (tRAG s') x" - proof(unfold tRAG_s, rule children_union_kept) - have start: "(Th th, Th th') \ tRAG s" - by (unfold tRAG_s, auto) - note x_u = 1(2) - show "x \ Range {(Th th, Th th')}" - proof - assume "x \ Range {(Th th, Th th')}" - hence eq_x: "x = Th th'" using RangeE by auto - show False - proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start]) - case 1 - from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG - show ?thesis by (auto simp:ancestors_def acyclic_def) - next - case 2 - with x_u[unfolded eq_x] - have "(Th th', Th th') \ (tRAG s)^+" by (auto simp:ancestors_def) - with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def) - qed - qed - qed - moreover have "cp_gen s ` RTree.children (tRAG s) x = - cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A") - proof(rule f_image_eq) - fix a - assume a_in: "a \ ?A" - from 1(2) - show "?f a = ?g a" - proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch]) - case in_ch - show ?thesis - proof(cases "a = u") - case True - from assms(2)[folded this] show ?thesis . - next - case False - have a_not_in: "a \ ancestors (tRAG s) (Th th)" - proof - assume a_in': "a \ ancestors (tRAG s) (Th th)" - have "a = u" - proof(rule vat_s.rtree_s.ancestors_children_unique) - from a_in' a_in show "a \ ancestors (tRAG s) (Th th) \ - RTree.children (tRAG s) x" by auto - next - from assms(1) in_ch show "u \ ancestors (tRAG s) (Th th) \ - RTree.children (tRAG s) x" by auto - qed - with False show False by simp - qed - from a_in obtain th_a where eq_a: "a = Th th_a" - by (unfold RTree.children_def tRAG_alt_def, auto) - from cp_kept[OF a_not_in[unfolded eq_a]] - have "cp s th_a = cp s' th_a" . - from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a] - show ?thesis . - qed - next - case (out_ch z) - hence h: "z \ ancestors (tRAG s) u" "z \ RTree.children (tRAG s) x" by auto - show ?thesis - proof(cases "a = z") - case True - from h(2) have zx_in: "(z, x) \ (tRAG s)" by (auto simp:RTree.children_def) - from 1(1)[rule_format, OF this h(1)] - have eq_cp_gen: "cp_gen s z = cp_gen s' z" . - with True show ?thesis by metis - next - case False - from a_in obtain th_a where eq_a: "a = Th th_a" - by (auto simp:RTree.children_def tRAG_alt_def) - have "a \ ancestors (tRAG s) (Th th)" - proof - assume a_in': "a \ ancestors (tRAG s) (Th th)" - have "a = z" - proof(rule vat_s.rtree_s.ancestors_children_unique) - from assms(1) h(1) have "z \ ancestors (tRAG s) (Th th)" - by (auto simp:ancestors_def) - with h(2) show " z \ ancestors (tRAG s) (Th th) \ - RTree.children (tRAG s) x" by auto - next - from a_in a_in' - show "a \ ancestors (tRAG s) (Th th) \ RTree.children (tRAG s) x" - by auto - qed - with False show False by auto - qed - from cp_kept[OF this[unfolded eq_a]] - have "cp s th_a = cp s' th_a" . - from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a] - show ?thesis . - qed - qed - qed - ultimately show ?thesis by metis - qed - ultimately show ?thesis by simp - qed - also have "... = ?R" - by (fold vat_s'.cp_gen_rec[OF eq_x], simp) - finally show ?thesis . - qed -qed - -lemma cp_up: - assumes "(Th th') \ ancestors (tRAG s) (Th th)" - and "cp s th' = cp s' th'" - and "(Th th'') \ ancestors (tRAG s) (Th th')" - shows "cp s th'' = cp s' th''" -proof - - have "cp_gen s (Th th'') = cp_gen s' (Th th'')" - proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)]) - from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]] - show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis - qed - with cp_gen_def_cond[OF refl[of "Th th''"]] - show ?thesis by metis -qed - -end - -locale step_create_cps = - fixes s' th prio s - defines s_def : "s \ (Create th prio#s')" - assumes vt_s: "vt s" - -sublocale step_create_cps < vat_s: valid_trace "s" - by (unfold_locales, insert vt_s, simp) - -sublocale step_create_cps < vat_s': valid_trace "s'" - by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp) - -context step_create_cps -begin - -lemma RAG_kept: "RAG s = RAG s'" - by (unfold s_def RAG_create_unchanged, auto) - -lemma tRAG_kept: "tRAG s = tRAG s'" - by (unfold tRAG_alt_def RAG_kept, auto) - -lemma preced_kept: - assumes "th' \ th" - shows "the_preced s th' = the_preced s' th'" - by (unfold s_def the_preced_def preced_def, insert assms, auto) - -lemma th_not_in: "Th th \ Field (tRAG s')" -proof - - from vt_s[unfolded s_def] - have "PIP s' (Create th prio)" by (cases, simp) - hence "th \ threads s'" by(cases, simp) - from vat_s'.not_in_thread_isolated[OF this] - have "Th th \ Field (RAG s')" . - with tRAG_Field show ?thesis by auto -qed - -lemma eq_cp: - assumes neq_th: "th' \ th" - shows "cp s th' = cp s' th'" -proof - - have "(the_preced s \ the_thread) ` subtree (tRAG s) (Th th') = - (the_preced s' \ the_thread) ` subtree (tRAG s') (Th th')" - proof(unfold tRAG_kept, rule f_image_eq) - fix a - assume a_in: "a \ subtree (tRAG s') (Th th')" - then obtain th_a where eq_a: "a = Th th_a" - proof(cases rule:subtreeE) - case 2 - from ancestors_Field[OF 2(2)] - and that show ?thesis by (unfold tRAG_alt_def, auto) - qed auto - have neq_th_a: "th_a \ th" - proof - - have "(Th th) \ subtree (tRAG s') (Th th')" - proof - assume "Th th \ subtree (tRAG s') (Th th')" - thus False - proof(cases rule:subtreeE) - case 2 - from ancestors_Field[OF this(2)] - and th_not_in[unfolded Field_def] - show ?thesis by auto - qed (insert assms, auto) - qed - with a_in[unfolded eq_a] show ?thesis by auto - qed - from preced_kept[OF this] - show "(the_preced s \ the_thread) a = (the_preced s' \ the_thread) a" - by (unfold eq_a, simp) - qed - thus ?thesis by (unfold cp_alt_def1, simp) -qed - -lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}" -proof - - { fix a - assume "a \ RTree.children (tRAG s) (Th th)" - hence "(a, Th th) \ tRAG s" by (auto simp:RTree.children_def) - with th_not_in have False - by (unfold Field_def tRAG_kept, auto) - } thus ?thesis by auto -qed - -lemma eq_cp_th: "cp s th = preced th s" - by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def) - -end - -locale step_exit_cps = - fixes s' th prio s - defines s_def : "s \ Exit th # s'" - assumes vt_s: "vt s" - -sublocale step_exit_cps < vat_s: valid_trace "s" - by (unfold_locales, insert vt_s, simp) - -sublocale step_exit_cps < vat_s': valid_trace "s'" - by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp) - -context step_exit_cps -begin - -lemma preced_kept: - assumes "th' \ th" - shows "the_preced s th' = the_preced s' th'" - by (unfold s_def the_preced_def preced_def, insert assms, auto) - -lemma RAG_kept: "RAG s = RAG s'" - by (unfold s_def RAG_exit_unchanged, auto) - -lemma tRAG_kept: "tRAG s = tRAG s'" - by (unfold tRAG_alt_def RAG_kept, auto) - -lemma th_ready: "th \ readys s'" -proof - - from vt_s[unfolded s_def] - have "PIP s' (Exit th)" by (cases, simp) - hence h: "th \ runing s' \ holdents s' th = {}" by (cases, metis) - thus ?thesis by (unfold runing_def, auto) -qed - -lemma th_holdents: "holdents s' th = {}" -proof - - from vt_s[unfolded s_def] - have "PIP s' (Exit th)" by (cases, simp) - thus ?thesis by (cases, metis) -qed - -lemma th_RAG: "Th th \ Field (RAG s')" -proof - - have "Th th \ Range (RAG s')" - proof - assume "Th th \ Range (RAG s')" - then obtain cs where "holding (wq s') th cs" - by (unfold Range_iff s_RAG_def, auto) - with th_holdents[unfolded holdents_def] - show False by (unfold eq_holding, auto) - qed - moreover have "Th th \ Domain (RAG s')" - proof - assume "Th th \ Domain (RAG s')" - then obtain cs where "waiting (wq s') th cs" - by (unfold Domain_iff s_RAG_def, auto) - with th_ready show False by (unfold readys_def eq_waiting, auto) - qed - ultimately show ?thesis by (auto simp:Field_def) -qed - -lemma th_tRAG: "(Th th) \ Field (tRAG s')" - using th_RAG tRAG_Field[of s'] by auto - -lemma eq_cp: - assumes neq_th: "th' \ th" - shows "cp s th' = cp s' th'" -proof - - have "(the_preced s \ the_thread) ` subtree (tRAG s) (Th th') = - (the_preced s' \ the_thread) ` subtree (tRAG s') (Th th')" - proof(unfold tRAG_kept, rule f_image_eq) - fix a - assume a_in: "a \ subtree (tRAG s') (Th th')" - then obtain th_a where eq_a: "a = Th th_a" - proof(cases rule:subtreeE) - case 2 - from ancestors_Field[OF 2(2)] - and that show ?thesis by (unfold tRAG_alt_def, auto) - qed auto - have neq_th_a: "th_a \ th" - proof - - from vat_s'.readys_in_no_subtree[OF th_ready assms] - have "(Th th) \ subtree (RAG s') (Th th')" . - with tRAG_subtree_RAG[of s' "Th th'"] - have "(Th th) \ subtree (tRAG s') (Th th')" by auto - with a_in[unfolded eq_a] show ?thesis by auto - qed - from preced_kept[OF this] - show "(the_preced s \ the_thread) a = (the_preced s' \ the_thread) a" - by (unfold eq_a, simp) - qed - thus ?thesis by (unfold cp_alt_def1, simp) -qed - -end - -end - diff -r ed938e2246b9 -r 0525670d8e6a Moment.thy~ --- a/Moment.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,225 +0,0 @@ -theory Moment -imports Main -begin - -definition moment :: "nat \ 'a list \ 'a list" -where "moment n s = rev (take n (rev s))" - -value "moment 3 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9::int]" -value "moment 2 [5, 4, 3, 2, 1, 0::int]" - -(* -lemma length_moment_le: - assumes le_k: "k \ length s" - shows "length (moment k s) = k" -using le_k unfolding moment_def by auto -*) - -(* -lemma length_moment_ge: - assumes le_k: "length s \ k" - shows "length (moment k s) = (length s)" -using assms unfolding moment_def by simp -*) - -lemma moment_app [simp]: - assumes ile: "i \ length s" - shows "moment i (s' @ s) = moment i s" -using assms unfolding moment_def by simp - -lemma moment_eq [simp]: "moment (length s) (s' @ s) = s" - unfolding moment_def by simp - -lemma moment_ge [simp]: "length s \ n \ moment n s = s" - by (unfold moment_def, simp) - -lemma moment_zero [simp]: "moment 0 s = []" - by (simp add:moment_def) - -lemma p_split_gen: - "\Q s; \ Q (moment k s)\ \ - (\ i. i < length s \ k \ i \ \ Q (moment i s) \ (\ i' > i. Q (moment i' s)))" -proof (induct s, simp) - fix a s - assume ih: "\Q s; \ Q (moment k s)\ - \ \i i \ \ Q (moment i s) \ (\i'>i. Q (moment i' s))" - and nq: "\ Q (moment k (a # s))" and qa: "Q (a # s)" - have le_k: "k \ length s" - proof - - { assume "length s < k" - hence "length (a#s) \ k" by simp - from moment_ge [OF this] and nq and qa - have "False" by auto - } thus ?thesis by arith - qed - have nq_k: "\ Q (moment k s)" - proof - - have "moment k (a#s) = moment k s" - proof - - from moment_app [OF le_k, of "[a]"] show ?thesis by simp - qed - with nq show ?thesis by simp - qed - show "\i i \ \ Q (moment i (a # s)) \ (\i'>i. Q (moment i' (a # s)))" - proof - - { assume "Q s" - from ih [OF this nq_k] - obtain i where lti: "i < length s" - and nq: "\ Q (moment i s)" - and rst: "\i'>i. Q (moment i' s)" - and lki: "k \ i" by auto - have ?thesis - proof - - from lti have "i < length (a # s)" by auto - moreover have " \ Q (moment i (a # s))" - proof - - from lti have "i \ (length s)" by simp - from moment_app [OF this, of "[a]"] - have "moment i (a # s) = moment i s" by simp - with nq show ?thesis by auto - qed - moreover have " (\i'>i. Q (moment i' (a # s)))" - proof - - { - fix i' - assume lti': "i < i'" - have "Q (moment i' (a # s))" - proof(cases "length (a#s) \ i'") - case True - from True have "moment i' (a#s) = a#s" by simp - with qa show ?thesis by simp - next - case False - from False have "i' \ length s" by simp - from moment_app [OF this, of "[a]"] - have "moment i' (a#s) = moment i' s" by simp - with rst lti' show ?thesis by auto - qed - } thus ?thesis by auto - qed - moreover note lki - ultimately show ?thesis by auto - qed - } moreover { - assume ns: "\ Q s" - have ?thesis - proof - - let ?i = "length s" - have "\ Q (moment ?i (a#s))" - proof - - have "?i \ length s" by simp - from moment_app [OF this, of "[a]"] - have "moment ?i (a#s) = moment ?i s" by simp - moreover have "\ = s" by simp - ultimately show ?thesis using ns by auto - qed - moreover have "\ i' > ?i. Q (moment i' (a#s))" - proof - - { fix i' - assume "i' > ?i" - hence "length (a#s) \ i'" by simp - from moment_ge [OF this] - have " moment i' (a # s) = a # s" . - with qa have "Q (moment i' (a#s))" by simp - } thus ?thesis by auto - qed - moreover have "?i < length (a#s)" by simp - moreover note le_k - ultimately show ?thesis by auto - qed - } ultimately show ?thesis by auto - qed -qed - -lemma p_split: - "\Q s; \ Q []\ \ - (\ i. i < length s \ \ Q (moment i s) \ (\ i' > i. Q (moment i' s)))" -proof - - fix s Q - assume qs: "Q s" and nq: "\ Q []" - from nq have "\ Q (moment 0 s)" by simp - from p_split_gen [of Q s 0, OF qs this] - show "(\ i. i < length s \ \ Q (moment i s) \ (\ i' > i. Q (moment i' s)))" - by auto -qed - -lemma moment_Suc_tl: - assumes "Suc i \ length s" - shows "tl (moment (Suc i) s) = moment i s" - using assms unfolding moment_def rev_take -by (simp, metis Suc_diff_le diff_Suc_Suc drop_Suc tl_drop) - -lemma moment_plus: - assumes "Suc i \ length s" - shows "(moment (Suc i) s) = (hd (moment (Suc i) s)) # (moment i s)" -proof - - have "(moment (Suc i) s) \ []" - using assms by (auto simp add: moment_def) - hence "(moment (Suc i) s) = (hd (moment (Suc i) s)) # tl (moment (Suc i) s)" - by auto -<<<<<<< local - have "moment (i + length s) (t @ s) = moment i t @ moment (length s) (t @ s)" - by (simp add: moment_def) - with moment_app show ?thesis by auto -qed - -lemma moment_Suc_tl: - assumes "Suc i \ length s" - shows "tl (moment (Suc i) s) = moment i s" - using assms unfolding moment_def rev_take - by (simp, metis Suc_diff_le diff_Suc_Suc drop_Suc tl_drop) - -lemma moment_plus': - assumes "Suc i \ length s" - shows "(moment (Suc i) s) = (hd (moment (Suc i) s)) # (moment i s)" -proof - - have "(moment (Suc i) s) \ []" - using assms length_moment_le by fastforce - hence "(moment (Suc i) s) = (hd (moment (Suc i) s)) # tl (moment (Suc i) s)" - by auto - with moment_Suc_tl[OF assms] - show ?thesis by metis -qed - -lemma moment_plus: - "Suc i \ length s \ moment (Suc i) s = (hd (moment (Suc i) s)) # (moment i s)" -proof(induct s, simp+) - fix a s - assume ih: "Suc i \ length s \ moment (Suc i) s = hd (moment (Suc i) s) # moment i s" - and le_i: "i \ length s" - show "moment (Suc i) (a # s) = hd (moment (Suc i) (a # s)) # moment i (a # s)" - proof(cases "i= length s") - case True - hence "Suc i = length (a#s)" by simp - with moment_eq have "moment (Suc i) (a#s) = a#s" by auto - moreover have "moment i (a#s) = s" - proof - - from moment_app [OF le_i, of "[a]"] - and True show ?thesis by simp - qed - ultimately show ?thesis by auto - next - case False - from False and le_i have lti: "i < length s" by arith - hence les_i: "Suc i \ length s" by arith - show ?thesis - proof - - from moment_app [OF les_i, of "[a]"] - have "moment (Suc i) (a # s) = moment (Suc i) s" by simp - moreover have "moment i (a#s) = moment i s" - proof - - from lti have "i \ length s" by simp - from moment_app [OF this, of "[a]"] show ?thesis by simp - qed - moreover note ih [OF les_i] - ultimately show ?thesis by auto - qed - qed -======= - with moment_Suc_tl[OF assms] - show ?thesis by metis ->>>>>>> other -qed - -end - diff -r ed938e2246b9 -r 0525670d8e6a PIPBasics.thy~ --- a/PIPBasics.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,3788 +0,0 @@ -theory PIPBasics -imports PIPDefs -begin - -locale valid_trace = - fixes s - assumes vt : "vt s" - -locale valid_trace_e = valid_trace + - fixes e - assumes vt_e: "vt (e#s)" -begin - -lemma pip_e: "PIP s e" - using vt_e by (cases, simp) - -end - -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 runing_head: - assumes "th \ runing s" - and "th \ set (wq_fun (schs s) cs)" - shows "th = hd (wq_fun (schs s) cs)" - using assms - by (simp add:runing_def readys_def s_waiting_def wq_def) - -context valid_trace -begin - -lemma actor_inv: - assumes "PIP s e" - and "\ isCreate e" - shows "actor e \ runing s" - using assms - by (induct, auto) - -lemma ind [consumes 0, case_names Nil Cons, induct type]: - assumes "PP []" - and "(\s e. valid_trace s \ valid_trace (e#s) \ - PP s \ PIP s e \ PP (e # s))" - shows "PP s" -proof(rule vt.induct[OF vt]) - from assms(1) show "PP []" . -next - fix s e - assume h: "vt s" "PP s" "PIP s e" - show "PP (e # s)" - proof(cases rule:assms(2)) - from h(1) show v1: "valid_trace s" by (unfold_locales, simp) - next - from h(1,3) have "vt (e#s)" by auto - thus "valid_trace (e # s)" by (unfold_locales, simp) - qed (insert h, auto) -qed - -lemma wq_distinct: "distinct (wq s cs)" -proof(induct rule:ind) - case (Cons s e) - from Cons(4,3) - show ?case - proof(induct) - case (thread_P th s cs1) - show ?case - proof(cases "cs = cs1") - case True - thus ?thesis (is "distinct ?L") - proof - - have "?L = wq_fun (schs s) cs1 @ [th]" using True - by (simp add:wq_def wf_def Let_def split:list.splits) - moreover have "distinct ..." - proof - - have "th \ set (wq_fun (schs s) cs1)" - proof - assume otherwise: "th \ set (wq_fun (schs s) cs1)" - from runing_head[OF thread_P(1) this] - have "th = hd (wq_fun (schs s) cs1)" . - hence "(Cs cs1, Th th) \ (RAG s)" using otherwise - by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def) - with thread_P(2) show False by auto - qed - moreover have "distinct (wq_fun (schs s) cs1)" - using True thread_P wq_def by auto - ultimately show ?thesis by auto - qed - ultimately show ?thesis by simp - qed - next - case False - with thread_P(3) - show ?thesis - by (auto simp:wq_def wf_def Let_def split:list.splits) - qed - next - case (thread_V th s cs1) - thus ?case - proof(cases "cs = cs1") - case True - show ?thesis (is "distinct ?L") - proof(cases "(wq s cs)") - case Nil - thus ?thesis - by (auto simp:wq_def wf_def Let_def split:list.splits) - next - case (Cons w_hd w_tl) - moreover have "distinct (SOME q. distinct q \ set q = set w_tl)" - proof(rule someI2) - from thread_V(3)[unfolded Cons] - show "distinct w_tl \ set w_tl = set w_tl" by auto - qed auto - ultimately show ?thesis - by (auto simp:wq_def wf_def Let_def True split:list.splits) - qed - next - case False - with thread_V(3) - show ?thesis - by (auto simp:wq_def wf_def Let_def split:list.splits) - qed - qed (insert Cons, auto simp: wq_def Let_def split:list.splits) -qed (unfold wq_def Let_def, simp) - -end - - -context valid_trace_e -begin - -text {* - The following lemma shows that only the @{text "P"} - operation can add new thread into waiting queues. - Such kind of lemmas are very obvious, but need to be checked formally. - This is a kind of confirmation that our modelling is correct. -*} - -lemma block_pre: - assumes s_ni: "thread \ set (wq s cs)" - and s_i: "thread \ set (wq (e#s) cs)" - shows "e = P thread cs" -proof(cases e) - -- {* This is the only non-trivial case: *} - case (V th cs1) - have False - proof(cases "cs1 = cs") - case True - show ?thesis - proof(cases "(wq s cs1)") - case (Cons w_hd w_tl) - have "set (wq (e#s) cs) \ set (wq s cs)" - proof - - have "(wq (e#s) cs) = (SOME q. distinct q \ set q = set w_tl)" - using Cons V by (auto simp:wq_def Let_def True split:if_splits) - moreover have "set ... \ set (wq s cs)" - proof(rule someI2) - show "distinct w_tl \ set w_tl = set w_tl" - by (metis distinct.simps(2) local.Cons wq_distinct) - qed (insert Cons True, auto) - ultimately show ?thesis by simp - qed - with assms show ?thesis by auto - qed (insert assms V True, auto simp:wq_def Let_def split:if_splits) - qed (insert assms V, auto simp:wq_def Let_def split:if_splits) - thus ?thesis by auto -qed (insert assms, auto simp:wq_def Let_def split:if_splits) - -end - -text {* - The following lemmas is also obvious and shallow. It says - that only running thread can request for a critical resource - and that the requested resource must be one which is - not current held by the thread. -*} - -lemma p_pre: "\vt ((P thread cs)#s)\ \ - thread \ runing s \ (Cs cs, Th thread) \ (RAG s)^+" -apply (ind_cases "vt ((P thread cs)#s)") -apply (ind_cases "step s (P thread cs)") -by auto - -lemma abs1: - 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)" - -context valid_trace_e -begin - -lemma abs2: - assumes 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 vt_e 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 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 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 - -end - - -context valid_trace -begin -lemma vt_moment: "\ t. vt (moment t s)" -proof(induct rule:ind) - case Nil - thus ?case by (simp add:vt_nil) -next - case (Cons s e t) - show ?case - proof(cases "t \ length (e#s)") - case True - from True have "moment t (e#s) = e#s" by simp - thus ?thesis using Cons - by (simp add:valid_trace_def) - next - case False - from Cons have "vt (moment t s)" by simp - moreover have "moment t (e#s) = moment t s" - proof - - from False have "t \ length s" by simp - from moment_app [OF this, of "[e]"] - show ?thesis by simp - qed - ultimately show ?thesis by simp - qed -qed -end - -locale valid_moment = valid_trace + - fixes i :: nat - -sublocale valid_moment < vat_moment: valid_trace "(moment i s)" - by (unfold_locales, insert vt_moment, auto) - -context valid_trace -begin - - -text {* (* ddd *) - The nature of the work is like this: since it starts from a very simple and basic - model, even intuitively very `basic` and `obvious` properties need to derived from scratch. - For instance, the fact - that one thread can not be blocked by two critical resources at the same time - is obvious, because only running threads can make new requests, if one is waiting for - a critical resource and get blocked, it can not make another resource request and get - blocked the second time (because it is not running). - - To derive this fact, one needs to prove by contraction and - reason about time (or @{text "moement"}). The reasoning is based on a generic theorem - named @{text "p_split"}, which is about status changing along the time axis. It says if - a condition @{text "Q"} is @{text "True"} at a state @{text "s"}, - but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} - in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history - of events leading to it), such that @{text "Q"} switched - from being @{text "False"} to @{text "True"} and kept being @{text "True"} - till the last moment of @{text "s"}. - - Suppose a thread @{text "th"} is blocked - on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, - since no thread is blocked at the very beginning, by applying - @{text "p_split"} to these two blocking facts, there exist - two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that - @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} - and kept on blocked on them respectively ever since. - - Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}. - However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still - in blocked state at moment @{text "t2"} and could not - make any request and get blocked the second time: Contradiction. -*} - -lemma waiting_unique_pre: (* ccc *) - assumes 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#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - 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 vt_e.abs2 [OF True eq_th h2 h1] - show ?thesis by auto - next - case False - from vt_e.block_pre[OF False h1] - have "e = P thread cs2" . - with vt_e.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#moment t1 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t1 s" e - by (unfold_locales, auto, cases, auto) - 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 vt_e.abs2 True eq_th h2 h1 - show ?thesis by auto - next - case False - from vt_e.block_pre [OF False h1] - have "e = P thread cs1" . - with vt_e.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 - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t1 s" e - by (unfold_locales, auto, cases, auto) - 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 vt_e.abs2 [OF True eq_th h2 h1] - show ?thesis by auto - next - case False - from vt_e.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 (e#moment t2 s)" by simp - then interpret vt_e2: valid_trace_e "moment t2 s" e - by (unfold_locales, auto, cases, auto) - from vt_e2.abs2 [OF True eq_th h2 h1] - show ?thesis . - next - case False - have "vt (e#moment t2 s)" - proof - - from vt_moment eqt12 - have "vt (moment (Suc t2) s)" by auto - with eq_m eqt12 show ?thesis by simp - qed - then interpret vt_e2: valid_trace_e "moment t2 s" e - by (unfold_locales, auto, cases, auto) - from vt_e2.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 - -text {* - This lemma is a simple corrolary of @{text "waiting_unique_pre"}. -*} - -lemma waiting_unique: - assumes "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 - -end - -(* not used *) -text {* - Every thread can only be blocked on one critical resource, - symmetrically, every critical resource can only be held by one thread. - This fact is much more easier according to our definition. -*} -lemma held_unique: - assumes "holding (s::event list) th1 cs" - and "holding s th2 cs" - shows "th1 = th2" - by (insert assms, unfold s_holding_def, auto) - - -lemma last_set_lt: "th \ threads s \ last_set th s < length s" - apply (induct s, auto) - by (case_tac a, auto split:if_splits) - -lemma last_set_unique: - "\last_set th1 s = last_set th2 s; th1 \ threads s; th2 \ threads s\ - \ th1 = th2" - apply (induct s, auto) - by (case_tac a, auto split:if_splits dest:last_set_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 "last_set th1 s = last_set th2 s" by (simp add:preced_def) - from last_set_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 - -(* An aux lemma used later *) -lemma unique_minus: - 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: - 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: - 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 - -text {* - The following three lemmas show that @{text "RAG"} does not change - by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"} - events, respectively. -*} - -lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - - -text {* - The following lemmas are used in the proof of - lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed - by @{text "V"}-events. - However, since our model is very concise, such seemingly obvious lemmas need to be derived from scratch, - starting from the model definitions. -*} -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 - -text {* - The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be - derived from scratch, which confirms the correctness of the definition of @{text "next_th"}. -*} -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" - from vt interpret vt_v: valid_trace_e s "V th cs" - by (cases, unfold_locales, simp) - 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 vt_v.wq_distinct[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 vt_v.wq_distinct[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 vt interpret vt_v: valid_trace_e s "V th cs" - by (cases, unfold_locales, simp+) - 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 vt_v.wq_distinct[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 vt_v.wq_distinct[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)" - from vt interpret vt_v: valid_trace_e s "V th cs" - by (cases, unfold_locales, simp+) - have "(SOME q. distinct q \ set q = set rest) \ []" - proof(rule someI2) - from vt_v.wq_distinct[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" - from vt interpret vt_v: valid_trace_e s "V th cs" - by (cases, unfold_locales, simp+) - 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 vt_v.wq_distinct [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 vt_v.wq_distinct [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 vt_v.wq_distinct [of cs, unfolded wq_def] - show False by auto - qed - qed -qed - -text {* (* ddd *) - The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed - with the happening of @{text "V"}-events: -*} -lemma step_RAG_v: -assumes vt: - "vt (V th cs#s)" -shows " - RAG (V th cs # s) = - RAG 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_RAG_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 - -text {* - The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed - with the happening of @{text "P"}-events: -*} -lemma step_RAG_p: - "vt (P th cs#s) \ - RAG (P th cs # s) = (if (wq s cs = []) then RAG s \ {(Cs cs, Th th)} - else RAG s \ {(Th th, Cs cs)})" - apply(simp only: s_RAG_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(simp add:s_RAG_def wq_def cs_holding_def) - apply(auto) - done - - -lemma RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" - by (unfold s_RAG_def, auto) - -context valid_trace -begin - -text {* - The following lemma shows that @{text "RAG"} is acyclic. - The overall structure is by induction on the formation of @{text "vt s"} - and then case analysis on event @{text "e"}, where the non-trivial cases - for those for @{text "V"} and @{text "P"} events. -*} -lemma acyclic_RAG: - shows "acyclic (RAG s)" -using vt -proof(induct) - case (vt_cons s e) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - assume ih: "acyclic (RAG 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:RAG_create_unchanged) - next - case (Exit th) - with ih show ?thesis by (simp add:RAG_exit_unchanged) - next - case (V th cs) - from V vt stp have vtt: "vt (V th cs#s)" by auto - from step_RAG_v [OF this] - have eq_de: - "RAG (e # s) = - RAG 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 RAG_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_RAG_def s_waiting_def cs_waiting_def wq_def by simp - hence "cs' = cs" - proof(rule vt_s.waiting_unique) - from eq_wq vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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 vt_s.wq_distinct[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_RAG_p [OF this] P - have "RAG (e # s) = - (if wq s cs = [] then RAG s \ {(Cs cs, Th th)} else - RAG 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 = RAG s \ {(Cs cs, Th th)}" by simp - have "(Th th, Cs cs) \ (RAG s)\<^sup>*" - proof - assume "(Th th, Cs cs) \ (RAG s)\<^sup>*" - hence "(Th th, Cs cs) \ (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl) - from tranclD2 [OF this] - obtain x where "(x, Cs cs) \ RAG s" by auto - with True show False by (auto simp:s_RAG_def cs_waiting_def) - qed - with acyclic_insert ih eq_r show ?thesis by auto - next - case False - hence eq_r: "?R = RAG s \ {(Th th, Cs cs)}" by simp - have "(Cs cs, Th th) \ (RAG s)\<^sup>*" - proof - assume "(Cs cs, Th th) \ (RAG s)\<^sup>*" - hence "(Cs cs, Th th) \ (RAG 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) \ (RAG 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 RAG_set_unchanged - show ?thesis by (simp add:RAG_set_unchanged) - qed - next - case vt_nil - show "acyclic (RAG ([]::state))" - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -qed - - -lemma finite_RAG: - shows "finite (RAG s)" -proof - - from vt show ?thesis - proof(induct) - case (vt_cons s e) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - assume ih: "finite (RAG 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:RAG_create_unchanged) - next - case (Exit th) - with ih show ?thesis by (simp add:RAG_exit_unchanged) - next - case (V th cs) - from V vt stp have vtt: "vt (V th cs#s)" by auto - from step_RAG_v [OF this] - have eq_de: "RAG (e # s) = - RAG 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 (metis finite.simps) - 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_RAG_p [OF this] P - have "RAG (e # s) = - (if wq s cs = [] then RAG s \ {(Cs cs, Th th)} else - RAG 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 = RAG s \ {(Cs cs, Th th)}" by simp - with True and ih show ?thesis by auto - next - case False - hence "?R = RAG 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:RAG_set_unchanged) - qed - next - case vt_nil - show "finite (RAG ([]::state))" - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) - qed -qed - -text {* Several useful lemmas *} - -lemma wf_dep_converse: - shows "wf ((RAG s)^-1)" -proof(rule finite_acyclic_wf_converse) - from finite_RAG - show "finite (RAG s)" . -next - from acyclic_RAG - show "acyclic (RAG s)" . -qed - -end - -lemma hd_np_in: "x \ set l \ hd l \ set l" - by (induct l, auto) - -lemma th_chasing: "(Th th, Cs cs) \ RAG (s::state) \ \ th'. (Cs cs, Th th') \ RAG s" - by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in) - -context valid_trace -begin - -lemma wq_threads: - assumes 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) - interpret vt_s: valid_trace s - using vt_cons(1) by (unfold_locales, auto) - 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_RAG_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 vt_s.wq_distinct[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: "\(Th th) \ Range (RAG (s::state))\ \ th \ threads s" - apply(unfold s_RAG_def cs_waiting_def cs_holding_def) - by (auto intro:wq_threads) - -lemma readys_v_eq: - assumes neq_th: "th \ thread" - and eq_wq: "wq s cs = thread#rest" - and not_in: "th \ set rest" - shows "(th \ readys (V thread cs#s)) = (th \ readys s)" -proof - - from assms show ?thesis - apply (auto simp:readys_def) - apply(simp add:s_waiting_def[folded wq_def]) - apply (erule_tac x = csa in allE) - apply (simp add:s_waiting_def wq_def Let_def split:if_splits) - apply (case_tac "csa = cs", simp) - apply (erule_tac x = cs in allE) - apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits) - apply(auto simp add: wq_def) - apply (auto simp:s_waiting_def wq_def Let_def split:list.splits) - proof - - assume th_nin: "th \ set rest" - and th_in: "th \ set (SOME q. distinct q \ set q = set rest)" - and eq_wq: "wq_fun (schs s) cs = thread # rest" - have "set (SOME q. distinct q \ set q = set rest) = set rest" - proof(rule someI2) - from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def] - show "distinct rest \ set rest = set rest" by auto - next - show "\x. distinct x \ set x = set rest \ set x = set rest" by auto - qed - with th_nin th_in show False by auto - qed -qed - -text {* \noindent - The following lemmas shows that: starting from any node in @{text "RAG"}, - by chasing out-going edges, it is always possible to reach a node representing a ready - thread. In this lemma, it is the @{text "th'"}. -*} - -lemma chain_building: - shows "node \ Domain (RAG s) \ (\ th'. th' \ readys s \ (node, Th th') \ (RAG s)^+)" -proof - - from wf_dep_converse - have h: "wf ((RAG s)\)" . - show ?thesis - proof(induct rule:wf_induct [OF h]) - fix x - assume ih [rule_format]: - "\y. (y, x) \ (RAG s)\ \ - y \ Domain (RAG s) \ (\th'. th' \ readys s \ (y, Th th') \ (RAG s)\<^sup>+)" - show "x \ Domain (RAG s) \ (\th'. th' \ readys s \ (x, Th th') \ (RAG s)\<^sup>+)" - proof - assume x_d: "x \ Domain (RAG s)" - show "\th'. th' \ readys s \ (x, Th th') \ (RAG s)\<^sup>+" - proof(cases x) - case (Th th) - from x_d Th obtain cs where x_in: "(Th th, Cs cs) \ RAG s" by (auto simp:s_RAG_def) - with Th have x_in_r: "(Cs cs, x) \ (RAG s)^-1" by simp - from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \ RAG s" by blast - hence "Cs cs \ Domain (RAG 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') \ (RAG s)\<^sup>+" by auto - have "(x, Th th') \ (RAG 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) \ (RAG s)^-1" by (auto simp:s_RAG_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 have "th' \ threads s" by auto - with False have "Th th' \ Domain (RAG s)" - by (auto simp:readys_def wq_def s_waiting_def s_RAG_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'') \ (RAG s)\<^sup>+" by auto - from th'_d and th''_in - have "(x, Th th'') \ (RAG s)\<^sup>+" by auto - with th''_r show ?thesis by auto - qed - qed - qed - qed -qed - -text {* \noindent - The following is just an instance of @{text "chain_building"}. -*} -lemma th_chain_to_ready: - assumes th_in: "th \ threads s" - shows "th \ readys s \ (\ th'. th' \ readys s \ (Th th, Th th') \ (RAG s)^+)" -proof(cases "th \ readys s") - case True - thus ?thesis by auto -next - case False - from False and th_in have "Th th \ Domain (RAG s)" - by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def) - from chain_building [rule_format, OF this] - show ?thesis by auto -qed - -end - -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) - -context valid_trace -begin - -lemma unique_RAG: "\(n, n1) \ RAG s; (n, n2) \ RAG s\ \ n1 = n2" - apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq) - by(auto elim:waiting_unique holding_unique) - -end - - -lemma trancl_split: "(a, b) \ r^+ \ \ c. (a, c) \ r" -by (induct rule:trancl_induct, auto) - -context valid_trace -begin - -lemma dchain_unique: - assumes th1_d: "(n, Th th1) \ (RAG s)^+" - and th1_r: "th1 \ readys s" - and th2_d: "(n, Th th2) \ (RAG 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_RAG - have "(Th th1, Th th2) \ (RAG s)\<^sup>+ \ (Th th2, Th th1) \ (RAG s)\<^sup>+" by auto - hence "False" - proof - assume "(Th th1, Th th2) \ (RAG s)\<^sup>+" - from trancl_split [OF this] - obtain n where dd: "(Th th1, n) \ RAG s" by auto - then obtain cs where eq_n: "n = Cs cs" - by (auto simp:s_RAG_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_RAG_def wq_def s_waiting_def cs_waiting_def) - with th1_r show ?thesis by auto - next - assume "(Th th2, Th th1) \ (RAG s)\<^sup>+" - from trancl_split [OF this] - obtain n where dd: "(Th th2, n) \ RAG s" by auto - then obtain cs where eq_n: "n = Cs cs" - by (auto simp:s_RAG_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_RAG_def s_waiting_def cs_waiting_def) - with th2_r show ?thesis by auto - qed - } thus ?thesis by auto -qed - -end - - -lemma step_holdents_p_add: - 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_RAG_p[OF vt] by (auto) -qed - -lemma step_holdents_p_eq: - 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_RAG_p[OF vt] by auto -qed - - -lemma (in valid_trace) finite_holding : - shows "finite (holdents s th)" -proof - - let ?F = "\ (x, y). the_cs x" - from finite_RAG - have "finite (RAG s)" . - hence "finite (?F `(RAG s))" by simp - moreover have "{cs . (Cs cs, Th th) \ RAG s} \ \" - proof - - { have h: "\ a A f. a \ A \ f a \ f ` A" by auto - fix x assume "(Cs x, Th th) \ RAG s" - hence "?F (Cs x, Th th) \ ?F `(RAG s)" by (rule h) - moreover have "?F (Cs x, Th th) = x" by simp - ultimately have "x \ (\(x, y). the_cs x) ` RAG s" by simp - } thus ?thesis by auto - qed - ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset) -qed - -lemma cntCS_v_dec: - assumes vtv: "vt (V thread cs#s)" - shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread" -proof - - from vtv interpret vt_s: valid_trace s - by (cases, unfold_locales, simp) - from vtv interpret vt_v: valid_trace "V thread cs#s" - by (unfold_locales, simp) - from step_back_step[OF vtv] - have cs_in: "cs \ holdents s thread" - apply (cases, unfold holdents_test s_RAG_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 vt_s.wq_distinct[of cs]) - apply (unfold holdents_test, unfold step_RAG_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))) \ RAG 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 vt_v.finite_holding - 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 count_rec1 [simp]: - assumes "Q e" - shows "count Q (e#es) = Suc (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec2 [simp]: - assumes "\Q e" - shows "count Q (e#es) = (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec3 [simp]: - shows "count Q [] = 0" - by (unfold count_def, auto) - -lemma cntP_diff_inv: - assumes "cntP (e#s) th \ cntP s th" - shows "isP e \ actor e = th" -proof(cases e) - case (P th' pty) - show ?thesis - by (cases "(\e. \cs. e = P th cs) (P th' pty)", - insert assms P, auto simp:cntP_def) -qed (insert assms, auto simp:cntP_def) - -lemma isP_E: - assumes "isP e" - obtains cs where "e = P (actor e) cs" - using assms by (cases e, auto) - -lemma isV_E: - assumes "isV e" - obtains cs where "e = V (actor e) cs" - using assms by (cases e, auto) (* ccc *) - -lemma cntV_diff_inv: - assumes "cntV (e#s) th \ cntV s th" - shows "isV e \ actor e = th" -proof(cases e) - case (V th' pty) - show ?thesis - by (cases "(\e. \cs. e = V th cs) (V th' pty)", - insert assms V, auto simp:cntV_def) -qed (insert assms, auto simp:cntV_def) - -context valid_trace -begin - -text {* (* ddd *) \noindent - The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} - of one particular thread. -*} - -lemma cnp_cnv_cncs: - 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) - interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp) - 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 vt_s.wq_threads [OF 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:RAG_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:RAG_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) \ (RAG s)\<^sup>+" - from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto - then interpret vt_p: valid_trace "(P thread cs#s)" - by (unfold_locales, simp) - 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) - 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_RAG_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_RAG_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) \ RAG s} = - Suc (card {cs. (Cs cs, Th thread) \ RAG 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 vt_s.finite_holding [of thread] - show " finite {cs. (Cs cs, Th thread) \ RAG s}" - by (unfold holdents_test, simp) - qed - moreover have "?R - {cs} = ?R" - proof - - have "cs \ ?R" - proof - assume "cs \ {cs. (Cs cs, Th thread) \ RAG 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_RAG_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 vt_p.wq_distinct [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_RAG_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 - then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp) - 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 vt_v.wq_distinct[of cs] and eq_wq - show "distinct rest \ set rest = set rest" - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 vt_v.wq_distinct[of cs] - and eq_wq show False - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq) - moreover have "cntCS (e#s) th = cntCS s th" - apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto) - proof - - have "{csa. (Cs csa, Th th) \ RAG s \ csa = cs \ next_th s thread cs th} = - {cs. (Cs cs, Th th) \ RAG s}" - proof - - from False eq_wq - have " next_th s thread cs th \ (Cs cs, Th th) \ RAG 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 vt_s.wq_distinct[ 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))) \ RAG s" - by auto - qed - thus ?thesis by auto - qed - thus "card {csa. (Cs csa, Th th) \ RAG s \ csa = cs \ next_th s thread cs th} = - card {cs. (Cs cs, Th th) \ RAG 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 vt_s.wq_distinct[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_RAG_v[OF vtv], - auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def - Let_def cs_holding_def) - by (insert vt_s.wq_distinct[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 vt_s.wq_threads) - 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 vt_s.wq_distinct[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 vt_s.waiting_unique[OF 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 vt_s.wq_threads [OF 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_RAG_v[OF vtv], auto) - proof - - show "card {csa. (Cs csa, Th th) \ RAG s \ csa = cs} = - Suc (card {cs. (Cs cs, Th th) \ RAG 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) ` RAG s)" - apply (auto simp:image_def) - by (rule_tac x = "(Cs x, Th th)" in bexI, auto) - with vt_s.finite_RAG - show "finite {cs. (Cs cs, Th th) \ RAG s}" by (auto intro:finite_subset) - next - show "cs \ {cs. (Cs cs, Th th) \ RAG s}" - proof - assume "cs \ {cs. (Cs cs, Th th) \ RAG s}" - hence "(Cs cs, Th th) \ RAG s" by simp - with True neq_th eq_wq show False - by (auto simp:next_th_def s_RAG_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:RAG_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_RAG_def wq_def cs_holding_def) - qed -qed - -lemma not_thread_cncs: - assumes 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) - interpret vt_s: valid_trace s using vt_cons(1) - by (unfold_locales, simp) - 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:RAG_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:RAG_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_RAG_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 - then interpret vt_v: valid_trace "(V thread cs#s)" - by (unfold_locales, simp) - 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 vt_v.wq_distinct[of cs] and eq_wq - show "distinct rest \ set rest = set rest" - by (metis distinct.simps(2) vt_s.wq_distinct) - 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 vt_s.wq_threads[OF this] and ni - show False - using `hd (SOME q. distinct q \ set q = set rest) \ set (wq s cs)` - ni vt_s.wq_threads by blast - 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_RAG_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:RAG_set_unchanged) - qed - next - case vt_nil - show ?case - by (unfold cntCS_def, - auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def) - qed -qed - -end - -lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs" - by (auto simp:s_waiting_def cs_waiting_def wq_def) - -context valid_trace -begin - -lemma dm_RAG_threads: - assumes in_dom: "(Th th) \ Domain (RAG s)" - shows "th \ threads s" -proof - - from in_dom obtain n where "(Th th, n) \ RAG s" by auto - moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto - ultimately have "(Th th, Cs cs) \ RAG s" by simp - hence "th \ set (wq s cs)" - by (unfold s_RAG_def, auto simp:cs_waiting_def) - from wq_threads [OF this] show ?thesis . -qed - -end - -lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th" -unfolding cp_def wq_def -apply(induct s rule: schs.induct) -thm cpreced_initial -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 - -context valid_trace -begin - -lemma runing_unique: - assumes 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" - unfolding runing_def - apply(simp) - done - hence eq_max: "Max ((\th. preced th s) ` ({th1} \ dependants (wq s) th1)) = - Max ((\th. preced th s) ` ({th2} \ dependants (wq s) th2))" - (is "Max (?f ` ?A) = Max (?f ` ?B)") - unfolding cp_eq_cpreced - unfolding 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 (dependants (wq s) th1)" - proof- - have "finite {th'. (Th th', Th th1) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th1) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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 - thm cpreced_def - unfolding cpreced_def[symmetric] - unfolding cp_eq_cpreced[symmetric] - unfolding cpreced_def - using that[intro] by (auto) - 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 (dependants (wq s) th2)" - proof- - have "finite {th'. (Th th', Th th2) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th2) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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' \ dependants (wq s) th1)" by simp - thus "th1' \ threads s" - proof - assume "th1' \ dependants (wq s) th1" - hence "(Th th1') \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "(Th th1') \ Domain (RAG s)" by (simp add:trancl_domain) - from dm_RAG_threads[OF 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' \ dependants (wq s) th2)" by simp - thus "th2' \ threads s" - proof - assume "th2' \ dependants (wq s) th2" - hence "(Th th2') \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "(Th th2') \ Domain (RAG s)" by (simp add:trancl_domain) - from dm_RAG_threads[OF 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' \ dependants (wq s) th1" by simp - thus ?thesis - proof - assume eq_th': "th1' = th1" - from th2_in have "th2' = th2 \ th2' \ dependants (wq s) th2" by simp - thus ?thesis - proof - assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp - next - assume "th2' \ dependants (wq s) th2" - with eq_th12 eq_th' have "th1 \ dependants (wq s) th2" by simp - hence "(Th th1, Th th2) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - hence "Th th1 \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "Th th1 \ Domain (RAG s)" by (simp add:trancl_domain) - then obtain n where d: "(Th th1, n) \ RAG s" by (auto simp:Domain_def) - from RAG_target_th [OF this] - obtain cs' where "n = Cs cs'" by auto - with d have "(Th th1, Cs cs') \ RAG s" by simp - with runing_1 have "False" - apply (unfold runing_def readys_def s_RAG_def) - by (auto simp:eq_waiting) - thus ?thesis by simp - qed - next - assume th1'_in: "th1' \ dependants (wq s) th1" - from th2_in have "th2' = th2 \ th2' \ dependants (wq s) th2" by simp - thus ?thesis - proof - assume "th2' = th2" - with th1'_in eq_th12 have "th2 \ dependants (wq s) th1" by simp - hence "(Th th2, Th th1) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - hence "Th th2 \ Domain ((RAG s)^+)" - apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) - by (auto simp:Domain_def) - hence "Th th2 \ Domain (RAG s)" by (simp add:trancl_domain) - then obtain n where d: "(Th th2, n) \ RAG s" by (auto simp:Domain_def) - from RAG_target_th [OF this] - obtain cs' where "n = Cs cs'" by auto - with d have "(Th th2, Cs cs') \ RAG s" by simp - with runing_2 have "False" - apply (unfold runing_def readys_def s_RAG_def) - by (auto simp:eq_waiting) - thus ?thesis by simp - next - assume "th2' \ dependants (wq s) th2" - with eq_th12 have "th1' \ dependants (wq s) th2" by simp - hence h1: "(Th th1', Th th2) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - from th1'_in have h2: "(Th th1', Th th1) \ (RAG s)^+" - by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) - show ?thesis - proof(rule dchain_unique[OF 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 "card (runing s) \ 1" -apply(subgoal_tac "finite (runing s)") -prefer 2 -apply (metis finite_nat_set_iff_bounded lessI runing_unique) -apply(rule ccontr) -apply(simp) -apply(case_tac "Suc (Suc 0) \ card (runing s)") -apply(subst (asm) card_le_Suc_iff) -apply(simp) -apply(auto)[1] -apply (metis insertCI runing_unique) -apply(auto) -done - -end - - -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 - - -context valid_trace -begin - -lemma cnp_cnv_eq: - assumes "th \ threads s" - shows "cntP s th = cntV s th" - using assms - using cnp_cnv_cncs not_thread_cncs by auto - -end - - -lemma eq_RAG: - "RAG (wq s) = RAG s" -by (unfold cs_RAG_def s_RAG_def, auto) - -context valid_trace -begin - -lemma count_eq_dependants: - assumes eq_pv: "cntP s th = cntV s th" - shows "dependants (wq s) th = {}" -proof - - from cnp_cnv_cncs and eq_pv - have "cntCS s th = 0" - by (auto split:if_splits) - moreover have "finite {cs. (Cs cs, Th th) \ RAG s}" - proof - - from finite_holding[of th] show ?thesis - by (simp add:holdents_test) - qed - ultimately have h: "{cs. (Cs cs, Th th) \ RAG s} = {}" - by (unfold cntCS_def holdents_test cs_dependants_def, auto) - show ?thesis - proof(unfold cs_dependants_def) - { assume "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ {}" - then obtain th' where "(Th th', Th th) \ (RAG (wq s))\<^sup>+" by auto - hence "False" - proof(cases) - assume "(Th th', Th th) \ RAG (wq s)" - thus "False" by (auto simp:cs_RAG_def) - next - fix c - assume "(c, Th th) \ RAG (wq s)" - with h and eq_RAG show "False" - by (cases c, auto simp:cs_RAG_def) - qed - } thus "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} = {}" by auto - qed -qed - -lemma dependants_threads: - shows "dependants (wq s) th \ threads s" -proof - { fix th th' - assume h: "th \ {th'a. (Th th'a, Th th') \ (RAG (wq s))\<^sup>+}" - have "Th th \ Domain (RAG s)" - proof - - from h obtain th' where "(Th th, Th th') \ (RAG (wq s))\<^sup>+" by auto - hence "(Th th) \ Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def) - with trancl_domain have "(Th th) \ Domain (RAG (wq s))" by simp - thus ?thesis using eq_RAG by simp - qed - from dm_RAG_threads[OF this] - have "th \ threads s" . - } note hh = this - fix th1 - assume "th1 \ dependants (wq s) th" - hence "th1 \ {th'a. (Th th'a, Th th) \ (RAG (wq s))\<^sup>+}" - by (unfold cs_dependants_def, simp) - from hh [OF this] show "th1 \ threads s" . -qed - -lemma finite_threads: - shows "finite (threads s)" -using vt by (induct) (auto elim: step.cases) - -end - -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 - -context valid_trace -begin - -lemma cp_le: - assumes th_in: "th \ threads s" - shows "cp s th \ Max ((\ th. (preced th s)) ` threads s)" -proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def) - show "Max ((\th. preced th s) ` ({th} \ {th'. (Th th', Th th) \ (RAG (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) \ (RAG (wq s))\<^sup>+} \ {}" by simp - next - from finite_threads - show "finite (threads s)" . - next - from th_in - show "{th} \ {th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ threads s" - apply (auto simp:Domain_def) - apply (rule_tac dm_RAG_threads) - apply (unfold trancl_domain [of "RAG s", symmetric]) - by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def) - qed -qed - -lemma le_cp: - shows "preced th s \ cp s th" -proof(unfold cp_eq_cpreced preced_def cpreced_def, simp) - show "Prc (priority th s) (last_set th s) - \ Max (insert (Prc (priority th s) (last_set th s)) - ((\th. Prc (priority th s) (last_set th s)) ` dependants (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) \ (RAG (wq s))\<^sup>+}" - proof - - let ?F = "\ (x, y). the_th x" - have "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (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_RAG have "finite (RAG s)" . - hence "finite ((RAG (wq s))\<^sup>+)" - apply (unfold finite_trancl) - by (auto simp: s_RAG_def cs_RAG_def wq_def) - thus ?thesis by auto - qed - ultimately show ?thesis by (auto intro:finite_subset) - qed - thus ?thesis by (simp add:cs_dependants_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: - 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 - 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 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 - 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 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 - 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 np: "threads s \ {}" - shows "Max (cp s ` readys s) = Max (cp s ` threads s)" -proof(unfold max_cp_eq) - 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 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 tm_in] - have "tm \ readys s \ (\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+)" . - thus ?thesis - proof - assume "\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+ " - then obtain th' where th'_in: "th' \ readys s" - and tm_chain:"(Th tm, Th th') \ (RAG s)\<^sup>+" by auto - have "cp s th' = ?f tm" - proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI) - from dependants_threads finite_threads - show "finite ((\th. preced th s) ` ({th'} \ dependants (wq s) th'))" - by (auto intro:finite_subset) - next - fix p assume p_in: "p \ (\th. preced th s) ` ({th'} \ dependants (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 - show "finite ((\th. preced th s) ` threads s)" by simp - next - from p_in and th'_in and dependants_threads[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'} \ dependants (wq s) th')" - proof - - from tm_chain - have "tm \ dependants (wq s) th'" - by (unfold cs_dependants_def s_RAG_def cs_RAG_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 dependants_threads [of th1] th1_in - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ - (\th. preced th s) ` threads s" - by (auto simp:readys_def) - next - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" by simp - next - from finite_threads - show " finite ((\th. preced th s) ` threads s)" by simp - qed - next - from finite_threads - 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} \ dependants (wq s) tm)" - show "y \ preced tm s" - proof - - { fix y' - assume hy' : "y' \ ((\th. preced th s) ` dependants (wq s) tm)" - have "y' \ preced tm s" - proof(unfold tm_max, rule Max_ge) - from hy' dependants_threads[of tm] - show "y' \ (\th. preced th s) ` threads s" by auto - next - from finite_threads - show "finite ((\th. preced th s) ` threads s)" by simp - qed - } with hy show ?thesis by auto - qed - next - from dependants_threads[of tm] finite_threads - show "finite ((\th. preced th s) ` ({tm} \ dependants (wq s) tm))" - by (auto intro:finite_subset) - next - show "preced tm s \ (\th. preced th s) ` ({tm} \ dependants (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 - 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 - show "finite ((\th. preced th s) ` threads s)" by simp - next - show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" - by simp - next - from dependants_threads[of th1] th1_readys - show "(\th. preced th s) ` ({th1} \ dependants (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 - -text {* (* ccc *) \noindent - Since the current precedence of the threads in ready queue will always be boosted, - there must be one inside it has the maximum precedence of the whole system. -*} -lemma max_cp_readys_threads: - 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 False]) -qed - -end - -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 (RAG s))" -apply(simp add: detached_def Field_def) -apply(simp add: s_RAG_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 - -context valid_trace -begin - -lemma detached_intro: - assumes eq_pv: "cntP s th = cntV s th" - shows "detached s th" -proof - - from cnp_cnv_cncs - 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 dm_RAG_threads - show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff) - next - assume "th \ readys s" - moreover have "Th th \ Range (RAG s)" - proof - - from card_0_eq [OF finite_holding] 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_RAG_def) - done - qed - ultimately show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def) - qed -qed - -lemma detached_elim: - assumes dtc: "detached s th" - shows "cntP s th = cntV s th" -proof - - from cnp_cnv_cncs - 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_RAG_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_RAG_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: - shows "(detached s th) = (cntP s th = cntV s th)" - by (insert vt, auto intro:detached_intro detached_elim) - -end - -text {* - The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived - from the concise and miniature model of PIP given in PrioGDef.thy. -*} - -lemma eq_dependants: "dependants (wq s) = dependants s" - by (simp add: s_dependants_abv wq_def) - -lemma next_th_unique: - assumes nt1: "next_th s th cs th1" - and nt2: "next_th s th cs th2" - shows "th1 = th2" -using assms by (unfold next_th_def, auto) - -lemma birth_time_lt: "s \ [] \ last_set th s < length s" - apply (induct s, simp) -proof - - fix a s - assume ih: "s \ [] \ last_set th s < length s" - and eq_as: "a # s \ []" - show "last_set th (a # s) < length (a # s)" - proof(cases "s \ []") - case False - from False show ?thesis - by (cases a, auto simp:last_set.simps) - next - case True - from ih [OF True] show ?thesis - by (cases a, auto simp:last_set.simps) - qed -qed - -lemma th_in_ne: "th \ threads s \ s \ []" - by (induct s, auto simp:threads.simps) - -lemma preced_tm_lt: "th \ threads s \ preced th s = Prc x y \ y < length s" - apply (drule_tac th_in_ne) - by (unfold preced_def, auto intro: birth_time_lt) - -lemma inj_the_preced: - "inj_on (the_preced s) (threads s)" - by (metis inj_onI preced_unique the_preced_def) - -lemma tRAG_alt_def: - "tRAG s = {(Th th1, Th th2) | th1 th2. - \ cs. (Th th1, Cs cs) \ RAG s \ (Cs cs, Th th2) \ RAG s}" - by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def) - -lemma tRAG_Field: - "Field (tRAG s) \ Field (RAG s)" - by (unfold tRAG_alt_def Field_def, auto) - -lemma tRAG_ancestorsE: - assumes "x \ ancestors (tRAG s) u" - obtains th where "x = Th th" -proof - - from assms have "(u, x) \ (tRAG s)^+" - by (unfold ancestors_def, auto) - from tranclE[OF this] obtain c where "(c, x) \ tRAG s" by auto - then obtain th where "x = Th th" - by (unfold tRAG_alt_def, auto) - from that[OF this] show ?thesis . -qed - -lemma tRAG_mono: - assumes "RAG s' \ RAG s" - shows "tRAG s' \ tRAG s" - using assms - by (unfold tRAG_alt_def, auto) - -lemma holding_next_thI: - assumes "holding s th cs" - and "length (wq s cs) > 1" - obtains th' where "next_th s th cs th'" -proof - - from assms(1)[folded eq_holding, unfolded cs_holding_def] - have " th \ set (wq s cs) \ th = hd (wq s cs)" . - then obtain rest where h1: "wq s cs = th#rest" - by (cases "wq s cs", auto) - with assms(2) have h2: "rest \ []" by auto - let ?th' = "hd (SOME q. distinct q \ set q = set rest)" - have "next_th s th cs ?th'" using h1(1) h2 - by (unfold next_th_def, auto) - from that[OF this] show ?thesis . -qed - -lemma RAG_tRAG_transfer: - assumes "vt s'" - assumes "RAG s = RAG s' \ {(Th th, Cs cs)}" - and "(Cs cs, Th th'') \ RAG s'" - shows "tRAG s = tRAG s' \ {(Th th, Th th'')}" (is "?L = ?R") -proof - - interpret vt_s': valid_trace "s'" using assms(1) - by (unfold_locales, simp) - interpret rtree: rtree "RAG s'" - proof - show "single_valued (RAG s')" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:vt_s'.unique_RAG) - - show "acyclic (RAG s')" - by (rule vt_s'.acyclic_RAG) - qed - { fix n1 n2 - assume "(n1, n2) \ ?L" - from this[unfolded tRAG_alt_def] - obtain th1 th2 cs' where - h: "n1 = Th th1" "n2 = Th th2" - "(Th th1, Cs cs') \ RAG s" - "(Cs cs', Th th2) \ RAG s" by auto - from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \ RAG s'" by auto - from h(3) and assms(2) - have "(Th th1, Cs cs') = (Th th, Cs cs) \ - (Th th1, Cs cs') \ RAG s'" by auto - hence "(n1, n2) \ ?R" - proof - assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)" - hence eq_th1: "th1 = th" by simp - moreover have "th2 = th''" - proof - - from h1 have "cs' = cs" by simp - from assms(3) cs_in[unfolded this] rtree.sgv - show ?thesis - by (unfold single_valued_def, auto) - qed - ultimately show ?thesis using h(1,2) by auto - next - assume "(Th th1, Cs cs') \ RAG s'" - with cs_in have "(Th th1, Th th2) \ tRAG s'" - by (unfold tRAG_alt_def, auto) - from this[folded h(1, 2)] show ?thesis by auto - qed - } moreover { - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \tRAG s' \ (n1, n2) = (Th th, Th th'')" by auto - hence "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ tRAG s'" - moreover have "... \ ?L" - proof(rule tRAG_mono) - show "RAG s' \ RAG s" by (unfold assms(2), auto) - qed - ultimately show ?thesis by auto - next - assume eq_n: "(n1, n2) = (Th th, Th th'')" - from assms(2, 3) have "(Cs cs, Th th'') \ RAG s" by auto - moreover have "(Th th, Cs cs) \ RAG s" using assms(2) by auto - ultimately show ?thesis - by (unfold eq_n tRAG_alt_def, auto) - qed - } ultimately show ?thesis by auto -qed - -context valid_trace -begin - -lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt] - -end - -lemma cp_alt_def: - "cp s th = - Max ((the_preced s) ` {th'. Th th' \ (subtree (RAG s) (Th th))})" -proof - - have "Max (the_preced s ` ({th} \ dependants (wq s) th)) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "Max (_ ` ?L) = Max (_ ` ?R)") - proof - - have "?L = ?R" - by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def) - thus ?thesis by simp - qed - thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp) -qed - -lemma cp_gen_alt_def: - "cp_gen s = (Max \ (\x. (the_preced s \ the_thread) ` subtree (tRAG s) x))" - by (auto simp:cp_gen_def) - -lemma tRAG_nodeE: - assumes "(n1, n2) \ tRAG s" - obtains th1 th2 where "n1 = Th th1" "n2 = Th th2" - using assms - by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def) - -lemma subtree_nodeE: - assumes "n \ subtree (tRAG s) (Th th)" - obtains th1 where "n = Th th1" -proof - - show ?thesis - proof(rule subtreeE[OF assms]) - assume "n = Th th" - from that[OF this] show ?thesis . - next - assume "Th th \ ancestors (tRAG s) n" - hence "(n, Th th) \ (tRAG s)^+" by (auto simp:ancestors_def) - hence "\ th1. n = Th th1" - proof(induct) - case (base y) - from tRAG_nodeE[OF this] show ?case by metis - next - case (step y z) - thus ?case by auto - qed - with that show ?thesis by auto - qed -qed - -lemma tRAG_star_RAG: "(tRAG s)^* \ (RAG s)^*" -proof - - have "(wRAG s O hRAG s)^* \ (RAG s O RAG s)^*" - by (rule rtrancl_mono, auto simp:RAG_split) - also have "... \ ((RAG s)^*)^*" - by (rule rtrancl_mono, auto) - also have "... = (RAG s)^*" by simp - finally show ?thesis by (unfold tRAG_def, simp) -qed - -lemma tRAG_subtree_RAG: "subtree (tRAG s) x \ subtree (RAG s) x" -proof - - { fix a - assume "a \ subtree (tRAG s) x" - hence "(a, x) \ (tRAG s)^*" by (auto simp:subtree_def) - with tRAG_star_RAG[of s] - have "(a, x) \ (RAG s)^*" by auto - hence "a \ subtree (RAG s) x" by (auto simp:subtree_def) - } thus ?thesis by auto -qed - -lemma tRAG_trancl_eq: - "{th'. (Th th', Th th) \ (tRAG s)^+} = - {th'. (Th th', Th th) \ (RAG s)^+}" - (is "?L = ?R") -proof - - { fix th' - assume "th' \ ?L" - hence "(Th th', Th th) \ (tRAG s)^+" by auto - from tranclD[OF this] - obtain z where h: "(Th th', z) \ tRAG s" "(z, Th th) \ (tRAG s)\<^sup>*" by auto - from tRAG_subtree_RAG[of s] and this(2) - have "(z, Th th) \ (RAG s)^*" by (meson subsetCE tRAG_star_RAG) - moreover from h(1) have "(Th th', z) \ (RAG s)^+" using tRAG_alt_def by auto - ultimately have "th' \ ?R" by auto - } moreover - { fix th' - assume "th' \ ?R" - hence "(Th th', Th th) \ (RAG s)^+" by (auto) - from plus_rpath[OF this] - obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \ []" by auto - hence "(Th th', Th th) \ (tRAG s)^+" - proof(induct xs arbitrary:th' th rule:length_induct) - case (1 xs th' th) - then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto) - show ?case - proof(cases "xs1") - case Nil - from 1(2)[unfolded Cons1 Nil] - have rp: "rpath (RAG s) (Th th') [x1] (Th th)" . - hence "(Th th', x1) \ (RAG s)" by (cases, simp) - then obtain cs where "x1 = Cs cs" - by (unfold s_RAG_def, auto) - from rpath_nnl_lastE[OF rp[unfolded this]] - show ?thesis by auto - next - case (Cons x2 xs2) - from 1(2)[unfolded Cons1[unfolded this]] - have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" . - from rpath_edges_on[OF this] - have eds: "edges_on (Th th' # x1 # x2 # xs2) \ RAG s" . - have "(Th th', x1) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - with eds have rg1: "(Th th', x1) \ RAG s" by auto - then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto) - have "(x1, x2) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - from this eds - have rg2: "(x1, x2) \ RAG s" by auto - from this[unfolded eq_x1] - obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto) - from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2] - have rt1: "(Th th', Th th1) \ tRAG s" by (unfold tRAG_alt_def, auto) - from rp have "rpath (RAG s) x2 xs2 (Th th)" - by (elim rpath_ConsE, simp) - from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" . - show ?thesis - proof(cases "xs2 = []") - case True - from rpath_nilE[OF rp'[unfolded this]] - have "th1 = th" by auto - from rt1[unfolded this] show ?thesis by auto - next - case False - from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons] - have "(Th th1, Th th) \ (tRAG s)\<^sup>+" by simp - with rt1 show ?thesis by auto - qed - qed - qed - hence "th' \ ?L" by auto - } ultimately show ?thesis by blast -qed - -lemma tRAG_trancl_eq_Th: - "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = - {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" - using tRAG_trancl_eq by auto - -lemma dependants_alt_def: - "dependants s th = {th'. (Th th', Th th) \ (tRAG s)^+}" - by (metis eq_RAG s_dependants_def tRAG_trancl_eq) - -context valid_trace -begin - -lemma count_eq_tRAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using assms count_eq_dependants dependants_alt_def eq_dependants by auto - -lemma count_eq_RAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (RAG s)^+} = {}" - using assms count_eq_dependants cs_dependants_def eq_RAG by auto - -lemma count_eq_RAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (RAG s)^+} = {}" - using count_eq_RAG_plus[OF assms] by auto - -lemma count_eq_tRAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using count_eq_tRAG_plus[OF assms] by auto - -end - -lemma tRAG_subtree_eq: - "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \ (subtree (RAG s) (Th th))}" - (is "?L = ?R") -proof - - { fix n - assume h: "n \ ?L" - hence "n \ ?R" - by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) - } moreover { - fix n - assume "n \ ?R" - then obtain th' where h: "n = Th th'" "(Th th', Th th) \ (RAG s)^*" - by (auto simp:subtree_def) - from rtranclD[OF this(2)] - have "n \ ?L" - proof - assume "Th th' \ Th th \ (Th th', Th th) \ (RAG s)\<^sup>+" - with h have "n \ {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" by auto - thus ?thesis using subtree_def tRAG_trancl_eq by fastforce - qed (insert h, auto simp:subtree_def) - } ultimately show ?thesis by auto -qed - -lemma threads_set_eq: - "the_thread ` (subtree (tRAG s) (Th th)) = - {th'. Th th' \ (subtree (RAG s) (Th th))}" (is "?L = ?R") - by (auto intro:rev_image_eqI simp:tRAG_subtree_eq) - -lemma cp_alt_def1: - "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))" -proof - - have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) = - ((the_preced s \ the_thread) ` subtree (tRAG s) (Th th))" - by auto - thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto) -qed - -lemma cp_gen_def_cond: - assumes "x = Th th" - shows "cp s th = cp_gen s (Th th)" -by (unfold cp_alt_def1 cp_gen_def, simp) - -lemma cp_gen_over_set: - assumes "\ x \ A. \ th. x = Th th" - shows "cp_gen s ` A = (cp s \ the_thread) ` A" -proof(rule f_image_eq) - fix a - assume "a \ A" - from assms[rule_format, OF this] - obtain th where eq_a: "a = Th th" by auto - show "cp_gen s a = (cp s \ the_thread) a" - by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp) -qed - - -context valid_trace -begin - -lemma RAG_threads: - assumes "(Th th) \ Field (RAG s)" - shows "th \ threads s" - using assms - by (metis Field_def UnE dm_RAG_threads range_in vt) - -lemma subtree_tRAG_thread: - assumes "th \ threads s" - shows "subtree (tRAG s) (Th th) \ Th ` threads s" (is "?L \ ?R") -proof - - have "?L = {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - by (unfold tRAG_subtree_eq, simp) - also have "... \ ?R" - proof - fix x - assume "x \ {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - then obtain th' where h: "x = Th th'" "Th th' \ subtree (RAG s) (Th th)" by auto - from this(2) - show "x \ ?R" - proof(cases rule:subtreeE) - case 1 - thus ?thesis by (simp add: assms h(1)) - next - case 2 - thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) - qed - qed - finally show ?thesis . -qed - -lemma readys_root: - assumes "th \ readys s" - shows "root (RAG s) (Th th)" -proof - - { fix x - assume "x \ ancestors (RAG s) (Th th)" - hence h: "(Th th, x) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - obtain z where "(Th th, z) \ RAG s" by auto - with assms(1) have False - apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def) - by (fold wq_def, blast) - } thus ?thesis by (unfold root_def, auto) -qed - -lemma readys_in_no_subtree: - assumes "th \ readys s" - and "th' \ th" - shows "Th th \ subtree (RAG s) (Th th')" -proof - assume "Th th \ subtree (RAG s) (Th th')" - thus False - proof(cases rule:subtreeE) - case 1 - with assms show ?thesis by auto - next - case 2 - with readys_root[OF assms(1)] - show ?thesis by (auto simp:root_def) - qed -qed - -lemma not_in_thread_isolated: - assumes "th \ threads s" - shows "(Th th) \ Field (RAG s)" -proof - assume "(Th th) \ Field (RAG s)" - with dm_RAG_threads and range_in assms - show False by (unfold Field_def, blast) -qed - -lemma wf_RAG: "wf (RAG s)" -proof(rule finite_acyclic_wf) - from finite_RAG show "finite (RAG s)" . -next - from acyclic_RAG show "acyclic (RAG s)" . -qed - -lemma sgv_wRAG: "single_valued (wRAG s)" - using waiting_unique - by (unfold single_valued_def wRAG_def, auto) - -lemma sgv_hRAG: "single_valued (hRAG s)" - using holding_unique - by (unfold single_valued_def hRAG_def, auto) - -lemma sgv_tRAG: "single_valued (tRAG s)" - by (unfold tRAG_def, rule single_valued_relcomp, - insert sgv_wRAG sgv_hRAG, auto) - -lemma acyclic_tRAG: "acyclic (tRAG s)" -proof(unfold tRAG_def, rule acyclic_compose) - show "acyclic (RAG s)" using acyclic_RAG . -next - show "wRAG s \ RAG s" unfolding RAG_split by auto -next - show "hRAG s \ RAG s" unfolding RAG_split by auto -qed - -lemma sgv_RAG: "single_valued (RAG s)" - using unique_RAG by (auto simp:single_valued_def) - -lemma rtree_RAG: "rtree (RAG s)" - using sgv_RAG acyclic_RAG - by (unfold rtree_def rtree_axioms_def sgv_def, auto) - -end - -sublocale valid_trace < rtree_RAG: rtree "RAG s" -proof - show "single_valued (RAG s)" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:unique_RAG) - - show "acyclic (RAG s)" - by (rule acyclic_RAG) -qed - -sublocale valid_trace < rtree_s: rtree "tRAG s" -proof(unfold_locales) - from sgv_tRAG show "single_valued (tRAG s)" . -next - from acyclic_tRAG show "acyclic (tRAG s)" . -qed - -sublocale valid_trace < fsbtRAGs : fsubtree "RAG s" -proof - - show "fsubtree (RAG s)" - proof(intro_locales) - show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] . - next - show "fsubtree_axioms (RAG s)" - proof(unfold fsubtree_axioms_def) - from wf_RAG show "wf (RAG s)" . - qed - qed -qed - -sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s" -proof - - have "fsubtree (tRAG s)" - proof - - have "fbranch (tRAG s)" - proof(unfold tRAG_def, rule fbranch_compose) - show "fbranch (wRAG s)" - proof(rule finite_fbranchI) - from finite_RAG show "finite (wRAG s)" - by (unfold RAG_split, auto) - qed - next - show "fbranch (hRAG s)" - proof(rule finite_fbranchI) - from finite_RAG - show "finite (hRAG s)" by (unfold RAG_split, auto) - qed - qed - moreover have "wf (tRAG s)" - proof(rule wf_subset) - show "wf (RAG s O RAG s)" using wf_RAG - by (fold wf_comp_self, simp) - next - show "tRAG s \ (RAG s O RAG s)" - by (unfold tRAG_alt_def, auto) - qed - ultimately show ?thesis - by (unfold fsubtree_def fsubtree_axioms_def,auto) - qed - from this[folded tRAG_def] show "fsubtree (tRAG s)" . -qed - -lemma Max_UNION: - assumes "finite A" - and "A \ {}" - and "\ M \ f ` A. finite M" - and "\ M \ f ` A. M \ {}" - shows "Max (\x\ A. f x) = Max (Max ` f ` A)" (is "?L = ?R") - using assms[simp] -proof - - have "?L = Max (\(f ` A))" - by (fold Union_image_eq, simp) - also have "... = ?R" - by (subst Max_Union, simp+) - finally show ?thesis . -qed - -lemma max_Max_eq: - assumes "finite A" - and "A \ {}" - and "x = y" - shows "max x (Max A) = Max ({y} \ A)" (is "?L = ?R") -proof - - have "?R = Max (insert y A)" by simp - also from assms have "... = ?L" - by (subst Max.insert, simp+) - finally show ?thesis by simp -qed - -context valid_trace -begin - -(* ddd *) -lemma cp_gen_rec: - assumes "x = Th th" - shows "cp_gen s x = Max ({the_preced s th} \ (cp_gen s) ` children (tRAG s) x)" -proof(cases "children (tRAG s) x = {}") - case True - show ?thesis - by (unfold True cp_gen_def subtree_children, simp add:assms) -next - case False - hence [simp]: "children (tRAG s) x \ {}" by auto - note fsbttRAGs.finite_subtree[simp] - have [simp]: "finite (children (tRAG s) x)" - by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], - rule children_subtree) - { fix r x - have "subtree r x \ {}" by (auto simp:subtree_def) - } note this[simp] - have [simp]: "\x\children (tRAG s) x. subtree (tRAG s) x \ {}" - proof - - from False obtain q where "q \ children (tRAG s) x" by blast - moreover have "subtree (tRAG s) q \ {}" by simp - ultimately show ?thesis by blast - qed - have h: "Max ((the_preced s \ the_thread) ` - ({x} \ \(subtree (tRAG s) ` children (tRAG s) x))) = - Max ({the_preced s th} \ cp_gen s ` children (tRAG s) x)" - (is "?L = ?R") - proof - - let "Max (?f ` (?A \ \ (?g ` ?B)))" = ?L - let "Max (_ \ (?h ` ?B))" = ?R - let ?L1 = "?f ` \(?g ` ?B)" - have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)" - proof - - have "?L1 = ?f ` (\ x \ ?B.(?g x))" by simp - also have "... = (\ x \ ?B. ?f ` (?g x))" by auto - finally have "Max ?L1 = Max ..." by simp - also have "... = Max (Max ` (\x. ?f ` subtree (tRAG s) x) ` ?B)" - by (subst Max_UNION, simp+) - also have "... = Max (cp_gen s ` children (tRAG s) x)" - by (unfold image_comp cp_gen_alt_def, simp) - finally show ?thesis . - qed - show ?thesis - proof - - have "?L = Max (?f ` ?A \ ?L1)" by simp - also have "... = max (the_preced s (the_thread x)) (Max ?L1)" - by (subst Max_Un, simp+) - also have "... = max (?f x) (Max (?h ` ?B))" - by (unfold eq_Max_L1, simp) - also have "... =?R" - by (rule max_Max_eq, (simp)+, unfold assms, simp) - finally show ?thesis . - qed - qed thus ?thesis - by (fold h subtree_children, unfold cp_gen_def, simp) -qed - -lemma cp_rec: - "cp s th = Max ({the_preced s th} \ - (cp s o the_thread) ` children (tRAG s) (Th th))" -proof - - have "Th th = Th th" by simp - note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this] - show ?thesis - proof - - have "cp_gen s ` children (tRAG s) (Th th) = - (cp s \ the_thread) ` children (tRAG s) (Th th)" - proof(rule cp_gen_over_set) - show " \x\children (tRAG s) (Th th). \th. x = Th th" - by (unfold tRAG_alt_def, auto simp:children_def) - qed - thus ?thesis by (subst (1) h(1), unfold h(2), simp) - qed -qed - -end - -(* keep *) -lemma next_th_holding: - assumes vt: "vt s" - and nxt: "next_th s th cs th'" - shows "holding (wq s) th cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - thus ?thesis - by (unfold cs_holding_def, auto) -qed - -context valid_trace -begin - -lemma next_th_waiting: - assumes nxt: "next_th s th cs th'" - shows "waiting (wq s) th' cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - from wq_distinct[of cs, unfolded h] - have dst: "distinct (th # rest)" . - have in_rest: "th' \ set rest" - proof(unfold h, rule someI2) - show "distinct rest \ set rest = set rest" using dst by auto - next - fix x assume "distinct x \ set x = set rest" - with h(2) - show "hd x \ set (rest)" by (cases x, auto) - qed - hence "th' \ set (wq s cs)" by (unfold h(1), auto) - moreover have "th' \ hd (wq s cs)" - by (unfold h(1), insert in_rest dst, auto) - ultimately show ?thesis by (auto simp:cs_waiting_def) -qed - -lemma next_th_RAG: - assumes nxt: "next_th (s::event list) th cs th'" - shows "{(Cs cs, Th th), (Th th', Cs cs)} \ RAG s" - using vt assms next_th_holding next_th_waiting - by (unfold s_RAG_def, simp) - -end - --- {* A useless definition *} -definition cps:: "state \ (thread \ precedence) set" -where "cps s = {(th, cp s th) | th . th \ threads s}" - - -find_theorems "waiting" holding -context valid_trace -begin - -find_theorems "waiting" holding - -end - - -end diff -r ed938e2246b9 -r 0525670d8e6a PIPDefs.thy~ --- a/PIPDefs.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,665 +0,0 @@ -chapter {* Definitions *} -(*<*) -theory PIPDefs -imports Precedence_ord Moment RTree Max -begin -(*>*) - -text {* - In this section, the formal model of Priority Inheritance Protocol (PIP) is presented. - The model is based on Paulson's inductive protocol verification method, where - the state of the system is modelled as a list of events happened so far with the latest - event put at the head. -*} - -text {* - To define events, the identifiers of {\em threads}, - {\em priority} and {\em critical resources } (abbreviated as @{text "cs"}) - need to be represented. All three are represetned using standard - Isabelle/HOL type @{typ "nat"}: -*} - -type_synonym thread = nat -- {* Type for thread identifiers. *} -type_synonym priority = nat -- {* Type for priorities. *} -type_synonym cs = nat -- {* Type for critical sections (or critical resources). *} - -text {* - \noindent - The abstraction of Priority Inheritance Protocol (PIP) is set at the system call level. - Every system call is represented as an event. The format of events is defined - defined as follows: - *} - -datatype event = - Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *} - Exit thread | -- {* Thread @{text "thread"} finishing its execution. *} - P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *} - V thread cs | -- {* Thread @{text "thread"} releasing critical resource @{text "cs"}. *} - Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *} - -fun actor :: "event \ thread" where - "actor (Create th pty) = th" | - "actor (Exit th) = th" | - "actor (P th cs) = th" | - "actor (V th cs) = th" | - "actor (Set th pty) = th" - -fun isCreate :: "event \ bool" where - "isCreate (Create th pty) = True" | - "isCreate _ = False" - -fun isP :: "event \ bool" where - "isP (P th cs) = True" | - "isP _ = False" - -fun isV :: "event \ bool" where - "isV (V th cs) = True" | - "isV _ = False" - -text {* - As mentioned earlier, in Paulson's inductive method, the states of system are represented as lists of events, - which is defined by the following type @{text "state"}: - *} -type_synonym state = "event list" - - -text {* -\noindent - Resource Allocation Graph (RAG for short) is used extensively in our formal analysis. - The following type @{text "node"} is used to represent nodes in RAG. - *} -datatype node = - Th "thread" | -- {* Node for thread. *} - Cs "cs" -- {* Node for critical resource. *} - -text {* - \noindent - The following function - @{text "threads"} is used to calculate the set of live threads (@{text "threads s"}) - in state @{text "s"}. - *} -fun threads :: "state \ thread set" - where - -- {* At the start of the system, the set of threads is empty: *} - "threads [] = {}" | - -- {* New thread is added to the @{text "threads"}: *} - "threads (Create thread prio#s) = {thread} \ threads s" | - -- {* Finished thread is removed: *} - "threads (Exit thread # s) = (threads s) - {thread}" | - -- {* Other kind of events does not affect the value of @{text "threads"}: *} - "threads (e#s) = threads s" - -text {* - \noindent - The function @{text "threads"} defined above is one of - the so called {\em observation function}s which forms - the very basis of Paulson's inductive protocol verification method. - Each observation function {\em observes} one particular aspect (or attribute) - of the system. For example, the attribute observed by @{text "threads s"} - is the set of threads living in state @{text "s"}. - The protocol being modelled - The decision made the protocol being modelled is based on the {\em observation}s - returned by {\em observation function}s. Since {\observation function}s forms - the very basis on which Paulson's inductive method is based, there will be - a lot of such observation functions introduced in the following. In fact, any function - which takes event list as argument is a {\em observation function}. - *} - -text {* \noindent - Observation @{text "priority th s"} is - the {\em original priority} of thread @{text "th"} in state @{text "s"}. - The {\em original priority} is the priority - assigned to a thread when it is created or when it is reset by system call - (represented by event @{text "Set thread priority"}). -*} - -fun priority :: "thread \ state \ priority" - where - -- {* @{text "0"} is assigned to threads which have never been created: *} - "priority thread [] = 0" | - "priority thread (Create thread' prio#s) = - (if thread' = thread then prio else priority thread s)" | - "priority thread (Set thread' prio#s) = - (if thread' = thread then prio else priority thread s)" | - "priority thread (e#s) = priority thread s" - -text {* - \noindent - Observation @{text "last_set th s"} is the last time when the priority of thread @{text "th"} is set, - observed from state @{text "s"}. - The time in the system is measured by the number of events happened so far since the very beginning. -*} -fun last_set :: "thread \ state \ nat" - where - "last_set thread [] = 0" | - "last_set thread ((Create thread' prio)#s) = - (if (thread = thread') then length s else last_set thread s)" | - "last_set thread ((Set thread' prio)#s) = - (if (thread = thread') then length s else last_set thread s)" | - "last_set thread (_#s) = last_set thread s" - -text {* - \noindent - The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of - a thread is the combination of its {\em original priority} and {\em time} the priority is set. - The intention is to discriminate threads with the same priority by giving threads whose priority - is assigned earlier higher precedences, becasue such threads are more urgent to finish. - This explains the following definition: - *} -definition preced :: "thread \ state \ precedence" - where "preced thread s \ Prc (priority thread s) (last_set thread s)" - - -text {* - \noindent - A number of important notions in PIP are represented as the following functions, - defined in terms of the waiting queues of the system, where the waiting queues - , as a whole, is represented by the @{text "wq"} argument of every notion function. - The @{text "wq"} argument is itself a functions which maps every critical resource - @{text "cs"} to the list of threads which are holding or waiting for it. - The thread at the head of this list is designated as the thread which is current - holding the resrouce, which is slightly different from tradition where - all threads in the waiting queue are considered as waiting for the resource. - *} - -consts - holding :: "'b \ thread \ cs \ bool" - waiting :: "'b \ thread \ cs \ bool" - RAG :: "'b \ (node \ node) set" - dependants :: "'b \ thread \ thread set" - -defs (overloaded) - -- {* - \begin{minipage}{0.9\textwidth} - This meaning of @{text "wq"} is reflected in the following definition of @{text "holding wq th cs"}, - where @{text "holding wq th cs"} means thread @{text "th"} is holding the critical - resource @{text "cs"}. This decision is based on @{text "wq"}. - \end{minipage} - *} - - cs_holding_def: - "holding wq thread cs \ (thread \ set (wq cs) \ thread = hd (wq cs))" - -- {* - \begin{minipage}{0.9\textwidth} - In accordance with the definition of @{text "holding wq th cs"}, - a thread @{text "th"} is considered waiting for @{text "cs"} if - it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head. - This is reflected in the definition of @{text "waiting wq th cs"} as follows: - \end{minipage} - *} - cs_waiting_def: - "waiting wq thread cs \ (thread \ set (wq cs) \ thread \ hd (wq cs))" - -- {* - \begin{minipage}{0.9\textwidth} - @{text "RAG wq"} generates RAG (a binary relations on @{text "node"}) - out of waiting queues of the system (represented by the @{text "wq"} argument): - \end{minipage} - *} - cs_RAG_def: - "RAG (wq::cs \ thread list) \ - {(Th th, Cs cs) | th cs. waiting wq th cs} \ {(Cs cs, Th th) | cs th. holding wq th cs}" - -- {* - \begin{minipage}{0.9\textwidth} - The following @{text "dependants wq th"} represents the set of threads which are RAGing on - thread @{text "th"} in Resource Allocation Graph @{text "RAG wq"}. - Here, "RAGing" means waiting directly or indirectly on the critical resource. - \end{minipage} - *} - cs_dependants_def: - "dependants (wq::cs \ thread list) th \ {th' . (Th th', Th th) \ (RAG wq)^+}" - - -text {* \noindent - The following - @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under - state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of - Priority Inheritance that the {\em current precedence} of a thread is the precedence - inherited from the maximum of all its dependants, i.e. the threads which are waiting - directly or indirectly waiting for some resources from it. If no such thread exits, - @{text "th"}'s {\em current precedence} equals its original precedence, i.e. - @{text "preced th s"}. - *} - -definition cpreced :: "(cs \ thread list) \ state \ thread \ precedence" - where "cpreced wq s = (\th. Max ((\th'. preced th' s) ` ({th} \ dependants wq th)))" - -text {* - Notice that the current precedence (@{text "cpreced"}) of one thread @{text "th"} can be boosted - (becoming larger than its own precedence) by those threads in - the @{text "dependants wq th"}-set. If one thread get boosted, we say - it inherits the priority (or, more precisely, the precedence) of - its dependants. This is how the word "Inheritance" in - Priority Inheritance Protocol comes. -*} - -(*<*) -lemma - cpreced_def2: - "cpreced wq s th \ Max ({preced th s} \ {preced th' s | th'. th' \ dependants wq th})" - unfolding cpreced_def image_def - apply(rule eq_reflection) - apply(rule_tac f="Max" in arg_cong) - by (auto) -(*>*) - - -text {* \noindent - Assuming @{text "qs"} be the waiting queue of a critical resource, - the following abbreviation "release qs" is the waiting queue after the thread - holding the resource (which is thread at the head of @{text "qs"}) released - the resource: -*} -abbreviation - "release qs \ case qs of - [] => [] - | (_#qs') => (SOME q. distinct q \ set q = set qs')" -text {* \noindent - It can be seen from the definition that the thread at the head of @{text "qs"} is removed - from the return value, and the value @{term "q"} is an reordering of @{text "qs'"}, the - tail of @{text "qs"}. Through this reordering, one of the waiting threads (those in @{text "qs'"} } - is chosen nondeterministically to be the head of the new queue @{text "q"}. - Therefore, this thread is the one who takes over the resource. This is a little better different - from common sense that the thread who comes the earliest should take over. - The intention of this definition is to show that the choice of which thread to take over the - release resource does not affect the correctness of the PIP protocol. -*} - -text {* - The data structure used by the operating system for scheduling is referred to as - {\em schedule state}. It is represented as a record consisting of - a function assigning waiting queue to resources - (to be used as the @{text "wq"} argument in @{text "holding"}, @{text "waiting"} - and @{text "RAG"}, etc) and a function assigning precedence to threads: - *} - -record schedule_state = - wq_fun :: "cs \ thread list" -- {* The function assigning waiting queue. *} - cprec_fun :: "thread \ precedence" -- {* The function assigning precedence. *} - -text {* \noindent - The following two abbreviations (@{text "all_unlocked"} and @{text "initial_cprec"}) - are used to set the initial values of the @{text "wq_fun"} @{text "cprec_fun"} fields - respectively of the @{text "schedule_state"} record by the following function @{text "sch"}, - which is used to calculate the system's {\em schedule state}. - - Since there is no thread at the very beginning to make request, all critical resources - are free (or unlocked). This status is represented by the abbreviation - @{text "all_unlocked"}. - *} -abbreviation - "all_unlocked \ \_::cs. ([]::thread list)" - - -text {* \noindent - The initial current precedence for a thread can be anything, because there is no thread then. - We simply assume every thread has precedence @{text "Prc 0 0"}. - *} - -abbreviation - "initial_cprec \ \_::thread. Prc 0 0" - - -text {* \noindent - The following function @{text "schs"} is used to calculate the system's schedule state @{text "schs s"} - out of the current system state @{text "s"}. It is the central function to model Priority Inheritance: - *} -fun schs :: "state \ schedule_state" - where - -- {* - \begin{minipage}{0.9\textwidth} - Setting the initial value of the @{text "schedule_state"} record (see the explanations above). - \end{minipage} - *} - "schs [] = (| wq_fun = all_unlocked, cprec_fun = initial_cprec |)" | - - -- {* - \begin{minipage}{0.9\textwidth} - \begin{enumerate} - \item @{text "ps"} is the schedule state of last moment. - \item @{text "pwq"} is the waiting queue function of last moment. - \item @{text "pcp"} is the precedence function of last moment (NOT USED). - \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement: - \begin{enumerate} - \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to - the end of @{text "cs"}'s waiting queue. - \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state, - @{text "th'"} must equal to @{text "thread"}, - because @{text "thread"} is the one currently holding @{text "cs"}. - The case @{text "[] \ []"} may never be executed in a legal state. - the @{text "(SOME q. distinct q \ set q = set qs)"} is used to choose arbitrarily one - thread in waiting to take over the released resource @{text "cs"}. In our representation, - this amounts to rearrange elements in waiting queue, so that one of them is put at the head. - \item For other happening event, the schedule state just does not change. - \end{enumerate} - \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue - function. The RAGency of precedence function on waiting queue function is the reason to - put them in the same record so that they can evolve together. - \end{enumerate} - - - The calculation of @{text "cprec_fun"} depends on the value of @{text "wq_fun"}. - Therefore, in the following cases, @{text "wq_fun"} is always calculated first, in - the name of @{text "wq"} (if @{text "wq_fun"} is not changed - by the happening event) or @{text "new_wq"} (if the value of @{text "wq_fun"} is changed). - \end{minipage} - *} - "schs (Create th prio # s) = - (let wq = wq_fun (schs s) in - (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))" -| "schs (Exit th # s) = - (let wq = wq_fun (schs s) in - (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))" -| "schs (Set th prio # s) = - (let wq = wq_fun (schs s) in - (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))" - -- {* - \begin{minipage}{0.9\textwidth} - Different from the forth coming cases, the @{text "wq_fun"} field of the schedule state - is changed. So, the new value is calculated first, in the name of @{text "new_wq"}. - \end{minipage} - *} -| "schs (P th cs # s) = - (let wq = wq_fun (schs s) in - let new_wq = wq(cs := (wq cs @ [th])) in - (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))" -| "schs (V th cs # s) = - (let wq = wq_fun (schs s) in - let new_wq = wq(cs := release (wq cs)) in - (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))" - -lemma cpreced_initial: - "cpreced (\ cs. []) [] = (\_. (Prc 0 0))" -apply(simp add: cpreced_def) -apply(simp add: cs_dependants_def cs_RAG_def cs_waiting_def cs_holding_def) -apply(simp add: preced_def) -done - -lemma sch_old_def: - "schs (e#s) = (let ps = schs s in - let pwq = wq_fun ps in - let nwq = case e of - P th cs \ pwq(cs:=(pwq cs @ [th])) | - V th cs \ let nq = case (pwq cs) of - [] \ [] | - (_#qs) \ (SOME q. distinct q \ set q = set qs) - in pwq(cs:=nq) | - _ \ pwq - in let ncp = cpreced nwq (e#s) in - \wq_fun = nwq, cprec_fun = ncp\ - )" -apply(cases e) -apply(simp_all) -done - - -text {* - \noindent - The following @{text "wq"} is a shorthand for @{text "wq_fun"}. - *} -definition wq :: "state \ cs \ thread list" - where "wq s = wq_fun (schs s)" - -text {* \noindent - The following @{text "cp"} is a shorthand for @{text "cprec_fun"}. - *} -definition cp :: "state \ thread \ precedence" - where "cp s \ cprec_fun (schs s)" - -text {* \noindent - Functions @{text "holding"}, @{text "waiting"}, @{text "RAG"} and - @{text "dependants"} still have the - same meaning, but redefined so that they no longer RAG on the - fictitious {\em waiting queue function} - @{text "wq"}, but on system state @{text "s"}. - *} -defs (overloaded) - s_holding_abv: - "holding (s::state) \ holding (wq_fun (schs s))" - s_waiting_abv: - "waiting (s::state) \ waiting (wq_fun (schs s))" - s_RAG_abv: - "RAG (s::state) \ RAG (wq_fun (schs s))" - s_dependants_abv: - "dependants (s::state) \ dependants (wq_fun (schs s))" - - -text {* - The following lemma can be proved easily, and the meaning is obvious. - *} -lemma - s_holding_def: - "holding (s::state) th cs \ (th \ set (wq_fun (schs s) cs) \ th = hd (wq_fun (schs s) cs))" - by (auto simp:s_holding_abv wq_def cs_holding_def) - -lemma s_waiting_def: - "waiting (s::state) th cs \ (th \ set (wq_fun (schs s) cs) \ th \ hd (wq_fun (schs s) cs))" - by (auto simp:s_waiting_abv wq_def cs_waiting_def) - -lemma s_RAG_def: - "RAG (s::state) = - {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \ {(Cs cs, Th th) | cs th. holding (wq s) th cs}" - by (auto simp:s_RAG_abv wq_def cs_RAG_def) - -lemma - s_dependants_def: - "dependants (s::state) th \ {th' . (Th th', Th th) \ (RAG (wq s))^+}" - by (auto simp:s_dependants_abv wq_def cs_dependants_def) - -text {* - The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready} - for running if it is a live thread and it is not waiting for any critical resource. - *} -definition readys :: "state \ thread set" - where "readys s \ {th . th \ threads s \ (\ cs. \ waiting s th cs)}" - -text {* \noindent - The following function @{text "runing"} calculates the set of running thread, which is the ready - thread with the highest precedence. - *} -definition runing :: "state \ thread set" - where "runing s \ {th . th \ readys s \ cp s th = Max ((cp s) ` (readys s))}" - -text {* \noindent - Notice that the definition of @{text "running"} reflects the preemptive scheduling strategy, - because, if the @{text "running"}-thread (the one in @{text "runing"} set) - lowered its precedence by resetting its own priority to a lower - one, it will lose its status of being the max in @{text "ready"}-set and be superseded. -*} - -text {* \noindent - The following function @{text "holdents s th"} returns the set of resources held by thread - @{text "th"} in state @{text "s"}. - *} -definition holdents :: "state \ thread \ cs set" - where "holdents s th \ {cs . holding s th cs}" - -lemma holdents_test: - "holdents s th = {cs . (Cs cs, Th th) \ RAG s}" -unfolding holdents_def -unfolding s_RAG_def -unfolding s_holding_abv -unfolding wq_def -by (simp) - -text {* \noindent - Observation @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in - state @{text "s"}: - *} -definition cntCS :: "state \ thread \ nat" - where "cntCS s th = card (holdents s th)" - -text {* \noindent - According to the convention of Paulson's inductive method, - the decision made by a protocol that event @{text "e"} is eligible to happen next under state @{text "s"} - is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as - follows (notice how the decision is based on the {\em observation function}s - defined above, and also notice how a complicated protocol is modeled by a few simple - observations, and how such a kind of simplicity gives rise to improved trust on - faithfulness): - *} -inductive step :: "state \ event \ bool" - where - -- {* - A thread can be created if it is not a live thread: - *} - thread_create: "\thread \ threads s\ \ step s (Create thread prio)" | - -- {* - A thread can exit if it no longer hold any resource: - *} - thread_exit: "\thread \ runing s; holdents s thread = {}\ \ step s (Exit thread)" | - -- {* - \begin{minipage}{0.9\textwidth} - A thread can request for an critical resource @{text "cs"}, if it is running and - the request does not form a loop in the current RAG. The latter condition - is set up to avoid deadlock. The condition also reflects our assumption all threads are - carefully programmed so that deadlock can not happen: - \end{minipage} - *} - thread_P: "\thread \ runing s; (Cs cs, Th thread) \ (RAG s)^+\ \ - step s (P thread cs)" | - -- {* - \begin{minipage}{0.9\textwidth} - A thread can release a critical resource @{text "cs"} - if it is running and holding that resource: - \end{minipage} - *} - thread_V: "\thread \ runing s; holding s thread cs\ \ step s (V thread cs)" | - -- {* - \begin{minipage}{0.9\textwidth} - A thread can adjust its own priority as long as it is current running. - With the resetting of one thread's priority, its precedence may change. - If this change lowered the precedence, according to the definition of @{text "running"} - function, - \end{minipage} - *} - thread_set: "\thread \ runing s\ \ step s (Set thread prio)" - -text {* - In Paulson's inductive method, every protocol is defined by such a @{text "step"} - predicate. For instance, the predicate @{text "step"} given above - defines the PIP protocol. So, it can also be called "PIP". -*} - -abbreviation - "PIP \ step" - - -text {* \noindent - For any protocol defined by a @{text "step"} predicate, - the fact that @{text "s"} is a legal state in - the protocol is expressed as: @{text "vt step s"}, where - the predicate @{text "vt"} can be defined as the following: - *} -inductive vt :: "state \ bool" - where - -- {* Empty list @{text "[]"} is a legal state in any protocol:*} - vt_nil[intro]: "vt []" | - -- {* - \begin{minipage}{0.9\textwidth} - If @{text "s"} a legal state of the protocol defined by predicate @{text "step"}, - and event @{text "e"} is allowed to happen under state @{text "s"} by the protocol - predicate @{text "step"}, then @{text "e#s"} is a new legal state rendered by the - happening of @{text "e"}: - \end{minipage} - *} - vt_cons[intro]: "\vt s; step s e\ \ vt (e#s)" - -text {* \noindent - It is easy to see that the definition of @{text "vt"} is generic. It can be applied to - any specific protocol specified by a @{text "step"}-predicate to get the set of - legal states of that particular protocol. - *} - -text {* - The following are two very basic properties of @{text "vt"}. -*} - -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) - -text {* \noindent - The following two auxiliary functions @{text "the_cs"} and @{text "the_th"} are used to extract - critical resource and thread respectively out of RAG nodes. - *} -fun the_cs :: "node \ cs" - where "the_cs (Cs cs) = cs" - -fun the_th :: "node \ thread" - where "the_th (Th th) = th" - -text {* \noindent - The following predicate @{text "next_th"} describe the next thread to - take over when a critical resource is released. In @{text "next_th s th cs t"}, - @{text "th"} is the thread to release, @{text "t"} is the one to take over. - Notice how this definition is backed up by the @{text "release"} function and its use - in the @{text "V"}-branch of @{text "schs"} function. This @{text "next_th"} function - is not needed for the execution of PIP. It is introduced as an auxiliary function - to state lemmas. The correctness of this definition will be confirmed by - lemmas @{text "step_v_hold_inv"}, @{text " step_v_wait_inv"}, - @{text "step_v_get_hold"} and @{text "step_v_not_wait"}. - *} -definition next_th:: "state \ thread \ cs \ thread \ bool" - where "next_th s th cs t = (\ rest. wq s cs = th#rest \ rest \ [] \ - t = hd (SOME q. distinct q \ set q = set rest))" - -text {* \noindent - The aux function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"} - in list @{text "l"}: - *} -definition count :: "('a \ bool) \ 'a list \ nat" - where "count Q l = length (filter Q l)" - -text {* \noindent - The following observation @{text "cntP s"} returns the number of operation @{text "P"} happened - before reaching state @{text "s"}. - *} -definition cntP :: "state \ thread \ nat" - where "cntP s th = count (\ e. \ cs. e = P th cs) s" - -text {* \noindent - The following observation @{text "cntV s"} returns the number of operation @{text "V"} happened - before reaching state @{text "s"}. - *} -definition cntV :: "state \ thread \ nat" - where "cntV s th = count (\ e. \ cs. e = V th cs) s" - -definition "pvD s th = (if (th \ readys s \ th \ threads s) then 0 else (1::nat))" - -text {* @{text "the_preced"} is also the same as @{text "preced"}, the only - difference is the order of arguemts. *} -definition "the_preced s th = preced th s" - -text {* @{term "the_thread"} extracts thread out of RAG node. *} -fun the_thread :: "node \ thread" where - "the_thread (Th th) = th" - -text {* The following @{text "wRAG"} is the waiting sub-graph of @{text "RAG"}. *} -definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}" - -text {* The following @{text "hRAG"} is the holding sub-graph of @{text "RAG"}. *} -definition "hRAG (s::state) = {(Cs cs, Th th) | th cs. holding s th cs}" - -text {* - The following @{text "tRAG"} is the thread-graph derived from @{term "RAG"}. - It characterizes the dependency between threads when calculating current - precedences. It is defined as the composition of the above two sub-graphs, - names @{term "wRAG"} and @{term "hRAG"}. - *} -definition "tRAG s = wRAG s O hRAG s" - -text {* The following lemma splits @{term "RAG"} graph into the above two sub-graphs. *} -lemma RAG_split: "RAG s = (wRAG s \ hRAG s)" - by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv - s_holding_abv cs_RAG_def, auto) - -definition "cp_gen s x = - Max ((the_preced s \ the_thread) ` subtree (tRAG s) x)" - -(*<*) - -end -(*>*) - diff -r ed938e2246b9 -r 0525670d8e6a Precedence_ord.thy~ --- a/Precedence_ord.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,45 +0,0 @@ -header {* Order on product types *} - -theory Precedence_ord -imports Main -begin - -datatype precedence = Prc nat nat - -instantiation precedence :: order -begin - -definition - precedence_le_def: "x \ y \ (case (x, y) of - (Prc fx sx, Prc fy sy) \ - fx < fy \ (fx \ fy \ sy \ sx))" - -definition - precedence_less_def: "x < y \ (case (x, y) of - (Prc fx sx, Prc fy sy) \ - fx < fy \ (fx \ fy \ sy < sx))" - -instance -proof -qed (auto simp: precedence_le_def precedence_less_def - intro: order_trans split:precedence.splits) -end - -instance precedence :: preorder .. - -instance precedence :: linorder -proof -qed (auto simp: precedence_le_def precedence_less_def - intro: order_trans split:precedence.splits) - -instantiation precedence :: zero -begin - -definition Zero_precedence_def: - "0 = Prc 0 0" - -instance .. - -end - -end diff -r ed938e2246b9 -r 0525670d8e6a PrioG.thy~ --- a/PrioG.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1611 +0,0 @@ -theory PrioG -imports CpsG -begin - - -text {* - The following two auxiliary lemmas are used to reason about @{term Max}. -*} -lemma image_Max_eqI: - assumes "finite B" - and "b \ B" - and "\ x \ B. f x \ f b" - shows "Max (f ` B) = f b" - using assms - using Max_eqI by blast - -lemma image_Max_subset: - assumes "finite A" - and "B \ A" - and "a \ B" - and "Max (f ` A) = f a" - shows "Max (f ` B) = f a" -proof(rule image_Max_eqI) - show "finite B" - using assms(1) assms(2) finite_subset by auto -next - show "a \ B" using assms by simp -next - show "\x\B. f x \ f a" - by (metis Max_ge assms(1) assms(2) assms(4) - finite_imageI image_eqI subsetCE) -qed - -text {* - The following locale @{text "highest_gen"} sets the basic context for our - investigation: supposing thread @{text th} holds the highest @{term cp}-value - in state @{text s}, which means the task for @{text th} is the - most urgent. We want to show that - @{text th} is treated correctly by PIP, which means - @{text th} will not be blocked unreasonably by other less urgent - threads. -*} -locale highest_gen = - fixes s th prio tm - assumes vt_s: "vt s" - and threads_s: "th \ threads s" - and highest: "preced th s = Max ((cp s)`threads s)" - -- {* The internal structure of @{term th}'s precedence is exposed:*} - and preced_th: "preced th s = Prc prio tm" - --- {* @{term s} is a valid trace, so it will inherit all results derived for - a valid trace: *} -sublocale highest_gen < vat_s: valid_trace "s" - by (unfold_locales, insert vt_s, simp) - -context highest_gen -begin - -text {* - @{term tm} is the time when the precedence of @{term th} is set, so - @{term tm} must be a valid moment index into @{term s}. -*} -lemma lt_tm: "tm < length s" - by (insert preced_tm_lt[OF threads_s preced_th], simp) - -text {* - Since @{term th} holds the highest precedence and @{text "cp"} - is the highest precedence of all threads in the sub-tree of - @{text "th"} and @{text th} is among these threads, - its @{term cp} must equal to its precedence: -*} -lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R") -proof - - have "?L \ ?R" - by (unfold highest, rule Max_ge, - auto simp:threads_s finite_threads) - moreover have "?R \ ?L" - by (unfold vat_s.cp_rec, rule Max_ge, - auto simp:the_preced_def vat_s.fsbttRAGs.finite_children) - ultimately show ?thesis by auto -qed - -lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)" - using eq_cp_s_th highest max_cp_eq the_preced_def by presburger - - -lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)" - by (fold eq_cp_s_th, unfold highest_cp_preced, simp) - -lemma highest': "cp s th = Max (cp s ` threads s)" - by (simp add: eq_cp_s_th highest) - -end - -locale extend_highest_gen = highest_gen + - fixes t - assumes vt_t: "vt (t@s)" - and create_low: "Create th' prio' \ set t \ prio' \ prio" - and set_diff_low: "Set th' prio' \ set t \ th' \ th \ prio' \ prio" - and exit_diff: "Exit th' \ set t \ th' \ th" - -sublocale extend_highest_gen < vat_t: valid_trace "t@s" - by (unfold_locales, insert vt_t, simp) - -lemma step_back_vt_app: - assumes vt_ts: "vt (t@s)" - shows "vt s" -proof - - from vt_ts show ?thesis - proof(induct t) - case Nil - from Nil show ?case by auto - next - case (Cons e t) - assume ih: " vt (t @ s) \ vt s" - and vt_et: "vt ((e # t) @ s)" - show ?case - proof(rule ih) - show "vt (t @ s)" - proof(rule step_back_vt) - from vt_et show "vt (e # t @ s)" by simp - qed - qed - qed -qed - -(* locale red_extend_highest_gen = extend_highest_gen + - fixes i::nat -*) - -(* -sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)" - apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric]) - apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp) - by (unfold highest_gen_def, auto dest:step_back_vt_app) -*) - -context extend_highest_gen -begin - - lemma ind [consumes 0, case_names Nil Cons, induct type]: - assumes - h0: "R []" - and h2: "\ e t. \vt (t@s); step (t@s) e; - extend_highest_gen s th prio tm t; - extend_highest_gen s th prio tm (e#t); R t\ \ R (e#t)" - shows "R t" -proof - - from vt_t extend_highest_gen_axioms show ?thesis - proof(induct t) - from h0 show "R []" . - next - case (Cons e t') - assume ih: "\vt (t' @ s); extend_highest_gen s th prio tm t'\ \ R t'" - and vt_e: "vt ((e # t') @ s)" - and et: "extend_highest_gen s th prio tm (e # t')" - from vt_e and step_back_step have stp: "step (t'@s) e" by auto - from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto - show ?case - proof(rule h2 [OF vt_ts stp _ _ _ ]) - show "R t'" - proof(rule ih) - from et show ext': "extend_highest_gen s th prio tm t'" - by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) - next - from vt_ts show "vt (t' @ s)" . - qed - next - from et show "extend_highest_gen s th prio tm (e # t')" . - next - from et show ext': "extend_highest_gen s th prio tm t'" - by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) - qed - qed -qed - - -lemma th_kept: "th \ threads (t @ s) \ - preced th (t@s) = preced th s" (is "?Q t") -proof - - show ?thesis - proof(induct rule:ind) - case Nil - from threads_s - show ?case - by auto - next - case (Cons e t) - interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto - interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto - show ?case - proof(cases e) - case (Create thread prio) - show ?thesis - proof - - from Cons and Create have "step (t@s) (Create thread prio)" by auto - hence "th \ thread" - proof(cases) - case thread_create - with Cons show ?thesis by auto - qed - hence "preced th ((e # t) @ s) = preced th (t @ s)" - by (unfold Create, auto simp:preced_def) - moreover note Cons - ultimately show ?thesis - by (auto simp:Create) - qed - next - case (Exit thread) - from h_e.exit_diff and Exit - have neq_th: "thread \ th" by auto - with Cons - show ?thesis - by (unfold Exit, auto simp:preced_def) - next - case (P thread cs) - with Cons - show ?thesis - by (auto simp:P preced_def) - next - case (V thread cs) - with Cons - show ?thesis - by (auto simp:V preced_def) - next - case (Set thread prio') - show ?thesis - proof - - from h_e.set_diff_low and Set - have "th \ thread" by auto - hence "preced th ((e # t) @ s) = preced th (t @ s)" - by (unfold Set, auto simp:preced_def) - moreover note Cons - ultimately show ?thesis - by (auto simp:Set) - qed - qed - qed -qed - -text {* - According to @{thm th_kept}, thread @{text "th"} has its living status - and precedence kept along the way of @{text "t"}. The following lemma - shows that this preserved precedence of @{text "th"} remains as the highest - along the way of @{text "t"}. - - The proof goes by induction over @{text "t"} using the specialized - induction rule @{thm ind}, followed by case analysis of each possible - operations of PIP. All cases follow the same pattern rendered by the - generalized introduction rule @{thm "image_Max_eqI"}. - - The very essence is to show that precedences, no matter whether they - are newly introduced or modified, are always lower than the one held - by @{term "th"}, which by @{thm th_kept} is preserved along the way. -*} -lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s" -proof(induct rule:ind) - case Nil - from highest_preced_thread - show ?case by simp -next - case (Cons e t) - interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto - interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto - show ?case - proof(cases e) - case (Create thread prio') - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - -- {* The following is the common pattern of each branch of the case analysis. *} - -- {* The major part is to show that @{text "th"} holds the highest precedence: *} - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume "x \ ?A" - hence "x = thread \ x \ threads (t@s)" by (auto simp:Create) - thus "?f x \ ?f th" - proof - assume "x = thread" - thus ?thesis - apply (simp add:Create the_preced_def preced_def, fold preced_def) - using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 - preced_th by force - next - assume h: "x \ threads (t @ s)" - from Cons(2)[unfolded Create] - have "x \ thread" using h by (cases, auto) - hence "?f x = the_preced (t@s) x" - by (simp add:Create the_preced_def preced_def) - hence "?f x \ Max (the_preced (t@s) ` threads (t@s))" - by (simp add: h_t.finite_threads h) - also have "... = ?f th" - by (metis Cons.hyps(5) h_e.th_kept the_preced_def) - finally show ?thesis . - qed - qed - qed - -- {* The minor part is to show that the precedence of @{text "th"} - equals to preserved one, given by the foregoing lemma @{thm th_kept} *} - also have "... = ?t" using h_e.th_kept the_preced_def by auto - -- {* Then it follows trivially that the precedence preserved - for @{term "th"} remains the maximum of all living threads along the way. *} - finally show ?thesis . - qed - next - case (Exit thread) - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume "x \ ?A" - hence "x \ threads (t@s)" by (simp add: Exit) - hence "?f x \ Max (?f ` threads (t@s))" - by (simp add: h_t.finite_threads) - also have "... \ ?f th" - apply (simp add:Exit the_preced_def preced_def, fold preced_def) - using Cons.hyps(5) h_t.th_kept the_preced_def by auto - finally show "?f x \ ?f th" . - qed - qed - also have "... = ?t" using h_e.th_kept the_preced_def by auto - finally show ?thesis . - qed - next - case (P thread cs) - with Cons - show ?thesis by (auto simp:preced_def the_preced_def) - next - case (V thread cs) - with Cons - show ?thesis by (auto simp:preced_def the_preced_def) - next - case (Set thread prio') - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume h: "x \ ?A" - show "?f x \ ?f th" - proof(cases "x = thread") - case True - moreover have "the_preced (Set thread prio' # t @ s) thread \ the_preced (t @ s) th" - proof - - have "the_preced (t @ s) th = Prc prio tm" - using h_t.th_kept preced_th by (simp add:the_preced_def) - moreover have "prio' \ prio" using Set h_e.set_diff_low by auto - ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def) - qed - ultimately show ?thesis - by (unfold Set, simp add:the_preced_def preced_def) - next - case False - then have "?f x = the_preced (t@s) x" - by (simp add:the_preced_def preced_def Set) - also have "... \ Max (the_preced (t@s) ` threads (t@s))" - using Set h h_t.finite_threads by auto - also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) - finally show ?thesis . - qed - qed - qed - also have "... = ?t" using h_e.th_kept the_preced_def by auto - finally show ?thesis . - qed - qed -qed - -lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))" - by (insert th_kept max_kept, auto) - -text {* - The reason behind the following lemma is that: - Since @{term "cp"} is defined as the maximum precedence - of those threads contained in the sub-tree of node @{term "Th th"} - in @{term "RAG (t@s)"}, and all these threads are living threads, and - @{term "th"} is also among them, the maximum precedence of - them all must be the one for @{text "th"}. -*} -lemma th_cp_max_preced: - "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") -proof - - let ?f = "the_preced (t@s)" - have "?L = ?f th" - proof(unfold cp_alt_def, rule image_Max_eqI) - show "finite {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - proof - - have "{th'. Th th' \ subtree (RAG (t @ s)) (Th th)} = - the_thread ` {n . n \ subtree (RAG (t @ s)) (Th th) \ - (\ th'. n = Th th')}" - by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps) - moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) - ultimately show ?thesis by simp - qed - next - show "th \ {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - by (auto simp:subtree_def) - next - show "\x\{th'. Th th' \ subtree (RAG (t @ s)) (Th th)}. - the_preced (t @ s) x \ the_preced (t @ s) th" - proof - fix th' - assume "th' \ {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - hence "Th th' \ subtree (RAG (t @ s)) (Th th)" by auto - moreover have "... \ Field (RAG (t @ s)) \ {Th th}" - by (meson subtree_Field) - ultimately have "Th th' \ ..." by auto - hence "th' \ threads (t@s)" - proof - assume "Th th' \ {Th th}" - thus ?thesis using th_kept by auto - next - assume "Th th' \ Field (RAG (t @ s))" - thus ?thesis using vat_t.not_in_thread_isolated by blast - qed - thus "the_preced (t @ s) th' \ the_preced (t @ s) th" - by (metis Max_ge finite_imageI finite_threads image_eqI - max_kept th_kept the_preced_def) - qed - qed - also have "... = ?R" by (simp add: max_preced the_preced_def) - finally show ?thesis . -qed - -lemma th_cp_max[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th" - using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger - -lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))" - by (simp add: th_cp_max_preced) - -lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th" - using max_kept th_kept the_preced_def by auto - -lemma [simp]: "the_preced (t@s) th = preced th (t@s)" - using the_preced_def by auto - -lemma [simp]: "preced th (t@s) = preced th s" - by (simp add: th_kept) - -lemma [simp]: "cp s th = preced th s" - by (simp add: eq_cp_s_th) - -lemma th_cp_preced [simp]: "cp (t@s) th = preced th s" - by (fold max_kept, unfold th_cp_max_preced, simp) - -lemma preced_less: - assumes th'_in: "th' \ threads s" - and neq_th': "th' \ th" - shows "preced th' s < preced th s" - using assms -by (metis Max.coboundedI finite_imageI highest not_le order.trans - preced_linorder rev_image_eqI threads_s vat_s.finite_threads - vat_s.le_cp) - -section {* The `blocking thread` *} - -text {* - The purpose of PIP is to ensure that the most - urgent thread @{term th} is not blocked unreasonably. - Therefore, a clear picture of the blocking thread is essential - to assure people that the purpose is fulfilled. - - In this section, we are going to derive a series of lemmas - with finally give rise to a picture of the blocking thread. - - By `blocking thread`, we mean a thread in running state but - different from thread @{term th}. -*} - -text {* - The following lemmas shows that the @{term cp}-value - of the blocking thread @{text th'} equals to the highest - precedence in the whole system. -*} -lemma runing_preced_inversion: - assumes runing': "th' \ runing (t@s)" - shows "cp (t@s) th' = preced th s" (is "?L = ?R") -proof - - have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms - by (unfold runing_def, auto) - also have "\ = ?R" - by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) - finally show ?thesis . -qed - -text {* - - The following lemma shows how the counters for @{term "P"} and - @{term "V"} operations relate to the running threads in the states - @{term s} and @{term "t @ s"}. The lemma shows that if a thread's - @{term "P"}-count equals its @{term "V"}-count (which means it no - longer has any resource in its possession), it cannot be a running - thread. - - The proof is by contraction with the assumption @{text "th' \ th"}. - The key is the use of @{thm count_eq_dependants} to derive the - emptiness of @{text th'}s @{term dependants}-set from the balance of - its @{term P} and @{term V} counts. From this, it can be shown - @{text th'}s @{term cp}-value equals to its own precedence. - - On the other hand, since @{text th'} is running, by @{thm - runing_preced_inversion}, its @{term cp}-value equals to the - precedence of @{term th}. - - Combining the above two resukts we have that @{text th'} and @{term - th} have the same precedence. By uniqueness of precedences, we have - @{text "th' = th"}, which is in contradiction with the assumption - @{text "th' \ th"}. - -*} - -lemma eq_pv_blocked: (* ddd *) - assumes neq_th': "th' \ th" - and eq_pv: "cntP (t@s) th' = cntV (t@s) th'" - shows "th' \ runing (t@s)" -proof - assume otherwise: "th' \ runing (t@s)" - show False - proof - - have th'_in: "th' \ threads (t@s)" - using otherwise readys_threads runing_def by auto - have "th' = th" - proof(rule preced_unique) - -- {* The proof goes like this: - it is first shown that the @{term preced}-value of @{term th'} - equals to that of @{term th}, then by uniqueness - of @{term preced}-values (given by lemma @{thm preced_unique}), - @{term th'} equals to @{term th}: *} - show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R") - proof - - -- {* Since the counts of @{term th'} are balanced, the subtree - of it contains only itself, so, its @{term cp}-value - equals its @{term preced}-value: *} - have "?L = cp (t@s) th'" - by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp) - -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion}, - its @{term cp}-value equals @{term "preced th s"}, - which equals to @{term "?R"} by simplification: *} - also have "... = ?R" - thm runing_preced_inversion - using runing_preced_inversion[OF otherwise] by simp - finally show ?thesis . - qed - qed (auto simp: th'_in th_kept) - with `th' \ th` show ?thesis by simp - qed -qed - -text {* - The following lemma is the extrapolation of @{thm eq_pv_blocked}. - It says if a thread, different from @{term th}, - does not hold any resource at the very beginning, - it will keep hand-emptied in the future @{term "t@s"}. -*} -lemma eq_pv_persist: (* ddd *) - assumes neq_th': "th' \ th" - and eq_pv: "cntP s th' = cntV s th'" - shows "cntP (t@s) th' = cntV (t@s) th'" -proof(induction rule:ind) -- {* The proof goes by induction. *} - -- {* The nontrivial case is for the @{term Cons}: *} - case (Cons e t) - -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *} - interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp - interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp - show ?case - proof - - -- {* It can be proved that @{term cntP}-value of @{term th'} does not change - by the happening of event @{term e}: *} - have "cntP ((e#t)@s) th' = cntP (t@s) th'" - proof(rule ccontr) -- {* Proof by contradiction. *} - -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *} - assume otherwise: "cntP ((e # t) @ s) th' \ cntP (t @ s) th'" - -- {* Then the actor of @{term e} must be @{term th'} and @{term e} - must be a @{term P}-event: *} - hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv) - with vat_t.actor_inv[OF Cons(2)] - -- {* According to @{thm actor_inv}, @{term th'} must be running at - the moment @{term "t@s"}: *} - have "th' \ runing (t@s)" by (cases e, auto) - -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis - shows @{term th'} can not be running at moment @{term "t@s"}: *} - moreover have "th' \ runing (t@s)" - using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] . - -- {* Contradiction is finally derived: *} - ultimately show False by simp - qed - -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change - by the happening of event @{term e}: *} - -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *} - moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'" - proof(rule ccontr) -- {* Proof by contradiction. *} - assume otherwise: "cntV ((e # t) @ s) th' \ cntV (t @ s) th'" - hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv) - with vat_t.actor_inv[OF Cons(2)] - have "th' \ runing (t@s)" by (cases e, auto) - moreover have "th' \ runing (t@s)" - using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] . - ultimately show False by simp - qed - -- {* Finally, it can be shown that the @{term cntP} and @{term cntV} - value for @{term th'} are still in balance, so @{term th'} - is still hand-emptied after the execution of event @{term e}: *} - ultimately show ?thesis using Cons(5) by metis - qed -qed (auto simp:eq_pv) - -text {* - By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist}, - it can be derived easily that @{term th'} can not be running in the future: -*} -lemma eq_pv_blocked_persist: - assumes neq_th': "th' \ th" - and eq_pv: "cntP s th' = cntV s th'" - shows "th' \ runing (t@s)" - using assms - by (simp add: eq_pv_blocked eq_pv_persist) - -text {* - The following lemma shows the blocking thread @{term th'} - must hold some resource in the very beginning. -*} -lemma runing_cntP_cntV_inv: (* ddd *) - assumes is_runing: "th' \ runing (t@s)" - and neq_th': "th' \ th" - shows "cntP s th' > cntV s th'" - using assms -proof - - -- {* First, it can be shown that the number of @{term P} and - @{term V} operations can not be equal for thred @{term th'} *} - have "cntP s th' \ cntV s th'" - proof - -- {* The proof goes by contradiction, suppose otherwise: *} - assume otherwise: "cntP s th' = cntV s th'" - -- {* By applying @{thm eq_pv_blocked_persist} to this: *} - from eq_pv_blocked_persist[OF neq_th' otherwise] - -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *} - have "th' \ runing (t@s)" . - -- {* This is obvious in contradiction with assumption @{thm is_runing} *} - thus False using is_runing by simp - qed - -- {* However, the number of @{term V} is always less or equal to @{term P}: *} - moreover have "cntV s th' \ cntP s th'" using vat_s.cnp_cnv_cncs by auto - -- {* Thesis is finally derived by combining the these two results: *} - ultimately show ?thesis by auto -qed - - -text {* - The following lemmas shows the blocking thread @{text th'} must be live - at the very beginning, i.e. the moment (or state) @{term s}. - - The proof is a simple combination of the results above: -*} -lemma runing_threads_inv: - assumes runing': "th' \ runing (t@s)" - and neq_th': "th' \ th" - shows "th' \ threads s" -proof(rule ccontr) -- {* Proof by contradiction: *} - assume otherwise: "th' \ threads s" - have "th' \ runing (t @ s)" - proof - - from vat_s.cnp_cnv_eq[OF otherwise] - have "cntP s th' = cntV s th'" . - from eq_pv_blocked_persist[OF neq_th' this] - show ?thesis . - qed - with runing' show False by simp -qed - -text {* - The following lemma summarizes several foregoing - lemmas to give an overall picture of the blocking thread @{text "th'"}: -*} -lemma runing_inversion: (* ddd, one of the main lemmas to present *) - assumes runing': "th' \ runing (t@s)" - and neq_th: "th' \ th" - shows "th' \ threads s" - and "\detached s th'" - and "cp (t@s) th' = preced th s" -proof - - from runing_threads_inv[OF assms] - show "th' \ threads s" . -next - from runing_cntP_cntV_inv[OF runing' neq_th] - show "\detached s th'" using vat_s.detached_eq by simp -next - from runing_preced_inversion[OF runing'] - show "cp (t@s) th' = preced th s" . -qed - -section {* The existence of `blocking thread` *} - -text {* - Suppose @{term th} is not running, it is first shown that - there is a path in RAG leading from node @{term th} to another thread @{text "th'"} - in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}). - - Now, since @{term readys}-set is non-empty, there must be - one in it which holds the highest @{term cp}-value, which, by definition, - is the @{term runing}-thread. However, we are going to show more: this running thread - is exactly @{term "th'"}. - *} -lemma th_blockedE: (* ddd, the other main lemma to be presented: *) - assumes "th \ runing (t@s)" - obtains th' where "Th th' \ ancestors (RAG (t @ s)) (Th th)" - "th' \ runing (t@s)" -proof - - -- {* According to @{thm vat_t.th_chain_to_ready}, either - @{term "th"} is in @{term "readys"} or there is path leading from it to - one thread in @{term "readys"}. *} - have "th \ readys (t @ s) \ (\th'. th' \ readys (t @ s) \ (Th th, Th th') \ (RAG (t @ s))\<^sup>+)" - using th_kept vat_t.th_chain_to_ready by auto - -- {* However, @{term th} can not be in @{term readys}, because otherwise, since - @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *} - moreover have "th \ readys (t@s)" - using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto - -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in - term @{term readys}: *} - ultimately obtain th' where th'_in: "th' \ readys (t@s)" - and dp: "(Th th, Th th') \ (RAG (t @ s))\<^sup>+" by auto - -- {* We are going to show that this @{term th'} is running. *} - have "th' \ runing (t@s)" - proof - - -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *} - have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R") - proof - - have "?L = Max ((the_preced (t @ s) \ the_thread) ` subtree (tRAG (t @ s)) (Th th'))" - by (unfold cp_alt_def1, simp) - also have "... = (the_preced (t @ s) \ the_thread) (Th th)" - proof(rule image_Max_subset) - show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads) - next - show "subtree (tRAG (t @ s)) (Th th') \ Th ` threads (t @ s)" - by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) - next - show "Th th \ subtree (tRAG (t @ s)) (Th th')" using dp - by (unfold tRAG_subtree_eq, auto simp:subtree_def) - next - show "Max ((the_preced (t @ s) \ the_thread) ` Th ` threads (t @ s)) = - (the_preced (t @ s) \ the_thread) (Th th)" (is "Max ?L = _") - proof - - have "?L = the_preced (t @ s) ` threads (t @ s)" - by (unfold image_comp, rule image_cong, auto) - thus ?thesis using max_preced the_preced_def by auto - qed - qed - also have "... = ?R" - using th_cp_max th_cp_preced th_kept - the_preced_def vat_t.max_cp_readys_threads by auto - finally show ?thesis . - qed - -- {* Now, since @{term th'} holds the highest @{term cp} - and we have already show it is in @{term readys}, - it is @{term runing} by definition. *} - with `th' \ readys (t@s)` show ?thesis by (simp add: runing_def) - qed - -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *} - moreover have "Th th' \ ancestors (RAG (t @ s)) (Th th)" - using `(Th th, Th th') \ (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def) - ultimately show ?thesis using that by metis -qed - -text {* - Now it is easy to see there is always a thread to run by case analysis - on whether thread @{term th} is running: if the answer is Yes, the - the running thread is obviously @{term th} itself; otherwise, the running - thread is the @{text th'} given by lemma @{thm th_blockedE}. -*} -lemma live: "runing (t@s) \ {}" -proof(cases "th \ runing (t@s)") - case True thus ?thesis by auto -next - case False - thus ?thesis using th_blockedE by auto -qed - - -end -end -======= -theory Correctness -imports PIPBasics -begin - - -text {* - The following two auxiliary lemmas are used to reason about @{term Max}. -*} -lemma image_Max_eqI: - assumes "finite B" - and "b \ B" - and "\ x \ B. f x \ f b" - shows "Max (f ` B) = f b" - using assms - using Max_eqI by blast - -lemma image_Max_subset: - assumes "finite A" - and "B \ A" - and "a \ B" - and "Max (f ` A) = f a" - shows "Max (f ` B) = f a" -proof(rule image_Max_eqI) - show "finite B" - using assms(1) assms(2) finite_subset by auto -next - show "a \ B" using assms by simp -next - show "\x\B. f x \ f a" - by (metis Max_ge assms(1) assms(2) assms(4) - finite_imageI image_eqI subsetCE) -qed - -text {* - The following locale @{text "highest_gen"} sets the basic context for our - investigation: supposing thread @{text th} holds the highest @{term cp}-value - in state @{text s}, which means the task for @{text th} is the - most urgent. We want to show that - @{text th} is treated correctly by PIP, which means - @{text th} will not be blocked unreasonably by other less urgent - threads. -*} -locale highest_gen = - fixes s th prio tm - assumes vt_s: "vt s" - and threads_s: "th \ threads s" - and highest: "preced th s = Max ((cp s)`threads s)" - -- {* The internal structure of @{term th}'s precedence is exposed:*} - and preced_th: "preced th s = Prc prio tm" - --- {* @{term s} is a valid trace, so it will inherit all results derived for - a valid trace: *} -sublocale highest_gen < vat_s: valid_trace "s" - by (unfold_locales, insert vt_s, simp) - -context highest_gen -begin - -text {* - @{term tm} is the time when the precedence of @{term th} is set, so - @{term tm} must be a valid moment index into @{term s}. -*} -lemma lt_tm: "tm < length s" - by (insert preced_tm_lt[OF threads_s preced_th], simp) - -text {* - Since @{term th} holds the highest precedence and @{text "cp"} - is the highest precedence of all threads in the sub-tree of - @{text "th"} and @{text th} is among these threads, - its @{term cp} must equal to its precedence: -*} -lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R") -proof - - have "?L \ ?R" - by (unfold highest, rule Max_ge, - auto simp:threads_s finite_threads) - moreover have "?R \ ?L" - by (unfold vat_s.cp_rec, rule Max_ge, - auto simp:the_preced_def vat_s.fsbttRAGs.finite_children) - ultimately show ?thesis by auto -qed - -lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)" - using eq_cp_s_th highest max_cp_eq the_preced_def by presburger - - -lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)" - by (fold eq_cp_s_th, unfold highest_cp_preced, simp) - -lemma highest': "cp s th = Max (cp s ` threads s)" - by (simp add: eq_cp_s_th highest) - -end - -locale extend_highest_gen = highest_gen + - fixes t - assumes vt_t: "vt (t@s)" - and create_low: "Create th' prio' \ set t \ prio' \ prio" - and set_diff_low: "Set th' prio' \ set t \ th' \ th \ prio' \ prio" - and exit_diff: "Exit th' \ set t \ th' \ th" - -sublocale extend_highest_gen < vat_t: valid_trace "t@s" - by (unfold_locales, insert vt_t, simp) - -lemma step_back_vt_app: - assumes vt_ts: "vt (t@s)" - shows "vt s" -proof - - from vt_ts show ?thesis - proof(induct t) - case Nil - from Nil show ?case by auto - next - case (Cons e t) - assume ih: " vt (t @ s) \ vt s" - and vt_et: "vt ((e # t) @ s)" - show ?case - proof(rule ih) - show "vt (t @ s)" - proof(rule step_back_vt) - from vt_et show "vt (e # t @ s)" by simp - qed - qed - qed -qed - -(* locale red_extend_highest_gen = extend_highest_gen + - fixes i::nat -*) - -(* -sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)" - apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric]) - apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp) - by (unfold highest_gen_def, auto dest:step_back_vt_app) -*) - -context extend_highest_gen -begin - - lemma ind [consumes 0, case_names Nil Cons, induct type]: - assumes - h0: "R []" - and h2: "\ e t. \vt (t@s); step (t@s) e; - extend_highest_gen s th prio tm t; - extend_highest_gen s th prio tm (e#t); R t\ \ R (e#t)" - shows "R t" -proof - - from vt_t extend_highest_gen_axioms show ?thesis - proof(induct t) - from h0 show "R []" . - next - case (Cons e t') - assume ih: "\vt (t' @ s); extend_highest_gen s th prio tm t'\ \ R t'" - and vt_e: "vt ((e # t') @ s)" - and et: "extend_highest_gen s th prio tm (e # t')" - from vt_e and step_back_step have stp: "step (t'@s) e" by auto - from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto - show ?case - proof(rule h2 [OF vt_ts stp _ _ _ ]) - show "R t'" - proof(rule ih) - from et show ext': "extend_highest_gen s th prio tm t'" - by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) - next - from vt_ts show "vt (t' @ s)" . - qed - next - from et show "extend_highest_gen s th prio tm (e # t')" . - next - from et show ext': "extend_highest_gen s th prio tm t'" - by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) - qed - qed -qed - - -lemma th_kept: "th \ threads (t @ s) \ - preced th (t@s) = preced th s" (is "?Q t") -proof - - show ?thesis - proof(induct rule:ind) - case Nil - from threads_s - show ?case - by auto - next - case (Cons e t) - interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto - interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto - show ?case - proof(cases e) - case (Create thread prio) - show ?thesis - proof - - from Cons and Create have "step (t@s) (Create thread prio)" by auto - hence "th \ thread" - proof(cases) - case thread_create - with Cons show ?thesis by auto - qed - hence "preced th ((e # t) @ s) = preced th (t @ s)" - by (unfold Create, auto simp:preced_def) - moreover note Cons - ultimately show ?thesis - by (auto simp:Create) - qed - next - case (Exit thread) - from h_e.exit_diff and Exit - have neq_th: "thread \ th" by auto - with Cons - show ?thesis - by (unfold Exit, auto simp:preced_def) - next - case (P thread cs) - with Cons - show ?thesis - by (auto simp:P preced_def) - next - case (V thread cs) - with Cons - show ?thesis - by (auto simp:V preced_def) - next - case (Set thread prio') - show ?thesis - proof - - from h_e.set_diff_low and Set - have "th \ thread" by auto - hence "preced th ((e # t) @ s) = preced th (t @ s)" - by (unfold Set, auto simp:preced_def) - moreover note Cons - ultimately show ?thesis - by (auto simp:Set) - qed - qed - qed -qed - -text {* - According to @{thm th_kept}, thread @{text "th"} has its living status - and precedence kept along the way of @{text "t"}. The following lemma - shows that this preserved precedence of @{text "th"} remains as the highest - along the way of @{text "t"}. - - The proof goes by induction over @{text "t"} using the specialized - induction rule @{thm ind}, followed by case analysis of each possible - operations of PIP. All cases follow the same pattern rendered by the - generalized introduction rule @{thm "image_Max_eqI"}. - - The very essence is to show that precedences, no matter whether they - are newly introduced or modified, are always lower than the one held - by @{term "th"}, which by @{thm th_kept} is preserved along the way. -*} -lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s" -proof(induct rule:ind) - case Nil - from highest_preced_thread - show ?case by simp -next - case (Cons e t) - interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto - interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto - show ?case - proof(cases e) - case (Create thread prio') - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - -- {* The following is the common pattern of each branch of the case analysis. *} - -- {* The major part is to show that @{text "th"} holds the highest precedence: *} - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume "x \ ?A" - hence "x = thread \ x \ threads (t@s)" by (auto simp:Create) - thus "?f x \ ?f th" - proof - assume "x = thread" - thus ?thesis - apply (simp add:Create the_preced_def preced_def, fold preced_def) - using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 - preced_th by force - next - assume h: "x \ threads (t @ s)" - from Cons(2)[unfolded Create] - have "x \ thread" using h by (cases, auto) - hence "?f x = the_preced (t@s) x" - by (simp add:Create the_preced_def preced_def) - hence "?f x \ Max (the_preced (t@s) ` threads (t@s))" - by (simp add: h_t.finite_threads h) - also have "... = ?f th" - by (metis Cons.hyps(5) h_e.th_kept the_preced_def) - finally show ?thesis . - qed - qed - qed - -- {* The minor part is to show that the precedence of @{text "th"} - equals to preserved one, given by the foregoing lemma @{thm th_kept} *} - also have "... = ?t" using h_e.th_kept the_preced_def by auto - -- {* Then it follows trivially that the precedence preserved - for @{term "th"} remains the maximum of all living threads along the way. *} - finally show ?thesis . - qed - next - case (Exit thread) - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume "x \ ?A" - hence "x \ threads (t@s)" by (simp add: Exit) - hence "?f x \ Max (?f ` threads (t@s))" - by (simp add: h_t.finite_threads) - also have "... \ ?f th" - apply (simp add:Exit the_preced_def preced_def, fold preced_def) - using Cons.hyps(5) h_t.th_kept the_preced_def by auto - finally show "?f x \ ?f th" . - qed - qed - also have "... = ?t" using h_e.th_kept the_preced_def by auto - finally show ?thesis . - qed - next - case (P thread cs) - with Cons - show ?thesis by (auto simp:preced_def the_preced_def) - next - case (V thread cs) - with Cons - show ?thesis by (auto simp:preced_def the_preced_def) - next - case (Set thread prio') - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume h: "x \ ?A" - show "?f x \ ?f th" - proof(cases "x = thread") - case True - moreover have "the_preced (Set thread prio' # t @ s) thread \ the_preced (t @ s) th" - proof - - have "the_preced (t @ s) th = Prc prio tm" - using h_t.th_kept preced_th by (simp add:the_preced_def) - moreover have "prio' \ prio" using Set h_e.set_diff_low by auto - ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def) - qed - ultimately show ?thesis - by (unfold Set, simp add:the_preced_def preced_def) - next - case False - then have "?f x = the_preced (t@s) x" - by (simp add:the_preced_def preced_def Set) - also have "... \ Max (the_preced (t@s) ` threads (t@s))" - using Set h h_t.finite_threads by auto - also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) - finally show ?thesis . - qed - qed - qed - also have "... = ?t" using h_e.th_kept the_preced_def by auto - finally show ?thesis . - qed - qed -qed - -lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))" - by (insert th_kept max_kept, auto) - -text {* - The reason behind the following lemma is that: - Since @{term "cp"} is defined as the maximum precedence - of those threads contained in the sub-tree of node @{term "Th th"} - in @{term "RAG (t@s)"}, and all these threads are living threads, and - @{term "th"} is also among them, the maximum precedence of - them all must be the one for @{text "th"}. -*} -lemma th_cp_max_preced: - "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") -proof - - let ?f = "the_preced (t@s)" - have "?L = ?f th" - proof(unfold cp_alt_def, rule image_Max_eqI) - show "finite {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - proof - - have "{th'. Th th' \ subtree (RAG (t @ s)) (Th th)} = - the_thread ` {n . n \ subtree (RAG (t @ s)) (Th th) \ - (\ th'. n = Th th')}" - by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps) - moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) - ultimately show ?thesis by simp - qed - next - show "th \ {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - by (auto simp:subtree_def) - next - show "\x\{th'. Th th' \ subtree (RAG (t @ s)) (Th th)}. - the_preced (t @ s) x \ the_preced (t @ s) th" - proof - fix th' - assume "th' \ {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - hence "Th th' \ subtree (RAG (t @ s)) (Th th)" by auto - moreover have "... \ Field (RAG (t @ s)) \ {Th th}" - by (meson subtree_Field) - ultimately have "Th th' \ ..." by auto - hence "th' \ threads (t@s)" - proof - assume "Th th' \ {Th th}" - thus ?thesis using th_kept by auto - next - assume "Th th' \ Field (RAG (t @ s))" - thus ?thesis using vat_t.not_in_thread_isolated by blast - qed - thus "the_preced (t @ s) th' \ the_preced (t @ s) th" - by (metis Max_ge finite_imageI finite_threads image_eqI - max_kept th_kept the_preced_def) - qed - qed - also have "... = ?R" by (simp add: max_preced the_preced_def) - finally show ?thesis . -qed - -lemma th_cp_max[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th" - using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger - -lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))" - by (simp add: th_cp_max_preced) - -lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th" - using max_kept th_kept the_preced_def by auto - -lemma [simp]: "the_preced (t@s) th = preced th (t@s)" - using the_preced_def by auto - -lemma [simp]: "preced th (t@s) = preced th s" - by (simp add: th_kept) - -lemma [simp]: "cp s th = preced th s" - by (simp add: eq_cp_s_th) - -lemma th_cp_preced [simp]: "cp (t@s) th = preced th s" - by (fold max_kept, unfold th_cp_max_preced, simp) - -lemma preced_less: - assumes th'_in: "th' \ threads s" - and neq_th': "th' \ th" - shows "preced th' s < preced th s" - using assms -by (metis Max.coboundedI finite_imageI highest not_le order.trans - preced_linorder rev_image_eqI threads_s vat_s.finite_threads - vat_s.le_cp) - -section {* The `blocking thread` *} - -text {* - - The purpose of PIP is to ensure that the most urgent thread @{term - th} is not blocked unreasonably. Therefore, below, we will derive - properties of the blocking thread. By blocking thread, we mean a - thread in running state t @ s, but is different from thread @{term - th}. - - The first lemmas shows that the @{term cp}-value of the blocking - thread @{text th'} equals to the highest precedence in the whole - system. - -*} - -lemma runing_preced_inversion: - assumes runing': "th' \ runing (t @ s)" - shows "cp (t @ s) th' = preced th s" -proof - - have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))" - using assms by (unfold runing_def, auto) - also have "\ = preced th s" - by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) - finally show ?thesis . -qed - -text {* - - The next lemma shows how the counters for @{term "P"} and @{term - "V"} operations relate to the running threads in the states @{term - s} and @{term "t @ s"}: if a thread's @{term "P"}-count equals its - @{term "V"}-count (which means it no longer has any resource in its - possession), it cannot be a running thread. - - The proof is by contraction with the assumption @{text "th' \ th"}. - The key is the use of @{thm count_eq_dependants} to derive the - emptiness of @{text th'}s @{term dependants}-set from the balance of - its @{term P} and @{term V} counts. From this, it can be shown - @{text th'}s @{term cp}-value equals to its own precedence. - - On the other hand, since @{text th'} is running, by @{thm - runing_preced_inversion}, its @{term cp}-value equals to the - precedence of @{term th}. - - Combining the above two results we have that @{text th'} and @{term - th} have the same precedence. By uniqueness of precedences, we have - @{text "th' = th"}, which is in contradiction with the assumption - @{text "th' \ th"}. - -*} - -lemma eq_pv_blocked: (* ddd *) - assumes neq_th': "th' \ th" - and eq_pv: "cntP (t @ s) th' = cntV (t @ s) th'" - shows "th' \ runing (t @ s)" -proof - assume otherwise: "th' \ runing (t @ s)" - show False - proof - - have th'_in: "th' \ threads (t @ s)" - using otherwise readys_threads runing_def by auto - have "th' = th" - proof(rule preced_unique) - -- {* The proof goes like this: - it is first shown that the @{term preced}-value of @{term th'} - equals to that of @{term th}, then by uniqueness - of @{term preced}-values (given by lemma @{thm preced_unique}), - @{term th'} equals to @{term th}: *} - show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R") - proof - - -- {* Since the counts of @{term th'} are balanced, the subtree - of it contains only itself, so, its @{term cp}-value - equals its @{term preced}-value: *} - have "?L = cp (t @ s) th'" - by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp) - -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion}, - its @{term cp}-value equals @{term "preced th s"}, - which equals to @{term "?R"} by simplification: *} - also have "... = ?R" - using runing_preced_inversion[OF otherwise] by simp - finally show ?thesis . - qed - qed (auto simp: th'_in th_kept) - with `th' \ th` show ?thesis by simp - qed -qed - -text {* - The following lemma is the extrapolation of @{thm eq_pv_blocked}. - It says if a thread, different from @{term th}, - does not hold any resource at the very beginning, - it will keep hand-emptied in the future @{term "t@s"}. -*} -lemma eq_pv_persist: (* ddd *) - assumes neq_th': "th' \ th" - and eq_pv: "cntP s th' = cntV s th'" - shows "cntP (t @ s) th' = cntV (t @ s) th'" -proof(induction rule: ind) - -- {* The nontrivial case is for the @{term Cons}: *} - case (Cons e t) - -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *} - interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp - interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp - show ?case - proof - - -- {* It can be proved that @{term cntP}-value of @{term th'} does not change - by the happening of event @{term e}: *} - have "cntP ((e#t)@s) th' = cntP (t@s) th'" - proof(rule ccontr) -- {* Proof by contradiction. *} - -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *} - assume otherwise: "cntP ((e # t) @ s) th' \ cntP (t @ s) th'" - -- {* Then the actor of @{term e} must be @{term th'} and @{term e} - must be a @{term P}-event: *} - hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv) - with vat_t.actor_inv[OF Cons(2)] - -- {* According to @{thm actor_inv}, @{term th'} must be running at - the moment @{term "t@s"}: *} - have "th' \ runing (t@s)" by (cases e, auto) - -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis - shows @{term th'} can not be running at moment @{term "t@s"}: *} - moreover have "th' \ runing (t@s)" - using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] . - -- {* Contradiction is finally derived: *} - ultimately show False by simp - qed - -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change - by the happening of event @{term e}: *} - -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *} - moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'" - proof(rule ccontr) -- {* Proof by contradiction. *} - assume otherwise: "cntV ((e # t) @ s) th' \ cntV (t @ s) th'" - hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv) - with vat_t.actor_inv[OF Cons(2)] - have "th' \ runing (t@s)" by (cases e, auto) - moreover have "th' \ runing (t@s)" - using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] . - ultimately show False by simp - qed - -- {* Finally, it can be shown that the @{term cntP} and @{term cntV} - value for @{term th'} are still in balance, so @{term th'} - is still hand-emptied after the execution of event @{term e}: *} - ultimately show ?thesis using Cons(5) by metis - qed -qed (auto simp:eq_pv) - -text {* - - By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist}, it can - be derived easily that @{term th'} can not be running in the future: - -*} - -lemma eq_pv_blocked_persist: - assumes neq_th': "th' \ th" - and eq_pv: "cntP s th' = cntV s th'" - shows "th' \ runing (t @ s)" - using assms - by (simp add: eq_pv_blocked eq_pv_persist) - -text {* - - The following lemma shows the blocking thread @{term th'} must hold - some resource in the very beginning. - -*} - -lemma runing_cntP_cntV_inv: (* ddd *) - assumes is_runing: "th' \ runing (t @ s)" - and neq_th': "th' \ th" - shows "cntP s th' > cntV s th'" - using assms -proof - - -- {* First, it can be shown that the number of @{term P} and - @{term V} operations can not be equal for thred @{term th'} *} - have "cntP s th' \ cntV s th'" - proof - -- {* The proof goes by contradiction, suppose otherwise: *} - assume otherwise: "cntP s th' = cntV s th'" - -- {* By applying @{thm eq_pv_blocked_persist} to this: *} - from eq_pv_blocked_persist[OF neq_th' otherwise] - -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *} - have "th' \ runing (t@s)" . - -- {* This is obvious in contradiction with assumption @{thm is_runing} *} - thus False using is_runing by simp - qed - -- {* However, the number of @{term V} is always less or equal to @{term P}: *} - moreover have "cntV s th' \ cntP s th'" using vat_s.cnp_cnv_cncs by auto - -- {* Thesis is finally derived by combining the these two results: *} - ultimately show ?thesis by auto -qed - - -text {* - - The following lemmas shows the blocking thread @{text th'} must be - live at the very beginning, i.e. the moment (or state) @{term s}. - The proof is a simple combination of the results above: - -*} - -lemma runing_threads_inv: - assumes runing': "th' \ runing (t@s)" - and neq_th': "th' \ th" - shows "th' \ threads s" -proof(rule ccontr) -- {* Proof by contradiction: *} - assume otherwise: "th' \ threads s" - have "th' \ runing (t @ s)" - proof - - from vat_s.cnp_cnv_eq[OF otherwise] - have "cntP s th' = cntV s th'" . - from eq_pv_blocked_persist[OF neq_th' this] - show ?thesis . - qed - with runing' show False by simp -qed - -text {* - - The following lemma summarises the above lemmas to give an overall - characterisationof the blocking thread @{text "th'"}: - -*} - -lemma runing_inversion: (* ddd, one of the main lemmas to present *) - assumes runing': "th' \ runing (t@s)" - and neq_th: "th' \ th" - shows "th' \ threads s" - and "\detached s th'" - and "cp (t@s) th' = preced th s" -proof - - from runing_threads_inv[OF assms] - show "th' \ threads s" . -next - from runing_cntP_cntV_inv[OF runing' neq_th] - show "\detached s th'" using vat_s.detached_eq by simp -next - from runing_preced_inversion[OF runing'] - show "cp (t@s) th' = preced th s" . -qed - - -section {* The existence of `blocking thread` *} - -text {* - - Suppose @{term th} is not running, it is first shown that there is a - path in RAG leading from node @{term th} to another thread @{text - "th'"} in the @{term readys}-set (So @{text "th'"} is an ancestor of - @{term th}}). - - Now, since @{term readys}-set is non-empty, there must be one in it - which holds the highest @{term cp}-value, which, by definition, is - the @{term runing}-thread. However, we are going to show more: this - running thread is exactly @{term "th'"}. - -*} - -lemma th_blockedE: (* ddd, the other main lemma to be presented: *) - assumes "th \ runing (t@s)" - obtains th' where "Th th' \ ancestors (RAG (t @ s)) (Th th)" - "th' \ runing (t@s)" -proof - - -- {* According to @{thm vat_t.th_chain_to_ready}, either - @{term "th"} is in @{term "readys"} or there is path leading from it to - one thread in @{term "readys"}. *} - have "th \ readys (t @ s) \ (\th'. th' \ readys (t @ s) \ (Th th, Th th') \ (RAG (t @ s))\<^sup>+)" - using th_kept vat_t.th_chain_to_ready by auto - -- {* However, @{term th} can not be in @{term readys}, because otherwise, since - @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *} - moreover have "th \ readys (t@s)" - using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto - -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in - term @{term readys}: *} - ultimately obtain th' where th'_in: "th' \ readys (t@s)" - and dp: "(Th th, Th th') \ (RAG (t @ s))\<^sup>+" by auto - -- {* We are going to show that this @{term th'} is running. *} - have "th' \ runing (t@s)" - proof - - -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *} - have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R") - proof - - have "?L = Max ((the_preced (t @ s) \ the_thread) ` subtree (tRAG (t @ s)) (Th th'))" - by (unfold cp_alt_def1, simp) - also have "... = (the_preced (t @ s) \ the_thread) (Th th)" - proof(rule image_Max_subset) - show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads) - next - show "subtree (tRAG (t @ s)) (Th th') \ Th ` threads (t @ s)" - by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) - next - show "Th th \ subtree (tRAG (t @ s)) (Th th')" using dp - by (unfold tRAG_subtree_eq, auto simp:subtree_def) - next - show "Max ((the_preced (t @ s) \ the_thread) ` Th ` threads (t @ s)) = - (the_preced (t @ s) \ the_thread) (Th th)" (is "Max ?L = _") - proof - - have "?L = the_preced (t @ s) ` threads (t @ s)" - by (unfold image_comp, rule image_cong, auto) - thus ?thesis using max_preced the_preced_def by auto - qed - qed - also have "... = ?R" - using th_cp_max th_cp_preced th_kept - the_preced_def vat_t.max_cp_readys_threads by auto - finally show ?thesis . - qed - -- {* Now, since @{term th'} holds the highest @{term cp} - and we have already show it is in @{term readys}, - it is @{term runing} by definition. *} - with `th' \ readys (t@s)` show ?thesis by (simp add: runing_def) - qed - -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *} - moreover have "Th th' \ ancestors (RAG (t @ s)) (Th th)" - using `(Th th, Th th') \ (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def) - ultimately show ?thesis using that by metis -qed - -text {* - - Now it is easy to see there is always a thread to run by case - analysis on whether thread @{term th} is running: if the answer is - yes, the the running thread is obviously @{term th} itself; - otherwise, the running thread is the @{text th'} given by lemma - @{thm th_blockedE}. - -*} - -lemma live: "runing (t@s) \ {}" -proof(cases "th \ runing (t@s)") - case True thus ?thesis by auto -next - case False - thus ?thesis using th_blockedE by auto -qed - - -end -end diff -r ed938e2246b9 -r 0525670d8e6a PrioGDef.thy~ --- a/PrioGDef.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,616 +0,0 @@ -chapter {* Definitions *} -(*<*) -theory PrioGDef -imports Precedence_ord Moment -begin -(*>*) - -text {* - In this section, the formal model of Priority Inheritance Protocol (PIP) is presented. - The model is based on Paulson's inductive protocol verification method, where - the state of the system is modelled as a list of events happened so far with the latest - event put at the head. -*} - -text {* - To define events, the identifiers of {\em threads}, - {\em priority} and {\em critical resources } (abbreviated as @{text "cs"}) - need to be represented. All three are represetned using standard - Isabelle/HOL type @{typ "nat"}: -*} - -type_synonym thread = nat -- {* Type for thread identifiers. *} -type_synonym priority = nat -- {* Type for priorities. *} -type_synonym cs = nat -- {* Type for critical sections (or critical resources). *} - -text {* - \noindent - The abstraction of Priority Inheritance Protocol (PIP) is set at the system call level. - Every system call is represented as an event. The format of events is defined - defined as follows: - *} - -datatype event = - Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *} - Exit thread | -- {* Thread @{text "thread"} finishing its execution. *} - P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *} - V thread cs | -- {* Thread @{text "thread"} releasing critical resource @{text "cs"}. *} - Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *} - - -text {* - As mentioned earlier, in Paulson's inductive method, the states of system are represented as lists of events, - which is defined by the following type @{text "state"}: - *} -type_synonym state = "event list" - - -text {* -\noindent - Resource Allocation Graph (RAG for short) is used extensively in our formal analysis. - The following type @{text "node"} is used to represent nodes in RAG. - *} -datatype node = - Th "thread" | -- {* Node for thread. *} - Cs "cs" -- {* Node for critical resource. *} - -text {* - \noindent - The following function - @{text "threads"} is used to calculate the set of live threads (@{text "threads s"}) - in state @{text "s"}. - *} -fun threads :: "state \ thread set" - where - -- {* At the start of the system, the set of threads is empty: *} - "threads [] = {}" | - -- {* New thread is added to the @{text "threads"}: *} - "threads (Create thread prio#s) = {thread} \ threads s" | - -- {* Finished thread is removed: *} - "threads (Exit thread # s) = (threads s) - {thread}" | - -- {* Other kind of events does not affect the value of @{text "threads"}: *} - "threads (e#s) = threads s" - -text {* - \noindent - The function @{text "threads"} defined above is one of - the so called {\em observation function}s which forms - the very basis of Paulson's inductive protocol verification method. - Each observation function {\em observes} one particular aspect (or attribute) - of the system. For example, the attribute observed by @{text "threads s"} - is the set of threads living in state @{text "s"}. - The protocol being modelled - The decision made the protocol being modelled is based on the {\em observation}s - returned by {\em observation function}s. Since {\observation function}s forms - the very basis on which Paulson's inductive method is based, there will be - a lot of such observation functions introduced in the following. In fact, any function - which takes event list as argument is a {\em observation function}. - *} - -text {* \noindent - Observation @{text "priority th s"} is - the {\em original priority} of thread @{text "th"} in state @{text "s"}. - The {\em original priority} is the priority - assigned to a thread when it is created or when it is reset by system call - (represented by event @{text "Set thread priority"}). -*} - -fun priority :: "thread \ state \ priority" - where - -- {* @{text "0"} is assigned to threads which have never been created: *} - "priority thread [] = 0" | - "priority thread (Create thread' prio#s) = - (if thread' = thread then prio else priority thread s)" | - "priority thread (Set thread' prio#s) = - (if thread' = thread then prio else priority thread s)" | - "priority thread (e#s) = priority thread s" - -text {* - \noindent - Observation @{text "last_set th s"} is the last time when the priority of thread @{text "th"} is set, - observed from state @{text "s"}. - The time in the system is measured by the number of events happened so far since the very beginning. -*} -fun last_set :: "thread \ state \ nat" - where - "last_set thread [] = 0" | - "last_set thread ((Create thread' prio)#s) = - (if (thread = thread') then length s else last_set thread s)" | - "last_set thread ((Set thread' prio)#s) = - (if (thread = thread') then length s else last_set thread s)" | - "last_set thread (_#s) = last_set thread s" - -text {* - \noindent - The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of - a thread is the combination of its {\em original priority} and {\em time} the priority is set. - The intention is to discriminate threads with the same priority by giving threads whose priority - is assigned earlier higher precedences, becasue such threads are more urgent to finish. - This explains the following definition: - *} -definition preced :: "thread \ state \ precedence" - where "preced thread s \ Prc (priority thread s) (last_set thread s)" - - -text {* - \noindent - A number of important notions in PIP are represented as the following functions, - defined in terms of the waiting queues of the system, where the waiting queues - , as a whole, is represented by the @{text "wq"} argument of every notion function. - The @{text "wq"} argument is itself a functions which maps every critical resource - @{text "cs"} to the list of threads which are holding or waiting for it. - The thread at the head of this list is designated as the thread which is current - holding the resrouce, which is slightly different from tradition where - all threads in the waiting queue are considered as waiting for the resource. - *} - -consts - holding :: "'b \ thread \ cs \ bool" - waiting :: "'b \ thread \ cs \ bool" - RAG :: "'b \ (node \ node) set" - dependants :: "'b \ thread \ thread set" - -defs (overloaded) - -- {* - \begin{minipage}{0.9\textwidth} - This meaning of @{text "wq"} is reflected in the following definition of @{text "holding wq th cs"}, - where @{text "holding wq th cs"} means thread @{text "th"} is holding the critical - resource @{text "cs"}. This decision is based on @{text "wq"}. - \end{minipage} - *} - - cs_holding_def: - "holding wq thread cs \ (thread \ set (wq cs) \ thread = hd (wq cs))" - -- {* - \begin{minipage}{0.9\textwidth} - In accordance with the definition of @{text "holding wq th cs"}, - a thread @{text "th"} is considered waiting for @{text "cs"} if - it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head. - This is reflected in the definition of @{text "waiting wq th cs"} as follows: - \end{minipage} - *} - cs_waiting_def: - "waiting wq thread cs \ (thread \ set (wq cs) \ thread \ hd (wq cs))" - -- {* - \begin{minipage}{0.9\textwidth} - @{text "RAG wq"} generates RAG (a binary relations on @{text "node"}) - out of waiting queues of the system (represented by the @{text "wq"} argument): - \end{minipage} - *} - cs_RAG_def: - "RAG (wq::cs \ thread list) \ - {(Th th, Cs cs) | th cs. waiting wq th cs} \ {(Cs cs, Th th) | cs th. holding wq th cs}" - -- {* - \begin{minipage}{0.9\textwidth} - The following @{text "dependants wq th"} represents the set of threads which are RAGing on - thread @{text "th"} in Resource Allocation Graph @{text "RAG wq"}. - Here, "RAGing" means waiting directly or indirectly on the critical resource. - \end{minipage} - *} - cs_dependants_def: - "dependants (wq::cs \ thread list) th \ {th' . (Th th', Th th) \ (RAG wq)^+}" - - -text {* \noindent - The following - @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under - state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of - Priority Inheritance that the {\em current precedence} of a thread is the precedence - inherited from the maximum of all its dependants, i.e. the threads which are waiting - directly or indirectly waiting for some resources from it. If no such thread exits, - @{text "th"}'s {\em current precedence} equals its original precedence, i.e. - @{text "preced th s"}. - *} - -definition cpreced :: "(cs \ thread list) \ state \ thread \ precedence" - where "cpreced wq s = (\th. Max ((\th'. preced th' s) ` ({th} \ dependants wq th)))" - -text {* - Notice that the current precedence (@{text "cpreced"}) of one thread @{text "th"} can be boosted - (becoming larger than its own precedence) by those threads in - the @{text "dependants wq th"}-set. If one thread get boosted, we say - it inherits the priority (or, more precisely, the precedence) of - its dependants. This is how the word "Inheritance" in - Priority Inheritance Protocol comes. -*} - -(*<*) -lemma - cpreced_def2: - "cpreced wq s th \ Max ({preced th s} \ {preced th' s | th'. th' \ dependants wq th})" - unfolding cpreced_def image_def - apply(rule eq_reflection) - apply(rule_tac f="Max" in arg_cong) - by (auto) -(*>*) - - -text {* \noindent - Assuming @{text "qs"} be the waiting queue of a critical resource, - the following abbreviation "release qs" is the waiting queue after the thread - holding the resource (which is thread at the head of @{text "qs"}) released - the resource: -*} -abbreviation - "release qs \ case qs of - [] => [] - | (_#qs') => (SOME q. distinct q \ set q = set qs')" -text {* \noindent - It can be seen from the definition that the thread at the head of @{text "qs"} is removed - from the return value, and the value @{term "q"} is an reordering of @{text "qs'"}, the - tail of @{text "qs"}. Through this reordering, one of the waiting threads (those in @{text "qs'"} } - is chosen nondeterministically to be the head of the new queue @{text "q"}. - Therefore, this thread is the one who takes over the resource. This is a little better different - from common sense that the thread who comes the earliest should take over. - The intention of this definition is to show that the choice of which thread to take over the - release resource does not affect the correctness of the PIP protocol. -*} - -text {* - The data structure used by the operating system for scheduling is referred to as - {\em schedule state}. It is represented as a record consisting of - a function assigning waiting queue to resources - (to be used as the @{text "wq"} argument in @{text "holding"}, @{text "waiting"} - and @{text "RAG"}, etc) and a function assigning precedence to threads: - *} - -record schedule_state = - wq_fun :: "cs \ thread list" -- {* The function assigning waiting queue. *} - cprec_fun :: "thread \ precedence" -- {* The function assigning precedence. *} - -text {* \noindent - The following two abbreviations (@{text "all_unlocked"} and @{text "initial_cprec"}) - are used to set the initial values of the @{text "wq_fun"} @{text "cprec_fun"} fields - respectively of the @{text "schedule_state"} record by the following function @{text "sch"}, - which is used to calculate the system's {\em schedule state}. - - Since there is no thread at the very beginning to make request, all critical resources - are free (or unlocked). This status is represented by the abbreviation - @{text "all_unlocked"}. - *} -abbreviation - "all_unlocked \ \_::cs. ([]::thread list)" - - -text {* \noindent - The initial current precedence for a thread can be anything, because there is no thread then. - We simply assume every thread has precedence @{text "Prc 0 0"}. - *} - -abbreviation - "initial_cprec \ \_::thread. Prc 0 0" - - -text {* \noindent - The following function @{text "schs"} is used to calculate the system's schedule state @{text "schs s"} - out of the current system state @{text "s"}. It is the central function to model Priority Inheritance: - *} -fun schs :: "state \ schedule_state" - where - -- {* - \begin{minipage}{0.9\textwidth} - Setting the initial value of the @{text "schedule_state"} record (see the explanations above). - \end{minipage} - *} - "schs [] = (| wq_fun = all_unlocked, cprec_fun = initial_cprec |)" | - - -- {* - \begin{minipage}{0.9\textwidth} - \begin{enumerate} - \item @{text "ps"} is the schedule state of last moment. - \item @{text "pwq"} is the waiting queue function of last moment. - \item @{text "pcp"} is the precedence function of last moment (NOT USED). - \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement: - \begin{enumerate} - \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to - the end of @{text "cs"}'s waiting queue. - \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state, - @{text "th'"} must equal to @{text "thread"}, - because @{text "thread"} is the one currently holding @{text "cs"}. - The case @{text "[] \ []"} may never be executed in a legal state. - the @{text "(SOME q. distinct q \ set q = set qs)"} is used to choose arbitrarily one - thread in waiting to take over the released resource @{text "cs"}. In our representation, - this amounts to rearrange elements in waiting queue, so that one of them is put at the head. - \item For other happening event, the schedule state just does not change. - \end{enumerate} - \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue - function. The RAGency of precedence function on waiting queue function is the reason to - put them in the same record so that they can evolve together. - \end{enumerate} - - - The calculation of @{text "cprec_fun"} depends on the value of @{text "wq_fun"}. - Therefore, in the following cases, @{text "wq_fun"} is always calculated first, in - the name of @{text "wq"} (if @{text "wq_fun"} is not changed - by the happening event) or @{text "new_wq"} (if the value of @{text "wq_fun"} is changed). - \end{minipage} - *} - "schs (Create th prio # s) = - (let wq = wq_fun (schs s) in - (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))" -| "schs (Exit th # s) = - (let wq = wq_fun (schs s) in - (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))" -| "schs (Set th prio # s) = - (let wq = wq_fun (schs s) in - (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))" - -- {* - \begin{minipage}{0.9\textwidth} - Different from the forth coming cases, the @{text "wq_fun"} field of the schedule state - is changed. So, the new value is calculated first, in the name of @{text "new_wq"}. - \end{minipage} - *} -| "schs (P th cs # s) = - (let wq = wq_fun (schs s) in - let new_wq = wq(cs := (wq cs @ [th])) in - (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))" -| "schs (V th cs # s) = - (let wq = wq_fun (schs s) in - let new_wq = wq(cs := release (wq cs)) in - (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))" - -lemma cpreced_initial: - "cpreced (\ cs. []) [] = (\_. (Prc 0 0))" -apply(simp add: cpreced_def) -apply(simp add: cs_dependants_def cs_RAG_def cs_waiting_def cs_holding_def) -apply(simp add: preced_def) -done - -lemma sch_old_def: - "schs (e#s) = (let ps = schs s in - let pwq = wq_fun ps in - let nwq = case e of - P th cs \ pwq(cs:=(pwq cs @ [th])) | - V th cs \ let nq = case (pwq cs) of - [] \ [] | - (_#qs) \ (SOME q. distinct q \ set q = set qs) - in pwq(cs:=nq) | - _ \ pwq - in let ncp = cpreced nwq (e#s) in - \wq_fun = nwq, cprec_fun = ncp\ - )" -apply(cases e) -apply(simp_all) -done - - -text {* - \noindent - The following @{text "wq"} is a shorthand for @{text "wq_fun"}. - *} -definition wq :: "state \ cs \ thread list" - where "wq s = wq_fun (schs s)" - -text {* \noindent - The following @{text "cp"} is a shorthand for @{text "cprec_fun"}. - *} -definition cp :: "state \ thread \ precedence" - where "cp s \ cprec_fun (schs s)" - -definition "cp_gen s x = Max ((the_preced s \ the_thread) ` subtree (tRAG s) x)" - -text {* \noindent - Functions @{text "holding"}, @{text "waiting"}, @{text "RAG"} and - @{text "dependants"} still have the - same meaning, but redefined so that they no longer RAG on the - fictitious {\em waiting queue function} - @{text "wq"}, but on system state @{text "s"}. - *} -defs (overloaded) - s_holding_abv: - "holding (s::state) \ holding (wq_fun (schs s))" - s_waiting_abv: - "waiting (s::state) \ waiting (wq_fun (schs s))" - s_RAG_abv: - "RAG (s::state) \ RAG (wq_fun (schs s))" - s_dependants_abv: - "dependants (s::state) \ dependants (wq_fun (schs s))" - - -text {* - The following lemma can be proved easily, and the meaning is obvious. - *} -lemma - s_holding_def: - "holding (s::state) th cs \ (th \ set (wq_fun (schs s) cs) \ th = hd (wq_fun (schs s) cs))" - by (auto simp:s_holding_abv wq_def cs_holding_def) - -lemma s_waiting_def: - "waiting (s::state) th cs \ (th \ set (wq_fun (schs s) cs) \ th \ hd (wq_fun (schs s) cs))" - by (auto simp:s_waiting_abv wq_def cs_waiting_def) - -lemma s_RAG_def: - "RAG (s::state) = - {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \ {(Cs cs, Th th) | cs th. holding (wq s) th cs}" - by (auto simp:s_RAG_abv wq_def cs_RAG_def) - -lemma - s_dependants_def: - "dependants (s::state) th \ {th' . (Th th', Th th) \ (RAG (wq s))^+}" - by (auto simp:s_dependants_abv wq_def cs_dependants_def) - -text {* - The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready} - for running if it is a live thread and it is not waiting for any critical resource. - *} -definition readys :: "state \ thread set" - where "readys s \ {th . th \ threads s \ (\ cs. \ waiting s th cs)}" - -text {* \noindent - The following function @{text "runing"} calculates the set of running thread, which is the ready - thread with the highest precedence. - *} -definition runing :: "state \ thread set" - where "runing s \ {th . th \ readys s \ cp s th = Max ((cp s) ` (readys s))}" - -text {* \noindent - Notice that the definition of @{text "running"} reflects the preemptive scheduling strategy, - because, if the @{text "running"}-thread (the one in @{text "runing"} set) - lowered its precedence by resetting its own priority to a lower - one, it will lose its status of being the max in @{text "ready"}-set and be superseded. -*} - -text {* \noindent - The following function @{text "holdents s th"} returns the set of resources held by thread - @{text "th"} in state @{text "s"}. - *} -definition holdents :: "state \ thread \ cs set" - where "holdents s th \ {cs . holding s th cs}" - -lemma holdents_test: - "holdents s th = {cs . (Cs cs, Th th) \ RAG s}" -unfolding holdents_def -unfolding s_RAG_def -unfolding s_holding_abv -unfolding wq_def -by (simp) - -text {* \noindent - Observation @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in - state @{text "s"}: - *} -definition cntCS :: "state \ thread \ nat" - where "cntCS s th = card (holdents s th)" - -text {* \noindent - According to the convention of Paulson's inductive method, - the decision made by a protocol that event @{text "e"} is eligible to happen next under state @{text "s"} - is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as - follows (notice how the decision is based on the {\em observation function}s - defined above, and also notice how a complicated protocol is modeled by a few simple - observations, and how such a kind of simplicity gives rise to improved trust on - faithfulness): - *} -inductive step :: "state \ event \ bool" - where - -- {* - A thread can be created if it is not a live thread: - *} - thread_create: "\thread \ threads s\ \ step s (Create thread prio)" | - -- {* - A thread can exit if it no longer hold any resource: - *} - thread_exit: "\thread \ runing s; holdents s thread = {}\ \ step s (Exit thread)" | - -- {* - \begin{minipage}{0.9\textwidth} - A thread can request for an critical resource @{text "cs"}, if it is running and - the request does not form a loop in the current RAG. The latter condition - is set up to avoid deadlock. The condition also reflects our assumption all threads are - carefully programmed so that deadlock can not happen: - \end{minipage} - *} - thread_P: "\thread \ runing s; (Cs cs, Th thread) \ (RAG s)^+\ \ - step s (P thread cs)" | - -- {* - \begin{minipage}{0.9\textwidth} - A thread can release a critical resource @{text "cs"} - if it is running and holding that resource: - \end{minipage} - *} - thread_V: "\thread \ runing s; holding s thread cs\ \ step s (V thread cs)" | - -- {* - \begin{minipage}{0.9\textwidth} - A thread can adjust its own priority as long as it is current running. - With the resetting of one thread's priority, its precedence may change. - If this change lowered the precedence, according to the definition of @{text "running"} - function, - \end{minipage} - *} - thread_set: "\thread \ runing s\ \ step s (Set thread prio)" - -text {* - In Paulson's inductive method, every protocol is defined by such a @{text "step"} - predicate. For instance, the predicate @{text "step"} given above - defines the PIP protocol. So, it can also be called "PIP". -*} - -abbreviation - "PIP \ step" - - -text {* \noindent - For any protocol defined by a @{text "step"} predicate, - the fact that @{text "s"} is a legal state in - the protocol is expressed as: @{text "vt step s"}, where - the predicate @{text "vt"} can be defined as the following: - *} -inductive vt :: "state \ bool" - where - -- {* Empty list @{text "[]"} is a legal state in any protocol:*} - vt_nil[intro]: "vt []" | - -- {* - \begin{minipage}{0.9\textwidth} - If @{text "s"} a legal state of the protocol defined by predicate @{text "step"}, - and event @{text "e"} is allowed to happen under state @{text "s"} by the protocol - predicate @{text "step"}, then @{text "e#s"} is a new legal state rendered by the - happening of @{text "e"}: - \end{minipage} - *} - vt_cons[intro]: "\vt s; step s e\ \ vt (e#s)" - -text {* \noindent - It is easy to see that the definition of @{text "vt"} is generic. It can be applied to - any specific protocol specified by a @{text "step"}-predicate to get the set of - legal states of that particular protocol. - *} - -text {* - The following are two very basic properties of @{text "vt"}. -*} - -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) - -text {* \noindent - The following two auxiliary functions @{text "the_cs"} and @{text "the_th"} are used to extract - critical resource and thread respectively out of RAG nodes. - *} -fun the_cs :: "node \ cs" - where "the_cs (Cs cs) = cs" - -fun the_th :: "node \ thread" - where "the_th (Th th) = th" - -text {* \noindent - The following predicate @{text "next_th"} describe the next thread to - take over when a critical resource is released. In @{text "next_th s th cs t"}, - @{text "th"} is the thread to release, @{text "t"} is the one to take over. - Notice how this definition is backed up by the @{text "release"} function and its use - in the @{text "V"}-branch of @{text "schs"} function. This @{text "next_th"} function - is not needed for the execution of PIP. It is introduced as an auxiliary function - to state lemmas. The correctness of this definition will be confirmed by - lemmas @{text "step_v_hold_inv"}, @{text " step_v_wait_inv"}, - @{text "step_v_get_hold"} and @{text "step_v_not_wait"}. - *} -definition next_th:: "state \ thread \ cs \ thread \ bool" - where "next_th s th cs t = (\ rest. wq s cs = th#rest \ rest \ [] \ - t = hd (SOME q. distinct q \ set q = set rest))" - -text {* \noindent - The aux function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"} - in list @{text "l"}: - *} -definition count :: "('a \ bool) \ 'a list \ nat" - where "count Q l = length (filter Q l)" - -text {* \noindent - The following observation @{text "cntP s"} returns the number of operation @{text "P"} happened - before reaching state @{text "s"}. - *} -definition cntP :: "state \ thread \ nat" - where "cntP s th = count (\ e. \ cs. e = P th cs) s" - -text {* \noindent - The following observation @{text "cntV s"} returns the number of operation @{text "V"} happened - before reaching state @{text "s"}. - *} -definition cntV :: "state \ thread \ nat" - where "cntV s th = count (\ e. \ cs. e = V th cs) s" -(*<*) - -end -(*>*) - diff -r ed938e2246b9 -r 0525670d8e6a RTree.thy~ --- a/RTree.thy~ Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1880 +0,0 @@ -theory RTree -imports "~~/src/HOL/Library/Transitive_Closure_Table" Max -begin - -section {* A theory of relational trees *} - -inductive_cases path_nilE [elim!]: "rtrancl_path r x [] y" -inductive_cases path_consE [elim!]: "rtrancl_path r x (z#zs) y" - -subsection {* Definitions *} - -text {* - In this theory, we are going to give a notion of of `Relational Graph` and - its derived notion `Relational Tree`. Given a binary relation @{text "r"}, - the `Relational Graph of @{text "r"}` is the graph, the edges of which - are those in @{text "r"}. In this way, any binary relation can be viewed - as a `Relational Graph`. Note, this notion of graph includes infinite graphs. - - A `Relation Graph` @{text "r"} is said to be a `Relational Tree` if it is both - {\em single valued} and {\em acyclic}. -*} - -text {* - The following @{text "sgv"} specifies that relation @{text "r"} is {\em single valued}. -*} -locale sgv = - fixes r - assumes sgv: "single_valued r" - -text {* - The following @{text "rtree"} specifies that @{text "r"} is a - {\em Relational Tree}. -*} -locale rtree = sgv + - assumes acl: "acyclic r" - -text {* - The following two auxiliary functions @{text "rel_of"} and @{text "pred_of"} - transfer between the predicate and set representation of binary relations. -*} - -definition "rel_of r = {(x, y) | x y. r x y}" - -definition "pred_of r = (\ x y. (x, y) \ r)" - -text {* - To reason about {\em Relational Graph}, a notion of path is - needed, which is given by the following @{text "rpath"} (short - for `relational path`). - The path @{text "xs"} in proposition @{text "rpath r x xs y"} is - a path leading from @{text "x"} to @{text "y"}, which serves as a - witness of the fact @{text "(x, y) \ r^*"}. - - @{text "rpath"} - is simply a wrapper of the @{text "rtrancl_path"} defined in the imported - theory @{text "Transitive_Closure_Table"}, which defines - a notion of path for the predicate form of binary relations. -*} -definition "rpath r x xs y = rtrancl_path (pred_of r) x xs y" - -text {* - Given a path @{text "ps"}, @{text "edges_on ps"} is the - set of edges along the path, which is defined as follows: -*} - -definition "edges_on ps = {(a,b) | a b. \ xs ys. ps = xs@[a,b]@ys}" - -text {* - The following @{text "indep"} defines a notion of independence. - Two nodes @{text "x"} and @{text "y"} are said to be independent - (expressed as @{text "indep x y"}), if neither one is reachable - from the other in relational graph @{text "r"}. -*} -definition "indep r x y = (((x, y) \ r^*) \ ((y, x) \ r^*))" - -text {* - In relational tree @{text "r"}, the sub tree of node @{text "x"} is written - @{text "subtree r x"}, which is defined to be the set of nodes (including itself) - which can reach @{text "x"} by following some path in @{text "r"}: -*} - -definition "subtree r x = {y . (y, x) \ r^*}" - -definition "ancestors r x = {y. (x, y) \ r^+}" - -definition "root r x = (ancestors r x = {})" - -text {* - The following @{text "edge_in r x"} is the set of edges - contained in the sub-tree of @{text "x"}, with @{text "r"} as the underlying graph. -*} - -definition "edges_in r x = {(a, b) | a b. (a, b) \ r \ b \ subtree r x}" - -text {* - The following lemma @{text "edges_in_meaning"} shows the intuitive meaning - of `an edge @{text "(a, b)"} is in the sub-tree of @{text "x"}`, - i.e., both @{text "a"} and @{text "b"} are in the sub-tree. -*} -lemma edges_in_meaning: - "edges_in r x = {(a, b) | a b. (a, b) \ r \ a \ subtree r x \ b \ subtree r x}" -proof - - { fix a b - assume h: "(a, b) \ r" "b \ subtree r x" - moreover have "a \ subtree r x" - proof - - from h(2)[unfolded subtree_def] have "(b, x) \ r^*" by simp - with h(1) have "(a, x) \ r^*" by auto - thus ?thesis by (auto simp:subtree_def) - qed - ultimately have "((a, b) \ r \ a \ subtree r x \ b \ subtree r x)" - by (auto) - } thus ?thesis by (auto simp:edges_in_def) -qed - -text {* - The following lemma shows the meaning of @{term "edges_in"} from the other side, - which says: for the edge @{text "(a,b)"} to be outside of the sub-tree of @{text "x"}, - it is sufficient to show that @{text "b"} is. -*} -lemma edges_in_refutation: - assumes "b \ subtree r x" - shows "(a, b) \ edges_in r x" - using assms by (unfold edges_in_def subtree_def, auto) - -definition "children r x = {y. (y, x) \ r}" - -locale fbranch = - fixes r - assumes fb: "\ x \ Range r . finite (children r x)" -begin - -lemma finite_children: "finite (children r x)" -proof(cases "children r x = {}") - case True - thus ?thesis by auto -next - case False - then obtain y where "(y, x) \ r" by (auto simp:children_def) - hence "x \ Range r" by auto - from fb[rule_format, OF this] - show ?thesis . -qed - -end - -locale fsubtree = fbranch + - assumes wf: "wf r" - -(* ccc *) - -subsection {* Auxiliary lemmas *} - -lemma index_minimize: - assumes "P (i::nat)" - obtains j where "P j" and "\ k < j. \ P k" -using assms -proof - - have "\ j. P j \ (\ k < j. \ P k)" - using assms - proof(induct i rule:less_induct) - case (less t) - show ?case - proof(cases "\ j < t. \ P j") - case True - with less (2) show ?thesis by blast - next - case False - then obtain j where "j < t" "P j" by auto - from less(1)[OF this] - show ?thesis . - qed - qed - with that show ?thesis by metis -qed - -subsection {* Properties of Relational Graphs and Relational Trees *} - -subsubsection {* Properties of @{text "rel_of"} and @{text "pred_of"} *} - -text {* The following lemmas establish bijectivity of the two functions *} - -lemma pred_rel_eq: "pred_of (rel_of r) = r" by (auto simp:rel_of_def pred_of_def) - -lemma rel_pred_eq: "rel_of (pred_of r) = r" by (auto simp:rel_of_def pred_of_def) - -lemma rel_of_star: "rel_of (r^**) = (rel_of r)^*" - by (unfold rel_of_def rtranclp_rtrancl_eq, auto) - -lemma pred_of_star: "pred_of (r^*) = (pred_of r)^**" -proof - - { fix x y - have "pred_of (r^*) x y = (pred_of r)^** x y" - by (unfold pred_of_def rtranclp_rtrancl_eq, auto) - } thus ?thesis by auto -qed - -lemma star_2_pstar: "(x, y) \ r^* = (pred_of (r^*)) x y" - by (simp add: pred_of_def) - -subsubsection {* Properties of @{text "rpath"} *} - -text {* Induction rule for @{text "rpath"}: *} - -lemma rpath_induct [consumes 1, case_names rbase rstep, induct pred: rpath]: - assumes "rpath r x1 x2 x3" - and "\x. P x [] x" - and "\x y ys z. (x, y) \ r \ rpath r y ys z \ P y ys z \ P x (y # ys) z" - shows "P x1 x2 x3" - using assms[unfolded rpath_def] - by (induct, auto simp:pred_of_def rpath_def) - -lemma rpathE: - assumes "rpath r x xs y" - obtains (base) "y = x" "xs = []" - | (step) z zs where "(x, z) \ r" "rpath r z zs y" "xs = z#zs" - using assms - by (induct, auto) - -text {* Introduction rule for empty path *} -lemma rbaseI [intro!]: - assumes "x = y" - shows "rpath r x [] y" - by (unfold rpath_def assms, - rule Transitive_Closure_Table.rtrancl_path.base) - -text {* Introduction rule for non-empty path *} -lemma rstepI [intro!]: - assumes "(x, y) \ r" - and "rpath r y ys z" - shows "rpath r x (y#ys) z" -proof(unfold rpath_def, rule Transitive_Closure_Table.rtrancl_path.step) - from assms(1) show "pred_of r x y" by (auto simp:pred_of_def) -next - from assms(2) show "rtrancl_path (pred_of r) y ys z" - by (auto simp:pred_of_def rpath_def) -qed - -text {* Introduction rule for @{text "@"}-path *} -lemma rpath_appendI [intro]: - assumes "rpath r x xs a" and "rpath r a ys y" - shows "rpath r x (xs @ ys) y" - using assms - by (unfold rpath_def, auto intro:rtrancl_path_trans) - -text {* Elimination rule for empty path *} - -lemma rpath_cases [cases pred:rpath]: - assumes "rpath r a1 a2 a3" - obtains (rbase) "a1 = a3" and "a2 = []" - | (rstep) y :: "'a" and ys :: "'a list" - where "(a1, y) \ r" and "a2 = y # ys" and "rpath r y ys a3" - using assms [unfolded rpath_def] - by (cases, auto simp:rpath_def pred_of_def) - -lemma rpath_nilE [elim!, cases pred:rpath]: - assumes "rpath r x [] y" - obtains "y = x" - using assms[unfolded rpath_def] by auto - --- {* This is a auxiliary lemmas used only in the proof of @{text "rpath_nnl_lastE"} *} -lemma rpath_nnl_last: - assumes "rtrancl_path r x xs y" - and "xs \ []" - obtains xs' where "xs = xs'@[y]" -proof - - from append_butlast_last_id[OF `xs \ []`, symmetric] - obtain xs' y' where eq_xs: "xs = (xs' @ y' # [])" by simp - with assms(1) - have "rtrancl_path r x ... y" by simp - hence "y = y'" by (rule rtrancl_path_appendE, auto) - with eq_xs have "xs = xs'@[y]" by simp - from that[OF this] show ?thesis . -qed - -text {* - Elimination rule for non-empty paths constructed with @{text "#"}. -*} - -lemma rpath_ConsE [elim!, cases pred:rpath]: - assumes "rpath r x (y # ys) x2" - obtains (rstep) "(x, y) \ r" and "rpath r y ys x2" - using assms[unfolded rpath_def] - by (cases, auto simp:rpath_def pred_of_def) - -text {* - Elimination rule for non-empty path, where the destination node - @{text "y"} is shown to be at the end of the path. -*} -lemma rpath_nnl_lastE: - assumes "rpath r x xs y" - and "xs \ []" - obtains xs' where "xs = xs'@[y]" - using assms[unfolded rpath_def] - by (rule rpath_nnl_last, auto) - -text {* Other elimination rules of @{text "rpath"} *} - -lemma rpath_appendE: - assumes "rpath r x (xs @ [a] @ ys) y" - obtains "rpath r x (xs @ [a]) a" and "rpath r a ys y" - using rtrancl_path_appendE[OF assms[unfolded rpath_def, simplified], folded rpath_def] - by auto - -lemma rpath_subE: - assumes "rpath r x (xs @ [a] @ ys @ [b] @ zs) y" - obtains "rpath r x (xs @ [a]) a" and "rpath r a (ys @ [b]) b" and "rpath r b zs y" - using assms - by (elim rpath_appendE, auto) - -text {* Every path has a unique end point. *} -lemma rpath_dest_eq: - assumes "rpath r x xs x1" - and "rpath r x xs x2" - shows "x1 = x2" - using assms - by (induct, auto) - -subsubsection {* Properites of @{text "edges_on"} *} - -lemma edges_on_unfold: - "edges_on (a # b # xs) = {(a, b)} \ edges_on (b # xs)" (is "?L = ?R") -proof - - { fix c d - assume "(c, d) \ ?L" - then obtain l1 l2 where h: "(a # b # xs) = l1 @ [c, d] @ l2" - by (auto simp:edges_on_def) - have "(c, d) \ ?R" - proof(cases "l1") - case Nil - with h have "(c, d) = (a, b)" by auto - thus ?thesis by auto - next - case (Cons e es) - from h[unfolded this] have "b#xs = es@[c, d]@l2" by auto - thus ?thesis by (auto simp:edges_on_def) - qed - } moreover - { fix c d - assume "(c, d) \ ?R" - moreover have "(a, b) \ ?L" - proof - - have "(a # b # xs) = []@[a,b]@xs" by simp - hence "\ l1 l2. (a # b # xs) = l1@[a,b]@l2" by auto - thus ?thesis by (unfold edges_on_def, simp) - qed - moreover { - assume "(c, d) \ edges_on (b#xs)" - then obtain l1 l2 where "b#xs = l1@[c, d]@l2" by (unfold edges_on_def, auto) - hence "a#b#xs = (a#l1)@[c,d]@l2" by simp - hence "\ l1 l2. (a # b # xs) = l1@[c,d]@l2" by metis - hence "(c,d) \ ?L" by (unfold edges_on_def, simp) - } - ultimately have "(c, d) \ ?L" by auto - } ultimately show ?thesis by auto -qed - -lemma edges_on_len: - assumes "(a,b) \ edges_on l" - shows "length l \ 2" - using assms - by (unfold edges_on_def, auto) - -text {* Elimination of @{text "edges_on"} for non-empty path *} - -lemma edges_on_consE [elim, cases set:edges_on]: - assumes "(a,b) \ edges_on (x#xs)" - obtains (head) xs' where "x = a" and "xs = b#xs'" - | (tail) "(a,b) \ edges_on xs" -proof - - from assms obtain l1 l2 - where h: "(x#xs) = l1 @ [a,b] @ l2" by (unfold edges_on_def, blast) - have "(\ xs'. x = a \ xs = b#xs') \ ((a,b) \ edges_on xs)" - proof(cases "l1") - case Nil with h - show ?thesis by auto - next - case (Cons e el) - from h[unfolded this] - have "xs = el @ [a,b] @ l2" by auto - thus ?thesis - by (unfold edges_on_def, auto) - qed - thus ?thesis - proof - assume "(\xs'. x = a \ xs = b # xs')" - then obtain xs' where "x = a" "xs = b#xs'" by blast - from that(1)[OF this] show ?thesis . - next - assume "(a, b) \ edges_on xs" - from that(2)[OF this] show ?thesis . - qed -qed - -text {* - Every edges on the path is a graph edges: -*} -lemma rpath_edges_on: - assumes "rpath r x xs y" - shows "(edges_on (x#xs)) \ r" - using assms -proof(induct arbitrary:y) - case (rbase x) - thus ?case by (unfold edges_on_def, auto) -next - case (rstep x y ys z) - show ?case - proof - - { fix a b - assume "(a, b) \ edges_on (x # y # ys)" - hence "(a, b) \ r" by (cases, insert rstep, auto) - } thus ?thesis by auto - qed -qed - -text {* @{text "edges_on"} is mono with respect to @{text "#"}-operation: *} -lemma edges_on_Cons_mono: - shows "edges_on xs \ edges_on (x#xs)" -proof - - { fix a b - assume "(a, b) \ edges_on xs" - then obtain l1 l2 where "xs = l1 @ [a,b] @ l2" - by (auto simp:edges_on_def) - hence "x # xs = (x#l1) @ [a, b] @ l2" by auto - hence "(a, b) \ edges_on (x#xs)" - by (unfold edges_on_def, blast) - } thus ?thesis by auto -qed - -text {* - The following rule @{text "rpath_transfer"} is used to show - that one path is intact as long as all the edges on it are intact - with the change of graph. - - If @{text "x#xs"} is path in graph @{text "r1"} and - every edges along the path is also in @{text "r2"}, - then @{text "x#xs"} is also a edge in graph @{text "r2"}: -*} - -lemma rpath_transfer: - assumes "rpath r1 x xs y" - and "edges_on (x#xs) \ r2" - shows "rpath r2 x xs y" - using assms -proof(induct) - case (rstep x y ys z) - show ?case - proof(rule rstepI) - show "(x, y) \ r2" - proof - - have "(x, y) \ edges_on (x # y # ys)" - by (unfold edges_on_def, auto) - with rstep(4) show ?thesis by auto - qed - next - show "rpath r2 y ys z" - using rstep edges_on_Cons_mono[of "y#ys" "x"] by (auto) - qed -qed (unfold rpath_def, auto intro!:Transitive_Closure_Table.rtrancl_path.base) - -lemma edges_on_rpathI: - assumes "edges_on (a#xs@[b]) \ r" - shows "rpath r a (xs@[b]) b" - using assms -proof(induct xs arbitrary: a b) - case Nil - moreover have "(a, b) \ edges_on (a # [] @ [b])" - by (unfold edges_on_def, auto) - ultimately have "(a, b) \ r" by auto - thus ?case by auto -next - case (Cons x xs a b) - from this(2) have "edges_on (x # xs @ [b]) \ r" by (simp add:edges_on_unfold) - from Cons(1)[OF this] have " rpath r x (xs @ [b]) b" . - moreover from Cons(2) have "(a, x) \ r" by (auto simp:edges_on_unfold) - ultimately show ?case by (auto) -qed - -text {* - The following lemma extracts the path from @{text "x"} to @{text "y"} - from proposition @{text "(x, y) \ r^*"} -*} -lemma star_rpath: - assumes "(x, y) \ r^*" - obtains xs where "rpath r x xs y" -proof - - have "\ xs. rpath r x xs y" - proof(unfold rpath_def, rule iffD1[OF rtranclp_eq_rtrancl_path]) - from assms - show "(pred_of r)\<^sup>*\<^sup>* x y" - apply (fold pred_of_star) - by (auto simp:pred_of_def) - qed - from that and this show ?thesis by blast -qed - -text {* - The following lemma uses the path @{text "xs"} from @{text "x"} to @{text "y"} - as a witness to show @{text "(x, y) \ r^*"}. -*} -lemma rpath_star: - assumes "rpath r x xs y" - shows "(x, y) \ r^*" -proof - - from iffD2[OF rtranclp_eq_rtrancl_path] and assms[unfolded rpath_def] - have "(pred_of r)\<^sup>*\<^sup>* x y" by metis - thus ?thesis by (simp add: pred_of_star star_2_pstar) -qed - -lemma subtree_transfer: - assumes "a \ subtree r1 a'" - and "r1 \ r2" - shows "a \ subtree r2 a'" -proof - - from assms(1)[unfolded subtree_def] - have "(a, a') \ r1^*" by auto - from star_rpath[OF this] - obtain xs where rp: "rpath r1 a xs a'" by blast - hence "rpath r2 a xs a'" - proof(rule rpath_transfer) - from rpath_edges_on[OF rp] and assms(2) - show "edges_on (a # xs) \ r2" by simp - qed - from rpath_star[OF this] - show ?thesis by (auto simp:subtree_def) -qed - -lemma subtree_rev_transfer: - assumes "a \ subtree r2 a'" - and "r1 \ r2" - shows "a \ subtree r1 a'" - using assms and subtree_transfer by metis - -text {* - The following lemmas establishes a relation from paths in @{text "r"} - to @{text "r^+"} relation. -*} -lemma rpath_plus: - assumes "rpath r x xs y" - and "xs \ []" - shows "(x, y) \ r^+" -proof - - from assms(2) obtain e es where "xs = e#es" by (cases xs, auto) - from assms(1)[unfolded this] - show ?thesis - proof(cases) - case rstep - show ?thesis - proof - - from rpath_star[OF rstep(2)] have "(e, y) \ r\<^sup>*" . - with rstep(1) show "(x, y) \ r^+" by auto - qed - qed -qed - -lemma plus_rpath: - assumes "(x, y) \ r^+" - obtains xs where "rpath r x xs y" and "xs \ []" -proof - - from assms - show ?thesis - proof(cases rule:converse_tranclE[consumes 1]) - case 1 - hence "rpath r x [y] y" by auto - from that[OF this] show ?thesis by auto - next - case (2 z) - from 2(2) have "(z, y) \ r^*" by auto - from star_rpath[OF this] obtain xs where "rpath r z xs y" by auto - from rstepI[OF 2(1) this] - have "rpath r x (z # xs) y" . - from that[OF this] show ?thesis by auto - qed -qed - -subsubsection {* Properties of @{text "subtree"} and @{term "ancestors"}*} - -lemma ancestors_subtreeI: - assumes "b \ ancestors r a" - shows "a \ subtree r b" - using assms by (auto simp:subtree_def ancestors_def) - -lemma ancestors_Field: - assumes "b \ ancestors r a" - obtains "a \ Domain r" "b \ Range r" - using assms - apply (unfold ancestors_def, simp) - by (metis Domain.DomainI Range.intros trancl_domain trancl_range) - -lemma subtreeE: - assumes "a \ subtree r b" - obtains "a = b" - | "a \ b" and "b \ ancestors r a" -proof - - from assms have "(a, b) \ r^*" by (auto simp:subtree_def) - from rtranclD[OF this] - have " a = b \ a \ b \ (a, b) \ r\<^sup>+" . - with that[unfolded ancestors_def] show ?thesis by auto -qed - - -lemma subtree_Field: - "subtree r x \ Field r \ {x}" -proof - fix y - assume "y \ subtree r x" - thus "y \ Field r \ {x}" - proof(cases rule:subtreeE) - case 1 - thus ?thesis by auto - next - case 2 - thus ?thesis apply (auto simp:ancestors_def) - using Field_def tranclD by fastforce - qed -qed - -lemma subtree_ancestorsI: - assumes "a \ subtree r b" - and "a \ b" - shows "b \ ancestors r a" - using assms - by (auto elim!:subtreeE) - -text {* - @{text "subtree"} is mono with respect to the underlying graph. -*} -lemma subtree_mono: - assumes "r1 \ r2" - shows "subtree r1 x \ subtree r2 x" -proof - fix c - assume "c \ subtree r1 x" - hence "(c, x) \ r1^*" by (auto simp:subtree_def) - from star_rpath[OF this] obtain xs - where rp:"rpath r1 c xs x" by metis - hence "rpath r2 c xs x" - proof(rule rpath_transfer) - from rpath_edges_on[OF rp] have "edges_on (c # xs) \ r1" . - with assms show "edges_on (c # xs) \ r2" by auto - qed - thus "c \ subtree r2 x" - by (rule rpath_star[elim_format], auto simp:subtree_def) -qed - -text {* - The following lemma characterizes the change of sub-tree of @{text "x"} - with the removal of an outside edge @{text "(a,b)"}. - - Note that, according to lemma @{thm edges_in_refutation}, the assumption - @{term "b \ subtree r x"} amounts to saying @{text "(a, b)"} - is outside the sub-tree of @{text "x"}. -*} -lemma subtree_del_outside: (* ddd *) - assumes "b \ subtree r x" - shows "subtree (r - {(a, b)}) x = (subtree r x)" -proof - - { fix c - assume "c \ (subtree r x)" - hence "(c, x) \ r^*" by (auto simp:subtree_def) - hence "c \ subtree (r - {(a, b)}) x" - proof(rule star_rpath) - fix xs - assume rp: "rpath r c xs x" - show ?thesis - proof - - from rp - have "rpath (r - {(a, b)}) c xs x" - proof(rule rpath_transfer) - from rpath_edges_on[OF rp] have "edges_on (c # xs) \ r" . - moreover have "(a, b) \ edges_on (c#xs)" - proof - assume "(a, b) \ edges_on (c # xs)" - then obtain l1 l2 where h: "c#xs = l1@[a,b]@l2" by (auto simp:edges_on_def) - hence "tl (c#xs) = tl (l1@[a,b]@l2)" by simp - then obtain l1' where eq_xs_b: "xs = l1'@[b]@l2" by (cases l1, auto) - from rp[unfolded this] - show False - proof(rule rpath_appendE) - assume "rpath r b l2 x" - thus ?thesis - by(rule rpath_star[elim_format], insert assms(1), auto simp:subtree_def) - qed - qed - ultimately show "edges_on (c # xs) \ r - {(a,b)}" by auto - qed - thus ?thesis by (rule rpath_star[elim_format], auto simp:subtree_def) - qed - qed - } moreover { - fix c - assume "c \ subtree (r - {(a, b)}) x" - moreover have "... \ (subtree r x)" by (rule subtree_mono, auto) - ultimately have "c \ (subtree r x)" by auto - } ultimately show ?thesis by auto -qed - -(* ddd *) -lemma subset_del_subtree_outside: (* ddd *) - assumes "Range r' \ subtree r x = {}" - shows "subtree (r - r') x = (subtree r x)" -proof - - { fix c - assume "c \ (subtree r x)" - hence "(c, x) \ r^*" by (auto simp:subtree_def) - hence "c \ subtree (r - r') x" - proof(rule star_rpath) - fix xs - assume rp: "rpath r c xs x" - show ?thesis - proof - - from rp - have "rpath (r - r') c xs x" - proof(rule rpath_transfer) - from rpath_edges_on[OF rp] have "edges_on (c # xs) \ r" . - moreover { - fix a b - assume h: "(a, b) \ r'" - have "(a, b) \ edges_on (c#xs)" - proof - assume "(a, b) \ edges_on (c # xs)" - then obtain l1 l2 where "c#xs = (l1@[a])@[b]@l2" by (auto simp:edges_on_def) - hence "tl (c#xs) = tl (l1@[a,b]@l2)" by simp - then obtain l1' where eq_xs_b: "xs = l1'@[b]@l2" by (cases l1, auto) - from rp[unfolded this] - show False - proof(rule rpath_appendE) - assume "rpath r b l2 x" - from rpath_star[OF this] - have "b \ subtree r x" by (auto simp:subtree_def) - with assms (1) and h show ?thesis by (auto) - qed - qed - } ultimately show "edges_on (c # xs) \ r - r'" by auto - qed - thus ?thesis by (rule rpath_star[elim_format], auto simp:subtree_def) - qed - qed - } moreover { - fix c - assume "c \ subtree (r - r') x" - moreover have "... \ (subtree r x)" by (rule subtree_mono, auto) - ultimately have "c \ (subtree r x)" by auto - } ultimately show ?thesis by auto -qed - -lemma subtree_insert_ext: - assumes "b \ subtree r x" - shows "subtree (r \ {(a, b)}) x = (subtree r x) \ (subtree r a)" - using assms by (auto simp:subtree_def rtrancl_insert) - -lemma subtree_insert_next: - assumes "b \ subtree r x" - shows "subtree (r \ {(a, b)}) x = (subtree r x)" - using assms - by (auto simp:subtree_def rtrancl_insert) - -lemma set_add_rootI: - assumes "root r a" - and "a \ Domain r1" - shows "root (r \ r1) a" -proof - - let ?r = "r \ r1" - { fix a' - assume "a' \ ancestors ?r a" - hence "(a, a') \ ?r^+" by (auto simp:ancestors_def) - from tranclD[OF this] obtain z where "(a, z) \ ?r" by auto - moreover have "(a, z) \ r" - proof - assume "(a, z) \ r" - with assms(1) show False - by (auto simp:root_def ancestors_def) - qed - ultimately have "(a, z) \ r1" by auto - with assms(2) - have False by (auto) - } thus ?thesis by (auto simp:root_def) -qed - -lemma ancestors_mono: - assumes "r1 \ r2" - shows "ancestors r1 x \ ancestors r2 x" -proof - fix a - assume "a \ ancestors r1 x" - hence "(x, a) \ r1^+" by (auto simp:ancestors_def) - from plus_rpath[OF this] obtain xs where - h: "rpath r1 x xs a" "xs \ []" . - have "rpath r2 x xs a" - proof(rule rpath_transfer[OF h(1)]) - from rpath_edges_on[OF h(1)] and assms - show "edges_on (x # xs) \ r2" by auto - qed - from rpath_plus[OF this h(2)] - show "a \ ancestors r2 x" by (auto simp:ancestors_def) -qed - -lemma subtree_refute: - assumes "x \ ancestors r y" - and "x \ y" - shows "y \ subtree r x" -proof - assume "y \ subtree r x" - thus False - by(elim subtreeE, insert assms, auto) -qed - -subsubsection {* Properties about relational trees *} - -context rtree -begin - -lemma ancestors_headE: - assumes "c \ ancestors r a" - assumes "(a, b) \ r" - obtains "b = c" - | "c \ ancestors r b" -proof - - from assms(1) - have "(a, c) \ r^+" by (auto simp:ancestors_def) - hence "b = c \ c \ ancestors r b" - proof(cases rule:converse_tranclE[consumes 1]) - case 1 - with assms(2) and sgv have "b = c" by (auto simp:single_valued_def) - thus ?thesis by auto - next - case (2 y) - from 2(1) and assms(2) and sgv have "y = b" by (auto simp:single_valued_def) - from 2(2)[unfolded this] have "c \ ancestors r b" by (auto simp:ancestors_def) - thus ?thesis by auto - qed - with that show ?thesis by metis -qed - -lemma ancestors_accum: - assumes "(a, b) \ r" - shows "ancestors r a = ancestors r b \ {b}" -proof - - { fix c - assume "c \ ancestors r a" - hence "(a, c) \ r^+" by (auto simp:ancestors_def) - hence "c \ ancestors r b \ {b}" - proof(cases rule:converse_tranclE[consumes 1]) - case 1 - with sgv assms have "c = b" by (unfold single_valued_def, auto) - thus ?thesis by auto - next - case (2 c') - with sgv assms have "c' = b" by (unfold single_valued_def, auto) - from 2(2)[unfolded this] - show ?thesis by (auto simp:ancestors_def) - qed - } moreover { - fix c - assume "c \ ancestors r b \ {b}" - hence "c = b \ c \ ancestors r b" by auto - hence "c \ ancestors r a" - proof - assume "c = b" - from assms[folded this] - show ?thesis by (auto simp:ancestors_def) - next - assume "c \ ancestors r b" - with assms show ?thesis by (auto simp:ancestors_def) - qed - } ultimately show ?thesis by auto -qed - -lemma rootI: - assumes h: "\ x'. x' \ x \ x \ subtree r' x'" - and "r' \ r" - shows "root r' x" -proof - - from acyclic_subset[OF acl assms(2)] - have acl': "acyclic r'" . - { fix x' - assume "x' \ ancestors r' x" - hence h1: "(x, x') \ r'^+" by (auto simp:ancestors_def) - have "x' \ x" - proof - assume eq_x: "x' = x" - from h1[unfolded this] and acl' - show False by (auto simp:acyclic_def) - qed - moreover from h1 have "x \ subtree r' x'" by (auto simp:subtree_def) - ultimately have False using h by auto - } thus ?thesis by (auto simp:root_def) -qed - -lemma rpath_overlap_oneside: (* ddd *) - assumes "rpath r x xs1 x1" (* ccc *) - and "rpath r x xs2 x2" - and "length xs1 \ length xs2" - obtains xs3 where - "xs2 = xs1 @ xs3" "rpath r x xs1 x1" "rpath r x1 xs3 x2" -proof(cases "xs1 = []") - case True - with that show ?thesis by auto -next - case False - have "\ i \ length xs1. take i xs1 = take i xs2" - proof - - { assume "\ (\ i \ length xs1. take i xs1 = take i xs2)" - then obtain i where "i \ length xs1 \ take i xs1 \ take i xs2" by auto - from this(1) have "False" - proof(rule index_minimize) - fix j - assume h1: "j \ length xs1 \ take j xs1 \ take j xs2" - and h2: " \k (k \ length xs1 \ take k xs1 \ take k xs2)" - -- {* @{text "j - 1"} is the branch point between @{text "xs1"} and @{text "xs2"} *} - let ?idx = "j - 1" - -- {* A number of inequalities concerning @{text "j - 1"} are derived first *} - have lt_i: "?idx < length xs1" using False h1 - by (metis Suc_diff_1 le_neq_implies_less length_greater_0_conv lessI less_imp_diff_less) - have lt_i': "?idx < length xs2" using lt_i and assms(3) by auto - have lt_j: "?idx < j" using h1 by (cases j, auto) - -- {* From thesis inequalities, a number of equations concerning @{text "xs1"} - and @{text "xs2"} are derived *} - have eq_take: "take ?idx xs1 = take ?idx xs2" - using h2[rule_format, OF lt_j] and h1 by auto - have eq_xs1: " xs1 = take ?idx xs1 @ xs1 ! (?idx) # drop (Suc (?idx)) xs1" - using id_take_nth_drop[OF lt_i] . - have eq_xs2: "xs2 = take ?idx xs2 @ xs2 ! (?idx) # drop (Suc (?idx)) xs2" - using id_take_nth_drop[OF lt_i'] . - -- {* The branch point along the path is finally pinpointed *} - have neq_idx: "xs1!?idx \ xs2!?idx" - proof - - have "take j xs1 = take ?idx xs1 @ [xs1 ! ?idx]" - using eq_xs1 Suc_diff_1 lt_i lt_j take_Suc_conv_app_nth by fastforce - moreover have eq_tk2: "take j xs2 = take ?idx xs2 @ [xs2 ! ?idx]" - using Suc_diff_1 lt_i' lt_j take_Suc_conv_app_nth by fastforce - ultimately show ?thesis using eq_take h1 by auto - qed - show ?thesis - proof(cases " take (j - 1) xs1 = []") - case True - have "(x, xs1!?idx) \ r" - proof - - from eq_xs1[unfolded True, simplified, symmetric] assms(1) - have "rpath r x ( xs1 ! ?idx # drop (Suc ?idx) xs1) x1" by simp - from this[unfolded rpath_def] - show ?thesis by (auto simp:pred_of_def) - qed - moreover have "(x, xs2!?idx) \ r" - proof - - from eq_xs2[folded eq_take, unfolded True, simplified, symmetric] assms(2) - have "rpath r x ( xs2 ! ?idx # drop (Suc ?idx) xs2) x2" by simp - from this[unfolded rpath_def] - show ?thesis by (auto simp:pred_of_def) - qed - ultimately show ?thesis using neq_idx sgv[unfolded single_valued_def] by metis - next - case False - then obtain e es where eq_es: "take ?idx xs1 = es@[e]" - using rev_exhaust by blast - have "(e, xs1!?idx) \ r" - proof - - from eq_xs1[unfolded eq_es] - have "xs1 = es@[e, xs1!?idx]@drop (Suc ?idx) xs1" by simp - hence "(e, xs1!?idx) \ edges_on xs1" by (simp add:edges_on_def, metis) - with rpath_edges_on[OF assms(1)] edges_on_Cons_mono[of xs1 x] - show ?thesis by auto - qed moreover have "(e, xs2!?idx) \ r" - proof - - from eq_xs2[folded eq_take, unfolded eq_es] - have "xs2 = es@[e, xs2!?idx]@drop (Suc ?idx) xs2" by simp - hence "(e, xs2!?idx) \ edges_on xs2" by (simp add:edges_on_def, metis) - with rpath_edges_on[OF assms(2)] edges_on_Cons_mono[of xs2 x] - show ?thesis by auto - qed - ultimately show ?thesis - using sgv[unfolded single_valued_def] neq_idx by metis - qed - qed - } thus ?thesis by auto - qed - from this[rule_format, of "length xs1"] - have "take (length xs1) xs1 = take (length xs1) xs2" by simp - moreover have "xs2 = take (length xs1) xs2 @ drop (length xs1) xs2" by simp - ultimately have "xs2 = xs1 @ drop (length xs1) xs2" by auto - from that[OF this] show ?thesis . -qed - -lemma rpath_overlap_oneside: (* ddd *) - assumes "rpath r x xs1 x1" - and "rpath r x xs2 x2" - and "length xs1 \ length xs2" - obtains xs3 where "xs2 = xs1 @ xs3" -proof(cases "xs1 = []") - case True - with that show ?thesis by auto -next - case False - have "\ i \ length xs1. take i xs1 = take i xs2" - proof - - { assume "\ (\ i \ length xs1. take i xs1 = take i xs2)" - then obtain i where "i \ length xs1 \ take i xs1 \ take i xs2" by auto - from this(1) have "False" - proof(rule index_minimize) - fix j - assume h1: "j \ length xs1 \ take j xs1 \ take j xs2" - and h2: " \k (k \ length xs1 \ take k xs1 \ take k xs2)" - -- {* @{text "j - 1"} is the branch point between @{text "xs1"} and @{text "xs2"} *} - let ?idx = "j - 1" - -- {* A number of inequalities concerning @{text "j - 1"} are derived first *} - have lt_i: "?idx < length xs1" using False h1 - by (metis Suc_diff_1 le_neq_implies_less length_greater_0_conv lessI less_imp_diff_less) - have lt_i': "?idx < length xs2" using lt_i and assms(3) by auto - have lt_j: "?idx < j" using h1 by (cases j, auto) - -- {* From thesis inequalities, a number of equations concerning @{text "xs1"} - and @{text "xs2"} are derived *} - have eq_take: "take ?idx xs1 = take ?idx xs2" - using h2[rule_format, OF lt_j] and h1 by auto - have eq_xs1: " xs1 = take ?idx xs1 @ xs1 ! (?idx) # drop (Suc (?idx)) xs1" - using id_take_nth_drop[OF lt_i] . - have eq_xs2: "xs2 = take ?idx xs2 @ xs2 ! (?idx) # drop (Suc (?idx)) xs2" - using id_take_nth_drop[OF lt_i'] . - -- {* The branch point along the path is finally pinpointed *} - have neq_idx: "xs1!?idx \ xs2!?idx" - proof - - have "take j xs1 = take ?idx xs1 @ [xs1 ! ?idx]" - using eq_xs1 Suc_diff_1 lt_i lt_j take_Suc_conv_app_nth by fastforce - moreover have eq_tk2: "take j xs2 = take ?idx xs2 @ [xs2 ! ?idx]" - using Suc_diff_1 lt_i' lt_j take_Suc_conv_app_nth by fastforce - ultimately show ?thesis using eq_take h1 by auto - qed - show ?thesis - proof(cases " take (j - 1) xs1 = []") - case True - have "(x, xs1!?idx) \ r" - proof - - from eq_xs1[unfolded True, simplified, symmetric] assms(1) - have "rpath r x ( xs1 ! ?idx # drop (Suc ?idx) xs1) x1" by simp - from this[unfolded rpath_def] - show ?thesis by (auto simp:pred_of_def) - qed - moreover have "(x, xs2!?idx) \ r" - proof - - from eq_xs2[folded eq_take, unfolded True, simplified, symmetric] assms(2) - have "rpath r x ( xs2 ! ?idx # drop (Suc ?idx) xs2) x2" by simp - from this[unfolded rpath_def] - show ?thesis by (auto simp:pred_of_def) - qed - ultimately show ?thesis using neq_idx sgv[unfolded single_valued_def] by metis - next - case False - then obtain e es where eq_es: "take ?idx xs1 = es@[e]" - using rev_exhaust by blast - have "(e, xs1!?idx) \ r" - proof - - from eq_xs1[unfolded eq_es] - have "xs1 = es@[e, xs1!?idx]@drop (Suc ?idx) xs1" by simp - hence "(e, xs1!?idx) \ edges_on xs1" by (simp add:edges_on_def, metis) - with rpath_edges_on[OF assms(1)] edges_on_Cons_mono[of xs1 x] - show ?thesis by auto - qed moreover have "(e, xs2!?idx) \ r" - proof - - from eq_xs2[folded eq_take, unfolded eq_es] - have "xs2 = es@[e, xs2!?idx]@drop (Suc ?idx) xs2" by simp - hence "(e, xs2!?idx) \ edges_on xs2" by (simp add:edges_on_def, metis) - with rpath_edges_on[OF assms(2)] edges_on_Cons_mono[of xs2 x] - show ?thesis by auto - qed - ultimately show ?thesis - using sgv[unfolded single_valued_def] neq_idx by metis - qed - qed - } thus ?thesis by auto - qed - from this[rule_format, of "length xs1"] - have "take (length xs1) xs1 = take (length xs1) xs2" by simp - moreover have "xs2 = take (length xs1) xs2 @ drop (length xs1) xs2" by simp - ultimately have "xs2 = xs1 @ drop (length xs1) xs2" by auto - from that[OF this] show ?thesis . -qed - -lemma rpath_overlap [consumes 2, cases pred:rpath]: - assumes "rpath r x xs1 x1" - and "rpath r x xs2 x2" - obtains (less_1) xs3 where "xs2 = xs1 @ xs3" - | (less_2) xs3 where "xs1 = xs2 @ xs3" -proof - - have "length xs1 \ length xs2 \ length xs2 \ length xs1" by auto - with assms rpath_overlap_oneside that show ?thesis by metis -qed - -text {* - As a corollary of @{thm "rpath_overlap_oneside"}, - the following two lemmas gives one important property of relation tree, - i.e. there is at most one path between any two nodes. - Similar to the proof of @{thm rpath_overlap}, we starts with - the one side version first. -*} - -lemma rpath_unique_oneside: - assumes "rpath r x xs1 y" - and "rpath r x xs2 y" - and "length xs1 \ length xs2" - shows "xs1 = xs2" -proof - - from rpath_overlap_oneside[OF assms] - obtain xs3 where less_1: "xs2 = xs1 @ xs3" by blast - show ?thesis - proof(cases "xs3 = []") - case True - from less_1[unfolded this] show ?thesis by simp - next - case False - note FalseH = this - show ?thesis - proof(cases "xs1 = []") - case True - have "(x, x) \ r^+" - proof(rule rpath_plus) - from assms(1)[unfolded True] - have "y = x" by (cases rule:rpath_nilE, simp) - from assms(2)[unfolded this] show "rpath r x xs2 x" . - next - from less_1 and False show "xs2 \ []" by simp - qed - with acl show ?thesis by (unfold acyclic_def, auto) - next - case False - then obtain e es where eq_xs1: "xs1 = es@[e]" using rev_exhaust by auto - from assms(2)[unfolded less_1 this] - have "rpath r x (es @ [e] @ xs3) y" by simp - thus ?thesis - proof(cases rule:rpath_appendE) - case 1 - from rpath_dest_eq [OF 1(1)[folded eq_xs1] assms(1)] - have "e = y" . - from rpath_plus [OF 1(2)[unfolded this] FalseH] - have "(y, y) \ r^+" . - with acl show ?thesis by (unfold acyclic_def, auto) - qed - qed - qed -qed - -text {* - The following is the full version of path uniqueness. -*} -lemma rpath_unique: - assumes "rpath r x xs1 y" - and "rpath r x xs2 y" - shows "xs1 = xs2" -proof(cases "length xs1 \ length xs2") - case True - from rpath_unique_oneside[OF assms this] show ?thesis . -next - case False - hence "length xs2 \ length xs1" by simp - from rpath_unique_oneside[OF assms(2,1) this] - show ?thesis by simp -qed - -text {* - The following lemma shows that the `independence` relation is symmetric. - It is an obvious auxiliary lemma which will be used later. -*} -lemma sym_indep: "indep r x y \ indep r y x" - by (unfold indep_def, auto) - -text {* - This is another `obvious` lemma about trees, which says trees rooted at - independent nodes are disjoint. -*} -lemma subtree_disjoint: - assumes "indep r x y" - shows "subtree r x \ subtree r y = {}" -proof - - { fix z x y xs1 xs2 xs3 - assume ind: "indep r x y" - and rp1: "rpath r z xs1 x" - and rp2: "rpath r z xs2 y" - and h: "xs2 = xs1 @ xs3" - have False - proof(cases "xs1 = []") - case True - from rp1[unfolded this] have "x = z" by auto - from rp2[folded this] rpath_star ind[unfolded indep_def] - show ?thesis by metis - next - case False - then obtain e es where eq_xs1: "xs1 = es@[e]" using rev_exhaust by blast - from rp2[unfolded h this] - have "rpath r z (es @ [e] @ xs3) y" by simp - thus ?thesis - proof(cases rule:rpath_appendE) - case 1 - have "e = x" using 1(1)[folded eq_xs1] rp1 rpath_dest_eq by metis - from rpath_star[OF 1(2)[unfolded this]] ind[unfolded indep_def] - show ?thesis by auto - qed - qed - } note my_rule = this - { fix z - assume h: "z \ subtree r x" "z \ subtree r y" - from h(1) have "(z, x) \ r^*" by (unfold subtree_def, auto) - then obtain xs1 where rp1: "rpath r z xs1 x" using star_rpath by metis - from h(2) have "(z, y) \ r^*" by (unfold subtree_def, auto) - then obtain xs2 where rp2: "rpath r z xs2 y" using star_rpath by metis - from rp1 rp2 - have False - by (cases, insert my_rule[OF sym_indep[OF assms(1)] rp2 rp1] - my_rule[OF assms(1) rp1 rp2], auto) - } thus ?thesis by auto -qed - -text {* - The following lemma @{text "subtree_del"} characterizes the change of sub-tree of - @{text "x"} with the removal of an inside edge @{text "(a, b)"}. - Note that, the case for the removal of an outside edge has already been dealt with - in lemma @{text "subtree_del_outside"}). - - This lemma is underpinned by the following two `obvious` facts: - \begin{enumearte} - \item - In graph @{text "r"}, for an inside edge @{text "(a,b) \ edges_in r x"}, - every node @{text "c"} in the sub-tree of @{text "a"} has a path - which goes first from @{text "c"} to @{text "a"}, then through edge @{text "(a, b)"}, and - finally reaches @{text "x"}. By the uniqueness of path in a tree, - all paths from sub-tree of @{text "a"} to @{text "x"} are such constructed, therefore - must go through @{text "(a, b)"}. The consequence is: with the removal of @{text "(a,b)"}, - all such paths will be broken. - - \item - On the other hand, all paths not originate from within the sub-tree of @{text "a"} - will not be affected by the removal of edge @{text "(a, b)"}. - The reason is simple: if the path is affected by the removal, it must - contain @{text "(a, b)"}, then it must originate from within the sub-tree of @{text "a"}. - \end{enumearte} -*} - -lemma subtree_del_inside: (* ddd *) - assumes "(a,b) \ edges_in r x" - shows "subtree (r - {(a, b)}) x = (subtree r x) - subtree r a" -proof - - from assms have asm: "b \ subtree r x" "(a, b) \ r" by (auto simp:edges_in_def) - -- {* The proof follows a common pattern to prove the equality of sets. *} - { -- {* The `left to right` direction. - *} - fix c - -- {* Assuming @{text "c"} is inside the sub-tree of @{text "x"} in the reduced graph *} - assume h: "c \ subtree (r - {(a, b)}) x" - -- {* We are going to show that @{text "c"} can not be in the sub-tree of @{text "a"} in - the original graph. *} - -- {* In other words, all nodes inside the sub-tree of @{text "a"} in the original - graph will be removed from the sub-tree of @{text "x"} in the reduced graph. *} - -- {* The reason, as analyzed before, is that all paths from within the - sub-tree of @{text "a"} are broken with the removal of edge @{text "(a,b)"}. - *} - have "c \ (subtree r x) - subtree r a" - proof - - let ?r' = "r - {(a, b)}" -- {* The reduced graph is abbreviated as @{text "?r'"} *} - from h have "(c, x) \ ?r'^*" by (auto simp:subtree_def) - -- {* Extract from the reduced graph the path @{text "xs"} from @{text "c"} to @{text "x"}. *} - then obtain xs where rp0: "rpath ?r' c xs x" by (rule star_rpath, auto) - -- {* It is easy to show @{text "xs"} is also a path in the original graph *} - hence rp1: "rpath r c xs x" - proof(rule rpath_transfer) - from rpath_edges_on[OF rp0] - show "edges_on (c # xs) \ r" by auto - qed - -- {* @{text "xs"} is used as the witness to show that @{text "c"} - in the sub-tree of @{text "x"} in the original graph. *} - hence "c \ subtree r x" - by (rule rpath_star[elim_format], auto simp:subtree_def) - -- {* The next step is to show that @{text "c"} can not be in the sub-tree of @{text "a"} - in the original graph. *} - -- {* We need to use the fact that all paths originate from within sub-tree of @{text "a"} - are broken. *} - moreover have "c \ subtree r a" - proof - -- {* Proof by contradiction, suppose otherwise *} - assume otherwise: "c \ subtree r a" - -- {* Then there is a path in original graph leading from @{text "c"} to @{text "a"} *} - obtain xs1 where rp_c: "rpath r c xs1 a" - proof - - from otherwise have "(c, a) \ r^*" by (auto simp:subtree_def) - thus ?thesis by (rule star_rpath, auto intro!:that) - qed - -- {* Starting from this path, we are going to construct a fictional - path from @{text "c"} to @{text "x"}, which, as explained before, - is broken, so that contradiction can be derived. *} - -- {* First, there is a path from @{text "b"} to @{text "x"} *} - obtain ys where rp_b: "rpath r b ys x" - proof - - from asm have "(b, x) \ r^*" by (auto simp:subtree_def) - thus ?thesis by (rule star_rpath, auto intro!:that) - qed - -- {* The paths @{text "xs1"} and @{text "ys"} can be - tied together using @{text "(a,b)"} to form a path - from @{text "c"} to @{text "x"}: *} - have "rpath r c (xs1 @ b # ys) x" - proof - - from rstepI[OF asm(2) rp_b] have "rpath r a (b # ys) x" . - from rpath_appendI[OF rp_c this] - show ?thesis . - qed - -- {* By the uniqueness of path between two nodes of a tree, we have: *} - from rpath_unique[OF rp1 this] have eq_xs: "xs = xs1 @ b # ys" . - -- {* Contradiction can be derived from from this fictional path . *} - show False - proof - - -- {* It can be shown that @{term "(a,b)"} is on this fictional path. *} - have "(a, b) \ edges_on (c#xs)" - proof(cases "xs1 = []") - case True - from rp_c[unfolded this] have "rpath r c [] a" . - hence eq_c: "c = a" by (rule rpath_nilE, simp) - hence "c#xs = a#xs" by simp - from this and eq_xs have "c#xs = a # xs1 @ b # ys" by simp - from this[unfolded True] have "c#xs = []@[a,b]@ys" by simp - thus ?thesis by (auto simp:edges_on_def) - next - case False - from rpath_nnl_lastE[OF rp_c this] - obtain xs' where "xs1 = xs'@[a]" by auto - from eq_xs[unfolded this] have "c#xs = (c#xs')@[a,b]@ys" by simp - thus ?thesis by (unfold edges_on_def, blast) - qed - -- {* It can also be shown that @{term "(a,b)"} is not on this fictional path. *} - moreover have "(a, b) \ edges_on (c#xs)" - using rpath_edges_on[OF rp0] by auto - -- {* Contradiction is thus derived. *} - ultimately show False by auto - qed - qed - ultimately show ?thesis by auto - qed - } moreover { - -- {* The `right to left` direction. - *} - fix c - -- {* Assuming that @{text "c"} is in the sub-tree of @{text "x"}, but - outside of the sub-tree of @{text "a"} in the original graph, *} - assume h: "c \ (subtree r x) - subtree r a" - -- {* we need to show that in the reduced graph, @{text "c"} is still in - the sub-tree of @{text "x"}. *} - have "c \ subtree (r - {(a, b)}) x" - proof - - -- {* The proof goes by showing that the path from @{text "c"} to @{text "x"} - in the original graph is not affected by the removal of @{text "(a,b)"}. - *} - from h have "(c, x) \ r^*" by (unfold subtree_def, auto) - -- {* Extract the path @{text "xs"} from @{text "c"} to @{text "x"} in the original graph. *} - from star_rpath[OF this] obtain xs where rp: "rpath r c xs x" by auto - -- {* Show that it is also a path in the reduced graph. *} - hence "rpath (r - {(a, b)}) c xs x" - -- {* The proof goes by using rule @{thm rpath_transfer} *} - proof(rule rpath_transfer) - -- {* We need to show all edges on the path are still in the reduced graph. *} - show "edges_on (c # xs) \ r - {(a, b)}" - proof - - -- {* It is easy to show that all the edges are in the original graph. *} - from rpath_edges_on [OF rp] have " edges_on (c # xs) \ r" . - -- {* The essential part is to show that @{text "(a, b)"} is not on the path. *} - moreover have "(a,b) \ edges_on (c#xs)" - proof - -- {* Proof by contradiction, suppose otherwise: *} - assume otherwise: "(a, b) \ edges_on (c#xs)" - -- {* Then @{text "(a, b)"} is in the middle of the path. - with @{text "l1"} and @{text "l2"} be the nodes in - the front and rear respectively. *} - then obtain l1 l2 where eq_xs: - "c#xs = l1 @ [a, b] @ l2" by (unfold edges_on_def, blast) - -- {* From this, it can be shown that @{text "c"} is - in the sub-tree of @{text "a"} *} - have "c \ subtree r a" - proof(cases "l1 = []") - case True - -- {* If @{text "l1"} is null, it can be derived that @{text "c = a"}. *} - with eq_xs have "c = a" by auto - -- {* So, @{text "c"} is obviously in the sub-tree of @{text "a"}. *} - thus ?thesis by (unfold subtree_def, auto) - next - case False - -- {* When @{text "l1"} is not null, it must have a tail @{text "es"}: *} - then obtain e es where "l1 = e#es" by (cases l1, auto) - -- {* The relation of this tail with @{text "xs"} is derived: *} - with eq_xs have "xs = es@[a,b]@l2" by auto - -- {* From this, a path from @{text "c"} to @{text "a"} is made visible: *} - from rp[unfolded this] have "rpath r c (es @ [a] @ (b#l2)) x" by simp - thus ?thesis - proof(cases rule:rpath_appendE) - -- {* The path from @{text "c"} to @{text "a"} is extraced - using @{thm "rpath_appendE"}: *} - case 1 - from rpath_star[OF this(1)] - -- {* The extracted path servers as a witness that @{text "c"} is - in the sub-tree of @{text "a"}: *} - show ?thesis by (simp add:subtree_def) - qed - qed with h show False by auto - qed ultimately show ?thesis by auto - qed - qed - -- {* From , it is shown that @{text "c"} is in the sub-tree of @{text "x"} - inthe reduced graph. *} - from rpath_star[OF this] show ?thesis by (auto simp:subtree_def) - qed - } - -- {* The equality of sets is derived from the two directions just proved. *} - ultimately show ?thesis by auto -qed - -lemma set_del_rootI: - assumes "r1 \ r" - and "a \ Domain r1" - shows "root (r - r1) a" -proof - - let ?r = "r - r1" - { fix a' - assume neq: "a' \ a" - have "a \ subtree ?r a'" - proof - assume "a \ subtree ?r a'" - hence "(a, a') \ ?r^*" by (auto simp:subtree_def) - from star_rpath[OF this] obtain xs - where rp: "rpath ?r a xs a'" by auto - from rpathE[OF this] and neq - obtain z zs where h: "(a, z) \ ?r" "rpath ?r z zs a'" "xs = z#zs" by auto - from assms(2) obtain z' where z'_in: "(a, z') \ r1" by (auto simp:DomainE) - with assms(1) have "(a, z') \ r" by auto - moreover from h(1) have "(a, z) \ r" by simp - ultimately have "z' = z" using sgv by (auto simp:single_valued_def) - from z'_in[unfolded this] and h(1) show False by auto - qed - } thus ?thesis by (intro rootI, auto) -qed - -lemma edge_del_no_rootI: - assumes "(a, b) \ r" - shows "root (r - {(a, b)}) a" - by (rule set_del_rootI, insert assms, auto) - -lemma ancestors_children_unique: - assumes "z1 \ ancestors r x \ children r y" - and "z2 \ ancestors r x \ children r y" - shows "z1 = z2" -proof - - from assms have h: - "(x, z1) \ r^+" "(z1, y) \ r" - "(x, z2) \ r^+" "(z2, y) \ r" - by (auto simp:ancestors_def children_def) - - -- {* From this, a path containing @{text "z1"} is obtained. *} - from plus_rpath[OF h(1)] obtain xs1 - where h1: "rpath r x xs1 z1" "xs1 \ []" by auto - from rpath_nnl_lastE[OF this] obtain xs1' where eq_xs1: "xs1 = xs1' @ [z1]" - by auto - from h(2) have h2: "rpath r z1 [y] y" by auto - from rpath_appendI[OF h1(1) h2, unfolded eq_xs1] - have rp1: "rpath r x (xs1' @ [z1, y]) y" by simp - - -- {* Then, another path containing @{text "z2"} is obtained. *} - from plus_rpath[OF h(3)] obtain xs2 - where h3: "rpath r x xs2 z2" "xs2 \ []" by auto - from rpath_nnl_lastE[OF this] obtain xs2' where eq_xs2: "xs2 = xs2' @ [z2]" - by auto - from h(4) have h4: "rpath r z2 [y] y" by auto - from rpath_appendI[OF h3(1) h4, unfolded eq_xs2] - have "rpath r x (xs2' @ [z2, y]) y" by simp - - -- {* Finally @{text "z1 = z2"} is proved by uniqueness of path. *} - from rpath_unique[OF rp1 this] - have "xs1' @ [z1, y] = xs2' @ [z2, y]" . - thus ?thesis by auto -qed - -lemma ancestors_childrenE: - assumes "y \ ancestors r x" - obtains "x \ children r y" - | z where "z \ ancestors r x \ children r y" -proof - - from assms(1) have "(x, y) \ r^+" by (auto simp:ancestors_def) - from tranclD2[OF this] obtain z where - h: "(x, z) \ r\<^sup>*" "(z, y) \ r" by auto - from h(1) - show ?thesis - proof(cases rule:rtranclE) - case base - from h(2)[folded this] have "x \ children r y" - by (auto simp:children_def) - thus ?thesis by (intro that, auto) - next - case (step u) - hence "z \ ancestors r x" by (auto simp:ancestors_def) - moreover from h(2) have "z \ children r y" - by (auto simp:children_def) - ultimately show ?thesis by (intro that, auto) - qed -qed - - -end (* of rtree *) - -lemma subtree_children: - "subtree r x = {x} \ (\ (subtree r ` (children r x)))" (is "?L = ?R") -proof - - { fix z - assume "z \ ?L" - hence "z \ ?R" - proof(cases rule:subtreeE[consumes 1]) - case 2 - hence "(z, x) \ r^+" by (auto simp:ancestors_def) - thus ?thesis - proof(rule tranclE) - assume "(z, x) \ r" - hence "z \ children r x" by (unfold children_def, auto) - moreover have "z \ subtree r z" by (auto simp:subtree_def) - ultimately show ?thesis by auto - next - fix c - assume h: "(z, c) \ r\<^sup>+" "(c, x) \ r" - hence "c \ children r x" by (auto simp:children_def) - moreover from h have "z \ subtree r c" by (auto simp:subtree_def) - ultimately show ?thesis by auto - qed - qed auto - } moreover { - fix z - assume h: "z \ ?R" - have "x \ subtree r x" by (auto simp:subtree_def) - moreover { - assume "z \ \(subtree r ` children r x)" - then obtain y where "(y, x) \ r" "(z, y) \ r^*" - by (auto simp:subtree_def children_def) - hence "(z, x) \ r^*" by auto - hence "z \ ?L" by (auto simp:subtree_def) - } ultimately have "z \ ?L" using h by auto - } ultimately show ?thesis by auto -qed - -context fsubtree -begin - -lemma finite_subtree: - shows "finite (subtree r x)" -proof(induct rule:wf_induct[OF wf]) - case (1 x) - have "finite (\(subtree r ` children r x))" - proof(rule finite_Union) - show "finite (subtree r ` children r x)" - proof(cases "children r x = {}") - case True - thus ?thesis by auto - next - case False - hence "x \ Range r" by (auto simp:children_def) - from fb[rule_format, OF this] - have "finite (children r x)" . - thus ?thesis by (rule finite_imageI) - qed - next - fix M - assume "M \ subtree r ` children r x" - then obtain y where h: "y \ children r x" "M = subtree r y" by auto - hence "(y, x) \ r" by (auto simp:children_def) - from 1[rule_format, OF this, folded h(2)] - show "finite M" . - qed - thus ?case - by (unfold subtree_children finite_Un, auto) -qed - -end - -definition "pairself f = (\(a, b). (f a, f b))" - -definition "rel_map f r = (pairself f ` r)" - -lemma rel_mapE: - assumes "(a, b) \ rel_map f r" - obtains c d - where "(c, d) \ r" "(a, b) = (f c, f d)" - using assms - by (unfold rel_map_def pairself_def, auto) - -lemma rel_mapI: - assumes "(a, b) \ r" - and "c = f a" - and "d = f b" - shows "(c, d) \ rel_map f r" - using assms - by (unfold rel_map_def pairself_def, auto) - -lemma map_appendE: - assumes "map f zs = xs @ ys" - obtains xs' ys' - where "zs = xs' @ ys'" "xs = map f xs'" "ys = map f ys'" -proof - - have "\ xs' ys'. zs = xs' @ ys' \ xs = map f xs' \ ys = map f ys'" - using assms - proof(induct xs arbitrary:zs ys) - case (Nil zs ys) - thus ?case by auto - next - case (Cons x xs zs ys) - note h = this - show ?case - proof(cases zs) - case (Cons e es) - with h have eq_x: "map f es = xs @ ys" "x = f e" by auto - from h(1)[OF this(1)] - obtain xs' ys' where "es = xs' @ ys'" "xs = map f xs'" "ys = map f ys'" - by blast - with Cons eq_x - have "zs = (e#xs') @ ys' \ x # xs = map f (e#xs') \ ys = map f ys'" by auto - thus ?thesis by metis - qed (insert h, auto) - qed - thus ?thesis by (auto intro!:that) -qed - -lemma rel_map_mono: - assumes "r1 \ r2" - shows "rel_map f r1 \ rel_map f r2" - using assms - by (auto simp:rel_map_def pairself_def) - -lemma rel_map_compose [simp]: - shows "rel_map f1 (rel_map f2 r) = rel_map (f1 o f2) r" - by (auto simp:rel_map_def pairself_def) - -lemma edges_on_map: "edges_on (map f xs) = rel_map f (edges_on xs)" -proof - - { fix a b - assume "(a, b) \ edges_on (map f xs)" - then obtain l1 l2 where eq_map: "map f xs = l1 @ [a, b] @ l2" - by (unfold edges_on_def, auto) - hence "(a, b) \ rel_map f (edges_on xs)" - by (auto elim!:map_appendE intro!:rel_mapI simp:edges_on_def) - } moreover { - fix a b - assume "(a, b) \ rel_map f (edges_on xs)" - then obtain c d where - h: "(c, d) \ edges_on xs" "(a, b) = (f c, f d)" - by (elim rel_mapE, auto) - then obtain l1 l2 where - eq_xs: "xs = l1 @ [c, d] @ l2" - by (auto simp:edges_on_def) - hence eq_map: "map f xs = map f l1 @ [f c, f d] @ map f l2" by auto - have "(a, b) \ edges_on (map f xs)" - proof - - from h(2) have "[f c, f d] = [a, b]" by simp - from eq_map[unfolded this] show ?thesis by (auto simp:edges_on_def) - qed - } ultimately show ?thesis by auto -qed - -lemma image_id: - assumes "\ x. x \ A \ f x = x" - shows "f ` A = A" - using assms by (auto simp:image_def) - -lemma rel_map_inv_id: - assumes "inj_on f ((Domain r) \ (Range r))" - shows "(rel_map (inv_into ((Domain r) \ (Range r)) f \ f) r) = r" -proof - - let ?f = "(inv_into (Domain r \ Range r) f \ f)" - { - fix a b - assume h0: "(a, b) \ r" - have "pairself ?f (a, b) = (a, b)" - proof - - from assms h0 have "?f a = a" by (auto intro:inv_into_f_f) - moreover have "?f b = b" - by (insert h0, simp, intro inv_into_f_f[OF assms], auto intro!:RangeI) - ultimately show ?thesis by (auto simp:pairself_def) - qed - } thus ?thesis by (unfold rel_map_def, intro image_id, case_tac x, auto) -qed - -lemma rel_map_acyclic: - assumes "acyclic r" - and "inj_on f ((Domain r) \ (Range r))" - shows "acyclic (rel_map f r)" -proof - - let ?D = "Domain r \ Range r" - { fix a - assume "(a, a) \ (rel_map f r)^+" - from plus_rpath[OF this] - obtain xs where rp: "rpath (rel_map f r) a xs a" "xs \ []" by auto - from rpath_nnl_lastE[OF this] obtain xs' where eq_xs: "xs = xs'@[a]" by auto - from rpath_edges_on[OF rp(1)] - have h: "edges_on (a # xs) \ rel_map f r" . - from edges_on_map[of "inv_into ?D f" "a#xs"] - have "edges_on (map (inv_into ?D f) (a # xs)) = rel_map (inv_into ?D f) (edges_on (a # xs))" . - with rel_map_mono[OF h, of "inv_into ?D f"] - have "edges_on (map (inv_into ?D f) (a # xs)) \ rel_map ((inv_into ?D f) o f) r" by simp - from this[unfolded eq_xs] - have subr: "edges_on (map (inv_into ?D f) (a # xs' @ [a])) \ rel_map (inv_into ?D f \ f) r" . - have "(map (inv_into ?D f) (a # xs' @ [a])) = (inv_into ?D f a) # map (inv_into ?D f) xs' @ [inv_into ?D f a]" - by simp - from edges_on_rpathI[OF subr[unfolded this]] - have "rpath (rel_map (inv_into ?D f \ f) r) - (inv_into ?D f a) (map (inv_into ?D f) xs' @ [inv_into ?D f a]) (inv_into ?D f a)" . - hence "(inv_into ?D f a, inv_into ?D f a) \ (rel_map (inv_into ?D f \ f) r)^+" - by (rule rpath_plus, simp) - moreover have "(rel_map (inv_into ?D f \ f) r) = r" by (rule rel_map_inv_id[OF assms(2)]) - moreover note assms(1) - ultimately have False by (unfold acyclic_def, auto) - } thus ?thesis by (auto simp:acyclic_def) -qed - -lemma relpow_mult: - "((r::'a rel) ^^ m) ^^ n = r ^^ (m*n)" -proof(induct n arbitrary:m) - case (Suc k m) - thus ?case - proof - - have h: "(m * k + m) = (m + m * k)" by auto - show ?thesis - apply (simp add:Suc relpow_add[symmetric]) - by (unfold h, simp) - qed -qed simp - -lemma compose_relpow_2: - assumes "r1 \ r" - and "r2 \ r" - shows "r1 O r2 \ r ^^ (2::nat)" -proof - - { fix a b - assume "(a, b) \ r1 O r2" - then obtain e where "(a, e) \ r1" "(e, b) \ r2" - by auto - with assms have "(a, e) \ r" "(e, b) \ r" by auto - hence "(a, b) \ r ^^ (Suc (Suc 0))" by auto - } thus ?thesis by (auto simp:numeral_2_eq_2) -qed - -lemma acyclic_compose: - assumes "acyclic r" - and "r1 \ r" - and "r2 \ r" - shows "acyclic (r1 O r2)" -proof - - { fix a - assume "(a, a) \ (r1 O r2)^+" - from trancl_mono[OF this compose_relpow_2[OF assms(2, 3)]] - have "(a, a) \ (r ^^ 2) ^+" . - from trancl_power[THEN iffD1, OF this] - obtain n where h: "(a, a) \ (r ^^ 2) ^^ n" "n > 0" by blast - from this(1)[unfolded relpow_mult] have h2: "(a, a) \ r ^^ (2 * n)" . - have "(a, a) \ r^+" - proof(cases rule:trancl_power[THEN iffD2]) - from h(2) h2 show "\n>0. (a, a) \ r ^^ n" - by (rule_tac x = "2*n" in exI, auto) - qed - with assms have "False" by (auto simp:acyclic_def) - } thus ?thesis by (auto simp:acyclic_def) -qed - -lemma children_compose_unfold: - "children (r1 O r2) x = \ (children r1 ` (children r2 x))" - by (auto simp:children_def) - -lemma fbranch_compose: - assumes "fbranch r1" - and "fbranch r2" - shows "fbranch (r1 O r2)" -proof - - { fix x - assume "x\Range (r1 O r2)" - then obtain y z where h: "(y, z) \ r1" "(z, x) \ r2" by auto - have "finite (children (r1 O r2) x)" - proof(unfold children_compose_unfold, rule finite_Union) - show "finite (children r1 ` children r2 x)" - proof(rule finite_imageI) - from h(2) have "x \ Range r2" by auto - from assms(2)[unfolded fbranch_def, rule_format, OF this] - show "finite (children r2 x)" . - qed - next - fix M - assume "M \ children r1 ` children r2 x" - then obtain y where h1: "y \ children r2 x" "M = children r1 y" by auto - show "finite M" - proof(cases "children r1 y = {}") - case True - with h1(2) show ?thesis by auto - next - case False - hence "y \ Range r1" by (unfold children_def, auto) - from assms(1)[unfolded fbranch_def, rule_format, OF this, folded h1(2)] - show ?thesis . - qed - qed - } thus ?thesis by (unfold fbranch_def, auto) -qed - -lemma finite_fbranchI: - assumes "finite r" - shows "fbranch r" -proof - - { fix x - assume "x \Range r" - have "finite (children r x)" - proof - - have "{y. (y, x) \ r} \ Domain r" by (auto) - from rev_finite_subset[OF finite_Domain[OF assms] this] - have "finite {y. (y, x) \ r}" . - thus ?thesis by (unfold children_def, simp) - qed - } thus ?thesis by (auto simp:fbranch_def) -qed - -lemma subset_fbranchI: - assumes "fbranch r1" - and "r2 \ r1" - shows "fbranch r2" -proof - - { fix x - assume "x \Range r2" - with assms(2) have "x \ Range r1" by auto - from assms(1)[unfolded fbranch_def, rule_format, OF this] - have "finite (children r1 x)" . - hence "finite (children r2 x)" - proof(rule rev_finite_subset) - from assms(2) - show "children r2 x \ children r1 x" by (auto simp:children_def) - qed - } thus ?thesis by (auto simp:fbranch_def) -qed - -lemma children_subtree: - shows "children r x \ subtree r x" - by (auto simp:children_def subtree_def) - -lemma children_union_kept: - assumes "x \ Range r'" - shows "children (r \ r') x = children r x" - using assms - by (auto simp:children_def) - -lemma wf_rbase: - assumes "wf r" - obtains b where "(b, a) \ r^*" "\ c. (c, b) \ r" -proof - - from assms - have "\ b. (b, a) \ r^* \ (\ c. (c, b) \ r)" - proof(induct) - case (less x) - thus ?case - proof(cases "\ z. (z, x) \ r") - case False - moreover have "(x, x) \ r^*" by auto - ultimately show ?thesis by metis - next - case True - then obtain z where h_z: "(z, x) \ r" by auto - from less[OF this] - obtain b where "(b, z) \ r^*" "(\c. (c, b) \ r)" - by auto - moreover from this(1) h_z have "(b, x) \ r^*" by auto - ultimately show ?thesis by metis - qed - qed - with that show ?thesis by metis -qed - -lemma wf_base: - assumes "wf r" - and "a \ Range r" - obtains b where "(b, a) \ r^+" "\ c. (c, b) \ r" -proof - - from assms(2) obtain a' where h_a: "(a', a) \ r" by auto - from wf_rbase[OF assms(1), of a] - obtain b where h_b: "(b, a) \ r\<^sup>*" "\c. (c, b) \ r" by auto - from rtranclD[OF this(1)] - have "b = a \ b \ a \ (b, a) \ r\<^sup>+" by auto - moreover have "b \ a" using h_a h_b(2) by auto - ultimately have "(b, a) \ r\<^sup>+" by auto - with h_b(2) and that show ?thesis by metis -qed - -end \ No newline at end of file diff -r ed938e2246b9 -r 0525670d8e6a log --- a/log Thu Jan 28 21:14:17 2016 +0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,464 +0,0 @@ -修改集: 89:2056d9f481e2 -标签: tip -用户: zhangx -日期: Thu Jan 28 16:36:46 2016 +0800 -摘要: Slightly modified ExtGG.thy and PrioG.thy. - -修改集: 88:83dd5345d5d0 -父亲: 83:d239aa953315 -父亲: 87:33cb65e00ac0 -用户: zhangx -日期: Thu Jan 28 16:33:49 2016 +0800 -摘要: Merged back ExtGG.thy and PrioG.thy. - -修改集: 87:33cb65e00ac0 -用户: zhangx -日期: Thu Jan 28 15:36:48 2016 +0800 -摘要: Tracking ExtGG.thy etc., so that a update to 83 is possible. - -修改集: 86:2106021bae53 -用户: zhangx -日期: Thu Jan 28 07:46:05 2016 +0800 -摘要: Added PrioG.thy again - -修改集: 85:61a4429e7d4d -父亲: 84:c0a4e840aefe -父亲: 33:7f87232d9424 -用户: Christian Urban -日期: Wed Jan 27 13:50:02 2016 +0000 -摘要: merged - -修改集: 84:c0a4e840aefe -父亲: 77:b6ea51cd2e88 -用户: Christian Urban -日期: Wed Jan 27 13:47:08 2016 +0000 -摘要: some small changes to Correctness and Paper - -修改集: 83:d239aa953315 -用户: zhangx -日期: Thu Jan 28 07:43:05 2016 +0800 -摘要: Added PrioG.thy as a parallel copy of Correctness.thy - -修改集: 82:cfd644dfc3b4 -用户: zhangx -日期: Wed Jan 27 23:34:23 2016 +0800 -摘要: The parallel of Implementation.thy, i.e. ExtGG.thy has been updated. 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-修改集: 38:c89013dca1aa -用户: Christian Urban -日期: Fri May 30 07:56:39 2014 +0100 -摘要: finished proof of acyclity - -修改集: 37:c820ac0f3088 -用户: Christian Urban -日期: Sat May 24 12:39:12 2014 +0100 -摘要: simplified RAG_acyclic proof - -修改集: 36:af38526275f8 -用户: Christian Urban -日期: Fri May 23 15:19:32 2014 +0100 -摘要: added a test theory for polishing teh proofs - -修改集: 35:92f61f6a0fe7 -用户: Christian Urban -日期: Thu May 22 17:40:39 2014 +0100 -摘要: added a bit more text to the paper and separated a theory about Max - -修改集: 34:313acffe63b6 -父亲: 32:9b9f2117561f -用户: Christian Urban -日期: Tue May 20 12:49:21 2014 +0100 -摘要: updated ROOT file - -修改集: 33:7f87232d9424 -父亲: 29:408ff78ce28f -用户: Christian Urban -日期: Wed May 14 11:52:53 2014 +0100 -摘要: test - -修改集: 32:9b9f2117561f -用户: Christian Urban -日期: Thu May 15 16:02:44 2014 +0100 -摘要: simplified the cp_rec proof - -修改集: 31:e861aff29655 -用户: Christian Urban -日期: Tue May 06 14:36:40 2014 +0100 -摘要: made some modifications. - -修改集: 30:8f026b608378 -用户: Christian Urban -日期: Wed Mar 12 10:08:20 2014 +0000 -摘要: added paper - -修改集: 29:408ff78ce28f -用户: Christian Urban -日期: Tue Mar 04 16:47:54 2014 +0000 -摘要: updated readme - -修改集: 28:7fa738a9615a -用户: Christian Urban -日期: Tue Mar 04 16:38:38 2014 +0000 -摘要: updated - -修改集: 27:6b1141c5e24c -用户: Christian Urban -日期: Tue Mar 04 15:49:36 2014 +0000 -摘要: cleaned up - -修改集: 26:da7a6ccfa7a9 -用户: Christian Urban -日期: Tue Mar 04 15:30:24 2014 +0000 -摘要: updated - -修改集: 25:a9c0eeb00cc3 -用户: Christian Urban -日期: Tue Mar 04 15:27:59 2014 +0000 -摘要: added two more references - -修改集: 24:6f50e6a8c6e0 -用户: Christian Urban -日期: Tue Mar 04 09:40:40 2014 +0000 -摘要: some additions - -修改集: 23:24e6884d9258 -用户: Christian Urban -日期: Tue Mar 04 08:45:11 2014 +0000 -摘要: made some small chages - -修改集: 22:9f0b78fcc894 -用户: Christian Urban -日期: Mon Mar 03 16:22:48 2014 +0000 -摘要: updated - -修改集: 21:55d1591b17f0 -用户: Christian Urban -日期: Fri Feb 28 12:49:58 2014 +0000 -摘要: added llncs to journal - -修改集: 20:b56616fd88dd -用户: Christian Urban -日期: Tue Feb 25 20:01:47 2014 +0000 -摘要: added - -修改集: 19:3cc70bd49588 -用户: Christian Urban -日期: Thu Jun 20 23:28:26 2013 -0400 -摘要: added paper - -修改集: 18:598409a21f4c -用户: Christian Urban -日期: Thu Jun 20 13:50:01 2013 -0400 -摘要: added nasa talk - -修改集: 17:105715a0a807 -用户: Christian Urban -日期: Sat Dec 22 14:50:29 2012 +0000 -摘要: updated - -修改集: 16:9764023f719e -用户: Christian Urban -日期: Sat Dec 22 01:58:45 2012 +0000 -摘要: added - -修改集: 15:9e664c268e25 -用户: Christian Urban -日期: Fri Dec 21 23:32:58 2012 +0000 -摘要: added - -修改集: 14:1bf194825a4e -用户: Christian Urban -日期: Fri Dec 21 18:06:00 2012 +0000 -摘要: more one the implementation - -修改集: 13:735e36c64a71 -用户: Christian Urban -日期: Fri Dec 21 13:30:14 2012 +0000 -摘要: added explanation of the code - -修改集: 12:85116bc854c0 -用户: Christian Urban -日期: Fri Dec 21 00:24:30 2012 +0000 -摘要: updated - -修改集: 11:8e02fb168350 -用户: Christian Urban -日期: Thu Dec 20 14:54:06 2012 +0000 -摘要: added - -修改集: 10:242a781135ba -用户: Christian Urban -日期: Thu Dec 20 12:23:44 2012 +0000 -摘要: added two papers about PIP on multiprocs - -修改集: 9:a8e8ec87a933 -用户: Christian Urban -日期: Thu Dec 20 10:53:31 2012 +0000 -摘要: added original Sha paper - -修改集: 8:5ba3d79622da -用户: Christian Urban -日期: Wed Dec 19 23:46:36 2012 +0000 -摘要: added a paragraph about RAGS - -修改集: 7:0514be2ad83e -用户: Christian Urban -日期: Wed Dec 19 12:51:06 2012 +0000 -摘要: started code explanation - -修改集: 6:7f2493296c39 -用户: Christian Urban -日期: Mon Dec 17 12:34:24 2012 +0000 -摘要: updated - -修改集: 5:0f2d4b78f839 -用户: Christian Urban -日期: Mon Dec 10 21:27:22 2012 +0000 -摘要: updated - -修改集: 4:9d667d545e32 -用户: Christian Urban -日期: Thu Dec 06 16:30:57 2012 +0000 -摘要: added - -修改集: 3:51019d035a79 -用户: Christian Urban -日期: Thu Dec 06 15:49:20 2012 +0000 -摘要: made everything working - -修改集: 2:a04084de4946 -用户: Christian Urban -日期: Thu Dec 06 15:12:49 2012 +0000 -摘要: added - -修改集: 1:c4783e4ef43f -用户: Christian Urban -日期: Thu Dec 06 15:11:51 2012 +0000 -摘要: added - -修改集: 0:110247f9d47e -用户: Christian Urban -日期: Thu Dec 06 15:11:21 2012 +0000 -摘要: added -