theory CpsGimports PIPDefsbeginsection {* Generic aulxiliary lemmas *}lemma f_image_eq: assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a" shows "f ` A = g ` A"proof show "f ` A \<subseteq> g ` A" by(rule image_subsetI, auto intro:h)next show "g ` A \<subseteq> f ` A" by (rule image_subsetI, auto intro:h[symmetric])qedlemma Max_fg_mono: assumes "finite A" and "\<forall> a \<in> A. f a \<le> g a" shows "Max (f ` A) \<le> Max (g ` A)"proof(cases "A = {}") case True thus ?thesis by autonext case False show ?thesis proof(rule Max.boundedI) from assms show "finite (f ` A)" by auto next from False show "f ` A \<noteq> {}" by auto next fix fa assume "fa \<in> f ` A" then obtain a where h_fa: "a \<in> A" "fa = f a" by auto show "fa \<le> Max (g ` A)" proof(rule Max_ge_iff[THEN iffD2]) from assms show "finite (g ` A)" by auto next from False show "g ` A \<noteq> {}" by auto next from h_fa have "g a \<in> g ` A" by auto moreover have "fa \<le> g a" using h_fa assms(2) by auto ultimately show "\<exists>a\<in>g ` A. fa \<le> a" by auto qed qedqed lemma Max_f_mono: assumes seq: "A \<subseteq> B" and np: "A \<noteq> {}" and fnt: "finite B" shows "Max (f ` A) \<le> Max (f ` B)"proof(rule Max_mono) from seq show "f ` A \<subseteq> f ` B" by autonext from np show "f ` A \<noteq> {}" by autonext from fnt and seq show "finite (f ` B)" by autoqedlemma Max_UNION: assumes "finite A" and "A \<noteq> {}" and "\<forall> M \<in> f ` A. finite M" and "\<forall> M \<in> f ` A. M \<noteq> {}" shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R") using assms[simp]proof - have "?L = Max (\<Union>(f ` A))" by (fold Union_image_eq, simp) also have "... = ?R" by (subst Max_Union, simp+) finally show ?thesis .qedlemma max_Max_eq: assumes "finite A" and "A \<noteq> {}" and "x = y" shows "max x (Max A) = Max ({y} \<union> 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 simpqedlemma rel_eqI: assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B" and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A" shows "A = B" using assms by autosection {* Lemmas do not depend on trace validity *}lemma birth_time_lt: assumes "s \<noteq> []" shows "last_set th s < length s" using assmsproof(induct s) case (Cons a s) show ?case proof(cases "s \<noteq> []") case False thus ?thesis by (cases a, auto) next case True show ?thesis using Cons(1)[OF True] by (cases a, auto) qedqed simplemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []" by (induct s, auto)lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> 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' \<in> 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 .qedlemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"unfolding cp_def wq_defapply(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)donelemma cp_alt_def: "cp s th = Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"proof - have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) = Max (the_preced s ` {th'. Th th' \<in> 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)qedlemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> 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 \<subseteq> readys s" unfolding runing_def readys_def by auto lemma readys_threads: shows "readys s \<subseteq> threads s" unfolding readys_def by autolemma wq_v_neq [simp]: "cs \<noteq> cs' \<Longrightarrow> 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 \<in> runing s" and "th \<in> 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 \<in> runing s" and "th \<in> 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' \<noteq> th" hence "th \<noteq> 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 metisqedlemma 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 \<in> threads s \<Longrightarrow> last_set th s < length s" apply (induct s, auto) by (case_tac a, auto split:if_splits)lemma last_set_unique: "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk> \<Longrightarrow> 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 \<in> threads s" and th_in2: " th2 \<in> 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 .qedlemma preced_linorder: assumes neq_12: "th1 \<noteq> th2" and th_in1: "th1 \<in> threads s" and th_in2: " th2 \<in> threads s" shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"proof - from preced_unique [OF _ th_in1 th_in2] and neq_12 have "preced th1 s \<noteq> preced th2 s" by auto thus ?thesis by autoqedlemma in_RAG_E: assumes "(n1, n2) \<in> 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 autolemma 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 "\<not>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' \<noteq> th" shows "cntP (P th cs'#s) th' = cntP s th'" using assms by (unfold cntP_def, simp)lemma cntP_simp3[simp]: assumes "\<not> 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' \<noteq> th" shows "cntV (V th cs'#s) th' = cntV s th'" using assms by (unfold cntV_def, simp)lemma cntV_simp3[simp]: assumes "\<not> 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 \<noteq> cntP s th" shows "isP e \<and> actor e = th"proof(cases e) case (P th' pty) show ?thesis by (cases "(\<lambda>e. \<exists>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 \<noteq> cntV s th" shows "isV e \<and> actor e = th"proof(cases e) case (V th' pty) show ?thesis by (cases "(\<lambda>e. \<exists>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 \<in> set (wq s cs) \<and> 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 \<noteq> []" by auto let ?th' = "hd (SOME q. distinct q \<and> 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)"begintext {* 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) endtext {* 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 \<noteq> []"}, 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 \<noteq> []"begintext {* 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)"endtext {* @{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 \<noteq> []"}, 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 \<and> 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'"endtext {* 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 \<noteq> []"begintext {* The following @{text "taker"} is the thread to take over @{text "cs"}. *} definition "taker = hd wq'"endlocale valid_trace_set = valid_trace_e + fixes th prio assumes is_set: "e = Set th prio"context valid_tracebegintext {* 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 "(\<And>s e. valid_trace_e s e \<Longrightarrow> PP s \<Longrightarrow> PIP s e \<Longrightarrow> 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 qedqedtext {* 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: "\<And> 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 \<ge> 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 \<le> length s" by simp from moment_app [OF this, of "[e]"] show ?thesis by simp qed ultimately show ?thesis by simp qedqedendtext {* 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 :: natsublocale 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 \<le> length s" by auto from moment_plus[OF this, folded next_e_def] show ?thesis . qedendsublocale 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_createbeginlemma 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 simpendcontext valid_trace_exitbeginlemma 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 simpendcontext valid_trace_p beginlemma wq_neq_simp [simp]: assumes "cs' \<noteq> 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 \<in> runing s"proof - from pip_e[unfolded is_p] show ?thesis by (cases, simp)qedlemma th_not_in_wq: shows "th \<notin> set (wq s cs)"proof assume otherwise: "th \<in> 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) \<in> 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 qedqedlemma 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)endcontext valid_trace_vbeginlemma wq_neq_simp [simp]: assumes "cs' \<noteq> 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)) qedqedlemma 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 \<and> set rest = set rest" using assms[unfolded True wq_s_cs] by auto qed simpqed (insert assms, simp)endcontext valid_trace_setbeginlemma 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 simpendcontext valid_tracebeginlemma 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 blastlemma 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) qedqed (unfold wq_def Let_def, simp)endsection {* Waiting queues and threads *}context valid_trace_ebeginlemma wq_out_inv: assumes s_in: "thread \<in> set (wq s cs)" and s_hd: "thread = hd (wq s cs)" and s_i: "thread \<noteq> 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 \<notin> set (wq s cs)" and s_i: "thread \<in> 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) \<subseteq> set (wq s cs)" proof - have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)" using Cons V by (auto simp:wq_def Let_def True split:if_splits) moreover have "set ... \<subseteq> set (wq s cs)" proof(rule someI2) show "distinct w_tl \<and> 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 autoqed (insert assms, auto simp:wq_def Let_def split:if_splits)endlemma (in valid_trace_create) th_not_in_threads: "th \<notin> threads s"proof - from pip_e[unfolded is_create] show ?thesis by (cases, simp)qedlemma (in valid_trace_create) threads_es [simp]: "threads (e#s) = threads s \<union> {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 \<notin> set rest"proof assume otherwise: "th \<in> set rest" have "distinct (wq s cs)" by (simp add: wq_distinct) from this[unfolded wq_s_cs] and otherwise show False by autoqedlemma (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 \<and> set rest = set rest" by (simp add: distinct_rest) next fix x assume "distinct x \<and> set x = set rest" thus "set x = set (wq s cs) - {th}" by (unfold wq_s_cs, simp)qedlemma (in valid_trace_exit) th_not_in_wq: "th \<notin> 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)qedlemma (in valid_trace) wq_threads: assumes "th \<in> set (wq s cs)" shows "th \<in> threads s" using assmsproof(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) qedqed section {* RAG and threads *}context valid_tracebeginlemma dm_RAG_threads: assumes in_dom: "(Th th) \<in> Domain (RAG s)" shows "th \<in> threads s"proof - from in_dom obtain n where "(Th th, n) \<in> 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) \<in> RAG s" by simp hence "th \<in> set (wq s cs)" by (unfold s_RAG_def, auto simp:cs_waiting_def) from wq_threads [OF this] show ?thesis .qedlemma rg_RAG_threads: assumes "(Th th) \<in> Range (RAG s)" shows "th \<in> 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) \<in> Field (RAG s)" shows "th \<in> threads s" using assms by (metis Field_def UnE dm_RAG_threads rg_RAG_threads)endsection {* 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_vbeginlemma 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 qedqedlemma 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' \<in> set wq'" shows "th' \<in> set rest" using assms set_wq' by simplemma runing_th_s: shows "th \<in> runing s"proof - from pip_e[unfolded is_v] show ?thesis by (cases, simp)qedlemma neq_t_th: assumes "waiting (e#s) t c" shows "t \<noteq> th"proof assume otherwise: "t = th" show False proof(cases "c = cs") case True have "t \<in> 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 \<in> 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 \<notin> readys s" by (unfold readys_def, auto) hence "t \<notin> runing s" using runing_ready by auto with runing_th_s[folded otherwise] show ?thesis by auto qedqedlemma waiting_esI1: assumes "waiting s t c" and "c \<noteq> 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 qedlemma holding_esI2: assumes "c \<noteq> 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 .qedlemma holding_esI1: assumes "holding s t c" and "t \<noteq> th" shows "holding (e#s) t c"proof - have "c \<noteq> cs" using assms using holding_cs_eq_th by blast from holding_esI2[OF this assms(1)] show ?thesis .qedendcontext valid_trace_v_nbeginlemma neq_wq': "wq' \<noteq> []" proof (unfold wq'_def, rule someI2) show "distinct rest \<and> set rest = set rest" by (simp add: distinct_rest) next fix x assume " distinct x \<and> set x = set rest" thus "x \<noteq> []" using rest_nnl by autoqed 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 \<and> 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 qedlemma 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 fastforcelemma 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 \<noteq> taker" shows "waiting (e#s) t cs" proof - have "t \<in> set wq'" proof(unfold wq'_def, rule someI2) show "distinct rest \<and> set rest = set rest" by (simp add: distinct_rest) next fix x assume "distinct x \<and> set x = set rest" moreover have "t \<in> set rest" using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto ultimately show "t \<in> set x" by simp qed moreover have "t \<noteq> 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)qedlemma waiting_esE: assumes "waiting (e#s) t c" obtains "c \<noteq> cs" "waiting s t c" | "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> 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 \<noteq> hd wq'" "t \<in> set wq'" by auto hence "t \<noteq> taker" by (simp add: taker_def) moreover hence "t \<noteq> th" using assms neq_t_th by blast moreover have "t \<in> set rest" by (simp add: `t \<in> 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 \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto qedlemma 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 \<noteq> 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 .qedend context valid_trace_v_ebeginlemma nil_wq': "wq' = []" proof (unfold wq'_def, rule someI2) show "distinct rest \<and> set rest = set rest" by (simp add: distinct_rest) next fix x assume " distinct x \<and> set x = set rest" thus "x = []" using rest_nil by autoqed 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' \<noteq> []" by auto thus ?thesis using rest_def rest_nil by auto qedlemma waiting_set_eq: "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}" using no_taker by autolemma holding_set_eq: "{(Cs cs, Th th') |th'. next_th s th cs th'} = {}" using no_taker by autolemma no_holding: assumes "holding (e#s) taker cs" shows Falseproof - 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 autoqedlemma no_waiting: assumes "waiting (e#s) t cs" shows Falseproof - 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 autoqedlemma waiting_esI2: assumes "waiting s t c" shows "waiting (e#s) t c"proof - have "c \<noteq> cs" using assms using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto from waiting_esI1[OF assms this] show ?thesis .qedlemma waiting_esE: assumes "waiting (e#s) t c" obtains "c \<noteq> 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 autoqedlemma holding_esE: assumes "holding (e#s) t c" obtains "c \<noteq> cs" "holding s t c"proof(cases "c = cs") case True from no_holding[OF assms[unfolded True]] show ?thesis by autonext 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 .qedend context valid_trace_vbeginlemma RAG_es: "RAG (e # s) = RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union> {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = ?R")proof(rule rel_eqI) fix n1 n2 assume "(n1, n2) \<in> ?L" thus "(n1, n2) \<in> ?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 qednext fix n1 n2 assume h: "(n1, n2) \<in> ?R" show "(n1, n2) \<in> ?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) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or> (n2 = Th h_n.taker \<and> n1 = Cs cs)" by auto thus ?thesis proof assume "n2 = Th h_n.taker \<and> n1 = Cs cs" with h_n.holding_taker show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) next assume h: "(n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)" hence "(n1, n2) \<in> RAG s" by simp thus ?thesis proof(cases rule:in_RAG_E) case (waiting th' cs') from h and this(1,2) have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto hence "waiting (e#s) th' cs'" proof assume "cs' \<noteq> cs" from waiting_esI1[OF waiting(3) this] show ?thesis . next assume neq_th': "th' \<noteq> 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' \<noteq> cs \<or> th' \<noteq> th" by auto hence "holding (e#s) th' cs'" proof assume "cs' \<noteq> cs" from holding_esI2[OF this holding(3)] show ?thesis . next assume "th' \<noteq> 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) \<in> RAG s" "(n1, n2) \<noteq> (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' \<noteq> cs \<or> th' \<noteq> th" by auto thus ?thesis proof assume neq_cs: "cs' \<noteq> 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' \<noteq> 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 qedqedlemma 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)qedendcontext valid_trace_pbeginlemma 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) qedend lemma (in valid_trace_p) th_not_waiting: "\<not> waiting s th c"proof - have "th \<in> readys s" using runing_ready runing_th_s by blast thus ?thesis by (unfold readys_def, auto)qedcontext valid_trace_p_hbeginlemma wq_es_cs': "wq (e#s) cs = [th]" using wq_es_cs[unfolded we] by simplemma holding_es_th_cs: shows "holding (e#s) th cs"proof - from wq_es_cs' have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto thus ?thesis using cs_holding_def holding_eq by blast qedlemma RAG_edge: "(Cs cs, Th th) \<in> 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' \<noteq> 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 .qedlemma RAG_es: "RAG (e # s) = RAG s \<union> {(Cs cs, Th th)}" (is "?L = ?R")proof(rule rel_eqI) fix n1 n2 assume "(n1, n2) \<in> ?L" thus "(n1, n2) \<in> ?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 qednext fix n1 n2 assume "(n1, n2) \<in> ?R" hence "(n1, n2) \<in> RAG s \<or> (n1 = Cs cs \<and> n2 = Th th)" by auto thus "(n1, n2) \<in> ?L" proof assume "(n1, n2) \<in> 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 \<and> n2 = Th th" with holding_es_th_cs show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) qedqedendcontext valid_trace_p_wbeginlemma 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 autolemma RAG_edge: "(Th th, Cs cs) \<in> 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) qedlemma waiting_esE: assumes "waiting (e#s) th' cs'" obtains "th' \<noteq> th" "waiting s th' cs'" | "th' = th" "cs' = cs"proof(cases "waiting s th' cs'") case True have "th' \<noteq> 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 \<and> 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 metisqedlemma RAG_es: "RAG (e # s) = RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")proof(rule rel_eqI) fix n1 n2 assume "(n1, n2) \<in> ?L" thus "(n1, n2) \<in> ?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 qednext fix n1 n2 assume "(n1, n2) \<in> ?R" hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto thus "(n1, n2) \<in> ?L" proof assume "(n1, n2) \<in> 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 \<and> n2 = Cs cs" thus ?thesis using RAG_edge by auto qedqedendcontext valid_trace_pbeginlemma RAG_es: "RAG (e # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)} else RAG s \<union> {(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) qedendsection {* Finiteness of RAG *}context valid_tracebeginlemma 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 qedqedendsection {* 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_tracebeginlemma waiting_unique_pre: (* ddd *) assumes h11: "thread \<in> set (wq s cs1)" and h12: "thread \<noteq> hd (wq s cs1)" assumes h21: "thread \<in> set (wq s cs2)" and h22: "thread \<noteq> hd (wq s cs2)" and neq12: "cs1 \<noteq> cs2" shows "False"proof - let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> 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: "\<not> ?Q cs1 []" by (simp add:wq_def) have nq2: "\<not> ?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: "\<not> ?Q cs1 (moment t1 s)" and nn1: "(\<forall>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: "\<not> ?Q cs2 (moment t2 s)" and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto { fix s cs assume q: "?Q cs s" have "thread \<notin> runing s" proof assume "thread \<in> runing s" hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and> thread \<noteq> 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: "\<not> ?Q cs1 (moment t1 s)" and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" and lt2: "t2 < length s" and np2: "\<not> ?Q cs2 (moment t2 s)" and nn2: "(\<forall>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 \<in> set (wq (?e#moment t2 s) cs2)" and h2: "thread \<noteq> hd (wq (?e#moment t2 s) cs2)" by auto have ?thesis proof - have "thread \<in> runing (moment t2 s)" proof(cases "thread \<in> 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 \<notin> 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 \<in> 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 \<in> set (wq (?e#moment t2 s) cs2)" and h2: "thread \<noteq> 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 \<in> set (wq (?e#moment t1 s) cs1)" and g2: "thread \<noteq> hd (wq (?e#moment t1 s) cs1)" by auto have "?e = V thread cs2 \<or> ?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 \<or> ?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 qedqedtext {* 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 autoendlemma (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)qedlemma (in valid_trace_v) the_preced_es: "the_preced (e#s) = the_preced s" by (unfold is_v preced_def, simp)context valid_trace_pbeginlemma not_holding_s_th_cs: "\<not> 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) \<in> RAG s" using otherwise cs_holding_def holding_eq th_not_in_wq by auto ultimately show ?thesis by auto qedqedendlemma (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_ebegin lemma acylic_RAG_kept: assumes "acyclic (RAG s)" shows "acyclic (RAG (e#s))"proof(rule acyclic_subset[OF assms]) show "RAG (e # s) \<subseteq> RAG s" by (unfold RAG_es waiting_set_eq holding_set_eq, auto)qedendcontext valid_trace_v_nbegin 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 fastforcelemma 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)}) \<union> {(Cs cs, Th taker)})" (is "acyclic (?A \<union> _)") proof - from assms have "acyclic ?A" by (rule acyclic_subset, auto) moreover have "(Th taker, Cs cs) \<notin> ?A^*" proof assume otherwise: "(Th taker, Cs cs) \<in> ?A^*" hence "(Th taker, Cs cs) \<in> ?A^+" by (unfold rtrancl_eq_or_trancl, auto) from tranclD[OF this] obtain cs' where h: "(Th taker, Cs cs') \<in> ?A" "(Th taker, Cs cs') \<in> 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)qedendcontext valid_trace_p_hbeginlemma acylic_RAG_kept: assumes "acyclic (RAG s)" shows "acyclic (RAG (e#s))"proof - have "acyclic (RAG s \<union> {(Cs cs, Th th)})" (is "acyclic (?A \<union> _)") proof - from assms have "acyclic ?A" by (rule acyclic_subset, auto) moreover have "(Th th, Cs cs) \<notin> ?A^*" proof assume otherwise: "(Th th, Cs cs) \<in> ?A^*" hence "(Th th, Cs cs) \<in> ?A^+" by (unfold rtrancl_eq_or_trancl, auto) from tranclD[OF this] obtain cs' where h: "(Th th, Cs cs') \<in> 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)qedendcontext valid_trace_p_wbeginlemma acylic_RAG_kept: assumes "acyclic (RAG s)" shows "acyclic (RAG (e#s))"proof - have "acyclic (RAG s \<union> {(Th th, Cs cs)})" (is "acyclic (?A \<union> _)") proof - from assms have "acyclic ?A" by (rule acyclic_subset, auto) moreover have "(Cs cs, Th th) \<notin> ?A^*" proof assume otherwise: "(Cs cs, Th th) \<in> ?A^*" from pip_e[unfolded is_p] show False proof(cases) case (thread_P) moreover from otherwise have "(Cs cs, Th th) \<in> ?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)qedendcontext valid_tracebeginlemma 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 qedqedendsection {* RAG is single-valued *}context valid_tracebeginlemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> 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)endsection {* RAG is well-founded *}context valid_tracebeginlemma 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)" .qedlemma 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)" .qedendsection {* RAG forms a forest (or tree) *}context valid_tracebeginlemma rtree_RAG: "rtree (RAG s)" using sgv_RAG acyclic_RAG by (unfold rtree_def rtree_axioms_def sgv_def, auto)endsublocale 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 qedqedsection {* Derived properties for parts of RAG *}context valid_tracebeginlemma acyclic_tRAG: "acyclic (tRAG s)"proof(unfold tRAG_def, rule acyclic_compose) show "acyclic (RAG s)" using acyclic_RAG .next show "wRAG s \<subseteq> RAG s" unfolding RAG_split by autonext show "hRAG s \<subseteq> RAG s" unfolding RAG_split by autoqedlemma 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)endsublocale 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)" .qedsublocale 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 \<subseteq> (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)" .qedlemma tRAG_nodeE: assumes "(n1, n2) \<in> 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 \<in> ancestors (tRAG s) u" obtains th where "x = Th th"proof - from assms have "(u, x) \<in> (tRAG s)^+" by (unfold ancestors_def, auto) from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto then obtain th where "x = Th th" by (unfold tRAG_alt_def, auto) from that[OF this] show ?thesis .qedlemma subtree_nodeE: assumes "n \<in> 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 \<in> ancestors (tRAG s) n" hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def) hence "\<exists> 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 qedqedlemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"proof - have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*" by (rule rtrancl_mono, auto simp:RAG_split) also have "... \<subseteq> ((RAG s)^*)^*" by (rule rtrancl_mono, auto) also have "... = (RAG s)^*" by simp finally show ?thesis by (unfold tRAG_def, simp)qedlemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"proof - { fix a assume "a \<in> subtree (tRAG s) x" hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def) with tRAG_star_RAG have "(a, x) \<in> (RAG s)^*" by auto hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def) } thus ?thesis by autoqedlemma tRAG_trancl_eq: "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {th'. (Th th', Th th) \<in> (RAG s)^+}" (is "?L = ?R")proof - { fix th' assume "th' \<in> ?L" hence "(Th th', Th th) \<in> (tRAG s)^+" by auto from tranclD[OF this] obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto from tRAG_subtree_RAG and this(2) have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG) moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto ultimately have "th' \<in> ?R" by auto } moreover { fix th' assume "th' \<in> ?R" hence "(Th th', Th th) \<in> (RAG s)^+" by (auto) from plus_rpath[OF this] obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto hence "(Th th', Th th) \<in> (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) \<in> (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) \<subseteq> RAG s" . have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)" by (simp add: edges_on_unfold) with eds have rg1: "(Th th', x1) \<in> RAG s" by auto then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto) have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)" by (simp add: edges_on_unfold) from this eds have rg2: "(x1, x2) \<in> 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) \<in> 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) \<in> (tRAG s)\<^sup>+" by simp with rt1 show ?thesis by auto qed qed qed hence "th' \<in> ?L" by auto } ultimately show ?thesis by blastqedlemma tRAG_trancl_eq_Th: "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}" using tRAG_trancl_eq by autolemma tRAG_Field: "Field (tRAG s) \<subseteq> Field (RAG s)" by (unfold tRAG_alt_def Field_def, auto)lemma tRAG_mono: assumes "RAG s' \<subseteq> RAG s" shows "tRAG s' \<subseteq> tRAG s" using assms by (unfold tRAG_alt_def, auto)lemma tRAG_subtree_eq: "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")proof - { fix n assume h: "n \<in> ?L" hence "n \<in> ?R" by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) } moreover { fix n assume "n \<in> ?R" then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*" by (auto simp:subtree_def) from rtranclD[OF this(2)] have "n \<in> ?L" proof assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+" with h have "n \<in> {Th th' | th'. (Th th', Th th) \<in> (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 autoqedlemma threads_set_eq: "the_thread ` (subtree (tRAG s) (Th th)) = {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R") by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)context valid_tracebeginlemma RAG_tRAG_transfer: assumes "RAG s' = RAG s \<union> {(Th th, Cs cs)}" and "(Cs cs, Th th'') \<in> RAG s" shows "tRAG s' = tRAG s \<union> {(Th th, Th th'')}" (is "?L = ?R")proof - { fix n1 n2 assume "(n1, n2) \<in> ?L" from this[unfolded tRAG_alt_def] obtain th1 th2 cs' where h: "n1 = Th th1" "n2 = Th th2" "(Th th1, Cs cs') \<in> RAG s'" "(Cs cs', Th th2) \<in> RAG s'" by auto from h(4) and assms(1) have cs_in: "(Cs cs', Th th2) \<in> RAG s" by auto from h(3) and assms(1) have "(Th th1, Cs cs') = (Th th, Cs cs) \<or> (Th th1, Cs cs') \<in> RAG s" by auto hence "(n1, n2) \<in> ?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') \<in> RAG s" with cs_in have "(Th th1, Th th2) \<in> 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) \<in> ?R" hence "(n1, n2) \<in>tRAG s \<or> (n1, n2) = (Th th, Th th'')" by auto hence "(n1, n2) \<in> ?L" proof assume "(n1, n2) \<in> tRAG s" moreover have "... \<subseteq> ?L" proof(rule tRAG_mono) show "RAG s \<subseteq> 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'') \<in> RAG s'" by auto moreover have "(Th th, Cs cs) \<in> RAG s'" using assms(1) by auto ultimately show ?thesis by (unfold eq_n tRAG_alt_def, auto) qed } ultimately show ?thesis by autoqedlemma subtree_tRAG_thread: assumes "th \<in> threads s" shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")proof - have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}" by (unfold tRAG_subtree_eq, simp) also have "... \<subseteq> ?R" proof fix x assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}" then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto from this(2) show "x \<in> ?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 .qedlemma dependants_alt_def: "dependants s th = {th'. (Th th', Th th) \<in> (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) \<in> (RAG s)^+}" using dependants_alt_def tRAG_trancl_eq by autoendsection {* Chain to readys *}context valid_tracebeginlemma chain_building: assumes "node \<in> Domain (RAG s)" obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"proof - from assms have "node \<in> Range ((RAG s)^-1)" by auto from wf_base[OF wf_RAG_converse this] obtain b where h_b: "(b, node) \<in> ((RAG s)\<inverse>)\<^sup>+" "\<forall>c. (c, b) \<notin> (RAG s)\<inverse>" 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) \<in> ((RAG s)\<^sup>+)" by auto from tranclE[OF this] obtain n where "(n, b) \<in> 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) \<in> 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' \<in> readys s" proof - from h_b(2)[unfolded eq_b] have "\<forall>cs. \<not> waiting s th' cs" by (unfold s_RAG_def, fold waiting_eq, auto) moreover have "th' \<in> threads s" proof(rule rg_RAG_threads) from tranclD[OF h_b(1), unfolded eq_b] obtain z where "(z, Th th') \<in> (RAG s)" by auto thus "Th th' \<in> Range (RAG s)" by auto qed ultimately show ?thesis by (auto simp:readys_def) qed moreover have "(node, Th th') \<in> (RAG s)^+" using h_b(1)[unfolded trancl_converse] eq_b by auto ultimately show ?thesis using that by metisqedtext {* \noindent The following is just an instance of @{text "chain_building"}.*} lemma th_chain_to_ready: assumes th_in: "th \<in> threads s" shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"proof(cases "th \<in> readys s") case True thus ?thesis by autonext case False from False and th_in have "Th th \<in> 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 autoqedlemma finite_subtree_threads: "finite {th'. Th th' \<in> subtree (RAG s) (Th th)}" (is "finite ?A")proof - have "?A = the_thread ` {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}" by (auto, insert image_iff, fastforce) moreover have "finite {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}" (is "finite ?B") proof - have "?B = (subtree (RAG s) (Th th)) \<inter> {Th th' | th'. True}" by auto moreover have "... \<subseteq> (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 autoqedlemma runing_unique: assumes runing_1: "th1 \<in> runing s" and runing_2: "th2 \<in> 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' \<in> subtree (RAG s) (Th th1)}) = Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th2)})" (is "Max ?L = Max ?R") . have "Max ?L \<in> ?L" proof(rule Max_in) show "finite ?L" by (simp add: finite_subtree_threads) next show "?L \<noteq> {}" using subtree_def by fastforce qed then obtain th1' where h_1: "Th th1' \<in> subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L" by auto have "Max ?R \<in> ?R" proof(rule Max_in) show "finite ?R" by (simp add: finite_subtree_threads) next show "?R \<noteq> {}" using subtree_def by fastforce qed then obtain th2' where h_2: "Th th2' \<in> 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' \<in> 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) \<in> (RAG s)^+" by (auto simp:ancestors_def) from tranclD[OF this] have "(Th th1') \<in> Domain (RAG s)" by auto from dm_RAG_threads[OF this] show ?thesis . qed next from h_2(1) show "th2' \<in> 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) \<in> (RAG s)^+" by (auto simp:ancestors_def) from tranclD[OF this] have "(Th th2') \<in> 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) \<in> (RAG s)^*" by (auto simp:subtree_def) from h_2(1)[unfolded this] have star2: "(Th th2', Th th2) \<in> (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' \<noteq> []" from rpath_plus[OF less_1(3) this] have "(Th th1, Th th2) \<in> (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' \<noteq> []" from rpath_plus[OF less_2(3) this] have "(Th th2, Th th1) \<in> (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 qedqedlemma card_runing: "card (runing s) \<le> 1"proof(cases "runing s = {}") case True thus ?thesis by autonext case False then obtain th where [simp]: "th \<in> runing s" by auto from runing_unique[OF this] have "runing s = {th}" by auto thus ?thesis by autoqedendsection {* Relating @{term cp} and @{term the_preced} and @{term preced} *}context valid_tracebeginlemma le_cp: shows "preced th s \<le> cp s th" proof(unfold cp_alt_def, rule Max_ge) show "finite (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})" by (simp add: finite_subtree_threads) next show "preced th s \<in> the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}" by (simp add: subtree_def the_preced_def) qedlemma cp_le: assumes th_in: "th \<in> threads s" shows "cp s th \<le> 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' \<in> subtree (RAG s) (Th th)} \<noteq> {}" using subtree_def by fastforcenext show "{th'. Th th' \<in> subtree (RAG s) (Th th)} \<subseteq> threads s" using assms by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq node.inject(1) rtranclD subsetI subtree_def trancl_domain) qedlemma max_cp_eq: shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)" (is "?L = ?R")proof - have "?L \<le> ?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 \<le> ?L" by (rule Max_fg_mono, simp add: finite_threads, simp add: le_cp the_preced_def) ultimately show ?thesis by autoqedlemma threads_alt_def: "(threads s) = (\<Union> th \<in> readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)})" (is "?L = ?R")proof - { fix th1 assume "th1 \<in> ?L" from th_chain_to_ready[OF this] have "th1 \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th th1, Th th') \<in> (RAG s)\<^sup>+)" . hence "th1 \<in> ?R" by (auto simp:subtree_def) } moreover { fix th' assume "th' \<in> ?R" then obtain th where h: "th \<in> readys s" " Th th' \<in> subtree (RAG s) (Th th)" by auto from this(2) have "th' \<in> ?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' \<in> Domain (RAG s)" by auto from dm_RAG_threads[OF this] show ?thesis . qed } ultimately show ?thesis by autoqedtext {* (* 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 ` (\<Union>th\<in>readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)}))" by (unfold threads_alt_def, simp) also have "... = Max ((\<Union>th\<in>readys s. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}))" by (unfold image_UN, simp) also have "... = Max (Max ` (\<lambda>th. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}) ` readys s)" proof(rule Max_UNION) show "\<forall>M\<in>(\<lambda>x. the_preced s ` {th'. Th th' \<in> 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 \<circ> (\<lambda>th. the_preced s ` {th'. Th th' \<in> 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 \<in> ?A" thus "?f th1 = ?g th1" by (unfold cp_alt_def, simp) qed thus ?thesis by simp qed finally show ?thesis by simpqed (auto simp:threads_alt_def)endsection {* Relating @{term cntP}, @{term cntV}, @{term cntCS} and @{term pvD} *}context valid_trace_p_wbeginlemma 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' \<in> ?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' \<in> ?R" by (auto simp:holdents_def) } moreover { fix cs' assume "cs' \<in> ?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' \<in> ?L" by (auto simp:holdents_def) } ultimately show ?thesis by autoqedlemma 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 \<notin> readys (e#s)" using waiting_es_th_cs by (unfold readys_def, auto)endlemma (in valid_trace) finite_holdents: "finite (holdents s th)" by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)context valid_trace_p beginlemma ready_th_s: "th \<in> readys s" using runing_th_s by (unfold runing_def, auto)lemma live_th_s: "th \<in> threads s" using readys_threads ready_th_s by autolemma live_th_es: "th \<in> threads (e#s)" using live_th_s by (unfold is_p, simp)lemma waiting_neq_th: assumes "waiting s t c" shows "t \<noteq> th" using assms using th_not_waiting by blast endcontext valid_trace_p_hbeginlemma th_not_waiting': "\<not> 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)qedlemma ready_th_es: shows "th \<in> 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) \<union> {cs}" (is "?L = ?R")proof - { fix cs' assume "cs' \<in> ?L" hence "holding (e#s) th cs'" by (unfold holdents_def, auto) hence "cs' \<in> ?R" by (cases rule:holding_esE, auto simp:holdents_def) } moreover { fix cs' assume "cs' \<in> ?R" hence "holding s th cs' \<or> cs' = cs" by (auto simp:holdents_def) hence "cs' \<in> ?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 autoqedlemma cntCS_es_th: "cntCS (e#s) th = cntCS s th + 1"proof - have "card (holdents s th \<union> {cs}) = card (holdents s th) + 1" proof(subst card_Un_disjoint) show "holdents s th \<inter> {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)qedlemma no_holder: "\<not> 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 autoqedlemma holdents_es_th': assumes "th' \<noteq> th" shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")proof - { fix cs' assume "cs' \<in> ?L" hence h_e: "holding (e#s) th' cs'" by (auto simp:holdents_def) have "cs' \<noteq> 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' \<in> set (wq s cs') \<and> th' = hd (wq s cs')" . hence "cs' \<in> ?R" by (unfold holdents_def s_holding_def, fold wq_def, auto) } moreover { fix cs' assume "cs' \<in> ?R" hence "holding s th' cs'" by (auto simp:holdents_def) from holding_kept[OF this] have "holding (e # s) th' cs'" . hence "cs' \<in> ?L" by (unfold holdents_def, auto) } ultimately show ?thesis by autoqedlemma cntCS_es_th'[simp]: assumes "th' \<noteq> th" shows "cntCS (e#s) th' = cntCS s th'" by (unfold cntCS_def holdents_es_th'[OF assms], simp)endcontext valid_trace_pbeginlemma readys_kept1: assumes "th' \<noteq> th" and "th' \<in> readys (e#s)" shows "th' \<in> readys s"proof - { fix cs' assume wait: "waiting s th' cs'" have n_wait: "\<not> 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)qedlemma readys_kept2: assumes "th' \<noteq> th" and "th' \<in> readys s" shows "th' \<in> readys (e#s)"proof - { fix cs' assume wait: "waiting (e#s) th' cs'" have n_wait: "\<not> 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)qedlemma readys_simp [simp]: assumes "th' \<noteq> th" shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)" using readys_kept1[OF assms] readys_kept2[OF assms] by metislemma 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) qednext 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) qedqedendcontext valid_trace_v beginlemma 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 \<in> readys s" using runing_th_s by (unfold runing_def readys_def, auto)lemma th_live_s [simp]: "th \<in> threads s" using th_ready_s by (unfold readys_def, auto)lemma th_ready_es [simp]: "th \<in> 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 \<in> 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 \<in> 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])qedendcontext valid_trace_vbeginlemma th_not_waiting: "\<not> waiting s th c"proof - have "th \<in> readys s" using runing_ready runing_th_s by blast thus ?thesis by (unfold readys_def, auto)qedlemma waiting_neq_th: assumes "waiting s t c" shows "t \<noteq> th" using assms using th_not_waiting by blast endcontext valid_trace_v_nbeginlemma not_ready_taker_s[simp]: "taker \<notin> readys s" using waiting_taker by (unfold readys_def, auto)lemma taker_live_s [simp]: "taker \<in> threads s"proof - have "taker \<in> set wq'" by (simp add: eq_wq') from th'_in_inv[OF this] have "taker \<in> set rest" . hence "taker \<in> set (wq s cs)" by (simp add: wq_s_cs) thus ?thesis using wq_threads by auto qedlemma taker_live_es [simp]: "taker \<in> threads (e#s)" using taker_live_s threads_es by blastlemma taker_ready_es [simp]: shows "taker \<in> 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)qedlemma neq_taker_th: "taker \<noteq> th" using th_not_waiting waiting_taker by blast lemma not_holding_taker_s_cs: shows "\<not> holding s taker cs" using holding_cs_eq_th neq_taker_th by autolemma holdents_es_taker: "holdents (e#s) taker = holdents s taker \<union> {cs}" (is "?L = ?R")proof - { fix cs' assume "cs' \<in> ?L" hence "holding (e#s) taker cs'" by (auto simp:holdents_def) hence "cs' \<in> ?R" proof(cases rule:holding_esE) case 2 thus ?thesis by (auto simp:holdents_def) qed auto } moreover { fix cs' assume "cs' \<in> ?R" hence "holding s taker cs' \<or> cs' = cs" by (auto simp:holdents_def) hence "cs' \<in> ?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 autoqedlemma cntCS_es_taker [simp]: "cntCS (e#s) taker = cntCS s taker + 1"proof - have "card (holdents s taker \<union> {cs}) = card (holdents s taker) + 1" proof(subst card_Un_disjoint) show "holdents s taker \<inter> {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)qedlemma 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' \<in> ?L" hence "holding (e#s) th cs'" by (auto simp:holdents_def) hence "cs' \<in> ?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' \<in> ?R" hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def) from holding_esI2[OF this] have "cs' \<in> ?L" by (auto simp:holdents_def) } ultimately show ?thesis by autoqedlemma 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 \<in> 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)qedlemma holdents_kept: assumes "th' \<noteq> taker" and "th' \<noteq> th" shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")proof - { fix cs' assume h: "cs' \<in> ?L" have "cs' \<in> ?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' \<in> ?R" have "cs' \<in> ?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 autoqedlemma cntCS_kept [simp]: assumes "th' \<noteq> taker" and "th' \<noteq> th" shows "cntCS (e#s) th' = cntCS s th'" by (unfold cntCS_def holdents_kept[OF assms], simp)lemma readys_kept1: assumes "th' \<noteq> taker" and "th' \<in> readys (e#s)" shows "th' \<in> readys s"proof - { fix cs' assume wait: "waiting s th' cs'" have n_wait: "\<not> 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' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)" using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . moreover have "\<not> (th' \<in> set rest \<and> th' \<noteq> 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)qedlemma readys_kept2: assumes "th' \<noteq> taker" and "th' \<in> readys s" shows "th' \<in> readys (e#s)"proof - { fix cs' assume wait: "waiting (e#s) th' cs'" have n_wait: "\<not> 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' \<in> set rest \<and> th' \<noteq> hd (taker # rest')" using wait [unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs set_wq', unfolded eq_wq'] . moreover have "\<not> (th' \<in> set (th # rest) \<and> th' \<noteq> 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)qedlemma readys_simp [simp]: assumes "th' \<noteq> taker" shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)" using readys_kept1[OF assms] readys_kept2[OF assms] by metislemma 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' \<noteq> taker" "th' \<noteq> 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 metisqedendcontext valid_trace_v_ebeginlemma holdents_es_th: "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")proof - { fix cs' assume "cs' \<in> ?L" hence "holding (e#s) th cs'" by (auto simp:holdents_def) hence "cs' \<in> ?R" proof(cases rule:holding_esE) case 1 thus ?thesis by (auto simp:holdents_def) qed } moreover { fix cs' assume "cs' \<in> ?R" hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def) from holding_esI2[OF this] have "cs' \<in> ?L" by (auto simp:holdents_def) } ultimately show ?thesis by autoqedlemma 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 \<in> 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)qedlemma holdents_kept: assumes "th' \<noteq> th" shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")proof - { fix cs' assume h: "cs' \<in> ?L" have "cs' \<in> ?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' \<in> ?R" have "cs' \<in> ?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 autoqedlemma cntCS_kept [simp]: assumes "th' \<noteq> th" shows "cntCS (e#s) th' = cntCS s th'" by (unfold cntCS_def holdents_kept[OF assms], simp)lemma readys_kept1: assumes "th' \<in> readys (e#s)" shows "th' \<in> readys s"proof - { fix cs' assume wait: "waiting s th' cs'" have n_wait: "\<not> 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' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)" using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . hence "th' \<in> set rest" by auto with set_wq' have "th' \<in> set wq'" by metis with nil_wq' show ?thesis by simp qed } thus ?thesis using assms by (unfold readys_def, auto)qedlemma readys_kept2: assumes "th' \<in> readys s" shows "th' \<in> readys (e#s)"proof - { fix cs' assume wait: "waiting (e#s) th' cs'" have n_wait: "\<not> 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' \<in> set [] \<and> th' \<noteq> 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)qedlemma readys_simp [simp]: shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)" using readys_kept1[OF assms] readys_kept2[OF assms] by metislemma 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' \<noteq> 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 metisqedendcontext valid_trace_vbeginlemma 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 qedendcontext valid_trace_createbeginlemma th_not_live_s [simp]: "th \<notin> threads s"proof - from pip_e[unfolded is_create] show ?thesis by (cases, simp)qedlemma th_not_ready_s [simp]: "th \<notin> readys s" using th_not_live_s by (unfold readys_def, simp)lemma th_live_es [simp]: "th \<in> threads (e#s)" by (unfold is_create, simp)lemma not_waiting_th_s [simp]: "\<not> waiting s th cs'"proof assume "waiting s th cs'" from this[unfolded s_waiting_def, folded wq_def, unfolded wq_kept] have "th \<in> set (wq s cs')" by auto from wq_threads[OF this] have "th \<in> threads s" . with th_not_live_s show False by simpqedlemma not_holding_th_s [simp]: "\<not> holding s th cs'"proof assume "holding s th cs'" from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept] have "th \<in> set (wq s cs')" by auto from wq_threads[OF this] have "th \<in> threads s" . with th_not_live_s show False by simpqedlemma not_waiting_th_es [simp]: "\<not> 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 \<in> set (wq s cs')" by auto from wq_threads[OF this] have "th \<in> threads s" . with th_not_live_s show False by simpqedlemma not_holding_th_es [simp]: "\<not> 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 \<in> set (wq s cs')" by auto from wq_threads[OF this] have "th \<in> threads s" . with th_not_live_s show False by simpqedlemma ready_th_es [simp]: "th \<in> 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' \<noteq> th" shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")proof - { fix cs' assume h: "cs' \<in> ?L" hence "cs' \<in> ?R" by (unfold holdents_def s_holding_def, fold wq_def, unfold wq_kept, auto) } moreover { fix cs' assume h: "cs' \<in> ?R" hence "cs' \<in> ?L" by (unfold holdents_def s_holding_def, fold wq_def, unfold wq_kept, auto) } ultimately show ?thesis by autoqedlemma cntCS_kept [simp]: assumes "th' \<noteq> 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' \<noteq> th" and "th' \<in> readys (e#s)" shows "th' \<in> readys s"proof - { fix cs' assume wait: "waiting s th' cs'" have n_wait: "\<not> 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)qedlemma readys_kept2: assumes "th' \<noteq> th" and "th' \<in> readys s" shows "th' \<in> readys (e#s)"proof - { fix cs' assume wait: "waiting (e#s) th' cs'" have n_wait: "\<not> 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)qedlemma readys_simp [simp]: assumes "th' \<noteq> th" shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)" using readys_kept1[OF assms] readys_kept2[OF assms] by metislemma pvD_kept [simp]: assumes "th' \<noteq> 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' \<noteq> th" hence ?thesis using assms by (simp, simp add:is_create) } ultimately show ?thesis by metisqedendcontext valid_trace_exitbeginlemma th_live_s [simp]: "th \<in> threads s"proof - from pip_e[unfolded is_exit] show ?thesis by (cases, unfold runing_def readys_def, simp)qedlemma th_ready_s [simp]: "th \<in> readys s"proof - from pip_e[unfolded is_exit] show ?thesis by (cases, unfold runing_def, simp)qedlemma th_not_live_es [simp]: "th \<notin> threads (e#s)" by (unfold is_exit, simp)lemma not_holding_th_s [simp]: "\<not> holding s th cs'"proof - from pip_e[unfolded is_exit] show ?thesis by (cases, unfold holdents_def, auto)qedlemma cntCS_th_s [simp]: "cntCS s th = 0"proof - from pip_e[unfolded is_exit] show ?thesis by (cases, unfold cntCS_def, simp)qedlemma not_holding_th_es [simp]: "\<not> 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 simpqedlemma ready_th_es [simp]: "th \<notin> 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' \<noteq> th" shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")proof - { fix cs' assume h: "cs' \<in> ?L" hence "cs' \<in> ?R" by (unfold holdents_def s_holding_def, fold wq_def, unfold wq_kept, auto) } moreover { fix cs' assume h: "cs' \<in> ?R" hence "cs' \<in> ?L" by (unfold holdents_def s_holding_def, fold wq_def, unfold wq_kept, auto) } ultimately show ?thesis by autoqedlemma cntCS_kept [simp]: assumes "th' \<noteq> 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' \<noteq> th" and "th' \<in> readys (e#s)" shows "th' \<in> readys s"proof - { fix cs' assume wait: "waiting s th' cs'" have n_wait: "\<not> 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)qedlemma readys_kept2: assumes "th' \<noteq> th" and "th' \<in> readys s" shows "th' \<in> readys (e#s)"proof - { fix cs' assume wait: "waiting (e#s) th' cs'" have n_wait: "\<not> 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)qedlemma readys_simp [simp]: assumes "th' \<noteq> th" shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)" using readys_kept1[OF assms] readys_kept2[OF assms] by metislemma pvD_kept [simp]: assumes "th' \<noteq> 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' \<noteq> th" hence ?thesis using assms by (simp, simp add:is_exit) } ultimately show ?thesis by metisqedendcontext valid_trace_setbeginlemma th_live_s [simp]: "th \<in> threads s"proof - from pip_e[unfolded is_set] show ?thesis by (cases, unfold runing_def readys_def, simp)qedlemma th_ready_s [simp]: "th \<in> readys s"proof - from pip_e[unfolded is_set] show ?thesis by (cases, unfold runing_def, simp)qedlemma th_not_live_es [simp]: "th \<in> 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' \<in> ?L" hence "cs' \<in> ?R" by (unfold holdents_def s_holding_def, fold wq_def, unfold wq_kept, auto) } moreover { fix cs' assume h: "cs' \<in> ?R" hence "cs' \<in> ?L" by (unfold holdents_def s_holding_def, fold wq_def, unfold wq_kept, auto) } ultimately show ?thesis by autoqedlemma 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' \<in> readys (e#s)" shows "th' \<in> readys s"proof - { fix cs' assume wait: "waiting s th' cs'" have n_wait: "\<not> 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' \<in> threads s" using assms[unfolded readys_def] by auto ultimately show ?thesis by (unfold readys_def, auto)qedlemma readys_kept2: assumes "th' \<in> readys s" shows "th' \<in> readys (e#s)"proof - { fix cs' assume wait: "waiting (e#s) th' cs'" have n_wait: "\<not> 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)qedlemma readys_simp [simp]: shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)" using readys_kept1 readys_kept2 by metislemma 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)endcontext valid_tracebeginlemma 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) qedqedendsection {* Corollaries of @{thm valid_trace.cnp_cnv_cncs} *}context valid_tracebeginlemma not_thread_holdents: assumes not_in: "th \<notin> threads s" shows "holdents s th = {}"proof - { fix cs assume "cs \<in> holdents s th" hence "holding s th cs" by (auto simp:holdents_def) from this[unfolded s_holding_def, folded wq_def] have "th \<in> set (wq s cs)" by auto with wq_threads have "th \<in> threads s" by auto with assms have False by simp } thus ?thesis by autoqedlemma not_thread_cncs: assumes not_in: "th \<notin> threads s" shows "cntCS s th = 0" using not_thread_holdents[OF assms] by (simp add:cntCS_def)lemma cnp_cnv_eq: assumes "th \<notin> 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 autoqedlemma 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) \<in> (RAG s)^+} = {}"proof(rule ccontr) assume otherwise: "{th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} \<noteq> {}" then obtain th' where "(Th th', Th th) \<in> (RAG s)^+" by auto from tranclD2[OF this] obtain z where "z \<in> children (RAG s) (Th th)" by (auto simp:children_def) with eq_pv_children[OF assms] show False by simpqedlemma 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 .qedlemma count_eq_tRAG_plus: assumes "cntP s th = cntV s th" shows "{th'. (Th th', Th th) \<in> (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) \<in> (RAG s)^+} = {}" using count_eq_RAG_plus[OF assms] by autolemma count_eq_tRAG_plus_Th: assumes "cntP s th = cntV s th" shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}" using count_eq_tRAG_plus[OF assms] by autoenddefinition detached :: "state \<Rightarrow> thread \<Rightarrow> bool" where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"lemma detached_test: shows "detached s th = (Th th \<notin> 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)donecontext valid_tracebeginlemma detached_intro: assumes eq_pv: "cntP s th = cntV s th" shows "detached s th"proof - from eq_pv cnp_cnv_cncs have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:pvD_def) thus ?thesis proof assume "th \<notin> 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 \<in> readys s" moreover have "Th th \<notin> 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) qedqedlemma 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 \<in> threads s") case True with dtc have "th \<in> 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) qedqedlemma detached_eq: shows "(detached s th) = (cntP s th = cntV s th)" by (insert vt, auto intro:detached_intro detached_elim)endsection {* 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 \<circ> the_thread) ` subtree (tRAG s) (Th th))" by auto thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)qedlemma 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 "\<forall> x \<in> A. \<exists> th. x = Th th" shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"proof(rule f_image_eq) fix a assume "a \<in> A" from assms[rule_format, OF this] obtain th where eq_a: "a = Th th" by auto show "cp_gen s a = (cp s \<circ> the_thread) a" by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)qedcontext valid_tracebegin(* ddd *)lemma cp_gen_rec: assumes "x = Th th" shows "cp_gen s x = Max ({the_preced s th} \<union> (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 \<noteq> {}" 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 \<noteq> {}" by (auto simp:subtree_def) } note this[simp] have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}" proof - from False obtain q where "q \<in> children (tRAG s) x" by blast moreover have "subtree (tRAG s) q \<noteq> {}" by simp ultimately show ?thesis by blast qed have h: "Max ((the_preced s \<circ> the_thread) ` ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) = Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)" (is "?L = ?R") proof - let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L let "Max (_ \<union> (?h ` ?B))" = ?R let ?L1 = "?f ` \<Union>(?g ` ?B)" have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)" proof - have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp also have "... = (\<Union> x \<in> ?B. ?f ` (?g x))" by auto finally have "Max ?L1 = Max ..." by simp also have "... = Max (Max ` (\<lambda>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 \<union> ?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) qedlemma cp_rec: "cp s th = Max ({the_preced s th} \<union> (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 \<circ> the_thread) ` children (tRAG s) (Th th)" proof(rule cp_gen_over_set) show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>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) qedqedendsection {* Other properties useful in Implementation.thy or Correctness.thy *}context valid_trace_e beginlemma actor_inv: assumes "\<not> isCreate e" shows "actor e \<in> runing s" using pip_e assms by (induct, auto)endcontext valid_tracebeginlemma readys_root: assumes "th \<in> readys s" shows "root (RAG s) (Th th)"proof - { fix x assume "x \<in> ancestors (RAG s) (Th th)" hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def) from tranclD[OF this] obtain z where "(Th th, z) \<in> 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)qedlemma readys_in_no_subtree: assumes "th \<in> readys s" and "th' \<noteq> th" shows "Th th \<notin> subtree (RAG s) (Th th')" proof assume "Th th \<in> 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) qedqedlemma not_in_thread_isolated: assumes "th \<notin> threads s" shows "(Th th) \<notin> Field (RAG s)"proof assume "(Th th) \<in> Field (RAG s)" with dm_RAG_threads and rg_RAG_threads assms show False by (unfold Field_def, blast)qedlemma 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 \<noteq> []" "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto thus ?thesis by (unfold cs_holding_def, auto)qedlemma 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 \<noteq> []" "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto from wq_distinct[of cs, unfolded h] have dst: "distinct (th # rest)" . have in_rest: "th' \<in> set rest" proof(unfold h, rule someI2) show "distinct rest \<and> set rest = set rest" using dst by auto next fix x assume "distinct x \<and> set x = set rest" with h(2) show "hd x \<in> set (rest)" by (cases x, auto) qed hence "th' \<in> set (wq s cs)" by (unfold h(1), auto) moreover have "th' \<noteq> hd (wq s cs)" by (unfold h(1), insert in_rest dst, auto) ultimately show ?thesis by (auto simp:cs_waiting_def)qedlemma next_th_RAG: assumes nxt: "next_th (s::event list) th cs th'" shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s" using vt assms next_th_holding next_th_waiting by (unfold s_RAG_def, simp)end context valid_trace_pbeginfind_theorems readys thendend