--- a/prio/Attic/Ext.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1057 +0,0 @@
-theory Ext
-imports Prio
-begin
-
-locale highest_create =
- fixes s' th prio fixes s
- defines s_def : "s \<equiv> (Create th prio#s')"
- assumes vt_s: "vt step s"
- and highest: "cp s th = Max ((cp s)`threads s)"
-
-context highest_create
-begin
-
-lemma threads_s: "threads s = threads s' \<union> {th}"
- by (unfold s_def, simp)
-
-lemma vt_s': "vt step s'"
- by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
-
-lemma step_create: "step s' (Create th prio)"
- by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
-
-lemma step_create_elim:
- "\<lbrakk>\<And>max_prio. \<lbrakk>prio \<le> max_prio; th \<notin> threads s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
- by (insert step_create, ind_cases "step s' (Create th prio)", auto)
-
-lemma eq_cp_s:
- assumes th'_in: "th' \<in> threads s'"
- shows "cp s th' = cp s' th'"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def s_def
- eq_depend depend_create_unchanged)
- show "Max ((\<lambda>tha. preced tha (Create th prio # s')) `
- ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+})) =
- Max ((\<lambda>th. preced th s') ` ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+}))"
- (is "Max (?f ` ?A) = Max (?g ` ?A)")
- proof -
- have "?f ` ?A = ?g ` ?A"
- proof(rule f_image_eq)
- fix a
- assume a_in: "a \<in> ?A"
- thus "?f a = ?g a"
- proof -
- from a_in
- have "a = th' \<or> (Th a, Th th') \<in> (depend s')\<^sup>+" by auto
- hence "a \<noteq> th"
- proof
- assume "a = th'"
- moreover have "th' \<noteq> th"
- proof(rule step_create_elim)
- assume th_not_in: "th \<notin> threads s'" with th'_in
- show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- next
- assume "(Th a, Th th') \<in> (depend s')\<^sup>+"
- hence "Th a \<in> Domain \<dots>"
- by (auto simp:Domain_def)
- hence "Th a \<in> Domain (depend s')"
- by (simp add:trancl_domain)
- from dm_depend_threads[OF vt_s' this]
- have h: "a \<in> threads s'" .
- show ?thesis
- proof(rule step_create_elim)
- assume "th \<notin> threads s'" with h
- show ?thesis by auto
- qed
- qed
- thus ?thesis
- by (unfold preced_def, auto)
- qed
- qed
- thus ?thesis by auto
- qed
-qed
-
-lemma same_depend: "depend s = depend s'"
- by (insert depend_create_unchanged, unfold s_def, simp)
-
-lemma same_dependents:
- "dependents (wq s) th = dependents (wq s') th"
- apply (unfold cs_dependents_def)
- by (unfold eq_depend same_depend, simp)
-
-lemma nil_dependents_s': "dependents (wq s') th = {}"
-proof -
- { assume ne: "dependents (wq s') th \<noteq> {}"
- then obtain th' where "th' \<in> dependents (wq s') th"
- by (unfold cs_dependents_def, auto)
- hence "(Th th', Th th) \<in> (depend (wq s'))^+"
- by (unfold cs_dependents_def, auto)
- hence "(Th th', Th th) \<in> (depend s')^+"
- by (simp add:eq_depend)
- hence "Th th \<in> Range ((depend s')^+)" by (auto simp:Range_def Domain_def)
- hence "Th th \<in> Range (depend s')" by (simp add:trancl_range)
- from range_in [OF vt_s' this]
- have h: "th \<in> threads s'" .
- have "False"
- proof(rule step_create_elim)
- assume "th \<notin> threads s'" with h show ?thesis by auto
- qed
- } thus ?thesis by auto
-qed
-
-lemma nil_dependents: "dependents (wq s) th = {}"
-proof -
- have "wq s' = wq s"
- by (unfold wq_def s_def, auto simp:Let_def)
- with nil_dependents_s' show ?thesis by auto
-qed
-
-lemma eq_cp_s_th: "cp s th = preced th s"
- by (unfold cp_eq_cpreced cpreced_def nil_dependents, auto)
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold max_cp_eq[OF vt_s], unfold highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma is_ready: "th \<in> readys s"
-proof -
- { assume "th \<notin> readys s"
- with threads_s obtain cs where
- "waiting s th cs"
- by (unfold readys_def, auto)
- hence "(Th th, Cs cs) \<in> depend s"
- by (unfold s_depend_def, unfold eq_waiting, simp)
- hence "Th th \<in> Domain (depend s')"
- by (unfold same_depend, auto simp:Domain_def)
- from dm_depend_threads [OF vt_s' this]
- have h: "th \<in> threads s'" .
- have "False"
- proof (rule_tac step_create_elim)
- assume "th \<notin> threads s'" with h show ?thesis by simp
- qed
- } thus ?thesis by auto
-qed
-
-lemma is_runing: "th \<in> runing s"
-proof -
- have "Max (cp s ` threads s) = Max (cp s ` readys s)"
- proof -
- have " Max (cp s ` readys s) = cp s th"
- proof(rule Max_eqI)
- from finite_threads[OF vt_s] readys_threads finite_subset
- have "finite (readys s)" by blast
- thus "finite (cp s ` readys s)" by auto
- next
- from is_ready show "cp s th \<in> cp s ` readys s" by auto
- next
- fix y
- assume h: "y \<in> cp s ` readys s"
- have "y \<le> Max (cp s ` readys s)"
- proof(rule Max_ge [OF _ h])
- from finite_threads[OF vt_s] readys_threads finite_subset
- have "finite (readys s)" by blast
- thus "finite (cp s ` readys s)" by auto
- qed
- moreover have "\<dots> \<le> Max (cp s ` threads s)"
- proof(rule Max_mono)
- from readys_threads
- show "cp s ` readys s \<subseteq> cp s ` threads s" by auto
- next
- from is_ready show "cp s ` readys s \<noteq> {}" by auto
- next
- from finite_threads [OF vt_s]
- show "finite (cp s ` threads s)" by auto
- qed
- moreover note highest
- ultimately show "y \<le> cp s th" by auto
- qed
- with highest show ?thesis by auto
- qed
- thus ?thesis
- by (unfold runing_def, insert highest is_ready, auto)
-qed
-
-end
-
-locale extend_highest = highest_create +
- fixes t
- assumes vt_t: "vt step (t@s)"
- and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
- and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
- and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app:
- assumes vt_ts: "vt cs (t@s)"
- shows "vt cs s"
-proof -
- from vt_ts show ?thesis
- proof(induct t)
- case Nil
- from Nil show ?case by auto
- next
- case (Cons e t)
- assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
- and vt_et: "vt cs ((e # t) @ s)"
- show ?case
- proof(rule ih)
- show "vt cs (t @ s)"
- proof(rule step_back_vt)
- from vt_et show "vt cs (e # t @ s)" by simp
- qed
- qed
- qed
-qed
-
-context extend_highest
-begin
-
-lemma red_moment:
- "extend_highest s' th prio (moment i t)"
- apply (insert extend_highest_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
- apply (unfold extend_highest_def extend_highest_axioms_def, clarsimp)
- by (unfold highest_create_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes
- h0: "R []"
- and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
- extend_highest s' th prio t;
- extend_highest s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
- shows "R t"
-proof -
- from vt_t extend_highest_axioms show ?thesis
- proof(induct t)
- from h0 show "R []" .
- next
- case (Cons e t')
- assume ih: "\<lbrakk>vt step (t' @ s); extend_highest s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
- and vt_e: "vt step ((e # t') @ s)"
- and et: "extend_highest s' th prio (e # t')"
- from vt_e and step_back_step have stp: "step (t'@s) e" by auto
- from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
- show ?case
- proof(rule h2 [OF vt_ts stp _ _ _ ])
- show "R t'"
- proof(rule ih)
- from et show ext': "extend_highest s' th prio t'"
- by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
- next
- from vt_ts show "vt step (t' @ s)" .
- qed
- next
- from et show "extend_highest s' th prio (e # t')" .
- next
- from et show ext': "extend_highest s' th prio t'"
- by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
- qed
- qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
-proof -
- show ?thesis
- proof(induct rule:ind)
- case Nil
- from threads_s
- show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
- by auto
- next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio)
- assume eq_e: " e = Create thread prio"
- show ?thesis
- proof -
- from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
- hence "th \<noteq> thread"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- with Cons show ?thesis by auto
- qed
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have "extend_highest s' th prio (e # t)" by auto
- from extend_highest.exit_diff [OF this] and eq_e
- have neq_th: "thread \<noteq> th" by auto
- with Cons
- show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- next
- case (P thread cs)
- assume eq_e: "e = P thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (V thread cs)
- assume eq_e: "e = V thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (Set thread prio')
- assume eq_e: " e = Set thread prio'"
- show ?thesis
- proof -
- from Cons have "extend_highest s' th prio (e # t)" by auto
- from extend_highest.set_diff_low[OF this] and eq_e
- have "th \<noteq> thread" by auto
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- qed
- qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
- case Nil
- from highest_preced_thread
- show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
- by simp
-next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio')
- assume eq_e: " e = Create thread prio'"
- from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
- hence neq_thread: "thread \<noteq> th"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th \<in> threads (t@s)"
- proof -
- from Cons have "extend_highest s' th prio t" by auto
- from extend_highest.th_kept[OF this] show ?thesis by (simp add:s_def)
- qed
- ultimately show ?thesis by auto
- qed
- from Cons have "extend_highest s' th prio t" by auto
- from extend_highest.th_kept[OF this]
- have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
- by (auto simp:s_def)
- from stp
- have thread_ts: "thread \<notin> threads (t @ s)"
- by (cases, auto)
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
- by (unfold eq_e, simp)
- moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
- proof(rule Max_insert)
- from Cons have "vt step (t @ s)" by auto
- from finite_threads[OF this]
- show "finite (?f ` (threads (t@s)))" by simp
- next
- from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
- qed
- moreover have "(Max (?f ` (threads (t@s)))) = ?t"
- proof -
- have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
- (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
- proof -
- { fix th'
- assume "th' \<in> ?B"
- with thread_ts eq_e
- have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
- } thus ?thesis
- apply (auto simp:Image_def)
- proof -
- fix th'
- assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
- preced th' (e # t @ s) = preced th' (t @ s)"
- and h1: "th' \<in> threads (t @ s)"
- show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
- proof -
- from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
- moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
- ultimately show ?thesis by simp
- qed
- qed
- qed
- with Cons show ?thesis by auto
- qed
- moreover have "?f thread < ?t"
- proof -
- from Cons have " extend_highest s' th prio (e # t)" by auto
- from extend_highest.create_low[OF this] and eq_e
- have "prio' \<le> prio" by auto
- thus ?thesis
- by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
- qed
- ultimately show ?thesis by (auto simp:max_def)
- qed
-next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have vt_e: "vt step (e#(t @ s))" by auto
- from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
- from stp have thread_ts: "thread \<in> threads (t @ s)"
- by(cases, unfold runing_def readys_def, auto)
- from Cons have "extend_highest s' th prio (e # t)" by auto
- from extend_highest.exit_diff[OF this] and eq_e
- have neq_thread: "thread \<noteq> th" by auto
- from Cons have "extend_highest s' th prio t" by auto
- from extend_highest.th_kept[OF this, folded s_def]
- have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "threads (t@s) = insert thread ?A"
- by (insert stp thread_ts, unfold eq_e, auto)
- hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
- also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
- also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
- proof(rule Max_insert)
- from finite_threads [OF vt_e]
- show "finite (?f ` ?A)" by simp
- next
- from Cons have "extend_highest s' th prio (e # t)" by auto
- from extend_highest.th_kept[OF this]
- show "?f ` ?A \<noteq> {}" by (auto simp:s_def)
- qed
- finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
- moreover have "Max (?f ` (threads (t@s))) = ?t"
- proof -
- from Cons show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- qed
- ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
- moreover have "?f thread < ?t"
- proof(unfold eq_e, simp add:preced_def, fold preced_def)
- show "preced thread (t @ s) < ?t"
- proof -
- have "preced thread (t @ s) \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "?t = Max (?g ` ?B)") by simp
- moreover have "?g thread \<le> \<dots>"
- proof(rule Max_ge)
- have "vt step (t@s)" by fact
- from finite_threads [OF this]
- show "finite (?g ` ?B)" by simp
- next
- from thread_ts
- show "?g thread \<in> (?g ` ?B)" by auto
- qed
- ultimately show ?thesis by auto
- qed
- moreover have "preced thread (t @ s) \<noteq> ?t"
- proof
- assume "preced thread (t @ s) = preced th s"
- with h' have "preced thread (t @ s) = preced th (t@s)" by simp
- from preced_unique [OF this] have "thread = th"
- proof
- from h' show "th \<in> threads (t @ s)" by simp
- next
- from thread_ts show "thread \<in> threads (t @ s)" .
- qed(simp)
- with neq_thread show "False" by simp
- qed
- ultimately show ?thesis by auto
- qed
- qed
- ultimately show ?thesis
- by (auto simp:max_def split:if_splits)
- qed
- next
- case (P thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (Set thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- let ?B = "threads (t@s)"
- from Cons have "extend_highest s' th prio (e # t)" by auto
- from extend_highest.set_diff_low[OF this] and Set
- have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
- from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
- also have "\<dots> = ?t"
- proof(rule Max_eqI)
- fix y
- assume y_in: "y \<in> ?f ` ?B"
- then obtain th1 where
- th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
- show "y \<le> ?t"
- proof(cases "th1 = thread")
- case True
- with neq_thread le_p eq_y s_def Set
- show ?thesis
- by (auto simp:preced_def precedence_le_def)
- next
- case False
- with Set eq_y
- have "y = preced th1 (t@s)"
- by (simp add:preced_def)
- moreover have "\<dots> \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
- by auto
- moreover have "preced th1 (t@s) \<le> \<dots>"
- proof(rule Max_ge)
- from th1_in
- show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
- by simp
- next
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- proof -
- from Cons have "vt step (t @ s)" by auto
- from finite_threads[OF this] show ?thesis by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- next
- from Cons and finite_threads
- show "finite (?f ` ?B)" by auto
- next
- from Cons have "extend_highest s' th prio t" by auto
- from extend_highest.th_kept [OF this, folded s_def]
- have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show "?t \<in> (?f ` ?B)"
- proof -
- from neq_thread Set h
- have "?t = ?f th" by (auto simp:preced_def)
- with h show ?thesis by auto
- qed
- qed
- finally show ?thesis .
- qed
- qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- (is "?L = ?R")
-proof -
- have "?L = cpreced (t@s) (wq (t@s)) th"
- by (unfold cp_eq_cpreced, simp)
- also have "\<dots> = ?R"
- proof(unfold cpreced_def)
- show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
- Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
- proof(cases "?A = {}")
- case False
- have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
- moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
- proof(rule Max_insert)
- show "finite (?f ` ?A)"
- proof -
- from dependents_threads[OF vt_t]
- have "?A \<subseteq> threads (t@s)" .
- moreover from finite_threads[OF vt_t] have "finite \<dots>" .
- ultimately show ?thesis
- by (auto simp:finite_subset)
- qed
- next
- from False show "(?f ` ?A) \<noteq> {}" by simp
- qed
- moreover have "\<dots> = Max (?f ` ?B)"
- proof -
- from max_preced have "?f th = Max (?f ` ?B)" .
- moreover have "Max (?f ` ?A) \<le> \<dots>"
- proof(rule Max_mono)
- from False show "(?f ` ?A) \<noteq> {}" by simp
- next
- show "?f ` ?A \<subseteq> ?f ` ?B"
- proof -
- have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
- thus ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t]
- show "finite (?f ` ?B)" by simp
- qed
- ultimately show ?thesis
- by (auto simp:max_def)
- qed
- ultimately show ?thesis by auto
- next
- case True
- with max_preced show ?thesis by auto
- qed
- qed
- finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
- by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
- by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less':
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- shows "preced th' s < preced th s"
-proof -
- have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
- proof(rule Max_ge)
- from finite_threads [OF vt_s]
- show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
- next
- from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
- by simp
- qed
- moreover have "preced th' s \<noteq> preced th s"
- proof
- assume "preced th' s = preced th s"
- from preced_unique[OF this th'_in] neq_th' is_ready
- show "False" by (auto simp:readys_def)
- qed
- ultimately show ?thesis using highest_preced_thread
- by auto
-qed
-
-lemma pv_blocked:
- fixes th'
- assumes th'_in: "th' \<in> threads (t@s)"
- and neq_th': "th' \<noteq> th"
- and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
- shows "th' \<notin> runing (t@s)"
-proof
- assume "th' \<in> runing (t@s)"
- hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
- by (auto simp:runing_def)
- with max_cp_readys_threads [OF vt_t]
- have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
- by auto
- moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
- ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
- moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
- by simp
- finally have h: "cp (t @ s) th' = preced th (t @ s)" .
- show False
- proof -
- have "dependents (wq (t @ s)) th' = {}"
- by (rule count_eq_dependents [OF vt_t eq_pv])
- moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
- proof
- assume "preced th' (t @ s) = preced th (t @ s)"
- hence "th' = th"
- proof(rule preced_unique)
- from th_kept show "th \<in> threads (t @ s)" by simp
- next
- from th'_in show "th' \<in> threads (t @ s)" by simp
- qed
- with assms show False by simp
- qed
- ultimately show ?thesis
- by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
- qed
-qed
-
-lemma runing_precond_pre:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads (t@s) \<and>
- cntP (t@s) th' = cntV (t@s) th'"
-proof -
- show ?thesis
- proof(induct rule:ind)
- case (Cons e t)
- from Cons
- have in_thread: "th' \<in> threads (t @ s)"
- and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- have "extend_highest s' th prio t" by fact
- from extend_highest.pv_blocked
- [OF this, folded s_def, OF in_thread neq_th' not_holding]
- have not_runing: "th' \<notin> runing (t @ s)" .
- show ?case
- proof(cases e)
- case (V thread cs)
- from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
-
- show ?thesis
- proof -
- from Cons and V have "step (t@s) (V thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (unfold V, simp add:cntP_def cntV_def count_def)
- moreover from in_thread
- have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (P thread cs)
- from Cons and P have "step (t@s) (P thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Create thread prio')
- from Cons and Create have "step (t@s) (Create thread prio')" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th' \<in> threads (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with Cons and Create
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Create
- have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
- ultimately show ?thesis by auto
- next
- case (Exit thread)
- from Cons and Exit have "step (t@s) (Exit thread)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t @ s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and Exit
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Exit and neq_th'
- have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Set thread prio')
- with Cons
- show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case Nil
- with assms
- show ?case by auto
- qed
-qed
-
-(*
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<notin> runing (t@s)"
-proof -
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2]
- show ?thesis .
-qed
-*)
-
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- and is_runing: "th' \<in> runing (t@s)"
- shows "cntP s th' > cntV s th'"
-proof -
- have "cntP s th' \<noteq> cntV s th'"
- proof
- assume eq_pv: "cntP s th' = cntV s th'"
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)"
- and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
- with is_runing show "False" by simp
- qed
- moreover from cnp_cnv_cncs[OF vt_s, of th']
- have "cntV s th' \<le> cntP s th'" by auto
- ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
- th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
- case (Suc k)
- show ?case
- proof -
- { assume True: "Suc (i+k) \<le> length t"
- from moment_head [OF this]
- obtain e where
- eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
- by blast
- from red_moment[of "Suc(i+k)"]
- and eq_me have "extend_highest s' th prio (e # moment (i + k) t)" by simp
- hence vt_e: "vt step (e#(moment (i + k) t)@s)"
- by (unfold extend_highest_def extend_highest_axioms_def
- highest_create_def s_def, auto)
- have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
- proof(unfold s_def)
- show "th' \<notin> runing (moment (i + k) t @ Create th prio # s')"
- proof(rule extend_highest.pv_blocked)
- from Suc show "th' \<in> threads (moment (i + k) t @ Create th prio # s')"
- by (simp add:s_def)
- next
- from neq_th' show "th' \<noteq> th" .
- next
- from red_moment show "extend_highest s' th prio (moment (i + k) t)" .
- next
- from Suc show "cntP (moment (i + k) t @ Create th prio # s') th' =
- cntV (moment (i + k) t @ Create th prio # s') th'"
- by (auto simp:s_def)
- qed
- qed
- from step_back_step[OF vt_e]
- have "step ((moment (i + k) t)@s) e" .
- hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
- th' \<in> threads (e#(moment (i + k) t)@s)
- "
- proof(cases)
- case (thread_create thread prio)
- with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_exit thread)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_P thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_V thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_set thread prio')
- with Suc show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- with eq_me have ?thesis using eq_me by auto
- } note h = this
- show ?thesis
- proof(cases "Suc (i+k) \<le> length t")
- case True
- from h [OF this] show ?thesis .
- next
- case False
- with moment_ge
- have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
- with Suc show ?thesis by auto
- qed
- qed
-next
- case 0
- from assms show ?case by auto
-qed
-
-lemma moment_blocked:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- and le_ij: "i \<le> j"
- shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
- have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
- and h2: "th' \<in> threads ((moment j t)@s)" by auto
- with extend_highest.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1]
- show ?thesis by auto
-qed
-
-lemma runing_inversion_1:
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
- case True
- with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
- case False
- let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
- let ?q = "moment 0 t"
- from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
- from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
- from p_split_gen [of ?Q, OF this not_thread]
- obtain i where lt_its: "i < length t"
- and le_i: "0 \<le> i"
- and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
- and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
- from lt_its have "Suc i \<le> length t" by auto
- from moment_head[OF this] obtain e where
- eq_me: "moment (Suc i) t = e # moment i t" by blast
- from red_moment[of "Suc i"] and eq_me
- have "extend_highest s' th prio (e # moment i t)" by simp
- hence vt_e: "vt step (e#(moment i t)@s)"
- by (unfold extend_highest_def extend_highest_axioms_def
- highest_create_def s_def, auto)
- from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
- from post[rule_format, of "Suc i"] and eq_me
- have not_in': "th' \<in> threads (e # moment i t@s)" by auto
- from create_pre[OF stp_i pre this]
- obtain prio where eq_e: "e = Create th' prio" .
- have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
- proof(rule cnp_cnv_eq)
- from step_back_vt [OF vt_e]
- show "vt step (moment i t @ s)" .
- next
- from eq_e and stp_i
- have "step (moment i t @ s) (Create th' prio)" by simp
- thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
- qed
- with eq_e
- have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
- by (simp add:cntP_def cntV_def count_def)
- with eq_me[symmetric]
- have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
- by simp
- from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
- with eq_me [symmetric]
- have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
- from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
- and moment_ge
- have "th' \<notin> runing (t @ s)" by auto
- with runing'
- show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
- assumes runing': "th' \<in> runing (t@s)"
- shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
- from runing_inversion_1[OF _ runing']
- show ?thesis by auto
-qed
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
- case True thus ?thesis by auto
-next
- case False
- then have not_ready: "th \<notin> readys (t@s)"
- apply (unfold runing_def,
- insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
- by auto
- from th_kept have "th \<in> threads (t@s)" by auto
- from th_chain_to_ready[OF vt_t this] and not_ready
- obtain th' where th'_in: "th' \<in> readys (t@s)"
- and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
- have "th' \<in> runing (t@s)"
- proof -
- have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
- preced th (t@s)"
- proof(rule Max_eqI)
- fix y
- assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- then obtain th1 where
- h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
- and eq_y: "y = preced th1 (t@s)" by auto
- show "y \<le> preced th (t @ s)"
- proof -
- from max_preced
- have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
- moreover have "y \<le> \<dots>"
- proof(rule Max_ge)
- from h1
- have "th1 \<in> threads (t@s)"
- proof
- assume "th1 = th'"
- with th'_in show ?thesis by (simp add:readys_def)
- next
- assume "th1 \<in> dependents (wq (t @ s)) th'"
- with dependents_threads [OF vt_t]
- show "th1 \<in> threads (t @ s)" by auto
- qed
- with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
- next
- from finite_threads[OF vt_t]
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
- qed
- ultimately show ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
- show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
- by (auto intro:finite_subset)
- next
- from dp
- have "th \<in> dependents (wq (t @ s)) th'"
- by (unfold cs_dependents_def, auto simp:eq_depend)
- thus "preced th (t @ s) \<in>
- (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- by auto
- qed
- moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- from max_preced and max_cp_eq[OF vt_t, symmetric]
- have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
- with max_cp_readys_threads[OF vt_t] show ?thesis by simp
- qed
- ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
- qed
- with th'_in show ?thesis by (auto simp:runing_def)
- qed
- thus ?thesis by auto
-qed
-
-end
-
-end
-
--- a/prio/Attic/ExtGG_1.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,973 +0,0 @@
-theory ExtGG
-imports PrioG
-begin
-
-lemma birth_time_lt: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
- apply (induct s, simp)
-proof -
- fix a s
- assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
- and eq_as: "a # s \<noteq> []"
- show "birthtime th (a # s) < length (a # s)"
- proof(cases "s \<noteq> []")
- case False
- from False show ?thesis
- by (cases a, auto simp:birthtime.simps)
- next
- case True
- from ih [OF True] show ?thesis
- by (cases a, auto simp:birthtime.simps)
- qed
-qed
-
-lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
- by (induct s, auto simp:threads.simps)
-
-lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
- apply (drule_tac th_in_ne)
- by (unfold preced_def, auto intro: birth_time_lt)
-
-locale highest_gen =
- fixes s' th s e' prio tm
- defines s_def : "s \<equiv> (e'#s')"
- assumes vt_s: "vt step s"
- and threads_s: "th \<in> threads s"
- and highest: "preced th s = Max ((cp s)`threads s)"
- and nh: "preced th s' \<noteq> Max ((cp s)`threads s')"
- and preced_th: "preced th s = Prc prio tm"
-
-context highest_gen
-begin
-
-lemma lt_tm: "tm < length s"
- by (insert preced_tm_lt[OF threads_s preced_th], simp)
-
-lemma vt_s': "vt step s'"
- by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
-
-lemma eq_cp_s_th: "cp s th = preced th s"
-proof -
- from highest and max_cp_eq[OF vt_s]
- have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
- have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
- proof -
- from threads_s and dependents_threads[OF vt_s, of th]
- show ?thesis by auto
- qed
- show ?thesis
- proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
- show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
- next
- fix y
- assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
- then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
- and eq_y: "y = preced th1 s" by auto
- show "y \<le> preced th s"
- proof(unfold is_max, rule Max_ge)
- from finite_threads[OF vt_s]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- from sbs th1_in and eq_y
- show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
- qed
- next
- from sbs and finite_threads[OF vt_s]
- show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
- by (auto intro:finite_subset)
- qed
-qed
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
-proof -
- from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
- show ?thesis by simp
-qed
-
-end
-
-locale extend_highest_gen = highest_gen +
- fixes t
- assumes vt_t: "vt step (t@s)"
- and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
- and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
- and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app:
- assumes vt_ts: "vt cs (t@s)"
- shows "vt cs s"
-proof -
- from vt_ts show ?thesis
- proof(induct t)
- case Nil
- from Nil show ?case by auto
- next
- case (Cons e t)
- assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
- and vt_et: "vt cs ((e # t) @ s)"
- show ?case
- proof(rule ih)
- show "vt cs (t @ s)"
- proof(rule step_back_vt)
- from vt_et show "vt cs (e # t @ s)" by simp
- qed
- qed
- qed
-qed
-
-context extend_highest_gen
-begin
-
-lemma red_moment:
- "extend_highest_gen s' th e' prio tm (moment i t)"
- apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
- apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
- by (unfold highest_gen_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes
- h0: "R []"
- and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
- extend_highest_gen s' th e' prio tm t;
- extend_highest_gen s' th e' prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
- shows "R t"
-proof -
- from vt_t extend_highest_gen_axioms show ?thesis
- proof(induct t)
- from h0 show "R []" .
- next
- case (Cons e t')
- assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_gen s' th e' prio tm t'\<rbrakk> \<Longrightarrow> R t'"
- and vt_e: "vt step ((e # t') @ s)"
- and et: "extend_highest_gen s' th e' prio tm (e # t')"
- from vt_e and step_back_step have stp: "step (t'@s) e" by auto
- from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
- show ?case
- proof(rule h2 [OF vt_ts stp _ _ _ ])
- show "R t'"
- proof(rule ih)
- from et show ext': "extend_highest_gen s' th e' prio tm t'"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
- next
- from vt_ts show "vt step (t' @ s)" .
- qed
- next
- from et show "extend_highest_gen s' th e' prio tm (e # t')" .
- next
- from et show ext': "extend_highest_gen s' th e' prio tm t'"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
- qed
- qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
-proof -
- show ?thesis
- proof(induct rule:ind)
- case Nil
- from threads_s
- show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
- by auto
- next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio)
- assume eq_e: " e = Create thread prio"
- show ?thesis
- proof -
- from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
- hence "th \<noteq> thread"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- with Cons show ?thesis by auto
- qed
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
- from extend_highest_gen.exit_diff [OF this] and eq_e
- have neq_th: "thread \<noteq> th" by auto
- with Cons
- show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- next
- case (P thread cs)
- assume eq_e: "e = P thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (V thread cs)
- assume eq_e: "e = V thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (Set thread prio')
- assume eq_e: " e = Set thread prio'"
- show ?thesis
- proof -
- from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
- from extend_highest_gen.set_diff_low[OF this] and eq_e
- have "th \<noteq> thread" by auto
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- qed
- qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
- case Nil
- from highest_preced_thread
- show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
- by simp
-next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio')
- assume eq_e: " e = Create thread prio'"
- from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
- hence neq_thread: "thread \<noteq> th"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th \<in> threads (t@s)"
- proof -
- from Cons have "extend_highest_gen s' th e' prio tm t" by auto
- from extend_highest_gen.th_kept[OF this] show ?thesis by (simp add:s_def)
- qed
- ultimately show ?thesis by auto
- qed
- from Cons have "extend_highest_gen s' th e' prio tm t" by auto
- from extend_highest_gen.th_kept[OF this]
- have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
- by (auto simp:s_def)
- from stp
- have thread_ts: "thread \<notin> threads (t @ s)"
- by (cases, auto)
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
- by (unfold eq_e, simp)
- moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
- proof(rule Max_insert)
- from Cons have "vt step (t @ s)" by auto
- from finite_threads[OF this]
- show "finite (?f ` (threads (t@s)))" by simp
- next
- from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
- qed
- moreover have "(Max (?f ` (threads (t@s)))) = ?t"
- proof -
- have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
- (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
- proof -
- { fix th'
- assume "th' \<in> ?B"
- with thread_ts eq_e
- have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
- } thus ?thesis
- apply (auto simp:Image_def)
- proof -
- fix th'
- assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
- preced th' (e # t @ s) = preced th' (t @ s)"
- and h1: "th' \<in> threads (t @ s)"
- show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
- proof -
- from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
- moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
- ultimately show ?thesis by simp
- qed
- qed
- qed
- with Cons show ?thesis by auto
- qed
- moreover have "?f thread < ?t"
- proof -
- from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
- from extend_highest_gen.create_low[OF this] and eq_e
- have "prio' \<le> prio" by auto
- thus ?thesis
- by (unfold preced_th, unfold eq_e, insert lt_tm,
- auto simp:preced_def s_def precedence_less_def preced_th)
- qed
- ultimately show ?thesis by (auto simp:max_def)
- qed
-next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have vt_e: "vt step (e#(t @ s))" by auto
- from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
- from stp have thread_ts: "thread \<in> threads (t @ s)"
- by(cases, unfold runing_def readys_def, auto)
- from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
- from extend_highest_gen.exit_diff[OF this] and eq_e
- have neq_thread: "thread \<noteq> th" by auto
- from Cons have "extend_highest_gen s' th e' prio tm t" by auto
- from extend_highest_gen.th_kept[OF this, folded s_def]
- have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "threads (t@s) = insert thread ?A"
- by (insert stp thread_ts, unfold eq_e, auto)
- hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
- also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
- also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
- proof(rule Max_insert)
- from finite_threads [OF vt_e]
- show "finite (?f ` ?A)" by simp
- next
- from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
- from extend_highest_gen.th_kept[OF this]
- show "?f ` ?A \<noteq> {}" by (auto simp:s_def)
- qed
- finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
- moreover have "Max (?f ` (threads (t@s))) = ?t"
- proof -
- from Cons show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- qed
- ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
- moreover have "?f thread < ?t"
- proof(unfold eq_e, simp add:preced_def, fold preced_def)
- show "preced thread (t @ s) < ?t"
- proof -
- have "preced thread (t @ s) \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "?t = Max (?g ` ?B)") by simp
- moreover have "?g thread \<le> \<dots>"
- proof(rule Max_ge)
- have "vt step (t@s)" by fact
- from finite_threads [OF this]
- show "finite (?g ` ?B)" by simp
- next
- from thread_ts
- show "?g thread \<in> (?g ` ?B)" by auto
- qed
- ultimately show ?thesis by auto
- qed
- moreover have "preced thread (t @ s) \<noteq> ?t"
- proof
- assume "preced thread (t @ s) = preced th s"
- with h' have "preced thread (t @ s) = preced th (t@s)" by simp
- from preced_unique [OF this] have "thread = th"
- proof
- from h' show "th \<in> threads (t @ s)" by simp
- next
- from thread_ts show "thread \<in> threads (t @ s)" .
- qed(simp)
- with neq_thread show "False" by simp
- qed
- ultimately show ?thesis by auto
- qed
- qed
- ultimately show ?thesis
- by (auto simp:max_def split:if_splits)
- qed
- next
- case (P thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (Set thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- let ?B = "threads (t@s)"
- from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
- from extend_highest_gen.set_diff_low[OF this] and Set
- have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
- from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
- also have "\<dots> = ?t"
- proof(rule Max_eqI)
- fix y
- assume y_in: "y \<in> ?f ` ?B"
- then obtain th1 where
- th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
- show "y \<le> ?t"
- proof(cases "th1 = thread")
- case True
- with neq_thread le_p eq_y s_def Set
- show ?thesis
- apply (subst preced_th, insert lt_tm)
- by (auto simp:preced_def precedence_le_def)
- next
- case False
- with Set eq_y
- have "y = preced th1 (t@s)"
- by (simp add:preced_def)
- moreover have "\<dots> \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
- by auto
- moreover have "preced th1 (t@s) \<le> \<dots>"
- proof(rule Max_ge)
- from th1_in
- show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
- by simp
- next
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- proof -
- from Cons have "vt step (t @ s)" by auto
- from finite_threads[OF this] show ?thesis by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- next
- from Cons and finite_threads
- show "finite (?f ` ?B)" by auto
- next
- from Cons have "extend_highest_gen s' th e' prio tm t" by auto
- from extend_highest_gen.th_kept [OF this, folded s_def]
- have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show "?t \<in> (?f ` ?B)"
- proof -
- from neq_thread Set h
- have "?t = ?f th" by (auto simp:preced_def)
- with h show ?thesis by auto
- qed
- qed
- finally show ?thesis .
- qed
- qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- (is "?L = ?R")
-proof -
- have "?L = cpreced (t@s) (wq (t@s)) th"
- by (unfold cp_eq_cpreced, simp)
- also have "\<dots> = ?R"
- proof(unfold cpreced_def)
- show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
- Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
- proof(cases "?A = {}")
- case False
- have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
- moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
- proof(rule Max_insert)
- show "finite (?f ` ?A)"
- proof -
- from dependents_threads[OF vt_t]
- have "?A \<subseteq> threads (t@s)" .
- moreover from finite_threads[OF vt_t] have "finite \<dots>" .
- ultimately show ?thesis
- by (auto simp:finite_subset)
- qed
- next
- from False show "(?f ` ?A) \<noteq> {}" by simp
- qed
- moreover have "\<dots> = Max (?f ` ?B)"
- proof -
- from max_preced have "?f th = Max (?f ` ?B)" .
- moreover have "Max (?f ` ?A) \<le> \<dots>"
- proof(rule Max_mono)
- from False show "(?f ` ?A) \<noteq> {}" by simp
- next
- show "?f ` ?A \<subseteq> ?f ` ?B"
- proof -
- have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
- thus ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t]
- show "finite (?f ` ?B)" by simp
- qed
- ultimately show ?thesis
- by (auto simp:max_def)
- qed
- ultimately show ?thesis by auto
- next
- case True
- with max_preced show ?thesis by auto
- qed
- qed
- finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
- by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
- by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less':
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- shows "preced th' s < preced th s"
-proof -
- have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
- proof(rule Max_ge)
- from finite_threads [OF vt_s]
- show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
- next
- from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
- by simp
- qed
- moreover have "preced th' s \<noteq> preced th s"
- proof
- assume "preced th' s = preced th s"
- from preced_unique[OF this th'_in] neq_th' threads_s
- show "False" by (auto simp:readys_def)
- qed
- ultimately show ?thesis using highest_preced_thread
- by auto
-qed
-
-lemma pv_blocked:
- fixes th'
- assumes th'_in: "th' \<in> threads (t@s)"
- and neq_th': "th' \<noteq> th"
- and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
- shows "th' \<notin> runing (t@s)"
-proof
- assume "th' \<in> runing (t@s)"
- hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
- by (auto simp:runing_def)
- with max_cp_readys_threads [OF vt_t]
- have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
- by auto
- moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
- ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
- moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
- by simp
- finally have h: "cp (t @ s) th' = preced th (t @ s)" .
- show False
- proof -
- have "dependents (wq (t @ s)) th' = {}"
- by (rule count_eq_dependents [OF vt_t eq_pv])
- moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
- proof
- assume "preced th' (t @ s) = preced th (t @ s)"
- hence "th' = th"
- proof(rule preced_unique)
- from th_kept show "th \<in> threads (t @ s)" by simp
- next
- from th'_in show "th' \<in> threads (t @ s)" by simp
- qed
- with assms show False by simp
- qed
- ultimately show ?thesis
- by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
- qed
-qed
-
-lemma runing_precond_pre:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads (t@s) \<and>
- cntP (t@s) th' = cntV (t@s) th'"
-proof -
- show ?thesis
- proof(induct rule:ind)
- case (Cons e t)
- from Cons
- have in_thread: "th' \<in> threads (t @ s)"
- and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from Cons have "extend_highest_gen s' th e' prio tm t" by auto
- from extend_highest_gen.pv_blocked
- [OF this, folded s_def, OF in_thread neq_th' not_holding]
- have not_runing: "th' \<notin> runing (t @ s)" .
- show ?case
- proof(cases e)
- case (V thread cs)
- from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
-
- show ?thesis
- proof -
- from Cons and V have "step (t@s) (V thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (unfold V, simp add:cntP_def cntV_def count_def)
- moreover from in_thread
- have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (P thread cs)
- from Cons and P have "step (t@s) (P thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Create thread prio')
- from Cons and Create have "step (t@s) (Create thread prio')" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th' \<in> threads (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with Cons and Create
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Create
- have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
- ultimately show ?thesis by auto
- next
- case (Exit thread)
- from Cons and Exit have "step (t@s) (Exit thread)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t @ s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and Exit
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Exit and neq_th'
- have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Set thread prio')
- with Cons
- show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case Nil
- with assms
- show ?case by auto
- qed
-qed
-
-(*
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<notin> runing (t@s)"
-proof -
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2]
- show ?thesis .
-qed
-*)
-
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- and is_runing: "th' \<in> runing (t@s)"
- shows "cntP s th' > cntV s th'"
-proof -
- have "cntP s th' \<noteq> cntV s th'"
- proof
- assume eq_pv: "cntP s th' = cntV s th'"
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)"
- and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
- with is_runing show "False" by simp
- qed
- moreover from cnp_cnv_cncs[OF vt_s, of th']
- have "cntV s th' \<le> cntP s th'" by auto
- ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
- th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
- case (Suc k)
- show ?case
- proof -
- { assume True: "Suc (i+k) \<le> length t"
- from moment_head [OF this]
- obtain e where
- eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
- by blast
- from red_moment[of "Suc(i+k)"]
- and eq_me have "extend_highest_gen s' th e' prio tm (e # moment (i + k) t)" by simp
- hence vt_e: "vt step (e#(moment (i + k) t)@s)"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
- highest_gen_def s_def, auto)
- have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
- proof(unfold s_def)
- show "th' \<notin> runing (moment (i + k) t @ e' # s')"
- proof(rule extend_highest_gen.pv_blocked)
- from Suc show "th' \<in> threads (moment (i + k) t @ e' # s')"
- by (simp add:s_def)
- next
- from neq_th' show "th' \<noteq> th" .
- next
- from red_moment show "extend_highest_gen s' th e' prio tm (moment (i + k) t)" .
- next
- from Suc show "cntP (moment (i + k) t @ e' # s') th' = cntV (moment (i + k) t @ e' # s') th'"
- by (auto simp:s_def)
- qed
- qed
- from step_back_step[OF vt_e]
- have "step ((moment (i + k) t)@s) e" .
- hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
- th' \<in> threads (e#(moment (i + k) t)@s)
- "
- proof(cases)
- case (thread_create thread prio)
- with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_exit thread)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_P thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_V thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_set thread prio')
- with Suc show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- with eq_me have ?thesis using eq_me by auto
- } note h = this
- show ?thesis
- proof(cases "Suc (i+k) \<le> length t")
- case True
- from h [OF this] show ?thesis .
- next
- case False
- with moment_ge
- have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
- with Suc show ?thesis by auto
- qed
- qed
-next
- case 0
- from assms show ?case by auto
-qed
-
-lemma moment_blocked:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- and le_ij: "i \<le> j"
- shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
- have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
- and h2: "th' \<in> threads ((moment j t)@s)" by auto
- with extend_highest_gen.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1]
- show ?thesis by auto
-qed
-
-lemma runing_inversion_1:
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
- case True
- with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
- case False
- let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
- let ?q = "moment 0 t"
- from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
- from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
- from p_split_gen [of ?Q, OF this not_thread]
- obtain i where lt_its: "i < length t"
- and le_i: "0 \<le> i"
- and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
- and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
- from lt_its have "Suc i \<le> length t" by auto
- from moment_head[OF this] obtain e where
- eq_me: "moment (Suc i) t = e # moment i t" by blast
- from red_moment[of "Suc i"] and eq_me
- have "extend_highest_gen s' th e' prio tm (e # moment i t)" by simp
- hence vt_e: "vt step (e#(moment i t)@s)"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
- highest_gen_def s_def, auto)
- from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
- from post[rule_format, of "Suc i"] and eq_me
- have not_in': "th' \<in> threads (e # moment i t@s)" by auto
- from create_pre[OF stp_i pre this]
- obtain prio where eq_e: "e = Create th' prio" .
- have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
- proof(rule cnp_cnv_eq)
- from step_back_vt [OF vt_e]
- show "vt step (moment i t @ s)" .
- next
- from eq_e and stp_i
- have "step (moment i t @ s) (Create th' prio)" by simp
- thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
- qed
- with eq_e
- have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
- by (simp add:cntP_def cntV_def count_def)
- with eq_me[symmetric]
- have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
- by simp
- from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
- with eq_me [symmetric]
- have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
- from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
- and moment_ge
- have "th' \<notin> runing (t @ s)" by auto
- with runing'
- show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
- assumes runing': "th' \<in> runing (t@s)"
- shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
- from runing_inversion_1[OF _ runing']
- show ?thesis by auto
-qed
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
- case True thus ?thesis by auto
-next
- case False
- then have not_ready: "th \<notin> readys (t@s)"
- apply (unfold runing_def,
- insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
- by auto
- from th_kept have "th \<in> threads (t@s)" by auto
- from th_chain_to_ready[OF vt_t this] and not_ready
- obtain th' where th'_in: "th' \<in> readys (t@s)"
- and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
- have "th' \<in> runing (t@s)"
- proof -
- have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
- preced th (t@s)"
- proof(rule Max_eqI)
- fix y
- assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- then obtain th1 where
- h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
- and eq_y: "y = preced th1 (t@s)" by auto
- show "y \<le> preced th (t @ s)"
- proof -
- from max_preced
- have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
- moreover have "y \<le> \<dots>"
- proof(rule Max_ge)
- from h1
- have "th1 \<in> threads (t@s)"
- proof
- assume "th1 = th'"
- with th'_in show ?thesis by (simp add:readys_def)
- next
- assume "th1 \<in> dependents (wq (t @ s)) th'"
- with dependents_threads [OF vt_t]
- show "th1 \<in> threads (t @ s)" by auto
- qed
- with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
- next
- from finite_threads[OF vt_t]
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
- qed
- ultimately show ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
- show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
- by (auto intro:finite_subset)
- next
- from dp
- have "th \<in> dependents (wq (t @ s)) th'"
- by (unfold cs_dependents_def, auto simp:eq_depend)
- thus "preced th (t @ s) \<in>
- (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- by auto
- qed
- moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- from max_preced and max_cp_eq[OF vt_t, symmetric]
- have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
- with max_cp_readys_threads[OF vt_t] show ?thesis by simp
- qed
- ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
- qed
- with th'_in show ?thesis by (auto simp:runing_def)
- qed
- thus ?thesis by auto
-qed
-
-end
-
-end
-
-
--- a/prio/Attic/ExtS.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1019 +0,0 @@
-theory ExtS
-imports Prio
-begin
-
-locale highest_set =
- fixes s' th prio fixes s
- defines s_def : "s \<equiv> (Set th prio#s')"
- assumes vt_s: "vt step s"
- and highest: "preced th s = Max ((cp s)`threads s)"
-
-context highest_set
-begin
-
-
-lemma vt_s': "vt step s'"
- by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
-
-lemma step_set: "step s' (Set th prio)"
- by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
-
-lemma step_set_elim:
- "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
- by (insert step_set, ind_cases "step s' (Set th prio)", auto)
-
-
-lemma threads_s: "th \<in> threads s"
- by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
-
-lemma same_depend: "depend s = depend s'"
- by (insert depend_set_unchanged, unfold s_def, simp)
-
-lemma same_dependents:
- "dependents (wq s) th = dependents (wq s') th"
- apply (unfold cs_dependents_def)
- by (unfold eq_depend same_depend, simp)
-
-lemma eq_cp_s_th: "cp s th = preced th s"
-proof -
- from highest and max_cp_eq[OF vt_s]
- have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
- have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
- proof -
- from threads_s and dependents_threads[OF vt_s, of th]
- show ?thesis by auto
- qed
- show ?thesis
- proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
- show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
- next
- fix y
- assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
- then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
- and eq_y: "y = preced th1 s" by auto
- show "y \<le> preced th s"
- proof(unfold is_max, rule Max_ge)
- from finite_threads[OF vt_s]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- from sbs th1_in and eq_y
- show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
- qed
- next
- from sbs and finite_threads[OF vt_s]
- show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
- by (auto intro:finite_subset)
- qed
-qed
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma is_ready: "th \<in> readys s"
-proof -
- have "\<forall>cs. \<not> waiting s th cs"
- apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
- apply (unfold s_depend_def, unfold runing_def readys_def)
- apply (auto, fold s_def)
- apply (erule_tac x = cs in allE, auto simp:waiting_eq)
- proof -
- fix cs
- assume h:
- "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
- {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
- (is "?L = ?R")
- and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
- from wt have "(Th th, Cs cs) \<in> ?L" by auto
- with h have "(Th th, Cs cs) \<in> ?R" by simp
- hence "waiting (wq s') th cs" by auto with nwt
- show False by auto
- qed
- with threads_s show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
-proof -
- from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
- show ?thesis by simp
-qed
-
-lemma is_runing: "th \<in> runing s"
-proof -
- have "Max (cp s ` threads s) = Max (cp s ` readys s)"
- proof -
- have " Max (cp s ` readys s) = cp s th"
- proof(rule Max_eqI)
- from finite_threads[OF vt_s] readys_threads finite_subset
- have "finite (readys s)" by blast
- thus "finite (cp s ` readys s)" by auto
- next
- from is_ready show "cp s th \<in> cp s ` readys s" by auto
- next
- fix y
- assume "y \<in> cp s ` readys s"
- then obtain th1 where
- eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
- show "y \<le> cp s th"
- proof -
- have "y \<le> Max (cp s ` threads s)"
- proof(rule Max_ge)
- from eq_y and th1_in
- show "y \<in> cp s ` threads s"
- by (auto simp:readys_def)
- next
- from finite_threads[OF vt_s]
- show "finite (cp s ` threads s)" by auto
- qed
- with highest' show ?thesis by auto
- qed
- qed
- with highest' show ?thesis by auto
- qed
- thus ?thesis
- by (unfold runing_def, insert highest' is_ready, auto)
-qed
-
-end
-
-locale extend_highest_set = highest_set +
- fixes t
- assumes vt_t: "vt step (t@s)"
- and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
- and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
- and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app:
- assumes vt_ts: "vt cs (t@s)"
- shows "vt cs s"
-proof -
- from vt_ts show ?thesis
- proof(induct t)
- case Nil
- from Nil show ?case by auto
- next
- case (Cons e t)
- assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
- and vt_et: "vt cs ((e # t) @ s)"
- show ?case
- proof(rule ih)
- show "vt cs (t @ s)"
- proof(rule step_back_vt)
- from vt_et show "vt cs (e # t @ s)" by simp
- qed
- qed
- qed
-qed
-
-context extend_highest_set
-begin
-
-lemma red_moment:
- "extend_highest_set s' th prio (moment i t)"
- apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
- apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
- by (unfold highest_set_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes
- h0: "R []"
- and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
- extend_highest_set s' th prio t;
- extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
- shows "R t"
-proof -
- from vt_t extend_highest_set_axioms show ?thesis
- proof(induct t)
- from h0 show "R []" .
- next
- case (Cons e t')
- assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
- and vt_e: "vt step ((e # t') @ s)"
- and et: "extend_highest_set s' th prio (e # t')"
- from vt_e and step_back_step have stp: "step (t'@s) e" by auto
- from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
- show ?case
- proof(rule h2 [OF vt_ts stp _ _ _ ])
- show "R t'"
- proof(rule ih)
- from et show ext': "extend_highest_set s' th prio t'"
- by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
- next
- from vt_ts show "vt step (t' @ s)" .
- qed
- next
- from et show "extend_highest_set s' th prio (e # t')" .
- next
- from et show ext': "extend_highest_set s' th prio t'"
- by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
- qed
- qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
-proof -
- show ?thesis
- proof(induct rule:ind)
- case Nil
- from threads_s
- show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
- by auto
- next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio)
- assume eq_e: " e = Create thread prio"
- show ?thesis
- proof -
- from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
- hence "th \<noteq> thread"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- with Cons show ?thesis by auto
- qed
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.exit_diff [OF this] and eq_e
- have neq_th: "thread \<noteq> th" by auto
- with Cons
- show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- next
- case (P thread cs)
- assume eq_e: "e = P thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (V thread cs)
- assume eq_e: "e = V thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (Set thread prio')
- assume eq_e: " e = Set thread prio'"
- show ?thesis
- proof -
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.set_diff_low[OF this] and eq_e
- have "th \<noteq> thread" by auto
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- qed
- qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
- case Nil
- from highest_preced_thread
- show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
- by simp
-next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio')
- assume eq_e: " e = Create thread prio'"
- from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
- hence neq_thread: "thread \<noteq> th"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th \<in> threads (t@s)"
- proof -
- from Cons have "extend_highest_set s' th prio t" by auto
- from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
- qed
- ultimately show ?thesis by auto
- qed
- from Cons have "extend_highest_set s' th prio t" by auto
- from extend_highest_set.th_kept[OF this]
- have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
- by (auto simp:s_def)
- from stp
- have thread_ts: "thread \<notin> threads (t @ s)"
- by (cases, auto)
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
- by (unfold eq_e, simp)
- moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
- proof(rule Max_insert)
- from Cons have "vt step (t @ s)" by auto
- from finite_threads[OF this]
- show "finite (?f ` (threads (t@s)))" by simp
- next
- from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
- qed
- moreover have "(Max (?f ` (threads (t@s)))) = ?t"
- proof -
- have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
- (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
- proof -
- { fix th'
- assume "th' \<in> ?B"
- with thread_ts eq_e
- have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
- } thus ?thesis
- apply (auto simp:Image_def)
- proof -
- fix th'
- assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
- preced th' (e # t @ s) = preced th' (t @ s)"
- and h1: "th' \<in> threads (t @ s)"
- show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
- proof -
- from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
- moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
- ultimately show ?thesis by simp
- qed
- qed
- qed
- with Cons show ?thesis by auto
- qed
- moreover have "?f thread < ?t"
- proof -
- from Cons have " extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.create_low[OF this] and eq_e
- have "prio' \<le> prio" by auto
- thus ?thesis
- by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
- qed
- ultimately show ?thesis by (auto simp:max_def)
- qed
-next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have vt_e: "vt step (e#(t @ s))" by auto
- from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
- from stp have thread_ts: "thread \<in> threads (t @ s)"
- by(cases, unfold runing_def readys_def, auto)
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.exit_diff[OF this] and eq_e
- have neq_thread: "thread \<noteq> th" by auto
- from Cons have "extend_highest_set s' th prio t" by auto
- from extend_highest_set.th_kept[OF this, folded s_def]
- have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "threads (t@s) = insert thread ?A"
- by (insert stp thread_ts, unfold eq_e, auto)
- hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
- also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
- also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
- proof(rule Max_insert)
- from finite_threads [OF vt_e]
- show "finite (?f ` ?A)" by simp
- next
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.th_kept[OF this]
- show "?f ` ?A \<noteq> {}" by (auto simp:s_def)
- qed
- finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
- moreover have "Max (?f ` (threads (t@s))) = ?t"
- proof -
- from Cons show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- qed
- ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
- moreover have "?f thread < ?t"
- proof(unfold eq_e, simp add:preced_def, fold preced_def)
- show "preced thread (t @ s) < ?t"
- proof -
- have "preced thread (t @ s) \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "?t = Max (?g ` ?B)") by simp
- moreover have "?g thread \<le> \<dots>"
- proof(rule Max_ge)
- have "vt step (t@s)" by fact
- from finite_threads [OF this]
- show "finite (?g ` ?B)" by simp
- next
- from thread_ts
- show "?g thread \<in> (?g ` ?B)" by auto
- qed
- ultimately show ?thesis by auto
- qed
- moreover have "preced thread (t @ s) \<noteq> ?t"
- proof
- assume "preced thread (t @ s) = preced th s"
- with h' have "preced thread (t @ s) = preced th (t@s)" by simp
- from preced_unique [OF this] have "thread = th"
- proof
- from h' show "th \<in> threads (t @ s)" by simp
- next
- from thread_ts show "thread \<in> threads (t @ s)" .
- qed(simp)
- with neq_thread show "False" by simp
- qed
- ultimately show ?thesis by auto
- qed
- qed
- ultimately show ?thesis
- by (auto simp:max_def split:if_splits)
- qed
- next
- case (P thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (Set thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- let ?B = "threads (t@s)"
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.set_diff_low[OF this] and Set
- have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
- from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
- also have "\<dots> = ?t"
- proof(rule Max_eqI)
- fix y
- assume y_in: "y \<in> ?f ` ?B"
- then obtain th1 where
- th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
- show "y \<le> ?t"
- proof(cases "th1 = thread")
- case True
- with neq_thread le_p eq_y s_def Set
- show ?thesis
- by (auto simp:preced_def precedence_le_def)
- next
- case False
- with Set eq_y
- have "y = preced th1 (t@s)"
- by (simp add:preced_def)
- moreover have "\<dots> \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
- by auto
- moreover have "preced th1 (t@s) \<le> \<dots>"
- proof(rule Max_ge)
- from th1_in
- show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
- by simp
- next
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- proof -
- from Cons have "vt step (t @ s)" by auto
- from finite_threads[OF this] show ?thesis by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- next
- from Cons and finite_threads
- show "finite (?f ` ?B)" by auto
- next
- from Cons have "extend_highest_set s' th prio t" by auto
- from extend_highest_set.th_kept [OF this, folded s_def]
- have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show "?t \<in> (?f ` ?B)"
- proof -
- from neq_thread Set h
- have "?t = ?f th" by (auto simp:preced_def)
- with h show ?thesis by auto
- qed
- qed
- finally show ?thesis .
- qed
- qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- (is "?L = ?R")
-proof -
- have "?L = cpreced (t@s) (wq (t@s)) th"
- by (unfold cp_eq_cpreced, simp)
- also have "\<dots> = ?R"
- proof(unfold cpreced_def)
- show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
- Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
- proof(cases "?A = {}")
- case False
- have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
- moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
- proof(rule Max_insert)
- show "finite (?f ` ?A)"
- proof -
- from dependents_threads[OF vt_t]
- have "?A \<subseteq> threads (t@s)" .
- moreover from finite_threads[OF vt_t] have "finite \<dots>" .
- ultimately show ?thesis
- by (auto simp:finite_subset)
- qed
- next
- from False show "(?f ` ?A) \<noteq> {}" by simp
- qed
- moreover have "\<dots> = Max (?f ` ?B)"
- proof -
- from max_preced have "?f th = Max (?f ` ?B)" .
- moreover have "Max (?f ` ?A) \<le> \<dots>"
- proof(rule Max_mono)
- from False show "(?f ` ?A) \<noteq> {}" by simp
- next
- show "?f ` ?A \<subseteq> ?f ` ?B"
- proof -
- have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
- thus ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t]
- show "finite (?f ` ?B)" by simp
- qed
- ultimately show ?thesis
- by (auto simp:max_def)
- qed
- ultimately show ?thesis by auto
- next
- case True
- with max_preced show ?thesis by auto
- qed
- qed
- finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
- by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
- by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less':
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- shows "preced th' s < preced th s"
-proof -
- have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
- proof(rule Max_ge)
- from finite_threads [OF vt_s]
- show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
- next
- from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
- by simp
- qed
- moreover have "preced th' s \<noteq> preced th s"
- proof
- assume "preced th' s = preced th s"
- from preced_unique[OF this th'_in] neq_th' is_ready
- show "False" by (auto simp:readys_def)
- qed
- ultimately show ?thesis using highest_preced_thread
- by auto
-qed
-
-lemma pv_blocked:
- fixes th'
- assumes th'_in: "th' \<in> threads (t@s)"
- and neq_th': "th' \<noteq> th"
- and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
- shows "th' \<notin> runing (t@s)"
-proof
- assume "th' \<in> runing (t@s)"
- hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
- by (auto simp:runing_def)
- with max_cp_readys_threads [OF vt_t]
- have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
- by auto
- moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
- ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
- moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
- by simp
- finally have h: "cp (t @ s) th' = preced th (t @ s)" .
- show False
- proof -
- have "dependents (wq (t @ s)) th' = {}"
- by (rule count_eq_dependents [OF vt_t eq_pv])
- moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
- proof
- assume "preced th' (t @ s) = preced th (t @ s)"
- hence "th' = th"
- proof(rule preced_unique)
- from th_kept show "th \<in> threads (t @ s)" by simp
- next
- from th'_in show "th' \<in> threads (t @ s)" by simp
- qed
- with assms show False by simp
- qed
- ultimately show ?thesis
- by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
- qed
-qed
-
-lemma runing_precond_pre:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads (t@s) \<and>
- cntP (t@s) th' = cntV (t@s) th'"
-proof -
- show ?thesis
- proof(induct rule:ind)
- case (Cons e t)
- from Cons
- have in_thread: "th' \<in> threads (t @ s)"
- and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- have "extend_highest_set s' th prio t" by fact
- from extend_highest_set.pv_blocked
- [OF this, folded s_def, OF in_thread neq_th' not_holding]
- have not_runing: "th' \<notin> runing (t @ s)" .
- show ?case
- proof(cases e)
- case (V thread cs)
- from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
-
- show ?thesis
- proof -
- from Cons and V have "step (t@s) (V thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (unfold V, simp add:cntP_def cntV_def count_def)
- moreover from in_thread
- have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (P thread cs)
- from Cons and P have "step (t@s) (P thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Create thread prio')
- from Cons and Create have "step (t@s) (Create thread prio')" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th' \<in> threads (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with Cons and Create
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Create
- have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
- ultimately show ?thesis by auto
- next
- case (Exit thread)
- from Cons and Exit have "step (t@s) (Exit thread)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t @ s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and Exit
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Exit and neq_th'
- have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Set thread prio')
- with Cons
- show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case Nil
- with assms
- show ?case by auto
- qed
-qed
-
-(*
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<notin> runing (t@s)"
-proof -
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2]
- show ?thesis .
-qed
-*)
-
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- and is_runing: "th' \<in> runing (t@s)"
- shows "cntP s th' > cntV s th'"
-proof -
- have "cntP s th' \<noteq> cntV s th'"
- proof
- assume eq_pv: "cntP s th' = cntV s th'"
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)"
- and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
- with is_runing show "False" by simp
- qed
- moreover from cnp_cnv_cncs[OF vt_s, of th']
- have "cntV s th' \<le> cntP s th'" by auto
- ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
- th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
- case (Suc k)
- show ?case
- proof -
- { assume True: "Suc (i+k) \<le> length t"
- from moment_head [OF this]
- obtain e where
- eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
- by blast
- from red_moment[of "Suc(i+k)"]
- and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
- hence vt_e: "vt step (e#(moment (i + k) t)@s)"
- by (unfold extend_highest_set_def extend_highest_set_axioms_def
- highest_set_def s_def, auto)
- have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
- proof(unfold s_def)
- show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
- proof(rule extend_highest_set.pv_blocked)
- from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
- by (simp add:s_def)
- next
- from neq_th' show "th' \<noteq> th" .
- next
- from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
- next
- from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
- cntV (moment (i + k) t @ Set th prio # s') th'"
- by (auto simp:s_def)
- qed
- qed
- from step_back_step[OF vt_e]
- have "step ((moment (i + k) t)@s) e" .
- hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
- th' \<in> threads (e#(moment (i + k) t)@s)
- "
- proof(cases)
- case (thread_create thread prio)
- with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_exit thread)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_P thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_V thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_set thread prio')
- with Suc show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- with eq_me have ?thesis using eq_me by auto
- } note h = this
- show ?thesis
- proof(cases "Suc (i+k) \<le> length t")
- case True
- from h [OF this] show ?thesis .
- next
- case False
- with moment_ge
- have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
- with Suc show ?thesis by auto
- qed
- qed
-next
- case 0
- from assms show ?case by auto
-qed
-
-lemma moment_blocked:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- and le_ij: "i \<le> j"
- shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
- have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
- and h2: "th' \<in> threads ((moment j t)@s)" by auto
- with extend_highest_set.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1]
- show ?thesis by auto
-qed
-
-lemma runing_inversion_1:
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
- case True
- with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
- case False
- let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
- let ?q = "moment 0 t"
- from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
- from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
- from p_split_gen [of ?Q, OF this not_thread]
- obtain i where lt_its: "i < length t"
- and le_i: "0 \<le> i"
- and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
- and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
- from lt_its have "Suc i \<le> length t" by auto
- from moment_head[OF this] obtain e where
- eq_me: "moment (Suc i) t = e # moment i t" by blast
- from red_moment[of "Suc i"] and eq_me
- have "extend_highest_set s' th prio (e # moment i t)" by simp
- hence vt_e: "vt step (e#(moment i t)@s)"
- by (unfold extend_highest_set_def extend_highest_set_axioms_def
- highest_set_def s_def, auto)
- from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
- from post[rule_format, of "Suc i"] and eq_me
- have not_in': "th' \<in> threads (e # moment i t@s)" by auto
- from create_pre[OF stp_i pre this]
- obtain prio where eq_e: "e = Create th' prio" .
- have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
- proof(rule cnp_cnv_eq)
- from step_back_vt [OF vt_e]
- show "vt step (moment i t @ s)" .
- next
- from eq_e and stp_i
- have "step (moment i t @ s) (Create th' prio)" by simp
- thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
- qed
- with eq_e
- have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
- by (simp add:cntP_def cntV_def count_def)
- with eq_me[symmetric]
- have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
- by simp
- from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
- with eq_me [symmetric]
- have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
- from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
- and moment_ge
- have "th' \<notin> runing (t @ s)" by auto
- with runing'
- show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
- assumes runing': "th' \<in> runing (t@s)"
- shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
- from runing_inversion_1[OF _ runing']
- show ?thesis by auto
-qed
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
- case True thus ?thesis by auto
-next
- case False
- then have not_ready: "th \<notin> readys (t@s)"
- apply (unfold runing_def,
- insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
- by auto
- from th_kept have "th \<in> threads (t@s)" by auto
- from th_chain_to_ready[OF vt_t this] and not_ready
- obtain th' where th'_in: "th' \<in> readys (t@s)"
- and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
- have "th' \<in> runing (t@s)"
- proof -
- have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
- preced th (t@s)"
- proof(rule Max_eqI)
- fix y
- assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- then obtain th1 where
- h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
- and eq_y: "y = preced th1 (t@s)" by auto
- show "y \<le> preced th (t @ s)"
- proof -
- from max_preced
- have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
- moreover have "y \<le> \<dots>"
- proof(rule Max_ge)
- from h1
- have "th1 \<in> threads (t@s)"
- proof
- assume "th1 = th'"
- with th'_in show ?thesis by (simp add:readys_def)
- next
- assume "th1 \<in> dependents (wq (t @ s)) th'"
- with dependents_threads [OF vt_t]
- show "th1 \<in> threads (t @ s)" by auto
- qed
- with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
- next
- from finite_threads[OF vt_t]
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
- qed
- ultimately show ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
- show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
- by (auto intro:finite_subset)
- next
- from dp
- have "th \<in> dependents (wq (t @ s)) th'"
- by (unfold cs_dependents_def, auto simp:eq_depend)
- thus "preced th (t @ s) \<in>
- (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- by auto
- qed
- moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- from max_preced and max_cp_eq[OF vt_t, symmetric]
- have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
- with max_cp_readys_threads[OF vt_t] show ?thesis by simp
- qed
- ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
- qed
- with th'_in show ?thesis by (auto simp:runing_def)
- qed
- thus ?thesis by auto
-qed
-
-end
-
-end
-
--- a/prio/Attic/ExtSG.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1019 +0,0 @@
-theory ExtSG
-imports PrioG
-begin
-
-locale highest_set =
- fixes s' th prio fixes s
- defines s_def : "s \<equiv> (Set th prio#s')"
- assumes vt_s: "vt step s"
- and highest: "preced th s = Max ((cp s)`threads s)"
-
-context highest_set
-begin
-
-lemma vt_s': "vt step s'"
- by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
-
-lemma step_set: "step s' (Set th prio)"
- by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
-
-lemma step_set_elim:
- "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
- by (insert step_set, ind_cases "step s' (Set th prio)", auto)
-
-
-lemma threads_s: "th \<in> threads s"
- by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
-
-lemma same_depend: "depend s = depend s'"
- by (insert depend_set_unchanged, unfold s_def, simp)
-
-lemma same_dependents:
- "dependents (wq s) th = dependents (wq s') th"
- apply (unfold cs_dependents_def)
- by (unfold eq_depend same_depend, simp)
-
-lemma eq_cp_s_th: "cp s th = preced th s"
-proof -
- from highest and max_cp_eq[OF vt_s]
- have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
- have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
- proof -
- from threads_s and dependents_threads[OF vt_s, of th]
- show ?thesis by auto
- qed
- show ?thesis
- proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
- show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
- next
- fix y
- assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
- then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
- and eq_y: "y = preced th1 s" by auto
- show "y \<le> preced th s"
- proof(unfold is_max, rule Max_ge)
- from finite_threads[OF vt_s]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- from sbs th1_in and eq_y
- show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
- qed
- next
- from sbs and finite_threads[OF vt_s]
- show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
- by (auto intro:finite_subset)
- qed
-qed
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma is_ready: "th \<in> readys s"
-proof -
- have "\<forall>cs. \<not> waiting s th cs"
- apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
- apply (unfold s_depend_def, unfold runing_def readys_def)
- apply (auto, fold s_def)
- apply (erule_tac x = cs in allE, auto simp:waiting_eq)
- proof -
- fix cs
- assume h:
- "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
- {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
- (is "?L = ?R")
- and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
- from wt have "(Th th, Cs cs) \<in> ?L" by auto
- with h have "(Th th, Cs cs) \<in> ?R" by simp
- hence "waiting (wq s') th cs" by auto with nwt
- show False by auto
- qed
- with threads_s show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
-proof -
- from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
- show ?thesis by simp
-qed
-
-lemma is_runing: "th \<in> runing s"
-proof -
- have "Max (cp s ` threads s) = Max (cp s ` readys s)"
- proof -
- have " Max (cp s ` readys s) = cp s th"
- proof(rule Max_eqI)
- from finite_threads[OF vt_s] readys_threads finite_subset
- have "finite (readys s)" by blast
- thus "finite (cp s ` readys s)" by auto
- next
- from is_ready show "cp s th \<in> cp s ` readys s" by auto
- next
- fix y
- assume "y \<in> cp s ` readys s"
- then obtain th1 where
- eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
- show "y \<le> cp s th"
- proof -
- have "y \<le> Max (cp s ` threads s)"
- proof(rule Max_ge)
- from eq_y and th1_in
- show "y \<in> cp s ` threads s"
- by (auto simp:readys_def)
- next
- from finite_threads[OF vt_s]
- show "finite (cp s ` threads s)" by auto
- qed
- with highest' show ?thesis by auto
- qed
- qed
- with highest' show ?thesis by auto
- qed
- thus ?thesis
- by (unfold runing_def, insert highest' is_ready, auto)
-qed
-
-end
-
-locale extend_highest_set = highest_set +
- fixes t
- assumes vt_t: "vt step (t@s)"
- and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
- and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
- and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app:
- assumes vt_ts: "vt cs (t@s)"
- shows "vt cs s"
-proof -
- from vt_ts show ?thesis
- proof(induct t)
- case Nil
- from Nil show ?case by auto
- next
- case (Cons e t)
- assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
- and vt_et: "vt cs ((e # t) @ s)"
- show ?case
- proof(rule ih)
- show "vt cs (t @ s)"
- proof(rule step_back_vt)
- from vt_et show "vt cs (e # t @ s)" by simp
- qed
- qed
- qed
-qed
-
-context extend_highest_set
-begin
-
-lemma red_moment:
- "extend_highest_set s' th prio (moment i t)"
- apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
- apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
- by (unfold highest_set_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes
- h0: "R []"
- and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
- extend_highest_set s' th prio t;
- extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
- shows "R t"
-proof -
- from vt_t extend_highest_set_axioms show ?thesis
- proof(induct t)
- from h0 show "R []" .
- next
- case (Cons e t')
- assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
- and vt_e: "vt step ((e # t') @ s)"
- and et: "extend_highest_set s' th prio (e # t')"
- from vt_e and step_back_step have stp: "step (t'@s) e" by auto
- from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
- show ?case
- proof(rule h2 [OF vt_ts stp _ _ _ ])
- show "R t'"
- proof(rule ih)
- from et show ext': "extend_highest_set s' th prio t'"
- by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
- next
- from vt_ts show "vt step (t' @ s)" .
- qed
- next
- from et show "extend_highest_set s' th prio (e # t')" .
- next
- from et show ext': "extend_highest_set s' th prio t'"
- by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
- qed
- qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
-proof -
- show ?thesis
- proof(induct rule:ind)
- case Nil
- from threads_s
- show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
- by auto
- next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio)
- assume eq_e: " e = Create thread prio"
- show ?thesis
- proof -
- from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
- hence "th \<noteq> thread"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- with Cons show ?thesis by auto
- qed
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.exit_diff [OF this] and eq_e
- have neq_th: "thread \<noteq> th" by auto
- with Cons
- show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- next
- case (P thread cs)
- assume eq_e: "e = P thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (V thread cs)
- assume eq_e: "e = V thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (Set thread prio')
- assume eq_e: " e = Set thread prio'"
- show ?thesis
- proof -
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.set_diff_low[OF this] and eq_e
- have "th \<noteq> thread" by auto
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- qed
- qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
- case Nil
- from highest_preced_thread
- show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
- by simp
-next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio')
- assume eq_e: " e = Create thread prio'"
- from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
- hence neq_thread: "thread \<noteq> th"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th \<in> threads (t@s)"
- proof -
- from Cons have "extend_highest_set s' th prio t" by auto
- from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
- qed
- ultimately show ?thesis by auto
- qed
- from Cons have "extend_highest_set s' th prio t" by auto
- from extend_highest_set.th_kept[OF this]
- have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
- by (auto simp:s_def)
- from stp
- have thread_ts: "thread \<notin> threads (t @ s)"
- by (cases, auto)
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
- by (unfold eq_e, simp)
- moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
- proof(rule Max_insert)
- from Cons have "vt step (t @ s)" by auto
- from finite_threads[OF this]
- show "finite (?f ` (threads (t@s)))" by simp
- next
- from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
- qed
- moreover have "(Max (?f ` (threads (t@s)))) = ?t"
- proof -
- have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
- (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
- proof -
- { fix th'
- assume "th' \<in> ?B"
- with thread_ts eq_e
- have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
- } thus ?thesis
- apply (auto simp:Image_def)
- proof -
- fix th'
- assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
- preced th' (e # t @ s) = preced th' (t @ s)"
- and h1: "th' \<in> threads (t @ s)"
- show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
- proof -
- from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
- moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
- ultimately show ?thesis by simp
- qed
- qed
- qed
- with Cons show ?thesis by auto
- qed
- moreover have "?f thread < ?t"
- proof -
- from Cons have " extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.create_low[OF this] and eq_e
- have "prio' \<le> prio" by auto
- thus ?thesis
- by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
- qed
- ultimately show ?thesis by (auto simp:max_def)
- qed
-next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have vt_e: "vt step (e#(t @ s))" by auto
- from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
- from stp have thread_ts: "thread \<in> threads (t @ s)"
- by(cases, unfold runing_def readys_def, auto)
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.exit_diff[OF this] and eq_e
- have neq_thread: "thread \<noteq> th" by auto
- from Cons have "extend_highest_set s' th prio t" by auto
- from extend_highest_set.th_kept[OF this, folded s_def]
- have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "threads (t@s) = insert thread ?A"
- by (insert stp thread_ts, unfold eq_e, auto)
- hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
- also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
- also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
- proof(rule Max_insert)
- from finite_threads [OF vt_e]
- show "finite (?f ` ?A)" by simp
- next
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.th_kept[OF this]
- show "?f ` ?A \<noteq> {}" by (auto simp:s_def)
- qed
- finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
- moreover have "Max (?f ` (threads (t@s))) = ?t"
- proof -
- from Cons show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- qed
- ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
- moreover have "?f thread < ?t"
- proof(unfold eq_e, simp add:preced_def, fold preced_def)
- show "preced thread (t @ s) < ?t"
- proof -
- have "preced thread (t @ s) \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "?t = Max (?g ` ?B)") by simp
- moreover have "?g thread \<le> \<dots>"
- proof(rule Max_ge)
- have "vt step (t@s)" by fact
- from finite_threads [OF this]
- show "finite (?g ` ?B)" by simp
- next
- from thread_ts
- show "?g thread \<in> (?g ` ?B)" by auto
- qed
- ultimately show ?thesis by auto
- qed
- moreover have "preced thread (t @ s) \<noteq> ?t"
- proof
- assume "preced thread (t @ s) = preced th s"
- with h' have "preced thread (t @ s) = preced th (t@s)" by simp
- from preced_unique [OF this] have "thread = th"
- proof
- from h' show "th \<in> threads (t @ s)" by simp
- next
- from thread_ts show "thread \<in> threads (t @ s)" .
- qed(simp)
- with neq_thread show "False" by simp
- qed
- ultimately show ?thesis by auto
- qed
- qed
- ultimately show ?thesis
- by (auto simp:max_def split:if_splits)
- qed
- next
- case (P thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (Set thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- let ?B = "threads (t@s)"
- from Cons have "extend_highest_set s' th prio (e # t)" by auto
- from extend_highest_set.set_diff_low[OF this] and Set
- have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
- from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
- also have "\<dots> = ?t"
- proof(rule Max_eqI)
- fix y
- assume y_in: "y \<in> ?f ` ?B"
- then obtain th1 where
- th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
- show "y \<le> ?t"
- proof(cases "th1 = thread")
- case True
- with neq_thread le_p eq_y s_def Set
- show ?thesis
- by (auto simp:preced_def precedence_le_def)
- next
- case False
- with Set eq_y
- have "y = preced th1 (t@s)"
- by (simp add:preced_def)
- moreover have "\<dots> \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
- by auto
- moreover have "preced th1 (t@s) \<le> \<dots>"
- proof(rule Max_ge)
- from th1_in
- show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
- by simp
- next
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- proof -
- from Cons have "vt step (t @ s)" by auto
- from finite_threads[OF this] show ?thesis by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- next
- from Cons and finite_threads
- show "finite (?f ` ?B)" by auto
- next
- from Cons have "extend_highest_set s' th prio t" by auto
- from extend_highest_set.th_kept [OF this, folded s_def]
- have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show "?t \<in> (?f ` ?B)"
- proof -
- from neq_thread Set h
- have "?t = ?f th" by (auto simp:preced_def)
- with h show ?thesis by auto
- qed
- qed
- finally show ?thesis .
- qed
- qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- (is "?L = ?R")
-proof -
- have "?L = cpreced (t@s) (wq (t@s)) th"
- by (unfold cp_eq_cpreced, simp)
- also have "\<dots> = ?R"
- proof(unfold cpreced_def)
- show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
- Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
- proof(cases "?A = {}")
- case False
- have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
- moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
- proof(rule Max_insert)
- show "finite (?f ` ?A)"
- proof -
- from dependents_threads[OF vt_t]
- have "?A \<subseteq> threads (t@s)" .
- moreover from finite_threads[OF vt_t] have "finite \<dots>" .
- ultimately show ?thesis
- by (auto simp:finite_subset)
- qed
- next
- from False show "(?f ` ?A) \<noteq> {}" by simp
- qed
- moreover have "\<dots> = Max (?f ` ?B)"
- proof -
- from max_preced have "?f th = Max (?f ` ?B)" .
- moreover have "Max (?f ` ?A) \<le> \<dots>"
- proof(rule Max_mono)
- from False show "(?f ` ?A) \<noteq> {}" by simp
- next
- show "?f ` ?A \<subseteq> ?f ` ?B"
- proof -
- have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
- thus ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t]
- show "finite (?f ` ?B)" by simp
- qed
- ultimately show ?thesis
- by (auto simp:max_def)
- qed
- ultimately show ?thesis by auto
- next
- case True
- with max_preced show ?thesis by auto
- qed
- qed
- finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
- by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
- by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less':
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- shows "preced th' s < preced th s"
-proof -
- have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
- proof(rule Max_ge)
- from finite_threads [OF vt_s]
- show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
- next
- from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
- by simp
- qed
- moreover have "preced th' s \<noteq> preced th s"
- proof
- assume "preced th' s = preced th s"
- from preced_unique[OF this th'_in] neq_th' is_ready
- show "False" by (auto simp:readys_def)
- qed
- ultimately show ?thesis using highest_preced_thread
- by auto
-qed
-
-lemma pv_blocked:
- fixes th'
- assumes th'_in: "th' \<in> threads (t@s)"
- and neq_th': "th' \<noteq> th"
- and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
- shows "th' \<notin> runing (t@s)"
-proof
- assume "th' \<in> runing (t@s)"
- hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
- by (auto simp:runing_def)
- with max_cp_readys_threads [OF vt_t]
- have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
- by auto
- moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
- ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
- moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
- by simp
- finally have h: "cp (t @ s) th' = preced th (t @ s)" .
- show False
- proof -
- have "dependents (wq (t @ s)) th' = {}"
- by (rule count_eq_dependents [OF vt_t eq_pv])
- moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
- proof
- assume "preced th' (t @ s) = preced th (t @ s)"
- hence "th' = th"
- proof(rule preced_unique)
- from th_kept show "th \<in> threads (t @ s)" by simp
- next
- from th'_in show "th' \<in> threads (t @ s)" by simp
- qed
- with assms show False by simp
- qed
- ultimately show ?thesis
- by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
- qed
-qed
-
-lemma runing_precond_pre:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads (t@s) \<and>
- cntP (t@s) th' = cntV (t@s) th'"
-proof -
- show ?thesis
- proof(induct rule:ind)
- case (Cons e t)
- from Cons
- have in_thread: "th' \<in> threads (t @ s)"
- and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- have "extend_highest_set s' th prio t" by fact
- from extend_highest_set.pv_blocked
- [OF this, folded s_def, OF in_thread neq_th' not_holding]
- have not_runing: "th' \<notin> runing (t @ s)" .
- show ?case
- proof(cases e)
- case (V thread cs)
- from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
-
- show ?thesis
- proof -
- from Cons and V have "step (t@s) (V thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (unfold V, simp add:cntP_def cntV_def count_def)
- moreover from in_thread
- have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (P thread cs)
- from Cons and P have "step (t@s) (P thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Create thread prio')
- from Cons and Create have "step (t@s) (Create thread prio')" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th' \<in> threads (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with Cons and Create
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Create
- have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
- ultimately show ?thesis by auto
- next
- case (Exit thread)
- from Cons and Exit have "step (t@s) (Exit thread)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t @ s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and Exit
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Exit and neq_th'
- have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Set thread prio')
- with Cons
- show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case Nil
- with assms
- show ?case by auto
- qed
-qed
-
-(*
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<notin> runing (t@s)"
-proof -
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2]
- show ?thesis .
-qed
-*)
-
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- and is_runing: "th' \<in> runing (t@s)"
- shows "cntP s th' > cntV s th'"
-proof -
- have "cntP s th' \<noteq> cntV s th'"
- proof
- assume eq_pv: "cntP s th' = cntV s th'"
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)"
- and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
- with is_runing show "False" by simp
- qed
- moreover from cnp_cnv_cncs[OF vt_s, of th']
- have "cntV s th' \<le> cntP s th'" by auto
- ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
- th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
- case (Suc k)
- show ?case
- proof -
- { assume True: "Suc (i+k) \<le> length t"
- from moment_head [OF this]
- obtain e where
- eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
- by blast
- from red_moment[of "Suc(i+k)"]
- and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
- hence vt_e: "vt step (e#(moment (i + k) t)@s)"
- by (unfold extend_highest_set_def extend_highest_set_axioms_def
- highest_set_def s_def, auto)
- have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
- proof(unfold s_def)
- show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
- proof(rule extend_highest_set.pv_blocked)
- from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
- by (simp add:s_def)
- next
- from neq_th' show "th' \<noteq> th" .
- next
- from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
- next
- from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
- cntV (moment (i + k) t @ Set th prio # s') th'"
- by (auto simp:s_def)
- qed
- qed
- from step_back_step[OF vt_e]
- have "step ((moment (i + k) t)@s) e" .
- hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
- th' \<in> threads (e#(moment (i + k) t)@s)
- "
- proof(cases)
- case (thread_create thread prio)
- with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_exit thread)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_P thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_V thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_set thread prio')
- with Suc show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- with eq_me have ?thesis using eq_me by auto
- } note h = this
- show ?thesis
- proof(cases "Suc (i+k) \<le> length t")
- case True
- from h [OF this] show ?thesis .
- next
- case False
- with moment_ge
- have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
- with Suc show ?thesis by auto
- qed
- qed
-next
- case 0
- from assms show ?case by auto
-qed
-
-lemma moment_blocked:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- and le_ij: "i \<le> j"
- shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
- have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
- and h2: "th' \<in> threads ((moment j t)@s)" by auto
- with extend_highest_set.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1]
- show ?thesis by auto
-qed
-
-lemma runing_inversion_1:
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
- case True
- with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
- case False
- let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
- let ?q = "moment 0 t"
- from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
- from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
- from p_split_gen [of ?Q, OF this not_thread]
- obtain i where lt_its: "i < length t"
- and le_i: "0 \<le> i"
- and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
- and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
- from lt_its have "Suc i \<le> length t" by auto
- from moment_head[OF this] obtain e where
- eq_me: "moment (Suc i) t = e # moment i t" by blast
- from red_moment[of "Suc i"] and eq_me
- have "extend_highest_set s' th prio (e # moment i t)" by simp
- hence vt_e: "vt step (e#(moment i t)@s)"
- by (unfold extend_highest_set_def extend_highest_set_axioms_def
- highest_set_def s_def, auto)
- from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
- from post[rule_format, of "Suc i"] and eq_me
- have not_in': "th' \<in> threads (e # moment i t@s)" by auto
- from create_pre[OF stp_i pre this]
- obtain prio where eq_e: "e = Create th' prio" .
- have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
- proof(rule cnp_cnv_eq)
- from step_back_vt [OF vt_e]
- show "vt step (moment i t @ s)" .
- next
- from eq_e and stp_i
- have "step (moment i t @ s) (Create th' prio)" by simp
- thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
- qed
- with eq_e
- have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
- by (simp add:cntP_def cntV_def count_def)
- with eq_me[symmetric]
- have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
- by simp
- from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
- with eq_me [symmetric]
- have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
- from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
- and moment_ge
- have "th' \<notin> runing (t @ s)" by auto
- with runing'
- show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
- assumes runing': "th' \<in> runing (t@s)"
- shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
- from runing_inversion_1[OF _ runing']
- show ?thesis by auto
-qed
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
- case True thus ?thesis by auto
-next
- case False
- then have not_ready: "th \<notin> readys (t@s)"
- apply (unfold runing_def,
- insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
- by auto
- from th_kept have "th \<in> threads (t@s)" by auto
- from th_chain_to_ready[OF vt_t this] and not_ready
- obtain th' where th'_in: "th' \<in> readys (t@s)"
- and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
- have "th' \<in> runing (t@s)"
- proof -
- have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
- preced th (t@s)"
- proof(rule Max_eqI)
- fix y
- assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- then obtain th1 where
- h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
- and eq_y: "y = preced th1 (t@s)" by auto
- show "y \<le> preced th (t @ s)"
- proof -
- from max_preced
- have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
- moreover have "y \<le> \<dots>"
- proof(rule Max_ge)
- from h1
- have "th1 \<in> threads (t@s)"
- proof
- assume "th1 = th'"
- with th'_in show ?thesis by (simp add:readys_def)
- next
- assume "th1 \<in> dependents (wq (t @ s)) th'"
- with dependents_threads [OF vt_t]
- show "th1 \<in> threads (t @ s)" by auto
- qed
- with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
- next
- from finite_threads[OF vt_t]
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
- qed
- ultimately show ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
- show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
- by (auto intro:finite_subset)
- next
- from dp
- have "th \<in> dependents (wq (t @ s)) th'"
- by (unfold cs_dependents_def, auto simp:eq_depend)
- thus "preced th (t @ s) \<in>
- (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- by auto
- qed
- moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- from max_preced and max_cp_eq[OF vt_t, symmetric]
- have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
- with max_cp_readys_threads[OF vt_t] show ?thesis by simp
- qed
- ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
- qed
- with th'_in show ?thesis by (auto simp:runing_def)
- qed
- thus ?thesis by auto
-qed
-
-end
-
-end
-
-
--- a/prio/Attic/Happen_within.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,126 +0,0 @@
-theory Happen_within
-imports Main Moment
-begin
-
-(*
- lemma
- fixes P :: "('a list) \<Rightarrow> bool"
- and Q :: "('a list) \<Rightarrow> bool"
- and k :: nat
- and f :: "('a list) \<Rightarrow> nat"
- assumes "\<And> s t. \<lbrakk>P s; \<not> Q s; P (t@s); k < length t\<rbrakk> \<Longrightarrow> f (t@s) < f s"
- shows "\<And> s t. \<lbrakk> P s; P(t @ s); f(s) * k < length t\<rbrakk> \<Longrightarrow> Q (t@s)"
- sorry
-*)
-
-text {*
- The following two notions are introduced to improve the situation.
- *}
-
-definition all_future :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> ('a list) \<Rightarrow> bool"
-where "all_future G R s = (\<forall> t. G (t@s) \<longrightarrow> R t)"
-
-definition happen_within :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> ('a list) \<Rightarrow> bool"
-where "happen_within G R k s = all_future G (\<lambda> t. k < length t \<longrightarrow>
- (\<exists> i \<le> k. R (moment i t @ s) \<and> G (moment i t @ s))) s"
-
-lemma happen_within_intro:
- fixes P :: "('a list) \<Rightarrow> bool"
- and Q :: "('a list) \<Rightarrow> bool"
- and k :: nat
- and f :: "('a list) \<Rightarrow> nat"
- assumes
- lt_k: "0 < k"
- and step: "\<And> s. \<lbrakk>P s; \<not> Q s\<rbrakk> \<Longrightarrow> happen_within P (\<lambda> s'. f s' < f s) k s"
- shows "\<And> s. P s \<Longrightarrow> happen_within P Q ((f s + 1) * k) s"
-proof -
- fix s
- assume "P s"
- thus "happen_within P Q ((f s + 1) * k) s"
- proof(induct n == "f s + 1" arbitrary:s rule:nat_less_induct)
- fix s
- assume ih [rule_format]: "\<forall>m<f s + 1. \<forall>x. m = f x + 1 \<longrightarrow> P x
- \<longrightarrow> happen_within P Q ((f x + 1) * k) x"
- and ps: "P s"
- show "happen_within P Q ((f s + 1) * k) s"
- proof(cases "Q s")
- case True
- show ?thesis
- proof -
- { fix t
- from True and ps have "0 \<le> ((f s + 1)*k) \<and> Q (moment 0 t @ s) \<and> P (moment 0 t @ s)" by auto
- hence "\<exists>i\<le>(f s + 1) * k. Q (moment i t @ s) \<and> P (moment i t @ s)" by auto
- } thus ?thesis by (auto simp: happen_within_def all_future_def)
- qed
- next
- case False
- from step [OF ps False] have kk: "happen_within P (\<lambda>s'. f s' < f s) k s" .
- show ?thesis
- proof -
- { fix t
- assume pts: "P (t @ s)" and ltk: "(f s + 1) * k < length t"
- from ltk have lt_k_lt: "k < length t" by auto
- with kk pts obtain i
- where le_ik: "i \<le> k"
- and lt_f: "f (moment i t @ s) < f s"
- and p_m: "P (moment i t @ s)"
- by (auto simp:happen_within_def all_future_def)
- from ih [of "f (moment i t @ s) + 1" "(moment i t @ s)", OF _ _ p_m] and lt_f
- have hw: "happen_within P Q ((f (moment i t @ s) + 1) * k) (moment i t @ s)" by auto
- have "(\<exists>j\<le>(f s + 1) * k. Q (moment j t @ s) \<and> P (moment j t @ s))" (is "\<exists> j. ?T j")
- proof -
- let ?t = "restm i t"
- have eq_t: "t = ?t @ moment i t" by (simp add:moment_restm_s)
- have h1: "P (restm i t @ moment i t @ s)"
- proof -
- from pts and eq_t have "P ((restm i t @ moment i t) @ s)" by simp
- thus ?thesis by simp
- qed
- moreover have h2: "(f (moment i t @ s) + 1) * k < length (restm i t)"
- proof -
- have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
- from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
- from h [OF this, of k]
- have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
- moreover from le_ik have "\<dots> \<le> ((f s) * k + k - i)" by simp
- moreover from le_ik lt_k_lt and ltk have "(f s) * k + k - i < length t - i" by simp
- moreover have "length (restm i t) = length t - i" using length_restm by metis
- ultimately show ?thesis by simp
- qed
- from hw [unfolded happen_within_def all_future_def, rule_format, OF h1 h2]
- obtain m where le_m: "m \<le> (f (moment i t @ s) + 1) * k"
- and q_m: "Q (moment m ?t @ moment i t @ s)"
- and p_m: "P (moment m ?t @ moment i t @ s)" by auto
- have eq_mm: "moment m ?t @ moment i t @ s = (moment (m+i) t)@s"
- proof -
- have "moment m (restm i t) @ moment i t = moment (m + i) t"
- using moment_plus_split by metis
- thus ?thesis by simp
- qed
- let ?j = "m + i"
- have "?T ?j"
- proof -
- have "m + i \<le> (f s + 1) * k"
- proof -
- have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
- from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
- from h [OF this, of k]
- have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
- with le_m have "m \<le> f s * k" by simp
- hence "m + i \<le> f s * k + i" by simp
- with le_ik show ?thesis by simp
- qed
- moreover from eq_mm q_m have " Q (moment (m + i) t @ s)" by metis
- moreover from eq_mm p_m have " P (moment (m + i) t @ s)" by metis
- ultimately show ?thesis by blast
- qed
- thus ?thesis by blast
- qed
- } thus ?thesis by (simp add:happen_within_def all_future_def firstn.simps)
- qed
- qed
- qed
-qed
-
-end
-
--- a/prio/Attic/Lsp.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,323 +0,0 @@
-theory Lsp
-imports Main
-begin
-
-fun lsp :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list)"
-where
- "lsp f [] = ([], [], [])" |
- "lsp f [x] = ([], [x], [])" |
- "lsp f (x#xs) = (case (lsp f xs) of
- (l, [], r) \<Rightarrow> ([], [x], []) |
- (l, y#ys, r) \<Rightarrow> if f x \<ge> f y then ([], [x], xs) else (x#l, y#ys, r))"
-
-inductive lsp_p :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list) \<Rightarrow> bool"
-for f :: "('a \<Rightarrow> ('b::linorder))"
-where
- lsp_nil [intro]: "lsp_p f [] ([], [], [])" |
- lsp_single [intro]: "lsp_p f [x] ([], [x], [])" |
- lsp_cons_1 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x \<ge> f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) ([], [x], xs)" |
- lsp_cons_2 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) (x#l, [m], r)"
-
-lemma lsp_p_lsp_1: "lsp_p f x y \<Longrightarrow> y = lsp f x"
-proof (induct rule:lsp_p.induct)
- case (lsp_cons_1 xs l m r x)
- assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
- and le_mx: "f m \<le> f x"
- show ?case (is "?L = ?R")
- proof(cases xs, simp)
- case (Cons v vs)
- show ?thesis
- apply (simp add:Cons)
- apply (fold Cons)
- by (simp add:lsp_xs le_mx)
- qed
-next
- case (lsp_cons_2 xs l m r x)
- assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
- and lt_xm: "f x < f m"
- show ?case (is "?L = ?R")
- proof(cases xs)
- case (Cons v vs)
- show ?thesis
- apply (simp add:Cons)
- apply (fold Cons)
- apply (simp add:lsp_xs)
- by (insert lt_xm, auto)
- next
- case Nil
- from prems show ?thesis by simp
- qed
-qed auto
-
-lemma lsp_mid_nil: "lsp f xs = (a, [], c) \<Longrightarrow> xs = []"
- apply (induct xs arbitrary:a c, auto)
- apply (case_tac xs, auto)
- by (case_tac "(lsp f (ab # list))", auto split:if_splits list.splits)
-
-
-lemma lsp_mid_length: "lsp f x = (u, v, w) \<Longrightarrow> length v \<le> 1"
-proof(induct x arbitrary:u v w, simp)
- case (Cons x xs)
- assume ih: "\<And> u v w. lsp f xs = (u, v, w) \<Longrightarrow> length v \<le> 1"
- and h: "lsp f (x # xs) = (u, v, w)"
- show "length v \<le> 1" using h
- proof(cases xs, simp add:h)
- case (Cons z zs)
- assume eq_xs: "xs = z # zs"
- show ?thesis
- proof(cases "lsp f xs")
- fix l m r
- assume eq_lsp: "lsp f xs = (l, m, r)"
- show ?thesis
- proof(cases m)
- case Nil
- from Nil and eq_lsp have "lsp f xs = (l, [], r)" by simp
- from lsp_mid_nil [OF this] have "xs = []" .
- with h show ?thesis by auto
- next
- case (Cons y ys)
- assume eq_m: "m = y # ys"
- from ih [OF eq_lsp] have eq_xs_1: "length m \<le> 1" .
- show ?thesis
- proof(cases "f x \<ge> f y")
- case True
- from eq_xs eq_xs_1 True h eq_lsp show ?thesis
- by (auto split:list.splits if_splits)
- next
- case False
- from eq_xs eq_xs_1 False h eq_lsp show ?thesis
- by (auto split:list.splits if_splits)
- qed
- qed
- qed
- next
- assume "[] = u \<and> [x] = v \<and> [] = w"
- hence "v = [x]" by simp
- thus "length v \<le> Suc 0" by simp
- qed
-qed
-
-lemma lsp_p_lsp_2: "lsp_p f x (lsp f x)"
-proof(induct x, auto)
- case (Cons x xs)
- assume ih: "lsp_p f xs (lsp f xs)"
- show ?case
- proof(cases xs)
- case Nil
- thus ?thesis by auto
- next
- case (Cons v vs)
- show ?thesis
- proof(cases "xs")
- case Nil
- thus ?thesis by auto
- next
- case (Cons v vs)
- assume eq_xs: "xs = v # vs"
- show ?thesis
- proof(cases "lsp f xs")
- fix l m r
- assume eq_lsp_xs: "lsp f xs = (l, m, r)"
- show ?thesis
- proof(cases m)
- case Nil
- from eq_lsp_xs and Nil have "lsp f xs = (l, [], r)" by simp
- from lsp_mid_nil [OF this] have eq_xs: "xs = []" .
- hence "lsp f (x#xs) = ([], [x], [])" by simp
- with eq_xs show ?thesis by auto
- next
- case (Cons y ys)
- assume eq_m: "m = y # ys"
- show ?thesis
- proof(cases "f x \<ge> f y")
- case True
- from eq_xs eq_lsp_xs Cons True
- have eq_lsp: "lsp f (x#xs) = ([], [x], v # vs)" by simp
- show ?thesis
- proof (simp add:eq_lsp)
- show "lsp_p f (x # xs) ([], [x], v # vs)"
- proof(fold eq_xs, rule lsp_cons_1 [OF _])
- from eq_xs show "xs \<noteq> []" by simp
- next
- from lsp_mid_length [OF eq_lsp_xs] and Cons
- have "m = [y]" by simp
- with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
- with ih show "lsp_p f xs (l, [y], r)" by simp
- next
- from True show "f y \<le> f x" by simp
- qed
- qed
- next
- case False
- from eq_xs eq_lsp_xs Cons False
- have eq_lsp: "lsp f (x#xs) = (x # l, y # ys, r) " by simp
- show ?thesis
- proof (simp add:eq_lsp)
- from lsp_mid_length [OF eq_lsp_xs] and eq_m
- have "ys = []" by simp
- moreover have "lsp_p f (x # xs) (x # l, [y], r)"
- proof(rule lsp_cons_2)
- from eq_xs show "xs \<noteq> []" by simp
- next
- from lsp_mid_length [OF eq_lsp_xs] and Cons
- have "m = [y]" by simp
- with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
- with ih show "lsp_p f xs (l, [y], r)" by simp
- next
- from False show "f x < f y" by simp
- qed
- ultimately show "lsp_p f (x # xs) (x # l, y # ys, r)" by simp
- qed
- qed
- qed
- qed
- qed
- qed
-qed
-
-lemma lsp_induct:
- fixes f x1 x2 P
- assumes h: "lsp f x1 = x2"
- and p1: "P [] ([], [], [])"
- and p2: "\<And>x. P [x] ([], [x], [])"
- and p3: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f m \<le> f x\<rbrakk> \<Longrightarrow> P (x # xs) ([], [x], xs)"
- and p4: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> P (x # xs) (x # l, [m], r)"
- shows "P x1 x2"
-proof(rule lsp_p.induct)
- from lsp_p_lsp_2 and h
- show "lsp_p f x1 x2" by metis
-next
- from p1 show "P [] ([], [], [])" by metis
-next
- from p2 show "\<And>x. P [x] ([], [x], [])" by metis
-next
- fix xs l m r x
- assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f m \<le> f x"
- show "P (x # xs) ([], [x], xs)"
- proof(rule p3 [OF h1 _ h3 h4])
- from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
- qed
-next
- fix xs l m r x
- assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f x < f m"
- show "P (x # xs) (x # l, [m], r)"
- proof(rule p4 [OF h1 _ h3 h4])
- from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
- qed
-qed
-
-lemma lsp_set_eq:
- fixes f x u v w
- assumes h: "lsp f x = (u, v, w)"
- shows "x = u@v@w"
-proof -
- have "\<And> f x r. lsp f x = r \<Longrightarrow> \<forall> u v w. (r = (u, v, w) \<longrightarrow> x = u@v@w)"
- by (erule lsp_induct, simp+)
- from this [rule_format, OF h] show ?thesis by simp
-qed
-
-lemma lsp_set:
- assumes h: "(u, v, w) = lsp f x"
- shows "set (u@v@w) = set x"
-proof -
- from lsp_set_eq [OF h[symmetric]]
- show ?thesis by simp
-qed
-
-lemma max_insert_gt:
- fixes S fx
- assumes h: "fx < Max S"
- and np: "S \<noteq> {}"
- and fn: "finite S"
- shows "Max S = Max (insert fx S)"
-proof -
- from Max_insert [OF fn np]
- have "Max (insert fx S) = max fx (Max S)" .
- moreover have "\<dots> = Max S"
- proof(cases "fx \<le> Max S")
- case False
- with h
- show ?thesis by (simp add:max_def)
- next
- case True
- thus ?thesis by (simp add:max_def)
- qed
- ultimately show ?thesis by simp
-qed
-
-lemma max_insert_le:
- fixes S fx
- assumes h: "Max S \<le> fx"
- and fn: "finite S"
- shows "fx = Max (insert fx S)"
-proof(cases "S = {}")
- case True
- thus ?thesis by simp
-next
- case False
- from Max_insert [OF fn False]
- have "Max (insert fx S) = max fx (Max S)" .
- moreover have "\<dots> = fx"
- proof(cases "fx \<le> Max S")
- case False
- thus ?thesis by (simp add:max_def)
- next
- case True
- have hh: "\<And> x y. \<lbrakk> x \<le> (y::('a::linorder)); y \<le> x\<rbrakk> \<Longrightarrow> x = y" by auto
- from hh [OF True h]
- have "fx = Max S" .
- thus ?thesis by simp
- qed
- ultimately show ?thesis by simp
-qed
-
-lemma lsp_max:
- fixes f x u m w
- assumes h: "lsp f x = (u, [m], w)"
- shows "f m = Max (f ` (set x))"
-proof -
- { fix y
- have "lsp f x = y \<Longrightarrow> \<forall> u m w. y = (u, [m], w) \<longrightarrow> f m = Max (f ` (set x))"
- proof(erule lsp_induct, simp)
- { fix x u m w
- assume "(([]::'a list), ([x]::'a list), ([]::'a list)) = (u, [m], w)"
- hence "f m = Max (f ` set [x])" by simp
- } thus "\<And>x. \<forall>u m w. ([], [x], []) = (u, [m], w) \<longrightarrow> f m = Max (f ` set [x])" by simp
- next
- fix xs l m r x
- assume h1: "xs \<noteq> []"
- and h2: " lsp f xs = (l, [m], r)"
- and h3: "\<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
- and h4: "f m \<le> f x"
- show " \<forall>u m w. ([], [x], xs) = (u, [m], w) \<longrightarrow> f m = Max (f ` set (x # xs))"
- proof -
- have "f x = Max (f ` set (x # xs))"
- proof -
- from h2 h3 have "f m = Max (f ` set xs)" by simp
- with h4 show ?thesis
- apply auto
- by (rule_tac max_insert_le, auto)
- qed
- thus ?thesis by simp
- qed
- next
- fix xs l m r x
- assume h1: "xs \<noteq> []"
- and h2: " lsp f xs = (l, [m], r)"
- and h3: " \<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
- and h4: "f x < f m"
- show "\<forall>u ma w. (x # l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set (x # xs))"
- proof -
- from h2 h3 have "f m = Max (f ` set xs)" by simp
- with h4
- have "f m = Max (f ` set (x # xs))"
- apply auto
- apply (rule_tac max_insert_gt, simp+)
- by (insert h1, simp+)
- thus ?thesis by auto
- qed
- qed
- } with h show ?thesis by metis
-qed
-
-end
--- a/prio/Attic/Prio.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2813 +0,0 @@
-theory Prio
-imports Precedence_ord Moment Lsp Happen_within
-begin
-
-type_synonym thread = nat
-type_synonym priority = nat
-type_synonym cs = nat
-
-datatype event =
- Create thread priority |
- Exit thread |
- P thread cs |
- V thread cs |
- Set thread priority
-
-datatype node =
- Th "thread" |
- Cs "cs"
-
-type_synonym state = "event list"
-
-fun threads :: "state \<Rightarrow> thread set"
-where
- "threads [] = {}" |
- "threads (Create thread prio#s) = {thread} \<union> threads s" |
- "threads (Exit thread # s) = (threads s) - {thread}" |
- "threads (e#s) = threads s"
-
-fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> nat"
-where
- "original_priority thread [] = 0" |
- "original_priority thread (Create thread' prio#s) =
- (if thread' = thread then prio else original_priority thread s)" |
- "original_priority thread (Set thread' prio#s) =
- (if thread' = thread then prio else original_priority thread s)" |
- "original_priority thread (e#s) = original_priority thread s"
-
-fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
-where
- "birthtime thread [] = 0" |
- "birthtime thread ((Create thread' prio)#s) = (if (thread = thread') then length s
- else birthtime thread s)" |
- "birthtime thread ((Set thread' prio)#s) = (if (thread = thread') then length s
- else birthtime thread s)" |
- "birthtime thread (e#s) = birthtime thread s"
-
-definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
- where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
-
-consts holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
- waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
- depend :: "'b \<Rightarrow> (node \<times> node) set"
- dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
-
-defs (overloaded) cs_holding_def: "holding wq thread cs == (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
- cs_waiting_def: "waiting wq thread cs == (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
- cs_depend_def: "depend (wq::cs \<Rightarrow> thread list) == {(Th t, Cs c) | t c. waiting wq t c} \<union>
- {(Cs c, Th t) | c t. holding wq t c}"
- cs_dependents_def: "dependents (wq::cs \<Rightarrow> thread list) th == {th' . (Th th', Th th) \<in> (depend wq)^+}"
-
-record schedule_state =
- waiting_queue :: "cs \<Rightarrow> thread list"
- cur_preced :: "thread \<Rightarrow> precedence"
-
-
-definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
-where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
-
-fun schs :: "state \<Rightarrow> schedule_state"
-where
- "schs [] = \<lparr>waiting_queue = \<lambda> cs. [],
- cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
- "schs (e#s) = (let ps = schs s in
- let pwq = waiting_queue ps in
- let pcp = cur_preced ps in
- let nwq = case e of
- P thread cs \<Rightarrow> pwq(cs:=(pwq cs @ [thread])) |
- V thread cs \<Rightarrow> let nq = case (pwq cs) of
- [] \<Rightarrow> [] |
- (th#pq) \<Rightarrow> case (lsp pcp pq) of
- (l, [], r) \<Rightarrow> []
- | (l, m#ms, r) \<Rightarrow> m#(l@ms@r)
- in pwq(cs:=nq) |
- _ \<Rightarrow> pwq
- in let ncp = cpreced (e#s) nwq in
- \<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
- )"
-
-definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
-where "wq s == waiting_queue (schs s)"
-
-definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
-where "cp s = cur_preced (schs s)"
-
-defs (overloaded) s_holding_def: "holding (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
- s_waiting_def: "waiting (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
- s_depend_def: "depend (s::state) == {(Th t, Cs c) | t c. waiting (wq s) t c} \<union>
- {(Cs c, Th t) | c t. holding (wq s) t c}"
- s_dependents_def: "dependents (s::state) th == {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
-
-definition readys :: "state \<Rightarrow> thread set"
-where
- "readys s =
- {thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
-
-definition runing :: "state \<Rightarrow> thread set"
-where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
-
-definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
- where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
-
-inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
-where
- thread_create: "\<lbrakk>prio \<le> max_prio; thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
- thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
- thread_P: "\<lbrakk>thread \<in> runing s; (Cs cs, Th thread) \<notin> (depend s)^+\<rbrakk> \<Longrightarrow> step s (P thread cs)" |
- thread_V: "\<lbrakk>thread \<in> runing s; holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
- thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
-
-inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
- for cs
-where
- vt_nil[intro]: "vt cs []" |
- vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
-
-lemma runing_ready: "runing s \<subseteq> readys s"
- by (auto simp only:runing_def readys_def)
-
-lemma wq_v_eq_nil:
- fixes s cs thread rest
- assumes eq_wq: "wq s cs = thread # rest"
- and eq_lsp: "lsp (cp s) rest = (l, [], r)"
- shows "wq (V thread cs#s) cs = []"
-proof -
- from prems show ?thesis
- by (auto simp:wq_def Let_def cp_def split:list.splits)
-qed
-
-lemma wq_v_eq:
- fixes s cs thread rest
- assumes eq_wq: "wq s cs = thread # rest"
- and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
- shows "wq (V thread cs#s) cs = th'#l@r"
-proof -
- from prems show ?thesis
- by (auto simp:wq_def Let_def cp_def split:list.splits)
-qed
-
-lemma wq_v_neq:
- "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
- by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-lemma wq_distinct: "vt step s \<Longrightarrow> distinct (wq s cs)"
-proof(erule_tac vt.induct, simp add:wq_def)
- fix s e
- assume h1: "step s e"
- and h2: "distinct (wq s cs)"
- thus "distinct (wq (e # s) cs)"
- proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
- fix thread s
- assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
- and h2: "thread \<in> set (waiting_queue (schs s) cs)"
- and h3: "thread \<in> runing s"
- show "False"
- proof -
- from h3 have "\<And> cs. thread \<in> set (waiting_queue (schs s) cs) \<Longrightarrow>
- thread = hd ((waiting_queue (schs s) cs))"
- by (simp add:runing_def readys_def s_waiting_def wq_def)
- from this [OF h2] have "thread = hd (waiting_queue (schs s) cs)" .
- with h2
- have "(Cs cs, Th thread) \<in> (depend s)"
- by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
- with h1 show False by auto
- qed
- next
- fix thread s a list
- assume h1: "thread \<in> runing s"
- and h2: "holding s thread cs"
- and h3: "waiting_queue (schs s) cs = a # list"
- and h4: "a \<notin> set list"
- and h5: "distinct list"
- thus "distinct
- ((\<lambda>(l, a, r). case a of [] \<Rightarrow> [] | m # ms \<Rightarrow> m # l @ ms @ r)
- (lsp (cur_preced (schs s)) list))"
- apply (cases "(lsp (cur_preced (schs s)) list)", simp)
- apply (case_tac b, simp)
- by (drule_tac lsp_set_eq, simp)
- qed
-qed
-
-lemma block_pre:
- fixes thread cs s
- assumes s_ni: "thread \<notin> set (wq s cs)"
- and s_i: "thread \<in> set (wq (e#s) cs)"
- shows "e = P thread cs"
-proof -
- have ee: "\<And> x y. \<lbrakk>x = y\<rbrakk> \<Longrightarrow> set x = set y"
- by auto
- from s_ni s_i show ?thesis
- proof (cases e, auto split:if_splits simp add:Let_def wq_def)
- fix uu uub uuc uud uue
- assume h: "(uuc, thread # uu, uub) = lsp (cur_preced (schs s)) uud"
- and h1 [symmetric]: "uue # uud = waiting_queue (schs s) cs"
- and h2: "thread \<notin> set (waiting_queue (schs s) cs)"
- from lsp_set [OF h] have "set (uuc @ (thread # uu) @ uub) = set uud" .
- hence "thread \<in> set uud" by auto
- with h1 have "thread \<in> set (waiting_queue (schs s) cs)" by auto
- with h2 show False by auto
- next
- fix uu uua uub uuc uud uue
- assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
- and h2: "uue # uud = waiting_queue (schs s) cs"
- and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
- and h4: "thread \<in> set uuc"
- from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
- with h4 have "thread \<in> set uud" by auto
- with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
- apply (drule_tac ee) by auto
- with h1 show "False" by fast
- next
- fix uu uua uub uuc uud uue
- assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
- and h2: "uue # uud = waiting_queue (schs s) cs"
- and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
- and h4: "thread \<in> set uu"
- from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
- with h4 have "thread \<in> set uud" by auto
- with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
- apply (drule_tac ee) by auto
- with h1 show "False" by fast
- next
- fix uu uua uub uuc uud uue
- assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
- and h2: "uue # uud = waiting_queue (schs s) cs"
- and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
- and h4: "thread \<in> set uub"
- from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
- with h4 have "thread \<in> set uud" by auto
- with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
- apply (drule_tac ee) by auto
- with h1 show "False" by fast
- qed
-qed
-
-lemma p_pre: "\<lbrakk>vt step ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
- thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (depend s)^+"
-apply (ind_cases "vt step ((P thread cs)#s)")
-apply (ind_cases "step s (P thread cs)")
-by auto
-
-lemma abs1:
- fixes e es
- assumes ein: "e \<in> set es"
- and neq: "hd es \<noteq> hd (es @ [x])"
- shows "False"
-proof -
- from ein have "es \<noteq> []" 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) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
- by (cases es, auto)
-
-inductive_cases evt_cons: "vt cs (a#s)"
-
-lemma abs2:
- assumes vt: "vt step (e#s)"
- and inq: "thread \<in> set (wq s cs)"
- and nh: "thread = hd (wq s cs)"
- and qt: "thread \<noteq> hd (wq (e#s) cs)"
- and inq': "thread \<in> set (wq (e#s) cs)"
- shows "False"
-proof -
- have ee: "\<And> uuc thread uu uub s list. (uuc, thread # uu, uub) = lsp (cur_preced (schs s)) list \<Longrightarrow>
- lsp (cur_preced (schs s)) list = (uuc, thread # uu, uub)
- " by simp
- from prems show "False"
- apply (cases e)
- apply ((simp split:if_splits add:Let_def wq_def)[1])+
- apply (insert abs1, fast)[1]
- apply ((simp split:if_splits add:Let_def)[1])+
- apply (simp split:if_splits list.splits add:Let_def wq_def)
- apply (auto dest!:ee)
- apply (drule_tac lsp_set_eq, simp)
- apply (subgoal_tac "distinct (waiting_queue (schs s) cs)", simp, fold wq_def)
- apply (rule_tac wq_distinct, auto)
- apply (erule_tac evt_cons, auto)
- apply (drule_tac lsp_set_eq, simp)
- apply (subgoal_tac "distinct (wq s cs)", simp)
- apply (rule_tac wq_distinct, auto)
- apply (erule_tac evt_cons, auto)
- apply (drule_tac lsp_set_eq, simp)
- apply (subgoal_tac "distinct (wq s cs)", simp)
- apply (rule_tac wq_distinct, auto)
- apply (erule_tac evt_cons, auto)
- apply (auto simp:wq_def Let_def split:if_splits prod.splits)
- done
-qed
-
-lemma vt_moment: "\<And> t. \<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
-proof(induct s, simp)
- fix a s t
- assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
- and vt_a: "vt cs (a # s)"
- and le_t: "t \<le> length (a # s)"
- show "vt cs (moment t (a # s))"
- proof(cases "t = length (a#s)")
- case True
- from True have "moment t (a#s) = a#s" by simp
- with vt_a show ?thesis by simp
- next
- case False
- with le_t have le_t1: "t \<le> length s" by simp
- from vt_a have "vt cs s"
- by (erule_tac evt_cons, simp)
- from h [OF this le_t1] have "vt cs (moment t s)" .
- moreover have "moment t (a#s) = moment t s"
- proof -
- from moment_app [OF le_t1, of "[a]"]
- show ?thesis by simp
- qed
- ultimately show ?thesis by auto
- qed
-qed
-
-(* Wrong:
- lemma \<lbrakk>thread \<in> set (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
-*)
-
-lemma waiting_unique_pre:
- fixes cs1 cs2 s thread
- assumes vt: "vt step s"
- and h11: "thread \<in> set (wq s cs1)"
- and h12: "thread \<noteq> hd (wq s cs1)"
- assumes h21: "thread \<in> set (wq s cs2)"
- and h22: "thread \<noteq> hd (wq s cs2)"
- and neq12: "cs1 \<noteq> cs2"
- shows "False"
-proof -
- let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
- from h11 and h12 have q1: "?Q cs1 s" by simp
- from h21 and h22 have q2: "?Q cs2 s" by simp
- have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
- have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
- from p_split [of "?Q cs1", OF q1 nq1]
- obtain t1 where lt1: "t1 < length s"
- and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
- thread \<noteq> hd (wq (moment t1 s) cs1))"
- and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
- thread \<noteq> 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: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
- thread \<noteq> hd (wq (moment t2 s) cs2))"
- and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
- thread \<noteq> hd (wq (moment i' s) cs2))" by auto
- show ?thesis
- proof -
- {
- assume lt12: "t1 < t2"
- let ?t3 = "Suc t2"
- from lt2 have le_t3: "?t3 \<le> 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 \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- have vt_e: "vt step (e#moment t2 s)"
- proof -
- from vt_moment [OF vt le_t3]
- have "vt step (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- have ?thesis
- proof(cases "thread \<in> 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 abs2 [OF vt_e True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from block_pre [OF False h1]
- have "e = P thread cs2" .
- with vt_e have "vt step ((P thread cs2)# moment t2 s)" by simp
- from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
- with runing_ready have "thread \<in> readys (moment t2 s)" by auto
- with nn1 [rule_format, OF lt12]
- show ?thesis by (simp add:readys_def s_waiting_def, auto)
- qed
- } moreover {
- assume lt12: "t2 < t1"
- let ?t3 = "Suc t1"
- from lt1 have le_t3: "?t3 \<le> 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 \<in> set (wq (e#moment t1 s) cs1)" and
- h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have vt_e: "vt step (e#moment t1 s)"
- proof -
- from vt_moment [OF vt le_t3]
- have "vt step (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- have ?thesis
- proof(cases "thread \<in> 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 abs2 [OF vt_e True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from block_pre [OF False h1]
- have "e = P thread cs1" .
- with vt_e have "vt step ((P thread cs1)# moment t1 s)" by simp
- from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
- with runing_ready have "thread \<in> readys (moment t1 s)" by auto
- with nn2 [rule_format, OF lt12]
- show ?thesis by (simp add:readys_def s_waiting_def, auto)
- qed
- } moreover {
- assume eqt12: "t1 = t2"
- let ?t3 = "Suc t1"
- from lt1 have le_t3: "?t3 \<le> 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 \<in> set (wq (e#moment t1 s) cs1)" and
- h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have vt_e: "vt step (e#moment t1 s)"
- proof -
- from vt_moment [OF vt le_t3]
- have "vt step (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- have ?thesis
- proof(cases "thread \<in> 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 abs2 [OF vt_e True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from 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 \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- show ?thesis
- proof(cases "thread \<in> 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 step (e#moment t2 s)" by simp
- from abs2 [OF this True eq_th h2 h1]
- show ?thesis .
- next
- case False
- from block_pre [OF False h1]
- have "e = P thread cs2" .
- with eq_e1 neq12 show ?thesis by auto
- qed
- qed
- } ultimately show ?thesis by arith
- qed
-qed
-
-lemma waiting_unique:
- assumes "vt step s"
- and "waiting s th cs1"
- and "waiting s th cs2"
- shows "cs1 = cs2"
-proof -
- from waiting_unique_pre and prems
- show ?thesis
- by (auto simp add:s_waiting_def)
-qed
-
-lemma holded_unique:
- assumes "vt step s"
- and "holding s th1 cs"
- and "holding s th2 cs"
- shows "th1 = th2"
-proof -
- from prems show ?thesis
- unfolding s_holding_def
- by auto
-qed
-
-lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits)
-
-lemma birthtime_unique:
- "\<lbrakk>birthtime th1 s = birthtime th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
- \<Longrightarrow> th1 = th2"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits dest:birthtime_lt)
-
-lemma preced_unique :
- assumes pcd_eq: "preced th1 s = preced th2 s"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "th1 = th2"
-proof -
- from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
- from birthtime_unique [OF this th_in1 th_in2]
- show ?thesis .
-qed
-
-lemma preced_linorder:
- assumes neq_12: "th1 \<noteq> th2"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
- from preced_unique [OF _ th_in1 th_in2] and neq_12
- have "preced th1 s \<noteq> preced th2 s" by auto
- thus ?thesis by auto
-qed
-
-lemma unique_minus:
- fixes x y z r
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq: "y \<noteq> z"
- shows "(y, z) \<in> r^+"
-proof -
- from xz and neq show ?thesis
- proof(induct)
- case (base ya)
- have "(x, ya) \<in> 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:
- fixes r x y z
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> 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) \<in> r\<^sup>+" by auto
- with step show ?thesis by auto
- qed
- qed
-qed
-
-lemma unique_chain:
- fixes r x y z
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r^+"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
-proof -
- from xy xz neq_yz show ?thesis
- proof(induct)
- case (base y)
- have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> 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) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
- thus ?thesis
- proof
- assume "(z, y) \<in> r\<^sup>+"
- with step have "(z, za) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- next
- assume h: "(y, z) \<in> r\<^sup>+"
- from step have yza: "(y, za) \<in> r" by simp
- from step have "za \<noteq> z" by simp
- from unique_minus [OF _ yza h this] and unique
- have "(za, z) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- qed
- qed
- qed
-qed
-
-lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-definition head_of :: "('a \<Rightarrow> 'b::linorder) \<Rightarrow> 'a set \<Rightarrow> 'a set"
- where "head_of f A = {a . a \<in> A \<and> f a = Max (f ` A)}"
-
-definition wq_head :: "state \<Rightarrow> cs \<Rightarrow> thread set"
- where "wq_head s cs = head_of (cp s) (set (wq s cs))"
-
-lemma f_nil_simp: "\<lbrakk>f cs = []\<rbrakk> \<Longrightarrow> f(cs:=[]) = f"
-proof
- fix x
- assume h:"f cs = []"
- show "(f(cs := [])) x = f x"
- proof(cases "cs = x")
- case True
- with h show ?thesis by simp
- next
- case False
- with h show ?thesis by simp
- qed
-qed
-
-lemma step_back_vt: "vt ccs (e#s) \<Longrightarrow> vt ccs s"
- by(ind_cases "vt ccs (e#s)", simp)
-
-lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e"
- by(ind_cases "vt ccs (e#s)", simp)
-
-lemma holding_nil:
- "\<lbrakk>wq s cs = []; holding (wq s) th cs\<rbrakk> \<Longrightarrow> False"
- by (unfold cs_holding_def, auto)
-
-lemma waiting_kept_1: "
- \<lbrakk>vt step (V th cs#s); wq s cs = a # list; waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c;
- lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
- \<Longrightarrow> waiting (wq s) t c"
- apply (drule_tac step_back_vt, drule_tac wq_distinct[of _ cs])
- apply(drule_tac lsp_set_eq)
- by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_kept_2:
- "\<And>a list t c aa ca.
- \<lbrakk>wq s cs = a # list; waiting ((wq s)(cs := [])) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
- \<Longrightarrow> waiting (wq s) t c"
- apply(drule_tac lsp_set_eq)
- by (unfold cs_waiting_def, auto split:if_splits)
-
-
-lemma holding_nil_simp: "\<lbrakk>holding ((wq s)(cs := [])) t c\<rbrakk> \<Longrightarrow> holding (wq s) t c"
- by(unfold cs_holding_def, auto)
-
-lemma step_wq_elim: "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; a \<noteq> th\<rbrakk> \<Longrightarrow> False"
- apply(drule_tac step_back_step)
- apply(ind_cases "step s (V th cs)")
- by(unfold s_holding_def, auto)
-
-lemma holding_cs_neq_simp: "c \<noteq> cs \<Longrightarrow> holding ((wq s)(cs := u)) t c = holding (wq s) t c"
- by (unfold cs_holding_def, auto)
-
-lemma holding_th_neq_elim:
- "\<And>a list c t aa ca ab lista.
- \<lbrakk>\<not> holding (wq s) t c; holding ((wq s)(cs := ab # aa @ lista @ ca)) t c;
- ab \<noteq> t\<rbrakk>
- \<Longrightarrow> False"
- by (unfold cs_holding_def, auto split:if_splits)
-
-lemma holding_nil_abs:
- "\<not> holding ((wq s)(cs := [])) th cs"
- by (unfold cs_holding_def, auto split:if_splits)
-
-lemma holding_abs: "\<lbrakk>holding ((wq s)(cs := ab # aa @ lista @ c)) th cs; ab \<noteq> th\<rbrakk>
- \<Longrightarrow> False"
- by (unfold cs_holding_def, auto split:if_splits)
-
-lemma waiting_abs: "\<not> waiting ((wq s)(cs := t # l @ r)) t cs"
- by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_abs_1:
- "\<lbrakk>\<not> waiting ((wq s)(cs := [])) t c; waiting (wq s) t c; c \<noteq> cs\<rbrakk>
- \<Longrightarrow> False"
- by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_abs_2: "
- \<lbrakk>\<not> waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c; waiting (wq s) t c;
- c \<noteq> cs\<rbrakk>
- \<Longrightarrow> False"
- by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_abs_3:
- "\<lbrakk>wq s cs = a # list; \<not> waiting ((wq s)(cs := [])) t c;
- waiting (wq s) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
- \<Longrightarrow> False"
- apply(drule_tac lsp_mid_nil, simp)
- by(unfold cs_waiting_def, auto split:if_splits)
-
-lemma waiting_simp: "c \<noteq> cs \<Longrightarrow> waiting ((wq s)(cs:=z)) t c = waiting (wq s) t c"
- by(unfold cs_waiting_def, auto split:if_splits)
-
-lemma holding_cs_eq:
- "\<lbrakk>\<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> c = cs"
- by(unfold cs_holding_def, auto split:if_splits)
-
-lemma holding_cs_eq_1:
- "\<lbrakk>\<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c\<rbrakk>
- \<Longrightarrow> c = cs"
- by(unfold cs_holding_def, auto split:if_splits)
-
-lemma holding_th_eq:
- "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; \<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c;
- lsp (cp s) list = (aa, [], ca)\<rbrakk>
- \<Longrightarrow> t = th"
- apply(drule_tac lsp_mid_nil, simp)
- apply(unfold cs_holding_def, auto split:if_splits)
- apply(drule_tac step_back_step)
- apply(ind_cases "step s (V th cs)")
- by (unfold s_holding_def, auto split:if_splits)
-
-lemma holding_th_eq_1:
- "\<lbrakk>vt step (V th cs#s);
- wq s cs = a # list; \<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c;
- lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
- \<Longrightarrow> t = th"
- apply(drule_tac step_back_step)
- apply(ind_cases "step s (V th cs)")
- apply(unfold s_holding_def cs_holding_def)
- by (auto split:if_splits)
-
-lemma holding_th_eq_2: "\<lbrakk>holding ((wq s)(cs := ac # x)) th cs\<rbrakk>
- \<Longrightarrow> ac = th"
- by (unfold cs_holding_def, auto)
-
-lemma holding_th_eq_3: "
- \<lbrakk>\<not> holding (wq s) t c;
- holding ((wq s)(cs := ac # x)) t c\<rbrakk>
- \<Longrightarrow> ac = t"
- by (unfold cs_holding_def, auto)
-
-lemma holding_wq_eq: "holding ((wq s)(cs := th' # l @ r)) th' cs"
- by (unfold cs_holding_def, auto)
-
-lemma waiting_th_eq: "
- \<lbrakk>waiting (wq s) t c; wq s cs = a # list;
- lsp (cp s) list = (aa, ac # lista, ba); \<not> waiting ((wq s)(cs := ac # aa @ lista @ ba)) t c\<rbrakk>
- \<Longrightarrow> ac = t"
- apply(drule_tac lsp_set_eq, simp)
- by (unfold cs_waiting_def, auto split:if_splits)
-
-lemma step_depend_v:
- "vt step (V th cs#s) \<Longrightarrow>
- depend (V th cs # s) =
- depend s - {(Cs cs, Th th)} -
- {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
- {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
- apply (unfold s_depend_def wq_def,
- auto split:list.splits simp:Let_def f_nil_simp holding_wq_eq, fold wq_def cp_def)
- apply (auto split:list.splits prod.splits
- simp:Let_def f_nil_simp holding_nil_simp holding_cs_neq_simp holding_nil_abs
- waiting_abs waiting_simp holding_wq_eq
- elim:holding_nil waiting_kept_1 waiting_kept_2 step_wq_elim holding_th_neq_elim
- holding_abs waiting_abs_1 waiting_abs_3 holding_cs_eq holding_cs_eq_1
- holding_th_eq holding_th_eq_1 holding_th_eq_2 holding_th_eq_3 waiting_th_eq
- dest:lsp_mid_length)
- done
-
-lemma step_depend_p:
- "vt step (P th cs#s) \<Longrightarrow>
- depend (P th cs # s) = (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
- else depend s \<union> {(Th th, Cs cs)})"
- apply(unfold s_depend_def wq_def)
- apply (auto split:list.splits prod.splits simp:Let_def cs_waiting_def cs_holding_def)
- apply(case_tac "c = cs", auto)
- apply(fold wq_def)
- apply(drule_tac step_back_step)
- by (ind_cases " step s (P (hd (wq s cs)) cs)",
- auto simp:s_depend_def wq_def cs_holding_def)
-
-lemma simple_A:
- fixes A
- assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
- shows "A = {} \<or> (\<exists> a. A = {a})"
-proof(cases "A = {}")
- case True thus ?thesis by simp
-next
- case False then obtain a where "a \<in> A" by auto
- with h have "A = {a}" by auto
- thus ?thesis by simp
-qed
-
-lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
- by (unfold s_depend_def, auto)
-
-lemma acyclic_depend:
- fixes s
- assumes vt: "vt step s"
- shows "acyclic (depend s)"
-proof -
- from vt show ?thesis
- proof(induct)
- case (vt_cons s e)
- assume ih: "acyclic (depend s)"
- and stp: "step s e"
- and vt: "vt step s"
- show ?case
- proof(cases e)
- case (Create th prio)
- with ih
- show ?thesis by (simp add:depend_create_unchanged)
- next
- case (Exit th)
- with ih show ?thesis by (simp add:depend_exit_unchanged)
- next
- case (V th cs)
- from V vt stp have vtt: "vt step (V th cs#s)" by auto
- from step_depend_v [OF this]
- have eq_de: "depend (e # s) =
- depend s - {(Cs cs, Th th)} -
- {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
- {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
- (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
- from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
- have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
- thus ?thesis
- proof(cases "wq s cs")
- case Nil
- hence "?D = {}" by simp
- with ac and eq_de show ?thesis by simp
- next
- case (Cons tth rest)
- from stp and V have "step s (V th cs)" by simp
- hence eq_wq: "wq s cs = th # rest"
- proof -
- show "step s (V th cs) \<Longrightarrow> wq s cs = th # rest"
- apply(ind_cases "step s (V th cs)")
- by(insert Cons, unfold s_holding_def, simp)
- qed
- show ?thesis
- proof(cases "lsp (cp s) rest")
- fix l b r
- assume eq_lsp: "lsp (cp s) rest = (l, b, r) "
- show ?thesis
- proof(cases "b")
- case Nil
- with eq_lsp and eq_wq have "?D = {}" by simp
- with ac and eq_de show ?thesis by simp
- next
- case (Cons th' m)
- with eq_lsp
- have eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
- apply simp
- by (drule_tac lsp_mid_length, simp)
- from eq_wq and eq_lsp
- have eq_D: "?D = {(Cs cs, Th th')}" by auto
- from eq_wq and eq_lsp
- have eq_C: "?C = {(Th th', Cs cs)}" by auto
- let ?E = "(?A - ?B - ?C)"
- have "(Th th', Cs cs) \<notin> ?E\<^sup>*"
- proof
- assume "(Th th', Cs cs) \<in> ?E\<^sup>*"
- hence " (Th th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- from tranclD [OF this]
- obtain x where th'_e: "(Th th', x) \<in> ?E" by blast
- hence th_d: "(Th th', x) \<in> ?A" by simp
- from depend_target_th [OF this]
- obtain cs' where eq_x: "x = Cs cs'" by auto
- with th_d have "(Th th', Cs cs') \<in> ?A" by simp
- hence wt_th': "waiting s th' cs'"
- unfolding s_depend_def s_waiting_def cs_waiting_def by simp
- hence "cs' = cs"
- proof(rule waiting_unique [OF vt])
- from eq_wq eq_lsp wq_distinct[OF vt, of cs]
- show "waiting s th' cs" by(unfold s_waiting_def, auto dest:lsp_set_eq)
- qed
- with th'_e eq_x have "(Th th', Cs cs) \<in> ?E" by simp
- with eq_C show "False" by simp
- qed
- with acyclic_insert[symmetric] and ac and eq_D
- and eq_de show ?thesis by simp
- qed
- qed
- qed
- next
- case (P th cs)
- from P vt stp have vtt: "vt step (P th cs#s)" by auto
- from step_depend_p [OF this] P
- have "depend (e # s) =
- (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
- depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
- by simp
- moreover have "acyclic ?R"
- proof(cases "wq s cs = []")
- case True
- hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
- have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
- proof
- assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
- hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- from tranclD2 [OF this]
- obtain x where "(x, Cs cs) \<in> depend s" by auto
- with True show False by (auto simp:s_depend_def cs_waiting_def)
- qed
- with acyclic_insert ih eq_r show ?thesis by auto
- next
- case False
- hence eq_r: "?R = depend s \<union> {(Th th, Cs cs)}" by simp
- have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
- proof
- assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
- hence "(Cs cs, Th th) \<in> (depend 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 " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> 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 depend_set_unchanged
- show ?thesis by (simp add:depend_set_unchanged)
- qed
- next
- case vt_nil
- show "acyclic (depend ([]::state))"
- by (auto simp: s_depend_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
- qed
-qed
-
-lemma finite_depend:
- fixes s
- assumes vt: "vt step s"
- shows "finite (depend s)"
-proof -
- from vt show ?thesis
- proof(induct)
- case (vt_cons s e)
- assume ih: "finite (depend s)"
- and stp: "step s e"
- and vt: "vt step s"
- show ?case
- proof(cases e)
- case (Create th prio)
- with ih
- show ?thesis by (simp add:depend_create_unchanged)
- next
- case (Exit th)
- with ih show ?thesis by (simp add:depend_exit_unchanged)
- next
- case (V th cs)
- from V vt stp have vtt: "vt step (V th cs#s)" by auto
- from step_depend_v [OF this]
- have eq_de: "depend (e # s) =
- depend s - {(Cs cs, Th th)} -
- {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
- {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
- (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
- moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
- moreover have "finite ?D"
- proof -
- have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
- thus ?thesis
- proof
- assume h: "?D = {}"
- show ?thesis by (unfold h, simp)
- next
- assume "\<exists> a. ?D = {a}"
- thus ?thesis by auto
- qed
- qed
- ultimately show ?thesis by simp
- next
- case (P th cs)
- from P vt stp have vtt: "vt step (P th cs#s)" by auto
- from step_depend_p [OF this] P
- have "depend (e # s) =
- (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
- depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
- by simp
- moreover have "finite ?R"
- proof(cases "wq s cs = []")
- case True
- hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
- with True and ih show ?thesis by auto
- next
- case False
- hence "?R = depend s \<union> {(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:depend_set_unchanged)
- qed
- next
- case vt_nil
- show "finite (depend ([]::state))"
- by (auto simp: s_depend_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
- qed
-qed
-
-text {* Several useful lemmas *}
-
-thm wf_trancl
-thm finite_acyclic_wf
-thm finite_acyclic_wf_converse
-thm wf_induct
-
-
-lemma wf_dep_converse:
- fixes s
- assumes vt: "vt step s"
- shows "wf ((depend s)^-1)"
-proof(rule finite_acyclic_wf_converse)
- from finite_depend [OF vt]
- show "finite (depend s)" .
-next
- from acyclic_depend[OF vt]
- show "acyclic (depend s)" .
-qed
-
-lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
-by (induct l, auto)
-
-lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-
-lemma wq_threads:
- fixes s cs
- assumes vt: "vt step s"
- and h: "th \<in> set (wq s cs)"
- shows "th \<in> threads s"
-proof -
- from vt and h show ?thesis
- proof(induct arbitrary: th cs)
- case (vt_cons s e)
- assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
- and stp: "step s e"
- and vt: "vt step s"
- and h: "th \<in> 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_depend_def s_holding_def cs_holding_def)
- by (fold wq_def, auto)
- 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 \<in> set (waiting_queue (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
- show "th \<in> threads (V th' cs' # s)"
- proof(cases "cs = cs'")
- case False
- hence "?l = waiting_queue (schs s) cs" by (simp add:Let_def)
- with th_in have " th \<in> set (wq s cs)"
- by (fold wq_def, simp)
- from ih [OF this] show ?thesis by simp
- next
- case True
- show ?thesis
- proof(cases "waiting_queue (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: "waiting_queue (schs s) cs' = a # rest"
- with h V show ?thesis
- proof(cases "(lsp (cur_preced (schs s)) rest)", unfold V)
- fix l m r
- assume eq_lsp: "lsp (cur_preced (schs s)) rest = (l, m, r)"
- and eq_wq: "waiting_queue (schs s) cs' = a # rest"
- and th_in_set: "th \<in> set (wq (V th' cs' # s) cs)"
- show ?thesis
- proof(cases "m")
- case Nil
- with eq_lsp have "rest = []" using lsp_mid_nil by auto
- with eq_wq have "waiting_queue (schs s) cs' = [a]" by simp
- with h[unfolded V wq_def] True
- show ?thesis
- by (simp add:Let_def)
- next
- case (Cons b rb)
- with lsp_mid_length[OF eq_lsp] have eq_m: "m = [b]" by auto
- with eq_lsp have "lsp (cur_preced (schs s)) rest = (l, [b], r)" by simp
- with h[unfolded V wq_def] True lsp_set_eq [OF this] eq_wq
- show ?thesis
- apply (auto simp:Let_def, fold wq_def)
- by (rule_tac ih [of _ cs'], auto)+
- qed
- 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
- next
- case vt_nil
- thus ?case by (auto simp:wq_def)
- qed
-qed
-
-lemma range_in: "\<lbrakk>vt step s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
- apply(unfold s_depend_def cs_waiting_def cs_holding_def)
- by (auto intro:wq_threads)
-
-lemma readys_v_eq:
- fixes th thread cs rest
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and not_in: "th \<notin> set rest"
- shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
- from prems show ?thesis
- apply (auto simp:readys_def)
- apply (case_tac "cs = csa", simp add:s_waiting_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)
- by (auto simp:s_waiting_def wq_def Let_def split:list.splits prod.splits
- dest:lsp_set_eq)
-qed
-
-lemma readys_v_eq_1:
- fixes th thread cs rest
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
- and neq_th': "th \<noteq> th'"
- shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
- from prems show ?thesis
- apply (auto simp:readys_def)
- apply (case_tac "cs = csa", simp add:s_waiting_def)
- apply (erule_tac x = cs in allE)
- apply (simp add:s_waiting_def wq_def Let_def split:prod.splits list.splits)
- apply (drule_tac lsp_mid_nil,simp, clarify, fold cp_def, clarsimp)
- apply (frule_tac lsp_set_eq, simp)
- apply (erule_tac x = csa in allE)
- apply (subst (asm) (2) s_waiting_def, unfold wq_def)
- apply (auto simp:Let_def split:list.splits prod.splits if_splits
- dest:lsp_set_eq)
- apply (unfold s_waiting_def)
- apply (fold wq_def, clarsimp)
- apply (clarsimp)+
- apply (case_tac "csa = cs", simp)
- apply (erule_tac x = cs in allE, simp)
- apply (unfold wq_def)
- by (auto simp:Let_def split:list.splits prod.splits if_splits
- dest:lsp_set_eq)
-qed
-
-lemma readys_v_eq_2:
- fixes th thread cs rest
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
- and neq_th': "th = th'"
- and vt: "vt step s"
- shows "(th \<in> readys (V thread cs#s))"
-proof -
- from prems show ?thesis
- apply (auto simp:readys_def)
- apply (rule_tac wq_threads [of s _ cs], auto dest:lsp_set_eq)
- apply (unfold s_waiting_def wq_def)
- apply (auto simp:Let_def split:list.splits prod.splits if_splits
- dest:lsp_set_eq lsp_mid_nil lsp_mid_length)
- apply (fold cp_def, simp+, clarsimp)
- apply (frule_tac lsp_set_eq, simp)
- apply (fold wq_def)
- apply (subgoal_tac "csa = cs", simp)
- apply (rule_tac waiting_unique [of s th'], simp)
- by (auto simp:s_waiting_def)
-qed
-
-lemma chain_building:
- assumes vt: "vt step s"
- shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
-proof -
- from wf_dep_converse [OF vt]
- have h: "wf ((depend s)\<inverse>)" .
- show ?thesis
- proof(induct rule:wf_induct [OF h])
- fix x
- assume ih [rule_format]:
- "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow>
- y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
- show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
- proof
- assume x_d: "x \<in> Domain (depend s)"
- show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
- proof(cases x)
- case (Th th)
- from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
- with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
- from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
- hence "Cs cs \<in> Domain (depend s)" by auto
- from ih [OF x_in_r this] obtain th'
- where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto
- have "(x, Th th') \<in> (depend 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) \<in> (depend s)^-1" by (auto simp:s_depend_def)
- show ?thesis
- proof(cases "th' \<in> readys s")
- case True
- from True and th'_d show ?thesis by auto
- next
- case False
- from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto
- with False have "Th th' \<in> Domain (depend s)"
- by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
- from ih [OF th'_d this]
- obtain th'' where
- th''_r: "th'' \<in> readys s" and
- th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto
- from th'_d and th''_in
- have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
- with th''_r show ?thesis by auto
- qed
- qed
- qed
- qed
-qed
-
-lemma th_chain_to_ready:
- fixes s th
- assumes vt: "vt step s"
- and th_in: "th \<in> threads s"
- shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"
-proof(cases "th \<in> readys s")
- case True
- thus ?thesis by auto
-next
- case False
- from False and th_in have "Th th \<in> Domain (depend s)"
- by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
- from chain_building [rule_format, OF vt this]
- show ?thesis by auto
-qed
-
-lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
- by (unfold s_waiting_def cs_waiting_def, auto)
-
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
- by (unfold s_holding_def cs_holding_def, simp)
-
-lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
- by (unfold s_holding_def cs_holding_def, auto)
-
-lemma unique_depend: "\<lbrakk>vt step s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
- apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
- by(auto elim:waiting_unique holding_unique)
-
-lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
-by (induct rule:trancl_induct, auto)
-
-lemma dchain_unique:
- assumes vt: "vt step s"
- and th1_d: "(n, Th th1) \<in> (depend s)^+"
- and th1_r: "th1 \<in> readys s"
- and th2_d: "(n, Th th2) \<in> (depend s)^+"
- and th2_r: "th2 \<in> readys s"
- shows "th1 = th2"
-proof -
- { assume neq: "th1 \<noteq> th2"
- hence "Th th1 \<noteq> Th th2" by simp
- from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]
- have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
- hence "False"
- proof
- assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
- from trancl_split [OF this]
- obtain n where dd: "(Th th1, n) \<in> depend s" by auto
- then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
- from dd eq_n have "th1 \<notin> readys s"
- by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
- with th1_r show ?thesis by auto
- next
- assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
- from trancl_split [OF this]
- obtain n where dd: "(Th th2, n) \<in> depend s" by auto
- then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
- from dd eq_n have "th2 \<notin> readys s"
- by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
- with th2_r show ?thesis by auto
- qed
- } thus ?thesis by auto
-qed
-
-definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
-where "count Q l = length (filter Q l)"
-
-definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
-where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
-
-definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
-where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
-
-
-lemma step_holdents_p_add:
- fixes th cs s
- assumes vt: "vt step (P th cs#s)"
- and "wq s cs = []"
- shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
-proof -
- from prems show ?thesis
- unfolding holdents_def step_depend_p[OF vt] by auto
-qed
-
-lemma step_holdents_p_eq:
- fixes th cs s
- assumes vt: "vt step (P th cs#s)"
- and "wq s cs \<noteq> []"
- shows "holdents (P th cs#s) th = holdents s th"
-proof -
- from prems show ?thesis
- unfolding holdents_def step_depend_p[OF vt] by auto
-qed
-
-lemma step_holdents_v_minus:
- fixes th cs s
- assumes vt: "vt step (V th cs#s)"
- shows "holdents (V th cs#s) th = holdents s th - {cs}"
-proof -
- { fix rest l r
- assume eq_wq: "wq s cs = th # rest"
- and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
- have "False"
- proof -
- from lsp_set_eq [OF eq_lsp] have " rest = l @ [th] @ r" .
- with eq_wq have "wq s cs = th#\<dots>" by simp
- with wq_distinct [OF step_back_vt[OF vt], of cs]
- show ?thesis by auto
- qed
- } thus ?thesis unfolding holdents_def step_depend_v[OF vt] by auto
-qed
-
-lemma step_holdents_v_add:
- fixes th th' cs s rest l r
- assumes vt: "vt step (V th' cs#s)"
- and neq_th: "th \<noteq> th'"
- and eq_wq: "wq s cs = th' # rest"
- and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
- shows "holdents (V th' cs#s) th = holdents s th \<union> {cs}"
-proof -
- from prems show ?thesis
- unfolding holdents_def step_depend_v[OF vt] by auto
-qed
-
-lemma step_holdents_v_eq:
- fixes th th' cs s rest l r th''
- assumes vt: "vt step (V th' cs#s)"
- and neq_th: "th \<noteq> th'"
- and eq_wq: "wq s cs = th' # rest"
- and eq_lsp: "lsp (cp s) rest = (l, [th''], r)"
- and neq_th': "th \<noteq> th''"
- shows "holdents (V th' cs#s) th = holdents s th"
-proof -
- from prems show ?thesis
- unfolding holdents_def step_depend_v[OF vt] by auto
-qed
-
-definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
-where "cntCS s th = card (holdents s th)"
-
-lemma cntCS_v_eq:
- fixes th thread cs rest
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and not_in: "th \<notin> set rest"
- and vtv: "vt step (V thread cs#s)"
- shows "cntCS (V thread cs#s) th = cntCS s th"
-proof -
- from prems show ?thesis
- apply (unfold cntCS_def holdents_def step_depend_v)
- apply auto
- apply (subgoal_tac "\<not> (\<exists>l r. lsp (cp s) rest = (l, [th], r))", auto)
- by (drule_tac lsp_set_eq, auto)
-qed
-
-lemma cntCS_v_eq_1:
- fixes th thread cs rest
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
- and neq_th': "th \<noteq> th'"
- and vtv: "vt step (V thread cs#s)"
- shows "cntCS (V thread cs#s) th = cntCS s th"
-proof -
- from prems show ?thesis
- apply (unfold cntCS_def holdents_def step_depend_v)
- by auto
-qed
-
-fun the_cs :: "node \<Rightarrow> cs"
-where "the_cs (Cs cs) = cs"
-
-lemma cntCS_v_eq_2:
- fixes th thread cs rest
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
- and neq_th': "th = th'"
- and vtv: "vt step (V thread cs#s)"
- shows "cntCS (V thread cs#s) th = 1 + cntCS s th"
-proof -
- have "card {csa. csa = cs \<or> (Cs csa, Th th') \<in> depend s} =
- Suc (card {cs. (Cs cs, Th th') \<in> depend s})"
- (is "card ?A = Suc (card ?B)")
- proof -
- have h: "?A = insert cs ?B" by auto
- moreover have h1: "?B = ?B - {cs}"
- proof -
- { assume "(Cs cs, Th th') \<in> depend s"
- moreover have "(Th th', Cs cs) \<in> depend s"
- proof -
- from wq_distinct [OF step_back_vt[OF vtv], of cs]
- eq_wq lsp_set_eq [OF eq_lsp] show ?thesis
- apply (auto simp:s_depend_def)
- by (unfold cs_waiting_def, auto)
- qed
- moreover note acyclic_depend [OF step_back_vt[OF vtv]]
- ultimately have "False"
- apply (auto simp:acyclic_def)
- apply (erule_tac x="Cs cs" in allE)
- apply (subgoal_tac "(Cs cs, Cs cs) \<in> (depend s)\<^sup>+", simp)
- by (rule_tac trancl_into_trancl [where b = "Th th'"], auto)
- } thus ?thesis by auto
- qed
- moreover have "card (insert cs ?B) = Suc (card (?B - {cs}))"
- proof(rule card_insert)
- from finite_depend [OF step_back_vt [OF vtv]]
- have fnt: "finite (depend s)" .
- show " finite {cs. (Cs cs, Th th') \<in> depend s}" (is "finite ?B")
- proof -
- have "?B \<subseteq> (\<lambda> (a, b). the_cs a) ` (depend s)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Cs x, Th th')" in bexI, auto)
- with fnt show ?thesis by (auto intro:finite_subset)
- qed
- qed
- ultimately show ?thesis by simp
- qed
- with prems show ?thesis
- apply (unfold cntCS_def holdents_def step_depend_v[OF vtv])
- by auto
-qed
-
-lemma finite_holding:
- fixes s th cs
- assumes vt: "vt step s"
- shows "finite (holdents s th)"
-proof -
- let ?F = "\<lambda> (x, y). the_cs x"
- from finite_depend [OF vt]
- have "finite (depend s)" .
- hence "finite (?F `(depend s))" by simp
- moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>"
- proof -
- { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
- fix x assume "(Cs x, Th th) \<in> depend s"
- hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
- moreover have "?F (Cs x, Th th) = x" by simp
- ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp
- } thus ?thesis by auto
- qed
- ultimately show ?thesis by (unfold holdents_def, auto intro:finite_subset)
-qed
-
-inductive_cases case_step_v: "step s (V thread cs)"
-
-lemma cntCS_v_dec:
- fixes s thread cs
- assumes vtv: "vt step (V thread cs#s)"
- shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
-proof -
- have cs_in: "cs \<in> holdents s thread" using step_back_step[OF vtv]
- apply (erule_tac case_step_v)
- apply (unfold holdents_def s_depend_def, simp)
- by (unfold cs_holding_def s_holding_def, auto)
- moreover have cs_not_in:
- "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
- apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
- by (unfold holdents_def, unfold step_depend_v[OF vtv],
- auto dest:lsp_set_eq)
- ultimately
- have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
- by auto
- moreover have "card \<dots> =
- Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
- proof(rule card_insert)
- from finite_holding [OF vtv]
- show " finite (holdents (V thread cs # s) thread)" .
- qed
- moreover from cs_not_in
- have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
- ultimately show ?thesis by (simp add:cntCS_def)
-qed
-
-lemma cnp_cnv_cncs:
- fixes s th
- assumes vt: "vt step s"
- shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
- then cntCS s th else cntCS s th + 1)"
-proof -
- from vt show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e)
- assume vt: "vt step s"
- and ih: "\<And>th. cntP s th = cntV s th +
- (if (th \<in> readys s \<or> th \<notin> 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 prio max_prio thread)
- assume eq_e: "e = Create thread prio"
- and not_in: "thread \<notin> threads s"
- show ?thesis
- proof -
- { fix cs
- assume "thread \<in> set (wq s cs)"
- from wq_threads [OF vt this] have "thread \<in> threads s" .
- with not_in have "False" by simp
- } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {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_def
- by (simp add:depend_create_unchanged eq_e)
- { assume "th \<noteq> thread"
- with eq_readys eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> 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 \<in> 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 \<in> 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_def
- by (simp add:depend_exit_unchanged eq_e)
- { assume "th \<noteq> thread"
- with eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> threads (s))"
- apply (simp add:threads.simps readys_def)
- apply (subst s_waiting_def)
- apply (subst (1 2) wq_def)
- apply (simp add:Let_def)
- apply (subst s_waiting_def, simp)
- by (fold wq_def, simp)
- 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 \<notin> 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 \<in> runing s"
- and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
- from prems have vtp: "vt step (P thread cs#s)" by auto
- show ?thesis
- proof -
- { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
- assume neq_th: "th \<noteq> thread"
- with eq_e
- have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
- apply (simp add:readys_def s_waiting_def wq_def Let_def)
- apply (rule_tac hh, clarify)
- apply (intro iffI allI, clarify)
- apply (erule_tac x = csa in allE, auto)
- apply (subgoal_tac "waiting_queue (schs s) cs \<noteq> []", auto)
- apply (erule_tac x = cs in allE, auto)
- by (case_tac "(waiting_queue (schs s) cs)", auto)
- moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
- apply (simp add:cntCS_def holdents_def)
- by (unfold step_depend_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 \<in> readys (e#s)"
- apply (unfold eq_e wq_def, unfold readys_def s_depend_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 \<or> (Cs csa, Th thread) \<in> depend s} =
- Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")
- proof -
- have "?L = insert cs ?R" by auto
- moreover have "card \<dots> = Suc (card (?R - {cs}))"
- proof(rule card_insert)
- from finite_holding [OF vt, of thread]
- show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
- by (unfold holdents_def, simp)
- qed
- moreover have "?R - {cs} = ?R"
- proof -
- have "cs \<notin> ?R"
- proof
- assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend 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_def)
- by (unfold step_depend_p [OF vtp], auto simp:True)
- qed
- moreover from is_runing have "th \<in> 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 \<notin> readys (e#s)"
- proof
- assume "th \<in> readys (e#s)"
- hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
- from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
- hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
- by (simp add:s_waiting_def)
- moreover from eq_wq have "th \<in> 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 wq_distinct [OF vtp, of cs]
- show False by (fold eq_e, auto)
- qed
- moreover from is_runing have "th \<in> 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_def eq_e step_depend_p[OF vtp])
- by (auto simp:False)
- moreover note eq_cnp eq_cnv ih[of th]
- moreover from is_runing have "th \<in> readys s"
- by (simp add:runing_def eq_th)
- ultimately show ?thesis by auto
- qed
- qed
- } ultimately show ?thesis by blast
- qed
- next
- case (thread_V thread cs)
- from prems have vtv: "vt step (V thread cs # s)" by auto
- assume eq_e: "e = V thread cs"
- and is_runing: "thread \<in> 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:s_holding_def)
- have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
- 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 \<in> readys s"
- by (unfold runing_def, simp)
- moreover have "thread \<in> readys (e # s)"
- proof -
- from is_runing
- have "thread \<in> threads (e#s)"
- by (unfold eq_e, auto simp:runing_def readys_def)
- moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
- proof
- fix cs1
- { assume eq_cs: "cs1 = cs"
- have "\<not> waiting (e # s) thread cs1"
- proof -
- have "thread \<notin> set (wq (e#s) cs1)"
- proof(cases "lsp (cp s) rest")
- fix l m r
- assume h: "lsp (cp s) rest = (l, m, r)"
- show ?thesis
- proof(cases "m")
- case Nil
- from wq_v_eq_nil [OF eq_wq] h Nil eq_e
- have " wq (e # s) cs = []" by auto
- thus ?thesis using eq_cs by auto
- next
- case (Cons th' l')
- from lsp_mid_length [OF h] and Cons h
- have eqh: "lsp (cp s) rest = (l, [th'], r)" by auto
- from wq_v_eq [OF eq_wq this]
- have "wq (V thread cs # s) cs = th' # l @ r" .
- moreover from lsp_set_eq [OF eqh]
- have "set rest = set \<dots>" by auto
- moreover have "thread \<notin> set rest"
- proof -
- from wq_distinct [OF step_back_vt[OF vtv], of cs]
- and eq_wq show ?thesis by auto
- qed
- moreover note eq_e eq_cs
- ultimately show ?thesis by simp
- qed
- qed
- thus ?thesis by (simp add:s_waiting_def)
- qed
- } moreover {
- assume neq_cs: "cs1 \<noteq> cs"
- have "\<not> 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 "\<not> waiting s thread cs1"
- proof -
- from runing_ready and is_runing
- have "thread \<in> readys s" by auto
- thus ?thesis by (simp add:readys_def)
- qed
- ultimately show ?thesis
- by (auto simp:s_waiting_def eq_e)
- qed
- } ultimately show "\<not> 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 \<noteq> 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 \<in> set rest")
- case False
- have "(th \<in> readys (e # s)) = (th \<in> readys s)"
- by(unfold eq_e, rule readys_v_eq [OF neq_th eq_wq False])
- moreover have "cntCS (e#s) th = cntCS s th"
- by(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq False vtv])
- moreover note ih eq_cnp eq_cnv eq_threads
- ultimately show ?thesis by auto
- next
- case True
- obtain l m r where eq_lsp: "lsp (cp s) rest = (l, m, r)"
- by (cases "lsp (cp s) rest", auto)
- with True have "m \<noteq> []" by (auto dest:lsp_mid_nil)
- with eq_lsp obtain th' where eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
- by (case_tac m, auto dest:lsp_mid_length)
- show ?thesis
- proof(cases "th = th'")
- case False
- have "(th \<in> readys (e # s)) = (th \<in> readys s)"
- by (unfold eq_e, rule readys_v_eq_1 [OF neq_th eq_wq eq_lsp False])
- moreover have "cntCS (e#s) th = cntCS s th"
- by (unfold eq_e, rule cntCS_v_eq_1[OF neq_th eq_wq eq_lsp False vtv])
- moreover note ih eq_cnp eq_cnv eq_threads
- ultimately show ?thesis by auto
- next
- case True
- have "th \<in> readys (e # s)"
- by (unfold eq_e, rule readys_v_eq_2 [OF neq_th eq_wq eq_lsp True vt])
- moreover have "cntP s th = cntV s th + cntCS s th + 1"
- proof -
- have "th \<notin> readys s"
- proof -
- from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
- show ?thesis
- apply (unfold readys_def s_waiting_def, auto)
- by (rule_tac x = cs in exI, auto)
- qed
- moreover have "th \<in> threads s"
- proof -
- from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
- have "th \<in> set (wq s cs)" by simp
- from wq_threads [OF step_back_vt[OF vtv] this]
- show ?thesis .
- qed
- ultimately show ?thesis using ih by auto
- qed
- moreover have "cntCS (e # s) th = 1 + cntCS s th"
- by (unfold eq_e, rule cntCS_v_eq_2 [OF neq_th eq_wq eq_lsp True vtv])
- 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 \<in> 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_def
- by (simp add:depend_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 \<noteq> thread"
- with eq_readys eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> 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 \<in> 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
- next
- case vt_nil
- show ?case
- by (unfold cntP_def cntV_def cntCS_def,
- auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
- qed
-qed
-
-lemma not_thread_cncs:
- fixes th s
- assumes vt: "vt step s"
- and not_in: "th \<notin> threads s"
- shows "cntCS s th = 0"
-proof -
- from vt not_in show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e th)
- assume vt: "vt step s"
- and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
- and stp: "step s e"
- and not_in: "th \<notin> threads (e # s)"
- from stp show ?case
- proof(cases)
- case (thread_create prio max_prio thread)
- assume eq_e: "e = Create thread prio"
- and not_in': "thread \<notin> threads s"
- have "cntCS (e # s) th = cntCS s th"
- apply (unfold eq_e cntCS_def holdents_def)
- by (simp add:depend_create_unchanged)
- moreover have "th \<notin> threads s"
- proof -
- from not_in eq_e show ?thesis by simp
- qed
- moreover note ih ultimately show ?thesis by auto
- next
- 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_def)
- by (simp add:depend_exit_unchanged)
- show ?thesis
- proof(cases "th = thread")
- case True
- have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
- with eq_cns show ?thesis by simp
- next
- case False
- with not_in and eq_e
- have "th \<notin> 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 \<in> runing s"
- from prems have vtp: "vt step (P thread cs#s)" by auto
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> 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_def eq_e)
- by (unfold step_depend_p[OF vtp], auto)
- moreover have "cntCS s th = 0"
- proof(rule ih)
- from not_in eq_e show "th \<notin> 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 \<in> runing s"
- and hold: "holding s thread cs"
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- from prems have vtv: "vt step (V thread cs#s)" by auto
- from hold obtain rest
- where eq_wq: "wq s cs = thread # rest"
- by (case_tac "wq s cs", auto simp:s_holding_def)
- have "cntCS (e # s) th = cntCS s th"
- proof(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq _ vtv])
- show "th \<notin> set rest"
- proof
- assume "th \<in> set rest"
- with eq_wq have "th \<in> set (wq s cs)" by simp
- from wq_threads [OF vt this] eq_e not_in
- show False by simp
- qed
- qed
- moreover have "cntCS s th = 0"
- proof(rule ih)
- from not_in eq_e show "th \<notin> 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 \<in> runing s"
- from not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] and eq_e
- show ?thesis
- apply (unfold eq_e cntCS_def holdents_def)
- by (simp add:depend_set_unchanged)
- qed
- next
- case vt_nil
- show ?case
- by (unfold cntCS_def,
- auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
- qed
-qed
-
-lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
- by (auto simp:s_waiting_def cs_waiting_def)
-
-lemma dm_depend_threads:
- fixes th s
- assumes vt: "vt step s"
- and in_dom: "(Th th) \<in> Domain (depend s)"
- shows "th \<in> threads s"
-proof -
- from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
- moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
- ultimately have "(Th th, Cs cs) \<in> depend s" by simp
- hence "th \<in> set (wq s cs)"
- by (unfold s_depend_def, auto simp:cs_waiting_def)
- from wq_threads [OF vt this] show ?thesis .
-qed
-
-lemma cp_eq_cpreced: "cp s th = cpreced s (wq s) th"
-proof(unfold cp_def wq_def, induct s)
- case (Cons e s')
- show ?case
- by (auto simp:Let_def)
-next
- case Nil
- show ?case by (auto simp:Let_def)
-qed
-
-fun the_th :: "node \<Rightarrow> thread"
- where "the_th (Th th) = th"
-
-lemma runing_unique:
- fixes th1 th2 s
- assumes vt: "vt step s"
- and runing_1: "th1 \<in> runing s"
- and runing_2: "th2 \<in> runing s"
- shows "th1 = th2"
-proof -
- from runing_1 and runing_2 have "cp s th1 = cp s th2"
- by (unfold runing_def, simp)
- hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
- Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
- (is "Max (?f ` ?A) = Max (?f ` ?B)")
- by (unfold cp_eq_cpreced cpreced_def)
- obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
- proof -
- have h1: "finite (?f ` ?A)"
- proof -
- have "finite ?A"
- proof -
- have "finite (dependents (wq s) th1)"
- proof-
- have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependents_def)
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by auto
- qed
- moreover have h2: "(?f ` ?A) \<noteq> {}"
- proof -
- have "?A \<noteq> {}" by simp
- thus ?thesis by simp
- qed
- from Max_in [OF h1 h2]
- have "Max (?f ` ?A) \<in> (?f ` ?A)" .
- thus ?thesis by (auto intro:that)
- qed
- obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
- proof -
- have h1: "finite (?f ` ?B)"
- proof -
- have "finite ?B"
- proof -
- have "finite (dependents (wq s) th2)"
- proof-
- have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependents_def)
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by auto
- qed
- moreover have h2: "(?f ` ?B) \<noteq> {}"
- proof -
- have "?B \<noteq> {}" by simp
- thus ?thesis by simp
- qed
- from Max_in [OF h1 h2]
- have "Max (?f ` ?B) \<in> (?f ` ?B)" .
- thus ?thesis by (auto intro:that)
- qed
- 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 \<or> (th1' \<in> dependents (wq s) th1)" by simp
- thus "th1' \<in> threads s"
- proof
- assume "th1' \<in> dependents (wq s) th1"
- hence "(Th th1') \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependents_def cs_depend_def s_depend_def)
- by (auto simp:Domain_def)
- hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
- from dm_depend_threads[OF vt this] show ?thesis .
- next
- assume "th1' = th1"
- with runing_1 show ?thesis
- by (unfold runing_def readys_def, auto)
- qed
- next
- from th2_in have "th2' = th2 \<or> (th2' \<in> dependents (wq s) th2)" by simp
- thus "th2' \<in> threads s"
- proof
- assume "th2' \<in> dependents (wq s) th2"
- hence "(Th th2') \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependents_def cs_depend_def s_depend_def)
- by (auto simp:Domain_def)
- hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
- from dm_depend_threads[OF vt this] show ?thesis .
- next
- assume "th2' = th2"
- with runing_2 show ?thesis
- by (unfold runing_def readys_def, auto)
- qed
- qed
- from th1_in have "th1' = th1 \<or> th1' \<in> dependents (wq s) th1" by simp
- thus ?thesis
- proof
- assume eq_th': "th1' = th1"
- from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
- thus ?thesis
- proof
- assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
- next
- assume "th2' \<in> dependents (wq s) th2"
- with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
- hence "(Th th1, Th th2) \<in> (depend s)^+"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
- hence "Th th1 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
- by auto
- hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
- from depend_target_th [OF this]
- obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th1, Cs cs') \<in> depend s" by simp
- with runing_1 have "False"
- apply (unfold runing_def readys_def s_depend_def)
- by (auto simp:eq_waiting)
- thus ?thesis by simp
- qed
- next
- assume th1'_in: "th1' \<in> dependents (wq s) th1"
- from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
- thus ?thesis
- proof
- assume "th2' = th2"
- with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
- hence "(Th th2, Th th1) \<in> (depend s)^+"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
- hence "Th th2 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
- by auto
- hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
- from depend_target_th [OF this]
- obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th2, Cs cs') \<in> depend s" by simp
- with runing_2 have "False"
- apply (unfold runing_def readys_def s_depend_def)
- by (auto simp:eq_waiting)
- thus ?thesis by simp
- next
- assume "th2' \<in> dependents (wq s) th2"
- with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
- hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
- from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
- show ?thesis
- proof(rule dchain_unique[OF vt h1 _ h2, symmetric])
- from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
- from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
- qed
- qed
- qed
-qed
-
-lemma create_pre:
- assumes stp: "step s e"
- and not_in: "th \<notin> threads s"
- and is_in: "th \<in> threads (e#s)"
- obtains prio where "e = Create th prio"
-proof -
- from assms
- show ?thesis
- proof(cases)
- case (thread_create prio max_prio thread)
- with is_in not_in have "e = Create th prio" by simp
- from that[OF this] show ?thesis .
- next
- case (thread_exit thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_P thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_V thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_set thread)
- with assms show ?thesis by (auto intro!:that)
- qed
-qed
-
-lemma length_down_to_in:
- assumes le_ij: "i \<le> j"
- and le_js: "j \<le> length s"
- shows "length (down_to j i s) = j - i"
-proof -
- have "length (down_to j i s) = length (from_to i j (rev s))"
- by (unfold down_to_def, auto)
- also have "\<dots> = j - i"
- proof(rule length_from_to_in[OF le_ij])
- from le_js show "j \<le> length (rev s)" by simp
- qed
- finally show ?thesis .
-qed
-
-
-lemma moment_head:
- assumes le_it: "Suc i \<le> length t"
- obtains e where "moment (Suc i) t = e#moment i t"
-proof -
- have "i \<le> Suc i" by simp
- from length_down_to_in [OF this le_it]
- have "length (down_to (Suc i) i t) = 1" by auto
- then obtain e where "down_to (Suc i) i t = [e]"
- apply (cases "(down_to (Suc i) i t)") by auto
- moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
- by (rule down_to_conc[symmetric], auto)
- ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
- by (auto simp:down_to_moment)
- from that [OF this] show ?thesis .
-qed
-
-lemma cnp_cnv_eq:
- fixes th s
- assumes "vt step s"
- and "th \<notin> threads s"
- shows "cntP s th = cntV s th"
-proof -
- from assms show ?thesis
- proof(induct)
- case (vt_cons s e)
- have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
- have not_in: "th \<notin> threads (e # s)" by fact
- have "step s e" by fact
- thus ?case proof(cases)
- case (thread_create prio max_prio thread)
- assume eq_e: "e = Create thread prio"
- hence "thread \<in> threads (e#s)" by simp
- with not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] show ?thesis using eq_e
- by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- and not_holding: "holdents s thread = {}"
- have vt_s: "vt step s" by fact
- from finite_holding[OF vt_s] have "finite (holdents s thread)" .
- with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
- moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
- moreover note cnp_cnv_cncs[OF vt_s, of thread]
- ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
- show ?thesis
- proof(cases "th = thread")
- case True
- with eq_thread eq_e show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- next
- case False
- with not_in and eq_e have "th \<notin> threads s" by simp
- from ih[OF this] and eq_e show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- have "thread \<in> runing s" by fact
- with not_in eq_e have neq_th: "thread \<noteq> th"
- by (auto simp:runing_def readys_def)
- from not_in eq_e have "th \<notin> threads s" by simp
- from ih[OF this] and neq_th and eq_e show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_V thread cs)
- assume eq_e: "e = V thread cs"
- have "thread \<in> runing s" by fact
- with not_in eq_e have neq_th: "thread \<noteq> th"
- by (auto simp:runing_def readys_def)
- from not_in eq_e have "th \<notin> threads s" by simp
- from ih[OF this] and neq_th and eq_e show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_set thread prio)
- assume eq_e: "e = Set thread prio"
- and "thread \<in> runing s"
- hence "thread \<in> threads (e#s)"
- by (simp add:runing_def readys_def)
- with not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] show ?thesis using eq_e
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case vt_nil
- show ?case by (auto simp:cntP_def cntV_def count_def)
- qed
-qed
-
-lemma eq_depend:
- "depend (wq s) = depend s"
-by (unfold cs_depend_def s_depend_def, auto)
-
-lemma count_eq_dependents:
- assumes vt: "vt step s"
- and eq_pv: "cntP s th = cntV s th"
- shows "dependents (wq s) th = {}"
-proof -
- from cnp_cnv_cncs[OF vt] and eq_pv
- have "cntCS s th = 0"
- by (auto split:if_splits)
- moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
- proof -
- from finite_holding[OF vt, of th] show ?thesis
- by (simp add:holdents_def)
- qed
- ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
- by (unfold cntCS_def holdents_def cs_dependents_def, auto)
- show ?thesis
- proof(unfold cs_dependents_def)
- { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
- then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
- hence "False"
- proof(cases)
- assume "(Th th', Th th) \<in> depend (wq s)"
- thus "False" by (auto simp:cs_depend_def)
- next
- fix c
- assume "(c, Th th) \<in> depend (wq s)"
- with h and eq_depend show "False"
- by (cases c, auto simp:cs_depend_def)
- qed
- } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
- qed
-qed
-
-lemma dependents_threads:
- fixes s th
- assumes vt: "vt step s"
- shows "dependents (wq s) th \<subseteq> threads s"
-proof
- { fix th th'
- assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
- have "Th th \<in> Domain (depend s)"
- proof -
- from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
- hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
- with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
- thus ?thesis using eq_depend by simp
- qed
- from dm_depend_threads[OF vt this]
- have "th \<in> threads s" .
- } note hh = this
- fix th1
- assume "th1 \<in> dependents (wq s) th"
- hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
- by (unfold cs_dependents_def, simp)
- from hh [OF this] show "th1 \<in> threads s" .
-qed
-
-lemma finite_threads:
- assumes vt: "vt step s"
- shows "finite (threads s)"
-proof -
- from vt show ?thesis
- proof(induct)
- case (vt_cons s e)
- assume vt: "vt step s"
- and step: "step s e"
- and ih: "finite (threads s)"
- from step
- show ?case
- proof(cases)
- case (thread_create prio max_prio thread)
- assume eq_e: "e = Create thread prio"
- with ih
- show ?thesis by (unfold eq_e, auto)
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- with ih show ?thesis
- by (unfold eq_e, auto)
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- with ih show ?thesis by (unfold eq_e, auto)
- next
- case (thread_V thread cs)
- assume eq_e: "e = V thread cs"
- with ih show ?thesis by (unfold eq_e, auto)
- next
- case (thread_set thread prio)
- from vt_cons thread_set show ?thesis by simp
- qed
- next
- case vt_nil
- show ?case by (auto)
- qed
-qed
-
-lemma Max_f_mono:
- assumes seq: "A \<subseteq> B"
- and np: "A \<noteq> {}"
- and fnt: "finite B"
- shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
- from seq show "f ` A \<subseteq> f ` B" by auto
-next
- from np show "f ` A \<noteq> {}" by auto
-next
- from fnt and seq show "finite (f ` B)" by auto
-qed
-
-lemma cp_le:
- assumes vt: "vt step s"
- and th_in: "th \<in> threads s"
- shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def)
- show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))
- \<le> Max ((\<lambda>th. preced th s) ` threads s)"
- (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
- proof(rule Max_f_mono)
- show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
- next
- from finite_threads [OF vt]
- show "finite (threads s)" .
- next
- from th_in
- show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
- apply (auto simp:Domain_def)
- apply (rule_tac dm_depend_threads[OF vt])
- apply (unfold trancl_domain [of "depend s", symmetric])
- by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
- qed
-qed
-
-lemma le_cp:
- assumes vt: "vt step s"
- shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
- show "Prc (original_priority th s) (birthtime th s)
- \<le> Max (insert (Prc (original_priority th s) (birthtime th s))
- ((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))"
- (is "?l \<le> 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) \<in> (depend (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependents_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:
- assumes vt: "vt step s"
- shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
- (is "?l = ?r")
-proof(cases "threads s = {}")
- case True
- thus ?thesis by auto
-next
- case False
- have "?l \<in> ((cp s) ` threads s)"
- proof(rule Max_in)
- from finite_threads[OF vt]
- show "finite (cp s ` threads s)" by auto
- next
- from False show "cp s ` threads s \<noteq> {}" by auto
- qed
- then obtain th
- where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
- have "\<dots> \<le> ?r" by (rule cp_le[OF vt th_in])
- moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
- proof -
- have "?r \<in> (?f ` ?A)"
- proof(rule Max_in)
- from finite_threads[OF vt]
- show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
- next
- from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
- qed
- then obtain th' where
- th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
- from le_cp [OF vt, of th'] eq_r
- have "?r \<le> cp s th'" by auto
- moreover have "\<dots> \<le> cp s th"
- proof(fold eq_l)
- show " cp s th' \<le> Max (cp s ` threads s)"
- proof(rule Max_ge)
- from th_in' show "cp s th' \<in> cp s ` threads s"
- by auto
- next
- from finite_threads[OF vt]
- 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 max_cp_readys_threads_pre:
- assumes vt: "vt step s"
- and np: "threads s \<noteq> {}"
- shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(unfold max_cp_eq[OF vt])
- show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
- proof -
- let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
- let ?f = "(\<lambda>th. preced th s)"
- have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
- proof(rule Max_in)
- from finite_threads[OF vt] show "finite (?f ` threads s)" by simp
- next
- from np show "?f ` threads s \<noteq> {}" by simp
- qed
- then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
- by (auto simp:Image_def)
- from th_chain_to_ready [OF vt tm_in]
- have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
- thus ?thesis
- proof
- assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
- then obtain th' where th'_in: "th' \<in> readys s"
- and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
- have "cp s th' = ?f tm"
- proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
- from dependents_threads[OF vt] finite_threads[OF vt]
- show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))"
- by (auto intro:finite_subset)
- next
- fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
- from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
- moreover have "p \<le> \<dots>"
- proof(rule Max_ge)
- from finite_threads[OF vt]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- from p_in and th'_in and dependents_threads[OF vt, of th']
- show "p \<in> (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- qed
- ultimately show "p \<le> preced tm s" by auto
- next
- show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
- proof -
- from tm_chain
- have "tm \<in> dependents (wq s) th'"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, auto)
- thus ?thesis by auto
- qed
- qed
- with tm_max
- have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
- show ?thesis
- proof (fold h, rule Max_eqI)
- fix q
- assume "q \<in> cp s ` readys s"
- then obtain th1 where th1_in: "th1 \<in> readys s"
- and eq_q: "q = cp s th1" by auto
- show "q \<le> cp s th'"
- apply (unfold h eq_q)
- apply (unfold cp_eq_cpreced cpreced_def)
- apply (rule Max_mono)
- proof -
- from dependents_threads [OF vt, of th1] th1_in
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq>
- (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- next
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp
- next
- from finite_threads[OF vt]
- show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
- qed
- next
- from finite_threads[OF vt]
- show "finite (cp s ` readys s)" by (auto simp:readys_def)
- next
- from th'_in
- show "cp s th' \<in> cp s ` readys s" by simp
- qed
- next
- assume tm_ready: "tm \<in> 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 \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
- show "y \<le> preced tm s"
- proof -
- { fix y'
- assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
- have "y' \<le> preced tm s"
- proof(unfold tm_max, rule Max_ge)
- from hy' dependents_threads[OF vt, of tm]
- show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
- next
- from finite_threads[OF vt]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- qed
- } with hy show ?thesis by auto
- qed
- next
- from dependents_threads[OF vt, of tm] finite_threads[OF vt]
- show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
- by (auto intro:finite_subset)
- next
- show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (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 \<in> cp s ` readys s" by simp
- next
- from finite_threads[OF vt]
- show "finite (cp s ` readys s)" by (auto simp:readys_def)
- next
- fix y assume "y \<in> cp s ` readys s"
- then obtain th1 where th1_readys: "th1 \<in> readys s"
- and h: "y = cp s th1" by auto
- show "y \<le> 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[OF vt]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
- by simp
- next
- from dependents_threads[OF vt, of th1] th1_readys
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)
- \<subseteq> (\<lambda>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 max_cp_readys_threads:
- assumes vt: "vt step s"
- 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 vt False])
-qed
-
-lemma readys_threads:
- shows "readys s \<subseteq> threads s"
-proof
- fix th
- assume "th \<in> readys s"
- thus "th \<in> threads s"
- by (unfold readys_def, auto)
-qed
-
-lemma eq_holding: "holding (wq s) th cs = holding s th cs"
- apply (unfold s_holding_def cs_holding_def, simp)
- done
-
-lemma f_image_eq:
- assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
- shows "f ` A = g ` A"
-proof
- show "f ` A \<subseteq> g ` A"
- by(rule image_subsetI, auto intro:h)
-next
- show "g ` A \<subseteq> f ` A"
- by(rule image_subsetI, auto intro:h[symmetric])
-qed
-
-end
\ No newline at end of file
--- a/prio/CpsG.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1997 +0,0 @@
-theory CpsG
-imports PrioG
-begin
-
-lemma not_thread_holdents:
- fixes th s
- assumes vt: "vt s"
- and not_in: "th \<notin> threads s"
- shows "holdents s th = {}"
-proof -
- from vt not_in show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e th)
- assume vt: "vt s"
- and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> holdents s th = {}"
- and stp: "step s e"
- and not_in: "th \<notin> 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 \<notin> threads s"
- have "holdents (e # s) th = holdents s th"
- apply (unfold eq_e holdents_test)
- by (simp add:depend_create_unchanged)
- moreover have "th \<notin> threads s"
- proof -
- from not_in eq_e show ?thesis by simp
- qed
- moreover note ih ultimately show ?thesis by auto
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- and nh: "holdents s thread = {}"
- show ?thesis
- proof(cases "th = thread")
- case True
- with nh eq_e
- show ?thesis
- by (auto simp:holdents_test depend_exit_unchanged)
- next
- case False
- with not_in and eq_e
- have "th \<notin> threads s" by simp
- from ih[OF this] False eq_e show ?thesis
- by (auto simp:holdents_test depend_exit_unchanged)
- qed
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- and is_runing: "thread \<in> runing s"
- from assms thread_exit ih stp not_in vt eq_e have vtp: "vt (P thread cs#s)" by auto
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- hence "holdents (e # s) th = holdents s th "
- apply (unfold cntCS_def holdents_test eq_e)
- by (unfold step_depend_p[OF vtp], auto)
- moreover have "holdents s th = {}"
- proof(rule ih)
- from not_in eq_e show "th \<notin> 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 \<in> runing s"
- and hold: "holding s thread cs"
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- from assms thread_V eq_e ih stp not_in vt have vtv: "vt (V thread cs#s)" by auto
- 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 "\<not> next_th s thread cs th"
- apply (auto simp:next_th_def)
- proof -
- assume ne: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
- have "?t \<in> set rest"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest" with ne
- show "hd x \<in> set rest" by (cases x, auto)
- qed
- with eq_wq have "?t \<in> set (wq s cs)" by simp
- from wq_threads[OF step_back_vt[OF vtv], OF this] and ni
- show False by auto
- qed
- moreover note neq_th eq_wq
- ultimately have "holdents (e # s) th = holdents s th"
- by (unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
- moreover have "holdents s th = {}"
- proof(rule ih)
- from not_in eq_e show "th \<notin> 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 \<in> runing s"
- from not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] and eq_e
- show ?thesis
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:depend_set_unchanged)
- qed
- next
- case vt_nil
- show ?case
- by (auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
- qed
-qed
-
-
-
-lemma next_th_neq:
- assumes vt: "vt s"
- and nt: "next_th s th cs th'"
- shows "th' \<noteq> th"
-proof -
- from nt show ?thesis
- apply (auto simp:next_th_def)
- proof -
- fix rest
- assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
- and ne: "rest \<noteq> []"
- have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x
- assume "distinct x \<and> set x = set rest"
- hence eq_set: "set x = set rest" by auto
- with ne have "x \<noteq> []" by auto
- hence "hd x \<in> set x" by auto
- with eq_set show "hd x \<in> set rest" by auto
- qed
- with wq_distinct[OF vt, of cs] eq_wq show False by auto
- qed
-qed
-
-lemma next_th_unique:
- assumes nt1: "next_th s th cs th1"
- and nt2: "next_th s th cs th2"
- shows "th1 = th2"
-proof -
- from assms show ?thesis
- by (unfold next_th_def, auto)
-qed
-
-lemma pp_sub: "(r^+)^+ \<subseteq> r^+"
- by auto
-
-lemma wf_depend:
- assumes vt: "vt s"
- shows "wf (depend s)"
-proof(rule finite_acyclic_wf)
- from finite_depend[OF vt] show "finite (depend s)" .
-next
- from acyclic_depend[OF vt] show "acyclic (depend s)" .
-qed
-
-lemma Max_Union:
- assumes fc: "finite C"
- and ne: "C \<noteq> {}"
- and fa: "\<And> A. A \<in> C \<Longrightarrow> finite A \<and> A \<noteq> {}"
- shows "Max (\<Union> C) = Max (Max ` C)"
-proof -
- from fc ne fa show ?thesis
- proof(induct)
- case (insert x F)
- assume ih: "\<lbrakk>F \<noteq> {}; \<And>A. A \<in> F \<Longrightarrow> finite A \<and> A \<noteq> {}\<rbrakk> \<Longrightarrow> Max (\<Union>F) = Max (Max ` F)"
- and h: "\<And> A. A \<in> insert x F \<Longrightarrow> finite A \<and> A \<noteq> {}"
- show ?case (is "?L = ?R")
- proof(cases "F = {}")
- case False
- from Union_insert have "?L = Max (x \<union> (\<Union> F))" by simp
- also have "\<dots> = max (Max x) (Max(\<Union> F))"
- proof(rule Max_Un)
- from h[of x] show "finite x" by auto
- next
- from h[of x] show "x \<noteq> {}" by auto
- next
- show "finite (\<Union>F)"
- proof(rule finite_Union)
- show "finite F" by fact
- next
- from h show "\<And>M. M \<in> F \<Longrightarrow> finite M" by auto
- qed
- next
- from False and h show "\<Union>F \<noteq> {}" by auto
- qed
- also have "\<dots> = ?R"
- proof -
- have "?R = Max (Max ` ({x} \<union> F))" by simp
- also have "\<dots> = Max ({Max x} \<union> (Max ` F))" by simp
- also have "\<dots> = max (Max x) (Max (\<Union>F))"
- proof -
- have "Max ({Max x} \<union> Max ` F) = max (Max {Max x}) (Max (Max ` F))"
- proof(rule Max_Un)
- show "finite {Max x}" by simp
- next
- show "{Max x} \<noteq> {}" by simp
- next
- from insert show "finite (Max ` F)" by auto
- next
- from False show "Max ` F \<noteq> {}" by auto
- qed
- moreover have "Max {Max x} = Max x" by simp
- moreover have "Max (\<Union>F) = Max (Max ` F)"
- proof(rule ih)
- show "F \<noteq> {}" by fact
- next
- from h show "\<And>A. A \<in> F \<Longrightarrow> finite A \<and> A \<noteq> {}"
- by auto
- qed
- ultimately show ?thesis by auto
- qed
- finally show ?thesis by simp
- qed
- finally show ?thesis by simp
- next
- case True
- thus ?thesis by auto
- qed
- next
- case empty
- assume "{} \<noteq> {}" show ?case by auto
- qed
-qed
-
-definition child :: "state \<Rightarrow> (node \<times> node) set"
- where "child s \<equiv>
- {(Th th', Th th) | th th'. \<exists> cs. (Th th', Cs cs) \<in> depend s \<and> (Cs cs, Th th) \<in> depend s}"
-
-definition children :: "state \<Rightarrow> thread \<Rightarrow> thread set"
- where "children s th \<equiv> {th'. (Th th', Th th) \<in> child s}"
-
-lemma children_def2:
- "children s th \<equiv> {th'. \<exists> cs. (Th th', Cs cs) \<in> depend s \<and> (Cs cs, Th th) \<in> depend s}"
-unfolding child_def children_def by simp
-
-lemma children_dependents: "children s th \<subseteq> dependents (wq s) th"
- by (unfold children_def child_def cs_dependents_def, auto simp:eq_depend)
-
-lemma child_unique:
- assumes vt: "vt s"
- and ch1: "(Th th, Th th1) \<in> child s"
- and ch2: "(Th th, Th th2) \<in> child s"
- shows "th1 = th2"
-proof -
- from ch1 ch2 show ?thesis
- proof(unfold child_def, clarsimp)
- fix cs csa
- assume h1: "(Th th, Cs cs) \<in> depend s"
- and h2: "(Cs cs, Th th1) \<in> depend s"
- and h3: "(Th th, Cs csa) \<in> depend s"
- and h4: "(Cs csa, Th th2) \<in> depend s"
- from unique_depend[OF vt h1 h3] have "cs = csa" by simp
- with h4 have "(Cs cs, Th th2) \<in> depend s" by simp
- from unique_depend[OF vt h2 this]
- show "th1 = th2" by simp
- qed
-qed
-
-
-lemma cp_eq_cpreced_f: "cp s = cpreced (wq s) s"
-proof -
- from fun_eq_iff
- have h:"\<And>f g. (\<forall> x. f x = g x) \<Longrightarrow> f = g" by auto
- show ?thesis
- proof(rule h)
- from cp_eq_cpreced show "\<forall>x. cp s x = cpreced (wq s) s x" by auto
- qed
-qed
-
-lemma depend_children:
- assumes h: "(Th th1, Th th2) \<in> (depend s)^+"
- shows "th1 \<in> children s th2 \<or> (\<exists> th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)^+)"
-proof -
- from h show ?thesis
- proof(induct rule: tranclE)
- fix c th2
- assume h1: "(Th th1, c) \<in> (depend s)\<^sup>+"
- and h2: "(c, Th th2) \<in> depend s"
- from h2 obtain cs where eq_c: "c = Cs cs"
- by (case_tac c, auto simp:s_depend_def)
- show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)\<^sup>+)"
- proof(rule tranclE[OF h1])
- fix ca
- assume h3: "(Th th1, ca) \<in> (depend s)\<^sup>+"
- and h4: "(ca, c) \<in> depend s"
- show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)\<^sup>+)"
- proof -
- from eq_c and h4 obtain th3 where eq_ca: "ca = Th th3"
- by (case_tac ca, auto simp:s_depend_def)
- from eq_ca h4 h2 eq_c
- have "th3 \<in> children s th2" by (auto simp:children_def child_def)
- moreover from h3 eq_ca have "(Th th1, Th th3) \<in> (depend s)\<^sup>+" by simp
- ultimately show ?thesis by auto
- qed
- next
- assume "(Th th1, c) \<in> depend s"
- with h2 eq_c
- have "th1 \<in> children s th2"
- by (auto simp:children_def child_def)
- thus ?thesis by auto
- qed
- next
- assume "(Th th1, Th th2) \<in> depend s"
- thus ?thesis
- by (auto simp:s_depend_def)
- qed
-qed
-
-lemma sub_child: "child s \<subseteq> (depend s)^+"
- by (unfold child_def, auto)
-
-lemma wf_child:
- assumes vt: "vt s"
- shows "wf (child s)"
-proof(rule wf_subset)
- from wf_trancl[OF wf_depend[OF vt]]
- show "wf ((depend s)\<^sup>+)" .
-next
- from sub_child show "child s \<subseteq> (depend s)\<^sup>+" .
-qed
-
-lemma depend_child_pre:
- assumes vt: "vt s"
- shows
- "(Th th, n) \<in> (depend s)^+ \<longrightarrow> (\<forall> th'. n = (Th th') \<longrightarrow> (Th th, Th th') \<in> (child s)^+)" (is "?P n")
-proof -
- from wf_trancl[OF wf_depend[OF vt]]
- have wf: "wf ((depend s)^+)" .
- show ?thesis
- proof(rule wf_induct[OF wf, of ?P], clarsimp)
- fix th'
- assume ih[rule_format]: "\<forall>y. (y, Th th') \<in> (depend s)\<^sup>+ \<longrightarrow>
- (Th th, y) \<in> (depend s)\<^sup>+ \<longrightarrow> (\<forall>th'. y = Th th' \<longrightarrow> (Th th, Th th') \<in> (child s)\<^sup>+)"
- and h: "(Th th, Th th') \<in> (depend s)\<^sup>+"
- show "(Th th, Th th') \<in> (child s)\<^sup>+"
- proof -
- from depend_children[OF h]
- have "th \<in> children s th' \<or> (\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (depend s)\<^sup>+)" .
- thus ?thesis
- proof
- assume "th \<in> children s th'"
- thus "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def)
- next
- assume "\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (depend s)\<^sup>+"
- then obtain th3 where th3_in: "th3 \<in> children s th'"
- and th_dp: "(Th th, Th th3) \<in> (depend s)\<^sup>+" by auto
- from th3_in have "(Th th3, Th th') \<in> (depend s)^+" by (auto simp:children_def child_def)
- from ih[OF this th_dp, of th3] have "(Th th, Th th3) \<in> (child s)\<^sup>+" by simp
- with th3_in show "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def)
- qed
- qed
- qed
-qed
-
-lemma depend_child: "\<lbrakk>vt s; (Th th, Th th') \<in> (depend s)^+\<rbrakk> \<Longrightarrow> (Th th, Th th') \<in> (child s)^+"
- by (insert depend_child_pre, auto)
-
-lemma child_depend_p:
- assumes "(n1, n2) \<in> (child s)^+"
- shows "(n1, n2) \<in> (depend s)^+"
-proof -
- from assms show ?thesis
- proof(induct)
- case (base y)
- with sub_child show ?case by auto
- next
- case (step y z)
- assume "(y, z) \<in> child s"
- with sub_child have "(y, z) \<in> (depend s)^+" by auto
- moreover have "(n1, y) \<in> (depend s)^+" by fact
- ultimately show ?case by auto
- qed
-qed
-
-lemma child_depend_eq:
- assumes vt: "vt s"
- shows
- "((Th th1, Th th2) \<in> (child s)^+) =
- ((Th th1, Th th2) \<in> (depend s)^+)"
- by (auto intro: depend_child[OF vt] child_depend_p)
-
-lemma children_no_dep:
- fixes s th th1 th2 th3
- assumes vt: "vt s"
- and ch1: "(Th th1, Th th) \<in> child s"
- and ch2: "(Th th2, Th th) \<in> child s"
- and ch3: "(Th th1, Th th2) \<in> (depend s)^+"
- shows "False"
-proof -
- from depend_child[OF vt ch3]
- have "(Th th1, Th th2) \<in> (child s)\<^sup>+" .
- thus ?thesis
- proof(rule converse_tranclE)
- thm tranclD
- assume "(Th th1, Th th2) \<in> child s"
- from child_unique[OF vt ch1 this] have "th = th2" by simp
- with ch2 have "(Th th2, Th th2) \<in> child s" by simp
- with wf_child[OF vt] show ?thesis by auto
- next
- fix c
- assume h1: "(Th th1, c) \<in> child s"
- and h2: "(c, Th th2) \<in> (child s)\<^sup>+"
- from h1 obtain th3 where eq_c: "c = Th th3" by (unfold child_def, auto)
- with h1 have "(Th th1, Th th3) \<in> child s" by simp
- from child_unique[OF vt ch1 this] have eq_th3: "th3 = th" by simp
- with eq_c and h2 have "(Th th, Th th2) \<in> (child s)\<^sup>+" by simp
- with ch2 have "(Th th, Th th) \<in> (child s)\<^sup>+" by auto
- moreover have "wf ((child s)\<^sup>+)"
- proof(rule wf_trancl)
- from wf_child[OF vt] show "wf (child s)" .
- qed
- ultimately show False by auto
- qed
-qed
-
-lemma unique_depend_p:
- assumes vt: "vt s"
- and dp1: "(n, n1) \<in> (depend s)^+"
- and dp2: "(n, n2) \<in> (depend s)^+"
- and neq: "n1 \<noteq> n2"
- shows "(n1, n2) \<in> (depend s)^+ \<or> (n2, n1) \<in> (depend s)^+"
-proof(rule unique_chain [OF _ dp1 dp2 neq])
- from unique_depend[OF vt]
- show "\<And>a b c. \<lbrakk>(a, b) \<in> depend s; (a, c) \<in> depend s\<rbrakk> \<Longrightarrow> b = c" by auto
-qed
-
-lemma dependents_child_unique:
- fixes s th th1 th2 th3
- assumes vt: "vt s"
- and ch1: "(Th th1, Th th) \<in> child s"
- and ch2: "(Th th2, Th th) \<in> child s"
- and dp1: "th3 \<in> dependents s th1"
- and dp2: "th3 \<in> dependents s th2"
-shows "th1 = th2"
-proof -
- { assume neq: "th1 \<noteq> th2"
- from dp1 have dp1: "(Th th3, Th th1) \<in> (depend s)^+"
- by (simp add:s_dependents_def eq_depend)
- from dp2 have dp2: "(Th th3, Th th2) \<in> (depend s)^+"
- by (simp add:s_dependents_def eq_depend)
- from unique_depend_p[OF vt dp1 dp2] and neq
- have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
- hence False
- proof
- assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+ "
- from children_no_dep[OF vt ch1 ch2 this] show ?thesis .
- next
- assume " (Th th2, Th th1) \<in> (depend s)\<^sup>+"
- from children_no_dep[OF vt ch2 ch1 this] show ?thesis .
- qed
- } thus ?thesis by auto
-qed
-
-lemma cp_rec:
- fixes s th
- assumes vt: "vt s"
- shows "cp s th = Max ({preced th s} \<union> (cp s ` children s th))"
-proof(unfold cp_eq_cpreced_f cpreced_def)
- let ?f = "(\<lambda>th. preced th s)"
- show "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) =
- Max ({preced th s} \<union> (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th)"
- proof(cases " children s th = {}")
- case False
- have "(\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th =
- {Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) | th' . th' \<in> children s th}"
- (is "?L = ?R")
- by auto
- also have "\<dots> =
- Max ` {((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) | th' . th' \<in> children s th}"
- (is "_ = Max ` ?C")
- by auto
- finally have "Max ?L = Max (Max ` ?C)" by auto
- also have "\<dots> = Max (\<Union> ?C)"
- proof(rule Max_Union[symmetric])
- from children_dependents[of s th] finite_threads[OF vt] and dependents_threads[OF vt, of th]
- show "finite {(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
- by (auto simp:finite_subset)
- next
- from False
- show "{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th} \<noteq> {}"
- by simp
- next
- show "\<And>A. A \<in> {(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th} \<Longrightarrow>
- finite A \<and> A \<noteq> {}"
- apply (auto simp:finite_subset)
- proof -
- fix th'
- from finite_threads[OF vt] and dependents_threads[OF vt, of th']
- show "finite ((\<lambda>th. preced th s) ` dependents (wq s) th')" by (auto simp:finite_subset)
- qed
- qed
- also have "\<dots> = Max ((\<lambda>th. preced th s) ` dependents (wq s) th)"
- (is "Max ?A = Max ?B")
- proof -
- have "?A = ?B"
- proof
- show "\<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}
- \<subseteq> (\<lambda>th. preced th s) ` dependents (wq s) th"
- proof
- fix x
- assume "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
- then obtain th' where
- th'_in: "th' \<in> children s th"
- and x_in: "x \<in> ?f ` ({th'} \<union> dependents (wq s) th')" by auto
- hence "x = ?f th' \<or> x \<in> (?f ` dependents (wq s) th')" by auto
- thus "x \<in> ?f ` dependents (wq s) th"
- proof
- assume "x = preced th' s"
- with th'_in and children_dependents
- show "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th" by auto
- next
- assume "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th'"
- moreover note th'_in
- ultimately show " x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th"
- by (unfold cs_dependents_def children_def child_def, auto simp:eq_depend)
- qed
- qed
- next
- show "?f ` dependents (wq s) th
- \<subseteq> \<Union>{?f ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
- proof
- fix x
- assume x_in: "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th"
- then obtain th' where
- eq_x: "x = ?f th'" and dp: "(Th th', Th th) \<in> (depend s)^+"
- by (auto simp:cs_dependents_def eq_depend)
- from depend_children[OF dp]
- have "th' \<in> children s th \<or> (\<exists>th3. th3 \<in> children s th \<and> (Th th', Th th3) \<in> (depend s)\<^sup>+)" .
- thus "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
- proof
- assume "th' \<in> children s th"
- with eq_x
- show "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
- by auto
- next
- assume "\<exists>th3. th3 \<in> children s th \<and> (Th th', Th th3) \<in> (depend s)\<^sup>+"
- then obtain th3 where th3_in: "th3 \<in> children s th"
- and dp3: "(Th th', Th th3) \<in> (depend s)\<^sup>+" by auto
- show "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
- proof -
- from dp3
- have "th' \<in> dependents (wq s) th3"
- by (auto simp:cs_dependents_def eq_depend)
- with eq_x th3_in show ?thesis by auto
- qed
- qed
- qed
- qed
- thus ?thesis by simp
- qed
- finally have "Max ((\<lambda>th. preced th s) ` dependents (wq s) th) = Max (?L)"
- (is "?X = ?Y") by auto
- moreover have "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) =
- max (?f th) ?X"
- proof -
- have "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) =
- Max ({?f th} \<union> ?f ` (dependents (wq s) th))" by simp
- also have "\<dots> = max (Max {?f th}) (Max (?f ` (dependents (wq s) th)))"
- proof(rule Max_Un, auto)
- from finite_threads[OF vt] and dependents_threads[OF vt, of th]
- show "finite ((\<lambda>th. preced th s) ` dependents (wq s) th)" by (auto simp:finite_subset)
- next
- assume "dependents (wq s) th = {}"
- with False and children_dependents show False by auto
- qed
- also have "\<dots> = max (?f th) ?X" by simp
- finally show ?thesis .
- qed
- moreover have "Max ({preced th s} \<union>
- (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th) =
- max (?f th) ?Y"
- proof -
- have "Max ({preced th s} \<union>
- (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th) =
- max (Max {preced th s}) ?Y"
- proof(rule Max_Un, auto)
- from finite_threads[OF vt] dependents_threads[OF vt, of th] children_dependents [of s th]
- show "finite ((\<lambda>th. Max (insert (preced th s) ((\<lambda>th. preced th s) ` dependents (wq s) th))) `
- children s th)"
- by (auto simp:finite_subset)
- next
- assume "children s th = {}"
- with False show False by auto
- qed
- thus ?thesis by simp
- qed
- ultimately show ?thesis by auto
- next
- case True
- moreover have "dependents (wq s) th = {}"
- proof -
- { fix th'
- assume "th' \<in> dependents (wq s) th"
- hence " (Th th', Th th) \<in> (depend s)\<^sup>+" by (simp add:cs_dependents_def eq_depend)
- from depend_children[OF this] and True
- have "False" by auto
- } thus ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
-qed
-
-definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
-where "cps s = {(th, cp s th) | th . th \<in> threads s}"
-
-locale step_set_cps =
- fixes s' th prio s
- defines s_def : "s \<equiv> (Set th prio#s')"
- assumes vt_s: "vt s"
-
-context step_set_cps
-begin
-
-lemma eq_preced:
- fixes th'
- assumes "th' \<noteq> th"
- shows "preced th' s = preced th' s'"
-proof -
- from assms show ?thesis
- by (unfold s_def, auto simp:preced_def)
-qed
-
-lemma eq_dep: "depend s = depend s'"
- by (unfold s_def depend_set_unchanged, auto)
-
-lemma eq_cp_pre:
- fixes th'
- assumes neq_th: "th' \<noteq> th"
- and nd: "th \<notin> dependents s th'"
- shows "cp s th' = cp s' th'"
- apply (unfold cp_eq_cpreced cpreced_def)
-proof -
- have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
- by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
- moreover {
- fix th1
- assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
- hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
- hence "preced th1 s = preced th1 s'"
- proof
- assume "th1 = th'"
- with eq_preced[OF neq_th]
- show "preced th1 s = preced th1 s'" by simp
- next
- assume "th1 \<in> dependents (wq s') th'"
- with nd and eq_dp have "th1 \<noteq> th"
- by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
- from eq_preced[OF this] show "preced th1 s = preced th1 s'" by simp
- qed
- } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
- by (auto simp:image_def)
- thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
-qed
-
-lemma no_dependents:
- assumes "th' \<noteq> th"
- shows "th \<notin> dependents s th'"
-proof
- assume h: "th \<in> dependents s th'"
- from step_back_step [OF vt_s[unfolded s_def]]
- have "step s' (Set th prio)" .
- hence "th \<in> runing s'" by (cases, simp)
- hence rd_th: "th \<in> readys s'"
- by (simp add:readys_def runing_def)
- from h have "(Th th, Th th') \<in> (depend s')\<^sup>+"
- by (unfold s_dependents_def, unfold eq_depend, unfold eq_dep, auto)
- from tranclD[OF this]
- obtain z where "(Th th, z) \<in> depend s'" by auto
- with rd_th show "False"
- apply (case_tac z, auto simp:readys_def s_waiting_def s_depend_def s_waiting_def cs_waiting_def)
- by (fold wq_def, blast)
-qed
-
-(* Result improved *)
-lemma eq_cp:
- fixes th'
- assumes neq_th: "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
-proof(rule eq_cp_pre [OF neq_th])
- from no_dependents[OF neq_th]
- show "th \<notin> dependents s th'" .
-qed
-
-lemma eq_up:
- fixes th' th''
- assumes dp1: "th \<in> dependents s th'"
- and dp2: "th' \<in> dependents s th''"
- and eq_cps: "cp s th' = cp s' th'"
- shows "cp s th'' = cp s' th''"
-proof -
- from dp2
- have "(Th th', Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
- from depend_child[OF vt_s this[unfolded eq_depend]]
- have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" .
- moreover { fix n th''
- have "\<lbrakk>(Th th', n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
- (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
- proof(erule trancl_induct, auto)
- fix y th''
- assume y_ch: "(y, Th th'') \<in> child s"
- and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
- and ch': "(Th th', y) \<in> (child s)\<^sup>+"
- from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
- with ih have eq_cpy:"cp s thy = cp s' thy" by blast
- from dp1 have "(Th th, Th th') \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
- moreover from child_depend_p[OF ch'] and eq_y
- have "(Th th', Th thy) \<in> (depend s)^+" by simp
- ultimately have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by auto
- show "cp s th'' = cp s' th''"
- apply (subst cp_rec[OF vt_s])
- proof -
- have "preced th'' s = preced th'' s'"
- proof(rule eq_preced)
- show "th'' \<noteq> th"
- proof
- assume "th'' = th"
- with dp_thy y_ch[unfolded eq_y]
- have "(Th th, Th th) \<in> (depend s)^+"
- by (auto simp:child_def)
- with wf_trancl[OF wf_depend[OF vt_s]]
- show False by auto
- qed
- qed
- moreover {
- fix th1
- assume th1_in: "th1 \<in> children s th''"
- have "cp s th1 = cp s' th1"
- proof(cases "th1 = thy")
- case True
- with eq_cpy show ?thesis by simp
- next
- case False
- have neq_th1: "th1 \<noteq> th"
- proof
- assume eq_th1: "th1 = th"
- with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
- from children_no_dep[OF vt_s _ _ this] and
- th1_in y_ch eq_y show False by (auto simp:children_def)
- qed
- have "th \<notin> dependents s th1"
- proof
- assume h:"th \<in> dependents s th1"
- from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
- from dependents_child_unique[OF vt_s _ _ h this]
- th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
- with False show False by auto
- qed
- from eq_cp_pre[OF neq_th1 this]
- show ?thesis .
- qed
- }
- ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
- {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
- moreover have "children s th'' = children s' th''"
- by (unfold children_def child_def s_def depend_set_unchanged, simp)
- ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
- by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
- qed
- next
- fix th''
- assume dp': "(Th th', Th th'') \<in> child s"
- show "cp s th'' = cp s' th''"
- apply (subst cp_rec[OF vt_s])
- proof -
- have "preced th'' s = preced th'' s'"
- proof(rule eq_preced)
- show "th'' \<noteq> th"
- proof
- assume "th'' = th"
- with dp1 dp'
- have "(Th th, Th th) \<in> (depend s)^+"
- by (auto simp:child_def s_dependents_def eq_depend)
- with wf_trancl[OF wf_depend[OF vt_s]]
- show False by auto
- qed
- qed
- moreover {
- fix th1
- assume th1_in: "th1 \<in> children s th''"
- have "cp s th1 = cp s' th1"
- proof(cases "th1 = th'")
- case True
- with eq_cps show ?thesis by simp
- next
- case False
- have neq_th1: "th1 \<noteq> th"
- proof
- assume eq_th1: "th1 = th"
- with dp1 have "(Th th1, Th th') \<in> (depend s)^+"
- by (auto simp:s_dependents_def eq_depend)
- from children_no_dep[OF vt_s _ _ this]
- th1_in dp'
- show False by (auto simp:children_def)
- qed
- thus ?thesis
- proof(rule eq_cp_pre)
- show "th \<notin> dependents s th1"
- proof
- assume "th \<in> dependents s th1"
- from dependents_child_unique[OF vt_s _ _ this dp1]
- th1_in dp' have "th1 = th'"
- by (auto simp:children_def)
- with False show False by auto
- qed
- qed
- qed
- }
- ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
- {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
- moreover have "children s th'' = children s' th''"
- by (unfold children_def child_def s_def depend_set_unchanged, simp)
- ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
- by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
- qed
- qed
- }
- ultimately show ?thesis by auto
-qed
-
-lemma eq_up_self:
- fixes th' th''
- assumes dp: "th \<in> dependents s th''"
- and eq_cps: "cp s th = cp s' th"
- shows "cp s th'' = cp s' th''"
-proof -
- from dp
- have "(Th th, Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
- from depend_child[OF vt_s this[unfolded eq_depend]]
- have ch_th': "(Th th, Th th'') \<in> (child s)\<^sup>+" .
- moreover { fix n th''
- have "\<lbrakk>(Th th, n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
- (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
- proof(erule trancl_induct, auto)
- fix y th''
- assume y_ch: "(y, Th th'') \<in> child s"
- and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
- and ch': "(Th th, y) \<in> (child s)\<^sup>+"
- from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
- with ih have eq_cpy:"cp s thy = cp s' thy" by blast
- from child_depend_p[OF ch'] and eq_y
- have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by simp
- show "cp s th'' = cp s' th''"
- apply (subst cp_rec[OF vt_s])
- proof -
- have "preced th'' s = preced th'' s'"
- proof(rule eq_preced)
- show "th'' \<noteq> th"
- proof
- assume "th'' = th"
- with dp_thy y_ch[unfolded eq_y]
- have "(Th th, Th th) \<in> (depend s)^+"
- by (auto simp:child_def)
- with wf_trancl[OF wf_depend[OF vt_s]]
- show False by auto
- qed
- qed
- moreover {
- fix th1
- assume th1_in: "th1 \<in> children s th''"
- have "cp s th1 = cp s' th1"
- proof(cases "th1 = thy")
- case True
- with eq_cpy show ?thesis by simp
- next
- case False
- have neq_th1: "th1 \<noteq> th"
- proof
- assume eq_th1: "th1 = th"
- with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
- from children_no_dep[OF vt_s _ _ this] and
- th1_in y_ch eq_y show False by (auto simp:children_def)
- qed
- have "th \<notin> dependents s th1"
- proof
- assume h:"th \<in> dependents s th1"
- from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
- from dependents_child_unique[OF vt_s _ _ h this]
- th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
- with False show False by auto
- qed
- from eq_cp_pre[OF neq_th1 this]
- show ?thesis .
- qed
- }
- ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
- {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
- moreover have "children s th'' = children s' th''"
- by (unfold children_def child_def s_def depend_set_unchanged, simp)
- ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
- by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
- qed
- next
- fix th''
- assume dp': "(Th th, Th th'') \<in> child s"
- show "cp s th'' = cp s' th''"
- apply (subst cp_rec[OF vt_s])
- proof -
- have "preced th'' s = preced th'' s'"
- proof(rule eq_preced)
- show "th'' \<noteq> th"
- proof
- assume "th'' = th"
- with dp dp'
- have "(Th th, Th th) \<in> (depend s)^+"
- by (auto simp:child_def s_dependents_def eq_depend)
- with wf_trancl[OF wf_depend[OF vt_s]]
- show False by auto
- qed
- qed
- moreover {
- fix th1
- assume th1_in: "th1 \<in> children s th''"
- have "cp s th1 = cp s' th1"
- proof(cases "th1 = th")
- case True
- with eq_cps show ?thesis by simp
- next
- case False
- assume neq_th1: "th1 \<noteq> th"
- thus ?thesis
- proof(rule eq_cp_pre)
- show "th \<notin> dependents s th1"
- proof
- assume "th \<in> dependents s th1"
- hence "(Th th, Th th1) \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
- from children_no_dep[OF vt_s _ _ this]
- and th1_in dp' show False
- by (auto simp:children_def)
- qed
- qed
- qed
- }
- ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
- {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
- moreover have "children s th'' = children s' th''"
- by (unfold children_def child_def s_def depend_set_unchanged, simp)
- ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
- by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
- qed
- qed
- }
- ultimately show ?thesis by auto
-qed
-end
-
-lemma next_waiting:
- assumes vt: "vt s"
- and nxt: "next_th s th cs th'"
- shows "waiting s th' cs"
-proof -
- from assms show ?thesis
- apply (auto simp:next_th_def s_waiting_def[folded wq_def])
- proof -
- fix rest
- assume ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq s cs = th # rest"
- and ne: "rest \<noteq> []"
- have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- with ni
- have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set (SOME q. distinct q \<and> set q = set rest)"
- by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- from ne show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> x \<noteq> []" by auto
- qed
- ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
- next
- fix rest
- assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
- and ne: "rest \<noteq> []"
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- from ne show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> x \<noteq> []" by auto
- qed
- hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> set (SOME q. distinct q \<and> set q = set rest)"
- by auto
- moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest" by simp
- with eq_wq and wq_distinct[OF vt, of cs]
- show False by auto
- qed
-qed
-
-
-
-
-locale step_v_cps =
- fixes s' th cs s
- defines s_def : "s \<equiv> (V th cs#s')"
- assumes vt_s: "vt s"
-
-locale step_v_cps_nt = step_v_cps +
- fixes th'
- assumes nt: "next_th s' th cs th'"
-
-context step_v_cps_nt
-begin
-
-lemma depend_s:
- "depend s = (depend s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
- {(Cs cs, Th th')}"
-proof -
- from step_depend_v[OF vt_s[unfolded s_def], folded s_def]
- and nt show ?thesis by (auto intro:next_th_unique)
-qed
-
-lemma dependents_kept:
- fixes th''
- assumes neq1: "th'' \<noteq> th"
- and neq2: "th'' \<noteq> th'"
- shows "dependents (wq s) th'' = dependents (wq s') th''"
-proof(auto)
- fix x
- assume "x \<in> dependents (wq s) th''"
- hence dp: "(Th x, Th th'') \<in> (depend s)^+"
- by (auto simp:cs_dependents_def eq_depend)
- { fix n
- have "(n, Th th'') \<in> (depend s)^+ \<Longrightarrow> (n, Th th'') \<in> (depend s')^+"
- proof(induct rule:converse_trancl_induct)
- fix y
- assume "(y, Th th'') \<in> depend s"
- with depend_s neq1 neq2
- have "(y, Th th'') \<in> depend s'" by auto
- thus "(y, Th th'') \<in> (depend s')\<^sup>+" by auto
- next
- fix y z
- assume yz: "(y, z) \<in> depend s"
- and ztp: "(z, Th th'') \<in> (depend s)\<^sup>+"
- and ztp': "(z, Th th'') \<in> (depend s')\<^sup>+"
- have "y \<noteq> Cs cs \<and> y \<noteq> Th th'"
- proof
- show "y \<noteq> Cs cs"
- proof
- assume eq_y: "y = Cs cs"
- with yz have dp_yz: "(Cs cs, z) \<in> depend s" by simp
- from depend_s
- have cst': "(Cs cs, Th th') \<in> depend s" by simp
- from unique_depend[OF vt_s this dp_yz]
- have eq_z: "z = Th th'" by simp
- with ztp have "(Th th', Th th'') \<in> (depend s)^+" by simp
- from converse_tranclE[OF this]
- obtain cs' where dp'': "(Th th', Cs cs') \<in> depend s"
- by (auto simp:s_depend_def)
- with depend_s have dp': "(Th th', Cs cs') \<in> depend s'" by auto
- from dp'' eq_y yz eq_z have "(Cs cs, Cs cs') \<in> (depend s)^+" by auto
- moreover have "cs' = cs"
- proof -
- from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
- have "(Th th', Cs cs) \<in> depend s'"
- by (auto simp:s_waiting_def wq_def s_depend_def cs_waiting_def)
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this dp']
- show ?thesis by simp
- qed
- ultimately have "(Cs cs, Cs cs) \<in> (depend s)^+" by simp
- moreover note wf_trancl[OF wf_depend[OF vt_s]]
- ultimately show False by auto
- qed
- next
- show "y \<noteq> Th th'"
- proof
- assume eq_y: "y = Th th'"
- with yz have dps: "(Th th', z) \<in> depend s" by simp
- with depend_s have dps': "(Th th', z) \<in> depend s'" by auto
- have "z = Cs cs"
- proof -
- from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
- have "(Th th', Cs cs) \<in> depend s'"
- by (auto simp:s_waiting_def wq_def s_depend_def cs_waiting_def)
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] dps' this]
- show ?thesis .
- qed
- with dps depend_s show False by auto
- qed
- qed
- with depend_s yz have "(y, z) \<in> depend s'" by auto
- with ztp'
- show "(y, Th th'') \<in> (depend s')\<^sup>+" by auto
- qed
- }
- from this[OF dp]
- show "x \<in> dependents (wq s') th''"
- by (auto simp:cs_dependents_def eq_depend)
-next
- fix x
- assume "x \<in> dependents (wq s') th''"
- hence dp: "(Th x, Th th'') \<in> (depend s')^+"
- by (auto simp:cs_dependents_def eq_depend)
- { fix n
- have "(n, Th th'') \<in> (depend s')^+ \<Longrightarrow> (n, Th th'') \<in> (depend s)^+"
- proof(induct rule:converse_trancl_induct)
- fix y
- assume "(y, Th th'') \<in> depend s'"
- with depend_s neq1 neq2
- have "(y, Th th'') \<in> depend s" by auto
- thus "(y, Th th'') \<in> (depend s)\<^sup>+" by auto
- next
- fix y z
- assume yz: "(y, z) \<in> depend s'"
- and ztp: "(z, Th th'') \<in> (depend s')\<^sup>+"
- and ztp': "(z, Th th'') \<in> (depend s)\<^sup>+"
- have "y \<noteq> Cs cs \<and> y \<noteq> Th th'"
- proof
- show "y \<noteq> Cs cs"
- proof
- assume eq_y: "y = Cs cs"
- with yz have dp_yz: "(Cs cs, z) \<in> depend s'" by simp
- from this have eq_z: "z = Th th"
- proof -
- from step_back_step[OF vt_s[unfolded s_def]]
- have "(Cs cs, Th th) \<in> depend s'"
- by(cases, auto simp: wq_def s_depend_def cs_holding_def s_holding_def)
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this dp_yz]
- show ?thesis by simp
- qed
- from converse_tranclE[OF ztp]
- obtain u where "(z, u) \<in> depend s'" by auto
- moreover
- from step_back_step[OF vt_s[unfolded s_def]]
- have "th \<in> readys s'" by (cases, simp add:runing_def)
- moreover note eq_z
- ultimately show False
- by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def)
- qed
- next
- show "y \<noteq> Th th'"
- proof
- assume eq_y: "y = Th th'"
- with yz have dps: "(Th th', z) \<in> depend s'" by simp
- have "z = Cs cs"
- proof -
- from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
- have "(Th th', Cs cs) \<in> depend s'"
- by (auto simp:s_waiting_def wq_def s_depend_def cs_waiting_def)
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] dps this]
- show ?thesis .
- qed
- with ztp have cs_i: "(Cs cs, Th th'') \<in> (depend s')\<^sup>+" by simp
- from step_back_step[OF vt_s[unfolded s_def]]
- have cs_th: "(Cs cs, Th th) \<in> depend s'"
- by(cases, auto simp: s_depend_def wq_def cs_holding_def s_holding_def)
- have "(Cs cs, Th th'') \<notin> depend s'"
- proof
- assume "(Cs cs, Th th'') \<in> depend s'"
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this cs_th]
- and neq1 show "False" by simp
- qed
- with converse_tranclE[OF cs_i]
- obtain u where cu: "(Cs cs, u) \<in> depend s'"
- and u_t: "(u, Th th'') \<in> (depend s')\<^sup>+" by auto
- have "u = Th th"
- proof -
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] cu cs_th]
- show ?thesis .
- qed
- with u_t have "(Th th, Th th'') \<in> (depend s')\<^sup>+" by simp
- from converse_tranclE[OF this]
- obtain v where "(Th th, v) \<in> (depend s')" by auto
- moreover from step_back_step[OF vt_s[unfolded s_def]]
- have "th \<in> readys s'" by (cases, simp add:runing_def)
- ultimately show False
- by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def)
- qed
- qed
- with depend_s yz have "(y, z) \<in> depend s" by auto
- with ztp'
- show "(y, Th th'') \<in> (depend s)\<^sup>+" by auto
- qed
- }
- from this[OF dp]
- show "x \<in> dependents (wq s) th''"
- by (auto simp:cs_dependents_def eq_depend)
-qed
-
-lemma cp_kept:
- fixes th''
- assumes neq1: "th'' \<noteq> th"
- and neq2: "th'' \<noteq> th'"
- shows "cp s th'' = cp s' th''"
-proof -
- from dependents_kept[OF neq1 neq2]
- have "dependents (wq s) th'' = dependents (wq s') th''" .
- moreover {
- fix th1
- assume "th1 \<in> dependents (wq s) th''"
- have "preced th1 s = preced th1 s'"
- by (unfold s_def, auto simp:preced_def)
- }
- moreover have "preced th'' s = preced th'' s'"
- by (unfold s_def, auto simp:preced_def)
- ultimately have "((\<lambda>th. preced th s) ` ({th''} \<union> dependents (wq s) th'')) =
- ((\<lambda>th. preced th s') ` ({th''} \<union> dependents (wq s') th''))"
- by (auto simp:image_def)
- thus ?thesis
- by (unfold cp_eq_cpreced cpreced_def, simp)
-qed
-
-end
-
-locale step_v_cps_nnt = step_v_cps +
- assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
-
-context step_v_cps_nnt
-begin
-
-lemma nw_cs: "(Th th1, Cs cs) \<notin> depend s'"
-proof
- assume "(Th th1, Cs cs) \<in> depend s'"
- thus "False"
- apply (auto simp:s_depend_def cs_waiting_def)
- proof -
- assume h1: "th1 \<in> set (wq s' cs)"
- and h2: "th1 \<noteq> hd (wq s' cs)"
- from step_back_step[OF vt_s[unfolded s_def]]
- show "False"
- proof(cases)
- assume "holding s' th cs"
- then obtain rest where
- eq_wq: "wq s' cs = th#rest"
- apply (unfold s_holding_def wq_def[symmetric])
- by (case_tac "(wq s' cs)", auto)
- with h1 h2 have ne: "rest \<noteq> []" by auto
- with eq_wq
- have "next_th s' th cs (hd (SOME q. distinct q \<and> set q = set rest))"
- by(unfold next_th_def, rule_tac x = "rest" in exI, auto)
- with nnt show ?thesis by auto
- qed
- qed
-qed
-
-lemma depend_s: "depend s = depend s' - {(Cs cs, Th th)}"
-proof -
- from nnt and step_depend_v[OF vt_s[unfolded s_def], folded s_def]
- show ?thesis by auto
-qed
-
-lemma child_kept_left:
- assumes
- "(n1, n2) \<in> (child s')^+"
- shows "(n1, n2) \<in> (child s)^+"
-proof -
- from assms show ?thesis
- proof(induct rule: converse_trancl_induct)
- case (base y)
- from base obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s'"
- and h2: "(Cs cs1, Th th2) \<in> depend s'"
- and eq_y: "y = Th th1" and eq_n2: "n2 = Th th2" by (auto simp:child_def)
- have "cs1 \<noteq> cs"
- proof
- assume eq_cs: "cs1 = cs"
- with h1 have "(Th th1, Cs cs1) \<in> depend s'" by simp
- with nw_cs eq_cs show False by auto
- qed
- with h1 h2 depend_s have
- h1': "(Th th1, Cs cs1) \<in> depend s" and
- h2': "(Cs cs1, Th th2) \<in> depend s" by auto
- hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
- with eq_y eq_n2 have "(y, n2) \<in> child s" by simp
- thus ?case by auto
- next
- case (step y z)
- have "(y, z) \<in> child s'" by fact
- then obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s'"
- and h2: "(Cs cs1, Th th2) \<in> depend s'"
- and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def)
- have "cs1 \<noteq> cs"
- proof
- assume eq_cs: "cs1 = cs"
- with h1 have "(Th th1, Cs cs1) \<in> depend s'" by simp
- with nw_cs eq_cs show False by auto
- qed
- with h1 h2 depend_s have
- h1': "(Th th1, Cs cs1) \<in> depend s" and
- h2': "(Cs cs1, Th th2) \<in> depend s" by auto
- hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
- with eq_y eq_z have "(y, z) \<in> child s" by simp
- moreover have "(z, n2) \<in> (child s)^+" by fact
- ultimately show ?case by auto
- qed
-qed
-
-lemma child_kept_right:
- assumes
- "(n1, n2) \<in> (child s)^+"
- shows "(n1, n2) \<in> (child s')^+"
-proof -
- from assms show ?thesis
- proof(induct)
- case (base y)
- from base and depend_s
- have "(n1, y) \<in> child s'"
- by (auto simp:child_def)
- thus ?case by auto
- next
- case (step y z)
- have "(y, z) \<in> child s" by fact
- with depend_s have "(y, z) \<in> child s'"
- by (auto simp:child_def)
- moreover have "(n1, y) \<in> (child s')\<^sup>+" by fact
- ultimately show ?case by auto
- qed
-qed
-
-lemma eq_child: "(child s)^+ = (child s')^+"
- by (insert child_kept_left child_kept_right, auto)
-
-lemma eq_cp:
- fixes th'
- shows "cp s th' = cp s' th'"
- apply (unfold cp_eq_cpreced cpreced_def)
-proof -
- have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
- apply (unfold cs_dependents_def, unfold eq_depend)
- proof -
- from eq_child
- have "\<And>th. {th'. (Th th', Th th) \<in> (child s)\<^sup>+} = {th'. (Th th', Th th) \<in> (child s')\<^sup>+}"
- by simp
- with child_depend_eq[OF vt_s] child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
- show "\<And>th. {th'. (Th th', Th th) \<in> (depend s)\<^sup>+} = {th'. (Th th', Th th) \<in> (depend s')\<^sup>+}"
- by simp
- qed
- moreover {
- fix th1
- assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
- hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
- hence "preced th1 s = preced th1 s'"
- proof
- assume "th1 = th'"
- show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
- next
- assume "th1 \<in> dependents (wq s') th'"
- show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
- qed
- } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
- by (auto simp:image_def)
- thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
-qed
-
-end
-
-locale step_P_cps =
- fixes s' th cs s
- defines s_def : "s \<equiv> (P th cs#s')"
- assumes vt_s: "vt s"
-
-locale step_P_cps_ne =step_P_cps +
- assumes ne: "wq s' cs \<noteq> []"
-
-locale step_P_cps_e =step_P_cps +
- assumes ee: "wq s' cs = []"
-
-context step_P_cps_e
-begin
-
-lemma depend_s: "depend s = depend s' \<union> {(Cs cs, Th th)}"
-proof -
- from ee and step_depend_p[OF vt_s[unfolded s_def], folded s_def]
- show ?thesis by auto
-qed
-
-lemma child_kept_left:
- assumes
- "(n1, n2) \<in> (child s')^+"
- shows "(n1, n2) \<in> (child s)^+"
-proof -
- from assms show ?thesis
- proof(induct rule: converse_trancl_induct)
- case (base y)
- from base obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s'"
- and h2: "(Cs cs1, Th th2) \<in> depend s'"
- and eq_y: "y = Th th1" and eq_n2: "n2 = Th th2" by (auto simp:child_def)
- have "cs1 \<noteq> cs"
- proof
- assume eq_cs: "cs1 = cs"
- with h1 have "(Th th1, Cs cs) \<in> depend s'" by simp
- with ee show False
- by (auto simp:s_depend_def cs_waiting_def)
- qed
- with h1 h2 depend_s have
- h1': "(Th th1, Cs cs1) \<in> depend s" and
- h2': "(Cs cs1, Th th2) \<in> depend s" by auto
- hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
- with eq_y eq_n2 have "(y, n2) \<in> child s" by simp
- thus ?case by auto
- next
- case (step y z)
- have "(y, z) \<in> child s'" by fact
- then obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s'"
- and h2: "(Cs cs1, Th th2) \<in> depend s'"
- and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def)
- have "cs1 \<noteq> cs"
- proof
- assume eq_cs: "cs1 = cs"
- with h1 have "(Th th1, Cs cs) \<in> depend s'" by simp
- with ee show False
- by (auto simp:s_depend_def cs_waiting_def)
- qed
- with h1 h2 depend_s have
- h1': "(Th th1, Cs cs1) \<in> depend s" and
- h2': "(Cs cs1, Th th2) \<in> depend s" by auto
- hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
- with eq_y eq_z have "(y, z) \<in> child s" by simp
- moreover have "(z, n2) \<in> (child s)^+" by fact
- ultimately show ?case by auto
- qed
-qed
-
-lemma child_kept_right:
- assumes
- "(n1, n2) \<in> (child s)^+"
- shows "(n1, n2) \<in> (child s')^+"
-proof -
- from assms show ?thesis
- proof(induct)
- case (base y)
- from base and depend_s
- have "(n1, y) \<in> child s'"
- apply (auto simp:child_def)
- proof -
- fix th'
- assume "(Th th', Cs cs) \<in> depend s'"
- with ee have "False"
- by (auto simp:s_depend_def cs_waiting_def)
- thus "\<exists>cs. (Th th', Cs cs) \<in> depend s' \<and> (Cs cs, Th th) \<in> depend s'" by auto
- qed
- thus ?case by auto
- next
- case (step y z)
- have "(y, z) \<in> child s" by fact
- with depend_s have "(y, z) \<in> child s'"
- apply (auto simp:child_def)
- proof -
- fix th'
- assume "(Th th', Cs cs) \<in> depend s'"
- with ee have "False"
- by (auto simp:s_depend_def cs_waiting_def)
- thus "\<exists>cs. (Th th', Cs cs) \<in> depend s' \<and> (Cs cs, Th th) \<in> depend s'" by auto
- qed
- moreover have "(n1, y) \<in> (child s')\<^sup>+" by fact
- ultimately show ?case by auto
- qed
-qed
-
-lemma eq_child: "(child s)^+ = (child s')^+"
- by (insert child_kept_left child_kept_right, auto)
-
-lemma eq_cp:
- fixes th'
- shows "cp s th' = cp s' th'"
- apply (unfold cp_eq_cpreced cpreced_def)
-proof -
- have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
- apply (unfold cs_dependents_def, unfold eq_depend)
- proof -
- from eq_child
- have "\<And>th. {th'. (Th th', Th th) \<in> (child s)\<^sup>+} = {th'. (Th th', Th th) \<in> (child s')\<^sup>+}"
- by auto
- with child_depend_eq[OF vt_s] child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
- show "\<And>th. {th'. (Th th', Th th) \<in> (depend s)\<^sup>+} = {th'. (Th th', Th th) \<in> (depend s')\<^sup>+}"
- by simp
- qed
- moreover {
- fix th1
- assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
- hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
- hence "preced th1 s = preced th1 s'"
- proof
- assume "th1 = th'"
- show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
- next
- assume "th1 \<in> dependents (wq s') th'"
- show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
- qed
- } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
- by (auto simp:image_def)
- thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
-qed
-
-end
-
-context step_P_cps_ne
-begin
-
-lemma depend_s: "depend s = depend s' \<union> {(Th th, Cs cs)}"
-proof -
- from step_depend_p[OF vt_s[unfolded s_def]] and ne
- show ?thesis by (simp add:s_def)
-qed
-
-lemma eq_child_left:
- assumes nd: "(Th th, Th th') \<notin> (child s)^+"
- shows "(n1, Th th') \<in> (child s)^+ \<Longrightarrow> (n1, Th th') \<in> (child s')^+"
-proof(induct rule:converse_trancl_induct)
- case (base y)
- from base obtain th1 cs1
- where h1: "(Th th1, Cs cs1) \<in> depend s"
- and h2: "(Cs cs1, Th th') \<in> depend s"
- and eq_y: "y = Th th1" by (auto simp:child_def)
- have "th1 \<noteq> th"
- proof
- assume "th1 = th"
- with base eq_y have "(Th th, Th th') \<in> child s" by simp
- with nd show False by auto
- qed
- with h1 h2 depend_s
- have h1': "(Th th1, Cs cs1) \<in> depend s'" and
- h2': "(Cs cs1, Th th') \<in> depend s'" by auto
- with eq_y show ?case by (auto simp:child_def)
-next
- case (step y z)
- have yz: "(y, z) \<in> child s" by fact
- then obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s"
- and h2: "(Cs cs1, Th th2) \<in> depend s"
- and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def)
- have "th1 \<noteq> th"
- proof
- assume "th1 = th"
- with yz eq_y have "(Th th, z) \<in> child s" by simp
- moreover have "(z, Th th') \<in> (child s)^+" by fact
- ultimately have "(Th th, Th th') \<in> (child s)^+" by auto
- with nd show False by auto
- qed
- with h1 h2 depend_s have h1': "(Th th1, Cs cs1) \<in> depend s'"
- and h2': "(Cs cs1, Th th2) \<in> depend s'" by auto
- with eq_y eq_z have "(y, z) \<in> child s'" by (auto simp:child_def)
- moreover have "(z, Th th') \<in> (child s')^+" by fact
- ultimately show ?case by auto
-qed
-
-lemma eq_child_right:
- shows "(n1, Th th') \<in> (child s')^+ \<Longrightarrow> (n1, Th th') \<in> (child s)^+"
-proof(induct rule:converse_trancl_induct)
- case (base y)
- with depend_s show ?case by (auto simp:child_def)
-next
- case (step y z)
- have "(y, z) \<in> child s'" by fact
- with depend_s have "(y, z) \<in> child s" by (auto simp:child_def)
- moreover have "(z, Th th') \<in> (child s)^+" by fact
- ultimately show ?case by auto
-qed
-
-lemma eq_child:
- assumes nd: "(Th th, Th th') \<notin> (child s)^+"
- shows "((n1, Th th') \<in> (child s)^+) = ((n1, Th th') \<in> (child s')^+)"
- by (insert eq_child_left[OF nd] eq_child_right, auto)
-
-lemma eq_cp:
- fixes th'
- assumes nd: "th \<notin> dependents s th'"
- shows "cp s th' = cp s' th'"
- apply (unfold cp_eq_cpreced cpreced_def)
-proof -
- have nd': "(Th th, Th th') \<notin> (child s)^+"
- proof
- assume "(Th th, Th th') \<in> (child s)\<^sup>+"
- with child_depend_eq[OF vt_s]
- have "(Th th, Th th') \<in> (depend s)\<^sup>+" by simp
- with nd show False
- by (simp add:s_dependents_def eq_depend)
- qed
- have eq_dp: "dependents (wq s) th' = dependents (wq s') th'"
- proof(auto)
- fix x assume " x \<in> dependents (wq s) th'"
- thus "x \<in> dependents (wq s') th'"
- apply (auto simp:cs_dependents_def eq_depend)
- proof -
- assume "(Th x, Th th') \<in> (depend s)\<^sup>+"
- with child_depend_eq[OF vt_s] have "(Th x, Th th') \<in> (child s)\<^sup>+" by simp
- with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s')^+" by simp
- with child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
- show "(Th x, Th th') \<in> (depend s')\<^sup>+" by simp
- qed
- next
- fix x assume "x \<in> dependents (wq s') th'"
- thus "x \<in> dependents (wq s) th'"
- apply (auto simp:cs_dependents_def eq_depend)
- proof -
- assume "(Th x, Th th') \<in> (depend s')\<^sup>+"
- with child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
- have "(Th x, Th th') \<in> (child s')\<^sup>+" by simp
- with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s)^+" by simp
- with child_depend_eq[OF vt_s]
- show "(Th x, Th th') \<in> (depend s)\<^sup>+" by simp
- qed
- qed
- moreover {
- fix th1 have "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
- } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
- by (auto simp:image_def)
- thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
-qed
-
-lemma eq_up:
- fixes th' th''
- assumes dp1: "th \<in> dependents s th'"
- and dp2: "th' \<in> dependents s th''"
- and eq_cps: "cp s th' = cp s' th'"
- shows "cp s th'' = cp s' th''"
-proof -
- from dp2
- have "(Th th', Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
- from depend_child[OF vt_s this[unfolded eq_depend]]
- have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" .
- moreover {
- fix n th''
- have "\<lbrakk>(Th th', n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
- (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
- proof(erule trancl_induct, auto)
- fix y th''
- assume y_ch: "(y, Th th'') \<in> child s"
- and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
- and ch': "(Th th', y) \<in> (child s)\<^sup>+"
- from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
- with ih have eq_cpy:"cp s thy = cp s' thy" by blast
- from dp1 have "(Th th, Th th') \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
- moreover from child_depend_p[OF ch'] and eq_y
- have "(Th th', Th thy) \<in> (depend s)^+" by simp
- ultimately have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by auto
- show "cp s th'' = cp s' th''"
- apply (subst cp_rec[OF vt_s])
- proof -
- have "preced th'' s = preced th'' s'"
- by (simp add:s_def preced_def)
- moreover {
- fix th1
- assume th1_in: "th1 \<in> children s th''"
- have "cp s th1 = cp s' th1"
- proof(cases "th1 = thy")
- case True
- with eq_cpy show ?thesis by simp
- next
- case False
- have neq_th1: "th1 \<noteq> th"
- proof
- assume eq_th1: "th1 = th"
- with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
- from children_no_dep[OF vt_s _ _ this] and
- th1_in y_ch eq_y show False by (auto simp:children_def)
- qed
- have "th \<notin> dependents s th1"
- proof
- assume h:"th \<in> dependents s th1"
- from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
- from dependents_child_unique[OF vt_s _ _ h this]
- th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
- with False show False by auto
- qed
- from eq_cp[OF this]
- show ?thesis .
- qed
- }
- ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
- {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
- moreover have "children s th'' = children s' th''"
- apply (unfold children_def child_def s_def depend_set_unchanged, simp)
- apply (fold s_def, auto simp:depend_s)
- proof -
- assume "(Cs cs, Th th'') \<in> depend s'"
- with depend_s have cs_th': "(Cs cs, Th th'') \<in> depend s" by auto
- from dp1 have "(Th th, Th th') \<in> (depend s)^+"
- by (auto simp:s_dependents_def eq_depend)
- from converse_tranclE[OF this]
- obtain cs1 where h1: "(Th th, Cs cs1) \<in> depend s"
- and h2: "(Cs cs1 , Th th') \<in> (depend s)\<^sup>+"
- by (auto simp:s_depend_def)
- have eq_cs: "cs1 = cs"
- proof -
- from depend_s have "(Th th, Cs cs) \<in> depend s" by simp
- from unique_depend[OF vt_s this h1]
- show ?thesis by simp
- qed
- have False
- proof(rule converse_tranclE[OF h2])
- assume "(Cs cs1, Th th') \<in> depend s"
- with eq_cs have "(Cs cs, Th th') \<in> depend s" by simp
- from unique_depend[OF vt_s this cs_th']
- have "th' = th''" by simp
- with ch' y_ch have "(Th th'', Th th'') \<in> (child s)^+" by auto
- with wf_trancl[OF wf_child[OF vt_s]]
- show False by auto
- next
- fix y
- assume "(Cs cs1, y) \<in> depend s"
- and ytd: " (y, Th th') \<in> (depend s)\<^sup>+"
- with eq_cs have csy: "(Cs cs, y) \<in> depend s" by simp
- from unique_depend[OF vt_s this cs_th']
- have "y = Th th''" .
- with ytd have "(Th th'', Th th') \<in> (depend s)^+" by simp
- from depend_child[OF vt_s this]
- have "(Th th'', Th th') \<in> (child s)\<^sup>+" .
- moreover from ch' y_ch have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto
- ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto
- with wf_trancl[OF wf_child[OF vt_s]]
- show False by auto
- qed
- thus "\<exists>cs. (Th th, Cs cs) \<in> depend s' \<and> (Cs cs, Th th'') \<in> depend s'" by auto
- qed
- ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
- by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
- qed
- next
- fix th''
- assume dp': "(Th th', Th th'') \<in> child s"
- show "cp s th'' = cp s' th''"
- apply (subst cp_rec[OF vt_s])
- proof -
- have "preced th'' s = preced th'' s'"
- by (simp add:s_def preced_def)
- moreover {
- fix th1
- assume th1_in: "th1 \<in> children s th''"
- have "cp s th1 = cp s' th1"
- proof(cases "th1 = th'")
- case True
- with eq_cps show ?thesis by simp
- next
- case False
- have neq_th1: "th1 \<noteq> th"
- proof
- assume eq_th1: "th1 = th"
- with dp1 have "(Th th1, Th th') \<in> (depend s)^+"
- by (auto simp:s_dependents_def eq_depend)
- from children_no_dep[OF vt_s _ _ this]
- th1_in dp'
- show False by (auto simp:children_def)
- qed
- show ?thesis
- proof(rule eq_cp)
- show "th \<notin> dependents s th1"
- proof
- assume "th \<in> dependents s th1"
- from dependents_child_unique[OF vt_s _ _ this dp1]
- th1_in dp' have "th1 = th'"
- by (auto simp:children_def)
- with False show False by auto
- qed
- qed
- qed
- }
- ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
- {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
- moreover have "children s th'' = children s' th''"
- apply (unfold children_def child_def s_def depend_set_unchanged, simp)
- apply (fold s_def, auto simp:depend_s)
- proof -
- assume "(Cs cs, Th th'') \<in> depend s'"
- with depend_s have cs_th': "(Cs cs, Th th'') \<in> depend s" by auto
- from dp1 have "(Th th, Th th') \<in> (depend s)^+"
- by (auto simp:s_dependents_def eq_depend)
- from converse_tranclE[OF this]
- obtain cs1 where h1: "(Th th, Cs cs1) \<in> depend s"
- and h2: "(Cs cs1 , Th th') \<in> (depend s)\<^sup>+"
- by (auto simp:s_depend_def)
- have eq_cs: "cs1 = cs"
- proof -
- from depend_s have "(Th th, Cs cs) \<in> depend s" by simp
- from unique_depend[OF vt_s this h1]
- show ?thesis by simp
- qed
- have False
- proof(rule converse_tranclE[OF h2])
- assume "(Cs cs1, Th th') \<in> depend s"
- with eq_cs have "(Cs cs, Th th') \<in> depend s" by simp
- from unique_depend[OF vt_s this cs_th']
- have "th' = th''" by simp
- with dp' have "(Th th'', Th th'') \<in> (child s)^+" by auto
- with wf_trancl[OF wf_child[OF vt_s]]
- show False by auto
- next
- fix y
- assume "(Cs cs1, y) \<in> depend s"
- and ytd: " (y, Th th') \<in> (depend s)\<^sup>+"
- with eq_cs have csy: "(Cs cs, y) \<in> depend s" by simp
- from unique_depend[OF vt_s this cs_th']
- have "y = Th th''" .
- with ytd have "(Th th'', Th th') \<in> (depend s)^+" by simp
- from depend_child[OF vt_s this]
- have "(Th th'', Th th') \<in> (child s)\<^sup>+" .
- moreover from dp' have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto
- ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto
- with wf_trancl[OF wf_child[OF vt_s]]
- show False by auto
- qed
- thus "\<exists>cs. (Th th, Cs cs) \<in> depend s' \<and> (Cs cs, Th th'') \<in> depend s'" by auto
- qed
- ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
- by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
- qed
- qed
- }
- ultimately show ?thesis by auto
-qed
-
-end
-
-locale step_create_cps =
- fixes s' th prio s
- defines s_def : "s \<equiv> (Create th prio#s')"
- assumes vt_s: "vt s"
-
-context step_create_cps
-begin
-
-lemma eq_dep: "depend s = depend s'"
- by (unfold s_def depend_create_unchanged, auto)
-
-lemma eq_cp:
- fixes th'
- assumes neq_th: "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
- apply (unfold cp_eq_cpreced cpreced_def)
-proof -
- have nd: "th \<notin> dependents s th'"
- proof
- assume "th \<in> dependents s th'"
- hence "(Th th, Th th') \<in> (depend s)^+" by (simp add:s_dependents_def eq_depend)
- with eq_dep have "(Th th, Th th') \<in> (depend s')^+" by simp
- from converse_tranclE[OF this]
- obtain y where "(Th th, y) \<in> depend s'" by auto
- with dm_depend_threads[OF step_back_vt[OF vt_s[unfolded s_def]]]
- have in_th: "th \<in> threads s'" by auto
- from step_back_step[OF vt_s[unfolded s_def]]
- show False
- proof(cases)
- assume "th \<notin> threads s'"
- with in_th show ?thesis by simp
- qed
- qed
- have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
- by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
- moreover {
- fix th1
- assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
- hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
- hence "preced th1 s = preced th1 s'"
- proof
- assume "th1 = th'"
- with neq_th
- show "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
- next
- assume "th1 \<in> dependents (wq s') th'"
- with nd and eq_dp have "th1 \<noteq> th"
- by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
- thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
- qed
- } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
- by (auto simp:image_def)
- thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
-qed
-
-lemma nil_dependents: "dependents s th = {}"
-proof -
- from step_back_step[OF vt_s[unfolded s_def]]
- show ?thesis
- proof(cases)
- assume "th \<notin> threads s'"
- from not_thread_holdents[OF step_back_vt[OF vt_s[unfolded s_def]] this]
- have hdn: " holdents s' th = {}" .
- have "dependents s' th = {}"
- proof -
- { assume "dependents s' th \<noteq> {}"
- then obtain th' where dp: "(Th th', Th th) \<in> (depend s')^+"
- by (auto simp:s_dependents_def eq_depend)
- from tranclE[OF this] obtain cs' where
- "(Cs cs', Th th) \<in> depend s'" by (auto simp:s_depend_def)
- with hdn
- have False by (auto simp:holdents_test)
- } thus ?thesis by auto
- qed
- thus ?thesis
- by (unfold s_def s_dependents_def eq_depend depend_create_unchanged, simp)
- qed
-qed
-
-lemma eq_cp_th: "cp s th = preced th s"
- apply (unfold cp_eq_cpreced cpreced_def)
- by (insert nil_dependents, unfold s_dependents_def cs_dependents_def, auto)
-
-end
-
-
-locale step_exit_cps =
- fixes s' th prio s
- defines s_def : "s \<equiv> (Exit th#s')"
- assumes vt_s: "vt s"
-
-context step_exit_cps
-begin
-
-lemma eq_dep: "depend s = depend s'"
- by (unfold s_def depend_exit_unchanged, auto)
-
-lemma eq_cp:
- fixes th'
- assumes neq_th: "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
- apply (unfold cp_eq_cpreced cpreced_def)
-proof -
- have nd: "th \<notin> dependents s th'"
- proof
- assume "th \<in> dependents s th'"
- hence "(Th th, Th th') \<in> (depend s)^+" by (simp add:s_dependents_def eq_depend)
- with eq_dep have "(Th th, Th th') \<in> (depend s')^+" by simp
- from converse_tranclE[OF this]
- obtain cs' where bk: "(Th th, Cs cs') \<in> depend s'"
- by (auto simp:s_depend_def)
- from step_back_step[OF vt_s[unfolded s_def]]
- show False
- proof(cases)
- assume "th \<in> runing s'"
- with bk show ?thesis
- apply (unfold runing_def readys_def s_waiting_def s_depend_def)
- by (auto simp:cs_waiting_def wq_def)
- qed
- qed
- have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
- by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
- moreover {
- fix th1
- assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
- hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
- hence "preced th1 s = preced th1 s'"
- proof
- assume "th1 = th'"
- with neq_th
- show "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
- next
- assume "th1 \<in> dependents (wq s') th'"
- with nd and eq_dp have "th1 \<noteq> th"
- by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
- thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
- qed
- } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
- by (auto simp:image_def)
- thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
- Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
-qed
-
-end
-end
-
--- a/prio/ExtGG.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1046 +0,0 @@
-theory ExtGG
-imports PrioG
-begin
-
-lemma birth_time_lt: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
- apply (induct s, simp)
-proof -
- fix a s
- assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
- and eq_as: "a # s \<noteq> []"
- show "birthtime th (a # s) < length (a # s)"
- proof(cases "s \<noteq> []")
- case False
- from False show ?thesis
- by (cases a, auto simp:birthtime.simps)
- next
- case True
- from ih [OF True] show ?thesis
- by (cases a, auto simp:birthtime.simps)
- qed
-qed
-
-lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
- by (induct s, auto simp:threads.simps)
-
-lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
- apply (drule_tac th_in_ne)
- by (unfold preced_def, auto intro: birth_time_lt)
-
-locale highest_gen =
- fixes s th prio tm
- assumes vt_s: "vt s"
- and threads_s: "th \<in> threads s"
- and highest: "preced th s = Max ((cp s)`threads s)"
- and preced_th: "preced th s = Prc prio tm"
-
-context highest_gen
-begin
-
-
-
-lemma lt_tm: "tm < length s"
- by (insert preced_tm_lt[OF threads_s preced_th], simp)
-
-lemma eq_cp_s_th: "cp s th = preced th s"
-proof -
- from highest and max_cp_eq[OF vt_s]
- have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
- have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
- proof -
- from threads_s and dependents_threads[OF vt_s, of th]
- show ?thesis by auto
- qed
- show ?thesis
- proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
- show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
- next
- fix y
- assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
- then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
- and eq_y: "y = preced th1 s" by auto
- show "y \<le> preced th s"
- proof(unfold is_max, rule Max_ge)
- from finite_threads[OF vt_s]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- from sbs th1_in and eq_y
- show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
- qed
- next
- from sbs and finite_threads[OF vt_s]
- show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
- by (auto intro:finite_subset)
- qed
-qed
-
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
-proof -
- from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
- show ?thesis by simp
-qed
-
-end
-
-locale extend_highest_gen = highest_gen +
- fixes t
- assumes vt_t: "vt (t@s)"
- and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
- and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
- and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-lemma step_back_vt_app:
- assumes vt_ts: "vt (t@s)"
- shows "vt s"
-proof -
- from vt_ts show ?thesis
- proof(induct t)
- case Nil
- from Nil show ?case by auto
- next
- case (Cons e t)
- assume ih: " vt (t @ s) \<Longrightarrow> vt s"
- and vt_et: "vt ((e # t) @ s)"
- show ?case
- proof(rule ih)
- show "vt (t @ s)"
- proof(rule step_back_vt)
- from vt_et show "vt (e # t @ s)" by simp
- qed
- qed
- qed
-qed
-
-context extend_highest_gen
-begin
-
-thm extend_highest_gen_axioms_def
-
-lemma red_moment:
- "extend_highest_gen s th prio tm (moment i t)"
- apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
- apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
- by (unfold highest_gen_def, auto dest:step_back_vt_app)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes
- h0: "R []"
- and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e;
- extend_highest_gen s th prio tm t;
- extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
- shows "R t"
-proof -
- from vt_t extend_highest_gen_axioms show ?thesis
- proof(induct t)
- from h0 show "R []" .
- next
- case (Cons e t')
- assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
- and vt_e: "vt ((e # t') @ s)"
- and et: "extend_highest_gen s th prio tm (e # t')"
- from vt_e and step_back_step have stp: "step (t'@s) e" by auto
- from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
- show ?case
- proof(rule h2 [OF vt_ts stp _ _ _ ])
- show "R t'"
- proof(rule ih)
- from et show ext': "extend_highest_gen s th prio tm t'"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
- next
- from vt_ts show "vt (t' @ s)" .
- qed
- next
- from et show "extend_highest_gen s th prio tm (e # t')" .
- next
- from et show ext': "extend_highest_gen s th prio tm t'"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
- qed
- qed
-qed
-
-lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
-proof -
- show ?thesis
- proof(induct rule:ind)
- case Nil
- from threads_s
- show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
- by auto
- next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio)
- assume eq_e: " e = Create thread prio"
- show ?thesis
- proof -
- from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
- hence "th \<noteq> thread"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- with Cons show ?thesis by auto
- qed
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
- from extend_highest_gen.exit_diff [OF this] and eq_e
- have neq_th: "thread \<noteq> th" by auto
- with Cons
- show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- next
- case (P thread cs)
- assume eq_e: "e = P thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (V thread cs)
- assume eq_e: "e = V thread cs"
- with Cons
- show ?thesis
- by (auto simp:eq_e preced_def)
- next
- case (Set thread prio')
- assume eq_e: " e = Set thread prio'"
- show ?thesis
- proof -
- from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
- from extend_highest_gen.set_diff_low[OF this] and eq_e
- have "th \<noteq> thread" by auto
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:eq_e)
- qed
- qed
- qed
-qed
-
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
- case Nil
- from highest_preced_thread
- show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
- by simp
-next
- case (Cons e t)
- show ?case
- proof(cases e)
- case (Create thread prio')
- assume eq_e: " e = Create thread prio'"
- from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
- hence neq_thread: "thread \<noteq> th"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th \<in> threads (t@s)"
- proof -
- from Cons have "extend_highest_gen s th prio tm t" by auto
- from extend_highest_gen.th_kept[OF this] show ?thesis by (simp)
- qed
- ultimately show ?thesis by auto
- qed
- from Cons have "extend_highest_gen s th prio tm t" by auto
- from extend_highest_gen.th_kept[OF this]
- have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
- by (auto)
- from stp
- have thread_ts: "thread \<notin> threads (t @ s)"
- by (cases, auto)
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
- by (unfold eq_e, simp)
- moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
- proof(rule Max_insert)
- from Cons have "vt (t @ s)" by auto
- from finite_threads[OF this]
- show "finite (?f ` (threads (t@s)))" by simp
- next
- from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
- qed
- moreover have "(Max (?f ` (threads (t@s)))) = ?t"
- proof -
- have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
- (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
- proof -
- { fix th'
- assume "th' \<in> ?B"
- with thread_ts eq_e
- have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
- } thus ?thesis
- apply (auto simp:Image_def)
- proof -
- fix th'
- assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
- preced th' (e # t @ s) = preced th' (t @ s)"
- and h1: "th' \<in> threads (t @ s)"
- show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
- proof -
- from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
- moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
- ultimately show ?thesis by simp
- qed
- qed
- qed
- with Cons show ?thesis by auto
- qed
- moreover have "?f thread < ?t"
- proof -
- from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
- from extend_highest_gen.create_low[OF this] and eq_e
- have "prio' \<le> prio" by auto
- thus ?thesis
- by (unfold preced_th, unfold eq_e, insert lt_tm,
- auto simp:preced_def precedence_less_def preced_th)
- qed
- ultimately show ?thesis by (auto simp:max_def)
- qed
-next
- case (Exit thread)
- assume eq_e: "e = Exit thread"
- from Cons have vt_e: "vt (e#(t @ s))" by auto
- from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
- from stp have thread_ts: "thread \<in> threads (t @ s)"
- by(cases, unfold runing_def readys_def, auto)
- from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
- from extend_highest_gen.exit_diff[OF this] and eq_e
- have neq_thread: "thread \<noteq> th" by auto
- from Cons have "extend_highest_gen s th prio tm t" by auto
- from extend_highest_gen.th_kept[OF this]
- have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "threads (t@s) = insert thread ?A"
- by (insert stp thread_ts, unfold eq_e, auto)
- hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
- also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
- also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
- proof(rule Max_insert)
- from finite_threads [OF vt_e]
- show "finite (?f ` ?A)" by simp
- next
- from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
- from extend_highest_gen.th_kept[OF this]
- show "?f ` ?A \<noteq> {}" by auto
- qed
- finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
- moreover have "Max (?f ` (threads (t@s))) = ?t"
- proof -
- from Cons show ?thesis
- by (unfold eq_e, auto simp:preced_def)
- qed
- ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
- moreover have "?f thread < ?t"
- proof(unfold eq_e, simp add:preced_def, fold preced_def)
- show "preced thread (t @ s) < ?t"
- proof -
- have "preced thread (t @ s) \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "?t = Max (?g ` ?B)") by simp
- moreover have "?g thread \<le> \<dots>"
- proof(rule Max_ge)
- have "vt (t@s)" by fact
- from finite_threads [OF this]
- show "finite (?g ` ?B)" by simp
- next
- from thread_ts
- show "?g thread \<in> (?g ` ?B)" by auto
- qed
- ultimately show ?thesis by auto
- qed
- moreover have "preced thread (t @ s) \<noteq> ?t"
- proof
- assume "preced thread (t @ s) = preced th s"
- with h' have "preced thread (t @ s) = preced th (t@s)" by simp
- from preced_unique [OF this] have "thread = th"
- proof
- from h' show "th \<in> threads (t @ s)" by simp
- next
- from thread_ts show "thread \<in> threads (t @ s)" .
- qed(simp)
- with neq_thread show "False" by simp
- qed
- ultimately show ?thesis by auto
- qed
- qed
- ultimately show ?thesis
- by (auto simp:max_def split:if_splits)
- qed
- next
- case (P thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def)
- next
- case (Set thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- let ?B = "threads (t@s)"
- from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
- from extend_highest_gen.set_diff_low[OF this] and Set
- have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
- from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
- also have "\<dots> = ?t"
- proof(rule Max_eqI)
- fix y
- assume y_in: "y \<in> ?f ` ?B"
- then obtain th1 where
- th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
- show "y \<le> ?t"
- proof(cases "th1 = thread")
- case True
- with neq_thread le_p eq_y Set
- show ?thesis
- apply (subst preced_th, insert lt_tm)
- by (auto simp:preced_def precedence_le_def)
- next
- case False
- with Set eq_y
- have "y = preced th1 (t@s)"
- by (simp add:preced_def)
- moreover have "\<dots> \<le> ?t"
- proof -
- from Cons
- have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
- by auto
- moreover have "preced th1 (t@s) \<le> \<dots>"
- proof(rule Max_ge)
- from th1_in
- show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
- by simp
- next
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- proof -
- from Cons have "vt (t @ s)" by auto
- from finite_threads[OF this] show ?thesis by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- next
- from Cons and finite_threads
- show "finite (?f ` ?B)" by auto
- next
- from Cons have "extend_highest_gen s th prio tm t" by auto
- from extend_highest_gen.th_kept [OF this]
- have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
- show "?t \<in> (?f ` ?B)"
- proof -
- from neq_thread Set h
- have "?t = ?f th" by (auto simp:preced_def)
- with h show ?thesis by auto
- qed
- qed
- finally show ?thesis .
- qed
- qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- by (insert th_kept max_kept, auto)
-
-lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
- (is "?L = ?R")
-proof -
- have "?L = cpreced (wq (t@s)) (t@s) th"
- by (unfold cp_eq_cpreced, simp)
- also have "\<dots> = ?R"
- proof(unfold cpreced_def)
- show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
- Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
- (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
- proof(cases "?A = {}")
- case False
- have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
- moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
- proof(rule Max_insert)
- show "finite (?f ` ?A)"
- proof -
- from dependents_threads[OF vt_t]
- have "?A \<subseteq> threads (t@s)" .
- moreover from finite_threads[OF vt_t] have "finite \<dots>" .
- ultimately show ?thesis
- by (auto simp:finite_subset)
- qed
- next
- from False show "(?f ` ?A) \<noteq> {}" by simp
- qed
- moreover have "\<dots> = Max (?f ` ?B)"
- proof -
- from max_preced have "?f th = Max (?f ` ?B)" .
- moreover have "Max (?f ` ?A) \<le> \<dots>"
- proof(rule Max_mono)
- from False show "(?f ` ?A) \<noteq> {}" by simp
- next
- show "?f ` ?A \<subseteq> ?f ` ?B"
- proof -
- have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
- thus ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t]
- show "finite (?f ` ?B)" by simp
- qed
- ultimately show ?thesis
- by (auto simp:max_def)
- qed
- ultimately show ?thesis by auto
- next
- case True
- with max_preced show ?thesis by auto
- qed
- qed
- finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
- by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
- by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- shows "preced th' s < preced th s"
-proof -
- have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
- proof(rule Max_ge)
- from finite_threads [OF vt_s]
- show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
- next
- from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
- by simp
- qed
- moreover have "preced th' s \<noteq> preced th s"
- proof
- assume "preced th' s = preced th s"
- from preced_unique[OF this th'_in] neq_th' threads_s
- show "False" by (auto simp:readys_def)
- qed
- ultimately show ?thesis using highest_preced_thread
- by auto
-qed
-
-lemma pv_blocked_pre:
- fixes th'
- assumes th'_in: "th' \<in> threads (t@s)"
- and neq_th': "th' \<noteq> th"
- and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
- shows "th' \<notin> runing (t@s)"
-proof
- assume "th' \<in> runing (t@s)"
- hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
- by (auto simp:runing_def)
- with max_cp_readys_threads [OF vt_t]
- have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
- by auto
- moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
- ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
- moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
- by simp
- finally have h: "cp (t @ s) th' = preced th (t @ s)" .
- show False
- proof -
- have "dependents (wq (t @ s)) th' = {}"
- by (rule count_eq_dependents [OF vt_t eq_pv])
- moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
- proof
- assume "preced th' (t @ s) = preced th (t @ s)"
- hence "th' = th"
- proof(rule preced_unique)
- from th_kept show "th \<in> threads (t @ s)" by simp
- next
- from th'_in show "th' \<in> threads (t @ s)" by simp
- qed
- with assms show False by simp
- qed
- ultimately show ?thesis
- by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
- qed
-qed
-
-lemmas pv_blocked = pv_blocked_pre[folded detached_eq [OF vt_t]]
-
-lemma runing_precond_pre:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads (t@s) \<and>
- cntP (t@s) th' = cntV (t@s) th'"
-proof -
- show ?thesis
- proof(induct rule:ind)
- case (Cons e t)
- from Cons
- have in_thread: "th' \<in> threads (t @ s)"
- and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from Cons have "extend_highest_gen s th prio tm t" by auto
- then have not_runing: "th' \<notin> runing (t @ s)"
- apply(rule extend_highest_gen.pv_blocked)
- using Cons(1) in_thread neq_th' not_holding
- apply(simp_all add: detached_eq)
- done
- show ?case
- proof(cases e)
- case (V thread cs)
- from Cons and V have vt_v: "vt (V thread cs#(t@s))" by auto
-
- show ?thesis
- proof -
- from Cons and V have "step (t@s) (V thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (unfold V, simp add:cntP_def cntV_def count_def)
- moreover from in_thread
- have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (P thread cs)
- from Cons and P have "step (t@s) (P thread cs)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Create thread prio')
- from Cons and Create have "step (t@s) (Create thread prio')" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<notin> threads (t @ s)"
- moreover have "th' \<in> threads (t@s)" by fact
- ultimately show ?thesis by auto
- qed
- with Cons and Create
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Create
- have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
- ultimately show ?thesis by auto
- next
- case (Exit thread)
- from Cons and Exit have "step (t@s) (Exit thread)" by auto
- hence neq_th': "thread \<noteq> th'"
- proof(cases)
- assume "thread \<in> runing (t @ s)"
- moreover note not_runing
- ultimately show ?thesis by auto
- qed
- with Cons and Exit
- have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- by (auto simp:cntP_def cntV_def count_def)
- moreover from Cons and Exit and neq_th'
- have in_thread': "th' \<in> threads ((e # t) @ s)"
- by auto
- ultimately show ?thesis by auto
- next
- case (Set thread prio')
- with Cons
- show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case Nil
- with assms
- show ?case by auto
- qed
-qed
-
-(*
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<notin> runing (t@s)"
-proof -
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked[OF h1 neq_th' h2]
- show ?thesis .
-qed
-*)
-
-lemmas runing_precond_pre_dtc = runing_precond_pre[folded detached_eq[OF vt_t] detached_eq[OF vt_s]]
-
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- and is_runing: "th' \<in> runing (t@s)"
- shows "cntP s th' > cntV s th'"
-proof -
- have "cntP s th' \<noteq> cntV s th'"
- proof
- assume eq_pv: "cntP s th' = cntV s th'"
- from runing_precond_pre[OF th'_in eq_pv neq_th']
- have h1: "th' \<in> threads (t @ s)"
- and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
- from pv_blocked_pre[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
- with is_runing show "False" by simp
- qed
- moreover from cnp_cnv_cncs[OF vt_s, of th']
- have "cntV s th' \<le> cntP s th'" by auto
- ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
- th' \<in> threads ((moment (i+j) t)@s)"
-proof(induct j)
- case (Suc k)
- show ?case
- proof -
- { assume True: "Suc (i+k) \<le> length t"
- from moment_head [OF this]
- obtain e where
- eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
- by blast
- from red_moment[of "Suc(i+k)"]
- and eq_me have "extend_highest_gen s th prio tm (e # moment (i + k) t)" by simp
- hence vt_e: "vt (e#(moment (i + k) t)@s)"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
- highest_gen_def, auto)
- have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
- proof -
- show "th' \<notin> runing (moment (i + k) t @ s)"
- proof(rule extend_highest_gen.pv_blocked)
- from Suc show "th' \<in> threads (moment (i + k) t @ s)"
- by simp
- next
- from neq_th' show "th' \<noteq> th" .
- next
- from red_moment show "extend_highest_gen s th prio tm (moment (i + k) t)" .
- next
- from Suc vt_e show "detached (moment (i + k) t @ s) th'"
- apply(subst detached_eq)
- apply(auto intro: vt_e evt_cons)
- done
- qed
- qed
- from step_back_step[OF vt_e]
- have "step ((moment (i + k) t)@s) e" .
- hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
- th' \<in> threads (e#(moment (i + k) t)@s)"
- proof(cases)
- case (thread_create thread prio)
- with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_exit thread)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_P thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_V thread cs)
- moreover have "thread \<noteq> th'"
- proof -
- have "thread \<in> runing (moment (i + k) t @ s)" by fact
- moreover note not_runing'
- ultimately show ?thesis by auto
- qed
- moreover note Suc
- ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_set thread prio')
- with Suc show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- with eq_me have ?thesis using eq_me by auto
- } note h = this
- show ?thesis
- proof(cases "Suc (i+k) \<le> length t")
- case True
- from h [OF this] show ?thesis .
- next
- case False
- with moment_ge
- have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
- with Suc show ?thesis by auto
- qed
- qed
-next
- case 0
- from assms show ?case by auto
-qed
-
-lemma moment_blocked_eqpv:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- and le_ij: "i \<le> j"
- shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
- have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
- and h2: "th' \<in> threads ((moment j t)@s)" by auto
- with extend_highest_gen.pv_blocked
- show ?thesis
- using red_moment [of j] h2 neq_th' h1
- apply(auto)
- by (metis extend_highest_gen.pv_blocked_pre)
-qed
-
-lemma moment_blocked:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and dtc: "detached (moment i t @ s) th'"
- and le_ij: "i \<le> j"
- shows "detached (moment j t @ s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- from vt_moment[OF vt_t, of "i+length s"] moment_prefix[of i t s]
- have vt_i: "vt (moment i t @ s)" by auto
- from vt_moment[OF vt_t, of "j+length s"] moment_prefix[of j t s]
- have vt_j: "vt (moment j t @ s)" by auto
- from moment_blocked_eqpv [OF neq_th' th'_in detached_elim [OF vt_i dtc] le_ij,
- folded detached_eq[OF vt_j]]
- show ?thesis .
-qed
-
-lemma runing_inversion_1:
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof(cases "th' \<in> threads s")
- case True
- with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-next
- case False
- let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
- let ?q = "moment 0 t"
- from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
- from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
- from p_split_gen [of ?Q, OF this not_thread]
- obtain i where lt_its: "i < length t"
- and le_i: "0 \<le> i"
- and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
- and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
- from lt_its have "Suc i \<le> length t" by auto
- from moment_head[OF this] obtain e where
- eq_me: "moment (Suc i) t = e # moment i t" by blast
- from red_moment[of "Suc i"] and eq_me
- have "extend_highest_gen s th prio tm (e # moment i t)" by simp
- hence vt_e: "vt (e#(moment i t)@s)"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
- highest_gen_def, auto)
- from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
- from post[rule_format, of "Suc i"] and eq_me
- have not_in': "th' \<in> threads (e # moment i t@s)" by auto
- from create_pre[OF stp_i pre this]
- obtain prio where eq_e: "e = Create th' prio" .
- have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
- proof(rule cnp_cnv_eq)
- from step_back_vt [OF vt_e]
- show "vt (moment i t @ s)" .
- next
- from eq_e and stp_i
- have "step (moment i t @ s) (Create th' prio)" by simp
- thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
- qed
- with eq_e
- have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
- by (simp add:cntP_def cntV_def count_def)
- with eq_me[symmetric]
- have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
- by simp
- from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
- with eq_me [symmetric]
- have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
- from moment_blocked_eqpv [OF neq_th' h2 h1, of "length t"] and lt_its
- and moment_ge
- have "th' \<notin> runing (t @ s)" by auto
- with runing'
- show ?thesis by auto
-qed
-
-lemma runing_inversion_2:
- assumes runing': "th' \<in> runing (t@s)"
- shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
- from runing_inversion_1[OF _ runing']
- show ?thesis by auto
-qed
-
-lemma runing_preced_inversion:
- assumes runing': "th' \<in> runing (t@s)"
- shows "cp (t@s) th' = preced th s"
-proof -
- from runing' have "cp (t@s) th' = Max (cp (t @ s) ` readys (t @ s))"
- by (unfold runing_def, auto)
- also have "\<dots> = preced th s"
- proof -
- from max_cp_readys_threads[OF vt_t]
- have "\<dots> = Max (cp (t @ s) ` threads (t @ s))" .
- also have "\<dots> = preced th s"
- proof -
- from max_kept
- and max_cp_eq [OF vt_t]
- show ?thesis by auto
- qed
- finally show ?thesis .
- qed
- finally show ?thesis .
-qed
-
-lemma runing_inversion_3:
- assumes runing': "th' \<in> runing (t@s)"
- and neq_th: "th' \<noteq> th"
- shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
-proof -
- from runing_inversion_2 [OF runing']
- and neq_th
- and runing_preced_inversion[OF runing']
- show ?thesis by auto
-qed
-
-lemma runing_inversion_4:
- assumes runing': "th' \<in> runing (t@s)"
- and neq_th: "th' \<noteq> th"
- shows "th' \<in> threads s"
- and "\<not>detached s th'"
- and "cp (t@s) th' = preced th s"
-using runing_inversion_3 [OF runing']
- and neq_th
- and runing_preced_inversion[OF runing']
-apply(auto simp add: detached_eq[OF vt_s])
-done
-
-
-
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
- case True thus ?thesis by auto
-next
- case False
- then have not_ready: "th \<notin> readys (t@s)"
- apply (unfold runing_def,
- insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
- by auto
- from th_kept have "th \<in> threads (t@s)" by auto
- from th_chain_to_ready[OF vt_t this] and not_ready
- obtain th' where th'_in: "th' \<in> readys (t@s)"
- and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
- have "th' \<in> runing (t@s)"
- proof -
- have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
- preced th (t@s)"
- proof(rule Max_eqI)
- fix y
- assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- then obtain th1 where
- h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
- and eq_y: "y = preced th1 (t@s)" by auto
- show "y \<le> preced th (t @ s)"
- proof -
- from max_preced
- have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
- moreover have "y \<le> \<dots>"
- proof(rule Max_ge)
- from h1
- have "th1 \<in> threads (t@s)"
- proof
- assume "th1 = th'"
- with th'_in show ?thesis by (simp add:readys_def)
- next
- assume "th1 \<in> dependents (wq (t @ s)) th'"
- with dependents_threads [OF vt_t]
- show "th1 \<in> threads (t @ s)" by auto
- qed
- with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
- next
- from finite_threads[OF vt_t]
- show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
- qed
- ultimately show ?thesis by auto
- qed
- next
- from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
- show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
- by (auto intro:finite_subset)
- next
- from dp
- have "th \<in> dependents (wq (t @ s)) th'"
- by (unfold cs_dependents_def, auto simp:eq_depend)
- thus "preced th (t @ s) \<in>
- (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
- by auto
- qed
- moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
- proof -
- from max_preced and max_cp_eq[OF vt_t, symmetric]
- have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
- with max_cp_readys_threads[OF vt_t] show ?thesis by simp
- qed
- ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
- qed
- with th'_in show ?thesis by (auto simp:runing_def)
- qed
- thus ?thesis by auto
-qed
-
-end
-end
-
-
-
--- a/prio/IsaMakefile Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,51 +0,0 @@
-
-## targets
-
-default: itp
-all: session itp slides1
-
-## global settings
-
-SRC = $(ISABELLE_HOME)/src
-OUT = $(ISABELLE_OUTPUT)
-LOG = $(OUT)/log
-
-
-USEDIR = $(ISABELLE_TOOL) usedir -v true -t true
-
-
-## Slides
-
-session1: Slides/ROOT1.ML \
- Slides/document/root* \
- Slides/Slides1.thy
- @$(USEDIR) -D generated -f ROOT1.ML HOL Slides
-
-slides1: session1
- rm -f Slides/generated/*.aux # otherwise latex will fall over
- cd Slides/generated ; $(ISABELLE_TOOL) latex -o pdf root.beamer.tex
- cp Slides/generated/root.beamer.pdf Slides/slides.pdf
-
-# main files
-
-session: ./ROOT.ML ./*.thy
- @$(USEDIR) -b -D generated -f ROOT.ML HOL Prio
-
-
-# itp paper
-
-itp: Paper/*.thy Paper/*.ML
- @$(USEDIR) -D generated -f ROOT.ML Prio Paper
- rm -f Paper/generated/*.aux # otherwise latex will fall over
- cd Paper/generated ; $(ISABELLE_TOOL) latex -o pdf root.tex
- cd Paper/generated ; bibtex root
- cd Paper/generated ; $(ISABELLE_TOOL) latex -o pdf root.tex
- cp Paper/generated/root.pdf paper.pdf
-
-
-slides: Slides/ROOT1.ML Slides/*.thy
- @$(USEDIR) -D generated -f ROOT1.ML Prio Slides
- rm -f Slides/generated/*.aux # otherwise latex will fall over
- cd Slides/generated ; $(ISABELLE_TOOL) latex -o pdf root.beamer.tex
- cp Slides/generated/root.beamer.pdf Slides/slides.pdf
-
--- a/prio/Moment.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,783 +0,0 @@
-theory Moment
-imports Main
-begin
-
-fun firstn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
- "firstn 0 s = []" |
- "firstn (Suc n) [] = []" |
- "firstn (Suc n) (e#s) = e#(firstn n s)"
-
-fun restn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "restn n s = rev (firstn (length s - n) (rev s))"
-
-definition moment :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "moment n s = rev (firstn n (rev s))"
-
-definition restm :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "restm n s = rev (restn n (rev s))"
-
-definition from_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
- where "from_to i j s = firstn (j - i) (restn i s)"
-
-definition down_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where "down_to j i s = rev (from_to i j (rev s))"
-
-(*
-value "down_to 6 2 [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]"
-value "from_to 2 6 [0, 1, 2, 3, 4, 5, 6, 7]"
-*)
-
-lemma length_eq_elim_l: "\<lbrakk>length xs = length ys; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"
- by auto
-
-lemma length_eq_elim_r: "\<lbrakk>length us = length vs; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"
- by simp
-
-lemma firstn_nil [simp]: "firstn n [] = []"
- by (cases n, simp+)
-
-(*
-value "from_to 0 2 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] @
- from_to 2 5 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"
-*)
-
-lemma firstn_le: "\<And> n s'. n \<le> length s \<Longrightarrow> firstn n (s@s') = firstn n s"
-proof (induct s, simp)
- fix a s n s'
- assume ih: "\<And>n s'. n \<le> length s \<Longrightarrow> firstn n (s @ s') = firstn n s"
- and le_n: " n \<le> length (a # s)"
- show "firstn n ((a # s) @ s') = firstn n (a # s)"
- proof(cases n, simp)
- fix k
- assume eq_n: "n = Suc k"
- with le_n have "k \<le> length s" by auto
- from ih [OF this] and eq_n
- show "firstn n ((a # s) @ s') = firstn n (a # s)" by auto
- qed
-qed
-
-lemma firstn_ge [simp]: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"
-proof(induct s, simp)
- fix a s n
- assume ih: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"
- and le: "length (a # s) \<le> n"
- show "firstn n (a # s) = a # s"
- proof(cases n)
- assume eq_n: "n = 0" with le show ?thesis by simp
- next
- fix k
- assume eq_n: "n = Suc k"
- with le have le_k: "length s \<le> k" by simp
- from ih [OF this] have "firstn k s = s" .
- from eq_n and this
- show ?thesis by simp
- qed
-qed
-
-lemma firstn_eq [simp]: "firstn (length s) s = s"
- by simp
-
-lemma firstn_restn_s: "(firstn n (s::'a list)) @ (restn n s) = s"
-proof(induct n arbitrary:s, simp)
- fix n s
- assume ih: "\<And>t. firstn n (t::'a list) @ restn n t = t"
- show "firstn (Suc n) (s::'a list) @ restn (Suc n) s = s"
- proof(cases s, simp)
- fix x xs
- assume eq_s: "s = x#xs"
- show "firstn (Suc n) s @ restn (Suc n) s = s"
- proof -
- have "firstn (Suc n) s @ restn (Suc n) s = x # (firstn n xs @ restn n xs)"
- proof -
- from eq_s have "firstn (Suc n) s = x # firstn n xs" by simp
- moreover have "restn (Suc n) s = restn n xs"
- proof -
- from eq_s have "restn (Suc n) s = rev (firstn (length xs - n) (rev xs @ [x]))" by simp
- also have "\<dots> = restn n xs"
- proof -
- have "(firstn (length xs - n) (rev xs @ [x])) = (firstn (length xs - n) (rev xs))"
- by(rule firstn_le, simp)
- hence "rev (firstn (length xs - n) (rev xs @ [x])) =
- rev (firstn (length xs - n) (rev xs))" by simp
- also have "\<dots> = rev (firstn (length (rev xs) - n) (rev xs))" by simp
- finally show ?thesis by simp
- qed
- finally show ?thesis by simp
- qed
- ultimately show ?thesis by simp
- qed with ih eq_s show ?thesis by simp
- qed
- qed
-qed
-
-lemma moment_restm_s: "(restm n s)@(moment n s) = s"
-proof -
- have " rev ((firstn n (rev s)) @ (restn n (rev s))) = s" (is "rev ?x = s")
- proof -
- have "?x = rev s" by (simp only:firstn_restn_s)
- thus ?thesis by auto
- qed
- thus ?thesis
- by (auto simp:restm_def moment_def)
-qed
-
-declare restn.simps [simp del] firstn.simps[simp del]
-
-lemma length_firstn_ge: "length s \<le> n \<Longrightarrow> length (firstn n s) = length s"
-proof(induct n arbitrary:s, simp add:firstn.simps)
- case (Suc k)
- assume ih: "\<And> s. length (s::'a list) \<le> k \<Longrightarrow> length (firstn k s) = length s"
- and le: "length s \<le> Suc k"
- show ?case
- proof(cases s)
- case Nil
- from Nil show ?thesis by simp
- next
- case (Cons x xs)
- from le and Cons have "length xs \<le> k" by simp
- from ih [OF this] have "length (firstn k xs) = length xs" .
- moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))"
- by (simp add:firstn.simps)
- moreover note Cons
- ultimately show ?thesis by simp
- qed
-qed
-
-lemma length_firstn_le: "n \<le> length s \<Longrightarrow> length (firstn n s) = n"
-proof(induct n arbitrary:s, simp add:firstn.simps)
- case (Suc k)
- assume ih: "\<And>s. k \<le> length (s::'a list) \<Longrightarrow> length (firstn k s) = k"
- and le: "Suc k \<le> length s"
- show ?case
- proof(cases s)
- case Nil
- from Nil and le show ?thesis by auto
- next
- case (Cons x xs)
- from le and Cons have "k \<le> length xs" by simp
- from ih [OF this] have "length (firstn k xs) = k" .
- moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))"
- by (simp add:firstn.simps)
- ultimately show ?thesis by simp
- qed
-qed
-
-lemma app_firstn_restn:
- fixes s1 s2
- shows "s1 = firstn (length s1) (s1 @ s2) \<and> s2 = restn (length s1) (s1 @ s2)"
-proof(rule length_eq_elim_l)
- have "length s1 \<le> length (s1 @ s2)" by simp
- from length_firstn_le [OF this]
- show "length s1 = length (firstn (length s1) (s1 @ s2))" by simp
-next
- from firstn_restn_s
- show "s1 @ s2 = firstn (length s1) (s1 @ s2) @ restn (length s1) (s1 @ s2)"
- by metis
-qed
-
-
-lemma length_moment_le:
- fixes k s
- assumes le_k: "k \<le> length s"
- shows "length (moment k s) = k"
-proof -
- have "length (rev (firstn k (rev s))) = k"
- proof -
- have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
- also have "\<dots> = k"
- proof(rule length_firstn_le)
- from le_k show "k \<le> length (rev s)" by simp
- qed
- finally show ?thesis .
- qed
- thus ?thesis by (simp add:moment_def)
-qed
-
-lemma app_moment_restm:
- fixes s1 s2
- shows "s1 = restm (length s2) (s1 @ s2) \<and> s2 = moment (length s2) (s1 @ s2)"
-proof(rule length_eq_elim_r)
- have "length s2 \<le> length (s1 @ s2)" by simp
- from length_moment_le [OF this]
- show "length s2 = length (moment (length s2) (s1 @ s2))" by simp
-next
- from moment_restm_s
- show "s1 @ s2 = restm (length s2) (s1 @ s2) @ moment (length s2) (s1 @ s2)"
- by metis
-qed
-
-lemma length_moment_ge:
- fixes k s
- assumes le_k: "length s \<le> k"
- shows "length (moment k s) = (length s)"
-proof -
- have "length (rev (firstn k (rev s))) = length s"
- proof -
- have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
- also have "\<dots> = length s"
- proof -
- have "\<dots> = length (rev s)"
- proof(rule length_firstn_ge)
- from le_k show "length (rev s) \<le> k" by simp
- qed
- also have "\<dots> = length s" by simp
- finally show ?thesis .
- qed
- finally show ?thesis .
- qed
- thus ?thesis by (simp add:moment_def)
-qed
-
-lemma length_firstn: "(length (firstn n s) = length s) \<or> (length (firstn n s) = n)"
-proof(cases "n \<le> length s")
- case True
- from length_firstn_le [OF True] show ?thesis by auto
-next
- case False
- from False have "length s \<le> n" by simp
- from firstn_ge [OF this] show ?thesis by auto
-qed
-
-lemma firstn_conc:
- fixes m n
- assumes le_mn: "m \<le> n"
- shows "firstn m s = firstn m (firstn n s)"
-proof(cases "m \<le> length s")
- case True
- have "s = (firstn n s) @ (restn n s)" by (simp add:firstn_restn_s)
- hence "firstn m s = firstn m \<dots>" by simp
- also have "\<dots> = firstn m (firstn n s)"
- proof -
- from length_firstn [of n s]
- have "m \<le> length (firstn n s)"
- proof
- assume "length (firstn n s) = length s" with True show ?thesis by simp
- next
- assume "length (firstn n s) = n " with le_mn show ?thesis by simp
- qed
- from firstn_le [OF this, of "restn n s"]
- show ?thesis .
- qed
- finally show ?thesis by simp
-next
- case False
- from False and le_mn have "length s \<le> n" by simp
- from firstn_ge [OF this] show ?thesis by simp
-qed
-
-lemma restn_conc:
- fixes i j k s
- assumes eq_k: "j + i = k"
- shows "restn k s = restn j (restn i s)"
-proof -
- have "(firstn (length s - k) (rev s)) =
- (firstn (length (rev (firstn (length s - i) (rev s))) - j)
- (rev (rev (firstn (length s - i) (rev s)))))"
- proof -
- have "(firstn (length s - k) (rev s)) =
- (firstn (length (rev (firstn (length s - i) (rev s))) - j)
- (firstn (length s - i) (rev s)))"
- proof -
- have " (length (rev (firstn (length s - i) (rev s))) - j) = length s - k"
- proof -
- have "(length (rev (firstn (length s - i) (rev s))) - j) = (length s - i) - j"
- proof -
- have "(length (rev (firstn (length s - i) (rev s))) - j) =
- length ((firstn (length s - i) (rev s))) - j"
- by simp
- also have "\<dots> = length ((firstn (length (rev s) - i) (rev s))) - j" by simp
- also have "\<dots> = (length (rev s) - i) - j"
- proof -
- have "length ((firstn (length (rev s) - i) (rev s))) = (length (rev s) - i)"
- by (rule length_firstn_le, simp)
- thus ?thesis by simp
- qed
- also have "\<dots> = (length s - i) - j" by simp
- finally show ?thesis .
- qed
- with eq_k show ?thesis by auto
- qed
- moreover have "(firstn (length s - k) (rev s)) =
- (firstn (length s - k) (firstn (length s - i) (rev s)))"
- proof(rule firstn_conc)
- from eq_k show "length s - k \<le> length s - i" by simp
- qed
- ultimately show ?thesis by simp
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by (simp only:restn.simps)
-qed
-
-(*
-value "down_to 2 0 [5, 4, 3, 2, 1, 0]"
-value "moment 2 [5, 4, 3, 2, 1, 0]"
-*)
-
-lemma from_to_firstn: "from_to 0 k s = firstn k s"
-by (simp add:from_to_def restn.simps)
-
-lemma moment_app [simp]:
- assumes
- ile: "i \<le> length s"
- shows "moment i (s'@s) = moment i s"
-proof -
- have "moment i (s'@s) = rev (firstn i (rev (s'@s)))" by (simp add:moment_def)
- moreover have "firstn i (rev (s'@s)) = firstn i (rev s @ rev s')" by simp
- moreover have "\<dots> = firstn i (rev s)"
- proof(rule firstn_le)
- have "length (rev s) = length s" by simp
- with ile show "i \<le> length (rev s)" by simp
- qed
- ultimately show ?thesis by (simp add:moment_def)
-qed
-
-lemma moment_eq [simp]: "moment (length s) (s'@s) = s"
-proof -
- have "length s \<le> length s" by simp
- from moment_app [OF this, of s']
- have " moment (length s) (s' @ s) = moment (length s) s" .
- moreover have "\<dots> = s" by (simp add:moment_def)
- ultimately show ?thesis by simp
-qed
-
-lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"
- by (unfold moment_def, simp)
-
-lemma moment_zero [simp]: "moment 0 s = []"
- by (simp add:moment_def firstn.simps)
-
-lemma p_split_gen:
- "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk> \<Longrightarrow>
- (\<exists> i. i < length s \<and> k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
-proof (induct s, simp)
- fix a s
- assume ih: "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk>
- \<Longrightarrow> \<exists>i<length s. k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall>i'>i. Q (moment i' s))"
- and nq: "\<not> Q (moment k (a # s))" and qa: "Q (a # s)"
- have le_k: "k \<le> length s"
- proof -
- { assume "length s < k"
- hence "length (a#s) \<le> k" by simp
- from moment_ge [OF this] and nq and qa
- have "False" by auto
- } thus ?thesis by arith
- qed
- have nq_k: "\<not> Q (moment k s)"
- proof -
- have "moment k (a#s) = moment k s"
- proof -
- from moment_app [OF le_k, of "[a]"] show ?thesis by simp
- qed
- with nq show ?thesis by simp
- qed
- show "\<exists>i<length (a # s). k \<le> i \<and> \<not> Q (moment i (a # s)) \<and> (\<forall>i'>i. Q (moment i' (a # s)))"
- proof -
- { assume "Q s"
- from ih [OF this nq_k]
- obtain i where lti: "i < length s"
- and nq: "\<not> Q (moment i s)"
- and rst: "\<forall>i'>i. Q (moment i' s)"
- and lki: "k \<le> i" by auto
- have ?thesis
- proof -
- from lti have "i < length (a # s)" by auto
- moreover have " \<not> Q (moment i (a # s))"
- proof -
- from lti have "i \<le> (length s)" by simp
- from moment_app [OF this, of "[a]"]
- have "moment i (a # s) = moment i s" by simp
- with nq show ?thesis by auto
- qed
- moreover have " (\<forall>i'>i. Q (moment i' (a # s)))"
- proof -
- {
- fix i'
- assume lti': "i < i'"
- have "Q (moment i' (a # s))"
- proof(cases "length (a#s) \<le> i'")
- case True
- from True have "moment i' (a#s) = a#s" by simp
- with qa show ?thesis by simp
- next
- case False
- from False have "i' \<le> length s" by simp
- from moment_app [OF this, of "[a]"]
- have "moment i' (a#s) = moment i' s" by simp
- with rst lti' show ?thesis by auto
- qed
- } thus ?thesis by auto
- qed
- moreover note lki
- ultimately show ?thesis by auto
- qed
- } moreover {
- assume ns: "\<not> Q s"
- have ?thesis
- proof -
- let ?i = "length s"
- have "\<not> Q (moment ?i (a#s))"
- proof -
- have "?i \<le> length s" by simp
- from moment_app [OF this, of "[a]"]
- have "moment ?i (a#s) = moment ?i s" by simp
- moreover have "\<dots> = s" by simp
- ultimately show ?thesis using ns by auto
- qed
- moreover have "\<forall> i' > ?i. Q (moment i' (a#s))"
- proof -
- { fix i'
- assume "i' > ?i"
- hence "length (a#s) \<le> i'" by simp
- from moment_ge [OF this]
- have " moment i' (a # s) = a # s" .
- with qa have "Q (moment i' (a#s))" by simp
- } thus ?thesis by auto
- qed
- moreover have "?i < length (a#s)" by simp
- moreover note le_k
- ultimately show ?thesis by auto
- qed
- } ultimately show ?thesis by auto
- qed
-qed
-
-lemma p_split:
- "\<And> s Q. \<lbrakk>Q s; \<not> Q []\<rbrakk> \<Longrightarrow>
- (\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
-proof -
- fix s Q
- assume qs: "Q s" and nq: "\<not> Q []"
- from nq have "\<not> Q (moment 0 s)" by simp
- from p_split_gen [of Q s 0, OF qs this]
- show "(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
- by auto
-qed
-
-lemma moment_plus:
- "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = (hd (moment (Suc i) s)) # (moment i s)"
-proof(induct s, simp+)
- fix a s
- assume ih: "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = hd (moment (Suc i) s) # moment i s"
- and le_i: "i \<le> length s"
- show "moment (Suc i) (a # s) = hd (moment (Suc i) (a # s)) # moment i (a # s)"
- proof(cases "i= length s")
- case True
- hence "Suc i = length (a#s)" by simp
- with moment_eq have "moment (Suc i) (a#s) = a#s" by auto
- moreover have "moment i (a#s) = s"
- proof -
- from moment_app [OF le_i, of "[a]"]
- and True show ?thesis by simp
- qed
- ultimately show ?thesis by auto
- next
- case False
- from False and le_i have lti: "i < length s" by arith
- hence les_i: "Suc i \<le> length s" by arith
- show ?thesis
- proof -
- from moment_app [OF les_i, of "[a]"]
- have "moment (Suc i) (a # s) = moment (Suc i) s" by simp
- moreover have "moment i (a#s) = moment i s"
- proof -
- from lti have "i \<le> length s" by simp
- from moment_app [OF this, of "[a]"] show ?thesis by simp
- qed
- moreover note ih [OF les_i]
- ultimately show ?thesis by auto
- qed
- qed
-qed
-
-lemma from_to_conc:
- fixes i j k s
- assumes le_ij: "i \<le> j"
- and le_jk: "j \<le> k"
- shows "from_to i j s @ from_to j k s = from_to i k s"
-proof -
- let ?ris = "restn i s"
- have "firstn (j - i) (restn i s) @ firstn (k - j) (restn j s) =
- firstn (k - i) (restn i s)" (is "?x @ ?y = ?z")
- proof -
- let "firstn (k-j) ?u" = "?y"
- let ?rst = " restn (k - j) (restn (j - i) ?ris)"
- let ?rst' = "restn (k - i) ?ris"
- have "?u = restn (j-i) ?ris"
- proof(rule restn_conc)
- from le_ij show "j - i + i = j" by simp
- qed
- hence "?x @ ?y = ?x @ firstn (k-j) (restn (j-i) ?ris)" by simp
- moreover have "firstn (k - j) (restn (j - i) (restn i s)) @ ?rst =
- restn (j-i) ?ris" by (simp add:firstn_restn_s)
- ultimately have "?x @ ?y @ ?rst = ?x @ (restn (j-i) ?ris)" by simp
- also have "\<dots> = ?ris" by (simp add:firstn_restn_s)
- finally have "?x @ ?y @ ?rst = ?ris" .
- moreover have "?z @ ?rst = ?ris"
- proof -
- have "?z @ ?rst' = ?ris" by (simp add:firstn_restn_s)
- moreover have "?rst' = ?rst"
- proof(rule restn_conc)
- from le_ij le_jk show "k - j + (j - i) = k - i" by auto
- qed
- ultimately show ?thesis by simp
- qed
- ultimately have "?x @ ?y @ ?rst = ?z @ ?rst" by simp
- thus ?thesis by auto
- qed
- thus ?thesis by (simp only:from_to_def)
-qed
-
-lemma down_to_conc:
- fixes i j k s
- assumes le_ij: "i \<le> j"
- and le_jk: "j \<le> k"
- shows "down_to k j s @ down_to j i s = down_to k i s"
-proof -
- have "rev (from_to j k (rev s)) @ rev (from_to i j (rev s)) = rev (from_to i k (rev s))"
- (is "?L = ?R")
- proof -
- have "?L = rev (from_to i j (rev s) @ from_to j k (rev s))" by simp
- also have "\<dots> = ?R" (is "rev ?x = rev ?y")
- proof -
- have "?x = ?y" by (rule from_to_conc[OF le_ij le_jk])
- thus ?thesis by simp
- qed
- finally show ?thesis .
- qed
- thus ?thesis by (simp add:down_to_def)
-qed
-
-lemma restn_ge:
- fixes s k
- assumes le_k: "length s \<le> k"
- shows "restn k s = []"
-proof -
- from firstn_restn_s [of k s, symmetric] have "s = (firstn k s) @ (restn k s)" .
- hence "length s = length \<dots>" by simp
- also have "\<dots> = length (firstn k s) + length (restn k s)" by simp
- finally have "length s = ..." by simp
- moreover from length_firstn_ge and le_k
- have "length (firstn k s) = length s" by simp
- ultimately have "length (restn k s) = 0" by auto
- thus ?thesis by auto
-qed
-
-lemma from_to_ge: "length s \<le> k \<Longrightarrow> from_to k j s = []"
-proof(simp only:from_to_def)
- assume "length s \<le> k"
- from restn_ge [OF this]
- show "firstn (j - k) (restn k s) = []" by simp
-qed
-
-(*
-value "from_to 2 5 [0, 1, 2, 3, 4]"
-value "restn 2 [0, 1, 2, 3, 4]"
-*)
-
-lemma from_to_restn:
- fixes k j s
- assumes le_j: "length s \<le> j"
- shows "from_to k j s = restn k s"
-proof -
- have "from_to 0 k s @ from_to k j s = from_to 0 j s"
- proof(cases "k \<le> j")
- case True
- from from_to_conc True show ?thesis by auto
- next
- case False
- from False le_j have lek: "length s \<le> k" by auto
- from from_to_ge [OF this] have "from_to k j s = []" .
- hence "from_to 0 k s @ from_to k j s = from_to 0 k s" by simp
- also have "\<dots> = s"
- proof -
- from from_to_firstn [of k s]
- have "\<dots> = firstn k s" .
- also have "\<dots> = s" by (rule firstn_ge [OF lek])
- finally show ?thesis .
- qed
- finally have "from_to 0 k s @ from_to k j s = s" .
- moreover have "from_to 0 j s = s"
- proof -
- have "from_to 0 j s = firstn j s" by (simp add:from_to_firstn)
- also have "\<dots> = s"
- proof(rule firstn_ge)
- from le_j show "length s \<le> j " by simp
- qed
- finally show ?thesis .
- qed
- ultimately show ?thesis by auto
- qed
- also have "\<dots> = s"
- proof -
- from from_to_firstn have "\<dots> = firstn j s" .
- also have "\<dots> = s"
- proof(rule firstn_ge)
- from le_j show "length s \<le> j" by simp
- qed
- finally show ?thesis .
- qed
- finally have "from_to 0 k s @ from_to k j s = s" .
- moreover have "from_to 0 k s @ restn k s = s"
- proof -
- from from_to_firstn [of k s]
- have "from_to 0 k s = firstn k s" .
- thus ?thesis by (simp add:firstn_restn_s)
- qed
- ultimately have "from_to 0 k s @ from_to k j s =
- from_to 0 k s @ restn k s" by simp
- thus ?thesis by auto
-qed
-
-lemma down_to_moment: "down_to k 0 s = moment k s"
-proof -
- have "rev (from_to 0 k (rev s)) = rev (firstn k (rev s))"
- using from_to_firstn by metis
- thus ?thesis by (simp add:down_to_def moment_def)
-qed
-
-lemma down_to_restm:
- assumes le_s: "length s \<le> j"
- shows "down_to j k s = restm k s"
-proof -
- have "rev (from_to k j (rev s)) = rev (restn k (rev s))" (is "?L = ?R")
- proof -
- from le_s have "length (rev s) \<le> j" by simp
- from from_to_restn [OF this, of k] show ?thesis by simp
- qed
- thus ?thesis by (simp add:down_to_def restm_def)
-qed
-
-lemma moment_split: "moment (m+i) s = down_to (m+i) i s @down_to i 0 s"
-proof -
- have "moment (m + i) s = down_to (m+i) 0 s" using down_to_moment by metis
- also have "\<dots> = (down_to (m+i) i s) @ (down_to i 0 s)"
- by(rule down_to_conc[symmetric], auto)
- finally show ?thesis .
-qed
-
-lemma length_restn: "length (restn i s) = length s - i"
-proof(cases "i \<le> length s")
- case True
- from length_firstn_le [OF this] have "length (firstn i s) = i" .
- moreover have "length s = length (firstn i s) + length (restn i s)"
- proof -
- have "s = firstn i s @ restn i s" using firstn_restn_s by metis
- hence "length s = length \<dots>" by simp
- thus ?thesis by simp
- qed
- ultimately show ?thesis by simp
-next
- case False
- hence "length s \<le> i" by simp
- from restn_ge [OF this] have "restn i s = []" .
- with False show ?thesis by simp
-qed
-
-lemma length_from_to_in:
- fixes i j s
- assumes le_ij: "i \<le> j"
- and le_j: "j \<le> length s"
- shows "length (from_to i j s) = j - i"
-proof -
- have "from_to 0 j s = from_to 0 i s @ from_to i j s"
- by (rule from_to_conc[symmetric, OF _ le_ij], simp)
- moreover have "length (from_to 0 j s) = j"
- proof -
- have "from_to 0 j s = firstn j s" using from_to_firstn by metis
- moreover have "length \<dots> = j" by (rule length_firstn_le [OF le_j])
- ultimately show ?thesis by simp
- qed
- moreover have "length (from_to 0 i s) = i"
- proof -
- have "from_to 0 i s = firstn i s" using from_to_firstn by metis
- moreover have "length \<dots> = i"
- proof (rule length_firstn_le)
- from le_ij le_j show "i \<le> length s" by simp
- qed
- ultimately show ?thesis by simp
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma firstn_restn_from_to: "from_to i (m + i) s = firstn m (restn i s)"
-proof(cases "m+i \<le> length s")
- case True
- have "restn i s = from_to i (m+i) s @ from_to (m+i) (length s) s"
- proof -
- have "restn i s = from_to i (length s) s"
- by(rule from_to_restn[symmetric], simp)
- also have "\<dots> = from_to i (m+i) s @ from_to (m+i) (length s) s"
- by(rule from_to_conc[symmetric, OF _ True], simp)
- finally show ?thesis .
- qed
- hence "firstn m (restn i s) = firstn m \<dots>" by simp
- moreover have "\<dots> = firstn (length (from_to i (m+i) s))
- (from_to i (m+i) s @ from_to (m+i) (length s) s)"
- proof -
- have "length (from_to i (m+i) s) = m"
- proof -
- have "length (from_to i (m+i) s) = (m+i) - i"
- by(rule length_from_to_in [OF _ True], simp)
- thus ?thesis by simp
- qed
- thus ?thesis by simp
- qed
- ultimately show ?thesis using app_firstn_restn by metis
-next
- case False
- hence "length s \<le> m + i" by simp
- from from_to_restn [OF this]
- have "from_to i (m + i) s = restn i s" .
- moreover have "firstn m (restn i s) = restn i s"
- proof(rule firstn_ge)
- show "length (restn i s) \<le> m"
- proof -
- have "length (restn i s) = length s - i" using length_restn by metis
- with False show ?thesis by simp
- qed
- qed
- ultimately show ?thesis by simp
-qed
-
-lemma down_to_moment_restm:
- fixes m i s
- shows "down_to (m + i) i s = moment m (restm i s)"
- by (simp add:firstn_restn_from_to down_to_def moment_def restm_def)
-
-lemma moment_plus_split:
- fixes m i s
- shows "moment (m + i) s = moment m (restm i s) @ moment i s"
-proof -
- from moment_split [of m i s]
- have "moment (m + i) s = down_to (m + i) i s @ down_to i 0 s" .
- also have "\<dots> = down_to (m+i) i s @ moment i s" using down_to_moment by simp
- also from down_to_moment_restm have "\<dots> = moment m (restm i s) @ moment i s"
- by simp
- finally show ?thesis .
-qed
-
-lemma length_restm: "length (restm i s) = length s - i"
-proof -
- have "length (rev (restn i (rev s))) = length s - i" (is "?L = ?R")
- proof -
- have "?L = length (restn i (rev s))" by simp
- also have "\<dots> = length (rev s) - i" using length_restn by metis
- also have "\<dots> = ?R" by simp
- finally show ?thesis .
- qed
- thus ?thesis by (simp add:restm_def)
-qed
-
-lemma moment_prefix: "(moment i t @ s) = moment (i + length s) (t @ s)"
-proof -
- from moment_plus_split [of i "length s" "t@s"]
- have " moment (i + length s) (t @ s) = moment i (restm (length s) (t @ s)) @ moment (length s) (t @ s)"
- by auto
- with app_moment_restm[of t s]
- have "moment (i + length s) (t @ s) = moment i t @ moment (length s) (t @ s)" by simp
- with moment_app show ?thesis by auto
-qed
-
-end
\ No newline at end of file
--- a/prio/Paper/Paper.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1345 +0,0 @@
-(*<*)
-theory Paper
-imports "../CpsG" "../ExtGG" "~~/src/HOL/Library/LaTeXsugar"
-begin
-
-(*
-find_unused_assms CpsG
-find_unused_assms ExtGG
-find_unused_assms Moment
-find_unused_assms Precedence_ord
-find_unused_assms PrioG
-find_unused_assms PrioGDef
-*)
-
-ML {*
- open Printer;
- show_question_marks_default := false;
- *}
-
-notation (latex output)
- Cons ("_::_" [78,77] 73) and
- vt ("valid'_state") and
- runing ("running") and
- birthtime ("last'_set") and
- If ("(\<^raw:\textrm{>if\<^raw:}> (_)/ \<^raw:\textrm{>then\<^raw:}> (_)/ \<^raw:\textrm{>else\<^raw:}> (_))" 10) and
- Prc ("'(_, _')") and
- holding ("holds") and
- waiting ("waits") and
- Th ("T") and
- Cs ("C") and
- readys ("ready") and
- depend ("RAG") and
- preced ("prec") and
- cpreced ("cprec") and
- dependents ("dependants") and
- cp ("cprec") and
- holdents ("resources") and
- original_priority ("priority") and
- DUMMY ("\<^raw:\mbox{$\_\!\_$}>")
-
-(*abbreviation
- "detached s th \<equiv> cntP s th = cntV s th"
-*)
-(*>*)
-
-section {* Introduction *}
-
-text {*
- Many real-time systems need to support threads involving priorities and
- locking of resources. Locking of resources ensures mutual exclusion
- when accessing shared data or devices that cannot be
- preempted. Priorities allow scheduling of threads that need to
- finish their work within deadlines. Unfortunately, both features
- can interact in subtle ways leading to a problem, called
- \emph{Priority Inversion}. Suppose three threads having priorities
- $H$(igh), $M$(edium) and $L$(ow). We would expect that the thread
- $H$ blocks any other thread with lower priority and the thread itself cannot
- be blocked indefinitely by threads with lower priority. Alas, in a naive
- implementation of resource locking and priorities this property can
- be violated. For this let $L$ be in the
- possession of a lock for a resource that $H$ also needs. $H$ must
- therefore wait for $L$ to exit the critical section and release this
- lock. The problem is that $L$ might in turn be blocked by any
- thread with priority $M$, and so $H$ sits there potentially waiting
- indefinitely. Since $H$ is blocked by threads with lower
- priorities, the problem is called Priority Inversion. It was first
- described in \cite{Lampson80} in the context of the
- Mesa programming language designed for concurrent programming.
-
- If the problem of Priority Inversion is ignored, real-time systems
- can become unpredictable and resulting bugs can be hard to diagnose.
- The classic example where this happened is the software that
- controlled the Mars Pathfinder mission in 1997 \cite{Reeves98}.
- Once the spacecraft landed, the software shut down at irregular
- intervals leading to loss of project time as normal operation of the
- craft could only resume the next day (the mission and data already
- collected were fortunately not lost, because of a clever system
- design). The reason for the shutdowns was that the scheduling
- software fell victim to Priority Inversion: a low priority thread
- locking a resource prevented a high priority thread from running in
- time, leading to a system reset. Once the problem was found, it was
- rectified by enabling the \emph{Priority Inheritance Protocol} (PIP)
- \cite{Sha90}\footnote{Sha et al.~call it the \emph{Basic Priority
- Inheritance Protocol} \cite{Sha90} and others sometimes also call it
- \emph{Priority Boosting} or \emph{Priority Donation}.} in the scheduling software.
-
- The idea behind PIP is to let the thread $L$ temporarily inherit
- the high priority from $H$ until $L$ leaves the critical section
- unlocking the resource. This solves the problem of $H$ having to
- wait indefinitely, because $L$ cannot be blocked by threads having
- priority $M$. While a few other solutions exist for the Priority
- Inversion problem, PIP is one that is widely deployed and
- implemented. This includes VxWorks (a proprietary real-time OS used
- in the Mars Pathfinder mission, in Boeing's 787 Dreamliner, Honda's
- ASIMO robot, etc.), but also the POSIX 1003.1c Standard realised for
- example in libraries for FreeBSD, Solaris and Linux.
-
- One advantage of PIP is that increasing the priority of a thread
- can be dynamically calculated by the scheduler. This is in contrast
- to, for example, \emph{Priority Ceiling} \cite{Sha90}, another
- solution to the Priority Inversion problem, which requires static
- analysis of the program in order to prevent Priority
- Inversion. However, there has also been strong criticism against
- PIP. For instance, PIP cannot prevent deadlocks when lock
- dependencies are circular, and also blocking times can be
- substantial (more than just the duration of a critical section).
- Though, most criticism against PIP centres around unreliable
- implementations and PIP being too complicated and too inefficient.
- For example, Yodaiken writes in \cite{Yodaiken02}:
-
- \begin{quote}
- \it{}``Priority inheritance is neither efficient nor reliable. Implementations
- are either incomplete (and unreliable) or surprisingly complex and intrusive.''
- \end{quote}
-
- \noindent
- He suggests avoiding PIP altogether by designing the system so that no
- priority inversion may happen in the first place. However, such ideal designs may
- not always be achievable in practice.
-
- In our opinion, there is clearly a need for investigating correct
- algorithms for PIP. A few specifications for PIP exist (in English)
- and also a few high-level descriptions of implementations (e.g.~in
- the textbook \cite[Section 5.6.5]{Vahalia96}), but they help little
- with actual implementations. That this is a problem in practice is
- proved by an email by Baker, who wrote on 13 July 2009 on the Linux
- Kernel mailing list:
-
- \begin{quote}
- \it{}``I observed in the kernel code (to my disgust), the Linux PIP
- implementation is a nightmare: extremely heavy weight, involving
- maintenance of a full wait-for graph, and requiring updates for a
- range of events, including priority changes and interruptions of
- wait operations.''
- \end{quote}
-
- \noindent
- The criticism by Yodaiken, Baker and others suggests another look
- at PIP from a more abstract level (but still concrete enough
- to inform an implementation), and makes PIP a good candidate for a
- formal verification. An additional reason is that the original
- presentation of PIP~\cite{Sha90}, despite being informally
- ``proved'' correct, is actually \emph{flawed}.
-
- Yodaiken \cite{Yodaiken02} points to a subtlety that had been
- overlooked in the informal proof by Sha et al. They specify in
- \cite{Sha90} that after the thread (whose priority has been raised)
- completes its critical section and releases the lock, it ``returns
- to its original priority level.'' This leads them to believe that an
- implementation of PIP is ``rather straightforward''~\cite{Sha90}.
- Unfortunately, as Yodaiken points out, this behaviour is too
- simplistic. Consider the case where the low priority thread $L$
- locks \emph{two} resources, and two high-priority threads $H$ and
- $H'$ each wait for one of them. If $L$ releases one resource
- so that $H$, say, can proceed, then we still have Priority Inversion
- with $H'$ (which waits for the other resource). The correct
- behaviour for $L$ is to switch to the highest remaining priority of
- the threads that it blocks. The advantage of formalising the
- correctness of a high-level specification of PIP in a theorem prover
- is that such issues clearly show up and cannot be overlooked as in
- informal reasoning (since we have to analyse all possible behaviours
- of threads, i.e.~\emph{traces}, that could possibly happen).\medskip
-
- \noindent
- {\bf Contributions:} There have been earlier formal investigations
- into PIP \cite{Faria08,Jahier09,Wellings07}, but they employ model
- checking techniques. This paper presents a formalised and
- mechanically checked proof for the correctness of PIP (to our
- knowledge the first one). In contrast to model checking, our
- formalisation provides insight into why PIP is correct and allows us
- to prove stronger properties that, as we will show, can help with an
- efficient implementation of PIP in the educational PINTOS operating
- system \cite{PINTOS}. For example, we found by ``playing'' with the
- formalisation that the choice of the next thread to take over a lock
- when a resource is released is irrelevant for PIP being correct---a
- fact that has not been mentioned in the literature and not been used
- in the reference implementation of PIP in PINTOS. This fact, however, is important
- for an efficient implementation of PIP, because we can give the lock
- to the thread with the highest priority so that it terminates more
- quickly.
-*}
-
-section {* Formal Model of the Priority Inheritance Protocol *}
-
-text {*
- The Priority Inheritance Protocol, short PIP, is a scheduling
- algorithm for a single-processor system.\footnote{We shall come back
- later to the case of PIP on multi-processor systems.}
- Following good experience in earlier work \cite{Wang09},
- our model of PIP is based on Paulson's inductive approach to protocol
- verification \cite{Paulson98}. In this approach a \emph{state} of a system is
- given by a list of events that happened so far (with new events prepended to the list).
- \emph{Events} of PIP fall
- into five categories defined as the datatype:
-
- \begin{isabelle}\ \ \ \ \ %%%
- \mbox{\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{7mm}}l}
- \isacommand{datatype} event
- & @{text "="} & @{term "Create thread priority"}\\
- & @{text "|"} & @{term "Exit thread"} \\
- & @{text "|"} & @{term "Set thread priority"} & {\rm reset of the priority for} @{text thread}\\
- & @{text "|"} & @{term "P thread cs"} & {\rm request of resource} @{text "cs"} {\rm by} @{text "thread"}\\
- & @{text "|"} & @{term "V thread cs"} & {\rm release of resource} @{text "cs"} {\rm by} @{text "thread"}
- \end{tabular}}
- \end{isabelle}
-
- \noindent
- whereby threads, priorities and (critical) resources are represented
- as natural numbers. The event @{term Set} models the situation that
- a thread obtains a new priority given by the programmer or
- user (for example via the {\tt nice} utility under UNIX). As in Paulson's work, we
- need to define functions that allow us to make some observations
- about states. One, called @{term threads}, calculates the set of
- ``live'' threads that we have seen so far:
-
- \begin{isabelle}\ \ \ \ \ %%%
- \mbox{\begin{tabular}{lcl}
- @{thm (lhs) threads.simps(1)} & @{text "\<equiv>"} &
- @{thm (rhs) threads.simps(1)}\\
- @{thm (lhs) threads.simps(2)[where thread="th"]} & @{text "\<equiv>"} &
- @{thm (rhs) threads.simps(2)[where thread="th"]}\\
- @{thm (lhs) threads.simps(3)[where thread="th"]} & @{text "\<equiv>"} &
- @{thm (rhs) threads.simps(3)[where thread="th"]}\\
- @{term "threads (DUMMY#s)"} & @{text "\<equiv>"} & @{term "threads s"}\\
- \end{tabular}}
- \end{isabelle}
-
- \noindent
- In this definition @{term "DUMMY # DUMMY"} stands for list-cons.
- Another function calculates the priority for a thread @{text "th"}, which is
- defined as
-
- \begin{isabelle}\ \ \ \ \ %%%
- \mbox{\begin{tabular}{lcl}
- @{thm (lhs) original_priority.simps(1)[where thread="th"]} & @{text "\<equiv>"} &
- @{thm (rhs) original_priority.simps(1)[where thread="th"]}\\
- @{thm (lhs) original_priority.simps(2)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
- @{thm (rhs) original_priority.simps(2)[where thread="th" and thread'="th'"]}\\
- @{thm (lhs) original_priority.simps(3)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
- @{thm (rhs) original_priority.simps(3)[where thread="th" and thread'="th'"]}\\
- @{term "original_priority th (DUMMY#s)"} & @{text "\<equiv>"} & @{term "original_priority th s"}\\
- \end{tabular}}
- \end{isabelle}
-
- \noindent
- In this definition we set @{text 0} as the default priority for
- threads that have not (yet) been created. The last function we need
- calculates the ``time'', or index, at which time a process had its
- priority last set.
-
- \begin{isabelle}\ \ \ \ \ %%%
- \mbox{\begin{tabular}{lcl}
- @{thm (lhs) birthtime.simps(1)[where thread="th"]} & @{text "\<equiv>"} &
- @{thm (rhs) birthtime.simps(1)[where thread="th"]}\\
- @{thm (lhs) birthtime.simps(2)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
- @{thm (rhs) birthtime.simps(2)[where thread="th" and thread'="th'"]}\\
- @{thm (lhs) birthtime.simps(3)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
- @{thm (rhs) birthtime.simps(3)[where thread="th" and thread'="th'"]}\\
- @{term "birthtime th (DUMMY#s)"} & @{text "\<equiv>"} & @{term "birthtime th s"}\\
- \end{tabular}}
- \end{isabelle}
-
- \noindent
- In this definition @{term "length s"} stands for the length of the list
- of events @{text s}. Again the default value in this function is @{text 0}
- for threads that have not been created yet. A \emph{precedence} of a thread @{text th} in a
- state @{text s} is the pair of natural numbers defined as
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{thm preced_def[where thread="th"]}
- \end{isabelle}
-
- \noindent
- The point of precedences is to schedule threads not according to priorities (because what should
- we do in case two threads have the same priority), but according to precedences.
- Precedences allow us to always discriminate between two threads with equal priority by
- taking into account the time when the priority was last set. We order precedences so
- that threads with the same priority get a higher precedence if their priority has been
- set earlier, since for such threads it is more urgent to finish their work. In an implementation
- this choice would translate to a quite natural FIFO-scheduling of processes with
- the same priority.
-
- Next, we introduce the concept of \emph{waiting queues}. They are
- lists of threads associated with every resource. The first thread in
- this list (i.e.~the head, or short @{term hd}) is chosen to be the one
- that is in possession of the
- ``lock'' of the corresponding resource. We model waiting queues as
- functions, below abbreviated as @{text wq}. They take a resource as
- argument and return a list of threads. This allows us to define
- when a thread \emph{holds}, respectively \emph{waits} for, a
- resource @{text cs} given a waiting queue function @{text wq}.
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm cs_holding_def[where thread="th"]}\\
- @{thm cs_waiting_def[where thread="th"]}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- In this definition we assume @{text "set"} converts a list into a set.
- At the beginning, that is in the state where no thread is created yet,
- the waiting queue function will be the function that returns the
- empty list for every resource.
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{abbrev all_unlocked}\hfill\numbered{allunlocked}
- \end{isabelle}
-
- \noindent
- Using @{term "holding"} and @{term waiting}, we can introduce \emph{Resource Allocation Graphs}
- (RAG), which represent the dependencies between threads and resources.
- We represent RAGs as relations using pairs of the form
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{term "(Th th, Cs cs)"} \hspace{5mm}{\rm and}\hspace{5mm}
- @{term "(Cs cs, Th th)"}
- \end{isabelle}
-
- \noindent
- where the first stands for a \emph{waiting edge} and the second for a
- \emph{holding edge} (@{term Cs} and @{term Th} are constructors of a
- datatype for vertices). Given a waiting queue function, a RAG is defined
- as the union of the sets of waiting and holding edges, namely
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{thm cs_depend_def}
- \end{isabelle}
-
- \noindent
- Given four threads and three resources, an instance of a RAG can be pictured
- as follows:
-
- \begin{center}
- \newcommand{\fnt}{\fontsize{7}{8}\selectfont}
- \begin{tikzpicture}[scale=1]
- %%\draw[step=2mm] (-3,2) grid (1,-1);
-
- \node (A) at (0,0) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>0"}};
- \node (B) at (2,0) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>1"}};
- \node (C) at (4,0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>1"}};
- \node (D) at (4,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>2"}};
- \node (E) at (6,-0.7) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>2"}};
- \node (E1) at (6, 0.2) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>3"}};
- \node (F) at (8,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>3"}};
-
- \draw [<-,line width=0.6mm] (A) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (B);
- \draw [->,line width=0.6mm] (C) to node [pos=0.4,sloped,above=-0.5mm] {\fnt{}waiting} (B);
- \draw [->,line width=0.6mm] (D) to node [pos=0.4,sloped,below=-0.5mm] {\fnt{}waiting} (B);
- \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,below=-0.5mm] {\fnt{}holding} (E);
- \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (E1);
- \draw [->,line width=0.6mm] (F) to node [pos=0.45,sloped,below=-0.5mm] {\fnt{}waiting} (E);
- \end{tikzpicture}
- \end{center}
-
- \noindent
- The use of relations for representing RAGs allows us to conveniently define
- the notion of the \emph{dependants} of a thread using the transitive closure
- operation for relations. This gives
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{thm cs_dependents_def}
- \end{isabelle}
-
- \noindent
- This definition needs to account for all threads that wait for a thread to
- release a resource. This means we need to include threads that transitively
- wait for a resource being released (in the picture above this means the dependants
- of @{text "th\<^isub>0"} are @{text "th\<^isub>1"} and @{text "th\<^isub>2"}, which wait for resource @{text "cs\<^isub>1"},
- but also @{text "th\<^isub>3"},
- which cannot make any progress unless @{text "th\<^isub>2"} makes progress, which
- in turn needs to wait for @{text "th\<^isub>0"} to finish). If there is a circle of dependencies
- in a RAG, then clearly
- we have a deadlock. Therefore when a thread requests a resource,
- we must ensure that the resulting RAG is not circular. In practice, the
- programmer has to ensure this.
-
-
- Next we introduce the notion of the \emph{current precedence} of a thread @{text th} in a
- state @{text s}. It is defined as
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{thm cpreced_def2}\hfill\numbered{cpreced}
- \end{isabelle}
-
- \noindent
- where the dependants of @{text th} are given by the waiting queue function.
- While the precedence @{term prec} of a thread is determined statically
- (for example when the thread is
- created), the point of the current precedence is to let the scheduler increase this
- precedence, if needed according to PIP. Therefore the current precedence of @{text th} is
- given as the maximum of the precedence @{text th} has in state @{text s} \emph{and} all
- threads that are dependants of @{text th}. Since the notion @{term "dependants"} is
- defined as the transitive closure of all dependent threads, we deal correctly with the
- problem in the informal algorithm by Sha et al.~\cite{Sha90} where a priority of a thread is
- lowered prematurely.
-
- The next function, called @{term schs}, defines the behaviour of the scheduler. It will be defined
- by recursion on the state (a list of events); this function returns a \emph{schedule state}, which
- we represent as a record consisting of two
- functions:
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{text "\<lparr>wq_fun, cprec_fun\<rparr>"}
- \end{isabelle}
-
- \noindent
- The first function is a waiting queue function (that is, it takes a
- resource @{text "cs"} and returns the corresponding list of threads
- that lock, respectively wait for, it); the second is a function that
- takes a thread and returns its current precedence (see
- the definition in \eqref{cpreced}). We assume the usual getter and setter methods for
- such records.
-
- In the initial state, the scheduler starts with all resources unlocked (the corresponding
- function is defined in \eqref{allunlocked}) and the
- current precedence of every thread is initialised with @{term "Prc 0 0"}; that means
- \mbox{@{abbrev initial_cprec}}. Therefore
- we have for the initial shedule state
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm (lhs) schs.simps(1)} @{text "\<equiv>"}\\
- \hspace{5mm}@{term "(|wq_fun = all_unlocked, cprec_fun = (\<lambda>_::thread. Prc 0 0)|)"}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- The cases for @{term Create}, @{term Exit} and @{term Set} are also straightforward:
- we calculate the waiting queue function of the (previous) state @{text s};
- this waiting queue function @{text wq} is unchanged in the next schedule state---because
- none of these events lock or release any resource;
- for calculating the next @{term "cprec_fun"}, we use @{text wq} and
- @{term cpreced}. This gives the following three clauses for @{term schs}:
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm (lhs) schs.simps(2)} @{text "\<equiv>"}\\
- \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
- \hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Create th prio # s)|)"}\smallskip\\
- @{thm (lhs) schs.simps(3)} @{text "\<equiv>"}\\
- \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
- \hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Exit th # s)|)"}\smallskip\\
- @{thm (lhs) schs.simps(4)} @{text "\<equiv>"}\\
- \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
- \hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Set th prio # s)|)"}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- More interesting are the cases where a resource, say @{text cs}, is locked or released. In these cases
- we need to calculate a new waiting queue function. For the event @{term "P th cs"}, we have to update
- the function so that the new thread list for @{text cs} is the old thread list plus the thread @{text th}
- appended to the end of that list (remember the head of this list is assigned to be in the possession of this
- resource). This gives the clause
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm (lhs) schs.simps(5)} @{text "\<equiv>"}\\
- \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
- \hspace{5mm}@{text "let"} @{text "new_wq = wq(cs := (wq cs @ [th]))"} @{text "in"}\\
- \hspace{8mm}@{term "(|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|)"}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- The clause for event @{term "V th cs"} is similar, except that we need to update the waiting queue function
- so that the thread that possessed the lock is deleted from the corresponding thread list. For this
- list transformation, we use
- the auxiliary function @{term release}. A simple version of @{term release} would
- just delete this thread and return the remaining threads, namely
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}lcl}
- @{term "release []"} & @{text "\<equiv>"} & @{term "[]"}\\
- @{term "release (DUMMY # qs)"} & @{text "\<equiv>"} & @{term "qs"}\\
- \end{tabular}
- \end{isabelle}
-
- \noindent
- In practice, however, often the thread with the highest precedence in the list will get the
- lock next. We have implemented this choice, but later found out that the choice
- of which thread is chosen next is actually irrelevant for the correctness of PIP.
- Therefore we prove the stronger result where @{term release} is defined as
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}lcl}
- @{term "release []"} & @{text "\<equiv>"} & @{term "[]"}\\
- @{term "release (DUMMY # qs)"} & @{text "\<equiv>"} & @{term "SOME qs'. distinct qs' \<and> set qs' = set qs"}\\
- \end{tabular}
- \end{isabelle}
-
- \noindent
- where @{text "SOME"} stands for Hilbert's epsilon and implements an arbitrary
- choice for the next waiting list. It just has to be a list of distinctive threads and
- contain the same elements as @{text "qs"}. This gives for @{term V} the clause:
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm (lhs) schs.simps(6)} @{text "\<equiv>"}\\
- \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
- \hspace{5mm}@{text "let"} @{text "new_wq = release (wq cs)"} @{text "in"}\\
- \hspace{8mm}@{term "(|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|)"}
- \end{tabular}
- \end{isabelle}
-
- Having the scheduler function @{term schs} at our disposal, we can ``lift'', or
- overload, the notions
- @{term waiting}, @{term holding}, @{term depend} and @{term cp} to operate on states only.
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}rcl}
- @{thm (lhs) s_holding_abv} & @{text "\<equiv>"} & @{thm (rhs) s_holding_abv}\\
- @{thm (lhs) s_waiting_abv} & @{text "\<equiv>"} & @{thm (rhs) s_waiting_abv}\\
- @{thm (lhs) s_depend_abv} & @{text "\<equiv>"} & @{thm (rhs) s_depend_abv}\\
- @{thm (lhs) cp_def} & @{text "\<equiv>"} & @{thm (rhs) cp_def}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- With these abbreviations in place we can introduce
- the notion of a thread being @{term ready} in a state (i.e.~threads
- that do not wait for any resource) and the running thread.
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm readys_def}\\
- @{thm runing_def}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- In the second definition @{term "DUMMY ` DUMMY"} stands for the image of a set under a function.
- Note that in the initial state, that is where the list of events is empty, the set
- @{term threads} is empty and therefore there is neither a thread ready nor running.
- If there is one or more threads ready, then there can only be \emph{one} thread
- running, namely the one whose current precedence is equal to the maximum of all ready
- threads. We use sets to capture both possibilities.
- We can now also conveniently define the set of resources that are locked by a thread in a
- given state and also when a thread is detached that state (meaning the thread neither
- holds nor waits for a resource):
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm holdents_def}\\
- @{thm detached_def}
- \end{tabular}
- \end{isabelle}
-
- %\noindent
- %The second definition states that @{text th} in @{text s}.
-
- Finally we can define what a \emph{valid state} is in our model of PIP. For
- example we cannot expect to be able to exit a thread, if it was not
- created yet.
- These validity constraints on states are characterised by the
- inductive predicate @{term "step"} and @{term vt}. We first give five inference rules
- for @{term step} relating a state and an event that can happen next.
-
- \begin{center}
- \begin{tabular}{c}
- @{thm[mode=Rule] thread_create[where thread=th]}\hspace{1cm}
- @{thm[mode=Rule] thread_exit[where thread=th]}
- \end{tabular}
- \end{center}
-
- \noindent
- The first rule states that a thread can only be created, if it is not alive yet.
- Similarly, the second rule states that a thread can only be terminated if it was
- running and does not lock any resources anymore (this simplifies slightly our model;
- in practice we would expect the operating system releases all locks held by a
- thread that is about to exit). The event @{text Set} can happen
- if the corresponding thread is running.
-
- \begin{center}
- @{thm[mode=Rule] thread_set[where thread=th]}
- \end{center}
-
- \noindent
- If a thread wants to lock a resource, then the thread needs to be
- running and also we have to make sure that the resource lock does
- not lead to a cycle in the RAG. In practice, ensuring the latter
- is the responsibility of the programmer. In our formal
- model we brush aside these problematic cases in order to be able to make
- some meaningful statements about PIP.\footnote{This situation is
- similar to the infamous \emph{occurs check} in Prolog: In order to say
- anything meaningful about unification, one needs to perform an occurs
- check. But in practice the occurs check is omitted and the
- responsibility for avoiding problems rests with the programmer.}
-
-
- \begin{center}
- @{thm[mode=Rule] thread_P[where thread=th]}
- \end{center}
-
- \noindent
- Similarly, if a thread wants to release a lock on a resource, then
- it must be running and in the possession of that lock. This is
- formally given by the last inference rule of @{term step}.
-
- \begin{center}
- @{thm[mode=Rule] thread_V[where thread=th]}
- \end{center}
-
- \noindent
- A valid state of PIP can then be conveniently be defined as follows:
-
- \begin{center}
- \begin{tabular}{c}
- @{thm[mode=Axiom] vt_nil}\hspace{1cm}
- @{thm[mode=Rule] vt_cons}
- \end{tabular}
- \end{center}
-
- \noindent
- This completes our formal model of PIP. In the next section we present
- properties that show our model of PIP is correct.
-*}
-
-section {* The Correctness Proof *}
-
-(*<*)
-context extend_highest_gen
-begin
-(*>*)
-text {*
- Sha et al.~state their first correctness criterion for PIP in terms
- of the number of low-priority threads \cite[Theorem 3]{Sha90}: if
- there are @{text n} low-priority threads, then a blocked job with
- high priority can only be blocked a maximum of @{text n} times.
- Their second correctness criterion is given
- in terms of the number of critical resources \cite[Theorem 6]{Sha90}: if there are
- @{text m} critical resources, then a blocked job with high priority
- can only be blocked a maximum of @{text m} times. Both results on their own, strictly speaking, do
- \emph{not} prevent indefinite, or unbounded, Priority Inversion,
- because if a low-priority thread does not give up its critical
- resource (the one the high-priority thread is waiting for), then the
- high-priority thread can never run. The argument of Sha et al.~is
- that \emph{if} threads release locked resources in a finite amount
- of time, then indefinite Priority Inversion cannot occur---the high-priority
- thread is guaranteed to run eventually. The assumption is that
- programmers must ensure that threads are programmed in this way. However, even
- taking this assumption into account, the correctness properties of
- Sha et al.~are
- \emph{not} true for their version of PIP---despite being ``proved''. As Yodaiken
- \cite{Yodaiken02} pointed out: If a low-priority thread possesses
- locks to two resources for which two high-priority threads are
- waiting for, then lowering the priority prematurely after giving up
- only one lock, can cause indefinite Priority Inversion for one of the
- high-priority threads, invalidating their two bounds.
-
- Even when fixed, their proof idea does not seem to go through for
- us, because of the way we have set up our formal model of PIP. One
- reason is that we allow critical sections, which start with a @{text P}-event
- and finish with a corresponding @{text V}-event, to arbitrarily overlap
- (something Sha et al.~explicitly exclude). Therefore we have
- designed a different correctness criterion for PIP. The idea behind
- our criterion is as follows: for all states @{text s}, we know the
- corresponding thread @{text th} with the highest precedence; we show
- that in every future state (denoted by @{text "s' @ s"}) in which
- @{text th} is still alive, either @{text th} is running or it is
- blocked by a thread that was alive in the state @{text s} and was waiting
- for or in the possession of a lock in @{text s}. Since in @{text s}, as in
- every state, the set of alive threads is finite, @{text th} can only
- be blocked a finite number of times. This is independent of how many
- threads of lower priority are created in @{text "s'"}. We will actually prove a
- stronger statement where we also provide the current precedence of
- the blocking thread. However, this correctness criterion hinges upon
- a number of assumptions about the states @{text s} and @{text "s' @
- s"}, the thread @{text th} and the events happening in @{text
- s'}. We list them next:
-
- \begin{quote}
- {\bf Assumptions on the states {\boldmath@{text s}} and
- {\boldmath@{text "s' @ s"}:}} We need to require that @{text "s"} and
- @{text "s' @ s"} are valid states:
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{l}
- @{term "vt s"}, @{term "vt (s' @ s)"}
- \end{tabular}
- \end{isabelle}
- \end{quote}
-
- \begin{quote}
- {\bf Assumptions on the thread {\boldmath@{text "th"}:}}
- The thread @{text th} must be alive in @{text s} and
- has the highest precedence of all alive threads in @{text s}. Furthermore the
- priority of @{text th} is @{text prio} (we need this in the next assumptions).
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{l}
- @{term "th \<in> threads s"}\\
- @{term "prec th s = Max (cprec s ` threads s)"}\\
- @{term "prec th s = (prio, DUMMY)"}
- \end{tabular}
- \end{isabelle}
- \end{quote}
-
- \begin{quote}
- {\bf Assumptions on the events in {\boldmath@{text "s'"}:}} We want to prove that @{text th} cannot
- be blocked indefinitely. Of course this can happen if threads with higher priority
- than @{text th} are continuously created in @{text s'}. Therefore we have to assume that
- events in @{text s'} can only create (respectively set) threads with equal or lower
- priority than @{text prio} of @{text th}. We also need to assume that the
- priority of @{text "th"} does not get reset and also that @{text th} does
- not get ``exited'' in @{text "s'"}. This can be ensured by assuming the following three implications.
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{l}
- {If}~~@{text "Create th' prio' \<in> set s'"}~~{then}~~@{text "prio' \<le> prio"}\\
- {If}~~@{text "Set th' prio' \<in> set s'"}~~{then}~~@{text "th' \<noteq> th"}~~{and}~~@{text "prio' \<le> prio"}\\
- {If}~~@{text "Exit th' \<in> set s'"}~~{then}~~@{text "th' \<noteq> th"}\\
- \end{tabular}
- \end{isabelle}
- \end{quote}
-
- \noindent
- The locale mechanism of Isabelle helps us to manage conveniently such assumptions~\cite{Haftmann08}.
- Under these assumptions we shall prove the following correctness property:
-
- \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' \<in> running (s' @ s)"} and @{text "th' \<noteq> th"} then
- @{text "th' \<in> threads s"}, @{text "\<not> detached s th'"} and
- @{term "cp (s' @ s) th' = prec th s"}.
- \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 current 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 further have to assume that every thread
- gives up its resources after a finite amount of time. We found that
- this assumption is awkward to formalise in our model. Therefore we
- leave it out and let the programmer assume the responsibility to
- program threads in such a benign manner (in addition to causing no
- circularity in the @{text RAG}). In this detail, we do not
- make any progress in comparison with the work by Sha et al.
- However, we are able to combine their two separate bounds into a
- single theorem improving their bound.
-
- In what follows we will describe properties of PIP that allow us to prove
- Theorem~\ref{mainthm} and, when instructive, briefly describe our argument.
- It is relatively easy to see that
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{text "running s \<subseteq> ready s \<subseteq> threads s"}\\
- @{thm[mode=IfThen] finite_threads}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- The second property is by induction of @{term vt}. The next three
- properties are
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm[mode=IfThen] waiting_unique[of _ _ "cs\<^isub>1" "cs\<^isub>2"]}\\
- @{thm[mode=IfThen] held_unique[of _ "th\<^isub>1" _ "th\<^isub>2"]}\\
- @{thm[mode=IfThen] runing_unique[of _ "th\<^isub>1" "th\<^isub>2"]}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- The first property states that every waiting thread can only wait for a single
- resource (because it gets suspended after requesting that resource); the second
- that every resource can only be held by a single thread;
- the third property establishes that in every given valid state, there is
- at most one running thread. We can also show the following properties
- about the @{term RAG} in @{text "s"}.
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{text If}~@{thm (prem 1) acyclic_depend}~@{text "then"}:\\
- \hspace{5mm}@{thm (concl) acyclic_depend},
- @{thm (concl) finite_depend} and
- @{thm (concl) wf_dep_converse},\\
- \hspace{5mm}@{text "if"}~@{thm (prem 2) dm_depend_threads}~@{text "then"}~@{thm (concl) dm_depend_threads}
- and\\
- \hspace{5mm}@{text "if"}~@{thm (prem 2) range_in}~@{text "then"}~@{thm (concl) range_in}.
- \end{tabular}
- \end{isabelle}
-
- \noindent
- The acyclicity property follows from how we restricted the events in
- @{text step}; similarly the finiteness and well-foundedness property.
- The last two properties establish that every thread in a @{text "RAG"}
- (either holding or waiting for a resource) is a live thread.
-
- The key lemma in our proof of Theorem~\ref{mainthm} is as follows:
-
- \begin{lemma}\label{mainlem}
- Given the assumptions about states @{text "s"} and @{text "s' @ s"},
- the thread @{text th} and the events in @{text "s'"},
- if @{term "th' \<in> threads (s' @ s)"}, @{text "th' \<noteq> th"} and @{text "detached (s' @ s) th'"}\\
- then @{text "th' \<notin> running (s' @ s)"}.
- \end{lemma}
-
- \noindent
- The point of this lemma is that a thread different from @{text th} (which has the highest
- precedence in @{text s}) and not holding any resource, cannot be running
- in the state @{text "s' @ s"}.
-
- \begin{proof}
- Since thread @{text "th'"} does not hold any resource, no thread can depend on it.
- Therefore its current precedence @{term "cp (s' @ s) th'"} equals its own precedence
- @{term "prec th' (s' @ s)"}. Since @{text "th"} has the highest precedence in the
- state @{text "(s' @ s)"} and precedences are distinct among threads, we have
- @{term "prec th' (s' @s ) < prec th (s' @ s)"}. From this
- we have @{term "cp (s' @ s) th' < prec th (s' @ s)"}.
- Since @{text "prec th (s' @ s)"} is already the highest
- @{term "cp (s' @ s) th"} can not be higher than this and can not be lower either (by
- definition of @{term "cp"}). Consequently, we have @{term "prec th (s' @ s) = cp (s' @ s) th"}.
- Finally we have @{term "cp (s' @ s) th' < cp (s' @ s) th"}.
- By defintion of @{text "running"}, @{text "th'"} can not be running in state
- @{text "s' @ s"}, as we had to show.\qed
- \end{proof}
-
- \noindent
- Since @{text "th'"} is not able to run in state @{text "s' @ s"}, it is not able to
- issue a @{text "P"} or @{text "V"} event. Therefore if @{text "s' @ s"} is extended
- one step further, @{text "th'"} still cannot hold any resource. The situation will
- not change in further extensions as long as @{text "th"} holds the highest precedence.
-
- From this lemma we can deduce Theorem~\ref{mainthm}: that @{text th} can only be
- blocked by a thread @{text th'} that
- held some resource in state @{text s} (that is not @{text "detached"}). And furthermore
- that the current precedence of @{text th'} in state @{text "(s' @ s)"} must be equal to the
- precedence of @{text th} in @{text "s"}.
- We show this theorem by induction on @{text "s'"} using Lemma~\ref{mainlem}.
- This theorem gives a stricter bound on the threads that can block @{text th} than the
- one obtained by Sha et al.~\cite{Sha90}:
- only threads that were alive in state @{text s} and moreover held a resource.
- This means our bound is in terms of both---alive threads in state @{text s}
- and number of critical resources. Finally, the theorem establishes that the blocking threads have the
- current precedence raised to the precedence of @{text th}.
-
- We can furthermore prove that under our assumptions no deadlock exists in the state @{text "s' @ s"}
- by showing that @{text "running (s' @ s)"} is not empty.
-
- \begin{lemma}
- Given the assumptions about states @{text "s"} and @{text "s' @ s"},
- the thread @{text th} and the events in @{text "s'"},
- @{term "running (s' @ s) \<noteq> {}"}.
- \end{lemma}
-
- \begin{proof}
- If @{text th} is blocked, then by following its dependants graph, we can always
- reach a ready thread @{text th'}, and that thread must have inherited the
- precedence of @{text th}.\qed
- \end{proof}
-
-
- %The following lemmas show how every node in RAG can be chased to ready threads:
- %\begin{enumerate}
- %\item Every node in RAG can be chased to a ready thread (@{text "chain_building"}):
- % @ {thm [display] chain_building[rule_format]}
- %\item The ready thread chased to is unique (@{text "dchain_unique"}):
- % @ {thm [display] dchain_unique[of _ _ "th\<^isub>1" "th\<^isub>2"]}
- %\end{enumerate}
-
- %Some deeper results about the system:
- %\begin{enumerate}
- %\item The maximum of @{term "cp"} and @{term "preced"} are equal (@{text "max_cp_eq"}):
- %@ {thm [display] max_cp_eq}
- %\item There must be one ready thread having the max @{term "cp"}-value
- %(@{text "max_cp_readys_threads"}):
- %@ {thm [display] max_cp_readys_threads}
- %\end{enumerate}
-
- %The relationship between the count of @{text "P"} and @{text "V"} and the number of
- %critical resources held by a thread is given as follows:
- %\begin{enumerate}
- %\item The @{term "V"}-operation decreases the number of critical resources
- % one thread holds (@{text "cntCS_v_dec"})
- % @ {thm [display] cntCS_v_dec}
- %\item The number of @{text "V"} never exceeds the number of @{text "P"}
- % (@ {text "cnp_cnv_cncs"}):
- % @ {thm [display] cnp_cnv_cncs}
- %\item The number of @{text "V"} equals the number of @{text "P"} when
- % the relevant thread is not living:
- % (@{text "cnp_cnv_eq"}):
- % @ {thm [display] cnp_cnv_eq}
- %\item When a thread is not living, it does not hold any critical resource
- % (@{text "not_thread_holdents"}):
- % @ {thm [display] not_thread_holdents}
- %\item When the number of @{text "P"} equals the number of @{text "V"}, the relevant
- % thread does not hold any critical resource, therefore no thread can depend on it
- % (@{text "count_eq_dependents"}):
- % @ {thm [display] count_eq_dependents}
- %\end{enumerate}
-
- %The reason that only threads which already held some resoures
- %can be runing and block @{text "th"} is that if , otherwise, one thread
- %does not hold any resource, it may never have its prioirty raised
- %and will not get a chance to run. This fact is supported by
- %lemma @{text "moment_blocked"}:
- %@ {thm [display] moment_blocked}
- %When instantiating @{text "i"} to @{text "0"}, the lemma means threads which did not hold any
- %resource in state @{text "s"} will not have a change to run latter. Rephrased, it means
- %any thread which is running after @{text "th"} became the highest must have already held
- %some resource at state @{text "s"}.
-
-
- %When instantiating @{text "i"} to a number larger than @{text "0"}, the lemma means
- %if a thread releases all its resources at some moment in @{text "t"}, after that,
- %it may never get a change to run. If every thread releases its resource in finite duration,
- %then after a while, only thread @{text "th"} is left running. This shows how indefinite
- %priority inversion can be avoided.
-
- %All these assumptions are put into a predicate @{term "extend_highest_gen"}.
- %It can be proved that @{term "extend_highest_gen"} holds
- %for any moment @{text "i"} in it @{term "t"} (@{text "red_moment"}):
- %@ {thm [display] red_moment}
-
- %From this, an induction principle can be derived for @{text "t"}, so that
- %properties already derived for @{term "t"} can be applied to any prefix
- %of @{text "t"} in the proof of new properties
- %about @{term "t"} (@{text "ind"}):
- %\begin{center}
- %@ {thm[display] ind}
- %\end{center}
-
- %The following properties can be proved about @{term "th"} in @{term "t"}:
- %\begin{enumerate}
- %\item In @{term "t"}, thread @{term "th"} is kept live and its
- % precedence is preserved as well
- % (@{text "th_kept"}):
- % @ {thm [display] th_kept}
- %\item In @{term "t"}, thread @{term "th"}'s precedence is always the maximum among
- % all living threads
- % (@{text "max_preced"}):
- % @ {thm [display] max_preced}
- %\item In @{term "t"}, thread @{term "th"}'s current precedence is always the maximum precedence
- % among all living threads
- % (@{text "th_cp_max_preced"}):
- % @ {thm [display] th_cp_max_preced}
- %\item In @{term "t"}, thread @{term "th"}'s current precedence is always the maximum current
- % precedence among all living threads
- % (@{text "th_cp_max"}):
- % @ {thm [display] th_cp_max}
- %\item In @{term "t"}, thread @{term "th"}'s current precedence equals its precedence at moment
- % @{term "s"}
- % (@{text "th_cp_preced"}):
- % @ {thm [display] th_cp_preced}
- %\end{enumerate}
-
- %The main theorem of this part is to characterizing the running thread during @{term "t"}
- %(@{text "runing_inversion_2"}):
- %@ {thm [display] runing_inversion_2}
- %According to this, if a thread is running, it is either @{term "th"} or was
- %already live and held some resource
- %at moment @{text "s"} (expressed by: @{text "cntV s th' < cntP s th'"}).
-
- %Since there are only finite many threads live and holding some resource at any moment,
- %if every such thread can release all its resources in finite duration, then after finite
- %duration, none of them may block @{term "th"} anymore. So, no priority inversion may happen
- %then.
- *}
-(*<*)
-end
-(*>*)
-
-section {* Properties for an Implementation\label{implement} *}
-
-text {*
- While our formalised proof gives us confidence about the correctness of our model of PIP,
- we found that the formalisation can even help us with efficiently implementing it.
-
- For example Baker complained that calculating the current precedence
- in PIP is quite ``heavy weight'' in Linux (see the Introduction).
- In our model of PIP the current precedence of a thread in a state @{text s}
- depends on all its dependants---a ``global'' transitive notion,
- which is indeed heavy weight (see Def.~shown in \eqref{cpreced}).
- We can however improve upon this. For this let us define the notion
- of @{term children} of a thread @{text th} in a state @{text s} as
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm children_def2}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- where a child is a thread that is only one ``hop'' away from the thread
- @{text th} in the @{term RAG} (and waiting for @{text th} to release
- a resource). We can prove the following lemma.
-
- \begin{lemma}\label{childrenlem}
- @{text "If"} @{thm (prem 1) cp_rec} @{text "then"}
- \begin{center}
- @{thm (concl) cp_rec}.
- \end{center}
- \end{lemma}
-
- \noindent
- That means the current precedence of a thread @{text th} can be
- computed locally by considering only the children of @{text th}. In
- effect, it only needs to be recomputed for @{text th} when one of
- its children changes its current precedence. Once the current
- precedence is computed in this more efficient manner, the selection
- of the thread with highest precedence from a set of ready threads is
- a standard scheduling operation implemented in most operating
- systems.
-
- Of course the main work for implementing PIP involves the
- scheduler and coding how it should react to events. Below we
- outline how our formalisation guides this implementation for each
- kind of events.\smallskip
-*}
-
-(*<*)
-context step_create_cps
-begin
-(*>*)
-text {*
- \noindent
- \colorbox{mygrey}{@{term "Create th prio"}:} We assume that the current state @{text s'} and
- the next state @{term "s \<equiv> Create th prio#s'"} are both valid (meaning the event
- is allowed to occur). In this situation we can show that
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm eq_dep},\\
- @{thm eq_cp_th}, and\\
- @{thm[mode=IfThen] eq_cp}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- This means in an implementation we do not have recalculate the @{text RAG} and also none of the
- current precedences of the other threads. The current precedence of the created
- thread @{text th} is just its precedence, namely the pair @{term "(prio, length (s::event list))"}.
- \smallskip
- *}
-(*<*)
-end
-context step_exit_cps
-begin
-(*>*)
-text {*
- \noindent
- \colorbox{mygrey}{@{term "Exit th"}:} We again assume that the current state @{text s'} and
- the next state @{term "s \<equiv> Exit th#s'"} are both valid. We can show that
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm eq_dep}, and\\
- @{thm[mode=IfThen] eq_cp}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- This means again we do not have to recalculate the @{text RAG} and
- also not the current precedences for the other threads. Since @{term th} is not
- alive anymore in state @{term "s"}, there is no need to calculate its
- current precedence.
- \smallskip
-*}
-(*<*)
-end
-context step_set_cps
-begin
-(*>*)
-text {*
- \noindent
- \colorbox{mygrey}{@{term "Set th prio"}:} We assume that @{text s'} and
- @{term "s \<equiv> Set th prio#s'"} are both valid. We can show that
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm[mode=IfThen] eq_dep}, and\\
- @{thm[mode=IfThen] eq_cp_pre}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- The first property is again telling us we do not need to change the @{text RAG}.
- The second shows that the @{term cp}-values of all threads other than @{text th}
- are unchanged. The reason is that @{text th} is running; therefore it is not in
- the @{term dependants} relation of any other thread. This in turn means that the
- change of its priority cannot affect other threads.
-
- %The second
- %however states that only threads that are \emph{not} dependants of @{text th} have their
- %current precedence unchanged. For the others we have to recalculate the current
- %precedence. To do this we can start from @{term "th"}
- %and follow the @{term "depend"}-edges to recompute using Lemma~\ref{childrenlem}
- %the @{term "cp"} of every
- %thread encountered on the way. Since the @{term "depend"}
- %is assumed to be loop free, this procedure will always stop. The following two lemmas show, however,
- %that this procedure can actually stop often earlier without having to consider all
- %dependants.
- %
- %\begin{isabelle}\ \ \ \ \ %%%
- %\begin{tabular}{@ {}l}
- %@{thm[mode=IfThen] eq_up_self}\\
- %@{text "If"} @{thm (prem 1) eq_up}, @{thm (prem 2) eq_up} and @{thm (prem 3) eq_up}\\
- %@{text "then"} @{thm (concl) eq_up}.
- %\end{tabular}
- %\end{isabelle}
- %
- %\noindent
- %The first lemma states that if the current precedence of @{text th} is unchanged,
- %then the procedure can stop immediately (all dependent threads have their @{term cp}-value unchanged).
- %The second states that if an intermediate @{term cp}-value does not change, then
- %the procedure can also stop, because none of its dependent threads will
- %have their current precedence changed.
- \smallskip
- *}
-(*<*)
-end
-context step_v_cps_nt
-begin
-(*>*)
-text {*
- \noindent
- \colorbox{mygrey}{@{term "V th cs"}:} We assume that @{text s'} and
- @{term "s \<equiv> V th cs#s'"} are both valid. We have to consider two
- subcases: one where there is a thread to ``take over'' the released
- resource @{text cs}, and one where there is not. Let us consider them
- in turn. Suppose in state @{text s}, the thread @{text th'} takes over
- resource @{text cs} from thread @{text th}. We can prove
-
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{thm depend_s}
- \end{isabelle}
-
- \noindent
- which shows how the @{text RAG} needs to be changed. The next lemma suggests
- how the current precedences need to be recalculated. For threads that are
- not @{text "th"} and @{text "th'"} nothing needs to be changed, since we
- can show
-
- \begin{isabelle}\ \ \ \ \ %%%
- @{thm[mode=IfThen] cp_kept}
- \end{isabelle}
-
- \noindent
- For @{text th} and @{text th'} we need to use Lemma~\ref{childrenlem} to
- recalculate their current precedence since their children have changed. *}(*<*)end context step_v_cps_nnt begin (*>*)text {*
- \noindent
- In the other case where there is no thread that takes over @{text cs}, we can show how
- to recalculate the @{text RAG} and also show that no current precedence needs
- to be recalculated.
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm depend_s}\\
- @{thm eq_cp}
- \end{tabular}
- \end{isabelle}
- *}
-(*<*)
-end
-context step_P_cps_e
-begin
-(*>*)
-text {*
- \noindent
- \colorbox{mygrey}{@{term "P th cs"}:} We assume that @{text s'} and
- @{term "s \<equiv> P th cs#s'"} are both valid. We again have to analyse two subcases, namely
- the one where @{text cs} is not locked, and one where it is. We treat the former case
- first by showing that
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm depend_s}\\
- @{thm eq_cp}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- This means we need to add a holding edge to the @{text RAG} and no
- current precedence needs to be recalculated.*}(*<*)end context step_P_cps_ne begin(*>*) text {*
- \noindent
- In the second case we know that resource @{text cs} is locked. We can show that
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- @{thm depend_s}\\
- @{thm[mode=IfThen] eq_cp}
- \end{tabular}
- \end{isabelle}
-
- \noindent
- That means we have to add a waiting edge to the @{text RAG}. Furthermore
- the current precedence for all threads that are not dependants of @{text th}
- are unchanged. For the others we need to follow the edges
- in the @{text RAG} and recompute the @{term "cp"}. To do this we can start from @{term "th"}
- and follow the @{term "depend"}-edges to recompute using Lemma~\ref{childrenlem}
- the @{term "cp"} of every
- thread encountered on the way. Since the @{term "depend"}
- is loop free, this procedure will always stop. The following lemma shows, however,
- that this procedure can actually stop often earlier without having to consider all
- dependants.
-
- \begin{isabelle}\ \ \ \ \ %%%
- \begin{tabular}{@ {}l}
- %%@ {t hm[mode=IfThen] eq_up_self}\\
- @{text "If"} @{thm (prem 1) eq_up}, @{thm (prem 2) eq_up} and @{thm (prem 3) eq_up}\\
- @{text "then"} @{thm (concl) eq_up}.
- \end{tabular}
- \end{isabelle}
-
- \noindent
- This lemma states that if an intermediate @{term cp}-value does not change, then
- the procedure can also stop, because none of its dependent threads will
- have their current precedence changed.
- *}
-(*<*)
-end
-(*>*)
-text {*
- \noindent
- As can be seen, a pleasing byproduct of our formalisation is that the properties in
- this section closely inform an implementation of PIP, namely whether the
- @{text RAG} needs to be reconfigured or current precedences need to
- be recalculated for an event. This information is provided by the lemmas we proved.
- We confirmed that our observations translate into practice by implementing
- our version of PIP on top of PINTOS, a small operating system written in C and used for teaching at
- Stanford University \cite{PINTOS}. To implement PIP, we only need to modify the kernel
- functions corresponding to the events in our formal model. The events translate to the following
- function interface in PINTOS:
-
- \begin{center}
- \begin{tabular}{|l@ {\hspace{2mm}}|l@ {\hspace{2mm}}|}
- \hline
- {\bf Event} & {\bf PINTOS function} \\
- \hline
- @{text Create} & @{text "thread_create"}\\
- @{text Exit} & @{text "thread_exit"}\\
- @{text Set} & @{text "thread_set_priority"}\\
- @{text P} & @{text "lock_acquire"}\\
- @{text V} & @{text "lock_release"}\\
- \hline
- \end{tabular}
- \end{center}
-
- \noindent
- Our implicit assumption that every event is an atomic operation is ensured by the architecture of
- PINTOS. The case where an unlocked resource is given next to the waiting thread with the
- highest precedence is realised in our implementation by priority queues. We implemented
- them as \emph{Braun trees} \cite{Paulson96}, which provide efficient @{text "O(log n)"}-operations
- for accessing and updating. Apart from having to implement relatively complex data\-structures in C
- using pointers, our experience with the implementation has been very positive: our specification
- and formalisation of PIP translates smoothly to an efficent implementation in PINTOS.
-*}
-
-section {* Conclusion *}
-
-text {*
- The Priority Inheritance Protocol (PIP) is a classic textbook
- algorithm used in many real-time operating systems in order to avoid the problem of
- Priority Inversion. Although classic and widely used, PIP does have
- its faults: for example it does not prevent deadlocks in cases where threads
- have circular lock dependencies.
-
- We had two goals in mind with our formalisation of PIP: One is to
- make the notions in the correctness proof by Sha et al.~\cite{Sha90}
- precise so that they can be processed by a theorem prover. The reason is
- that a mechanically checked proof avoids the flaws that crept into their
- informal reasoning. We achieved this goal: The correctness of PIP now
- only hinges on the assumptions behind our formal model. The reasoning, which is
- sometimes quite intricate and tedious, has been checked by Isabelle/HOL.
- We can also confirm that Paulson's
- inductive method for protocol verification~\cite{Paulson98} is quite
- suitable for our formal model and proof. The traditional application
- area of this method is security protocols.
-
- The second goal of our formalisation is to provide a specification for actually
- implementing PIP. Textbooks, for example \cite[Section 5.6.5]{Vahalia96},
- explain how to use various implementations of PIP and abstractly
- discuss their properties, but surprisingly lack most details important for a
- programmer who wants to implement PIP (similarly Sha et al.~\cite{Sha90}).
- That this is an issue in practice is illustrated by the
- email from Baker we cited in the Introduction. We achieved also this
- goal: The formalisation allowed us to efficently implement our version
- of PIP on top of PINTOS \cite{PINTOS}, a simple instructional operating system for the x86
- architecture. It also gives the first author enough data to enable
- his undergraduate students to implement PIP (as part of their OS course).
- A byproduct of our formalisation effort is that nearly all
- design choices for the PIP scheduler are backed up with a proved
- lemma. We were also able to establish the property that the choice of
- the next thread which takes over a lock is irrelevant for the correctness
- of PIP.
-
- PIP is a scheduling algorithm for single-processor systems. We are
- now living in a multi-processor world. Priority Inversion certainly
- occurs also there. However, there is very little ``foundational''
- work about PIP-algorithms on multi-processor systems. We are not
- aware of any correctness proofs, not even informal ones. There is an
- implementation of a PIP-algorithm for multi-processors as part of the
- ``real-time'' effort in Linux, including an informal description of the implemented scheduling
- algorithm given in \cite{LINUX}. We estimate that the formal
- verification of this algorithm, involving more fine-grained events,
- is a magnitude harder than the one we presented here, but still
- within reach of current theorem proving technology. We leave this
- for future work.
-
- The most closely related work to ours is the formal verification in
- PVS of the Priority Ceiling Protocol done by Dutertre
- \cite{dutertre99b}---another solution to the Priority Inversion
- problem, which however needs static analysis of programs in order to
- avoid it. There have been earlier formal investigations
- into PIP \cite{Faria08,Jahier09,Wellings07}, but they employ model
- checking techniques. The results obtained by them apply,
- however, only to systems with a fixed size, such as a fixed number of
- events and threads. In contrast, our result applies to systems of arbitrary
- size. Moreover, our result is a good
- witness for one of the major reasons to be interested in machine checked
- reasoning: gaining deeper understanding of the subject matter.
-
- Our formalisation
- consists of around 210 lemmas and overall 6950 lines of readable Isabelle/Isar
- code with a few apply-scripts interspersed. The formal model of PIP
- is 385 lines long; the formal correctness proof 3800 lines. Some auxiliary
- definitions and proofs span over 770 lines of code. The properties relevant
- for an implementation require 2000 lines.
- %The code of our formalisation
- %can be downloaded from
- %\url{http://www.inf.kcl.ac.uk/staff/urbanc/pip.html}.\medskip
-
- \noindent
- {\bf Acknowledgements:}
- We are grateful for the comments we received from anonymous
- referees.
-
- \bibliographystyle{plain}
- \bibliography{root}
-*}
-
-
-(*<*)
-end
-(*>*)
\ No newline at end of file
--- a/prio/Paper/PrioGDef.tex Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,488 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{PrioGDef}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-In this section, the formal model of Priority Inheritance is presented. First, the identifiers of {\em threads},
- {\em priority} and {\em critical resources } (abbreviated as \isa{cs}) are all represented as natural numbers,
- i.e. standard Isabelle/HOL type \isa{nat}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
-\ thread\ {\isaliteral{3D}{\isacharequal}}\ nat\ %
-\isamarkupcmt{Type for thread identifiers.%
-}
-\isanewline
-\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
-\ priority\ {\isaliteral{3D}{\isacharequal}}\ nat\ \ %
-\isamarkupcmt{Type for priorities.%
-}
-\isanewline
-\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
-\ cs\ {\isaliteral{3D}{\isacharequal}}\ nat\ %
-\isamarkupcmt{Type for critical sections (or critical resources).%
-}
-%
-\begin{isamarkuptext}%
-Priority Inheritance protocol is modeled as an event driven system, where every event represents an
- system call. Event format is given by the following type definition:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ event\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
-\ \ Create\ thread\ priority\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{Thread \isa{thread} is created with priority \isa{priority}.%
-}
-\isanewline
-\ \ Exit\ thread\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{Thread \isa{thread} finishing its execution.%
-}
-\isanewline
-\ \ P\ thread\ cs\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{Thread \isa{thread} requesting critical resource \isa{cs}.%
-}
-\isanewline
-\ \ V\ thread\ cs\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{Thread \isa{thread} releasing critical resource \isa{cs}.%
-}
-\isanewline
-\ \ Set\ thread\ priority\ %
-\isamarkupcmt{Thread \isa{thread} resets its priority to \isa{priority}.%
-}
-%
-\begin{isamarkuptext}%
-Resource Allocation Graph (RAG for short) is used extensively in the analysis of Priority Inheritance.
- The following type \isa{node} is used to model nodes in RAG.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ node\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
-\ \ \ Th\ {\isaliteral{22}{\isachardoublequoteopen}}thread{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{Node for thread.%
-}
-\isanewline
-\ \ \ Cs\ {\isaliteral{22}{\isachardoublequoteopen}}cs{\isaliteral{22}{\isachardoublequoteclose}}\ %
-\isamarkupcmt{Node for critical resource.%
-}
-%
-\begin{isamarkuptext}%
-The protocol is analyzed using Paulson's inductive protocol verification method, where
- the state of the system is modelled as the list of events happened so far with the latest
- event at the head. Therefore, the state of the system is represented by the following
- type \isa{state}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
-\ state\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}event\ list{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The following \isa{threads} is used to calculate the set of live threads (\isa{threads\ s})
- in state \isa{s}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ threads\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ \isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{At the start of the system, the set of threads is empty.%
-}
-\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}Create\ thread\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}thread{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ threads\ s{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{New thread is added to the \isa{threads}.%
-}
-\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}Exit\ thread\ {\isaliteral{23}{\isacharhash}}\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}threads\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{7B}{\isacharbraceleft}}thread{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{Finished thread is removed.%
-}
-\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ threads\ s{\isaliteral{22}{\isachardoublequoteclose}}\ %
-\isamarkupcmt{other kind of events does not affect the value of \isa{threads}.%
-}
-%
-\begin{isamarkuptext}%
-Functions such as \isa{threads}, which extract information out of system states, are called
- {\em observing functions}. A series of observing functions will be defined in the sequel in order to
- model the protocol.
- Observing function \isa{original{\isaliteral{5F}{\isacharunderscore}}priority} calculates
- the {\em original priority} of thread \isa{th} in state \isa{s}, expressed as
- : \isa{original{\isaliteral{5F}{\isacharunderscore}}priority\ th\ s}. The {\em original priority} is the priority
- assigned to a thread when it is created or when it is reset by system call \isa{Set\ thread\ priority}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ original{\isaliteral{5F}{\isacharunderscore}}priority\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ priority{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
-\isamarkupcmt{\isa{{\isadigit{0}}} is assigned to threads which have never been created.%
-}
-\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}Create\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
-\ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ thread{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ thread\ then\ prio\ else\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}Set\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
-\ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ thread{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ thread\ then\ prio\ else\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\isa{birthtime\ th\ s} is the time when thread \isa{th} is created, observed from state \isa{s}.
- The time in the system is measured by the number of events happened so far since the very beginning.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ birthtime\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Create\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{3D}{\isacharequal}}\ thread{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ then\ length\ s\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Set\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{3D}{\isacharequal}}\ thread{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ then\ length\ s\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ birthtime\ thread\ s{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of
- a thread is the combination of its {\em original priority} and {\em birth time}. The intention is
- to discriminate threads with the same priority by giving threads with the earlier assigned priority
- higher precedence in scheduling. This explains the following definition:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ preced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}preced\ thread\ s\ {\isaliteral{3D}{\isacharequal}}\ Prc\ {\isaliteral{28}{\isacharparenleft}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-A number of important notions are defined here:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{consts}\isamarkupfalse%
-\ \isanewline
-\ \ holding\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isanewline
-\ \ \ \ \ \ \ waiting\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ depend\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}node\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ node{\isaliteral{29}{\isacharparenright}}\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ dependents\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The definition of the following several functions, it is supposed that
- the waiting queue of every critical resource is given by a waiting queue
- function \isa{wq}, which servers as arguments of these functions.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{defs}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{overloaded}{\isaliteral{29}{\isacharparenright}}\ \isanewline
-\ \ %
-\isamarkupcmt{\begin{minipage}{0.8\textwidth}
- We define that the thread which is at the head of waiting queue of resource \isa{cs}
- is holding the resource. This definition is slightly different from tradition where
- all threads in the waiting queue are considered as waiting for the resource.
- This notion is reflected in the definition of \isa{holding\ wq\ th\ cs} as follows:
- \end{minipage}%
-}
-\isanewline
-\ \ cs{\isaliteral{5F}{\isacharunderscore}}holding{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}holding\ wq\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ %
-\isamarkupcmt{\begin{minipage}{0.8\textwidth}
- In accordance with the definition of \isa{holding\ wq\ th\ cs},
- a thread \isa{th} is considered waiting for \isa{cs} if
- it is in the {\em waiting queue} of critical resource \isa{cs}, but not at the head.
- This is reflected in the definition of \isa{waiting\ wq\ th\ cs} as follows:
- \end{minipage}%
-}
-\isanewline
-\ \ cs{\isaliteral{5F}{\isacharunderscore}}waiting{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}waiting\ wq\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ %
-\isamarkupcmt{\begin{minipage}{0.8\textwidth}
- \isa{depend\ wq} represents the Resource Allocation Graph of the system under the waiting
- queue function \isa{wq}.
- \end{minipage}%
-}
-\isanewline
-\ \ cs{\isaliteral{5F}{\isacharunderscore}}depend{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}depend\ {\isaliteral{28}{\isacharparenleft}}wq{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Th\ t{\isaliteral{2C}{\isacharcomma}}\ Cs\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ t\ c{\isaliteral{2E}{\isachardot}}\ waiting\ wq\ t\ c{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Cs\ c{\isaliteral{2C}{\isacharcomma}}\ Th\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ c\ t{\isaliteral{2E}{\isachardot}}\ holding\ wq\ t\ c{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ %
-\isamarkupcmt{\begin{minipage}{0.8\textwidth}
- \isa{dependents\ wq\ th} represents the set of threads which are depending on
- thread \isa{th} in Resource Allocation Graph \isa{depend\ wq}:
- \end{minipage}%
-}
-\isanewline
-\ \ cs{\isaliteral{5F}{\isacharunderscore}}dependents{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dependents\ {\isaliteral{28}{\isacharparenleft}}wq{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ th\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{27}{\isacharprime}}\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ wq{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The data structure used by the operating system for scheduling is referred to as
- {\em schedule state}. It is represented as a record consisting of
- a function assigning waiting queue to resources and a function assigning precedence to
- threads:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{record}\isamarkupfalse%
-\ schedule{\isaliteral{5F}{\isacharunderscore}}state\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
-\ \ \ \ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{22}{\isachardoublequoteclose}}\ %
-\isamarkupcmt{The function assigning waiting queue.%
-}
-\isanewline
-\ \ \ \ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\ %
-\isamarkupcmt{The function assigning precedence.%
-}
-%
-\begin{isamarkuptext}%
-\isa{cpreced\ s\ th} gives the {\em current precedence} of thread \isa{th} under
- state \isa{s}. The definition of \isa{cpreced} reflects the basic idea of
- Priority Inheritance that the {\em current precedence} of a thread is the precedence
- inherited from the maximum of all its dependents, i.e. the threads which are waiting
- directly or indirectly waiting for some resources from it. If no such thread exits,
- \isa{th}'s {\em current precedence} equals its original precedence, i.e.
- \isa{preced\ th\ s}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ cpreced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cpreced\ s\ wq\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ th{\isaliteral{2E}{\isachardot}}\ Max\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ th{\isaliteral{2E}{\isachardot}}\ preced\ th\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ dependents\ wq\ th{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The following function \isa{schs} is used to calculate the schedule state \isa{schs\ s}.
- It is the key function to model Priority Inheritance:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ schs\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ schedule{\isaliteral{5F}{\isacharunderscore}}state{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ \ {\isaliteral{22}{\isachardoublequoteopen}}schs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{2C}{\isacharcomma}}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3D}{\isacharequal}}\ cpreced\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ %
-\isamarkupcmt{\begin{minipage}{0.8\textwidth}
- \begin{enumerate}
- \item \isa{ps} is the schedule state of last moment.
- \item \isa{pwq} is the waiting queue function of last moment.
- \item \isa{pcp} is the precedence function of last moment.
- \item \isa{nwq} is the new waiting queue function. It is calculated using a \isa{case} statement:
- \begin{enumerate}
- \item If the happening event is \isa{P\ thread\ cs}, \isa{thread} is added to
- the end of \isa{cs}'s waiting queue.
- \item If the happening event is \isa{V\ thread\ cs} and \isa{s} is a legal state,
- \isa{th{\isaliteral{27}{\isacharprime}}} must equal to \isa{thread},
- because \isa{thread} is the one currently holding \isa{cs}.
- The case \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} may never be executed in a legal state.
- the \isa{{\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ qs{\isaliteral{29}{\isacharparenright}}} is used to choose arbitrarily one
- thread in waiting to take over the released resource \isa{cs}. In our representation,
- this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
- \item For other happening event, the schedule state just does not change.
- \end{enumerate}
- \item \isa{ncp} is new precedence function, it is calculated from the newly updated waiting queue
- function. The dependency of precedence function on waiting queue function is the reason to
- put them in the same record so that they can evolve together.
- \end{enumerate}
- \end{minipage}%
-}
-\isanewline
-\ \ \ {\isaliteral{22}{\isachardoublequoteopen}}schs\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}let\ ps\ {\isaliteral{3D}{\isacharequal}}\ schs\ s\ in\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ pwq\ {\isaliteral{3D}{\isacharequal}}\ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ ps\ in\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ pcp\ {\isaliteral{3D}{\isacharequal}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ ps\ in\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ nwq\ {\isaliteral{3D}{\isacharequal}}\ case\ e\ of\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P\ thread\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ \ pwq{\isaliteral{28}{\isacharparenleft}}cs{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{28}{\isacharparenleft}}pwq\ cs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}thread{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ V\ thread\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ let\ nq\ {\isaliteral{3D}{\isacharequal}}\ case\ {\isaliteral{28}{\isacharparenleft}}pwq\ cs{\isaliteral{29}{\isacharparenright}}\ of\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7C}{\isacharbar}}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}th{\isaliteral{27}{\isacharprime}}{\isaliteral{23}{\isacharhash}}qs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ qs{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ pwq{\isaliteral{28}{\isacharparenleft}}cs{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}nq{\isaliteral{29}{\isacharparenright}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ pwq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ let\ ncp\ {\isaliteral{3D}{\isacharequal}}\ cpreced\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ nwq\ in\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3D}{\isacharequal}}\ nwq{\isaliteral{2C}{\isacharcomma}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3D}{\isacharequal}}\ ncp{\isaliteral{5C3C72706172723E}{\isasymrparr}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\isa{wq} is a shorthand for \isa{waiting{\isaliteral{5F}{\isacharunderscore}}queue}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ wq\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{22}{\isachardoublequoteclose}}\ \isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}wq\ s\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{28}{\isacharparenleft}}schs\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\isa{cp} is a shorthand for \isa{cur{\isaliteral{5F}{\isacharunderscore}}preced}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ cp\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cp\ s\ {\isaliteral{3D}{\isacharequal}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{28}{\isacharparenleft}}schs\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-Functions \isa{holding}, \isa{waiting}, \isa{depend} and \isa{dependents} still have the
- same meaning, but redefined so that they no longer depend on the fictitious {\em waiting queue function}
- \isa{wq}, but on system state \isa{s}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{defs}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{overloaded}{\isaliteral{29}{\isacharparenright}}\ \isanewline
-\ \ s{\isaliteral{5F}{\isacharunderscore}}holding{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}holding\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ s{\isaliteral{5F}{\isacharunderscore}}waiting{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}waiting\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ s{\isaliteral{5F}{\isacharunderscore}}depend{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}depend\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Th\ t{\isaliteral{2C}{\isacharcomma}}\ Cs\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ t\ c{\isaliteral{2E}{\isachardot}}\ waiting\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}\ t\ c{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Cs\ c{\isaliteral{2C}{\isacharcomma}}\ Th\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ c\ t{\isaliteral{2E}{\isachardot}}\ holding\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}\ t\ c{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ s{\isaliteral{5F}{\isacharunderscore}}dependents{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dependents\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ th\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{27}{\isacharprime}}\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The following function \isa{readys} calculates the set of ready threads. A thread is {\em ready}
- for running if it is a live thread and it is not waiting for any critical resource.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ readys\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}readys\ s\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
-\ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}thread\ {\isaliteral{2E}{\isachardot}}\ thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ threads\ s\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ waiting\ s\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The following function \isa{runing} calculates the set of running thread, which is the ready
- thread with the highest precedence.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ runing\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}runing\ s\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th\ {\isaliteral{2E}{\isachardot}}\ th\ {\isaliteral{5C3C696E3E}{\isasymin}}\ readys\ s\ {\isaliteral{5C3C616E643E}{\isasymand}}\ cp\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ Max\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}cp\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{28}{\isacharparenleft}}readys\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The following function \isa{holdents\ s\ th} returns the set of resources held by thread
- \isa{th} in state \isa{s}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ holdents\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}holdents\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}cs\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ depend\ s{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\isa{cntCS\ s\ th} returns the number of resources held by thread \isa{th} in
- state \isa{s}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ cntCS\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntCS\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ card\ {\isaliteral{28}{\isacharparenleft}}holdents\ s\ th{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The fact that event \isa{e} is eligible to happen next in state \isa{s}
- is expressed as \isa{step\ s\ e}. The predicate \isa{step} is inductively defined as
- follows:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{inductive}\isamarkupfalse%
-\ step\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ event\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ %
-\isamarkupcmt{A thread can be created if it is not a live thread:%
-}
-\isanewline
-\ \ thread{\isaliteral{5F}{\isacharunderscore}}create{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ threads\ s{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Create\ thread\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ %
-\isamarkupcmt{A thread can exit if it no longer hold any resource:%
-}
-\isanewline
-\ \ thread{\isaliteral{5F}{\isacharunderscore}}exit{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ holdents\ s\ thread\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Exit\ thread{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ %
-\isamarkupcmt{A thread can request for an critical resource \isa{cs}, if it is running and
- the request does not form a loop in the current RAG. The latter condition
- is set up to avoid deadlock. The condition also reflects our assumption all threads are
- carefully programmed so that deadlock can not happen.%
-}
-\isanewline
-\ \ thread{\isaliteral{5F}{\isacharunderscore}}P{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ \ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{2C}{\isacharcomma}}\ Th\ thread{\isaliteral{29}{\isacharparenright}}\ \ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}P\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ %
-\isamarkupcmt{A thread can release a critical resource \isa{cs} if it is running and holding that resource.%
-}
-\isanewline
-\ \ thread{\isaliteral{5F}{\isacharunderscore}}V{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ \ holding\ s\ thread\ cs{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}V\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ %
-\isamarkupcmt{A thread can adjust its own priority as long as it is current running.%
-}
-\ \ \isanewline
-\ \ thread{\isaliteral{5F}{\isacharunderscore}}set{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Set\ thread\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-With predicate \isa{step}, the fact that \isa{s} is a legal state in
- Priority Inheritance protocol can be expressed as: \isa{vt\ step\ s}, where
- the predicate \isa{vt} can be defined as the following:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{inductive}\isamarkupfalse%
-\ vt\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ event\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \isakeyword{for}\ cs\ %
-\isamarkupcmt{\isa{cs} is an argument representing any step predicate.%
-}
-\isanewline
-\isakeyword{where}\isanewline
-\ \ %
-\isamarkupcmt{Empty list \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} is a legal state in any protocol:%
-}
-\isanewline
-\ \ vt{\isaliteral{5F}{\isacharunderscore}}nil{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}vt\ cs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-\ \ %
-\isamarkupcmt{If \isa{s} a legal state, and event \isa{e} is eligible to happen
- in state \isa{s}, then \isa{e{\isaliteral{23}{\isacharhash}}{\isaliteral{23}{\isacharhash}}s} is a legal state as well:%
-}
-\isanewline
-\ \ vt{\isaliteral{5F}{\isacharunderscore}}cons{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}vt\ cs\ s{\isaliteral{3B}{\isacharsemicolon}}\ cs\ s\ e{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ vt\ cs\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-It is easy to see that the definition of \isa{vt} is generic. It can be applied to
- any step predicate to get the set of legal states.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The following two functions \isa{the{\isaliteral{5F}{\isacharunderscore}}cs} and \isa{the{\isaliteral{5F}{\isacharunderscore}}th} are used to extract
- critical resource and thread respectively out of RAG nodes.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ the{\isaliteral{5F}{\isacharunderscore}}cs\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}node\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}the{\isaliteral{5F}{\isacharunderscore}}cs\ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ cs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ the{\isaliteral{5F}{\isacharunderscore}}th\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}node\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}the{\isaliteral{5F}{\isacharunderscore}}th\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ th{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The following predicate \isa{next{\isaliteral{5F}{\isacharunderscore}}th} describe the next thread to
- take over when a critical resource is released. In \isa{next{\isaliteral{5F}{\isacharunderscore}}th\ s\ th\ cs\ t},
- \isa{th} is the thread to release, \isa{t} is the one to take over.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ next{\isaliteral{5F}{\isacharunderscore}}th{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}next{\isaliteral{5F}{\isacharunderscore}}th\ s\ th\ cs\ t\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}\ rest{\isaliteral{2E}{\isachardot}}\ wq\ s\ cs\ {\isaliteral{3D}{\isacharequal}}\ th{\isaliteral{23}{\isacharhash}}rest\ {\isaliteral{5C3C616E643E}{\isasymand}}\ rest\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ rest{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The function \isa{count\ Q\ l} is used to count the occurrence of situation \isa{Q}
- in list \isa{l}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ count\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}count\ Q\ l\ {\isaliteral{3D}{\isacharequal}}\ length\ {\isaliteral{28}{\isacharparenleft}}filter\ Q\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\isa{cntP\ s} returns the number of operation \isa{P} happened
- before reaching state \isa{s}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ cntP\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntP\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ count\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ e{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}\ cs{\isaliteral{2E}{\isachardot}}\ e\ {\isaliteral{3D}{\isacharequal}}\ P\ th\ cs{\isaliteral{29}{\isacharparenright}}\ s{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\isa{cntV\ s} returns the number of operation \isa{V} happened
- before reaching state \isa{s}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{definition}\isamarkupfalse%
-\ cntV\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntV\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ count\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ e{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}\ cs{\isaliteral{2E}{\isachardot}}\ e\ {\isaliteral{3D}{\isacharequal}}\ V\ th\ cs{\isaliteral{29}{\isacharparenright}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/prio/Paper/ROOT.ML Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1 +0,0 @@
-use_thy "Paper";
\ No newline at end of file
--- a/prio/Paper/document/llncs.cls Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1189 +0,0 @@
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-\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
- {-18\p@ \@plus -4\p@ \@minus -4\p@}%
- {-0.5em \@plus -0.22em \@minus -0.1em}%
- {\normalfont\normalsize\bfseries\boldmath}}
-\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
- {-12\p@ \@plus -4\p@ \@minus -4\p@}%
- {-0.5em \@plus -0.22em \@minus -0.1em}%
- {\normalfont\normalsize\itshape}}
-\renewcommand\subparagraph[1]{\typeout{LLNCS warning: You should not use
- \string\subparagraph\space with this class}\vskip0.5cm
-You should not use \verb|\subparagraph| with this class.\vskip0.5cm}
-
-\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
-\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
-\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
-\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
-\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
-\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
-\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
-\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
-\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
-\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
-\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}
-
-\let\footnotesize\small
-
-\if@custvec
-\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}}
-{\mbox{\boldmath$\textstyle#1$}}
-{\mbox{\boldmath$\scriptstyle#1$}}
-{\mbox{\boldmath$\scriptscriptstyle#1$}}}
-\fi
-
-\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}}
-\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil
-\penalty50\hskip1em\null\nobreak\hfil\squareforqed
-\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
-
-\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
-\halign{\hfil
-$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
-\cr\to\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
-\cr\to\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
-\gets\cr\to\cr}}}}}
-\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
-$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
-\noalign{\vskip1.2pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
-\noalign{\vskip1pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
-<\cr
-\noalign{\vskip0.9pt}=\cr}}}}}
-\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
-$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
-\noalign{\vskip1.2pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
-\noalign{\vskip1pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
->\cr
-\noalign{\vskip0.9pt}=\cr}}}}}
-\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
-\halign{\hfil
-$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
->\cr\noalign{\vskip-1pt}<\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
->\cr\noalign{\vskip-0.8pt}<\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
->\cr\noalign{\vskip-0.3pt}<\cr}}}}}
-\def\bbbr{{\rm I\!R}} %reelle Zahlen
-\def\bbbm{{\rm I\!M}}
-\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
-\def\bbbf{{\rm I\!F}}
-\def\bbbh{{\rm I\!H}}
-\def\bbbk{{\rm I\!K}}
-\def\bbbp{{\rm I\!P}}
-\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
-{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
-\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
-to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
-to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
-to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
-to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
-\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
-Q$}\hbox{\raise
-0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
-{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
-0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
-0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
-0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
-\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
-T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
-to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
-to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
-to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
-\def\bbbs{{\mathchoice
-{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox
-to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
-to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
-{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox
-to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
-to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox
-to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
-to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
-to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
-to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
-\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
-{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
-{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
-{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
-
-\let\ts\,
-
-\setlength\leftmargini {17\p@}
-\setlength\leftmargin {\leftmargini}
-\setlength\leftmarginii {\leftmargini}
-\setlength\leftmarginiii {\leftmargini}
-\setlength\leftmarginiv {\leftmargini}
-\setlength \labelsep {.5em}
-\setlength \labelwidth{\leftmargini}
-\addtolength\labelwidth{-\labelsep}
-
-\def\@listI{\leftmargin\leftmargini
- \parsep 0\p@ \@plus1\p@ \@minus\p@
- \topsep 8\p@ \@plus2\p@ \@minus4\p@
- \itemsep0\p@}
-\let\@listi\@listI
-\@listi
-\def\@listii {\leftmargin\leftmarginii
- \labelwidth\leftmarginii
- \advance\labelwidth-\labelsep
- \topsep 0\p@ \@plus2\p@ \@minus\p@}
-\def\@listiii{\leftmargin\leftmarginiii
- \labelwidth\leftmarginiii
- \advance\labelwidth-\labelsep
- \topsep 0\p@ \@plus\p@\@minus\p@
- \parsep \z@
- \partopsep \p@ \@plus\z@ \@minus\p@}
-
-\renewcommand\labelitemi{\normalfont\bfseries --}
-\renewcommand\labelitemii{$\m@th\bullet$}
-
-\setlength\arraycolsep{1.4\p@}
-\setlength\tabcolsep{1.4\p@}
-
-\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
- {{\contentsname}}}
- \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
- \def\lastand{\ifnum\value{auco}=2\relax
- \unskip{} \andname\
- \else
- \unskip \lastandname\
- \fi}%
- \def\and{\stepcounter{@auth}\relax
- \ifnum\value{@auth}=\value{auco}%
- \lastand
- \else
- \unskip,
- \fi}%
- \@starttoc{toc}\if@restonecol\twocolumn\fi}
-
-\def\l@part#1#2{\addpenalty{\@secpenalty}%
- \addvspace{2em plus\p@}% % space above part line
- \begingroup
- \parindent \z@
- \rightskip \z@ plus 5em
- \hrule\vskip5pt
- \large % same size as for a contribution heading
- \bfseries\boldmath % set line in boldface
- \leavevmode % TeX command to enter horizontal mode.
- #1\par
- \vskip5pt
- \hrule
- \vskip1pt
- \nobreak % Never break after part entry
- \endgroup}
-
-\def\@dotsep{2}
-
-\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
-{chapter.\thechapter}\fi}
-
-\def\addnumcontentsmark#1#2#3{%
-\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
- {\thechapter}#3}{\thepage}\hyperhrefextend}}
-\def\addcontentsmark#1#2#3{%
-\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}
-\def\addcontentsmarkwop#1#2#3{%
-\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}
-
-\def\@adcmk[#1]{\ifcase #1 \or
-\def\@gtempa{\addnumcontentsmark}%
- \or \def\@gtempa{\addcontentsmark}%
- \or \def\@gtempa{\addcontentsmarkwop}%
- \fi\@gtempa{toc}{chapter}}
-\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
-
-\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
- \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
- \parindent \z@ \rightskip \@tocrmarg
- \advance\rightskip by 0pt plus 2cm
- \parfillskip -\rightskip \pretolerance=10000
- \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
- {\large\bfseries\boldmath#1}\ifx0#2\hfil\null
- \else
- \nobreak
- \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
- \@dotsep mu$}\hfill
- \nobreak\hbox to\@pnumwidth{\hss #2}%
- \fi\par
- \penalty\@highpenalty \endgroup}
-
-\def\l@title#1#2{\addpenalty{-\@highpenalty}
- \addvspace{8pt plus 1pt}
- \@tempdima \z@
- \begingroup
- \parindent \z@ \rightskip \@tocrmarg
- \advance\rightskip by 0pt plus 2cm
- \parfillskip -\rightskip \pretolerance=10000
- \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
- #1\nobreak
- \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
- \@dotsep mu$}\hfill
- \nobreak\hbox to\@pnumwidth{\hss #2}\par
- \penalty\@highpenalty \endgroup}
-
-\def\l@author#1#2{\addpenalty{\@highpenalty}
- \@tempdima=\z@ %15\p@
- \begingroup
- \parindent \z@ \rightskip \@tocrmarg
- \advance\rightskip by 0pt plus 2cm
- \pretolerance=10000
- \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
- \textit{#1}\par
- \penalty\@highpenalty \endgroup}
-
-%\setcounter{tocdepth}{0}
-\newdimen\tocchpnum
-\newdimen\tocsecnum
-\newdimen\tocsectotal
-\newdimen\tocsubsecnum
-\newdimen\tocsubsectotal
-\newdimen\tocsubsubsecnum
-\newdimen\tocsubsubsectotal
-\newdimen\tocparanum
-\newdimen\tocparatotal
-\newdimen\tocsubparanum
-\tocchpnum=\z@ % no chapter numbers
-\tocsecnum=15\p@ % section 88. plus 2.222pt
-\tocsubsecnum=23\p@ % subsection 88.8 plus 2.222pt
-\tocsubsubsecnum=27\p@ % subsubsection 88.8.8 plus 1.444pt
-\tocparanum=35\p@ % paragraph 88.8.8.8 plus 1.666pt
-\tocsubparanum=43\p@ % subparagraph 88.8.8.8.8 plus 1.888pt
-\def\calctocindent{%
-\tocsectotal=\tocchpnum
-\advance\tocsectotal by\tocsecnum
-\tocsubsectotal=\tocsectotal
-\advance\tocsubsectotal by\tocsubsecnum
-\tocsubsubsectotal=\tocsubsectotal
-\advance\tocsubsubsectotal by\tocsubsubsecnum
-\tocparatotal=\tocsubsubsectotal
-\advance\tocparatotal by\tocparanum}
-\calctocindent
-
-\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
-\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
-\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
-\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
-\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
-
-\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
- \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
- \@starttoc{lof}\if@restonecol\twocolumn\fi}
-\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
-
-\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
- \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
- \@starttoc{lot}\if@restonecol\twocolumn\fi}
-\let\l@table\l@figure
-
-\renewcommand\listoffigures{%
- \section*{\listfigurename
- \@mkboth{\listfigurename}{\listfigurename}}%
- \@starttoc{lof}%
- }
-
-\renewcommand\listoftables{%
- \section*{\listtablename
- \@mkboth{\listtablename}{\listtablename}}%
- \@starttoc{lot}%
- }
-
-\ifx\oribibl\undefined
-\ifx\citeauthoryear\undefined
-\renewenvironment{thebibliography}[1]
- {\section*{\refname}
- \def\@biblabel##1{##1.}
- \small
- \list{\@biblabel{\@arabic\c@enumiv}}%
- {\settowidth\labelwidth{\@biblabel{#1}}%
- \leftmargin\labelwidth
- \advance\leftmargin\labelsep
- \if@openbib
- \advance\leftmargin\bibindent
- \itemindent -\bibindent
- \listparindent \itemindent
- \parsep \z@
- \fi
- \usecounter{enumiv}%
- \let\p@enumiv\@empty
- \renewcommand\theenumiv{\@arabic\c@enumiv}}%
- \if@openbib
- \renewcommand\newblock{\par}%
- \else
- \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
- \fi
- \sloppy\clubpenalty4000\widowpenalty4000%
- \sfcode`\.=\@m}
- {\def\@noitemerr
- {\@latex@warning{Empty `thebibliography' environment}}%
- \endlist}
-\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
- {\let\protect\noexpand\immediate
- \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
-\newcount\@tempcntc
-\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
- \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
- {\@ifundefined
- {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
- ?}\@warning
- {Citation `\@citeb' on page \thepage \space undefined}}%
- {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
- \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
- \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
- \else
- \advance\@tempcntb\@ne
- \ifnum\@tempcntb=\@tempcntc
- \else\advance\@tempcntb\m@ne\@citeo
- \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
-\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
- \@citea\def\@citea{,\,\hskip\z@skip}%
- \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
- {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
- \def\@citea{--}\fi
- \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
-\else
-\renewenvironment{thebibliography}[1]
- {\section*{\refname}
- \small
- \list{}%
- {\settowidth\labelwidth{}%
- \leftmargin\parindent
- \itemindent=-\parindent
- \labelsep=\z@
- \if@openbib
- \advance\leftmargin\bibindent
- \itemindent -\bibindent
- \listparindent \itemindent
- \parsep \z@
- \fi
- \usecounter{enumiv}%
- \let\p@enumiv\@empty
- \renewcommand\theenumiv{}}%
- \if@openbib
- \renewcommand\newblock{\par}%
- \else
- \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
- \fi
- \sloppy\clubpenalty4000\widowpenalty4000%
- \sfcode`\.=\@m}
- {\def\@noitemerr
- {\@latex@warning{Empty `thebibliography' environment}}%
- \endlist}
- \def\@cite#1{#1}%
- \def\@lbibitem[#1]#2{\item[]\if@filesw
- {\def\protect##1{\string ##1\space}\immediate
- \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
- \fi
-\else
-\@cons\@openbib@code{\noexpand\small}
-\fi
-
-\def\idxquad{\hskip 10\p@}% space that divides entry from number
-
-\def\@idxitem{\par\hangindent 10\p@}
-
-\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
- \noindent\hangindent\wd0\box0}% index entry
-
-\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
- \noindent\hangindent\wd0\box0}% order index entry
-
-\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
-
-\renewenvironment{theindex}
- {\@mkboth{\indexname}{\indexname}%
- \thispagestyle{empty}\parindent\z@
- \parskip\z@ \@plus .3\p@\relax
- \let\item\par
- \def\,{\relax\ifmmode\mskip\thinmuskip
- \else\hskip0.2em\ignorespaces\fi}%
- \normalfont\small
- \begin{multicols}{2}[\@makeschapterhead{\indexname}]%
- }
- {\end{multicols}}
-
-\renewcommand\footnoterule{%
- \kern-3\p@
- \hrule\@width 2truecm
- \kern2.6\p@}
- \newdimen\fnindent
- \fnindent1em
-\long\def\@makefntext#1{%
- \parindent \fnindent%
- \leftskip \fnindent%
- \noindent
- \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
-
-\long\def\@makecaption#1#2{%
- \vskip\abovecaptionskip
- \sbox\@tempboxa{{\bfseries #1.} #2}%
- \ifdim \wd\@tempboxa >\hsize
- {\bfseries #1.} #2\par
- \else
- \global \@minipagefalse
- \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
- \fi
- \vskip\belowcaptionskip}
-
-\def\fps@figure{htbp}
-\def\fnum@figure{\figurename\thinspace\thefigure}
-\def \@floatboxreset {%
- \reset@font
- \small
- \@setnobreak
- \@setminipage
-}
-\def\fps@table{htbp}
-\def\fnum@table{\tablename~\thetable}
-\renewenvironment{table}
- {\setlength\abovecaptionskip{0\p@}%
- \setlength\belowcaptionskip{10\p@}%
- \@float{table}}
- {\end@float}
-\renewenvironment{table*}
- {\setlength\abovecaptionskip{0\p@}%
- \setlength\belowcaptionskip{10\p@}%
- \@dblfloat{table}}
- {\end@dblfloat}
-
-\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
- ext@#1\endcsname}{#1}{\protect\numberline{\csname
- the#1\endcsname}{\ignorespaces #2}}\begingroup
- \@parboxrestore
- \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
- \endgroup}
-
-% LaTeX does not provide a command to enter the authors institute
-% addresses. The \institute command is defined here.
-
-\newcounter{@inst}
-\newcounter{@auth}
-\newcounter{auco}
-\newdimen\instindent
-\newbox\authrun
-\newtoks\authorrunning
-\newtoks\tocauthor
-\newbox\titrun
-\newtoks\titlerunning
-\newtoks\toctitle
-
-\def\clearheadinfo{\gdef\@author{No Author Given}%
- \gdef\@title{No Title Given}%
- \gdef\@subtitle{}%
- \gdef\@institute{No Institute Given}%
- \gdef\@thanks{}%
- \global\titlerunning={}\global\authorrunning={}%
- \global\toctitle={}\global\tocauthor={}}
-
-\def\institute#1{\gdef\@institute{#1}}
-
-\def\institutename{\par
- \begingroup
- \parskip=\z@
- \parindent=\z@
- \setcounter{@inst}{1}%
- \def\and{\par\stepcounter{@inst}%
- \noindent$^{\the@inst}$\enspace\ignorespaces}%
- \setbox0=\vbox{\def\thanks##1{}\@institute}%
- \ifnum\c@@inst=1\relax
- \gdef\fnnstart{0}%
- \else
- \xdef\fnnstart{\c@@inst}%
- \setcounter{@inst}{1}%
- \noindent$^{\the@inst}$\enspace
- \fi
- \ignorespaces
- \@institute\par
- \endgroup}
-
-\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
- {\star\star\star}\or \dagger\or \ddagger\or
- \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
- \or \ddagger\ddagger \else\@ctrerr\fi}}
-
-\def\inst#1{\unskip$^{#1}$}
-\def\fnmsep{\unskip$^,$}
-\def\email#1{{\tt#1}}
-\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
-\@ifpackageloaded{babel}{%
-\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
-\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
-\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
-}{\switcht@@therlang}%
-}
-\def\homedir{\~{ }}
-
-\def\subtitle#1{\gdef\@subtitle{#1}}
-\clearheadinfo
-
-\renewcommand\maketitle{\newpage
- \refstepcounter{chapter}%
- \stepcounter{section}%
- \setcounter{section}{0}%
- \setcounter{subsection}{0}%
- \setcounter{figure}{0}
- \setcounter{table}{0}
- \setcounter{equation}{0}
- \setcounter{footnote}{0}%
- \begingroup
- \parindent=\z@
- \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
- \if@twocolumn
- \ifnum \col@number=\@ne
- \@maketitle
- \else
- \twocolumn[\@maketitle]%
- \fi
- \else
- \newpage
- \global\@topnum\z@ % Prevents figures from going at top of page.
- \@maketitle
- \fi
- \thispagestyle{empty}\@thanks
-%
- \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
- \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
- \instindent=\hsize
- \advance\instindent by-\headlineindent
-% \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
-% \addcontentsline{toc}{title}{\the\toctitle}\fi
- \if@runhead
- \if!\the\titlerunning!\else
- \edef\@title{\the\titlerunning}%
- \fi
- \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
- \ifdim\wd\titrun>\instindent
- \typeout{Title too long for running head. Please supply}%
- \typeout{a shorter form with \string\titlerunning\space prior to
- \string\maketitle}%
- \global\setbox\titrun=\hbox{\small\rm
- Title Suppressed Due to Excessive Length}%
- \fi
- \xdef\@title{\copy\titrun}%
- \fi
-%
- \if!\the\tocauthor!\relax
- {\def\and{\noexpand\protect\noexpand\and}%
- \protected@xdef\toc@uthor{\@author}}%
- \else
- \def\\{\noexpand\protect\noexpand\newline}%
- \protected@xdef\scratch{\the\tocauthor}%
- \protected@xdef\toc@uthor{\scratch}%
- \fi
-% \addcontentsline{toc}{author}{\toc@uthor}%
- \if@runhead
- \if!\the\authorrunning!
- \value{@inst}=\value{@auth}%
- \setcounter{@auth}{1}%
- \else
- \edef\@author{\the\authorrunning}%
- \fi
- \global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}%
- \ifdim\wd\authrun>\instindent
- \typeout{Names of authors too long for running head. Please supply}%
- \typeout{a shorter form with \string\authorrunning\space prior to
- \string\maketitle}%
- \global\setbox\authrun=\hbox{\small\rm
- Authors Suppressed Due to Excessive Length}%
- \fi
- \xdef\@author{\copy\authrun}%
- \markboth{\@author}{\@title}%
- \fi
- \endgroup
- \setcounter{footnote}{\fnnstart}%
- \clearheadinfo}
-%
-\def\@maketitle{\newpage
- \markboth{}{}%
- \def\lastand{\ifnum\value{@inst}=2\relax
- \unskip{} \andname\
- \else
- \unskip \lastandname\
- \fi}%
- \def\and{\stepcounter{@auth}\relax
- \ifnum\value{@auth}=\value{@inst}%
- \lastand
- \else
- \unskip,
- \fi}%
- \begin{center}%
- \let\newline\\
- {\Large \bfseries\boldmath
- \pretolerance=10000
- \@title \par}\vskip .8cm
-\if!\@subtitle!\else {\large \bfseries\boldmath
- \vskip -.65cm
- \pretolerance=10000
- \@subtitle \par}\vskip .8cm\fi
- \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
- \def\thanks##1{}\@author}%
- \global\value{@inst}=\value{@auth}%
- \global\value{auco}=\value{@auth}%
- \setcounter{@auth}{1}%
-{\lineskip .5em
-\noindent\ignorespaces
-\@author\vskip.35cm}
- {\small\institutename}
- \end{center}%
- }
-
-% definition of the "\spnewtheorem" command.
-%
-% Usage:
-%
-% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
-% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
-% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
-%
-% New is "cap_font" and "body_font". It stands for
-% fontdefinition of the caption and the text itself.
-%
-% "\spnewtheorem*" gives a theorem without number.
-%
-% A defined spnewthoerem environment is used as described
-% by Lamport.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\@thmcountersep{}
-\def\@thmcounterend{.}
-
-\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
-
-% definition of \spnewtheorem with number
-
-\def\@spnthm#1#2{%
- \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
-\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
-
-\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
- {\@definecounter{#1}\@addtoreset{#1}{#3}%
- \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
- \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
- \expandafter\xdef\csname #1name\endcsname{#2}%
- \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
- \global\@namedef{end#1}{\@endtheorem}}}
-
-\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
- {\@definecounter{#1}%
- \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
- \expandafter\xdef\csname #1name\endcsname{#2}%
- \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
- \global\@namedef{end#1}{\@endtheorem}}}
-
-\def\@spothm#1[#2]#3#4#5{%
- \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
- {\expandafter\@ifdefinable\csname #1\endcsname
- {\global\@namedef{the#1}{\@nameuse{the#2}}%
- \expandafter\xdef\csname #1name\endcsname{#3}%
- \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
- \global\@namedef{end#1}{\@endtheorem}}}}
-
-\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
-\refstepcounter{#1}%
-\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
-
-\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
- \ignorespaces}
-
-\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
- the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
-
-\def\@spbegintheorem#1#2#3#4{\trivlist
- \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
-
-\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
- \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
-
-% definition of \spnewtheorem* without number
-
-\def\@sthm#1#2{\@Ynthm{#1}{#2}}
-
-\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
- {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
- \expandafter\xdef\csname #1name\endcsname{#2}%
- \global\@namedef{end#1}{\@endtheorem}}}
-
-\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
-\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
-
-\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
-
-\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
- {#4}{#2}{#3}\ignorespaces}
-
-\def\@Begintheorem#1#2#3{#3\trivlist
- \item[\hskip\labelsep{#2#1\@thmcounterend}]}
-
-\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
- \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
-
-\if@envcntsect
- \def\@thmcountersep{.}
- \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
-\else
- \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
- \if@envcntreset
- \@addtoreset{theorem}{section}
- \else
- \@addtoreset{theorem}{chapter}
- \fi
-\fi
-
-%definition of divers theorem environments
-\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
-\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
-\if@envcntsame % alle Umgebungen wie Theorem.
- \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
-\else % alle Umgebungen mit eigenem Zaehler
- \if@envcntsect % mit section numeriert
- \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
- \else % nicht mit section numeriert
- \if@envcntreset
- \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
- \@addtoreset{#1}{section}}
- \else
- \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
- \@addtoreset{#1}{chapter}}%
- \fi
- \fi
-\fi
-\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
-\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
-\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
-\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
-\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
-\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
-\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
-\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
-\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
-\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
-\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
-\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
-\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
-\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
-
-\def\@takefromreset#1#2{%
- \def\@tempa{#1}%
- \let\@tempd\@elt
- \def\@elt##1{%
- \def\@tempb{##1}%
- \ifx\@tempa\@tempb\else
- \@addtoreset{##1}{#2}%
- \fi}%
- \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
- \expandafter\def\csname cl@#2\endcsname{}%
- \@tempc
- \let\@elt\@tempd}
-
-\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
- \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
- \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
- \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
- }
-
-\renewenvironment{abstract}{%
- \list{}{\advance\topsep by0.35cm\relax\small
- \leftmargin=1cm
- \labelwidth=\z@
- \listparindent=\z@
- \itemindent\listparindent
- \rightmargin\leftmargin}\item[\hskip\labelsep
- \bfseries\abstractname]}
- {\endlist}
-
-\newdimen\headlineindent % dimension for space between
-\headlineindent=1.166cm % number and text of headings.
-
-\def\ps@headings{\let\@mkboth\@gobbletwo
- \let\@oddfoot\@empty\let\@evenfoot\@empty
- \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
- \leftmark\hfil}
- \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
- \llap{\thepage}}
- \def\chaptermark##1{}%
- \def\sectionmark##1{}%
- \def\subsectionmark##1{}}
-
-\def\ps@titlepage{\let\@mkboth\@gobbletwo
- \let\@oddfoot\@empty\let\@evenfoot\@empty
- \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
- \hfil}
- \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
- \llap{\thepage}}
- \def\chaptermark##1{}%
- \def\sectionmark##1{}%
- \def\subsectionmark##1{}}
-
-\if@runhead\ps@headings\else
-\ps@empty\fi
-
-\setlength\arraycolsep{1.4\p@}
-\setlength\tabcolsep{1.4\p@}
-
-\endinput
-%end of file llncs.cls
--- a/prio/Paper/document/mathpartir.sty Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,446 +0,0 @@
-% Mathpartir --- Math Paragraph for Typesetting Inference Rules
-%
-% Copyright (C) 2001, 2002, 2003, 2004, 2005 Didier Rémy
-%
-% Author : Didier Remy
-% Version : 1.2.0
-% Bug Reports : to author
-% Web Site : http://pauillac.inria.fr/~remy/latex/
-%
-% Mathpartir is free software; you can redistribute it and/or modify
-% it under the terms of the GNU General Public License as published by
-% the Free Software Foundation; either version 2, or (at your option)
-% any later version.
-%
-% Mathpartir is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-% GNU General Public License for more details
-% (http://pauillac.inria.fr/~remy/license/GPL).
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% File mathpartir.sty (LaTeX macros)
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{mathpartir}
- [2005/12/20 version 1.2.0 Math Paragraph for Typesetting Inference Rules]
-
-%%
-
-%% Identification
-%% Preliminary declarations
-
-\RequirePackage {keyval}
-
-%% Options
-%% More declarations
-
-%% PART I: Typesetting maths in paragraphe mode
-
-%% \newdimen \mpr@tmpdim
-%% Dimens are a precious ressource. Uses seems to be local.
-\let \mpr@tmpdim \@tempdima
-
-% To ensure hevea \hva compatibility, \hva should expands to nothing
-% in mathpar or in inferrule
-\let \mpr@hva \empty
-
-%% normal paragraph parametters, should rather be taken dynamically
-\def \mpr@savepar {%
- \edef \MathparNormalpar
- {\noexpand \lineskiplimit \the\lineskiplimit
- \noexpand \lineskip \the\lineskip}%
- }
-
-\def \mpr@rulelineskip {\lineskiplimit=0.3em\lineskip=0.2em plus 0.1em}
-\def \mpr@lesslineskip {\lineskiplimit=0.6em\lineskip=0.5em plus 0.2em}
-\def \mpr@lineskip {\lineskiplimit=1.2em\lineskip=1.2em plus 0.2em}
-\let \MathparLineskip \mpr@lineskip
-\def \mpr@paroptions {\MathparLineskip}
-\let \mpr@prebindings \relax
-
-\newskip \mpr@andskip \mpr@andskip 2em plus 0.5fil minus 0.5em
-
-\def \mpr@goodbreakand
- {\hskip -\mpr@andskip \penalty -1000\hskip \mpr@andskip}
-\def \mpr@and {\hskip \mpr@andskip}
-\def \mpr@andcr {\penalty 50\mpr@and}
-\def \mpr@cr {\penalty -10000\mpr@and}
-\def \mpr@eqno #1{\mpr@andcr #1\hskip 0em plus -1fil \penalty 10}
-
-\def \mpr@bindings {%
- \let \and \mpr@andcr
- \let \par \mpr@andcr
- \let \\\mpr@cr
- \let \eqno \mpr@eqno
- \let \hva \mpr@hva
- }
-\let \MathparBindings \mpr@bindings
-
-% \@ifundefined {ignorespacesafterend}
-% {\def \ignorespacesafterend {\aftergroup \ignorespaces}
-
-\newenvironment{mathpar}[1][]
- {$$\mpr@savepar \parskip 0em \hsize \linewidth \centering
- \vbox \bgroup \mpr@prebindings \mpr@paroptions #1\ifmmode $\else
- \noindent $\displaystyle\fi
- \MathparBindings}
- {\unskip \ifmmode $\fi\egroup $$\ignorespacesafterend}
-
-\newenvironment{mathparpagebreakable}[1][]
- {\begingroup
- \par
- \mpr@savepar \parskip 0em \hsize \linewidth \centering
- \mpr@prebindings \mpr@paroptions #1%
- \vskip \abovedisplayskip \vskip -\lineskip%
- \ifmmode \else $\displaystyle\fi
- \MathparBindings
- }
- {\unskip
- \ifmmode $\fi \par\endgroup
- \vskip \belowdisplayskip
- \noindent
- \ignorespacesafterend}
-
-% \def \math@mathpar #1{\setbox0 \hbox {$\displaystyle #1$}\ifnum
-% \wd0 < \hsize $$\box0$$\else \bmathpar #1\emathpar \fi}
-
-%%% HOV BOXES
-
-\def \mathvbox@ #1{\hbox \bgroup \mpr@normallineskip
- \vbox \bgroup \tabskip 0em \let \\ \cr
- \halign \bgroup \hfil $##$\hfil\cr #1\crcr \egroup \egroup
- \egroup}
-
-\def \mathhvbox@ #1{\setbox0 \hbox {\let \\\qquad $#1$}\ifnum \wd0 < \hsize
- \box0\else \mathvbox {#1}\fi}
-
-
-%% Part II -- operations on lists
-
-\newtoks \mpr@lista
-\newtoks \mpr@listb
-
-\long \def\mpr@cons #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
-{#2}\edef #2{\the \mpr@lista \the \mpr@listb}}
-
-\long \def\mpr@snoc #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
-{#2}\edef #2{\the \mpr@listb\the\mpr@lista}}
-
-\long \def \mpr@concat#1=#2\mpr@to#3{\mpr@lista \expandafter {#2}\mpr@listb
-\expandafter {#3}\edef #1{\the \mpr@listb\the\mpr@lista}}
-
-\def \mpr@head #1\mpr@to #2{\expandafter \mpr@head@ #1\mpr@head@ #1#2}
-\long \def \mpr@head@ #1#2\mpr@head@ #3#4{\def #4{#1}\def#3{#2}}
-
-\def \mpr@flatten #1\mpr@to #2{\expandafter \mpr@flatten@ #1\mpr@flatten@ #1#2}
-\long \def \mpr@flatten@ \\#1\\#2\mpr@flatten@ #3#4{\def #4{#1}\def #3{\\#2}}
-
-\def \mpr@makelist #1\mpr@to #2{\def \mpr@all {#1}%
- \mpr@lista {\\}\mpr@listb \expandafter {\mpr@all}\edef \mpr@all {\the
- \mpr@lista \the \mpr@listb \the \mpr@lista}\let #2\empty
- \def \mpr@stripof ##1##2\mpr@stripend{\def \mpr@stripped{##2}}\loop
- \mpr@flatten \mpr@all \mpr@to \mpr@one
- \expandafter \mpr@snoc \mpr@one \mpr@to #2\expandafter \mpr@stripof
- \mpr@all \mpr@stripend
- \ifx \mpr@stripped \empty \let \mpr@isempty 0\else \let \mpr@isempty 1\fi
- \ifx 1\mpr@isempty
- \repeat
-}
-
-\def \mpr@rev #1\mpr@to #2{\let \mpr@tmp \empty
- \def \\##1{\mpr@cons ##1\mpr@to \mpr@tmp}#1\let #2\mpr@tmp}
-
-%% Part III -- Type inference rules
-
-\newif \if@premisse
-\newbox \mpr@hlist
-\newbox \mpr@vlist
-\newif \ifmpr@center \mpr@centertrue
-\def \mpr@htovlist {%
- \setbox \mpr@hlist
- \hbox {\strut
- \ifmpr@center \hskip -0.5\wd\mpr@hlist\fi
- \unhbox \mpr@hlist}%
- \setbox \mpr@vlist
- \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
- \else \unvbox \mpr@vlist \box \mpr@hlist
- \fi}%
-}
-% OLD version
-% \def \mpr@htovlist {%
-% \setbox \mpr@hlist
-% \hbox {\strut \hskip -0.5\wd\mpr@hlist \unhbox \mpr@hlist}%
-% \setbox \mpr@vlist
-% \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
-% \else \unvbox \mpr@vlist \box \mpr@hlist
-% \fi}%
-% }
-
-\def \mpr@item #1{$\displaystyle #1$}
-\def \mpr@sep{2em}
-\def \mpr@blank { }
-\def \mpr@hovbox #1#2{\hbox
- \bgroup
- \ifx #1T\@premissetrue
- \else \ifx #1B\@premissefalse
- \else
- \PackageError{mathpartir}
- {Premisse orientation should either be T or B}
- {Fatal error in Package}%
- \fi \fi
- \def \@test {#2}\ifx \@test \mpr@blank\else
- \setbox \mpr@hlist \hbox {}%
- \setbox \mpr@vlist \vbox {}%
- \if@premisse \let \snoc \mpr@cons \else \let \snoc \mpr@snoc \fi
- \let \@hvlist \empty \let \@rev \empty
- \mpr@tmpdim 0em
- \expandafter \mpr@makelist #2\mpr@to \mpr@flat
- \if@premisse \mpr@rev \mpr@flat \mpr@to \@rev \else \let \@rev \mpr@flat \fi
- \def \\##1{%
- \def \@test {##1}\ifx \@test \empty
- \mpr@htovlist
- \mpr@tmpdim 0em %%% last bug fix not extensively checked
- \else
- \setbox0 \hbox{\mpr@item {##1}}\relax
- \advance \mpr@tmpdim by \wd0
- %\mpr@tmpdim 1.02\mpr@tmpdim
- \ifnum \mpr@tmpdim < \hsize
- \ifnum \wd\mpr@hlist > 0
- \if@premisse
- \setbox \mpr@hlist
- \hbox {\unhbox0 \hskip \mpr@sep \unhbox \mpr@hlist}%
- \else
- \setbox \mpr@hlist
- \hbox {\unhbox \mpr@hlist \hskip \mpr@sep \unhbox0}%
- \fi
- \else
- \setbox \mpr@hlist \hbox {\unhbox0}%
- \fi
- \else
- \ifnum \wd \mpr@hlist > 0
- \mpr@htovlist
- \mpr@tmpdim \wd0
- \fi
- \setbox \mpr@hlist \hbox {\unhbox0}%
- \fi
- \advance \mpr@tmpdim by \mpr@sep
- \fi
- }%
- \@rev
- \mpr@htovlist
- \ifmpr@center \hskip \wd\mpr@vlist\fi \box \mpr@vlist
- \fi
- \egroup
-}
-
-%%% INFERENCE RULES
-
-\@ifundefined{@@over}{%
- \let\@@over\over % fallback if amsmath is not loaded
- \let\@@overwithdelims\overwithdelims
- \let\@@atop\atop \let\@@atopwithdelims\atopwithdelims
- \let\@@above\above \let\@@abovewithdelims\abovewithdelims
- }{}
-
-%% The default
-
-\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
- $\displaystyle {#1\mpr@over #2}$}}
-\def \mpr@@nofraction #1#2{\hbox {\advance \hsize by -0.5em
- $\displaystyle {#1\@@atop #2}$}}
-
-\let \mpr@fraction \mpr@@fraction
-
-%% A generic solution to arrow
-
-\def \mpr@make@fraction #1#2#3#4#5{\hbox {%
- \def \mpr@tail{#1}%
- \def \mpr@body{#2}%
- \def \mpr@head{#3}%
- \setbox1=\hbox{$#4$}\setbox2=\hbox{$#5$}%
- \setbox3=\hbox{$\mkern -3mu\mpr@body\mkern -3mu$}%
- \setbox3=\hbox{$\mkern -3mu \mpr@body\mkern -3mu$}%
- \dimen0=\dp1\advance\dimen0 by \ht3\relax\dp1\dimen0\relax
- \dimen0=\ht2\advance\dimen0 by \dp3\relax\ht2\dimen0\relax
- \setbox0=\hbox {$\box1 \@@atop \box2$}%
- \dimen0=\wd0\box0
- \box0 \hskip -\dimen0\relax
- \hbox to \dimen0 {$%
- \mathrel{\mpr@tail}\joinrel
- \xleaders\hbox{\copy3}\hfil\joinrel\mathrel{\mpr@head}%
- $}}}
-
-%% Old stuff should be removed in next version
-\def \mpr@@nothing #1#2
- {$\lower 0.01pt \mpr@@nofraction {#1}{#2}$}
-\def \mpr@@reduce #1#2{\hbox
- {$\lower 0.01pt \mpr@@fraction {#1}{#2}\mkern -15mu\rightarrow$}}
-\def \mpr@@rewrite #1#2#3{\hbox
- {$\lower 0.01pt \mpr@@fraction {#2}{#3}\mkern -8mu#1$}}
-\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
-
-\def \mpr@empty {}
-\def \mpr@inferrule
- {\bgroup
- \ifnum \linewidth<\hsize \hsize \linewidth\fi
- \mpr@rulelineskip
- \let \and \qquad
- \let \hva \mpr@hva
- \let \@rulename \mpr@empty
- \let \@rule@options \mpr@empty
- \let \mpr@over \@@over
- \mpr@inferrule@}
-\newcommand {\mpr@inferrule@}[3][]
- {\everymath={\displaystyle}%
- \def \@test {#2}\ifx \empty \@test
- \setbox0 \hbox {$\vcenter {\mpr@hovbox{B}{#3}}$}%
- \else
- \def \@test {#3}\ifx \empty \@test
- \setbox0 \hbox {$\vcenter {\mpr@hovbox{T}{#2}}$}%
- \else
- \setbox0 \mpr@fraction {\mpr@hovbox{T}{#2}}{\mpr@hovbox{B}{#3}}%
- \fi \fi
- \def \@test {#1}\ifx \@test\empty \box0
- \else \vbox
-%%% Suggestion de Francois pour les etiquettes longues
-%%% {\hbox to \wd0 {\RefTirName {#1}\hfil}\box0}\fi
- {\hbox {\RefTirName {#1}}\box0}\fi
- \egroup}
-
-\def \mpr@vdotfil #1{\vbox to #1{\leaders \hbox{$\cdot$} \vfil}}
-
-% They are two forms
-% \inferrule [label]{[premisses}{conclusions}
-% or
-% \inferrule* [options]{[premisses}{conclusions}
-%
-% Premisses and conclusions are lists of elements separated by \\
-% Each \\ produces a break, attempting horizontal breaks if possible,
-% and vertical breaks if needed.
-%
-% An empty element obtained by \\\\ produces a vertical break in all cases.
-%
-% The former rule is aligned on the fraction bar.
-% The optional label appears on top of the rule
-% The second form to be used in a derivation tree is aligned on the last
-% line of its conclusion
-%
-% The second form can be parameterized, using the key=val interface. The
-% folloiwng keys are recognized:
-%
-% width set the width of the rule to val
-% narrower set the width of the rule to val\hsize
-% before execute val at the beginning/left
-% lab put a label [Val] on top of the rule
-% lskip add negative skip on the right
-% left put a left label [Val]
-% Left put a left label [Val], ignoring its width
-% right put a right label [Val]
-% Right put a right label [Val], ignoring its width
-% leftskip skip negative space on the left-hand side
-% rightskip skip negative space on the right-hand side
-% vdots lift the rule by val and fill vertical space with dots
-% after execute val at the end/right
-%
-% Note that most options must come in this order to avoid strange
-% typesetting (in particular leftskip must preceed left and Left and
-% rightskip must follow Right or right; vdots must come last
-% or be only followed by rightskip.
-%
-
-%% Keys that make sence in all kinds of rules
-\def \mprset #1{\setkeys{mprset}{#1}}
-\define@key {mprset}{andskip}[]{\mpr@andskip=#1}
-\define@key {mprset}{lineskip}[]{\lineskip=#1}
-\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
-\define@key {mprset}{center}[]{\mpr@centertrue}
-\define@key {mprset}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
-\define@key {mprset}{atop}[]{\let \mpr@fraction \mpr@@nofraction}
-\define@key {mprset}{myfraction}[]{\let \mpr@fraction #1}
-\define@key {mprset}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
-\define@key {mprset}{sep}{\def\mpr@sep{#1}}
-
-\newbox \mpr@right
-\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
-\define@key {mpr}{center}[]{\mpr@centertrue}
-\define@key {mpr}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
-\define@key {mpr}{myfraction}[]{\let \mpr@fraction #1}
-\define@key {mpr}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
-\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{width}{\hsize #1}
-\define@key {mpr}{sep}{\def\mpr@sep{#1}}
-\define@key {mpr}{before}{#1}
-\define@key {mpr}{lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
-\define@key {mpr}{Lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
-\define@key {mpr}{narrower}{\hsize #1\hsize}
-\define@key {mpr}{leftskip}{\hskip -#1}
-\define@key {mpr}{reduce}[]{\let \mpr@fraction \mpr@@reduce}
-\define@key {mpr}{rightskip}
- {\setbox \mpr@right \hbox {\unhbox \mpr@right \hskip -#1}}
-\define@key {mpr}{LEFT}{\setbox0 \hbox {$#1$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{Left}{\llap{$\TirName {#1}\;$}}
-\define@key {mpr}{right}
- {\setbox0 \hbox {$\;\TirName {#1}$}\relax \advance \hsize by -\wd0
- \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
-\define@key {mpr}{RIGHT}
- {\setbox0 \hbox {$#1$}\relax \advance \hsize by -\wd0
- \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
-\define@key {mpr}{Right}
- {\setbox \mpr@right \hbox {\unhbox \mpr@right \rlap {$\;\TirName {#1}$}}}
-\define@key {mpr}{vdots}{\def \mpr@vdots {\@@atop \mpr@vdotfil{#1}}}
-\define@key {mpr}{after}{\edef \mpr@after {\mpr@after #1}}
-
-\newcommand \mpr@inferstar@ [3][]{\setbox0
- \hbox {\let \mpr@rulename \mpr@empty \let \mpr@vdots \relax
- \setbox \mpr@right \hbox{}%
- $\setkeys{mpr}{#1}%
- \ifx \mpr@rulename \mpr@empty \mpr@inferrule {#2}{#3}\else
- \mpr@inferrule [{\mpr@rulename}]{#2}{#3}\fi
- \box \mpr@right \mpr@vdots$}
- \setbox1 \hbox {\strut}
- \@tempdima \dp0 \advance \@tempdima by -\dp1
- \raise \@tempdima \box0}
-
-\def \mpr@infer {\@ifnextchar *{\mpr@inferstar}{\mpr@inferrule}}
-\newcommand \mpr@err@skipargs[3][]{}
-\def \mpr@inferstar*{\ifmmode
- \let \@do \mpr@inferstar@
- \else
- \let \@do \mpr@err@skipargs
- \PackageError {mathpartir}
- {\string\inferrule* can only be used in math mode}{}%
- \fi \@do}
-
-
-%%% Exports
-
-% Envirnonment mathpar
-
-\let \inferrule \mpr@infer
-
-% make a short name \infer is not already defined
-\@ifundefined {infer}{\let \infer \mpr@infer}{}
-
-\def \TirNameStyle #1{\small \textsc{#1}}
-\def \tir@name #1{\hbox {\small \TirNameStyle{#1}}}
-\let \TirName \tir@name
-\let \DefTirName \TirName
-\let \RefTirName \TirName
-
-%%% Other Exports
-
-% \let \listcons \mpr@cons
-% \let \listsnoc \mpr@snoc
-% \let \listhead \mpr@head
-% \let \listmake \mpr@makelist
-
-
-
-
-\endinput
--- a/prio/Paper/document/root.bib Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,205 +0,0 @@
-
-@Book{Paulson96,
- author = {L.~C.~Paulson},
- title = {{ML} for the {W}orking {P}rogrammer},
- publisher = {Cambridge University Press},
- year = {1996}
-}
-
-
-@Manual{LINUX,
- author = {S.~Rostedt},
- title = {{RT}-{M}utex {I}mplementation {D}esign},
- note = {Linux Kernel Distribution at,
- www.kernel.org/doc/Documentation/rt-mutex-design.txt}
-}
-
-@Misc{PINTOS,
- author = {B.~Pfaff},
- title = {{PINTOS}},
- note = {\url{http://www.stanford.edu/class/cs140/projects/}},
-}
-
-
-@inproceedings{Haftmann08,
- author = {F.~Haftmann and M.~Wenzel},
- title = {{L}ocal {T}heory {S}pecifications in {I}sabelle/{I}sar},
- booktitle = {Proc.~of the International Conference on Types, Proofs and Programs (TYPES)},
- year = {2008},
- pages = {153-168},
- series = {LNCS},
- volume = {5497}
-}
-
-
-@TechReport{Yodaiken02,
- author = {V.~Yodaiken},
- title = {{A}gainst {P}riority {I}nheritance},
- institution = {Finite State Machine Labs (FSMLabs)},
- year = {2004}
-}
-
-
-@Book{Vahalia96,
- author = {U.~Vahalia},
- title = {{UNIX} {I}nternals: {T}he {N}ew {F}rontiers},
- publisher = {Prentice-Hall},
- year = {1996}
-}
-
-@Article{Reeves98,
- title = "{R}e: {W}hat {R}eally {H}appened on {M}ars?",
- author = "G.~E.~Reeves",
- journal = "Risks Forum",
- year = "1998",
- number = "54",
- volume = "19"
-}
-
-@Article{Sha90,
- title = "{P}riority {I}nheritance {P}rotocols: {A}n {A}pproach to
- {R}eal-{T}ime {S}ynchronization",
- author = "L.~Sha and R.~Rajkumar and J.~P.~Lehoczky",
- journal = "IEEE Transactions on Computers",
- year = "1990",
- number = "9",
- volume = "39",
- pages = "1175--1185"
-}
-
-
-@Article{Lampson80,
- author = "B.~W.~Lampson and D.~D.~Redell",
- title = "{{E}xperiences with {P}rocesses and {M}onitors in {M}esa}",
- journal = "Communications of the ACM",
- volume = "23",
- number = "2",
- pages = "105--117",
- year = "1980"
-}
-
-@inproceedings{Wang09,
- author = {J.~Wang and H.~Yang and X.~Zhang},
- title = {{L}iveness {R}easoning with {I}sabelle/{HOL}},
- booktitle = {Proc.~of the 22nd International Conference on Theorem Proving in
- Higher Order Logics (TPHOLs)},
- year = {2009},
- pages = {485--499},
- volume = {5674},
- series = {LNCS}
-}
-
-@PhdThesis{Faria08,
- author = {J.~M.~S.~Faria},
- title = {{F}ormal {D}evelopment of {S}olutions for {R}eal-{T}ime {O}perating {S}ystems
- with {TLA+/TLC}},
- school = {University of Porto},
- year = {2008}
-}
-
-
-@Article{Paulson98,
- author = {L.~C.~Paulson},
- title = {{T}he {I}nductive {A}pproach to {V}erifying {C}ryptographic {P}rotocols},
- journal = {Journal of Computer Security},
- year = {1998},
- volume = {6},
- number = {1--2},
- pages = {85--128}
-}
-
-@MISC{locke-july02,
-author = {D. Locke},
-title = {Priority Inheritance: The Real Story},
-month = July,
-year = {2002},
-howpublished={\url{http://www.math.unipd.it/~tullio/SCD/2007/Materiale/Locke.pdf}},
-}
-
-
-@InProceedings{Steinberg10,
- author = {U.~Steinberg and A.~B\"otcher and B.~Kauer},
- title = {{T}imeslice {D}onation in {C}omponent-{B}ased {S}ystems},
- booktitle = {Proc.~of the 6th International Workshop on Operating Systems
- Platforms for Embedded Real-Time Applications (OSPERT)},
- pages = {16--23},
- year = {2010}
-}
-
-@inproceedings{dutertre99b,
- title = "{T}he {P}riority {C}eiling {P}rotocol: {F}ormalization and
- {A}nalysis {U}sing {PVS}",
- author = "B.~Dutertre",
- booktitle = {Proc.~of the 21st IEEE Conference on Real-Time Systems Symposium (RTSS)},
- year = {2000},
- pages = {151--160},
- publisher = {IEEE Computer Society}
-}
-
-@InProceedings{Jahier09,
- title = "{S}ynchronous {M}odeling and {V}alidation of {P}riority
- {I}nheritance {S}chedulers",
- author = "E.~Jahier and B.~Halbwachs and P.~Raymond",
- booktitle = "Proc.~of the 12th International Conference on Fundamental
- Approaches to Software Engineering (FASE)",
- year = "2009",
- volume = "5503",
- series = "LNCS",
- pages = "140--154"
-}
-
-@InProceedings{Wellings07,
- title = "{I}ntegrating {P}riority {I}nheritance {A}lgorithms in the {R}eal-{T}ime
- {S}pecification for {J}ava",
- author = "A.~Wellings and A.~Burns and O.~M.~Santos and B.~M.~Brosgol",
- publisher = "IEEE Computer Society",
- year = "2007",
- booktitle = "Proc.~of the 10th IEEE International Symposium on Object
- and Component-Oriented Real-Time Distributed Computing (ISORC)",
- pages = "115--123"
-}
-
-@Article{Wang:2002:SGP,
- author = "Y. Wang and E. Anceaume and F. Brasileiro and F.
- Greve and M. Hurfin",
- title = "Solving the group priority inversion problem in a
- timed asynchronous system",
- journal = "IEEE Transactions on Computers",
- volume = "51",
- number = "8",
- pages = "900--915",
- month = aug,
- year = "2002",
- CODEN = "ITCOB4",
- doi = "http://dx.doi.org/10.1109/TC.2002.1024738",
- ISSN = "0018-9340 (print), 1557-9956 (electronic)",
- issn-l = "0018-9340",
- bibdate = "Tue Jul 5 09:41:56 MDT 2011",
- bibsource = "http://www.computer.org/tc/;
- http://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib",
- URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1024738",
- acknowledgement = "Nelson H. F. Beebe, University of Utah, Department
- of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake
- City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1
- 801 581 4148, e-mail: \path|beebe@math.utah.edu|,
- \path|beebe@acm.org|, \path|beebe@computer.org|
- (Internet), URL:
- \path|http://www.math.utah.edu/~beebe/|",
- fjournal = "IEEE Transactions on Computers",
- doi-url = "http://dx.doi.org/10.1109/TC.2002.1024738",
-}
-
-@Article{journals/rts/BabaogluMS93,
- title = "A Formalization of Priority Inversion",
- author = "{\"O} Babaoglu and K. Marzullo and F. B. Schneider",
- journal = "Real-Time Systems",
- year = "1993",
- number = "4",
- volume = "5",
- bibdate = "2011-06-03",
- bibsource = "DBLP,
- http://dblp.uni-trier.de/db/journals/rts/rts5.html#BabaogluMS93",
- pages = "285--303",
- URL = "http://dx.doi.org/10.1007/BF01088832",
-}
-
--- a/prio/Paper/document/root.tex Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,79 +0,0 @@
-\documentclass[runningheads]{llncs}
-\usepackage{isabelle}
-\usepackage{isabellesym}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{mathpartir}
-\usepackage{tikz}
-\usepackage{pgf}
-%\usetikzlibrary{arrows,automata,decorations,fit,calc}
-%\usetikzlibrary{shapes,shapes.arrows,snakes,positioning}
-%\usepgflibrary{shapes.misc} % LATEX and plain TEX and pure pgf
-%\usetikzlibrary{matrix}
-\usepackage{pdfsetup}
-\usepackage{ot1patch}
-\usepackage{times}
-%%\usepackage{proof}
-%%\usepackage{mathabx}
-\usepackage{stmaryrd}
-\usepackage{url}
-\usepackage{color}
-\titlerunning{Proving the Priority Inheritance Protocol Correct}
-
-
-\urlstyle{rm}
-\isabellestyle{it}
-\renewcommand{\isastyleminor}{\it}%
-\renewcommand{\isastyle}{\normalsize\it}%
-
-
-\def\dn{\,\stackrel{\mbox{\scriptsize def}}{=}\,}
-\renewcommand{\isasymequiv}{$\dn$}
-\renewcommand{\isasymemptyset}{$\varnothing$}
-\renewcommand{\isacharunderscore}{\mbox{$\_\!\_$}}
-\renewcommand{\isasymiota}{}
-
-\newcommand{\numbered}[1]{\refstepcounter{equation}{\rm(\arabic{equation})}\label{#1}}
-\definecolor{mygrey}{rgb}{.80,.80,.80}
-
-\begin{document}
-
-\title{Priority Inheritance Protocol Proved Correct}
-\author{Xingyuan Zhang\inst{1} \and Christian Urban\inst{2} \and Chunhan Wu\inst{1}}
-\institute{PLA University of Science and Technology, China \and
- King's College London, United Kingdom}
-\maketitle
-
-\begin{abstract}
-In real-time systems with threads, resource locking and
-priority sched\-uling, one faces the problem of Priority
-Inversion. This problem can make the behaviour of threads
-unpredictable and the resulting bugs can be hard to find. The
-Priority Inheritance Protocol is one solution implemented in many
-systems for solving this problem, but the correctness of this solution
-has never been formally verified in a theorem prover. As already
-pointed out in the literature, the original informal investigation of
-the Property Inheritance Protocol presents a correctness ``proof'' for
-an \emph{incorrect} algorithm. In this paper we fix the problem of
-this proof by making all notions precise and implementing a variant of
-a solution proposed earlier. Our formalisation in Isabelle/HOL
-uncovers facts not mentioned in the literature, but also shows how to
-efficiently implement this protocol. Earlier correct implementations
-were criticised as too inefficient. Our formalisation is based on
-Paulson's inductive approach to verifying protocols.\medskip
-
-{\bf Keywords:} Priority Inheritance Protocol, formal correctness proof,
-real-time systems, Isabelle/HOL
-\end{abstract}
-
-\input{session}
-
-%\bibliographystyle{plain}
-%\bibliography{root}
-
-\end{document}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:
--- a/prio/Paper/tt.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,94 +0,0 @@
-
-There are several works on inversion avoidance:
-\begin{enumerate}
-\item {\em Solving the group priority inversion problem in a timed asynchronous system}.
-The notion of \<exclamdown>\<degree>Group Priority Inversion\<exclamdown>\<plusminus> is introduced. The main strategy is still inversion avoidance.
-The method is by reordering requests in the setting of Client-Server.
-\item {\em Examples of inaccurate specification of the protocol}.
-\end{enumerate}
-
-
-
-
-
-
-section{* Related works *}
-
-text {*
-1. <<Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java>> models and
-verifies the combination of Priority Inheritance (PI) and Priority Ceiling Emulation (PCE) protocols in
-the setting of Java virtual machine using extended Timed Automata(TA) formalism of the UPPAAL tool.
-Although a detailed formal model of combined PI and PCE is given, the number of properties is quite
-small and the focus is put on the harmonious working of PI and PCE. Most key features of PI
-(as well as PCE) are not shown. Because of the limitation of the model checking technique
- used there, properties are shown only for a small number of scenarios. Therefore, the verification
-does not show the correctness of the formal model itself in a convincing way.
-2. << Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC>>. A formal model
-of PI is given in TLA+. Only 3 properties are shown for PI using model checking. The limitation of
-model checking is intrinsic to the work.
-3. <<Synchronous modeling and validation of priority inheritance schedulers>>. Gives a formal model
-of PI and PCE in AADL (Architecture Analysis & Design Language) and checked several properties
-using model checking. The number of properties shown there is less than here and the scale
-is also limited by the model checking technique.
-
-
-There are several works on inversion avoidance:
-1. <<Solving the group priority inversion problem in a timed asynchronous system>>.
-The notion of \<exclamdown>\<degree>Group Priority Inversion\<exclamdown>\<plusminus> is introduced. The main strategy is still inversion avoidance.
-The method is by reordering requests in the setting of Client-Server.
-
-
-<<Examples of inaccurate specification of the protocol>>.
-
-*}
-
-text {*
-
-\section{An overview of priority inversion and priority inheritance}
-
-Priority inversion refers to the phenomenon when a thread with high priority is blocked
-by a thread with low priority. Priority happens when the high priority thread requests
-for some critical resource already taken by the low priority thread. Since the high
-priority thread has to wait for the low priority thread to complete, it is said to be
-blocked by the low priority thread. Priority inversion might prevent high priority
-thread from fulfill its task in time if the duration of priority inversion is indefinite
-and unpredictable. Indefinite priority inversion happens when indefinite number
-of threads with medium priorities is activated during the period when the high
-priority thread is blocked by the low priority thread. Although these medium
-priority threads can not preempt the high priority thread directly, they are able
-to preempt the low priority threads and cause it to stay in critical section for
-an indefinite long duration. In this way, the high priority thread may be blocked indefinitely.
-
-Priority inheritance is one protocol proposed to avoid indefinite priority inversion.
-The basic idea is to let the high priority thread donate its priority to the low priority
-thread holding the critical resource, so that it will not be preempted by medium priority
-threads. The thread with highest priority will not be blocked unless it is requesting
-some critical resource already taken by other threads. Viewed from a different angle,
-any thread which is able to block the highest priority threads must already hold some
-critical resource. Further more, it must have hold some critical resource at the
-moment the highest priority is created, otherwise, it may never get change to run and
-get hold. Since the number of such resource holding lower priority threads is finite,
-if every one of them finishes with its own critical section in a definite duration,
-the duration the highest priority thread is blocked is definite as well. The key to
-guarantee lower priority threads to finish in definite is to donate them the highest
-priority. In such cases, the lower priority threads is said to have inherited the
-highest priority. And this explains the name of the protocol:
-{\em Priority Inheritance} and how Priority Inheritance prevents indefinite delay.
-
-The objectives of this paper are:
-\begin{enumerate}
-\item Build the above mentioned idea into formal model and prove a series of properties
-until we are convinced that the formal model does fulfill the original idea.
-\item Show how formally derived properties can be used as guidelines for correct
-and efficient implementation.
-\end{enumerate}.
-The proof is totally formal in the sense that every detail is reduced to the
-very first principles of Higher Order Logic. The nature of interactive theorem
-proving is for the human user to persuade computer program to accept its arguments.
-A clear and simple understanding of the problem at hand is both a prerequisite and a
-byproduct of such an effort, because everything has finally be reduced to the very
-first principle to be checked mechanically. The former intuitive explanation of
-Priority Inheritance is just such a byproduct.
-*}
-
-*)
--- a/prio/Precedence_ord.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,34 +0,0 @@
-header {* Order on product types *}
-
-theory Precedence_ord
-imports Main
-begin
-
-datatype precedence = Prc nat nat
-
-instantiation precedence :: order
-begin
-
-definition
- precedence_le_def: "x \<le> y \<longleftrightarrow> (case (x, y) of
- (Prc fx sx, Prc fy sy) \<Rightarrow>
- fx < fy \<or> (fx \<le> fy \<and> sy \<le> sx))"
-
-definition
- precedence_less_def: "x < y \<longleftrightarrow> (case (x, y) of
- (Prc fx sx, Prc fy sy) \<Rightarrow>
- fx < fy \<or> (fx \<le> fy \<and> sy < sx))"
-
-instance
-proof
-qed (auto simp: precedence_le_def precedence_less_def
- intro: order_trans split:precedence.splits)
-end
-
-instance precedence :: preorder ..
-
-instance precedence :: linorder proof
-qed (auto simp: precedence_le_def precedence_less_def
- intro: order_trans split:precedence.splits)
-
-end
--- a/prio/PrioG.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2864 +0,0 @@
-theory PrioG
-imports PrioGDef
-begin
-
-lemma runing_ready:
- shows "runing s \<subseteq> readys s"
- unfolding runing_def readys_def
- by auto
-
-lemma readys_threads:
- shows "readys s \<subseteq> threads s"
- unfolding readys_def
- by auto
-
-lemma wq_v_neq:
- "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
- by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-lemma wq_distinct: "vt s \<Longrightarrow> distinct (wq s cs)"
-proof(erule_tac vt.induct, simp add:wq_def)
- fix s e
- assume h1: "step s e"
- and h2: "distinct (wq s cs)"
- thus "distinct (wq (e # s) cs)"
- proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
- fix thread s
- assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
- and h2: "thread \<in> set (wq_fun (schs s) cs)"
- and h3: "thread \<in> runing s"
- show "False"
- proof -
- from h3 have "\<And> cs. thread \<in> set (wq_fun (schs s) cs) \<Longrightarrow>
- thread = hd ((wq_fun (schs s) cs))"
- by (simp add:runing_def readys_def s_waiting_def wq_def)
- from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" .
- with h2
- have "(Cs cs, Th thread) \<in> (depend s)"
- by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
- with h1 show False by auto
- qed
- next
- fix thread s a list
- assume dst: "distinct list"
- show "distinct (SOME q. distinct q \<and> set q = set list)"
- proof(rule someI2)
- from dst show "distinct list \<and> set list = set list" by auto
- next
- fix q assume "distinct q \<and> set q = set list"
- thus "distinct q" by auto
- qed
- qed
-qed
-
-lemma step_back_vt: "vt (e#s) \<Longrightarrow> vt s"
- by(ind_cases "vt (e#s)", simp)
-
-lemma step_back_step: "vt (e#s) \<Longrightarrow> step s e"
- by(ind_cases "vt (e#s)", simp)
-
-lemma block_pre:
- fixes thread cs s
- assumes vt_e: "vt (e#s)"
- and s_ni: "thread \<notin> set (wq s cs)"
- and s_i: "thread \<in> set (wq (e#s) cs)"
- shows "e = P thread cs"
-proof -
- show ?thesis
- proof(cases e)
- case (P th cs)
- with assms
- show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Create th prio)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Exit th)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Set th prio)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (V th cs)
- with assms show ?thesis
- apply (auto simp:wq_def Let_def split:if_splits)
- proof -
- fix q qs
- assume h1: "thread \<notin> set (wq_fun (schs s) cs)"
- and h2: "q # qs = wq_fun (schs s) cs"
- and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
- and vt: "vt (V th cs # s)"
- from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp
- moreover have "thread \<in> set qs"
- proof -
- have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
- proof(rule someI2)
- from wq_distinct [OF step_back_vt[OF vt], of cs]
- and h2[symmetric, folded wq_def]
- show "distinct qs \<and> set qs = set qs" by auto
- next
- fix x assume "distinct x \<and> set x = set qs"
- thus "set x = set qs" by auto
- qed
- with h3 show ?thesis by simp
- qed
- ultimately show "False" by auto
- qed
- qed
-qed
-
-lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
- thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (depend s)^+"
-apply (ind_cases "vt ((P thread cs)#s)")
-apply (ind_cases "step s (P thread cs)")
-by auto
-
-lemma abs1:
- fixes e es
- assumes ein: "e \<in> set es"
- and neq: "hd es \<noteq> hd (es @ [x])"
- shows "False"
-proof -
- from ein have "es \<noteq> []" 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) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
- by (cases es, auto)
-
-inductive_cases evt_cons: "vt (a#s)"
-
-lemma abs2:
- assumes vt: "vt (e#s)"
- and inq: "thread \<in> set (wq s cs)"
- and nh: "thread = hd (wq s cs)"
- and qt: "thread \<noteq> hd (wq (e#s) cs)"
- and inq': "thread \<in> set (wq (e#s) cs)"
- shows "False"
-proof -
- from 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 \<in> set (SOME q. distinct q \<and> set q = set qs)"
- and eq_wq: "wq_fun (schs s) cs = thread # qs"
- show "False"
- proof -
- from wq_distinct[OF step_back_vt[OF vt], of cs]
- and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
- moreover have "thread \<in> set qs"
- proof -
- have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
- proof(rule someI2)
- from wq_distinct [OF step_back_vt[OF vt], of cs]
- and eq_wq [folded wq_def]
- show "distinct qs \<and> set qs = set qs" by auto
- next
- fix x assume "distinct x \<and> 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
-qed
-
-lemma vt_moment: "\<And> t. \<lbrakk>vt s\<rbrakk> \<Longrightarrow> vt (moment t s)"
-proof(induct s, simp)
- fix a s t
- assume h: "\<And>t.\<lbrakk>vt s\<rbrakk> \<Longrightarrow> vt (moment t s)"
- and vt_a: "vt (a # s)"
- show "vt (moment t (a # s))"
- proof(cases "t \<ge> length (a#s)")
- case True
- from True have "moment t (a#s) = a#s" by simp
- with vt_a show ?thesis by simp
- next
- case False
- hence le_t1: "t \<le> length s" by simp
- from vt_a have "vt s"
- by (erule_tac evt_cons, simp)
- from h [OF this] have "vt (moment t s)" .
- moreover have "moment t (a#s) = moment t s"
- proof -
- from moment_app [OF le_t1, of "[a]"]
- show ?thesis by simp
- qed
- ultimately show ?thesis by auto
- qed
-qed
-
-(* Wrong:
- lemma \<lbrakk>thread \<in> set (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
-*)
-
-lemma waiting_unique_pre:
- fixes cs1 cs2 s thread
- assumes vt: "vt s"
- and h11: "thread \<in> set (wq s cs1)"
- and h12: "thread \<noteq> hd (wq s cs1)"
- assumes h21: "thread \<in> set (wq s cs2)"
- and h22: "thread \<noteq> hd (wq s cs2)"
- and neq12: "cs1 \<noteq> cs2"
- shows "False"
-proof -
- let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
- from h11 and h12 have q1: "?Q cs1 s" by simp
- from h21 and h22 have q2: "?Q cs2 s" by simp
- have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
- have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
- from p_split [of "?Q cs1", OF q1 nq1]
- obtain t1 where lt1: "t1 < length s"
- and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
- thread \<noteq> hd (wq (moment t1 s) cs1))"
- and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
- thread \<noteq> 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: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
- thread \<noteq> hd (wq (moment t2 s) cs2))"
- and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
- thread \<noteq> hd (wq (moment i' s) cs2))" by auto
- show ?thesis
- proof -
- {
- assume lt12: "t1 < t2"
- let ?t3 = "Suc t2"
- from lt2 have le_t3: "?t3 \<le> 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 \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- have vt_e: "vt (e#moment t2 s)"
- proof -
- from vt_moment [OF vt]
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- have ?thesis
- proof(cases "thread \<in> 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 abs2 [OF vt_e True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from block_pre [OF vt_e False h1]
- have "e = P thread cs2" .
- with vt_e have "vt ((P thread cs2)# moment t2 s)" by simp
- from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
- with runing_ready have "thread \<in> 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 \<le> 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 \<in> set (wq (e#moment t1 s) cs1)" and
- h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have vt_e: "vt (e#moment t1 s)"
- proof -
- from vt_moment [OF vt]
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- have ?thesis
- proof(cases "thread \<in> 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 abs2 [OF vt_e True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from block_pre [OF vt_e False h1]
- have "e = P thread cs1" .
- with vt_e have "vt ((P thread cs1)# moment t1 s)" by simp
- from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
- with runing_ready have "thread \<in> 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 {
- assume eqt12: "t1 = t2"
- let ?t3 = "Suc t1"
- from lt1 have le_t3: "?t3 \<le> 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 \<in> set (wq (e#moment t1 s) cs1)" and
- h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have vt_e: "vt (e#moment t1 s)"
- proof -
- from vt_moment [OF vt]
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- have ?thesis
- proof(cases "thread \<in> 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 abs2 [OF vt_e True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from block_pre [OF vt_e 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 \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- show ?thesis
- proof(cases "thread \<in> 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
- from abs2 [OF this True eq_th h2 h1]
- show ?thesis .
- next
- case False
- have vt_e: "vt (e#moment t2 s)"
- proof -
- from vt_moment [OF vt] eqt12
- have "vt (moment (Suc t2) s)" by auto
- with eq_m eqt12 show ?thesis by simp
- qed
- from block_pre [OF vt_e False h1]
- have "e = P thread cs2" .
- with eq_e1 neq12 show ?thesis by auto
- qed
- qed
- } ultimately show ?thesis by arith
- qed
-qed
-
-lemma waiting_unique:
- fixes s cs1 cs2
- assumes "vt s"
- and "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
-
-(* not used *)
-lemma held_unique:
- fixes s::"state"
- assumes "holding s th1 cs"
- and "holding s th2 cs"
- shows "th1 = th2"
-using assms
-unfolding s_holding_def
-by auto
-
-
-lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits)
-
-lemma birthtime_unique:
- "\<lbrakk>birthtime th1 s = birthtime th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
- \<Longrightarrow> th1 = th2"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits dest:birthtime_lt)
-
-lemma preced_unique :
- assumes pcd_eq: "preced th1 s = preced th2 s"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "th1 = th2"
-proof -
- from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
- from birthtime_unique [OF this th_in1 th_in2]
- show ?thesis .
-qed
-
-lemma preced_linorder:
- assumes neq_12: "th1 \<noteq> th2"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
- from preced_unique [OF _ th_in1 th_in2] and neq_12
- have "preced th1 s \<noteq> preced th2 s" by auto
- thus ?thesis by auto
-qed
-
-lemma unique_minus:
- fixes x y z r
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq: "y \<noteq> z"
- shows "(y, z) \<in> r^+"
-proof -
- from xz and neq show ?thesis
- proof(induct)
- case (base ya)
- have "(x, ya) \<in> 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:
- fixes r x y z
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> 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) \<in> r\<^sup>+" by auto
- with step show ?thesis by auto
- qed
- qed
-qed
-
-lemma unique_chain:
- fixes r x y z
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r^+"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
-proof -
- from xy xz neq_yz show ?thesis
- proof(induct)
- case (base y)
- have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> 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) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
- thus ?thesis
- proof
- assume "(z, y) \<in> r\<^sup>+"
- with step have "(z, za) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- next
- assume h: "(y, z) \<in> r\<^sup>+"
- from step have yza: "(y, za) \<in> r" by simp
- from step have "za \<noteq> z" by simp
- from unique_minus [OF _ yza h this] and unique
- have "(za, z) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- qed
- qed
- qed
-qed
-
-lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-
-
-lemma step_v_hold_inv[elim_format]:
- "\<And>c t. \<lbrakk>vt (V th cs # s);
- \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> next_th s th cs t \<and> c = cs"
-proof -
- fix c t
- assume vt: "vt (V th cs # s)"
- and nhd: "\<not> holding (wq s) t c"
- and hd: "holding (wq (V th cs # s)) t c"
- show "next_th s th cs t \<and> 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 \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
- moreover have "\<dots> = set []"
- proof(rule someI2)
- show "distinct [] \<and> [] = []" by auto
- next
- fix x assume "distinct x \<and> x = []"
- thus "set x = set []" by auto
- qed
- ultimately show False by auto
- next
- assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
- moreover have "\<dots> = set []"
- proof(rule someI2)
- show "distinct [] \<and> [] = []" by auto
- next
- fix x assume "distinct x \<and> x = []"
- thus "set x = set []" by auto
- qed
- ultimately show False by auto
- qed
- qed
- with True show ?thesis by auto
- qed
-qed
-
-lemma step_v_wait_inv[elim_format]:
- "\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
- \<rbrakk>
- \<Longrightarrow> (next_th s th cs t \<and> cs = c)"
-proof -
- fix t c
- assume vt: "vt (V th cs # s)"
- and nw: "\<not> waiting (wq (V th cs # s)) t c"
- and wt: "waiting (wq s) t c"
- show "next_th s th cs t \<and> 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
- 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 \<in> set list"
- and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have " set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> 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 \<in> set list"
- and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have " set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
- by auto
- qed
- with t_ni and t_in show "t = hd (SOME q. distinct q \<and> 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 step_v_not_wait[consumes 3]:
- "\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
- by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
-
-lemma step_v_release:
- "\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
-proof -
- assume vt: "vt (V th cs # s)"
- and hd: "holding (wq (V th cs # s)) th cs"
- 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 \<and> set q = set list) # list"
- and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
- \<in> set (SOME q. distinct q \<and> set q = set list)"
- have "set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
- by auto
- qed
- moreover have "distinct (hd (SOME q. distinct q \<and> set q = set list) # list)"
- proof -
- from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
- show ?thesis by auto
- qed
- moreover note eq_wq and hd_in
- ultimately show "False" by auto
- qed
- qed
-qed
-
-lemma step_v_get_hold:
- "\<And>th'. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
- apply (unfold cs_holding_def next_th_def wq_def,
- auto simp:Let_def)
-proof -
- 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 \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest)
- \<notin> set (SOME q. distinct q \<and> set q = set rest)"
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- hence "set x = set rest" by auto
- with nrest
- show "x \<noteq> []" by (case_tac x, auto)
- qed
- with ni show "False" by auto
-qed
-
-lemma step_v_release_inv[elim_format]:
-"\<And>c t. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow>
- c = cs \<and> 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
- by (unfold s_holding_def wq_def, auto)
- qed
- next
- 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
- by (unfold s_holding_def wq_def, auto)
- qed
- qed
-
-lemma step_v_waiting_mono:
- "\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> 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"
- show "waiting (wq s) t c"
- proof(cases "c = cs")
- case False
- assume neq_cs: "c \<noteq> 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 \<notin> set list"
- and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have "set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from wq_distinct [OF step_back_vt[OF vt], of cs]
- and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- fix x assume "distinct x \<and> set x = set list"
- thus "set x = set list" by auto
- qed
- with not_in is_in show "t = a" by auto
- next
- 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 \<in> set list"
- apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
- proof -
- assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
- moreover have "\<dots> = set list"
- proof(rule someI2)
- from wq_distinct [OF step_back_vt[OF vt], of cs]
- and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- fix x assume "distinct x \<and> set x = set list"
- thus "set x = set list" by auto
- qed
- ultimately show "t \<in> set list" by simp
- qed
- with eq_wq and wq_distinct [OF step_back_vt[OF vt], of cs, unfolded wq_def]
- show False by auto
- qed
- qed
-qed
-
-lemma step_depend_v:
-fixes th::thread
-assumes vt:
- "vt (V th cs#s)"
-shows "
- depend (V th cs # s) =
- depend s - {(Cs cs, Th th)} -
- {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}"
- apply (insert vt, unfold s_depend_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 step_depend_p:
- "vt (P th cs#s) \<Longrightarrow>
- depend (P th cs # s) = (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
- else depend s \<union> {(Th th, Cs cs)})"
- apply(simp only: s_depend_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(auto simp:s_depend_def wq_def cs_holding_def)
- done
-
-lemma simple_A:
- fixes A
- assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
- shows "A = {} \<or> (\<exists> a. A = {a})"
-proof(cases "A = {}")
- case True thus ?thesis by simp
-next
- case False then obtain a where "a \<in> A" by auto
- with h have "A = {a}" by auto
- thus ?thesis by simp
-qed
-
-lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
- by (unfold s_depend_def, auto)
-
-lemma acyclic_depend:
- fixes s
- assumes vt: "vt s"
- shows "acyclic (depend s)"
-proof -
- from vt show ?thesis
- proof(induct)
- case (vt_cons s e)
- assume ih: "acyclic (depend 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:depend_create_unchanged)
- next
- case (Exit th)
- with ih show ?thesis by (simp add:depend_exit_unchanged)
- next
- case (V th cs)
- from V vt stp have vtt: "vt (V th cs#s)" by auto
- from step_depend_v [OF this]
- have eq_de:
- "depend (e # s) =
- depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}"
- (is "?L = (?A - ?B - ?C) \<union> ?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 \<in> 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 \<and> 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) \<notin> ?E\<^sup>*"
- proof
- assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
- hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- from tranclD [OF this]
- obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
- hence th_d: "(Th ?th', x) \<in> ?A" by simp
- from depend_target_th [OF this]
- obtain cs' where eq_x: "x = Cs cs'" by auto
- with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
- hence wt_th': "waiting s ?th' cs'"
- unfolding s_depend_def s_waiting_def cs_waiting_def wq_def by simp
- hence "cs' = cs"
- proof(rule waiting_unique [OF vt])
- from eq_wq wq_distinct[OF vt, of cs]
- show "waiting s ?th' cs"
- apply (unfold s_waiting_def wq_def, auto)
- proof -
- assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq_fun (schs s) cs = th # rest"
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with False show "x \<noteq> []" by auto
- qed
- hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
- set (SOME q. distinct q \<and> set q = set rest)" by auto
- moreover have "\<dots> = set rest"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- moreover note hd_in
- ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
- next
- assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with False show "x \<noteq> []" by auto
- qed
- hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
- set (SOME q. distinct q \<and> set q = set rest)" by auto
- moreover have "\<dots> = set rest"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> 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) \<in> ?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_depend_p [OF this] P
- have "depend (e # s) =
- (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
- depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
- by simp
- moreover have "acyclic ?R"
- proof(cases "wq s cs = []")
- case True
- hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
- have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
- proof
- assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
- hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- from tranclD2 [OF this]
- obtain x where "(x, Cs cs) \<in> depend s" by auto
- with True show False by (auto simp:s_depend_def cs_waiting_def)
- qed
- with acyclic_insert ih eq_r show ?thesis by auto
- next
- case False
- hence eq_r: "?R = depend s \<union> {(Th th, Cs cs)}" by simp
- have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
- proof
- assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
- hence "(Cs cs, Th th) \<in> (depend 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 " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> 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 depend_set_unchanged
- show ?thesis by (simp add:depend_set_unchanged)
- qed
- next
- case vt_nil
- show "acyclic (depend ([]::state))"
- by (auto simp: s_depend_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
- qed
-qed
-
-lemma finite_depend:
- fixes s
- assumes vt: "vt s"
- shows "finite (depend s)"
-proof -
- from vt show ?thesis
- proof(induct)
- case (vt_cons s e)
- assume ih: "finite (depend 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:depend_create_unchanged)
- next
- case (Exit th)
- with ih show ?thesis by (simp add:depend_exit_unchanged)
- next
- case (V th cs)
- from V vt stp have vtt: "vt (V th cs#s)" by auto
- from step_depend_v [OF this]
- have eq_de: "depend (e # s) =
- depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}
-"
- (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
- moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
- moreover have "finite ?D"
- proof -
- have "?D = {} \<or> (\<exists> a. ?D = {a})"
- by (unfold next_th_def, auto)
- thus ?thesis
- proof
- assume h: "?D = {}"
- show ?thesis by (unfold h, simp)
- next
- assume "\<exists> a. ?D = {a}"
- thus ?thesis by auto
- qed
- qed
- ultimately show ?thesis by simp
- next
- case (P th cs)
- from P vt stp have vtt: "vt (P th cs#s)" by auto
- from step_depend_p [OF this] P
- have "depend (e # s) =
- (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
- depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
- by simp
- moreover have "finite ?R"
- proof(cases "wq s cs = []")
- case True
- hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
- with True and ih show ?thesis by auto
- next
- case False
- hence "?R = depend s \<union> {(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:depend_set_unchanged)
- qed
- next
- case vt_nil
- show "finite (depend ([]::state))"
- by (auto simp: s_depend_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
- qed
-qed
-
-text {* Several useful lemmas *}
-
-lemma wf_dep_converse:
- fixes s
- assumes vt: "vt s"
- shows "wf ((depend s)^-1)"
-proof(rule finite_acyclic_wf_converse)
- from finite_depend [OF vt]
- show "finite (depend s)" .
-next
- from acyclic_depend[OF vt]
- show "acyclic (depend s)" .
-qed
-
-lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
-by (induct l, auto)
-
-lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-
-lemma wq_threads:
- fixes s cs
- assumes vt: "vt s"
- and h: "th \<in> set (wq s cs)"
- shows "th \<in> threads s"
-proof -
- from vt and h show ?thesis
- proof(induct arbitrary: th cs)
- case (vt_cons s e)
- assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
- and stp: "step s e"
- and vt: "vt s"
- and h: "th \<in> 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_depend_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 \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
- show "th \<in> 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 \<in> 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 \<in> set (SOME q. distinct q \<and> set q = set rest)"
- have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs'] and eq_wq[folded wq_def]
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
- by auto
- qed
- with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
- from ih[OF this[folded wq_def]] show "th \<in> threads s" .
- next
- assume th_in: "th \<in> set (wq_fun (schs s) cs)"
- from ih[OF this[folded wq_def]]
- show "th \<in> 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
- next
- case vt_nil
- thus ?case by (auto simp:wq_def)
- qed
-qed
-
-lemma range_in: "\<lbrakk>vt s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
- apply(unfold s_depend_def cs_waiting_def cs_holding_def)
- by (auto intro:wq_threads)
-
-lemma readys_v_eq:
- fixes th thread cs rest
- assumes vt: "vt s"
- and neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and not_in: "th \<notin> set rest"
- shows "(th \<in> readys (V thread cs#s)) = (th \<in> 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 \<notin> set rest"
- and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
- and eq_wq: "wq_fun (schs s) cs = thread # rest"
- have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- with th_nin th_in show False by auto
- qed
-qed
-
-lemma chain_building:
- assumes vt: "vt s"
- shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
-proof -
- from wf_dep_converse [OF vt]
- have h: "wf ((depend s)\<inverse>)" .
- show ?thesis
- proof(induct rule:wf_induct [OF h])
- fix x
- assume ih [rule_format]:
- "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow>
- y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
- show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
- proof
- assume x_d: "x \<in> Domain (depend s)"
- show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
- proof(cases x)
- case (Th th)
- from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
- with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
- from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
- hence "Cs cs \<in> Domain (depend s)" by auto
- from ih [OF x_in_r this] obtain th'
- where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto
- have "(x, Th th') \<in> (depend 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) \<in> (depend s)^-1" by (auto simp:s_depend_def)
- show ?thesis
- proof(cases "th' \<in> readys s")
- case True
- from True and th'_d show ?thesis by auto
- next
- case False
- from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto
- with False have "Th th' \<in> Domain (depend s)"
- by (auto simp:readys_def wq_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
- from ih [OF th'_d this]
- obtain th'' where
- th''_r: "th'' \<in> readys s" and
- th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto
- from th'_d and th''_in
- have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
- with th''_r show ?thesis by auto
- qed
- qed
- qed
- qed
-qed
-
-lemma th_chain_to_ready:
- fixes s th
- assumes vt: "vt s"
- and th_in: "th \<in> threads s"
- shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"
-proof(cases "th \<in> readys s")
- case True
- thus ?thesis by auto
-next
- case False
- from False and th_in have "Th th \<in> Domain (depend s)"
- by (auto simp:readys_def s_waiting_def s_depend_def wq_def cs_waiting_def Domain_def)
- from chain_building [rule_format, OF vt this]
- show ?thesis by auto
-qed
-
-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: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
- by (unfold s_holding_def cs_holding_def, auto)
-
-lemma unique_depend: "\<lbrakk>vt s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
- apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
- by(auto elim:waiting_unique holding_unique)
-
-lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
-by (induct rule:trancl_induct, auto)
-
-lemma dchain_unique:
- assumes vt: "vt s"
- and th1_d: "(n, Th th1) \<in> (depend s)^+"
- and th1_r: "th1 \<in> readys s"
- and th2_d: "(n, Th th2) \<in> (depend s)^+"
- and th2_r: "th2 \<in> readys s"
- shows "th1 = th2"
-proof -
- { assume neq: "th1 \<noteq> th2"
- hence "Th th1 \<noteq> Th th2" by simp
- from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]
- have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
- hence "False"
- proof
- assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
- from trancl_split [OF this]
- obtain n where dd: "(Th th1, n) \<in> depend s" by auto
- then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
- from dd eq_n have "th1 \<notin> readys s"
- by (auto simp:readys_def s_depend_def wq_def s_waiting_def cs_waiting_def)
- with th1_r show ?thesis by auto
- next
- assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
- from trancl_split [OF this]
- obtain n where dd: "(Th th2, n) \<in> depend s" by auto
- then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
- from dd eq_n have "th2 \<notin> readys s"
- by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def)
- with th2_r show ?thesis by auto
- qed
- } thus ?thesis by auto
-qed
-
-
-lemma step_holdents_p_add:
- fixes th cs s
- assumes vt: "vt (P th cs#s)"
- and "wq s cs = []"
- shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
-proof -
- from assms show ?thesis
- unfolding holdents_test step_depend_p[OF vt] by (auto)
-qed
-
-lemma step_holdents_p_eq:
- fixes th cs s
- assumes vt: "vt (P th cs#s)"
- and "wq s cs \<noteq> []"
- shows "holdents (P th cs#s) th = holdents s th"
-proof -
- from assms show ?thesis
- unfolding holdents_test step_depend_p[OF vt] by auto
-qed
-
-
-lemma finite_holding:
- fixes s th cs
- assumes vt: "vt s"
- shows "finite (holdents s th)"
-proof -
- let ?F = "\<lambda> (x, y). the_cs x"
- from finite_depend [OF vt]
- have "finite (depend s)" .
- hence "finite (?F `(depend s))" by simp
- moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>"
- proof -
- { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
- fix x assume "(Cs x, Th th) \<in> depend s"
- hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
- moreover have "?F (Cs x, Th th) = x" by simp
- ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp
- } thus ?thesis by auto
- qed
- ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
-qed
-
-lemma cntCS_v_dec:
- fixes s thread cs
- assumes vtv: "vt (V thread cs#s)"
- shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
-proof -
- from step_back_step[OF vtv]
- have cs_in: "cs \<in> holdents s thread"
- apply (cases, unfold holdents_test s_depend_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 wq_distinct[OF step_back_vt[OF vtv], of cs])
- apply (unfold holdents_test, unfold step_depend_v[OF vtv],
- auto simp:next_th_def)
- proof -
- fix rest
- assume dst: "distinct (rest::thread list)"
- and ne: "rest \<noteq> []"
- and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)" by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume " distinct x \<and> set x = set rest" with ne
- show "x \<noteq> []" by auto
- qed
- ultimately
- show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"
- by auto
- next
- fix rest
- assume dst: "distinct (rest::thread list)"
- and ne: "rest \<noteq> []"
- and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)" by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume " distinct x \<and> set x = set rest" with ne
- show "x \<noteq> []" 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 \<dots> =
- Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
- proof(rule card_insert)
- from finite_holding [OF vtv]
- show " finite (holdents (V thread cs # s) thread)" .
- qed
- moreover from cs_not_in
- have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
- ultimately show ?thesis by (simp add:cntCS_def)
-qed
-
-lemma cnp_cnv_cncs:
- fixes s th
- assumes vt: "vt s"
- shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
- then cntCS s th else cntCS s th + 1)"
-proof -
- from vt show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e)
- assume vt: "vt s"
- and ih: "\<And>th. cntP s th = cntV s th +
- (if (th \<in> readys s \<or> th \<notin> 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 \<notin> threads s"
- show ?thesis
- proof -
- { fix cs
- assume "thread \<in> set (wq s cs)"
- from wq_threads [OF vt this] have "thread \<in> threads s" .
- with not_in have "False" by simp
- } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {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:depend_create_unchanged eq_e)
- { assume "th \<noteq> thread"
- with eq_readys eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> 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 \<in> 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 \<in> 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:depend_exit_unchanged eq_e)
- { assume "th \<noteq> thread"
- with eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> 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 \<notin> 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 \<in> runing s"
- and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
- from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto
- show ?thesis
- proof -
- { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
- assume neq_th: "th \<noteq> thread"
- with eq_e
- have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
- apply (simp add:readys_def s_waiting_def wq_def Let_def)
- apply (rule_tac hh, clarify)
- apply (intro iffI allI, clarify)
- apply (erule_tac x = csa in allE, auto)
- apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", 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_depend_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 \<in> readys (e#s)"
- apply (unfold eq_e wq_def, unfold readys_def s_depend_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 \<or> (Cs csa, Th thread) \<in> depend s} =
- Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")
- proof -
- have "?L = insert cs ?R" by auto
- moreover have "card \<dots> = Suc (card (?R - {cs}))"
- proof(rule card_insert)
- from finite_holding [OF vt, of thread]
- show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
- by (unfold holdents_test, simp)
- qed
- moreover have "?R - {cs} = ?R"
- proof -
- have "cs \<notin> ?R"
- proof
- assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend 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_depend_p [OF vtp], auto simp:True)
- qed
- moreover from is_runing have "th \<in> 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 \<notin> readys (e#s)"
- proof
- assume "th \<in> readys (e#s)"
- hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
- from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
- hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
- by (simp add:s_waiting_def wq_def)
- moreover from eq_wq have "th \<in> 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 wq_distinct [OF vtp, of cs]
- show False by (fold eq_e, auto)
- qed
- moreover from is_runing have "th \<in> 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_depend_p[OF vtp])
- by (auto simp:False)
- moreover note eq_cnp eq_cnv ih[of th]
- moreover from is_runing have "th \<in> readys s"
- by (simp add:runing_def eq_th)
- ultimately show ?thesis by auto
- qed
- qed
- } ultimately show ?thesis by blast
- qed
- next
- case (thread_V thread cs)
- from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
- assume eq_e: "e = V thread cs"
- and is_runing: "thread \<in> 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 \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> 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 \<in> readys s"
- by (unfold runing_def, simp)
- moreover have "thread \<in> readys (e # s)"
- proof -
- from is_runing
- have "thread \<in> threads (e#s)"
- by (unfold eq_e, auto simp:runing_def readys_def)
- moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
- proof
- fix cs1
- { assume eq_cs: "cs1 = cs"
- have "\<not> waiting (e # s) thread cs1"
- proof -
- from eq_wq
- have "thread \<notin> set (wq (e#s) cs1)"
- apply(unfold eq_e wq_def eq_cs s_holding_def)
- apply (auto simp:Let_def)
- proof -
- assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
- with eq_set have "thread \<in> set rest" by simp
- with wq_distinct[OF step_back_vt[OF vtv], of cs]
- and eq_wq show False by auto
- qed
- thus ?thesis by (simp add:wq_def s_waiting_def)
- qed
- } moreover {
- assume neq_cs: "cs1 \<noteq> cs"
- have "\<not> 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 "\<not> waiting s thread cs1"
- proof -
- from runing_ready and is_runing
- have "thread \<in> 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 "\<not> 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 \<noteq> 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 \<in> set rest")
- case False
- have "(th \<in> readys (e # s)) = (th \<in> readys s)"
- apply (insert step_back_vt[OF vtv])
- by (unfold eq_e, rule readys_v_eq [OF _ neq_th eq_wq False], auto)
- moreover have "cntCS (e#s) th = cntCS s th"
- apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
- proof -
- have "{csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
- {cs. (Cs cs, Th th) \<in> depend s}"
- proof -
- from False eq_wq
- have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> depend s"
- apply (unfold next_th_def, auto)
- proof -
- assume ne: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq s cs = thread # rest"
- from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)
- " by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with ne show "x \<noteq> []" by auto
- qed
- ultimately show
- "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"
- by auto
- qed
- thus ?thesis by auto
- qed
- thus "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
- card {cs. (Cs cs, Th th) \<in> depend 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 \<in> set rest"
- show ?thesis
- proof(cases "next_th s thread cs th")
- case False
- with eq_wq and th_in have
- neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
- by (auto simp:next_th_def)
- have "(th \<in> readys (e # s)) = (th \<in> readys s)"
- proof -
- from eq_wq and th_in
- have "\<not> th \<in> readys s"
- apply (auto simp:readys_def s_waiting_def)
- apply (rule_tac x = cs in exI, auto)
- by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto simp add: wq_def)
- moreover
- from eq_wq and th_in and neq_hd
- have "\<not> (th \<in> 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_depend_v[OF vtv],
- auto simp:next_th_def eq_set s_depend_def holdents_test wq_def
- Let_def cs_holding_def)
- by (insert wq_distinct[OF step_back_vt[OF vtv], 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 \<and> set q = set rest)"
- let ?t = "hd ?rest"
- from True eq_wq th_in neq_th
- have "th \<in> 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 \<in> set rest"
- show "?t \<in> threads s"
- proof(rule wq_threads[OF step_back_vt[OF vtv]])
- from eq_wq and t_in
- show "?t \<in> 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 \<in> set rest"
- and neq_cs: "csa \<noteq> cs"
- and t_in': "?t \<in> set (wq_fun (schs s) csa)"
- show "?t = hd (wq_fun (schs s) csa)"
- proof -
- { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
- from wq_distinct[OF step_back_vt[OF vtv], of cs] and
- eq_wq[folded wq_def] and t_in eq_wq
- have "?t \<noteq> 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 waiting_unique[OF step_back_vt[OF vtv] 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 \<notin> 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 \<in> threads s"
- proof -
- from th_in eq_wq
- have "th \<in> set (wq s cs)" by simp
- from wq_threads [OF step_back_vt[OF vtv] 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_depend_v[OF vtv], auto)
- proof -
- show "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs} =
- Suc (card {cs. (Cs cs, Th th) \<in> depend 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 "\<dots> = Suc (card ?B)"
- proof(rule card_insert_disjoint)
- have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` depend s)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
- with finite_depend[OF step_back_vt[OF vtv]]
- show "finite {cs. (Cs cs, Th th) \<in> depend s}" by (auto intro:finite_subset)
- next
- show "cs \<notin> {cs. (Cs cs, Th th) \<in> depend s}"
- proof
- assume "cs \<in> {cs. (Cs cs, Th th) \<in> depend s}"
- hence "(Cs cs, Th th) \<in> depend s" by simp
- with True neq_th eq_wq show False
- by (auto simp:next_th_def s_depend_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 \<in> 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:depend_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 \<noteq> thread"
- with eq_readys eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> 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 \<in> 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
- next
- case vt_nil
- show ?case
- by (unfold cntP_def cntV_def cntCS_def,
- auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
- qed
-qed
-
-lemma not_thread_cncs:
- fixes th s
- assumes vt: "vt s"
- and not_in: "th \<notin> threads s"
- shows "cntCS s th = 0"
-proof -
- from vt not_in show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e th)
- assume vt: "vt s"
- and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
- and stp: "step s e"
- and not_in: "th \<notin> 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 \<notin> threads s"
- have "cntCS (e # s) th = cntCS s th"
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:depend_create_unchanged)
- moreover have "th \<notin> threads s"
- proof -
- from not_in eq_e show ?thesis by simp
- qed
- moreover note ih ultimately show ?thesis by auto
- next
- 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:depend_exit_unchanged)
- show ?thesis
- proof(cases "th = thread")
- case True
- have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
- with eq_cns show ?thesis by simp
- next
- case False
- with not_in and eq_e
- have "th \<notin> 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 \<in> runing s"
- from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> 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_depend_p[OF vtp], auto)
- moreover have "cntCS s th = 0"
- proof(rule ih)
- from not_in eq_e show "th \<notin> 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 \<in> runing s"
- and hold: "holding s thread cs"
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- from assms thread_V vt stp ih have vtv: "vt (V thread cs#s)" by auto
- 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 "\<not> next_th s thread cs th"
- apply (auto simp:next_th_def)
- proof -
- assume ne: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
- have "?t \<in> set rest"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest" with ne
- show "hd x \<in> set rest" by (cases x, auto)
- qed
- with eq_wq have "?t \<in> set (wq s cs)" by simp
- from wq_threads[OF step_back_vt[OF vtv], OF this] and ni
- show False by auto
- 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_depend_v[OF vtv], auto)
- moreover have "cntCS s th = 0"
- proof(rule ih)
- from not_in eq_e show "th \<notin> 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 \<in> runing s"
- from not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] and eq_e
- show ?thesis
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:depend_set_unchanged)
- qed
- next
- case vt_nil
- show ?case
- by (unfold cntCS_def,
- auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
- qed
-qed
-
-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 dm_depend_threads:
- fixes th s
- assumes vt: "vt s"
- and in_dom: "(Th th) \<in> Domain (depend s)"
- shows "th \<in> threads s"
-proof -
- from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
- moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
- ultimately have "(Th th, Cs cs) \<in> depend s" by simp
- hence "th \<in> set (wq s cs)"
- by (unfold s_depend_def, auto simp:cs_waiting_def)
- from wq_threads [OF vt 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 runing_unique:
- fixes th1 th2 s
- assumes vt: "vt s"
- and runing_1: "th1 \<in> runing s"
- and runing_2: "th2 \<in> runing s"
- shows "th1 = th2"
-proof -
- from runing_1 and runing_2 have "cp s th1 = cp s th2"
- by (unfold runing_def, simp)
- hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
- Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
- (is "Max (?f ` ?A) = Max (?f ` ?B)")
- by (unfold cp_eq_cpreced cpreced_def)
- obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
- proof -
- have h1: "finite (?f ` ?A)"
- proof -
- have "finite ?A"
- proof -
- have "finite (dependents (wq s) th1)"
- proof-
- have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependents_def)
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by auto
- qed
- moreover have h2: "(?f ` ?A) \<noteq> {}"
- proof -
- have "?A \<noteq> {}" by simp
- thus ?thesis by simp
- qed
- from Max_in [OF h1 h2]
- have "Max (?f ` ?A) \<in> (?f ` ?A)" .
- thus ?thesis by (auto intro:that)
- qed
- obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
- proof -
- have h1: "finite (?f ` ?B)"
- proof -
- have "finite ?B"
- proof -
- have "finite (dependents (wq s) th2)"
- proof-
- have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependents_def)
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by auto
- qed
- moreover have h2: "(?f ` ?B) \<noteq> {}"
- proof -
- have "?B \<noteq> {}" by simp
- thus ?thesis by simp
- qed
- from Max_in [OF h1 h2]
- have "Max (?f ` ?B) \<in> (?f ` ?B)" .
- thus ?thesis by (auto intro:that)
- qed
- 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 \<or> (th1' \<in> dependents (wq s) th1)" by simp
- thus "th1' \<in> threads s"
- proof
- assume "th1' \<in> dependents (wq s) th1"
- hence "(Th th1') \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependents_def cs_depend_def s_depend_def)
- by (auto simp:Domain_def)
- hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
- from dm_depend_threads[OF vt this] show ?thesis .
- next
- assume "th1' = th1"
- with runing_1 show ?thesis
- by (unfold runing_def readys_def, auto)
- qed
- next
- from th2_in have "th2' = th2 \<or> (th2' \<in> dependents (wq s) th2)" by simp
- thus "th2' \<in> threads s"
- proof
- assume "th2' \<in> dependents (wq s) th2"
- hence "(Th th2') \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependents_def cs_depend_def s_depend_def)
- by (auto simp:Domain_def)
- hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
- from dm_depend_threads[OF vt this] show ?thesis .
- next
- assume "th2' = th2"
- with runing_2 show ?thesis
- by (unfold runing_def readys_def, auto)
- qed
- qed
- from th1_in have "th1' = th1 \<or> th1' \<in> dependents (wq s) th1" by simp
- thus ?thesis
- proof
- assume eq_th': "th1' = th1"
- from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
- thus ?thesis
- proof
- assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
- next
- assume "th2' \<in> dependents (wq s) th2"
- with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
- hence "(Th th1, Th th2) \<in> (depend s)^+"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
- hence "Th th1 \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependents_def cs_depend_def s_depend_def)
- by (auto simp:Domain_def)
- hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
- from depend_target_th [OF this]
- obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th1, Cs cs') \<in> depend s" by simp
- with runing_1 have "False"
- apply (unfold runing_def readys_def s_depend_def)
- by (auto simp:eq_waiting)
- thus ?thesis by simp
- qed
- next
- assume th1'_in: "th1' \<in> dependents (wq s) th1"
- from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
- thus ?thesis
- proof
- assume "th2' = th2"
- with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
- hence "(Th th2, Th th1) \<in> (depend s)^+"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
- hence "Th th2 \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependents_def cs_depend_def s_depend_def)
- by (auto simp:Domain_def)
- hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
- from depend_target_th [OF this]
- obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th2, Cs cs') \<in> depend s" by simp
- with runing_2 have "False"
- apply (unfold runing_def readys_def s_depend_def)
- by (auto simp:eq_waiting)
- thus ?thesis by simp
- next
- assume "th2' \<in> dependents (wq s) th2"
- with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
- hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
- from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
- show ?thesis
- proof(rule dchain_unique[OF vt h1 _ h2, symmetric])
- from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
- from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
- qed
- qed
- qed
-qed
-
-lemma create_pre:
- assumes stp: "step s e"
- and not_in: "th \<notin> threads s"
- and is_in: "th \<in> threads (e#s)"
- obtains prio where "e = Create th prio"
-proof -
- from assms
- show ?thesis
- proof(cases)
- case (thread_create thread prio)
- with is_in not_in have "e = Create th prio" by simp
- from that[OF this] show ?thesis .
- next
- case (thread_exit thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_P thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_V thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_set thread)
- with assms show ?thesis by (auto intro!:that)
- qed
-qed
-
-lemma length_down_to_in:
- assumes le_ij: "i \<le> j"
- and le_js: "j \<le> length s"
- shows "length (down_to j i s) = j - i"
-proof -
- have "length (down_to j i s) = length (from_to i j (rev s))"
- by (unfold down_to_def, auto)
- also have "\<dots> = j - i"
- proof(rule length_from_to_in[OF le_ij])
- from le_js show "j \<le> length (rev s)" by simp
- qed
- finally show ?thesis .
-qed
-
-
-lemma moment_head:
- assumes le_it: "Suc i \<le> length t"
- obtains e where "moment (Suc i) t = e#moment i t"
-proof -
- have "i \<le> Suc i" by simp
- from length_down_to_in [OF this le_it]
- have "length (down_to (Suc i) i t) = 1" by auto
- then obtain e where "down_to (Suc i) i t = [e]"
- apply (cases "(down_to (Suc i) i t)") by auto
- moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
- by (rule down_to_conc[symmetric], auto)
- ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
- by (auto simp:down_to_moment)
- from that [OF this] show ?thesis .
-qed
-
-lemma cnp_cnv_eq:
- fixes th s
- assumes "vt s"
- and "th \<notin> threads s"
- shows "cntP s th = cntV s th"
-proof -
- from assms show ?thesis
- proof(induct)
- case (vt_cons s e)
- have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
- have not_in: "th \<notin> threads (e # s)" by fact
- have "step s e" by fact
- thus ?case proof(cases)
- case (thread_create thread prio)
- assume eq_e: "e = Create thread prio"
- hence "thread \<in> threads (e#s)" by simp
- with not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] show ?thesis using eq_e
- by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- and not_holding: "holdents s thread = {}"
- have vt_s: "vt s" by fact
- from finite_holding[OF vt_s] have "finite (holdents s thread)" .
- with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
- moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
- moreover note cnp_cnv_cncs[OF vt_s, of thread]
- ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
- show ?thesis
- proof(cases "th = thread")
- case True
- with eq_thread eq_e show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- next
- case False
- with not_in and eq_e have "th \<notin> threads s" by simp
- from ih[OF this] and eq_e show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- have "thread \<in> runing s" by fact
- with not_in eq_e have neq_th: "thread \<noteq> th"
- by (auto simp:runing_def readys_def)
- from not_in eq_e have "th \<notin> threads s" by simp
- from ih[OF this] and neq_th and eq_e show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_V thread cs)
- assume eq_e: "e = V thread cs"
- have "thread \<in> runing s" by fact
- with not_in eq_e have neq_th: "thread \<noteq> th"
- by (auto simp:runing_def readys_def)
- from not_in eq_e have "th \<notin> threads s" by simp
- from ih[OF this] and neq_th and eq_e show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- next
- case (thread_set thread prio)
- assume eq_e: "e = Set thread prio"
- and "thread \<in> runing s"
- hence "thread \<in> threads (e#s)"
- by (simp add:runing_def readys_def)
- with not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] show ?thesis using eq_e
- by (auto simp:cntP_def cntV_def count_def)
- qed
- next
- case vt_nil
- show ?case by (auto simp:cntP_def cntV_def count_def)
- qed
-qed
-
-lemma eq_depend:
- "depend (wq s) = depend s"
-by (unfold cs_depend_def s_depend_def, auto)
-
-lemma count_eq_dependents:
- assumes vt: "vt s"
- and eq_pv: "cntP s th = cntV s th"
- shows "dependents (wq s) th = {}"
-proof -
- from cnp_cnv_cncs[OF vt] and eq_pv
- have "cntCS s th = 0"
- by (auto split:if_splits)
- moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
- proof -
- from finite_holding[OF vt, of th] show ?thesis
- by (simp add:holdents_test)
- qed
- ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
- by (unfold cntCS_def holdents_test cs_dependents_def, auto)
- show ?thesis
- proof(unfold cs_dependents_def)
- { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
- then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
- hence "False"
- proof(cases)
- assume "(Th th', Th th) \<in> depend (wq s)"
- thus "False" by (auto simp:cs_depend_def)
- next
- fix c
- assume "(c, Th th) \<in> depend (wq s)"
- with h and eq_depend show "False"
- by (cases c, auto simp:cs_depend_def)
- qed
- } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
- qed
-qed
-
-lemma dependents_threads:
- fixes s th
- assumes vt: "vt s"
- shows "dependents (wq s) th \<subseteq> threads s"
-proof
- { fix th th'
- assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
- have "Th th \<in> Domain (depend s)"
- proof -
- from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
- hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
- with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
- thus ?thesis using eq_depend by simp
- qed
- from dm_depend_threads[OF vt this]
- have "th \<in> threads s" .
- } note hh = this
- fix th1
- assume "th1 \<in> dependents (wq s) th"
- hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
- by (unfold cs_dependents_def, simp)
- from hh [OF this] show "th1 \<in> threads s" .
-qed
-
-lemma finite_threads:
- assumes vt: "vt s"
- shows "finite (threads s)"
-using vt
-by (induct) (auto elim: step.cases)
-
-lemma Max_f_mono:
- assumes seq: "A \<subseteq> B"
- and np: "A \<noteq> {}"
- and fnt: "finite B"
- shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
- from seq show "f ` A \<subseteq> f ` B" by auto
-next
- from np show "f ` A \<noteq> {}" by auto
-next
- from fnt and seq show "finite (f ` B)" by auto
-qed
-
-lemma cp_le:
- assumes vt: "vt s"
- and th_in: "th \<in> threads s"
- shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def)
- show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))
- \<le> Max ((\<lambda>th. preced th s) ` threads s)"
- (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
- proof(rule Max_f_mono)
- show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
- next
- from finite_threads [OF vt]
- show "finite (threads s)" .
- next
- from th_in
- show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
- apply (auto simp:Domain_def)
- apply (rule_tac dm_depend_threads[OF vt])
- apply (unfold trancl_domain [of "depend s", symmetric])
- by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
- qed
-qed
-
-lemma le_cp:
- assumes vt: "vt s"
- shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
- show "Prc (original_priority th s) (birthtime th s)
- \<le> Max (insert (Prc (original_priority th s) (birthtime th s))
- ((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))"
- (is "?l \<le> 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) \<in> (depend (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependents_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:
- assumes vt: "vt s"
- shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
- (is "?l = ?r")
-proof(cases "threads s = {}")
- case True
- thus ?thesis by auto
-next
- case False
- have "?l \<in> ((cp s) ` threads s)"
- proof(rule Max_in)
- from finite_threads[OF vt]
- show "finite (cp s ` threads s)" by auto
- next
- from False show "cp s ` threads s \<noteq> {}" by auto
- qed
- then obtain th
- where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
- have "\<dots> \<le> ?r" by (rule cp_le[OF vt th_in])
- moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
- proof -
- have "?r \<in> (?f ` ?A)"
- proof(rule Max_in)
- from finite_threads[OF vt]
- show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
- next
- from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
- qed
- then obtain th' where
- th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
- from le_cp [OF vt, of th'] eq_r
- have "?r \<le> cp s th'" by auto
- moreover have "\<dots> \<le> cp s th"
- proof(fold eq_l)
- show " cp s th' \<le> Max (cp s ` threads s)"
- proof(rule Max_ge)
- from th_in' show "cp s th' \<in> cp s ` threads s"
- by auto
- next
- from finite_threads[OF vt]
- 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 max_cp_readys_threads_pre:
- assumes vt: "vt s"
- and np: "threads s \<noteq> {}"
- shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(unfold max_cp_eq[OF vt])
- show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
- proof -
- let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
- let ?f = "(\<lambda>th. preced th s)"
- have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
- proof(rule Max_in)
- from finite_threads[OF vt] show "finite (?f ` threads s)" by simp
- next
- from np show "?f ` threads s \<noteq> {}" by simp
- qed
- then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
- by (auto simp:Image_def)
- from th_chain_to_ready [OF vt tm_in]
- have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
- thus ?thesis
- proof
- assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
- then obtain th' where th'_in: "th' \<in> readys s"
- and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
- have "cp s th' = ?f tm"
- proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
- from dependents_threads[OF vt] finite_threads[OF vt]
- show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))"
- by (auto intro:finite_subset)
- next
- fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
- from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
- moreover have "p \<le> \<dots>"
- proof(rule Max_ge)
- from finite_threads[OF vt]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- from p_in and th'_in and dependents_threads[OF vt, of th']
- show "p \<in> (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- qed
- ultimately show "p \<le> preced tm s" by auto
- next
- show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
- proof -
- from tm_chain
- have "tm \<in> dependents (wq s) th'"
- by (unfold cs_dependents_def s_depend_def cs_depend_def, auto)
- thus ?thesis by auto
- qed
- qed
- with tm_max
- have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
- show ?thesis
- proof (fold h, rule Max_eqI)
- fix q
- assume "q \<in> cp s ` readys s"
- then obtain th1 where th1_in: "th1 \<in> readys s"
- and eq_q: "q = cp s th1" by auto
- show "q \<le> cp s th'"
- apply (unfold h eq_q)
- apply (unfold cp_eq_cpreced cpreced_def)
- apply (rule Max_mono)
- proof -
- from dependents_threads [OF vt, of th1] th1_in
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq>
- (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- next
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp
- next
- from finite_threads[OF vt]
- show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
- qed
- next
- from finite_threads[OF vt]
- show "finite (cp s ` readys s)" by (auto simp:readys_def)
- next
- from th'_in
- show "cp s th' \<in> cp s ` readys s" by simp
- qed
- next
- assume tm_ready: "tm \<in> 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 \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
- show "y \<le> preced tm s"
- proof -
- { fix y'
- assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
- have "y' \<le> preced tm s"
- proof(unfold tm_max, rule Max_ge)
- from hy' dependents_threads[OF vt, of tm]
- show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
- next
- from finite_threads[OF vt]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- qed
- } with hy show ?thesis by auto
- qed
- next
- from dependents_threads[OF vt, of tm] finite_threads[OF vt]
- show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
- by (auto intro:finite_subset)
- next
- show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (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 \<in> cp s ` readys s" by simp
- next
- from finite_threads[OF vt]
- show "finite (cp s ` readys s)" by (auto simp:readys_def)
- next
- fix y assume "y \<in> cp s ` readys s"
- then obtain th1 where th1_readys: "th1 \<in> readys s"
- and h: "y = cp s th1" by auto
- show "y \<le> 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[OF vt]
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
- by simp
- next
- from dependents_threads[OF vt, of th1] th1_readys
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)
- \<subseteq> (\<lambda>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 max_cp_readys_threads:
- assumes vt: "vt s"
- 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 vt False])
-qed
-
-
-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: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
- shows "f ` A = g ` A"
-proof
- show "f ` A \<subseteq> g ` A"
- by(rule image_subsetI, auto intro:h)
-next
- show "g ` A \<subseteq> f ` A"
- by (rule image_subsetI, auto intro:h[symmetric])
-qed
-
-
-definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
- where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
-
-
-lemma detached_test:
- shows "detached s th = (Th th \<notin> Field (depend s))"
-apply(simp add: detached_def Field_def)
-apply(simp add: s_depend_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
-
-lemma detached_intro:
- fixes s th
- assumes vt: "vt s"
- and eq_pv: "cntP s th = cntV s th"
- shows "detached s th"
-proof -
- from cnp_cnv_cncs[OF vt]
- have eq_cnt: "cntP s th =
- cntV s th + (if th \<in> readys s \<or> th \<notin> 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 \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
- thus ?thesis
- proof
- assume "th \<notin> threads s"
- with range_in[OF vt] dm_depend_threads[OF vt]
- show ?thesis
- by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
- next
- assume "th \<in> readys s"
- moreover have "Th th \<notin> Range (depend s)"
- proof -
- from card_0_eq [OF finite_holding [OF vt]] 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_depend_def)
- done
- qed
- ultimately show ?thesis
- by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def readys_def)
- qed
-qed
-
-lemma detached_elim:
- fixes s th
- assumes vt: "vt s"
- and dtc: "detached s th"
- shows "cntP s th = cntV s th"
-proof -
- from cnp_cnv_cncs[OF vt]
- have eq_pv: " cntP s th =
- cntV s th + (if th \<in> readys s \<or> th \<notin> 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_depend_def
- by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
- thus ?thesis by (auto simp:cntCS_def)
- qed
- show ?thesis
- proof(cases "th \<in> threads s")
- case True
- with dtc
- have "th \<in> readys s"
- by (unfold readys_def detached_def Field_def Domain_def Range_def,
- auto simp:eq_waiting s_depend_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:
- fixes s th
- assumes vt: "vt s"
- shows "(detached s th) = (cntP s th = cntV s th)"
- by (insert vt, auto intro:detached_intro detached_elim)
-
-end
\ No newline at end of file
--- a/prio/PrioGDef.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,483 +0,0 @@
-(*<*)
-theory PrioGDef
-imports Precedence_ord Moment
-begin
-(*>*)
-
-text {*
- In this section, the formal model of Priority Inheritance is presented.
- The model is based on Paulson's inductive protocol verification method, where
- the state of the system is modelled as a list of events happened so far with the latest
- event put at the head.
-
- To define events, the identifiers of {\em threads},
- {\em priority} and {\em critical resources } (abbreviated as @{text "cs"})
- need to be represented. All three are represetned using standard
- Isabelle/HOL type @{typ "nat"}:
-*}
-
-type_synonym thread = nat -- {* Type for thread identifiers. *}
-type_synonym priority = nat -- {* Type for priorities. *}
-type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
-
-text {*
- \noindent
- Every event in the system corresponds to a system call, the formats of which are
- defined as follows:
- *}
-
-datatype event =
- Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *}
- Exit thread | -- {* Thread @{text "thread"} finishing its execution. *}
- P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *}
- V thread cs | -- {* Thread @{text "thread"} releasing critical resource @{text "cs"}. *}
- Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
-
-text {*
-\noindent
- Resource Allocation Graph (RAG for short) is used extensively in our formal analysis.
- The following type @{text "node"} is used to represent nodes in RAG.
- *}
-datatype node =
- Th "thread" | -- {* Node for thread. *}
- Cs "cs" -- {* Node for critical resource. *}
-
-text {*
- In Paulson's inductive method, the states of system are represented as lists of events,
- which is defined by the following type @{text "state"}:
- *}
-type_synonym state = "event list"
-
-text {*
- \noindent
- The following function
- @{text "threads"} is used to calculate the set of live threads (@{text "threads s"})
- in state @{text "s"}.
- *}
-fun threads :: "state \<Rightarrow> thread set"
- where
- -- {* At the start of the system, the set of threads is empty: *}
- "threads [] = {}" |
- -- {* New thread is added to the @{text "threads"}: *}
- "threads (Create thread prio#s) = {thread} \<union> threads s" |
- -- {* Finished thread is removed: *}
- "threads (Exit thread # s) = (threads s) - {thread}" |
- -- {* Other kind of events does not affect the value of @{text "threads"}: *}
- "threads (e#s) = threads s"
-text {* \noindent
- Functions such as @{text "threads"}, which extract information out of system states, are called
- {\em observing functions}. A series of observing functions will be defined in the sequel in order to
- model the protocol.
- Observing function @{text "original_priority"} calculates
- the {\em original priority} of thread @{text "th"} in state @{text "s"}, expressed as
- : @{text "original_priority th s" }. The {\em original priority} is the priority
- assigned to a thread when it is created or when it is reset by system call
- @{text "Set thread priority"}.
-*}
-
-fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
- where
- -- {* @{text "0"} is assigned to threads which have never been created: *}
- "original_priority thread [] = 0" |
- "original_priority thread (Create thread' prio#s) =
- (if thread' = thread then prio else original_priority thread s)" |
- "original_priority thread (Set thread' prio#s) =
- (if thread' = thread then prio else original_priority thread s)" |
- "original_priority thread (e#s) = original_priority thread s"
-
-text {*
- \noindent
- In the following,
- @{text "birthtime th s"} is the time when thread @{text "th"} is created,
- observed from state @{text "s"}.
- The time in the system is measured by the number of events happened so far since the very beginning.
-*}
-fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
- where
- "birthtime thread [] = 0" |
- "birthtime thread ((Create thread' prio)#s) =
- (if (thread = thread') then length s else birthtime thread s)" |
- "birthtime thread ((Set thread' prio)#s) =
- (if (thread = thread') then length s else birthtime thread s)" |
- "birthtime thread (e#s) = birthtime thread s"
-
-text {*
- \noindent
- The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of
- a thread is the combination of its {\em original priority} and {\em birth time}. The intention is
- to discriminate threads with the same priority by giving threads whose priority
- is assigned earlier higher precedences, becasue such threads are more urgent to finish.
- This explains the following definition:
- *}
-definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
- where "preced thread s \<equiv> Prc (original_priority thread s) (birthtime thread s)"
-
-
-text {*
- \noindent
- A number of important notions are defined here:
- *}
-
-consts
- holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
- waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
- depend :: "'b \<Rightarrow> (node \<times> node) set"
- dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
-
-text {*
- \noindent
- In the definition of the following several functions, it is supposed that
- the waiting queue of every critical resource is given by a waiting queue
- function @{text "wq"}, which servers as arguments of these functions.
- *}
-defs (overloaded)
- -- {*
- \begin{minipage}{0.9\textwidth}
- We define that the thread which is at the head of waiting queue of resource @{text "cs"}
- is holding the resource. This definition is slightly different from tradition where
- all threads in the waiting queue are considered as waiting for the resource.
- This notion is reflected in the definition of @{text "holding wq th cs"} as follows:
- \end{minipage}
- *}
- cs_holding_def:
- "holding wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
- -- {*
- \begin{minipage}{0.9\textwidth}
- In accordance with the definition of @{text "holding wq th cs"},
- a thread @{text "th"} is considered waiting for @{text "cs"} if
- it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head.
- This is reflected in the definition of @{text "waiting wq th cs"} as follows:
- \end{minipage}
- *}
- cs_waiting_def:
- "waiting wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
- -- {*
- \begin{minipage}{0.9\textwidth}
- @{text "depend wq"} represents the Resource Allocation Graph of the system under the waiting
- queue function @{text "wq"}.
- \end{minipage}
- *}
- cs_depend_def:
- "depend (wq::cs \<Rightarrow> thread list) \<equiv>
- {(Th th, Cs cs) | th cs. waiting wq th cs} \<union> {(Cs cs, Th th) | cs th. holding wq th cs}"
- -- {*
- \begin{minipage}{0.9\textwidth}
- The following @{text "dependents wq th"} represents the set of threads which are depending on
- thread @{text "th"} in Resource Allocation Graph @{text "depend wq"}:
- \end{minipage}
- *}
- cs_dependents_def:
- "dependents (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (depend wq)^+}"
-
-text {*
- The data structure used by the operating system for scheduling is referred to as
- {\em schedule state}. It is represented as a record consisting of
- a function assigning waiting queue to resources and a function assigning precedence to
- threads:
- *}
-record schedule_state =
- wq_fun :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
- cprec_fun :: "thread \<Rightarrow> precedence" -- {* The function assigning precedence. *}
-
-text {* \noindent
- The following
- @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under
- state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of
- Priority Inheritance that the {\em current precedence} of a thread is the precedence
- inherited from the maximum of all its dependents, i.e. the threads which are waiting
- directly or indirectly waiting for some resources from it. If no such thread exits,
- @{text "th"}'s {\em current precedence} equals its original precedence, i.e.
- @{text "preced th s"}.
- *}
-definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
- where "cpreced wq s = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
-
-(*<*)
-lemma
- cpreced_def2:
- "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependents wq th})"
- unfolding cpreced_def image_def
- apply(rule eq_reflection)
- apply(rule_tac f="Max" in arg_cong)
- by (auto)
-(*>*)
-
-abbreviation
- "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
-
-abbreviation
- "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
-
-abbreviation
- "release qs \<equiv> case qs of
- [] => []
- | (_#qs) => (SOME q. distinct q \<and> set q = set qs)"
-
-text {* \noindent
- The following function @{text "schs"} is used to calculate the schedule state @{text "schs s"}.
- It is the key function to model Priority Inheritance:
- *}
-fun schs :: "state \<Rightarrow> schedule_state"
- where
- "schs [] = (| wq_fun = \<lambda> cs. [], cprec_fun = (\<lambda>_. Prc 0 0) |)" |
-
- -- {*
- \begin{minipage}{0.9\textwidth}
- \begin{enumerate}
- \item @{text "ps"} is the schedule state of last moment.
- \item @{text "pwq"} is the waiting queue function of last moment.
- \item @{text "pcp"} is the precedence function of last moment (NOT USED).
- \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement:
- \begin{enumerate}
- \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to
- the end of @{text "cs"}'s waiting queue.
- \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state,
- @{text "th'"} must equal to @{text "thread"},
- because @{text "thread"} is the one currently holding @{text "cs"}.
- The case @{text "[] \<Longrightarrow> []"} may never be executed in a legal state.
- the @{text "(SOME q. distinct q \<and> set q = set qs)"} is used to choose arbitrarily one
- thread in waiting to take over the released resource @{text "cs"}. In our representation,
- this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
- \item For other happening event, the schedule state just does not change.
- \end{enumerate}
- \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue
- function. The dependency of precedence function on waiting queue function is the reason to
- put them in the same record so that they can evolve together.
- \end{enumerate}
- \end{minipage}
- *}
- "schs (Create th prio # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"
-| "schs (Exit th # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"
-| "schs (Set th prio # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"
-| "schs (P th cs # s) =
- (let wq = wq_fun (schs s) in
- let new_wq = wq(cs := (wq cs @ [th])) in
- (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"
-| "schs (V th cs # s) =
- (let wq = wq_fun (schs s) in
- let new_wq = wq(cs := release (wq cs)) in
- (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"
-
-lemma cpreced_initial:
- "cpreced (\<lambda> cs. []) [] = (\<lambda>_. (Prc 0 0))"
-apply(simp add: cpreced_def)
-apply(simp add: cs_dependents_def cs_depend_def cs_waiting_def cs_holding_def)
-apply(simp add: preced_def)
-done
-
-lemma sch_old_def:
- "schs (e#s) = (let ps = schs s in
- let pwq = wq_fun ps in
- let nwq = case e of
- P th cs \<Rightarrow> pwq(cs:=(pwq cs @ [th])) |
- V th cs \<Rightarrow> let nq = case (pwq cs) of
- [] \<Rightarrow> [] |
- (_#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
- in pwq(cs:=nq) |
- _ \<Rightarrow> pwq
- in let ncp = cpreced nwq (e#s) in
- \<lparr>wq_fun = nwq, cprec_fun = ncp\<rparr>
- )"
-apply(cases e)
-apply(simp_all)
-done
-
-
-text {*
- \noindent
- The following @{text "wq"} is a shorthand for @{text "wq_fun"}.
- *}
-definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
- where "wq s = wq_fun (schs s)"
-
-text {* \noindent
- The following @{text "cp"} is a shorthand for @{text "cprec_fun"}.
- *}
-definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
- where "cp s \<equiv> cprec_fun (schs s)"
-
-text {* \noindent
- Functions @{text "holding"}, @{text "waiting"}, @{text "depend"} and
- @{text "dependents"} still have the
- same meaning, but redefined so that they no longer depend on the
- fictitious {\em waiting queue function}
- @{text "wq"}, but on system state @{text "s"}.
- *}
-defs (overloaded)
- s_holding_abv:
- "holding (s::state) \<equiv> holding (wq_fun (schs s))"
- s_waiting_abv:
- "waiting (s::state) \<equiv> waiting (wq_fun (schs s))"
- s_depend_abv:
- "depend (s::state) \<equiv> depend (wq_fun (schs s))"
- s_dependents_abv:
- "dependents (s::state) \<equiv> dependents (wq_fun (schs s))"
-
-
-text {*
- The following lemma can be proved easily:
- *}
-lemma
- s_holding_def:
- "holding (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th = hd (wq_fun (schs s) cs))"
- by (auto simp:s_holding_abv wq_def cs_holding_def)
-
-lemma s_waiting_def:
- "waiting (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th \<noteq> hd (wq_fun (schs s) cs))"
- by (auto simp:s_waiting_abv wq_def cs_waiting_def)
-
-lemma s_depend_def:
- "depend (s::state) =
- {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \<union> {(Cs cs, Th th) | cs th. holding (wq s) th cs}"
- by (auto simp:s_depend_abv wq_def cs_depend_def)
-
-lemma
- s_dependents_def:
- "dependents (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
- by (auto simp:s_dependents_abv wq_def cs_dependents_def)
-
-text {*
- The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready}
- for running if it is a live thread and it is not waiting for any critical resource.
- *}
-definition readys :: "state \<Rightarrow> thread set"
- where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waiting s th cs)}"
-
-text {* \noindent
- The following function @{text "runing"} calculates the set of running thread, which is the ready
- thread with the highest precedence.
- *}
-definition runing :: "state \<Rightarrow> thread set"
- where "runing s \<equiv> {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
-
-text {* \noindent
- The following function @{text "holdents s th"} returns the set of resources held by thread
- @{text "th"} in state @{text "s"}.
- *}
-definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
- where "holdents s th \<equiv> {cs . holding s th cs}"
-
-lemma holdents_test:
- "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
-unfolding holdents_def
-unfolding s_depend_def
-unfolding s_holding_abv
-unfolding wq_def
-by (simp)
-
-text {* \noindent
- @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in
- state @{text "s"}:
- *}
-definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
- where "cntCS s th = card (holdents s th)"
-
-text {* \noindent
- The fact that event @{text "e"} is eligible to happen next in state @{text "s"}
- is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as
- follows:
- *}
-inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
- where
- -- {*
- A thread can be created if it is not a live thread:
- *}
- thread_create: "\<lbrakk>thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
- -- {*
- A thread can exit if it no longer hold any resource:
- *}
- thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
- -- {*
- \begin{minipage}{0.9\textwidth}
- A thread can request for an critical resource @{text "cs"}, if it is running and
- the request does not form a loop in the current RAG. The latter condition
- is set up to avoid deadlock. The condition also reflects our assumption all threads are
- carefully programmed so that deadlock can not happen:
- \end{minipage}
- *}
- thread_P: "\<lbrakk>thread \<in> runing s; (Cs cs, Th thread) \<notin> (depend s)^+\<rbrakk> \<Longrightarrow>
- step s (P thread cs)" |
- -- {*
- \begin{minipage}{0.9\textwidth}
- A thread can release a critical resource @{text "cs"}
- if it is running and holding that resource:
- \end{minipage}
- *}
- thread_V: "\<lbrakk>thread \<in> runing s; holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
- -- {*
- A thread can adjust its own priority as long as it is current running:
- *}
- thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
-
-text {* \noindent
- With predicate @{text "step"}, the fact that @{text "s"} is a legal state in
- Priority Inheritance protocol can be expressed as: @{text "vt step s"}, where
- the predicate @{text "vt"} can be defined as the following:
- *}
-inductive vt :: "state \<Rightarrow> bool"
- where
- -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
- vt_nil[intro]: "vt []" |
- -- {*
- \begin{minipage}{0.9\textwidth}
- If @{text "s"} a legal state, and event @{text "e"} is eligible to happen
- in state @{text "s"}, then @{text "e#s"} is a legal state as well:
- \end{minipage}
- *}
- vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
-
-text {* \noindent
- It is easy to see that the definition of @{text "vt"} is generic. It can be applied to
- any step predicate to get the set of legal states.
- *}
-
-text {* \noindent
- The following two functions @{text "the_cs"} and @{text "the_th"} are used to extract
- critical resource and thread respectively out of RAG nodes.
- *}
-fun the_cs :: "node \<Rightarrow> cs"
- where "the_cs (Cs cs) = cs"
-
-fun the_th :: "node \<Rightarrow> thread"
- where "the_th (Th th) = th"
-
-text {* \noindent
- The following predicate @{text "next_th"} describe the next thread to
- take over when a critical resource is released. In @{text "next_th s th cs t"},
- @{text "th"} is the thread to release, @{text "t"} is the one to take over.
- *}
-definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
- where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and>
- t = hd (SOME q. distinct q \<and> set q = set rest))"
-
-text {* \noindent
- The function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
- in list @{text "l"}:
- *}
-definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
- where "count Q l = length (filter Q l)"
-
-text {* \noindent
- The following @{text "cntP s"} returns the number of operation @{text "P"} happened
- before reaching state @{text "s"}.
- *}
-definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
- where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
-
-text {* \noindent
- The following @{text "cntV s"} returns the number of operation @{text "V"} happened
- before reaching state @{text "s"}.
- *}
-definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
- where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
-(*<*)
-
-end
-(*>*)
-
--- a/prio/README Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,14 +0,0 @@
-Theories:
-=========
-
- Precedence_ord.thy A theory of precedences.
- Moment.thy The notion of moment.
- PrioGDef.thy The formal definition of the PIP-model.
- PrioG.thy Basic properties of the PIP-model.
- ExtGG.thy The correctness proof of the PIP-model.
- CpsG.thy Properties interesting for an implementation.
-
-The repository can be checked using Isabelle 2011-1.
-
- isabelle make session
-
--- a/prio/ROOT.ML Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2 +0,0 @@
-use_thy "CpsG";
-use_thy "ExtGG";
--- a/prio/Slides/ROOT1.ML Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,7 +0,0 @@
-(*show_question_marks := false;*)
-
-no_document use_thy "../CpsG";
-no_document use_thy "../ExtGG";
-no_document use_thy "~~/src/HOL/Library/LaTeXsugar";
-quick_and_dirty := true;
-use_thy "Slides1"
\ No newline at end of file
--- a/prio/Slides/Slides1.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,669 +0,0 @@
-(*<*)
-theory Slides1
-imports "../CpsG" "../ExtGG" "~~/src/HOL/Library/LaTeXsugar"
-begin
-
-notation (latex output)
- set ("_") and
- Cons ("_::/_" [66,65] 65)
-
-ML {*
- open Printer;
- show_question_marks_default := false;
- *}
-
-notation (latex output)
- Cons ("_::_" [78,77] 73) and
- vt ("valid'_state") and
- runing ("running") and
- birthtime ("last'_set") and
- If ("(\<^raw:\textrm{>if\<^raw:}> (_)/ \<^raw:\textrm{>then\<^raw:}> (_)/ \<^raw:\textrm{>else\<^raw:}> (_))" 10) and
- Prc ("'(_, _')") and
- holding ("holds") and
- waiting ("waits") and
- Th ("T") and
- Cs ("C") and
- readys ("ready") and
- depend ("RAG") and
- preced ("prec") and
- cpreced ("cprec") and
- dependents ("dependants") and
- cp ("cprec") and
- holdents ("resources") and
- original_priority ("priority") and
- DUMMY ("\<^raw:\mbox{$\_\!\_$}>")
-
-(*>*)
-
-
-
-text_raw {*
- \renewcommand{\slidecaption}{Nanjing, P.R. China, 1 August 2012}
- \newcommand{\bl}[1]{#1}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}
- \frametitle{%
- \begin{tabular}{@ {}c@ {}}
- \\[-3mm]
- \Large Priority Inheritance Protocol \\[-3mm]
- \Large Proved Correct \\[0mm]
- \end{tabular}}
-
- \begin{center}
- \small Xingyuan Zhang \\
- \small \mbox{PLA University of Science and Technology} \\
- \small \mbox{Nanjing, China}
- \end{center}
-
- \begin{center}
- \small joint work with \\
- Christian Urban \\
- Kings College, University of London, U.K.\\
- Chunhan Wu \\
- My Ph.D. student now working for Christian\\
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\large Prioirty Inheritance Protocol (PIP)}
- \large
-
- \begin{itemize}
- \item Widely used in Real-Time OSs \pause
- \item One solution of \textcolor{red}{`Priority Inversion'} \pause
- \item A flawed manual correctness proof (1990)\pause
- \begin{itemize} \large
- \item {Notations with no precise definition}
- \item {Resorts to intuitions}
- \end{itemize} \pause
- \item Formal treatments using model-checking \pause
- \begin{itemize} \large
- \item {Applicable to small size system models}
- \item { Unhelpful for human understanding }
- \end{itemize} \pause
- \item Verification of PCP in PVS (2000)\pause
- \begin{itemize} \large
- \item {A related protocol}
- \item {Priority Ceiling Protocol}
- \end{itemize}
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Our Motivation}
- \large
-
- \begin{itemize}
- \item Undergraduate OS course in our university \pause
- \begin{itemize}
- \item {\large Experiments using instrutional OSs }
- \item {\large PINTOS (Stanford) is chosen }
- \item {\large Core project: Implementing PIP in it}
- \end{itemize} \pause
- \item Understanding is crucial for the implemention \pause
- \item Existing literature of little help \pause
- \item Some mention the complication
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\mbox{Some excerpts}}
-
- \begin{quote}
- ``Priority inheritance is neither ef$\!$ficient nor reliable.
- Implementations are either incomplete (and unreliable)
- or surprisingly complex and intrusive.''
- \end{quote}\medskip
-
- \pause
-
- \begin{quote}
- ``I observed in the kernel code (to my disgust), the Linux
- PIP implementation is a nightmare: extremely heavy weight,
- involving maintenance of a full wait-for graph, and requiring
- updates for a range of events, including priority changes and
- interruptions of wait operations.''
- \end{quote}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Our Aims}
- \large
-
- \begin{itemize}
- \item Formal specification at appropriate abstract level,
- convenient for:
- \begin{itemize} \large
- \item Constructing interactive proofs
- \item Clarifying the underlying ideas
- \end{itemize} \pause
- \item Theorems usable to guide implementation, critical point:
- \begin{itemize} \large
- \item Understanding the relationship with real OS code \pause
- \item Not yet formalized
- \end{itemize}
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Real-Time OSes}
- \large
-
- \begin{itemize}
- \item Purpose: gurantee the most urgent task to be processed in time
- \item Processes have priorities\\
- \item Resources can be locked and unlocked
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Problem}
- \Large
-
- \begin{center}
- \begin{tabular}{l}
- \alert{H}igh-priority process\\[4mm]
- \onslide<2->{\alert{M}edium-priority process}\\[4mm]
- \alert{L}ow-priority process\\[4mm]
- \end{tabular}
- \end{center}
-
- \onslide<3->{
- \begin{itemize}
- \item \alert{Priority Inversion} @{text "\<equiv>"} \alert{H $<$ L}
- \item<4> avoid indefinite priority inversion
- \end{itemize}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Priority Inversion}
-
- \begin{center}
- \includegraphics[scale=0.4]{PriorityInversion.png}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Mars Pathfinder Mission 1997}
- \Large
-
- \begin{center}
- \includegraphics[scale=0.2]{marspath1.png}
- \includegraphics[scale=0.22]{marspath3.png}
- \includegraphics[scale=0.4]{marsrover.png}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Solution}
- \Large
-
- \alert{Priority Inheritance Protocol (PIP):}
-
- \begin{center}
- \begin{tabular}{l}
- \alert{H}igh-priority process\\[4mm]
- \textcolor{gray}{Medium-priority process}\\[4mm]
- \alert{L}ow-priority process\\[21mm]
- {\normalsize (temporarily raise its priority)}
- \end{tabular}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{A Correctness ``Proof'' in 1990}
- \Large
-
- \begin{itemize}
- \item a paper$^\star$
- in 1990 ``proved'' the correctness of an algorithm for PIP\\[5mm]
- \end{itemize}
-
- \normalsize
- \begin{quote}
- \ldots{}after the thread (whose priority has been raised) completes its
- critical section and releases the lock, it ``returns to its original
- priority level''.
- \end{quote}\bigskip
-
- \small
- $^\star$ in IEEE Transactions on Computers
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{}
- \Large
-
- \begin{center}
- \begin{tabular}{l}
- \alert{H}igh-priority process 1\\[2mm]
- \alert{H}igh-priority process 2\\[8mm]
- \alert{L}ow-priority process
- \end{tabular}
- \end{center}
-
- \onslide<2->{
- \begin{itemize}
- \item Solution: \\Return to highest \alert{remaining} priority
- \end{itemize}}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Event Abstraction}
-
- \begin{itemize}\large
- \item Use Inductive Approach of L. Paulson \pause
- \item System is event-driven \pause
- \item A \alert{state} is a list of events
- \end{itemize}
-
- \pause
-
- \begin{center}
- \includegraphics[scale=0.4]{EventAbstract.png}
- \end{center}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Events}
- \Large
-
- \begin{center}
- \begin{tabular}{l}
- Create \textcolor{gray}{thread priority}\\
- Exit \textcolor{gray}{thread}\\
- Set \textcolor{gray}{thread priority}\\
- Lock \textcolor{gray}{thread cs}\\
- Unlock \textcolor{gray}{thread cs}\\
- \end{tabular}
- \end{center}\medskip
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Precedences}
- \large
-
- \begin{center}
- \begin{tabular}{l}
- @{thm preced_def[where thread="th"]}
- \end{tabular}
- \end{center}
-
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{RAGs}
-
-\begin{center}
- \newcommand{\fnt}{\fontsize{7}{8}\selectfont}
- \begin{tikzpicture}[scale=1]
- %%\draw[step=2mm] (-3,2) grid (1,-1);
-
- \node (A) at (0,0) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>0"}};
- \node (B) at (2,0) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>1"}};
- \node (C) at (4,0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>1"}};
- \node (D) at (4,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>2"}};
- \node (E) at (6,-0.7) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>2"}};
- \node (E1) at (6, 0.2) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>3"}};
- \node (F) at (8,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>3"}};
-
- \draw [<-,line width=0.6mm] (A) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (B);
- \draw [->,line width=0.6mm] (C) to node [pos=0.4,sloped,above=-0.5mm] {\fnt{}waiting} (B);
- \draw [->,line width=0.6mm] (D) to node [pos=0.4,sloped,below=-0.5mm] {\fnt{}waiting} (B);
- \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,below=-0.5mm] {\fnt{}holding} (E);
- \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (E1);
- \draw [->,line width=0.6mm] (F) to node [pos=0.45,sloped,below=-0.5mm] {\fnt{}waiting} (E);
- \end{tikzpicture}
- \end{center}\bigskip
-
- \begin{center}
- \begin{minipage}{0.8\linewidth}
- \raggedleft
- @{thm cs_depend_def}
- \end{minipage}
- \end{center}\pause
-
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Good Next Events}
- %%\large
-
- \begin{center}
- @{thm[mode=Rule] thread_create[where thread=th]}\bigskip
-
- @{thm[mode=Rule] thread_exit[where thread=th]}\bigskip
-
- @{thm[mode=Rule] thread_set[where thread=th]}
- \end{center}
-
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Good Next Events}
- %%\large
-
- \begin{center}
- @{thm[mode=Rule] thread_P[where thread=th]}\bigskip
-
- @{thm[mode=Rule] thread_V[where thread=th]}\bigskip
- \end{center}
-
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-(*<*)
-context extend_highest_gen
-begin
-(*>*)
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{\mbox{\large Theorem: ``No indefinite priority inversion''}}
-
- \pause
-
- Theorem $^\star$: If th is the thread with the highest precedence in state
- @{text "s"}: \pause
- \begin{center}
- \textcolor{red}{@{thm highest})}
- \end{center}
- \pause
- and @{text "th"} is blocked by a thread @{text "th'"} in
- a future state @{text "s'"} (with @{text "s' = t@s"}): \pause
- \begin{center}
- \textcolor{red}{@{text "th' \<in> running (t@s)"} and @{text "th' \<noteq> th"}} \pause
- \end{center}
- \fbox{ \hspace{1em} \pause
- \begin{minipage}{0.95\textwidth}
- \begin{itemize}
- \item @{text "th'"} did not hold or wait for a resource in s:
- \begin{center}
- \textcolor{red}{@{text "\<not>detached s th'"}}
- \end{center} \pause
- \item @{text "th'"} is running with the precedence of @{text "th"}:
- \begin{center}
- \textcolor{red}{@{text "cp (t@s) th' = preced th s"}}
- \end{center}
- \end{itemize}
- \end{minipage}}
- \pause
- \small
- $^\star$ modulo some further assumptions\bigskip\pause
- It does not matter which process gets a released lock.
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[t]
- \frametitle{Implementation}
-
- s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
-
- When @{text "e"} = \alert{Create th prio}, \alert{Exit th}
-
- \begin{itemize}
- \item @{text "RAG s' = RAG s"}
- \item No precedence needs to be recomputed
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[t]
- \frametitle{Implementation}
-
- s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
-
-
- When @{text "e"} = \alert{Set th prio}
-
- \begin{itemize}
- \item @{text "RAG s' = RAG s"}
- \item No precedence needs to be recomputed
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[t]
- \frametitle{Implementation}
-
- s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
-
- When @{text "e"} = \alert{Unlock th cs} where there is a thread to take over
-
- \begin{itemize}
- \item @{text "RAG s' = RAG s - {(C cs, T th), (T th', C cs)} \<union> {(C cs, T th')}"}
- \item we have to recalculate the precedence of the direct descendants
- \end{itemize}\bigskip
-
- \pause
-
- When @{text "e"} = \alert{Unlock th cs} where no thread takes over
-
- \begin{itemize}
- \item @{text "RAG s' = RAG s - {(C cs, T th)}"}
- \item no recalculation of precedences
- \end{itemize}
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[t]
- \frametitle{Implementation}
-
- s $=$ current state; @{text "s'"} $=$ next state $=$ @{text "e#s"}\bigskip\bigskip
-
- When @{text "e"} = \alert{Lock th cs} where cs is not locked
-
- \begin{itemize}
- \item @{text "RAG s' = RAG s \<union> {(C cs, T th')}"}
- \item no recalculation of precedences
- \end{itemize}\bigskip
-
- \pause
-
- When @{text "e"} = \alert{Lock th cs} where cs is locked
-
- \begin{itemize}
- \item @{text "RAG s' = RAG s - {(T th, C cs)}"}
- \item we have to recalculate the precedence of the descendants
- \end{itemize}
-
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Conclusion}
-
- \begin{itemize} \large
- \item Aims fulfilled \medskip \pause
- \item Alternative way \pause
- \begin{itemize}
- \item using Isabelle/HOL in OS code development \medskip
- \item through the Inductive Approach
- \end{itemize} \pause
- \item Future research \pause
- \begin{itemize}
- \item scheduler in RT-Linux\medskip
- \item multiprocessor case\medskip
- \item other ``nails'' ? (networks, \ldots) \medskip \pause
- \item Refinement to real code and relation between implementations
- \end{itemize}
- \end{itemize}
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-text_raw {*
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mode<presentation>{
- \begin{frame}[c]
- \frametitle{Questions?}
-
- \begin{itemize} \large
- \item Thank you for listening!
- \end{itemize}
-
- \end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-*}
-
-
-(*<*)
-end
-end
-(*>*)
\ No newline at end of file
--- a/prio/Slides/document/beamerthemeplaincu.sty Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,126 +0,0 @@
-\ProvidesPackage{beamerthemeplaincu}[2003/11/07 ver 0.93]
-\NeedsTeXFormat{LaTeX2e}[1995/12/01]
-
-% Copyright 2003 by Till Tantau <tantau@cs.tu-berlin.de>.
-%
-% This program can be redistributed and/or modified under the terms
-% of the LaTeX Project Public License Distributed from CTAN
-% archives in directory macros/latex/base/lppl.txt.
-
-\newcommand{\slidecaption}{}
-
-\mode<presentation>
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% comic fonts fonts
-\DeclareFontFamily{T1}{comic}{}%
-\DeclareFontShape{T1}{comic}{m}{n}{<->s*[.9]comic8t}{}%
-\DeclareFontShape{T1}{comic}{m}{it}{<->s*[.9]comic8t}{}%
-\DeclareFontShape{T1}{comic}{m}{sc}{<->s*[.9]comic8t}{}%
-\DeclareFontShape{T1}{comic}{b}{n}{<->s*[.9]comicbd8t}{}%
-\DeclareFontShape{T1}{comic}{b}{it}{<->s*[.9]comicbd8t}{}%
-\DeclareFontShape{T1}{comic}{m}{sl}{<->ssub * comic/m/it}{}%
-\DeclareFontShape{T1}{comic}{b}{sc}{<->sub * comic/m/sc}{}%
-\DeclareFontShape{T1}{comic}{b}{sl}{<->ssub * comic/b/it}{}%
-\DeclareFontShape{T1}{comic}{bx}{n}{<->ssub * comic/b/n}{}%
-\DeclareFontShape{T1}{comic}{bx}{it}{<->ssub * comic/b/it}{}%
-\DeclareFontShape{T1}{comic}{bx}{sc}{<->sub * comic/m/sc}{}%
-\DeclareFontShape{T1}{comic}{bx}{sl}{<->ssub * comic/b/it}{}%
-%
-\renewcommand{\rmdefault}{comic}%
-\renewcommand{\sfdefault}{comic}%
-\renewcommand{\mathfamilydefault}{cmr}% mathfont should be still the old one
-%
-\DeclareMathVersion{bold}% mathfont needs to be bold
-\DeclareSymbolFont{operators}{OT1}{cmr}{b}{n}%
-\SetSymbolFont{operators}{bold}{OT1}{cmr}{b}{n}%
-\DeclareSymbolFont{letters}{OML}{cmm}{b}{it}%
-\SetSymbolFont{letters}{bold}{OML}{cmm}{b}{it}%
-\DeclareSymbolFont{symbols}{OMS}{cmsy}{b}{n}%
-\SetSymbolFont{symbols}{bold}{OMS}{cmsy}{b}{n}%
-\DeclareSymbolFont{largesymbols}{OMX}{cmex}{b}{n}%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% % Title page
-%\usetitlepagetemplate{
-% \vbox{}
-% \vfill
-% \begin{centering}
-% \Large\structure{\textrm{\textit{{\inserttitle}}}}
-% \vskip1em\par
-% \normalsize\insertauthor\vskip1em\par
-% {\scriptsize\insertinstitute\par}\par\vskip1em
-% \insertdate\par\vskip1.5em
-% \inserttitlegraphic
-% \end{centering}
-% \vfill
-% }
-
- % Part page
-%\usepartpagetemplate{
-% \begin{centering}
-% \Large\structure{\textrm{\textit{\partname~\@Roman\c@part}}}
-% \vskip1em\par
-% \textrm{\textit{\insertpart}}\par
-% \end{centering}
-% }
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Frametitles
-\setbeamerfont{frametitle}{size={\huge}}
-\setbeamerfont{frametitle}{family={\usefont{T1}{ptm}{b}{n}}}
-\setbeamercolor{frametitle}{fg=gray,bg=white}
-
-\setbeamertemplate{frametitle}{%
-\vskip 2mm % distance from the top margin
-\hskip -3mm % distance from left margin
-\vbox{%
-\begin{minipage}{1.05\textwidth}%
-\centering%
-\insertframetitle%
-\end{minipage}}%
-}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Foot
-%
-\setbeamertemplate{navigation symbols}{}
-\usefoottemplate{%
-\vbox{%
- \tinyline{%
- \tiny\hfill\textcolor{gray!50}{\slidecaption{} --
- p.~\insertframenumber/\inserttotalframenumber}}}%
-}
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\beamertemplateballitem
-\setlength\leftmargini{2mm}
-\setlength\leftmarginii{0.6cm}
-\setlength\leftmarginiii{1.5cm}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% blocks
-%\definecolor{cream}{rgb}{1,1,.65}
-\definecolor{cream}{rgb}{1,1,.8}
-\setbeamerfont{block title}{size=\normalsize}
-\setbeamercolor{block title}{fg=black,bg=cream}
-\setbeamercolor{block body}{fg=black,bg=cream}
-
-\setbeamertemplate{blocks}[rounded][shadow=true]
-
-\setbeamercolor{boxcolor}{fg=black,bg=cream}
-
-\mode
-<all>
-
-
-
-
-
-
--- a/prio/Slides/document/mathpartir.sty Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,446 +0,0 @@
-% Mathpartir --- Math Paragraph for Typesetting Inference Rules
-%
-% Copyright (C) 2001, 2002, 2003, 2004, 2005 Didier Rémy
-%
-% Author : Didier Remy
-% Version : 1.2.0
-% Bug Reports : to author
-% Web Site : http://pauillac.inria.fr/~remy/latex/
-%
-% Mathpartir is free software; you can redistribute it and/or modify
-% it under the terms of the GNU General Public License as published by
-% the Free Software Foundation; either version 2, or (at your option)
-% any later version.
-%
-% Mathpartir is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-% GNU General Public License for more details
-% (http://pauillac.inria.fr/~remy/license/GPL).
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% File mathpartir.sty (LaTeX macros)
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{mathpartir}
- [2005/12/20 version 1.2.0 Math Paragraph for Typesetting Inference Rules]
-
-%%
-
-%% Identification
-%% Preliminary declarations
-
-\RequirePackage {keyval}
-
-%% Options
-%% More declarations
-
-%% PART I: Typesetting maths in paragraphe mode
-
-%% \newdimen \mpr@tmpdim
-%% Dimens are a precious ressource. Uses seems to be local.
-\let \mpr@tmpdim \@tempdima
-
-% To ensure hevea \hva compatibility, \hva should expands to nothing
-% in mathpar or in inferrule
-\let \mpr@hva \empty
-
-%% normal paragraph parametters, should rather be taken dynamically
-\def \mpr@savepar {%
- \edef \MathparNormalpar
- {\noexpand \lineskiplimit \the\lineskiplimit
- \noexpand \lineskip \the\lineskip}%
- }
-
-\def \mpr@rulelineskip {\lineskiplimit=0.3em\lineskip=0.2em plus 0.1em}
-\def \mpr@lesslineskip {\lineskiplimit=0.6em\lineskip=0.5em plus 0.2em}
-\def \mpr@lineskip {\lineskiplimit=1.2em\lineskip=1.2em plus 0.2em}
-\let \MathparLineskip \mpr@lineskip
-\def \mpr@paroptions {\MathparLineskip}
-\let \mpr@prebindings \relax
-
-\newskip \mpr@andskip \mpr@andskip 2em plus 0.5fil minus 0.5em
-
-\def \mpr@goodbreakand
- {\hskip -\mpr@andskip \penalty -1000\hskip \mpr@andskip}
-\def \mpr@and {\hskip \mpr@andskip}
-\def \mpr@andcr {\penalty 50\mpr@and}
-\def \mpr@cr {\penalty -10000\mpr@and}
-\def \mpr@eqno #1{\mpr@andcr #1\hskip 0em plus -1fil \penalty 10}
-
-\def \mpr@bindings {%
- \let \and \mpr@andcr
- \let \par \mpr@andcr
- \let \\\mpr@cr
- \let \eqno \mpr@eqno
- \let \hva \mpr@hva
- }
-\let \MathparBindings \mpr@bindings
-
-% \@ifundefined {ignorespacesafterend}
-% {\def \ignorespacesafterend {\aftergroup \ignorespaces}
-
-\newenvironment{mathpar}[1][]
- {$$\mpr@savepar \parskip 0em \hsize \linewidth \centering
- \vbox \bgroup \mpr@prebindings \mpr@paroptions #1\ifmmode $\else
- \noindent $\displaystyle\fi
- \MathparBindings}
- {\unskip \ifmmode $\fi\egroup $$\ignorespacesafterend}
-
-\newenvironment{mathparpagebreakable}[1][]
- {\begingroup
- \par
- \mpr@savepar \parskip 0em \hsize \linewidth \centering
- \mpr@prebindings \mpr@paroptions #1%
- \vskip \abovedisplayskip \vskip -\lineskip%
- \ifmmode \else $\displaystyle\fi
- \MathparBindings
- }
- {\unskip
- \ifmmode $\fi \par\endgroup
- \vskip \belowdisplayskip
- \noindent
- \ignorespacesafterend}
-
-% \def \math@mathpar #1{\setbox0 \hbox {$\displaystyle #1$}\ifnum
-% \wd0 < \hsize $$\box0$$\else \bmathpar #1\emathpar \fi}
-
-%%% HOV BOXES
-
-\def \mathvbox@ #1{\hbox \bgroup \mpr@normallineskip
- \vbox \bgroup \tabskip 0em \let \\ \cr
- \halign \bgroup \hfil $##$\hfil\cr #1\crcr \egroup \egroup
- \egroup}
-
-\def \mathhvbox@ #1{\setbox0 \hbox {\let \\\qquad $#1$}\ifnum \wd0 < \hsize
- \box0\else \mathvbox {#1}\fi}
-
-
-%% Part II -- operations on lists
-
-\newtoks \mpr@lista
-\newtoks \mpr@listb
-
-\long \def\mpr@cons #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
-{#2}\edef #2{\the \mpr@lista \the \mpr@listb}}
-
-\long \def\mpr@snoc #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
-{#2}\edef #2{\the \mpr@listb\the\mpr@lista}}
-
-\long \def \mpr@concat#1=#2\mpr@to#3{\mpr@lista \expandafter {#2}\mpr@listb
-\expandafter {#3}\edef #1{\the \mpr@listb\the\mpr@lista}}
-
-\def \mpr@head #1\mpr@to #2{\expandafter \mpr@head@ #1\mpr@head@ #1#2}
-\long \def \mpr@head@ #1#2\mpr@head@ #3#4{\def #4{#1}\def#3{#2}}
-
-\def \mpr@flatten #1\mpr@to #2{\expandafter \mpr@flatten@ #1\mpr@flatten@ #1#2}
-\long \def \mpr@flatten@ \\#1\\#2\mpr@flatten@ #3#4{\def #4{#1}\def #3{\\#2}}
-
-\def \mpr@makelist #1\mpr@to #2{\def \mpr@all {#1}%
- \mpr@lista {\\}\mpr@listb \expandafter {\mpr@all}\edef \mpr@all {\the
- \mpr@lista \the \mpr@listb \the \mpr@lista}\let #2\empty
- \def \mpr@stripof ##1##2\mpr@stripend{\def \mpr@stripped{##2}}\loop
- \mpr@flatten \mpr@all \mpr@to \mpr@one
- \expandafter \mpr@snoc \mpr@one \mpr@to #2\expandafter \mpr@stripof
- \mpr@all \mpr@stripend
- \ifx \mpr@stripped \empty \let \mpr@isempty 0\else \let \mpr@isempty 1\fi
- \ifx 1\mpr@isempty
- \repeat
-}
-
-\def \mpr@rev #1\mpr@to #2{\let \mpr@tmp \empty
- \def \\##1{\mpr@cons ##1\mpr@to \mpr@tmp}#1\let #2\mpr@tmp}
-
-%% Part III -- Type inference rules
-
-\newif \if@premisse
-\newbox \mpr@hlist
-\newbox \mpr@vlist
-\newif \ifmpr@center \mpr@centertrue
-\def \mpr@htovlist {%
- \setbox \mpr@hlist
- \hbox {\strut
- \ifmpr@center \hskip -0.5\wd\mpr@hlist\fi
- \unhbox \mpr@hlist}%
- \setbox \mpr@vlist
- \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
- \else \unvbox \mpr@vlist \box \mpr@hlist
- \fi}%
-}
-% OLD version
-% \def \mpr@htovlist {%
-% \setbox \mpr@hlist
-% \hbox {\strut \hskip -0.5\wd\mpr@hlist \unhbox \mpr@hlist}%
-% \setbox \mpr@vlist
-% \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
-% \else \unvbox \mpr@vlist \box \mpr@hlist
-% \fi}%
-% }
-
-\def \mpr@item #1{$\displaystyle #1$}
-\def \mpr@sep{2em}
-\def \mpr@blank { }
-\def \mpr@hovbox #1#2{\hbox
- \bgroup
- \ifx #1T\@premissetrue
- \else \ifx #1B\@premissefalse
- \else
- \PackageError{mathpartir}
- {Premisse orientation should either be T or B}
- {Fatal error in Package}%
- \fi \fi
- \def \@test {#2}\ifx \@test \mpr@blank\else
- \setbox \mpr@hlist \hbox {}%
- \setbox \mpr@vlist \vbox {}%
- \if@premisse \let \snoc \mpr@cons \else \let \snoc \mpr@snoc \fi
- \let \@hvlist \empty \let \@rev \empty
- \mpr@tmpdim 0em
- \expandafter \mpr@makelist #2\mpr@to \mpr@flat
- \if@premisse \mpr@rev \mpr@flat \mpr@to \@rev \else \let \@rev \mpr@flat \fi
- \def \\##1{%
- \def \@test {##1}\ifx \@test \empty
- \mpr@htovlist
- \mpr@tmpdim 0em %%% last bug fix not extensively checked
- \else
- \setbox0 \hbox{\mpr@item {##1}}\relax
- \advance \mpr@tmpdim by \wd0
- %\mpr@tmpdim 1.02\mpr@tmpdim
- \ifnum \mpr@tmpdim < \hsize
- \ifnum \wd\mpr@hlist > 0
- \if@premisse
- \setbox \mpr@hlist
- \hbox {\unhbox0 \hskip \mpr@sep \unhbox \mpr@hlist}%
- \else
- \setbox \mpr@hlist
- \hbox {\unhbox \mpr@hlist \hskip \mpr@sep \unhbox0}%
- \fi
- \else
- \setbox \mpr@hlist \hbox {\unhbox0}%
- \fi
- \else
- \ifnum \wd \mpr@hlist > 0
- \mpr@htovlist
- \mpr@tmpdim \wd0
- \fi
- \setbox \mpr@hlist \hbox {\unhbox0}%
- \fi
- \advance \mpr@tmpdim by \mpr@sep
- \fi
- }%
- \@rev
- \mpr@htovlist
- \ifmpr@center \hskip \wd\mpr@vlist\fi \box \mpr@vlist
- \fi
- \egroup
-}
-
-%%% INFERENCE RULES
-
-\@ifundefined{@@over}{%
- \let\@@over\over % fallback if amsmath is not loaded
- \let\@@overwithdelims\overwithdelims
- \let\@@atop\atop \let\@@atopwithdelims\atopwithdelims
- \let\@@above\above \let\@@abovewithdelims\abovewithdelims
- }{}
-
-%% The default
-
-\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
- $\displaystyle {#1\mpr@over #2}$}}
-\def \mpr@@nofraction #1#2{\hbox {\advance \hsize by -0.5em
- $\displaystyle {#1\@@atop #2}$}}
-
-\let \mpr@fraction \mpr@@fraction
-
-%% A generic solution to arrow
-
-\def \mpr@make@fraction #1#2#3#4#5{\hbox {%
- \def \mpr@tail{#1}%
- \def \mpr@body{#2}%
- \def \mpr@head{#3}%
- \setbox1=\hbox{$#4$}\setbox2=\hbox{$#5$}%
- \setbox3=\hbox{$\mkern -3mu\mpr@body\mkern -3mu$}%
- \setbox3=\hbox{$\mkern -3mu \mpr@body\mkern -3mu$}%
- \dimen0=\dp1\advance\dimen0 by \ht3\relax\dp1\dimen0\relax
- \dimen0=\ht2\advance\dimen0 by \dp3\relax\ht2\dimen0\relax
- \setbox0=\hbox {$\box1 \@@atop \box2$}%
- \dimen0=\wd0\box0
- \box0 \hskip -\dimen0\relax
- \hbox to \dimen0 {$%
- \mathrel{\mpr@tail}\joinrel
- \xleaders\hbox{\copy3}\hfil\joinrel\mathrel{\mpr@head}%
- $}}}
-
-%% Old stuff should be removed in next version
-\def \mpr@@nothing #1#2
- {$\lower 0.01pt \mpr@@nofraction {#1}{#2}$}
-\def \mpr@@reduce #1#2{\hbox
- {$\lower 0.01pt \mpr@@fraction {#1}{#2}\mkern -15mu\rightarrow$}}
-\def \mpr@@rewrite #1#2#3{\hbox
- {$\lower 0.01pt \mpr@@fraction {#2}{#3}\mkern -8mu#1$}}
-\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
-
-\def \mpr@empty {}
-\def \mpr@inferrule
- {\bgroup
- \ifnum \linewidth<\hsize \hsize \linewidth\fi
- \mpr@rulelineskip
- \let \and \qquad
- \let \hva \mpr@hva
- \let \@rulename \mpr@empty
- \let \@rule@options \mpr@empty
- \let \mpr@over \@@over
- \mpr@inferrule@}
-\newcommand {\mpr@inferrule@}[3][]
- {\everymath={\displaystyle}%
- \def \@test {#2}\ifx \empty \@test
- \setbox0 \hbox {$\vcenter {\mpr@hovbox{B}{#3}}$}%
- \else
- \def \@test {#3}\ifx \empty \@test
- \setbox0 \hbox {$\vcenter {\mpr@hovbox{T}{#2}}$}%
- \else
- \setbox0 \mpr@fraction {\mpr@hovbox{T}{#2}}{\mpr@hovbox{B}{#3}}%
- \fi \fi
- \def \@test {#1}\ifx \@test\empty \box0
- \else \vbox
-%%% Suggestion de Francois pour les etiquettes longues
-%%% {\hbox to \wd0 {\RefTirName {#1}\hfil}\box0}\fi
- {\hbox {\RefTirName {#1}}\box0}\fi
- \egroup}
-
-\def \mpr@vdotfil #1{\vbox to #1{\leaders \hbox{$\cdot$} \vfil}}
-
-% They are two forms
-% \inferrule [label]{[premisses}{conclusions}
-% or
-% \inferrule* [options]{[premisses}{conclusions}
-%
-% Premisses and conclusions are lists of elements separated by \\
-% Each \\ produces a break, attempting horizontal breaks if possible,
-% and vertical breaks if needed.
-%
-% An empty element obtained by \\\\ produces a vertical break in all cases.
-%
-% The former rule is aligned on the fraction bar.
-% The optional label appears on top of the rule
-% The second form to be used in a derivation tree is aligned on the last
-% line of its conclusion
-%
-% The second form can be parameterized, using the key=val interface. The
-% folloiwng keys are recognized:
-%
-% width set the width of the rule to val
-% narrower set the width of the rule to val\hsize
-% before execute val at the beginning/left
-% lab put a label [Val] on top of the rule
-% lskip add negative skip on the right
-% left put a left label [Val]
-% Left put a left label [Val], ignoring its width
-% right put a right label [Val]
-% Right put a right label [Val], ignoring its width
-% leftskip skip negative space on the left-hand side
-% rightskip skip negative space on the right-hand side
-% vdots lift the rule by val and fill vertical space with dots
-% after execute val at the end/right
-%
-% Note that most options must come in this order to avoid strange
-% typesetting (in particular leftskip must preceed left and Left and
-% rightskip must follow Right or right; vdots must come last
-% or be only followed by rightskip.
-%
-
-%% Keys that make sence in all kinds of rules
-\def \mprset #1{\setkeys{mprset}{#1}}
-\define@key {mprset}{andskip}[]{\mpr@andskip=#1}
-\define@key {mprset}{lineskip}[]{\lineskip=#1}
-\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
-\define@key {mprset}{center}[]{\mpr@centertrue}
-\define@key {mprset}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
-\define@key {mprset}{atop}[]{\let \mpr@fraction \mpr@@nofraction}
-\define@key {mprset}{myfraction}[]{\let \mpr@fraction #1}
-\define@key {mprset}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
-\define@key {mprset}{sep}{\def\mpr@sep{#1}}
-
-\newbox \mpr@right
-\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
-\define@key {mpr}{center}[]{\mpr@centertrue}
-\define@key {mpr}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
-\define@key {mpr}{myfraction}[]{\let \mpr@fraction #1}
-\define@key {mpr}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
-\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{width}{\hsize #1}
-\define@key {mpr}{sep}{\def\mpr@sep{#1}}
-\define@key {mpr}{before}{#1}
-\define@key {mpr}{lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
-\define@key {mpr}{Lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
-\define@key {mpr}{narrower}{\hsize #1\hsize}
-\define@key {mpr}{leftskip}{\hskip -#1}
-\define@key {mpr}{reduce}[]{\let \mpr@fraction \mpr@@reduce}
-\define@key {mpr}{rightskip}
- {\setbox \mpr@right \hbox {\unhbox \mpr@right \hskip -#1}}
-\define@key {mpr}{LEFT}{\setbox0 \hbox {$#1$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
- \advance \hsize by -\wd0\box0}
-\define@key {mpr}{Left}{\llap{$\TirName {#1}\;$}}
-\define@key {mpr}{right}
- {\setbox0 \hbox {$\;\TirName {#1}$}\relax \advance \hsize by -\wd0
- \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
-\define@key {mpr}{RIGHT}
- {\setbox0 \hbox {$#1$}\relax \advance \hsize by -\wd0
- \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
-\define@key {mpr}{Right}
- {\setbox \mpr@right \hbox {\unhbox \mpr@right \rlap {$\;\TirName {#1}$}}}
-\define@key {mpr}{vdots}{\def \mpr@vdots {\@@atop \mpr@vdotfil{#1}}}
-\define@key {mpr}{after}{\edef \mpr@after {\mpr@after #1}}
-
-\newcommand \mpr@inferstar@ [3][]{\setbox0
- \hbox {\let \mpr@rulename \mpr@empty \let \mpr@vdots \relax
- \setbox \mpr@right \hbox{}%
- $\setkeys{mpr}{#1}%
- \ifx \mpr@rulename \mpr@empty \mpr@inferrule {#2}{#3}\else
- \mpr@inferrule [{\mpr@rulename}]{#2}{#3}\fi
- \box \mpr@right \mpr@vdots$}
- \setbox1 \hbox {\strut}
- \@tempdima \dp0 \advance \@tempdima by -\dp1
- \raise \@tempdima \box0}
-
-\def \mpr@infer {\@ifnextchar *{\mpr@inferstar}{\mpr@inferrule}}
-\newcommand \mpr@err@skipargs[3][]{}
-\def \mpr@inferstar*{\ifmmode
- \let \@do \mpr@inferstar@
- \else
- \let \@do \mpr@err@skipargs
- \PackageError {mathpartir}
- {\string\inferrule* can only be used in math mode}{}%
- \fi \@do}
-
-
-%%% Exports
-
-% Envirnonment mathpar
-
-\let \inferrule \mpr@infer
-
-% make a short name \infer is not already defined
-\@ifundefined {infer}{\let \infer \mpr@infer}{}
-
-\def \TirNameStyle #1{\small \textsc{#1}}
-\def \tir@name #1{\hbox {\small \TirNameStyle{#1}}}
-\let \TirName \tir@name
-\let \DefTirName \TirName
-\let \RefTirName \TirName
-
-%%% Other Exports
-
-% \let \listcons \mpr@cons
-% \let \listsnoc \mpr@snoc
-% \let \listhead \mpr@head
-% \let \listmake \mpr@makelist
-
-
-
-
-\endinput
--- a/prio/Slides/document/root.beamer.tex Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,12 +0,0 @@
-\documentclass[14pt,t]{beamer}
-%%%\usepackage{pstricks}
-
-\input{root.tex}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% TeX-command-default: "Slides"
-%%% TeX-view-style: (("." "kghostview --landscape --scale 0.45 --geometry 605x505 %f"))
-%%% End:
-
--- a/prio/Slides/document/root.notes.tex Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,18 +0,0 @@
-\documentclass[11pt]{article}
-%%\usepackage{pstricks}
-\usepackage{dina4}
-\usepackage{beamerarticle}
-\usepackage{times}
-\usepackage{hyperref}
-\usepackage{pgf}
-\usepackage{amssymb}
-\setjobnamebeamerversion{root.beamer}
-\input{root.tex}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% TeX-command-default: "Slides"
-%%% TeX-view-style: (("." "kghostview --landscape --scale 0.45 --geometry 605x505 %f"))
-%%% End:
-
--- a/prio/Slides/document/root.tex Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,147 +0,0 @@
-\usepackage{beamerthemeplaincu}
-%%\usepackage{ulem}
-\usepackage[T1]{fontenc}
-\usepackage{proof}
-\usepackage[latin1]{inputenc}
-\usepackage{isabelle}
-\usepackage{isabellesym}
-\usepackage{mathpartir}
-\usepackage[absolute, overlay]{textpos}
-\usepackage{proof}
-\usepackage{ifthen}
-\usepackage{animate}
-\usepackage{tikz}
-\usepackage{pgf}
-\usetikzlibrary{arrows}
-\usetikzlibrary{automata}
-\usetikzlibrary{shapes}
-\usetikzlibrary{shadows}
-\usetikzlibrary{calc}
-
-% Isabelle configuration
-%%\urlstyle{rm}
-\isabellestyle{rm}
-\renewcommand{\isastyle}{\rm}%
-\renewcommand{\isastyleminor}{\rm}%
-\renewcommand{\isastylescript}{\footnotesize\rm\slshape}%
-\renewcommand{\isatagproof}{}
-\renewcommand{\endisatagproof}{}
-\renewcommand{\isamarkupcmt}[1]{#1}
-
-% Isabelle characters
-\renewcommand{\isacharunderscore}{\_}
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-\renewcommand{\isasymiota}{}
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-\renewcommand{\isacharbraceright}{\}}
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-\renewcommand{\isasymsharp}{\isamath{\#}}
-\renewcommand{\isasymdots}{\isamath{...}}
-\renewcommand{\isasymbullet}{\act}
-
-% mathpatir
-\mprset{sep=1em}
-
-% general math stuff
-\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
-\newcommand{\dnn}{\stackrel{\mbox{\Large def}}{=}}
-\renewcommand{\isasymequiv}{$\dn$}
-\renewcommand{\emptyset}{\varnothing}% nice round empty set
-\renewcommand{\Gamma}{\varGamma}
-\DeclareRobustCommand{\flqq}{\mbox{\guillemotleft}}
-\DeclareRobustCommand{\frqq}{\mbox{\guillemotright}}
-\newcommand{\smath}[1]{\textcolor{blue}{\ensuremath{#1}}}
-\newcommand{\fresh}{\mathrel{\#}}
-\newcommand{\act}{{\raisebox{-0.5mm}{\Large$\boldsymbol{\cdot}$}}}% swapping action
-\newcommand{\swap}[2]{(#1\,#2)}% swapping operation
-
-% beamer stuff
-\renewcommand{\slidecaption}{Salvador, 26.~August 2008}
-
-
-% colours for Isar Code (in article mode everything is black and white)
-\mode<presentation>{
-\definecolor{isacol:brown}{rgb}{.823,.411,.117}
-\definecolor{isacol:lightblue}{rgb}{.274,.509,.705}
-\definecolor{isacol:green}{rgb}{0,.51,0.14}
-\definecolor{isacol:red}{rgb}{.803,0,0}
-\definecolor{isacol:blue}{rgb}{0,0,.803}
-\definecolor{isacol:darkred}{rgb}{.545,0,0}
-\definecolor{isacol:black}{rgb}{0,0,0}}
-\mode<article>{
-\definecolor{isacol:brown}{rgb}{0,0,0}
-\definecolor{isacol:lightblue}{rgb}{0,0,0}
-\definecolor{isacol:green}{rgb}{0,0,0}
-\definecolor{isacol:red}{rgb}{0,0,0}
-\definecolor{isacol:blue}{rgb}{0,0,0}
-\definecolor{isacol:darkred}{rgb}{0,0,0}
-\definecolor{isacol:black}{rgb}{0,0,0}
-}
-
-
-\newcommand{\strong}[1]{{\bfseries {#1}}}
-\newcommand{\bluecmd}[1]{{\color{isacol:lightblue}{\strong{#1}}}}
-\newcommand{\browncmd}[1]{{\color{isacol:brown}{\strong{#1}}}}
-\newcommand{\redcmd}[1]{{\color{isacol:red}{\strong{#1}}}}
-
-\renewcommand{\isakeyword}[1]{%
-\ifthenelse{\equal{#1}{show}}{\browncmd{#1}}{%
-\ifthenelse{\equal{#1}{case}}{\browncmd{#1}}{%
-\ifthenelse{\equal{#1}{assume}}{\browncmd{#1}}{%
-\ifthenelse{\equal{#1}{obtain}}{\browncmd{#1}}{%
-\ifthenelse{\equal{#1}{fix}}{\browncmd{#1}}{%
-\ifthenelse{\equal{#1}{oops}}{\redcmd{#1}}{%
-\ifthenelse{\equal{#1}{thm}}{\redcmd{#1}}{%
-{\bluecmd{#1}}}}}}}}}}%
-
-% inner syntax colour
-\chardef\isachardoublequoteopen=`\"%
-\chardef\isachardoublequoteclose=`\"%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-\newenvironment{innersingle}%
-{\isacharbackquoteopen\color{isacol:green}}%
-{\color{isacol:black}\isacharbackquoteclose}
-\newenvironment{innerdouble}%
-{\isachardoublequoteopen\color{isacol:green}}%
-{\color{isacol:black}\isachardoublequoteclose}
-
-%% misc
-\newcommand{\gb}[1]{\textcolor{isacol:green}{#1}}
-\newcommand{\rb}[1]{\textcolor{red}{#1}}
-
-%% animations
-\newcounter{growcnt}
-\newcommand{\grow}[2]
-{\begin{tikzpicture}[baseline=(n.base)]%
- \node[scale=(0.1 *#1 + 0.001),inner sep=0pt] (n) {#2};
- \end{tikzpicture}%
-}
-
-%% isatabbing
-%\renewcommand{\isamarkupcmt}[1]%
-%{\ifthenelse{\equal{TABSET}{#1}}{\=}%
-% {\ifthenelse{\equal{TAB}{#1}}{\>}%
-% {\ifthenelse{\equal{NEWLINE}{#1}}{\\}%
-% {\ifthenelse{\equal{DOTS}{#1}}{\ldots}{\isastylecmt--- {#1}}}%
-% }%
-% }%
-%}%
-
-
-\newenvironment{isatabbing}%
-{\renewcommand{\isanewline}{\\}\begin{tabbing}}%
-{\end{tabbing}}
-
-\begin{document}
-\input{session}
-\end{document}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% TeX-command-default: "Slides"
-%%% TeX-view-style: (("." "kghostview --landscape --scale 0.45 --geometry 605x505 %f"))
-%%% End:
-
Binary file prio/Slides/slides.pdf has changed
--- a/prio/document/beamerthemeplaincu.sty Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,126 +0,0 @@
-\ProvidesPackage{beamerthemeplaincu}[2003/11/07 ver 0.93]
-\NeedsTeXFormat{LaTeX2e}[1995/12/01]
-
-% Copyright 2003 by Till Tantau <tantau@cs.tu-berlin.de>.
-%
-% This program can be redistributed and/or modified under the terms
-% of the LaTeX Project Public License Distributed from CTAN
-% archives in directory macros/latex/base/lppl.txt.
-
-\newcommand{\slidecaption}{}
-
-\mode<presentation>
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% comic fonts fonts
-\DeclareFontFamily{T1}{comic}{}%
-\DeclareFontShape{T1}{comic}{m}{n}{<->s*[.9]comic8t}{}%
-\DeclareFontShape{T1}{comic}{m}{it}{<->s*[.9]comic8t}{}%
-\DeclareFontShape{T1}{comic}{m}{sc}{<->s*[.9]comic8t}{}%
-\DeclareFontShape{T1}{comic}{b}{n}{<->s*[.9]comicbd8t}{}%
-\DeclareFontShape{T1}{comic}{b}{it}{<->s*[.9]comicbd8t}{}%
-\DeclareFontShape{T1}{comic}{m}{sl}{<->ssub * comic/m/it}{}%
-\DeclareFontShape{T1}{comic}{b}{sc}{<->sub * comic/m/sc}{}%
-\DeclareFontShape{T1}{comic}{b}{sl}{<->ssub * comic/b/it}{}%
-\DeclareFontShape{T1}{comic}{bx}{n}{<->ssub * comic/b/n}{}%
-\DeclareFontShape{T1}{comic}{bx}{it}{<->ssub * comic/b/it}{}%
-\DeclareFontShape{T1}{comic}{bx}{sc}{<->sub * comic/m/sc}{}%
-\DeclareFontShape{T1}{comic}{bx}{sl}{<->ssub * comic/b/it}{}%
-%
-\renewcommand{\rmdefault}{comic}%
-\renewcommand{\sfdefault}{comic}%
-\renewcommand{\mathfamilydefault}{cmr}% mathfont should be still the old one
-%
-\DeclareMathVersion{bold}% mathfont needs to be bold
-\DeclareSymbolFont{operators}{OT1}{cmr}{b}{n}%
-\SetSymbolFont{operators}{bold}{OT1}{cmr}{b}{n}%
-\DeclareSymbolFont{letters}{OML}{cmm}{b}{it}%
-\SetSymbolFont{letters}{bold}{OML}{cmm}{b}{it}%
-\DeclareSymbolFont{symbols}{OMS}{cmsy}{b}{n}%
-\SetSymbolFont{symbols}{bold}{OMS}{cmsy}{b}{n}%
-\DeclareSymbolFont{largesymbols}{OMX}{cmex}{b}{n}%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% % Title page
-%\usetitlepagetemplate{
-% \vbox{}
-% \vfill
-% \begin{centering}
-% \Large\structure{\textrm{\textit{{\inserttitle}}}}
-% \vskip1em\par
-% \normalsize\insertauthor\vskip1em\par
-% {\scriptsize\insertinstitute\par}\par\vskip1em
-% \insertdate\par\vskip1.5em
-% \inserttitlegraphic
-% \end{centering}
-% \vfill
-% }
-
- % Part page
-%\usepartpagetemplate{
-% \begin{centering}
-% \Large\structure{\textrm{\textit{\partname~\@Roman\c@part}}}
-% \vskip1em\par
-% \textrm{\textit{\insertpart}}\par
-% \end{centering}
-% }
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Frametitles
-\setbeamerfont{frametitle}{size={\huge}}
-\setbeamerfont{frametitle}{family={\usefont{T1}{ptm}{b}{n}}}
-\setbeamercolor{frametitle}{fg=gray,bg=white}
-
-\setbeamertemplate{frametitle}{%
-\vskip 2mm % distance from the top margin
-\hskip -3mm % distance from left margin
-\vbox{%
-\begin{minipage}{1.05\textwidth}%
-\centering%
-\insertframetitle%
-\end{minipage}}%
-}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Foot
-%
-\setbeamertemplate{navigation symbols}{}
-\usefoottemplate{%
-\vbox{%
- \tinyline{%
- \tiny\hfill\textcolor{gray!50}{\slidecaption{} --
- p.~\insertframenumber/\inserttotalframenumber}}}%
-}
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\beamertemplateballitem
-\setlength\leftmargini{2mm}
-\setlength\leftmarginii{0.6cm}
-\setlength\leftmarginiii{1.5cm}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% blocks
-%\definecolor{cream}{rgb}{1,1,.65}
-\definecolor{cream}{rgb}{1,1,.8}
-\setbeamerfont{block title}{size=\normalsize}
-\setbeamercolor{block title}{fg=black,bg=cream}
-\setbeamercolor{block body}{fg=black,bg=cream}
-
-\setbeamertemplate{blocks}[rounded][shadow=true]
-
-\setbeamercolor{boxcolor}{fg=black,bg=cream}
-
-\mode
-<all>
-
-
-
-
-
-
--- a/prio/document/llncs.cls Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1189 +0,0 @@
-% LLNCS DOCUMENT CLASS -- version 2.13 (28-Jan-2002)
-% Springer Verlag LaTeX2e support for Lecture Notes in Computer Science
-%
-%%
-%% \CharacterTable
-%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
-%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
-%% Digits \0\1\2\3\4\5\6\7\8\9
-%% Exclamation \! Double quote \" Hash (number) \#
-%% Dollar \$ Percent \% Ampersand \&
-%% Acute accent \' Left paren \( Right paren \)
-%% Asterisk \* Plus \+ Comma \,
-%% Minus \- Point \. Solidus \/
-%% Colon \: Semicolon \; Less than \<
-%% Equals \= Greater than \> Question mark \?
-%% Commercial at \@ Left bracket \[ Backslash \\
-%% Right bracket \] Circumflex \^ Underscore \_
-%% Grave accent \` Left brace \{ Vertical bar \|
-%% Right brace \} Tilde \~}
-%%
-\NeedsTeXFormat{LaTeX2e}[1995/12/01]
-\ProvidesClass{llncs}[2002/01/28 v2.13
-^^J LaTeX document class for Lecture Notes in Computer Science]
-% Options
-\let\if@envcntreset\iffalse
-\DeclareOption{envcountreset}{\let\if@envcntreset\iftrue}
-\DeclareOption{citeauthoryear}{\let\citeauthoryear=Y}
-\DeclareOption{oribibl}{\let\oribibl=Y}
-\let\if@custvec\iftrue
-\DeclareOption{orivec}{\let\if@custvec\iffalse}
-\let\if@envcntsame\iffalse
-\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue}
-\let\if@envcntsect\iffalse
-\DeclareOption{envcountsect}{\let\if@envcntsect\iftrue}
-\let\if@runhead\iffalse
-\DeclareOption{runningheads}{\let\if@runhead\iftrue}
-
-\let\if@openbib\iffalse
-\DeclareOption{openbib}{\let\if@openbib\iftrue}
-
-% languages
-\let\switcht@@therlang\relax
-\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}}
-\def\ds@francais{\def\switcht@@therlang{\switcht@francais}}
-
-\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
-
-\ProcessOptions
-
-\LoadClass[twoside]{article}
-\RequirePackage{multicol} % needed for the list of participants, index
-
-\setlength{\textwidth}{12.2cm}
-\setlength{\textheight}{19.3cm}
-\renewcommand\@pnumwidth{2em}
-\renewcommand\@tocrmarg{3.5em}
-%
-\def\@dottedtocline#1#2#3#4#5{%
- \ifnum #1>\c@tocdepth \else
- \vskip \z@ \@plus.2\p@
- {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by 0pt plus 2cm
- \parfillskip -\rightskip \pretolerance=10000
- \parindent #2\relax\@afterindenttrue
- \interlinepenalty\@M
- \leavevmode
- \@tempdima #3\relax
- \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
- {#4}\nobreak
- \leaders\hbox{$\m@th
- \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
- mu$}\hfill
- \nobreak
- \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}%
- \par}%
- \fi}
-%
-\def\switcht@albion{%
-\def\abstractname{Abstract.}
-\def\ackname{Acknowledgement.}
-\def\andname{and}
-\def\lastandname{\unskip, and}
-\def\appendixname{Appendix}
-\def\chaptername{Chapter}
-\def\claimname{Claim}
-\def\conjecturename{Conjecture}
-\def\contentsname{Table of Contents}
-\def\corollaryname{Corollary}
-\def\definitionname{Definition}
-\def\examplename{Example}
-\def\exercisename{Exercise}
-\def\figurename{Fig.}
-\def\keywordname{{\bf Key words:}}
-\def\indexname{Index}
-\def\lemmaname{Lemma}
-\def\contriblistname{List of Contributors}
-\def\listfigurename{List of Figures}
-\def\listtablename{List of Tables}
-\def\mailname{{\it Correspondence to\/}:}
-\def\noteaddname{Note added in proof}
-\def\notename{Note}
-\def\partname{Part}
-\def\problemname{Problem}
-\def\proofname{Proof}
-\def\propertyname{Property}
-\def\propositionname{Proposition}
-\def\questionname{Question}
-\def\remarkname{Remark}
-\def\seename{see}
-\def\solutionname{Solution}
-\def\subclassname{{\it Subject Classifications\/}:}
-\def\tablename{Table}
-\def\theoremname{Theorem}}
-\switcht@albion
-% Names of theorem like environments are already defined
-% but must be translated if another language is chosen
-%
-% French section
-\def\switcht@francais{%\typeout{On parle francais.}%
- \def\abstractname{R\'esum\'e.}%
- \def\ackname{Remerciements.}%
- \def\andname{et}%
- \def\lastandname{ et}%
- \def\appendixname{Appendice}
- \def\chaptername{Chapitre}%
- \def\claimname{Pr\'etention}%
- \def\conjecturename{Hypoth\`ese}%
- \def\contentsname{Table des mati\`eres}%
- \def\corollaryname{Corollaire}%
- \def\definitionname{D\'efinition}%
- \def\examplename{Exemple}%
- \def\exercisename{Exercice}%
- \def\figurename{Fig.}%
- \def\keywordname{{\bf Mots-cl\'e:}}
- \def\indexname{Index}
- \def\lemmaname{Lemme}%
- \def\contriblistname{Liste des contributeurs}
- \def\listfigurename{Liste des figures}%
- \def\listtablename{Liste des tables}%
- \def\mailname{{\it Correspondence to\/}:}
- \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}%
- \def\notename{Remarque}%
- \def\partname{Partie}%
- \def\problemname{Probl\`eme}%
- \def\proofname{Preuve}%
- \def\propertyname{Caract\'eristique}%
-%\def\propositionname{Proposition}%
- \def\questionname{Question}%
- \def\remarkname{Remarque}%
- \def\seename{voir}
- \def\solutionname{Solution}%
- \def\subclassname{{\it Subject Classifications\/}:}
- \def\tablename{Tableau}%
- \def\theoremname{Th\'eor\`eme}%
-}
-%
-% German section
-\def\switcht@deutsch{%\typeout{Man spricht deutsch.}%
- \def\abstractname{Zusammenfassung.}%
- \def\ackname{Danksagung.}%
- \def\andname{und}%
- \def\lastandname{ und}%
- \def\appendixname{Anhang}%
- \def\chaptername{Kapitel}%
- \def\claimname{Behauptung}%
- \def\conjecturename{Hypothese}%
- \def\contentsname{Inhaltsverzeichnis}%
- \def\corollaryname{Korollar}%
-%\def\definitionname{Definition}%
- \def\examplename{Beispiel}%
- \def\exercisename{\"Ubung}%
- \def\figurename{Abb.}%
- \def\keywordname{{\bf Schl\"usselw\"orter:}}
- \def\indexname{Index}
-%\def\lemmaname{Lemma}%
- \def\contriblistname{Mitarbeiter}
- \def\listfigurename{Abbildungsverzeichnis}%
- \def\listtablename{Tabellenverzeichnis}%
- \def\mailname{{\it Correspondence to\/}:}
- \def\noteaddname{Nachtrag}%
- \def\notename{Anmerkung}%
- \def\partname{Teil}%
-%\def\problemname{Problem}%
- \def\proofname{Beweis}%
- \def\propertyname{Eigenschaft}%
-%\def\propositionname{Proposition}%
- \def\questionname{Frage}%
- \def\remarkname{Anmerkung}%
- \def\seename{siehe}
- \def\solutionname{L\"osung}%
- \def\subclassname{{\it Subject Classifications\/}:}
- \def\tablename{Tabelle}%
-%\def\theoremname{Theorem}%
-}
-
-% Ragged bottom for the actual page
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- \def\@listi{\leftmargin\leftmargini
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-
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-\newcounter {chapter}
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- \@mainmatterfalse\pagenumbering{Roman}}
-\newcommand\mainmatter{\cleardoublepage
- \@mainmattertrue\pagenumbering{arabic}}
-\newcommand\backmatter{\if@openright\cleardoublepage\else\clearpage\fi
- \@mainmatterfalse}
-
-\renewcommand\part{\cleardoublepage
- \thispagestyle{empty}%
- \if@twocolumn
- \onecolumn
- \@tempswatrue
- \else
- \@tempswafalse
- \fi
- \null\vfil
- \secdef\@part\@spart}
-
-\def\@part[#1]#2{%
- \ifnum \c@secnumdepth >-2\relax
- \refstepcounter{part}%
- \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
- \else
- \addcontentsline{toc}{part}{#1}%
- \fi
- \markboth{}{}%
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- \interlinepenalty \@M
- \normalfont
- \ifnum \c@secnumdepth >-2\relax
- \huge\bfseries \partname~\thepart
- \par
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- \fi
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- {\centering
- \interlinepenalty \@M
- \normalfont
- \Huge \bfseries #1\par}%
- \@endpart}
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- \null
- \thispagestyle{empty}%
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- \if@tempswa
- \twocolumn
- \fi}
-
-\newcommand\chapter{\clearpage
- \thispagestyle{empty}%
- \global\@topnum\z@
- \@afterindentfalse
- \secdef\@chapter\@schapter}
-\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
- \if@mainmatter
- \refstepcounter{chapter}%
- \typeout{\@chapapp\space\thechapter.}%
- \addcontentsline{toc}{chapter}%
- {\protect\numberline{\thechapter}#1}%
- \else
- \addcontentsline{toc}{chapter}{#1}%
- \fi
- \else
- \addcontentsline{toc}{chapter}{#1}%
- \fi
- \chaptermark{#1}%
- \addtocontents{lof}{\protect\addvspace{10\p@}}%
- \addtocontents{lot}{\protect\addvspace{10\p@}}%
- \if@twocolumn
- \@topnewpage[\@makechapterhead{#2}]%
- \else
- \@makechapterhead{#2}%
- \@afterheading
- \fi}
-\def\@makechapterhead#1{%
-% \vspace*{50\p@}%
- {\centering
- \ifnum \c@secnumdepth >\m@ne
- \if@mainmatter
- \large\bfseries \@chapapp{} \thechapter
- \par\nobreak
- \vskip 20\p@
- \fi
- \fi
- \interlinepenalty\@M
- \Large \bfseries #1\par\nobreak
- \vskip 40\p@
- }}
-\def\@schapter#1{\if@twocolumn
- \@topnewpage[\@makeschapterhead{#1}]%
- \else
- \@makeschapterhead{#1}%
- \@afterheading
- \fi}
-\def\@makeschapterhead#1{%
-% \vspace*{50\p@}%
- {\centering
- \normalfont
- \interlinepenalty\@M
- \Large \bfseries #1\par\nobreak
- \vskip 40\p@
- }}
-
-\renewcommand\section{\@startsection{section}{1}{\z@}%
- {-18\p@ \@plus -4\p@ \@minus -4\p@}%
- {12\p@ \@plus 4\p@ \@minus 4\p@}%
- {\normalfont\large\bfseries\boldmath
- \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
-\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
- {-18\p@ \@plus -4\p@ \@minus -4\p@}%
- {8\p@ \@plus 4\p@ \@minus 4\p@}%
- {\normalfont\normalsize\bfseries\boldmath
- \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
-\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
- {-18\p@ \@plus -4\p@ \@minus -4\p@}%
- {-0.5em \@plus -0.22em \@minus -0.1em}%
- {\normalfont\normalsize\bfseries\boldmath}}
-\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
- {-12\p@ \@plus -4\p@ \@minus -4\p@}%
- {-0.5em \@plus -0.22em \@minus -0.1em}%
- {\normalfont\normalsize\itshape}}
-\renewcommand\subparagraph[1]{\typeout{LLNCS warning: You should not use
- \string\subparagraph\space with this class}\vskip0.5cm
-You should not use \verb|\subparagraph| with this class.\vskip0.5cm}
-
-\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
-\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
-\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
-\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
-\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
-\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
-\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
-\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
-\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
-\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
-\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}
-
-\let\footnotesize\small
-
-\if@custvec
-\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}}
-{\mbox{\boldmath$\textstyle#1$}}
-{\mbox{\boldmath$\scriptstyle#1$}}
-{\mbox{\boldmath$\scriptscriptstyle#1$}}}
-\fi
-
-\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}}
-\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil
-\penalty50\hskip1em\null\nobreak\hfil\squareforqed
-\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
-
-\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
-\halign{\hfil
-$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
-\cr\to\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
-\cr\to\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
-\gets\cr\to\cr}}}}}
-\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
-$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
-\noalign{\vskip1.2pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
-\noalign{\vskip1pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
-<\cr
-\noalign{\vskip0.9pt}=\cr}}}}}
-\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
-$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
-\noalign{\vskip1.2pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
-\noalign{\vskip1pt}=\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
->\cr
-\noalign{\vskip0.9pt}=\cr}}}}}
-\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
-\halign{\hfil
-$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
->\cr\noalign{\vskip-1pt}<\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
->\cr\noalign{\vskip-0.8pt}<\cr}}}
-{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
->\cr\noalign{\vskip-0.3pt}<\cr}}}}}
-\def\bbbr{{\rm I\!R}} %reelle Zahlen
-\def\bbbm{{\rm I\!M}}
-\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
-\def\bbbf{{\rm I\!F}}
-\def\bbbh{{\rm I\!H}}
-\def\bbbk{{\rm I\!K}}
-\def\bbbp{{\rm I\!P}}
-\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
-{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
-\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
-to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
-to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
-to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
-to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
-\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
-Q$}\hbox{\raise
-0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
-{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
-0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
-0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
-0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
-\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
-T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
-to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
-to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
-to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
-\def\bbbs{{\mathchoice
-{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox
-to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
-to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
-{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox
-to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
-to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox
-to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
-to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
-{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
-to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
-to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
-\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
-{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
-{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
-{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
-
-\let\ts\,
-
-\setlength\leftmargini {17\p@}
-\setlength\leftmargin {\leftmargini}
-\setlength\leftmarginii {\leftmargini}
-\setlength\leftmarginiii {\leftmargini}
-\setlength\leftmarginiv {\leftmargini}
-\setlength \labelsep {.5em}
-\setlength \labelwidth{\leftmargini}
-\addtolength\labelwidth{-\labelsep}
-
-\def\@listI{\leftmargin\leftmargini
- \parsep 0\p@ \@plus1\p@ \@minus\p@
- \topsep 8\p@ \@plus2\p@ \@minus4\p@
- \itemsep0\p@}
-\let\@listi\@listI
-\@listi
-\def\@listii {\leftmargin\leftmarginii
- \labelwidth\leftmarginii
- \advance\labelwidth-\labelsep
- \topsep 0\p@ \@plus2\p@ \@minus\p@}
-\def\@listiii{\leftmargin\leftmarginiii
- \labelwidth\leftmarginiii
- \advance\labelwidth-\labelsep
- \topsep 0\p@ \@plus\p@\@minus\p@
- \parsep \z@
- \partopsep \p@ \@plus\z@ \@minus\p@}
-
-\renewcommand\labelitemi{\normalfont\bfseries --}
-\renewcommand\labelitemii{$\m@th\bullet$}
-
-\setlength\arraycolsep{1.4\p@}
-\setlength\tabcolsep{1.4\p@}
-
-\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
- {{\contentsname}}}
- \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
- \def\lastand{\ifnum\value{auco}=2\relax
- \unskip{} \andname\
- \else
- \unskip \lastandname\
- \fi}%
- \def\and{\stepcounter{@auth}\relax
- \ifnum\value{@auth}=\value{auco}%
- \lastand
- \else
- \unskip,
- \fi}%
- \@starttoc{toc}\if@restonecol\twocolumn\fi}
-
-\def\l@part#1#2{\addpenalty{\@secpenalty}%
- \addvspace{2em plus\p@}% % space above part line
- \begingroup
- \parindent \z@
- \rightskip \z@ plus 5em
- \hrule\vskip5pt
- \large % same size as for a contribution heading
- \bfseries\boldmath % set line in boldface
- \leavevmode % TeX command to enter horizontal mode.
- #1\par
- \vskip5pt
- \hrule
- \vskip1pt
- \nobreak % Never break after part entry
- \endgroup}
-
-\def\@dotsep{2}
-
-\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
-{chapter.\thechapter}\fi}
-
-\def\addnumcontentsmark#1#2#3{%
-\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
- {\thechapter}#3}{\thepage}\hyperhrefextend}}
-\def\addcontentsmark#1#2#3{%
-\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}
-\def\addcontentsmarkwop#1#2#3{%
-\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}
-
-\def\@adcmk[#1]{\ifcase #1 \or
-\def\@gtempa{\addnumcontentsmark}%
- \or \def\@gtempa{\addcontentsmark}%
- \or \def\@gtempa{\addcontentsmarkwop}%
- \fi\@gtempa{toc}{chapter}}
-\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
-
-\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
- \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
- \parindent \z@ \rightskip \@tocrmarg
- \advance\rightskip by 0pt plus 2cm
- \parfillskip -\rightskip \pretolerance=10000
- \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
- {\large\bfseries\boldmath#1}\ifx0#2\hfil\null
- \else
- \nobreak
- \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
- \@dotsep mu$}\hfill
- \nobreak\hbox to\@pnumwidth{\hss #2}%
- \fi\par
- \penalty\@highpenalty \endgroup}
-
-\def\l@title#1#2{\addpenalty{-\@highpenalty}
- \addvspace{8pt plus 1pt}
- \@tempdima \z@
- \begingroup
- \parindent \z@ \rightskip \@tocrmarg
- \advance\rightskip by 0pt plus 2cm
- \parfillskip -\rightskip \pretolerance=10000
- \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
- #1\nobreak
- \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
- \@dotsep mu$}\hfill
- \nobreak\hbox to\@pnumwidth{\hss #2}\par
- \penalty\@highpenalty \endgroup}
-
-\def\l@author#1#2{\addpenalty{\@highpenalty}
- \@tempdima=\z@ %15\p@
- \begingroup
- \parindent \z@ \rightskip \@tocrmarg
- \advance\rightskip by 0pt plus 2cm
- \pretolerance=10000
- \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
- \textit{#1}\par
- \penalty\@highpenalty \endgroup}
-
-%\setcounter{tocdepth}{0}
-\newdimen\tocchpnum
-\newdimen\tocsecnum
-\newdimen\tocsectotal
-\newdimen\tocsubsecnum
-\newdimen\tocsubsectotal
-\newdimen\tocsubsubsecnum
-\newdimen\tocsubsubsectotal
-\newdimen\tocparanum
-\newdimen\tocparatotal
-\newdimen\tocsubparanum
-\tocchpnum=\z@ % no chapter numbers
-\tocsecnum=15\p@ % section 88. plus 2.222pt
-\tocsubsecnum=23\p@ % subsection 88.8 plus 2.222pt
-\tocsubsubsecnum=27\p@ % subsubsection 88.8.8 plus 1.444pt
-\tocparanum=35\p@ % paragraph 88.8.8.8 plus 1.666pt
-\tocsubparanum=43\p@ % subparagraph 88.8.8.8.8 plus 1.888pt
-\def\calctocindent{%
-\tocsectotal=\tocchpnum
-\advance\tocsectotal by\tocsecnum
-\tocsubsectotal=\tocsectotal
-\advance\tocsubsectotal by\tocsubsecnum
-\tocsubsubsectotal=\tocsubsectotal
-\advance\tocsubsubsectotal by\tocsubsubsecnum
-\tocparatotal=\tocsubsubsectotal
-\advance\tocparatotal by\tocparanum}
-\calctocindent
-
-\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
-\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
-\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
-\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
-\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
-
-\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
- \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
- \@starttoc{lof}\if@restonecol\twocolumn\fi}
-\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
-
-\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
- \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
- \@starttoc{lot}\if@restonecol\twocolumn\fi}
-\let\l@table\l@figure
-
-\renewcommand\listoffigures{%
- \section*{\listfigurename
- \@mkboth{\listfigurename}{\listfigurename}}%
- \@starttoc{lof}%
- }
-
-\renewcommand\listoftables{%
- \section*{\listtablename
- \@mkboth{\listtablename}{\listtablename}}%
- \@starttoc{lot}%
- }
-
-\ifx\oribibl\undefined
-\ifx\citeauthoryear\undefined
-\renewenvironment{thebibliography}[1]
- {\section*{\refname}
- \def\@biblabel##1{##1.}
- \small
- \list{\@biblabel{\@arabic\c@enumiv}}%
- {\settowidth\labelwidth{\@biblabel{#1}}%
- \leftmargin\labelwidth
- \advance\leftmargin\labelsep
- \if@openbib
- \advance\leftmargin\bibindent
- \itemindent -\bibindent
- \listparindent \itemindent
- \parsep \z@
- \fi
- \usecounter{enumiv}%
- \let\p@enumiv\@empty
- \renewcommand\theenumiv{\@arabic\c@enumiv}}%
- \if@openbib
- \renewcommand\newblock{\par}%
- \else
- \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
- \fi
- \sloppy\clubpenalty4000\widowpenalty4000%
- \sfcode`\.=\@m}
- {\def\@noitemerr
- {\@latex@warning{Empty `thebibliography' environment}}%
- \endlist}
-\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
- {\let\protect\noexpand\immediate
- \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
-\newcount\@tempcntc
-\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
- \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
- {\@ifundefined
- {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
- ?}\@warning
- {Citation `\@citeb' on page \thepage \space undefined}}%
- {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
- \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
- \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
- \else
- \advance\@tempcntb\@ne
- \ifnum\@tempcntb=\@tempcntc
- \else\advance\@tempcntb\m@ne\@citeo
- \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
-\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
- \@citea\def\@citea{,\,\hskip\z@skip}%
- \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
- {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
- \def\@citea{--}\fi
- \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
-\else
-\renewenvironment{thebibliography}[1]
- {\section*{\refname}
- \small
- \list{}%
- {\settowidth\labelwidth{}%
- \leftmargin\parindent
- \itemindent=-\parindent
- \labelsep=\z@
- \if@openbib
- \advance\leftmargin\bibindent
- \itemindent -\bibindent
- \listparindent \itemindent
- \parsep \z@
- \fi
- \usecounter{enumiv}%
- \let\p@enumiv\@empty
- \renewcommand\theenumiv{}}%
- \if@openbib
- \renewcommand\newblock{\par}%
- \else
- \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
- \fi
- \sloppy\clubpenalty4000\widowpenalty4000%
- \sfcode`\.=\@m}
- {\def\@noitemerr
- {\@latex@warning{Empty `thebibliography' environment}}%
- \endlist}
- \def\@cite#1{#1}%
- \def\@lbibitem[#1]#2{\item[]\if@filesw
- {\def\protect##1{\string ##1\space}\immediate
- \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
- \fi
-\else
-\@cons\@openbib@code{\noexpand\small}
-\fi
-
-\def\idxquad{\hskip 10\p@}% space that divides entry from number
-
-\def\@idxitem{\par\hangindent 10\p@}
-
-\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
- \noindent\hangindent\wd0\box0}% index entry
-
-\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
- \noindent\hangindent\wd0\box0}% order index entry
-
-\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
-
-\renewenvironment{theindex}
- {\@mkboth{\indexname}{\indexname}%
- \thispagestyle{empty}\parindent\z@
- \parskip\z@ \@plus .3\p@\relax
- \let\item\par
- \def\,{\relax\ifmmode\mskip\thinmuskip
- \else\hskip0.2em\ignorespaces\fi}%
- \normalfont\small
- \begin{multicols}{2}[\@makeschapterhead{\indexname}]%
- }
- {\end{multicols}}
-
-\renewcommand\footnoterule{%
- \kern-3\p@
- \hrule\@width 2truecm
- \kern2.6\p@}
- \newdimen\fnindent
- \fnindent1em
-\long\def\@makefntext#1{%
- \parindent \fnindent%
- \leftskip \fnindent%
- \noindent
- \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
-
-\long\def\@makecaption#1#2{%
- \vskip\abovecaptionskip
- \sbox\@tempboxa{{\bfseries #1.} #2}%
- \ifdim \wd\@tempboxa >\hsize
- {\bfseries #1.} #2\par
- \else
- \global \@minipagefalse
- \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
- \fi
- \vskip\belowcaptionskip}
-
-\def\fps@figure{htbp}
-\def\fnum@figure{\figurename\thinspace\thefigure}
-\def \@floatboxreset {%
- \reset@font
- \small
- \@setnobreak
- \@setminipage
-}
-\def\fps@table{htbp}
-\def\fnum@table{\tablename~\thetable}
-\renewenvironment{table}
- {\setlength\abovecaptionskip{0\p@}%
- \setlength\belowcaptionskip{10\p@}%
- \@float{table}}
- {\end@float}
-\renewenvironment{table*}
- {\setlength\abovecaptionskip{0\p@}%
- \setlength\belowcaptionskip{10\p@}%
- \@dblfloat{table}}
- {\end@dblfloat}
-
-\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
- ext@#1\endcsname}{#1}{\protect\numberline{\csname
- the#1\endcsname}{\ignorespaces #2}}\begingroup
- \@parboxrestore
- \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
- \endgroup}
-
-% LaTeX does not provide a command to enter the authors institute
-% addresses. The \institute command is defined here.
-
-\newcounter{@inst}
-\newcounter{@auth}
-\newcounter{auco}
-\newdimen\instindent
-\newbox\authrun
-\newtoks\authorrunning
-\newtoks\tocauthor
-\newbox\titrun
-\newtoks\titlerunning
-\newtoks\toctitle
-
-\def\clearheadinfo{\gdef\@author{No Author Given}%
- \gdef\@title{No Title Given}%
- \gdef\@subtitle{}%
- \gdef\@institute{No Institute Given}%
- \gdef\@thanks{}%
- \global\titlerunning={}\global\authorrunning={}%
- \global\toctitle={}\global\tocauthor={}}
-
-\def\institute#1{\gdef\@institute{#1}}
-
-\def\institutename{\par
- \begingroup
- \parskip=\z@
- \parindent=\z@
- \setcounter{@inst}{1}%
- \def\and{\par\stepcounter{@inst}%
- \noindent$^{\the@inst}$\enspace\ignorespaces}%
- \setbox0=\vbox{\def\thanks##1{}\@institute}%
- \ifnum\c@@inst=1\relax
- \gdef\fnnstart{0}%
- \else
- \xdef\fnnstart{\c@@inst}%
- \setcounter{@inst}{1}%
- \noindent$^{\the@inst}$\enspace
- \fi
- \ignorespaces
- \@institute\par
- \endgroup}
-
-\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
- {\star\star\star}\or \dagger\or \ddagger\or
- \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
- \or \ddagger\ddagger \else\@ctrerr\fi}}
-
-\def\inst#1{\unskip$^{#1}$}
-\def\fnmsep{\unskip$^,$}
-\def\email#1{{\tt#1}}
-\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
-\@ifpackageloaded{babel}{%
-\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
-\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
-\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
-}{\switcht@@therlang}%
-}
-\def\homedir{\~{ }}
-
-\def\subtitle#1{\gdef\@subtitle{#1}}
-\clearheadinfo
-
-\renewcommand\maketitle{\newpage
- \refstepcounter{chapter}%
- \stepcounter{section}%
- \setcounter{section}{0}%
- \setcounter{subsection}{0}%
- \setcounter{figure}{0}
- \setcounter{table}{0}
- \setcounter{equation}{0}
- \setcounter{footnote}{0}%
- \begingroup
- \parindent=\z@
- \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
- \if@twocolumn
- \ifnum \col@number=\@ne
- \@maketitle
- \else
- \twocolumn[\@maketitle]%
- \fi
- \else
- \newpage
- \global\@topnum\z@ % Prevents figures from going at top of page.
- \@maketitle
- \fi
- \thispagestyle{empty}\@thanks
-%
- \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
- \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
- \instindent=\hsize
- \advance\instindent by-\headlineindent
-% \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
-% \addcontentsline{toc}{title}{\the\toctitle}\fi
- \if@runhead
- \if!\the\titlerunning!\else
- \edef\@title{\the\titlerunning}%
- \fi
- \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
- \ifdim\wd\titrun>\instindent
- \typeout{Title too long for running head. Please supply}%
- \typeout{a shorter form with \string\titlerunning\space prior to
- \string\maketitle}%
- \global\setbox\titrun=\hbox{\small\rm
- Title Suppressed Due to Excessive Length}%
- \fi
- \xdef\@title{\copy\titrun}%
- \fi
-%
- \if!\the\tocauthor!\relax
- {\def\and{\noexpand\protect\noexpand\and}%
- \protected@xdef\toc@uthor{\@author}}%
- \else
- \def\\{\noexpand\protect\noexpand\newline}%
- \protected@xdef\scratch{\the\tocauthor}%
- \protected@xdef\toc@uthor{\scratch}%
- \fi
-% \addcontentsline{toc}{author}{\toc@uthor}%
- \if@runhead
- \if!\the\authorrunning!
- \value{@inst}=\value{@auth}%
- \setcounter{@auth}{1}%
- \else
- \edef\@author{\the\authorrunning}%
- \fi
- \global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}%
- \ifdim\wd\authrun>\instindent
- \typeout{Names of authors too long for running head. Please supply}%
- \typeout{a shorter form with \string\authorrunning\space prior to
- \string\maketitle}%
- \global\setbox\authrun=\hbox{\small\rm
- Authors Suppressed Due to Excessive Length}%
- \fi
- \xdef\@author{\copy\authrun}%
- \markboth{\@author}{\@title}%
- \fi
- \endgroup
- \setcounter{footnote}{\fnnstart}%
- \clearheadinfo}
-%
-\def\@maketitle{\newpage
- \markboth{}{}%
- \def\lastand{\ifnum\value{@inst}=2\relax
- \unskip{} \andname\
- \else
- \unskip \lastandname\
- \fi}%
- \def\and{\stepcounter{@auth}\relax
- \ifnum\value{@auth}=\value{@inst}%
- \lastand
- \else
- \unskip,
- \fi}%
- \begin{center}%
- \let\newline\\
- {\Large \bfseries\boldmath
- \pretolerance=10000
- \@title \par}\vskip .8cm
-\if!\@subtitle!\else {\large \bfseries\boldmath
- \vskip -.65cm
- \pretolerance=10000
- \@subtitle \par}\vskip .8cm\fi
- \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
- \def\thanks##1{}\@author}%
- \global\value{@inst}=\value{@auth}%
- \global\value{auco}=\value{@auth}%
- \setcounter{@auth}{1}%
-{\lineskip .5em
-\noindent\ignorespaces
-\@author\vskip.35cm}
- {\small\institutename}
- \end{center}%
- }
-
-% definition of the "\spnewtheorem" command.
-%
-% Usage:
-%
-% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
-% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
-% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
-%
-% New is "cap_font" and "body_font". It stands for
-% fontdefinition of the caption and the text itself.
-%
-% "\spnewtheorem*" gives a theorem without number.
-%
-% A defined spnewthoerem environment is used as described
-% by Lamport.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\@thmcountersep{}
-\def\@thmcounterend{.}
-
-\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
-
-% definition of \spnewtheorem with number
-
-\def\@spnthm#1#2{%
- \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
-\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
-
-\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
- {\@definecounter{#1}\@addtoreset{#1}{#3}%
- \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
- \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
- \expandafter\xdef\csname #1name\endcsname{#2}%
- \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
- \global\@namedef{end#1}{\@endtheorem}}}
-
-\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
- {\@definecounter{#1}%
- \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
- \expandafter\xdef\csname #1name\endcsname{#2}%
- \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
- \global\@namedef{end#1}{\@endtheorem}}}
-
-\def\@spothm#1[#2]#3#4#5{%
- \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
- {\expandafter\@ifdefinable\csname #1\endcsname
- {\global\@namedef{the#1}{\@nameuse{the#2}}%
- \expandafter\xdef\csname #1name\endcsname{#3}%
- \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
- \global\@namedef{end#1}{\@endtheorem}}}}
-
-\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
-\refstepcounter{#1}%
-\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
-
-\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
- \ignorespaces}
-
-\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
- the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
-
-\def\@spbegintheorem#1#2#3#4{\trivlist
- \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
-
-\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
- \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
-
-% definition of \spnewtheorem* without number
-
-\def\@sthm#1#2{\@Ynthm{#1}{#2}}
-
-\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
- {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
- \expandafter\xdef\csname #1name\endcsname{#2}%
- \global\@namedef{end#1}{\@endtheorem}}}
-
-\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
-\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
-
-\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
-
-\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
- {#4}{#2}{#3}\ignorespaces}
-
-\def\@Begintheorem#1#2#3{#3\trivlist
- \item[\hskip\labelsep{#2#1\@thmcounterend}]}
-
-\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
- \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
-
-\if@envcntsect
- \def\@thmcountersep{.}
- \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
-\else
- \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
- \if@envcntreset
- \@addtoreset{theorem}{section}
- \else
- \@addtoreset{theorem}{chapter}
- \fi
-\fi
-
-%definition of divers theorem environments
-\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
-\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
-\if@envcntsame % alle Umgebungen wie Theorem.
- \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
-\else % alle Umgebungen mit eigenem Zaehler
- \if@envcntsect % mit section numeriert
- \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
- \else % nicht mit section numeriert
- \if@envcntreset
- \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
- \@addtoreset{#1}{section}}
- \else
- \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
- \@addtoreset{#1}{chapter}}%
- \fi
- \fi
-\fi
-\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
-\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
-\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
-\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
-\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
-\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
-\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
-\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
-\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
-\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
-\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
-\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
-\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
-\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
-
-\def\@takefromreset#1#2{%
- \def\@tempa{#1}%
- \let\@tempd\@elt
- \def\@elt##1{%
- \def\@tempb{##1}%
- \ifx\@tempa\@tempb\else
- \@addtoreset{##1}{#2}%
- \fi}%
- \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
- \expandafter\def\csname cl@#2\endcsname{}%
- \@tempc
- \let\@elt\@tempd}
-
-\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
- \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
- \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
- \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
- }
-
-\renewenvironment{abstract}{%
- \list{}{\advance\topsep by0.35cm\relax\small
- \leftmargin=1cm
- \labelwidth=\z@
- \listparindent=\z@
- \itemindent\listparindent
- \rightmargin\leftmargin}\item[\hskip\labelsep
- \bfseries\abstractname]}
- {\endlist}
-
-\newdimen\headlineindent % dimension for space between
-\headlineindent=1.166cm % number and text of headings.
-
-\def\ps@headings{\let\@mkboth\@gobbletwo
- \let\@oddfoot\@empty\let\@evenfoot\@empty
- \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
- \leftmark\hfil}
- \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
- \llap{\thepage}}
- \def\chaptermark##1{}%
- \def\sectionmark##1{}%
- \def\subsectionmark##1{}}
-
-\def\ps@titlepage{\let\@mkboth\@gobbletwo
- \let\@oddfoot\@empty\let\@evenfoot\@empty
- \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
- \hfil}
- \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
- \llap{\thepage}}
- \def\chaptermark##1{}%
- \def\sectionmark##1{}%
- \def\subsectionmark##1{}}
-
-\if@runhead\ps@headings\else
-\ps@empty\fi
-
-\setlength\arraycolsep{1.4\p@}
-\setlength\tabcolsep{1.4\p@}
-
-\endinput
-%end of file llncs.cls
--- a/prio/document/root.bib Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,111 +0,0 @@
-@article{OwensReppyTuron09,
- author = {S.~Owens and J.~Reppy and A.~Turon},
- title = {{R}egular-{E}xpression {D}erivatives {R}e-{E}xamined},
- journal = {Journal of Functional Programming},
- volume = 19,
- number = {2},
- year = 2009,
- pages = {173--190}
-}
-
-
-
-@Unpublished{KraussNipkow11,
- author = {A.~Kraus and T.~Nipkow},
- title = {{P}roof {P}earl: {R}egular {E}xpression {E}quivalence and {R}elation {A}lgebra},
- note = {To appear in Journal of Automated Reasoning},
- year = {2011}
-}
-
-@Book{Kozen97,
- author = {D.~Kozen},
- title = {{A}utomata and {C}omputability},
- publisher = {Springer Verlag},
- year = {1997}
-}
-
-
-@incollection{Constable00,
- author = {R.~L.~Constable and
- P.~B.~Jackson and
- P.~Naumov and
- J.~C.~Uribe},
- title = {{C}onstructively {F}ormalizing {A}utomata {T}heory},
- booktitle = {Proof, Language, and Interaction},
- year = {2000},
- publisher = {MIT Press},
- pages = {213-238}
-}
-
-
-@techreport{Filliatre97,
- author = {J.-C. Filli\^atre},
- institution = {LIP - ENS Lyon},
- number = {97--04},
- title = {{F}inite {A}utomata {T}heory in {C}oq:
- {A} {C}onstructive {P}roof of {K}leene's {T}heorem},
- type = {Research Report},
- year = {1997}
-}
-
-@article{OwensSlind08,
- author = {S.~Owens and K.~Slind},
- title = {{A}dapting {F}unctional {P}rograms to {H}igher {O}rder {L}ogic},
- journal = {Higher-Order and Symbolic Computation},
- volume = {21},
- number = {4},
- year = {2008},
- pages = {377--409}
-}
-
-@article{Brzozowski64,
- author = {J.~A.~Brzozowski},
- title = {{D}erivatives of {R}egular {E}xpressions},
- journal = {J.~ACM},
- volume = {11},
- issue = {4},
- year = {1964},
- pages = {481--494},
- publisher = {ACM}
-}
-
-@inproceedings{Nipkow98,
- author={T.~Nipkow},
- title={{V}erified {L}exical {A}nalysis},
- booktitle={Proc.~of the 11th International Conference on Theorem Proving in Higher Order Logics},
- series={LNCS},
- volume=1479,
- pages={1--15},
- year=1998
-}
-
-@inproceedings{BerghoferNipkow00,
- author={S.~Berghofer and T.~Nipkow},
- title={{E}xecuting {H}igher {O}rder {L}ogic},
- booktitle={Proc.~of the International Workshop on Types for Proofs and Programs},
- year=2002,
- series={LNCS},
- volume=2277,
- pages="24--40"
-}
-
-@book{HopcroftUllman69,
- author = {J.~E.~Hopcroft and
- J.~D.~Ullman},
- title = {{F}ormal {L}anguages and {T}heir {R}elation to {A}utomata},
- publisher = {Addison-Wesley},
- year = {1969}
-}
-
-
-@inproceedings{BerghoferReiter09,
- author = {S.~Berghofer and
- M.~Reiter},
- title = {{F}ormalizing the {L}ogic-{A}utomaton {C}onnection},
- booktitle = {Proc.~of the 22nd International
- Conference on Theorem Proving in Higher Order Logics},
- year = {2009},
- pages = {147-163},
- series = {LNCS},
- volume = {5674}
-}
\ No newline at end of file
--- a/prio/document/root.tex Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,73 +0,0 @@
-\documentclass[runningheads]{llncs}
-\usepackage{isabelle}
-\usepackage{isabellesym}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{tikz}
-\usepackage{pgf}
-\usetikzlibrary{arrows,automata,decorations,fit,calc}
-\usetikzlibrary{shapes,shapes.arrows,snakes,positioning}
-\usepgflibrary{shapes.misc} % LATEX and plain TEX and pure pgf
-\usetikzlibrary{matrix}
-\usepackage{pdfsetup}
-\usepackage{ot1patch}
-\usepackage{times}
-%%\usepackage{proof}
-%%\usepackage{mathabx}
-\usepackage{stmaryrd}
-
-\titlerunning{Proving the Priority Inheritance Protocol Correct}
-
-
-\urlstyle{rm}
-\isabellestyle{it}
-\renewcommand{\isastyleminor}{\it}%
-\renewcommand{\isastyle}{\normalsize\it}%
-
-
-\def\dn{\,\stackrel{\mbox{\scriptsize def}}{=}\,}
-\renewcommand{\isasymequiv}{$\dn$}
-\renewcommand{\isasymemptyset}{$\varnothing$}
-\renewcommand{\isacharunderscore}{\mbox{$\_\!\_$}}
-
-\newcommand{\isasymcalL}{\ensuremath{\cal{L}}}
-\newcommand{\isasymbigplus}{\ensuremath{\bigplus}}
-
-\newcommand{\bigplus}{\mbox{\Large\bf$+$}}
-\begin{document}
-
-\title{A Formalisation of the Myhill-Nerode Theorem\\ based on Regular
- Expressions (Proof Pearl)}
-\author{Chunhan Wu\inst{1} \and Xingyuan Zhang\inst{1} \and Christian Urban\inst{2}}
-\institute{PLA University of Science and Technology, China \and TU Munich, Germany}
-\maketitle
-
-%\mbox{}\\[-10mm]
-\begin{abstract}
-There are numerous textbooks on regular languages. Nearly all of them
-introduce the subject by describing finite automata and only mentioning on the
-side a connection with regular expressions. Unfortunately, automata are difficult
-to formalise in HOL-based theorem provers. The reason is that
-they need to be represented as graphs, matrices or functions, none of which
-are inductive datatypes. Also convenient operations for disjoint unions of
-graphs and functions are not easily formalisiable in HOL. In contrast, regular
-expressions can be defined conveniently as a datatype and a corresponding
-reasoning infrastructure comes for free. We show in this paper that a central
-result from formal language theory---the Myhill-Nerode theorem---can be
-recreated using only regular expressions.
-
-\end{abstract}
-
-
-\input{session}
-
-%%\mbox{}\\[-10mm]
-\bibliographystyle{plain}
-\bibliography{root}
-
-\end{document}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:
Binary file prio/paper.pdf has changed