diff -r 5d8ec128518b -r e3cf792db636 CpsG.thy --- a/CpsG.thy Tue Jun 14 13:56:51 2016 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,4669 +0,0 @@ -theory CpsG -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" -proof - show "f ` A \ g ` A" - by(rule image_subsetI, auto intro:h) -next - show "g ` A \ f ` A" - by (rule image_subsetI, auto intro:h[symmetric]) -qed - -lemma Max_fg_mono: - assumes "finite A" - and "\ a \ A. f a \ g a" - shows "Max (f ` A) \ Max (g ` A)" -proof(cases "A = {}") - case True - thus ?thesis by auto -next - case False - show ?thesis - proof(rule Max.boundedI) - from assms show "finite (f ` A)" by auto - next - from False show "f ` A \ {}" by auto - next - fix fa - assume "fa \ f ` A" - then obtain a where h_fa: "a \ A" "fa = f a" by auto - show "fa \ Max (g ` A)" - proof(rule Max_ge_iff[THEN iffD2]) - from assms show "finite (g ` A)" by auto - next - from False show "g ` A \ {}" by auto - next - from h_fa have "g a \ g ` A" by auto - moreover have "fa \ g a" using h_fa assms(2) by auto - ultimately show "\a\g ` A. fa \ a" by auto - qed - qed -qed - -lemma Max_f_mono: - assumes seq: "A \ B" - and np: "A \ {}" - and fnt: "finite B" - shows "Max (f ` A) \ Max (f ` B)" -proof(rule Max_mono) - from seq show "f ` A \ f ` B" by auto -next - from np show "f ` A \ {}" by auto -next - from fnt and seq show "finite (f ` B)" by auto -qed - -lemma Max_UNION: - assumes "finite A" - and "A \ {}" - and "\ M \ f ` A. finite M" - and "\ M \ f ` A. M \ {}" - shows "Max (\x\ A. f x) = Max (Max ` f ` A)" (is "?L = ?R") - using assms[simp] -proof - - have "?L = Max (\(f ` A))" - by (fold Union_image_eq, simp) - also have "... = ?R" - by (subst Max_Union, simp+) - finally show ?thesis . -qed - -lemma max_Max_eq: - assumes "finite A" - and "A \ {}" - and "x = y" - shows "max x (Max A) = Max ({y} \ A)" (is "?L = ?R") -proof - - have "?R = Max (insert y A)" by simp - also from assms have "... = ?L" - by (subst Max.insert, simp+) - finally show ?thesis by simp -qed - -lemma 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" - using assms -proof(induct s) - case (Cons a s) - show ?case - proof(cases "s \ []") - case False - thus ?thesis - by (cases a, auto) - next - case True - show ?thesis using Cons(1)[OF True] - by (cases a, auto) - qed -qed simp - -lemma th_in_ne: "th \ threads s \ s \ []" - by (induct s, auto) - -lemma preced_tm_lt: "th \ threads s \ preced th s = Prc x y \ y < length s" - by (drule_tac th_in_ne, unfold preced_def, auto intro: birth_time_lt) - -lemma eq_RAG: - "RAG (wq s) = RAG s" - by (unfold cs_RAG_def s_RAG_def, auto) - -lemma waiting_holding: - assumes "waiting (s::state) th cs" - obtains th' where "holding s th' cs" -proof - - from assms[unfolded s_waiting_def, folded wq_def] - obtain th' where "th' \ set (wq s cs)" "th' = hd (wq s cs)" - by (metis empty_iff hd_in_set list.set(1)) - hence "holding s th' cs" - by (unfold s_holding_def, fold wq_def, auto) - from that[OF this] show ?thesis . -qed - -lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th" -unfolding cp_def wq_def -apply(induct s rule: schs.induct) -apply(simp add: Let_def cpreced_initial) -apply(simp add: Let_def) -apply(simp add: Let_def) -apply(simp add: Let_def) -apply(subst (2) schs.simps) -apply(simp add: Let_def) -apply(subst (2) schs.simps) -apply(simp add: Let_def) -done - -lemma cp_alt_def: - "cp s th = - Max ((the_preced s) ` {th'. Th th' \ (subtree (RAG s) (Th th))})" -proof - - have "Max (the_preced s ` ({th} \ dependants (wq s) th)) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "Max (_ ` ?L) = Max (_ ` ?R)") - proof - - have "?L = ?R" - by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def) - thus ?thesis by simp - qed - thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp) -qed - -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" - -locale valid_trace_exit = valid_trace_e + - fixes th - assumes is_exit: "e = Exit th" - -locale valid_trace_p = valid_trace_e + - fixes th cs - assumes is_p: "e = P th cs" - -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 \ []" -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 \ - PP s \ PIP s e \ PP (e # s))" - shows "PP s" -proof(induct rule:vt.induct[OF vt, case_names Init Step]) - case Init - from assms(1) show ?case . -next - case (Step s e) - show ?case - proof(rule assms(2)) - show "valid_trace_e s e" using Step by (unfold_locales, auto) - next - show "PP s" using Step by simp - next - show "PIP s e" using Step by simp - qed -qed - -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 - thus ?case by (simp add:vt_nil) -next - case (Cons s e t) - show ?case - proof(cases "t \ length (e#s)") - case True - from True have "moment t (e#s) = e#s" by simp - thus ?thesis using Cons - by (simp add:valid_trace_def valid_trace_e_def, auto) - next - case False - from Cons have "vt (moment t s)" by simp - moreover have "moment t (e#s) = moment t s" - proof - - from False have "t \ length s" by simp - from moment_app [OF this, of "[e]"] - show ?thesis by simp - qed - ultimately show ?thesis by simp - qed -qed -end - -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)" - by (unfold_locales, insert vt_moment, auto) - -locale valid_moment_e = valid_moment + - assumes less_i: "i < length s" -begin - 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_kept [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_create wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace_exit -begin - -lemma wq_kept [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_exit wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace_p -begin - -lemma wq_neq_simp [simp]: - assumes "cs' \ cs" - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_p wq_def - by (auto simp:Let_def) - -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_p] - show ?thesis by (cases, simp) -qed - -lemma th_not_in_wq: - shows "th \ set (wq s cs)" -proof - assume otherwise: "th \ set (wq s cs)" - from runing_wqE[OF runing_th_s this] - obtain rest where eq_wq: "wq s cs = th#rest" by blast - with otherwise - have "holding s th cs" - by (unfold s_holding_def, fold wq_def, simp) - hence cs_th_RAG: "(Cs cs, Th th) \ RAG s" - by (unfold s_RAG_def, fold holding_eq, auto) - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - with cs_th_RAG show ?thesis by auto - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq s cs @ [th]" - by (unfold is_p wq_def, auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" -proof(cases "cs' = cs") - case True - show ?thesis using True assms th_not_in_wq - by (unfold True wq_es_cs, auto) -qed (insert assms, simp) - -end - -context valid_trace_v -begin - -lemma wq_neq_simp [simp]: - assumes "cs' \ cs" - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_v wq_def - by (auto simp:Let_def) - -lemma wq_s_cs: - "wq s cs = th#rest" -proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from this(2) show ?thesis - by (unfold rest_def s_holding_def, fold wq_def, - metis empty_iff list.collapse list.set(1)) - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq'" - using wq_s_cs[unfolded wq_def] - by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" -proof(cases "cs' = cs") - case True - show ?thesis - proof(unfold True wq_es_cs wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - using assms[unfolded True wq_s_cs] by auto - qed simp -qed (insert assms, simp) - -end - -context valid_trace_set -begin - -lemma wq_kept [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_set wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace -begin - -lemma 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) - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt_create: valid_trace_create s e th prio - using Create by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept) - next - case (Exit th) - interpret vt_exit: valid_trace_exit s e th - using Exit by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept) - next - case (P th cs) - interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) - next - case (V th cs) - interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) - next - case (Set th prio) - interpret vt_set: valid_trace_set s e th prio - using Set by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) - qed -qed (unfold wq_def Let_def, simp) - -end - -section {* Waiting queues and threads *} - -context valid_trace_e -begin - -lemma wq_out_inv: - assumes s_in: "thread \ set (wq s cs)" - and s_hd: "thread = hd (wq s cs)" - and s_i: "thread \ hd (wq (e#s) cs)" - shows "e = V thread cs" -proof(cases e) --- {* There are only two non-trivial cases: *} - case (V th cs1) - show ?thesis - proof(cases "cs1 = cs") - case True - have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] . - thus ?thesis - proof(cases) - case (thread_V) - moreover have "th = thread" using thread_V(2) s_hd - by (unfold s_holding_def wq_def, simp) - ultimately show ?thesis using V True by simp - qed - qed (insert assms V, auto simp:wq_def Let_def split:if_splits) -next - case (P th cs1) - show ?thesis - proof(cases "cs1 = cs") - case True - with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]" - by (auto simp:wq_def Let_def split:if_splits) - with s_i s_hd s_in have False - by (metis empty_iff hd_append2 list.set(1) wq_def) - thus ?thesis by simp - qed (insert assms P, auto simp:wq_def Let_def split:if_splits) -qed (insert assms, auto simp:wq_def Let_def split:if_splits) - -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 - -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 - -section {* The change of @{term RAG} *} - -text {* - The following three lemmas show that @{text "RAG"} does not change - by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"} - events, respectively. -*} - -lemma (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 holding_cs_eq_th: - assumes "holding s t cs" - shows "t = th" -proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from held_unique[OF this(2) assms] - show ?thesis by simp - qed -qed - -lemma distinct_wq': "distinct wq'" - by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def) - -lemma set_wq': "set wq' = set rest" - by (metis (mono_tags, lifting) distinct_rest 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" -proof - assume otherwise: "t = th" - show False - proof(cases "c = cs") - case True - have "t \ set wq'" - using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs] - by simp - from th'_in_inv[OF this] have "t \ set rest" . - with wq_s_cs[folded otherwise] wq_distinct[of cs] - show ?thesis by simp - next - case False - have "wq (e#s) c = wq s c" using False - by (unfold is_v, simp) - hence "waiting s t c" using assms - by (simp add: cs_waiting_def waiting_eq) - hence "t \ readys s" by (unfold readys_def, auto) - hence "t \ runing s" using runing_ready by auto - with runing_th_s[folded otherwise] show ?thesis by auto - qed -qed - -lemma waiting_esI1: - assumes "waiting s t c" - and "c \ cs" - shows "waiting (e#s) t c" -proof - - have "wq (e#s) c = wq s c" - using assms(2) is_v by auto - with assms(1) show ?thesis - using cs_waiting_def waiting_eq by auto -qed - -lemma holding_esI2: - assumes "c \ cs" - and "holding s t c" - shows "holding (e#s) t c" -proof - - from assms(1) have "wq (e#s) c = wq s c" using is_v by auto - from assms(2)[unfolded s_holding_def, folded wq_def, - folded this, unfolded wq_def, folded s_holding_def] - show ?thesis . -qed - -lemma holding_esI1: - assumes "holding s t c" - and "t \ th" - shows "holding (e#s) t c" -proof - - have "c \ cs" using assms using holding_cs_eq_th by blast - from holding_esI2[OF this assms(1)] - show ?thesis . -qed - -end - -context valid_trace_v_n -begin - -lemma neq_wq': "wq' \ []" -proof (unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume " distinct x \ set x = set rest" - thus "x \ []" using rest_nnl by auto -qed - -lemma eq_wq': "wq' = taker # rest'" - by (simp add: neq_wq' rest'_def taker_def) - -lemma next_th_taker: - shows "next_th s th cs taker" - using rest_nnl taker_def wq'_def wq_s_cs - by (auto simp:next_th_def) - -lemma taker_unique: - assumes "next_th s th cs taker'" - shows "taker' = taker" -proof - - from assms - obtain rest' where - h: "wq s cs = th # rest'" - "taker' = hd (SOME q. distinct q \ set q = set rest')" - by (unfold next_th_def, auto) - with wq_s_cs have "rest' = rest" by auto - thus ?thesis using h(2) taker_def wq'_def by auto -qed - -lemma waiting_set_eq: - "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}" - by (smt all_not_in_conv bot.extremum insertI1 insert_subset - mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique) - -lemma holding_set_eq: - "{(Cs cs, Th th') |th'. next_th s th cs th'} = {(Cs cs, Th taker)}" - using next_th_taker taker_def waiting_set_eq - by fastforce - -lemma holding_taker: - shows "holding (e#s) taker cs" - by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, - auto simp:neq_wq' taker_def) - -lemma waiting_esI2: - assumes "waiting s t cs" - and "t \ taker" - shows "waiting (e#s) t cs" -proof - - have "t \ set wq'" - proof(unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) - next - fix x - assume "distinct x \ set x = set rest" - moreover have "t \ set rest" - using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto - ultimately show "t \ set x" by simp - qed - moreover have "t \ hd wq'" - using assms(2) taker_def by auto - ultimately show ?thesis - by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp) -qed - -lemma waiting_esE: - assumes "waiting (e#s) t c" - obtains "c \ cs" "waiting s t c" - | "c = cs" "t \ taker" "waiting s t cs" "t \ set rest'" -proof(cases "c = cs") - case False - hence "wq (e#s) c = wq s c" using is_v by auto - with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto - from that(1)[OF False this] show ?thesis . -next - case True - from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs] - have "t \ hd wq'" "t \ set wq'" by auto - hence "t \ taker" by (simp add: taker_def) - moreover hence "t \ th" using assms neq_t_th by blast - moreover have "t \ set rest" by (simp add: `t \ set wq'` th'_in_inv) - ultimately have "waiting s t cs" - by (metis cs_waiting_def list.distinct(2) list.sel(1) - list.set_sel(2) rest_def waiting_eq wq_s_cs) - show ?thesis using that(2) - using True `t \ set wq'` `t \ taker` `waiting s t cs` eq_wq' by auto -qed - -lemma holding_esI1: - assumes "c = cs" - and "t = taker" - shows "holding (e#s) t c" - by (unfold assms, simp add: holding_taker) - -lemma holding_esE: - assumes "holding (e#s) t c" - obtains "c = cs" "t = taker" - | "c \ cs" "holding s t c" -proof(cases "c = cs") - case True - from assms[unfolded True, unfolded s_holding_def, - folded wq_def, unfolded wq_es_cs] - have "t = taker" by (simp add: taker_def) - from that(1)[OF True this] show ?thesis . -next - case False - hence "wq (e#s) c = wq s c" using is_v by auto - from assms[unfolded s_holding_def, folded wq_def, - unfolded this, unfolded wq_def, folded s_holding_def] - have "holding s t c" . - from that(2)[OF False this] show ?thesis . -qed - -end - - -context valid_trace_v_e -begin - -lemma nil_wq': "wq' = []" -proof (unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume " distinct x \ set x = set rest" - thus "x = []" using rest_nil by auto -qed - -lemma no_taker: - assumes "next_th s th cs taker" - shows "False" -proof - - from assms[unfolded next_th_def] - obtain rest' where "wq s cs = th # rest'" "rest' \ []" - by auto - thus ?thesis using rest_def rest_nil by auto -qed - -lemma waiting_set_eq: - "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}" - using no_taker by auto - -lemma holding_set_eq: - "{(Cs cs, Th th') |th'. next_th s th cs th'} = {}" - using no_taker by auto - -lemma no_holding: - assumes "holding (e#s) taker cs" - shows False -proof - - from wq_es_cs[unfolded nil_wq'] - have " wq (e # s) cs = []" . - from assms[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis by auto -qed - -lemma no_waiting: - assumes "waiting (e#s) t cs" - shows False -proof - - from wq_es_cs[unfolded nil_wq'] - have " wq (e # s) cs = []" . - from assms[unfolded s_waiting_def, folded wq_def, unfolded this] - show ?thesis by auto -qed - -lemma waiting_esI2: - assumes "waiting s t c" - shows "waiting (e#s) t c" -proof - - have "c \ cs" using assms - using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto - from waiting_esI1[OF assms this] - show ?thesis . -qed - -lemma waiting_esE: - assumes "waiting (e#s) t c" - obtains "c \ cs" "waiting s t c" -proof(cases "c = cs") - case False - hence "wq (e#s) c = wq s c" using is_v by auto - with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto - from that(1)[OF False this] show ?thesis . -next - case True - from no_waiting[OF assms[unfolded True]] - show ?thesis by auto -qed - -lemma holding_esE: - assumes "holding (e#s) t c" - obtains "c \ cs" "holding s t c" -proof(cases "c = cs") - case True - from no_holding[OF assms[unfolded True]] - show ?thesis by auto -next - case False - hence "wq (e#s) c = wq s c" using is_v by auto - from assms[unfolded s_holding_def, folded wq_def, - unfolded this, unfolded wq_def, folded s_holding_def] - have "holding s t c" . - from that[OF False this] show ?thesis . -qed - -end - - -context valid_trace_v -begin - -lemma RAG_es: - "RAG (e # s) = - RAG s - {(Cs cs, Th th)} - - {(Th th', Cs cs) |th'. next_th s th cs th'} \ - {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - show ?thesis - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from waiting(3) - show ?thesis - proof(cases rule:h_n.waiting_esE) - case 1 - with waiting(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - next - case 2 - with waiting(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from waiting(3) - show ?thesis - proof(cases rule:h_e.waiting_esE) - case 1 - with waiting(1,2) - show ?thesis - by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - qed - qed - next - case (holding th' cs') - show ?thesis - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from holding(3) - show ?thesis - proof(cases rule:h_n.holding_esE) - case 1 - with holding(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold waiting_eq, auto) - next - case 2 - with holding(1,2) - show ?thesis - by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def, - fold holding_eq, auto) - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from holding(3) - show ?thesis - proof(cases rule:h_e.holding_esE) - case 1 - with holding(1,2) - show ?thesis - by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def, - fold holding_eq, auto) - qed - qed - qed -next - fix n1 n2 - assume h: "(n1, n2) \ ?R" - show "(n1, n2) \ ?L" - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq] - have "((n1, n2) \ RAG s \ (n1 \ Cs cs \ n2 \ Th th) - \ (n1 \ Th h_n.taker \ n2 \ Cs cs)) \ - (n2 = Th h_n.taker \ n1 = Cs cs)" - by auto - thus ?thesis - proof - assume "n2 = Th h_n.taker \ n1 = Cs cs" - with h_n.holding_taker - show ?thesis - by (unfold s_RAG_def, fold holding_eq, auto) - next - assume h: "(n1, n2) \ RAG s \ - (n1 \ Cs cs \ n2 \ Th th) \ (n1 \ Th h_n.taker \ n2 \ Cs cs)" - hence "(n1, n2) \ RAG s" by simp - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from h and this(1,2) - have "th' \ h_n.taker \ cs' \ cs" by auto - hence "waiting (e#s) th' cs'" - proof - assume "cs' \ cs" - from waiting_esI1[OF waiting(3) this] - show ?thesis . - next - assume neq_th': "th' \ h_n.taker" - show ?thesis - proof(cases "cs' = cs") - case False - from waiting_esI1[OF waiting(3) this] - show ?thesis . - next - case True - from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True] - show ?thesis . - qed - qed - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - from h this(1,2) - have "cs' \ cs \ th' \ th" by auto - hence "holding (e#s) th' cs'" - proof - assume "cs' \ cs" - from holding_esI2[OF this holding(3)] - show ?thesis . - next - assume "th' \ th" - from holding_esI1[OF holding(3) this] - show ?thesis . - qed - thus ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq] - have h_s: "(n1, n2) \ RAG s" "(n1, n2) \ (Cs cs, Th th)" - by auto - from h_s(1) - show ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from h_e.waiting_esI2[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - with h_s(2) - have "cs' \ cs \ th' \ th" by auto - thus ?thesis - proof - assume neq_cs: "cs' \ cs" - from holding_esI2[OF this holding(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - next - assume "th' \ th" - from holding_esI1[OF holding(3) this] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed - qed -qed - -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_p -begin - -lemma waiting_kept: - assumes "waiting s th' cs'" - shows "waiting (e#s) th' cs'" - using assms - by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2) - rotate1.simps(2) self_append_conv2 set_rotate1 - th_not_in_wq waiting_eq wq_es_cs wq_neq_simp) - -lemma holding_kept: - assumes "holding s th' cs'" - shows "holding (e#s) th' cs'" -proof(cases "cs' = cs") - case False - hence "wq (e#s) cs' = wq s cs'" by simp - with assms show ?thesis using cs_holding_def holding_eq by auto -next - case True - from assms[unfolded s_holding_def, folded wq_def] - obtain rest where eq_wq: "wq s cs' = th'#rest" - by (metis empty_iff list.collapse list.set(1)) - hence "wq (e#s) cs' = th'#(rest@[th])" - by (simp add: True wq_es_cs) - thus ?thesis - by (simp add: cs_holding_def holding_eq) -qed -end - -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 - -context valid_trace_p_h -begin - -lemma wq_es_cs': "wq (e#s) cs = [th]" - using wq_es_cs[unfolded we] by simp - -lemma holding_es_th_cs: - shows "holding (e#s) th cs" -proof - - from wq_es_cs' - have "th \ set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto - thus ?thesis using cs_holding_def holding_eq by blast -qed - -lemma RAG_edge: "(Cs cs, Th th) \ RAG (e#s)" - by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto) - -lemma waiting_esE: - assumes "waiting (e#s) th' cs'" - obtains "waiting s th' cs'" - using assms - by (metis cs_waiting_def event.distinct(15) is_p list.sel(1) - set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv) - -lemma holding_esE: - assumes "holding (e#s) th' cs'" - obtains "cs' \ cs" "holding s th' cs'" - | "cs' = cs" "th' = th" -proof(cases "cs' = cs") - case True - from held_unique[OF holding_es_th_cs assms[unfolded True]] - have "th' = th" by simp - from that(2)[OF True this] show ?thesis . -next - case False - have "holding s th' cs'" using assms - using False cs_holding_def holding_eq by auto - from that(1)[OF False this] show ?thesis . -qed - -lemma RAG_es: "RAG (e # s) = RAG s \ {(Cs cs, Th th)}" (is "?L = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from this(3) - show ?thesis - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - qed - next - case (holding th' cs') - from this(3) - show ?thesis - proof(cases rule:holding_esE) - case 1 - with holding(1,2) - show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) - next - case 2 - with holding(1,2) show ?thesis by auto - qed - qed -next - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \ RAG s \ (n1 = Cs cs \ n2 = Th th)" by auto - thus "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ RAG s" - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from waiting_kept[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - from holding_kept[OF this(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - next - assume "n1 = Cs cs \ n2 = Th th" - with holding_es_th_cs - show ?thesis - by (unfold s_RAG_def, fold holding_eq, auto) - qed -qed - -end - -context valid_trace_p_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)} - else RAG s \ {(Th th, Cs cs)})" -proof(cases "wq s cs = []") - case True - interpret vt_p: valid_trace_p_h using True - by (unfold_locales, simp) - show ?thesis by (simp add: vt_p.RAG_es vt_p.we) -next - case False - interpret vt_p: valid_trace_p_w using False - by (unfold_locales, simp) - show ?thesis by (simp add: vt_p.RAG_es vt_p.wne) -qed - -end - -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) - -lemma (in valid_trace_v_n) finite_holding_set: - "finite {(Cs cs, Th th') |th'. next_th s th cs th'}" - by (simp add: holding_set_eq) - -lemma (in valid_trace_v_e) finite_waiting_set: - "finite {(Th th', Cs cs) |th'. next_th s th cs th'}" - by (simp add: waiting_set_eq) - -lemma (in valid_trace_v_e) finite_holding_set: - "finite {(Cs cs, Th th') |th'. next_th s th cs th'}" - by (simp add: holding_set_eq) - - -context valid_trace_v_e -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof(rule acyclic_subset[OF assms]) - show "RAG (e # s) \ RAG s" - by (unfold RAG_es waiting_set_eq holding_set_eq, auto) -qed - -end - -context valid_trace_v_n -begin - -lemma waiting_taker: "waiting s taker cs" - apply (unfold s_waiting_def, fold wq_def, unfold wq_s_cs taker_def) - using eq_wq' th'_in_inv wq'_def by fastforce - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic ((RAG s - {(Cs cs, Th th)} - {(Th taker, Cs cs)}) \ - {(Cs cs, Th taker)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Th taker, Cs cs) \ ?A^*" - proof - assume otherwise: "(Th taker, Cs cs) \ ?A^*" - hence "(Th taker, Cs cs) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - from tranclD[OF this] - obtain cs' where h: "(Th taker, Cs cs') \ ?A" - "(Th taker, Cs cs') \ RAG s" - by (unfold s_RAG_def, auto) - from this(2) have "waiting s taker cs'" - by (unfold s_RAG_def, fold waiting_eq, auto) - from waiting_unique[OF this waiting_taker] - have "cs' = cs" . - from h(1)[unfolded this] show False by auto - qed - ultimately show ?thesis by auto - qed - thus ?thesis - by (unfold RAG_es waiting_set_eq holding_set_eq, simp) -qed - -end - -context valid_trace_p_h -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic (RAG s \ {(Cs cs, Th th)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Th th, Cs cs) \ ?A^*" - proof - assume otherwise: "(Th th, Cs cs) \ ?A^*" - hence "(Th th, Cs cs) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - from tranclD[OF this] - obtain cs' where h: "(Th th, Cs cs') \ RAG s" - by (unfold s_RAG_def, auto) - hence "waiting s th cs'" - by (unfold s_RAG_def, fold waiting_eq, auto) - with th_not_waiting show False by auto - qed - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold RAG_es, simp) -qed - -end - -context valid_trace_p_w -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic (RAG s \ {(Th th, Cs cs)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Cs cs, Th th) \ ?A^*" - proof - assume otherwise: "(Cs cs, Th th) \ ?A^*" - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - moreover from otherwise have "(Cs cs, Th th) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - ultimately show ?thesis by auto - qed - qed - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold RAG_es, simp) -qed - -end - -context valid_trace -begin - -lemma acyclic_RAG: - shows "acyclic (RAG s)" -proof(induct rule:ind) - case Nil - show ?case - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt: valid_trace_create s e th prio using Create - by (unfold_locales, simp) - show ?thesis using Cons by simp - 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 - proof(cases "wq s cs = []") - case True - then interpret vt_h: valid_trace_p_h s e th cs - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_h.acylic_RAG_kept) - next - case False - then interpret vt_w: valid_trace_p_w s e th cs - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_w.acylic_RAG_kept) - qed - next - case (V th cs) - interpret vt: valid_trace_v s e th cs using V - by (unfold_locales, simp) - show ?thesis - proof(cases "vt.rest = []") - case True - then interpret vt_e: valid_trace_v_e s e th cs - by (unfold_locales, simp) - show ?thesis by (simp add: Cons.hyps(2) vt_e.acylic_RAG_kept) - next - case False - then interpret vt_n: valid_trace_v_n s e th cs - by (unfold_locales, simp) - show ?thesis by (simp add: Cons.hyps(2) vt_n.acylic_RAG_kept) - qed - next - case (Set th prio) - interpret vt: valid_trace_set s e th prio using Set - by (unfold_locales, simp) - show ?thesis using Cons by simp - 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)" . -next - 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) - -lemma sgv_hRAG: "single_valued (hRAG s)" - using held_unique - by (unfold single_valued_def hRAG_def, auto) - -lemma sgv_tRAG: "single_valued (tRAG s)" - by (unfold tRAG_def, rule single_valued_relcomp, - insert sgv_wRAG sgv_hRAG, auto) - -end - -sublocale valid_trace < rtree_s: rtree "tRAG s" -proof(unfold_locales) - from sgv_tRAG show "single_valued (tRAG s)" . -next - from acyclic_tRAG show "acyclic (tRAG s)" . -qed - -sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s" -proof - - have "fsubtree (tRAG s)" - proof - - have "fbranch (tRAG s)" - proof(unfold tRAG_def, rule fbranch_compose) - show "fbranch (wRAG s)" - proof(rule finite_fbranchI) - from finite_RAG show "finite (wRAG s)" - by (unfold RAG_split, auto) - qed - next - show "fbranch (hRAG s)" - proof(rule finite_fbranchI) - from finite_RAG - show "finite (hRAG s)" by (unfold RAG_split, auto) - qed - qed - moreover have "wf (tRAG s)" - proof(rule wf_subset) - show "wf (RAG s O RAG s)" using wf_RAG - by (fold wf_comp_self, simp) - next - show "tRAG s \ (RAG s O RAG s)" - by (unfold tRAG_alt_def, auto) - qed - ultimately show ?thesis - by (unfold fsubtree_def fsubtree_axioms_def,auto) - qed - from this[folded tRAG_def] show "fsubtree (tRAG s)" . -qed - -lemma 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 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 - - { 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 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 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)" - obtains th' where "th' \ readys s" "(node, Th th') \ (RAG s)^+" -proof - - from assms have "node \ Range ((RAG s)^-1)" by auto - from wf_base[OF wf_RAG_converse this] - obtain b where h_b: "(b, node) \ ((RAG s)\)\<^sup>+" "\c. (c, b) \ (RAG s)\" by auto - obtain th' where eq_b: "b = Th th'" - proof(cases b) - case (Cs cs) - from h_b(1)[unfolded trancl_converse] - have "(node, b) \ ((RAG s)\<^sup>+)" by auto - from tranclE[OF this] - obtain n where "(n, b) \ RAG s" by auto - from this[unfolded Cs] - obtain th1 where "waiting s th1 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - from waiting_holding[OF this] - obtain th2 where "holding s th2 cs" . - hence "(Cs cs, Th th2) \ RAG s" - by (unfold s_RAG_def, fold holding_eq, auto) - with h_b(2)[unfolded Cs, rule_format] - have False by auto - thus ?thesis by auto - qed auto - have "th' \ readys s" - proof - - from h_b(2)[unfolded eq_b] - have "\cs. \ waiting s th' cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - moreover have "th' \ threads s" - proof(rule rg_RAG_threads) - from tranclD[OF h_b(1), unfolded eq_b] - obtain z where "(z, Th th') \ (RAG s)" by auto - thus "Th th' \ Range (RAG s)" by auto - qed - ultimately show ?thesis by (auto simp:readys_def) - qed - moreover have "(node, Th th') \ (RAG s)^+" - using h_b(1)[unfolded trancl_converse] eq_b by auto - ultimately show ?thesis using that by metis -qed - -text {* \noindent - The following is just an instance of @{text "chain_building"}. -*} -lemma th_chain_to_ready: - assumes th_in: "th \ threads s" - shows "th \ readys s \ (\ th'. th' \ readys s \ (Th th, Th th') \ (RAG s)^+)" -proof(cases "th \ readys s") - case True - thus ?thesis by auto -next - case False - from False and th_in have "Th th \ Domain (RAG s)" - by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def) - from chain_building [rule_format, OF this] - show ?thesis by auto -qed - -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 - - -section {* Relating @{term cp} and @{term the_preced} and @{term preced} *} - -context valid_trace -begin - -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 - -lemma holding_s_holder: "holding s holder cs" - by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto) - -lemma holding_es_holder: "holding (e#s) holder cs" - by (unfold s_holding_def, fold wq_def, unfold wq_es_cs wq_s_cs, auto) - -lemma holdents_es: - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence h: "holding (e#s) th' cs'" by (auto simp:holdents_def) - have "holding s th' cs'" - proof(cases "cs' = cs") - case True - from held_unique[OF h[unfolded True] holding_es_holder] - have "th' = holder" . - thus ?thesis - by (unfold True holdents_def, insert holding_s_holder, simp) - next - case False - hence "wq (e#s) cs' = wq s cs'" by simp - from h[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis - by (unfold s_holding_def, fold wq_def, auto) - qed - hence "cs' \ ?R" by (auto simp:holdents_def) - } moreover { - fix cs' - assume "cs' \ ?R" - hence h: "holding s th' cs'" by (auto simp:holdents_def) - have "holding (e#s) th' cs'" - proof(cases "cs' = cs") - case True - from held_unique[OF h[unfolded True] holding_s_holder] - have "th' = holder" . - thus ?thesis - by (unfold True holdents_def, insert holding_es_holder, simp) - next - case False - hence "wq s cs' = wq (e#s) cs'" by simp - from h[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis - by (unfold s_holding_def, fold wq_def, auto) - qed - hence "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th[simp]: "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_es, simp) - -lemma th_not_ready_es: - shows "th \ readys (e#s)" - using waiting_es_th_cs - by (unfold readys_def, auto) - -end - -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 - -lemma th_not_waiting': - "\ waiting (e#s) th cs'" -proof(cases "cs' = cs") - case True - show ?thesis - by (unfold True s_waiting_def, fold wq_def, unfold wq_es_cs', auto) -next - case False - from th_not_waiting[of cs', unfolded s_waiting_def, folded wq_def] - show ?thesis - by (unfold s_waiting_def, fold wq_def, insert False, simp) -qed - -lemma ready_th_es: - shows "th \ readys (e#s)" - using th_not_waiting' - by (unfold readys_def, insert live_th_es, auto) - -lemma holdents_es_th: - "holdents (e#s) th = (holdents s th) \ {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" - by (unfold holdents_def, auto) - hence "cs' \ ?R" - by (cases rule:holding_esE, auto simp:holdents_def) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s th cs' \ cs' = cs" - by (auto simp:holdents_def) - hence "cs' \ ?L" - proof - assume "holding s th cs'" - from holding_kept[OF this] - show ?thesis by (auto simp:holdents_def) - next - assume "cs' = cs" - thus ?thesis using holding_es_th_cs - by (unfold holdents_def, auto) - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th: "cntCS (e#s) th = cntCS s th + 1" -proof - - have "card (holdents s th \ {cs}) = card (holdents s th) + 1" - proof(subst card_Un_disjoint) - show "holdents s th \ {cs} = {}" - using not_holding_s_th_cs by (auto simp:holdents_def) - qed (auto simp:finite_holdents) - thus ?thesis - by (unfold cntCS_def holdents_es_th, simp) -qed - -lemma no_holder: - "\ holding s th' cs" -proof - assume otherwise: "holding s th' cs" - from this[unfolded s_holding_def, folded wq_def, unfolded we] - show False by auto -qed - -lemma holdents_es_th': - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence h_e: "holding (e#s) th' cs'" by (auto simp:holdents_def) - have "cs' \ cs" - proof - assume "cs' = cs" - from held_unique[OF h_e[unfolded this] holding_es_th_cs] - have "th' = th" . - with assms show False by simp - qed - from h_e[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp[OF this]] - have "th' \ set (wq s cs') \ th' = hd (wq s cs')" . - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, auto) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s th' cs'" by (auto simp:holdents_def) - from holding_kept[OF this] - have "holding (e # s) th' cs'" . - hence "cs' \ ?L" - by (unfold holdents_def, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th'[simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_es_th'[OF assms], simp) - -end - -context valid_trace_p -begin - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - show ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h - by (unfold_locales, simp) - show ?thesis using n_wait wait waiting_kept by auto - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis using n_wait wait waiting_kept by blast - qed - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - show ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h - by (unfold_locales, simp) - show ?thesis using n_wait vt.waiting_esE wait by blast - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis using assms(1) n_wait vt.waiting_esE wait by auto - qed - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: (* ddd *) - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof(cases "th' = th") - case True - note eq_th' = this - show ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h by (unfold_locales, simp) - show ?thesis - using assms eq_th' is_p ready_th_s vt.cntCS_es_th vt.ready_th_es pvD_def by auto - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis - using add.commute add.left_commute assms eq_th' is_p live_th_s - ready_th_s vt.th_not_ready_es pvD_def - apply (auto) - by (fold is_p, simp) - qed -next - case False - note h_False = False - thus ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h by (unfold_locales, simp) - show ?thesis using assms - by (insert True h_False pvD_def, auto split:if_splits,unfold is_p, auto) - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis using assms - by (insert False h_False pvD_def, auto split:if_splits,unfold is_p, auto) - qed -qed - -end - - -context valid_trace_v -begin - -lemma holding_th_cs_s: - "holding s th cs" - by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto) - -lemma th_ready_s [simp]: "th \ readys s" - using runing_th_s - by (unfold runing_def readys_def, auto) - -lemma th_live_s [simp]: "th \ threads s" - using th_ready_s by (unfold readys_def, auto) - -lemma th_ready_es [simp]: "th \ readys (e#s)" - using runing_th_s neq_t_th - by (unfold is_v runing_def readys_def, auto) - -lemma th_live_es [simp]: "th \ threads (e#s)" - using th_ready_es by (unfold readys_def, auto) - -lemma pvD_th_s[simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es[simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma cntCS_s_th [simp]: "cntCS s th > 0" -proof - - have "cs \ holdents s th" using holding_th_cs_s - by (unfold holdents_def, simp) - moreover have "finite (holdents s th)" using finite_holdents - by simp - ultimately show ?thesis - by (unfold cntCS_def, - auto intro!:card_gt_0_iff[symmetric, THEN iffD1]) -qed - -end - -context valid_trace_v -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 - -lemma not_ready_taker_s[simp]: - "taker \ readys s" - using waiting_taker - by (unfold readys_def, auto) - -lemma taker_live_s [simp]: "taker \ threads s" -proof - - have "taker \ set wq'" by (simp add: eq_wq') - from th'_in_inv[OF this] - have "taker \ set rest" . - hence "taker \ set (wq s cs)" by (simp add: wq_s_cs) - thus ?thesis using wq_threads by auto -qed - -lemma taker_live_es [simp]: "taker \ threads (e#s)" - using taker_live_s threads_es by blast - -lemma taker_ready_es [simp]: - shows "taker \ readys (e#s)" -proof - - { fix cs' - assume "waiting (e#s) taker cs'" - hence False - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting_taker waiting_unique by auto - qed simp - } thus ?thesis by (unfold readys_def, auto) -qed - -lemma neq_taker_th: "taker \ th" - using th_not_waiting waiting_taker by blast - -lemma not_holding_taker_s_cs: - shows "\ holding s taker cs" - using holding_cs_eq_th neq_taker_th by auto - -lemma holdents_es_taker: - "holdents (e#s) taker = holdents s taker \ {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) taker cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 2 - thus ?thesis by (auto simp:holdents_def) - qed auto - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s taker cs' \ cs' = cs" by (auto simp:holdents_def) - hence "cs' \ ?L" - proof - assume "holding s taker cs'" - hence "holding (e#s) taker cs'" - using holding_esI2 holding_taker by fastforce - thus ?thesis by (auto simp:holdents_def) - next - assume "cs' = cs" - with holding_taker - show ?thesis by (auto simp:holdents_def) - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_taker [simp]: "cntCS (e#s) taker = cntCS s taker + 1" -proof - - have "card (holdents s taker \ {cs}) = card (holdents s taker) + 1" - proof(subst card_Un_disjoint) - show "holdents s taker \ {cs} = {}" - using not_holding_taker_s_cs by (auto simp:holdents_def) - qed (auto simp:finite_holdents) - thus ?thesis - by (unfold cntCS_def, insert holdents_es_taker, simp) -qed - -lemma pvD_taker_s[simp]: "pvD s taker = 1" - by (unfold pvD_def, simp) - -lemma pvD_taker_es[simp]: "pvD (e#s) taker = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_s[simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es[simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_es_th: - "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 2 - thus ?thesis by (auto simp:holdents_def) - qed (insert neq_taker_th, auto) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "cs' \ cs" "holding s th cs'" by (auto simp:holdents_def) - from holding_esI2[OF this] - have "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1" -proof - - have "card (holdents s th - {cs}) = card (holdents s th) - 1" - proof - - have "cs \ holdents s th" using holding_th_cs_s - by (auto simp:holdents_def) - moreover have "finite (holdents s th)" - by (simp add: finite_holdents) - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold cntCS_def holdents_es_th) -qed - -lemma holdents_kept: - assumes "th' \ taker" - and "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - have "cs' \ ?R" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding (e#s) th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, auto) - next - case True - from h[unfolded this] - have "holding (e#s) th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_taker] - have "th' = taker" . - with assms show ?thesis by auto - qed - } moreover { - fix cs' - assume h: "cs' \ ?R" - have "cs' \ ?L" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding s th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp) - next - case True - from h[unfolded this] - have "holding s th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_th_cs_s] - have "th' = th" . - with assms show ?thesis by auto - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ taker" - and "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_kept[OF assms], simp) - -lemma readys_kept1: - assumes "th' \ taker" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set (th # rest) \ th' \ hd (th # rest)" - using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - moreover have "\ (th' \ set rest \ th' \ hd (taker # rest'))" - using n_wait[unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs set_wq', unfolded eq_wq'] . - ultimately have "th' = taker" by auto - with assms(1) - show ?thesis by simp - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ taker" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set rest \ th' \ hd (taker # rest')" - using wait [unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs set_wq', unfolded eq_wq'] . - moreover have "\ (th' \ set (th # rest) \ th' \ hd (th # rest))" - using n_wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - ultimately have "th' = taker" by auto - with assms(1) - show ?thesis by simp - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ taker" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { assume eq_th': "th' = taker" - have ?thesis - apply (unfold eq_th' pvD_taker_es cntCS_es_taker) - by (insert neq_taker_th assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume eq_th': "th' = th" - have ?thesis - apply (unfold eq_th' pvD_th_es cntCS_es_th) - by (insert assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume h: "th' \ taker" "th' \ th" - have ?thesis using assms - apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp) - by (fold is_v, unfold pvD_def, simp) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_v_e -begin - -lemma holdents_es_th: - "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 1 - thus ?thesis by (auto simp:holdents_def) - qed - } moreover { - fix cs' - assume "cs' \ ?R" - hence "cs' \ cs" "holding s th cs'" by (auto simp:holdents_def) - from holding_esI2[OF this] - have "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1" -proof - - have "card (holdents s th - {cs}) = card (holdents s th) - 1" - proof - - have "cs \ holdents s th" using holding_th_cs_s - by (auto simp:holdents_def) - moreover have "finite (holdents s th)" - by (simp add: finite_holdents) - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold cntCS_def holdents_es_th) -qed - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - have "cs' \ ?R" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding (e#s) th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, auto) - next - case True - from h[unfolded this] - have "holding (e#s) th' cs" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, - unfolded wq_es_cs nil_wq'] - show ?thesis by auto - qed - } moreover { - fix cs' - assume h: "cs' \ ?R" - have "cs' \ ?L" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding s th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp) - next - case True - from h[unfolded this] - have "holding s th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_th_cs_s] - have "th' = th" . - with assms show ?thesis by auto - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_kept[OF assms], simp) - -lemma readys_kept1: - assumes "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(1)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set (th # rest) \ th' \ hd (th # rest)" - using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - hence "th' \ set rest" by auto - with set_wq' have "th' \ set wq'" by metis - with nil_wq' show ?thesis by simp - qed - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set [] \ th' \ hd []" - using wait[unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs nil_wq'] . - thus ?thesis by simp - qed - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis - apply (unfold eq_th' pvD_th_es cntCS_es_th) - by (insert assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume h: "th' \ th" - have ?thesis using assms - apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp) - by (fold is_v, unfold pvD_def, simp) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_v -begin - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof(cases "rest = []") - case True - then interpret vt: valid_trace_v_e by (unfold_locales, simp) - show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast -next - case False - then interpret vt: valid_trace_v_n by (unfold_locales, simp) - show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast -qed - -end - -context valid_trace_create -begin - -lemma th_not_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_create] - show ?thesis by (cases, simp) -qed - -lemma th_not_ready_s [simp]: "th \ readys s" - using th_not_live_s by (unfold readys_def, simp) - -lemma th_live_es [simp]: "th \ threads (e#s)" - by (unfold is_create, simp) - -lemma not_waiting_th_s [simp]: "\ waiting s th cs'" -proof - assume "waiting s th cs'" - from this[unfolded s_waiting_def, folded wq_def, unfolded wq_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 -qed - -lemma not_holding_th_s [simp]: "\ holding s th cs'" -proof - assume "holding s th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_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 -qed - -lemma not_waiting_th_es [simp]: "\ waiting (e#s) th cs'" -proof - assume "waiting (e # s) th cs'" - from this[unfolded s_waiting_def, folded wq_def, unfolded wq_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 -qed - -lemma not_holding_th_es [simp]: "\ holding (e#s) th cs'" -proof - assume "holding (e # s) th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_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 -qed - -lemma ready_th_es [simp]: "th \ readys (e#s)" - by (simp add:readys_def) - -lemma holdents_th_s: "holdents s th = {}" - by (unfold holdents_def, auto) - -lemma holdents_th_es: "holdents (e#s) th = {}" - by (unfold holdents_def, auto) - -lemma cntCS_th_s [simp]: "cntCS s th = 0" - by (unfold cntCS_def, simp add:holdents_th_s) - -lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0" - by (unfold cntCS_def, simp add:holdents_th_es) - -lemma pvD_th_s [simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es [simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_kept, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_kept, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept[OF assms] - by (unfold cntCS_def, simp) - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] - have False by auto - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2) by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma pvD_kept [simp]: - assumes "th' \ th" - shows "pvD (e#s) th' = pvD s th'" - using assms - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis using assms - by (unfold eq_th', simp, unfold is_create, simp) - } moreover { - assume h: "th' \ th" - hence ?thesis using assms - by (simp, simp add:is_create) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_exit -begin - -lemma th_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold runing_def readys_def, simp) -qed - -lemma th_ready_s [simp]: "th \ readys s" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold runing_def, simp) -qed - -lemma th_not_live_es [simp]: "th \ threads (e#s)" - by (unfold is_exit, simp) - -lemma not_holding_th_s [simp]: "\ holding s th cs'" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold holdents_def, auto) -qed - -lemma cntCS_th_s [simp]: "cntCS s th = 0" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold cntCS_def, simp) -qed - -lemma not_holding_th_es [simp]: "\ holding (e#s) th cs'" -proof - assume "holding (e # s) th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept] - have "holding s th cs'" - by (unfold s_holding_def, fold wq_def, auto) - with not_holding_th_s - show False by simp -qed - -lemma ready_th_es [simp]: "th \ readys (e#s)" - by (simp add:readys_def) - -lemma holdents_th_s: "holdents s th = {}" - by (unfold holdents_def, auto) - -lemma holdents_th_es: "holdents (e#s) th = {}" - by (unfold holdents_def, auto) - -lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0" - by (unfold cntCS_def, simp add:holdents_th_es) - -lemma pvD_th_s [simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es [simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_kept, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_kept, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept[OF assms] - by (unfold cntCS_def, simp) - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] - have False by auto - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2) by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma pvD_kept [simp]: - assumes "th' \ th" - shows "pvD (e#s) th' = pvD s th'" - using assms - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis using assms - by (unfold eq_th', simp, unfold is_exit, simp) - } moreover { - assume h: "th' \ th" - hence ?thesis using assms - by (simp, simp add:is_exit) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_set -begin - -lemma th_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_set] - show ?thesis - by (cases, unfold runing_def readys_def, simp) -qed - -lemma th_ready_s [simp]: "th \ readys s" -proof - - from pip_e[unfolded is_set] - show ?thesis - by (cases, unfold runing_def, simp) -qed - -lemma th_not_live_es [simp]: "th \ threads (e#s)" - by (unfold is_set, simp) - - -lemma holdents_kept: - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_kept, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_kept, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept - by (unfold cntCS_def, simp) - -lemma threads_kept[simp]: - "threads (e#s) = threads s" - by (unfold is_set, simp) - -lemma readys_kept1: - assumes "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] - have False by auto - } moreover have "th' \ threads s" - using assms[unfolded readys_def] by auto - ultimately show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1 readys_kept2 - by metis - -lemma pvD_kept [simp]: - shows "pvD (e#s) th' = pvD s th'" - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" - using assms - by (unfold is_set, simp, fold is_set, simp) - -end - -context valid_trace -begin - -lemma cnp_cnv_cncs: "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" -proof(induct rule:ind) - case Nil - thus ?case - by (unfold cntP_def cntV_def pvD_def cntCS_def holdents_def - s_holding_def, simp) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt_create: valid_trace_create s e th prio - using Create by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_create.cnp_cnv_cncs_kept) - next - case (Exit th) - interpret vt_exit: valid_trace_exit s e th - using Exit by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_exit.cnp_cnv_cncs_kept) - next - case (P th cs) - interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_p.cnp_cnv_cncs_kept) - next - case (V th cs) - interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_v.cnp_cnv_cncs_kept) - next - case (Set th prio) - interpret vt_set: valid_trace_set s e th prio - using Set by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_set.cnp_cnv_cncs_kept) - qed -qed - -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 = {}" -proof - - { fix cs - assume "cs \ holdents s th" - hence "holding s th cs" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def] - have "th \ set (wq s cs)" by auto - with wq_threads have "th \ threads s" by auto - with assms - have False by simp - } thus ?thesis by auto -qed - -lemma not_thread_cncs: - assumes not_in: "th \ threads s" - shows "cntCS s th = 0" - using not_thread_holdents[OF assms] - by (simp add:cntCS_def) - -lemma cnp_cnv_eq: - assumes "th \ threads s" - shows "cntP s th = cntV s th" - using assms cnp_cnv_cncs not_thread_cncs pvD_def - by (auto) - -lemma eq_pv_children: - assumes eq_pv: "cntP s th = cntV s th" - shows "children (RAG s) (Th th) = {}" -proof - - from cnp_cnv_cncs and eq_pv - have "cntCS s th = 0" - by (auto split:if_splits) - from this[unfolded cntCS_def holdents_alt_def] - have card_0: "card (the_cs ` children (RAG s) (Th th)) = 0" . - have "finite (the_cs ` children (RAG s) (Th th))" - by (simp add: fsbtRAGs.finite_children) - from card_0[unfolded card_0_eq[OF this]] - show ?thesis by auto -qed - -lemma eq_pv_holdents: - assumes eq_pv: "cntP s th = cntV s th" - shows "holdents s th = {}" - by (unfold holdents_alt_def eq_pv_children[OF assms], simp) - -lemma eq_pv_subtree: - assumes eq_pv: "cntP s th = cntV s th" - shows "subtree (RAG s) (Th th) = {Th th}" - using eq_pv_children[OF assms] - by (unfold subtree_children, simp) - -lemma count_eq_RAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (RAG s)^+} = {}" -proof(rule ccontr) - assume otherwise: "{th'. (Th th', Th th) \ (RAG s)\<^sup>+} \ {}" - then obtain th' where "(Th th', Th th) \ (RAG s)^+" by auto - from tranclD2[OF this] - obtain z where "z \ children (RAG s) (Th th)" - by (auto simp:children_def) - with eq_pv_children[OF assms] - show False by simp -qed - -lemma eq_pv_dependants: - assumes eq_pv: "cntP s th = cntV s th" - shows "dependants s th = {}" -proof - - from count_eq_RAG_plus[OF assms, folded dependants_alt_def1] - show ?thesis . -qed - -lemma count_eq_tRAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using assms eq_pv_dependants dependants_alt_def eq_dependants by auto - -lemma count_eq_RAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (RAG s)^+} = {}" - using count_eq_RAG_plus[OF assms] by auto - -lemma count_eq_tRAG_plus_Th: - assumes "cntP s th = cntV s th" - shows "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using count_eq_tRAG_plus[OF assms] by auto - -end - -definition detached :: "state \ thread \ bool" - where "detached s th \ (\(\ cs. holding s th cs)) \ (\(\cs. waiting s th cs))" - -lemma detached_test: - shows "detached s th = (Th th \ Field (RAG s))" -apply(simp add: detached_def Field_def) -apply(simp add: s_RAG_def) -apply(simp add: s_holding_abv s_waiting_abv) -apply(simp add: Domain_iff Range_iff) -apply(simp add: wq_def) -apply(auto) -done - -context valid_trace -begin - -lemma detached_intro: - assumes eq_pv: "cntP s th = cntV s th" - shows "detached s th" -proof - - from eq_pv cnp_cnv_cncs - have "th \ readys s \ th \ threads s" by (auto simp:pvD_def) - thus ?thesis - proof - assume "th \ threads s" - with rg_RAG_threads dm_RAG_threads - show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv - s_holding_abv wq_def Domain_iff Range_iff) - next - assume "th \ readys s" - moreover have "Th th \ Range (RAG s)" - proof - - from eq_pv_children[OF assms] - have "children (RAG s) (Th th) = {}" . - thus ?thesis - by (unfold children_def, auto) - qed - ultimately show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv - s_holding_abv wq_def readys_def) - qed -qed - -lemma detached_elim: - assumes dtc: "detached s th" - shows "cntP s th = cntV s th" -proof - - have cncs_z: "cntCS s th = 0" - proof - - from dtc have "holdents s th = {}" - unfolding detached_def holdents_test s_RAG_def - by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff) - thus ?thesis by (auto simp:cntCS_def) - qed - show ?thesis - proof(cases "th \ threads s") - case True - with dtc - have "th \ readys s" - by (unfold readys_def detached_def Field_def Domain_def Range_def, - auto simp:waiting_eq s_RAG_def) - with cncs_z show ?thesis using cnp_cnv_cncs by (simp add:pvD_def) - next - case False - with cncs_z and cnp_cnv_cncs show ?thesis by (simp add:pvD_def) - qed -qed - -lemma detached_eq: - shows "(detached s th) = (cntP s th = cntV s th)" - by (insert vt, auto intro:detached_intro detached_elim) - -end - -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 *) -lemma cp_gen_rec: - assumes "x = Th th" - shows "cp_gen s x = Max ({the_preced s th} \ (cp_gen s) ` children (tRAG s) x)" -proof(cases "children (tRAG s) x = {}") - case True - show ?thesis - by (unfold True cp_gen_def subtree_children, simp add:assms) -next - case False - hence [simp]: "children (tRAG s) x \ {}" by auto - note fsbttRAGs.finite_subtree[simp] - have [simp]: "finite (children (tRAG s) x)" - by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], - rule children_subtree) - { fix r x - have "subtree r x \ {}" by (auto simp:subtree_def) - } note this[simp] - have [simp]: "\x\children (tRAG s) x. subtree (tRAG s) x \ {}" - proof - - from False obtain q where "q \ children (tRAG s) x" by blast - moreover have "subtree (tRAG s) q \ {}" by simp - ultimately show ?thesis by blast - qed - have h: "Max ((the_preced s \ the_thread) ` - ({x} \ \(subtree (tRAG s) ` children (tRAG s) x))) = - Max ({the_preced s th} \ cp_gen s ` children (tRAG s) x)" - (is "?L = ?R") - proof - - let "Max (?f ` (?A \ \ (?g ` ?B)))" = ?L - let "Max (_ \ (?h ` ?B))" = ?R - let ?L1 = "?f ` \(?g ` ?B)" - have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)" - proof - - have "?L1 = ?f ` (\ x \ ?B.(?g x))" by simp - also have "... = (\ x \ ?B. ?f ` (?g x))" by auto - finally have "Max ?L1 = Max ..." by simp - also have "... = Max (Max ` (\x. ?f ` subtree (tRAG s) x) ` ?B)" - by (subst Max_UNION, simp+) - also have "... = Max (cp_gen s ` children (tRAG s) x)" - by (unfold image_comp cp_gen_alt_def, simp) - finally show ?thesis . - qed - show ?thesis - proof - - have "?L = Max (?f ` ?A \ ?L1)" by simp - also have "... = max (the_preced s (the_thread x)) (Max ?L1)" - by (subst Max_Un, simp+) - also have "... = max (?f x) (Max (?h ` ?B))" - by (unfold eq_Max_L1, simp) - also have "... =?R" - by (rule max_Max_eq, (simp)+, unfold assms, simp) - finally show ?thesis . - qed - qed thus ?thesis - by (fold h subtree_children, unfold cp_gen_def, simp) -qed - -lemma cp_rec: - "cp s th = Max ({the_preced s th} \ - (cp s o the_thread) ` children (tRAG s) (Th th))" -proof - - have "Th th = Th th" by simp - note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this] - show ?thesis - proof - - have "cp_gen s ` children (tRAG s) (Th th) = - (cp s \ the_thread) ` children (tRAG s) (Th th)" - proof(rule cp_gen_over_set) - show " \x\children (tRAG s) (Th th). \th. x = Th th" - by (unfold tRAG_alt_def, auto simp:children_def) - qed - thus ?thesis by (subst (1) h(1), unfold h(2), simp) - qed -qed -end - -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'" - shows "holding (wq s) th cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - thus ?thesis - by (unfold cs_holding_def, auto) -qed - -lemma next_th_waiting: - assumes nxt: "next_th s th cs th'" - shows "waiting (wq s) th' cs" -proof - - from nxt[unfolded next_th_def] - obtain rest where h: "wq s cs = th # rest" - "rest \ []" - "th' = hd (SOME q. distinct q \ set q = set rest)" by auto - from wq_distinct[of cs, unfolded h] - have dst: "distinct (th # rest)" . - have in_rest: "th' \ set rest" - proof(unfold h, rule someI2) - show "distinct rest \ set rest = set rest" using dst by auto - next - fix x assume "distinct x \ set x = set rest" - with h(2) - show "hd x \ set (rest)" by (cases x, auto) - qed - hence "th' \ set (wq s cs)" by (unfold h(1), auto) - moreover have "th' \ hd (wq s cs)" - by (unfold h(1), insert in_rest dst, auto) - ultimately show ?thesis by (auto simp:cs_waiting_def) -qed - -lemma next_th_RAG: - assumes nxt: "next_th (s::event list) th cs th'" - shows "{(Cs cs, Th th), (Th th', Cs cs)} \ RAG s" - using vt assms next_th_holding next_th_waiting - by (unfold s_RAG_def, simp) - -end - -context valid_trace_p -begin - -find_theorems readys th - -end - -end