--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/PrioG.thy Fri Jun 17 09:46:25 2016 +0100
@@ -0,0 +1,797 @@
+theory Correctness
+imports PIPBasics
+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
+
+lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)"
+ using eq_cp_s_th highest max_cp_eq the_preced_def by presburger
+
+
+lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+ by (simp add: eq_cp_s_th highest)
+
+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 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[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th"
+ using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
+
+lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))"
+ by (simp add: th_cp_max_preced)
+
+lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th"
+ using max_kept th_kept the_preced_def by auto
+
+lemma [simp]: "the_preced (t@s) th = preced th (t@s)"
+ using the_preced_def by auto
+
+lemma [simp]: "preced th (t@s) = preced th s"
+ by (simp add: th_kept)
+
+lemma [simp]: "cp s th = preced th s"
+ by (simp add: eq_cp_s_th)
+
+lemma th_cp_preced [simp]: "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)
+
+section {* The `blocking thread` *}
+
+text {*
+ The purpose of PIP is to ensure that the most
+ urgent thread @{term th} is not blocked unreasonably.
+ Therefore, a clear picture of the blocking thread is essential
+ to assure people that the purpose is fulfilled.
+
+ In this section, we are going to derive a series of lemmas
+ with finally give rise to a picture of the blocking thread.
+
+ By `blocking thread`, we mean a thread in running state but
+ different from thread @{term th}.
+*}
+
+text {*
+ The following lemmas shows that the @{term cp}-value
+ of the blocking thread @{text th'} equals to the highest
+ precedence in the whole system.
+*}
+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 following lemma shows how the counters for @{term "P"} and
+ @{term "V"} operations relate to the running threads in the states
+ @{term s} and @{term "t @ s"}. The lemma shows that if a thread's
+ @{term "P"}-count equals its @{term "V"}-count (which means it no
+ longer has any resource in its possession), it cannot be a running
+ thread.
+
+ The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
+ The key is the use of @{thm eq_pv_dependants} to derive the
+ emptiness of @{text th'}s @{term dependants}-set from the balance of
+ its @{term P} and @{term V} counts. From this, it can be shown
+ @{text th'}s @{term cp}-value equals to its own precedence.
+
+ On the other hand, since @{text th'} is running, by @{thm
+ runing_preced_inversion}, its @{term cp}-value equals to the
+ precedence of @{term th}.
+
+ Combining the above two resukts we have that @{text th'} and @{term
+ th} have the same precedence. By uniqueness of precedences, we have
+ @{text "th' = th"}, which is in contradiction with the assumption
+ @{text "th' \<noteq> th"}.
+
+*}
+
+lemma eq_pv_blocked: (* ddd *)
+ assumes 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'_in: "th' \<in> threads (t@s)"
+ using otherwise readys_threads runing_def by auto
+ have "th' = th"
+ proof(rule preced_unique)
+ -- {* The proof goes like this:
+ it is first shown that the @{term preced}-value of @{term th'}
+ equals to that of @{term th}, then by uniqueness
+ of @{term preced}-values (given by lemma @{thm preced_unique}),
+ @{term th'} equals to @{term th}: *}
+ show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+ proof -
+ -- {* Since the counts of @{term th'} are balanced, the subtree
+ of it contains only itself, so, its @{term cp}-value
+ equals its @{term preced}-value: *}
+ have "?L = cp (t@s) th'"
+ by (unfold cp_eq_cpreced cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp)
+ -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion},
+ its @{term cp}-value equals @{term "preced th s"},
+ which equals to @{term "?R"} by simplification: *}
+ also have "... = ?R"
+ thm runing_preced_inversion
+ using runing_preced_inversion[OF otherwise] by simp
+ finally show ?thesis .
+ qed
+ qed (auto simp: th'_in th_kept)
+ with `th' \<noteq> th` show ?thesis by simp
+ qed
+qed
+
+text {*
+ The following lemma is the extrapolation of @{thm eq_pv_blocked}.
+ It says if a thread, different from @{term th},
+ does not hold any resource at the very beginning,
+ it will keep hand-emptied in the future @{term "t@s"}.
+*}
+lemma eq_pv_persist: (* ddd *)
+ assumes neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP s th' = cntV s th'"
+ shows "cntP (t@s) th' = cntV (t@s) th'"
+proof(induction rule:ind) -- {* The proof goes by induction. *}
+ -- {* The nontrivial case is for the @{term Cons}: *}
+ case (Cons e t)
+ -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *}
+ 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
+ interpret vat_es: valid_trace_e "t@s" e using Cons(1,2) by (unfold_locales, auto)
+ show ?case
+ proof -
+ -- {* It can be proved that @{term cntP}-value of @{term th'} does not change
+ by the happening of event @{term e}: *}
+ have "cntP ((e#t)@s) th' = cntP (t@s) th'"
+ proof(rule ccontr) -- {* Proof by contradiction. *}
+ -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *}
+ assume otherwise: "cntP ((e # t) @ s) th' \<noteq> cntP (t @ s) th'"
+ -- {* Then the actor of @{term e} must be @{term th'} and @{term e}
+ must be a @{term P}-event: *}
+ hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv)
+ with vat_es.actor_inv
+ -- {* According to @{thm vat_es.actor_inv}, @{term th'} must be running at
+ the moment @{term "t@s"}: *}
+ have "th' \<in> runing (t@s)" by (cases e, auto)
+ -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis
+ shows @{term th'} can not be running at moment @{term "t@s"}: *}
+ moreover have "th' \<notin> runing (t@s)"
+ using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
+ -- {* Contradiction is finally derived: *}
+ ultimately show False by simp
+ qed
+ -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change
+ by the happening of event @{term e}: *}
+ -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *}
+ moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'"
+ proof(rule ccontr) -- {* Proof by contradiction. *}
+ assume otherwise: "cntV ((e # t) @ s) th' \<noteq> cntV (t @ s) th'"
+ hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv)
+ with vat_es.actor_inv
+ have "th' \<in> runing (t@s)" by (cases e, auto)
+ moreover have "th' \<notin> runing (t@s)"
+ using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
+ ultimately show False by simp
+ qed
+ -- {* Finally, it can be shown that the @{term cntP} and @{term cntV}
+ value for @{term th'} are still in balance, so @{term th'}
+ is still hand-emptied after the execution of event @{term e}: *}
+ ultimately show ?thesis using Cons(5) by metis
+ qed
+qed (auto simp:eq_pv)
+
+text {*
+ By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist},
+ it can be derived easily that @{term th'} can not be running in the future:
+*}
+lemma eq_pv_blocked_persist:
+ assumes neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP s th' = cntV s th'"
+ shows "th' \<notin> runing (t@s)"
+ using assms
+ by (simp add: eq_pv_blocked eq_pv_persist)
+
+text {*
+ The following lemma shows the blocking thread @{term th'}
+ must hold some resource in the very beginning.
+*}
+lemma runing_cntP_cntV_inv: (* ddd *)
+ assumes is_runing: "th' \<in> runing (t@s)"
+ and neq_th': "th' \<noteq> th"
+ shows "cntP s th' > cntV s th'"
+ using assms
+proof -
+ -- {* First, it can be shown that the number of @{term P} and
+ @{term V} operations can not be equal for thred @{term th'} *}
+ have "cntP s th' \<noteq> cntV s th'"
+ proof
+ -- {* The proof goes by contradiction, suppose otherwise: *}
+ assume otherwise: "cntP s th' = cntV s th'"
+ -- {* By applying @{thm eq_pv_blocked_persist} to this: *}
+ from eq_pv_blocked_persist[OF neq_th' otherwise]
+ -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *}
+ have "th' \<notin> runing (t@s)" .
+ -- {* This is obvious in contradiction with assumption @{thm is_runing} *}
+ thus False using is_runing by simp
+ qed
+ -- {* However, the number of @{term V} is always less or equal to @{term P}: *}
+ moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
+ -- {* Thesis is finally derived by combining the these two results: *}
+ ultimately show ?thesis by auto
+qed
+
+
+text {*
+ The following lemmas shows the blocking thread @{text th'} must be live
+ at the very beginning, i.e. the moment (or state) @{term s}.
+
+ The proof is a simple combination of the results above:
+*}
+lemma runing_threads_inv:
+ assumes runing': "th' \<in> runing (t@s)"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads s"
+proof(rule ccontr) -- {* Proof by contradiction: *}
+ assume otherwise: "th' \<notin> threads s"
+ have "th' \<notin> runing (t @ s)"
+ proof -
+ from vat_s.cnp_cnv_eq[OF otherwise]
+ have "cntP s th' = cntV s th'" .
+ from eq_pv_blocked_persist[OF neq_th' this]
+ show ?thesis .
+ qed
+ with runing' show False by simp
+qed
+
+text {*
+ The following lemma summarizes several foregoing
+ lemmas to give an overall picture of the blocking thread @{text "th'"}:
+*}
+lemma runing_inversion: (* ddd, one of the main lemmas to present *)
+ 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"
+proof -
+ from runing_threads_inv[OF assms]
+ show "th' \<in> threads s" .
+next
+ from runing_cntP_cntV_inv[OF runing' neq_th]
+ show "\<not>detached s th'" using vat_s.detached_eq by simp
+next
+ from runing_preced_inversion[OF runing']
+ show "cp (t@s) th' = preced th s" .
+qed
+
+section {* The existence of `blocking thread` *}
+
+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, the other main lemma to be presented: *)
+ 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.rg_RAG_threads 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