# HG changeset patch # User zhangx # Date 1454509049 -28800 # Node ID 0c89419b474203528a85e014027922c4b052e7c7 # Parent 43482ab313418c7baf5f0eb176d34aae6cc87f3a Commit to revert diff -r 43482ab31341 -r 0c89419b4742 CpsG.thy --- a/CpsG.thy Wed Feb 03 21:51:57 2016 +0800 +++ b/CpsG.thy Wed Feb 03 22:17:29 2016 +0800 @@ -1,7 +1,9 @@ -theory CpsG +theory PIPBasics imports PIPDefs begin +section {* Generic aulxiliary lemmas *} + lemma f_image_eq: assumes h: "\ a. a \ A \ f a = g a" shows "f ` A = g ` A" @@ -84,6 +86,14 @@ finally show ?thesis by simp qed +lemma rel_eqI: + assumes "\ x y. (x,y) \ A \ (x,y) \ B" + and "\ x y. (x,y) \ B \ (x, y) \ A" + shows "A = B" + using assms by auto + +section {* Lemmas do not depend on trace validity *} + lemma birth_time_lt: assumes "s \ []" shows "last_set th s < length s" @@ -152,23 +162,286 @@ thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp) qed +lemma RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" + by (unfold s_RAG_def, auto) + +lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs" + by (unfold s_waiting_def cs_waiting_def wq_def, auto) + +lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs" + by (unfold s_holding_def wq_def cs_holding_def, simp) + +lemma children_RAG_alt_def: + "children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}" + by (unfold s_RAG_def, auto simp:children_def holding_eq) + +lemma holdents_alt_def: + "holdents s th = the_cs ` (children (RAG (s::state)) (Th th))" + by (unfold children_RAG_alt_def holdents_def, simp add: image_image) + +lemma cntCS_alt_def: + "cntCS s th = card (children (RAG s) (Th th))" + apply (unfold children_RAG_alt_def cntCS_def holdents_def) + by (rule card_image[symmetric], auto simp:inj_on_def) + +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) + +lemma runing_wqE: + assumes "th \ runing s" + and "th \ set (wq s cs)" + obtains rest where "wq s cs = th#rest" +proof - + from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest" + by (meson list.set_cases) + have "th' = th" + proof(rule ccontr) + assume "th' \ th" + hence "th \ hd (wq s cs)" using eq_wq by auto + with assms(2) + have "waiting s th cs" + by (unfold s_waiting_def, fold wq_def, auto) + with assms show False + by (unfold runing_def readys_def, auto) + qed + with eq_wq that show ?thesis by metis +qed + +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) + + +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 + +lemma in_RAG_E: + assumes "(n1, n2) \ RAG (s::state)" + obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs" + | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs" + using assms[unfolded s_RAG_def, folded waiting_eq holding_eq] + by auto + +lemma count_rec1 [simp]: + assumes "Q e" + shows "count Q (e#es) = Suc (count Q es)" + using assms + by (unfold count_def, auto) + +lemma count_rec2 [simp]: + assumes "\Q e" + shows "count Q (e#es) = (count Q es)" + using assms + by (unfold count_def, auto) + +lemma count_rec3 [simp]: + shows "count Q [] = 0" + by (unfold count_def, auto) + +lemma cntP_simp1[simp]: + "cntP (P th cs'#s) th = cntP s th + 1" + by (unfold cntP_def, simp) + +lemma cntP_simp2[simp]: + assumes "th' \ th" + shows "cntP (P th cs'#s) th' = cntP s th'" + using assms + by (unfold cntP_def, simp) + +lemma cntP_simp3[simp]: + assumes "\ isP e" + shows "cntP (e#s) th' = cntP s th'" + using assms + by (unfold cntP_def, cases e, simp+) + +lemma cntV_simp1[simp]: + "cntV (V th cs'#s) th = cntV s th + 1" + by (unfold cntV_def, simp) + +lemma cntV_simp2[simp]: + assumes "th' \ th" + shows "cntV (V th cs'#s) th' = cntV s th'" + using assms + by (unfold cntV_def, simp) + +lemma cntV_simp3[simp]: + assumes "\ isV e" + shows "cntV (e#s) th' = cntV s th'" + using assms + by (unfold cntV_def, cases e, simp+) + +lemma cntP_diff_inv: + assumes "cntP (e#s) th \ cntP s th" + shows "isP e \ actor e = th" +proof(cases e) + case (P th' pty) + show ?thesis + by (cases "(\e. \cs. e = P th cs) (P th' pty)", + insert assms P, auto simp:cntP_def) +qed (insert assms, auto simp:cntP_def) + +lemma cntV_diff_inv: + assumes "cntV (e#s) th \ cntV s th" + shows "isV e \ actor e = th" +proof(cases e) + case (V th' pty) + show ?thesis + by (cases "(\e. \cs. e = V th cs) (V th' pty)", + insert assms V, auto simp:cntV_def) +qed (insert assms, auto simp:cntV_def) + +lemma eq_dependants: "dependants (wq s) = dependants s" + by (simp add: s_dependants_abv wq_def) + +lemma inj_the_preced: + "inj_on (the_preced s) (threads s)" + by (metis inj_onI preced_unique the_preced_def) + +lemma holding_next_thI: + assumes "holding s th cs" + and "length (wq s cs) > 1" + obtains th' where "next_th s th cs th'" +proof - + from assms(1)[folded holding_eq, unfolded cs_holding_def] + have " th \ set (wq s cs) \ th = hd (wq s cs)" + by (unfold s_holding_def, fold wq_def, auto) + then obtain rest where h1: "wq s cs = th#rest" + by (cases "wq s cs", auto) + with assms(2) have h2: "rest \ []" by auto + let ?th' = "hd (SOME q. distinct q \ set q = set rest)" + have "next_th s th cs ?th'" using h1(1) h2 + by (unfold next_th_def, auto) + from that[OF this] show ?thesis . +qed + (* ccc *) - +section {* Locales used to investigate the execution of PIP *} + +text {* + The following locale @{text valid_trace} is used to constrain the + trace to be valid. All properties hold for valid traces are + derived under this locale. +*} locale valid_trace = fixes s assumes vt : "vt s" +text {* + The following locale @{text valid_trace_e} describes + the valid extension of a valid trace. The event @{text "e"} + represents an event in the system, which corresponds + to a one step operation of the PIP protocol. + It is required that @{text "e"} is an event eligible to happen + under state @{text "s"}, which is already required to be valid + by the parent locale @{text "valid_trace"}. + + This locale is used to investigate one step execution of PIP, + properties concerning the effects of @{text "e"}'s execution, + for example, how the values of observation functions are changed, + or how desirable properties are kept invariant, are derived + under this locale. The state before execution is @{text "s"}, while + the state after execution is @{text "e#s"}. Therefore, the lemmas + derived usually relate observations on @{text "e#s"} to those + on @{text "s"}. +*} + locale valid_trace_e = valid_trace + fixes e assumes vt_e: "vt (e#s)" begin +text {* + The following lemma shows that @{text "e"} must be a + eligible event (or a valid step) to be taken under + the state represented by @{text "s"}. +*} lemma pip_e: "PIP s e" using vt_e by (cases, simp) end +text {* + Because @{term "e#s"} is also a valid trace, properties + derived for valid trace @{term s} also hold on @{term "e#s"}. +*} +sublocale valid_trace_e < vat_es!: valid_trace "e#s" + using vt_e + by (unfold_locales, simp) + +text {* + For each specific event (or operation), there is a sublocale + further constraining that the event @{text e} to be that + particular event. + + For example, the following + locale @{text "valid_trace_create"} is the sublocale for + event @{term "Create"}: +*} locale valid_trace_create = valid_trace_e + fixes th prio assumes is_create: "e = Create th prio" @@ -181,27 +454,145 @@ fixes th cs assumes is_p: "e = P th cs" +text {* + locale @{text "valid_trace_p"} is divided further into two + sublocales, namely, @{text "valid_trace_p_h"} + and @{text "valid_trace_p_w"}. +*} + +text {* + The following two sublocales @{text "valid_trace_p_h"} + and @{text "valid_trace_p_w"} represent two complementary + cases under @{text "valid_trace_p"}, where + @{text "valid_trace_p_h"} further constraints that + @{text "wq s cs = []"}, which means the waiting queue of + the requested resource @{text "cs"} is empty, in which + case, the requesting thread @{text "th"} + will take hold of @{text "cs"}. + + Opposite to @{text "valid_trace_p_h"}, + @{text "valid_trace_p_w"} constraints that + @{text "wq s cs \ []"}, which means the waiting queue of + the requested resource @{text "cs"} is nonempty, in which + case, the requesting thread @{text "th"} will be blocked + on @{text "cs"}: + + Peculiar properties will be derived under respective + locales. +*} + +locale valid_trace_p_h = valid_trace_p + + assumes we: "wq s cs = []" + +locale valid_trace_p_w = valid_trace_p + + assumes wne: "wq s cs \ []" +begin + +text {* + The following @{text "holder"} designates + the holder of @{text "cs"} before the @{text "P"}-operation. +*} +definition "holder = hd (wq s cs)" + +text {* + The following @{text "waiters"} designates + the list of threads waiting for @{text "cs"} + before the @{text "P"}-operation. +*} +definition "waiters = tl (wq s cs)" +end + +text {* + @{text "valid_trace_v"} is set for the @{term V}-operation. +*} locale valid_trace_v = valid_trace_e + fixes th cs assumes is_v: "e = V th cs" begin + -- {* The following @{text "rest"} is the tail of + waiting queue of the resource @{text "cs"} + to be released by this @{text "V"}-operation. + *} definition "rest = tl (wq s cs)" + + text {* + The following @{text "wq'"} is the waiting + queue of @{term "cs"} + after the @{text "V"}-operation, which + is simply a reordering of @{term "rest"}. + + The effect of this reordering needs to be + understood by two cases: + \begin{enumerate} + \item When @{text "rest = []"}, + the reordering gives rise to an empty list as well, + which means there is no thread holding or waiting + for resource @{term "cs"}, therefore, it is free. + + \item When @{text "rest \ []"}, the effect of + this reordering is to arbitrarily + switch one thread in @{term "rest"} to the + head, which, by definition take over the hold + of @{term "cs"} and is designated by @{text "taker"} + in the following sublocale @{text "valid_trace_v_n"}. + *} definition "wq' = (SOME q. distinct q \ set q = set rest)" + + text {* + The following @{text "rest'"} is the tail of the + waiting queue after the @{text "V"}-operation. + It plays only auxiliary role to ease reasoning. + *} + definition "rest' = tl wq'" + end +text {* + In the following, @{text "valid_trace_v"} is also + divided into two + sublocales: when @{text "rest"} is empty (represented + by @{text "valid_trace_v_e"}), which means, there is no thread waiting + for @{text "cs"}, therefore, after the @{text "V"}-operation, + it will become free; otherwise (represented + by @{text "valid_trace_v_n"}), one thread + will be picked from those in @{text "rest"} to take + over @{text "cs"}. +*} + +locale valid_trace_v_e = valid_trace_v + + assumes rest_nil: "rest = []" + locale valid_trace_v_n = valid_trace_v + assumes rest_nnl: "rest \ []" - -locale valid_trace_v_e = valid_trace_v + - assumes rest_nil: "rest = []" - -locale valid_trace_set= valid_trace_e + +begin + +text {* + The following @{text "taker"} is the thread to + take over @{text "cs"}. +*} + definition "taker = hd wq'" + +end + + +locale valid_trace_set = valid_trace_e + fixes th prio assumes is_set: "e = Set th prio" context valid_trace begin +text {* + Induction rule introduced to easy the + derivation of properties for valid trace @{term "s"}. + One more premises, namely @{term "valid_trace_e s e"} + is added, so that an interpretation of + @{text "valid_trace_e"} can be instantiated + so that all properties derived so far becomes + available in the proof of induction step. + + You will see its use in the proofs that follows. +*} lemma ind [consumes 0, case_names Nil Cons, induct type]: assumes "PP []" and "(\s e. valid_trace_e s e \ @@ -222,6 +613,11 @@ qed qed +text {* + The following lemma says that if @{text "s"} is a valid state, so + is its any postfix. Where @{term "monent t s"} is the postfix of + @{term "s"} with length @{term "t"}. +*} lemma vt_moment: "\ t. vt (moment t s)" proof(induct rule:ind) case Nil @@ -246,78 +642,43 @@ ultimately show ?thesis by simp qed qed - -lemma finite_threads: - shows "finite (threads s)" -using vt by (induct) (auto elim: step.cases) - end -lemma RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" - by (unfold s_RAG_def, auto) - +text {* + The following locale @{text "valid_moment"} is to inherit the properties + derived on any valid state to the prefix of it, with length @{text "i"}. +*} locale valid_moment = valid_trace + fixes i :: nat -sublocale valid_moment < vat_moment: valid_trace "(moment i s)" +sublocale valid_moment < vat_moment!: valid_trace "(moment i s)" by (unfold_locales, insert vt_moment, auto) -lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs" - by (unfold s_waiting_def cs_waiting_def wq_def, auto) - -lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs" - by (unfold s_holding_def wq_def cs_holding_def, simp) - -lemma runing_ready: - shows "runing s \ readys s" - unfolding runing_def readys_def - by auto - -lemma readys_threads: - shows "readys s \ threads s" - unfolding readys_def - by auto - -lemma wq_v_neq [simp]: - "cs \ cs' \ wq (V thread cs#s) cs' = wq s cs'" - by (auto simp:wq_def Let_def cp_def split:list.splits) - -lemma runing_head: - assumes "th \ runing s" - and "th \ set (wq_fun (schs s) cs)" - shows "th = hd (wq_fun (schs s) cs)" - using assms - by (simp add:runing_def readys_def s_waiting_def wq_def) - -context valid_trace +locale valid_moment_e = valid_moment + + assumes less_i: "i < length s" begin - -lemma runing_wqE: - assumes "th \ runing s" - and "th \ set (wq s cs)" - obtains rest where "wq s cs = th#rest" -proof - - from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest" - by (meson list.set_cases) - have "th' = th" - proof(rule ccontr) - assume "th' \ th" - hence "th \ hd (wq s cs)" using eq_wq by auto - with assms(2) - have "waiting s th cs" - by (unfold s_waiting_def, fold wq_def, auto) - with assms show False - by (unfold runing_def readys_def, auto) - qed - with eq_wq that show ?thesis by metis -qed + definition "next_e = hd (moment (Suc i) s)" + + lemma trace_e: + "moment (Suc i) s = next_e#moment i s" + proof - + from less_i have "Suc i \ length s" by auto + from moment_plus[OF this, folded next_e_def] + show ?thesis . + qed end +sublocale valid_moment_e < vat_moment_e!: valid_trace_e "moment i s" "next_e" + using vt_moment[of "Suc i", unfolded trace_e] + by (unfold_locales, simp) + +section {* Distinctiveness of waiting queues *} + context valid_trace_create begin -lemma wq_neq_simp [simp]: +lemma wq_kept [simp]: shows "wq (e#s) cs' = wq s cs'" using assms unfolding is_create wq_def by (auto simp:Let_def) @@ -331,7 +692,7 @@ context valid_trace_exit begin -lemma wq_neq_simp [simp]: +lemma wq_kept [simp]: shows "wq (e#s) cs' = wq s cs'" using assms unfolding is_exit wq_def by (auto simp:Let_def) @@ -342,7 +703,7 @@ using assms by simp end -context valid_trace_p +context valid_trace_p begin lemma wq_neq_simp [simp]: @@ -358,31 +719,6 @@ show ?thesis by (cases, simp) qed -lemma ready_th_s: "th \ readys s" - using runing_th_s - by (unfold runing_def, auto) - -lemma live_th_s: "th \ threads s" - using readys_threads ready_th_s by auto - -lemma live_th_es: "th \ threads (e#s)" - using live_th_s - by (unfold is_p, simp) - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - lemma th_not_in_wq: shows "th \ set (wq s cs)" proof @@ -426,27 +762,6 @@ using assms unfolding is_v wq_def by (auto simp:Let_def) -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_v] - show ?thesis by (cases, simp) -qed - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - lemma wq_s_cs: "wq s cs = th#rest" proof - @@ -482,7 +797,7 @@ context valid_trace_set begin -lemma wq_neq_simp [simp]: +lemma wq_kept [simp]: shows "wq (e#s) cs' = wq s cs'" using assms unfolding is_set wq_def by (auto simp:Let_def) @@ -496,22 +811,12 @@ 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 finite_threads: + shows "finite (threads s)" + using vt by (induct) (auto elim: step.cases) + +lemma finite_readys [simp]: "finite (readys s)" + using finite_threads readys_threads rev_finite_subset by blast lemma wq_distinct: "distinct (wq s cs)" proof(induct rule:ind) @@ -546,46 +851,11 @@ end +section {* Waiting queues and threads *} + 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)" @@ -619,248 +889,173 @@ 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) +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) + end +lemma (in valid_trace_create) + th_not_in_threads: "th \ threads s" +proof - + from pip_e[unfolded is_create] + show ?thesis by (cases, simp) +qed + +lemma (in valid_trace_create) + threads_es [simp]: "threads (e#s) = threads s \ {th}" + by (unfold is_create, simp) + +lemma (in valid_trace_exit) + threads_es [simp]: "threads (e#s) = threads s - {th}" + by (unfold is_exit, simp) + +lemma (in valid_trace_p) + threads_es [simp]: "threads (e#s) = threads s" + by (unfold is_p, simp) + +lemma (in valid_trace_v) + threads_es [simp]: "threads (e#s) = threads s" + by (unfold is_v, simp) + +lemma (in valid_trace_v) + th_not_in_rest[simp]: "th \ set rest" +proof + assume otherwise: "th \ set rest" + have "distinct (wq s cs)" by (simp add: wq_distinct) + from this[unfolded wq_s_cs] and otherwise + show False by auto +qed + +lemma (in valid_trace_v) distinct_rest: "distinct rest" + by (simp add: distinct_tl rest_def wq_distinct) + +lemma (in valid_trace_v) + set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}" +proof(unfold wq_es_cs wq'_def, rule someI2) + show "distinct rest \ set rest = set rest" + by (simp add: distinct_rest) +next + fix x + assume "distinct x \ set x = set rest" + thus "set x = set (wq s cs) - {th}" + by (unfold wq_s_cs, simp) +qed + +lemma (in valid_trace_exit) + th_not_in_wq: "th \ set (wq s cs)" +proof - + from pip_e[unfolded is_exit] + show ?thesis + by (cases, unfold holdents_def s_holding_def, fold wq_def, + auto elim!:runing_wqE) +qed + +lemma (in valid_trace) wq_threads: + assumes "th \ set (wq s cs)" + shows "th \ threads s" + using assms +proof(induct rule:ind) + case (Nil) + thus ?case by (auto simp:wq_def) +next + case (Cons s e) + interpret vt_e: valid_trace_e s e using Cons by simp + show ?case + proof(cases e) + case (Create th' prio') + interpret vt: valid_trace_create s e th' prio' + using Create by (unfold_locales, simp) + show ?thesis + using Cons.hyps(2) Cons.prems by auto + next + case (Exit th') + interpret vt: valid_trace_exit s e th' + using Exit by (unfold_locales, simp) + show ?thesis + using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto + next + case (P th' cs') + interpret vt: valid_trace_p s e th' cs' + using P by (unfold_locales, simp) + show ?thesis + using Cons.hyps(2) Cons.prems readys_threads + runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv + by fastforce + next + case (V th' cs') + interpret vt: valid_trace_v s e th' cs' + using V by (unfold_locales, simp) + show ?thesis using Cons + using vt.is_v vt.threads_es vt_e.wq_in_inv by blast + next + case (Set th' prio) + interpret vt: valid_trace_set s e th' prio + using Set by (unfold_locales, simp) + show ?thesis using Cons.hyps(2) Cons.prems vt.is_set + by (auto simp:wq_def Let_def) + qed +qed + +section {* RAG and threads *} context valid_trace begin - -text {* (* ddd *) - The nature of the work is like this: since it starts from a very simple and basic - model, even intuitively very `basic` and `obvious` properties need to derived from scratch. - For instance, the fact - that one thread can not be blocked by two critical resources at the same time - is obvious, because only running threads can make new requests, if one is waiting for - a critical resource and get blocked, it can not make another resource request and get - blocked the second time (because it is not running). - - To derive this fact, one needs to prove by contraction and - reason about time (or @{text "moement"}). The reasoning is based on a generic theorem - named @{text "p_split"}, which is about status changing along the time axis. It says if - a condition @{text "Q"} is @{text "True"} at a state @{text "s"}, - but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} - in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history - of events leading to it), such that @{text "Q"} switched - from being @{text "False"} to @{text "True"} and kept being @{text "True"} - till the last moment of @{text "s"}. - - Suppose a thread @{text "th"} is blocked - on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, - since no thread is blocked at the very beginning, by applying - @{text "p_split"} to these two blocking facts, there exist - two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that - @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} - and kept on blocked on them respectively ever since. - - Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}. - However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still - in blocked state at moment @{text "t2"} and could not - make any request and get blocked the second time: Contradiction. -*} - -lemma waiting_unique_pre: (* ddd *) - assumes h11: "thread \ set (wq s cs1)" - and h12: "thread \ hd (wq s cs1)" - assumes h21: "thread \ set (wq s cs2)" - and h22: "thread \ hd (wq s cs2)" - and neq12: "cs1 \ cs2" - shows "False" +lemma dm_RAG_threads: + assumes in_dom: "(Th th) \ Domain (RAG s)" + shows "th \ threads s" proof - - let "?Q" = "\ cs s. thread \ set (wq s cs) \ thread \ hd (wq s cs)" - from h11 and h12 have q1: "?Q cs1 s" by simp - from h21 and h22 have q2: "?Q cs2 s" by simp - have nq1: "\ ?Q cs1 []" by (simp add:wq_def) - have nq2: "\ ?Q cs2 []" by (simp add:wq_def) - from p_split [of "?Q cs1", OF q1 nq1] - obtain t1 where lt1: "t1 < length s" - and np1: "\ ?Q cs1 (moment t1 s)" - and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" by auto - from p_split [of "?Q cs2", OF q2 nq2] - obtain t2 where lt2: "t2 < length s" - and np2: "\ ?Q cs2 (moment t2 s)" - and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" by auto - { fix s cs - assume q: "?Q cs s" - have "thread \ runing s" - proof - assume "thread \ runing s" - hence " \cs. \ (thread \ set (wq_fun (schs s) cs) \ - thread \ hd (wq_fun (schs s) cs))" - by (unfold runing_def s_waiting_def readys_def, auto) - from this[rule_format, of cs] q - show False by (simp add: wq_def) - qed - } note q_not_runing = this - { fix t1 t2 cs1 cs2 - assume lt1: "t1 < length s" - and np1: "\ ?Q cs1 (moment t1 s)" - and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" - and lt2: "t2 < length s" - and np2: "\ ?Q cs2 (moment t2 s)" - and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" - and lt12: "t1 < t2" - let ?t3 = "Suc t2" - from lt2 have le_t3: "?t3 \ length s" by auto - from moment_plus [OF this] - obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto - have "t2 < ?t3" by simp - from nn2 [rule_format, OF this] and eq_m - have h1: "thread \ set (wq (e#moment t2 s) cs2)" and - h2: "thread \ hd (wq (e#moment t2 s) cs2)" by auto - have "vt (e#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - have ?thesis - proof - - have "thread \ runing (moment t2 s)" - proof(cases "thread \ set (wq (moment t2 s) cs2)") - case True - have "e = V thread cs2" - proof - - have eq_th: "thread = hd (wq (moment t2 s) cs2)" - using True and np2 by auto - from vt_e.wq_out_inv[OF True this h2] - show ?thesis . - qed - thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto - next - case False - have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] . - with vt_e.actor_inv[OF vt_e.pip_e] - show ?thesis by auto - qed - moreover have "thread \ runing (moment t2 s)" - by (rule q_not_runing[OF nn1[rule_format, OF lt12]]) - ultimately show ?thesis by simp - qed - } note lt_case = this - show ?thesis - proof - - { assume "t1 < t2" - from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this] - have ?thesis . - } moreover { - assume "t2 < t1" - from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this] - have ?thesis . - } moreover { - assume eq_12: "t1 = t2" - let ?t3 = "Suc t2" - from lt2 have le_t3: "?t3 \ length s" by auto - from moment_plus [OF this] - obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto - have lt_2: "t2 < ?t3" by simp - from nn2 [rule_format, OF this] and eq_m - have h1: "thread \ set (wq (e#moment t2 s) cs2)" and - h2: "thread \ hd (wq (e#moment t2 s) cs2)" by auto - from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12] - have g1: "thread \ set (wq (e#moment t1 s) cs1)" and - g2: "thread \ hd (wq (e#moment t1 s) cs1)" by auto - have "vt (e#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - have "e = V thread cs2 \ e = P thread cs2" - proof(cases "thread \ set (wq (moment t2 s) cs2)") - case True - have "e = V thread cs2" - proof - - have eq_th: "thread = hd (wq (moment t2 s) cs2)" - using True and np2 by auto - from vt_e.wq_out_inv[OF True this h2] - show ?thesis . - qed - thus ?thesis by auto - next - case False - have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] . - thus ?thesis by auto - qed - moreover have "e = V thread cs1 \ e = P thread cs1" - proof(cases "thread \ set (wq (moment t1 s) cs1)") - case True - have eq_th: "thread = hd (wq (moment t1 s) cs1)" - using True and np1 by auto - from vt_e.wq_out_inv[folded eq_12, OF True this g2] - have "e = V thread cs1" . - thus ?thesis by auto - next - case False - have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] . - thus ?thesis by auto - qed - ultimately have ?thesis using neq12 by auto - } ultimately show ?thesis using nat_neq_iff by blast - qed + 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 -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 +lemma rg_RAG_threads: + assumes "(Th th) \ Range (RAG s)" + shows "th \ threads s" + using assms + by (unfold s_RAG_def cs_waiting_def cs_holding_def, + auto intro:wq_threads) + +lemma RAG_threads: + assumes "(Th th) \ Field (RAG s)" + shows "th \ threads s" + using assms + by (metis Field_def UnE dm_RAG_threads rg_RAG_threads) end -(* 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 +section {* The change of @{term RAG} *} text {* The following three lemmas show that @{text "RAG"} does not change @@ -868,36 +1063,18 @@ events, respectively. *} -lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma (in valid_trace_set) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_set RAG_set_unchanged, simp) - -lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma (in valid_trace_create) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_create RAG_create_unchanged, simp) - -lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s" -apply (unfold s_RAG_def s_waiting_def wq_def) -by (simp add:Let_def) - -lemma (in valid_trace_exit) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_exit RAG_exit_unchanged, simp) +lemma (in valid_trace_set) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s" + by (unfold is_set s_RAG_def s_waiting_def wq_def, simp add:Let_def) + +lemma (in valid_trace_create) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s" + by (unfold is_create s_RAG_def s_waiting_def wq_def, simp add:Let_def) + +lemma (in valid_trace_exit) RAG_unchanged[simp]: "(RAG (e # s)) = RAG s" + by (unfold is_exit s_RAG_def s_waiting_def wq_def, simp add:Let_def) context valid_trace_v begin -lemma distinct_rest: "distinct rest" - by (simp add: distinct_tl rest_def wq_distinct) - lemma holding_cs_eq_th: assumes "holding s t cs" shows "t = th" @@ -915,14 +1092,20 @@ by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def) lemma set_wq': "set wq' = set rest" - by (metis (mono_tags, lifting) distinct_rest rest_def - some_eq_ex wq'_def) + by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def) lemma th'_in_inv: assumes "th' \ set wq'" shows "th' \ set rest" using assms set_wq' by simp +lemma runing_th_s: + shows "th \ runing s" +proof - + from pip_e[unfolded is_v] + show ?thesis by (cases, simp) +qed + lemma neq_t_th: assumes "waiting (e#s) t c" shows "t \ th" @@ -945,7 +1128,7 @@ by (simp add: cs_waiting_def waiting_eq) hence "t \ readys s" by (unfold readys_def, auto) hence "t \ runing s" using runing_ready by auto - with runing_th_s[folded otherwise] show ?thesis by auto + with runing_th_s[folded otherwise] show ?thesis by auto qed qed @@ -996,10 +1179,6 @@ thus "x \ []" using rest_nnl by auto qed -definition "taker = hd wq'" - -definition "rest' = tl wq'" - lemma eq_wq': "wq' = taker # rest'" by (simp add: neq_wq' rest'_def taker_def) @@ -1202,18 +1381,6 @@ end -lemma rel_eqI: - assumes "\ x y. (x,y) \ A \ (x,y) \ B" - and "\ x y. (x,y) \ B \ (x, y) \ A" - shows "A = B" - using assms by auto - -lemma in_RAG_E: - assumes "(n1, n2) \ RAG (s::state)" - obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs" - | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs" - using assms[unfolded s_RAG_def, folded waiting_eq holding_eq] - by auto context valid_trace_v begin @@ -1400,183 +1567,29 @@ qed qed -end - -lemma step_RAG_v: -assumes vt: - "vt (V th cs#s)" -shows " - RAG (V th cs # s) = - RAG s - {(Cs cs, Th th)} - - {(Th th', Cs cs) |th'. next_th s th cs th'} \ - {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = ?R") -proof - - interpret vt_v: valid_trace_v s "V th cs" - using assms step_back_vt by (unfold_locales, auto) - show ?thesis using vt_v.RAG_es . -qed - -lemma (in valid_trace_create) - th_not_in_threads: "th \ threads s" -proof - - from pip_e[unfolded is_create] - show ?thesis by (cases, simp) -qed - -lemma (in valid_trace_create) - threads_es [simp]: "threads (e#s) = threads s \ {th}" - by (unfold is_create, simp) - -lemma (in valid_trace_exit) - threads_es [simp]: "threads (e#s) = threads s - {th}" - by (unfold is_exit, simp) - -lemma (in valid_trace_p) - threads_es [simp]: "threads (e#s) = threads s" - by (unfold is_p, simp) - -lemma (in valid_trace_v) - threads_es [simp]: "threads (e#s) = threads s" - by (unfold is_v, simp) - -lemma (in valid_trace_v) - th_not_in_rest[simp]: "th \ set rest" -proof - assume otherwise: "th \ set rest" - have "distinct (wq s cs)" by (simp add: wq_distinct) - from this[unfolded wq_s_cs] and otherwise - show False by auto -qed - -lemma (in valid_trace_v) - set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}" -proof(unfold wq_es_cs wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) +lemma + finite_RAG_kept: + assumes "finite (RAG s)" + shows "finite (RAG (e#s))" +proof(cases "rest = []") + case True + interpret vt: valid_trace_v_e using True + by (unfold_locales, simp) + show ?thesis using assms + by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp) next - fix x - assume "distinct x \ set x = set rest" - thus "set x = set (wq s cs) - {th}" - by (unfold wq_s_cs, simp) -qed - -lemma (in valid_trace_exit) - th_not_in_wq: "th \ set (wq s cs)" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold holdents_def s_holding_def, fold wq_def, - auto elim!:runing_wqE) + case False + interpret vt: valid_trace_v_n using False + by (unfold_locales, simp) + show ?thesis using assms + by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp) qed -lemma (in valid_trace) wq_threads: - assumes "th \ set (wq s cs)" - shows "th \ threads s" - using assms -proof(induct rule:ind) - case (Nil) - thus ?case by (auto simp:wq_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th' prio') - interpret vt: valid_trace_create s e th' prio' - using Create by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems by auto - next - case (Exit th') - interpret vt: valid_trace_exit s e th' - using Exit by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto - next - case (P th' cs') - interpret vt: valid_trace_p s e th' cs' - using P by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems readys_threads - runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv - by fastforce - next - case (V th' cs') - interpret vt: valid_trace_v s e th' cs' - using V by (unfold_locales, simp) - show ?thesis using Cons - using vt.is_v vt.threads_es vt_e.wq_in_inv by blast - next - case (Set th' prio) - interpret vt: valid_trace_set s e th' prio - using Set by (unfold_locales, simp) - show ?thesis using Cons.hyps(2) Cons.prems vt.is_set - by (auto simp:wq_def Let_def) - qed -qed - -context valid_trace -begin - -lemma dm_RAG_threads: - assumes in_dom: "(Th th) \ Domain (RAG s)" - shows "th \ threads s" -proof - - from in_dom obtain n where "(Th th, n) \ RAG s" by auto - moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto - ultimately have "(Th th, Cs cs) \ RAG s" by simp - hence "th \ set (wq s cs)" - by (unfold s_RAG_def, auto simp:cs_waiting_def) - from wq_threads [OF this] show ?thesis . -qed - -lemma rg_RAG_threads: - assumes "(Th th) \ Range (RAG s)" - shows "th \ threads s" - using assms - by (unfold s_RAG_def cs_waiting_def cs_holding_def, - auto intro:wq_threads) - -lemma RAG_threads: - assumes "(Th th) \ Field (RAG s)" - shows "th \ threads s" - using assms - by (metis Field_def UnE dm_RAG_threads rg_RAG_threads) - end -lemma (in valid_trace_v) - preced_es [simp]: "preced th (e#s) = preced th s" - by (unfold is_v preced_def, simp) - -lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s" -proof - fix th' - show "the_preced (V th cs # s) th' = the_preced s th'" - by (unfold the_preced_def preced_def, simp) -qed - -lemma (in valid_trace_v) - the_preced_es: "the_preced (e#s) = the_preced s" - by (unfold is_v preced_def, simp) - context valid_trace_p begin -lemma not_holding_s_th_cs: "\ holding s th cs" -proof - assume otherwise: "holding s th cs" - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - moreover have "(Cs cs, Th th) \ RAG s" - using otherwise cs_holding_def - holding_eq th_not_in_wq by auto - ultimately show ?thesis by auto - qed -qed - lemma waiting_kept: assumes "waiting s th' cs'" shows "waiting (e#s) th' cs'" @@ -1584,7 +1597,7 @@ by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2) rotate1.simps(2) self_append_conv2 set_rotate1 th_not_in_wq waiting_eq wq_es_cs wq_neq_simp) - + lemma holding_kept: assumes "holding s th' cs'" shows "holding (e#s) th' cs'" @@ -1602,123 +1615,16 @@ thus ?thesis by (simp add: cs_holding_def holding_eq) qed - -end - -locale valid_trace_p_h = valid_trace_p + - assumes we: "wq s cs = []" - -locale valid_trace_p_w = valid_trace_p + - assumes wne: "wq s cs \ []" -begin - -definition "holder = hd (wq s cs)" -definition "waiters = tl (wq s cs)" -definition "waiters' = waiters @ [th]" - -lemma wq_s_cs: "wq s cs = holder#waiters" - by (simp add: holder_def waiters_def wne) - -lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]" - by (simp add: wq_es_cs wq_s_cs) - -lemma waiting_es_th_cs: "waiting (e#s) th cs" - using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto - -lemma RAG_edge: "(Th th, Cs cs) \ RAG (e#s)" - by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto) - -lemma holding_esE: - assumes "holding (e#s) th' cs'" - obtains "holding s th' cs'" - using assms -proof(cases "cs' = cs") - case False - hence "wq (e#s) cs' = wq s cs'" by simp - with assms show ?thesis - using cs_holding_def holding_eq that by auto -next - case True - with assms show ?thesis - by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that - wq_es_cs' wq_s_cs) +end + +lemma (in valid_trace_p) th_not_waiting: "\ waiting s th c" +proof - + have "th \ readys s" + using runing_ready runing_th_s by blast + thus ?thesis + by (unfold readys_def, auto) qed -lemma waiting_esE: - assumes "waiting (e#s) th' cs'" - obtains "th' \ th" "waiting s th' cs'" - | "th' = th" "cs' = cs" -proof(cases "waiting s th' cs'") - case True - have "th' \ th" - proof - assume otherwise: "th' = th" - from True[unfolded this] - show False by (simp add: th_not_waiting) - qed - from that(1)[OF this True] show ?thesis . -next - case False - hence "th' = th \ cs' = cs" - by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) - set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp) - with that(2) show ?thesis by metis -qed - -lemma RAG_es: "RAG (e # s) = RAG s \ {(Th th, Cs cs)}" (is "?L = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from this(3) - show ?thesis - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case 2 - thus ?thesis using waiting(1,2) by auto - qed - next - case (holding th' cs') - from this(3) - show ?thesis - proof(cases rule:holding_esE) - case 1 - with holding(1,2) - show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed -next - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \ RAG s \ (n1 = Th th \ n2 = Cs cs)" by auto - thus "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ RAG s" - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from waiting_kept[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - from holding_kept[OF this(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - next - assume "n1 = Th th \ n2 = Cs cs" - thus ?thesis using RAG_edge by auto - qed -qed - -end - context valid_trace_p_h begin @@ -1815,10 +1721,116 @@ end +context valid_trace_p_w +begin + +lemma wq_s_cs: "wq s cs = holder#waiters" + by (simp add: holder_def waiters_def wne) + +lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]" + by (simp add: wq_es_cs wq_s_cs) + +lemma waiting_es_th_cs: "waiting (e#s) th cs" + using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto + +lemma RAG_edge: "(Th th, Cs cs) \ RAG (e#s)" + by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto) + +lemma holding_esE: + assumes "holding (e#s) th' cs'" + obtains "holding s th' cs'" + using assms +proof(cases "cs' = cs") + case False + hence "wq (e#s) cs' = wq s cs'" by simp + with assms show ?thesis + using cs_holding_def holding_eq that by auto +next + case True + with assms show ?thesis + by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that + wq_es_cs' wq_s_cs) +qed + +lemma waiting_esE: + assumes "waiting (e#s) th' cs'" + obtains "th' \ th" "waiting s th' cs'" + | "th' = th" "cs' = cs" +proof(cases "waiting s th' cs'") + case True + have "th' \ th" + proof + assume otherwise: "th' = th" + from True[unfolded this] + show False by (simp add: th_not_waiting) + qed + from that(1)[OF this True] show ?thesis . +next + case False + hence "th' = th \ cs' = cs" + by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) + set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp) + with that(2) show ?thesis by metis +qed + +lemma RAG_es: "RAG (e # s) = RAG s \ {(Th th, Cs cs)}" (is "?L = ?R") +proof(rule rel_eqI) + fix n1 n2 + assume "(n1, n2) \ ?L" + thus "(n1, n2) \ ?R" + proof(cases rule:in_RAG_E) + case (waiting th' cs') + from this(3) + show ?thesis + proof(cases rule:waiting_esE) + case 1 + thus ?thesis using waiting(1,2) + by (unfold s_RAG_def, fold waiting_eq, auto) + next + case 2 + thus ?thesis using waiting(1,2) by auto + qed + next + case (holding th' cs') + from this(3) + show ?thesis + proof(cases rule:holding_esE) + case 1 + with holding(1,2) + show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) + qed + qed +next + fix n1 n2 + assume "(n1, n2) \ ?R" + hence "(n1, n2) \ RAG s \ (n1 = Th th \ n2 = Cs cs)" by auto + thus "(n1, n2) \ ?L" + proof + assume "(n1, n2) \ RAG s" + thus ?thesis + proof(cases rule:in_RAG_E) + case (waiting th' cs') + from waiting_kept[OF this(3)] + show ?thesis using waiting(1,2) + by (unfold s_RAG_def, fold waiting_eq, auto) + next + case (holding th' cs') + from holding_kept[OF this(3)] + show ?thesis using holding(1,2) + by (unfold s_RAG_def, fold holding_eq, auto) + qed + next + assume "n1 = Th th \ n2 = Cs cs" + thus ?thesis using RAG_edge by auto + qed +qed + +end + context valid_trace_p begin -lemma RAG_es': "RAG (e # s) = (if (wq s cs = []) then RAG s \ {(Cs cs, Th th)} +lemma RAG_es: "RAG (e # s) = (if (wq s cs = []) then RAG s \ {(Cs cs, Th th)} else RAG s \ {(Th th, Cs cs)})" proof(cases "wq s cs = []") case True @@ -1834,6 +1846,252 @@ end +section {* Finiteness of RAG *} + +context valid_trace +begin + +lemma finite_RAG: + shows "finite (RAG s)" +proof(induct rule:ind) + case Nil + show ?case + by (auto simp: s_RAG_def cs_waiting_def + cs_holding_def wq_def acyclic_def) +next + case (Cons s e) + interpret vt_e: valid_trace_e s e using Cons by simp + show ?case + proof(cases e) + case (Create th prio) + interpret vt: valid_trace_create s e th prio using Create + by (unfold_locales, simp) + show ?thesis using Cons by simp + next + case (Exit th) + interpret vt: valid_trace_exit s e th using Exit + by (unfold_locales, simp) + show ?thesis using Cons by simp + next + case (P th cs) + interpret vt: valid_trace_p s e th cs using P + by (unfold_locales, simp) + show ?thesis using Cons using vt.RAG_es by auto + next + case (V th cs) + interpret vt: valid_trace_v s e th cs using V + by (unfold_locales, simp) + show ?thesis using Cons by (simp add: vt.finite_RAG_kept) + next + case (Set th prio) + interpret vt: valid_trace_set s e th prio using Set + by (unfold_locales, simp) + show ?thesis using Cons by simp + qed +qed +end + +section {* RAG is acyclic *} + +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. +*} + + +context valid_trace +begin + +lemma waiting_unique_pre: (* ddd *) + assumes h11: "thread \ set (wq s cs1)" + and h12: "thread \ hd (wq s cs1)" + assumes h21: "thread \ set (wq s cs2)" + and h22: "thread \ hd (wq s cs2)" + and neq12: "cs1 \ cs2" + shows "False" +proof - + let "?Q" = "\ cs s. thread \ set (wq s cs) \ thread \ hd (wq s cs)" + from h11 and h12 have q1: "?Q cs1 s" by simp + from h21 and h22 have q2: "?Q cs2 s" by simp + have nq1: "\ ?Q cs1 []" by (simp add:wq_def) + have nq2: "\ ?Q cs2 []" by (simp add:wq_def) + from p_split [of "?Q cs1", OF q1 nq1] + obtain t1 where lt1: "t1 < length s" + and np1: "\ ?Q cs1 (moment t1 s)" + and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" by auto + from p_split [of "?Q cs2", OF q2 nq2] + obtain t2 where lt2: "t2 < length s" + and np2: "\ ?Q cs2 (moment t2 s)" + and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" by auto + { fix s cs + assume q: "?Q cs s" + have "thread \ runing s" + proof + assume "thread \ runing s" + hence " \cs. \ (thread \ set (wq_fun (schs s) cs) \ + thread \ hd (wq_fun (schs s) cs))" + by (unfold runing_def s_waiting_def readys_def, auto) + from this[rule_format, of cs] q + show False by (simp add: wq_def) + qed + } note q_not_runing = this + { fix t1 t2 cs1 cs2 + assume lt1: "t1 < length s" + and np1: "\ ?Q cs1 (moment t1 s)" + and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" + and lt2: "t2 < length s" + and np2: "\ ?Q cs2 (moment t2 s)" + and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" + and lt12: "t1 < t2" + let ?t3 = "Suc t2" + interpret ve2: valid_moment_e _ t2 using lt2 + by (unfold_locales, simp) + let ?e = ve2.next_e + have "t2 < ?t3" by simp + from nn2 [rule_format, OF this] and ve2.trace_e + have h1: "thread \ set (wq (?e#moment t2 s) cs2)" and + h2: "thread \ hd (wq (?e#moment t2 s) cs2)" by auto + have ?thesis + proof - + have "thread \ runing (moment t2 s)" + proof(cases "thread \ set (wq (moment t2 s) cs2)") + case True + have "?e = V thread cs2" + proof - + have eq_th: "thread = hd (wq (moment t2 s) cs2)" + using True and np2 by auto + thus ?thesis + using True h2 ve2.vat_moment_e.wq_out_inv by blast + qed + thus ?thesis + using step.cases ve2.vat_moment_e.pip_e by auto + next + case False + hence "?e = P thread cs2" + using h1 ve2.vat_moment_e.wq_in_inv by blast + thus ?thesis + using step.cases ve2.vat_moment_e.pip_e by auto + qed + moreover have "thread \ runing (moment t2 s)" + by (rule q_not_runing[OF nn1[rule_format, OF lt12]]) + ultimately show ?thesis by simp + qed + } note lt_case = this + show ?thesis + proof - + { assume "t1 < t2" + from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this] + have ?thesis . + } moreover { + assume "t2 < t1" + from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this] + have ?thesis . + } moreover { + assume eq_12: "t1 = t2" + let ?t3 = "Suc t2" + interpret ve2: valid_moment_e _ t2 using lt2 + by (unfold_locales, simp) + let ?e = ve2.next_e + have "t2 < ?t3" by simp + from nn2 [rule_format, OF this] and ve2.trace_e + have h1: "thread \ set (wq (?e#moment t2 s) cs2)" by auto + have lt_2: "t2 < ?t3" by simp + from nn2 [rule_format, OF this] and ve2.trace_e + have h1: "thread \ set (wq (?e#moment t2 s) cs2)" and + h2: "thread \ hd (wq (?e#moment t2 s) cs2)" by auto + from nn1[rule_format, OF lt_2[folded eq_12], unfolded ve2.trace_e[folded eq_12]] + eq_12[symmetric] + have g1: "thread \ set (wq (?e#moment t1 s) cs1)" and + g2: "thread \ hd (wq (?e#moment t1 s) cs1)" by auto + have "?e = V thread cs2 \ ?e = P thread cs2" + using h1 h2 np2 ve2.vat_moment_e.wq_in_inv + ve2.vat_moment_e.wq_out_inv by blast + moreover have "?e = V thread cs1 \ ?e = P thread cs1" + using eq_12 g1 g2 np1 ve2.vat_moment_e.wq_in_inv + ve2.vat_moment_e.wq_out_inv by blast + ultimately have ?thesis using neq12 by auto + } ultimately show ?thesis using nat_neq_iff by blast + qed +qed + +text {* + This lemma is a simple corrolary of @{text "waiting_unique_pre"}. +*} + +lemma waiting_unique: + assumes "waiting s th cs1" + and "waiting s th cs2" + shows "cs1 = cs2" + using waiting_unique_pre assms + unfolding wq_def s_waiting_def + by auto + +end + +lemma (in valid_trace_v) + preced_es [simp]: "preced th (e#s) = preced th s" + by (unfold is_v preced_def, simp) + +lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s" +proof + fix th' + show "the_preced (V th cs # s) th' = the_preced s th'" + by (unfold the_preced_def preced_def, simp) +qed + + +lemma (in valid_trace_v) + the_preced_es: "the_preced (e#s) = the_preced s" + by (unfold is_v preced_def, simp) + +context valid_trace_p +begin + +lemma not_holding_s_th_cs: "\ holding s th cs" +proof + assume otherwise: "holding s th cs" + from pip_e[unfolded is_p] + show False + proof(cases) + case (thread_P) + moreover have "(Cs cs, Th th) \ RAG s" + using otherwise cs_holding_def + holding_eq th_not_in_wq by auto + ultimately show ?thesis by auto + qed +qed + +end + + lemma (in valid_trace_v_n) finite_waiting_set: "finite {(Th th', Cs cs) |th'. next_th s th cs th'}" by (simp add: waiting_set_eq) @@ -1850,28 +2108,6 @@ "finite {(Cs cs, Th th') |th'. next_th s th cs th'}" by (simp add: holding_set_eq) -context valid_trace_v -begin - -lemma - finite_RAG_kept: - assumes "finite (RAG s)" - shows "finite (RAG (e#s))" -proof(cases "rest = []") - case True - interpret vt: valid_trace_v_e using True - by (unfold_locales, simp) - show ?thesis using assms - by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp) -next - case False - interpret vt: valid_trace_v_n using False - by (unfold_locales, simp) - show ?thesis using assms - by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp) -qed - -end context valid_trace_v_e begin @@ -1916,7 +2152,7 @@ by (unfold s_RAG_def, auto) from this(2) have "waiting s taker cs'" by (unfold s_RAG_def, fold waiting_eq, auto) - from waiting_unique[OF this waiting_taker] + from waiting_unique[OF this waiting_taker] have "cs' = cs" . from h(1)[unfolded this] show False by auto qed @@ -1951,7 +2187,7 @@ by (unfold s_RAG_def, auto) hence "waiting s th cs'" by (unfold s_RAG_def, fold waiting_eq, auto) - with th_not_waiting show False by auto + with th_not_waiting show False by auto qed ultimately show ?thesis by auto qed @@ -1995,45 +2231,6 @@ context valid_trace begin -lemma finite_RAG: - shows "finite (RAG s)" -proof(induct rule:ind) - case Nil - show ?case - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt: valid_trace_create s e th prio using Create - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (Exit th) - interpret vt: valid_trace_exit s e th using Exit - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (P th cs) - interpret vt: valid_trace_p s e th cs using P - by (unfold_locales, simp) - show ?thesis using Cons using vt.RAG_es' by auto - next - case (V th cs) - interpret vt: valid_trace_v s e th cs using V - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.finite_RAG_kept) - next - case (Set th prio) - interpret vt: valid_trace_set s e th prio using Set - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - qed -qed - lemma acyclic_RAG: shows "acyclic (RAG s)" proof(induct rule:ind) @@ -2049,12 +2246,12 @@ case (Create th prio) interpret vt: valid_trace_create s e th prio using Create by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) + show ?thesis using Cons by simp next case (Exit th) interpret vt: valid_trace_exit s e th using Exit by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) + show ?thesis using Cons by simp next case (P th cs) interpret vt: valid_trace_p s e th cs using P @@ -2091,10 +2288,31 @@ case (Set th prio) interpret vt: valid_trace_set s e th prio using Set by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) + show ?thesis using Cons by simp qed qed +end + +section {* RAG is single-valued *} + +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 held_unique) + +lemma sgv_RAG: "single_valued (RAG s)" + using unique_RAG by (auto simp:single_valued_def) + +end + +section {* RAG is well-founded *} + +context valid_trace +begin + lemma wf_RAG: "wf (RAG s)" proof(rule finite_acyclic_wf) from finite_RAG show "finite (RAG s)" . @@ -2102,6 +2320,60 @@ from acyclic_RAG show "acyclic (RAG s)" . qed +lemma wf_RAG_converse: + shows "wf ((RAG s)^-1)" +proof(rule finite_acyclic_wf_converse) + from finite_RAG + show "finite (RAG s)" . +next + from acyclic_RAG + show "acyclic (RAG s)" . +qed + +end + +section {* RAG forms a forest (or tree) *} + +context valid_trace +begin + +lemma rtree_RAG: "rtree (RAG s)" + using sgv_RAG acyclic_RAG + by (unfold rtree_def rtree_axioms_def sgv_def, auto) + +end + +sublocale valid_trace < rtree_RAG: rtree "RAG s" + using rtree_RAG . + +sublocale valid_trace < fsbtRAGs : fsubtree "RAG s" +proof - + show "fsubtree (RAG s)" + proof(intro_locales) + show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] . + next + show "fsubtree_axioms (RAG s)" + proof(unfold fsubtree_axioms_def) + from wf_RAG show "wf (RAG s)" . + qed + qed +qed + + +section {* Derived properties for parts of RAG *} + +context valid_trace +begin + +lemma acyclic_tRAG: "acyclic (tRAG s)" +proof(unfold tRAG_def, rule acyclic_compose) + show "acyclic (RAG s)" using acyclic_RAG . +next + show "wRAG s \ RAG s" unfolding RAG_split by auto +next + show "hRAG s \ RAG s" unfolding RAG_split by auto +qed + lemma sgv_wRAG: "single_valued (wRAG s)" using waiting_unique by (unfold single_valued_def wRAG_def, auto) @@ -2114,39 +2386,8 @@ by (unfold tRAG_def, rule single_valued_relcomp, insert sgv_wRAG sgv_hRAG, auto) -lemma acyclic_tRAG: "acyclic (tRAG s)" -proof(unfold tRAG_def, rule acyclic_compose) - show "acyclic (RAG s)" using acyclic_RAG . -next - show "wRAG s \ RAG s" unfolding RAG_split by auto -next - show "hRAG s \ RAG s" unfolding RAG_split by auto -qed - -lemma unique_RAG: "\(n, n1) \ RAG s; (n, n2) \ RAG s\ \ n1 = n2" - apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq) - by(auto elim:waiting_unique held_unique) - -lemma sgv_RAG: "single_valued (RAG s)" - using unique_RAG by (auto simp:single_valued_def) - -lemma rtree_RAG: "rtree (RAG s)" - using sgv_RAG acyclic_RAG - by (unfold rtree_def rtree_axioms_def sgv_def, auto) - end -sublocale valid_trace < rtree_RAG: rtree "RAG s" -proof - show "single_valued (RAG s)" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:unique_RAG) - - show "acyclic (RAG s)" - by (rule acyclic_RAG) -qed - sublocale valid_trace < rtree_s: rtree "tRAG s" proof(unfold_locales) from sgv_tRAG show "single_valued (tRAG s)" . @@ -2154,24 +2395,6 @@ from acyclic_tRAG show "acyclic (tRAG s)" . qed -sublocale valid_trace < fsbtRAGs : fsubtree "RAG s" -proof - - show "fsubtree (RAG s)" - proof(intro_locales) - show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] . - next - show "fsubtree_axioms (RAG s)" - proof(unfold fsubtree_axioms_def) - from wf_RAG show "wf (RAG s)" . - qed - qed -qed - -lemma tRAG_alt_def: - "tRAG s = {(Th th1, Th th2) | th1 th2. - \ cs. (Th th1, Cs cs) \ RAG s \ (Cs cs, Th th2) \ RAG s}" - by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def) - sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s" proof - have "fsubtree (tRAG s)" @@ -2204,80 +2427,282 @@ from this[folded tRAG_def] show "fsubtree (tRAG s)" . qed +lemma tRAG_nodeE: + assumes "(n1, n2) \ tRAG s" + obtains th1 th2 where "n1 = Th th1" "n2 = Th th2" + using assms + by (auto simp: tRAG_def wRAG_def hRAG_def) + +lemma tRAG_ancestorsE: + assumes "x \ ancestors (tRAG s) u" + obtains th where "x = Th th" +proof - + from assms have "(u, x) \ (tRAG s)^+" + by (unfold ancestors_def, auto) + from tranclE[OF this] obtain c where "(c, x) \ tRAG s" by auto + then obtain th where "x = Th th" + by (unfold tRAG_alt_def, auto) + from that[OF this] show ?thesis . +qed + +lemma subtree_nodeE: + assumes "n \ subtree (tRAG s) (Th th)" + obtains th1 where "n = Th th1" +proof - + show ?thesis + proof(rule subtreeE[OF assms]) + assume "n = Th th" + from that[OF this] show ?thesis . + next + assume "Th th \ ancestors (tRAG s) n" + hence "(n, Th th) \ (tRAG s)^+" by (auto simp:ancestors_def) + hence "\ th1. n = Th th1" + proof(induct) + case (base y) + from tRAG_nodeE[OF this] show ?case by metis + next + case (step y z) + thus ?case by auto + qed + with that show ?thesis by auto + qed +qed + +lemma tRAG_star_RAG: "(tRAG s)^* \ (RAG s)^*" +proof - + have "(wRAG s O hRAG s)^* \ (RAG s O RAG s)^*" + by (rule rtrancl_mono, auto simp:RAG_split) + also have "... \ ((RAG s)^*)^*" + by (rule rtrancl_mono, auto) + also have "... = (RAG s)^*" by simp + finally show ?thesis by (unfold tRAG_def, simp) +qed + +lemma tRAG_subtree_RAG: "subtree (tRAG s) x \ subtree (RAG s) x" +proof - + { fix a + assume "a \ subtree (tRAG s) x" + hence "(a, x) \ (tRAG s)^*" by (auto simp:subtree_def) + with tRAG_star_RAG + have "(a, x) \ (RAG s)^*" by auto + hence "a \ subtree (RAG s) x" by (auto simp:subtree_def) + } thus ?thesis by auto +qed + +lemma tRAG_trancl_eq: + "{th'. (Th th', Th th) \ (tRAG s)^+} = + {th'. (Th th', Th th) \ (RAG s)^+}" + (is "?L = ?R") +proof - + { fix th' + assume "th' \ ?L" + hence "(Th th', Th th) \ (tRAG s)^+" by auto + from tranclD[OF this] + obtain z where h: "(Th th', z) \ tRAG s" "(z, Th th) \ (tRAG s)\<^sup>*" by auto + from tRAG_subtree_RAG and this(2) + have "(z, Th th) \ (RAG s)^*" by (meson subsetCE tRAG_star_RAG) + moreover from h(1) have "(Th th', z) \ (RAG s)^+" using tRAG_alt_def by auto + ultimately have "th' \ ?R" by auto + } moreover + { fix th' + assume "th' \ ?R" + hence "(Th th', Th th) \ (RAG s)^+" by (auto) + from plus_rpath[OF this] + obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \ []" by auto + hence "(Th th', Th th) \ (tRAG s)^+" + proof(induct xs arbitrary:th' th rule:length_induct) + case (1 xs th' th) + then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto) + show ?case + proof(cases "xs1") + case Nil + from 1(2)[unfolded Cons1 Nil] + have rp: "rpath (RAG s) (Th th') [x1] (Th th)" . + hence "(Th th', x1) \ (RAG s)" + by (cases, auto) + then obtain cs where "x1 = Cs cs" + by (unfold s_RAG_def, auto) + from rpath_nnl_lastE[OF rp[unfolded this]] + show ?thesis by auto + next + case (Cons x2 xs2) + from 1(2)[unfolded Cons1[unfolded this]] + have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" . + from rpath_edges_on[OF this] + have eds: "edges_on (Th th' # x1 # x2 # xs2) \ RAG s" . + have "(Th th', x1) \ edges_on (Th th' # x1 # x2 # xs2)" + by (simp add: edges_on_unfold) + with eds have rg1: "(Th th', x1) \ RAG s" by auto + then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto) + have "(x1, x2) \ edges_on (Th th' # x1 # x2 # xs2)" + by (simp add: edges_on_unfold) + from this eds + have rg2: "(x1, x2) \ RAG s" by auto + from this[unfolded eq_x1] + obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto) + from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2] + have rt1: "(Th th', Th th1) \ tRAG s" by (unfold tRAG_alt_def, auto) + from rp have "rpath (RAG s) x2 xs2 (Th th)" + by (elim rpath_ConsE, simp) + from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" . + show ?thesis + proof(cases "xs2 = []") + case True + from rpath_nilE[OF rp'[unfolded this]] + have "th1 = th" by auto + from rt1[unfolded this] show ?thesis by auto + next + case False + from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons] + have "(Th th1, Th th) \ (tRAG s)\<^sup>+" by simp + with rt1 show ?thesis by auto + qed + qed + qed + hence "th' \ ?L" by auto + } ultimately show ?thesis by blast +qed + +lemma tRAG_trancl_eq_Th: + "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = + {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" + using tRAG_trancl_eq by auto + + +lemma tRAG_Field: + "Field (tRAG s) \ Field (RAG s)" + by (unfold tRAG_alt_def Field_def, auto) + +lemma tRAG_mono: + assumes "RAG s' \ RAG s" + shows "tRAG s' \ tRAG s" + using assms + by (unfold tRAG_alt_def, auto) + +lemma tRAG_subtree_eq: + "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \ (subtree (RAG s) (Th th))}" + (is "?L = ?R") +proof - + { fix n + assume h: "n \ ?L" + hence "n \ ?R" + by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) + } moreover { + fix n + assume "n \ ?R" + then obtain th' where h: "n = Th th'" "(Th th', Th th) \ (RAG s)^*" + by (auto simp:subtree_def) + from rtranclD[OF this(2)] + have "n \ ?L" + proof + assume "Th th' \ Th th \ (Th th', Th th) \ (RAG s)\<^sup>+" + with h have "n \ {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" by auto + thus ?thesis using subtree_def tRAG_trancl_eq by fastforce + qed (insert h, auto simp:subtree_def) + } ultimately show ?thesis by auto +qed + +lemma threads_set_eq: + "the_thread ` (subtree (tRAG s) (Th th)) = + {th'. Th th' \ (subtree (RAG s) (Th th))}" (is "?L = ?R") + by (auto intro:rev_image_eqI simp:tRAG_subtree_eq) context valid_trace begin -lemma finite_subtree_threads: - "finite {th'. Th th' \ subtree (RAG s) (Th th)}" (is "finite ?A") +lemma RAG_tRAG_transfer: + assumes "RAG s' = RAG s \ {(Th th, Cs cs)}" + and "(Cs cs, Th th'') \ RAG s" + shows "tRAG s' = tRAG s \ {(Th th, Th th'')}" (is "?L = ?R") proof - - have "?A = the_thread ` {Th th' | th' . Th th' \ subtree (RAG s) (Th th)}" - by (auto, insert image_iff, fastforce) - moreover have "finite {Th th' | th' . Th th' \ subtree (RAG s) (Th th)}" - (is "finite ?B") - proof - - have "?B = (subtree (RAG s) (Th th)) \ {Th th' | th'. True}" - by auto - moreover have "... \ (subtree (RAG s) (Th th))" by auto - moreover have "finite ..." by (simp add: fsbtRAGs.finite_subtree) - ultimately show ?thesis by auto - qed - ultimately show ?thesis by auto + { fix n1 n2 + assume "(n1, n2) \ ?L" + from this[unfolded tRAG_alt_def] + obtain th1 th2 cs' where + h: "n1 = Th th1" "n2 = Th th2" + "(Th th1, Cs cs') \ RAG s'" + "(Cs cs', Th th2) \ RAG s'" by auto + from h(4) and assms(1) have cs_in: "(Cs cs', Th th2) \ RAG s" by auto + from h(3) and assms(1) + have "(Th th1, Cs cs') = (Th th, Cs cs) \ + (Th th1, Cs cs') \ RAG s" by auto + hence "(n1, n2) \ ?R" + proof + assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)" + hence eq_th1: "th1 = th" by simp + moreover have "th2 = th''" + proof - + from h1 have "cs' = cs" by simp + from assms(2) cs_in[unfolded this] + show ?thesis using unique_RAG by auto + qed + ultimately show ?thesis using h(1,2) by auto + next + assume "(Th th1, Cs cs') \ RAG s" + with cs_in have "(Th th1, Th th2) \ tRAG s" + by (unfold tRAG_alt_def, auto) + from this[folded h(1, 2)] show ?thesis by auto + qed + } moreover { + fix n1 n2 + assume "(n1, n2) \ ?R" + hence "(n1, n2) \tRAG s \ (n1, n2) = (Th th, Th th'')" by auto + hence "(n1, n2) \ ?L" + proof + assume "(n1, n2) \ tRAG s" + moreover have "... \ ?L" + proof(rule tRAG_mono) + show "RAG s \ RAG s'" by (unfold assms(1), auto) + qed + ultimately show ?thesis by auto + next + assume eq_n: "(n1, n2) = (Th th, Th th'')" + from assms(1, 2) have "(Cs cs, Th th'') \ RAG s'" by auto + moreover have "(Th th, Cs cs) \ RAG s'" using assms(1) by auto + ultimately show ?thesis + by (unfold eq_n tRAG_alt_def, auto) + qed + } ultimately show ?thesis by auto qed -lemma le_cp: - shows "preced th s \ cp s th" - proof(unfold cp_alt_def, rule Max_ge) - show "finite (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - by (simp add: finite_subtree_threads) - next - show "preced th s \ the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}" - by (simp add: subtree_def the_preced_def) +lemma subtree_tRAG_thread: + assumes "th \ threads s" + shows "subtree (tRAG s) (Th th) \ Th ` threads s" (is "?L \ ?R") +proof - + have "?L = {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" + by (unfold tRAG_subtree_eq, simp) + also have "... \ ?R" + proof + fix x + assume "x \ {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" + then obtain th' where h: "x = Th th'" "Th th' \ subtree (RAG s) (Th th)" by auto + from this(2) + show "x \ ?R" + proof(cases rule:subtreeE) + case 1 + thus ?thesis by (simp add: assms h(1)) + next + case 2 + thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) + qed qed - -lemma cp_le: - assumes th_in: "th \ threads s" - shows "cp s th \ Max (the_preced s ` threads s)" -proof(unfold cp_alt_def, rule Max_f_mono) - show "finite (threads s)" by (simp add: finite_threads) -next - show " {th'. Th th' \ subtree (RAG s) (Th th)} \ {}" - using subtree_def by fastforce -next - show "{th'. Th th' \ subtree (RAG s) (Th th)} \ threads s" - using assms - by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq - node.inject(1) rtranclD subsetI subtree_def trancl_domain) + finally show ?thesis . qed -lemma max_cp_eq: - shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)" - (is "?L = ?R") -proof - - have "?L \ ?R" - proof(cases "threads s = {}") - case False - show ?thesis - by (rule Max.boundedI, - insert cp_le, - auto simp:finite_threads False) - qed auto - moreover have "?R \ ?L" - by (rule Max_fg_mono, - simp add: finite_threads, - simp add: le_cp the_preced_def) - ultimately show ?thesis by auto -qed - -lemma wf_RAG_converse: - shows "wf ((RAG s)^-1)" -proof(rule finite_acyclic_wf_converse) - from finite_RAG - show "finite (RAG s)" . -next - from acyclic_RAG - show "acyclic (RAG s)" . -qed +lemma dependants_alt_def: + "dependants s th = {th'. (Th th', Th th) \ (tRAG s)^+}" + by (metis eq_RAG s_dependants_def tRAG_trancl_eq) + +lemma dependants_alt_def1: + "dependants (s::state) th = {th'. (Th th', Th th) \ (RAG s)^+}" + using dependants_alt_def tRAG_trancl_eq by auto + +end + +section {* Chain to readys *} + +context valid_trace +begin lemma chain_building: assumes "node \ Domain (RAG s)" @@ -2324,7 +2749,7 @@ 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)^+)" @@ -2339,97 +2764,274 @@ show ?thesis by auto qed +lemma finite_subtree_threads: + "finite {th'. Th th' \ subtree (RAG s) (Th th)}" (is "finite ?A") +proof - + have "?A = the_thread ` {Th th' | th' . Th th' \ subtree (RAG s) (Th th)}" + by (auto, insert image_iff, fastforce) + moreover have "finite {Th th' | th' . Th th' \ subtree (RAG s) (Th th)}" + (is "finite ?B") + proof - + have "?B = (subtree (RAG s) (Th th)) \ {Th th' | th'. True}" + by auto + moreover have "... \ (subtree (RAG s) (Th th))" by auto + moreover have "finite ..." by (simp add: fsbtRAGs.finite_subtree) + ultimately show ?thesis by auto + qed + ultimately show ?thesis by auto +qed + +lemma runing_unique: + assumes runing_1: "th1 \ runing s" + and runing_2: "th2 \ runing s" + shows "th1 = th2" +proof - + from runing_1 and runing_2 have "cp s th1 = cp s th2" + unfolding runing_def by auto + from this[unfolded cp_alt_def] + have eq_max: + "Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th1)}) = + Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th2)})" + (is "Max ?L = Max ?R") . + have "Max ?L \ ?L" + proof(rule Max_in) + show "finite ?L" by (simp add: finite_subtree_threads) + next + show "?L \ {}" using subtree_def by fastforce + qed + then obtain th1' where + h_1: "Th th1' \ subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L" + by auto + have "Max ?R \ ?R" + proof(rule Max_in) + show "finite ?R" by (simp add: finite_subtree_threads) + next + show "?R \ {}" using subtree_def by fastforce + qed + then obtain th2' where + h_2: "Th th2' \ subtree (RAG s) (Th th2)" "the_preced s th2' = Max ?R" + by auto + have "th1' = th2'" + proof(rule preced_unique) + from h_1(1) + show "th1' \ threads s" + proof(cases rule:subtreeE) + case 1 + hence "th1' = th1" by simp + with runing_1 show ?thesis by (auto simp:runing_def readys_def) + next + case 2 + from this(2) + have "(Th th1', Th th1) \ (RAG s)^+" by (auto simp:ancestors_def) + from tranclD[OF this] + have "(Th th1') \ Domain (RAG s)" by auto + from dm_RAG_threads[OF this] show ?thesis . + qed + next + from h_2(1) + show "th2' \ threads s" + proof(cases rule:subtreeE) + case 1 + hence "th2' = th2" by simp + with runing_2 show ?thesis by (auto simp:runing_def readys_def) + next + case 2 + from this(2) + have "(Th th2', Th th2) \ (RAG s)^+" by (auto simp:ancestors_def) + from tranclD[OF this] + have "(Th th2') \ Domain (RAG s)" by auto + from dm_RAG_threads[OF this] show ?thesis . + qed + next + have "the_preced s th1' = the_preced s th2'" + using eq_max h_1(2) h_2(2) by metis + thus "preced th1' s = preced th2' s" by (simp add:the_preced_def) + qed + from h_1(1)[unfolded this] + have star1: "(Th th2', Th th1) \ (RAG s)^*" by (auto simp:subtree_def) + from h_2(1)[unfolded this] + have star2: "(Th th2', Th th2) \ (RAG s)^*" by (auto simp:subtree_def) + from star_rpath[OF star1] obtain xs1 + where rp1: "rpath (RAG s) (Th th2') xs1 (Th th1)" + by auto + from star_rpath[OF star2] obtain xs2 + where rp2: "rpath (RAG s) (Th th2') xs2 (Th th2)" + by auto + from rp1 rp2 + show ?thesis + proof(cases) + case (less_1 xs') + moreover have "xs' = []" + proof(rule ccontr) + assume otherwise: "xs' \ []" + from rpath_plus[OF less_1(3) this] + have "(Th th1, Th th2) \ (RAG s)\<^sup>+" . + from tranclD[OF this] + obtain cs where "waiting s th1 cs" + by (unfold s_RAG_def, fold waiting_eq, auto) + with runing_1 show False + by (unfold runing_def readys_def, auto) + qed + ultimately have "xs2 = xs1" by simp + from rpath_dest_eq[OF rp1 rp2[unfolded this]] + show ?thesis by simp + next + case (less_2 xs') + moreover have "xs' = []" + proof(rule ccontr) + assume otherwise: "xs' \ []" + from rpath_plus[OF less_2(3) this] + have "(Th th2, Th th1) \ (RAG s)\<^sup>+" . + from tranclD[OF this] + obtain cs where "waiting s th2 cs" + by (unfold s_RAG_def, fold waiting_eq, auto) + with runing_2 show False + by (unfold runing_def readys_def, auto) + qed + ultimately have "xs2 = xs1" by simp + from rpath_dest_eq[OF rp1 rp2[unfolded this]] + show ?thesis by simp + qed +qed + +lemma card_runing: "card (runing s) \ 1" +proof(cases "runing s = {}") + case True + thus ?thesis by auto +next + case False + then obtain th where [simp]: "th \ runing s" by auto + from runing_unique[OF this] + have "runing s = {th}" by auto + thus ?thesis by auto +qed + end -lemma count_rec1 [simp]: - assumes "Q e" - shows "count Q (e#es) = Suc (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec2 [simp]: - assumes "\Q e" - shows "count Q (e#es) = (count Q es)" - using assms - by (unfold count_def, auto) - -lemma count_rec3 [simp]: - shows "count Q [] = 0" - by (unfold count_def, auto) - -lemma cntP_simp1[simp]: - "cntP (P th cs'#s) th = cntP s th + 1" - by (unfold cntP_def, simp) - -lemma cntP_simp2[simp]: - assumes "th' \ th" - shows "cntP (P th cs'#s) th' = cntP s th'" - using assms - by (unfold cntP_def, simp) - -lemma cntP_simp3[simp]: - assumes "\ isP e" - shows "cntP (e#s) th' = cntP s th'" - using assms - by (unfold cntP_def, cases e, simp+) - -lemma cntV_simp1[simp]: - "cntV (V th cs'#s) th = cntV s th + 1" - by (unfold cntV_def, simp) - -lemma cntV_simp2[simp]: - assumes "th' \ th" - shows "cntV (V th cs'#s) th' = cntV s th'" - using assms - by (unfold cntV_def, simp) - -lemma cntV_simp3[simp]: - assumes "\ isV e" - shows "cntV (e#s) th' = cntV s th'" - using assms - by (unfold cntV_def, cases e, simp+) - -lemma cntP_diff_inv: - assumes "cntP (e#s) th \ cntP s th" - shows "isP e \ actor e = th" -proof(cases e) - case (P th' pty) - show ?thesis - by (cases "(\e. \cs. e = P th cs) (P th' pty)", - insert assms P, auto simp:cntP_def) -qed (insert assms, auto simp:cntP_def) - -lemma cntV_diff_inv: - assumes "cntV (e#s) th \ cntV s th" - shows "isV e \ actor e = th" -proof(cases e) - case (V th' pty) - show ?thesis - by (cases "(\e. \cs. e = V th cs) (V th' pty)", - insert assms V, auto simp:cntV_def) -qed (insert assms, auto simp:cntV_def) - -lemma children_RAG_alt_def: - "children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}" - by (unfold s_RAG_def, auto simp:children_def holding_eq) - -lemma holdents_alt_def: - "holdents s th = the_cs ` (children (RAG (s::state)) (Th th))" - by (unfold children_RAG_alt_def holdents_def, simp add: image_image) - -lemma cntCS_alt_def: - "cntCS s th = card (children (RAG s) (Th th))" - apply (unfold children_RAG_alt_def cntCS_def holdents_def) - by (rule card_image[symmetric], auto simp:inj_on_def) + +section {* Relating @{term cp} and @{term the_preced} and @{term preced} *} context valid_trace begin -lemma finite_holdents: "finite (holdents s th)" - by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto) - +lemma le_cp: + shows "preced th s \ cp s th" + proof(unfold cp_alt_def, rule Max_ge) + show "finite (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" + by (simp add: finite_subtree_threads) + next + show "preced th s \ the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}" + by (simp add: subtree_def the_preced_def) + qed + + +lemma cp_le: + assumes th_in: "th \ threads s" + shows "cp s th \ Max (the_preced s ` threads s)" +proof(unfold cp_alt_def, rule Max_f_mono) + show "finite (threads s)" by (simp add: finite_threads) +next + show " {th'. Th th' \ subtree (RAG s) (Th th)} \ {}" + using subtree_def by fastforce +next + show "{th'. Th th' \ subtree (RAG s) (Th th)} \ threads s" + using assms + by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq + node.inject(1) rtranclD subsetI subtree_def trancl_domain) +qed + +lemma max_cp_eq: + shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)" + (is "?L = ?R") +proof - + have "?L \ ?R" + proof(cases "threads s = {}") + case False + show ?thesis + by (rule Max.boundedI, + insert cp_le, + auto simp:finite_threads False) + qed auto + moreover have "?R \ ?L" + by (rule Max_fg_mono, + simp add: finite_threads, + simp add: le_cp the_preced_def) + ultimately show ?thesis by auto +qed + +lemma threads_alt_def: + "(threads s) = (\ th \ readys s. {th'. Th th' \ subtree (RAG s) (Th th)})" + (is "?L = ?R") +proof - + { fix th1 + assume "th1 \ ?L" + from th_chain_to_ready[OF this] + have "th1 \ readys s \ (\th'. th' \ readys s \ (Th th1, Th th') \ (RAG s)\<^sup>+)" . + hence "th1 \ ?R" by (auto simp:subtree_def) + } moreover + { fix th' + assume "th' \ ?R" + then obtain th where h: "th \ readys s" " Th th' \ subtree (RAG s) (Th th)" + by auto + from this(2) + have "th' \ ?L" + proof(cases rule:subtreeE) + case 1 + with h(1) show ?thesis by (auto simp:readys_def) + next + case 2 + from tranclD[OF this(2)[unfolded ancestors_def, simplified]] + have "Th th' \ Domain (RAG s)" by auto + from dm_RAG_threads[OF this] + show ?thesis . + qed + } ultimately show ?thesis by auto +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)" (is "?L = ?R") +proof(cases "readys s = {}") + case False + have "?R = Max (the_preced s ` threads s)" by (unfold max_cp_eq, simp) + also have "... = + Max (the_preced s ` (\th\readys s. {th'. Th th' \ subtree (RAG s) (Th th)}))" + by (unfold threads_alt_def, simp) + also have "... = + Max ((\th\readys s. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}))" + by (unfold image_UN, simp) + also have "... = + Max (Max ` (\th. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}) ` readys s)" + proof(rule Max_UNION) + show "\M\(\x. the_preced s ` + {th'. Th th' \ subtree (RAG s) (Th x)}) ` readys s. finite M" + using finite_subtree_threads by auto + qed (auto simp:False subtree_def) + also have "... = + Max ((Max \ (\th. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})) ` readys s)" + by (unfold image_comp, simp) + also have "... = ?L" (is "Max (?f ` ?A) = Max (?g ` ?A)") + proof - + have "(?f ` ?A) = (?g ` ?A)" + proof(rule f_image_eq) + fix th1 + assume "th1 \ ?A" + thus "?f th1 = ?g th1" + by (unfold cp_alt_def, simp) + qed + thus ?thesis by simp + qed + finally show ?thesis by simp +qed (auto simp:threads_alt_def) + end +section {* Relating @{term cntP}, @{term cntV}, @{term cntCS} and @{term pvD} *} + context valid_trace_p_w begin @@ -2492,6 +3094,30 @@ end +lemma (in valid_trace) finite_holdents: "finite (holdents s th)" + by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto) + +context valid_trace_p +begin + +lemma ready_th_s: "th \ readys s" + using runing_th_s + by (unfold runing_def, auto) + +lemma live_th_s: "th \ threads s" + using readys_threads ready_th_s by auto + +lemma live_th_es: "th \ threads (e#s)" + using live_th_s + by (unfold is_p, simp) + +lemma waiting_neq_th: + assumes "waiting s t c" + shows "t \ th" + using assms using th_not_waiting by blast + +end + context valid_trace_p_h begin @@ -2711,7 +3337,7 @@ end -context valid_trace_v (* ccc *) +context valid_trace_v begin lemma holding_th_cs_s: @@ -2742,7 +3368,7 @@ proof - have "cs \ holdents s th" using holding_th_cs_s by (unfold holdents_def, simp) - moreover have "finite (holdents s th)" using finite_holdents + moreover have "finite (holdents s th)" using finite_holdents by simp ultimately show ?thesis by (unfold cntCS_def, @@ -2751,6 +3377,25 @@ end +context valid_trace_v +begin + +lemma th_not_waiting: + "\ waiting s th c" +proof - + have "th \ readys s" + using runing_ready runing_th_s by blast + thus ?thesis + by (unfold readys_def, auto) +qed + +lemma waiting_neq_th: + assumes "waiting s t c" + shows "t \ th" + using assms using th_not_waiting by blast + +end + context valid_trace_v_n begin @@ -2785,7 +3430,7 @@ qed lemma neq_taker_th: "taker \ th" - using th_not_waiting waiting_taker by blast + using th_not_waiting waiting_taker by blast lemma not_holding_taker_s_cs: shows "\ holding s taker cs" @@ -3212,7 +3857,7 @@ lemma not_waiting_th_s [simp]: "\ waiting s th cs'" proof assume "waiting s th cs'" - from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] + from this[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] have "th \ set (wq s cs')" by auto from wq_threads[OF this] have "th \ threads s" . with th_not_live_s show False by simp @@ -3221,7 +3866,7 @@ lemma not_holding_th_s [simp]: "\ holding s th cs'" proof assume "holding s th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] + from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept] have "th \ set (wq s cs')" by auto from wq_threads[OF this] have "th \ threads s" . with th_not_live_s show False by simp @@ -3230,7 +3875,7 @@ lemma not_waiting_th_es [simp]: "\ waiting (e#s) th cs'" proof assume "waiting (e # s) th cs'" - from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] + from this[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] have "th \ set (wq s cs')" by auto from wq_threads[OF this] have "th \ threads s" . with th_not_live_s show False by simp @@ -3239,7 +3884,7 @@ lemma not_holding_th_es [simp]: "\ holding (e#s) th cs'" proof assume "holding (e # s) th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] + from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept] have "th \ set (wq s cs')" by auto from wq_threads[OF this] have "th \ threads s" . with th_not_live_s show False by simp @@ -3274,13 +3919,13 @@ assume h: "cs' \ ?L" hence "cs' \ ?R" by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) + unfold wq_kept, auto) } moreover { fix cs' assume h: "cs' \ ?R" hence "cs' \ ?L" by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) + unfold wq_kept, auto) } ultimately show ?thesis by auto qed @@ -3300,7 +3945,7 @@ have n_wait: "\ waiting (e#s) th' cs'" using assms by (auto simp:readys_def) from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] + n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] have False by auto } thus ?thesis using assms by (unfold readys_def, auto) @@ -3315,7 +3960,7 @@ assume wait: "waiting (e#s) th' cs'" have n_wait: "\ waiting s th' cs'" using assms(2) by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] + from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] n_wait[unfolded s_waiting_def, folded wq_def] have False by auto } with assms show ?thesis @@ -3388,7 +4033,7 @@ lemma not_holding_th_es [simp]: "\ holding (e#s) th cs'" proof assume "holding (e # s) th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] + from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept] have "holding s th cs'" by (unfold s_holding_def, fold wq_def, auto) with not_holding_th_s @@ -3421,13 +4066,13 @@ assume h: "cs' \ ?L" hence "cs' \ ?R" by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) + unfold wq_kept, auto) } moreover { fix cs' assume h: "cs' \ ?R" hence "cs' \ ?L" by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) + unfold wq_kept, auto) } ultimately show ?thesis by auto qed @@ -3447,7 +4092,7 @@ have n_wait: "\ waiting (e#s) th' cs'" using assms by (auto simp:readys_def) from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] + n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] have False by auto } thus ?thesis using assms by (unfold readys_def, auto) @@ -3462,7 +4107,7 @@ assume wait: "waiting (e#s) th' cs'" have n_wait: "\ waiting s th' cs'" using assms(2) by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] + from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] n_wait[unfolded s_waiting_def, folded wq_def] have False by auto } with assms show ?thesis @@ -3526,13 +4171,13 @@ assume h: "cs' \ ?L" hence "cs' \ ?R" by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) + unfold wq_kept, auto) } moreover { fix cs' assume h: "cs' \ ?R" hence "cs' \ ?L" by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) + unfold wq_kept, auto) } ultimately show ?thesis by auto qed @@ -3554,7 +4199,7 @@ have n_wait: "\ waiting (e#s) th' cs'" using assms by (auto simp:readys_def) from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] + n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] have False by auto } moreover have "th' \ threads s" using assms[unfolded readys_def] by auto @@ -3570,7 +4215,7 @@ assume wait: "waiting (e#s) th' cs'" have n_wait: "\ waiting s th' cs'" using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] + from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] n_wait[unfolded s_waiting_def, folded wq_def] have False by auto } with assms show ?thesis @@ -3633,6 +4278,13 @@ qed qed +end + +section {* Corollaries of @{thm valid_trace.cnp_cnv_cncs} *} + +context valid_trace +begin + lemma not_thread_holdents: assumes not_in: "th \ threads s" shows "holdents s th = {}" @@ -3660,158 +4312,6 @@ using assms cnp_cnv_cncs not_thread_cncs pvD_def by (auto) -lemma runing_unique: - assumes runing_1: "th1 \ runing s" - and runing_2: "th2 \ runing s" - shows "th1 = th2" -proof - - from runing_1 and runing_2 have "cp s th1 = cp s th2" - unfolding runing_def by auto - from this[unfolded cp_alt_def] - have eq_max: - "Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th1)}) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th2)})" - (is "Max ?L = Max ?R") . - have "Max ?L \ ?L" - proof(rule Max_in) - show "finite ?L" by (simp add: finite_subtree_threads) - next - show "?L \ {}" using subtree_def by fastforce - qed - then obtain th1' where - h_1: "Th th1' \ subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L" - by auto - have "Max ?R \ ?R" - proof(rule Max_in) - show "finite ?R" by (simp add: finite_subtree_threads) - next - show "?R \ {}" using subtree_def by fastforce - qed - then obtain th2' where - h_2: "Th th2' \ subtree (RAG s) (Th th2)" "the_preced s th2' = Max ?R" - by auto - have "th1' = th2'" - proof(rule preced_unique) - from h_1(1) - show "th1' \ threads s" - proof(cases rule:subtreeE) - case 1 - hence "th1' = th1" by simp - with runing_1 show ?thesis by (auto simp:runing_def readys_def) - next - case 2 - from this(2) - have "(Th th1', Th th1) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - have "(Th th1') \ Domain (RAG s)" by auto - from dm_RAG_threads[OF this] show ?thesis . - qed - next - from h_2(1) - show "th2' \ threads s" - proof(cases rule:subtreeE) - case 1 - hence "th2' = th2" by simp - with runing_2 show ?thesis by (auto simp:runing_def readys_def) - next - case 2 - from this(2) - have "(Th th2', Th th2) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - have "(Th th2') \ Domain (RAG s)" by auto - from dm_RAG_threads[OF this] show ?thesis . - qed - next - have "the_preced s th1' = the_preced s th2'" - using eq_max h_1(2) h_2(2) by metis - thus "preced th1' s = preced th2' s" by (simp add:the_preced_def) - qed - from h_1(1)[unfolded this] - have star1: "(Th th2', Th th1) \ (RAG s)^*" by (auto simp:subtree_def) - from h_2(1)[unfolded this] - have star2: "(Th th2', Th th2) \ (RAG s)^*" by (auto simp:subtree_def) - from star_rpath[OF star1] obtain xs1 - where rp1: "rpath (RAG s) (Th th2') xs1 (Th th1)" - by auto - from star_rpath[OF star2] obtain xs2 - where rp2: "rpath (RAG s) (Th th2') xs2 (Th th2)" - by auto - from rp1 rp2 - show ?thesis - proof(cases) - case (less_1 xs') - moreover have "xs' = []" - proof(rule ccontr) - assume otherwise: "xs' \ []" - from rpath_plus[OF less_1(3) this] - have "(Th th1, Th th2) \ (RAG s)\<^sup>+" . - from tranclD[OF this] - obtain cs where "waiting s th1 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - with runing_1 show False - by (unfold runing_def readys_def, auto) - qed - ultimately have "xs2 = xs1" by simp - from rpath_dest_eq[OF rp1 rp2[unfolded this]] - show ?thesis by simp - next - case (less_2 xs') - moreover have "xs' = []" - proof(rule ccontr) - assume otherwise: "xs' \ []" - from rpath_plus[OF less_2(3) this] - have "(Th th2, Th th1) \ (RAG s)\<^sup>+" . - from tranclD[OF this] - obtain cs where "waiting s th2 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - with runing_2 show False - by (unfold runing_def readys_def, auto) - qed - ultimately have "xs2 = xs1" by simp - from rpath_dest_eq[OF rp1 rp2[unfolded this]] - show ?thesis by simp - qed -qed - -lemma card_runing: "card (runing s) \ 1" -proof(cases "runing s = {}") - case True - thus ?thesis by auto -next - case False - then obtain th where [simp]: "th \ runing s" by auto - from runing_unique[OF this] - have "runing s = {th}" by auto - thus ?thesis by auto -qed - -lemma create_pre: - assumes stp: "step s e" - and not_in: "th \ threads s" - and is_in: "th \ threads (e#s)" - obtains prio where "e = Create th prio" -proof - - from assms - show ?thesis - proof(cases) - case (thread_create thread prio) - with is_in not_in have "e = Create th prio" by simp - from that[OF this] show ?thesis . - next - case (thread_exit thread) - with assms show ?thesis by (auto intro!:that) - next - case (thread_P thread) - with assms show ?thesis by (auto intro!:that) - next - case (thread_V thread) - with assms show ?thesis by (auto intro!:that) - next - case (thread_set thread) - with assms show ?thesis by (auto intro!:that) - qed -qed - lemma eq_pv_children: assumes eq_pv: "cntP s th = cntV s th" shows "children (RAG s) (Th th) = {}" @@ -3838,151 +4338,6 @@ using eq_pv_children[OF assms] by (unfold subtree_children, simp) -end - -lemma cp_gen_alt_def: - "cp_gen s = (Max \ (\x. (the_preced s \ the_thread) ` subtree (tRAG s) x))" - by (auto simp:cp_gen_def) - -lemma tRAG_nodeE: - assumes "(n1, n2) \ tRAG s" - obtains th1 th2 where "n1 = Th th1" "n2 = Th th2" - using assms - by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def) - -lemma subtree_nodeE: - assumes "n \ subtree (tRAG s) (Th th)" - obtains th1 where "n = Th th1" -proof - - show ?thesis - proof(rule subtreeE[OF assms]) - assume "n = Th th" - from that[OF this] show ?thesis . - next - assume "Th th \ ancestors (tRAG s) n" - hence "(n, Th th) \ (tRAG s)^+" by (auto simp:ancestors_def) - hence "\ th1. n = Th th1" - proof(induct) - case (base y) - from tRAG_nodeE[OF this] show ?case by metis - next - case (step y z) - thus ?case by auto - qed - with that show ?thesis by auto - qed -qed - -lemma tRAG_star_RAG: "(tRAG s)^* \ (RAG s)^*" -proof - - have "(wRAG s O hRAG s)^* \ (RAG s O RAG s)^*" - by (rule rtrancl_mono, auto simp:RAG_split) - also have "... \ ((RAG s)^*)^*" - by (rule rtrancl_mono, auto) - also have "... = (RAG s)^*" by simp - finally show ?thesis by (unfold tRAG_def, simp) -qed - -lemma tRAG_subtree_RAG: "subtree (tRAG s) x \ subtree (RAG s) x" -proof - - { fix a - assume "a \ subtree (tRAG s) x" - hence "(a, x) \ (tRAG s)^*" by (auto simp:subtree_def) - with tRAG_star_RAG - have "(a, x) \ (RAG s)^*" by auto - hence "a \ subtree (RAG s) x" by (auto simp:subtree_def) - } thus ?thesis by auto -qed - -lemma tRAG_trancl_eq: - "{th'. (Th th', Th th) \ (tRAG s)^+} = - {th'. (Th th', Th th) \ (RAG s)^+}" - (is "?L = ?R") -proof - - { fix th' - assume "th' \ ?L" - hence "(Th th', Th th) \ (tRAG s)^+" by auto - from tranclD[OF this] - obtain z where h: "(Th th', z) \ tRAG s" "(z, Th th) \ (tRAG s)\<^sup>*" by auto - from tRAG_subtree_RAG and this(2) - have "(z, Th th) \ (RAG s)^*" by (meson subsetCE tRAG_star_RAG) - moreover from h(1) have "(Th th', z) \ (RAG s)^+" using tRAG_alt_def by auto - ultimately have "th' \ ?R" by auto - } moreover - { fix th' - assume "th' \ ?R" - hence "(Th th', Th th) \ (RAG s)^+" by (auto) - from plus_rpath[OF this] - obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \ []" by auto - hence "(Th th', Th th) \ (tRAG s)^+" - proof(induct xs arbitrary:th' th rule:length_induct) - case (1 xs th' th) - then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto) - show ?case - proof(cases "xs1") - case Nil - from 1(2)[unfolded Cons1 Nil] - have rp: "rpath (RAG s) (Th th') [x1] (Th th)" . - hence "(Th th', x1) \ (RAG s)" - by (cases, auto) - then obtain cs where "x1 = Cs cs" - by (unfold s_RAG_def, auto) - from rpath_nnl_lastE[OF rp[unfolded this]] - show ?thesis by auto - next - case (Cons x2 xs2) - from 1(2)[unfolded Cons1[unfolded this]] - have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" . - from rpath_edges_on[OF this] - have eds: "edges_on (Th th' # x1 # x2 # xs2) \ RAG s" . - have "(Th th', x1) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - with eds have rg1: "(Th th', x1) \ RAG s" by auto - then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto) - have "(x1, x2) \ edges_on (Th th' # x1 # x2 # xs2)" - by (simp add: edges_on_unfold) - from this eds - have rg2: "(x1, x2) \ RAG s" by auto - from this[unfolded eq_x1] - obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto) - from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2] - have rt1: "(Th th', Th th1) \ tRAG s" by (unfold tRAG_alt_def, auto) - from rp have "rpath (RAG s) x2 xs2 (Th th)" - by (elim rpath_ConsE, simp) - from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" . - show ?thesis - proof(cases "xs2 = []") - case True - from rpath_nilE[OF rp'[unfolded this]] - have "th1 = th" by auto - from rt1[unfolded this] show ?thesis by auto - next - case False - from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons] - have "(Th th1, Th th) \ (tRAG s)\<^sup>+" by simp - with rt1 show ?thesis by auto - qed - qed - qed - hence "th' \ ?L" by auto - } ultimately show ?thesis by blast -qed - -lemma tRAG_trancl_eq_Th: - "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = - {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" - using tRAG_trancl_eq by auto - -lemma dependants_alt_def: - "dependants s th = {th'. (Th th', Th th) \ (tRAG s)^+}" - by (metis eq_RAG s_dependants_def tRAG_trancl_eq) - -lemma dependants_alt_def1: - "dependants (s::state) th = {th'. (Th th', Th th) \ (RAG s)^+}" - using dependants_alt_def tRAG_trancl_eq by auto - -context valid_trace -begin lemma count_eq_RAG_plus: assumes "cntP s th = cntV s th" shows "{th'. (Th th', Th th) \ (RAG s)^+} = {}" @@ -4004,14 +4359,6 @@ show ?thesis . qed -end - -lemma eq_dependants: "dependants (wq s) = dependants s" - by (simp add: s_dependants_abv wq_def) - -context valid_trace -begin - lemma count_eq_tRAG_plus: assumes "cntP s th = cntV s th" shows "{th'. (Th th', Th th) \ (tRAG s)^+} = {}" @@ -4026,244 +4373,12 @@ assumes "cntP s th = cntV s th" shows "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = {}" using count_eq_tRAG_plus[OF assms] by auto -end - -lemma inj_the_preced: - "inj_on (the_preced s) (threads s)" - by (metis inj_onI preced_unique the_preced_def) - -lemma tRAG_Field: - "Field (tRAG s) \ Field (RAG s)" - by (unfold tRAG_alt_def Field_def, auto) - -lemma tRAG_ancestorsE: - assumes "x \ ancestors (tRAG s) u" - obtains th where "x = Th th" -proof - - from assms have "(u, x) \ (tRAG s)^+" - by (unfold ancestors_def, auto) - from tranclE[OF this] obtain c where "(c, x) \ tRAG s" by auto - then obtain th where "x = Th th" - by (unfold tRAG_alt_def, auto) - from that[OF this] show ?thesis . -qed - -lemma tRAG_mono: - assumes "RAG s' \ RAG s" - shows "tRAG s' \ tRAG s" - using assms - by (unfold tRAG_alt_def, auto) - -lemma holding_next_thI: - assumes "holding s th cs" - and "length (wq s cs) > 1" - obtains th' where "next_th s th cs th'" -proof - - from assms(1)[folded holding_eq, unfolded cs_holding_def] - have " th \ set (wq s cs) \ th = hd (wq s cs)" - by (unfold s_holding_def, fold wq_def, auto) - then obtain rest where h1: "wq s cs = th#rest" - by (cases "wq s cs", auto) - with assms(2) have h2: "rest \ []" by auto - let ?th' = "hd (SOME q. distinct q \ set q = set rest)" - have "next_th s th cs ?th'" using h1(1) h2 - by (unfold next_th_def, auto) - from that[OF this] show ?thesis . -qed - -lemma RAG_tRAG_transfer: - assumes "vt s'" - assumes "RAG s = RAG s' \ {(Th th, Cs cs)}" - and "(Cs cs, Th th'') \ RAG s'" - shows "tRAG s = tRAG s' \ {(Th th, Th th'')}" (is "?L = ?R") -proof - - interpret vt_s': valid_trace "s'" using assms(1) - by (unfold_locales, simp) - { fix n1 n2 - assume "(n1, n2) \ ?L" - from this[unfolded tRAG_alt_def] - obtain th1 th2 cs' where - h: "n1 = Th th1" "n2 = Th th2" - "(Th th1, Cs cs') \ RAG s" - "(Cs cs', Th th2) \ RAG s" by auto - from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \ RAG s'" by auto - from h(3) and assms(2) - have "(Th th1, Cs cs') = (Th th, Cs cs) \ - (Th th1, Cs cs') \ RAG s'" by auto - hence "(n1, n2) \ ?R" - proof - assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)" - hence eq_th1: "th1 = th" by simp - moreover have "th2 = th''" - proof - - from h1 have "cs' = cs" by simp - from assms(3) cs_in[unfolded this] - show ?thesis using vt_s'.unique_RAG by auto - qed - ultimately show ?thesis using h(1,2) by auto - next - assume "(Th th1, Cs cs') \ RAG s'" - with cs_in have "(Th th1, Th th2) \ tRAG s'" - by (unfold tRAG_alt_def, auto) - from this[folded h(1, 2)] show ?thesis by auto - qed - } moreover { - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \tRAG s' \ (n1, n2) = (Th th, Th th'')" by auto - hence "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ tRAG s'" - moreover have "... \ ?L" - proof(rule tRAG_mono) - show "RAG s' \ RAG s" by (unfold assms(2), auto) - qed - ultimately show ?thesis by auto - next - assume eq_n: "(n1, n2) = (Th th, Th th'')" - from assms(2, 3) have "(Cs cs, Th th'') \ RAG s" by auto - moreover have "(Th th, Cs cs) \ RAG s" using assms(2) by auto - ultimately show ?thesis - by (unfold eq_n tRAG_alt_def, auto) - qed - } ultimately show ?thesis by auto -qed - -context valid_trace -begin - -lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt] - -end - -lemma tRAG_subtree_eq: - "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \ (subtree (RAG s) (Th th))}" - (is "?L = ?R") -proof - - { fix n - assume h: "n \ ?L" - hence "n \ ?R" - by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) - } moreover { - fix n - assume "n \ ?R" - then obtain th' where h: "n = Th th'" "(Th th', Th th) \ (RAG s)^*" - by (auto simp:subtree_def) - from rtranclD[OF this(2)] - have "n \ ?L" - proof - assume "Th th' \ Th th \ (Th th', Th th) \ (RAG s)\<^sup>+" - with h have "n \ {Th th' | th'. (Th th', Th th) \ (RAG s)^+}" by auto - thus ?thesis using subtree_def tRAG_trancl_eq by fastforce - qed (insert h, auto simp:subtree_def) - } ultimately show ?thesis by auto -qed - -lemma threads_set_eq: - "the_thread ` (subtree (tRAG s) (Th th)) = - {th'. Th th' \ (subtree (RAG s) (Th th))}" (is "?L = ?R") - by (auto intro:rev_image_eqI simp:tRAG_subtree_eq) - -lemma cp_alt_def1: - "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))" -proof - - have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) = - ((the_preced s \ the_thread) ` subtree (tRAG s) (Th th))" - by auto - thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto) -qed - -lemma cp_gen_def_cond: - assumes "x = Th th" - shows "cp s th = cp_gen s (Th th)" -by (unfold cp_alt_def1 cp_gen_def, simp) - -lemma cp_gen_over_set: - assumes "\ x \ A. \ th. x = Th th" - shows "cp_gen s ` A = (cp s \ the_thread) ` A" -proof(rule f_image_eq) - fix a - assume "a \ A" - from assms[rule_format, OF this] - obtain th where eq_a: "a = Th th" by auto - show "cp_gen s a = (cp s \ the_thread) a" - by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp) -qed - -context valid_trace -begin - -lemma subtree_tRAG_thread: - assumes "th \ threads s" - shows "subtree (tRAG s) (Th th) \ Th ` threads s" (is "?L \ ?R") -proof - - have "?L = {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - by (unfold tRAG_subtree_eq, simp) - also have "... \ ?R" - proof - fix x - assume "x \ {Th th' |th'. Th th' \ subtree (RAG s) (Th th)}" - then obtain th' where h: "x = Th th'" "Th th' \ subtree (RAG s) (Th th)" by auto - from this(2) - show "x \ ?R" - proof(cases rule:subtreeE) - case 1 - thus ?thesis by (simp add: assms h(1)) - next - case 2 - thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) - qed - qed - finally show ?thesis . -qed - -lemma readys_root: - assumes "th \ readys s" - shows "root (RAG s) (Th th)" -proof - - { fix x - assume "x \ ancestors (RAG s) (Th th)" - hence h: "(Th th, x) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - obtain z where "(Th th, z) \ RAG s" by auto - with assms(1) have False - apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def) - by (fold wq_def, blast) - } thus ?thesis by (unfold root_def, auto) -qed - -lemma readys_in_no_subtree: - assumes "th \ readys s" - and "th' \ th" - shows "Th th \ subtree (RAG s) (Th th')" -proof - assume "Th th \ subtree (RAG s) (Th th')" - thus False - proof(cases rule:subtreeE) - case 1 - with assms show ?thesis by auto - next - case 2 - with readys_root[OF assms(1)] - show ?thesis by (auto simp:root_def) - qed -qed - -lemma not_in_thread_isolated: - assumes "th \ threads s" - shows "(Th th) \ Field (RAG s)" -proof - assume "(Th th) \ Field (RAG s)" - with dm_RAG_threads and rg_RAG_threads assms - show False by (unfold Field_def, blast) -qed end definition detached :: "state \ thread \ bool" where "detached s th \ (\(\ cs. holding s th cs)) \ (\(\cs. waiting s th cs))" - lemma detached_test: shows "detached s th = (Th th \ Field (RAG s))" apply(simp add: detached_def Field_def) @@ -4336,6 +4451,35 @@ end +section {* Recursive definition of @{term "cp"} *} + +lemma cp_alt_def1: + "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))" +proof - + have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) = + ((the_preced s \ the_thread) ` subtree (tRAG s) (Th th))" + by auto + thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto) +qed + +lemma cp_gen_def_cond: + assumes "x = Th th" + shows "cp s th = cp_gen s (Th th)" +by (unfold cp_alt_def1 cp_gen_def, simp) + +lemma cp_gen_over_set: + assumes "\ x \ A. \ th. x = Th th" + shows "cp_gen s ` A = (cp s \ the_thread) ` A" +proof(rule f_image_eq) + fix a + assume "a \ A" + from assms[rule_format, OF this] + obtain th where eq_a: "a = Th th" by auto + show "cp_gen s a = (cp s \ the_thread) a" + by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp) +qed + + context valid_trace begin (* ddd *) @@ -4413,6 +4557,63 @@ thus ?thesis by (subst (1) h(1), unfold h(2), simp) qed qed +end + +section {* Other properties useful in Implementation.thy or Correctness.thy *} + +context valid_trace_e +begin + +lemma actor_inv: + assumes "\ isCreate e" + shows "actor e \ runing s" + using pip_e assms + by (induct, auto) +end + +context valid_trace +begin + +lemma readys_root: + assumes "th \ readys s" + shows "root (RAG s) (Th th)" +proof - + { fix x + assume "x \ ancestors (RAG s) (Th th)" + hence h: "(Th th, x) \ (RAG s)^+" by (auto simp:ancestors_def) + from tranclD[OF this] + obtain z where "(Th th, z) \ RAG s" by auto + with assms(1) have False + apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def) + by (fold wq_def, blast) + } thus ?thesis by (unfold root_def, auto) +qed + +lemma readys_in_no_subtree: + assumes "th \ readys s" + and "th' \ th" + shows "Th th \ subtree (RAG s) (Th th')" +proof + assume "Th th \ subtree (RAG s) (Th th')" + thus False + proof(cases rule:subtreeE) + case 1 + with assms show ?thesis by auto + next + case 2 + with readys_root[OF assms(1)] + show ?thesis by (auto simp:root_def) + qed +qed + +lemma not_in_thread_isolated: + assumes "th \ threads s" + shows "(Th th) \ Field (RAG s)" +proof + assume "(Th th) \ Field (RAG s)" + with dm_RAG_threads and rg_RAG_threads assms + show False by (unfold Field_def, blast) +qed lemma next_th_holding: assumes nxt: "next_th s th cs th'" @@ -4458,91 +4659,4 @@ end -lemma next_th_unique: - assumes nt1: "next_th s th cs th1" - and nt2: "next_th s th cs th2" - shows "th1 = th2" -using assms by (unfold next_th_def, auto) - -context valid_trace -begin - -thm th_chain_to_ready - -find_theorems subtree Th RAG - -lemma threads_alt_def: - "(threads s) = (\ th \ readys s. {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "?L = ?R") -proof - - { fix th1 - assume "th1 \ ?L" - from th_chain_to_ready[OF this] - have "th1 \ readys s \ (\th'. th' \ readys s \ (Th th1, Th th') \ (RAG s)\<^sup>+)" . - hence "th1 \ ?R" by (auto simp:subtree_def) - } moreover - { fix th' - assume "th' \ ?R" - then obtain th where h: "th \ readys s" " Th th' \ subtree (RAG s) (Th th)" - by auto - from this(2) - have "th' \ ?L" - proof(cases rule:subtreeE) - case 1 - with h(1) show ?thesis by (auto simp:readys_def) - next - case 2 - from tranclD[OF this(2)[unfolded ancestors_def, simplified]] - have "Th th' \ Domain (RAG s)" by auto - from dm_RAG_threads[OF this] - show ?thesis . - qed - } ultimately show ?thesis by auto -qed - -lemma finite_readys [simp]: "finite (readys s)" - using finite_threads readys_threads rev_finite_subset by blast - -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)" (is "?L = ?R") -proof(cases "readys s = {}") - case False - have "?R = Max (the_preced s ` threads s)" by (unfold max_cp_eq, simp) - also have "... = - Max (the_preced s ` (\th\readys s. {th'. Th th' \ subtree (RAG s) (Th th)}))" - by (unfold threads_alt_def, simp) - also have "... = - Max ((\th\readys s. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}))" - by (unfold image_UN, simp) - also have "... = - Max (Max ` (\th. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}) ` readys s)" - proof(rule Max_UNION) - show "\M\(\x. the_preced s ` - {th'. Th th' \ subtree (RAG s) (Th x)}) ` readys s. finite M" - using finite_subtree_threads by auto - qed (auto simp:False subtree_def) - also have "... = - Max ((Max \ (\th. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})) ` readys s)" - by (unfold image_comp, simp) - also have "... = ?L" (is "Max (?f ` ?A) = Max (?g ` ?A)") - proof - - have "(?f ` ?A) = (?g ` ?A)" - proof(rule f_image_eq) - fix th1 - assume "th1 \ ?A" - thus "?f th1 = ?g th1" - by (unfold cp_alt_def, simp) - qed - thus ?thesis by simp - qed - finally show ?thesis by simp -qed (auto simp:threads_alt_def) - -end - -end - +end \ No newline at end of file diff -r 43482ab31341 -r 0c89419b4742 ExtGG.thy --- a/ExtGG.thy Wed Feb 03 21:51:57 2016 +0800 +++ b/ExtGG.thy Wed Feb 03 22:17:29 2016 +0800 @@ -2,8 +2,8 @@ This file contains lemmas used to guide the recalculation of current precedence after every system call (or system operation) *} -theory ExtGG -imports CpsG +theory Implementation +imports PIPBasics begin text {* (* ddd *) @@ -376,9 +376,6 @@ context valid_trace_p_w begin -interpretation vat_e: valid_trace "e#s" - by (unfold_locales, insert vt_e, simp) - lemma cs_held: "(Cs cs, Th holder) \ RAG s" using holding_s_holder by (unfold s_RAG_def, fold holding_eq, auto) @@ -428,13 +425,13 @@ and "y \ ancestors (tRAG (e#s)) u" shows "cp_gen (e#s) y = cp_gen s y" using assms(3) -proof(induct rule:wf_induct[OF vat_e.fsbttRAGs.wf]) +proof(induct rule:wf_induct[OF vat_es.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_e.cp_gen_rec[OF this] + from vat_es.cp_gen_rec[OF this] have "?L = Max ({the_preced (e#s) th2} \ cp_gen (e#s) ` RTree.children (tRAG (e#s)) x)" . also have "... = @@ -454,15 +451,15 @@ assume "x \ Range {(Th th, Th holder)}" hence eq_x: "x = Th holder" using RangeE by auto show False - proof(cases rule:vat_e.ancestors_headE[OF assms(1) start]) + proof(cases rule:vat_es.ancestors_headE[OF assms(1) start]) case 1 - from x_u[folded this, unfolded eq_x] vat_e.acyclic_tRAG + from x_u[folded this, unfolded eq_x] vat_es.acyclic_tRAG show ?thesis by (auto simp:ancestors_def acyclic_def) next case 2 with x_u[unfolded eq_x] have "(Th holder, Th holder) \ (tRAG (e#s))^+" by (auto simp:ancestors_def) - with vat_e.acyclic_tRAG show ?thesis by (auto simp:acyclic_def) + with vat_es.acyclic_tRAG show ?thesis by (auto simp:acyclic_def) qed qed qed @@ -473,7 +470,7 @@ assume a_in: "a \ ?A" from 1(2) show "?f a = ?g a" - proof(cases rule:vat_e.rtree_s.ancestors_childrenE[case_names in_ch out_ch]) + proof(cases rule:vat_es.rtree_s.ancestors_childrenE[case_names in_ch out_ch]) case in_ch show ?thesis proof(cases "a = u") @@ -485,7 +482,7 @@ proof assume a_in': "a \ ancestors (tRAG (e#s)) (Th th)" have "a = u" - proof(rule vat_e.rtree_s.ancestors_children_unique) + proof(rule vat_es.rtree_s.ancestors_children_unique) from a_in' a_in show "a \ ancestors (tRAG (e#s)) (Th th) \ RTree.children (tRAG (e#s)) x" by auto next @@ -519,7 +516,7 @@ proof assume a_in': "a \ ancestors (tRAG (e#s)) (Th th)" have "a = z" - proof(rule vat_e.rtree_s.ancestors_children_unique) + proof(rule vat_es.rtree_s.ancestors_children_unique) from assms(1) h(1) have "z \ ancestors (tRAG (e#s)) (Th th)" by (auto simp:ancestors_def) with h(2) show " z \ ancestors (tRAG (e#s)) (Th th) \ @@ -570,9 +567,6 @@ context valid_trace_create begin -interpretation vat_e: valid_trace "e#s" - by (unfold_locales, insert vt_e, simp) - lemma tRAG_kept: "tRAG (e#s) = tRAG s" by (unfold tRAG_alt_def RAG_unchanged, auto) @@ -632,7 +626,7 @@ qed lemma eq_cp_th: "cp (e#s) th = preced th (e#s)" - by (unfold vat_e.cp_rec children_of_th, simp add:the_preced_def) + by (unfold vat_es.cp_rec children_of_th, simp add:the_preced_def) end @@ -706,924 +700,3 @@ end -======= -theory ExtGG -imports PrioG CpsG -begin - -text {* - The following two auxiliary lemmas are used to reason about @{term Max}. -*} -lemma image_Max_eqI: - assumes "finite B" - and "b \ B" - and "\ x \ B. f x \ f b" - shows "Max (f ` B) = f b" - using assms - using Max_eqI by blast - -lemma image_Max_subset: - assumes "finite A" - and "B \ A" - and "a \ B" - and "Max (f ` A) = f a" - shows "Max (f ` B) = f a" -proof(rule image_Max_eqI) - show "finite B" - using assms(1) assms(2) finite_subset by auto -next - show "a \ B" using assms by simp -next - show "\x\B. f x \ f a" - by (metis Max_ge assms(1) assms(2) assms(4) - finite_imageI image_eqI subsetCE) -qed - -text {* - The following locale @{text "highest_gen"} sets the basic context for our - investigation: supposing thread @{text th} holds the highest @{term cp}-value - in state @{text s}, which means the task for @{text th} is the - most urgent. We want to show that - @{text th} is treated correctly by PIP, which means - @{text th} will not be blocked unreasonably by other less urgent - threads. -*} -locale highest_gen = - fixes s th prio tm - assumes vt_s: "vt s" - and threads_s: "th \ threads s" - and highest: "preced th s = Max ((cp s)`threads s)" - -- {* The internal structure of @{term th}'s precedence is exposed:*} - and preced_th: "preced th s = Prc prio tm" - --- {* @{term s} is a valid trace, so it will inherit all results derived for - a valid trace: *} -sublocale highest_gen < vat_s: valid_trace "s" - by (unfold_locales, insert vt_s, simp) - -context highest_gen -begin - -text {* - @{term tm} is the time when the precedence of @{term th} is set, so - @{term tm} must be a valid moment index into @{term s}. -*} -lemma lt_tm: "tm < length s" - by (insert preced_tm_lt[OF threads_s preced_th], simp) - -text {* - Since @{term th} holds the highest precedence and @{text "cp"} - is the highest precedence of all threads in the sub-tree of - @{text "th"} and @{text th} is among these threads, - its @{term cp} must equal to its precedence: -*} -lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R") -proof - - have "?L \ ?R" - by (unfold highest, rule Max_ge, - auto simp:threads_s finite_threads) - moreover have "?R \ ?L" - by (unfold vat_s.cp_rec, rule Max_ge, - auto simp:the_preced_def vat_s.fsbttRAGs.finite_children) - ultimately show ?thesis by auto -qed - -(* ccc *) -lemma highest_cp_preced: "cp s th = Max ((\ th'. preced th' s) ` threads s)" - by (fold max_cp_eq, unfold eq_cp_s_th, insert highest, simp) - -lemma highest_preced_thread: "preced th s = Max ((\ th'. preced th' s) ` threads s)" - by (fold eq_cp_s_th, unfold highest_cp_preced, simp) - -lemma highest': "cp s th = Max (cp s ` threads s)" -proof - - from highest_cp_preced max_cp_eq[symmetric] - show ?thesis by simp -qed - -end - -locale extend_highest_gen = highest_gen + - fixes t - assumes vt_t: "vt (t@s)" - and create_low: "Create th' prio' \ set t \ prio' \ prio" - and set_diff_low: "Set th' prio' \ set t \ th' \ th \ prio' \ prio" - and exit_diff: "Exit th' \ set t \ th' \ th" - -sublocale extend_highest_gen < vat_t: valid_trace "t@s" - by (unfold_locales, insert vt_t, simp) - -lemma step_back_vt_app: - assumes vt_ts: "vt (t@s)" - shows "vt s" -proof - - from vt_ts show ?thesis - proof(induct t) - case Nil - from Nil show ?case by auto - next - case (Cons e t) - assume ih: " vt (t @ s) \ vt s" - and vt_et: "vt ((e # t) @ s)" - show ?case - proof(rule ih) - show "vt (t @ s)" - proof(rule step_back_vt) - from vt_et show "vt (e # t @ s)" by simp - qed - qed - qed -qed - - -locale red_extend_highest_gen = extend_highest_gen + - fixes i::nat - -sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)" - apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric]) - apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp) - by (unfold highest_gen_def, auto dest:step_back_vt_app) - - -context extend_highest_gen -begin - - lemma ind [consumes 0, case_names Nil Cons, induct type]: - assumes - h0: "R []" - and h2: "\ e t. \vt (t@s); step (t@s) e; - extend_highest_gen s th prio tm t; - extend_highest_gen s th prio tm (e#t); R t\ \ R (e#t)" - shows "R t" -proof - - from vt_t extend_highest_gen_axioms show ?thesis - proof(induct t) - from h0 show "R []" . - next - case (Cons e t') - assume ih: "\vt (t' @ s); extend_highest_gen s th prio tm t'\ \ R t'" - and vt_e: "vt ((e # t') @ s)" - and et: "extend_highest_gen s th prio tm (e # t')" - from vt_e and step_back_step have stp: "step (t'@s) e" by auto - from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto - show ?case - proof(rule h2 [OF vt_ts stp _ _ _ ]) - show "R t'" - proof(rule ih) - from et show ext': "extend_highest_gen s th prio tm t'" - by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) - next - from vt_ts show "vt (t' @ s)" . - qed - next - from et show "extend_highest_gen s th prio tm (e # t')" . - next - from et show ext': "extend_highest_gen s th prio tm t'" - by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) - qed - qed -qed - - -lemma th_kept: "th \ threads (t @ s) \ - preced th (t@s) = preced th s" (is "?Q t") -proof - - show ?thesis - proof(induct rule:ind) - case Nil - from threads_s - show ?case - by auto - next - case (Cons e t) - interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto - interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto - show ?case - proof(cases e) - case (Create thread prio) - show ?thesis - proof - - from Cons and Create have "step (t@s) (Create thread prio)" by auto - hence "th \ thread" - proof(cases) - case thread_create - with Cons show ?thesis by auto - qed - hence "preced th ((e # t) @ s) = preced th (t @ s)" - by (unfold Create, auto simp:preced_def) - moreover note Cons - ultimately show ?thesis - by (auto simp:Create) - qed - next - case (Exit thread) - from h_e.exit_diff and Exit - have neq_th: "thread \ th" by auto - with Cons - show ?thesis - by (unfold Exit, auto simp:preced_def) - next - case (P thread cs) - with Cons - show ?thesis - by (auto simp:P preced_def) - next - case (V thread cs) - with Cons - show ?thesis - by (auto simp:V preced_def) - next - case (Set thread prio') - show ?thesis - proof - - from h_e.set_diff_low and Set - have "th \ thread" by auto - hence "preced th ((e # t) @ s) = preced th (t @ s)" - by (unfold Set, auto simp:preced_def) - moreover note Cons - ultimately show ?thesis - by (auto simp:Set) - qed - qed - qed -qed - -text {* - According to @{thm th_kept}, thread @{text "th"} has its living status - and precedence kept along the way of @{text "t"}. The following lemma - shows that this preserved precedence of @{text "th"} remains as the highest - along the way of @{text "t"}. - - The proof goes by induction over @{text "t"} using the specialized - induction rule @{thm ind}, followed by case analysis of each possible - operations of PIP. All cases follow the same pattern rendered by the - generalized introduction rule @{thm "image_Max_eqI"}. - - The very essence is to show that precedences, no matter whether they are newly introduced - or modified, are always lower than the one held by @{term "th"}, - which by @{thm th_kept} is preserved along the way. -*} -lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s" -proof(induct rule:ind) - case Nil - from highest_preced_thread - show ?case - by (unfold the_preced_def, simp) -next - case (Cons e t) - interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto - interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto - show ?case - proof(cases e) - case (Create thread prio') - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - -- {* The following is the common pattern of each branch of the case analysis. *} - -- {* The major part is to show that @{text "th"} holds the highest precedence: *} - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume "x \ ?A" - hence "x = thread \ x \ threads (t@s)" by (auto simp:Create) - thus "?f x \ ?f th" - proof - assume "x = thread" - thus ?thesis - apply (simp add:Create the_preced_def preced_def, fold preced_def) - using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force - next - assume h: "x \ threads (t @ s)" - from Cons(2)[unfolded Create] - have "x \ thread" using h by (cases, auto) - hence "?f x = the_preced (t@s) x" - by (simp add:Create the_preced_def preced_def) - hence "?f x \ Max (the_preced (t@s) ` threads (t@s))" - by (simp add: h_t.finite_threads h) - also have "... = ?f th" - by (metis Cons.hyps(5) h_e.th_kept the_preced_def) - finally show ?thesis . - qed - qed - qed - -- {* The minor part is to show that the precedence of @{text "th"} - equals to preserved one, given by the foregoing lemma @{thm th_kept} *} - also have "... = ?t" using h_e.th_kept the_preced_def by auto - -- {* Then it follows trivially that the precedence preserved - for @{term "th"} remains the maximum of all living threads along the way. *} - finally show ?thesis . - qed - next - case (Exit thread) - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume "x \ ?A" - hence "x \ threads (t@s)" by (simp add: Exit) - hence "?f x \ Max (?f ` threads (t@s))" - by (simp add: h_t.finite_threads) - also have "... \ ?f th" - apply (simp add:Exit the_preced_def preced_def, fold preced_def) - using Cons.hyps(5) h_t.th_kept the_preced_def by auto - finally show "?f x \ ?f th" . - qed - qed - also have "... = ?t" using h_e.th_kept the_preced_def by auto - finally show ?thesis . - qed - next - case (P thread cs) - with Cons - show ?thesis by (auto simp:preced_def the_preced_def) - next - case (V thread cs) - with Cons - show ?thesis by (auto simp:preced_def the_preced_def) - next - case (Set thread prio') - show ?thesis (is "Max (?f ` ?A) = ?t") - proof - - have "Max (?f ` ?A) = ?f th" - proof(rule image_Max_eqI) - show "finite ?A" using h_e.finite_threads by auto - next - show "th \ ?A" using h_e.th_kept by auto - next - show "\x\?A. ?f x \ ?f th" - proof - fix x - assume h: "x \ ?A" - show "?f x \ ?f th" - proof(cases "x = thread") - case True - moreover have "the_preced (Set thread prio' # t @ s) thread \ the_preced (t @ s) th" - proof - - have "the_preced (t @ s) th = Prc prio tm" - using h_t.th_kept preced_th by (simp add:the_preced_def) - moreover have "prio' \ prio" using Set h_e.set_diff_low by auto - ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def) - qed - ultimately show ?thesis - by (unfold Set, simp add:the_preced_def preced_def) - next - case False - then have "?f x = the_preced (t@s) x" - by (simp add:the_preced_def preced_def Set) - also have "... \ Max (the_preced (t@s) ` threads (t@s))" - using Set h h_t.finite_threads by auto - also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) - finally show ?thesis . - qed - qed - qed - also have "... = ?t" using h_e.th_kept the_preced_def by auto - finally show ?thesis . - qed - qed -qed - -lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))" - by (insert th_kept max_kept, auto) - -text {* - The reason behind the following lemma is that: - Since @{term "cp"} is defined as the maximum precedence - of those threads contained in the sub-tree of node @{term "Th th"} - in @{term "RAG (t@s)"}, and all these threads are living threads, and - @{term "th"} is also among them, the maximum precedence of - them all must be the one for @{text "th"}. -*} -lemma th_cp_max_preced: - "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") -proof - - let ?f = "the_preced (t@s)" - have "?L = ?f th" - proof(unfold cp_alt_def, rule image_Max_eqI) - show "finite {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - proof - - have "{th'. Th th' \ subtree (RAG (t @ s)) (Th th)} = - the_thread ` {n . n \ subtree (RAG (t @ s)) (Th th) \ - (\ th'. n = Th th')}" - by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps) - moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) - ultimately show ?thesis by simp - qed - next - show "th \ {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - by (auto simp:subtree_def) - next - show "\x\{th'. Th th' \ subtree (RAG (t @ s)) (Th th)}. - the_preced (t @ s) x \ the_preced (t @ s) th" - proof - fix th' - assume "th' \ {th'. Th th' \ subtree (RAG (t @ s)) (Th th)}" - hence "Th th' \ subtree (RAG (t @ s)) (Th th)" by auto - moreover have "... \ Field (RAG (t @ s)) \ {Th th}" - by (meson subtree_Field) - ultimately have "Th th' \ ..." by auto - hence "th' \ threads (t@s)" - proof - assume "Th th' \ {Th th}" - thus ?thesis using th_kept by auto - next - assume "Th th' \ Field (RAG (t @ s))" - thus ?thesis using vat_t.not_in_thread_isolated by blast - qed - thus "the_preced (t @ s) th' \ the_preced (t @ s) th" - by (metis Max_ge finite_imageI finite_threads image_eqI - max_kept th_kept the_preced_def) - qed - qed - also have "... = ?R" by (simp add: max_preced the_preced_def) - finally show ?thesis . -qed - -lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))" - using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger - -lemma th_cp_preced: "cp (t@s) th = preced th s" - by (fold max_kept, unfold th_cp_max_preced, simp) - -lemma preced_less: - assumes th'_in: "th' \ threads s" - and neq_th': "th' \ th" - shows "preced th' s < preced th s" - using assms -by (metis Max.coboundedI finite_imageI highest not_le order.trans - preced_linorder rev_image_eqI threads_s vat_s.finite_threads - vat_s.le_cp) - -text {* - Counting of the number of @{term "P"} and @{term "V"} operations - is the cornerstone of a large number of the following proofs. - The reason is that this counting is quite easy to calculate and - convenient to use in the reasoning. - - The following lemma shows that the counting controls whether - a thread is running or not. -*} - -lemma pv_blocked_pre: - assumes th'_in: "th' \ threads (t@s)" - and neq_th': "th' \ th" - and eq_pv: "cntP (t@s) th' = cntV (t@s) th'" - shows "th' \ runing (t@s)" -proof - assume otherwise: "th' \ runing (t@s)" - show False - proof - - have "th' = th" - proof(rule preced_unique) - show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R") - proof - - have "?L = cp (t@s) th'" - by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp) - also have "... = cp (t @ s) th" using otherwise - by (metis (mono_tags, lifting) mem_Collect_eq - runing_def th_cp_max vat_t.max_cp_readys_threads) - also have "... = ?R" by (metis th_cp_preced th_kept) - finally show ?thesis . - qed - qed (auto simp: th'_in th_kept) - moreover have "th' \ th" using neq_th' . - ultimately show ?thesis by simp - qed -qed - -lemmas pv_blocked = pv_blocked_pre[folded detached_eq] - -lemma runing_precond_pre: - fixes th' - assumes th'_in: "th' \ threads s" - and eq_pv: "cntP s th' = cntV s th'" - and neq_th': "th' \ th" - shows "th' \ threads (t@s) \ - cntP (t@s) th' = cntV (t@s) th'" -proof(induct rule:ind) - case (Cons e t) - interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp - interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp - show ?case - proof(cases e) - case (P thread cs) - show ?thesis - proof - - have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" - proof - - have "thread \ th'" - proof - - have "step (t@s) (P thread cs)" using Cons P by auto - thus ?thesis - proof(cases) - assume "thread \ runing (t@s)" - moreover have "th' \ runing (t@s)" using Cons(5) - by (metis neq_th' vat_t.pv_blocked_pre) - ultimately show ?thesis by auto - qed - qed with Cons show ?thesis - by (unfold P, simp add:cntP_def cntV_def count_def) - qed - moreover have "th' \ threads ((e # t) @ s)" using Cons by (unfold P, simp) - ultimately show ?thesis by auto - qed - next - case (V thread cs) - show ?thesis - proof - - have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" - proof - - have "thread \ th'" - proof - - have "step (t@s) (V thread cs)" using Cons V by auto - thus ?thesis - proof(cases) - assume "thread \ runing (t@s)" - moreover have "th' \ runing (t@s)" using Cons(5) - by (metis neq_th' vat_t.pv_blocked_pre) - ultimately show ?thesis by auto - qed - qed with Cons show ?thesis - by (unfold V, simp add:cntP_def cntV_def count_def) - qed - moreover have "th' \ threads ((e # t) @ s)" using Cons by (unfold V, simp) - ultimately show ?thesis by auto - qed - next - case (Create thread prio') - show ?thesis - proof - - have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" - proof - - have "thread \ th'" - proof - - have "step (t@s) (Create thread prio')" using Cons Create by auto - thus ?thesis using Cons(5) by (cases, auto) - qed with Cons show ?thesis - by (unfold Create, simp add:cntP_def cntV_def count_def) - qed - moreover have "th' \ threads ((e # t) @ s)" using Cons by (unfold Create, simp) - ultimately show ?thesis by auto - qed - next - case (Exit thread) - show ?thesis - proof - - have neq_thread: "thread \ th'" - proof - - have "step (t@s) (Exit thread)" using Cons Exit by auto - thus ?thesis apply (cases) using Cons(5) - by (metis neq_th' vat_t.pv_blocked_pre) - qed - hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons - by (unfold Exit, simp add:cntP_def cntV_def count_def) - moreover have "th' \ threads ((e # t) @ s)" using Cons neq_thread - by (unfold Exit, simp) - ultimately show ?thesis by auto - qed - next - case (Set thread prio') - with Cons - show ?thesis - by (auto simp:cntP_def cntV_def count_def) - qed -next - case Nil - with assms - show ?case by auto -qed - -text {* Changing counting balance to detachedness *} -lemmas runing_precond_pre_dtc = runing_precond_pre - [folded vat_t.detached_eq vat_s.detached_eq] - -lemma runing_precond: - fixes th' - assumes th'_in: "th' \ threads s" - and neq_th': "th' \ th" - and is_runing: "th' \ runing (t@s)" - shows "cntP s th' > cntV s th'" - using assms -proof - - have "cntP s th' \ cntV s th'" - by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in) - moreover have "cntV s th' \ cntP s th'" using vat_s.cnp_cnv_cncs by auto - ultimately show ?thesis by auto -qed - -lemma moment_blocked_pre: - assumes neq_th': "th' \ th" - and th'_in: "th' \ threads ((moment i t)@s)" - and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'" - shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \ - th' \ threads ((moment (i+j) t)@s)" -proof - - interpret h_i: red_extend_highest_gen _ _ _ _ _ i - by (unfold_locales) - interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j" - by (unfold_locales) - interpret h: extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)" - proof(unfold_locales) - show "vt (moment i t @ s)" by (metis h_i.vt_t) - next - show "th \ threads (moment i t @ s)" by (metis h_i.th_kept) - next - show "preced th (moment i t @ s) = - Max (cp (moment i t @ s) ` threads (moment i t @ s))" - by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept) - next - show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th) - next - show "vt (moment j (restm i t) @ moment i t @ s)" - using moment_plus_split by (metis add.commute append_assoc h_j.vt_t) - next - fix th' prio' - assume "Create th' prio' \ set (moment j (restm i t))" - thus "prio' \ prio" using assms - by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append) - next - fix th' prio' - assume "Set th' prio' \ set (moment j (restm i t))" - thus "th' \ th \ prio' \ prio" - by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append) - next - fix th' - assume "Exit th' \ set (moment j (restm i t))" - thus "th' \ th" - by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append) - qed - show ?thesis - by (metis add.commute append_assoc eq_pv h.runing_precond_pre - moment_plus_split neq_th' th'_in) -qed - -lemma moment_blocked_eqpv: - assumes neq_th': "th' \ th" - and th'_in: "th' \ threads ((moment i t)@s)" - and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'" - and le_ij: "i \ j" - shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \ - th' \ threads ((moment j t)@s) \ - th' \ runing ((moment j t)@s)" -proof - - from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij - have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'" - and h2: "th' \ threads ((moment j t)@s)" by auto - moreover have "th' \ runing ((moment j t)@s)" - proof - - interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales) - show ?thesis - using h.pv_blocked_pre h1 h2 neq_th' by auto - qed - ultimately show ?thesis by auto -qed - -(* The foregoing two lemmas are preparation for this one, but - in long run can be combined. Maybe I am wrong. -*) -lemma moment_blocked: - assumes neq_th': "th' \ th" - and th'_in: "th' \ threads ((moment i t)@s)" - and dtc: "detached (moment i t @ s) th'" - and le_ij: "i \ j" - shows "detached (moment j t @ s) th' \ - th' \ threads ((moment j t)@s) \ - th' \ runing ((moment j t)@s)" -proof - - interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales) - interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales) - have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'" - by (metis dtc h_i.detached_elim) - from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij] - show ?thesis by (metis h_j.detached_intro) -qed - -lemma runing_preced_inversion: - assumes runing': "th' \ runing (t@s)" - shows "cp (t@s) th' = preced th s" (is "?L = ?R") -proof - - have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms - by (unfold runing_def, auto) - also have "\ = ?R" - by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) - finally show ?thesis . -qed - -text {* - The situation when @{term "th"} is blocked is analyzed by the following lemmas. -*} - -text {* - The following lemmas shows the running thread @{text "th'"}, if it is different from - @{term th}, must be live at the very beginning. By the term {\em the very beginning}, - we mean the moment where the formal investigation starts, i.e. the moment (or state) - @{term s}. -*} - -lemma runing_inversion_0: - assumes neq_th': "th' \ th" - and runing': "th' \ runing (t@s)" - shows "th' \ threads s" -proof - - -- {* The proof is by contradiction: *} - { assume otherwise: "\ ?thesis" - have "th' \ runing (t @ s)" - proof - - -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *} - have th'_in: "th' \ threads (t@s)" using runing' by (simp add:runing_def readys_def) - -- {* However, @{text "th'"} does not exist at very beginning. *} - have th'_notin: "th' \ threads (moment 0 t @ s)" using otherwise - by (metis append.simps(1) moment_zero) - -- {* Therefore, there must be a moment during @{text "t"}, when - @{text "th'"} came into being. *} - -- {* Let us suppose the moment being @{text "i"}: *} - from p_split_gen[OF th'_in th'_notin] - obtain i where lt_its: "i < length t" - and le_i: "0 \ i" - and pre: " th' \ threads (moment i t @ s)" (is "th' \ threads ?pre") - and post: "(\i'>i. th' \ threads (moment i' t @ s))" by (auto) - interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales) - interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales) - from lt_its have "Suc i \ length t" by auto - -- {* Let us also suppose the event which makes this change is @{text e}: *} - from moment_head[OF this] obtain e where - eq_me: "moment (Suc i) t = e # moment i t" by blast - hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t) - hence "PIP (moment i t @ s) e" by (cases, simp) - -- {* It can be derived that this event @{text "e"}, which - gives birth to @{term "th'"} must be a @{term "Create"}: *} - from create_pre[OF this, of th'] - obtain prio where eq_e: "e = Create th' prio" - by (metis append_Cons eq_me lessI post pre) - have h1: "th' \ threads (moment (Suc i) t @ s)" using post by auto - have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'" - proof - - have "cntP (moment i t@s) th' = cntV (moment i t@s) th'" - by (metis h_i.cnp_cnv_eq pre) - thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def) - qed - show ?thesis - using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge - by auto - qed - with `th' \ runing (t@s)` - have False by simp - } thus ?thesis by auto -qed - -text {* - The second lemma says, if the running thread @{text th'} is different from - @{term th}, then this @{text th'} must in the possession of some resources - at the very beginning. - - To ease the reasoning of resource possession of one particular thread, - we used two auxiliary functions @{term cntV} and @{term cntP}, - which are the counters of @{term P}-operations and - @{term V}-operations respectively. - If the number of @{term V}-operation is less than the number of - @{term "P"}-operations, the thread must have some unreleased resource. -*} - -lemma runing_inversion_1: (* ddd *) - assumes neq_th': "th' \ th" - and runing': "th' \ runing (t@s)" - -- {* thread @{term "th'"} is a live on in state @{term "s"} and - it has some unreleased resource. *} - shows "th' \ threads s \ cntV s th' < cntP s th'" -proof - - -- {* The proof is a simple composition of @{thm runing_inversion_0} and - @{thm runing_precond}: *} - -- {* By applying @{thm runing_inversion_0} to assumptions, - it can be shown that @{term th'} is live in state @{term s}: *} - have "th' \ threads s" using runing_inversion_0[OF assms(1,2)] . - -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *} - with runing_precond [OF this neq_th' runing'] show ?thesis by simp -qed - -text {* - The following lemma is just a rephrasing of @{thm runing_inversion_1}: -*} -lemma runing_inversion_2: - assumes runing': "th' \ runing (t@s)" - shows "th' = th \ (th' \ th \ th' \ threads s \ cntV s th' < cntP s th')" -proof - - from runing_inversion_1[OF _ runing'] - show ?thesis by auto -qed - -lemma runing_inversion_3: - assumes runing': "th' \ runing (t@s)" - and neq_th: "th' \ th" - shows "th' \ threads s \ (cntV s th' < cntP s th' \ cp (t@s) th' = preced th s)" - by (metis neq_th runing' runing_inversion_2 runing_preced_inversion) - -lemma runing_inversion_4: - assumes runing': "th' \ runing (t@s)" - and neq_th: "th' \ th" - shows "th' \ threads s" - and "\detached s th'" - and "cp (t@s) th' = preced th s" - apply (metis neq_th runing' runing_inversion_2) - apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc) - by (metis neq_th runing' runing_inversion_3) - - -text {* - Suppose @{term th} is not running, it is first shown that - there is a path in RAG leading from node @{term th} to another thread @{text "th'"} - in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}). - - Now, since @{term readys}-set is non-empty, there must be - one in it which holds the highest @{term cp}-value, which, by definition, - is the @{term runing}-thread. However, we are going to show more: this running thread - is exactly @{term "th'"}. - *} -lemma th_blockedE: (* ddd *) - assumes "th \ runing (t@s)" - obtains th' where "Th th' \ ancestors (RAG (t @ s)) (Th th)" - "th' \ runing (t@s)" -proof - - -- {* According to @{thm vat_t.th_chain_to_ready}, either - @{term "th"} is in @{term "readys"} or there is path leading from it to - one thread in @{term "readys"}. *} - have "th \ readys (t @ s) \ (\th'. th' \ readys (t @ s) \ (Th th, Th th') \ (RAG (t @ s))\<^sup>+)" - using th_kept vat_t.th_chain_to_ready by auto - -- {* However, @{term th} can not be in @{term readys}, because otherwise, since - @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *} - moreover have "th \ readys (t@s)" - using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto - -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in - term @{term readys}: *} - ultimately obtain th' where th'_in: "th' \ readys (t@s)" - and dp: "(Th th, Th th') \ (RAG (t @ s))\<^sup>+" by auto - -- {* We are going to show that this @{term th'} is running. *} - have "th' \ runing (t@s)" - proof - - -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *} - have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R") - proof - - have "?L = Max ((the_preced (t @ s) \ the_thread) ` subtree (tRAG (t @ s)) (Th th'))" - by (unfold cp_alt_def1, simp) - also have "... = (the_preced (t @ s) \ the_thread) (Th th)" - proof(rule image_Max_subset) - show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads) - next - show "subtree (tRAG (t @ s)) (Th th') \ Th ` threads (t @ s)" - by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) - next - show "Th th \ subtree (tRAG (t @ s)) (Th th')" using dp - by (unfold tRAG_subtree_eq, auto simp:subtree_def) - next - show "Max ((the_preced (t @ s) \ the_thread) ` Th ` threads (t @ s)) = - (the_preced (t @ s) \ the_thread) (Th th)" (is "Max ?L = _") - proof - - have "?L = the_preced (t @ s) ` threads (t @ s)" - by (unfold image_comp, rule image_cong, auto) - thus ?thesis using max_preced the_preced_def by auto - qed - qed - also have "... = ?R" - using th_cp_max th_cp_preced th_kept - the_preced_def vat_t.max_cp_readys_threads by auto - finally show ?thesis . - qed - -- {* Now, since @{term th'} holds the highest @{term cp} - and we have already show it is in @{term readys}, - it is @{term runing} by definition. *} - with `th' \ readys (t@s)` show ?thesis by (simp add: runing_def) - qed - -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *} - moreover have "Th th' \ ancestors (RAG (t @ s)) (Th th)" - using `(Th th, Th th') \ (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def) - ultimately show ?thesis using that by metis -qed - -text {* - Now it is easy to see there is always a thread to run by case analysis - on whether thread @{term th} is running: if the answer is Yes, the - the running thread is obviously @{term th} itself; otherwise, the running - thread is the @{text th'} given by lemma @{thm th_blockedE}. -*} -lemma live: "runing (t@s) \ {}" -proof(cases "th \ runing (t@s)") - case True thus ?thesis by auto -next - case False - thus ?thesis using th_blockedE by auto -qed - -end -end - diff -r 43482ab31341 -r 0c89419b4742 PrioG.thy --- a/PrioG.thy Wed Feb 03 21:51:57 2016 +0800 +++ b/PrioG.thy Wed Feb 03 22:17:29 2016 +0800 @@ -1,5 +1,5 @@ -theory PrioG -imports CpsG +theory Correctness +imports PIPBasics begin text {* @@ -580,6 +580,7 @@ -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *} interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp + interpret vat_es: valid_trace_e "t@s" e using Cons(1,2) by (unfold_locales, auto) show ?case proof - -- {* It can be proved that @{term cntP}-value of @{term th'} does not change @@ -591,8 +592,8 @@ -- {* Then the actor of @{term e} must be @{term th'} and @{term e} must be a @{term P}-event: *} hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv) - with vat_t.actor_inv[OF Cons(2)] - -- {* According to @{thm actor_inv}, @{term th'} must be running at + with vat_es.actor_inv + -- {* According to @{thm vat_es.actor_inv}, @{term th'} must be running at the moment @{term "t@s"}: *} have "th' \ runing (t@s)" by (cases e, auto) -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis @@ -609,7 +610,7 @@ proof(rule ccontr) -- {* Proof by contradiction. *} assume otherwise: "cntV ((e # t) @ s) th' \ cntV (t @ s) th'" hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv) - with vat_t.actor_inv[OF Cons(2)] + with vat_es.actor_inv have "th' \ runing (t@s)" by (cases e, auto) moreover have "th' \ runing (t@s)" using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .