--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ExtGG.thy Thu Jan 28 15:36:48 2016 +0800
@@ -0,0 +1,922 @@
+theory ExtGG
+imports PrioG CpsG
+begin
+
+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
+
+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)"
+ -- {* 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)
+ 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, 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[symmetric]
+ show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen +
+ fixes t
+ assumes vt_t: "vt (t@s)"
+ and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+ and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+ and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+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"
+proof -
+ from vt_ts show ?thesis
+ proof(induct t)
+ case Nil
+ from Nil show ?case by auto
+ next
+ case (Cons e t)
+ assume ih: " vt (t @ s) \<Longrightarrow> vt s"
+ and vt_et: "vt ((e # t) @ s)"
+ show ?case
+ proof(rule ih)
+ show "vt (t @ s)"
+ proof(rule step_back_vt)
+ from vt_et show "vt (e # t @ s)" by simp
+ qed
+ qed
+ qed
+qed
+
+
+locale red_extend_highest_gen = extend_highest_gen +
+ fixes i::nat
+
+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)
+
+
+context extend_highest_gen
+begin
+
+ lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes
+ h0: "R []"
+ and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e;
+ extend_highest_gen s th prio tm t;
+ extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+ shows "R t"
+proof -
+ from vt_t extend_highest_gen_axioms show ?thesis
+ proof(induct t)
+ from h0 show "R []" .
+ next
+ case (Cons e t')
+ assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+ and vt_e: "vt ((e # t') @ s)"
+ and et: "extend_highest_gen s th prio tm (e # t')"
+ from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+ from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
+ show ?case
+ proof(rule h2 [OF vt_ts stp _ _ _ ])
+ show "R t'"
+ proof(rule ih)
+ from et show ext': "extend_highest_gen s th prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ next
+ from vt_ts show "vt (t' @ s)" .
+ qed
+ next
+ from et show "extend_highest_gen s th prio tm (e # t')" .
+ next
+ from et show ext': "extend_highest_gen s th prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ qed
+ qed
+qed
+
+
+lemma th_kept: "th \<in> threads (t @ s) \<and>
+ preced th (t@s) = preced th s" (is "?Q t")
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case Nil
+ from threads_s
+ show ?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)
+ show ?thesis
+ proof -
+ from Cons and Create have "step (t@s) (Create thread prio)" by auto
+ hence "th \<noteq> thread"
+ proof(cases)
+ case thread_create
+ with Cons show ?thesis by auto
+ qed
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold Create, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:Create)
+ qed
+ next
+ case (Exit thread)
+ from h_e.exit_diff and Exit
+ have neq_th: "thread \<noteq> th" by auto
+ with Cons
+ show ?thesis
+ by (unfold Exit, auto simp:preced_def)
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis
+ by (auto simp:P preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis
+ by (auto simp:V preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis
+ proof -
+ 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 Set, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:Set)
+ qed
+ qed
+ qed
+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
+ from highest_preced_thread
+ 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')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ -- {* 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
+ -- {* 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 (is "Max (?f ` ?A) = ?t")
+ proof -
+ 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 \<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
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ finally show ?thesis .
+ qed
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def the_preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def the_preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ 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 (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
+ qed
+ qed
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
+ by (insert th_kept max_kept, auto)
+
+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 -
+ 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
+ assume "Th th' \<in> Field (RAG (t @ s))"
+ thus ?thesis using vat_t.not_in_thread_isolated by blast
+ qed
+ 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))"
+ 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:
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ shows "preced th' s < preced th s"
+ 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:
+ 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 otherwise: "th' \<in> runing (t@s)"
+ show False
+ proof -
+ 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
+ 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]
+
+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(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 (P 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) (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
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ 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
+ 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)
+ 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
+ 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
+
+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'
+ 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'"
+ using assms
+proof -
+ have "cntP s th' \<noteq> cntV s th'"
+ 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
+
+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 -
+ 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
+ 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:
+ 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
+ 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)"
+ and dtc: "detached (moment i t @ s) th'"
+ and le_ij: "i \<le> j"
+ shows "detached (moment j t @ s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ 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_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"
+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')"
+proof -
+ from runing_inversion_1[OF _ runing']
+ show ?thesis by auto
+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)"
+ by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
+
+lemma runing_inversion_4:
+ assumes runing': "th' \<in> runing (t@s)"
+ and neq_th: "th' \<noteq> th"
+ shows "th' \<in> threads s"
+ and "\<not>detached s th'"
+ and "cp (t@s) th' = preced th s"
+ 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)")
+ case True thus ?thesis by auto
+next
+ case False
+ thus ?thesis using th_blockedE by auto
+qed
+
+end
+end
+
+
+