--- a/ExtGG.thy Fri Dec 18 22:47:32 2015 +0800
+++ b/ExtGG.thy Tue Dec 22 23:13:31 2015 +0800
@@ -1,5 +1,5 @@
theory ExtGG
-imports PrioG
+imports PrioG CpsG
begin
lemma birth_time_lt: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
@@ -34,44 +34,24 @@
and highest: "preced th s = Max ((cp s)`threads s)"
and preced_th: "preced th s = Prc prio tm"
+sublocale highest_gen < vat_s: valid_trace "s"
+ by (unfold_locales, insert vt_s, simp)
+
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"
+lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
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> dependants (wq s) th) \<subseteq> threads s"
- proof -
- from threads_s and dependants_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> dependants (wq s) th)" by simp
- next
- fix y
- assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependants (wq s) th)"
- then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependants (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> dependants (wq s) th))"
- by (auto intro:finite_subset)
- qed
+ have "?L \<le> ?R"
+ by (unfold highest, rule Max_ge,
+ auto simp:threads_s finite_threads[OF vt_s])
+ moreover have "?R \<le> ?L"
+ by (unfold vat_s.cp_rec, rule Max_ge,
+ auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
+ ultimately show ?thesis by auto
qed
lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
@@ -117,18 +97,28 @@
qed
qed
-context extend_highest_gen
-begin
-thm extend_highest_gen_axioms_def
+locale red_extend_highest_gen = extend_highest_gen +
+ fixes i::nat
-lemma red_moment:
- "extend_highest_gen s th prio tm (moment i t)"
+sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
by (unfold highest_gen_def, auto dest:step_back_vt_app)
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
+
+context extend_highest_gen
+begin
+
+(*
+ 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;
@@ -164,237 +154,195 @@
qed
qed
+
lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
+ 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"
+ show ?case
by auto
next
case (Cons e t)
+ interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+ interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
show ?case
proof(cases e)
case (Create thread prio)
- assume eq_e: " e = Create thread prio"
show ?thesis
proof -
- from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+ from Cons and Create have "step (t@s) (Create thread prio)" by auto
hence "th \<noteq> thread"
proof(cases)
- assume "thread \<notin> threads (t @ s)"
+ case thread_create
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)
+ by (unfold Create, auto simp:preced_def)
moreover note Cons
ultimately show ?thesis
- by (auto simp:eq_e)
+ by (auto simp:Create)
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
+ from h_e.exit_diff and Exit
have neq_th: "thread \<noteq> th" by auto
with Cons
show ?thesis
- by (unfold eq_e, auto simp:preced_def)
+ by (unfold Exit, 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)
+ by (auto simp:P 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)
+ by (auto simp:V 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
+ from h_e.set_diff_low and Set
have "th \<noteq> thread" by auto
hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold eq_e, auto simp:preced_def)
+ by (unfold Set, auto simp:preced_def)
moreover note Cons
ultimately show ?thesis
- by (auto simp:eq_e)
+ by (auto simp:Set)
qed
qed
qed
qed
-lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+lemma Max_remove_less:
+ assumes "finite A"
+ and "a \<in> A"
+ and "b \<in> A"
+ and "a \<noteq> b"
+ and "inj_on f A"
+ and "f a = Max (f ` A)"
+ shows "Max (f ` (A - {b})) = (Max (f ` A))"
+proof -
+ have "Max (f ` (A - {b})) = Max (f`A - {f b})"
+ proof -
+ have "f ` (A - {b}) = f ` A - f ` {b}"
+ by (rule inj_on_image_set_diff[OF assms(5)], insert assms(3), auto)
+ thus ?thesis by simp
+ qed
+ also have "... =
+ (if f ` A - {f b} - {f a} = {} then f a else max (f a) (Max (f ` A - {f b} - {f a})))"
+ proof(subst Max.remove)
+ from assms show "f a \<in> f ` A - {f b}"
+ by (meson DiffI empty_iff imageI inj_on_eq_iff insert_iff)
+ next
+ from assms(1) show "finite (f ` A - {f b})" by auto
+ qed auto
+ also have "... = Max (f ` A)" (is "?L = ?R")
+ proof(cases "f ` A - {f b} - {f a} = {}")
+ case True
+ with assms show ?thesis by auto
+ next
+ case False
+ hence "?L = max (f a) (Max (f ` A - {f b} - {f a}))"
+ by simp
+ also have "... = ?R"
+ proof -
+ from assms False
+ have "(Max (f ` A - {f b} - {f a})) \<le> f a" by auto
+ thus ?thesis by (simp add: assms(6) max_def)
+ qed
+ finally show ?thesis .
+ qed
+ finally show ?thesis .
+qed
+
+lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
proof(induct rule:ind)
case Nil
from highest_preced_thread
- show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
- by simp
+ show ?case
+ by (unfold the_preced_def, simp)
next
case (Cons e t)
+ interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+ interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
show ?case
proof(cases e)
case (Create thread prio')
- 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)
+ from Cons(2)[unfolded this]
+ have thread_not_in: "thread \<notin> threads (t@s)" by (cases, simp)
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)
+ have "Max (?f ` ?A) = Max (insert (?f thread) (?f ` (threads (t@s))))"
+ by (unfold Create, 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]
+ proof(rule Max.insert)
+ from finite_threads[OF Cons(1)]
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"
+ qed (insert h_t.th_kept, auto)
+ 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
+ (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+ by (intro f_image_eq, insert thread_not_in, auto simp:Create preced_def)
+ with Cons show ?thesis by (auto simp:the_preced_def)
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
+ from h_e.create_low and Create
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)
+ by (unfold preced_th, unfold Create, insert lt_tm,
+ auto simp:preced_def precedence_less_def preced_th the_preced_def)
+ qed
+ ultimately show ?thesis by (auto simp:max_def)
qed
- ultimately show ?thesis by (auto simp:max_def)
- qed
-next
+ 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")
+ show ?thesis
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
+ have "Max (the_preced (t @ s) ` (threads (t @ s) - {thread})) =
+ Max (the_preced (t @ s) ` (threads (t @ s)))"
+ proof(rule Max_remove_less)
+ show "th \<noteq> thread" using Exit h_e.exit_diff by auto
+ next
+ from Cons(2)[unfolded Exit]
+ show "thread \<in> threads (t @ s)"
+ by (cases, simp add: readys_def runing_def)
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)
+ show "finite (threads (t @ s))" by (simp add: finite_threads h_t.vt_t)
+ next
+ show "th \<in> threads (t @ s)" by (simp add: h_t.th_kept)
+ next
+ show "inj_on (the_preced (t @ s)) (threads (t @ s))" by (simp add: inj_the_preced)
+ next
+ show "the_preced (t @ s) th = Max (the_preced (t @ s) ` threads (t @ s))"
+ by (simp add: Cons.hyps(5) h_t.th_kept the_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)
+ from this[unfolded Cons(5)]
+ have "Max (the_preced (t @ s) ` (threads (t @ s) - {thread})) = preced th s" .
+ moreover have "the_preced ((e # t) @ s) = the_preced (t@s)"
+ by (auto simp:Exit the_preced_def preced_def)
+ ultimately show ?thesis by (simp add:Exit)
qed
next
case (P thread cs)
with Cons
- show ?thesis by (auto simp:preced_def)
+ show ?thesis by (auto simp:preced_def the_preced_def)
next
case (V thread cs)
with Cons
- show ?thesis by (auto simp:preced_def)
- next
+ show ?thesis by (auto simp:preced_def the_preced_def)
+ next (* ccc *)
case (Set thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
+ show ?thesis
+ apply (unfold Set, simp, insert Cons(5)) (* ccc *)
+ find_theorems priority Set
+ find_theorems preced Set
proof -
let ?B = "threads (t@s)"
from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
@@ -964,8 +912,6 @@
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