--- a/ExtGG.thy Tue Dec 22 23:13:31 2015 +0800
+++ b/ExtGG.thy Wed Jan 06 20:46:14 2016 +0800
@@ -2,67 +2,93 @@
imports PrioG CpsG
begin
-lemma birth_time_lt: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
- apply (induct s, simp)
-proof -
- fix a s
- assume ih: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
- and eq_as: "a # s \<noteq> []"
- show "last_set th (a # s) < length (a # s)"
- proof(cases "s \<noteq> []")
- case False
- from False show ?thesis
- by (cases a, auto simp:last_set.simps)
- next
- case True
- from ih [OF True] show ?thesis
- by (cases a, auto simp:last_set.simps)
- qed
+text {*
+ The following two auxiliary lemmas are used to reason about @{term Max}.
+*}
+lemma image_Max_eqI:
+ assumes "finite B"
+ and "b \<in> B"
+ and "\<forall> x \<in> B. f x \<le> f b"
+ shows "Max (f ` B) = f b"
+ using assms
+ using Max_eqI by blast
+
+lemma image_Max_subset:
+ assumes "finite A"
+ and "B \<subseteq> A"
+ and "a \<in> B"
+ and "Max (f ` A) = f a"
+ shows "Max (f ` B) = f a"
+proof(rule image_Max_eqI)
+ show "finite B"
+ using assms(1) assms(2) finite_subset by auto
+next
+ show "a \<in> B" using assms by simp
+next
+ show "\<forall>x\<in>B. f x \<le> f a"
+ by (metis Max_ge assms(1) assms(2) assms(4)
+ finite_imageI image_eqI subsetCE)
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)
-
+text {*
+ The following locale @{text "highest_gen"} sets the basic context for our
+ investigation: supposing thread @{text th} holds the highest @{term cp}-value
+ in state @{text s}, which means the task for @{text th} is the
+ most urgent. We want to show that
+ @{text th} is treated correctly by PIP, which means
+ @{text th} will not be blocked unreasonably by other less urgent
+ threads.
+*}
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"
+ -- {* The internal structure of @{term th}'s precedence is exposed:*}
+ and preced_th: "preced th s = Prc prio tm"
+-- {* @{term s} is a valid trace, so it will inherit all results derived for
+ a valid trace: *}
sublocale highest_gen < vat_s: valid_trace "s"
by (unfold_locales, insert vt_s, simp)
context highest_gen
begin
+text {*
+ @{term tm} is the time when the precedence of @{term th} is set, so
+ @{term tm} must be a valid moment index into @{term s}.
+*}
lemma lt_tm: "tm < length s"
by (insert preced_tm_lt[OF threads_s preced_th], simp)
+text {*
+ Since @{term th} holds the highest precedence and @{text "cp"}
+ is the highest precedence of all threads in the sub-tree of
+ @{text "th"} and @{text th} is among these threads,
+ its @{term cp} must equal to its precedence:
+*}
lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
proof -
have "?L \<le> ?R"
by (unfold highest, rule Max_ge,
- auto simp:threads_s finite_threads[OF vt_s])
+ auto simp:threads_s finite_threads)
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
+(* ccc *)
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)
+ by (fold max_cp_eq, 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]
+ from highest_cp_preced max_cp_eq[symmetric]
show ?thesis by simp
qed
@@ -75,6 +101,9 @@
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"
+sublocale extend_highest_gen < vat_t: valid_trace "t@s"
+ by (unfold_locales, insert vt_t, simp)
+
lemma step_back_vt_app:
assumes vt_ts: "vt (t@s)"
shows "vt s"
@@ -110,14 +139,6 @@
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 []"
@@ -218,48 +239,21 @@
qed
qed
-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
+text {*
+ According to @{thm th_kept}, thread @{text "th"} has its living status
+ and precedence kept along the way of @{text "t"}. The following lemma
+ shows that this preserved precedence of @{text "th"} remains as the highest
+ along the way of @{text "t"}.
+ The proof goes by induction over @{text "t"} using the specialized
+ induction rule @{thm ind}, followed by case analysis of each possible
+ operations of PIP. All cases follow the same pattern rendered by the
+ generalized introduction rule @{thm "image_Max_eqI"}.
+
+ The very essence is to show that precedences, no matter whether they are newly introduced
+ or modified, are always lower than the one held by @{term "th"},
+ which by @{thm th_kept} is preserved along the way.
+*}
lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
proof(induct rule:ind)
case Nil
@@ -273,62 +267,74 @@
show ?case
proof(cases e)
case (Create thread prio')
- 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 Create, simp)
- moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
- proof(rule Max.insert)
- from finite_threads[OF Cons(1)]
- show "finite (?f ` (threads (t@s)))" by simp
- 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)"
- by (intro f_image_eq, insert thread_not_in, auto simp:Create preced_def)
- with Cons show ?thesis by (auto simp:the_preced_def)
+ -- {* The following is the common pattern of each branch of the case analysis. *}
+ -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
+ next
+ show "th \<in> ?A" using h_e.th_kept by auto
+ next
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume "x \<in> ?A"
+ hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
+ thus "?f x \<le> ?f th"
+ proof
+ assume "x = thread"
+ thus ?thesis
+ apply (simp add:Create the_preced_def preced_def, fold preced_def)
+ using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force
+ next
+ assume h: "x \<in> threads (t @ s)"
+ from Cons(2)[unfolded Create]
+ have "x \<noteq> thread" using h by (cases, auto)
+ hence "?f x = the_preced (t@s) x"
+ by (simp add:Create the_preced_def preced_def)
+ hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
+ by (simp add: h_t.finite_threads h)
+ also have "... = ?f th"
+ by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
+ finally show ?thesis .
+ qed
+ qed
qed
- moreover have "?f thread < ?t"
- proof -
- from h_e.create_low and Create
- have "prio' \<le> prio" by auto
- thus ?thesis
- 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
+ -- {* The minor part is to show that the precedence of @{text "th"}
+ equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ -- {* Then it follows trivially that the precedence preserved
+ for @{term "th"} remains the maximum of all living threads along the way. *}
+ finally show ?thesis .
+ qed
next
case (Exit thread)
- show ?thesis
+ show ?thesis (is "Max (?f ` ?A) = ?t")
proof -
- 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
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
next
- from Cons(2)[unfolded Exit]
- show "thread \<in> threads (t @ s)"
- by (cases, simp add: readys_def runing_def)
- next
- show "finite (threads (t @ s))" by (simp add: finite_threads h_t.vt_t)
+ show "th \<in> ?A" using h_e.th_kept by auto
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)
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume "x \<in> ?A"
+ hence "x \<in> threads (t@s)" by (simp add: Exit)
+ hence "?f x \<le> Max (?f ` threads (t@s))"
+ by (simp add: h_t.finite_threads)
+ also have "... \<le> ?f th"
+ apply (simp add:Exit the_preced_def preced_def, fold preced_def)
+ using Cons.hyps(5) h_t.th_kept the_preced_def by auto
+ finally show "?f x \<le> ?f th" .
+ qed
qed
- 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
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ finally show ?thesis .
+ qed
next
case (P thread cs)
with Cons
@@ -337,202 +343,158 @@
case (V thread cs)
with Cons
show ?thesis by (auto simp:preced_def the_preced_def)
- next (* ccc *)
+ next
case (Set thread prio')
- show ?thesis
- apply (unfold Set, simp, insert Cons(5)) (* ccc *)
- find_theorems priority Set
- find_theorems preced Set
+ 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
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
+ next
+ show "th \<in> ?A" using h_e.th_kept by auto
+ next
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume h: "x \<in> ?A"
+ show "?f x \<le> ?f th"
+ proof(cases "x = thread")
+ case True
+ moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
+ proof -
+ have "the_preced (t @ s) th = Prc prio tm"
+ using h_t.th_kept preced_th by (simp add:the_preced_def)
+ moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
+ ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
qed
- ultimately show ?thesis by auto
+ ultimately show ?thesis
+ by (unfold Set, simp add:the_preced_def preced_def)
+ next
+ case False
+ then have "?f x = the_preced (t@s) x"
+ by (simp add:the_preced_def preced_def Set)
+ also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
+ using Set h h_t.finite_threads by auto
+ also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
+ finally show ?thesis .
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
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
finally show ?thesis .
- qed
+ qed
qed
qed
-lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+lemma max_preced: "preced th (t@s) = Max (the_preced (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")
+text {*
+ The reason behind the following lemma is that:
+ Since @{term "cp"} is defined as the maximum precedence
+ of those threads contained in the sub-tree of node @{term "Th th"}
+ in @{term "RAG (t@s)"}, and all these threads are living threads, and
+ @{term "th"} is also among them, the maximum precedence of
+ them all must be the one for @{text "th"}.
+*}
+lemma th_cp_max_preced:
+ "cp (t@s) th = Max (the_preced (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> dependants (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 dependants_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
+ let ?f = "the_preced (t@s)"
+ have "?L = ?f th"
+ proof(unfold cp_alt_def, rule image_Max_eqI)
+ show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ proof -
+ have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} =
+ the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
+ (\<exists> th'. n = Th th')}"
+ by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
+ moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree)
+ ultimately show ?thesis by simp
+ qed
+ next
+ show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ by (auto simp:subtree_def)
+ next
+ show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
+ the_preced (t @ s) x \<le> the_preced (t @ s) th"
+ proof
+ fix th'
+ assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
+ moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
+ by (meson subtree_Field)
+ ultimately have "Th th' \<in> ..." by auto
+ hence "th' \<in> threads (t@s)"
+ proof
+ assume "Th th' \<in> {Th th}"
+ thus ?thesis using th_kept by auto
next
- from False show "(?f ` ?A) \<noteq> {}" by simp
+ assume "Th th' \<in> Field (RAG (t @ s))"
+ thus ?thesis using vat_t.not_in_thread_isolated by blast
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 dependants_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
+ thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
+ by (metis Max_ge finite_imageI finite_threads image_eqI
+ max_kept th_kept the_preced_def)
qed
qed
+ also have "... = ?R" by (simp add: max_preced the_preced_def)
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)
+ using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
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
+ using assms
+by (metis Max.coboundedI finite_imageI highest not_le order.trans
+ preced_linorder rev_image_eqI threads_s vat_s.finite_threads
+ vat_s.le_cp)
+
+text {*
+ Counting of the number of @{term "P"} and @{term "V"} operations
+ is the cornerstone of a large number of the following proofs.
+ The reason is that this counting is quite easy to calculate and
+ convenient to use in the reasoning.
+
+ The following lemma shows that the counting controls whether
+ a thread is running or not.
+*}
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)" .
+ assume otherwise: "th' \<in> runing (t@s)"
show False
proof -
- have "dependants (wq (t @ s)) th' = {}"
- by (rule count_eq_dependants [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
+ have "th' = th"
+ proof(rule preced_unique)
+ show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+ proof -
+ have "?L = cp (t@s) th'"
+ by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
+ also have "... = cp (t @ s) th" using otherwise
+ by (metis (mono_tags, lifting) mem_Collect_eq
+ runing_def th_cp_max vat_t.max_cp_readys_threads)
+ also have "... = ?R" by (metis th_cp_preced th_kept)
+ finally show ?thesis .
qed
- with assms show False by simp
- qed
- ultimately show ?thesis
- by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
- qed
+ qed (auto simp: th'_in th_kept)
+ moreover have "th' \<noteq> th" using neq_th' .
+ ultimately show ?thesis by simp
+ qed
qed
-lemmas pv_blocked = pv_blocked_pre[folded detached_eq [OF vt_t]]
+lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
lemma runing_precond_pre:
fixes th'
@@ -541,113 +503,102 @@
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
+proof(induct rule:ind)
+ case (Cons e t)
+ interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
+ interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
show ?case
proof(cases e)
- case (V thread cs)
- from Cons and V have vt_v: "vt (V thread cs#(t@s))" by auto
-
+ case (P thread cs)
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
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (P thread cs)" using Cons P by auto
+ thus ?thesis
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ ultimately show ?thesis by auto
+ qed
+ qed with Cons show ?thesis
+ by (unfold P, simp add:cntP_def cntV_def count_def)
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)
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, 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
+ case (V thread cs)
+ show ?thesis
+ proof -
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (V thread cs)" using Cons V by auto
+ thus ?thesis
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ ultimately show ?thesis by auto
+ qed
+ qed with Cons show ?thesis
+ by (unfold V, simp add:cntP_def cntV_def count_def)
+ qed
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)
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 (Create thread prio')
+ show ?thesis
+ proof -
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (Create thread prio')" using Cons Create by auto
+ thus ?thesis using Cons(5) by (cases, auto)
+ qed with Cons show ?thesis
+ by (unfold Create, simp add:cntP_def cntV_def count_def)
+ qed
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)
+ ultimately show ?thesis by auto
+ qed
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
+ show ?thesis
+ proof -
+ have neq_thread: "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (Exit thread)" using Cons Exit by auto
+ thus ?thesis apply (cases) using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ qed
+ hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons
+ by (unfold Exit, simp add:cntP_def cntV_def count_def)
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread
+ by (unfold Exit, simp)
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
+next
+ case Nil
+ with assms
+ show ?case by auto
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]]
+text {* Changing counting balance to detachedness *}
+lemmas runing_precond_pre_dtc = runing_precond_pre
+ [folded vat_t.detached_eq vat_s.detached_eq]
lemma runing_precond:
fixes th'
@@ -655,18 +606,11 @@
and neq_th': "th' \<noteq> th"
and is_runing: "th' \<in> runing (t@s)"
shows "cntP s th' > cntV s th'"
+ using assms
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
+ by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)
+ moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
ultimately show ?thesis by auto
qed
@@ -676,95 +620,44 @@
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
+proof -
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i
+ by (unfold_locales)
+ interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"
+ by (unfold_locales)
+ interpret h: extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"
+ proof(unfold_locales)
+ show "vt (moment i t @ s)" by (metis h_i.vt_t)
+ next
+ show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)
+ next
+ show "preced th (moment i t @ s) =
+ Max (cp (moment i t @ s) ` threads (moment i t @ s))"
+ by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)
+ next
+ show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th)
+ next
+ show "vt (moment j (restm i t) @ moment i t @ s)"
+ using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)
+ next
+ fix th' prio'
+ assume "Create th' prio' \<in> set (moment j (restm i t))"
+ thus "prio' \<le> prio" using assms
+ by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)
+ next
+ fix th' prio'
+ assume "Set th' prio' \<in> set (moment j (restm i t))"
+ thus "th' \<noteq> th \<and> prio' \<le> prio"
+ by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)
+ next
+ fix th'
+ assume "Exit th' \<in> set (moment j (restm i t))"
+ thus "th' \<noteq> th"
+ by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)
qed
-next
- case 0
- from assms show ?case by auto
+ show ?thesis
+ by (metis add.commute append_assoc eq_pv h.runing_precond_pre
+ moment_plus_split neq_th' th'_in)
qed
lemma moment_blocked_eqpv:
@@ -778,14 +671,19 @@
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)
+ and h2: "th' \<in> threads ((moment j t)@s)" by auto
+ moreover have "th' \<notin> runing ((moment j t)@s)"
+ proof -
+ interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+ show ?thesis
+ using h.pv_blocked_pre h1 h2 neq_th' by auto
+ qed
+ ultimately show ?thesis by auto
qed
+(* The foregoing two lemmas are preparation for this one, but
+ in long run can be combined. Maybe I am wrong.
+*)
lemma moment_blocked:
assumes neq_th': "th' \<noteq> th"
and th'_in: "th' \<in> threads ((moment i t)@s)"
@@ -795,71 +693,119 @@
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 .
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+ interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+ have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"
+ by (metis dtc h_i.detached_elim)
+ from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]
+ show ?thesis by (metis h_j.detached_intro)
qed
-lemma runing_inversion_1:
+lemma runing_preced_inversion:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "cp (t@s) th' = preced th s" (is "?L = ?R")
+proof -
+ have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
+ by (unfold runing_def, auto)
+ also have "\<dots> = ?R"
+ by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads)
+ finally show ?thesis .
+qed
+
+text {*
+ The situation when @{term "th"} is blocked is analyzed by the following lemmas.
+*}
+
+text {*
+ The following lemmas shows the running thread @{text "th'"}, if it is different from
+ @{term th}, must be live at the very beginning. By the term {\em the very beginning},
+ we mean the moment where the formal investigation starts, i.e. the moment (or state)
+ @{term s}.
+*}
+
+lemma runing_inversion_0:
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
+ shows "th' \<in> threads s"
+proof -
+ -- {* The proof is by contradiction: *}
+ { assume otherwise: "\<not> ?thesis"
+ have "th' \<notin> runing (t @ s)"
+ proof -
+ -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}
+ have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)
+ -- {* However, @{text "th'"} does not exist at very beginning. *}
+ have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise
+ by (metis append.simps(1) moment_zero)
+ -- {* Therefore, there must be a moment during @{text "t"}, when
+ @{text "th'"} came into being. *}
+ -- {* Let us suppose the moment being @{text "i"}: *}
+ from p_split_gen[OF th'_in th'_notin]
+ 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)
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+ interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)
+ from lt_its have "Suc i \<le> length t" by auto
+ -- {* Let us also suppose the event which makes this change is @{text e}: *}
+ from moment_head[OF this] obtain e where
+ eq_me: "moment (Suc i) t = e # moment i t" by blast
+ hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t)
+ hence "PIP (moment i t @ s) e" by (cases, simp)
+ -- {* It can be derived that this event @{text "e"}, which
+ gives birth to @{term "th'"} must be a @{term "Create"}: *}
+ from create_pre[OF this, of th']
+ obtain prio where eq_e: "e = Create th' prio"
+ by (metis append_Cons eq_me lessI post pre)
+ have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto
+ have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+ proof -
+ have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+ by (metis h_i.cnp_cnv_eq pre)
+ thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)
+ qed
+ show ?thesis
+ using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge
+ by auto
+ qed
+ with `th' \<in> runing (t@s)`
+ have False by simp
+ } thus ?thesis by auto
qed
+text {*
+ The second lemma says, if the running thread @{text th'} is different from
+ @{term th}, then this @{text th'} must in the possession of some resources
+ at the very beginning.
+
+ To ease the reasoning of resource possession of one particular thread,
+ we used two auxiliary functions @{term cntV} and @{term cntP},
+ which are the counters of @{term P}-operations and
+ @{term V}-operations respectively.
+ If the number of @{term V}-operation is less than the number of
+ @{term "P"}-operations, the thread must have some unreleased resource.
+*}
+
+lemma runing_inversion_1: (* ddd *)
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ -- {* thread @{term "th'"} is a live on in state @{term "s"} and
+ it has some unreleased resource. *}
+ shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof -
+ -- {* The proof is a simple composition of @{thm runing_inversion_0} and
+ @{thm runing_precond}: *}
+ -- {* By applying @{thm runing_inversion_0} to assumptions,
+ it can be shown that @{term th'} is live in state @{term s}: *}
+ have "th' \<in> threads s" using runing_inversion_0[OF assms(1,2)] .
+ -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}
+ with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+qed
+
+text {*
+ The following lemma is just a rephrasing of @{thm runing_inversion_1}:
+*}
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')"
@@ -868,37 +814,11 @@
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
+ by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
lemma runing_inversion_4:
assumes runing': "th' \<in> runing (t@s)"
@@ -906,83 +826,93 @@
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
+ apply (metis neq_th runing' runing_inversion_2)
+ apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
+ by (metis neq_th runing' runing_inversion_3)
+
+
+text {*
+ Suppose @{term th} is not running, it is first shown that
+ there is a path in RAG leading from node @{term th} to another thread @{text "th'"}
+ in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
+ Now, since @{term readys}-set is non-empty, there must be
+ one in it which holds the highest @{term cp}-value, which, by definition,
+ is the @{term runing}-thread. However, we are going to show more: this running thread
+ is exactly @{term "th'"}.
+ *}
+lemma th_blockedE: (* ddd *)
+ assumes "th \<notin> runing (t@s)"
+ obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ "th' \<in> runing (t@s)"
+proof -
+ -- {* According to @{thm vat_t.th_chain_to_ready}, either
+ @{term "th"} is in @{term "readys"} or there is path leading from it to
+ one thread in @{term "readys"}. *}
+ have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)"
+ using th_kept vat_t.th_chain_to_ready by auto
+ -- {* However, @{term th} can not be in @{term readys}, because otherwise, since
+ @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
+ moreover have "th \<notin> readys (t@s)"
+ using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto
+ -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in
+ term @{term readys}: *}
+ ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
+ and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
+ -- {* We are going to show that this @{term th'} is running. *}
+ have "th' \<in> runing (t@s)"
+ proof -
+ -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
+ have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
+ proof -
+ have "?L = Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
+ by (unfold cp_alt_def1, simp)
+ also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
+ proof(rule image_Max_subset)
+ show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
+ next
+ show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
+ by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread)
+ next
+ show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
+ by (unfold tRAG_subtree_eq, auto simp:subtree_def)
+ next
+ show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
+ (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
+ proof -
+ have "?L = the_preced (t @ s) ` threads (t @ s)"
+ by (unfold image_comp, rule image_cong, auto)
+ thus ?thesis using max_preced the_preced_def by auto
+ qed
+ qed
+ also have "... = ?R"
+ using th_cp_max th_cp_preced th_kept
+ the_preced_def vat_t.max_cp_readys_threads by auto
+ finally show ?thesis .
+ qed
+ -- {* Now, since @{term th'} holds the highest @{term cp}
+ and we have already show it is in @{term readys},
+ it is @{term runing} by definition. *}
+ with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def)
+ qed
+ -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
+ moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
+ ultimately show ?thesis using that by metis
+qed
+
+text {*
+ Now it is easy to see there is always a thread to run by case analysis
+ on whether thread @{term th} is running: if the answer is Yes, the
+ the running thread is obviously @{term th} itself; otherwise, the running
+ thread is the @{text th'} given by lemma @{thm th_blockedE}.
+*}
lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
+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> (RAG (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> dependants (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> dependants (wq (t @ s)) th')"
- then obtain th1 where
- h1: "th1 = th' \<or> th1 \<in> dependants (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> dependants (wq (t @ s)) th'"
- with dependants_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] dependants_threads [OF vt_t, of th']
- show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependants (wq (t @ s)) th'))"
- by (auto intro:finite_subset)
- next
- from dp
- have "th \<in> dependants (wq (t @ s)) th'"
- by (unfold cs_dependants_def, auto simp:eq_RAG)
- thus "preced th (t @ s) \<in>
- (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependants (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
+ thus ?thesis using th_blockedE by auto
qed
end