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+ −
+ −
+ −