diff -r 3fa70b12c117 -r b0054fb0d1ce CpsG.thy~ --- a/CpsG.thy~ Wed Jan 13 15:22:14 2016 +0000 +++ b/CpsG.thy~ Wed Jan 13 23:39:59 2016 +0800 @@ -1,47 +1,3208 @@ -section {* - This file contains lemmas used to guide the recalculation of current precedence - after every system call (or system operation) +theory CpsG +imports PIPDefs +begin + +(* 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 + +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 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 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 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 + +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: "\ ?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 i1 i2 + let ?i3 = "Suc i2" + assume lt12: "i1 < i2" + and "i1 < length s" "i2 < length s" + hence le_i3: "?i3 \ length s" by auto + from moment_plus [OF this] + obtain e where eq_m: "moment ?i3 s = e#moment i2 s" by auto + have "i2 < ?i3" 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 + } + 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. *} -theory CpsG -imports PrioG Max RTree +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 -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" +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 + +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 + +lemma length_down_to_in: + assumes le_ij: "i \ j" + and le_js: "j \ length s" + shows "length (down_to j i s) = j - i" +proof - + have "length (down_to j i s) = length (from_to i j (rev s))" + by (unfold down_to_def, auto) + also have "\ = j - i" + proof(rule length_from_to_in[OF le_ij]) + from le_js show "j \ length (rev s)" by simp + qed + finally show ?thesis . +qed + + +lemma moment_head: + assumes le_it: "Suc i \ length t" + obtains e where "moment (Suc i) t = e#moment i t" +proof - + have "i \ Suc i" by simp + from length_down_to_in [OF this le_it] + have "length (down_to (Suc i) i t) = 1" by auto + then obtain e where "down_to (Suc i) i t = [e]" + apply (cases "(down_to (Suc i) i t)") by auto + moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t" + by (rule down_to_conc[symmetric], auto) + ultimately have eq_me: "moment (Suc i) t = e#(moment i t)" + by (auto simp:down_to_moment) + from that [OF this] show ?thesis . +qed + +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) -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 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) - -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" - -(* ccc *) - -definition "cp_gen s x = - Max ((the_preced s \ the_thread) ` subtree (tRAG s) x)" - 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}" @@ -503,7 +3664,6 @@ end - sublocale valid_trace < rtree_RAG: rtree "RAG s" proof show "single_valued (RAG s)" @@ -530,7 +3690,6 @@ next show "fsubtree_axioms (RAG s)" proof(unfold fsubtree_axioms_def) - find_theorems wf RAG from wf_RAG show "wf (RAG s)" . qed qed @@ -729,912 +3888,4 @@ definition cps:: "state \ (thread \ precedence) set" where "cps s = {(th, cp s th) | th . th \ threads s}" - -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 - find_theorems readys subtree - 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}" -find_theorems "subtree" "_ - _" RAG -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 - - find_theorems readys subtree s' - 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 -