--- a/prio/ExtGG_1.thy Sun Feb 05 14:29:08 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
-
-