theory Correctnessimports PIPBasicsbegintext {* The following two auxiliary lemmas are used to reason about @{term Max}.*}lemma image_Max_eqI: assumes "finite B" and "b \<in> B" and "\<forall> x \<in> B. f x \<le> f b" shows "Max (f ` B) = f b" using assms using Max_eqI by blast lemma image_Max_subset: assumes "finite A" and "B \<subseteq> A" and "a \<in> B" and "Max (f ` A) = f a" shows "Max (f ` B) = f a"proof(rule image_Max_eqI) show "finite B" using assms(1) assms(2) finite_subset by auto next show "a \<in> B" using assms by simpnext show "\<forall>x\<in>B. f x \<le> f a" by (metis Max_ge assms(1) assms(2) assms(4) finite_imageI image_eqI subsetCE) qedtext {* The following locale @{text "highest_gen"} sets the basic context for our investigation: supposing thread @{text th} holds the highest @{term cp}-value in state @{text s}, which means the task for @{text th} is the most urgent. We want to show that @{text th} is treated correctly by PIP, which means @{text th} will not be blocked unreasonably by other less urgent threads. *}locale highest_gen = fixes s th prio tm assumes vt_s: "vt s" and threads_s: "th \<in> threads s" and highest: "preced th s = Max ((cp s)`threads s)" -- {* The internal structure of @{term th}'s precedence is exposed:*} and preced_th: "preced th s = Prc prio tm" -- {* @{term s} is a valid trace, so it will inherit all results derived for a valid trace: *}sublocale highest_gen < vat_s?: valid_trace "s" by (unfold_locales, insert vt_s, simp)context highest_genbegintext {* @{term tm} is the time when the precedence of @{term th} is set, so @{term tm} must be a valid moment index into @{term s}.*}lemma lt_tm: "tm < length s" by (insert preced_tm_lt[OF threads_s preced_th], simp)text {* Since @{term th} holds the highest precedence and @{text "cp"} is the highest precedence of all threads in the sub-tree of @{text "th"} and @{text th} is among these threads, its @{term cp} must equal to its precedence:*}lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")proof - have "?L \<le> ?R" by (unfold highest, rule Max_ge, auto simp:threads_s finite_threads) moreover have "?R \<le> ?L" by (unfold vat_s.cp_rec, rule Max_ge, auto simp:the_preced_def vat_s.fsbttRAGs.finite_children) ultimately show ?thesis by autoqedlemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)" using eq_cp_s_th highest max_cp_eq the_preced_def by presburgerlemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)" by (fold eq_cp_s_th, unfold highest_cp_preced, simp)lemma highest': "cp s th = Max (cp s ` threads s)" by (simp add: eq_cp_s_th highest)endlocale extend_highest_gen = highest_gen + fixes t assumes vt_t: "vt (t@s)" and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio" and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio" and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"sublocale extend_highest_gen < vat_t?: valid_trace "t@s" by (unfold_locales, insert vt_t, simp)lemma step_back_vt_app: assumes vt_ts: "vt (t@s)" shows "vt s"proof - from vt_ts show ?thesis proof(induct t) case Nil from Nil show ?case by auto next case (Cons e t) assume ih: " vt (t @ s) \<Longrightarrow> vt s" and vt_et: "vt ((e # t) @ s)" show ?case proof(rule ih) show "vt (t @ s)" proof(rule step_back_vt) from vt_et show "vt (e # t @ s)" by simp qed qed qedqed(* locale red_extend_highest_gen = extend_highest_gen + fixes i::nat*)(*sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)" apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric]) apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp) by (unfold highest_gen_def, auto dest:step_back_vt_app)*)context extend_highest_genbegin lemma ind [consumes 0, case_names Nil Cons, induct type]: assumes h0: "R []" and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e; extend_highest_gen s th prio tm t; extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)" shows "R t"proof - from vt_t extend_highest_gen_axioms show ?thesis proof(induct t) from h0 show "R []" . next case (Cons e t') assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'" and vt_e: "vt ((e # t') @ s)" and et: "extend_highest_gen s th prio tm (e # t')" from vt_e and step_back_step have stp: "step (t'@s) e" by auto from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto show ?case proof(rule h2 [OF vt_ts stp _ _ _ ]) show "R t'" proof(rule ih) from et show ext': "extend_highest_gen s th prio tm t'" by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) next from vt_ts show "vt (t' @ s)" . qed next from et show "extend_highest_gen s th prio tm (e # t')" . next from et show ext': "extend_highest_gen s th prio tm t'" by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) qed qedqedlemma th_kept: "th \<in> threads (t @ s) \<and> preced th (t@s) = preced th s" (is "?Q t") proof - show ?thesis proof(induct rule:ind) case Nil from threads_s show ?case by auto next case (Cons e t) interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto show ?case proof(cases e) case (Create thread prio) show ?thesis proof - from Cons and Create have "step (t@s) (Create thread prio)" by auto hence "th \<noteq> thread" proof(cases) case thread_create with Cons show ?thesis by auto qed hence "preced th ((e # t) @ s) = preced th (t @ s)" by (unfold Create, auto simp:preced_def) moreover note Cons ultimately show ?thesis by (auto simp:Create) qed next case (Exit thread) from h_e.exit_diff and Exit have neq_th: "thread \<noteq> th" by auto with Cons show ?thesis by (unfold Exit, auto simp:preced_def) next case (P thread cs) with Cons show ?thesis by (auto simp:P preced_def) next case (V thread cs) with Cons show ?thesis by (auto simp:V preced_def) next case (Set thread prio') show ?thesis proof - from h_e.set_diff_low and Set have "th \<noteq> thread" by auto hence "preced th ((e # t) @ s) = preced th (t @ s)" by (unfold Set, auto simp:preced_def) moreover note Cons ultimately show ?thesis by (auto simp:Set) qed qed qedqedtext {* According to @{thm th_kept}, thread @{text "th"} has its living status and precedence kept along the way of @{text "t"}. The following lemma shows that this preserved precedence of @{text "th"} remains as the highest along the way of @{text "t"}. The proof goes by induction over @{text "t"} using the specialized induction rule @{thm ind}, followed by case analysis of each possible operations of PIP. All cases follow the same pattern rendered by the generalized introduction rule @{thm "image_Max_eqI"}. The very essence is to show that precedences, no matter whether they are newly introduced or modified, are always lower than the one held by @{term "th"}, which by @{thm th_kept} is preserved along the way.*}lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"proof(induct rule:ind) case Nil from highest_preced_thread show ?case by simpnext case (Cons e t) interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto show ?case proof(cases e) case (Create thread prio') show ?thesis (is "Max (?f ` ?A) = ?t") proof - -- {* The following is the common pattern of each branch of the case analysis. *} -- {* The major part is to show that @{text "th"} holds the highest precedence: *} have "Max (?f ` ?A) = ?f th" proof(rule image_Max_eqI) show "finite ?A" using h_e.finite_threads by auto next show "th \<in> ?A" using h_e.th_kept by auto next show "\<forall>x\<in>?A. ?f x \<le> ?f th" proof fix x assume "x \<in> ?A" hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create) thus "?f x \<le> ?f th" proof assume "x = thread" thus ?thesis apply (simp add:Create the_preced_def preced_def, fold preced_def) using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force next assume h: "x \<in> threads (t @ s)" from Cons(2)[unfolded Create] have "x \<noteq> thread" using h by (cases, auto) hence "?f x = the_preced (t@s) x" by (simp add:Create the_preced_def preced_def) hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))" by (simp add: h_t.finite_threads h) also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) finally show ?thesis . qed qed qed -- {* The minor part is to show that the precedence of @{text "th"} equals to preserved one, given by the foregoing lemma @{thm th_kept} *} also have "... = ?t" using h_e.th_kept the_preced_def by auto -- {* Then it follows trivially that the precedence preserved for @{term "th"} remains the maximum of all living threads along the way. *} finally show ?thesis . qed next case (Exit thread) show ?thesis (is "Max (?f ` ?A) = ?t") proof - have "Max (?f ` ?A) = ?f th" proof(rule image_Max_eqI) show "finite ?A" using h_e.finite_threads by auto next show "th \<in> ?A" using h_e.th_kept by auto next show "\<forall>x\<in>?A. ?f x \<le> ?f th" proof fix x assume "x \<in> ?A" hence "x \<in> threads (t@s)" by (simp add: Exit) hence "?f x \<le> Max (?f ` threads (t@s))" by (simp add: h_t.finite_threads) also have "... \<le> ?f th" apply (simp add:Exit the_preced_def preced_def, fold preced_def) using Cons.hyps(5) h_t.th_kept the_preced_def by auto finally show "?f x \<le> ?f th" . qed qed also have "... = ?t" using h_e.th_kept the_preced_def by auto finally show ?thesis . qed next case (P thread cs) with Cons show ?thesis by (auto simp:preced_def the_preced_def) next case (V thread cs) with Cons show ?thesis by (auto simp:preced_def the_preced_def) next case (Set thread prio') show ?thesis (is "Max (?f ` ?A) = ?t") proof - have "Max (?f ` ?A) = ?f th" proof(rule image_Max_eqI) show "finite ?A" using h_e.finite_threads by auto next show "th \<in> ?A" using h_e.th_kept by auto next show "\<forall>x\<in>?A. ?f x \<le> ?f th" proof fix x assume h: "x \<in> ?A" show "?f x \<le> ?f th" proof(cases "x = thread") case True moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th" proof - have "the_preced (t @ s) th = Prc prio tm" using h_t.th_kept preced_th by (simp add:the_preced_def) moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def) qed ultimately show ?thesis by (unfold Set, simp add:the_preced_def preced_def) next case False then have "?f x = the_preced (t@s) x" by (simp add:the_preced_def preced_def Set) also have "... \<le> Max (the_preced (t@s) ` threads (t@s))" using Set h h_t.finite_threads by auto also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) finally show ?thesis . qed qed qed also have "... = ?t" using h_e.th_kept the_preced_def by auto finally show ?thesis . qed qedqedlemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))" by (insert th_kept max_kept, auto)text {* The reason behind the following lemma is that: Since @{term "cp"} is defined as the maximum precedence of those threads contained in the sub-tree of node @{term "Th th"} in @{term "RAG (t@s)"}, and all these threads are living threads, and @{term "th"} is also among them, the maximum precedence of them all must be the one for @{text "th"}.*}lemma th_cp_max_preced: "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") proof - let ?f = "the_preced (t@s)" have "?L = ?f th" proof(unfold cp_alt_def, rule image_Max_eqI) show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}" proof - have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} = the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and> (\<exists> th'. n = Th th')}" by (force) moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) ultimately show ?thesis by simp qed next show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}" by (auto simp:subtree_def) next show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}. the_preced (t @ s) x \<le> the_preced (t @ s) th" proof fix th' assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}" hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}" by (meson subtree_Field) ultimately have "Th th' \<in> ..." by auto hence "th' \<in> threads (t@s)" proof assume "Th th' \<in> {Th th}" thus ?thesis using th_kept by auto next assume "Th th' \<in> Field (RAG (t @ s))" thus ?thesis using vat_t.not_in_thread_isolated by blast qed thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th" by (metis Max_ge finite_imageI finite_threads image_eqI max_kept th_kept the_preced_def) qed qed also have "... = ?R" by (simp add: max_preced the_preced_def) finally show ?thesis .qedlemma th_cp_max[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th" using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburgerlemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))" by (simp add: th_cp_max_preced)lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th" using max_kept th_kept the_preced_def by autolemma [simp]: "the_preced (t@s) th = preced th (t@s)" using the_preced_def by autolemma [simp]: "preced th (t@s) = preced th s" by (simp add: th_kept)lemma [simp]: "cp s th = preced th s" by (simp add: eq_cp_s_th)lemma th_cp_preced [simp]: "cp (t@s) th = preced th s" by (fold max_kept, unfold th_cp_max_preced, simp)lemma preced_less: assumes th'_in: "th' \<in> threads s" and neq_th': "th' \<noteq> th" shows "preced th' s < preced th s" using assmsby (metis Max.coboundedI finite_imageI highest not_le order.trans preced_linorder rev_image_eqI threads_s vat_s.finite_threads vat_s.le_cp)section {* The `blocking thread` *}text {* The purpose of PIP is to ensure that the most urgent thread @{term th} is not blocked unreasonably. Therefore, a clear picture of the blocking thread is essential to assure people that the purpose is fulfilled. In this section, we are going to derive a series of lemmas with finally give rise to a picture of the blocking thread. By `blocking thread`, we mean a thread in running state but different from thread @{term th}.*}text {* The following lemmas shows that the @{term cp}-value of the blocking thread @{text th'} equals to the highest precedence in the whole system.*}lemma runing_preced_inversion: assumes runing': "th' \<in> runing (t@s)" shows "cp (t@s) th' = preced th s" (is "?L = ?R")proof - have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms by (unfold runing_def, auto) also have "\<dots> = ?R" by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) finally show ?thesis .qedtext {* The following lemma shows how the counters for @{term "P"} and @{term "V"} operations relate to the running threads in the states @{term s} and @{term "t @ s"}. The lemma shows that if a thread's @{term "P"}-count equals its @{term "V"}-count (which means it no longer has any resource in its possession), it cannot be a running thread. The proof is by contraction with the assumption @{text "th' \<noteq> th"}. The key is the use of @{thm eq_pv_dependants} to derive the emptiness of @{text th'}s @{term dependants}-set from the balance of its @{term P} and @{term V} counts. From this, it can be shown @{text th'}s @{term cp}-value equals to its own precedence. On the other hand, since @{text th'} is running, by @{thm runing_preced_inversion}, its @{term cp}-value equals to the precedence of @{term th}. Combining the above two resukts we have that @{text th'} and @{term th} have the same precedence. By uniqueness of precedences, we have @{text "th' = th"}, which is in contradiction with the assumption @{text "th' \<noteq> th"}.*} lemma eq_pv_blocked: (* ddd *) assumes neq_th': "th' \<noteq> th" and eq_pv: "cntP (t@s) th' = cntV (t@s) th'" shows "th' \<notin> runing (t@s)"proof assume otherwise: "th' \<in> runing (t@s)" show False proof - have th'_in: "th' \<in> threads (t@s)" using otherwise readys_threads runing_def by auto have "th' = th" proof(rule preced_unique) -- {* The proof goes like this: it is first shown that the @{term preced}-value of @{term th'} equals to that of @{term th}, then by uniqueness of @{term preced}-values (given by lemma @{thm preced_unique}), @{term th'} equals to @{term th}: *} show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R") proof - -- {* Since the counts of @{term th'} are balanced, the subtree of it contains only itself, so, its @{term cp}-value equals its @{term preced}-value: *} have "?L = cp (t@s) th'" by (unfold cp_eq cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp) -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion}, its @{term cp}-value equals @{term "preced th s"}, which equals to @{term "?R"} by simplification: *} also have "... = ?R" thm runing_preced_inversion using runing_preced_inversion[OF otherwise] by simp finally show ?thesis . qed qed (auto simp: th'_in th_kept) with `th' \<noteq> th` show ?thesis by simp qedqedtext {* The following lemma is the extrapolation of @{thm eq_pv_blocked}. It says if a thread, different from @{term th}, does not hold any resource at the very beginning, it will keep hand-emptied in the future @{term "t@s"}.*}lemma eq_pv_persist: (* ddd *) assumes neq_th': "th' \<noteq> th" and eq_pv: "cntP s th' = cntV s th'" shows "cntP (t@s) th' = cntV (t@s) th'"proof(induction rule:ind) -- {* The proof goes by induction. *} -- {* The nontrivial case is for the @{term Cons}: *} case (Cons e t) -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *} interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp interpret vat_es: valid_trace_e "t@s" e using Cons(1,2) by (unfold_locales, auto) show ?case proof - -- {* It can be proved that @{term cntP}-value of @{term th'} does not change by the happening of event @{term e}: *} have "cntP ((e#t)@s) th' = cntP (t@s) th'" proof(rule ccontr) -- {* Proof by contradiction. *} -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *} assume otherwise: "cntP ((e # t) @ s) th' \<noteq> cntP (t @ s) th'" from cntP_diff_inv[OF this[simplified]] obtain cs' where "e = P th' cs'" by auto from vat_es.pip_e[unfolded this] have "th' \<in> runing (t@s)" by (cases, simp) -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis shows @{term th'} can not be running at moment @{term "t@s"}: *} moreover have "th' \<notin> runing (t@s)" using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] . -- {* Contradiction is finally derived: *} ultimately show False by simp qed -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change by the happening of event @{term e}: *} -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *} moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'" proof(rule ccontr) -- {* Proof by contradiction. *} assume otherwise: "cntV ((e # t) @ s) th' \<noteq> cntV (t @ s) th'" from cntV_diff_inv[OF this[simplified]] obtain cs' where "e = V th' cs'" by auto from vat_es.pip_e[unfolded this] have "th' \<in> runing (t@s)" by (cases, auto) moreover have "th' \<notin> runing (t@s)" using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] . ultimately show False by simp qed -- {* Finally, it can be shown that the @{term cntP} and @{term cntV} value for @{term th'} are still in balance, so @{term th'} is still hand-emptied after the execution of event @{term e}: *} ultimately show ?thesis using Cons(5) by metis qedqed (auto simp:eq_pv)text {* By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist}, it can be derived easily that @{term th'} can not be running in the future:*}lemma eq_pv_blocked_persist: assumes neq_th': "th' \<noteq> th" and eq_pv: "cntP s th' = cntV s th'" shows "th' \<notin> runing (t@s)" using assms by (simp add: eq_pv_blocked eq_pv_persist) text {* The following lemma shows the blocking thread @{term th'} must hold some resource in the very beginning. *}lemma runing_cntP_cntV_inv: (* ddd *) assumes is_runing: "th' \<in> runing (t@s)" and neq_th': "th' \<noteq> th" shows "cntP s th' > cntV s th'" using assmsproof - -- {* First, it can be shown that the number of @{term P} and @{term V} operations can not be equal for thred @{term th'} *} have "cntP s th' \<noteq> cntV s th'" proof -- {* The proof goes by contradiction, suppose otherwise: *} assume otherwise: "cntP s th' = cntV s th'" -- {* By applying @{thm eq_pv_blocked_persist} to this: *} from eq_pv_blocked_persist[OF neq_th' otherwise] -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *} have "th' \<notin> runing (t@s)" . -- {* This is obvious in contradiction with assumption @{thm is_runing} *} thus False using is_runing by simp qed -- {* However, the number of @{term V} is always less or equal to @{term P}: *} moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto -- {* Thesis is finally derived by combining the these two results: *} ultimately show ?thesis by autoqedtext {* The following lemmas shows the blocking thread @{text th'} must be live at the very beginning, i.e. the moment (or state) @{term s}. The proof is a simple combination of the results above:*}lemma runing_threads_inv: assumes runing': "th' \<in> runing (t@s)" and neq_th': "th' \<noteq> th" shows "th' \<in> threads s"proof(rule ccontr) -- {* Proof by contradiction: *} assume otherwise: "th' \<notin> threads s" have "th' \<notin> runing (t @ s)" proof - from vat_s.cnp_cnv_eq[OF otherwise] have "cntP s th' = cntV s th'" . from eq_pv_blocked_persist[OF neq_th' this] show ?thesis . qed with runing' show False by simpqedtext {* The following lemma summarizes several foregoing lemmas to give an overall picture of the blocking thread @{text "th'"}:*}lemma runing_inversion: (* ddd, one of the main lemmas to present *) assumes runing': "th' \<in> runing (t@s)" and neq_th: "th' \<noteq> th" shows "th' \<in> threads s" and "\<not>detached s th'" and "cp (t@s) th' = preced th s"proof - from runing_threads_inv[OF assms] show "th' \<in> threads s" .next from runing_cntP_cntV_inv[OF runing' neq_th] show "\<not>detached s th'" using vat_s.detached_eq by simpnext from runing_preced_inversion[OF runing'] show "cp (t@s) th' = preced th s" .qedsection {* The existence of `blocking thread` *}text {* Suppose @{term th} is not running, it is first shown that there is a path in RAG leading from node @{term th} to another thread @{text "th'"} in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}). Now, since @{term readys}-set is non-empty, there must be one in it which holds the highest @{term cp}-value, which, by definition, is the @{term runing}-thread. However, we are going to show more: this running thread is exactly @{term "th'"}. *}lemma th_blockedE: (* ddd, the other main lemma to be presented: *) assumes "th \<notin> runing (t@s)" obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" "th' \<in> runing (t@s)"proof - -- {* According to @{thm vat_t.th_chain_to_ready}, either @{term "th"} is in @{term "readys"} or there is path leading from it to one thread in @{term "readys"}. *} have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)" using th_kept vat_t.th_chain_to_ready by auto -- {* However, @{term th} can not be in @{term readys}, because otherwise, since @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *} moreover have "th \<notin> readys (t@s)" using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in term @{term readys}: *} ultimately obtain th' where th'_in: "th' \<in> readys (t@s)" and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto -- {* We are going to show that this @{term th'} is running. *} have "th' \<in> runing (t@s)" proof - -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *} have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R") proof - have "?L = Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))" by (unfold cp_alt_def1, simp) also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)" proof(rule image_Max_subset) show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads) next show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)" by (metis Range.intros dp trancl_range vat_t.rg_RAG_threads vat_t.subtree_tRAG_thread) next show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp by (unfold tRAG_subtree_eq, auto simp:subtree_def) next show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) = (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _") proof - have "?L = the_preced (t @ s) ` threads (t @ s)" by (unfold image_comp, rule image_cong, auto) thus ?thesis using max_preced the_preced_def by auto qed qed also have "... = ?R" using th_cp_max th_cp_preced th_kept the_preced_def vat_t.max_cp_readys_threads by auto finally show ?thesis . qed -- {* Now, since @{term th'} holds the highest @{term cp} and we have already show it is in @{term readys}, it is @{term runing} by definition. *} with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def) qed -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *} moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def) ultimately show ?thesis using that by metisqedtext {* Now it is easy to see there is always a thread to run by case analysis on whether thread @{term th} is running: if the answer is Yes, the the running thread is obviously @{term th} itself; otherwise, the running thread is the @{text th'} given by lemma @{thm th_blockedE}.*}lemma live: "runing (t@s) \<noteq> {}"proof(cases "th \<in> runing (t@s)") case True thus ?thesis by autonext case False thus ?thesis using th_blockedE by autoqedendend