# HG changeset patch # User Christian Urban # Date 1454087162 0 # Node ID c7ba70dc49bd2b0e120166db3598d3552fdb8add # Parent 4805c6333fef688ed394f9390af1ee82683a9f7a# Parent 524bd3caa6b6de37e6a12591b05f9b0a911daa42 merged diff -r 524bd3caa6b6 -r c7ba70dc49bd Correctness.thy --- a/Correctness.thy Fri Jan 29 11:01:13 2016 +0800 +++ b/Correctness.thy Fri Jan 29 17:06:02 2016 +0000 @@ -2,6 +2,9 @@ imports PIPBasics begin +lemma Setcompr_eq_image: "{f x | x. x \ A} = f ` A" + by blast + text {* The following two auxiliary lemmas are used to reason about @{term Max}. *} @@ -473,45 +476,40 @@ 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}. + The purpose of PIP is to ensure that the most urgent thread @{term + th} is not blocked unreasonably. Therefore, below, we will derive + properties of the blocking thread. By blocking thread, we mean a + thread in running state t @ s, but is different from thread @{term + th}. + + The first lemmas shows that the @{term cp}-value of the blocking + thread @{text th'} equals to the highest precedence in the whole + system. + *} -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' \ runing (t@s)" - shows "cp (t@s) th' = preced th s" (is "?L = ?R") + assumes runing': "th' \ runing (t @ s)" + shows "cp (t @ s) th' = preced th s" proof - - have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms - by (unfold runing_def, auto) - also have "\ = ?R" - by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) + have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))" + using assms by (unfold runing_def, auto) + also have "\ = preced th s" + by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) finally show ?thesis . qed text {* - 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 next 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"}: 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' \ th"}. - The key is the use of @{thm eq_pv_dependants} to derive the + The key is the use of @{thm count_eq_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. @@ -520,7 +518,7 @@ 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 + Combining the above two results 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' \ th"}. @@ -529,13 +527,13 @@ lemma eq_pv_blocked: (* ddd *) assumes neq_th': "th' \ th" - and eq_pv: "cntP (t@s) th' = cntV (t@s) th'" - shows "th' \ runing (t@s)" + and eq_pv: "cntP (t @ s) th' = cntV (t @ s) th'" + shows "th' \ runing (t @ s)" proof - assume otherwise: "th' \ runing (t@s)" + assume otherwise: "th' \ runing (t @ s)" show False proof - - have th'_in: "th' \ threads (t@s)" + have th'_in: "th' \ threads (t @ s)" using otherwise readys_threads runing_def by auto have "th' = th" proof(rule preced_unique) @@ -549,13 +547,12 @@ -- {* 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 cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp) + have "?L = cp (t @ s) th'" + by (unfold cp_eq_cpreced cpreced_def count_eq_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 @@ -573,8 +570,8 @@ lemma eq_pv_persist: (* ddd *) assumes neq_th': "th' \ 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. *} + shows "cntP (t @ s) th' = cntV (t @ s) th'" +proof(induction rule: ind) -- {* 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"}: *} @@ -623,22 +620,28 @@ qed (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: + + 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' \ th" and eq_pv: "cntP s th' = cntV s th'" - shows "th' \ runing (t@s)" + shows "th' \ 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. + + 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' \ runing (t@s)" + assumes is_runing: "th' \ runing (t @ s)" and neq_th': "th' \ th" shows "cntP s th' > cntV s th'" using assms @@ -664,11 +667,13 @@ text {* - 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 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' \ runing (t@s)" and neq_th': "th' \ th" @@ -686,9 +691,12 @@ qed text {* - The following lemma summarizes several foregoing - lemmas to give an overall picture of the blocking thread @{text "th'"}: + + The following lemma summarises the above lemmas to give an overall + characterisationof the blocking thread @{text "th'"}: + *} + lemma runing_inversion: (* ddd, one of the main lemmas to present *) assumes runing': "th' \ runing (t@s)" and neq_th: "th' \ th" @@ -706,22 +714,27 @@ show "cp (t@s) th' = preced th s" . qed + section {* 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}}). + + 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'"}. - *} + 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 \ runing (t@s)" + assumes "th \ runing (t @ s)" obtains th' where "Th th' \ ancestors (RAG (t @ s)) (Th th)" - "th' \ runing (t@s)" + "th' \ runing (t @ s)" proof - -- {* According to @{thm vat_t.th_chain_to_ready}, either @{term "th"} is in @{term "readys"} or there is path leading from it to @@ -749,7 +762,7 @@ show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads) next show "subtree (tRAG (t @ s)) (Th th') \ Th ` threads (t @ s)" - by (metis Range.intros dp trancl_range vat_t.rg_RAG_threads vat_t.subtree_tRAG_thread) + by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) next show "Th th \ subtree (tRAG (t @ s)) (Th th')" using dp by (unfold tRAG_subtree_eq, auto simp:subtree_def) @@ -779,18 +792,23 @@ qed text {* - Now it is easy to see there is always a thread to run by case analysis - on whether thread @{term th} is running: if the answer is Yes, the - the running thread is obviously @{term th} itself; otherwise, the running - thread is the @{text th'} given by lemma @{thm th_blockedE}. + + Now it is easy to see there is always a thread to run by case + analysis on whether thread @{term th} is running: if the answer is + yes, the the running thread is obviously @{term th} itself; + otherwise, the running thread is the @{text th'} given by lemma + @{thm th_blockedE}. + *} -lemma live: "runing (t@s) \ {}" -proof(cases "th \ runing (t@s)") + +lemma live: "runing (t @ s) \ {}" +proof(cases "th \ runing (t @ s)") case True thus ?thesis by auto next case False thus ?thesis using th_blockedE by auto qed + end end diff -r 524bd3caa6b6 -r c7ba70dc49bd Implementation.thy --- a/Implementation.thy Fri Jan 29 11:01:13 2016 +0800 +++ b/Implementation.thy Fri Jan 29 17:06:02 2016 +0000 @@ -1,10 +1,12 @@ +(*<*) +theory Implementation +imports PIPBasics +begin +(*>*) section {* This file contains lemmas used to guide the recalculation of current precedence after every system call (or system operation) *} -theory Implementation -imports PIPBasics -begin text {* (* ddd *) One beauty of our modelling is that we follow the definitional extension tradition of HOL. diff -r 524bd3caa6b6 -r c7ba70dc49bd Journal/Paper.thy --- a/Journal/Paper.thy Fri Jan 29 11:01:13 2016 +0800 +++ b/Journal/Paper.thy Fri Jan 29 17:06:02 2016 +0000 @@ -883,20 +883,26 @@ \begin{theorem}\label{mainthm} Given the assumptions about states @{text "s"} and @{text "s' @ s"}, - the thread @{text th} and the events in @{text "s'"}, - if @{term "th' \ running (s' @ s)"} and @{text "th' \ th"} then - @{text "th' \ threads s"}, @{text "\ detached s th'"} and - @{term "cp (s' @ s) th' = prec th s"}. + the thread @{text th} and the events in @{text "s'"}, then either + \begin{itemize} + \item @{term "th \ running (s' @ s)"} or\medskip + + \item there exists a thread @{term "th'"} with @{term "th' \ th"} + and @{term "th' \ running (s' @ s)"} such that @{text "th' \ threads + s"}, @{text "\ detached s th'"} and @{term "cp (s' @ s) th' = prec + th s"}. + \end{itemize} \end{theorem} \noindent This theorem ensures that the thread @{text th}, which has - the highest precedence in the state @{text s}, can only be blocked - in the state @{text "s' @ s"} by a thread @{text th'} that already - existed in @{text s} and requested or had a lock on at least one - resource---that means the thread was not \emph{detached} in @{text - s}. As we shall see shortly, that means there are only finitely - many threads that can block @{text th} in this way and then they - need to run with the same precedence as @{text th}. + the highest precedence in the state @{text s}, is either running in + state @{term "s' @ s"}, or can only be blocked in the state @{text + "s' @ s"} by a thread @{text th'} that already existed in @{text s} + and requested or had a lock on at least one resource---that means + the thread was not \emph{detached} in @{text s}. As we shall see + shortly, that means there are only finitely many threads that can + block @{text th} in this way and then they need to run with the same + precedence as @{text th}. Like in the argument by Sha et al.~our finite bound does not guarantee absence of indefinite Priority Inversion. For this we diff -r 524bd3caa6b6 -r c7ba70dc49bd PIPBasics.thy --- a/PIPBasics.thy Fri Jan 29 11:01:13 2016 +0800 +++ b/PIPBasics.thy Fri Jan 29 17:06:02 2016 +0000 @@ -1,160 +1,7 @@ theory PIPBasics -imports PIPDefs +imports PIPDefs begin -lemma f_image_eq: - assumes h: "\ a. a \ A \ f a = g a" - shows "f ` A = g ` A" -proof - show "f ` A \ g ` A" - by(rule image_subsetI, auto intro:h) -next - show "g ` A \ f ` A" - by (rule image_subsetI, auto intro:h[symmetric]) -qed - -lemma Max_fg_mono: - assumes "finite A" - and "\ a \ A. f a \ g a" - shows "Max (f ` A) \ Max (g ` A)" -proof(cases "A = {}") - case True - thus ?thesis by auto -next - case False - show ?thesis - proof(rule Max.boundedI) - from assms show "finite (f ` A)" by auto - next - from False show "f ` A \ {}" by auto - next - fix fa - assume "fa \ f ` A" - then obtain a where h_fa: "a \ A" "fa = f a" by auto - show "fa \ Max (g ` A)" - proof(rule Max_ge_iff[THEN iffD2]) - from assms show "finite (g ` A)" by auto - next - from False show "g ` A \ {}" by auto - next - from h_fa have "g a \ g ` A" by auto - moreover have "fa \ g a" using h_fa assms(2) by auto - ultimately show "\a\g ` A. fa \ a" by auto - qed - qed -qed - -lemma Max_f_mono: - assumes seq: "A \ B" - and np: "A \ {}" - and fnt: "finite B" - shows "Max (f ` A) \ Max (f ` B)" -proof(rule Max_mono) - from seq show "f ` A \ f ` B" by auto -next - from np show "f ` A \ {}" by auto -next - from fnt and seq show "finite (f ` B)" by auto -qed - -lemma Max_UNION: - assumes "finite A" - and "A \ {}" - and "\ M \ f ` A. finite M" - and "\ M \ f ` A. M \ {}" - shows "Max (\x\ A. f x) = Max (Max ` f ` A)" (is "?L = ?R") - using assms[simp] -proof - - have "?L = Max (\(f ` A))" - by (fold Union_image_eq, simp) - also have "... = ?R" - by (subst Max_Union, simp+) - finally show ?thesis . -qed - -lemma max_Max_eq: - assumes "finite A" - and "A \ {}" - and "x = y" - shows "max x (Max A) = Max ({y} \ A)" (is "?L = ?R") -proof - - have "?R = Max (insert y A)" by simp - also from assms have "... = ?L" - by (subst Max.insert, simp+) - finally show ?thesis by simp -qed - -lemma birth_time_lt: - assumes "s \ []" - shows "last_set th s < length s" - using assms -proof(induct s) - case (Cons a s) - show ?case - proof(cases "s \ []") - case False - thus ?thesis - by (cases a, auto) - next - case True - show ?thesis using Cons(1)[OF True] - by (cases a, auto) - qed -qed simp - -lemma th_in_ne: "th \ threads s \ s \ []" - by (induct s, auto) - -lemma preced_tm_lt: "th \ threads s \ preced th s = Prc x y \ y < length s" - by (drule_tac th_in_ne, unfold preced_def, auto intro: birth_time_lt) - -lemma eq_RAG: - "RAG (wq s) = RAG s" - by (unfold cs_RAG_def s_RAG_def, auto) - -lemma waiting_holding: - assumes "waiting (s::state) th cs" - obtains th' where "holding s th' cs" -proof - - from assms[unfolded s_waiting_def, folded wq_def] - obtain th' where "th' \ set (wq s cs)" "th' = hd (wq s cs)" - by (metis empty_iff hd_in_set list.set(1)) - hence "holding s th' cs" - by (unfold s_holding_def, fold wq_def, auto) - from that[OF this] show ?thesis . -qed - -lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th" -unfolding cp_def wq_def -apply(induct s rule: schs.induct) -apply(simp add: Let_def cpreced_initial) -apply(simp add: Let_def) -apply(simp add: Let_def) -apply(simp add: Let_def) -apply(subst (2) schs.simps) -apply(simp add: Let_def) -apply(subst (2) schs.simps) -apply(simp add: Let_def) -done - -lemma cp_alt_def: - "cp s th = - Max ((the_preced s) ` {th'. Th th' \ (subtree (RAG s) (Th th))})" -proof - - have "Max (the_preced s ` ({th} \ dependants (wq s) th)) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "Max (_ ` ?L) = Max (_ ` ?R)") - proof - - have "?L = ?R" - by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def) - thus ?thesis by simp - qed - thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp) -qed - -(* ccc *) - - locale valid_trace = fixes s assumes vt : "vt s" @@ -169,105 +16,6 @@ end -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" - -locale valid_trace_v = valid_trace_e + - fixes th cs - assumes is_v: "e = V th cs" -begin - definition "rest = tl (wq s cs)" - definition "wq' = (SOME q. distinct q \ set q = set rest)" -end - -locale valid_trace_v_n = valid_trace_v + - assumes rest_nnl: "rest \ []" - -locale valid_trace_v_e = valid_trace_v + - assumes rest_nil: "rest = []" - -locale valid_trace_set= valid_trace_e + - fixes th prio - assumes is_set: "e = Set th prio" - -context valid_trace -begin - -lemma ind [consumes 0, case_names Nil Cons, induct type]: - assumes "PP []" - and "(\s e. valid_trace_e s e \ - PP s \ PIP s e \ PP (e # s))" - shows "PP s" -proof(induct rule:vt.induct[OF vt, case_names Init Step]) - case Init - from assms(1) show ?case . -next - case (Step s e) - show ?case - proof(rule assms(2)) - show "valid_trace_e s e" using Step by (unfold_locales, auto) - next - show "PP s" using Step by simp - next - show "PIP s e" using Step by simp - qed -qed - -lemma vt_moment: "\ t. vt (moment t s)" -proof(induct rule:ind) - case Nil - thus ?case by (simp add:vt_nil) -next - case (Cons s e t) - show ?case - proof(cases "t \ length (e#s)") - case True - from True have "moment t (e#s) = e#s" by simp - thus ?thesis using Cons - by (simp add:valid_trace_def valid_trace_e_def, auto) - next - case False - from Cons have "vt (moment t s)" by simp - moreover have "moment t (e#s) = moment t s" - proof - - from False have "t \ length s" by simp - from moment_app [OF this, of "[e]"] - show ?thesis by simp - qed - ultimately show ?thesis by simp - qed -qed - -lemma finite_threads: - shows "finite (threads s)" -using vt by (induct) (auto elim: step.cases) - -end - -lemma RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" - by (unfold s_RAG_def, auto) - -locale valid_moment = valid_trace + - fixes i :: nat - -sublocale valid_moment < vat_moment: valid_trace "(moment i s)" - by (unfold_locales, insert vt_moment, auto) - -lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs" - by (unfold s_waiting_def cs_waiting_def wq_def, auto) - -lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs" - by (unfold s_holding_def wq_def cs_holding_def, simp) - lemma runing_ready: shows "runing s \ readys s" unfolding runing_def readys_def @@ -278,7 +26,7 @@ unfolding readys_def by auto -lemma wq_v_neq [simp]: +lemma wq_v_neq: "cs \ cs' \ wq (V thread cs#s) cs' = wq s cs'" by (auto simp:wq_def Let_def cp_def split:list.splits) @@ -292,210 +40,6 @@ context valid_trace begin -lemma runing_wqE: - assumes "th \ runing s" - and "th \ set (wq s cs)" - obtains rest where "wq s cs = th#rest" -proof - - from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest" - by (meson list.set_cases) - have "th' = th" - proof(rule ccontr) - assume "th' \ th" - hence "th \ hd (wq s cs)" using eq_wq by auto - with assms(2) - have "waiting s th cs" - by (unfold s_waiting_def, fold wq_def, auto) - with assms show False - by (unfold runing_def readys_def, auto) - qed - with eq_wq that show ?thesis by metis -qed - -end - -context valid_trace_create -begin - -lemma wq_neq_simp [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_create wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace_exit -begin - -lemma wq_neq_simp [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_exit wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace_p -begin - -lemma wq_neq_simp [simp]: - assumes "cs' \ cs" - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_p wq_def - by (auto simp:Let_def) - -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_p] - show ?thesis by (cases, simp) -qed - -lemma ready_th_s: "th \ readys s" - using runing_th_s - by (unfold runing_def, auto) - -lemma live_th_s: "th \ threads s" - using readys_threads ready_th_s by auto - -lemma live_th_es: "th \ threads (e#s)" - using live_th_s - by (unfold is_p, simp) - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - -lemma th_not_in_wq: - shows "th \ set (wq s cs)" -proof - assume otherwise: "th \ set (wq s cs)" - from runing_wqE[OF runing_th_s this] - obtain rest where eq_wq: "wq s cs = th#rest" by blast - with otherwise - have "holding s th cs" - by (unfold s_holding_def, fold wq_def, simp) - hence cs_th_RAG: "(Cs cs, Th th) \ RAG s" - by (unfold s_RAG_def, fold holding_eq, auto) - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - with cs_th_RAG show ?thesis by auto - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq s cs @ [th]" - by (unfold is_p wq_def, auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" -proof(cases "cs' = cs") - case True - show ?thesis using True assms th_not_in_wq - by (unfold True wq_es_cs, auto) -qed (insert assms, simp) - -end - -context valid_trace_v -begin - -lemma wq_neq_simp [simp]: - assumes "cs' \ cs" - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_v wq_def - by (auto simp:Let_def) - -lemma runing_th_s: - shows "th \ runing s" -proof - - from pip_e[unfolded is_v] - show ?thesis by (cases, simp) -qed - -lemma th_not_waiting: - "\ waiting s th c" -proof - - have "th \ readys s" - using runing_ready runing_th_s by blast - thus ?thesis - by (unfold readys_def, auto) -qed - -lemma waiting_neq_th: - assumes "waiting s t c" - shows "t \ th" - using assms using th_not_waiting by blast - -lemma wq_s_cs: - "wq s cs = th#rest" -proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from this(2) show ?thesis - by (unfold rest_def s_holding_def, fold wq_def, - metis empty_iff list.collapse list.set(1)) - qed -qed - -lemma wq_es_cs: - "wq (e#s) cs = wq'" - using wq_s_cs[unfolded wq_def] - by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" -proof(cases "cs' = cs") - case True - show ?thesis - proof(unfold True wq_es_cs wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - using assms[unfolded True wq_s_cs] by auto - qed simp -qed (insert assms, simp) - -end - -context valid_trace_set -begin - -lemma wq_neq_simp [simp]: - shows "wq (e#s) cs' = wq s cs'" - using assms unfolding is_set wq_def - by (auto simp:Let_def) - -lemma wq_distinct_kept: - assumes "distinct (wq s cs')" - shows "distinct (wq (e#s) cs')" - using assms by simp -end - -context valid_trace -begin - lemma actor_inv: assumes "PIP s e" and "\ isCreate e" @@ -503,49 +47,94 @@ using assms by (induct, auto) -lemma isP_E: - assumes "isP e" - obtains cs where "e = P (actor e) cs" - using assms by (cases e, auto) - -lemma isV_E: - assumes "isV e" - obtains cs where "e = V (actor e) cs" - using assms by (cases e, auto) +lemma ind [consumes 0, case_names Nil Cons, induct type]: + assumes "PP []" + and "(\s e. valid_trace s \ valid_trace (e#s) \ + PP s \ PIP s e \ PP (e # s))" + shows "PP s" +proof(rule vt.induct[OF vt]) + from assms(1) show "PP []" . +next + fix s e + assume h: "vt s" "PP s" "PIP s e" + show "PP (e # s)" + proof(cases rule:assms(2)) + from h(1) show v1: "valid_trace s" by (unfold_locales, simp) + next + from h(1,3) have "vt (e#s)" by auto + thus "valid_trace (e # s)" by (unfold_locales, simp) + qed (insert h, auto) +qed lemma wq_distinct: "distinct (wq s cs)" proof(induct rule:ind) case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp + from Cons(4,3) 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) + proof(induct) + case (thread_P th s cs1) + show ?case + proof(cases "cs = cs1") + case True + thus ?thesis (is "distinct ?L") + proof - + have "?L = wq_fun (schs s) cs1 @ [th]" using True + by (simp add:wq_def wf_def Let_def split:list.splits) + moreover have "distinct ..." + proof - + have "th \ set (wq_fun (schs s) cs1)" + proof + assume otherwise: "th \ set (wq_fun (schs s) cs1)" + from runing_head[OF thread_P(1) this] + have "th = hd (wq_fun (schs s) cs1)" . + hence "(Cs cs1, Th th) \ (RAG s)" using otherwise + by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def) + with thread_P(2) show False by auto + qed + moreover have "distinct (wq_fun (schs s) cs1)" + using True thread_P wq_def by auto + ultimately show ?thesis by auto + qed + ultimately show ?thesis by simp + qed + next + case False + with thread_P(3) + show ?thesis + by (auto simp:wq_def wf_def Let_def split:list.splits) + qed next - case (P th cs) - interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept) - next - case (V th cs) - interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept) - next - case (Set th prio) - interpret vt_set: valid_trace_set s e th prio - using Set by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept) - qed + case (thread_V th s cs1) + thus ?case + proof(cases "cs = cs1") + case True + show ?thesis (is "distinct ?L") + proof(cases "(wq s cs)") + case Nil + thus ?thesis + by (auto simp:wq_def wf_def Let_def split:list.splits) + next + case (Cons w_hd w_tl) + moreover have "distinct (SOME q. distinct q \ set q = set w_tl)" + proof(rule someI2) + from thread_V(3)[unfolded Cons] + show "distinct w_tl \ set w_tl = set w_tl" by auto + qed auto + ultimately show ?thesis + by (auto simp:wq_def wf_def Let_def True split:list.splits) + qed + next + case False + with thread_V(3) + show ?thesis + by (auto simp:wq_def wf_def Let_def split:list.splits) + qed + qed (insert Cons, auto simp: wq_def Let_def split:list.splits) qed (unfold wq_def Let_def, simp) end + context valid_trace_e begin @@ -556,7 +145,7 @@ This is a kind of confirmation that our modelling is correct. *} -lemma wq_in_inv: +lemma block_pre: assumes s_ni: "thread \ set (wq s cs)" and s_i: "thread \ set (wq (e#s) cs)" shows "e = P thread cs" @@ -586,44 +175,117 @@ thus ?thesis by auto qed (insert assms, auto simp:wq_def Let_def split:if_splits) -lemma wq_out_inv: - assumes s_in: "thread \ set (wq s cs)" - and s_hd: "thread = hd (wq s cs)" - and s_i: "thread \ hd (wq (e#s) cs)" - shows "e = V thread cs" -proof(cases e) --- {* There are only two non-trivial cases: *} - case (V th cs1) - show ?thesis - proof(cases "cs1 = cs") - case True - have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] . - thus ?thesis - proof(cases) - case (thread_V) - moreover have "th = thread" using thread_V(2) s_hd - by (unfold s_holding_def wq_def, simp) - ultimately show ?thesis using V True by simp +end + +text {* + The following lemmas is also obvious and shallow. It says + that only running thread can request for a critical resource + and that the requested resource must be one which is + not current held by the thread. +*} + +lemma p_pre: "\vt ((P thread cs)#s)\ \ + thread \ runing s \ (Cs cs, Th thread) \ (RAG s)^+" +apply (ind_cases "vt ((P thread cs)#s)") +apply (ind_cases "step s (P thread cs)") +by auto + +lemma abs1: + assumes ein: "e \ set es" + and neq: "hd es \ hd (es @ [x])" + shows "False" +proof - + from ein have "es \ []" by auto + then obtain e ess where "es = e # ess" by (cases es, auto) + with neq show ?thesis by auto +qed + +lemma q_head: "Q (hd es) \ hd es = hd [th\es . Q th]" + by (cases es, auto) + +inductive_cases evt_cons: "vt (a#s)" + +context valid_trace_e +begin + +lemma abs2: + assumes inq: "thread \ set (wq s cs)" + and nh: "thread = hd (wq s cs)" + and qt: "thread \ hd (wq (e#s) cs)" + and inq': "thread \ set (wq (e#s) cs)" + shows "False" +proof - + from vt_e assms show "False" + apply (cases e) + apply ((simp split:if_splits add:Let_def wq_def)[1])+ + apply (insert abs1, fast)[1] + apply (auto simp:wq_def simp:Let_def split:if_splits list.splits) + proof - + fix th qs + assume vt: "vt (V th cs # s)" + and th_in: "thread \ set (SOME q. distinct q \ set q = set qs)" + and eq_wq: "wq_fun (schs s) cs = thread # qs" + show "False" + proof - + from wq_distinct[of cs] + and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp + moreover have "thread \ set qs" + proof - + have "set (SOME q. distinct q \ set q = set qs) = set qs" + proof(rule someI2) + from wq_distinct [of cs] + and eq_wq [folded wq_def] + show "distinct qs \ set qs = set qs" by auto + next + fix x assume "distinct x \ set x = set qs" + thus "set x = set qs" by auto + qed + with th_in show ?thesis by auto + qed + ultimately show ?thesis by auto qed - qed (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) + qed +qed end context valid_trace begin +lemma vt_moment: "\ t. vt (moment t s)" +proof(induct rule:ind) + case Nil + thus ?case by (simp add:vt_nil) +next + case (Cons s e t) + show ?case + proof(cases "t \ length (e#s)") + case True + from True have "moment t (e#s) = e#s" by simp + thus ?thesis using Cons + by (simp add:valid_trace_def) + next + case False + from Cons have "vt (moment t s)" by simp + moreover have "moment t (e#s) = moment t s" + proof - + from False have "t \ length s" by simp + from moment_app [OF this, of "[e]"] + show ?thesis by simp + qed + ultimately show ?thesis by simp + qed +qed +end + +locale valid_moment = valid_trace + + fixes i :: nat + +sublocale valid_moment < vat_moment: valid_trace "(moment i s)" + by (unfold_locales, insert vt_moment, auto) + +context valid_trace +begin text {* (* ddd *) @@ -659,7 +321,7 @@ make any request and get blocked the second time: Contradiction. *} -lemma waiting_unique_pre: (* ddd *) +lemma waiting_unique_pre: (* ccc *) assumes h11: "thread \ set (wq s cs1)" and h12: "thread \ hd (wq s cs1)" assumes h21: "thread \ set (wq s cs2)" @@ -667,101 +329,35 @@ and neq12: "cs1 \ cs2" shows "False" proof - - let "?Q" = "\ cs s. thread \ set (wq s cs) \ thread \ hd (wq s cs)" + let "?Q cs s" = "thread \ set (wq s cs) \ thread \ hd (wq s cs)" from h11 and h12 have q1: "?Q cs1 s" by simp from h21 and h22 have q2: "?Q cs2 s" by simp have nq1: "\ ?Q cs1 []" by (simp add:wq_def) have nq2: "\ ?Q cs2 []" by (simp add:wq_def) from p_split [of "?Q cs1", OF q1 nq1] obtain t1 where lt1: "t1 < length s" - and np1: "\ ?Q cs1 (moment t1 s)" - and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" by auto + and np1: "\(thread \ set (wq (moment t1 s) cs1) \ + thread \ hd (wq (moment t1 s) cs1))" + and nn1: "(\i'>t1. thread \ set (wq (moment i' s) cs1) \ + thread \ hd (wq (moment i' s) cs1))" by auto from p_split [of "?Q cs2", OF q2 nq2] obtain t2 where lt2: "t2 < length s" - and np2: "\ ?Q cs2 (moment t2 s)" - and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" by auto - { fix s cs - assume q: "?Q cs s" - have "thread \ runing s" - proof - assume "thread \ runing s" - hence " \cs. \ (thread \ set (wq_fun (schs s) cs) \ - thread \ hd (wq_fun (schs s) cs))" - by (unfold runing_def s_waiting_def readys_def, auto) - from this[rule_format, of cs] q - show False by (simp add: wq_def) - qed - } note q_not_runing = this - { fix t1 t2 cs1 cs2 - assume lt1: "t1 < length s" - and np1: "\ ?Q cs1 (moment t1 s)" - and nn1: "(\i'>t1. ?Q cs1 (moment i' s))" - and lt2: "t2 < length s" - and np2: "\ ?Q cs2 (moment t2 s)" - and nn2: "(\i'>t2. ?Q cs2 (moment i' s))" - and lt12: "t1 < t2" - let ?t3 = "Suc t2" - from lt2 have le_t3: "?t3 \ length s" by auto - from moment_plus [OF this] - obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto - have "t2 < ?t3" by simp - from nn2 [rule_format, OF this] and eq_m - have h1: "thread \ set (wq (e#moment t2 s) cs2)" and - h2: "thread \ hd (wq (e#moment t2 s) cs2)" by auto - have "vt (e#moment t2 s)" - proof - - from vt_moment - have "vt (moment ?t3 s)" . - with eq_m show ?thesis by simp - qed - then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - have ?thesis - proof - - have "thread \ runing (moment t2 s)" - proof(cases "thread \ set (wq (moment t2 s) cs2)") - case True - have "e = V thread cs2" - proof - - have eq_th: "thread = hd (wq (moment t2 s) cs2)" - using True and np2 by auto - from vt_e.wq_out_inv[OF True this h2] - show ?thesis . - qed - thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto - next - case False - have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] . - with vt_e.actor_inv[OF vt_e.pip_e] - show ?thesis by auto - qed - moreover have "thread \ runing (moment t2 s)" - by (rule q_not_runing[OF nn1[rule_format, OF lt12]]) - ultimately show ?thesis by simp - qed - } note lt_case = this + and np2: "\(thread \ set (wq (moment t2 s) cs2) \ + thread \ hd (wq (moment t2 s) cs2))" + and nn2: "(\i'>t2. thread \ set (wq (moment i' s) cs2) \ + thread \ hd (wq (moment i' s) cs2))" by auto 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" + { + assume lt12: "t1 < t2" let ?t3 = "Suc t2" from lt2 have le_t3: "?t3 \ length s" by auto from moment_plus [OF this] obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto - have lt_2: "t2 < ?t3" by simp + have "t2 < ?t3" by simp from nn2 [rule_format, OF this] and eq_m have h1: "thread \ set (wq (e#moment t2 s) cs2)" and - h2: "thread \ hd (wq (e#moment t2 s) cs2)" by auto - from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12] - have g1: "thread \ set (wq (e#moment t1 s) cs1)" and - g2: "thread \ hd (wq (e#moment t1 s) cs1)" by auto + h2: "thread \ hd (wq (e#moment t2 s) cs2)" by auto have "vt (e#moment t2 s)" proof - from vt_moment @@ -769,38 +365,119 @@ with eq_m show ?thesis by simp qed then interpret vt_e: valid_trace_e "moment t2 s" "e" - by (unfold_locales, auto, cases, simp) - have "e = V thread cs2 \ e = P thread cs2" + by (unfold_locales, auto, cases, simp) + have ?thesis proof(cases "thread \ set (wq (moment t2 s) cs2)") case True - have "e = V thread cs2" - proof - - have eq_th: "thread = hd (wq (moment t2 s) cs2)" - using True and np2 by auto - from vt_e.wq_out_inv[OF True this h2] - show ?thesis . - qed - thus ?thesis by auto + from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)" + by auto + from vt_e.abs2 [OF True eq_th h2 h1] + show ?thesis by auto + next + case False + from vt_e.block_pre[OF False h1] + have "e = P thread cs2" . + with vt_e.vt_e have "vt ((P thread cs2)# moment t2 s)" by simp + from p_pre [OF this] have "thread \ runing (moment t2 s)" by simp + with runing_ready have "thread \ readys (moment t2 s)" by auto + with nn1 [rule_format, OF lt12] + show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto) + qed + } moreover { + assume lt12: "t2 < t1" + let ?t3 = "Suc t1" + from lt1 have le_t3: "?t3 \ length s" by auto + from moment_plus [OF this] + obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto + have lt_t3: "t1 < ?t3" by simp + from nn1 [rule_format, OF this] and eq_m + have h1: "thread \ set (wq (e#moment t1 s) cs1)" and + h2: "thread \ hd (wq (e#moment t1 s) cs1)" by auto + have "vt (e#moment t1 s)" + proof - + from vt_moment + have "vt (moment ?t3 s)" . + with eq_m show ?thesis by simp + qed + then interpret vt_e: valid_trace_e "moment t1 s" e + by (unfold_locales, auto, cases, auto) + have ?thesis + proof(cases "thread \ set (wq (moment t1 s) cs1)") + case True + from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)" + by auto + from vt_e.abs2 True eq_th h2 h1 + show ?thesis by auto next case False - have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] . - thus ?thesis by auto + from vt_e.block_pre [OF False h1] + have "e = P thread cs1" . + with vt_e.vt_e have "vt ((P thread cs1)# moment t1 s)" by simp + from p_pre [OF this] have "thread \ runing (moment t1 s)" by simp + with runing_ready have "thread \ readys (moment t1 s)" by auto + with nn2 [rule_format, OF lt12] + show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto) qed - moreover have "e = V thread cs1 \ e = P thread cs1" + } moreover { + assume eqt12: "t1 = t2" + let ?t3 = "Suc t1" + from lt1 have le_t3: "?t3 \ length s" by auto + from moment_plus [OF this] + obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto + have lt_t3: "t1 < ?t3" by simp + from nn1 [rule_format, OF this] and eq_m + have h1: "thread \ set (wq (e#moment t1 s) cs1)" and + h2: "thread \ hd (wq (e#moment t1 s) cs1)" by auto + have vt_e: "vt (e#moment t1 s)" + proof - + from vt_moment + have "vt (moment ?t3 s)" . + with eq_m show ?thesis by simp + qed + then interpret vt_e: valid_trace_e "moment t1 s" e + by (unfold_locales, auto, cases, auto) + have ?thesis proof(cases "thread \ set (wq (moment t1 s) cs1)") case True - have eq_th: "thread = hd (wq (moment t1 s) cs1)" - using True and np1 by auto - from vt_e.wq_out_inv[folded eq_12, OF True this g2] - have "e = V thread cs1" . - thus ?thesis by auto + from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)" + by auto + from vt_e.abs2 [OF True eq_th h2 h1] + show ?thesis by auto next case False - have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] . - thus ?thesis by auto + from vt_e.block_pre [OF False h1] + have eq_e1: "e = P thread cs1" . + have lt_t3: "t1 < ?t3" by simp + with eqt12 have "t2 < ?t3" by simp + from nn2 [rule_format, OF this] and eq_m and eqt12 + have h1: "thread \ set (wq (e#moment t2 s) cs2)" and + h2: "thread \ hd (wq (e#moment t2 s) cs2)" by auto + show ?thesis + proof(cases "thread \ set (wq (moment t2 s) cs2)") + case True + from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)" + by auto + from vt_e and eqt12 have "vt (e#moment t2 s)" by simp + then interpret vt_e2: valid_trace_e "moment t2 s" e + by (unfold_locales, auto, cases, auto) + from vt_e2.abs2 [OF True eq_th h2 h1] + show ?thesis . + next + case False + have "vt (e#moment t2 s)" + proof - + from vt_moment eqt12 + have "vt (moment (Suc t2) s)" by auto + with eq_m eqt12 show ?thesis by simp + qed + then interpret vt_e2: valid_trace_e "moment t2 s" e + by (unfold_locales, auto, cases, auto) + from vt_e2.block_pre [OF False h1] + have "e = P thread cs2" . + with eq_e1 neq12 show ?thesis by auto + qed qed - ultimately have ?thesis using neq12 by auto - } ultimately show ?thesis using nat_neq_iff by blast + } ultimately show ?thesis by arith qed qed @@ -812,9 +489,9 @@ assumes "waiting s th cs1" and "waiting s th cs2" shows "cs1 = cs2" - using waiting_unique_pre assms - unfolding wq_def s_waiting_def - by auto +using waiting_unique_pre assms +unfolding wq_def s_waiting_def +by auto end @@ -830,6 +507,7 @@ shows "th1 = th2" by (insert assms, unfold s_holding_def, auto) + lemma last_set_lt: "th \ threads s \ last_set th s < length s" apply (induct s, auto) by (case_tac a, auto split:if_splits) @@ -850,7 +528,7 @@ from last_set_unique [OF this th_in1 th_in2] show ?thesis . qed - + lemma preced_linorder: assumes neq_12: "th1 \ th2" and th_in1: "th1 \ threads s" @@ -862,6 +540,98 @@ thus ?thesis by auto qed +(* An aux lemma used later *) +lemma unique_minus: + assumes unique: "\ a b c. \(a, b) \ r; (a, c) \ r\ \ b = c" + and xy: "(x, y) \ r" + and xz: "(x, z) \ r^+" + and neq: "y \ z" + shows "(y, z) \ r^+" +proof - + from xz and neq show ?thesis + proof(induct) + case (base ya) + have "(x, ya) \ r" by fact + from unique [OF xy this] have "y = ya" . + with base show ?case by auto + next + case (step ya z) + show ?case + proof(cases "y = ya") + case True + from step True show ?thesis by simp + next + case False + from step False + show ?thesis by auto + qed + qed +qed + +lemma unique_base: + assumes unique: "\ a b c. \(a, b) \ r; (a, c) \ r\ \ b = c" + and xy: "(x, y) \ r" + and xz: "(x, z) \ r^+" + and neq_yz: "y \ z" + shows "(y, z) \ r^+" +proof - + from xz neq_yz show ?thesis + proof(induct) + case (base ya) + from xy unique base show ?case by auto + next + case (step ya z) + show ?case + proof(cases "y = ya") + case True + from True step show ?thesis by auto + next + case False + from False step + have "(y, ya) \ r\<^sup>+" by auto + with step show ?thesis by auto + qed + qed +qed + +lemma unique_chain: + assumes unique: "\ a b c. \(a, b) \ r; (a, c) \ r\ \ b = c" + and xy: "(x, y) \ r^+" + and xz: "(x, z) \ r^+" + and neq_yz: "y \ z" + shows "(y, z) \ r^+ \ (z, y) \ r^+" +proof - + from xy xz neq_yz show ?thesis + proof(induct) + case (base y) + have h1: "(x, y) \ r" and h2: "(x, z) \ r\<^sup>+" and h3: "y \ z" using base by auto + from unique_base [OF _ h1 h2 h3] and unique show ?case by auto + next + case (step y za) + show ?case + proof(cases "y = z") + case True + from True step show ?thesis by auto + next + case False + from False step have "(y, z) \ r\<^sup>+ \ (z, y) \ r\<^sup>+" by auto + thus ?thesis + proof + assume "(z, y) \ r\<^sup>+" + with step have "(z, za) \ r\<^sup>+" by auto + thus ?thesis by auto + next + assume h: "(y, z) \ r\<^sup>+" + from step have yza: "(y, za) \ r" by simp + from step have "za \ z" by simp + from unique_minus [OF _ yza h this] and unique + have "(za, z) \ r\<^sup>+" by auto + thus ?thesis by auto + qed + qed + qed +qed + text {* The following three lemmas show that @{text "RAG"} does not change by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"} @@ -872,1404 +642,598 @@ apply (unfold s_RAG_def s_waiting_def wq_def) by (simp add:Let_def) -lemma (in valid_trace_set) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_set RAG_set_unchanged, simp) - lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s" apply (unfold s_RAG_def s_waiting_def wq_def) by (simp add:Let_def) -lemma (in valid_trace_create) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_create RAG_create_unchanged, simp) - lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s" apply (unfold s_RAG_def s_waiting_def wq_def) by (simp add:Let_def) -lemma (in valid_trace_exit) - RAG_unchanged: "(RAG (e # s)) = RAG s" - by (unfold is_exit RAG_exit_unchanged, simp) -context valid_trace_v -begin - -lemma distinct_rest: "distinct rest" - by (simp add: distinct_tl rest_def wq_distinct) - -lemma holding_cs_eq_th: - assumes "holding s t cs" - shows "t = th" +text {* + The following lemmas are used in the proof of + lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed + by @{text "V"}-events. + However, since our model is very concise, such seemingly obvious lemmas need to be derived from scratch, + starting from the model definitions. +*} +lemma step_v_hold_inv[elim_format]: + "\c t. \vt (V th cs # s); + \ holding (wq s) t c; holding (wq (V th cs # s)) t c\ \ + next_th s th cs t \ c = cs" proof - - from pip_e[unfolded is_v] - show ?thesis - proof(cases) - case (thread_V) - from held_unique[OF this(2) assms] - show ?thesis by simp + fix c t + assume vt: "vt (V th cs # s)" + and nhd: "\ holding (wq s) t c" + and hd: "holding (wq (V th cs # s)) t c" + show "next_th s th cs t \ c = cs" + proof(cases "c = cs") + case False + with nhd hd show ?thesis + by (unfold cs_holding_def wq_def, auto simp:Let_def) + next + case True + with step_back_step [OF vt] + have "step s (V th c)" by simp + hence "next_th s th cs t" + proof(cases) + assume "holding s th c" + with nhd hd show ?thesis + apply (unfold s_holding_def cs_holding_def wq_def next_th_def, + auto simp:Let_def split:list.splits if_splits) + proof - + assume " hd (SOME q. distinct q \ q = []) \ set (SOME q. distinct q \ q = [])" + moreover have "\ = set []" + proof(rule someI2) + show "distinct [] \ [] = []" by auto + next + fix x assume "distinct x \ x = []" + thus "set x = set []" by auto + qed + ultimately show False by auto + next + assume " hd (SOME q. distinct q \ q = []) \ set (SOME q. distinct q \ q = [])" + moreover have "\ = set []" + proof(rule someI2) + show "distinct [] \ [] = []" by auto + next + fix x assume "distinct x \ x = []" + thus "set x = set []" by auto + qed + ultimately show False by auto + qed + qed + with True show ?thesis by auto qed qed -lemma distinct_wq': "distinct wq'" - by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def) - -lemma set_wq': "set wq' = set rest" - by (metis (mono_tags, lifting) distinct_rest rest_def - some_eq_ex wq'_def) - -lemma th'_in_inv: - assumes "th' \ set wq'" - shows "th' \ set rest" - using assms set_wq' by simp - -lemma neq_t_th: - assumes "waiting (e#s) t c" - shows "t \ th" -proof - assume otherwise: "t = th" - show False - proof(cases "c = cs") +text {* + The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be + derived from scratch, which confirms the correctness of the definition of @{text "next_th"}. +*} +lemma step_v_wait_inv[elim_format]: + "\t c. \vt (V th cs # s); \ waiting (wq (V th cs # s)) t c; waiting (wq s) t c + \ + \ (next_th s th cs t \ cs = c)" +proof - + fix t c + assume vt: "vt (V th cs # s)" + and nw: "\ waiting (wq (V th cs # s)) t c" + and wt: "waiting (wq s) t c" + from vt interpret vt_v: valid_trace_e s "V th cs" + by (cases, unfold_locales, simp) + show "next_th s th cs t \ cs = c" + proof(cases "cs = c") + case False + with nw wt show ?thesis + by (auto simp:cs_waiting_def wq_def Let_def) + next case True - have "t \ set wq'" - using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs] - by simp - from th'_in_inv[OF this] have "t \ set rest" . - with wq_s_cs[folded otherwise] wq_distinct[of cs] - show ?thesis by simp - next - case False - have "wq (e#s) c = wq s c" using False - by (unfold is_v, simp) - hence "waiting s t c" using assms - by (simp add: cs_waiting_def waiting_eq) - hence "t \ readys s" by (unfold readys_def, auto) - hence "t \ runing s" using runing_ready by auto - with runing_th_s[folded otherwise] show ?thesis by auto + from nw[folded True] wt[folded True] + have "next_th s th cs t" + apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits) + proof - + fix a list + assume t_in: "t \ set list" + and t_ni: "t \ set (SOME q. distinct q \ set q = set list)" + and eq_wq: "wq_fun (schs s) cs = a # list" + have " set (SOME q. distinct q \ set q = set list) = set list" + proof(rule someI2) + from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def] + show "distinct list \ set list = set list" by auto + next + show "\x. distinct x \ set x = set list \ set x = set list" + by auto + qed + with t_ni and t_in show "a = th" by auto + next + fix a list + assume t_in: "t \ set list" + and t_ni: "t \ set (SOME q. distinct q \ set q = set list)" + and eq_wq: "wq_fun (schs s) cs = a # list" + have " set (SOME q. distinct q \ set q = set list) = set list" + proof(rule someI2) + from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def] + show "distinct list \ set list = set list" by auto + next + show "\x. distinct x \ set x = set list \ set x = set list" + by auto + qed + with t_ni and t_in show "t = hd (SOME q. distinct q \ set q = set list)" by auto + next + fix a list + assume eq_wq: "wq_fun (schs s) cs = a # list" + from step_back_step[OF vt] + show "a = th" + proof(cases) + assume "holding s th cs" + with eq_wq show ?thesis + by (unfold s_holding_def wq_def, auto) + qed + qed + with True show ?thesis by simp qed qed -lemma waiting_esI1: - assumes "waiting s t c" - and "c \ cs" - shows "waiting (e#s) t c" -proof - - have "wq (e#s) c = wq s c" - using assms(2) is_v by auto - with assms(1) show ?thesis - using cs_waiting_def waiting_eq by auto -qed - -lemma holding_esI2: - assumes "c \ cs" - and "holding s t c" - shows "holding (e#s) t c" -proof - - from assms(1) have "wq (e#s) c = wq s c" using is_v by auto - from assms(2)[unfolded s_holding_def, folded wq_def, - folded this, unfolded wq_def, folded s_holding_def] - show ?thesis . -qed - -lemma holding_esI1: - assumes "holding s t c" - and "t \ th" - shows "holding (e#s) t c" -proof - - have "c \ cs" using assms using holding_cs_eq_th by blast - from holding_esI2[OF this assms(1)] - show ?thesis . -qed - -end - -context valid_trace_v_n -begin - -lemma neq_wq': "wq' \ []" -proof (unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume " distinct x \ set x = set rest" - thus "x \ []" using rest_nnl by auto -qed - -definition "taker = hd wq'" - -definition "rest' = tl wq'" - -lemma eq_wq': "wq' = taker # rest'" - by (simp add: neq_wq' rest'_def taker_def) +lemma step_v_not_wait[consumes 3]: + "\vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\ \ False" + by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def) -lemma 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" +lemma step_v_release: + "\vt (V th cs # s); holding (wq (V th cs # s)) th cs\ \ False" proof - - from assms - obtain rest' where - h: "wq s cs = th # rest'" - "taker' = hd (SOME q. distinct q \ set q = set rest')" - by (unfold next_th_def, auto) - with wq_s_cs have "rest' = rest" by auto - thus ?thesis using h(2) taker_def wq'_def by auto -qed - -lemma waiting_set_eq: - "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}" - by (smt all_not_in_conv bot.extremum insertI1 insert_subset - mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique) - -lemma holding_set_eq: - "{(Cs cs, Th th') |th'. next_th s th cs th'} = {(Cs cs, Th taker)}" - using next_th_taker taker_def waiting_set_eq - by fastforce - -lemma holding_taker: - shows "holding (e#s) taker cs" - by (unfold s_holding_def, fold wq_def, unfold wq_es_cs, - auto simp:neq_wq' taker_def) - -lemma waiting_esI2: - assumes "waiting s t cs" - and "t \ taker" - shows "waiting (e#s) t cs" -proof - - have "t \ set wq'" - proof(unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) - next - fix x - assume "distinct x \ set x = set rest" - moreover have "t \ set rest" - using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto - ultimately show "t \ set x" by simp + assume vt: "vt (V th cs # s)" + and hd: "holding (wq (V th cs # s)) th cs" + from vt interpret vt_v: valid_trace_e s "V th cs" + by (cases, unfold_locales, simp+) + from step_back_step [OF vt] and hd + show "False" + proof(cases) + assume "holding (wq (V th cs # s)) th cs" and "holding s th cs" + thus ?thesis + apply (unfold s_holding_def wq_def cs_holding_def) + apply (auto simp:Let_def split:list.splits) + proof - + fix list + assume eq_wq[folded wq_def]: + "wq_fun (schs s) cs = hd (SOME q. distinct q \ set q = set list) # list" + and hd_in: "hd (SOME q. distinct q \ set q = set list) + \ set (SOME q. distinct q \ set q = set list)" + have "set (SOME q. distinct q \ set q = set list) = set list" + proof(rule someI2) + from vt_v.wq_distinct[of cs] and eq_wq + show "distinct list \ set list = set list" by auto + next + show "\x. distinct x \ set x = set list \ set x = set list" + by auto + qed + moreover have "distinct (hd (SOME q. distinct q \ set q = set list) # list)" + proof - + from vt_v.wq_distinct[of cs] and eq_wq + show ?thesis by auto + qed + moreover note eq_wq and hd_in + ultimately show "False" by auto + qed qed - moreover have "t \ hd wq'" - using assms(2) taker_def by auto - ultimately show ?thesis - by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp) qed -lemma waiting_esE: - assumes "waiting (e#s) t c" - obtains "c \ cs" "waiting s t c" - | "c = cs" "t \ taker" "waiting s t cs" "t \ set rest'" -proof(cases "c = cs") - case False - hence "wq (e#s) c = wq s c" using is_v by auto - with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto - from that(1)[OF False this] show ?thesis . -next - case True - from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs] - have "t \ hd wq'" "t \ set wq'" by auto - hence "t \ taker" by (simp add: taker_def) - moreover hence "t \ th" using assms neq_t_th by blast - moreover have "t \ set rest" by (simp add: `t \ set wq'` th'_in_inv) - ultimately have "waiting s t cs" - by (metis cs_waiting_def list.distinct(2) list.sel(1) - list.set_sel(2) rest_def waiting_eq wq_s_cs) - show ?thesis using that(2) - using True `t \ set wq'` `t \ taker` `waiting s t cs` eq_wq' by auto -qed - -lemma holding_esI1: - assumes "c = cs" - and "t = taker" - shows "holding (e#s) t c" - by (unfold assms, simp add: holding_taker) - -lemma holding_esE: - assumes "holding (e#s) t c" - obtains "c = cs" "t = taker" - | "c \ cs" "holding s t c" -proof(cases "c = cs") - case True - from assms[unfolded True, unfolded s_holding_def, - folded wq_def, unfolded wq_es_cs] - have "t = taker" by (simp add: taker_def) - from that(1)[OF True this] show ?thesis . -next - case False - hence "wq (e#s) c = wq s c" using is_v by auto - from assms[unfolded s_holding_def, folded wq_def, - unfolded this, unfolded wq_def, folded s_holding_def] - have "holding s t c" . - from that(2)[OF False this] show ?thesis . -qed - -end - - -context valid_trace_v_e -begin - -lemma nil_wq': "wq' = []" -proof (unfold wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume " distinct x \ set x = set rest" - thus "x = []" using rest_nil by auto -qed - -lemma no_taker: - assumes "next_th s th cs taker" - shows "False" +lemma step_v_get_hold: + "\th'. \vt (V th cs # s); \ holding (wq (V th cs # s)) th' cs; next_th s th cs th'\ \ False" + apply (unfold cs_holding_def next_th_def wq_def, + auto simp:Let_def) proof - - from assms[unfolded next_th_def] - obtain rest' where "wq s cs = th # rest'" "rest' \ []" - by auto - thus ?thesis using rest_def rest_nil by auto -qed - -lemma waiting_set_eq: - "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}" - using no_taker by auto - -lemma holding_set_eq: - "{(Cs cs, Th th') |th'. next_th s th cs th'} = {}" - using no_taker by auto - -lemma no_holding: - assumes "holding (e#s) taker cs" - shows False -proof - - from wq_es_cs[unfolded nil_wq'] - have " wq (e # s) cs = []" . - from assms[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis by auto -qed - -lemma no_waiting: - assumes "waiting (e#s) t cs" - shows False -proof - - from wq_es_cs[unfolded nil_wq'] - have " wq (e # s) cs = []" . - from assms[unfolded s_waiting_def, folded wq_def, unfolded this] - show ?thesis by auto -qed - -lemma waiting_esI2: - assumes "waiting s t c" - shows "waiting (e#s) t c" -proof - - have "c \ cs" using assms - using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto - from waiting_esI1[OF assms this] - show ?thesis . + fix rest + assume vt: "vt (V th cs # s)" + and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest" + and nrest: "rest \ []" + and ni: "hd (SOME q. distinct q \ set q = set rest) + \ set (SOME q. distinct q \ set q = set rest)" + from vt interpret vt_v: valid_trace_e s "V th cs" + by (cases, unfold_locales, simp+) + have "(SOME q. distinct q \ set q = set rest) \ []" + proof(rule someI2) + from vt_v.wq_distinct[of cs] and eq_wq + show "distinct rest \ set rest = set rest" by auto + next + fix x assume "distinct x \ set x = set rest" + hence "set x = set rest" by auto + with nrest + show "x \ []" by (case_tac x, auto) + qed + with ni show "False" by auto qed -lemma waiting_esE: - assumes "waiting (e#s) t c" - obtains "c \ cs" "waiting s t c" -proof(cases "c = cs") - case False - hence "wq (e#s) c = wq s c" using is_v by auto - with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto - from that(1)[OF False this] show ?thesis . -next - case True - from no_waiting[OF assms[unfolded True]] - show ?thesis by auto -qed - -lemma holding_esE: - assumes "holding (e#s) t c" - obtains "c \ cs" "holding s t c" -proof(cases "c = cs") - case True - from no_holding[OF assms[unfolded True]] - show ?thesis by auto -next - case False - hence "wq (e#s) c = wq s c" using is_v by auto - from assms[unfolded s_holding_def, folded wq_def, - unfolded this, unfolded wq_def, folded s_holding_def] - have "holding s t c" . - from that[OF False this] show ?thesis . -qed - -end - -lemma rel_eqI: - assumes "\ x y. (x,y) \ A \ (x,y) \ B" - and "\ x y. (x,y) \ B \ (x, y) \ A" - shows "A = B" - using assms by auto - -lemma in_RAG_E: - assumes "(n1, n2) \ RAG (s::state)" - obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs" - | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs" - using assms[unfolded s_RAG_def, folded waiting_eq holding_eq] - by auto - -context valid_trace_v -begin - -lemma RAG_es: - "RAG (e # s) = - RAG s - {(Cs cs, Th th)} - - {(Th th', Cs cs) |th'. next_th s th cs th'} \ - {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - show ?thesis - proof(cases "rest = []") - case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from waiting(3) +lemma step_v_release_inv[elim_format]: +"\c t. \vt (V th cs # s); \ holding (wq (V th cs # s)) t c; holding (wq s) t c\ \ + c = cs \ t = th" + apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits) + proof - + fix a list + assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list" + from step_back_step [OF vt] show "a = th" + proof(cases) + assume "holding s th cs" with eq_wq show ?thesis - 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 + by (unfold s_holding_def wq_def, auto) 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) + fix a list + assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list" + from step_back_step [OF vt] show "a = th" + proof(cases) + assume "holding s th cs" with eq_wq show ?thesis - 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 + by (unfold s_holding_def wq_def, auto) qed qed -next - fix n1 n2 - assume h: "(n1, n2) \ ?R" - show "(n1, n2) \ ?L" - proof(cases "rest = []") + +lemma step_v_waiting_mono: + "\t c. \vt (V th cs # s); waiting (wq (V th cs # s)) t c\ \ waiting (wq s) t c" +proof - + fix t c + let ?s' = "(V th cs # s)" + assume vt: "vt ?s'" + and wt: "waiting (wq ?s') t c" + from vt interpret vt_v: valid_trace_e s "V th cs" + by (cases, unfold_locales, simp+) + show "waiting (wq s) t c" + proof(cases "c = cs") case False - interpret h_n: valid_trace_v_n s e th cs - by (unfold_locales, insert False, simp) - from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq] - have "((n1, n2) \ RAG s \ (n1 \ Cs cs \ n2 \ Th th) - \ (n1 \ Th h_n.taker \ n2 \ Cs cs)) \ - (n2 = Th h_n.taker \ n1 = Cs cs)" - by auto - thus ?thesis - proof - assume "n2 = Th h_n.taker \ n1 = Cs cs" - with h_n.holding_taker - show ?thesis - by (unfold s_RAG_def, fold holding_eq, auto) - next - assume h: "(n1, n2) \ RAG s \ - (n1 \ Cs cs \ n2 \ Th th) \ (n1 \ Th h_n.taker \ n2 \ Cs cs)" - hence "(n1, n2) \ RAG s" by simp - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from h and this(1,2) - have "th' \ h_n.taker \ cs' \ cs" by auto - hence "waiting (e#s) th' cs'" - proof - assume "cs' \ cs" - from waiting_esI1[OF waiting(3) this] - show ?thesis . + assume neq_cs: "c \ cs" + hence "waiting (wq ?s') t c = waiting (wq s) t c" + by (unfold cs_waiting_def wq_def, auto simp:Let_def) + with wt show ?thesis by simp + next + case True + with wt show ?thesis + apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits) + proof - + fix a list + assume not_in: "t \ set list" + and is_in: "t \ set (SOME q. distinct q \ set q = set list)" + and eq_wq: "wq_fun (schs s) cs = a # list" + have "set (SOME q. distinct q \ set q = set list) = set list" + proof(rule someI2) + from vt_v.wq_distinct [of cs] + and eq_wq[folded wq_def] + show "distinct list \ set list = set list" by auto next - assume neq_th': "th' \ h_n.taker" - show ?thesis - proof(cases "cs' = cs") - case False - from waiting_esI1[OF waiting(3) this] - show ?thesis . - next - case True - from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True] - show ?thesis . - qed + fix x assume "distinct x \ set x = set list" + thus "set x = set list" by auto qed - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) + with not_in is_in show "t = a" by auto next - case (holding th' cs') - from h this(1,2) - have "cs' \ cs \ th' \ th" by auto - hence "holding (e#s) th' cs'" - proof - assume "cs' \ cs" - from holding_esI2[OF this holding(3)] - show ?thesis . - next - assume "th' \ th" - from holding_esI1[OF holding(3) this] - show ?thesis . + fix list + assume is_waiting: "waiting (wq (V th cs # s)) t cs" + and eq_wq: "wq_fun (schs s) cs = t # list" + hence "t \ set list" + apply (unfold wq_def, auto simp:Let_def cs_waiting_def) + proof - + assume " t \ set (SOME q. distinct q \ set q = set list)" + moreover have "\ = set list" + proof(rule someI2) + from vt_v.wq_distinct [of cs] + and eq_wq[folded wq_def] + show "distinct list \ set list = set list" by auto + next + fix x assume "distinct x \ set x = set list" + thus "set x = set list" by auto + qed + ultimately show "t \ set list" by simp qed - thus ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) + with eq_wq and vt_v.wq_distinct [of cs, unfolded wq_def] + show False by auto qed - qed - next - case True - interpret h_e: valid_trace_v_e s e th cs - by (unfold_locales, insert True, simp) - from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq] - have h_s: "(n1, n2) \ RAG s" "(n1, n2) \ (Cs cs, Th th)" - by auto - from h_s(1) - show ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from h_e.waiting_esI2[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - with h_s(2) - have "cs' \ cs \ th' \ th" by auto - thus ?thesis - proof - assume neq_cs: "cs' \ cs" - from holding_esI2[OF this holding(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - next - assume "th' \ th" - from holding_esI1[OF holding(3) this] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed - qed + qed qed -end - -lemma step_RAG_v: +text {* (* ddd *) + The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed + with the happening of @{text "V"}-events: +*} +lemma step_RAG_v: assumes vt: "vt (V th cs#s)" shows " RAG (V th cs # s) = RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \ - {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = ?R") -proof - - interpret vt_v: valid_trace_v s "V th cs" - using assms step_back_vt by (unfold_locales, auto) - show ?thesis using vt_v.RAG_es . -qed - -lemma (in valid_trace_create) - th_not_in_threads: "th \ threads s" -proof - - from pip_e[unfolded is_create] - show ?thesis by (cases, simp) -qed - -lemma (in valid_trace_create) - threads_es [simp]: "threads (e#s) = threads s \ {th}" - by (unfold is_create, simp) - -lemma (in valid_trace_exit) - threads_es [simp]: "threads (e#s) = threads s - {th}" - by (unfold is_exit, simp) - -lemma (in valid_trace_p) - threads_es [simp]: "threads (e#s) = threads s" - by (unfold is_p, simp) - -lemma (in valid_trace_v) - threads_es [simp]: "threads (e#s) = threads s" - by (unfold is_v, simp) - -lemma (in valid_trace_v) - th_not_in_rest[simp]: "th \ set rest" -proof - assume otherwise: "th \ set rest" - have "distinct (wq s cs)" by (simp add: wq_distinct) - from this[unfolded wq_s_cs] and otherwise - show False by auto -qed - -lemma (in valid_trace_v) - set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}" -proof(unfold wq_es_cs wq'_def, rule someI2) - show "distinct rest \ set rest = set rest" - by (simp add: distinct_rest) -next - fix x - assume "distinct x \ set x = set rest" - thus "set x = set (wq s cs) - {th}" - by (unfold wq_s_cs, simp) -qed + {(Cs cs, Th th') |th'. next_th s th cs th'}" + apply (insert vt, unfold s_RAG_def) + apply (auto split:if_splits list.splits simp:Let_def) + apply (auto elim: step_v_waiting_mono step_v_hold_inv + step_v_release step_v_wait_inv + step_v_get_hold step_v_release_inv) + apply (erule_tac step_v_not_wait, auto) + done -lemma (in valid_trace_exit) - th_not_in_wq: "th \ set (wq s cs)" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold holdents_def s_holding_def, fold wq_def, - auto elim!:runing_wqE) -qed +text {* + The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed + with the happening of @{text "P"}-events: +*} +lemma step_RAG_p: + "vt (P th cs#s) \ + RAG (P th cs # s) = (if (wq s cs = []) then RAG s \ {(Cs cs, Th th)} + else RAG s \ {(Th th, Cs cs)})" + apply(simp only: s_RAG_def wq_def) + apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def) + apply(case_tac "csa = cs", auto) + apply(fold wq_def) + apply(drule_tac step_back_step) + apply(ind_cases " step s (P (hd (wq s cs)) cs)") + apply(simp add:s_RAG_def wq_def cs_holding_def) + apply(auto) + done -lemma (in valid_trace) wq_threads: - assumes "th \ set (wq s cs)" - shows "th \ threads s" - using assms -proof(induct rule:ind) - case (Nil) - thus ?case by (auto simp:wq_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th' prio') - interpret vt: valid_trace_create s e th' prio' - using Create by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems by auto - next - case (Exit th') - interpret vt: valid_trace_exit s e th' - using Exit by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto - next - case (P th' cs') - interpret vt: valid_trace_p s e th' cs' - using P by (unfold_locales, simp) - show ?thesis - using Cons.hyps(2) Cons.prems readys_threads - runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv - by fastforce - next - case (V th' cs') - interpret vt: valid_trace_v s e th' cs' - using V by (unfold_locales, simp) - show ?thesis using Cons - using vt.is_v vt.threads_es vt_e.wq_in_inv by blast - next - case (Set th' prio) - interpret vt: valid_trace_set s e th' prio - using Set by (unfold_locales, simp) - show ?thesis using Cons.hyps(2) Cons.prems vt.is_set - by (auto simp:wq_def Let_def) - qed -qed + +lemma RAG_target_th: "(Th th, x) \ RAG (s::state) \ \ cs. x = Cs cs" + by (unfold s_RAG_def, auto) context valid_trace begin -lemma dm_RAG_threads: - assumes in_dom: "(Th th) \ Domain (RAG s)" - shows "th \ threads s" -proof - - from in_dom obtain n where "(Th th, n) \ RAG s" by auto - moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto - ultimately have "(Th th, Cs cs) \ RAG s" by simp - hence "th \ set (wq s cs)" - by (unfold s_RAG_def, auto simp:cs_waiting_def) - from wq_threads [OF this] show ?thesis . -qed - -lemma rg_RAG_threads: - assumes "(Th th) \ Range (RAG s)" - shows "th \ threads s" - using assms - by (unfold s_RAG_def cs_waiting_def cs_holding_def, - auto intro:wq_threads) - -lemma RAG_threads: - assumes "(Th th) \ Field (RAG s)" - shows "th \ threads s" - using assms - by (metis Field_def UnE dm_RAG_threads rg_RAG_threads) - -end - -lemma (in valid_trace_v) - preced_es [simp]: "preced th (e#s) = preced th s" - by (unfold is_v preced_def, simp) - -lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s" -proof - fix th' - show "the_preced (V th cs # s) th' = the_preced s th'" - by (unfold the_preced_def preced_def, simp) -qed - -lemma (in valid_trace_v) - the_preced_es: "the_preced (e#s) = the_preced s" - by (unfold is_v preced_def, simp) - -context valid_trace_p -begin - -lemma not_holding_s_th_cs: "\ holding s th cs" -proof - assume otherwise: "holding s th cs" - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - moreover have "(Cs cs, Th th) \ RAG s" - using otherwise cs_holding_def - holding_eq th_not_in_wq by auto - ultimately show ?thesis by auto - qed -qed - -lemma waiting_kept: - assumes "waiting s th' cs'" - shows "waiting (e#s) th' cs'" - using assms - by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2) - rotate1.simps(2) self_append_conv2 set_rotate1 - th_not_in_wq waiting_eq wq_es_cs wq_neq_simp) - -lemma holding_kept: - assumes "holding s th' cs'" - shows "holding (e#s) th' cs'" -proof(cases "cs' = cs") - case False - hence "wq (e#s) cs' = wq s cs'" by simp - with assms show ?thesis using cs_holding_def holding_eq by auto -next - case True - from assms[unfolded s_holding_def, folded wq_def] - obtain rest where eq_wq: "wq s cs' = th'#rest" - by (metis empty_iff list.collapse list.set(1)) - hence "wq (e#s) cs' = th'#(rest@[th])" - by (simp add: True wq_es_cs) - thus ?thesis - by (simp add: cs_holding_def holding_eq) -qed - -end - -locale valid_trace_p_h = valid_trace_p + - assumes we: "wq s cs = []" - -locale valid_trace_p_w = valid_trace_p + - assumes wne: "wq s cs \ []" -begin - -definition "holder = hd (wq s cs)" -definition "waiters = tl (wq s cs)" -definition "waiters' = waiters @ [th]" - -lemma wq_s_cs: "wq s cs = holder#waiters" - by (simp add: holder_def waiters_def wne) - -lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]" - by (simp add: wq_es_cs wq_s_cs) - -lemma waiting_es_th_cs: "waiting (e#s) th cs" - using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto - -lemma RAG_edge: "(Th th, Cs cs) \ RAG (e#s)" - by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto) - -lemma holding_esE: - assumes "holding (e#s) th' cs'" - obtains "holding s th' cs'" - using assms -proof(cases "cs' = cs") - case False - hence "wq (e#s) cs' = wq s cs'" by simp - with assms show ?thesis - using cs_holding_def holding_eq that by auto -next - case True - with assms show ?thesis - by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that - wq_es_cs' wq_s_cs) +text {* + The following lemma shows that @{text "RAG"} is acyclic. + The overall structure is by induction on the formation of @{text "vt s"} + and then case analysis on event @{text "e"}, where the non-trivial cases + for those for @{text "V"} and @{text "P"} events. +*} +lemma acyclic_RAG: + shows "acyclic (RAG s)" +using vt +proof(induct) + case (vt_cons s e) + interpret vt_s: valid_trace s using vt_cons(1) + by (unfold_locales, simp) + assume ih: "acyclic (RAG s)" + and stp: "step s e" + and vt: "vt s" + show ?case + proof(cases e) + case (Create th prio) + with ih + show ?thesis by (simp add:RAG_create_unchanged) + next + case (Exit th) + with ih show ?thesis by (simp add:RAG_exit_unchanged) + next + case (V th cs) + from V vt stp have vtt: "vt (V th cs#s)" by auto + from step_RAG_v [OF this] + have eq_de: + "RAG (e # s) = + RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \ + {(Cs cs, Th th') |th'. next_th s th cs th'}" + (is "?L = (?A - ?B - ?C) \ ?D") by (simp add:V) + from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset) + from step_back_step [OF vtt] + have "step s (V th cs)" . + thus ?thesis + proof(cases) + assume "holding s th cs" + hence th_in: "th \ set (wq s cs)" and + eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto + then obtain rest where + eq_wq: "wq s cs = th#rest" + by (cases "wq s cs", auto) + show ?thesis + proof(cases "rest = []") + case False + let ?th' = "hd (SOME q. distinct q \ set q = set rest)" + from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}" + by (unfold next_th_def, auto) + let ?E = "(?A - ?B - ?C)" + have "(Th ?th', Cs cs) \ ?E\<^sup>*" + proof + assume "(Th ?th', Cs cs) \ ?E\<^sup>*" + hence " (Th ?th', Cs cs) \ ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl) + from tranclD [OF this] + obtain x where th'_e: "(Th ?th', x) \ ?E" by blast + hence th_d: "(Th ?th', x) \ ?A" by simp + from RAG_target_th [OF this] + obtain cs' where eq_x: "x = Cs cs'" by auto + with th_d have "(Th ?th', Cs cs') \ ?A" by simp + hence wt_th': "waiting s ?th' cs'" + unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp + hence "cs' = cs" + proof(rule vt_s.waiting_unique) + from eq_wq vt_s.wq_distinct[of cs] + show "waiting s ?th' cs" + apply (unfold s_waiting_def wq_def, auto) + proof - + assume hd_in: "hd (SOME q. distinct q \ set q = set rest) \ set rest" + and eq_wq: "wq_fun (schs s) cs = th # rest" + have "(SOME q. distinct q \ set q = set rest) \ []" + proof(rule someI2) + from vt_s.wq_distinct[of cs] and eq_wq + show "distinct rest \ set rest = set rest" unfolding wq_def by auto + next + fix x assume "distinct x \ set x = set rest" + with False show "x \ []" by auto + qed + hence "hd (SOME q. distinct q \ set q = set rest) \ + set (SOME q. distinct q \ set q = set rest)" by auto + moreover have "\ = set rest" + proof(rule someI2) + from vt_s.wq_distinct[of cs] and eq_wq + show "distinct rest \ set rest = set rest" unfolding wq_def by auto + next + show "\x. distinct x \ set x = set rest \ set x = set rest" by auto + qed + moreover note hd_in + ultimately show "hd (SOME q. distinct q \ set q = set rest) = th" by auto + next + assume hd_in: "hd (SOME q. distinct q \ set q = set rest) \ set rest" + and eq_wq: "wq s cs = hd (SOME q. distinct q \ set q = set rest) # rest" + have "(SOME q. distinct q \ set q = set rest) \ []" + proof(rule someI2) + from vt_s.wq_distinct[of cs] and eq_wq + show "distinct rest \ set rest = set rest" by auto + next + fix x assume "distinct x \ set x = set rest" + with False show "x \ []" by auto + qed + hence "hd (SOME q. distinct q \ set q = set rest) \ + set (SOME q. distinct q \ set q = set rest)" by auto + moreover have "\ = set rest" + proof(rule someI2) + from vt_s.wq_distinct[of cs] and eq_wq + show "distinct rest \ set rest = set rest" by auto + next + show "\x. distinct x \ set x = set rest \ set x = set rest" by auto + qed + moreover note hd_in + ultimately show False by auto + qed + qed + with th'_e eq_x have "(Th ?th', Cs cs) \ ?E" by simp + with False + show "False" by (auto simp: next_th_def eq_wq) + qed + with acyclic_insert[symmetric] and ac + and eq_de eq_D show ?thesis by auto + next + case True + with eq_wq + have eq_D: "?D = {}" + by (unfold next_th_def, auto) + with eq_de ac + show ?thesis by auto + qed + qed + next + case (P th cs) + from P vt stp have vtt: "vt (P th cs#s)" by auto + from step_RAG_p [OF this] P + have "RAG (e # s) = + (if wq s cs = [] then RAG s \ {(Cs cs, Th th)} else + RAG s \ {(Th th, Cs cs)})" (is "?L = ?R") + by simp + moreover have "acyclic ?R" + proof(cases "wq s cs = []") + case True + hence eq_r: "?R = RAG s \ {(Cs cs, Th th)}" by simp + have "(Th th, Cs cs) \ (RAG s)\<^sup>*" + proof + assume "(Th th, Cs cs) \ (RAG s)\<^sup>*" + hence "(Th th, Cs cs) \ (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl) + from tranclD2 [OF this] + obtain x where "(x, Cs cs) \ RAG s" by auto + with True show False by (auto simp:s_RAG_def cs_waiting_def) + qed + with acyclic_insert ih eq_r show ?thesis by auto + next + case False + hence eq_r: "?R = RAG s \ {(Th th, Cs cs)}" by simp + have "(Cs cs, Th th) \ (RAG s)\<^sup>*" + proof + assume "(Cs cs, Th th) \ (RAG s)\<^sup>*" + hence "(Cs cs, Th th) \ (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl) + moreover from step_back_step [OF vtt] have "step s (P th cs)" . + ultimately show False + proof - + show " \(Cs cs, Th th) \ (RAG s)\<^sup>+; step s (P th cs)\ \ False" + by (ind_cases "step s (P th cs)", simp) + qed + qed + with acyclic_insert ih eq_r show ?thesis by auto + qed + ultimately show ?thesis by simp + next + case (Set thread prio) + with ih + thm RAG_set_unchanged + show ?thesis by (simp add:RAG_set_unchanged) + qed + next + case vt_nil + show "acyclic (RAG ([]::state))" + by (auto simp: s_RAG_def cs_waiting_def + cs_holding_def wq_def acyclic_def) qed -lemma waiting_esE: - assumes "waiting (e#s) th' cs'" - obtains "th' \ th" "waiting s th' cs'" - | "th' = th" "cs' = cs" -proof(cases "waiting s th' cs'") - case True - have "th' \ th" - proof - assume otherwise: "th' = th" - from True[unfolded this] - show False by (simp add: th_not_waiting) - qed - from that(1)[OF this True] show ?thesis . -next - case False - hence "th' = th \ cs' = cs" - by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2) - set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp) - with that(2) show ?thesis by metis -qed -lemma RAG_es: "RAG (e # s) = RAG s \ {(Th th, Cs cs)}" (is "?L = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from this(3) - show ?thesis - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) +lemma finite_RAG: + shows "finite (RAG s)" +proof - + from vt show ?thesis + proof(induct) + case (vt_cons s e) + interpret vt_s: valid_trace s using vt_cons(1) + by (unfold_locales, simp) + assume ih: "finite (RAG s)" + and stp: "step s e" + and vt: "vt s" + show ?case + proof(cases e) + case (Create th prio) + with ih + show ?thesis by (simp add:RAG_create_unchanged) + next + case (Exit th) + with ih show ?thesis by (simp add:RAG_exit_unchanged) next - case 2 - thus ?thesis using waiting(1,2) by auto - qed - next - case (holding th' cs') - from this(3) - show ?thesis - proof(cases rule:holding_esE) - case 1 - with holding(1,2) - show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) - qed - qed -next - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \ RAG s \ (n1 = Th th \ n2 = Cs cs)" by auto - thus "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ RAG s" - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from waiting_kept[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) + case (V th cs) + from V vt stp have vtt: "vt (V th cs#s)" by auto + from step_RAG_v [OF this] + have eq_de: "RAG (e # s) = + RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \ + {(Cs cs, Th th') |th'. next_th s th cs th'} +" + (is "?L = (?A - ?B - ?C) \ ?D") by (simp add:V) + moreover from ih have ac: "finite (?A - ?B - ?C)" by simp + moreover have "finite ?D" + proof - + have "?D = {} \ (\ a. ?D = {a})" + by (unfold next_th_def, auto) + thus ?thesis + proof + assume h: "?D = {}" + show ?thesis by (unfold h, simp) + next + assume "\ a. ?D = {a}" + thus ?thesis + by (metis finite.simps) + qed + qed + ultimately show ?thesis by simp next - case (holding th' cs') - from holding_kept[OF this(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) + case (P th cs) + from P vt stp have vtt: "vt (P th cs#s)" by auto + from step_RAG_p [OF this] P + have "RAG (e # s) = + (if wq s cs = [] then RAG s \ {(Cs cs, Th th)} else + RAG s \ {(Th th, Cs cs)})" (is "?L = ?R") + by simp + moreover have "finite ?R" + proof(cases "wq s cs = []") + case True + hence eq_r: "?R = RAG s \ {(Cs cs, Th th)}" by simp + with True and ih show ?thesis by auto + next + case False + hence "?R = RAG s \ {(Th th, Cs cs)}" by simp + with False and ih show ?thesis by auto + qed + ultimately show ?thesis by auto + next + case (Set thread prio) + with ih + show ?thesis by (simp add:RAG_set_unchanged) qed next - assume "n1 = Th th \ n2 = Cs cs" - thus ?thesis using RAG_edge by auto - qed -qed - -end - -context valid_trace_p_h -begin - -lemma wq_es_cs': "wq (e#s) cs = [th]" - using wq_es_cs[unfolded we] by simp - -lemma holding_es_th_cs: - shows "holding (e#s) th cs" -proof - - from wq_es_cs' - have "th \ set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto - thus ?thesis using cs_holding_def holding_eq by blast -qed - -lemma RAG_edge: "(Cs cs, Th th) \ RAG (e#s)" - by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto) - -lemma waiting_esE: - assumes "waiting (e#s) th' cs'" - obtains "waiting s th' cs'" - using assms - by (metis cs_waiting_def event.distinct(15) is_p list.sel(1) - set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv) - -lemma holding_esE: - assumes "holding (e#s) th' cs'" - obtains "cs' \ cs" "holding s th' cs'" - | "cs' = cs" "th' = th" -proof(cases "cs' = cs") - case True - from held_unique[OF holding_es_th_cs assms[unfolded True]] - have "th' = th" by simp - from that(2)[OF True this] show ?thesis . -next - case False - have "holding s th' cs'" using assms - using False cs_holding_def holding_eq by auto - from that(1)[OF False this] show ?thesis . -qed - -lemma RAG_es: "RAG (e # s) = RAG s \ {(Cs cs, Th th)}" (is "?L = ?R") -proof(rule rel_eqI) - fix n1 n2 - assume "(n1, n2) \ ?L" - thus "(n1, n2) \ ?R" - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from this(3) - show ?thesis - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - qed - next - case (holding th' cs') - from this(3) - show ?thesis - proof(cases rule:holding_esE) - case 1 - with holding(1,2) - show ?thesis by (unfold s_RAG_def, fold holding_eq, auto) - next - case 2 - with holding(1,2) show ?thesis by auto - qed - qed -next - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \ RAG s \ (n1 = Cs cs \ n2 = Th th)" by auto - thus "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ RAG s" - thus ?thesis - proof(cases rule:in_RAG_E) - case (waiting th' cs') - from waiting_kept[OF this(3)] - show ?thesis using waiting(1,2) - by (unfold s_RAG_def, fold waiting_eq, auto) - next - case (holding th' cs') - from holding_kept[OF this(3)] - show ?thesis using holding(1,2) - by (unfold s_RAG_def, fold holding_eq, auto) - qed - next - assume "n1 = Cs cs \ n2 = Th th" - with holding_es_th_cs - show ?thesis - by (unfold s_RAG_def, fold holding_eq, auto) + case vt_nil + show "finite (RAG ([]::state))" + by (auto simp: s_RAG_def cs_waiting_def + cs_holding_def wq_def acyclic_def) qed qed -end - -context valid_trace_p -begin - -lemma RAG_es': "RAG (e # s) = (if (wq s cs = []) then RAG s \ {(Cs cs, Th th)} - else RAG s \ {(Th th, Cs cs)})" -proof(cases "wq s cs = []") - case True - interpret vt_p: valid_trace_p_h using True - by (unfold_locales, simp) - show ?thesis by (simp add: vt_p.RAG_es vt_p.we) -next - case False - interpret vt_p: valid_trace_p_w using False - by (unfold_locales, simp) - show ?thesis by (simp add: vt_p.RAG_es vt_p.wne) -qed - -end - -lemma (in valid_trace_v_n) finite_waiting_set: - "finite {(Th th', Cs cs) |th'. next_th s th cs th'}" - by (simp add: waiting_set_eq) - -lemma (in valid_trace_v_n) finite_holding_set: - "finite {(Cs cs, Th th') |th'. next_th s th cs th'}" - by (simp add: holding_set_eq) - -lemma (in valid_trace_v_e) finite_waiting_set: - "finite {(Th th', Cs cs) |th'. next_th s th cs th'}" - by (simp add: waiting_set_eq) - -lemma (in valid_trace_v_e) finite_holding_set: - "finite {(Cs cs, Th th') |th'. next_th s th cs th'}" - by (simp add: holding_set_eq) - -context valid_trace_v -begin - -lemma - finite_RAG_kept: - assumes "finite (RAG s)" - shows "finite (RAG (e#s))" -proof(cases "rest = []") - case True - interpret vt: valid_trace_v_e using True - by (unfold_locales, simp) - show ?thesis using assms - by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp) -next - case False - interpret vt: valid_trace_v_n using False - by (unfold_locales, simp) - show ?thesis using assms - by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp) -qed - -end - -context valid_trace_v_e -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof(rule acyclic_subset[OF assms]) - show "RAG (e # s) \ RAG s" - by (unfold RAG_es waiting_set_eq holding_set_eq, auto) -qed - -end - -context valid_trace_v_n -begin - -lemma waiting_taker: "waiting s taker cs" - apply (unfold s_waiting_def, fold wq_def, unfold wq_s_cs taker_def) - using eq_wq' th'_in_inv wq'_def by fastforce - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic ((RAG s - {(Cs cs, Th th)} - {(Th taker, Cs cs)}) \ - {(Cs cs, Th taker)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Th taker, Cs cs) \ ?A^*" - proof - assume otherwise: "(Th taker, Cs cs) \ ?A^*" - hence "(Th taker, Cs cs) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - from tranclD[OF this] - obtain cs' where h: "(Th taker, Cs cs') \ ?A" - "(Th taker, Cs cs') \ RAG s" - by (unfold s_RAG_def, auto) - from this(2) have "waiting s taker cs'" - by (unfold s_RAG_def, fold waiting_eq, auto) - from waiting_unique[OF this waiting_taker] - have "cs' = cs" . - from h(1)[unfolded this] show False by auto - qed - ultimately show ?thesis by auto - qed - thus ?thesis - by (unfold RAG_es waiting_set_eq holding_set_eq, simp) -qed - -end - -context valid_trace_p_h -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic (RAG s \ {(Cs cs, Th th)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Th th, Cs cs) \ ?A^*" - proof - assume otherwise: "(Th th, Cs cs) \ ?A^*" - hence "(Th th, Cs cs) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - from tranclD[OF this] - obtain cs' where h: "(Th th, Cs cs') \ RAG s" - by (unfold s_RAG_def, auto) - hence "waiting s th cs'" - by (unfold s_RAG_def, fold waiting_eq, auto) - with th_not_waiting show False by auto - qed - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold RAG_es, simp) -qed - -end - -context valid_trace_p_w -begin - -lemma - acylic_RAG_kept: - assumes "acyclic (RAG s)" - shows "acyclic (RAG (e#s))" -proof - - have "acyclic (RAG s \ {(Th th, Cs cs)})" (is "acyclic (?A \ _)") - proof - - from assms - have "acyclic ?A" - by (rule acyclic_subset, auto) - moreover have "(Cs cs, Th th) \ ?A^*" - proof - assume otherwise: "(Cs cs, Th th) \ ?A^*" - from pip_e[unfolded is_p] - show False - proof(cases) - case (thread_P) - moreover from otherwise have "(Cs cs, Th th) \ ?A^+" - by (unfold rtrancl_eq_or_trancl, auto) - ultimately show ?thesis by auto - qed - qed - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold RAG_es, simp) -qed - -end - -context valid_trace -begin - -lemma finite_RAG: - shows "finite (RAG s)" -proof(induct rule:ind) - case Nil - show ?case - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt: valid_trace_create s e th prio using Create - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (Exit th) - interpret vt: valid_trace_exit s e th using Exit - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (P th cs) - interpret vt: valid_trace_p s e th cs using P - by (unfold_locales, simp) - show ?thesis using Cons using vt.RAG_es' by auto - next - case (V th cs) - interpret vt: valid_trace_v s e th cs using V - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.finite_RAG_kept) - next - case (Set th prio) - interpret vt: valid_trace_set s e th prio using Set - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - qed -qed +text {* Several useful lemmas *} -lemma acyclic_RAG: - shows "acyclic (RAG s)" -proof(induct rule:ind) - case Nil - show ?case - by (auto simp: s_RAG_def cs_waiting_def - cs_holding_def wq_def acyclic_def) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt: valid_trace_create s e th prio using Create - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (Exit th) - interpret vt: valid_trace_exit s e th using Exit - by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt.RAG_unchanged) - next - case (P th cs) - interpret vt: valid_trace_p s e th cs using P - by (unfold_locales, simp) - show ?thesis - 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 add: vt.RAG_unchanged) - qed -qed - -lemma wf_RAG: "wf (RAG s)" -proof(rule finite_acyclic_wf) - from finite_RAG show "finite (RAG s)" . -next - from acyclic_RAG show "acyclic (RAG s)" . -qed - -lemma 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) - -lemma acyclic_tRAG: "acyclic (tRAG s)" -proof(unfold tRAG_def, rule acyclic_compose) - show "acyclic (RAG s)" using acyclic_RAG . -next - show "wRAG s \ RAG s" unfolding RAG_split by auto -next - show "hRAG s \ RAG s" unfolding RAG_split by auto -qed - -lemma unique_RAG: "\(n, n1) \ RAG s; (n, n2) \ RAG s\ \ n1 = n2" - apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq) - by(auto elim:waiting_unique held_unique) - -lemma sgv_RAG: "single_valued (RAG s)" - using unique_RAG by (auto simp:single_valued_def) - -lemma rtree_RAG: "rtree (RAG s)" - using sgv_RAG acyclic_RAG - by (unfold rtree_def rtree_axioms_def sgv_def, auto) - -end - -sublocale valid_trace < rtree_RAG: rtree "RAG s" -proof - show "single_valued (RAG s)" - apply (intro_locales) - by (unfold single_valued_def, - auto intro:unique_RAG) - - show "acyclic (RAG s)" - by (rule acyclic_RAG) -qed - -sublocale valid_trace < rtree_s: rtree "tRAG s" -proof(unfold_locales) - from sgv_tRAG show "single_valued (tRAG s)" . -next - from acyclic_tRAG show "acyclic (tRAG s)" . -qed - -sublocale valid_trace < fsbtRAGs : fsubtree "RAG s" -proof - - show "fsubtree (RAG s)" - proof(intro_locales) - show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] . - next - show "fsubtree_axioms (RAG s)" - proof(unfold fsubtree_axioms_def) - from wf_RAG show "wf (RAG s)" . - qed - qed -qed - -lemma tRAG_alt_def: - "tRAG s = {(Th th1, Th th2) | th1 th2. - \ cs. (Th th1, Cs cs) \ RAG s \ (Cs cs, Th th2) \ RAG s}" - by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def) - -sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s" -proof - - have "fsubtree (tRAG s)" - proof - - have "fbranch (tRAG s)" - proof(unfold tRAG_def, rule fbranch_compose) - show "fbranch (wRAG s)" - proof(rule finite_fbranchI) - from finite_RAG show "finite (wRAG s)" - by (unfold RAG_split, auto) - qed - next - show "fbranch (hRAG s)" - proof(rule finite_fbranchI) - from finite_RAG - show "finite (hRAG s)" by (unfold RAG_split, auto) - qed - qed - moreover have "wf (tRAG s)" - proof(rule wf_subset) - show "wf (RAG s O RAG s)" using wf_RAG - by (fold wf_comp_self, simp) - next - show "tRAG s \ (RAG s O RAG s)" - by (unfold tRAG_alt_def, auto) - qed - ultimately show ?thesis - by (unfold fsubtree_def fsubtree_axioms_def,auto) - qed - from this[folded tRAG_def] show "fsubtree (tRAG s)" . -qed - - -context valid_trace -begin - -lemma finite_subtree_threads: - "finite {th'. Th th' \ subtree (RAG s) (Th th)}" (is "finite ?A") -proof - - have "?A = the_thread ` {Th th' | th' . Th th' \ subtree (RAG s) (Th th)}" - by (auto, insert image_iff, fastforce) - moreover have "finite {Th th' | th' . Th th' \ subtree (RAG s) (Th th)}" - (is "finite ?B") - proof - - have "?B = (subtree (RAG s) (Th th)) \ {Th th' | th'. True}" - by auto - moreover have "... \ (subtree (RAG s) (Th th))" by auto - moreover have "finite ..." by (simp add: fsbtRAGs.finite_subtree) - ultimately show ?thesis by auto - qed - ultimately show ?thesis by auto -qed - -lemma le_cp: - shows "preced th s \ cp s th" - proof(unfold cp_alt_def, rule Max_ge) - show "finite (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" - by (simp add: finite_subtree_threads) - next - show "preced th s \ the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}" - by (simp add: subtree_def the_preced_def) - qed - -lemma cp_le: - assumes th_in: "th \ threads s" - shows "cp s th \ Max (the_preced s ` threads s)" -proof(unfold cp_alt_def, rule Max_f_mono) - show "finite (threads s)" by (simp add: finite_threads) -next - show " {th'. Th th' \ subtree (RAG s) (Th th)} \ {}" - using subtree_def by fastforce -next - show "{th'. Th th' \ subtree (RAG s) (Th th)} \ threads s" - using assms - by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq - node.inject(1) rtranclD subsetI subtree_def trancl_domain) -qed - -lemma max_cp_eq: - shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)" - (is "?L = ?R") -proof - - have "?L \ ?R" - proof(cases "threads s = {}") - case False - show ?thesis - by (rule Max.boundedI, - insert cp_le, - auto simp:finite_threads False) - qed auto - moreover have "?R \ ?L" - by (rule Max_fg_mono, - simp add: finite_threads, - simp add: le_cp the_preced_def) - ultimately show ?thesis by auto -qed - -lemma wf_RAG_converse: +lemma wf_dep_converse: shows "wf ((RAG s)^-1)" proof(rule finite_acyclic_wf_converse) from finite_RAG @@ -2279,47 +1243,208 @@ show "acyclic (RAG s)" . qed -lemma chain_building: - assumes "node \ Domain (RAG s)" - obtains th' where "th' \ readys s" "(node, Th th') \ (RAG s)^+" +end + +lemma hd_np_in: "x \ set l \ hd l \ set l" + by (induct l, auto) + +lemma th_chasing: "(Th th, Cs cs) \ RAG (s::state) \ \ th'. (Cs cs, Th th') \ RAG s" + by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in) + +context valid_trace +begin + +lemma wq_threads: + assumes h: "th \ set (wq s cs)" + shows "th \ threads s" proof - - from assms have "node \ Range ((RAG s)^-1)" by auto - from wf_base[OF wf_RAG_converse this] - obtain b where h_b: "(b, node) \ ((RAG s)\)\<^sup>+" "\c. (c, b) \ (RAG s)\" by auto - obtain th' where eq_b: "b = Th th'" - proof(cases b) - case (Cs cs) - from h_b(1)[unfolded trancl_converse] - have "(node, b) \ ((RAG s)\<^sup>+)" by auto - from tranclE[OF this] - obtain n where "(n, b) \ RAG s" by auto - from this[unfolded Cs] - obtain th1 where "waiting s th1 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - from waiting_holding[OF this] - obtain th2 where "holding s th2 cs" . - hence "(Cs cs, Th th2) \ RAG s" - by (unfold s_RAG_def, fold holding_eq, auto) - with h_b(2)[unfolded Cs, rule_format] - have False by auto - thus ?thesis by auto - qed auto - have "th' \ readys s" - proof - - from h_b(2)[unfolded eq_b] - have "\cs. \ waiting s th' cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - moreover have "th' \ threads s" - proof(rule rg_RAG_threads) - from tranclD[OF h_b(1), unfolded eq_b] - obtain z where "(z, Th th') \ (RAG s)" by auto - thus "Th th' \ Range (RAG s)" by auto + from vt and h show ?thesis + proof(induct arbitrary: th cs) + case (vt_cons s e) + interpret vt_s: valid_trace s + using vt_cons(1) by (unfold_locales, auto) + assume ih: "\th cs. th \ set (wq s cs) \ th \ threads s" + and stp: "step s e" + and vt: "vt s" + and h: "th \ set (wq (e # s) cs)" + show ?case + proof(cases e) + case (Create th' prio) + with ih h show ?thesis + by (auto simp:wq_def Let_def) + next + case (Exit th') + with stp ih h show ?thesis + apply (auto simp:wq_def Let_def) + apply (ind_cases "step s (Exit th')") + apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def + s_RAG_def s_holding_def cs_holding_def) + done + next + case (V th' cs') + show ?thesis + proof(cases "cs' = cs") + case False + with h + show ?thesis + apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def) + by (drule_tac ih, simp) + next + case True + from h + show ?thesis + proof(unfold V wq_def) + assume th_in: "th \ set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \ set ?l") + show "th \ threads (V th' cs' # s)" + proof(cases "cs = cs'") + case False + hence "?l = wq_fun (schs s) cs" by (simp add:Let_def) + with th_in have " th \ set (wq s cs)" + by (fold wq_def, simp) + from ih [OF this] show ?thesis by simp + next + case True + show ?thesis + proof(cases "wq_fun (schs s) cs'") + case Nil + with h V show ?thesis + apply (auto simp:wq_def Let_def split:if_splits) + by (fold wq_def, drule_tac ih, simp) + next + case (Cons a rest) + assume eq_wq: "wq_fun (schs s) cs' = a # rest" + with h V show ?thesis + apply (auto simp:Let_def wq_def split:if_splits) + proof - + assume th_in: "th \ set (SOME q. distinct q \ set q = set rest)" + have "set (SOME q. distinct q \ set q = set rest) = set rest" + proof(rule someI2) + from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def] + show "distinct rest \ set rest = set rest" by auto + next + show "\x. distinct x \ set x = set rest \ set x = set rest" + by auto + qed + with eq_wq th_in have "th \ set (wq_fun (schs s) cs')" by auto + from ih[OF this[folded wq_def]] show "th \ threads s" . + next + assume th_in: "th \ set (wq_fun (schs s) cs)" + from ih[OF this[folded wq_def]] + show "th \ threads s" . + qed + qed + qed + qed + qed + next + case (P th' cs') + from h stp + show ?thesis + apply (unfold P wq_def) + apply (auto simp:Let_def split:if_splits, fold wq_def) + apply (auto intro:ih) + apply(ind_cases "step s (P th' cs')") + by (unfold runing_def readys_def, auto) + next + case (Set thread prio) + with ih h show ?thesis + by (auto simp:wq_def Let_def) qed - ultimately show ?thesis by (auto simp:readys_def) + next + case vt_nil + thus ?case by (auto simp:wq_def) qed - moreover have "(node, Th th') \ (RAG s)^+" - using h_b(1)[unfolded trancl_converse] eq_b by auto - ultimately show ?thesis using that by metis +qed + +lemma range_in: "\(Th th) \ Range (RAG (s::state))\ \ th \ threads s" + apply(unfold s_RAG_def cs_waiting_def cs_holding_def) + by (auto intro:wq_threads) + +lemma readys_v_eq: + assumes neq_th: "th \ thread" + and eq_wq: "wq s cs = thread#rest" + and not_in: "th \ set rest" + shows "(th \ readys (V thread cs#s)) = (th \ readys s)" +proof - + from assms show ?thesis + apply (auto simp:readys_def) + apply(simp add:s_waiting_def[folded wq_def]) + apply (erule_tac x = csa in allE) + apply (simp add:s_waiting_def wq_def Let_def split:if_splits) + apply (case_tac "csa = cs", simp) + apply (erule_tac x = cs in allE) + apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits) + apply(auto simp add: wq_def) + apply (auto simp:s_waiting_def wq_def Let_def split:list.splits) + proof - + assume th_nin: "th \ set rest" + and th_in: "th \ set (SOME q. distinct q \ set q = set rest)" + and eq_wq: "wq_fun (schs s) cs = thread # rest" + have "set (SOME q. distinct q \ set q = set rest) = set rest" + proof(rule someI2) + from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def] + show "distinct rest \ set rest = set rest" by auto + next + show "\x. distinct x \ set x = set rest \ set x = set rest" by auto + qed + with th_nin th_in show False by auto + qed +qed + +text {* \noindent + The following lemmas shows that: starting from any node in @{text "RAG"}, + by chasing out-going edges, it is always possible to reach a node representing a ready + thread. In this lemma, it is the @{text "th'"}. +*} + +lemma chain_building: + shows "node \ Domain (RAG s) \ (\ th'. th' \ readys s \ (node, Th th') \ (RAG s)^+)" +proof - + from wf_dep_converse + have h: "wf ((RAG s)\)" . + show ?thesis + proof(induct rule:wf_induct [OF h]) + fix x + assume ih [rule_format]: + "\y. (y, x) \ (RAG s)\ \ + y \ Domain (RAG s) \ (\th'. th' \ readys s \ (y, Th th') \ (RAG s)\<^sup>+)" + show "x \ Domain (RAG s) \ (\th'. th' \ readys s \ (x, Th th') \ (RAG s)\<^sup>+)" + proof + assume x_d: "x \ Domain (RAG s)" + show "\th'. th' \ readys s \ (x, Th th') \ (RAG s)\<^sup>+" + proof(cases x) + case (Th th) + from x_d Th obtain cs where x_in: "(Th th, Cs cs) \ RAG s" by (auto simp:s_RAG_def) + with Th have x_in_r: "(Cs cs, x) \ (RAG s)^-1" by simp + from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \ RAG s" by blast + hence "Cs cs \ Domain (RAG s)" by auto + from ih [OF x_in_r this] obtain th' + where th'_ready: " th' \ readys s" and cs_in: "(Cs cs, Th th') \ (RAG s)\<^sup>+" by auto + have "(x, Th th') \ (RAG s)\<^sup>+" using Th x_in cs_in by auto + with th'_ready show ?thesis by auto + next + case (Cs cs) + from x_d Cs obtain th' where th'_d: "(Th th', x) \ (RAG s)^-1" by (auto simp:s_RAG_def) + show ?thesis + proof(cases "th' \ readys s") + case True + from True and th'_d show ?thesis by auto + next + case False + from th'_d and range_in have "th' \ threads s" by auto + with False have "Th th' \ Domain (RAG s)" + by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def) + from ih [OF th'_d this] + obtain th'' where + th''_r: "th'' \ readys s" and + th''_in: "(Th th', Th th'') \ (RAG s)\<^sup>+" by auto + from th'_d and th''_in + have "(x, Th th'') \ (RAG s)\<^sup>+" by auto + with th''_r show ?thesis by auto + qed + qed + qed + qed qed text {* \noindent @@ -2341,6 +1466,182 @@ end +lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs" + by (unfold s_waiting_def cs_waiting_def wq_def, auto) + +lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs" + by (unfold s_holding_def wq_def cs_holding_def, simp) + +lemma holding_unique: "\holding (s::state) th1 cs; holding s th2 cs\ \ th1 = th2" + by (unfold s_holding_def cs_holding_def, auto) + +context valid_trace +begin + +lemma unique_RAG: "\(n, n1) \ RAG s; (n, n2) \ RAG s\ \ n1 = n2" + apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq) + by(auto elim:waiting_unique holding_unique) + +end + + +lemma trancl_split: "(a, b) \ r^+ \ \ c. (a, c) \ r" +by (induct rule:trancl_induct, auto) + +context valid_trace +begin + +lemma dchain_unique: + assumes th1_d: "(n, Th th1) \ (RAG s)^+" + and th1_r: "th1 \ readys s" + and th2_d: "(n, Th th2) \ (RAG s)^+" + and th2_r: "th2 \ readys s" + shows "th1 = th2" +proof - + { assume neq: "th1 \ th2" + hence "Th th1 \ Th th2" by simp + from unique_chain [OF _ th1_d th2_d this] and unique_RAG + have "(Th th1, Th th2) \ (RAG s)\<^sup>+ \ (Th th2, Th th1) \ (RAG s)\<^sup>+" by auto + hence "False" + proof + assume "(Th th1, Th th2) \ (RAG s)\<^sup>+" + from trancl_split [OF this] + obtain n where dd: "(Th th1, n) \ RAG s" by auto + then obtain cs where eq_n: "n = Cs cs" + by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in) + from dd eq_n have "th1 \ readys s" + by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def) + with th1_r show ?thesis by auto + next + assume "(Th th2, Th th1) \ (RAG s)\<^sup>+" + from trancl_split [OF this] + obtain n where dd: "(Th th2, n) \ RAG s" by auto + then obtain cs where eq_n: "n = Cs cs" + by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in) + from dd eq_n have "th2 \ readys s" + by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def) + with th2_r show ?thesis by auto + qed + } thus ?thesis by auto +qed + +end + + +lemma step_holdents_p_add: + assumes vt: "vt (P th cs#s)" + and "wq s cs = []" + shows "holdents (P th cs#s) th = holdents s th \ {cs}" +proof - + from assms show ?thesis + unfolding holdents_test step_RAG_p[OF vt] by (auto) +qed + +lemma step_holdents_p_eq: + assumes vt: "vt (P th cs#s)" + and "wq s cs \ []" + shows "holdents (P th cs#s) th = holdents s th" +proof - + from assms show ?thesis + unfolding holdents_test step_RAG_p[OF vt] by auto +qed + + +lemma (in valid_trace) finite_holding : + shows "finite (holdents s th)" +proof - + let ?F = "\ (x, y). the_cs x" + from finite_RAG + have "finite (RAG s)" . + hence "finite (?F `(RAG s))" by simp + moreover have "{cs . (Cs cs, Th th) \ RAG s} \ \" + proof - + { have h: "\ a A f. a \ A \ f a \ f ` A" by auto + fix x assume "(Cs x, Th th) \ RAG s" + hence "?F (Cs x, Th th) \ ?F `(RAG s)" by (rule h) + moreover have "?F (Cs x, Th th) = x" by simp + ultimately have "x \ (\(x, y). the_cs x) ` RAG s" by simp + } thus ?thesis by auto + qed + ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset) +qed + +lemma cntCS_v_dec: + assumes vtv: "vt (V thread cs#s)" + shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread" +proof - + from vtv interpret vt_s: valid_trace s + by (cases, unfold_locales, simp) + from vtv interpret vt_v: valid_trace "V thread cs#s" + by (unfold_locales, simp) + from step_back_step[OF vtv] + have cs_in: "cs \ holdents s thread" + apply (cases, unfold holdents_test s_RAG_def, simp) + by (unfold cs_holding_def s_holding_def wq_def, auto) + moreover have cs_not_in: + "(holdents (V thread cs#s) thread) = holdents s thread - {cs}" + apply (insert vt_s.wq_distinct[of cs]) + apply (unfold holdents_test, unfold step_RAG_v[OF vtv], + auto simp:next_th_def) + proof - + fix rest + assume dst: "distinct (rest::thread list)" + and ne: "rest \ []" + and hd_ni: "hd (SOME q. distinct q \ set q = set rest) \ set rest" + moreover have "set (SOME q. distinct q \ set q = set rest) = set rest" + proof(rule someI2) + from dst show "distinct rest \ set rest = set rest" by auto + next + show "\x. distinct x \ set x = set rest \ set x = set rest" by auto + qed + ultimately have "hd (SOME q. distinct q \ set q = set rest) \ + set (SOME q. distinct q \ set q = set rest)" by simp + moreover have "(SOME q. distinct q \ set q = set rest) \ []" + proof(rule someI2) + from dst show "distinct rest \ set rest = set rest" by auto + next + fix x assume " distinct x \ set x = set rest" with ne + show "x \ []" by auto + qed + ultimately + show "(Cs cs, Th (hd (SOME q. distinct q \ set q = set rest))) \ RAG s" + by auto + next + fix rest + assume dst: "distinct (rest::thread list)" + and ne: "rest \ []" + and hd_ni: "hd (SOME q. distinct q \ set q = set rest) \ set rest" + moreover have "set (SOME q. distinct q \ set q = set rest) = set rest" + proof(rule someI2) + from dst show "distinct rest \ set rest = set rest" by auto + next + show "\x. distinct x \ set x = set rest \ set x = set rest" by auto + qed + ultimately have "hd (SOME q. distinct q \ set q = set rest) \ + set (SOME q. distinct q \ set q = set rest)" by simp + moreover have "(SOME q. distinct q \ set q = set rest) \ []" + proof(rule someI2) + from dst show "distinct rest \ set rest = set rest" by auto + next + fix x assume " distinct x \ set x = set rest" with ne + show "x \ []" by auto + qed + ultimately show "False" by auto + qed + ultimately + have "holdents s thread = insert cs (holdents (V thread cs#s) thread)" + by auto + moreover have "card \ = + Suc (card ((holdents (V thread cs#s) thread) - {cs}))" + proof(rule card_insert) + from vt_v.finite_holding + show " finite (holdents (V thread cs # s) thread)" . + qed + moreover from cs_not_in + have "cs \ (holdents (V thread cs#s) thread)" by auto + ultimately show ?thesis by (simp add:cntCS_def) +qed + lemma count_rec1 [simp]: assumes "Q e" shows "count Q (e#es) = Suc (count Q es)" @@ -2356,39 +1657,7 @@ lemma count_rec3 [simp]: shows "count Q [] = 0" by (unfold count_def, auto) - -lemma cntP_simp1[simp]: - "cntP (P th cs'#s) th = cntP s th + 1" - by (unfold cntP_def, simp) - -lemma cntP_simp2[simp]: - assumes "th' \ th" - shows "cntP (P th cs'#s) th' = cntP s th'" - using assms - by (unfold cntP_def, simp) - -lemma cntP_simp3[simp]: - assumes "\ isP e" - shows "cntP (e#s) th' = cntP s th'" - using assms - by (unfold cntP_def, cases e, simp+) - -lemma cntV_simp1[simp]: - "cntV (V th cs'#s) th = cntV s th + 1" - by (unfold cntV_def, simp) - -lemma cntV_simp2[simp]: - assumes "th' \ th" - shows "cntV (V th cs'#s) th' = cntV s th'" - using assms - by (unfold cntV_def, simp) - -lemma cntV_simp3[simp]: - assumes "\ isV e" - shows "cntV (e#s) th' = cntV s th'" - using assms - by (unfold cntV_def, cases e, simp+) - + lemma cntP_diff_inv: assumes "cntP (e#s) th \ cntP s th" shows "isP e \ actor e = th" @@ -2398,7 +1667,17 @@ by (cases "(\e. \cs. e = P th cs) (P th' pty)", insert assms P, auto simp:cntP_def) qed (insert assms, auto simp:cntP_def) - + +lemma 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) (* ccc *) + lemma cntV_diff_inv: assumes "cntV (e#s) th \ cntV s th" shows "isV e \ actor e = th" @@ -2409,1381 +1688,871 @@ insert assms V, auto simp:cntV_def) qed (insert assms, auto simp:cntV_def) -lemma children_RAG_alt_def: - "children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}" - by (unfold s_RAG_def, auto simp:children_def holding_eq) - -lemma holdents_alt_def: - "holdents s th = the_cs ` (children (RAG (s::state)) (Th th))" - by (unfold children_RAG_alt_def holdents_def, simp add: image_image) - -lemma cntCS_alt_def: - "cntCS s th = card (children (RAG s) (Th th))" - apply (unfold children_RAG_alt_def cntCS_def holdents_def) - by (rule card_image[symmetric], auto simp:inj_on_def) - context valid_trace begin -lemma finite_holdents: "finite (holdents s th)" - by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto) - -end - -context valid_trace_p_w -begin - -lemma holding_s_holder: "holding s holder cs" - by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto) - -lemma holding_es_holder: "holding (e#s) holder cs" - by (unfold s_holding_def, fold wq_def, unfold wq_es_cs wq_s_cs, auto) - -lemma holdents_es: - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence h: "holding (e#s) th' cs'" by (auto simp:holdents_def) - have "holding s th' cs'" - proof(cases "cs' = cs") - case True - from held_unique[OF h[unfolded True] holding_es_holder] - have "th' = holder" . - thus ?thesis - by (unfold True holdents_def, insert holding_s_holder, simp) - next - case False - hence "wq (e#s) cs' = wq s cs'" by simp - from h[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis - by (unfold s_holding_def, fold wq_def, auto) - qed - hence "cs' \ ?R" by (auto simp:holdents_def) - } moreover { - fix cs' - assume "cs' \ ?R" - hence h: "holding s th' cs'" by (auto simp:holdents_def) - have "holding (e#s) th' cs'" - proof(cases "cs' = cs") - case True - from held_unique[OF h[unfolded True] holding_s_holder] - have "th' = holder" . - thus ?thesis - by (unfold True holdents_def, insert holding_es_holder, simp) - next - case False - hence "wq s cs' = wq (e#s) cs'" by simp - from h[unfolded s_holding_def, folded wq_def, unfolded this] - show ?thesis - by (unfold s_holding_def, fold wq_def, auto) - qed - hence "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th[simp]: "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_es, simp) - -lemma th_not_ready_es: - shows "th \ readys (e#s)" - using waiting_es_th_cs - by (unfold readys_def, auto) +text {* (* ddd *) \noindent + The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} + of one particular thread. +*} -end - -context valid_trace_p_h -begin - -lemma th_not_waiting': - "\ waiting (e#s) th cs'" -proof(cases "cs' = cs") - case True - show ?thesis - by (unfold True s_waiting_def, fold wq_def, unfold wq_es_cs', auto) -next - case False - from th_not_waiting[of cs', unfolded s_waiting_def, folded wq_def] - show ?thesis - by (unfold s_waiting_def, fold wq_def, insert False, simp) -qed - -lemma ready_th_es: - shows "th \ readys (e#s)" - using th_not_waiting' - by (unfold readys_def, insert live_th_es, auto) - -lemma holdents_es_th: - "holdents (e#s) th = (holdents s th) \ {cs}" (is "?L = ?R") +lemma cnp_cnv_cncs: + shows "cntP s th = cntV s th + (if (th \ readys s \ th \ threads s) + then cntCS s th else cntCS s th + 1)" proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" - by (unfold holdents_def, auto) - hence "cs' \ ?R" - by (cases rule:holding_esE, auto simp:holdents_def) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s th cs' \ cs' = cs" - by (auto simp:holdents_def) - hence "cs' \ ?L" - proof - assume "holding s th cs'" - from holding_kept[OF this] - show ?thesis by (auto simp:holdents_def) + from vt show ?thesis + proof(induct arbitrary:th) + case (vt_cons s e) + interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp) + assume vt: "vt s" + and ih: "\th. cntP s th = cntV s th + + (if (th \ readys s \ th \ threads s) then cntCS s th else cntCS s th + 1)" + and stp: "step s e" + from stp show ?case + proof(cases) + case (thread_create thread prio) + assume eq_e: "e = Create thread prio" + and not_in: "thread \ threads s" + show ?thesis + proof - + { fix cs + assume "thread \ set (wq s cs)" + from vt_s.wq_threads [OF this] have "thread \ threads s" . + with not_in have "False" by simp + } with eq_e have eq_readys: "readys (e#s) = readys s \ {thread}" + by (auto simp:readys_def threads.simps s_waiting_def + wq_def cs_waiting_def Let_def) + from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def) + from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def) + have eq_cncs: "cntCS (e#s) th = cntCS s th" + unfolding cntCS_def holdents_test + by (simp add:RAG_create_unchanged eq_e) + { assume "th \ thread" + with eq_readys eq_e + have "(th \ readys (e # s) \ th \ threads (e # s)) = + (th \ readys (s) \ th \ threads (s))" + by (simp add:threads.simps) + with eq_cnp eq_cnv eq_cncs ih not_in + have ?thesis by simp + } moreover { + assume eq_th: "th = thread" + with not_in ih have " cntP s th = cntV s th + cntCS s th" by simp + moreover from eq_th and eq_readys have "th \ readys (e#s)" by simp + moreover note eq_cnp eq_cnv eq_cncs + ultimately have ?thesis by auto + } ultimately show ?thesis by blast + qed + next + case (thread_exit thread) + assume eq_e: "e = Exit thread" + and is_runing: "thread \ runing s" + and no_hold: "holdents s thread = {}" + from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def) + from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def) + have eq_cncs: "cntCS (e#s) th = cntCS s th" + unfolding cntCS_def holdents_test + by (simp add:RAG_exit_unchanged eq_e) + { assume "th \ thread" + with eq_e + have "(th \ readys (e # s) \ th \ threads (e # s)) = + (th \ readys (s) \ th \ threads (s))" + apply (simp add:threads.simps readys_def) + apply (subst s_waiting_def) + apply (simp add:Let_def) + apply (subst s_waiting_def, simp) + done + with eq_cnp eq_cnv eq_cncs ih + have ?thesis by simp + } moreover { + assume eq_th: "th = thread" + with ih is_runing have " cntP s th = cntV s th + cntCS s th" + by (simp add:runing_def) + moreover from eq_th eq_e have "th \ threads (e#s)" + by simp + moreover note eq_cnp eq_cnv eq_cncs + ultimately have ?thesis by auto + } ultimately show ?thesis by blast + next + case (thread_P thread cs) + assume eq_e: "e = P thread cs" + and is_runing: "thread \ runing s" + and no_dep: "(Cs cs, Th thread) \ (RAG s)\<^sup>+" + from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto + then interpret vt_p: valid_trace "(P thread cs#s)" + by (unfold_locales, simp) + show ?thesis + proof - + { have hh: "\ A B C. (B = C) \ (A \ B) = (A \ C)" by blast + assume neq_th: "th \ thread" + with eq_e + have eq_readys: "(th \ readys (e#s)) = (th \ readys (s))" + apply (simp add:readys_def s_waiting_def wq_def Let_def) + apply (rule_tac hh) + apply (intro iffI allI, clarify) + apply (erule_tac x = csa in allE, auto) + apply (subgoal_tac "wq_fun (schs s) cs \ []", auto) + apply (erule_tac x = cs in allE, auto) + by (case_tac "(wq_fun (schs s) cs)", auto) + moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th" + apply (simp add:cntCS_def holdents_test) + by (unfold step_RAG_p [OF vtp], auto) + moreover from eq_e neq_th have "cntP (e # s) th = cntP s th" + by (simp add:cntP_def count_def) + moreover from eq_e neq_th have "cntV (e#s) th = cntV s th" + by (simp add:cntV_def count_def) + moreover from eq_e neq_th have "threads (e#s) = threads s" by simp + moreover note ih [of th] + ultimately have ?thesis by simp + } moreover { + assume eq_th: "th = thread" + have ?thesis + proof - + from eq_e eq_th have eq_cnp: "cntP (e # s) th = 1 + (cntP s th)" + by (simp add:cntP_def count_def) + from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th" + by (simp add:cntV_def count_def) + show ?thesis + proof (cases "wq s cs = []") + case True + with is_runing + have "th \ readys (e#s)" + apply (unfold eq_e wq_def, unfold readys_def s_RAG_def) + apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def) + by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def) + moreover have "cntCS (e # s) th = 1 + cntCS s th" + proof - + have "card {csa. csa = cs \ (Cs csa, Th thread) \ RAG s} = + Suc (card {cs. (Cs cs, Th thread) \ RAG s})" (is "card ?L = Suc (card ?R)") + proof - + have "?L = insert cs ?R" by auto + moreover have "card \ = Suc (card (?R - {cs}))" + proof(rule card_insert) + from vt_s.finite_holding [of thread] + show " finite {cs. (Cs cs, Th thread) \ RAG s}" + by (unfold holdents_test, simp) + qed + moreover have "?R - {cs} = ?R" + proof - + have "cs \ ?R" + proof + assume "cs \ {cs. (Cs cs, Th thread) \ RAG s}" + with no_dep show False by auto + qed + thus ?thesis by auto + qed + ultimately show ?thesis by auto + qed + thus ?thesis + apply (unfold eq_e eq_th cntCS_def) + apply (simp add: holdents_test) + by (unfold step_RAG_p [OF vtp], auto simp:True) + qed + moreover from is_runing have "th \ readys s" + by (simp add:runing_def eq_th) + moreover note eq_cnp eq_cnv ih [of th] + ultimately show ?thesis by auto + next + case False + have eq_wq: "wq (e#s) cs = wq s cs @ [th]" + by (unfold eq_th eq_e wq_def, auto simp:Let_def) + have "th \ readys (e#s)" + proof + assume "th \ readys (e#s)" + hence "\cs. \ waiting (e # s) th cs" by (simp add:readys_def) + from this[rule_format, of cs] have " \ waiting (e # s) th cs" . + hence "th \ set (wq (e#s) cs) \ th = hd (wq (e#s) cs)" + by (simp add:s_waiting_def wq_def) + moreover from eq_wq have "th \ set (wq (e#s) cs)" by auto + ultimately have "th = hd (wq (e#s) cs)" by blast + with eq_wq have "th = hd (wq s cs @ [th])" by simp + hence "th = hd (wq s cs)" using False by auto + with False eq_wq vt_p.wq_distinct [of cs] + show False by (fold eq_e, auto) + qed + moreover from is_runing have "th \ threads (e#s)" + by (unfold eq_e, auto simp:runing_def readys_def eq_th) + moreover have "cntCS (e # s) th = cntCS s th" + apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp]) + by (auto simp:False) + moreover note eq_cnp eq_cnv ih[of th] + moreover from is_runing have "th \ readys s" + by (simp add:runing_def eq_th) + ultimately show ?thesis by auto + qed + qed + } ultimately show ?thesis by blast + qed next - assume "cs' = cs" - thus ?thesis using holding_es_th_cs - by (unfold holdents_def, auto) + case (thread_V thread cs) + from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto + then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp) + assume eq_e: "e = V thread cs" + and is_runing: "thread \ runing s" + and hold: "holding s thread cs" + from hold obtain rest + where eq_wq: "wq s cs = thread # rest" + by (case_tac "wq s cs", auto simp: wq_def s_holding_def) + have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e) + have eq_set: "set (SOME q. distinct q \ set q = set rest) = set rest" + proof(rule someI2) + from vt_v.wq_distinct[of cs] and eq_wq + show "distinct rest \ set rest = set rest" + by (metis distinct.simps(2) vt_s.wq_distinct) + next + show "\x. distinct x \ set x = set rest \ set x = set rest" + by auto + qed + show ?thesis + proof - + { assume eq_th: "th = thread" + from eq_th have eq_cnp: "cntP (e # s) th = cntP s th" + by (unfold eq_e, simp add:cntP_def count_def) + moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th" + by (unfold eq_e, simp add:cntV_def count_def) + moreover from cntCS_v_dec [OF vtv] + have "cntCS (e # s) thread + 1 = cntCS s thread" + by (simp add:eq_e) + moreover from is_runing have rd_before: "thread \ readys s" + by (unfold runing_def, simp) + moreover have "thread \ readys (e # s)" + proof - + from is_runing + have "thread \ threads (e#s)" + by (unfold eq_e, auto simp:runing_def readys_def) + moreover have "\ cs1. \ waiting (e#s) thread cs1" + proof + fix cs1 + { assume eq_cs: "cs1 = cs" + have "\ waiting (e # s) thread cs1" + proof - + from eq_wq + have "thread \ set (wq (e#s) cs1)" + apply(unfold eq_e wq_def eq_cs s_holding_def) + apply (auto simp:Let_def) + proof - + assume "thread \ set (SOME q. distinct q \ set q = set rest)" + with eq_set have "thread \ set rest" by simp + with vt_v.wq_distinct[of cs] + and eq_wq show False + by (metis distinct.simps(2) vt_s.wq_distinct) + qed + thus ?thesis by (simp add:wq_def s_waiting_def) + qed + } moreover { + assume neq_cs: "cs1 \ cs" + have "\ waiting (e # s) thread cs1" + proof - + from wq_v_neq [OF neq_cs[symmetric]] + have "wq (V thread cs # s) cs1 = wq s cs1" . + moreover have "\ waiting s thread cs1" + proof - + from runing_ready and is_runing + have "thread \ readys s" by auto + thus ?thesis by (simp add:readys_def) + qed + ultimately show ?thesis + by (auto simp:wq_def s_waiting_def eq_e) + qed + } ultimately show "\ waiting (e # s) thread cs1" by blast + qed + ultimately show ?thesis by (simp add:readys_def) + qed + moreover note eq_th ih + ultimately have ?thesis by auto + } moreover { + assume neq_th: "th \ thread" + from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" + by (simp add:cntP_def count_def) + from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" + by (simp add:cntV_def count_def) + have ?thesis + proof(cases "th \ set rest") + case False + have "(th \ readys (e # s)) = (th \ readys s)" + apply (insert step_back_vt[OF vtv]) + by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq) + moreover have "cntCS (e#s) th = cntCS s th" + apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto) + proof - + have "{csa. (Cs csa, Th th) \ RAG s \ csa = cs \ next_th s thread cs th} = + {cs. (Cs cs, Th th) \ RAG s}" + proof - + from False eq_wq + have " next_th s thread cs th \ (Cs cs, Th th) \ RAG s" + apply (unfold next_th_def, auto) + proof - + assume ne: "rest \ []" + and ni: "hd (SOME q. distinct q \ set q = set rest) \ set rest" + and eq_wq: "wq s cs = thread # rest" + from eq_set ni have "hd (SOME q. distinct q \ set q = set rest) \ + set (SOME q. distinct q \ set q = set rest) + " by simp + moreover have "(SOME q. distinct q \ set q = set rest) \ []" + proof(rule someI2) + from vt_s.wq_distinct[ of cs] and eq_wq + show "distinct rest \ set rest = set rest" by auto + next + fix x assume "distinct x \ set x = set rest" + with ne show "x \ []" by auto + qed + ultimately show + "(Cs cs, Th (hd (SOME q. distinct q \ set q = set rest))) \ RAG s" + by auto + qed + thus ?thesis by auto + qed + thus "card {csa. (Cs csa, Th th) \ RAG s \ csa = cs \ next_th s thread cs th} = + card {cs. (Cs cs, Th th) \ RAG s}" by simp + qed + moreover note ih eq_cnp eq_cnv eq_threads + ultimately show ?thesis by auto + next + case True + assume th_in: "th \ set rest" + show ?thesis + proof(cases "next_th s thread cs th") + case False + with eq_wq and th_in have + neq_hd: "th \ hd (SOME q. distinct q \ set q = set rest)" (is "th \ hd ?rest") + by (auto simp:next_th_def) + have "(th \ readys (e # s)) = (th \ readys s)" + proof - + from eq_wq and th_in + have "\ th \ readys s" + apply (auto simp:readys_def s_waiting_def) + apply (rule_tac x = cs in exI, auto) + by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def) + moreover + from eq_wq and th_in and neq_hd + have "\ (th \ readys (e # s))" + apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits) + by (rule_tac x = cs in exI, auto simp:eq_set) + ultimately show ?thesis by auto + qed + moreover have "cntCS (e#s) th = cntCS s th" + proof - + from eq_wq and th_in and neq_hd + have "(holdents (e # s) th) = (holdents s th)" + apply (unfold eq_e step_RAG_v[OF vtv], + auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def + Let_def cs_holding_def) + by (insert vt_s.wq_distinct[of cs], auto simp:wq_def) + thus ?thesis by (simp add:cntCS_def) + qed + moreover note ih eq_cnp eq_cnv eq_threads + ultimately show ?thesis by auto + next + case True + let ?rest = " (SOME q. distinct q \ set q = set rest)" + let ?t = "hd ?rest" + from True eq_wq th_in neq_th + have "th \ readys (e # s)" + apply (auto simp:eq_e readys_def s_waiting_def wq_def + Let_def next_th_def) + proof - + assume eq_wq: "wq_fun (schs s) cs = thread # rest" + and t_in: "?t \ set rest" + show "?t \ threads s" + proof(rule vt_s.wq_threads) + from eq_wq and t_in + show "?t \ set (wq s cs)" by (auto simp:wq_def) + qed + next + fix csa + assume eq_wq: "wq_fun (schs s) cs = thread # rest" + and t_in: "?t \ set rest" + and neq_cs: "csa \ cs" + and t_in': "?t \ set (wq_fun (schs s) csa)" + show "?t = hd (wq_fun (schs s) csa)" + proof - + { assume neq_hd': "?t \ hd (wq_fun (schs s) csa)" + from vt_s.wq_distinct[of cs] and + eq_wq[folded wq_def] and t_in eq_wq + have "?t \ thread" by auto + with eq_wq and t_in + have w1: "waiting s ?t cs" + by (auto simp:s_waiting_def wq_def) + from t_in' neq_hd' + have w2: "waiting s ?t csa" + by (auto simp:s_waiting_def wq_def) + from vt_s.waiting_unique[OF w1 w2] + and neq_cs have "False" by auto + } thus ?thesis by auto + qed + qed + moreover have "cntP s th = cntV s th + cntCS s th + 1" + proof - + have "th \ readys s" + proof - + from True eq_wq neq_th th_in + show ?thesis + apply (unfold readys_def s_waiting_def, auto) + by (rule_tac x = cs in exI, auto simp add: wq_def) + qed + moreover have "th \ threads s" + proof - + from th_in eq_wq + have "th \ set (wq s cs)" by simp + from vt_s.wq_threads [OF this] + show ?thesis . + qed + ultimately show ?thesis using ih by auto + qed + moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th" + apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto) + proof - + show "card {csa. (Cs csa, Th th) \ RAG s \ csa = cs} = + Suc (card {cs. (Cs cs, Th th) \ RAG s})" + (is "card ?A = Suc (card ?B)") + proof - + have "?A = insert cs ?B" by auto + hence "card ?A = card (insert cs ?B)" by simp + also have "\ = Suc (card ?B)" + proof(rule card_insert_disjoint) + have "?B \ ((\ (x, y). the_cs x) ` RAG s)" + apply (auto simp:image_def) + by (rule_tac x = "(Cs x, Th th)" in bexI, auto) + with vt_s.finite_RAG + show "finite {cs. (Cs cs, Th th) \ RAG s}" by (auto intro:finite_subset) + next + show "cs \ {cs. (Cs cs, Th th) \ RAG s}" + proof + assume "cs \ {cs. (Cs cs, Th th) \ RAG s}" + hence "(Cs cs, Th th) \ RAG s" by simp + with True neq_th eq_wq show False + by (auto simp:next_th_def s_RAG_def cs_holding_def) + qed + qed + finally show ?thesis . + qed + qed + moreover note eq_cnp eq_cnv + ultimately show ?thesis by simp + qed + qed + } ultimately show ?thesis by blast + qed + next + case (thread_set thread prio) + assume eq_e: "e = Set thread prio" + and is_runing: "thread \ runing s" + show ?thesis + proof - + from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def) + from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def) + have eq_cncs: "cntCS (e#s) th = cntCS s th" + unfolding cntCS_def holdents_test + by (simp add:RAG_set_unchanged eq_e) + from eq_e have eq_readys: "readys (e#s) = readys s" + by (simp add:readys_def cs_waiting_def s_waiting_def wq_def, + auto simp:Let_def) + { assume "th \ thread" + with eq_readys eq_e + have "(th \ readys (e # s) \ th \ threads (e # s)) = + (th \ readys (s) \ th \ threads (s))" + by (simp add:threads.simps) + with eq_cnp eq_cnv eq_cncs ih is_runing + have ?thesis by simp + } moreover { + assume eq_th: "th = thread" + with is_runing ih have " cntP s th = cntV s th + cntCS s th" + by (unfold runing_def, auto) + moreover from eq_th and eq_readys is_runing have "th \ readys (e#s)" + by (simp add:runing_def) + moreover note eq_cnp eq_cnv eq_cncs + ultimately have ?thesis by auto + } ultimately show ?thesis by blast + qed qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th: "cntCS (e#s) th = cntCS s th + 1" -proof - - have "card (holdents s th \ {cs}) = card (holdents s th) + 1" - proof(subst card_Un_disjoint) - show "holdents s th \ {cs} = {}" - using not_holding_s_th_cs by (auto simp:holdents_def) - qed (auto simp:finite_holdents) - thus ?thesis - by (unfold cntCS_def holdents_es_th, simp) -qed - -lemma no_holder: - "\ holding s th' cs" -proof - assume otherwise: "holding s th' cs" - from this[unfolded s_holding_def, folded wq_def, unfolded we] - show False by auto + next + case vt_nil + show ?case + by (unfold cntP_def cntV_def cntCS_def, + auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def) + qed qed -lemma holdents_es_th': - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") +lemma not_thread_cncs: + assumes not_in: "th \ threads s" + shows "cntCS s th = 0" proof - - { fix cs' - assume "cs' \ ?L" - hence h_e: "holding (e#s) th' cs'" by (auto simp:holdents_def) - have "cs' \ cs" - proof - assume "cs' = cs" - from held_unique[OF h_e[unfolded this] holding_es_th_cs] - have "th' = th" . - with assms show False by simp - qed - from h_e[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp[OF this]] - have "th' \ set (wq s cs') \ th' = hd (wq s cs')" . - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, auto) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s th' cs'" by (auto simp:holdents_def) - from holding_kept[OF this] - have "holding (e # s) th' cs'" . - hence "cs' \ ?L" - by (unfold holdents_def, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th'[simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_es_th'[OF assms], simp) - -end - -context valid_trace_p -begin - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) + from vt not_in show ?thesis + proof(induct arbitrary:th) + case (vt_cons s e th) + interpret vt_s: valid_trace s using vt_cons(1) + by (unfold_locales, simp) + assume vt: "vt s" + and ih: "\th. th \ threads s \ cntCS s th = 0" + and stp: "step s e" + and not_in: "th \ threads (e # s)" + from stp show ?case + proof(cases) + case (thread_create thread prio) + assume eq_e: "e = Create thread prio" + and not_in': "thread \ threads s" + have "cntCS (e # s) th = cntCS s th" + apply (unfold eq_e cntCS_def holdents_test) + by (simp add:RAG_create_unchanged) + moreover have "th \ threads s" + proof - + from not_in eq_e show ?thesis by simp + qed + moreover note ih ultimately show ?thesis by auto next - case True + case (thread_exit thread) + assume eq_e: "e = Exit thread" + and nh: "holdents s thread = {}" + have eq_cns: "cntCS (e # s) th = cntCS s th" + apply (unfold eq_e cntCS_def holdents_test) + by (simp add:RAG_exit_unchanged) show ?thesis - proof(cases "wq s cs = []") + proof(cases "th = thread") case True - then interpret vt: valid_trace_p_h - by (unfold_locales, simp) - show ?thesis using n_wait wait waiting_kept by auto + have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True) + with eq_cns show ?thesis by simp next case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis using n_wait wait waiting_kept by blast + with not_in and eq_e + have "th \ threads s" by simp + from ih[OF this] and eq_cns show ?thesis by simp + qed + next + case (thread_P thread cs) + assume eq_e: "e = P thread cs" + and is_runing: "thread \ runing s" + from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto + have neq_th: "th \ thread" + proof - + from not_in eq_e have "th \ threads s" by simp + moreover from is_runing have "thread \ threads s" + by (simp add:runing_def readys_def) + ultimately show ?thesis by auto + qed + hence "cntCS (e # s) th = cntCS s th " + apply (unfold cntCS_def holdents_test eq_e) + by (unfold step_RAG_p[OF vtp], auto) + moreover have "cntCS s th = 0" + proof(rule ih) + from not_in eq_e show "th \ threads s" by simp + qed + ultimately show ?thesis by simp + next + case (thread_V thread cs) + assume eq_e: "e = V thread cs" + and is_runing: "thread \ runing s" + and hold: "holding s thread cs" + have neq_th: "th \ thread" + proof - + from not_in eq_e have "th \ threads s" by simp + moreover from is_runing have "thread \ threads s" + by (simp add:runing_def readys_def) + ultimately show ?thesis by auto qed - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait + from assms thread_V vt stp ih + have vtv: "vt (V thread cs#s)" by auto + then interpret vt_v: valid_trace "(V thread cs#s)" + by (unfold_locales, simp) + from hold obtain rest + where eq_wq: "wq s cs = thread # rest" + by (case_tac "wq s cs", auto simp: wq_def s_holding_def) + from not_in eq_e eq_wq + have "\ next_th s thread cs th" + apply (auto simp:next_th_def) + proof - + assume ne: "rest \ []" + and ni: "hd (SOME q. distinct q \ set q = set rest) \ threads s" (is "?t \ threads s") + have "?t \ set rest" + proof(rule someI2) + from vt_v.wq_distinct[of cs] and eq_wq + show "distinct rest \ set rest = set rest" + by (metis distinct.simps(2) vt_s.wq_distinct) + next + fix x assume "distinct x \ set x = set rest" with ne + show "hd x \ set rest" by (cases x, auto) + qed + with eq_wq have "?t \ set (wq s cs)" by simp + from vt_s.wq_threads[OF this] and ni + show False + using `hd (SOME q. distinct q \ set q = set rest) \ set (wq s cs)` + ni vt_s.wq_threads by blast + qed + moreover note neq_th eq_wq + ultimately have "cntCS (e # s) th = cntCS s th" + by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto) + moreover have "cntCS s th = 0" + proof(rule ih) + from not_in eq_e show "th \ threads s" by simp + qed + ultimately show ?thesis by simp + next + case (thread_set thread prio) + print_facts + assume eq_e: "e = Set thread prio" + and is_runing: "thread \ runing s" + from not_in and eq_e have "th \ threads s" by auto + from ih [OF this] and eq_e show ?thesis - 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 + apply (unfold eq_e cntCS_def holdents_test) + by (simp add:RAG_set_unchanged) qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: (* ddd *) - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof(cases "th' = th") - case True - note eq_th' = this - show ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h by (unfold_locales, simp) - show ?thesis - using assms eq_th' is_p ready_th_s vt.cntCS_es_th vt.ready_th_es pvD_def by auto - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis - using add.commute add.left_commute assms eq_th' is_p live_th_s - ready_th_s vt.th_not_ready_es pvD_def - apply (auto) - by (fold is_p, simp) - qed -next - case False - note h_False = False - thus ?thesis - proof(cases "wq s cs = []") - case True - then interpret vt: valid_trace_p_h by (unfold_locales, simp) - show ?thesis using assms - by (insert True h_False pvD_def, auto split:if_splits,unfold is_p, auto) - next - case False - then interpret vt: valid_trace_p_w by (unfold_locales, simp) - show ?thesis using assms - by (insert False h_False pvD_def, auto split:if_splits,unfold is_p, auto) + next + case vt_nil + show ?case + by (unfold cntCS_def, + auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def) qed qed end - -context valid_trace_v (* ccc *) -begin - -lemma holding_th_cs_s: - "holding s th cs" - by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto) - -lemma th_ready_s [simp]: "th \ readys s" - using runing_th_s - by (unfold runing_def readys_def, auto) - -lemma th_live_s [simp]: "th \ threads s" - using th_ready_s by (unfold readys_def, auto) - -lemma th_ready_es [simp]: "th \ readys (e#s)" - using runing_th_s neq_t_th - by (unfold is_v runing_def readys_def, auto) +lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs" + by (auto simp:s_waiting_def cs_waiting_def wq_def) -lemma th_live_es [simp]: "th \ threads (e#s)" - using th_ready_es by (unfold readys_def, auto) - -lemma pvD_th_s[simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es[simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma cntCS_s_th [simp]: "cntCS s th > 0" -proof - - have "cs \ holdents s th" using holding_th_cs_s - by (unfold holdents_def, simp) - moreover have "finite (holdents s th)" using finite_holdents - by simp - ultimately show ?thesis - by (unfold cntCS_def, - auto intro!:card_gt_0_iff[symmetric, THEN iffD1]) -qed - -end - -context valid_trace_v_n +context valid_trace begin -lemma not_ready_taker_s[simp]: - "taker \ readys s" - using waiting_taker - by (unfold readys_def, auto) - -lemma taker_live_s [simp]: "taker \ threads s" -proof - - have "taker \ set wq'" by (simp add: eq_wq') - from th'_in_inv[OF this] - have "taker \ set rest" . - hence "taker \ set (wq s cs)" by (simp add: wq_s_cs) - thus ?thesis using wq_threads by auto -qed - -lemma taker_live_es [simp]: "taker \ threads (e#s)" - using taker_live_s threads_es by blast - -lemma taker_ready_es [simp]: - shows "taker \ readys (e#s)" -proof - - { fix cs' - assume "waiting (e#s) taker cs'" - hence False - proof(cases rule:waiting_esE) - case 1 - thus ?thesis using waiting_taker waiting_unique by auto - qed simp - } thus ?thesis by (unfold readys_def, auto) -qed - -lemma neq_taker_th: "taker \ th" - using th_not_waiting waiting_taker by blast - -lemma not_holding_taker_s_cs: - shows "\ holding s taker cs" - using holding_cs_eq_th neq_taker_th by auto - -lemma holdents_es_taker: - "holdents (e#s) taker = holdents s taker \ {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) taker cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 2 - thus ?thesis by (auto simp:holdents_def) - qed auto - } moreover { - fix cs' - assume "cs' \ ?R" - hence "holding s taker cs' \ cs' = cs" by (auto simp:holdents_def) - hence "cs' \ ?L" - proof - assume "holding s taker cs'" - hence "holding (e#s) taker cs'" - using holding_esI2 holding_taker by fastforce - thus ?thesis by (auto simp:holdents_def) - next - assume "cs' = cs" - with holding_taker - show ?thesis by (auto simp:holdents_def) - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_taker [simp]: "cntCS (e#s) taker = cntCS s taker + 1" -proof - - have "card (holdents s taker \ {cs}) = card (holdents s taker) + 1" - proof(subst card_Un_disjoint) - show "holdents s taker \ {cs} = {}" - using not_holding_taker_s_cs by (auto simp:holdents_def) - qed (auto simp:finite_holdents) - thus ?thesis - by (unfold cntCS_def, insert holdents_es_taker, simp) -qed - -lemma pvD_taker_s[simp]: "pvD s taker = 1" - by (unfold pvD_def, simp) - -lemma pvD_taker_es[simp]: "pvD (e#s) taker = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_s[simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es[simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_es_th: - "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 2 - thus ?thesis by (auto simp:holdents_def) - qed (insert neq_taker_th, auto) - } moreover { - fix cs' - assume "cs' \ ?R" - hence "cs' \ cs" "holding s th cs'" by (auto simp:holdents_def) - from holding_esI2[OF this] - have "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1" -proof - - have "card (holdents s th - {cs}) = card (holdents s th) - 1" - proof - - have "cs \ holdents s th" using holding_th_cs_s - by (auto simp:holdents_def) - moreover have "finite (holdents s th)" - by (simp add: finite_holdents) - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold cntCS_def holdents_es_th) -qed - -lemma holdents_kept: - assumes "th' \ taker" - and "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - have "cs' \ ?R" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding (e#s) th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, auto) - next - case True - from h[unfolded this] - have "holding (e#s) th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_taker] - have "th' = taker" . - with assms show ?thesis by auto - qed - } moreover { - fix cs' - assume h: "cs' \ ?R" - have "cs' \ ?L" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding s th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp) - next - case True - from h[unfolded this] - have "holding s th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_th_cs_s] - have "th' = th" . - with assms show ?thesis by auto - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ taker" - and "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_kept[OF assms], simp) - -lemma readys_kept1: - assumes "th' \ taker" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set (th # rest) \ th' \ hd (th # rest)" - using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - moreover have "\ (th' \ set rest \ th' \ hd (taker # rest'))" - using n_wait[unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs set_wq', unfolded eq_wq'] . - ultimately have "th' = taker" by auto - with assms(1) - show ?thesis by simp - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ taker" - and "th' \ readys s" - shows "th' \ readys (e#s)" +lemma dm_RAG_threads: + assumes in_dom: "(Th th) \ Domain (RAG s)" + shows "th \ threads s" proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set rest \ th' \ hd (taker # rest')" - using wait [unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs set_wq', unfolded eq_wq'] . - moreover have "\ (th' \ set (th # rest) \ th' \ hd (th # rest))" - using n_wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - ultimately have "th' = taker" by auto - with assms(1) - show ?thesis by simp - qed - } with assms(2) show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ taker" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { assume eq_th': "th' = taker" - have ?thesis - apply (unfold eq_th' pvD_taker_es cntCS_es_taker) - by (insert neq_taker_th assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume eq_th': "th' = th" - have ?thesis - apply (unfold eq_th' pvD_th_es cntCS_es_th) - by (insert assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume h: "th' \ taker" "th' \ th" - have ?thesis using assms - apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp) - by (fold is_v, unfold pvD_def, simp) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_v_e -begin - -lemma holdents_es_th: - "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R") -proof - - { fix cs' - assume "cs' \ ?L" - hence "holding (e#s) th cs'" by (auto simp:holdents_def) - hence "cs' \ ?R" - proof(cases rule:holding_esE) - case 1 - thus ?thesis by (auto simp:holdents_def) - qed - } moreover { - fix cs' - assume "cs' \ ?R" - hence "cs' \ cs" "holding s th cs'" by (auto simp:holdents_def) - from holding_esI2[OF this] - have "cs' \ ?L" by (auto simp:holdents_def) - } ultimately show ?thesis by auto -qed - -lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1" -proof - - have "card (holdents s th - {cs}) = card (holdents s th) - 1" - proof - - have "cs \ holdents s th" using holding_th_cs_s - by (auto simp:holdents_def) - moreover have "finite (holdents s th)" - by (simp add: finite_holdents) - ultimately show ?thesis by auto - qed - thus ?thesis by (unfold cntCS_def holdents_es_th) -qed - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - have "cs' \ ?R" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding (e#s) th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, auto) - next - case True - from h[unfolded this] - have "holding (e#s) th' cs" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, - unfolded wq_es_cs nil_wq'] - show ?thesis by auto - qed - } moreover { - fix cs' - assume h: "cs' \ ?R" - have "cs' \ ?L" - proof(cases "cs' = cs") - case False - hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp - from h have "holding s th' cs'" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq] - show ?thesis - by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp) - next - case True - from h[unfolded this] - have "holding s th' cs" by (auto simp:holdents_def) - from held_unique[OF this holding_th_cs_s] - have "th' = th" . - with assms show ?thesis by auto - qed - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" - by (unfold cntCS_def holdents_kept[OF assms], simp) - -lemma readys_kept1: - assumes "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms(1)[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set (th # rest) \ th' \ hd (th # rest)" - using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] . - hence "th' \ set rest" by auto - with set_wq' have "th' \ set wq'" by metis - with nil_wq' show ?thesis by simp - qed - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms[unfolded readys_def] by auto - have False - proof(cases "cs' = cs") - case False - with n_wait wait - show ?thesis - by (unfold s_waiting_def, fold wq_def, auto) - next - case True - have "th' \ set [] \ th' \ hd []" - using wait[unfolded True s_waiting_def, folded wq_def, - unfolded wq_es_cs nil_wq'] . - thus ?thesis by simp - qed - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis - apply (unfold eq_th' pvD_th_es cntCS_es_th) - by (insert assms[unfolded eq_th'], unfold is_v, simp) - } moreover { - assume h: "th' \ th" - have ?thesis using assms - apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp) - by (fold is_v, unfold pvD_def, simp) - } ultimately show ?thesis by metis + from in_dom obtain n where "(Th th, n) \ RAG s" by auto + moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto + ultimately have "(Th th, Cs cs) \ RAG s" by simp + hence "th \ set (wq s cs)" + by (unfold s_RAG_def, auto simp:cs_waiting_def) + from wq_threads [OF this] show ?thesis . qed end -context valid_trace_v -begin - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof(cases "rest = []") - case True - then interpret vt: valid_trace_v_e by (unfold_locales, simp) - show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast -next - case False - then interpret vt: valid_trace_v_n by (unfold_locales, simp) - show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast -qed - -end - -context valid_trace_create -begin - -lemma th_not_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_create] - show ?thesis by (cases, simp) -qed - -lemma th_not_ready_s [simp]: "th \ readys s" - using th_not_live_s by (unfold readys_def, simp) - -lemma th_live_es [simp]: "th \ threads (e#s)" - by (unfold is_create, simp) - -lemma not_waiting_th_s [simp]: "\ waiting s th cs'" -proof - assume "waiting s th cs'" - from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have "th \ set (wq s cs')" by auto - from wq_threads[OF this] have "th \ threads s" . - with th_not_live_s show False by simp -qed - -lemma not_holding_th_s [simp]: "\ holding s th cs'" -proof - assume "holding s th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] - have "th \ set (wq s cs')" by auto - from wq_threads[OF this] have "th \ threads s" . - with th_not_live_s show False by simp -qed - -lemma not_waiting_th_es [simp]: "\ waiting (e#s) th cs'" -proof - assume "waiting (e # s) th cs'" - from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have "th \ set (wq s cs')" by auto - from wq_threads[OF this] have "th \ threads s" . - with th_not_live_s show False by simp -qed - -lemma not_holding_th_es [simp]: "\ holding (e#s) th cs'" -proof - assume "holding (e # s) th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] - have "th \ set (wq s cs')" by auto - from wq_threads[OF this] have "th \ threads s" . - with th_not_live_s show False by simp -qed - -lemma ready_th_es [simp]: "th \ readys (e#s)" - by (simp add:readys_def) - -lemma holdents_th_s: "holdents s th = {}" - by (unfold holdents_def, auto) - -lemma holdents_th_es: "holdents (e#s) th = {}" - by (unfold holdents_def, auto) - -lemma cntCS_th_s [simp]: "cntCS s th = 0" - by (unfold cntCS_def, simp add:holdents_th_s) - -lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0" - by (unfold cntCS_def, simp add:holdents_th_es) - -lemma pvD_th_s [simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es [simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept[OF assms] - by (unfold cntCS_def, simp) - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have False by auto - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2) by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma pvD_kept [simp]: - assumes "th' \ th" - shows "pvD (e#s) th' = pvD s th'" - using assms - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis using assms - by (unfold eq_th', simp, unfold is_create, simp) - } moreover { - assume h: "th' \ th" - hence ?thesis using assms - by (simp, simp add:is_create) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_exit -begin - -lemma th_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold runing_def readys_def, simp) -qed - -lemma th_ready_s [simp]: "th \ readys s" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold runing_def, simp) -qed - -lemma th_not_live_es [simp]: "th \ threads (e#s)" - by (unfold is_exit, simp) - -lemma not_holding_th_s [simp]: "\ holding s th cs'" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold holdents_def, auto) -qed - -lemma cntCS_th_s [simp]: "cntCS s th = 0" -proof - - from pip_e[unfolded is_exit] - show ?thesis - by (cases, unfold cntCS_def, simp) -qed - -lemma not_holding_th_es [simp]: "\ holding (e#s) th cs'" -proof - assume "holding (e # s) th cs'" - from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp] - have "holding s th cs'" - by (unfold s_holding_def, fold wq_def, auto) - with not_holding_th_s - show False by simp -qed - -lemma ready_th_es [simp]: "th \ readys (e#s)" - by (simp add:readys_def) - -lemma holdents_th_s: "holdents s th = {}" - by (unfold holdents_def, auto) - -lemma holdents_th_es: "holdents (e#s) th = {}" - by (unfold holdents_def, auto) - -lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0" - by (unfold cntCS_def, simp add:holdents_th_es) - -lemma pvD_th_s [simp]: "pvD s th = 0" - by (unfold pvD_def, simp) - -lemma pvD_th_es [simp]: "pvD (e#s) th = 0" - by (unfold pvD_def, simp) - -lemma holdents_kept: - assumes "th' \ th" - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - assumes "th' \ th" - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept[OF assms] - by (unfold cntCS_def, simp) - -lemma readys_kept1: - assumes "th' \ th" - and "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have False by auto - } thus ?thesis using assms - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ th" - and "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms(2) by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - assumes "th' \ th" - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1[OF assms] readys_kept2[OF assms] - by metis - -lemma pvD_kept [simp]: - assumes "th' \ th" - shows "pvD (e#s) th' = pvD s th'" - using assms - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" -proof - - { - assume eq_th': "th' = th" - have ?thesis using assms - by (unfold eq_th', simp, unfold is_exit, simp) - } moreover { - assume h: "th' \ th" - hence ?thesis using assms - by (simp, simp add:is_exit) - } ultimately show ?thesis by metis -qed - -end - -context valid_trace_set -begin - -lemma th_live_s [simp]: "th \ threads s" -proof - - from pip_e[unfolded is_set] - show ?thesis - by (cases, unfold runing_def readys_def, simp) -qed - -lemma th_ready_s [simp]: "th \ readys s" -proof - - from pip_e[unfolded is_set] - show ?thesis - by (cases, unfold runing_def, simp) -qed - -lemma th_not_live_es [simp]: "th \ threads (e#s)" - by (unfold is_set, simp) - - -lemma holdents_kept: - shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R") -proof - - { fix cs' - assume h: "cs' \ ?L" - hence "cs' \ ?R" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } moreover { - fix cs' - assume h: "cs' \ ?R" - hence "cs' \ ?L" - by (unfold holdents_def s_holding_def, fold wq_def, - unfold wq_neq_simp, auto) - } ultimately show ?thesis by auto -qed - -lemma cntCS_kept [simp]: - shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R") - using holdents_kept - by (unfold cntCS_def, simp) - -lemma threads_kept[simp]: - "threads (e#s) = threads s" - by (unfold is_set, simp) - -lemma readys_kept1: - assumes "th' \ readys (e#s)" - shows "th' \ readys s" -proof - - { fix cs' - assume wait: "waiting s th' cs'" - have n_wait: "\ waiting (e#s) th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def] - n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - have False by auto - } moreover have "th' \ threads s" - using assms[unfolded readys_def] by auto - ultimately show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_kept2: - assumes "th' \ readys s" - shows "th' \ readys (e#s)" -proof - - { fix cs' - assume wait: "waiting (e#s) th' cs'" - have n_wait: "\ waiting s th' cs'" - using assms by (auto simp:readys_def) - from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp] - n_wait[unfolded s_waiting_def, folded wq_def] - have False by auto - } with assms show ?thesis - by (unfold readys_def, auto) -qed - -lemma readys_simp [simp]: - shows "(th' \ readys (e#s)) = (th' \ readys s)" - using readys_kept1 readys_kept2 - by metis - -lemma pvD_kept [simp]: - shows "pvD (e#s) th' = pvD s th'" - by (unfold pvD_def, simp) - -lemma cnp_cnv_cncs_kept: - assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" - shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'" - using assms - by (unfold is_set, simp, fold is_set, simp) - -end +lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th" +unfolding cp_def wq_def +apply(induct s rule: schs.induct) +thm cpreced_initial +apply(simp add: Let_def cpreced_initial) +apply(simp add: Let_def) +apply(simp add: Let_def) +apply(simp add: Let_def) +apply(subst (2) schs.simps) +apply(simp add: Let_def) +apply(subst (2) schs.simps) +apply(simp add: Let_def) +done context valid_trace begin -lemma cnp_cnv_cncs: "cntP s th' = cntV s th' + cntCS s th' + pvD s th'" -proof(induct rule:ind) - case Nil - thus ?case - by (unfold cntP_def cntV_def pvD_def cntCS_def holdents_def - s_holding_def, simp) -next - case (Cons s e) - interpret vt_e: valid_trace_e s e using Cons by simp - show ?case - proof(cases e) - case (Create th prio) - interpret vt_create: valid_trace_create s e th prio - using Create by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_create.cnp_cnv_cncs_kept) - next - case (Exit th) - interpret vt_exit: valid_trace_exit s e th - using Exit by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_exit.cnp_cnv_cncs_kept) - next - case (P th cs) - interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_p.cnp_cnv_cncs_kept) - next - case (V th cs) - interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_v.cnp_cnv_cncs_kept) - next - case (Set th prio) - interpret vt_set: valid_trace_set s e th prio - using Set by (unfold_locales, simp) - show ?thesis using Cons by (simp add: vt_set.cnp_cnv_cncs_kept) - qed -qed - -lemma not_thread_holdents: - assumes not_in: "th \ threads s" - shows "holdents s th = {}" -proof - - { fix cs - assume "cs \ holdents s th" - hence "holding s th cs" by (auto simp:holdents_def) - from this[unfolded s_holding_def, folded wq_def] - have "th \ set (wq s cs)" by auto - with wq_threads have "th \ threads s" by auto - with assms - have False by simp - } thus ?thesis by auto -qed - -lemma not_thread_cncs: - assumes not_in: "th \ threads s" - shows "cntCS s th = 0" - using not_thread_holdents[OF assms] - by (simp add:cntCS_def) - -lemma cnp_cnv_eq: - assumes "th \ threads s" - shows "cntP s th = cntV s th" - using assms cnp_cnv_cncs not_thread_cncs pvD_def - by (auto) - lemma runing_unique: assumes runing_1: "th1 \ runing s" and runing_2: "th2 \ runing s" shows "th1 = th2" proof - from runing_1 and runing_2 have "cp s th1 = cp s th2" - unfolding runing_def by auto - from this[unfolded cp_alt_def] - have eq_max: - "Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th1)}) = - Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th2)})" - (is "Max ?L = Max ?R") . - have "Max ?L \ ?L" - proof(rule Max_in) - show "finite ?L" by (simp add: finite_subtree_threads) - next - show "?L \ {}" using subtree_def by fastforce + unfolding runing_def + apply(simp) + done + hence eq_max: "Max ((\th. preced th s) ` ({th1} \ dependants (wq s) th1)) = + Max ((\th. preced th s) ` ({th2} \ dependants (wq s) th2))" + (is "Max (?f ` ?A) = Max (?f ` ?B)") + unfolding cp_eq_cpreced + unfolding cpreced_def . + obtain th1' where th1_in: "th1' \ ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)" + proof - + have h1: "finite (?f ` ?A)" + proof - + have "finite ?A" + proof - + have "finite (dependants (wq s) th1)" + proof- + have "finite {th'. (Th th', Th th1) \ (RAG (wq s))\<^sup>+}" + proof - + let ?F = "\ (x, y). the_th x" + have "{th'. (Th th', Th th1) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (wq s))\<^sup>+)" + apply (auto simp:image_def) + by (rule_tac x = "(Th x, Th th1)" in bexI, auto) + moreover have "finite \" + proof - + from finite_RAG have "finite (RAG s)" . + hence "finite ((RAG (wq s))\<^sup>+)" + apply (unfold finite_trancl) + by (auto simp: s_RAG_def cs_RAG_def wq_def) + thus ?thesis by auto + qed + ultimately show ?thesis by (auto intro:finite_subset) + qed + thus ?thesis by (simp add:cs_dependants_def) + qed + thus ?thesis by simp + qed + thus ?thesis by auto + qed + moreover have h2: "(?f ` ?A) \ {}" + proof - + have "?A \ {}" by simp + thus ?thesis by simp + qed + from Max_in [OF h1 h2] + have "Max (?f ` ?A) \ (?f ` ?A)" . + thus ?thesis + thm cpreced_def + unfolding cpreced_def[symmetric] + unfolding cp_eq_cpreced[symmetric] + unfolding cpreced_def + using that[intro] by (auto) qed - then obtain th1' where - h_1: "Th th1' \ subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L" - by auto - have "Max ?R \ ?R" - proof(rule Max_in) - show "finite ?R" by (simp add: finite_subtree_threads) - next - show "?R \ {}" using subtree_def by fastforce + obtain th2' where th2_in: "th2' \ ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)" + proof - + have h1: "finite (?f ` ?B)" + proof - + have "finite ?B" + proof - + have "finite (dependants (wq s) th2)" + proof- + have "finite {th'. (Th th', Th th2) \ (RAG (wq s))\<^sup>+}" + proof - + let ?F = "\ (x, y). the_th x" + have "{th'. (Th th', Th th2) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (wq s))\<^sup>+)" + apply (auto simp:image_def) + by (rule_tac x = "(Th x, Th th2)" in bexI, auto) + moreover have "finite \" + proof - + from finite_RAG have "finite (RAG s)" . + hence "finite ((RAG (wq s))\<^sup>+)" + apply (unfold finite_trancl) + by (auto simp: s_RAG_def cs_RAG_def wq_def) + thus ?thesis by auto + qed + ultimately show ?thesis by (auto intro:finite_subset) + qed + thus ?thesis by (simp add:cs_dependants_def) + qed + thus ?thesis by simp + qed + thus ?thesis by auto + qed + moreover have h2: "(?f ` ?B) \ {}" + proof - + have "?B \ {}" by simp + thus ?thesis by simp + qed + from Max_in [OF h1 h2] + have "Max (?f ` ?B) \ (?f ` ?B)" . + thus ?thesis by (auto intro:that) qed - then obtain th2' where - h_2: "Th th2' \ subtree (RAG s) (Th th2)" "the_preced s th2' = Max ?R" - by auto - have "th1' = th2'" - proof(rule preced_unique) - from h_1(1) - show "th1' \ threads s" - proof(cases rule:subtreeE) - case 1 - hence "th1' = th1" by simp - with runing_1 show ?thesis by (auto simp:runing_def readys_def) + from eq_f_th1 eq_f_th2 eq_max + have eq_preced: "preced th1' s = preced th2' s" by auto + hence eq_th12: "th1' = th2'" + proof (rule preced_unique) + from th1_in have "th1' = th1 \ (th1' \ dependants (wq s) th1)" by simp + thus "th1' \ threads s" + proof + assume "th1' \ dependants (wq s) th1" + hence "(Th th1') \ Domain ((RAG s)^+)" + apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) + by (auto simp:Domain_def) + hence "(Th th1') \ Domain (RAG s)" by (simp add:trancl_domain) + from dm_RAG_threads[OF this] show ?thesis . next - case 2 - from this(2) - have "(Th th1', Th th1) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - have "(Th th1') \ Domain (RAG s)" by auto - from dm_RAG_threads[OF this] show ?thesis . + assume "th1' = th1" + with runing_1 show ?thesis + by (unfold runing_def readys_def, auto) qed next - from h_2(1) - show "th2' \ threads s" - proof(cases rule:subtreeE) - case 1 - hence "th2' = th2" by simp - with runing_2 show ?thesis by (auto simp:runing_def readys_def) + from th2_in have "th2' = th2 \ (th2' \ dependants (wq s) th2)" by simp + thus "th2' \ threads s" + proof + assume "th2' \ dependants (wq s) th2" + hence "(Th th2') \ Domain ((RAG s)^+)" + apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) + by (auto simp:Domain_def) + hence "(Th th2') \ Domain (RAG s)" by (simp add:trancl_domain) + from dm_RAG_threads[OF this] show ?thesis . next - case 2 - from this(2) - have "(Th th2', Th th2) \ (RAG s)^+" by (auto simp:ancestors_def) - from tranclD[OF this] - have "(Th th2') \ Domain (RAG s)" by auto - from dm_RAG_threads[OF this] show ?thesis . + assume "th2' = th2" + with runing_2 show ?thesis + by (unfold runing_def readys_def, auto) + qed + qed + from th1_in have "th1' = th1 \ th1' \ dependants (wq s) th1" by simp + thus ?thesis + proof + assume eq_th': "th1' = th1" + from th2_in have "th2' = th2 \ th2' \ dependants (wq s) th2" by simp + thus ?thesis + proof + assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp + next + assume "th2' \ dependants (wq s) th2" + with eq_th12 eq_th' have "th1 \ dependants (wq s) th2" by simp + hence "(Th th1, Th th2) \ (RAG s)^+" + by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) + hence "Th th1 \ Domain ((RAG s)^+)" + apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) + by (auto simp:Domain_def) + hence "Th th1 \ Domain (RAG s)" by (simp add:trancl_domain) + then obtain n where d: "(Th th1, n) \ RAG s" by (auto simp:Domain_def) + from RAG_target_th [OF this] + obtain cs' where "n = Cs cs'" by auto + with d have "(Th th1, Cs cs') \ RAG s" by simp + with runing_1 have "False" + apply (unfold runing_def readys_def s_RAG_def) + by (auto simp:eq_waiting) + thus ?thesis by simp qed next - have "the_preced s th1' = the_preced s th2'" - using eq_max h_1(2) h_2(2) by metis - thus "preced th1' s = preced th2' s" by (simp add:the_preced_def) - qed - from h_1(1)[unfolded this] - have star1: "(Th th2', Th th1) \ (RAG s)^*" by (auto simp:subtree_def) - from h_2(1)[unfolded this] - have star2: "(Th th2', Th th2) \ (RAG s)^*" by (auto simp:subtree_def) - from star_rpath[OF star1] obtain xs1 - where rp1: "rpath (RAG s) (Th th2') xs1 (Th th1)" - by auto - from star_rpath[OF star2] obtain xs2 - where rp2: "rpath (RAG s) (Th th2') xs2 (Th th2)" - by auto - from rp1 rp2 - show ?thesis - proof(cases) - case (less_1 xs') - moreover have "xs' = []" - proof(rule ccontr) - assume otherwise: "xs' \ []" - from rpath_plus[OF less_1(3) this] - have "(Th th1, Th th2) \ (RAG s)\<^sup>+" . - from tranclD[OF this] - obtain cs where "waiting s th1 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - with runing_1 show False - by (unfold runing_def readys_def, auto) + assume th1'_in: "th1' \ dependants (wq s) th1" + from th2_in have "th2' = th2 \ th2' \ dependants (wq s) th2" by simp + thus ?thesis + proof + assume "th2' = th2" + with th1'_in eq_th12 have "th2 \ dependants (wq s) th1" by simp + hence "(Th th2, Th th1) \ (RAG s)^+" + by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) + hence "Th th2 \ Domain ((RAG s)^+)" + apply (unfold cs_dependants_def cs_RAG_def s_RAG_def) + by (auto simp:Domain_def) + hence "Th th2 \ Domain (RAG s)" by (simp add:trancl_domain) + then obtain n where d: "(Th th2, n) \ RAG s" by (auto simp:Domain_def) + from RAG_target_th [OF this] + obtain cs' where "n = Cs cs'" by auto + with d have "(Th th2, Cs cs') \ RAG s" by simp + with runing_2 have "False" + apply (unfold runing_def readys_def s_RAG_def) + by (auto simp:eq_waiting) + thus ?thesis by simp + next + assume "th2' \ dependants (wq s) th2" + with eq_th12 have "th1' \ dependants (wq s) th2" by simp + hence h1: "(Th th1', Th th2) \ (RAG s)^+" + by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) + from th1'_in have h2: "(Th th1', Th th1) \ (RAG s)^+" + by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp) + show ?thesis + proof(rule dchain_unique[OF h1 _ h2, symmetric]) + from runing_1 show "th1 \ readys s" by (simp add:runing_def) + from runing_2 show "th2 \ readys s" by (simp add:runing_def) + qed qed - ultimately have "xs2 = xs1" by simp - from rpath_dest_eq[OF rp1 rp2[unfolded this]] - show ?thesis by simp - next - case (less_2 xs') - moreover have "xs' = []" - proof(rule ccontr) - assume otherwise: "xs' \ []" - from rpath_plus[OF less_2(3) this] - have "(Th th2, Th th1) \ (RAG s)\<^sup>+" . - from tranclD[OF this] - obtain cs where "waiting s th2 cs" - by (unfold s_RAG_def, fold waiting_eq, auto) - with runing_2 show False - by (unfold runing_def readys_def, auto) - qed - ultimately have "xs2 = xs1" by simp - from rpath_dest_eq[OF rp1 rp2[unfolded this]] - show ?thesis by simp qed qed -lemma card_runing: "card (runing s) \ 1" -proof(cases "runing s = {}") - case True - thus ?thesis by auto -next - case False - then obtain th where [simp]: "th \ runing s" by auto - from runing_unique[OF this] - have "runing s = {th}" by auto - thus ?thesis by auto -qed + +lemma "card (runing s) \ 1" +apply(subgoal_tac "finite (runing s)") +prefer 2 +apply (metis finite_nat_set_iff_bounded lessI runing_unique) +apply(rule ccontr) +apply(simp) +apply(case_tac "Suc (Suc 0) \ card (runing s)") +apply(subst (asm) card_le_Suc_iff) +apply(simp) +apply(auto)[1] +apply (metis insertCI runing_unique) +apply(auto) +done + +end + lemma create_pre: assumes stp: "step s e" @@ -3812,34 +2581,648 @@ qed qed -lemma eq_pv_children: + +context valid_trace +begin + +lemma cnp_cnv_eq: + assumes "th \ threads s" + shows "cntP s th = cntV s th" + using assms + using cnp_cnv_cncs not_thread_cncs by auto + +end + +lemma eq_RAG: + "RAG (wq s) = RAG s" +by (unfold cs_RAG_def s_RAG_def, auto) + +context valid_trace +begin + +lemma count_eq_dependants: assumes eq_pv: "cntP s th = cntV s th" - shows "children (RAG s) (Th th) = {}" + shows "dependants (wq s) th = {}" proof - - from cnp_cnv_cncs and eq_pv - have "cntCS s th = 0" - by (auto split:if_splits) - from this[unfolded cntCS_def holdents_alt_def] - have card_0: "card (the_cs ` children (RAG s) (Th th)) = 0" . - have "finite (the_cs ` children (RAG s) (Th th))" - by (simp add: fsbtRAGs.finite_children) - from card_0[unfolded card_0_eq[OF this]] - show ?thesis by auto + from cnp_cnv_cncs and eq_pv + have "cntCS s th = 0" + by (auto split:if_splits) + moreover have "finite {cs. (Cs cs, Th th) \ RAG s}" + proof - + from finite_holding[of th] show ?thesis + by (simp add:holdents_test) + qed + ultimately have h: "{cs. (Cs cs, Th th) \ RAG s} = {}" + by (unfold cntCS_def holdents_test cs_dependants_def, auto) + show ?thesis + proof(unfold cs_dependants_def) + { assume "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ {}" + then obtain th' where "(Th th', Th th) \ (RAG (wq s))\<^sup>+" by auto + hence "False" + proof(cases) + assume "(Th th', Th th) \ RAG (wq s)" + thus "False" by (auto simp:cs_RAG_def) + next + fix c + assume "(c, Th th) \ RAG (wq s)" + with h and eq_RAG show "False" + by (cases c, auto simp:cs_RAG_def) + qed + } thus "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} = {}" by auto + qed +qed + +lemma dependants_threads: + shows "dependants (wq s) th \ threads s" +proof + { fix th th' + assume h: "th \ {th'a. (Th th'a, Th th') \ (RAG (wq s))\<^sup>+}" + have "Th th \ Domain (RAG s)" + proof - + from h obtain th' where "(Th th, Th th') \ (RAG (wq s))\<^sup>+" by auto + hence "(Th th) \ Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def) + with trancl_domain have "(Th th) \ Domain (RAG (wq s))" by simp + thus ?thesis using eq_RAG by simp + qed + from dm_RAG_threads[OF this] + have "th \ threads s" . + } note hh = this + fix th1 + assume "th1 \ dependants (wq s) th" + hence "th1 \ {th'a. (Th th'a, Th th) \ (RAG (wq s))\<^sup>+}" + by (unfold cs_dependants_def, simp) + from hh [OF this] show "th1 \ threads s" . +qed + +lemma finite_threads: + shows "finite (threads s)" +using vt by (induct) (auto elim: step.cases) + +end + +lemma Max_f_mono: + assumes seq: "A \ B" + and np: "A \ {}" + and fnt: "finite B" + shows "Max (f ` A) \ Max (f ` B)" +proof(rule Max_mono) + from seq show "f ` A \ f ` B" by auto +next + from np show "f ` A \ {}" by auto +next + from fnt and seq show "finite (f ` B)" by auto +qed + +context valid_trace +begin + +lemma cp_le: + assumes th_in: "th \ threads s" + shows "cp s th \ Max ((\ th. (preced th s)) ` threads s)" +proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def) + show "Max ((\th. preced th s) ` ({th} \ {th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+})) + \ Max ((\th. preced th s) ` threads s)" + (is "Max (?f ` ?A) \ Max (?f ` ?B)") + proof(rule Max_f_mono) + show "{th} \ {th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ {}" by simp + next + from finite_threads + show "finite (threads s)" . + next + from th_in + show "{th} \ {th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ threads s" + apply (auto simp:Domain_def) + apply (rule_tac dm_RAG_threads) + apply (unfold trancl_domain [of "RAG s", symmetric]) + by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def) + qed +qed + +lemma le_cp: + shows "preced th s \ cp s th" +proof(unfold cp_eq_cpreced preced_def cpreced_def, simp) + show "Prc (priority th s) (last_set th s) + \ Max (insert (Prc (priority th s) (last_set th s)) + ((\th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))" + (is "?l \ Max (insert ?l ?A)") + proof(cases "?A = {}") + case False + have "finite ?A" (is "finite (?f ` ?B)") + proof - + have "finite ?B" + proof- + have "finite {th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+}" + proof - + let ?F = "\ (x, y). the_th x" + have "{th'. (Th th', Th th) \ (RAG (wq s))\<^sup>+} \ ?F ` ((RAG (wq s))\<^sup>+)" + apply (auto simp:image_def) + by (rule_tac x = "(Th x, Th th)" in bexI, auto) + moreover have "finite \" + proof - + from finite_RAG have "finite (RAG s)" . + hence "finite ((RAG (wq s))\<^sup>+)" + apply (unfold finite_trancl) + by (auto simp: s_RAG_def cs_RAG_def wq_def) + thus ?thesis by auto + qed + ultimately show ?thesis by (auto intro:finite_subset) + qed + thus ?thesis by (simp add:cs_dependants_def) + qed + thus ?thesis by simp + qed + from Max_insert [OF this False, of ?l] show ?thesis by auto + next + case True + thus ?thesis by auto + qed +qed + +lemma max_cp_eq: + shows "Max ((cp s) ` threads s) = Max ((\ th. (preced th s)) ` threads s)" + (is "?l = ?r") +proof(cases "threads s = {}") + case True + thus ?thesis by auto +next + case False + have "?l \ ((cp s) ` threads s)" + proof(rule Max_in) + from finite_threads + show "finite (cp s ` threads s)" by auto + next + from False show "cp s ` threads s \ {}" by auto + qed + then obtain th + where th_in: "th \ threads s" and eq_l: "?l = cp s th" by auto + have "\ \ ?r" by (rule cp_le[OF th_in]) + moreover have "?r \ cp s th" (is "Max (?f ` ?A) \ cp s th") + proof - + have "?r \ (?f ` ?A)" + proof(rule Max_in) + from finite_threads + show " finite ((\th. preced th s) ` threads s)" by auto + next + from False show " (\th. preced th s) ` threads s \ {}" by auto + qed + then obtain th' where + th_in': "th' \ ?A " and eq_r: "?r = ?f th'" by auto + from le_cp [of th'] eq_r + have "?r \ cp s th'" by auto + moreover have "\ \ cp s th" + proof(fold eq_l) + show " cp s th' \ Max (cp s ` threads s)" + proof(rule Max_ge) + from th_in' show "cp s th' \ cp s ` threads s" + by auto + next + from finite_threads + show "finite (cp s ` threads s)" by auto + qed + qed + ultimately show ?thesis by auto + qed + ultimately show ?thesis using eq_l by auto qed -lemma 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 max_cp_readys_threads_pre: + assumes np: "threads s \ {}" + shows "Max (cp s ` readys s) = Max (cp s ` threads s)" +proof(unfold max_cp_eq) + show "Max (cp s ` readys s) = Max ((\th. preced th s) ` threads s)" + proof - + let ?p = "Max ((\th. preced th s) ` threads s)" + let ?f = "(\th. preced th s)" + have "?p \ ((\th. preced th s) ` threads s)" + proof(rule Max_in) + from finite_threads show "finite (?f ` threads s)" by simp + next + from np show "?f ` threads s \ {}" by simp + qed + then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \ threads s" + by (auto simp:Image_def) + from th_chain_to_ready [OF tm_in] + have "tm \ readys s \ (\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+)" . + thus ?thesis + proof + assume "\th'. th' \ readys s \ (Th tm, Th th') \ (RAG s)\<^sup>+ " + then obtain th' where th'_in: "th' \ readys s" + and tm_chain:"(Th tm, Th th') \ (RAG s)\<^sup>+" by auto + have "cp s th' = ?f tm" + proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI) + from dependants_threads finite_threads + show "finite ((\th. preced th s) ` ({th'} \ dependants (wq s) th'))" + by (auto intro:finite_subset) + next + fix p assume p_in: "p \ (\th. preced th s) ` ({th'} \ dependants (wq s) th')" + from tm_max have " preced tm s = Max ((\th. preced th s) ` threads s)" . + moreover have "p \ \" + proof(rule Max_ge) + from finite_threads + show "finite ((\th. preced th s) ` threads s)" by simp + next + from p_in and th'_in and dependants_threads[of th'] + show "p \ (\th. preced th s) ` threads s" + by (auto simp:readys_def) + qed + ultimately show "p \ preced tm s" by auto + next + show "preced tm s \ (\th. preced th s) ` ({th'} \ dependants (wq s) th')" + proof - + from tm_chain + have "tm \ dependants (wq s) th'" + by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto) + thus ?thesis by auto + qed + qed + with tm_max + have h: "cp s th' = Max ((\th. preced th s) ` threads s)" by simp + show ?thesis + proof (fold h, rule Max_eqI) + fix q + assume "q \ cp s ` readys s" + then obtain th1 where th1_in: "th1 \ readys s" + and eq_q: "q = cp s th1" by auto + show "q \ cp s th'" + apply (unfold h eq_q) + apply (unfold cp_eq_cpreced cpreced_def) + apply (rule Max_mono) + proof - + from dependants_threads [of th1] th1_in + show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ + (\th. preced th s) ` threads s" + by (auto simp:readys_def) + next + show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" by simp + next + from finite_threads + show " finite ((\th. preced th s) ` threads s)" by simp + qed + next + from finite_threads + show "finite (cp s ` readys s)" by (auto simp:readys_def) + next + from th'_in + show "cp s th' \ cp s ` readys s" by simp + qed + next + assume tm_ready: "tm \ readys s" + show ?thesis + proof(fold tm_max) + have cp_eq_p: "cp s tm = preced tm s" + proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI) + fix y + assume hy: "y \ (\th. preced th s) ` ({tm} \ dependants (wq s) tm)" + show "y \ preced tm s" + proof - + { fix y' + assume hy' : "y' \ ((\th. preced th s) ` dependants (wq s) tm)" + have "y' \ preced tm s" + proof(unfold tm_max, rule Max_ge) + from hy' dependants_threads[of tm] + show "y' \ (\th. preced th s) ` threads s" by auto + next + from finite_threads + show "finite ((\th. preced th s) ` threads s)" by simp + qed + } with hy show ?thesis by auto + qed + next + from dependants_threads[of tm] finite_threads + show "finite ((\th. preced th s) ` ({tm} \ dependants (wq s) tm))" + by (auto intro:finite_subset) + next + show "preced tm s \ (\th. preced th s) ` ({tm} \ dependants (wq s) tm)" + by simp + qed + moreover have "Max (cp s ` readys s) = cp s tm" + proof(rule Max_eqI) + from tm_ready show "cp s tm \ cp s ` readys s" by simp + next + from finite_threads + show "finite (cp s ` readys s)" by (auto simp:readys_def) + next + fix y assume "y \ cp s ` readys s" + then obtain th1 where th1_readys: "th1 \ readys s" + and h: "y = cp s th1" by auto + show "y \ cp s tm" + apply(unfold cp_eq_p h) + apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono) + proof - + from finite_threads + show "finite ((\th. preced th s) ` threads s)" by simp + next + show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) \ {}" + by simp + next + from dependants_threads[of th1] th1_readys + show "(\th. preced th s) ` ({th1} \ dependants (wq s) th1) + \ (\th. preced th s) ` threads s" + by (auto simp:readys_def) + qed + qed + ultimately show " Max (cp s ` readys s) = preced tm s" by simp + qed + qed + qed +qed -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) +text {* (* ccc *) \noindent + Since the current precedence of the threads in ready queue will always be boosted, + there must be one inside it has the maximum precedence of the whole system. +*} +lemma max_cp_readys_threads: + shows "Max (cp s ` readys s) = Max (cp s ` threads s)" +proof(cases "threads s = {}") + case True + thus ?thesis + by (auto simp:readys_def) +next + case False + show ?thesis by (rule max_cp_readys_threads_pre[OF False]) +qed end +lemma eq_holding: "holding (wq s) th cs = holding s th cs" + apply (unfold s_holding_def cs_holding_def wq_def, simp) + done + +lemma f_image_eq: + assumes h: "\ a. a \ A \ f a = g a" + shows "f ` A = g ` A" +proof + show "f ` A \ g ` A" + by(rule image_subsetI, auto intro:h) +next + show "g ` A \ f ` A" + by (rule image_subsetI, auto intro:h[symmetric]) +qed + + +definition detached :: "state \ thread \ bool" + where "detached s th \ (\(\ cs. holding s th cs)) \ (\(\cs. waiting s th cs))" + + +lemma detached_test: + shows "detached s th = (Th th \ Field (RAG s))" +apply(simp add: detached_def Field_def) +apply(simp add: s_RAG_def) +apply(simp add: s_holding_abv s_waiting_abv) +apply(simp add: Domain_iff Range_iff) +apply(simp add: wq_def) +apply(auto) +done + +context valid_trace +begin + +lemma detached_intro: + assumes eq_pv: "cntP s th = cntV s th" + shows "detached s th" +proof - + from cnp_cnv_cncs + have eq_cnt: "cntP s th = + cntV s th + (if th \ readys s \ th \ threads s then cntCS s th else cntCS s th + 1)" . + hence cncs_zero: "cntCS s th = 0" + by (auto simp:eq_pv split:if_splits) + with eq_cnt + have "th \ readys s \ th \ threads s" by (auto simp:eq_pv) + thus ?thesis + proof + assume "th \ threads s" + with range_in dm_RAG_threads + show ?thesis + by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff) + next + assume "th \ readys s" + moreover have "Th th \ Range (RAG s)" + proof - + from card_0_eq [OF finite_holding] and cncs_zero + have "holdents s th = {}" + by (simp add:cntCS_def) + thus ?thesis + apply(auto simp:holdents_test) + apply(case_tac a) + apply(auto simp:holdents_test s_RAG_def) + done + qed + ultimately show ?thesis + by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def) + qed +qed + +lemma detached_elim: + assumes dtc: "detached s th" + shows "cntP s th = cntV s th" +proof - + from cnp_cnv_cncs + have eq_pv: " cntP s th = + cntV s th + (if th \ readys s \ th \ threads s then cntCS s th else cntCS s th + 1)" . + have cncs_z: "cntCS s th = 0" + proof - + from dtc have "holdents s th = {}" + unfolding detached_def holdents_test s_RAG_def + by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff) + thus ?thesis by (auto simp:cntCS_def) + qed + show ?thesis + proof(cases "th \ threads s") + case True + with dtc + have "th \ readys s" + by (unfold readys_def detached_def Field_def Domain_def Range_def, + auto simp:eq_waiting s_RAG_def) + with cncs_z and eq_pv show ?thesis by simp + next + case False + with cncs_z and eq_pv show ?thesis by simp + qed +qed + +lemma detached_eq: + shows "(detached s th) = (cntP s th = cntV s th)" + by (insert vt, auto intro:detached_intro detached_elim) + +end + +text {* + The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived + from the concise and miniature model of PIP given in PrioGDef.thy. +*} + +lemma eq_dependants: "dependants (wq s) = dependants s" + by (simp add: s_dependants_abv wq_def) + +lemma next_th_unique: + assumes nt1: "next_th s th cs th1" + and nt2: "next_th s th cs th2" + shows "th1 = th2" +using assms by (unfold next_th_def, auto) + +lemma birth_time_lt: "s \ [] \ last_set th s < length s" + apply (induct s, simp) +proof - + fix a s + assume ih: "s \ [] \ last_set th s < length s" + and eq_as: "a # s \ []" + show "last_set th (a # s) < length (a # s)" + proof(cases "s \ []") + case False + from False show ?thesis + by (cases a, auto simp:last_set.simps) + next + case True + from ih [OF True] show ?thesis + by (cases a, auto simp:last_set.simps) + qed +qed + +lemma th_in_ne: "th \ threads s \ s \ []" + by (induct s, auto simp:threads.simps) + +lemma preced_tm_lt: "th \ threads s \ preced th s = Prc x y \ y < length s" + apply (drule_tac th_in_ne) + by (unfold preced_def, auto intro: birth_time_lt) + +lemma inj_the_preced: + "inj_on (the_preced s) (threads s)" + by (metis inj_onI preced_unique the_preced_def) + +lemma tRAG_alt_def: + "tRAG s = {(Th th1, Th th2) | th1 th2. + \ cs. (Th th1, Cs cs) \ RAG s \ (Cs cs, Th th2) \ RAG s}" + by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def) + +lemma tRAG_Field: + "Field (tRAG s) \ Field (RAG s)" + by (unfold tRAG_alt_def Field_def, auto) + +lemma tRAG_ancestorsE: + assumes "x \ ancestors (tRAG s) u" + obtains th where "x = Th th" +proof - + from assms have "(u, x) \ (tRAG s)^+" + by (unfold ancestors_def, auto) + from tranclE[OF this] obtain c where "(c, x) \ tRAG s" by auto + then obtain th where "x = Th th" + by (unfold tRAG_alt_def, auto) + from that[OF this] show ?thesis . +qed + +lemma tRAG_mono: + assumes "RAG s' \ RAG s" + shows "tRAG s' \ tRAG s" + using assms + by (unfold tRAG_alt_def, auto) + +lemma holding_next_thI: + assumes "holding s th cs" + and "length (wq s cs) > 1" + obtains th' where "next_th s th cs th'" +proof - + from assms(1)[folded eq_holding, unfolded cs_holding_def] + have " th \ set (wq s cs) \ th = hd (wq s cs)" . + then obtain rest where h1: "wq s cs = th#rest" + by (cases "wq s cs", auto) + with assms(2) have h2: "rest \ []" by auto + let ?th' = "hd (SOME q. distinct q \ set q = set rest)" + have "next_th s th cs ?th'" using h1(1) h2 + by (unfold next_th_def, auto) + from that[OF this] show ?thesis . +qed + +lemma RAG_tRAG_transfer: + assumes "vt s'" + assumes "RAG s = RAG s' \ {(Th th, Cs cs)}" + and "(Cs cs, Th th'') \ RAG s'" + shows "tRAG s = tRAG s' \ {(Th th, Th th'')}" (is "?L = ?R") +proof - + interpret vt_s': valid_trace "s'" using assms(1) + by (unfold_locales, simp) + interpret rtree: rtree "RAG s'" + proof + show "single_valued (RAG s')" + apply (intro_locales) + by (unfold single_valued_def, + auto intro:vt_s'.unique_RAG) + + show "acyclic (RAG s')" + by (rule vt_s'.acyclic_RAG) + qed + { fix n1 n2 + assume "(n1, n2) \ ?L" + from this[unfolded tRAG_alt_def] + obtain th1 th2 cs' where + h: "n1 = Th th1" "n2 = Th th2" + "(Th th1, Cs cs') \ RAG s" + "(Cs cs', Th th2) \ RAG s" by auto + from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \ RAG s'" by auto + from h(3) and assms(2) + have "(Th th1, Cs cs') = (Th th, Cs cs) \ + (Th th1, Cs cs') \ RAG s'" by auto + hence "(n1, n2) \ ?R" + proof + assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)" + hence eq_th1: "th1 = th" by simp + moreover have "th2 = th''" + proof - + from h1 have "cs' = cs" by simp + from assms(3) cs_in[unfolded this] rtree.sgv + show ?thesis + by (unfold single_valued_def, auto) + qed + ultimately show ?thesis using h(1,2) by auto + next + assume "(Th th1, Cs cs') \ RAG s'" + with cs_in have "(Th th1, Th th2) \ tRAG s'" + by (unfold tRAG_alt_def, auto) + from this[folded h(1, 2)] show ?thesis by auto + qed + } moreover { + fix n1 n2 + assume "(n1, n2) \ ?R" + hence "(n1, n2) \tRAG s' \ (n1, n2) = (Th th, Th th'')" by auto + hence "(n1, n2) \ ?L" + proof + assume "(n1, n2) \ tRAG s'" + moreover have "... \ ?L" + proof(rule tRAG_mono) + show "RAG s' \ RAG s" by (unfold assms(2), auto) + qed + ultimately show ?thesis by auto + next + assume eq_n: "(n1, n2) = (Th th, Th th'')" + from assms(2, 3) have "(Cs cs, Th th'') \ RAG s" by auto + moreover have "(Th th, Cs cs) \ RAG s" using assms(2) by auto + ultimately show ?thesis + by (unfold eq_n tRAG_alt_def, auto) + qed + } ultimately show ?thesis by auto +qed + +context valid_trace +begin + +lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt] + +end + +lemma cp_alt_def: + "cp s th = + Max ((the_preced s) ` {th'. Th th' \ (subtree (RAG s) (Th th))})" +proof - + have "Max (the_preced s ` ({th} \ dependants (wq s) th)) = + Max (the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})" + (is "Max (_ ` ?L) = Max (_ ` ?R)") + proof - + have "?L = ?R" + by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def) + thus ?thesis by simp + qed + thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp) +qed + lemma cp_gen_alt_def: "cp_gen s = (Max \ (\x. (the_preced s \ the_thread) ` subtree (tRAG s) x))" by (auto simp:cp_gen_def) @@ -3888,7 +3271,7 @@ { fix a assume "a \ subtree (tRAG s) x" hence "(a, x) \ (tRAG s)^*" by (auto simp:subtree_def) - with tRAG_star_RAG + with tRAG_star_RAG[of s] have "(a, x) \ (RAG s)^*" by auto hence "a \ subtree (RAG s) x" by (auto simp:subtree_def) } thus ?thesis by auto @@ -3904,7 +3287,7 @@ hence "(Th th', Th th) \ (tRAG s)^+" by auto from tranclD[OF this] obtain z where h: "(Th th', z) \ tRAG s" "(z, Th th) \ (tRAG s)\<^sup>*" by auto - from tRAG_subtree_RAG and this(2) + from tRAG_subtree_RAG[of s] and this(2) have "(z, Th th) \ (RAG s)^*" by (meson subsetCE tRAG_star_RAG) moreover from h(1) have "(Th th', z) \ (RAG s)^+" using tRAG_alt_def by auto ultimately have "th' \ ?R" by auto @@ -3923,8 +3306,7 @@ case Nil from 1(2)[unfolded Cons1 Nil] have rp: "rpath (RAG s) (Th th') [x1] (Th th)" . - hence "(Th th', x1) \ (RAG s)" - by (cases, auto) + hence "(Th th', x1) \ (RAG s)" by (cases, simp) then obtain cs where "x1 = Cs cs" by (unfold s_RAG_def, auto) from rpath_nnl_lastE[OF rp[unfolded this]] @@ -3976,46 +3358,19 @@ lemma dependants_alt_def: "dependants s th = {th'. (Th th', Th th) \ (tRAG s)^+}" by (metis eq_RAG s_dependants_def tRAG_trancl_eq) - -lemma dependants_alt_def1: - "dependants (s::state) th = {th'. (Th th', Th th) \ (RAG s)^+}" - using dependants_alt_def tRAG_trancl_eq by auto - -context valid_trace -begin -lemma count_eq_RAG_plus: - assumes "cntP s th = cntV s th" - shows "{th'. (Th th', Th th) \ (RAG s)^+} = {}" -proof(rule ccontr) - assume otherwise: "{th'. (Th th', Th th) \ (RAG s)\<^sup>+} \ {}" - then obtain th' where "(Th th', Th th) \ (RAG s)^+" by auto - from tranclD2[OF this] - obtain z where "z \ children (RAG s) (Th th)" - by (auto simp:children_def) - with eq_pv_children[OF assms] - show False by simp -qed - -lemma eq_pv_dependants: - assumes eq_pv: "cntP s th = cntV s th" - shows "dependants s th = {}" -proof - - from count_eq_RAG_plus[OF assms, folded dependants_alt_def1] - show ?thesis . -qed - -end - -lemma eq_dependants: "dependants (wq s) = dependants s" - by (simp add: s_dependants_abv wq_def) - + context valid_trace begin lemma count_eq_tRAG_plus: assumes "cntP s th = cntV s th" shows "{th'. (Th th', Th th) \ (tRAG s)^+} = {}" - using assms eq_pv_dependants dependants_alt_def eq_dependants by auto + using assms count_eq_dependants dependants_alt_def eq_dependants by auto + +lemma count_eq_RAG_plus: + assumes "cntP s th = cntV s th" + shows "{th'. (Th th', Th th) \ (RAG s)^+} = {}" + using assms count_eq_dependants cs_dependants_def eq_RAG by auto lemma count_eq_RAG_plus_Th: assumes "cntP s th = cntV s th" @@ -4026,113 +3381,6 @@ assumes "cntP s th = cntV s th" shows "{Th th' | th'. (Th th', Th th) \ (tRAG s)^+} = {}" using count_eq_tRAG_plus[OF assms] by auto -end - -lemma inj_the_preced: - "inj_on (the_preced s) (threads s)" - by (metis inj_onI preced_unique the_preced_def) - -lemma tRAG_Field: - "Field (tRAG s) \ Field (RAG s)" - by (unfold tRAG_alt_def Field_def, auto) - -lemma tRAG_ancestorsE: - assumes "x \ ancestors (tRAG s) u" - obtains th where "x = Th th" -proof - - from assms have "(u, x) \ (tRAG s)^+" - by (unfold ancestors_def, auto) - from tranclE[OF this] obtain c where "(c, x) \ tRAG s" by auto - then obtain th where "x = Th th" - by (unfold tRAG_alt_def, auto) - from that[OF this] show ?thesis . -qed - -lemma tRAG_mono: - assumes "RAG s' \ RAG s" - shows "tRAG s' \ tRAG s" - using assms - by (unfold tRAG_alt_def, auto) - -lemma holding_next_thI: - assumes "holding s th cs" - and "length (wq s cs) > 1" - obtains th' where "next_th s th cs th'" -proof - - from assms(1)[folded holding_eq, unfolded cs_holding_def] - have " th \ set (wq s cs) \ th = hd (wq s cs)" - by (unfold s_holding_def, fold wq_def, auto) - then obtain rest where h1: "wq s cs = th#rest" - by (cases "wq s cs", auto) - with assms(2) have h2: "rest \ []" by auto - let ?th' = "hd (SOME q. distinct q \ set q = set rest)" - have "next_th s th cs ?th'" using h1(1) h2 - by (unfold next_th_def, auto) - from that[OF this] show ?thesis . -qed - -lemma RAG_tRAG_transfer: - assumes "vt s'" - assumes "RAG s = RAG s' \ {(Th th, Cs cs)}" - and "(Cs cs, Th th'') \ RAG s'" - shows "tRAG s = tRAG s' \ {(Th th, Th th'')}" (is "?L = ?R") -proof - - interpret vt_s': valid_trace "s'" using assms(1) - by (unfold_locales, simp) - { fix n1 n2 - assume "(n1, n2) \ ?L" - from this[unfolded tRAG_alt_def] - obtain th1 th2 cs' where - h: "n1 = Th th1" "n2 = Th th2" - "(Th th1, Cs cs') \ RAG s" - "(Cs cs', Th th2) \ RAG s" by auto - from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \ RAG s'" by auto - from h(3) and assms(2) - have "(Th th1, Cs cs') = (Th th, Cs cs) \ - (Th th1, Cs cs') \ RAG s'" by auto - hence "(n1, n2) \ ?R" - proof - assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)" - hence eq_th1: "th1 = th" by simp - moreover have "th2 = th''" - proof - - from h1 have "cs' = cs" by simp - from assms(3) cs_in[unfolded this] - show ?thesis using vt_s'.unique_RAG by auto - qed - ultimately show ?thesis using h(1,2) by auto - next - assume "(Th th1, Cs cs') \ RAG s'" - with cs_in have "(Th th1, Th th2) \ tRAG s'" - by (unfold tRAG_alt_def, auto) - from this[folded h(1, 2)] show ?thesis by auto - qed - } moreover { - fix n1 n2 - assume "(n1, n2) \ ?R" - hence "(n1, n2) \tRAG s' \ (n1, n2) = (Th th, Th th'')" by auto - hence "(n1, n2) \ ?L" - proof - assume "(n1, n2) \ tRAG s'" - moreover have "... \ ?L" - proof(rule tRAG_mono) - show "RAG s' \ RAG s" by (unfold assms(2), auto) - qed - ultimately show ?thesis by auto - next - assume eq_n: "(n1, n2) = (Th th, Th th'')" - from assms(2, 3) have "(Cs cs, Th th'') \ RAG s" by auto - moreover have "(Th th, Cs cs) \ RAG s" using assms(2) by auto - ultimately show ?thesis - by (unfold eq_n tRAG_alt_def, auto) - qed - } ultimately show ?thesis by auto -qed - -context valid_trace -begin - -lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt] end @@ -4190,9 +3438,16 @@ by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp) qed + context valid_trace begin +lemma RAG_threads: + assumes "(Th th) \ Field (RAG s)" + shows "th \ threads s" + using assms + by (metis Field_def UnE dm_RAG_threads range_in vt) + lemma subtree_tRAG_thread: assumes "th \ threads s" shows "subtree (tRAG s) (Th th) \ Th ` threads s" (is "?L \ ?R") @@ -4254,90 +3509,140 @@ shows "(Th th) \ Field (RAG s)" proof assume "(Th th) \ Field (RAG s)" - with dm_RAG_threads and rg_RAG_threads assms + with dm_RAG_threads and range_in assms show False by (unfold Field_def, blast) qed +lemma wf_RAG: "wf (RAG s)" +proof(rule finite_acyclic_wf) + from finite_RAG show "finite (RAG s)" . +next + from acyclic_RAG show "acyclic (RAG s)" . +qed + +lemma 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 holding_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) + +lemma acyclic_tRAG: "acyclic (tRAG s)" +proof(unfold tRAG_def, rule acyclic_compose) + show "acyclic (RAG s)" using acyclic_RAG . +next + show "wRAG s \ RAG s" unfolding RAG_split by auto +next + show "hRAG s \ RAG s" unfolding RAG_split by auto +qed + +lemma sgv_RAG: "single_valued (RAG s)" + using unique_RAG by (auto simp:single_valued_def) + +lemma rtree_RAG: "rtree (RAG s)" + using sgv_RAG acyclic_RAG + by (unfold rtree_def rtree_axioms_def sgv_def, auto) + end -definition detached :: "state \ thread \ bool" - where "detached s th \ (\(\ cs. holding s th cs)) \ (\(\cs. waiting s th cs))" +sublocale valid_trace < rtree_RAG: rtree "RAG s" +proof + show "single_valued (RAG s)" + apply (intro_locales) + by (unfold single_valued_def, + auto intro:unique_RAG) + + show "acyclic (RAG s)" + by (rule acyclic_RAG) +qed +sublocale valid_trace < rtree_s: rtree "tRAG s" +proof(unfold_locales) + from sgv_tRAG show "single_valued (tRAG s)" . +next + from acyclic_tRAG show "acyclic (tRAG s)" . +qed + +sublocale valid_trace < fsbtRAGs : fsubtree "RAG s" +proof - + show "fsubtree (RAG s)" + proof(intro_locales) + show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] . + next + show "fsubtree_axioms (RAG s)" + proof(unfold fsubtree_axioms_def) + from wf_RAG show "wf (RAG s)" . + qed + qed +qed -lemma detached_test: - shows "detached s th = (Th th \ Field (RAG s))" -apply(simp add: detached_def Field_def) -apply(simp add: s_RAG_def) -apply(simp add: s_holding_abv s_waiting_abv) -apply(simp add: Domain_iff Range_iff) -apply(simp add: wq_def) -apply(auto) -done +sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s" +proof - + have "fsubtree (tRAG s)" + proof - + have "fbranch (tRAG s)" + proof(unfold tRAG_def, rule fbranch_compose) + show "fbranch (wRAG s)" + proof(rule finite_fbranchI) + from finite_RAG show "finite (wRAG s)" + by (unfold RAG_split, auto) + qed + next + show "fbranch (hRAG s)" + proof(rule finite_fbranchI) + from finite_RAG + show "finite (hRAG s)" by (unfold RAG_split, auto) + qed + qed + moreover have "wf (tRAG s)" + proof(rule wf_subset) + show "wf (RAG s O RAG s)" using wf_RAG + by (fold wf_comp_self, simp) + next + show "tRAG s \ (RAG s O RAG s)" + by (unfold tRAG_alt_def, auto) + qed + ultimately show ?thesis + by (unfold fsubtree_def fsubtree_axioms_def,auto) + qed + from this[folded tRAG_def] show "fsubtree (tRAG s)" . +qed + +lemma Max_UNION: + assumes "finite A" + and "A \ {}" + and "\ M \ f ` A. finite M" + and "\ M \ f ` A. M \ {}" + shows "Max (\x\ A. f x) = Max (Max ` f ` A)" (is "?L = ?R") + using assms[simp] +proof - + have "?L = Max (\(f ` A))" + by (fold Union_image_eq, simp) + also have "... = ?R" + by (subst Max_Union, simp+) + finally show ?thesis . +qed + +lemma max_Max_eq: + assumes "finite A" + and "A \ {}" + and "x = y" + shows "max x (Max A) = Max ({y} \ A)" (is "?L = ?R") +proof - + have "?R = Max (insert y A)" by simp + also from assms have "... = ?L" + by (subst Max.insert, simp+) + finally show ?thesis by simp +qed context valid_trace begin -lemma detached_intro: - assumes eq_pv: "cntP s th = cntV s th" - shows "detached s th" -proof - - from eq_pv cnp_cnv_cncs - have "th \ readys s \ th \ threads s" by (auto simp:pvD_def) - thus ?thesis - proof - assume "th \ threads s" - with rg_RAG_threads dm_RAG_threads - show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv - s_holding_abv wq_def Domain_iff Range_iff) - next - assume "th \ readys s" - moreover have "Th th \ Range (RAG s)" - proof - - from eq_pv_children[OF assms] - have "children (RAG s) (Th th) = {}" . - thus ?thesis - by (unfold children_def, auto) - qed - ultimately show ?thesis - by (auto simp add: detached_def s_RAG_def s_waiting_abv - s_holding_abv wq_def readys_def) - qed -qed - -lemma detached_elim: - assumes dtc: "detached s th" - shows "cntP s th = cntV s th" -proof - - have cncs_z: "cntCS s th = 0" - proof - - from dtc have "holdents s th = {}" - unfolding detached_def holdents_test s_RAG_def - by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff) - thus ?thesis by (auto simp:cntCS_def) - qed - show ?thesis - proof(cases "th \ threads s") - case True - with dtc - have "th \ readys s" - by (unfold readys_def detached_def Field_def Domain_def Range_def, - auto simp:waiting_eq s_RAG_def) - with cncs_z show ?thesis using cnp_cnv_cncs by (simp add:pvD_def) - next - case False - with cncs_z and cnp_cnv_cncs show ?thesis by (simp add:pvD_def) - qed -qed - -lemma detached_eq: - shows "(detached s th) = (cntP s th = cntV s th)" - by (insert vt, auto intro:detached_intro detached_elim) - -end - -context valid_trace -begin (* ddd *) lemma cp_gen_rec: assumes "x = Th th" @@ -4414,8 +3719,12 @@ qed qed +end + +(* keep *) lemma next_th_holding: - assumes nxt: "next_th s th cs th'" + assumes vt: "vt s" + and nxt: "next_th s th cs th'" shows "holding (wq s) th cs" proof - from nxt[unfolded next_th_def] @@ -4426,6 +3735,9 @@ by (unfold cs_holding_def, auto) qed +context valid_trace +begin + lemma next_th_waiting: assumes nxt: "next_th s th cs th'" shows "waiting (wq s) th' cs" @@ -4458,91 +3770,8 @@ end -lemma next_th_unique: - assumes nt1: "next_th s th cs th1" - and nt2: "next_th s th cs th2" - shows "th1 = th2" -using assms by (unfold next_th_def, auto) - -context valid_trace -begin - -thm th_chain_to_ready - -find_theorems subtree Th RAG - -lemma threads_alt_def: - "(threads s) = (\ th \ readys s. {th'. Th th' \ subtree (RAG s) (Th th)})" - (is "?L = ?R") -proof - - { fix th1 - assume "th1 \ ?L" - from th_chain_to_ready[OF this] - have "th1 \ readys s \ (\th'. th' \ readys s \ (Th th1, Th th') \ (RAG s)\<^sup>+)" . - hence "th1 \ ?R" by (auto simp:subtree_def) - } moreover - { fix th' - assume "th' \ ?R" - then obtain th where h: "th \ readys s" " Th th' \ subtree (RAG s) (Th th)" - by auto - from this(2) - have "th' \ ?L" - proof(cases rule:subtreeE) - case 1 - with h(1) show ?thesis by (auto simp:readys_def) - next - case 2 - from tranclD[OF this(2)[unfolded ancestors_def, simplified]] - have "Th th' \ Domain (RAG s)" by auto - from dm_RAG_threads[OF this] - show ?thesis . - qed - } ultimately show ?thesis by auto -qed - -lemma finite_readys [simp]: "finite (readys s)" - using finite_threads readys_threads rev_finite_subset by blast - -text {* (* ccc *) \noindent - Since the current precedence of the threads in ready queue will always be boosted, - there must be one inside it has the maximum precedence of the whole system. -*} -lemma max_cp_readys_threads: - shows "Max (cp s ` readys s) = Max (cp s ` threads s)" (is "?L = ?R") -proof(cases "readys s = {}") - case False - have "?R = Max (the_preced s ` threads s)" by (unfold max_cp_eq, simp) - also have "... = - Max (the_preced s ` (\th\readys s. {th'. Th th' \ subtree (RAG s) (Th th)}))" - by (unfold threads_alt_def, simp) - also have "... = - Max ((\th\readys s. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}))" - by (unfold image_UN, simp) - also have "... = - Max (Max ` (\th. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)}) ` readys s)" - proof(rule Max_UNION) - show "\M\(\x. the_preced s ` - {th'. Th th' \ subtree (RAG s) (Th x)}) ` readys s. finite M" - using finite_subtree_threads by auto - qed (auto simp:False subtree_def) - also have "... = - Max ((Max \ (\th. the_preced s ` {th'. Th th' \ subtree (RAG s) (Th th)})) ` readys s)" - by (unfold image_comp, simp) - also have "... = ?L" (is "Max (?f ` ?A) = Max (?g ` ?A)") - proof - - have "(?f ` ?A) = (?g ` ?A)" - proof(rule f_image_eq) - fix th1 - assume "th1 \ ?A" - thus "?f th1 = ?g th1" - by (unfold cp_alt_def, simp) - qed - thus ?thesis by simp - qed - finally show ?thesis by simp -qed (auto simp:threads_alt_def) +-- {* A useless definition *} +definition cps:: "state \ (thread \ precedence) set" +where "cps s = {(th, cp s th) | th . th \ threads s}" end - -end - diff -r 524bd3caa6b6 -r c7ba70dc49bd PIPDefs.thy --- a/PIPDefs.thy Fri Jan 29 11:01:13 2016 +0800 +++ b/PIPDefs.thy Fri Jan 29 17:06:02 2016 +0000 @@ -1,10 +1,11 @@ -chapter {* Definitions *} (*<*) theory PIPDefs imports Precedence_ord Moment RTree Max begin (*>*) +chapter {* Definitions *} + text {* In this section, the formal model of Priority Inheritance Protocol (PIP) is presented. The model is based on Paulson's inductive protocol verification method, where diff -r 524bd3caa6b6 -r c7ba70dc49bd journal.pdf Binary file journal.pdf has changed