--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Correctness.thy Wed Jan 06 16:34:26 2016 +0000
@@ -0,0 +1,922 @@
+theory Correctness
+imports PIPBasics Implementation
+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
+
+
+
--- a/CpsG.thy Wed Jan 06 20:46:14 2016 +0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1640 +0,0 @@
-section {*
- This file contains lemmas used to guide the recalculation of current precedence
- after every system call (or system operation)
-*}
-theory CpsG
-imports PrioG Max RTree
-begin
-
-text {* @{text "the_preced"} is also the same as @{text "preced"}, the only
- difference is the order of arguemts. *}
-definition "the_preced s th = preced th s"
-
-lemma inj_the_preced:
- "inj_on (the_preced s) (threads s)"
- by (metis inj_onI preced_unique the_preced_def)
-
-text {* @{term "the_thread"} extracts thread out of RAG node. *}
-fun the_thread :: "node \<Rightarrow> thread" where
- "the_thread (Th th) = th"
-
-text {* The following @{text "wRAG"} is the waiting sub-graph of @{text "RAG"}. *}
-definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"
-
-text {* The following @{text "hRAG"} is the holding sub-graph of @{text "RAG"}. *}
-definition "hRAG (s::state) = {(Cs cs, Th th) | th cs. holding s th cs}"
-
-text {* The following lemma splits @{term "RAG"} graph into the above two sub-graphs. *}
-lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"
- by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv
- s_holding_abv cs_RAG_def, auto)
-
-text {*
- The following @{text "tRAG"} is the thread-graph derived from @{term "RAG"}.
- It characterizes the dependency between threads when calculating current
- precedences. It is defined as the composition of the above two sub-graphs,
- names @{term "wRAG"} and @{term "hRAG"}.
- *}
-definition "tRAG s = wRAG s O hRAG s"
-
-(* ccc *)
-
-definition "cp_gen s x =
- Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"
-
-lemma tRAG_alt_def:
- "tRAG s = {(Th th1, Th th2) | th1 th2.
- \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
- by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
-
-lemma tRAG_Field:
- "Field (tRAG s) \<subseteq> Field (RAG s)"
- by (unfold tRAG_alt_def Field_def, auto)
-
-lemma tRAG_ancestorsE:
- assumes "x \<in> ancestors (tRAG s) u"
- obtains th where "x = Th th"
-proof -
- from assms have "(u, x) \<in> (tRAG s)^+"
- by (unfold ancestors_def, auto)
- from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
- then obtain th where "x = Th th"
- by (unfold tRAG_alt_def, auto)
- from that[OF this] show ?thesis .
-qed
-
-lemma tRAG_mono:
- assumes "RAG s' \<subseteq> RAG s"
- shows "tRAG s' \<subseteq> tRAG s"
- using assms
- by (unfold tRAG_alt_def, auto)
-
-lemma holding_next_thI:
- assumes "holding s th cs"
- and "length (wq s cs) > 1"
- obtains th' where "next_th s th cs th'"
-proof -
- from assms(1)[folded eq_holding, unfolded cs_holding_def]
- have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
- then obtain rest where h1: "wq s cs = th#rest"
- by (cases "wq s cs", auto)
- with assms(2) have h2: "rest \<noteq> []" by auto
- let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
- have "next_th s th cs ?th'" using h1(1) h2
- by (unfold next_th_def, auto)
- from that[OF this] show ?thesis .
-qed
-
-lemma RAG_tRAG_transfer:
- assumes "vt s'"
- assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
- and "(Cs cs, Th th'') \<in> RAG s'"
- shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
-proof -
- interpret vt_s': valid_trace "s'" using assms(1)
- by (unfold_locales, simp)
- interpret rtree: rtree "RAG s'"
- proof
- show "single_valued (RAG s')"
- apply (intro_locales)
- by (unfold single_valued_def,
- auto intro:vt_s'.unique_RAG)
-
- show "acyclic (RAG s')"
- by (rule vt_s'.acyclic_RAG)
- qed
- { fix n1 n2
- assume "(n1, n2) \<in> ?L"
- from this[unfolded tRAG_alt_def]
- obtain th1 th2 cs' where
- h: "n1 = Th th1" "n2 = Th th2"
- "(Th th1, Cs cs') \<in> RAG s"
- "(Cs cs', Th th2) \<in> RAG s" by auto
- from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
- from h(3) and assms(2)
- have "(Th th1, Cs cs') = (Th th, Cs cs) \<or>
- (Th th1, Cs cs') \<in> RAG s'" by auto
- hence "(n1, n2) \<in> ?R"
- proof
- assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
- hence eq_th1: "th1 = th" by simp
- moreover have "th2 = th''"
- proof -
- from h1 have "cs' = cs" by simp
- from assms(3) cs_in[unfolded this] rtree.sgv
- show ?thesis
- by (unfold single_valued_def, auto)
- qed
- ultimately show ?thesis using h(1,2) by auto
- next
- assume "(Th th1, Cs cs') \<in> RAG s'"
- with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
- by (unfold tRAG_alt_def, auto)
- from this[folded h(1, 2)] show ?thesis by auto
- qed
- } moreover {
- fix n1 n2
- assume "(n1, n2) \<in> ?R"
- hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
- hence "(n1, n2) \<in> ?L"
- proof
- assume "(n1, n2) \<in> tRAG s'"
- moreover have "... \<subseteq> ?L"
- proof(rule tRAG_mono)
- show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
- qed
- ultimately show ?thesis by auto
- next
- assume eq_n: "(n1, n2) = (Th th, Th th'')"
- from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
- moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
- ultimately show ?thesis
- by (unfold eq_n tRAG_alt_def, auto)
- qed
- } ultimately show ?thesis by auto
-qed
-
-context valid_trace
-begin
-
-lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
-
-end
-
-lemma cp_alt_def:
- "cp s th =
- Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
-proof -
- have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
- Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
- (is "Max (_ ` ?L) = Max (_ ` ?R)")
- proof -
- have "?L = ?R"
- by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
- thus ?thesis by simp
- qed
- thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
-qed
-
-lemma cp_gen_alt_def:
- "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
- by (auto simp:cp_gen_def)
-
-lemma tRAG_nodeE:
- assumes "(n1, n2) \<in> tRAG s"
- obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
- using assms
- by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
-
-lemma subtree_nodeE:
- assumes "n \<in> subtree (tRAG s) (Th th)"
- obtains th1 where "n = Th th1"
-proof -
- show ?thesis
- proof(rule subtreeE[OF assms])
- assume "n = Th th"
- from that[OF this] show ?thesis .
- next
- assume "Th th \<in> ancestors (tRAG s) n"
- hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
- hence "\<exists> th1. n = Th th1"
- proof(induct)
- case (base y)
- from tRAG_nodeE[OF this] show ?case by metis
- next
- case (step y z)
- thus ?case by auto
- qed
- with that show ?thesis by auto
- qed
-qed
-
-lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
-proof -
- have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*"
- by (rule rtrancl_mono, auto simp:RAG_split)
- also have "... \<subseteq> ((RAG s)^*)^*"
- by (rule rtrancl_mono, auto)
- also have "... = (RAG s)^*" by simp
- finally show ?thesis by (unfold tRAG_def, simp)
-qed
-
-lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
-proof -
- { fix a
- assume "a \<in> subtree (tRAG s) x"
- hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
- with tRAG_star_RAG[of s]
- have "(a, x) \<in> (RAG s)^*" by auto
- hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
- } thus ?thesis by auto
-qed
-
-lemma tRAG_trancl_eq:
- "{th'. (Th th', Th th) \<in> (tRAG s)^+} =
- {th'. (Th th', Th th) \<in> (RAG s)^+}"
- (is "?L = ?R")
-proof -
- { fix th'
- assume "th' \<in> ?L"
- hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
- from tranclD[OF this]
- obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
- from tRAG_subtree_RAG[of s] and this(2)
- have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG)
- moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto
- ultimately have "th' \<in> ?R" by auto
- } moreover
- { fix th'
- assume "th' \<in> ?R"
- hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
- from plus_rpath[OF this]
- obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
- hence "(Th th', Th th) \<in> (tRAG s)^+"
- proof(induct xs arbitrary:th' th rule:length_induct)
- case (1 xs th' th)
- then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
- show ?case
- proof(cases "xs1")
- case Nil
- from 1(2)[unfolded Cons1 Nil]
- have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
- hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
- then obtain cs where "x1 = Cs cs"
- by (unfold s_RAG_def, auto)
- from rpath_nnl_lastE[OF rp[unfolded this]]
- show ?thesis by auto
- next
- case (Cons x2 xs2)
- from 1(2)[unfolded Cons1[unfolded this]]
- have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
- from rpath_edges_on[OF this]
- have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
- have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
- by (simp add: edges_on_unfold)
- with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
- then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
- have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
- by (simp add: edges_on_unfold)
- from this eds
- have rg2: "(x1, x2) \<in> RAG s" by auto
- from this[unfolded eq_x1]
- obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
- from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
- have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
- from rp have "rpath (RAG s) x2 xs2 (Th th)"
- by (elim rpath_ConsE, simp)
- from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
- show ?thesis
- proof(cases "xs2 = []")
- case True
- from rpath_nilE[OF rp'[unfolded this]]
- have "th1 = th" by auto
- from rt1[unfolded this] show ?thesis by auto
- next
- case False
- from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
- have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
- with rt1 show ?thesis by auto
- qed
- qed
- qed
- hence "th' \<in> ?L" by auto
- } ultimately show ?thesis by blast
-qed
-
-lemma tRAG_trancl_eq_Th:
- "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} =
- {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}"
- using tRAG_trancl_eq by auto
-
-lemma dependants_alt_def:
- "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
- by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
-
-context valid_trace
-begin
-
-lemma count_eq_tRAG_plus:
- assumes "cntP s th = cntV s th"
- shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
- using assms count_eq_dependants dependants_alt_def eq_dependants by auto
-
-lemma count_eq_RAG_plus:
- assumes "cntP s th = cntV s th"
- shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
- using assms count_eq_dependants cs_dependants_def eq_RAG by auto
-
-lemma count_eq_RAG_plus_Th:
- assumes "cntP s th = cntV s th"
- shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
- using count_eq_RAG_plus[OF assms] by auto
-
-lemma count_eq_tRAG_plus_Th:
- assumes "cntP s th = cntV s th"
- shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
- using count_eq_tRAG_plus[OF assms] by auto
-
-end
-
-lemma tRAG_subtree_eq:
- "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}"
- (is "?L = ?R")
-proof -
- { fix n
- assume h: "n \<in> ?L"
- hence "n \<in> ?R"
- by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG)
- } moreover {
- fix n
- assume "n \<in> ?R"
- then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
- by (auto simp:subtree_def)
- from rtranclD[OF this(2)]
- have "n \<in> ?L"
- proof
- assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
- with h have "n \<in> {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}" by auto
- thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
- qed (insert h, auto simp:subtree_def)
- } ultimately show ?thesis by auto
-qed
-
-lemma threads_set_eq:
- "the_thread ` (subtree (tRAG s) (Th th)) =
- {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
- by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
-
-lemma cp_alt_def1:
- "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
-proof -
- have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
- ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
- by auto
- thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
-qed
-
-lemma cp_gen_def_cond:
- assumes "x = Th th"
- shows "cp s th = cp_gen s (Th th)"
-by (unfold cp_alt_def1 cp_gen_def, simp)
-
-lemma cp_gen_over_set:
- assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
- shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
-proof(rule f_image_eq)
- fix a
- assume "a \<in> A"
- from assms[rule_format, OF this]
- obtain th where eq_a: "a = Th th" by auto
- show "cp_gen s a = (cp s \<circ> the_thread) a"
- by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
-qed
-
-
-context valid_trace
-begin
-
-lemma RAG_threads:
- assumes "(Th th) \<in> Field (RAG s)"
- shows "th \<in> threads s"
- using assms
- by (metis Field_def UnE dm_RAG_threads range_in vt)
-
-lemma subtree_tRAG_thread:
- assumes "th \<in> threads s"
- shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
-proof -
- have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
- by (unfold tRAG_subtree_eq, simp)
- also have "... \<subseteq> ?R"
- proof
- fix x
- assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
- then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
- from this(2)
- show "x \<in> ?R"
- proof(cases rule:subtreeE)
- case 1
- thus ?thesis by (simp add: assms h(1))
- next
- case 2
- thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI)
- qed
- qed
- finally show ?thesis .
-qed
-
-lemma readys_root:
- assumes "th \<in> readys s"
- shows "root (RAG s) (Th th)"
-proof -
- { fix x
- assume "x \<in> ancestors (RAG s) (Th th)"
- hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
- from tranclD[OF this]
- obtain z where "(Th th, z) \<in> RAG s" by auto
- with assms(1) have False
- apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
- by (fold wq_def, blast)
- } thus ?thesis by (unfold root_def, auto)
-qed
-
-lemma readys_in_no_subtree:
- assumes "th \<in> readys s"
- and "th' \<noteq> th"
- shows "Th th \<notin> subtree (RAG s) (Th th')"
-proof
- assume "Th th \<in> subtree (RAG s) (Th th')"
- thus False
- proof(cases rule:subtreeE)
- case 1
- with assms show ?thesis by auto
- next
- case 2
- with readys_root[OF assms(1)]
- show ?thesis by (auto simp:root_def)
- qed
-qed
-
-lemma not_in_thread_isolated:
- assumes "th \<notin> threads s"
- shows "(Th th) \<notin> Field (RAG s)"
-proof
- assume "(Th th) \<in> Field (RAG s)"
- with dm_RAG_threads and range_in assms
- show False by (unfold Field_def, blast)
-qed
-
-lemma wf_RAG: "wf (RAG s)"
-proof(rule finite_acyclic_wf)
- from finite_RAG show "finite (RAG s)" .
-next
- from acyclic_RAG show "acyclic (RAG s)" .
-qed
-
-lemma sgv_wRAG: "single_valued (wRAG s)"
- using waiting_unique
- by (unfold single_valued_def wRAG_def, auto)
-
-lemma sgv_hRAG: "single_valued (hRAG s)"
- using holding_unique
- by (unfold single_valued_def hRAG_def, auto)
-
-lemma sgv_tRAG: "single_valued (tRAG s)"
- by (unfold tRAG_def, rule single_valued_relcomp,
- insert sgv_wRAG sgv_hRAG, auto)
-
-lemma acyclic_tRAG: "acyclic (tRAG s)"
-proof(unfold tRAG_def, rule acyclic_compose)
- show "acyclic (RAG s)" using acyclic_RAG .
-next
- show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-next
- show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-qed
-
-lemma sgv_RAG: "single_valued (RAG s)"
- using unique_RAG by (auto simp:single_valued_def)
-
-lemma rtree_RAG: "rtree (RAG s)"
- using sgv_RAG acyclic_RAG
- by (unfold rtree_def rtree_axioms_def sgv_def, auto)
-
-end
-
-
-sublocale valid_trace < rtree_RAG: rtree "RAG s"
-proof
- show "single_valued (RAG s)"
- apply (intro_locales)
- by (unfold single_valued_def,
- auto intro:unique_RAG)
-
- show "acyclic (RAG s)"
- by (rule acyclic_RAG)
-qed
-
-sublocale valid_trace < rtree_s: rtree "tRAG s"
-proof(unfold_locales)
- from sgv_tRAG show "single_valued (tRAG s)" .
-next
- from acyclic_tRAG show "acyclic (tRAG s)" .
-qed
-
-sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
-proof -
- show "fsubtree (RAG s)"
- proof(intro_locales)
- show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
- next
- show "fsubtree_axioms (RAG s)"
- proof(unfold fsubtree_axioms_def)
- find_theorems wf RAG
- from wf_RAG show "wf (RAG s)" .
- qed
- qed
-qed
-
-sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
-proof -
- have "fsubtree (tRAG s)"
- proof -
- have "fbranch (tRAG s)"
- proof(unfold tRAG_def, rule fbranch_compose)
- show "fbranch (wRAG s)"
- proof(rule finite_fbranchI)
- from finite_RAG show "finite (wRAG s)"
- by (unfold RAG_split, auto)
- qed
- next
- show "fbranch (hRAG s)"
- proof(rule finite_fbranchI)
- from finite_RAG
- show "finite (hRAG s)" by (unfold RAG_split, auto)
- qed
- qed
- moreover have "wf (tRAG s)"
- proof(rule wf_subset)
- show "wf (RAG s O RAG s)" using wf_RAG
- by (fold wf_comp_self, simp)
- next
- show "tRAG s \<subseteq> (RAG s O RAG s)"
- by (unfold tRAG_alt_def, auto)
- qed
- ultimately show ?thesis
- by (unfold fsubtree_def fsubtree_axioms_def,auto)
- qed
- from this[folded tRAG_def] show "fsubtree (tRAG s)" .
-qed
-
-lemma Max_UNION:
- assumes "finite A"
- and "A \<noteq> {}"
- and "\<forall> M \<in> f ` A. finite M"
- and "\<forall> M \<in> f ` A. M \<noteq> {}"
- shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
- using assms[simp]
-proof -
- have "?L = Max (\<Union>(f ` A))"
- by (fold Union_image_eq, simp)
- also have "... = ?R"
- by (subst Max_Union, simp+)
- finally show ?thesis .
-qed
-
-lemma max_Max_eq:
- assumes "finite A"
- and "A \<noteq> {}"
- and "x = y"
- shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
-proof -
- have "?R = Max (insert y A)" by simp
- also from assms have "... = ?L"
- by (subst Max.insert, simp+)
- finally show ?thesis by simp
-qed
-
-context valid_trace
-begin
-
-(* ddd *)
-lemma cp_gen_rec:
- assumes "x = Th th"
- shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
-proof(cases "children (tRAG s) x = {}")
- case True
- show ?thesis
- by (unfold True cp_gen_def subtree_children, simp add:assms)
-next
- case False
- hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
- note fsbttRAGs.finite_subtree[simp]
- have [simp]: "finite (children (tRAG s) x)"
- by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree],
- rule children_subtree)
- { fix r x
- have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
- } note this[simp]
- have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
- proof -
- from False obtain q where "q \<in> children (tRAG s) x" by blast
- moreover have "subtree (tRAG s) q \<noteq> {}" by simp
- ultimately show ?thesis by blast
- qed
- have h: "Max ((the_preced s \<circ> the_thread) `
- ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
- Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
- (is "?L = ?R")
- proof -
- let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
- let "Max (_ \<union> (?h ` ?B))" = ?R
- let ?L1 = "?f ` \<Union>(?g ` ?B)"
- have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
- proof -
- have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
- also have "... = (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
- finally have "Max ?L1 = Max ..." by simp
- also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
- by (subst Max_UNION, simp+)
- also have "... = Max (cp_gen s ` children (tRAG s) x)"
- by (unfold image_comp cp_gen_alt_def, simp)
- finally show ?thesis .
- qed
- show ?thesis
- proof -
- have "?L = Max (?f ` ?A \<union> ?L1)" by simp
- also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
- by (subst Max_Un, simp+)
- also have "... = max (?f x) (Max (?h ` ?B))"
- by (unfold eq_Max_L1, simp)
- also have "... =?R"
- by (rule max_Max_eq, (simp)+, unfold assms, simp)
- finally show ?thesis .
- qed
- qed thus ?thesis
- by (fold h subtree_children, unfold cp_gen_def, simp)
-qed
-
-lemma cp_rec:
- "cp s th = Max ({the_preced s th} \<union>
- (cp s o the_thread) ` children (tRAG s) (Th th))"
-proof -
- have "Th th = Th th" by simp
- note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this]
- show ?thesis
- proof -
- have "cp_gen s ` children (tRAG s) (Th th) =
- (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
- proof(rule cp_gen_over_set)
- show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
- by (unfold tRAG_alt_def, auto simp:children_def)
- qed
- thus ?thesis by (subst (1) h(1), unfold h(2), simp)
- qed
-qed
-
-end
-
-(* keep *)
-lemma next_th_holding:
- assumes vt: "vt s"
- and nxt: "next_th s th cs th'"
- shows "holding (wq s) th cs"
-proof -
- from nxt[unfolded next_th_def]
- obtain rest where h: "wq s cs = th # rest"
- "rest \<noteq> []"
- "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
- thus ?thesis
- by (unfold cs_holding_def, auto)
-qed
-
-context valid_trace
-begin
-
-lemma next_th_waiting:
- assumes nxt: "next_th s th cs th'"
- shows "waiting (wq s) th' cs"
-proof -
- from nxt[unfolded next_th_def]
- obtain rest where h: "wq s cs = th # rest"
- "rest \<noteq> []"
- "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
- from wq_distinct[of cs, unfolded h]
- have dst: "distinct (th # rest)" .
- have in_rest: "th' \<in> set rest"
- proof(unfold h, rule someI2)
- show "distinct rest \<and> set rest = set rest" using dst by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with h(2)
- show "hd x \<in> set (rest)" by (cases x, auto)
- qed
- hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
- moreover have "th' \<noteq> hd (wq s cs)"
- by (unfold h(1), insert in_rest dst, auto)
- ultimately show ?thesis by (auto simp:cs_waiting_def)
-qed
-
-lemma next_th_RAG:
- assumes nxt: "next_th (s::event list) th cs th'"
- shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
- using vt assms next_th_holding next_th_waiting
- by (unfold s_RAG_def, simp)
-
-end
-
--- {* A useless definition *}
-definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
-where "cps s = {(th, cp s th) | th . th \<in> threads s}"
-
-
-text {* (* ddd *)
- One beauty of our modelling is that we follow the definitional extension tradition of HOL.
- The benefit of such a concise and miniature model is that large number of intuitively
- obvious facts are derived as lemmas, rather than asserted as axioms.
-*}
-
-text {*
- However, the lemmas in the forthcoming several locales are no longer
- obvious. These lemmas show how the current precedences should be recalculated
- after every execution step (in our model, every step is represented by an event,
- which in turn, represents a system call, or operation). Each operation is
- treated in a separate locale.
-
- The complication of current precedence recalculation comes
- because the changing of RAG needs to be taken into account,
- in addition to the changing of precedence.
- The reason RAG changing affects current precedence is that,
- according to the definition, current precedence
- of a thread is the maximum of the precedences of its dependants,
- where the dependants are defined in terms of RAG.
-
- Therefore, each operation, lemmas concerning the change of the precedences
- and RAG are derived first, so that the lemmas about
- current precedence recalculation can be based on.
-*}
-
-text {* (* ddd *)
- The following locale @{text "step_set_cps"} investigates the recalculation
- after the @{text "Set"} operation.
-*}
-locale step_set_cps =
- fixes s' th prio s
- -- {* @{text "s'"} is the system state before the operation *}
- -- {* @{text "s"} is the system state after the operation *}
- defines s_def : "s \<equiv> (Set th prio#s')"
- -- {* @{text "s"} is assumed to be a legitimate state, from which
- the legitimacy of @{text "s"} can be derived. *}
- assumes vt_s: "vt s"
-
-sublocale step_set_cps < vat_s : valid_trace "s"
-proof
- from vt_s show "vt s" .
-qed
-
-sublocale step_set_cps < vat_s' : valid_trace "s'"
-proof
- from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_set_cps
-begin
-
-text {* (* ddd *)
- The following two lemmas confirm that @{text "Set"}-operating only changes the precedence
- of the initiating thread.
-*}
-
-lemma eq_preced:
- assumes "th' \<noteq> th"
- shows "preced th' s = preced th' s'"
-proof -
- from assms show ?thesis
- by (unfold s_def, auto simp:preced_def)
-qed
-
-lemma eq_the_preced:
- fixes th'
- assumes "th' \<noteq> th"
- shows "the_preced s th' = the_preced s' th'"
- using assms
- by (unfold the_preced_def, intro eq_preced, simp)
-
-text {*
- The following lemma assures that the resetting of priority does not change the RAG.
-*}
-
-lemma eq_dep: "RAG s = RAG s'"
- by (unfold s_def RAG_set_unchanged, auto)
-
-text {* (* ddd *)
- Th following lemma @{text "eq_cp_pre"} says the priority change of @{text "th"}
- only affects those threads, which as @{text "Th th"} in their sub-trees.
-
- The proof of this lemma is simplified by using the alternative definition of @{text "cp"}.
-*}
-
-lemma eq_cp_pre:
- fixes th'
- assumes nd: "Th th \<notin> subtree (RAG s') (Th th')"
- shows "cp s th' = cp s' th'"
-proof -
- -- {* After unfolding using the alternative definition, elements
- affecting the @{term "cp"}-value of threads become explicit.
- We only need to prove the following: *}
- have "Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
- Max (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
- (is "Max (?f ` ?S1) = Max (?g ` ?S2)")
- proof -
- -- {* The base sets are equal. *}
- have "?S1 = ?S2" using eq_dep by simp
- -- {* The function values on the base set are equal as well. *}
- moreover have "\<forall> e \<in> ?S2. ?f e = ?g e"
- proof
- fix th1
- assume "th1 \<in> ?S2"
- with nd have "th1 \<noteq> th" by (auto)
- from eq_the_preced[OF this]
- show "the_preced s th1 = the_preced s' th1" .
- qed
- -- {* Therefore, the image of the functions are equal. *}
- ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq)
- thus ?thesis by simp
- qed
- thus ?thesis by (simp add:cp_alt_def)
-qed
-
-text {*
- The following lemma shows that @{term "th"} is not in the
- sub-tree of any other thread.
-*}
-lemma th_in_no_subtree:
- assumes "th' \<noteq> th"
- shows "Th th \<notin> subtree (RAG s') (Th th')"
-proof -
- have "th \<in> readys s'"
- proof -
- from step_back_step [OF vt_s[unfolded s_def]]
- have "step s' (Set th prio)" .
- hence "th \<in> runing s'" by (cases, simp)
- thus ?thesis by (simp add:readys_def runing_def)
- qed
- find_theorems readys subtree
- from vat_s'.readys_in_no_subtree[OF this assms(1)]
- show ?thesis by blast
-qed
-
-text {*
- By combining @{thm "eq_cp_pre"} and @{thm "th_in_no_subtree"},
- it is obvious that the change of priority only affects the @{text "cp"}-value
- of the initiating thread @{text "th"}.
-*}
-lemma eq_cp:
- fixes th'
- assumes "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
- by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])
-
-end
-
-text {*
- The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.
-*}
-
-locale step_v_cps =
- -- {* @{text "th"} is the initiating thread *}
- -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *}
- fixes s' th cs s -- {* @{text "s'"} is the state before operation*}
- defines s_def : "s \<equiv> (V th cs#s')" -- {* @{text "s"} is the state after operation*}
- -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *}
- assumes vt_s: "vt s"
-
-sublocale step_v_cps < vat_s : valid_trace "s"
-proof
- from vt_s show "vt s" .
-qed
-
-sublocale step_v_cps < vat_s' : valid_trace "s'"
-proof
- from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_v_cps
-begin
-
-lemma ready_th_s': "th \<in> readys s'"
- using step_back_step[OF vt_s[unfolded s_def]]
- by (cases, simp add:runing_def)
-
-lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"
-proof -
- from vat_s'.readys_root[OF ready_th_s']
- show ?thesis
- by (unfold root_def, simp)
-qed
-
-lemma holding_th: "holding s' th cs"
-proof -
- from vt_s[unfolded s_def]
- have " PIP s' (V th cs)" by (cases, simp)
- thus ?thesis by (cases, auto)
-qed
-
-lemma edge_of_th:
- "(Cs cs, Th th) \<in> RAG s'"
-proof -
- from holding_th
- show ?thesis
- by (unfold s_RAG_def holding_eq, auto)
-qed
-
-lemma ancestors_cs:
- "ancestors (RAG s') (Cs cs) = {Th th}"
-proof -
- have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th) \<union> {Th th}"
- proof(rule vat_s'.rtree_RAG.ancestors_accum)
- from vt_s[unfolded s_def]
- have " PIP s' (V th cs)" by (cases, simp)
- thus "(Cs cs, Th th) \<in> RAG s'"
- proof(cases)
- assume "holding s' th cs"
- from this[unfolded holding_eq]
- show ?thesis by (unfold s_RAG_def, auto)
- qed
- qed
- from this[unfolded ancestors_th] show ?thesis by simp
-qed
-
-lemma preced_kept: "the_preced s = the_preced s'"
- by (auto simp: s_def the_preced_def preced_def)
-
-end
-
-text {*
- The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation,
- which represents the case when there is another thread @{text "th'"}
- to take over the critical resource released by the initiating thread @{text "th"}.
-*}
-locale step_v_cps_nt = step_v_cps +
- fixes th'
- -- {* @{text "th'"} is assumed to take over @{text "cs"} *}
- assumes nt: "next_th s' th cs th'"
-
-context step_v_cps_nt
-begin
-
-text {*
- Lemma @{text "RAG_s"} confirms the change of RAG:
- two edges removed and one added, as shown by the following diagram.
-*}
-
-(*
- RAG before the V-operation
- th1 ----|
- |
- th' ----|
- |----> cs -----|
- th2 ----| |
- | |
- th3 ----| |
- |------> th
- th4 ----| |
- | |
- th5 ----| |
- |----> cs'-----|
- th6 ----|
- |
- th7 ----|
-
- RAG after the V-operation
- th1 ----|
- |
- |----> cs ----> th'
- th2 ----|
- |
- th3 ----|
-
- th4 ----|
- |
- th5 ----|
- |----> cs'----> th
- th6 ----|
- |
- th7 ----|
-*)
-
-lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"
- using next_th_RAG[OF nt] .
-
-lemma ancestors_th':
- "ancestors (RAG s') (Th th') = {Th th, Cs cs}"
-proof -
- have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"
- proof(rule vat_s'.rtree_RAG.ancestors_accum)
- from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto
- qed
- thus ?thesis using ancestors_th ancestors_cs by auto
-qed
-
-lemma RAG_s:
- "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
- {(Cs cs, Th th')}"
-proof -
- from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
- and nt show ?thesis by (auto intro:next_th_unique)
-qed
-
-lemma subtree_kept:
- assumes "th1 \<notin> {th, th'}"
- shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R")
-proof -
- let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})"
- let ?RAG'' = "?RAG' \<union> {(Cs cs, Th th')}"
- have "subtree ?RAG' (Th th1) = ?R"
- proof(rule subset_del_subtree_outside)
- show "Range {(Cs cs, Th th), (Th th', Cs cs)} \<inter> subtree (RAG s') (Th th1) = {}"
- proof -
- have "(Th th) \<notin> subtree (RAG s') (Th th1)"
- proof(rule subtree_refute)
- show "Th th1 \<notin> ancestors (RAG s') (Th th)"
- by (unfold ancestors_th, simp)
- next
- from assms show "Th th1 \<noteq> Th th" by simp
- qed
- moreover have "(Cs cs) \<notin> subtree (RAG s') (Th th1)"
- proof(rule subtree_refute)
- show "Th th1 \<notin> ancestors (RAG s') (Cs cs)"
- by (unfold ancestors_cs, insert assms, auto)
- qed simp
- ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s') (Th th1) = {}" by auto
- thus ?thesis by simp
- qed
- qed
- moreover have "subtree ?RAG'' (Th th1) = subtree ?RAG' (Th th1)"
- proof(rule subtree_insert_next)
- show "Th th' \<notin> subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)"
- proof(rule subtree_refute)
- show "Th th1 \<notin> ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')"
- (is "_ \<notin> ?R")
- proof -
- have "?R \<subseteq> ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto)
- moreover have "Th th1 \<notin> ..." using ancestors_th' assms by simp
- ultimately show ?thesis by auto
- qed
- next
- from assms show "Th th1 \<noteq> Th th'" by simp
- qed
- qed
- ultimately show ?thesis by (unfold RAG_s, simp)
-qed
-
-lemma cp_kept:
- assumes "th1 \<notin> {th, th'}"
- shows "cp s th1 = cp s' th1"
- by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
-
-end
-
-locale step_v_cps_nnt = step_v_cps +
- assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
-
-context step_v_cps_nnt
-begin
-
-lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"
-proof -
- from nnt and step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
- show ?thesis by auto
-qed
-
-lemma subtree_kept:
- assumes "th1 \<noteq> th"
- shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)"
-proof(unfold RAG_s, rule subset_del_subtree_outside)
- show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s') (Th th1) = {}"
- proof -
- have "(Th th) \<notin> subtree (RAG s') (Th th1)"
- proof(rule subtree_refute)
- show "Th th1 \<notin> ancestors (RAG s') (Th th)"
- by (unfold ancestors_th, simp)
- next
- from assms show "Th th1 \<noteq> Th th" by simp
- qed
- thus ?thesis by auto
- qed
-qed
-
-lemma cp_kept_1:
- assumes "th1 \<noteq> th"
- shows "cp s th1 = cp s' th1"
- by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
-
-lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}"
-proof -
- { fix n
- have "(Cs cs) \<notin> ancestors (RAG s') n"
- proof
- assume "Cs cs \<in> ancestors (RAG s') n"
- hence "(n, Cs cs) \<in> (RAG s')^+" by (auto simp:ancestors_def)
- from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s'" by auto
- then obtain th' where "nn = Th th'"
- by (unfold s_RAG_def, auto)
- from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s'" .
- from this[unfolded s_RAG_def]
- have "waiting (wq s') th' cs" by auto
- from this[unfolded cs_waiting_def]
- have "1 < length (wq s' cs)"
- by (cases "wq s' cs", auto)
- from holding_next_thI[OF holding_th this]
- obtain th' where "next_th s' th cs th'" by auto
- with nnt show False by auto
- qed
- } note h = this
- { fix n
- assume "n \<in> subtree (RAG s') (Cs cs)"
- hence "n = (Cs cs)"
- by (elim subtreeE, insert h, auto)
- } moreover have "(Cs cs) \<in> subtree (RAG s') (Cs cs)"
- by (auto simp:subtree_def)
- ultimately show ?thesis by auto
-qed
-
-lemma subtree_th:
- "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
-find_theorems "subtree" "_ - _" RAG
-proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
- from edge_of_th
- show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"
- by (unfold edges_in_def, auto simp:subtree_def)
-qed
-
-lemma cp_kept_2:
- shows "cp s th = cp s' th"
- by (unfold cp_alt_def subtree_th preced_kept, auto)
-
-lemma eq_cp:
- fixes th'
- shows "cp s th' = cp s' th'"
- using cp_kept_1 cp_kept_2
- by (cases "th' = th", auto)
-end
-
-
-locale step_P_cps =
- fixes s' th cs s
- defines s_def : "s \<equiv> (P th cs#s')"
- assumes vt_s: "vt s"
-
-sublocale step_P_cps < vat_s : valid_trace "s"
-proof
- from vt_s show "vt s" .
-qed
-
-sublocale step_P_cps < vat_s' : valid_trace "s'"
-proof
- from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_P_cps
-begin
-
-lemma readys_th: "th \<in> readys s'"
-proof -
- from step_back_step [OF vt_s[unfolded s_def]]
- have "PIP s' (P th cs)" .
- hence "th \<in> runing s'" by (cases, simp)
- thus ?thesis by (simp add:readys_def runing_def)
-qed
-
-lemma root_th: "root (RAG s') (Th th)"
- using readys_root[OF readys_th] .
-
-lemma in_no_others_subtree:
- assumes "th' \<noteq> th"
- shows "Th th \<notin> subtree (RAG s') (Th th')"
-proof
- assume "Th th \<in> subtree (RAG s') (Th th')"
- thus False
- proof(cases rule:subtreeE)
- case 1
- with assms show ?thesis by auto
- next
- case 2
- with root_th show ?thesis by (auto simp:root_def)
- qed
-qed
-
-lemma preced_kept: "the_preced s = the_preced s'"
- by (auto simp: s_def the_preced_def preced_def)
-
-end
-
-locale step_P_cps_ne =step_P_cps +
- fixes th'
- assumes ne: "wq s' cs \<noteq> []"
- defines th'_def: "th' \<equiv> hd (wq s' cs)"
-
-locale step_P_cps_e =step_P_cps +
- assumes ee: "wq s' cs = []"
-
-context step_P_cps_e
-begin
-
-lemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"
-proof -
- from ee and step_RAG_p[OF vt_s[unfolded s_def], folded s_def]
- show ?thesis by auto
-qed
-
-lemma subtree_kept:
- assumes "th' \<noteq> th"
- shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')"
-proof(unfold RAG_s, rule subtree_insert_next)
- from in_no_others_subtree[OF assms]
- show "Th th \<notin> subtree (RAG s') (Th th')" .
-qed
-
-lemma cp_kept:
- assumes "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
-proof -
- have "(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
- (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
- by (unfold preced_kept subtree_kept[OF assms], simp)
- thus ?thesis by (unfold cp_alt_def, simp)
-qed
-
-end
-
-context step_P_cps_ne
-begin
-
-lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
-proof -
- from step_RAG_p[OF vt_s[unfolded s_def]] and ne
- show ?thesis by (simp add:s_def)
-qed
-
-lemma cs_held: "(Cs cs, Th th') \<in> RAG s'"
-proof -
- have "(Cs cs, Th th') \<in> hRAG s'"
- proof -
- from ne
- have " holding s' th' cs"
- by (unfold th'_def holding_eq cs_holding_def, auto)
- thus ?thesis
- by (unfold hRAG_def, auto)
- qed
- thus ?thesis by (unfold RAG_split, auto)
-qed
-
-lemma tRAG_s:
- "tRAG s = tRAG s' \<union> {(Th th, Th th')}"
- using RAG_tRAG_transfer[OF RAG_s cs_held] .
-
-lemma cp_kept:
- assumes "Th th'' \<notin> ancestors (tRAG s) (Th th)"
- shows "cp s th'' = cp s' th''"
-proof -
- have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')"
- proof -
- have "Th th' \<notin> subtree (tRAG s') (Th th'')"
- proof
- assume "Th th' \<in> subtree (tRAG s') (Th th'')"
- thus False
- proof(rule subtreeE)
- assume "Th th' = Th th''"
- from assms[unfolded tRAG_s ancestors_def, folded this]
- show ?thesis by auto
- next
- assume "Th th'' \<in> ancestors (tRAG s') (Th th')"
- moreover have "... \<subseteq> ancestors (tRAG s) (Th th')"
- proof(rule ancestors_mono)
- show "tRAG s' \<subseteq> tRAG s" by (unfold tRAG_s, auto)
- qed
- ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th')" by auto
- moreover have "Th th' \<in> ancestors (tRAG s) (Th th)"
- by (unfold tRAG_s, auto simp:ancestors_def)
- ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th)"
- by (auto simp:ancestors_def)
- with assms show ?thesis by auto
- qed
- qed
- from subtree_insert_next[OF this]
- have "subtree (tRAG s' \<union> {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" .
- from this[folded tRAG_s] show ?thesis .
- qed
- show ?thesis by (unfold cp_alt_def1 h preced_kept, simp)
-qed
-
-lemma cp_gen_update_stop: (* ddd *)
- assumes "u \<in> ancestors (tRAG s) (Th th)"
- and "cp_gen s u = cp_gen s' u"
- and "y \<in> ancestors (tRAG s) u"
- shows "cp_gen s y = cp_gen s' y"
- using assms(3)
-proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf])
- case (1 x)
- show ?case (is "?L = ?R")
- proof -
- from tRAG_ancestorsE[OF 1(2)]
- obtain th2 where eq_x: "x = Th th2" by blast
- from vat_s.cp_gen_rec[OF this]
- have "?L =
- Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)" .
- also have "... =
- Max ({the_preced s' th2} \<union> cp_gen s' ` RTree.children (tRAG s') x)"
-
- proof -
- from preced_kept have "the_preced s th2 = the_preced s' th2" by simp
- moreover have "cp_gen s ` RTree.children (tRAG s) x =
- cp_gen s' ` RTree.children (tRAG s') x"
- proof -
- have "RTree.children (tRAG s) x = RTree.children (tRAG s') x"
- proof(unfold tRAG_s, rule children_union_kept)
- have start: "(Th th, Th th') \<in> tRAG s"
- by (unfold tRAG_s, auto)
- note x_u = 1(2)
- show "x \<notin> Range {(Th th, Th th')}"
- proof
- assume "x \<in> Range {(Th th, Th th')}"
- hence eq_x: "x = Th th'" using RangeE by auto
- show False
- proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start])
- case 1
- from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG
- show ?thesis by (auto simp:ancestors_def acyclic_def)
- next
- case 2
- with x_u[unfolded eq_x]
- have "(Th th', Th th') \<in> (tRAG s)^+" by (auto simp:ancestors_def)
- with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
- qed
- qed
- qed
- moreover have "cp_gen s ` RTree.children (tRAG s) x =
- cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A")
- proof(rule f_image_eq)
- fix a
- assume a_in: "a \<in> ?A"
- from 1(2)
- show "?f a = ?g a"
- proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
- case in_ch
- show ?thesis
- proof(cases "a = u")
- case True
- from assms(2)[folded this] show ?thesis .
- next
- case False
- have a_not_in: "a \<notin> ancestors (tRAG s) (Th th)"
- proof
- assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
- have "a = u"
- proof(rule vat_s.rtree_s.ancestors_children_unique)
- from a_in' a_in show "a \<in> ancestors (tRAG s) (Th th) \<inter>
- RTree.children (tRAG s) x" by auto
- next
- from assms(1) in_ch show "u \<in> ancestors (tRAG s) (Th th) \<inter>
- RTree.children (tRAG s) x" by auto
- qed
- with False show False by simp
- qed
- from a_in obtain th_a where eq_a: "a = Th th_a"
- by (unfold RTree.children_def tRAG_alt_def, auto)
- from cp_kept[OF a_not_in[unfolded eq_a]]
- have "cp s th_a = cp s' th_a" .
- from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
- show ?thesis .
- qed
- next
- case (out_ch z)
- hence h: "z \<in> ancestors (tRAG s) u" "z \<in> RTree.children (tRAG s) x" by auto
- show ?thesis
- proof(cases "a = z")
- case True
- from h(2) have zx_in: "(z, x) \<in> (tRAG s)" by (auto simp:RTree.children_def)
- from 1(1)[rule_format, OF this h(1)]
- have eq_cp_gen: "cp_gen s z = cp_gen s' z" .
- with True show ?thesis by metis
- next
- case False
- from a_in obtain th_a where eq_a: "a = Th th_a"
- by (auto simp:RTree.children_def tRAG_alt_def)
- have "a \<notin> ancestors (tRAG s) (Th th)"
- proof
- assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
- have "a = z"
- proof(rule vat_s.rtree_s.ancestors_children_unique)
- from assms(1) h(1) have "z \<in> ancestors (tRAG s) (Th th)"
- by (auto simp:ancestors_def)
- with h(2) show " z \<in> ancestors (tRAG s) (Th th) \<inter>
- RTree.children (tRAG s) x" by auto
- next
- from a_in a_in'
- show "a \<in> ancestors (tRAG s) (Th th) \<inter> RTree.children (tRAG s) x"
- by auto
- qed
- with False show False by auto
- qed
- from cp_kept[OF this[unfolded eq_a]]
- have "cp s th_a = cp s' th_a" .
- from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
- show ?thesis .
- qed
- qed
- qed
- ultimately show ?thesis by metis
- qed
- ultimately show ?thesis by simp
- qed
- also have "... = ?R"
- by (fold vat_s'.cp_gen_rec[OF eq_x], simp)
- finally show ?thesis .
- qed
-qed
-
-lemma cp_up:
- assumes "(Th th') \<in> ancestors (tRAG s) (Th th)"
- and "cp s th' = cp s' th'"
- and "(Th th'') \<in> ancestors (tRAG s) (Th th')"
- shows "cp s th'' = cp s' th''"
-proof -
- have "cp_gen s (Th th'') = cp_gen s' (Th th'')"
- proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)])
- from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]]
- show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis
- qed
- with cp_gen_def_cond[OF refl[of "Th th''"]]
- show ?thesis by metis
-qed
-
-end
-
-locale step_create_cps =
- fixes s' th prio s
- defines s_def : "s \<equiv> (Create th prio#s')"
- assumes vt_s: "vt s"
-
-sublocale step_create_cps < vat_s: valid_trace "s"
- by (unfold_locales, insert vt_s, simp)
-
-sublocale step_create_cps < vat_s': valid_trace "s'"
- by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
-
-context step_create_cps
-begin
-
-lemma RAG_kept: "RAG s = RAG s'"
- by (unfold s_def RAG_create_unchanged, auto)
-
-lemma tRAG_kept: "tRAG s = tRAG s'"
- by (unfold tRAG_alt_def RAG_kept, auto)
-
-lemma preced_kept:
- assumes "th' \<noteq> th"
- shows "the_preced s th' = the_preced s' th'"
- by (unfold s_def the_preced_def preced_def, insert assms, auto)
-
-lemma th_not_in: "Th th \<notin> Field (tRAG s')"
-proof -
- from vt_s[unfolded s_def]
- have "PIP s' (Create th prio)" by (cases, simp)
- hence "th \<notin> threads s'" by(cases, simp)
- from vat_s'.not_in_thread_isolated[OF this]
- have "Th th \<notin> Field (RAG s')" .
- with tRAG_Field show ?thesis by auto
-qed
-
-lemma eq_cp:
- assumes neq_th: "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
-proof -
- have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
- (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
- proof(unfold tRAG_kept, rule f_image_eq)
- fix a
- assume a_in: "a \<in> subtree (tRAG s') (Th th')"
- then obtain th_a where eq_a: "a = Th th_a"
- proof(cases rule:subtreeE)
- case 2
- from ancestors_Field[OF 2(2)]
- and that show ?thesis by (unfold tRAG_alt_def, auto)
- qed auto
- have neq_th_a: "th_a \<noteq> th"
- proof -
- have "(Th th) \<notin> subtree (tRAG s') (Th th')"
- proof
- assume "Th th \<in> subtree (tRAG s') (Th th')"
- thus False
- proof(cases rule:subtreeE)
- case 2
- from ancestors_Field[OF this(2)]
- and th_not_in[unfolded Field_def]
- show ?thesis by auto
- qed (insert assms, auto)
- qed
- with a_in[unfolded eq_a] show ?thesis by auto
- qed
- from preced_kept[OF this]
- show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
- by (unfold eq_a, simp)
- qed
- thus ?thesis by (unfold cp_alt_def1, simp)
-qed
-
-lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}"
-proof -
- { fix a
- assume "a \<in> RTree.children (tRAG s) (Th th)"
- hence "(a, Th th) \<in> tRAG s" by (auto simp:RTree.children_def)
- with th_not_in have False
- by (unfold Field_def tRAG_kept, auto)
- } thus ?thesis by auto
-qed
-
-lemma eq_cp_th: "cp s th = preced th s"
- by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def)
-
-end
-
-locale step_exit_cps =
- fixes s' th prio s
- defines s_def : "s \<equiv> Exit th # s'"
- assumes vt_s: "vt s"
-
-sublocale step_exit_cps < vat_s: valid_trace "s"
- by (unfold_locales, insert vt_s, simp)
-
-sublocale step_exit_cps < vat_s': valid_trace "s'"
- by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
-
-context step_exit_cps
-begin
-
-lemma preced_kept:
- assumes "th' \<noteq> th"
- shows "the_preced s th' = the_preced s' th'"
- by (unfold s_def the_preced_def preced_def, insert assms, auto)
-
-lemma RAG_kept: "RAG s = RAG s'"
- by (unfold s_def RAG_exit_unchanged, auto)
-
-lemma tRAG_kept: "tRAG s = tRAG s'"
- by (unfold tRAG_alt_def RAG_kept, auto)
-
-lemma th_ready: "th \<in> readys s'"
-proof -
- from vt_s[unfolded s_def]
- have "PIP s' (Exit th)" by (cases, simp)
- hence h: "th \<in> runing s' \<and> holdents s' th = {}" by (cases, metis)
- thus ?thesis by (unfold runing_def, auto)
-qed
-
-lemma th_holdents: "holdents s' th = {}"
-proof -
- from vt_s[unfolded s_def]
- have "PIP s' (Exit th)" by (cases, simp)
- thus ?thesis by (cases, metis)
-qed
-
-lemma th_RAG: "Th th \<notin> Field (RAG s')"
-proof -
- have "Th th \<notin> Range (RAG s')"
- proof
- assume "Th th \<in> Range (RAG s')"
- then obtain cs where "holding (wq s') th cs"
- by (unfold Range_iff s_RAG_def, auto)
- with th_holdents[unfolded holdents_def]
- show False by (unfold eq_holding, auto)
- qed
- moreover have "Th th \<notin> Domain (RAG s')"
- proof
- assume "Th th \<in> Domain (RAG s')"
- then obtain cs where "waiting (wq s') th cs"
- by (unfold Domain_iff s_RAG_def, auto)
- with th_ready show False by (unfold readys_def eq_waiting, auto)
- qed
- ultimately show ?thesis by (auto simp:Field_def)
-qed
-
-lemma th_tRAG: "(Th th) \<notin> Field (tRAG s')"
- using th_RAG tRAG_Field[of s'] by auto
-
-lemma eq_cp:
- assumes neq_th: "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
-proof -
- have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
- (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
- proof(unfold tRAG_kept, rule f_image_eq)
- fix a
- assume a_in: "a \<in> subtree (tRAG s') (Th th')"
- then obtain th_a where eq_a: "a = Th th_a"
- proof(cases rule:subtreeE)
- case 2
- from ancestors_Field[OF 2(2)]
- and that show ?thesis by (unfold tRAG_alt_def, auto)
- qed auto
- have neq_th_a: "th_a \<noteq> th"
- proof -
- find_theorems readys subtree s'
- from vat_s'.readys_in_no_subtree[OF th_ready assms]
- have "(Th th) \<notin> subtree (RAG s') (Th th')" .
- with tRAG_subtree_RAG[of s' "Th th'"]
- have "(Th th) \<notin> subtree (tRAG s') (Th th')" by auto
- with a_in[unfolded eq_a] show ?thesis by auto
- qed
- from preced_kept[OF this]
- show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
- by (unfold eq_a, simp)
- qed
- thus ?thesis by (unfold cp_alt_def1, simp)
-qed
-
-end
-
-end
-
--- a/ExtGG.thy Wed Jan 06 20:46:14 2016 +0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,922 +0,0 @@
-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
-
-
-
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Implementation.thy Wed Jan 06 16:34:26 2016 +0000
@@ -0,0 +1,1640 @@
+section {*
+ This file contains lemmas used to guide the recalculation of current precedence
+ after every system call (or system operation)
+*}
+theory Implementation
+imports PIPBasics Max RTree
+begin
+
+text {* @{text "the_preced"} is also the same as @{text "preced"}, the only
+ difference is the order of arguemts. *}
+definition "the_preced s th = preced th s"
+
+lemma inj_the_preced:
+ "inj_on (the_preced s) (threads s)"
+ by (metis inj_onI preced_unique the_preced_def)
+
+text {* @{term "the_thread"} extracts thread out of RAG node. *}
+fun the_thread :: "node \<Rightarrow> thread" where
+ "the_thread (Th th) = th"
+
+text {* The following @{text "wRAG"} is the waiting sub-graph of @{text "RAG"}. *}
+definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"
+
+text {* The following @{text "hRAG"} is the holding sub-graph of @{text "RAG"}. *}
+definition "hRAG (s::state) = {(Cs cs, Th th) | th cs. holding s th cs}"
+
+text {* The following lemma splits @{term "RAG"} graph into the above two sub-graphs. *}
+lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"
+ by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv
+ s_holding_abv cs_RAG_def, auto)
+
+text {*
+ The following @{text "tRAG"} is the thread-graph derived from @{term "RAG"}.
+ It characterizes the dependency between threads when calculating current
+ precedences. It is defined as the composition of the above two sub-graphs,
+ names @{term "wRAG"} and @{term "hRAG"}.
+ *}
+definition "tRAG s = wRAG s O hRAG s"
+
+(* ccc *)
+
+definition "cp_gen s x =
+ Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"
+
+lemma tRAG_alt_def:
+ "tRAG s = {(Th th1, Th th2) | th1 th2.
+ \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
+ by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
+
+lemma tRAG_Field:
+ "Field (tRAG s) \<subseteq> Field (RAG s)"
+ by (unfold tRAG_alt_def Field_def, auto)
+
+lemma tRAG_ancestorsE:
+ assumes "x \<in> ancestors (tRAG s) u"
+ obtains th where "x = Th th"
+proof -
+ from assms have "(u, x) \<in> (tRAG s)^+"
+ by (unfold ancestors_def, auto)
+ from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
+ then obtain th where "x = Th th"
+ by (unfold tRAG_alt_def, auto)
+ from that[OF this] show ?thesis .
+qed
+
+lemma tRAG_mono:
+ assumes "RAG s' \<subseteq> RAG s"
+ shows "tRAG s' \<subseteq> tRAG s"
+ using assms
+ by (unfold tRAG_alt_def, auto)
+
+lemma holding_next_thI:
+ assumes "holding s th cs"
+ and "length (wq s cs) > 1"
+ obtains th' where "next_th s th cs th'"
+proof -
+ from assms(1)[folded eq_holding, unfolded cs_holding_def]
+ have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
+ then obtain rest where h1: "wq s cs = th#rest"
+ by (cases "wq s cs", auto)
+ with assms(2) have h2: "rest \<noteq> []" by auto
+ let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+ have "next_th s th cs ?th'" using h1(1) h2
+ by (unfold next_th_def, auto)
+ from that[OF this] show ?thesis .
+qed
+
+lemma RAG_tRAG_transfer:
+ assumes "vt s'"
+ assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+ and "(Cs cs, Th th'') \<in> RAG s'"
+ shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
+proof -
+ interpret vt_s': valid_trace "s'" using assms(1)
+ by (unfold_locales, simp)
+ interpret rtree: rtree "RAG s'"
+ proof
+ show "single_valued (RAG s')"
+ apply (intro_locales)
+ by (unfold single_valued_def,
+ auto intro:vt_s'.unique_RAG)
+
+ show "acyclic (RAG s')"
+ by (rule vt_s'.acyclic_RAG)
+ qed
+ { fix n1 n2
+ assume "(n1, n2) \<in> ?L"
+ from this[unfolded tRAG_alt_def]
+ obtain th1 th2 cs' where
+ h: "n1 = Th th1" "n2 = Th th2"
+ "(Th th1, Cs cs') \<in> RAG s"
+ "(Cs cs', Th th2) \<in> RAG s" by auto
+ from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
+ from h(3) and assms(2)
+ have "(Th th1, Cs cs') = (Th th, Cs cs) \<or>
+ (Th th1, Cs cs') \<in> RAG s'" by auto
+ hence "(n1, n2) \<in> ?R"
+ proof
+ assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
+ hence eq_th1: "th1 = th" by simp
+ moreover have "th2 = th''"
+ proof -
+ from h1 have "cs' = cs" by simp
+ from assms(3) cs_in[unfolded this] rtree.sgv
+ show ?thesis
+ by (unfold single_valued_def, auto)
+ qed
+ ultimately show ?thesis using h(1,2) by auto
+ next
+ assume "(Th th1, Cs cs') \<in> RAG s'"
+ with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
+ by (unfold tRAG_alt_def, auto)
+ from this[folded h(1, 2)] show ?thesis by auto
+ qed
+ } moreover {
+ fix n1 n2
+ assume "(n1, n2) \<in> ?R"
+ hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
+ hence "(n1, n2) \<in> ?L"
+ proof
+ assume "(n1, n2) \<in> tRAG s'"
+ moreover have "... \<subseteq> ?L"
+ proof(rule tRAG_mono)
+ show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
+ qed
+ ultimately show ?thesis by auto
+ next
+ assume eq_n: "(n1, n2) = (Th th, Th th'')"
+ from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
+ moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
+ ultimately show ?thesis
+ by (unfold eq_n tRAG_alt_def, auto)
+ qed
+ } ultimately show ?thesis by auto
+qed
+
+context valid_trace
+begin
+
+lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
+
+end
+
+lemma cp_alt_def:
+ "cp s th =
+ Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
+proof -
+ have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
+ Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
+ (is "Max (_ ` ?L) = Max (_ ` ?R)")
+ proof -
+ have "?L = ?R"
+ by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
+ thus ?thesis by simp
+ qed
+ thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
+qed
+
+lemma cp_gen_alt_def:
+ "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
+ by (auto simp:cp_gen_def)
+
+lemma tRAG_nodeE:
+ assumes "(n1, n2) \<in> tRAG s"
+ obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
+ using assms
+ by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
+
+lemma subtree_nodeE:
+ assumes "n \<in> subtree (tRAG s) (Th th)"
+ obtains th1 where "n = Th th1"
+proof -
+ show ?thesis
+ proof(rule subtreeE[OF assms])
+ assume "n = Th th"
+ from that[OF this] show ?thesis .
+ next
+ assume "Th th \<in> ancestors (tRAG s) n"
+ hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+ hence "\<exists> th1. n = Th th1"
+ proof(induct)
+ case (base y)
+ from tRAG_nodeE[OF this] show ?case by metis
+ next
+ case (step y z)
+ thus ?case by auto
+ qed
+ with that show ?thesis by auto
+ qed
+qed
+
+lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
+proof -
+ have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*"
+ by (rule rtrancl_mono, auto simp:RAG_split)
+ also have "... \<subseteq> ((RAG s)^*)^*"
+ by (rule rtrancl_mono, auto)
+ also have "... = (RAG s)^*" by simp
+ finally show ?thesis by (unfold tRAG_def, simp)
+qed
+
+lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
+proof -
+ { fix a
+ assume "a \<in> subtree (tRAG s) x"
+ hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
+ with tRAG_star_RAG[of s]
+ have "(a, x) \<in> (RAG s)^*" by auto
+ hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
+ } thus ?thesis by auto
+qed
+
+lemma tRAG_trancl_eq:
+ "{th'. (Th th', Th th) \<in> (tRAG s)^+} =
+ {th'. (Th th', Th th) \<in> (RAG s)^+}"
+ (is "?L = ?R")
+proof -
+ { fix th'
+ assume "th' \<in> ?L"
+ hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
+ from tranclD[OF this]
+ obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
+ from tRAG_subtree_RAG[of s] and this(2)
+ have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG)
+ moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto
+ ultimately have "th' \<in> ?R" by auto
+ } moreover
+ { fix th'
+ assume "th' \<in> ?R"
+ hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
+ from plus_rpath[OF this]
+ obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
+ hence "(Th th', Th th) \<in> (tRAG s)^+"
+ proof(induct xs arbitrary:th' th rule:length_induct)
+ case (1 xs th' th)
+ then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
+ show ?case
+ proof(cases "xs1")
+ case Nil
+ from 1(2)[unfolded Cons1 Nil]
+ have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
+ hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
+ then obtain cs where "x1 = Cs cs"
+ by (unfold s_RAG_def, auto)
+ from rpath_nnl_lastE[OF rp[unfolded this]]
+ show ?thesis by auto
+ next
+ case (Cons x2 xs2)
+ from 1(2)[unfolded Cons1[unfolded this]]
+ have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
+ from rpath_edges_on[OF this]
+ have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
+ have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+ by (simp add: edges_on_unfold)
+ with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
+ then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
+ have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+ by (simp add: edges_on_unfold)
+ from this eds
+ have rg2: "(x1, x2) \<in> RAG s" by auto
+ from this[unfolded eq_x1]
+ obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
+ from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
+ have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
+ from rp have "rpath (RAG s) x2 xs2 (Th th)"
+ by (elim rpath_ConsE, simp)
+ from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
+ show ?thesis
+ proof(cases "xs2 = []")
+ case True
+ from rpath_nilE[OF rp'[unfolded this]]
+ have "th1 = th" by auto
+ from rt1[unfolded this] show ?thesis by auto
+ next
+ case False
+ from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
+ have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
+ with rt1 show ?thesis by auto
+ qed
+ qed
+ qed
+ hence "th' \<in> ?L" by auto
+ } ultimately show ?thesis by blast
+qed
+
+lemma tRAG_trancl_eq_Th:
+ "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} =
+ {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}"
+ using tRAG_trancl_eq by auto
+
+lemma dependants_alt_def:
+ "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
+ by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
+
+context valid_trace
+begin
+
+lemma count_eq_tRAG_plus:
+ assumes "cntP s th = cntV s th"
+ shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+ using assms count_eq_dependants dependants_alt_def eq_dependants by auto
+
+lemma count_eq_RAG_plus:
+ assumes "cntP s th = cntV s th"
+ shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+ using assms count_eq_dependants cs_dependants_def eq_RAG by auto
+
+lemma count_eq_RAG_plus_Th:
+ assumes "cntP s th = cntV s th"
+ shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+ using count_eq_RAG_plus[OF assms] by auto
+
+lemma count_eq_tRAG_plus_Th:
+ assumes "cntP s th = cntV s th"
+ shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+ using count_eq_tRAG_plus[OF assms] by auto
+
+end
+
+lemma tRAG_subtree_eq:
+ "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}"
+ (is "?L = ?R")
+proof -
+ { fix n
+ assume h: "n \<in> ?L"
+ hence "n \<in> ?R"
+ by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG)
+ } moreover {
+ fix n
+ assume "n \<in> ?R"
+ then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
+ by (auto simp:subtree_def)
+ from rtranclD[OF this(2)]
+ have "n \<in> ?L"
+ proof
+ assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
+ with h have "n \<in> {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}" by auto
+ thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
+ qed (insert h, auto simp:subtree_def)
+ } ultimately show ?thesis by auto
+qed
+
+lemma threads_set_eq:
+ "the_thread ` (subtree (tRAG s) (Th th)) =
+ {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
+ by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
+
+lemma cp_alt_def1:
+ "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
+proof -
+ have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
+ ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
+ by auto
+ thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
+qed
+
+lemma cp_gen_def_cond:
+ assumes "x = Th th"
+ shows "cp s th = cp_gen s (Th th)"
+by (unfold cp_alt_def1 cp_gen_def, simp)
+
+lemma cp_gen_over_set:
+ assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
+ shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
+proof(rule f_image_eq)
+ fix a
+ assume "a \<in> A"
+ from assms[rule_format, OF this]
+ obtain th where eq_a: "a = Th th" by auto
+ show "cp_gen s a = (cp s \<circ> the_thread) a"
+ by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
+qed
+
+
+context valid_trace
+begin
+
+lemma RAG_threads:
+ assumes "(Th th) \<in> Field (RAG s)"
+ shows "th \<in> threads s"
+ using assms
+ by (metis Field_def UnE dm_RAG_threads range_in vt)
+
+lemma subtree_tRAG_thread:
+ assumes "th \<in> threads s"
+ shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
+proof -
+ have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+ by (unfold tRAG_subtree_eq, simp)
+ also have "... \<subseteq> ?R"
+ proof
+ fix x
+ assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+ then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
+ from this(2)
+ show "x \<in> ?R"
+ proof(cases rule:subtreeE)
+ case 1
+ thus ?thesis by (simp add: assms h(1))
+ next
+ case 2
+ thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI)
+ qed
+ qed
+ finally show ?thesis .
+qed
+
+lemma readys_root:
+ assumes "th \<in> readys s"
+ shows "root (RAG s) (Th th)"
+proof -
+ { fix x
+ assume "x \<in> ancestors (RAG s) (Th th)"
+ hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+ from tranclD[OF this]
+ obtain z where "(Th th, z) \<in> RAG s" by auto
+ with assms(1) have False
+ apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
+ by (fold wq_def, blast)
+ } thus ?thesis by (unfold root_def, auto)
+qed
+
+lemma readys_in_no_subtree:
+ assumes "th \<in> readys s"
+ and "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s) (Th th')"
+proof
+ assume "Th th \<in> subtree (RAG s) (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 1
+ with assms show ?thesis by auto
+ next
+ case 2
+ with readys_root[OF assms(1)]
+ show ?thesis by (auto simp:root_def)
+ qed
+qed
+
+lemma not_in_thread_isolated:
+ assumes "th \<notin> threads s"
+ shows "(Th th) \<notin> Field (RAG s)"
+proof
+ assume "(Th th) \<in> Field (RAG s)"
+ with dm_RAG_threads and range_in assms
+ show False by (unfold Field_def, blast)
+qed
+
+lemma wf_RAG: "wf (RAG s)"
+proof(rule finite_acyclic_wf)
+ from finite_RAG show "finite (RAG s)" .
+next
+ from acyclic_RAG show "acyclic (RAG s)" .
+qed
+
+lemma sgv_wRAG: "single_valued (wRAG s)"
+ using waiting_unique
+ by (unfold single_valued_def wRAG_def, auto)
+
+lemma sgv_hRAG: "single_valued (hRAG s)"
+ using holding_unique
+ by (unfold single_valued_def hRAG_def, auto)
+
+lemma sgv_tRAG: "single_valued (tRAG s)"
+ by (unfold tRAG_def, rule single_valued_relcomp,
+ insert sgv_wRAG sgv_hRAG, auto)
+
+lemma acyclic_tRAG: "acyclic (tRAG s)"
+proof(unfold tRAG_def, rule acyclic_compose)
+ show "acyclic (RAG s)" using acyclic_RAG .
+next
+ show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+next
+ show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+qed
+
+lemma sgv_RAG: "single_valued (RAG s)"
+ using unique_RAG by (auto simp:single_valued_def)
+
+lemma rtree_RAG: "rtree (RAG s)"
+ using sgv_RAG acyclic_RAG
+ by (unfold rtree_def rtree_axioms_def sgv_def, auto)
+
+end
+
+
+sublocale valid_trace < rtree_RAG: rtree "RAG s"
+proof
+ show "single_valued (RAG s)"
+ apply (intro_locales)
+ by (unfold single_valued_def,
+ auto intro:unique_RAG)
+
+ show "acyclic (RAG s)"
+ by (rule acyclic_RAG)
+qed
+
+sublocale valid_trace < rtree_s: rtree "tRAG s"
+proof(unfold_locales)
+ from sgv_tRAG show "single_valued (tRAG s)" .
+next
+ from acyclic_tRAG show "acyclic (tRAG s)" .
+qed
+
+sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
+proof -
+ show "fsubtree (RAG s)"
+ proof(intro_locales)
+ show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
+ next
+ show "fsubtree_axioms (RAG s)"
+ proof(unfold fsubtree_axioms_def)
+ find_theorems wf RAG
+ from wf_RAG show "wf (RAG s)" .
+ qed
+ qed
+qed
+
+sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
+proof -
+ have "fsubtree (tRAG s)"
+ proof -
+ have "fbranch (tRAG s)"
+ proof(unfold tRAG_def, rule fbranch_compose)
+ show "fbranch (wRAG s)"
+ proof(rule finite_fbranchI)
+ from finite_RAG show "finite (wRAG s)"
+ by (unfold RAG_split, auto)
+ qed
+ next
+ show "fbranch (hRAG s)"
+ proof(rule finite_fbranchI)
+ from finite_RAG
+ show "finite (hRAG s)" by (unfold RAG_split, auto)
+ qed
+ qed
+ moreover have "wf (tRAG s)"
+ proof(rule wf_subset)
+ show "wf (RAG s O RAG s)" using wf_RAG
+ by (fold wf_comp_self, simp)
+ next
+ show "tRAG s \<subseteq> (RAG s O RAG s)"
+ by (unfold tRAG_alt_def, auto)
+ qed
+ ultimately show ?thesis
+ by (unfold fsubtree_def fsubtree_axioms_def,auto)
+ qed
+ from this[folded tRAG_def] show "fsubtree (tRAG s)" .
+qed
+
+lemma Max_UNION:
+ assumes "finite A"
+ and "A \<noteq> {}"
+ and "\<forall> M \<in> f ` A. finite M"
+ and "\<forall> M \<in> f ` A. M \<noteq> {}"
+ shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
+ using assms[simp]
+proof -
+ have "?L = Max (\<Union>(f ` A))"
+ by (fold Union_image_eq, simp)
+ also have "... = ?R"
+ by (subst Max_Union, simp+)
+ finally show ?thesis .
+qed
+
+lemma max_Max_eq:
+ assumes "finite A"
+ and "A \<noteq> {}"
+ and "x = y"
+ shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
+proof -
+ have "?R = Max (insert y A)" by simp
+ also from assms have "... = ?L"
+ by (subst Max.insert, simp+)
+ finally show ?thesis by simp
+qed
+
+context valid_trace
+begin
+
+(* ddd *)
+lemma cp_gen_rec:
+ assumes "x = Th th"
+ shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
+proof(cases "children (tRAG s) x = {}")
+ case True
+ show ?thesis
+ by (unfold True cp_gen_def subtree_children, simp add:assms)
+next
+ case False
+ hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
+ note fsbttRAGs.finite_subtree[simp]
+ have [simp]: "finite (children (tRAG s) x)"
+ by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree],
+ rule children_subtree)
+ { fix r x
+ have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
+ } note this[simp]
+ have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
+ proof -
+ from False obtain q where "q \<in> children (tRAG s) x" by blast
+ moreover have "subtree (tRAG s) q \<noteq> {}" by simp
+ ultimately show ?thesis by blast
+ qed
+ have h: "Max ((the_preced s \<circ> the_thread) `
+ ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
+ Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
+ (is "?L = ?R")
+ proof -
+ let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
+ let "Max (_ \<union> (?h ` ?B))" = ?R
+ let ?L1 = "?f ` \<Union>(?g ` ?B)"
+ have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
+ proof -
+ have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
+ also have "... = (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
+ finally have "Max ?L1 = Max ..." by simp
+ also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
+ by (subst Max_UNION, simp+)
+ also have "... = Max (cp_gen s ` children (tRAG s) x)"
+ by (unfold image_comp cp_gen_alt_def, simp)
+ finally show ?thesis .
+ qed
+ show ?thesis
+ proof -
+ have "?L = Max (?f ` ?A \<union> ?L1)" by simp
+ also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
+ by (subst Max_Un, simp+)
+ also have "... = max (?f x) (Max (?h ` ?B))"
+ by (unfold eq_Max_L1, simp)
+ also have "... =?R"
+ by (rule max_Max_eq, (simp)+, unfold assms, simp)
+ finally show ?thesis .
+ qed
+ qed thus ?thesis
+ by (fold h subtree_children, unfold cp_gen_def, simp)
+qed
+
+lemma cp_rec:
+ "cp s th = Max ({the_preced s th} \<union>
+ (cp s o the_thread) ` children (tRAG s) (Th th))"
+proof -
+ have "Th th = Th th" by simp
+ note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this]
+ show ?thesis
+ proof -
+ have "cp_gen s ` children (tRAG s) (Th th) =
+ (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
+ proof(rule cp_gen_over_set)
+ show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
+ by (unfold tRAG_alt_def, auto simp:children_def)
+ qed
+ thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+ qed
+qed
+
+end
+
+(* keep *)
+lemma next_th_holding:
+ assumes vt: "vt s"
+ and nxt: "next_th s th cs th'"
+ shows "holding (wq s) th cs"
+proof -
+ from nxt[unfolded next_th_def]
+ obtain rest where h: "wq s cs = th # rest"
+ "rest \<noteq> []"
+ "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+ thus ?thesis
+ by (unfold cs_holding_def, auto)
+qed
+
+context valid_trace
+begin
+
+lemma next_th_waiting:
+ assumes nxt: "next_th s th cs th'"
+ shows "waiting (wq s) th' cs"
+proof -
+ from nxt[unfolded next_th_def]
+ obtain rest where h: "wq s cs = th # rest"
+ "rest \<noteq> []"
+ "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+ from wq_distinct[of cs, unfolded h]
+ have dst: "distinct (th # rest)" .
+ have in_rest: "th' \<in> set rest"
+ proof(unfold h, rule someI2)
+ show "distinct rest \<and> set rest = set rest" using dst by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ with h(2)
+ show "hd x \<in> set (rest)" by (cases x, auto)
+ qed
+ hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
+ moreover have "th' \<noteq> hd (wq s cs)"
+ by (unfold h(1), insert in_rest dst, auto)
+ ultimately show ?thesis by (auto simp:cs_waiting_def)
+qed
+
+lemma next_th_RAG:
+ assumes nxt: "next_th (s::event list) th cs th'"
+ shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
+ using vt assms next_th_holding next_th_waiting
+ by (unfold s_RAG_def, simp)
+
+end
+
+-- {* A useless definition *}
+definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+
+
+text {* (* ddd *)
+ One beauty of our modelling is that we follow the definitional extension tradition of HOL.
+ The benefit of such a concise and miniature model is that large number of intuitively
+ obvious facts are derived as lemmas, rather than asserted as axioms.
+*}
+
+text {*
+ However, the lemmas in the forthcoming several locales are no longer
+ obvious. These lemmas show how the current precedences should be recalculated
+ after every execution step (in our model, every step is represented by an event,
+ which in turn, represents a system call, or operation). Each operation is
+ treated in a separate locale.
+
+ The complication of current precedence recalculation comes
+ because the changing of RAG needs to be taken into account,
+ in addition to the changing of precedence.
+ The reason RAG changing affects current precedence is that,
+ according to the definition, current precedence
+ of a thread is the maximum of the precedences of its dependants,
+ where the dependants are defined in terms of RAG.
+
+ Therefore, each operation, lemmas concerning the change of the precedences
+ and RAG are derived first, so that the lemmas about
+ current precedence recalculation can be based on.
+*}
+
+text {* (* ddd *)
+ The following locale @{text "step_set_cps"} investigates the recalculation
+ after the @{text "Set"} operation.
+*}
+locale step_set_cps =
+ fixes s' th prio s
+ -- {* @{text "s'"} is the system state before the operation *}
+ -- {* @{text "s"} is the system state after the operation *}
+ defines s_def : "s \<equiv> (Set th prio#s')"
+ -- {* @{text "s"} is assumed to be a legitimate state, from which
+ the legitimacy of @{text "s"} can be derived. *}
+ assumes vt_s: "vt s"
+
+sublocale step_set_cps < vat_s : valid_trace "s"
+proof
+ from vt_s show "vt s" .
+qed
+
+sublocale step_set_cps < vat_s' : valid_trace "s'"
+proof
+ from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_set_cps
+begin
+
+text {* (* ddd *)
+ The following two lemmas confirm that @{text "Set"}-operating only changes the precedence
+ of the initiating thread.
+*}
+
+lemma eq_preced:
+ assumes "th' \<noteq> th"
+ shows "preced th' s = preced th' s'"
+proof -
+ from assms show ?thesis
+ by (unfold s_def, auto simp:preced_def)
+qed
+
+lemma eq_the_preced:
+ fixes th'
+ assumes "th' \<noteq> th"
+ shows "the_preced s th' = the_preced s' th'"
+ using assms
+ by (unfold the_preced_def, intro eq_preced, simp)
+
+text {*
+ The following lemma assures that the resetting of priority does not change the RAG.
+*}
+
+lemma eq_dep: "RAG s = RAG s'"
+ by (unfold s_def RAG_set_unchanged, auto)
+
+text {* (* ddd *)
+ Th following lemma @{text "eq_cp_pre"} says the priority change of @{text "th"}
+ only affects those threads, which as @{text "Th th"} in their sub-trees.
+
+ The proof of this lemma is simplified by using the alternative definition of @{text "cp"}.
+*}
+
+lemma eq_cp_pre:
+ fixes th'
+ assumes nd: "Th th \<notin> subtree (RAG s') (Th th')"
+ shows "cp s th' = cp s' th'"
+proof -
+ -- {* After unfolding using the alternative definition, elements
+ affecting the @{term "cp"}-value of threads become explicit.
+ We only need to prove the following: *}
+ have "Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
+ Max (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+ (is "Max (?f ` ?S1) = Max (?g ` ?S2)")
+ proof -
+ -- {* The base sets are equal. *}
+ have "?S1 = ?S2" using eq_dep by simp
+ -- {* The function values on the base set are equal as well. *}
+ moreover have "\<forall> e \<in> ?S2. ?f e = ?g e"
+ proof
+ fix th1
+ assume "th1 \<in> ?S2"
+ with nd have "th1 \<noteq> th" by (auto)
+ from eq_the_preced[OF this]
+ show "the_preced s th1 = the_preced s' th1" .
+ qed
+ -- {* Therefore, the image of the functions are equal. *}
+ ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq)
+ thus ?thesis by simp
+ qed
+ thus ?thesis by (simp add:cp_alt_def)
+qed
+
+text {*
+ The following lemma shows that @{term "th"} is not in the
+ sub-tree of any other thread.
+*}
+lemma th_in_no_subtree:
+ assumes "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s') (Th th')"
+proof -
+ have "th \<in> readys s'"
+ proof -
+ from step_back_step [OF vt_s[unfolded s_def]]
+ have "step s' (Set th prio)" .
+ hence "th \<in> runing s'" by (cases, simp)
+ thus ?thesis by (simp add:readys_def runing_def)
+ qed
+ find_theorems readys subtree
+ from vat_s'.readys_in_no_subtree[OF this assms(1)]
+ show ?thesis by blast
+qed
+
+text {*
+ By combining @{thm "eq_cp_pre"} and @{thm "th_in_no_subtree"},
+ it is obvious that the change of priority only affects the @{text "cp"}-value
+ of the initiating thread @{text "th"}.
+*}
+lemma eq_cp:
+ fixes th'
+ assumes "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+ by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])
+
+end
+
+text {*
+ The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.
+*}
+
+locale step_v_cps =
+ -- {* @{text "th"} is the initiating thread *}
+ -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *}
+ fixes s' th cs s -- {* @{text "s'"} is the state before operation*}
+ defines s_def : "s \<equiv> (V th cs#s')" -- {* @{text "s"} is the state after operation*}
+ -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *}
+ assumes vt_s: "vt s"
+
+sublocale step_v_cps < vat_s : valid_trace "s"
+proof
+ from vt_s show "vt s" .
+qed
+
+sublocale step_v_cps < vat_s' : valid_trace "s'"
+proof
+ from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_v_cps
+begin
+
+lemma ready_th_s': "th \<in> readys s'"
+ using step_back_step[OF vt_s[unfolded s_def]]
+ by (cases, simp add:runing_def)
+
+lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"
+proof -
+ from vat_s'.readys_root[OF ready_th_s']
+ show ?thesis
+ by (unfold root_def, simp)
+qed
+
+lemma holding_th: "holding s' th cs"
+proof -
+ from vt_s[unfolded s_def]
+ have " PIP s' (V th cs)" by (cases, simp)
+ thus ?thesis by (cases, auto)
+qed
+
+lemma edge_of_th:
+ "(Cs cs, Th th) \<in> RAG s'"
+proof -
+ from holding_th
+ show ?thesis
+ by (unfold s_RAG_def holding_eq, auto)
+qed
+
+lemma ancestors_cs:
+ "ancestors (RAG s') (Cs cs) = {Th th}"
+proof -
+ have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th) \<union> {Th th}"
+ proof(rule vat_s'.rtree_RAG.ancestors_accum)
+ from vt_s[unfolded s_def]
+ have " PIP s' (V th cs)" by (cases, simp)
+ thus "(Cs cs, Th th) \<in> RAG s'"
+ proof(cases)
+ assume "holding s' th cs"
+ from this[unfolded holding_eq]
+ show ?thesis by (unfold s_RAG_def, auto)
+ qed
+ qed
+ from this[unfolded ancestors_th] show ?thesis by simp
+qed
+
+lemma preced_kept: "the_preced s = the_preced s'"
+ by (auto simp: s_def the_preced_def preced_def)
+
+end
+
+text {*
+ The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation,
+ which represents the case when there is another thread @{text "th'"}
+ to take over the critical resource released by the initiating thread @{text "th"}.
+*}
+locale step_v_cps_nt = step_v_cps +
+ fixes th'
+ -- {* @{text "th'"} is assumed to take over @{text "cs"} *}
+ assumes nt: "next_th s' th cs th'"
+
+context step_v_cps_nt
+begin
+
+text {*
+ Lemma @{text "RAG_s"} confirms the change of RAG:
+ two edges removed and one added, as shown by the following diagram.
+*}
+
+(*
+ RAG before the V-operation
+ th1 ----|
+ |
+ th' ----|
+ |----> cs -----|
+ th2 ----| |
+ | |
+ th3 ----| |
+ |------> th
+ th4 ----| |
+ | |
+ th5 ----| |
+ |----> cs'-----|
+ th6 ----|
+ |
+ th7 ----|
+
+ RAG after the V-operation
+ th1 ----|
+ |
+ |----> cs ----> th'
+ th2 ----|
+ |
+ th3 ----|
+
+ th4 ----|
+ |
+ th5 ----|
+ |----> cs'----> th
+ th6 ----|
+ |
+ th7 ----|
+*)
+
+lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"
+ using next_th_RAG[OF nt] .
+
+lemma ancestors_th':
+ "ancestors (RAG s') (Th th') = {Th th, Cs cs}"
+proof -
+ have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"
+ proof(rule vat_s'.rtree_RAG.ancestors_accum)
+ from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto
+ qed
+ thus ?thesis using ancestors_th ancestors_cs by auto
+qed
+
+lemma RAG_s:
+ "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
+ {(Cs cs, Th th')}"
+proof -
+ from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
+ and nt show ?thesis by (auto intro:next_th_unique)
+qed
+
+lemma subtree_kept:
+ assumes "th1 \<notin> {th, th'}"
+ shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R")
+proof -
+ let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})"
+ let ?RAG'' = "?RAG' \<union> {(Cs cs, Th th')}"
+ have "subtree ?RAG' (Th th1) = ?R"
+ proof(rule subset_del_subtree_outside)
+ show "Range {(Cs cs, Th th), (Th th', Cs cs)} \<inter> subtree (RAG s') (Th th1) = {}"
+ proof -
+ have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+ proof(rule subtree_refute)
+ show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+ by (unfold ancestors_th, simp)
+ next
+ from assms show "Th th1 \<noteq> Th th" by simp
+ qed
+ moreover have "(Cs cs) \<notin> subtree (RAG s') (Th th1)"
+ proof(rule subtree_refute)
+ show "Th th1 \<notin> ancestors (RAG s') (Cs cs)"
+ by (unfold ancestors_cs, insert assms, auto)
+ qed simp
+ ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s') (Th th1) = {}" by auto
+ thus ?thesis by simp
+ qed
+ qed
+ moreover have "subtree ?RAG'' (Th th1) = subtree ?RAG' (Th th1)"
+ proof(rule subtree_insert_next)
+ show "Th th' \<notin> subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)"
+ proof(rule subtree_refute)
+ show "Th th1 \<notin> ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')"
+ (is "_ \<notin> ?R")
+ proof -
+ have "?R \<subseteq> ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto)
+ moreover have "Th th1 \<notin> ..." using ancestors_th' assms by simp
+ ultimately show ?thesis by auto
+ qed
+ next
+ from assms show "Th th1 \<noteq> Th th'" by simp
+ qed
+ qed
+ ultimately show ?thesis by (unfold RAG_s, simp)
+qed
+
+lemma cp_kept:
+ assumes "th1 \<notin> {th, th'}"
+ shows "cp s th1 = cp s' th1"
+ by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+
+end
+
+locale step_v_cps_nnt = step_v_cps +
+ assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
+
+context step_v_cps_nnt
+begin
+
+lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"
+proof -
+ from nnt and step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
+ show ?thesis by auto
+qed
+
+lemma subtree_kept:
+ assumes "th1 \<noteq> th"
+ shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)"
+proof(unfold RAG_s, rule subset_del_subtree_outside)
+ show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s') (Th th1) = {}"
+ proof -
+ have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+ proof(rule subtree_refute)
+ show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+ by (unfold ancestors_th, simp)
+ next
+ from assms show "Th th1 \<noteq> Th th" by simp
+ qed
+ thus ?thesis by auto
+ qed
+qed
+
+lemma cp_kept_1:
+ assumes "th1 \<noteq> th"
+ shows "cp s th1 = cp s' th1"
+ by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+
+lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}"
+proof -
+ { fix n
+ have "(Cs cs) \<notin> ancestors (RAG s') n"
+ proof
+ assume "Cs cs \<in> ancestors (RAG s') n"
+ hence "(n, Cs cs) \<in> (RAG s')^+" by (auto simp:ancestors_def)
+ from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s'" by auto
+ then obtain th' where "nn = Th th'"
+ by (unfold s_RAG_def, auto)
+ from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s'" .
+ from this[unfolded s_RAG_def]
+ have "waiting (wq s') th' cs" by auto
+ from this[unfolded cs_waiting_def]
+ have "1 < length (wq s' cs)"
+ by (cases "wq s' cs", auto)
+ from holding_next_thI[OF holding_th this]
+ obtain th' where "next_th s' th cs th'" by auto
+ with nnt show False by auto
+ qed
+ } note h = this
+ { fix n
+ assume "n \<in> subtree (RAG s') (Cs cs)"
+ hence "n = (Cs cs)"
+ by (elim subtreeE, insert h, auto)
+ } moreover have "(Cs cs) \<in> subtree (RAG s') (Cs cs)"
+ by (auto simp:subtree_def)
+ ultimately show ?thesis by auto
+qed
+
+lemma subtree_th:
+ "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
+find_theorems "subtree" "_ - _" RAG
+proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
+ from edge_of_th
+ show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"
+ by (unfold edges_in_def, auto simp:subtree_def)
+qed
+
+lemma cp_kept_2:
+ shows "cp s th = cp s' th"
+ by (unfold cp_alt_def subtree_th preced_kept, auto)
+
+lemma eq_cp:
+ fixes th'
+ shows "cp s th' = cp s' th'"
+ using cp_kept_1 cp_kept_2
+ by (cases "th' = th", auto)
+end
+
+
+locale step_P_cps =
+ fixes s' th cs s
+ defines s_def : "s \<equiv> (P th cs#s')"
+ assumes vt_s: "vt s"
+
+sublocale step_P_cps < vat_s : valid_trace "s"
+proof
+ from vt_s show "vt s" .
+qed
+
+sublocale step_P_cps < vat_s' : valid_trace "s'"
+proof
+ from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_P_cps
+begin
+
+lemma readys_th: "th \<in> readys s'"
+proof -
+ from step_back_step [OF vt_s[unfolded s_def]]
+ have "PIP s' (P th cs)" .
+ hence "th \<in> runing s'" by (cases, simp)
+ thus ?thesis by (simp add:readys_def runing_def)
+qed
+
+lemma root_th: "root (RAG s') (Th th)"
+ using readys_root[OF readys_th] .
+
+lemma in_no_others_subtree:
+ assumes "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s') (Th th')"
+proof
+ assume "Th th \<in> subtree (RAG s') (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 1
+ with assms show ?thesis by auto
+ next
+ case 2
+ with root_th show ?thesis by (auto simp:root_def)
+ qed
+qed
+
+lemma preced_kept: "the_preced s = the_preced s'"
+ by (auto simp: s_def the_preced_def preced_def)
+
+end
+
+locale step_P_cps_ne =step_P_cps +
+ fixes th'
+ assumes ne: "wq s' cs \<noteq> []"
+ defines th'_def: "th' \<equiv> hd (wq s' cs)"
+
+locale step_P_cps_e =step_P_cps +
+ assumes ee: "wq s' cs = []"
+
+context step_P_cps_e
+begin
+
+lemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"
+proof -
+ from ee and step_RAG_p[OF vt_s[unfolded s_def], folded s_def]
+ show ?thesis by auto
+qed
+
+lemma subtree_kept:
+ assumes "th' \<noteq> th"
+ shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')"
+proof(unfold RAG_s, rule subtree_insert_next)
+ from in_no_others_subtree[OF assms]
+ show "Th th \<notin> subtree (RAG s') (Th th')" .
+qed
+
+lemma cp_kept:
+ assumes "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+proof -
+ have "(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
+ (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+ by (unfold preced_kept subtree_kept[OF assms], simp)
+ thus ?thesis by (unfold cp_alt_def, simp)
+qed
+
+end
+
+context step_P_cps_ne
+begin
+
+lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+proof -
+ from step_RAG_p[OF vt_s[unfolded s_def]] and ne
+ show ?thesis by (simp add:s_def)
+qed
+
+lemma cs_held: "(Cs cs, Th th') \<in> RAG s'"
+proof -
+ have "(Cs cs, Th th') \<in> hRAG s'"
+ proof -
+ from ne
+ have " holding s' th' cs"
+ by (unfold th'_def holding_eq cs_holding_def, auto)
+ thus ?thesis
+ by (unfold hRAG_def, auto)
+ qed
+ thus ?thesis by (unfold RAG_split, auto)
+qed
+
+lemma tRAG_s:
+ "tRAG s = tRAG s' \<union> {(Th th, Th th')}"
+ using RAG_tRAG_transfer[OF RAG_s cs_held] .
+
+lemma cp_kept:
+ assumes "Th th'' \<notin> ancestors (tRAG s) (Th th)"
+ shows "cp s th'' = cp s' th''"
+proof -
+ have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')"
+ proof -
+ have "Th th' \<notin> subtree (tRAG s') (Th th'')"
+ proof
+ assume "Th th' \<in> subtree (tRAG s') (Th th'')"
+ thus False
+ proof(rule subtreeE)
+ assume "Th th' = Th th''"
+ from assms[unfolded tRAG_s ancestors_def, folded this]
+ show ?thesis by auto
+ next
+ assume "Th th'' \<in> ancestors (tRAG s') (Th th')"
+ moreover have "... \<subseteq> ancestors (tRAG s) (Th th')"
+ proof(rule ancestors_mono)
+ show "tRAG s' \<subseteq> tRAG s" by (unfold tRAG_s, auto)
+ qed
+ ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th')" by auto
+ moreover have "Th th' \<in> ancestors (tRAG s) (Th th)"
+ by (unfold tRAG_s, auto simp:ancestors_def)
+ ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th)"
+ by (auto simp:ancestors_def)
+ with assms show ?thesis by auto
+ qed
+ qed
+ from subtree_insert_next[OF this]
+ have "subtree (tRAG s' \<union> {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" .
+ from this[folded tRAG_s] show ?thesis .
+ qed
+ show ?thesis by (unfold cp_alt_def1 h preced_kept, simp)
+qed
+
+lemma cp_gen_update_stop: (* ddd *)
+ assumes "u \<in> ancestors (tRAG s) (Th th)"
+ and "cp_gen s u = cp_gen s' u"
+ and "y \<in> ancestors (tRAG s) u"
+ shows "cp_gen s y = cp_gen s' y"
+ using assms(3)
+proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf])
+ case (1 x)
+ show ?case (is "?L = ?R")
+ proof -
+ from tRAG_ancestorsE[OF 1(2)]
+ obtain th2 where eq_x: "x = Th th2" by blast
+ from vat_s.cp_gen_rec[OF this]
+ have "?L =
+ Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)" .
+ also have "... =
+ Max ({the_preced s' th2} \<union> cp_gen s' ` RTree.children (tRAG s') x)"
+
+ proof -
+ from preced_kept have "the_preced s th2 = the_preced s' th2" by simp
+ moreover have "cp_gen s ` RTree.children (tRAG s) x =
+ cp_gen s' ` RTree.children (tRAG s') x"
+ proof -
+ have "RTree.children (tRAG s) x = RTree.children (tRAG s') x"
+ proof(unfold tRAG_s, rule children_union_kept)
+ have start: "(Th th, Th th') \<in> tRAG s"
+ by (unfold tRAG_s, auto)
+ note x_u = 1(2)
+ show "x \<notin> Range {(Th th, Th th')}"
+ proof
+ assume "x \<in> Range {(Th th, Th th')}"
+ hence eq_x: "x = Th th'" using RangeE by auto
+ show False
+ proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start])
+ case 1
+ from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG
+ show ?thesis by (auto simp:ancestors_def acyclic_def)
+ next
+ case 2
+ with x_u[unfolded eq_x]
+ have "(Th th', Th th') \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+ with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
+ qed
+ qed
+ qed
+ moreover have "cp_gen s ` RTree.children (tRAG s) x =
+ cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A")
+ proof(rule f_image_eq)
+ fix a
+ assume a_in: "a \<in> ?A"
+ from 1(2)
+ show "?f a = ?g a"
+ proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
+ case in_ch
+ show ?thesis
+ proof(cases "a = u")
+ case True
+ from assms(2)[folded this] show ?thesis .
+ next
+ case False
+ have a_not_in: "a \<notin> ancestors (tRAG s) (Th th)"
+ proof
+ assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+ have "a = u"
+ proof(rule vat_s.rtree_s.ancestors_children_unique)
+ from a_in' a_in show "a \<in> ancestors (tRAG s) (Th th) \<inter>
+ RTree.children (tRAG s) x" by auto
+ next
+ from assms(1) in_ch show "u \<in> ancestors (tRAG s) (Th th) \<inter>
+ RTree.children (tRAG s) x" by auto
+ qed
+ with False show False by simp
+ qed
+ from a_in obtain th_a where eq_a: "a = Th th_a"
+ by (unfold RTree.children_def tRAG_alt_def, auto)
+ from cp_kept[OF a_not_in[unfolded eq_a]]
+ have "cp s th_a = cp s' th_a" .
+ from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
+ show ?thesis .
+ qed
+ next
+ case (out_ch z)
+ hence h: "z \<in> ancestors (tRAG s) u" "z \<in> RTree.children (tRAG s) x" by auto
+ show ?thesis
+ proof(cases "a = z")
+ case True
+ from h(2) have zx_in: "(z, x) \<in> (tRAG s)" by (auto simp:RTree.children_def)
+ from 1(1)[rule_format, OF this h(1)]
+ have eq_cp_gen: "cp_gen s z = cp_gen s' z" .
+ with True show ?thesis by metis
+ next
+ case False
+ from a_in obtain th_a where eq_a: "a = Th th_a"
+ by (auto simp:RTree.children_def tRAG_alt_def)
+ have "a \<notin> ancestors (tRAG s) (Th th)"
+ proof
+ assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+ have "a = z"
+ proof(rule vat_s.rtree_s.ancestors_children_unique)
+ from assms(1) h(1) have "z \<in> ancestors (tRAG s) (Th th)"
+ by (auto simp:ancestors_def)
+ with h(2) show " z \<in> ancestors (tRAG s) (Th th) \<inter>
+ RTree.children (tRAG s) x" by auto
+ next
+ from a_in a_in'
+ show "a \<in> ancestors (tRAG s) (Th th) \<inter> RTree.children (tRAG s) x"
+ by auto
+ qed
+ with False show False by auto
+ qed
+ from cp_kept[OF this[unfolded eq_a]]
+ have "cp s th_a = cp s' th_a" .
+ from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
+ show ?thesis .
+ qed
+ qed
+ qed
+ ultimately show ?thesis by metis
+ qed
+ ultimately show ?thesis by simp
+ qed
+ also have "... = ?R"
+ by (fold vat_s'.cp_gen_rec[OF eq_x], simp)
+ finally show ?thesis .
+ qed
+qed
+
+lemma cp_up:
+ assumes "(Th th') \<in> ancestors (tRAG s) (Th th)"
+ and "cp s th' = cp s' th'"
+ and "(Th th'') \<in> ancestors (tRAG s) (Th th')"
+ shows "cp s th'' = cp s' th''"
+proof -
+ have "cp_gen s (Th th'') = cp_gen s' (Th th'')"
+ proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)])
+ from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]]
+ show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis
+ qed
+ with cp_gen_def_cond[OF refl[of "Th th''"]]
+ show ?thesis by metis
+qed
+
+end
+
+locale step_create_cps =
+ fixes s' th prio s
+ defines s_def : "s \<equiv> (Create th prio#s')"
+ assumes vt_s: "vt s"
+
+sublocale step_create_cps < vat_s: valid_trace "s"
+ by (unfold_locales, insert vt_s, simp)
+
+sublocale step_create_cps < vat_s': valid_trace "s'"
+ by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
+
+context step_create_cps
+begin
+
+lemma RAG_kept: "RAG s = RAG s'"
+ by (unfold s_def RAG_create_unchanged, auto)
+
+lemma tRAG_kept: "tRAG s = tRAG s'"
+ by (unfold tRAG_alt_def RAG_kept, auto)
+
+lemma preced_kept:
+ assumes "th' \<noteq> th"
+ shows "the_preced s th' = the_preced s' th'"
+ by (unfold s_def the_preced_def preced_def, insert assms, auto)
+
+lemma th_not_in: "Th th \<notin> Field (tRAG s')"
+proof -
+ from vt_s[unfolded s_def]
+ have "PIP s' (Create th prio)" by (cases, simp)
+ hence "th \<notin> threads s'" by(cases, simp)
+ from vat_s'.not_in_thread_isolated[OF this]
+ have "Th th \<notin> Field (RAG s')" .
+ with tRAG_Field show ?thesis by auto
+qed
+
+lemma eq_cp:
+ assumes neq_th: "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+proof -
+ have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
+ (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+ proof(unfold tRAG_kept, rule f_image_eq)
+ fix a
+ assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+ then obtain th_a where eq_a: "a = Th th_a"
+ proof(cases rule:subtreeE)
+ case 2
+ from ancestors_Field[OF 2(2)]
+ and that show ?thesis by (unfold tRAG_alt_def, auto)
+ qed auto
+ have neq_th_a: "th_a \<noteq> th"
+ proof -
+ have "(Th th) \<notin> subtree (tRAG s') (Th th')"
+ proof
+ assume "Th th \<in> subtree (tRAG s') (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 2
+ from ancestors_Field[OF this(2)]
+ and th_not_in[unfolded Field_def]
+ show ?thesis by auto
+ qed (insert assms, auto)
+ qed
+ with a_in[unfolded eq_a] show ?thesis by auto
+ qed
+ from preced_kept[OF this]
+ show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+ by (unfold eq_a, simp)
+ qed
+ thus ?thesis by (unfold cp_alt_def1, simp)
+qed
+
+lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}"
+proof -
+ { fix a
+ assume "a \<in> RTree.children (tRAG s) (Th th)"
+ hence "(a, Th th) \<in> tRAG s" by (auto simp:RTree.children_def)
+ with th_not_in have False
+ by (unfold Field_def tRAG_kept, auto)
+ } thus ?thesis by auto
+qed
+
+lemma eq_cp_th: "cp s th = preced th s"
+ by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def)
+
+end
+
+locale step_exit_cps =
+ fixes s' th prio s
+ defines s_def : "s \<equiv> Exit th # s'"
+ assumes vt_s: "vt s"
+
+sublocale step_exit_cps < vat_s: valid_trace "s"
+ by (unfold_locales, insert vt_s, simp)
+
+sublocale step_exit_cps < vat_s': valid_trace "s'"
+ by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
+
+context step_exit_cps
+begin
+
+lemma preced_kept:
+ assumes "th' \<noteq> th"
+ shows "the_preced s th' = the_preced s' th'"
+ by (unfold s_def the_preced_def preced_def, insert assms, auto)
+
+lemma RAG_kept: "RAG s = RAG s'"
+ by (unfold s_def RAG_exit_unchanged, auto)
+
+lemma tRAG_kept: "tRAG s = tRAG s'"
+ by (unfold tRAG_alt_def RAG_kept, auto)
+
+lemma th_ready: "th \<in> readys s'"
+proof -
+ from vt_s[unfolded s_def]
+ have "PIP s' (Exit th)" by (cases, simp)
+ hence h: "th \<in> runing s' \<and> holdents s' th = {}" by (cases, metis)
+ thus ?thesis by (unfold runing_def, auto)
+qed
+
+lemma th_holdents: "holdents s' th = {}"
+proof -
+ from vt_s[unfolded s_def]
+ have "PIP s' (Exit th)" by (cases, simp)
+ thus ?thesis by (cases, metis)
+qed
+
+lemma th_RAG: "Th th \<notin> Field (RAG s')"
+proof -
+ have "Th th \<notin> Range (RAG s')"
+ proof
+ assume "Th th \<in> Range (RAG s')"
+ then obtain cs where "holding (wq s') th cs"
+ by (unfold Range_iff s_RAG_def, auto)
+ with th_holdents[unfolded holdents_def]
+ show False by (unfold eq_holding, auto)
+ qed
+ moreover have "Th th \<notin> Domain (RAG s')"
+ proof
+ assume "Th th \<in> Domain (RAG s')"
+ then obtain cs where "waiting (wq s') th cs"
+ by (unfold Domain_iff s_RAG_def, auto)
+ with th_ready show False by (unfold readys_def eq_waiting, auto)
+ qed
+ ultimately show ?thesis by (auto simp:Field_def)
+qed
+
+lemma th_tRAG: "(Th th) \<notin> Field (tRAG s')"
+ using th_RAG tRAG_Field[of s'] by auto
+
+lemma eq_cp:
+ assumes neq_th: "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+proof -
+ have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
+ (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+ proof(unfold tRAG_kept, rule f_image_eq)
+ fix a
+ assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+ then obtain th_a where eq_a: "a = Th th_a"
+ proof(cases rule:subtreeE)
+ case 2
+ from ancestors_Field[OF 2(2)]
+ and that show ?thesis by (unfold tRAG_alt_def, auto)
+ qed auto
+ have neq_th_a: "th_a \<noteq> th"
+ proof -
+ find_theorems readys subtree s'
+ from vat_s'.readys_in_no_subtree[OF th_ready assms]
+ have "(Th th) \<notin> subtree (RAG s') (Th th')" .
+ with tRAG_subtree_RAG[of s' "Th th'"]
+ have "(Th th) \<notin> subtree (tRAG s') (Th th')" by auto
+ with a_in[unfolded eq_a] show ?thesis by auto
+ qed
+ from preced_kept[OF this]
+ show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+ by (unfold eq_a, simp)
+ qed
+ thus ?thesis by (unfold cp_alt_def1, simp)
+qed
+
+end
+
+end
+
--- a/Journal/Paper.thy Wed Jan 06 20:46:14 2016 +0800
+++ b/Journal/Paper.thy Wed Jan 06 16:34:26 2016 +0000
@@ -1,6 +1,8 @@
(*<*)
theory Paper
-imports "../CpsG" "../ExtGG" "~~/src/HOL/Library/LaTeXsugar"
+imports "../Implementation"
+ "../Correctness"
+ "~~/src/HOL/Library/LaTeXsugar"
begin
@@ -19,7 +21,7 @@
Cs ("C") and
readys ("ready") and
preced ("prec") and
- preceds ("precs") and
+(* preceds ("precs") and*)
cpreced ("cprec") and
cp ("cprec") and
holdents ("resources") and
@@ -361,7 +363,7 @@
We also use the abbreviation
\begin{isabelle}\ \ \ \ \ %%%
- @{abbrev "preceds s ths"}
+ ???preceds s ths
\end{isabelle}
\noindent
@@ -898,6 +900,53 @@
However, we are able to combine their two separate bounds into a
single theorem improving their bound.
+ \subsection*{\bf OUTLINE}
+
+ Since @{term "th"} is the most urgent thread, if it is somehow
+ blocked, people want to know why and wether this blocking is
+ reasonable.
+
+ @{thm [source] th_blockedE} @{thm th_blockedE}
+
+ if @{term "th"} is blocked, then there is a path leading from
+ @{term "th"} to @{term "th'"}, which means:
+ there is a chain of demand leading from @{term th} to @{term th'}.
+
+ %%% in other words
+ %%% th -> cs1 -> th1 -> cs2 -> th2 -> ... -> csn -> thn -> cs -> th'.
+ %%%
+ %%% We says that th is blocked by "th'".
+
+ THEN
+
+ @{thm [source] vat_t.th_chain_to_ready} @{thm vat_t.th_chain_to_ready}
+
+ It is basic propery with non-trival proof.
+
+ THEN
+
+ @{thm [source] max_preced} @{thm max_preced}
+
+ which says @{term "th"} holds the max precedence.
+
+ THEN
+
+ @{thm [source] th_cp_max th_cp_preced th_kept}
+ @{thm th_cp_max th_cp_preced th_kept}
+
+ THENTHEN
+
+ @{thm [source] runing_inversion_4} @{thm runing_inversion_4}
+
+ which explains what the @{term "th'"} looks like. Now, we have found the
+ @{term "th'"} which blocks @{term th}, we need to know more about it.
+ To see what kind of thread can block @{term th}.
+
+ From these two lemmas we can see the correctness of PIP, which is
+ that: the blockage of th is reasonable and under control.
+
+ \subsection*{END OUTLINE}
+
In what follows we will describe properties of PIP that allow us to prove
Theorem~\ref{mainthm} and, when instructive, briefly describe our argument.
It is relatively easy to see that:
@@ -915,9 +964,10 @@
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
- @{thm[mode=IfThen] waiting_unique[of _ _ "cs\<^sub>1" "cs\<^sub>2"]}\\
- @{thm[mode=IfThen] held_unique[of _ "th\<^sub>1" _ "th\<^sub>2"]}\\
- @{thm[mode=IfThen] runing_unique[of _ "th\<^sub>1" "th\<^sub>2"]}
+ HERE??
+ %@ {thm[mode=IfThen] waiting_unique[of _ _ "cs\<^sub>1" "cs\<^sub>2"]}\\
+ %@ {thm[mode=IfThen] held_unique[of _ "th\<^sub>1" _ "th\<^sub>2"]}\\
+ %@ {thm[mode=IfThen] runing_unique[of _ "th\<^sub>1" "th\<^sub>2"]}
\end{tabular}
\end{isabelle}
@@ -931,13 +981,13 @@
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
- @{text If}~@{thm (prem 1) acyclic_RAG}~@{text "then"}:\\
+ HERE?? %@{text If}~@ {thm (prem 1) acyclic_RAG}~@{text "then"}:\\
\hspace{5mm}@{thm (concl) acyclic_RAG},
@{thm (concl) finite_RAG} and
@{thm (concl) wf_dep_converse},\\
- \hspace{5mm}@{text "if"}~@{thm (prem 2) dm_RAG_threads}~@{text "then"}~@{thm (concl) dm_RAG_threads}
+ %\hspace{5mm}@{text "if"}~@ {thm (prem 2) dm_RAG_threads}~@{text "then"}~@{thm (concl) dm_RAG_threads}
and\\
- \hspace{5mm}@{text "if"}~@{thm (prem 2) range_in}~@{text "then"}~@{thm (concl) range_in}.
+ %\hspace{5mm}@{text "if"}~@ {thm (prem 2) range_in}~@{text "then"}~@{thm (concl) range_in}.
\end{tabular}
\end{isabelle}
@@ -1136,7 +1186,7 @@
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
- @{thm children_def2}
+ HERE?? %%@ {thm children_def2}
\end{tabular}
\end{isabelle}
@@ -1146,9 +1196,9 @@
a resource). We can prove the following lemma.
\begin{lemma}\label{childrenlem}
- @{text "If"} @{thm (prem 1) cp_rec} @{text "then"}
+ HERE %@{text "If"} @ {thm (prem 1) cp_rec} @{text "then"}
\begin{center}
- @{thm (concl) cp_rec}.
+ %@ {thm (concl) cp_rec}.
\end{center}
\end{lemma}
@@ -1184,7 +1234,7 @@
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
- @{thm eq_dep},\\
+ HERE ?? %@ {thm eq_dep},\\
@{thm eq_cp_th}, and\\
@{thm[mode=IfThen] eq_cp}
\end{tabular}
@@ -1208,7 +1258,7 @@
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
- @{thm eq_dep}, and\\
+ HERE %@ {thm eq_dep}, and\\
@{thm[mode=IfThen] eq_cp}
\end{tabular}
\end{isabelle}
@@ -1257,9 +1307,9 @@
%
%\begin{isabelle}\ \ \ \ \ %%%
%\begin{tabular}{@ {}l}
- %@{thm[mode=IfThen] eq_up_self}\\
- %@{text "If"} @{thm (prem 1) eq_up}, @{thm (prem 2) eq_up} and @{thm (prem 3) eq_up}\\
- %@{text "then"} @{thm (concl) eq_up}.
+ %@ {thm[mode=IfThen] eq_up_self}\\
+ %@{text "If"} @ {thm (prem 1) eq_up}, @ {thm (prem 2) eq_up} and @ {thm (prem 3) eq_up}\\
+ %@{text "then"} @ {thm (concl) eq_up}.
%\end{tabular}
%\end{isabelle}
%
@@ -1330,7 +1380,7 @@
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm RAG_s}\\
- @{thm eq_cp}
+ HERE %@ {thm eq_cp}
\end{tabular}
\end{isabelle}
@@ -1343,7 +1393,7 @@
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm RAG_s}\\
- @{thm[mode=IfThen] eq_cp}
+ HERE %@ {thm[mode=IfThen] eq_cp}
\end{tabular}
\end{isabelle}
@@ -1362,8 +1412,9 @@
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
%%@ {t hm[mode=IfThen] eq_up_self}\\
- @{text "If"} @{thm (prem 1) eq_up}, @{thm (prem 2) eq_up} and @{thm (prem 3) eq_up}\\
- @{text "then"} @{thm (concl) eq_up}.
+ HERE
+ %@{text "If"} @ {thm (prem 1) eq_up}, @ {thm (prem 2) eq_up} and @ {thm (prem 3) eq_up}\\
+ %@{text "then"} @ {thm (concl) eq_up}.
\end{tabular}
\end{isabelle}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/PIPBasics.thy Wed Jan 06 16:34:26 2016 +0000
@@ -0,0 +1,3051 @@
+theory PIPBasics
+imports PIPDefs
+begin
+
+locale valid_trace =
+ fixes s
+ assumes vt : "vt s"
+
+locale valid_trace_e = valid_trace +
+ fixes e
+ assumes vt_e: "vt (e#s)"
+begin
+
+lemma pip_e: "PIP s e"
+ using vt_e by (cases, simp)
+
+end
+
+lemma runing_ready:
+ shows "runing s \<subseteq> readys s"
+ unfolding runing_def readys_def
+ by auto
+
+lemma readys_threads:
+ shows "readys s \<subseteq> threads s"
+ unfolding readys_def
+ by auto
+
+lemma wq_v_neq:
+ "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
+ by (auto simp:wq_def Let_def cp_def split:list.splits)
+
+context valid_trace
+begin
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes "PP []"
+ and "(\<And>s e. valid_trace s \<Longrightarrow> valid_trace (e#s) \<Longrightarrow>
+ PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
+ shows "PP s"
+proof(rule vt.induct[OF vt])
+ from assms(1) show "PP []" .
+next
+ fix s e
+ assume h: "vt s" "PP s" "PIP s e"
+ show "PP (e # s)"
+ proof(cases rule:assms(2))
+ from h(1) show v1: "valid_trace s" by (unfold_locales, simp)
+ next
+ from h(1,3) have "vt (e#s)" by auto
+ thus "valid_trace (e # s)" by (unfold_locales, simp)
+ qed (insert h, auto)
+qed
+
+lemma wq_distinct: "distinct (wq s cs)"
+proof(rule ind, simp add:wq_def)
+ fix s e
+ assume h1: "step s e"
+ and h2: "distinct (wq s cs)"
+ thus "distinct (wq (e # s) cs)"
+ proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
+ fix thread s
+ assume h1: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
+ and h2: "thread \<in> set (wq_fun (schs s) cs)"
+ and h3: "thread \<in> runing s"
+ show "False"
+ proof -
+ from h3 have "\<And> cs. thread \<in> set (wq_fun (schs s) cs) \<Longrightarrow>
+ thread = hd ((wq_fun (schs s) cs))"
+ by (simp add:runing_def readys_def s_waiting_def wq_def)
+ from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" .
+ with h2
+ have "(Cs cs, Th thread) \<in> (RAG s)"
+ by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
+ with h1 show False by auto
+ qed
+ next
+ fix thread s a list
+ assume dst: "distinct list"
+ show "distinct (SOME q. distinct q \<and> set q = set list)"
+ proof(rule someI2)
+ from dst show "distinct list \<and> set list = set list" by auto
+ next
+ fix q assume "distinct q \<and> set q = set list"
+ thus "distinct q" by auto
+ qed
+ qed
+qed
+
+end
+
+
+context valid_trace_e
+begin
+
+text {*
+ The following lemma shows that only the @{text "P"}
+ operation can add new thread into waiting queues.
+ Such kind of lemmas are very obvious, but need to be checked formally.
+ This is a kind of confirmation that our modelling is correct.
+*}
+
+lemma block_pre:
+ assumes s_ni: "thread \<notin> set (wq s cs)"
+ and s_i: "thread \<in> set (wq (e#s) cs)"
+ shows "e = P thread cs"
+proof -
+ show ?thesis
+ proof(cases e)
+ case (P th cs)
+ with assms
+ show ?thesis
+ by (auto simp:wq_def Let_def split:if_splits)
+ next
+ case (Create th prio)
+ with assms show ?thesis
+ by (auto simp:wq_def Let_def split:if_splits)
+ next
+ case (Exit th)
+ with assms show ?thesis
+ by (auto simp:wq_def Let_def split:if_splits)
+ next
+ case (Set th prio)
+ with assms show ?thesis
+ by (auto simp:wq_def Let_def split:if_splits)
+ next
+ case (V th cs)
+ with vt_e assms show ?thesis
+ apply (auto simp:wq_def Let_def split:if_splits)
+ proof -
+ fix q qs
+ assume h1: "thread \<notin> set (wq_fun (schs s) cs)"
+ and h2: "q # qs = wq_fun (schs s) cs"
+ and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+ and vt: "vt (V th cs # s)"
+ from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp
+ moreover have "thread \<in> set qs"
+ proof -
+ have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
+ proof(rule someI2)
+ from wq_distinct [of cs]
+ and h2[symmetric, folded wq_def]
+ show "distinct qs \<and> set qs = set qs" by auto
+ next
+ fix x assume "distinct x \<and> set x = set qs"
+ thus "set x = set qs" by auto
+ qed
+ with h3 show ?thesis by simp
+ qed
+ ultimately show "False" by auto
+ qed
+ qed
+qed
+
+end
+
+text {*
+ The following lemmas is also obvious and shallow. It says
+ that only running thread can request for a critical resource
+ and that the requested resource must be one which is
+ not current held by the thread.
+*}
+
+lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
+ thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (RAG s)^+"
+apply (ind_cases "vt ((P thread cs)#s)")
+apply (ind_cases "step s (P thread cs)")
+by auto
+
+lemma abs1:
+ assumes ein: "e \<in> set es"
+ and neq: "hd es \<noteq> hd (es @ [x])"
+ shows "False"
+proof -
+ from ein have "es \<noteq> []" by auto
+ then obtain e ess where "es = e # ess" by (cases es, auto)
+ with neq show ?thesis by auto
+qed
+
+lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
+ by (cases es, auto)
+
+inductive_cases evt_cons: "vt (a#s)"
+
+context valid_trace_e
+begin
+
+lemma abs2:
+ assumes inq: "thread \<in> set (wq s cs)"
+ and nh: "thread = hd (wq s cs)"
+ and qt: "thread \<noteq> hd (wq (e#s) cs)"
+ and inq': "thread \<in> set (wq (e#s) cs)"
+ shows "False"
+proof -
+ from vt_e assms show "False"
+ apply (cases e)
+ apply ((simp split:if_splits add:Let_def wq_def)[1])+
+ apply (insert abs1, fast)[1]
+ apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)
+ proof -
+ fix th qs
+ assume vt: "vt (V th cs # s)"
+ and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+ and eq_wq: "wq_fun (schs s) cs = thread # qs"
+ show "False"
+ proof -
+ from wq_distinct[of cs]
+ and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
+ moreover have "thread \<in> set qs"
+ proof -
+ have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
+ proof(rule someI2)
+ from wq_distinct [of cs]
+ and eq_wq [folded wq_def]
+ show "distinct qs \<and> set qs = set qs" by auto
+ next
+ fix x assume "distinct x \<and> set x = set qs"
+ thus "set x = set qs" by auto
+ qed
+ with th_in show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ qed
+qed
+
+end
+
+context valid_trace
+begin
+
+lemma vt_moment: "\<And> t. vt (moment t s)"
+proof(induct rule:ind)
+ case Nil
+ thus ?case by (simp add:vt_nil)
+next
+ case (Cons s e t)
+ show ?case
+ proof(cases "t \<ge> length (e#s)")
+ case True
+ from True have "moment t (e#s) = e#s" by simp
+ thus ?thesis using Cons
+ by (simp add:valid_trace_def)
+ next
+ case False
+ from Cons have "vt (moment t s)" by simp
+ moreover have "moment t (e#s) = moment t s"
+ proof -
+ from False have "t \<le> length s" by simp
+ from moment_app [OF this, of "[e]"]
+ show ?thesis by simp
+ qed
+ ultimately show ?thesis by simp
+ qed
+qed
+
+(* Wrong:
+ lemma \<lbrakk>thread \<in> set (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
+*)
+
+text {* (* ddd *)
+ The nature of the work is like this: since it starts from a very simple and basic
+ model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
+ For instance, the fact
+ that one thread can not be blocked by two critical resources at the same time
+ is obvious, because only running threads can make new requests, if one is waiting for
+ a critical resource and get blocked, it can not make another resource request and get
+ blocked the second time (because it is not running).
+
+ To derive this fact, one needs to prove by contraction and
+ reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
+ named @{text "p_split"}, which is about status changing along the time axis. It says if
+ a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
+ but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"}
+ in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history
+ of events leading to it), such that @{text "Q"} switched
+ from being @{text "False"} to @{text "True"} and kept being @{text "True"}
+ till the last moment of @{text "s"}.
+
+ Suppose a thread @{text "th"} is blocked
+ on @{text "cs1"} and @{text "cs2"} in some state @{text "s"},
+ since no thread is blocked at the very beginning, by applying
+ @{text "p_split"} to these two blocking facts, there exist
+ two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that
+ @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"}
+ and kept on blocked on them respectively ever since.
+
+ Without lose of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
+ However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
+ in blocked state at moment @{text "t2"} and could not
+ make any request and get blocked the second time: Contradiction.
+*}
+
+lemma waiting_unique_pre:
+ assumes h11: "thread \<in> set (wq s cs1)"
+ and h12: "thread \<noteq> hd (wq s cs1)"
+ assumes h21: "thread \<in> set (wq s cs2)"
+ and h22: "thread \<noteq> hd (wq s cs2)"
+ and neq12: "cs1 \<noteq> cs2"
+ shows "False"
+proof -
+ let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+ from h11 and h12 have q1: "?Q cs1 s" by simp
+ from h21 and h22 have q2: "?Q cs2 s" by simp
+ have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+ have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+ from p_split [of "?Q cs1", OF q1 nq1]
+ obtain t1 where lt1: "t1 < length s"
+ and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
+ thread \<noteq> hd (wq (moment t1 s) cs1))"
+ and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
+ thread \<noteq> hd (wq (moment i' s) cs1))" by auto
+ from p_split [of "?Q cs2", OF q2 nq2]
+ obtain t2 where lt2: "t2 < length s"
+ and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
+ thread \<noteq> hd (wq (moment t2 s) cs2))"
+ and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
+ thread \<noteq> hd (wq (moment i' s) cs2))" by auto
+ show ?thesis
+ proof -
+ {
+ assume lt12: "t1 < t2"
+ let ?t3 = "Suc t2"
+ from lt2 have le_t3: "?t3 \<le> length s" by auto
+ from moment_plus [OF this]
+ obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+ have "t2 < ?t3" by simp
+ from nn2 [rule_format, OF this] and eq_m
+ have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+ h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+ have "vt (e#moment t2 s)"
+ proof -
+ from vt_moment
+ have "vt (moment ?t3 s)" .
+ with eq_m show ?thesis by simp
+ qed
+ then interpret vt_e: valid_trace_e "moment t2 s" "e"
+ by (unfold_locales, auto, cases, simp)
+ have ?thesis
+ proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+ case True
+ from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+ by auto
+ from vt_e.abs2 [OF True eq_th h2 h1]
+ show ?thesis by auto
+ next
+ case False
+ from vt_e.block_pre[OF False h1]
+ have "e = P thread cs2" .
+ with vt_e.vt_e have "vt ((P thread cs2)# moment t2 s)" by simp
+ from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
+ with runing_ready have "thread \<in> readys (moment t2 s)" by auto
+ with nn1 [rule_format, OF lt12]
+ show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto)
+ qed
+ } moreover {
+ assume lt12: "t2 < t1"
+ let ?t3 = "Suc t1"
+ from lt1 have le_t3: "?t3 \<le> length s" by auto
+ from moment_plus [OF this]
+ obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+ have lt_t3: "t1 < ?t3" by simp
+ from nn1 [rule_format, OF this] and eq_m
+ have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+ h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+ have "vt (e#moment t1 s)"
+ proof -
+ from vt_moment
+ have "vt (moment ?t3 s)" .
+ with eq_m show ?thesis by simp
+ qed
+ then interpret vt_e: valid_trace_e "moment t1 s" e
+ by (unfold_locales, auto, cases, auto)
+ have ?thesis
+ proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+ case True
+ from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+ by auto
+ from vt_e.abs2 True eq_th h2 h1
+ show ?thesis by auto
+ next
+ case False
+ from vt_e.block_pre [OF False h1]
+ have "e = P thread cs1" .
+ with vt_e.vt_e have "vt ((P thread cs1)# moment t1 s)" by simp
+ from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
+ with runing_ready have "thread \<in> readys (moment t1 s)" by auto
+ with nn2 [rule_format, OF lt12]
+ show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto)
+ qed
+ } moreover {
+ assume eqt12: "t1 = t2"
+ let ?t3 = "Suc t1"
+ from lt1 have le_t3: "?t3 \<le> length s" by auto
+ from moment_plus [OF this]
+ obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+ have lt_t3: "t1 < ?t3" by simp
+ from nn1 [rule_format, OF this] and eq_m
+ have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+ h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+ have vt_e: "vt (e#moment t1 s)"
+ proof -
+ from vt_moment
+ have "vt (moment ?t3 s)" .
+ with eq_m show ?thesis by simp
+ qed
+ then interpret vt_e: valid_trace_e "moment t1 s" e
+ by (unfold_locales, auto, cases, auto)
+ have ?thesis
+ proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+ case True
+ from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+ by auto
+ from vt_e.abs2 [OF True eq_th h2 h1]
+ show ?thesis by auto
+ next
+ case False
+ from vt_e.block_pre [OF False h1]
+ have eq_e1: "e = P thread cs1" .
+ have lt_t3: "t1 < ?t3" by simp
+ with eqt12 have "t2 < ?t3" by simp
+ from nn2 [rule_format, OF this] and eq_m and eqt12
+ have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+ h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+ show ?thesis
+ proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+ case True
+ from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+ by auto
+ from vt_e and eqt12 have "vt (e#moment t2 s)" by simp
+ then interpret vt_e2: valid_trace_e "moment t2 s" e
+ by (unfold_locales, auto, cases, auto)
+ from vt_e2.abs2 [OF True eq_th h2 h1]
+ show ?thesis .
+ next
+ case False
+ have "vt (e#moment t2 s)"
+ proof -
+ from vt_moment eqt12
+ have "vt (moment (Suc t2) s)" by auto
+ with eq_m eqt12 show ?thesis by simp
+ qed
+ then interpret vt_e2: valid_trace_e "moment t2 s" e
+ by (unfold_locales, auto, cases, auto)
+ from vt_e2.block_pre [OF False h1]
+ have "e = P thread cs2" .
+ with eq_e1 neq12 show ?thesis by auto
+ qed
+ qed
+ } ultimately show ?thesis by arith
+ qed
+qed
+
+text {*
+ This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
+*}
+
+lemma waiting_unique:
+ assumes "waiting s th cs1"
+ and "waiting s th cs2"
+ shows "cs1 = cs2"
+using waiting_unique_pre assms
+unfolding wq_def s_waiting_def
+by auto
+
+end
+
+(* not used *)
+text {*
+ Every thread can only be blocked on one critical resource,
+ symmetrically, every critical resource can only be held by one thread.
+ This fact is much more easier according to our definition.
+*}
+lemma held_unique:
+ assumes "holding (s::event list) th1 cs"
+ and "holding s th2 cs"
+ shows "th1 = th2"
+ by (insert assms, unfold s_holding_def, auto)
+
+
+lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+ apply (induct s, auto)
+ by (case_tac a, auto split:if_splits)
+
+lemma last_set_unique:
+ "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
+ \<Longrightarrow> th1 = th2"
+ apply (induct s, auto)
+ by (case_tac a, auto split:if_splits dest:last_set_lt)
+
+lemma preced_unique :
+ assumes pcd_eq: "preced th1 s = preced th2 s"
+ and th_in1: "th1 \<in> threads s"
+ and th_in2: " th2 \<in> threads s"
+ shows "th1 = th2"
+proof -
+ from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
+ from last_set_unique [OF this th_in1 th_in2]
+ show ?thesis .
+qed
+
+lemma preced_linorder:
+ assumes neq_12: "th1 \<noteq> th2"
+ and th_in1: "th1 \<in> threads s"
+ and th_in2: " th2 \<in> threads s"
+ shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
+proof -
+ from preced_unique [OF _ th_in1 th_in2] and neq_12
+ have "preced th1 s \<noteq> preced th2 s" by auto
+ thus ?thesis by auto
+qed
+
+(* An aux lemma used later *)
+lemma unique_minus:
+ fixes x y z r
+ assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+ and xy: "(x, y) \<in> r"
+ and xz: "(x, z) \<in> r^+"
+ and neq: "y \<noteq> z"
+ shows "(y, z) \<in> r^+"
+proof -
+ from xz and neq show ?thesis
+ proof(induct)
+ case (base ya)
+ have "(x, ya) \<in> r" by fact
+ from unique [OF xy this] have "y = ya" .
+ with base show ?case by auto
+ next
+ case (step ya z)
+ show ?case
+ proof(cases "y = ya")
+ case True
+ from step True show ?thesis by simp
+ next
+ case False
+ from step False
+ show ?thesis by auto
+ qed
+ qed
+qed
+
+lemma unique_base:
+ fixes r x y z
+ assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+ and xy: "(x, y) \<in> r"
+ and xz: "(x, z) \<in> r^+"
+ and neq_yz: "y \<noteq> z"
+ shows "(y, z) \<in> r^+"
+proof -
+ from xz neq_yz show ?thesis
+ proof(induct)
+ case (base ya)
+ from xy unique base show ?case by auto
+ next
+ case (step ya z)
+ show ?case
+ proof(cases "y = ya")
+ case True
+ from True step show ?thesis by auto
+ next
+ case False
+ from False step
+ have "(y, ya) \<in> r\<^sup>+" by auto
+ with step show ?thesis by auto
+ qed
+ qed
+qed
+
+lemma unique_chain:
+ fixes r x y z
+ assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+ and xy: "(x, y) \<in> r^+"
+ and xz: "(x, z) \<in> r^+"
+ and neq_yz: "y \<noteq> z"
+ shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
+proof -
+ from xy xz neq_yz show ?thesis
+ proof(induct)
+ case (base y)
+ have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
+ from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
+ next
+ case (step y za)
+ show ?case
+ proof(cases "y = z")
+ case True
+ from True step show ?thesis by auto
+ next
+ case False
+ from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
+ thus ?thesis
+ proof
+ assume "(z, y) \<in> r\<^sup>+"
+ with step have "(z, za) \<in> r\<^sup>+" by auto
+ thus ?thesis by auto
+ next
+ assume h: "(y, z) \<in> r\<^sup>+"
+ from step have yza: "(y, za) \<in> r" by simp
+ from step have "za \<noteq> z" by simp
+ from unique_minus [OF _ yza h this] and unique
+ have "(za, z) \<in> r\<^sup>+" by auto
+ thus ?thesis by auto
+ qed
+ qed
+ qed
+qed
+
+text {*
+ The following three lemmas show that @{text "RAG"} does not change
+ by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
+ events, respectively.
+*}
+
+lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+
+text {*
+ The following lemmas are used in the proof of
+ lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed
+ by @{text "V"}-events.
+ However, since our model is very concise, such seemingly obvious lemmas need to be derived from scratch,
+ starting from the model definitions.
+*}
+lemma step_v_hold_inv[elim_format]:
+ "\<And>c t. \<lbrakk>vt (V th cs # s);
+ \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow>
+ next_th s th cs t \<and> c = cs"
+proof -
+ fix c t
+ assume vt: "vt (V th cs # s)"
+ and nhd: "\<not> holding (wq s) t c"
+ and hd: "holding (wq (V th cs # s)) t c"
+ show "next_th s th cs t \<and> c = cs"
+ proof(cases "c = cs")
+ case False
+ with nhd hd show ?thesis
+ by (unfold cs_holding_def wq_def, auto simp:Let_def)
+ next
+ case True
+ with step_back_step [OF vt]
+ have "step s (V th c)" by simp
+ hence "next_th s th cs t"
+ proof(cases)
+ assume "holding s th c"
+ with nhd hd show ?thesis
+ apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
+ auto simp:Let_def split:list.splits if_splits)
+ proof -
+ assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
+ moreover have "\<dots> = set []"
+ proof(rule someI2)
+ show "distinct [] \<and> [] = []" by auto
+ next
+ fix x assume "distinct x \<and> x = []"
+ thus "set x = set []" by auto
+ qed
+ ultimately show False by auto
+ next
+ assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
+ moreover have "\<dots> = set []"
+ proof(rule someI2)
+ show "distinct [] \<and> [] = []" by auto
+ next
+ fix x assume "distinct x \<and> x = []"
+ thus "set x = set []" by auto
+ qed
+ ultimately show False by auto
+ qed
+ qed
+ with True show ?thesis by auto
+ qed
+qed
+
+text {*
+ The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be
+ derived from scratch, which confirms the correctness of the definition of @{text "next_th"}.
+*}
+lemma step_v_wait_inv[elim_format]:
+ "\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
+ \<rbrakk>
+ \<Longrightarrow> (next_th s th cs t \<and> cs = c)"
+proof -
+ fix t c
+ assume vt: "vt (V th cs # s)"
+ and nw: "\<not> waiting (wq (V th cs # s)) t c"
+ and wt: "waiting (wq s) t c"
+ from vt interpret vt_v: valid_trace_e s "V th cs"
+ by (cases, unfold_locales, simp)
+ show "next_th s th cs t \<and> cs = c"
+ proof(cases "cs = c")
+ case False
+ with nw wt show ?thesis
+ by (auto simp:cs_waiting_def wq_def Let_def)
+ next
+ case True
+ from nw[folded True] wt[folded True]
+ have "next_th s th cs t"
+ apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
+ proof -
+ fix a list
+ assume t_in: "t \<in> set list"
+ and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
+ and eq_wq: "wq_fun (schs s) cs = a # list"
+ have " set (SOME q. distinct q \<and> set q = set list) = set list"
+ proof(rule someI2)
+ from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
+ show "distinct list \<and> set list = set list" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+ by auto
+ qed
+ with t_ni and t_in show "a = th" by auto
+ next
+ fix a list
+ assume t_in: "t \<in> set list"
+ and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
+ and eq_wq: "wq_fun (schs s) cs = a # list"
+ have " set (SOME q. distinct q \<and> set q = set list) = set list"
+ proof(rule someI2)
+ from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
+ show "distinct list \<and> set list = set list" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+ by auto
+ qed
+ with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto
+ next
+ fix a list
+ assume eq_wq: "wq_fun (schs s) cs = a # list"
+ from step_back_step[OF vt]
+ show "a = th"
+ proof(cases)
+ assume "holding s th cs"
+ with eq_wq show ?thesis
+ by (unfold s_holding_def wq_def, auto)
+ qed
+ qed
+ with True show ?thesis by simp
+ qed
+qed
+
+lemma step_v_not_wait[consumes 3]:
+ "\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
+ by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
+
+lemma step_v_release:
+ "\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
+proof -
+ assume vt: "vt (V th cs # s)"
+ and hd: "holding (wq (V th cs # s)) th cs"
+ from vt interpret vt_v: valid_trace_e s "V th cs"
+ by (cases, unfold_locales, simp+)
+ from step_back_step [OF vt] and hd
+ show "False"
+ proof(cases)
+ assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"
+ thus ?thesis
+ apply (unfold s_holding_def wq_def cs_holding_def)
+ apply (auto simp:Let_def split:list.splits)
+ proof -
+ fix list
+ assume eq_wq[folded wq_def]:
+ "wq_fun (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
+ and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
+ \<in> set (SOME q. distinct q \<and> set q = set list)"
+ have "set (SOME q. distinct q \<and> set q = set list) = set list"
+ proof(rule someI2)
+ from vt_v.wq_distinct[of cs] and eq_wq
+ show "distinct list \<and> set list = set list" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+ by auto
+ qed
+ moreover have "distinct (hd (SOME q. distinct q \<and> set q = set list) # list)"
+ proof -
+ from vt_v.wq_distinct[of cs] and eq_wq
+ show ?thesis by auto
+ qed
+ moreover note eq_wq and hd_in
+ ultimately show "False" by auto
+ qed
+ qed
+qed
+
+lemma step_v_get_hold:
+ "\<And>th'. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
+ apply (unfold cs_holding_def next_th_def wq_def,
+ auto simp:Let_def)
+proof -
+ fix rest
+ assume vt: "vt (V th cs # s)"
+ and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest"
+ and nrest: "rest \<noteq> []"
+ and ni: "hd (SOME q. distinct q \<and> set q = set rest)
+ \<notin> set (SOME q. distinct q \<and> set q = set rest)"
+ from vt interpret vt_v: valid_trace_e s "V th cs"
+ by (cases, unfold_locales, simp+)
+ have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from vt_v.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ hence "set x = set rest" by auto
+ with nrest
+ show "x \<noteq> []" by (case_tac x, auto)
+ qed
+ with ni show "False" by auto
+qed
+
+lemma step_v_release_inv[elim_format]:
+"\<And>c t. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow>
+ c = cs \<and> t = th"
+ apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
+ proof -
+ fix a list
+ assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
+ from step_back_step [OF vt] show "a = th"
+ proof(cases)
+ assume "holding s th cs" with eq_wq
+ show ?thesis
+ by (unfold s_holding_def wq_def, auto)
+ qed
+ next
+ fix a list
+ assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
+ from step_back_step [OF vt] show "a = th"
+ proof(cases)
+ assume "holding s th cs" with eq_wq
+ show ?thesis
+ by (unfold s_holding_def wq_def, auto)
+ qed
+ qed
+
+lemma step_v_waiting_mono:
+ "\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
+proof -
+ fix t c
+ let ?s' = "(V th cs # s)"
+ assume vt: "vt ?s'"
+ and wt: "waiting (wq ?s') t c"
+ from vt interpret vt_v: valid_trace_e s "V th cs"
+ by (cases, unfold_locales, simp+)
+ show "waiting (wq s) t c"
+ proof(cases "c = cs")
+ case False
+ assume neq_cs: "c \<noteq> cs"
+ hence "waiting (wq ?s') t c = waiting (wq s) t c"
+ by (unfold cs_waiting_def wq_def, auto simp:Let_def)
+ with wt show ?thesis by simp
+ next
+ case True
+ with wt show ?thesis
+ apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
+ proof -
+ fix a list
+ assume not_in: "t \<notin> set list"
+ and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
+ and eq_wq: "wq_fun (schs s) cs = a # list"
+ have "set (SOME q. distinct q \<and> set q = set list) = set list"
+ proof(rule someI2)
+ from vt_v.wq_distinct [of cs]
+ and eq_wq[folded wq_def]
+ show "distinct list \<and> set list = set list" by auto
+ next
+ fix x assume "distinct x \<and> set x = set list"
+ thus "set x = set list" by auto
+ qed
+ with not_in is_in show "t = a" by auto
+ next
+ fix list
+ assume is_waiting: "waiting (wq (V th cs # s)) t cs"
+ and eq_wq: "wq_fun (schs s) cs = t # list"
+ hence "t \<in> set list"
+ apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
+ proof -
+ assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
+ moreover have "\<dots> = set list"
+ proof(rule someI2)
+ from vt_v.wq_distinct [of cs]
+ and eq_wq[folded wq_def]
+ show "distinct list \<and> set list = set list" by auto
+ next
+ fix x assume "distinct x \<and> set x = set list"
+ thus "set x = set list" by auto
+ qed
+ ultimately show "t \<in> set list" by simp
+ qed
+ with eq_wq and vt_v.wq_distinct [of cs, unfolded wq_def]
+ show False by auto
+ qed
+ qed
+qed
+
+text {* (* ddd *)
+ The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
+ with the happening of @{text "V"}-events:
+*}
+lemma step_RAG_v:
+fixes th::thread
+assumes vt:
+ "vt (V th cs#s)"
+shows "
+ RAG (V th cs # s) =
+ RAG s - {(Cs cs, Th th)} -
+ {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+ {(Cs cs, Th th') |th'. next_th s th cs th'}"
+ apply (insert vt, unfold s_RAG_def)
+ apply (auto split:if_splits list.splits simp:Let_def)
+ apply (auto elim: step_v_waiting_mono step_v_hold_inv
+ step_v_release step_v_wait_inv
+ step_v_get_hold step_v_release_inv)
+ apply (erule_tac step_v_not_wait, auto)
+ done
+
+text {*
+ The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
+ with the happening of @{text "P"}-events:
+*}
+lemma step_RAG_p:
+ "vt (P th cs#s) \<Longrightarrow>
+ RAG (P th cs # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+ else RAG s \<union> {(Th th, Cs cs)})"
+ apply(simp only: s_RAG_def wq_def)
+ apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
+ apply(case_tac "csa = cs", auto)
+ apply(fold wq_def)
+ apply(drule_tac step_back_step)
+ apply(ind_cases " step s (P (hd (wq s cs)) cs)")
+ apply(simp add:s_RAG_def wq_def cs_holding_def)
+ apply(auto)
+ done
+
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+ by (unfold s_RAG_def, auto)
+
+context valid_trace
+begin
+
+text {*
+ The following lemma shows that @{text "RAG"} is acyclic.
+ The overall structure is by induction on the formation of @{text "vt s"}
+ and then case analysis on event @{text "e"}, where the non-trivial cases
+ for those for @{text "V"} and @{text "P"} events.
+*}
+lemma acyclic_RAG:
+ shows "acyclic (RAG s)"
+using vt
+proof(induct)
+ case (vt_cons s e)
+ interpret vt_s: valid_trace s using vt_cons(1)
+ by (unfold_locales, simp)
+ assume ih: "acyclic (RAG s)"
+ and stp: "step s e"
+ and vt: "vt s"
+ show ?case
+ proof(cases e)
+ case (Create th prio)
+ with ih
+ show ?thesis by (simp add:RAG_create_unchanged)
+ next
+ case (Exit th)
+ with ih show ?thesis by (simp add:RAG_exit_unchanged)
+ next
+ case (V th cs)
+ from V vt stp have vtt: "vt (V th cs#s)" by auto
+ from step_RAG_v [OF this]
+ have eq_de:
+ "RAG (e # s) =
+ RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+ {(Cs cs, Th th') |th'. next_th s th cs th'}"
+ (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+ from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+ from step_back_step [OF vtt]
+ have "step s (V th cs)" .
+ thus ?thesis
+ proof(cases)
+ assume "holding s th cs"
+ hence th_in: "th \<in> set (wq s cs)" and
+ eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
+ then obtain rest where
+ eq_wq: "wq s cs = th#rest"
+ by (cases "wq s cs", auto)
+ show ?thesis
+ proof(cases "rest = []")
+ case False
+ let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+ from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
+ by (unfold next_th_def, auto)
+ let ?E = "(?A - ?B - ?C)"
+ have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
+ proof
+ assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
+ hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+ from tranclD [OF this]
+ obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
+ hence th_d: "(Th ?th', x) \<in> ?A" by simp
+ from RAG_target_th [OF this]
+ obtain cs' where eq_x: "x = Cs cs'" by auto
+ with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
+ hence wt_th': "waiting s ?th' cs'"
+ unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
+ hence "cs' = cs"
+ proof(rule vt_s.waiting_unique)
+ from eq_wq vt_s.wq_distinct[of cs]
+ show "waiting s ?th' cs"
+ apply (unfold s_waiting_def wq_def, auto)
+ proof -
+ assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ and eq_wq: "wq_fun (schs s) cs = th # rest"
+ have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ with False show "x \<noteq> []" by auto
+ qed
+ hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
+ set (SOME q. distinct q \<and> set q = set rest)" by auto
+ moreover have "\<dots> = set rest"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+ qed
+ moreover note hd_in
+ ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
+ next
+ assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+ have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ with False show "x \<noteq> []" by auto
+ qed
+ hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
+ set (SOME q. distinct q \<and> set q = set rest)" by auto
+ moreover have "\<dots> = set rest"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+ qed
+ moreover note hd_in
+ ultimately show False by auto
+ qed
+ qed
+ with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
+ with False
+ show "False" by (auto simp: next_th_def eq_wq)
+ qed
+ with acyclic_insert[symmetric] and ac
+ and eq_de eq_D show ?thesis by auto
+ next
+ case True
+ with eq_wq
+ have eq_D: "?D = {}"
+ by (unfold next_th_def, auto)
+ with eq_de ac
+ show ?thesis by auto
+ qed
+ qed
+ next
+ case (P th cs)
+ from P vt stp have vtt: "vt (P th cs#s)" by auto
+ from step_RAG_p [OF this] P
+ have "RAG (e # s) =
+ (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
+ RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+ by simp
+ moreover have "acyclic ?R"
+ proof(cases "wq s cs = []")
+ case True
+ hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
+ have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
+ proof
+ assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
+ hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+ from tranclD2 [OF this]
+ obtain x where "(x, Cs cs) \<in> RAG s" by auto
+ with True show False by (auto simp:s_RAG_def cs_waiting_def)
+ qed
+ with acyclic_insert ih eq_r show ?thesis by auto
+ next
+ case False
+ hence eq_r: "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
+ have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
+ proof
+ assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
+ hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+ moreover from step_back_step [OF vtt] have "step s (P th cs)" .
+ ultimately show False
+ proof -
+ show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+ by (ind_cases "step s (P th cs)", simp)
+ qed
+ qed
+ with acyclic_insert ih eq_r show ?thesis by auto
+ qed
+ ultimately show ?thesis by simp
+ next
+ case (Set thread prio)
+ with ih
+ thm RAG_set_unchanged
+ show ?thesis by (simp add:RAG_set_unchanged)
+ qed
+ next
+ case vt_nil
+ show "acyclic (RAG ([]::state))"
+ by (auto simp: s_RAG_def cs_waiting_def
+ cs_holding_def wq_def acyclic_def)
+qed
+
+
+lemma finite_RAG:
+ shows "finite (RAG s)"
+proof -
+ from vt show ?thesis
+ proof(induct)
+ case (vt_cons s e)
+ interpret vt_s: valid_trace s using vt_cons(1)
+ by (unfold_locales, simp)
+ assume ih: "finite (RAG s)"
+ and stp: "step s e"
+ and vt: "vt s"
+ show ?case
+ proof(cases e)
+ case (Create th prio)
+ with ih
+ show ?thesis by (simp add:RAG_create_unchanged)
+ next
+ case (Exit th)
+ with ih show ?thesis by (simp add:RAG_exit_unchanged)
+ next
+ case (V th cs)
+ from V vt stp have vtt: "vt (V th cs#s)" by auto
+ from step_RAG_v [OF this]
+ have eq_de: "RAG (e # s) =
+ RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+ {(Cs cs, Th th') |th'. next_th s th cs th'}
+"
+ (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+ moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
+ moreover have "finite ?D"
+ proof -
+ have "?D = {} \<or> (\<exists> a. ?D = {a})"
+ by (unfold next_th_def, auto)
+ thus ?thesis
+ proof
+ assume h: "?D = {}"
+ show ?thesis by (unfold h, simp)
+ next
+ assume "\<exists> a. ?D = {a}"
+ thus ?thesis
+ by (metis finite.simps)
+ qed
+ qed
+ ultimately show ?thesis by simp
+ next
+ case (P th cs)
+ from P vt stp have vtt: "vt (P th cs#s)" by auto
+ from step_RAG_p [OF this] P
+ have "RAG (e # s) =
+ (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
+ RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+ by simp
+ moreover have "finite ?R"
+ proof(cases "wq s cs = []")
+ case True
+ hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
+ with True and ih show ?thesis by auto
+ next
+ case False
+ hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
+ with False and ih show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ next
+ case (Set thread prio)
+ with ih
+ show ?thesis by (simp add:RAG_set_unchanged)
+ qed
+ next
+ case vt_nil
+ show "finite (RAG ([]::state))"
+ by (auto simp: s_RAG_def cs_waiting_def
+ cs_holding_def wq_def acyclic_def)
+ qed
+qed
+
+text {* Several useful lemmas *}
+
+lemma wf_dep_converse:
+ shows "wf ((RAG s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+ from finite_RAG
+ show "finite (RAG s)" .
+next
+ from acyclic_RAG
+ show "acyclic (RAG s)" .
+qed
+
+end
+
+lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
+ by (induct l, auto)
+
+lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
+ by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+
+context valid_trace
+begin
+
+lemma wq_threads:
+ assumes h: "th \<in> set (wq s cs)"
+ shows "th \<in> threads s"
+proof -
+ from vt and h show ?thesis
+ proof(induct arbitrary: th cs)
+ case (vt_cons s e)
+ interpret vt_s: valid_trace s
+ using vt_cons(1) by (unfold_locales, auto)
+ assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+ and stp: "step s e"
+ and vt: "vt s"
+ and h: "th \<in> set (wq (e # s) cs)"
+ show ?case
+ proof(cases e)
+ case (Create th' prio)
+ with ih h show ?thesis
+ by (auto simp:wq_def Let_def)
+ next
+ case (Exit th')
+ with stp ih h show ?thesis
+ apply (auto simp:wq_def Let_def)
+ apply (ind_cases "step s (Exit th')")
+ apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
+ s_RAG_def s_holding_def cs_holding_def)
+ done
+ next
+ case (V th' cs')
+ show ?thesis
+ proof(cases "cs' = cs")
+ case False
+ with h
+ show ?thesis
+ apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+ by (drule_tac ih, simp)
+ next
+ case True
+ from h
+ show ?thesis
+ proof(unfold V wq_def)
+ assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+ show "th \<in> threads (V th' cs' # s)"
+ proof(cases "cs = cs'")
+ case False
+ hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
+ with th_in have " th \<in> set (wq s cs)"
+ by (fold wq_def, simp)
+ from ih [OF this] show ?thesis by simp
+ next
+ case True
+ show ?thesis
+ proof(cases "wq_fun (schs s) cs'")
+ case Nil
+ with h V show ?thesis
+ apply (auto simp:wq_def Let_def split:if_splits)
+ by (fold wq_def, drule_tac ih, simp)
+ next
+ case (Cons a rest)
+ assume eq_wq: "wq_fun (schs s) cs' = a # rest"
+ with h V show ?thesis
+ apply (auto simp:Let_def wq_def split:if_splits)
+ proof -
+ assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+ have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+ by auto
+ qed
+ with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
+ from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+ next
+ assume th_in: "th \<in> set (wq_fun (schs s) cs)"
+ from ih[OF this[folded wq_def]]
+ show "th \<in> threads s" .
+ qed
+ qed
+ qed
+ qed
+ qed
+ next
+ case (P th' cs')
+ from h stp
+ show ?thesis
+ apply (unfold P wq_def)
+ apply (auto simp:Let_def split:if_splits, fold wq_def)
+ apply (auto intro:ih)
+ apply(ind_cases "step s (P th' cs')")
+ by (unfold runing_def readys_def, auto)
+ next
+ case (Set thread prio)
+ with ih h show ?thesis
+ by (auto simp:wq_def Let_def)
+ qed
+ next
+ case vt_nil
+ thus ?case by (auto simp:wq_def)
+ qed
+qed
+
+lemma range_in: "\<lbrakk>(Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+ apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
+ by (auto intro:wq_threads)
+
+lemma readys_v_eq:
+ fixes th thread cs rest
+ assumes neq_th: "th \<noteq> thread"
+ and eq_wq: "wq s cs = thread#rest"
+ and not_in: "th \<notin> set rest"
+ shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
+proof -
+ from assms show ?thesis
+ apply (auto simp:readys_def)
+ apply(simp add:s_waiting_def[folded wq_def])
+ apply (erule_tac x = csa in allE)
+ apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
+ apply (case_tac "csa = cs", simp)
+ apply (erule_tac x = cs in allE)
+ apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
+ apply(auto simp add: wq_def)
+ apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
+ proof -
+ assume th_nin: "th \<notin> set rest"
+ and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+ and eq_wq: "wq_fun (schs s) cs = thread # rest"
+ have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+ qed
+ with th_nin th_in show False by auto
+ qed
+qed
+
+text {* \noindent
+ The following lemmas shows that: starting from any node in @{text "RAG"},
+ by chasing out-going edges, it is always possible to reach a node representing a ready
+ thread. In this lemma, it is the @{text "th'"}.
+*}
+
+lemma chain_building:
+ shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
+proof -
+ from wf_dep_converse
+ have h: "wf ((RAG s)\<inverse>)" .
+ show ?thesis
+ proof(induct rule:wf_induct [OF h])
+ fix x
+ assume ih [rule_format]:
+ "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow>
+ y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
+ show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
+ proof
+ assume x_d: "x \<in> Domain (RAG s)"
+ show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
+ proof(cases x)
+ case (Th th)
+ from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
+ with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
+ from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
+ hence "Cs cs \<in> Domain (RAG s)" by auto
+ from ih [OF x_in_r this] obtain th'
+ where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
+ have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
+ with th'_ready show ?thesis by auto
+ next
+ case (Cs cs)
+ from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
+ show ?thesis
+ proof(cases "th' \<in> readys s")
+ case True
+ from True and th'_d show ?thesis by auto
+ next
+ case False
+ from th'_d and range_in have "th' \<in> threads s" by auto
+ with False have "Th th' \<in> Domain (RAG s)"
+ by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
+ from ih [OF th'_d this]
+ obtain th'' where
+ th''_r: "th'' \<in> readys s" and
+ th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
+ from th'_d and th''_in
+ have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
+ with th''_r show ?thesis by auto
+ qed
+ qed
+ qed
+ qed
+qed
+
+text {* \noindent
+ The following is just an instance of @{text "chain_building"}.
+*}
+lemma th_chain_to_ready:
+ assumes th_in: "th \<in> threads s"
+ shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
+proof(cases "th \<in> readys s")
+ case True
+ thus ?thesis by auto
+next
+ case False
+ from False and th_in have "Th th \<in> Domain (RAG s)"
+ by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
+ from chain_building [rule_format, OF this]
+ show ?thesis by auto
+qed
+
+end
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+ by (unfold s_waiting_def cs_waiting_def wq_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+ by (unfold s_holding_def wq_def cs_holding_def, simp)
+
+lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
+ by (unfold s_holding_def cs_holding_def, auto)
+
+context valid_trace
+begin
+
+lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
+ apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
+ by(auto elim:waiting_unique holding_unique)
+
+end
+
+
+lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
+by (induct rule:trancl_induct, auto)
+
+context valid_trace
+begin
+
+lemma dchain_unique:
+ assumes th1_d: "(n, Th th1) \<in> (RAG s)^+"
+ and th1_r: "th1 \<in> readys s"
+ and th2_d: "(n, Th th2) \<in> (RAG s)^+"
+ and th2_r: "th2 \<in> readys s"
+ shows "th1 = th2"
+proof -
+ { assume neq: "th1 \<noteq> th2"
+ hence "Th th1 \<noteq> Th th2" by simp
+ from unique_chain [OF _ th1_d th2_d this] and unique_RAG
+ have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
+ hence "False"
+ proof
+ assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
+ from trancl_split [OF this]
+ obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
+ then obtain cs where eq_n: "n = Cs cs"
+ by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+ from dd eq_n have "th1 \<notin> readys s"
+ by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
+ with th1_r show ?thesis by auto
+ next
+ assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
+ from trancl_split [OF this]
+ obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
+ then obtain cs where eq_n: "n = Cs cs"
+ by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+ from dd eq_n have "th2 \<notin> readys s"
+ by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
+ with th2_r show ?thesis by auto
+ qed
+ } thus ?thesis by auto
+qed
+
+end
+
+
+lemma step_holdents_p_add:
+ fixes th cs s
+ assumes vt: "vt (P th cs#s)"
+ and "wq s cs = []"
+ shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+ from assms show ?thesis
+ unfolding holdents_test step_RAG_p[OF vt] by (auto)
+qed
+
+lemma step_holdents_p_eq:
+ fixes th cs s
+ assumes vt: "vt (P th cs#s)"
+ and "wq s cs \<noteq> []"
+ shows "holdents (P th cs#s) th = holdents s th"
+proof -
+ from assms show ?thesis
+ unfolding holdents_test step_RAG_p[OF vt] by auto
+qed
+
+
+lemma (in valid_trace) finite_holding :
+ shows "finite (holdents s th)"
+proof -
+ let ?F = "\<lambda> (x, y). the_cs x"
+ from finite_RAG
+ have "finite (RAG s)" .
+ hence "finite (?F `(RAG s))" by simp
+ moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>"
+ proof -
+ { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
+ fix x assume "(Cs x, Th th) \<in> RAG s"
+ hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
+ moreover have "?F (Cs x, Th th) = x" by simp
+ ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp
+ } thus ?thesis by auto
+ qed
+ ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
+qed
+
+lemma cntCS_v_dec:
+ fixes s thread cs
+ assumes vtv: "vt (V thread cs#s)"
+ shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
+proof -
+ from vtv interpret vt_s: valid_trace s
+ by (cases, unfold_locales, simp)
+ from vtv interpret vt_v: valid_trace "V thread cs#s"
+ by (unfold_locales, simp)
+ from step_back_step[OF vtv]
+ have cs_in: "cs \<in> holdents s thread"
+ apply (cases, unfold holdents_test s_RAG_def, simp)
+ by (unfold cs_holding_def s_holding_def wq_def, auto)
+ moreover have cs_not_in:
+ "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
+ apply (insert vt_s.wq_distinct[of cs])
+ apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
+ auto simp:next_th_def)
+ proof -
+ fix rest
+ assume dst: "distinct (rest::thread list)"
+ and ne: "rest \<noteq> []"
+ and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from dst show "distinct rest \<and> set rest = set rest" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+ qed
+ ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
+ set (SOME q. distinct q \<and> set q = set rest)" by simp
+ moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from dst show "distinct rest \<and> set rest = set rest" by auto
+ next
+ fix x assume " distinct x \<and> set x = set rest" with ne
+ show "x \<noteq> []" by auto
+ qed
+ ultimately
+ show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
+ by auto
+ next
+ fix rest
+ assume dst: "distinct (rest::thread list)"
+ and ne: "rest \<noteq> []"
+ and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from dst show "distinct rest \<and> set rest = set rest" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+ qed
+ ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
+ set (SOME q. distinct q \<and> set q = set rest)" by simp
+ moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from dst show "distinct rest \<and> set rest = set rest" by auto
+ next
+ fix x assume " distinct x \<and> set x = set rest" with ne
+ show "x \<noteq> []" by auto
+ qed
+ ultimately show "False" by auto
+ qed
+ ultimately
+ have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
+ by auto
+ moreover have "card \<dots> =
+ Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
+ proof(rule card_insert)
+ from vt_v.finite_holding
+ show " finite (holdents (V thread cs # s) thread)" .
+ qed
+ moreover from cs_not_in
+ have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
+ ultimately show ?thesis by (simp add:cntCS_def)
+qed
+
+context valid_trace
+begin
+
+text {* (* ddd *) \noindent
+ The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"}
+ of one particular thread.
+*}
+
+lemma cnp_cnv_cncs:
+ shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
+ then cntCS s th else cntCS s th + 1)"
+proof -
+ from vt show ?thesis
+ proof(induct arbitrary:th)
+ case (vt_cons s e)
+ interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
+ assume vt: "vt s"
+ and ih: "\<And>th. cntP s th = cntV s th +
+ (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
+ and stp: "step s e"
+ from stp show ?case
+ proof(cases)
+ case (thread_create thread prio)
+ assume eq_e: "e = Create thread prio"
+ and not_in: "thread \<notin> threads s"
+ show ?thesis
+ proof -
+ { fix cs
+ assume "thread \<in> set (wq s cs)"
+ from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
+ with not_in have "False" by simp
+ } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
+ by (auto simp:readys_def threads.simps s_waiting_def
+ wq_def cs_waiting_def Let_def)
+ from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+ from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+ have eq_cncs: "cntCS (e#s) th = cntCS s th"
+ unfolding cntCS_def holdents_test
+ by (simp add:RAG_create_unchanged eq_e)
+ { assume "th \<noteq> thread"
+ with eq_readys eq_e
+ have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
+ (th \<in> readys (s) \<or> th \<notin> threads (s))"
+ by (simp add:threads.simps)
+ with eq_cnp eq_cnv eq_cncs ih not_in
+ have ?thesis by simp
+ } moreover {
+ assume eq_th: "th = thread"
+ with not_in ih have " cntP s th = cntV s th + cntCS s th" by simp
+ moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
+ moreover note eq_cnp eq_cnv eq_cncs
+ ultimately have ?thesis by auto
+ } ultimately show ?thesis by blast
+ qed
+ next
+ case (thread_exit thread)
+ assume eq_e: "e = Exit thread"
+ and is_runing: "thread \<in> runing s"
+ and no_hold: "holdents s thread = {}"
+ from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+ from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+ have eq_cncs: "cntCS (e#s) th = cntCS s th"
+ unfolding cntCS_def holdents_test
+ by (simp add:RAG_exit_unchanged eq_e)
+ { assume "th \<noteq> thread"
+ with eq_e
+ have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
+ (th \<in> readys (s) \<or> th \<notin> threads (s))"
+ apply (simp add:threads.simps readys_def)
+ apply (subst s_waiting_def)
+ apply (simp add:Let_def)
+ apply (subst s_waiting_def, simp)
+ done
+ with eq_cnp eq_cnv eq_cncs ih
+ have ?thesis by simp
+ } moreover {
+ assume eq_th: "th = thread"
+ with ih is_runing have " cntP s th = cntV s th + cntCS s th"
+ by (simp add:runing_def)
+ moreover from eq_th eq_e have "th \<notin> threads (e#s)"
+ by simp
+ moreover note eq_cnp eq_cnv eq_cncs
+ ultimately have ?thesis by auto
+ } ultimately show ?thesis by blast
+ next
+ case (thread_P thread cs)
+ assume eq_e: "e = P thread cs"
+ and is_runing: "thread \<in> runing s"
+ and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
+ from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto
+ then interpret vt_p: valid_trace "(P thread cs#s)"
+ by (unfold_locales, simp)
+ show ?thesis
+ proof -
+ { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
+ assume neq_th: "th \<noteq> thread"
+ with eq_e
+ have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
+ apply (simp add:readys_def s_waiting_def wq_def Let_def)
+ apply (rule_tac hh)
+ apply (intro iffI allI, clarify)
+ apply (erule_tac x = csa in allE, auto)
+ apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
+ apply (erule_tac x = cs in allE, auto)
+ by (case_tac "(wq_fun (schs s) cs)", auto)
+ moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+ apply (simp add:cntCS_def holdents_test)
+ by (unfold step_RAG_p [OF vtp], auto)
+ moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
+ by (simp add:cntP_def count_def)
+ moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
+ by (simp add:cntV_def count_def)
+ moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
+ moreover note ih [of th]
+ ultimately have ?thesis by simp
+ } moreover {
+ assume eq_th: "th = thread"
+ have ?thesis
+ proof -
+ from eq_e eq_th have eq_cnp: "cntP (e # s) th = 1 + (cntP s th)"
+ by (simp add:cntP_def count_def)
+ from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
+ by (simp add:cntV_def count_def)
+ show ?thesis
+ proof (cases "wq s cs = []")
+ case True
+ with is_runing
+ have "th \<in> readys (e#s)"
+ apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
+ apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
+ by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
+ moreover have "cntCS (e # s) th = 1 + cntCS s th"
+ proof -
+ have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
+ Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
+ proof -
+ have "?L = insert cs ?R" by auto
+ moreover have "card \<dots> = Suc (card (?R - {cs}))"
+ proof(rule card_insert)
+ from vt_s.finite_holding [of thread]
+ show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
+ by (unfold holdents_test, simp)
+ qed
+ moreover have "?R - {cs} = ?R"
+ proof -
+ have "cs \<notin> ?R"
+ proof
+ assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
+ with no_dep show False by auto
+ qed
+ thus ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ thus ?thesis
+ apply (unfold eq_e eq_th cntCS_def)
+ apply (simp add: holdents_test)
+ by (unfold step_RAG_p [OF vtp], auto simp:True)
+ qed
+ moreover from is_runing have "th \<in> readys s"
+ by (simp add:runing_def eq_th)
+ moreover note eq_cnp eq_cnv ih [of th]
+ ultimately show ?thesis by auto
+ next
+ case False
+ have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
+ by (unfold eq_th eq_e wq_def, auto simp:Let_def)
+ have "th \<notin> readys (e#s)"
+ proof
+ assume "th \<in> readys (e#s)"
+ hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
+ from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
+ hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
+ by (simp add:s_waiting_def wq_def)
+ moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
+ ultimately have "th = hd (wq (e#s) cs)" by blast
+ with eq_wq have "th = hd (wq s cs @ [th])" by simp
+ hence "th = hd (wq s cs)" using False by auto
+ with False eq_wq vt_p.wq_distinct [of cs]
+ show False by (fold eq_e, auto)
+ qed
+ moreover from is_runing have "th \<in> threads (e#s)"
+ by (unfold eq_e, auto simp:runing_def readys_def eq_th)
+ moreover have "cntCS (e # s) th = cntCS s th"
+ apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
+ by (auto simp:False)
+ moreover note eq_cnp eq_cnv ih[of th]
+ moreover from is_runing have "th \<in> readys s"
+ by (simp add:runing_def eq_th)
+ ultimately show ?thesis by auto
+ qed
+ qed
+ } ultimately show ?thesis by blast
+ qed
+ next
+ case (thread_V thread cs)
+ from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
+ then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
+ assume eq_e: "e = V thread cs"
+ and is_runing: "thread \<in> runing s"
+ and hold: "holding s thread cs"
+ from hold obtain rest
+ where eq_wq: "wq s cs = thread # rest"
+ by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+ have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+ have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from vt_v.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest"
+ by (metis distinct.simps(2) vt_s.wq_distinct)
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+ by auto
+ qed
+ show ?thesis
+ proof -
+ { assume eq_th: "th = thread"
+ from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
+ by (unfold eq_e, simp add:cntP_def count_def)
+ moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
+ by (unfold eq_e, simp add:cntV_def count_def)
+ moreover from cntCS_v_dec [OF vtv]
+ have "cntCS (e # s) thread + 1 = cntCS s thread"
+ by (simp add:eq_e)
+ moreover from is_runing have rd_before: "thread \<in> readys s"
+ by (unfold runing_def, simp)
+ moreover have "thread \<in> readys (e # s)"
+ proof -
+ from is_runing
+ have "thread \<in> threads (e#s)"
+ by (unfold eq_e, auto simp:runing_def readys_def)
+ moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
+ proof
+ fix cs1
+ { assume eq_cs: "cs1 = cs"
+ have "\<not> waiting (e # s) thread cs1"
+ proof -
+ from eq_wq
+ have "thread \<notin> set (wq (e#s) cs1)"
+ apply(unfold eq_e wq_def eq_cs s_holding_def)
+ apply (auto simp:Let_def)
+ proof -
+ assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
+ with eq_set have "thread \<in> set rest" by simp
+ with vt_v.wq_distinct[of cs]
+ and eq_wq show False
+ by (metis distinct.simps(2) vt_s.wq_distinct)
+ qed
+ thus ?thesis by (simp add:wq_def s_waiting_def)
+ qed
+ } moreover {
+ assume neq_cs: "cs1 \<noteq> cs"
+ have "\<not> waiting (e # s) thread cs1"
+ proof -
+ from wq_v_neq [OF neq_cs[symmetric]]
+ have "wq (V thread cs # s) cs1 = wq s cs1" .
+ moreover have "\<not> waiting s thread cs1"
+ proof -
+ from runing_ready and is_runing
+ have "thread \<in> readys s" by auto
+ thus ?thesis by (simp add:readys_def)
+ qed
+ ultimately show ?thesis
+ by (auto simp:wq_def s_waiting_def eq_e)
+ qed
+ } ultimately show "\<not> waiting (e # s) thread cs1" by blast
+ qed
+ ultimately show ?thesis by (simp add:readys_def)
+ qed
+ moreover note eq_th ih
+ ultimately have ?thesis by auto
+ } moreover {
+ assume neq_th: "th \<noteq> thread"
+ from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th"
+ by (simp add:cntP_def count_def)
+ from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th"
+ by (simp add:cntV_def count_def)
+ have ?thesis
+ proof(cases "th \<in> set rest")
+ case False
+ have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+ apply (insert step_back_vt[OF vtv])
+ by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
+ moreover have "cntCS (e#s) th = cntCS s th"
+ apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
+ proof -
+ have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+ {cs. (Cs cs, Th th) \<in> RAG s}"
+ proof -
+ from False eq_wq
+ have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
+ apply (unfold next_th_def, auto)
+ proof -
+ assume ne: "rest \<noteq> []"
+ and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ and eq_wq: "wq s cs = thread # rest"
+ from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
+ set (SOME q. distinct q \<and> set q = set rest)
+ " by simp
+ moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from vt_s.wq_distinct[ of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ with ne show "x \<noteq> []" by auto
+ qed
+ ultimately show
+ "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
+ by auto
+ qed
+ thus ?thesis by auto
+ qed
+ thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+ card {cs. (Cs cs, Th th) \<in> RAG s}" by simp
+ qed
+ moreover note ih eq_cnp eq_cnv eq_threads
+ ultimately show ?thesis by auto
+ next
+ case True
+ assume th_in: "th \<in> set rest"
+ show ?thesis
+ proof(cases "next_th s thread cs th")
+ case False
+ with eq_wq and th_in have
+ neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
+ by (auto simp:next_th_def)
+ have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+ proof -
+ from eq_wq and th_in
+ have "\<not> th \<in> readys s"
+ apply (auto simp:readys_def s_waiting_def)
+ apply (rule_tac x = cs in exI, auto)
+ by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
+ moreover
+ from eq_wq and th_in and neq_hd
+ have "\<not> (th \<in> readys (e # s))"
+ apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
+ by (rule_tac x = cs in exI, auto simp:eq_set)
+ ultimately show ?thesis by auto
+ qed
+ moreover have "cntCS (e#s) th = cntCS s th"
+ proof -
+ from eq_wq and th_in and neq_hd
+ have "(holdents (e # s) th) = (holdents s th)"
+ apply (unfold eq_e step_RAG_v[OF vtv],
+ auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
+ Let_def cs_holding_def)
+ by (insert vt_s.wq_distinct[of cs], auto simp:wq_def)
+ thus ?thesis by (simp add:cntCS_def)
+ qed
+ moreover note ih eq_cnp eq_cnv eq_threads
+ ultimately show ?thesis by auto
+ next
+ case True
+ let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
+ let ?t = "hd ?rest"
+ from True eq_wq th_in neq_th
+ have "th \<in> readys (e # s)"
+ apply (auto simp:eq_e readys_def s_waiting_def wq_def
+ Let_def next_th_def)
+ proof -
+ assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+ and t_in: "?t \<in> set rest"
+ show "?t \<in> threads s"
+ proof(rule vt_s.wq_threads)
+ from eq_wq and t_in
+ show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
+ qed
+ next
+ fix csa
+ assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+ and t_in: "?t \<in> set rest"
+ and neq_cs: "csa \<noteq> cs"
+ and t_in': "?t \<in> set (wq_fun (schs s) csa)"
+ show "?t = hd (wq_fun (schs s) csa)"
+ proof -
+ { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
+ from vt_s.wq_distinct[of cs] and
+ eq_wq[folded wq_def] and t_in eq_wq
+ have "?t \<noteq> thread" by auto
+ with eq_wq and t_in
+ have w1: "waiting s ?t cs"
+ by (auto simp:s_waiting_def wq_def)
+ from t_in' neq_hd'
+ have w2: "waiting s ?t csa"
+ by (auto simp:s_waiting_def wq_def)
+ from vt_s.waiting_unique[OF w1 w2]
+ and neq_cs have "False" by auto
+ } thus ?thesis by auto
+ qed
+ qed
+ moreover have "cntP s th = cntV s th + cntCS s th + 1"
+ proof -
+ have "th \<notin> readys s"
+ proof -
+ from True eq_wq neq_th th_in
+ show ?thesis
+ apply (unfold readys_def s_waiting_def, auto)
+ by (rule_tac x = cs in exI, auto simp add: wq_def)
+ qed
+ moreover have "th \<in> threads s"
+ proof -
+ from th_in eq_wq
+ have "th \<in> set (wq s cs)" by simp
+ from vt_s.wq_threads [OF this]
+ show ?thesis .
+ qed
+ ultimately show ?thesis using ih by auto
+ qed
+ moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
+ apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
+ proof -
+ show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
+ Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
+ (is "card ?A = Suc (card ?B)")
+ proof -
+ have "?A = insert cs ?B" by auto
+ hence "card ?A = card (insert cs ?B)" by simp
+ also have "\<dots> = Suc (card ?B)"
+ proof(rule card_insert_disjoint)
+ have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
+ with vt_s.finite_RAG
+ show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
+ next
+ show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
+ proof
+ assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
+ hence "(Cs cs, Th th) \<in> RAG s" by simp
+ with True neq_th eq_wq show False
+ by (auto simp:next_th_def s_RAG_def cs_holding_def)
+ qed
+ qed
+ finally show ?thesis .
+ qed
+ qed
+ moreover note eq_cnp eq_cnv
+ ultimately show ?thesis by simp
+ qed
+ qed
+ } ultimately show ?thesis by blast
+ qed
+ next
+ case (thread_set thread prio)
+ assume eq_e: "e = Set thread prio"
+ and is_runing: "thread \<in> runing s"
+ show ?thesis
+ proof -
+ from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+ from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+ have eq_cncs: "cntCS (e#s) th = cntCS s th"
+ unfolding cntCS_def holdents_test
+ by (simp add:RAG_set_unchanged eq_e)
+ from eq_e have eq_readys: "readys (e#s) = readys s"
+ by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
+ auto simp:Let_def)
+ { assume "th \<noteq> thread"
+ with eq_readys eq_e
+ have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
+ (th \<in> readys (s) \<or> th \<notin> threads (s))"
+ by (simp add:threads.simps)
+ with eq_cnp eq_cnv eq_cncs ih is_runing
+ have ?thesis by simp
+ } moreover {
+ assume eq_th: "th = thread"
+ with is_runing ih have " cntP s th = cntV s th + cntCS s th"
+ by (unfold runing_def, auto)
+ moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
+ by (simp add:runing_def)
+ moreover note eq_cnp eq_cnv eq_cncs
+ ultimately have ?thesis by auto
+ } ultimately show ?thesis by blast
+ qed
+ qed
+ next
+ case vt_nil
+ show ?case
+ by (unfold cntP_def cntV_def cntCS_def,
+ auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+ qed
+qed
+
+lemma not_thread_cncs:
+ assumes not_in: "th \<notin> threads s"
+ shows "cntCS s th = 0"
+proof -
+ from vt not_in show ?thesis
+ proof(induct arbitrary:th)
+ case (vt_cons s e th)
+ interpret vt_s: valid_trace s using vt_cons(1)
+ by (unfold_locales, simp)
+ assume vt: "vt s"
+ and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
+ and stp: "step s e"
+ and not_in: "th \<notin> threads (e # s)"
+ from stp show ?case
+ proof(cases)
+ case (thread_create thread prio)
+ assume eq_e: "e = Create thread prio"
+ and not_in': "thread \<notin> threads s"
+ have "cntCS (e # s) th = cntCS s th"
+ apply (unfold eq_e cntCS_def holdents_test)
+ by (simp add:RAG_create_unchanged)
+ moreover have "th \<notin> threads s"
+ proof -
+ from not_in eq_e show ?thesis by simp
+ qed
+ moreover note ih ultimately show ?thesis by auto
+ next
+ case (thread_exit thread)
+ assume eq_e: "e = Exit thread"
+ and nh: "holdents s thread = {}"
+ have eq_cns: "cntCS (e # s) th = cntCS s th"
+ apply (unfold eq_e cntCS_def holdents_test)
+ by (simp add:RAG_exit_unchanged)
+ show ?thesis
+ proof(cases "th = thread")
+ case True
+ have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
+ with eq_cns show ?thesis by simp
+ next
+ case False
+ with not_in and eq_e
+ have "th \<notin> threads s" by simp
+ from ih[OF this] and eq_cns show ?thesis by simp
+ qed
+ next
+ case (thread_P thread cs)
+ assume eq_e: "e = P thread cs"
+ and is_runing: "thread \<in> runing s"
+ from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
+ have neq_th: "th \<noteq> thread"
+ proof -
+ from not_in eq_e have "th \<notin> threads s" by simp
+ moreover from is_runing have "thread \<in> threads s"
+ by (simp add:runing_def readys_def)
+ ultimately show ?thesis by auto
+ qed
+ hence "cntCS (e # s) th = cntCS s th "
+ apply (unfold cntCS_def holdents_test eq_e)
+ by (unfold step_RAG_p[OF vtp], auto)
+ moreover have "cntCS s th = 0"
+ proof(rule ih)
+ from not_in eq_e show "th \<notin> threads s" by simp
+ qed
+ ultimately show ?thesis by simp
+ next
+ case (thread_V thread cs)
+ assume eq_e: "e = V thread cs"
+ and is_runing: "thread \<in> runing s"
+ and hold: "holding s thread cs"
+ have neq_th: "th \<noteq> thread"
+ proof -
+ from not_in eq_e have "th \<notin> threads s" by simp
+ moreover from is_runing have "thread \<in> threads s"
+ by (simp add:runing_def readys_def)
+ ultimately show ?thesis by auto
+ qed
+ from assms thread_V vt stp ih
+ have vtv: "vt (V thread cs#s)" by auto
+ then interpret vt_v: valid_trace "(V thread cs#s)"
+ by (unfold_locales, simp)
+ from hold obtain rest
+ where eq_wq: "wq s cs = thread # rest"
+ by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+ from not_in eq_e eq_wq
+ have "\<not> next_th s thread cs th"
+ apply (auto simp:next_th_def)
+ proof -
+ assume ne: "rest \<noteq> []"
+ and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
+ have "?t \<in> set rest"
+ proof(rule someI2)
+ from vt_v.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest"
+ by (metis distinct.simps(2) vt_s.wq_distinct)
+ next
+ fix x assume "distinct x \<and> set x = set rest" with ne
+ show "hd x \<in> set rest" by (cases x, auto)
+ qed
+ with eq_wq have "?t \<in> set (wq s cs)" by simp
+ from vt_s.wq_threads[OF this] and ni
+ show False
+ using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)`
+ ni vt_s.wq_threads by blast
+ qed
+ moreover note neq_th eq_wq
+ ultimately have "cntCS (e # s) th = cntCS s th"
+ by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
+ moreover have "cntCS s th = 0"
+ proof(rule ih)
+ from not_in eq_e show "th \<notin> threads s" by simp
+ qed
+ ultimately show ?thesis by simp
+ next
+ case (thread_set thread prio)
+ print_facts
+ assume eq_e: "e = Set thread prio"
+ and is_runing: "thread \<in> runing s"
+ from not_in and eq_e have "th \<notin> threads s" by auto
+ from ih [OF this] and eq_e
+ show ?thesis
+ apply (unfold eq_e cntCS_def holdents_test)
+ by (simp add:RAG_set_unchanged)
+ qed
+ next
+ case vt_nil
+ show ?case
+ by (unfold cntCS_def,
+ auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+ qed
+qed
+
+end
+
+lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
+ by (auto simp:s_waiting_def cs_waiting_def wq_def)
+
+context valid_trace
+begin
+
+lemma dm_RAG_threads:
+ assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+ shows "th \<in> threads s"
+proof -
+ from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
+ moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
+ ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
+ hence "th \<in> set (wq s cs)"
+ by (unfold s_RAG_def, auto simp:cs_waiting_def)
+ from wq_threads [OF this] show ?thesis .
+qed
+
+end
+
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+thm cpreced_initial
+apply(simp add: Let_def cpreced_initial)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+
+context valid_trace
+begin
+
+lemma runing_unique:
+ assumes runing_1: "th1 \<in> runing s"
+ and runing_2: "th2 \<in> runing s"
+ shows "th1 = th2"
+proof -
+ from runing_1 and runing_2 have "cp s th1 = cp s th2"
+ unfolding runing_def
+ apply(simp)
+ done
+ hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
+ Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
+ (is "Max (?f ` ?A) = Max (?f ` ?B)")
+ unfolding cp_eq_cpreced
+ unfolding cpreced_def .
+ obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
+ proof -
+ have h1: "finite (?f ` ?A)"
+ proof -
+ have "finite ?A"
+ proof -
+ have "finite (dependants (wq s) th1)"
+ proof-
+ have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
+ proof -
+ let ?F = "\<lambda> (x, y). the_th x"
+ have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
+ moreover have "finite \<dots>"
+ proof -
+ from finite_RAG have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
+ apply (unfold finite_trancl)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
+ thus ?thesis by auto
+ qed
+ ultimately show ?thesis by (auto intro:finite_subset)
+ qed
+ thus ?thesis by (simp add:cs_dependants_def)
+ qed
+ thus ?thesis by simp
+ qed
+ thus ?thesis by auto
+ qed
+ moreover have h2: "(?f ` ?A) \<noteq> {}"
+ proof -
+ have "?A \<noteq> {}" by simp
+ thus ?thesis by simp
+ qed
+ from Max_in [OF h1 h2]
+ have "Max (?f ` ?A) \<in> (?f ` ?A)" .
+ thus ?thesis
+ thm cpreced_def
+ unfolding cpreced_def[symmetric]
+ unfolding cp_eq_cpreced[symmetric]
+ unfolding cpreced_def
+ using that[intro] by (auto)
+ qed
+ obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
+ proof -
+ have h1: "finite (?f ` ?B)"
+ proof -
+ have "finite ?B"
+ proof -
+ have "finite (dependants (wq s) th2)"
+ proof-
+ have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
+ proof -
+ let ?F = "\<lambda> (x, y). the_th x"
+ have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
+ moreover have "finite \<dots>"
+ proof -
+ from finite_RAG have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
+ apply (unfold finite_trancl)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
+ thus ?thesis by auto
+ qed
+ ultimately show ?thesis by (auto intro:finite_subset)
+ qed
+ thus ?thesis by (simp add:cs_dependants_def)
+ qed
+ thus ?thesis by simp
+ qed
+ thus ?thesis by auto
+ qed
+ moreover have h2: "(?f ` ?B) \<noteq> {}"
+ proof -
+ have "?B \<noteq> {}" by simp
+ thus ?thesis by simp
+ qed
+ from Max_in [OF h1 h2]
+ have "Max (?f ` ?B) \<in> (?f ` ?B)" .
+ thus ?thesis by (auto intro:that)
+ qed
+ from eq_f_th1 eq_f_th2 eq_max
+ have eq_preced: "preced th1' s = preced th2' s" by auto
+ hence eq_th12: "th1' = th2'"
+ proof (rule preced_unique)
+ from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
+ thus "th1' \<in> threads s"
+ proof
+ assume "th1' \<in> dependants (wq s) th1"
+ hence "(Th th1') \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+ by (auto simp:Domain_def)
+ hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ from dm_RAG_threads[OF this] show ?thesis .
+ next
+ assume "th1' = th1"
+ with runing_1 show ?thesis
+ by (unfold runing_def readys_def, auto)
+ qed
+ next
+ from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
+ thus "th2' \<in> threads s"
+ proof
+ assume "th2' \<in> dependants (wq s) th2"
+ hence "(Th th2') \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+ by (auto simp:Domain_def)
+ hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ from dm_RAG_threads[OF this] show ?thesis .
+ next
+ assume "th2' = th2"
+ with runing_2 show ?thesis
+ by (unfold runing_def readys_def, auto)
+ qed
+ qed
+ from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
+ thus ?thesis
+ proof
+ assume eq_th': "th1' = th1"
+ from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+ thus ?thesis
+ proof
+ assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+ next
+ assume "th2' \<in> dependants (wq s) th2"
+ with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
+ hence "(Th th1, Th th2) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ hence "Th th1 \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+ by (auto simp:Domain_def)
+ hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
+ from RAG_target_th [OF this]
+ obtain cs' where "n = Cs cs'" by auto
+ with d have "(Th th1, Cs cs') \<in> RAG s" by simp
+ with runing_1 have "False"
+ apply (unfold runing_def readys_def s_RAG_def)
+ by (auto simp:eq_waiting)
+ thus ?thesis by simp
+ qed
+ next
+ assume th1'_in: "th1' \<in> dependants (wq s) th1"
+ from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+ thus ?thesis
+ proof
+ assume "th2' = th2"
+ with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
+ hence "(Th th2, Th th1) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ hence "Th th2 \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+ by (auto simp:Domain_def)
+ hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
+ from RAG_target_th [OF this]
+ obtain cs' where "n = Cs cs'" by auto
+ with d have "(Th th2, Cs cs') \<in> RAG s" by simp
+ with runing_2 have "False"
+ apply (unfold runing_def readys_def s_RAG_def)
+ by (auto simp:eq_waiting)
+ thus ?thesis by simp
+ next
+ assume "th2' \<in> dependants (wq s) th2"
+ with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
+ hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ show ?thesis
+ proof(rule dchain_unique[OF h1 _ h2, symmetric])
+ from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
+ from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
+ qed
+ qed
+ qed
+qed
+
+
+lemma "card (runing s) \<le> 1"
+apply(subgoal_tac "finite (runing s)")
+prefer 2
+apply (metis finite_nat_set_iff_bounded lessI runing_unique)
+apply(rule ccontr)
+apply(simp)
+apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
+apply(subst (asm) card_le_Suc_iff)
+apply(simp)
+apply(auto)[1]
+apply (metis insertCI runing_unique)
+apply(auto)
+done
+
+end
+
+
+lemma create_pre:
+ assumes stp: "step s e"
+ and not_in: "th \<notin> threads s"
+ and is_in: "th \<in> threads (e#s)"
+ obtains prio where "e = Create th prio"
+proof -
+ from assms
+ show ?thesis
+ proof(cases)
+ case (thread_create thread prio)
+ with is_in not_in have "e = Create th prio" by simp
+ from that[OF this] show ?thesis .
+ next
+ case (thread_exit thread)
+ with assms show ?thesis by (auto intro!:that)
+ next
+ case (thread_P thread)
+ with assms show ?thesis by (auto intro!:that)
+ next
+ case (thread_V thread)
+ with assms show ?thesis by (auto intro!:that)
+ next
+ case (thread_set thread)
+ with assms show ?thesis by (auto intro!:that)
+ qed
+qed
+
+lemma length_down_to_in:
+ assumes le_ij: "i \<le> j"
+ and le_js: "j \<le> length s"
+ shows "length (down_to j i s) = j - i"
+proof -
+ have "length (down_to j i s) = length (from_to i j (rev s))"
+ by (unfold down_to_def, auto)
+ also have "\<dots> = j - i"
+ proof(rule length_from_to_in[OF le_ij])
+ from le_js show "j \<le> length (rev s)" by simp
+ qed
+ finally show ?thesis .
+qed
+
+
+lemma moment_head:
+ assumes le_it: "Suc i \<le> length t"
+ obtains e where "moment (Suc i) t = e#moment i t"
+proof -
+ have "i \<le> Suc i" by simp
+ from length_down_to_in [OF this le_it]
+ have "length (down_to (Suc i) i t) = 1" by auto
+ then obtain e where "down_to (Suc i) i t = [e]"
+ apply (cases "(down_to (Suc i) i t)") by auto
+ moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
+ by (rule down_to_conc[symmetric], auto)
+ ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
+ by (auto simp:down_to_moment)
+ from that [OF this] show ?thesis .
+qed
+
+context valid_trace
+begin
+
+lemma cnp_cnv_eq:
+ assumes "th \<notin> threads s"
+ shows "cntP s th = cntV s th"
+ using assms
+ using cnp_cnv_cncs not_thread_cncs by auto
+
+end
+
+
+lemma eq_RAG:
+ "RAG (wq s) = RAG s"
+by (unfold cs_RAG_def s_RAG_def, auto)
+
+context valid_trace
+begin
+
+lemma count_eq_dependants:
+ assumes eq_pv: "cntP s th = cntV s th"
+ shows "dependants (wq s) th = {}"
+proof -
+ from cnp_cnv_cncs and eq_pv
+ have "cntCS s th = 0"
+ by (auto split:if_splits)
+ moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
+ proof -
+ from finite_holding[of th] show ?thesis
+ by (simp add:holdents_test)
+ qed
+ ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
+ by (unfold cntCS_def holdents_test cs_dependants_def, auto)
+ show ?thesis
+ proof(unfold cs_dependants_def)
+ { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
+ then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
+ hence "False"
+ proof(cases)
+ assume "(Th th', Th th) \<in> RAG (wq s)"
+ thus "False" by (auto simp:cs_RAG_def)
+ next
+ fix c
+ assume "(c, Th th) \<in> RAG (wq s)"
+ with h and eq_RAG show "False"
+ by (cases c, auto simp:cs_RAG_def)
+ qed
+ } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
+ qed
+qed
+
+lemma dependants_threads:
+ shows "dependants (wq s) th \<subseteq> threads s"
+proof
+ { fix th th'
+ assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
+ have "Th th \<in> Domain (RAG s)"
+ proof -
+ from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
+ hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
+ with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
+ thus ?thesis using eq_RAG by simp
+ qed
+ from dm_RAG_threads[OF this]
+ have "th \<in> threads s" .
+ } note hh = this
+ fix th1
+ assume "th1 \<in> dependants (wq s) th"
+ hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
+ by (unfold cs_dependants_def, simp)
+ from hh [OF this] show "th1 \<in> threads s" .
+qed
+
+lemma finite_threads:
+ shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+lemma Max_f_mono:
+ assumes seq: "A \<subseteq> B"
+ and np: "A \<noteq> {}"
+ and fnt: "finite B"
+ shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+ from seq show "f ` A \<subseteq> f ` B" by auto
+next
+ from np show "f ` A \<noteq> {}" by auto
+next
+ from fnt and seq show "finite (f ` B)" by auto
+qed
+
+context valid_trace
+begin
+
+lemma cp_le:
+ assumes th_in: "th \<in> threads s"
+ shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
+ show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
+ \<le> Max ((\<lambda>th. preced th s) ` threads s)"
+ (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
+ proof(rule Max_f_mono)
+ show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
+ next
+ from finite_threads
+ show "finite (threads s)" .
+ next
+ from th_in
+ show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
+ apply (auto simp:Domain_def)
+ apply (rule_tac dm_RAG_threads)
+ apply (unfold trancl_domain [of "RAG s", symmetric])
+ by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
+ qed
+qed
+
+lemma le_cp:
+ shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+ show "Prc (priority th s) (last_set th s)
+ \<le> Max (insert (Prc (priority th s) (last_set th s))
+ ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
+ (is "?l \<le> Max (insert ?l ?A)")
+ proof(cases "?A = {}")
+ case False
+ have "finite ?A" (is "finite (?f ` ?B)")
+ proof -
+ have "finite ?B"
+ proof-
+ have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
+ proof -
+ let ?F = "\<lambda> (x, y). the_th x"
+ have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Th x, Th th)" in bexI, auto)
+ moreover have "finite \<dots>"
+ proof -
+ from finite_RAG have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
+ apply (unfold finite_trancl)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
+ thus ?thesis by auto
+ qed
+ ultimately show ?thesis by (auto intro:finite_subset)
+ qed
+ thus ?thesis by (simp add:cs_dependants_def)
+ qed
+ thus ?thesis by simp
+ qed
+ from Max_insert [OF this False, of ?l] show ?thesis by auto
+ next
+ case True
+ thus ?thesis by auto
+ qed
+qed
+
+lemma max_cp_eq:
+ shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
+ (is "?l = ?r")
+proof(cases "threads s = {}")
+ case True
+ thus ?thesis by auto
+next
+ case False
+ have "?l \<in> ((cp s) ` threads s)"
+ proof(rule Max_in)
+ from finite_threads
+ show "finite (cp s ` threads s)" by auto
+ next
+ from False show "cp s ` threads s \<noteq> {}" by auto
+ qed
+ then obtain th
+ where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
+ have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
+ moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
+ proof -
+ have "?r \<in> (?f ` ?A)"
+ proof(rule Max_in)
+ from finite_threads
+ show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
+ next
+ from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
+ qed
+ then obtain th' where
+ th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
+ from le_cp [of th'] eq_r
+ have "?r \<le> cp s th'" by auto
+ moreover have "\<dots> \<le> cp s th"
+ proof(fold eq_l)
+ show " cp s th' \<le> Max (cp s ` threads s)"
+ proof(rule Max_ge)
+ from th_in' show "cp s th' \<in> cp s ` threads s"
+ by auto
+ next
+ from finite_threads
+ show "finite (cp s ` threads s)" by auto
+ qed
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis using eq_l by auto
+qed
+
+lemma max_cp_readys_threads_pre:
+ assumes np: "threads s \<noteq> {}"
+ shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq)
+ show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
+ proof -
+ let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
+ let ?f = "(\<lambda>th. preced th s)"
+ have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
+ proof(rule Max_in)
+ from finite_threads show "finite (?f ` threads s)" by simp
+ next
+ from np show "?f ` threads s \<noteq> {}" by simp
+ qed
+ then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
+ by (auto simp:Image_def)
+ from th_chain_to_ready [OF tm_in]
+ have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
+ thus ?thesis
+ proof
+ assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
+ then obtain th' where th'_in: "th' \<in> readys s"
+ and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
+ have "cp s th' = ?f tm"
+ proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
+ from dependants_threads finite_threads
+ show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))"
+ by (auto intro:finite_subset)
+ next
+ fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+ from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
+ moreover have "p \<le> \<dots>"
+ proof(rule Max_ge)
+ from finite_threads
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ next
+ from p_in and th'_in and dependants_threads[of th']
+ show "p \<in> (\<lambda>th. preced th s) ` threads s"
+ by (auto simp:readys_def)
+ qed
+ ultimately show "p \<le> preced tm s" by auto
+ next
+ show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+ proof -
+ from tm_chain
+ have "tm \<in> dependants (wq s) th'"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
+ thus ?thesis by auto
+ qed
+ qed
+ with tm_max
+ have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+ show ?thesis
+ proof (fold h, rule Max_eqI)
+ fix q
+ assume "q \<in> cp s ` readys s"
+ then obtain th1 where th1_in: "th1 \<in> readys s"
+ and eq_q: "q = cp s th1" by auto
+ show "q \<le> cp s th'"
+ apply (unfold h eq_q)
+ apply (unfold cp_eq_cpreced cpreced_def)
+ apply (rule Max_mono)
+ proof -
+ from dependants_threads [of th1] th1_in
+ show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq>
+ (\<lambda>th. preced th s) ` threads s"
+ by (auto simp:readys_def)
+ next
+ show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
+ next
+ from finite_threads
+ show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ qed
+ next
+ from finite_threads
+ show "finite (cp s ` readys s)" by (auto simp:readys_def)
+ next
+ from th'_in
+ show "cp s th' \<in> cp s ` readys s" by simp
+ qed
+ next
+ assume tm_ready: "tm \<in> readys s"
+ show ?thesis
+ proof(fold tm_max)
+ have cp_eq_p: "cp s tm = preced tm s"
+ proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+ fix y
+ assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+ show "y \<le> preced tm s"
+ proof -
+ { fix y'
+ assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
+ have "y' \<le> preced tm s"
+ proof(unfold tm_max, rule Max_ge)
+ from hy' dependants_threads[of tm]
+ show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+ next
+ from finite_threads
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ qed
+ } with hy show ?thesis by auto
+ qed
+ next
+ from dependants_threads[of tm] finite_threads
+ show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
+ by (auto intro:finite_subset)
+ next
+ show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+ by simp
+ qed
+ moreover have "Max (cp s ` readys s) = cp s tm"
+ proof(rule Max_eqI)
+ from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
+ next
+ from finite_threads
+ show "finite (cp s ` readys s)" by (auto simp:readys_def)
+ next
+ fix y assume "y \<in> cp s ` readys s"
+ then obtain th1 where th1_readys: "th1 \<in> readys s"
+ and h: "y = cp s th1" by auto
+ show "y \<le> cp s tm"
+ apply(unfold cp_eq_p h)
+ apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
+ proof -
+ from finite_threads
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ next
+ show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
+ by simp
+ next
+ from dependants_threads[of th1] th1_readys
+ show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)
+ \<subseteq> (\<lambda>th. preced th s) ` threads s"
+ by (auto simp:readys_def)
+ qed
+ qed
+ ultimately show " Max (cp s ` readys s) = preced tm s" by simp
+ qed
+ qed
+ qed
+qed
+
+text {* (* ccc *) \noindent
+ Since the current precedence of the threads in ready queue will always be boosted,
+ there must be one inside it has the maximum precedence of the whole system.
+*}
+lemma max_cp_readys_threads:
+ shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(cases "threads s = {}")
+ case True
+ thus ?thesis
+ by (auto simp:readys_def)
+next
+ case False
+ show ?thesis by (rule max_cp_readys_threads_pre[OF False])
+qed
+
+end
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+ apply (unfold s_holding_def cs_holding_def wq_def, simp)
+ done
+
+lemma f_image_eq:
+ assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
+ shows "f ` A = g ` A"
+proof
+ show "f ` A \<subseteq> g ` A"
+ by(rule image_subsetI, auto intro:h)
+next
+ show "g ` A \<subseteq> f ` A"
+ by (rule image_subsetI, auto intro:h[symmetric])
+qed
+
+
+definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
+ where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
+
+
+lemma detached_test:
+ shows "detached s th = (Th th \<notin> Field (RAG s))"
+apply(simp add: detached_def Field_def)
+apply(simp add: s_RAG_def)
+apply(simp add: s_holding_abv s_waiting_abv)
+apply(simp add: Domain_iff Range_iff)
+apply(simp add: wq_def)
+apply(auto)
+done
+
+context valid_trace
+begin
+
+lemma detached_intro:
+ assumes eq_pv: "cntP s th = cntV s th"
+ shows "detached s th"
+proof -
+ from cnp_cnv_cncs
+ have eq_cnt: "cntP s th =
+ cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+ hence cncs_zero: "cntCS s th = 0"
+ by (auto simp:eq_pv split:if_splits)
+ with eq_cnt
+ have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
+ thus ?thesis
+ proof
+ assume "th \<notin> threads s"
+ with range_in dm_RAG_threads
+ show ?thesis
+ by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
+ next
+ assume "th \<in> readys s"
+ moreover have "Th th \<notin> Range (RAG s)"
+ proof -
+ from card_0_eq [OF finite_holding] and cncs_zero
+ have "holdents s th = {}"
+ by (simp add:cntCS_def)
+ thus ?thesis
+ apply(auto simp:holdents_test)
+ apply(case_tac a)
+ apply(auto simp:holdents_test s_RAG_def)
+ done
+ qed
+ ultimately show ?thesis
+ by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
+ qed
+qed
+
+lemma detached_elim:
+ assumes dtc: "detached s th"
+ shows "cntP s th = cntV s th"
+proof -
+ from cnp_cnv_cncs
+ have eq_pv: " cntP s th =
+ cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+ have cncs_z: "cntCS s th = 0"
+ proof -
+ from dtc have "holdents s th = {}"
+ unfolding detached_def holdents_test s_RAG_def
+ by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
+ thus ?thesis by (auto simp:cntCS_def)
+ qed
+ show ?thesis
+ proof(cases "th \<in> threads s")
+ case True
+ with dtc
+ have "th \<in> readys s"
+ by (unfold readys_def detached_def Field_def Domain_def Range_def,
+ auto simp:eq_waiting s_RAG_def)
+ with cncs_z and eq_pv show ?thesis by simp
+ next
+ case False
+ with cncs_z and eq_pv show ?thesis by simp
+ qed
+qed
+
+lemma detached_eq:
+ shows "(detached s th) = (cntP s th = cntV s th)"
+ by (insert vt, auto intro:detached_intro detached_elim)
+
+end
+
+text {*
+ The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived
+ from the concise and miniature model of PIP given in PrioGDef.thy.
+*}
+
+lemma eq_dependants: "dependants (wq s) = dependants s"
+ by (simp add: s_dependants_abv wq_def)
+
+lemma next_th_unique:
+ assumes nt1: "next_th s th cs th1"
+ and nt2: "next_th s th cs th2"
+ shows "th1 = th2"
+using assms by (unfold next_th_def, auto)
+
+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
+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)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/PIPDefs.thy Wed Jan 06 16:34:26 2016 +0000
@@ -0,0 +1,614 @@
+chapter {* Definitions *}
+(*<*)
+theory PIPDefs
+imports Precedence_ord Moment
+begin
+(*>*)
+
+text {*
+ In this section, the formal model of Priority Inheritance Protocol (PIP) is presented.
+ The model is based on Paulson's inductive protocol verification method, where
+ the state of the system is modelled as a list of events happened so far with the latest
+ event put at the head.
+*}
+
+text {*
+ To define events, the identifiers of {\em threads},
+ {\em priority} and {\em critical resources } (abbreviated as @{text "cs"})
+ need to be represented. All three are represetned using standard
+ Isabelle/HOL type @{typ "nat"}:
+*}
+
+type_synonym thread = nat -- {* Type for thread identifiers. *}
+type_synonym priority = nat -- {* Type for priorities. *}
+type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
+
+text {*
+ \noindent
+ The abstraction of Priority Inheritance Protocol (PIP) is set at the system call level.
+ Every system call is represented as an event. The format of events is defined
+ defined as follows:
+ *}
+
+datatype event =
+ Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *}
+ Exit thread | -- {* Thread @{text "thread"} finishing its execution. *}
+ P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *}
+ V thread cs | -- {* Thread @{text "thread"} releasing critical resource @{text "cs"}. *}
+ Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
+
+
+text {*
+ As mentioned earlier, in Paulson's inductive method, the states of system are represented as lists of events,
+ which is defined by the following type @{text "state"}:
+ *}
+type_synonym state = "event list"
+
+
+text {*
+\noindent
+ Resource Allocation Graph (RAG for short) is used extensively in our formal analysis.
+ The following type @{text "node"} is used to represent nodes in RAG.
+ *}
+datatype node =
+ Th "thread" | -- {* Node for thread. *}
+ Cs "cs" -- {* Node for critical resource. *}
+
+text {*
+ \noindent
+ The following function
+ @{text "threads"} is used to calculate the set of live threads (@{text "threads s"})
+ in state @{text "s"}.
+ *}
+fun threads :: "state \<Rightarrow> thread set"
+ where
+ -- {* At the start of the system, the set of threads is empty: *}
+ "threads [] = {}" |
+ -- {* New thread is added to the @{text "threads"}: *}
+ "threads (Create thread prio#s) = {thread} \<union> threads s" |
+ -- {* Finished thread is removed: *}
+ "threads (Exit thread # s) = (threads s) - {thread}" |
+ -- {* Other kind of events does not affect the value of @{text "threads"}: *}
+ "threads (e#s) = threads s"
+
+text {*
+ \noindent
+ The function @{text "threads"} defined above is one of
+ the so called {\em observation function}s which forms
+ the very basis of Paulson's inductive protocol verification method.
+ Each observation function {\em observes} one particular aspect (or attribute)
+ of the system. For example, the attribute observed by @{text "threads s"}
+ is the set of threads living in state @{text "s"}.
+ The protocol being modelled
+ The decision made the protocol being modelled is based on the {\em observation}s
+ returned by {\em observation function}s. Since {\observation function}s forms
+ the very basis on which Paulson's inductive method is based, there will be
+ a lot of such observation functions introduced in the following. In fact, any function
+ which takes event list as argument is a {\em observation function}.
+ *}
+
+text {* \noindent
+ Observation @{text "priority th s"} is
+ the {\em original priority} of thread @{text "th"} in state @{text "s"}.
+ The {\em original priority} is the priority
+ assigned to a thread when it is created or when it is reset by system call
+ (represented by event @{text "Set thread priority"}).
+*}
+
+fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
+ where
+ -- {* @{text "0"} is assigned to threads which have never been created: *}
+ "priority thread [] = 0" |
+ "priority thread (Create thread' prio#s) =
+ (if thread' = thread then prio else priority thread s)" |
+ "priority thread (Set thread' prio#s) =
+ (if thread' = thread then prio else priority thread s)" |
+ "priority thread (e#s) = priority thread s"
+
+text {*
+ \noindent
+ Observation @{text "last_set th s"} is the last time when the priority of thread @{text "th"} is set,
+ observed from state @{text "s"}.
+ The time in the system is measured by the number of events happened so far since the very beginning.
+*}
+fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
+ where
+ "last_set thread [] = 0" |
+ "last_set thread ((Create thread' prio)#s) =
+ (if (thread = thread') then length s else last_set thread s)" |
+ "last_set thread ((Set thread' prio)#s) =
+ (if (thread = thread') then length s else last_set thread s)" |
+ "last_set thread (_#s) = last_set thread s"
+
+text {*
+ \noindent
+ The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of
+ a thread is the combination of its {\em original priority} and {\em time} the priority is set.
+ The intention is to discriminate threads with the same priority by giving threads whose priority
+ is assigned earlier higher precedences, becasue such threads are more urgent to finish.
+ This explains the following definition:
+ *}
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+ where "preced thread s \<equiv> Prc (priority thread s) (last_set thread s)"
+
+
+text {*
+ \noindent
+ A number of important notions in PIP are represented as the following functions,
+ defined in terms of the waiting queues of the system, where the waiting queues
+ , as a whole, is represented by the @{text "wq"} argument of every notion function.
+ The @{text "wq"} argument is itself a functions which maps every critical resource
+ @{text "cs"} to the list of threads which are holding or waiting for it.
+ The thread at the head of this list is designated as the thread which is current
+ holding the resrouce, which is slightly different from tradition where
+ all threads in the waiting queue are considered as waiting for the resource.
+ *}
+
+consts
+ holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ RAG :: "'b \<Rightarrow> (node \<times> node) set"
+ dependants :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+defs (overloaded)
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ This meaning of @{text "wq"} is reflected in the following definition of @{text "holding wq th cs"},
+ where @{text "holding wq th cs"} means thread @{text "th"} is holding the critical
+ resource @{text "cs"}. This decision is based on @{text "wq"}.
+ \end{minipage}
+ *}
+
+ cs_holding_def:
+ "holding wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ In accordance with the definition of @{text "holding wq th cs"},
+ a thread @{text "th"} is considered waiting for @{text "cs"} if
+ it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head.
+ This is reflected in the definition of @{text "waiting wq th cs"} as follows:
+ \end{minipage}
+ *}
+ cs_waiting_def:
+ "waiting wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ @{text "RAG wq"} generates RAG (a binary relations on @{text "node"})
+ out of waiting queues of the system (represented by the @{text "wq"} argument):
+ \end{minipage}
+ *}
+ cs_RAG_def:
+ "RAG (wq::cs \<Rightarrow> thread list) \<equiv>
+ {(Th th, Cs cs) | th cs. waiting wq th cs} \<union> {(Cs cs, Th th) | cs th. holding wq th cs}"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ The following @{text "dependants wq th"} represents the set of threads which are RAGing on
+ thread @{text "th"} in Resource Allocation Graph @{text "RAG wq"}.
+ Here, "RAGing" means waiting directly or indirectly on the critical resource.
+ \end{minipage}
+ *}
+ cs_dependants_def:
+ "dependants (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
+
+
+text {* \noindent
+ The following
+ @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under
+ state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of
+ Priority Inheritance that the {\em current precedence} of a thread is the precedence
+ inherited from the maximum of all its dependants, i.e. the threads which are waiting
+ directly or indirectly waiting for some resources from it. If no such thread exits,
+ @{text "th"}'s {\em current precedence} equals its original precedence, i.e.
+ @{text "preced th s"}.
+ *}
+
+definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
+ where "cpreced wq s = (\<lambda>th. Max ((\<lambda>th'. preced th' s) ` ({th} \<union> dependants wq th)))"
+
+text {*
+ Notice that the current precedence (@{text "cpreced"}) of one thread @{text "th"} can be boosted
+ (becoming larger than its own precedence) by those threads in
+ the @{text "dependants wq th"}-set. If one thread get boosted, we say
+ it inherits the priority (or, more precisely, the precedence) of
+ its dependants. This is how the word "Inheritance" in
+ Priority Inheritance Protocol comes.
+*}
+
+(*<*)
+lemma
+ cpreced_def2:
+ "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
+ unfolding cpreced_def image_def
+ apply(rule eq_reflection)
+ apply(rule_tac f="Max" in arg_cong)
+ by (auto)
+(*>*)
+
+
+text {* \noindent
+ Assuming @{text "qs"} be the waiting queue of a critical resource,
+ the following abbreviation "release qs" is the waiting queue after the thread
+ holding the resource (which is thread at the head of @{text "qs"}) released
+ the resource:
+*}
+abbreviation
+ "release qs \<equiv> case qs of
+ [] => []
+ | (_#qs') => (SOME q. distinct q \<and> set q = set qs')"
+text {* \noindent
+ It can be seen from the definition that the thread at the head of @{text "qs"} is removed
+ from the return value, and the value @{term "q"} is an reordering of @{text "qs'"}, the
+ tail of @{text "qs"}. Through this reordering, one of the waiting threads (those in @{text "qs'"} }
+ is chosen nondeterministically to be the head of the new queue @{text "q"}.
+ Therefore, this thread is the one who takes over the resource. This is a little better different
+ from common sense that the thread who comes the earliest should take over.
+ The intention of this definition is to show that the choice of which thread to take over the
+ release resource does not affect the correctness of the PIP protocol.
+*}
+
+text {*
+ The data structure used by the operating system for scheduling is referred to as
+ {\em schedule state}. It is represented as a record consisting of
+ a function assigning waiting queue to resources
+ (to be used as the @{text "wq"} argument in @{text "holding"}, @{text "waiting"}
+ and @{text "RAG"}, etc) and a function assigning precedence to threads:
+ *}
+
+record schedule_state =
+ wq_fun :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
+ cprec_fun :: "thread \<Rightarrow> precedence" -- {* The function assigning precedence. *}
+
+text {* \noindent
+ The following two abbreviations (@{text "all_unlocked"} and @{text "initial_cprec"})
+ are used to set the initial values of the @{text "wq_fun"} @{text "cprec_fun"} fields
+ respectively of the @{text "schedule_state"} record by the following function @{text "sch"},
+ which is used to calculate the system's {\em schedule state}.
+
+ Since there is no thread at the very beginning to make request, all critical resources
+ are free (or unlocked). This status is represented by the abbreviation
+ @{text "all_unlocked"}.
+ *}
+abbreviation
+ "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
+
+
+text {* \noindent
+ The initial current precedence for a thread can be anything, because there is no thread then.
+ We simply assume every thread has precedence @{text "Prc 0 0"}.
+ *}
+
+abbreviation
+ "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
+
+
+text {* \noindent
+ The following function @{text "schs"} is used to calculate the system's schedule state @{text "schs s"}
+ out of the current system state @{text "s"}. It is the central function to model Priority Inheritance:
+ *}
+fun schs :: "state \<Rightarrow> schedule_state"
+ where
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ Setting the initial value of the @{text "schedule_state"} record (see the explanations above).
+ \end{minipage}
+ *}
+ "schs [] = (| wq_fun = all_unlocked, cprec_fun = initial_cprec |)" |
+
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ \begin{enumerate}
+ \item @{text "ps"} is the schedule state of last moment.
+ \item @{text "pwq"} is the waiting queue function of last moment.
+ \item @{text "pcp"} is the precedence function of last moment (NOT USED).
+ \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement:
+ \begin{enumerate}
+ \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to
+ the end of @{text "cs"}'s waiting queue.
+ \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state,
+ @{text "th'"} must equal to @{text "thread"},
+ because @{text "thread"} is the one currently holding @{text "cs"}.
+ The case @{text "[] \<Longrightarrow> []"} may never be executed in a legal state.
+ the @{text "(SOME q. distinct q \<and> set q = set qs)"} is used to choose arbitrarily one
+ thread in waiting to take over the released resource @{text "cs"}. In our representation,
+ this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
+ \item For other happening event, the schedule state just does not change.
+ \end{enumerate}
+ \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue
+ function. The RAGency of precedence function on waiting queue function is the reason to
+ put them in the same record so that they can evolve together.
+ \end{enumerate}
+
+
+ The calculation of @{text "cprec_fun"} depends on the value of @{text "wq_fun"}.
+ Therefore, in the following cases, @{text "wq_fun"} is always calculated first, in
+ the name of @{text "wq"} (if @{text "wq_fun"} is not changed
+ by the happening event) or @{text "new_wq"} (if the value of @{text "wq_fun"} is changed).
+ \end{minipage}
+ *}
+ "schs (Create th prio # s) =
+ (let wq = wq_fun (schs s) in
+ (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"
+| "schs (Exit th # s) =
+ (let wq = wq_fun (schs s) in
+ (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"
+| "schs (Set th prio # s) =
+ (let wq = wq_fun (schs s) in
+ (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ Different from the forth coming cases, the @{text "wq_fun"} field of the schedule state
+ is changed. So, the new value is calculated first, in the name of @{text "new_wq"}.
+ \end{minipage}
+ *}
+| "schs (P th cs # s) =
+ (let wq = wq_fun (schs s) in
+ let new_wq = wq(cs := (wq cs @ [th])) in
+ (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"
+| "schs (V th cs # s) =
+ (let wq = wq_fun (schs s) in
+ let new_wq = wq(cs := release (wq cs)) in
+ (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"
+
+lemma cpreced_initial:
+ "cpreced (\<lambda> cs. []) [] = (\<lambda>_. (Prc 0 0))"
+apply(simp add: cpreced_def)
+apply(simp add: cs_dependants_def cs_RAG_def cs_waiting_def cs_holding_def)
+apply(simp add: preced_def)
+done
+
+lemma sch_old_def:
+ "schs (e#s) = (let ps = schs s in
+ let pwq = wq_fun ps in
+ let nwq = case e of
+ P th cs \<Rightarrow> pwq(cs:=(pwq cs @ [th])) |
+ V th cs \<Rightarrow> let nq = case (pwq cs) of
+ [] \<Rightarrow> [] |
+ (_#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
+ in pwq(cs:=nq) |
+ _ \<Rightarrow> pwq
+ in let ncp = cpreced nwq (e#s) in
+ \<lparr>wq_fun = nwq, cprec_fun = ncp\<rparr>
+ )"
+apply(cases e)
+apply(simp_all)
+done
+
+
+text {*
+ \noindent
+ The following @{text "wq"} is a shorthand for @{text "wq_fun"}.
+ *}
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
+ where "wq s = wq_fun (schs s)"
+
+text {* \noindent
+ The following @{text "cp"} is a shorthand for @{text "cprec_fun"}.
+ *}
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+ where "cp s \<equiv> cprec_fun (schs s)"
+
+text {* \noindent
+ Functions @{text "holding"}, @{text "waiting"}, @{text "RAG"} and
+ @{text "dependants"} still have the
+ same meaning, but redefined so that they no longer RAG on the
+ fictitious {\em waiting queue function}
+ @{text "wq"}, but on system state @{text "s"}.
+ *}
+defs (overloaded)
+ s_holding_abv:
+ "holding (s::state) \<equiv> holding (wq_fun (schs s))"
+ s_waiting_abv:
+ "waiting (s::state) \<equiv> waiting (wq_fun (schs s))"
+ s_RAG_abv:
+ "RAG (s::state) \<equiv> RAG (wq_fun (schs s))"
+ s_dependants_abv:
+ "dependants (s::state) \<equiv> dependants (wq_fun (schs s))"
+
+
+text {*
+ The following lemma can be proved easily, and the meaning is obvious.
+ *}
+lemma
+ s_holding_def:
+ "holding (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th = hd (wq_fun (schs s) cs))"
+ by (auto simp:s_holding_abv wq_def cs_holding_def)
+
+lemma s_waiting_def:
+ "waiting (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th \<noteq> hd (wq_fun (schs s) cs))"
+ by (auto simp:s_waiting_abv wq_def cs_waiting_def)
+
+lemma s_RAG_def:
+ "RAG (s::state) =
+ {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \<union> {(Cs cs, Th th) | cs th. holding (wq s) th cs}"
+ by (auto simp:s_RAG_abv wq_def cs_RAG_def)
+
+lemma
+ s_dependants_def:
+ "dependants (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (RAG (wq s))^+}"
+ by (auto simp:s_dependants_abv wq_def cs_dependants_def)
+
+text {*
+ The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready}
+ for running if it is a live thread and it is not waiting for any critical resource.
+ *}
+definition readys :: "state \<Rightarrow> thread set"
+ where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waiting s th cs)}"
+
+text {* \noindent
+ The following function @{text "runing"} calculates the set of running thread, which is the ready
+ thread with the highest precedence.
+ *}
+definition runing :: "state \<Rightarrow> thread set"
+ where "runing s \<equiv> {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
+
+text {* \noindent
+ Notice that the definition of @{text "running"} reflects the preemptive scheduling strategy,
+ because, if the @{text "running"}-thread (the one in @{text "runing"} set)
+ lowered its precedence by resetting its own priority to a lower
+ one, it will lose its status of being the max in @{text "ready"}-set and be superseded.
+*}
+
+text {* \noindent
+ The following function @{text "holdents s th"} returns the set of resources held by thread
+ @{text "th"} in state @{text "s"}.
+ *}
+definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+ where "holdents s th \<equiv> {cs . holding s th cs}"
+
+lemma holdents_test:
+ "holdents s th = {cs . (Cs cs, Th th) \<in> RAG s}"
+unfolding holdents_def
+unfolding s_RAG_def
+unfolding s_holding_abv
+unfolding wq_def
+by (simp)
+
+text {* \noindent
+ Observation @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in
+ state @{text "s"}:
+ *}
+definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntCS s th = card (holdents s th)"
+
+text {* \noindent
+ According to the convention of Paulson's inductive method,
+ the decision made by a protocol that event @{text "e"} is eligible to happen next under state @{text "s"}
+ is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as
+ follows (notice how the decision is based on the {\em observation function}s
+ defined above, and also notice how a complicated protocol is modeled by a few simple
+ observations, and how such a kind of simplicity gives rise to improved trust on
+ faithfulness):
+ *}
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+ where
+ -- {*
+ A thread can be created if it is not a live thread:
+ *}
+ thread_create: "\<lbrakk>thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
+ -- {*
+ A thread can exit if it no longer hold any resource:
+ *}
+ thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ A thread can request for an critical resource @{text "cs"}, if it is running and
+ the request does not form a loop in the current RAG. The latter condition
+ is set up to avoid deadlock. The condition also reflects our assumption all threads are
+ carefully programmed so that deadlock can not happen:
+ \end{minipage}
+ *}
+ thread_P: "\<lbrakk>thread \<in> runing s; (Cs cs, Th thread) \<notin> (RAG s)^+\<rbrakk> \<Longrightarrow>
+ step s (P thread cs)" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ A thread can release a critical resource @{text "cs"}
+ if it is running and holding that resource:
+ \end{minipage}
+ *}
+ thread_V: "\<lbrakk>thread \<in> runing s; holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ A thread can adjust its own priority as long as it is current running.
+ With the resetting of one thread's priority, its precedence may change.
+ If this change lowered the precedence, according to the definition of @{text "running"}
+ function,
+ \end{minipage}
+ *}
+ thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+text {*
+ In Paulson's inductive method, every protocol is defined by such a @{text "step"}
+ predicate. For instance, the predicate @{text "step"} given above
+ defines the PIP protocol. So, it can also be called "PIP".
+*}
+
+abbreviation
+ "PIP \<equiv> step"
+
+
+text {* \noindent
+ For any protocol defined by a @{text "step"} predicate,
+ the fact that @{text "s"} is a legal state in
+ the protocol is expressed as: @{text "vt step s"}, where
+ the predicate @{text "vt"} can be defined as the following:
+ *}
+inductive vt :: "state \<Rightarrow> bool"
+ where
+ -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
+ vt_nil[intro]: "vt []" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ If @{text "s"} a legal state of the protocol defined by predicate @{text "step"},
+ and event @{text "e"} is allowed to happen under state @{text "s"} by the protocol
+ predicate @{text "step"}, then @{text "e#s"} is a new legal state rendered by the
+ happening of @{text "e"}:
+ \end{minipage}
+ *}
+ vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
+
+text {* \noindent
+ It is easy to see that the definition of @{text "vt"} is generic. It can be applied to
+ any specific protocol specified by a @{text "step"}-predicate to get the set of
+ legal states of that particular protocol.
+ *}
+
+text {*
+ The following are two very basic properties of @{text "vt"}.
+*}
+
+lemma step_back_vt: "vt (e#s) \<Longrightarrow> vt s"
+ by(ind_cases "vt (e#s)", simp)
+
+lemma step_back_step: "vt (e#s) \<Longrightarrow> step s e"
+ by(ind_cases "vt (e#s)", simp)
+
+text {* \noindent
+ The following two auxiliary functions @{text "the_cs"} and @{text "the_th"} are used to extract
+ critical resource and thread respectively out of RAG nodes.
+ *}
+fun the_cs :: "node \<Rightarrow> cs"
+ where "the_cs (Cs cs) = cs"
+
+fun the_th :: "node \<Rightarrow> thread"
+ where "the_th (Th th) = th"
+
+text {* \noindent
+ The following predicate @{text "next_th"} describe the next thread to
+ take over when a critical resource is released. In @{text "next_th s th cs t"},
+ @{text "th"} is the thread to release, @{text "t"} is the one to take over.
+ Notice how this definition is backed up by the @{text "release"} function and its use
+ in the @{text "V"}-branch of @{text "schs"} function. This @{text "next_th"} function
+ is not needed for the execution of PIP. It is introduced as an auxiliary function
+ to state lemmas. The correctness of this definition will be confirmed by
+ lemmas @{text "step_v_hold_inv"}, @{text " step_v_wait_inv"},
+ @{text "step_v_get_hold"} and @{text "step_v_not_wait"}.
+ *}
+definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
+ where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and>
+ t = hd (SOME q. distinct q \<and> set q = set rest))"
+
+text {* \noindent
+ The aux function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
+ in list @{text "l"}:
+ *}
+definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
+ where "count Q l = length (filter Q l)"
+
+text {* \noindent
+ The following observation @{text "cntP s"} returns the number of operation @{text "P"} happened
+ before reaching state @{text "s"}.
+ *}
+definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
+
+text {* \noindent
+ The following observation @{text "cntV s"} returns the number of operation @{text "V"} happened
+ before reaching state @{text "s"}.
+ *}
+definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+(*<*)
+
+end
+(*>*)
+
--- a/PrioG.thy Wed Jan 06 20:46:14 2016 +0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,3051 +0,0 @@
-theory PrioG
-imports PrioGDef
-begin
-
-locale valid_trace =
- fixes s
- assumes vt : "vt s"
-
-locale valid_trace_e = valid_trace +
- fixes e
- assumes vt_e: "vt (e#s)"
-begin
-
-lemma pip_e: "PIP s e"
- using vt_e by (cases, simp)
-
-end
-
-lemma runing_ready:
- shows "runing s \<subseteq> readys s"
- unfolding runing_def readys_def
- by auto
-
-lemma readys_threads:
- shows "readys s \<subseteq> threads s"
- unfolding readys_def
- by auto
-
-lemma wq_v_neq:
- "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
- by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-context valid_trace
-begin
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes "PP []"
- and "(\<And>s e. valid_trace s \<Longrightarrow> valid_trace (e#s) \<Longrightarrow>
- PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
- shows "PP s"
-proof(rule vt.induct[OF vt])
- from assms(1) show "PP []" .
-next
- fix s e
- assume h: "vt s" "PP s" "PIP s e"
- show "PP (e # s)"
- proof(cases rule:assms(2))
- from h(1) show v1: "valid_trace s" by (unfold_locales, simp)
- next
- from h(1,3) have "vt (e#s)" by auto
- thus "valid_trace (e # s)" by (unfold_locales, simp)
- qed (insert h, auto)
-qed
-
-lemma wq_distinct: "distinct (wq s cs)"
-proof(rule ind, simp add:wq_def)
- fix s e
- assume h1: "step s e"
- and h2: "distinct (wq s cs)"
- thus "distinct (wq (e # s) cs)"
- proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
- fix thread s
- assume h1: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
- and h2: "thread \<in> set (wq_fun (schs s) cs)"
- and h3: "thread \<in> runing s"
- show "False"
- proof -
- from h3 have "\<And> cs. thread \<in> set (wq_fun (schs s) cs) \<Longrightarrow>
- thread = hd ((wq_fun (schs s) cs))"
- by (simp add:runing_def readys_def s_waiting_def wq_def)
- from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" .
- with h2
- have "(Cs cs, Th thread) \<in> (RAG s)"
- by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
- with h1 show False by auto
- qed
- next
- fix thread s a list
- assume dst: "distinct list"
- show "distinct (SOME q. distinct q \<and> set q = set list)"
- proof(rule someI2)
- from dst show "distinct list \<and> set list = set list" by auto
- next
- fix q assume "distinct q \<and> set q = set list"
- thus "distinct q" by auto
- qed
- qed
-qed
-
-end
-
-
-context valid_trace_e
-begin
-
-text {*
- The following lemma shows that only the @{text "P"}
- operation can add new thread into waiting queues.
- Such kind of lemmas are very obvious, but need to be checked formally.
- This is a kind of confirmation that our modelling is correct.
-*}
-
-lemma block_pre:
- assumes s_ni: "thread \<notin> set (wq s cs)"
- and s_i: "thread \<in> set (wq (e#s) cs)"
- shows "e = P thread cs"
-proof -
- show ?thesis
- proof(cases e)
- case (P th cs)
- with assms
- show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Create th prio)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Exit th)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Set th prio)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (V th cs)
- with vt_e assms show ?thesis
- apply (auto simp:wq_def Let_def split:if_splits)
- proof -
- fix q qs
- assume h1: "thread \<notin> set (wq_fun (schs s) cs)"
- and h2: "q # qs = wq_fun (schs s) cs"
- and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
- and vt: "vt (V th cs # s)"
- from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp
- moreover have "thread \<in> set qs"
- proof -
- have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
- proof(rule someI2)
- from wq_distinct [of cs]
- and h2[symmetric, folded wq_def]
- show "distinct qs \<and> set qs = set qs" by auto
- next
- fix x assume "distinct x \<and> set x = set qs"
- thus "set x = set qs" by auto
- qed
- with h3 show ?thesis by simp
- qed
- ultimately show "False" by auto
- qed
- qed
-qed
-
-end
-
-text {*
- The following lemmas is also obvious and shallow. It says
- that only running thread can request for a critical resource
- and that the requested resource must be one which is
- not current held by the thread.
-*}
-
-lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
- thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (RAG s)^+"
-apply (ind_cases "vt ((P thread cs)#s)")
-apply (ind_cases "step s (P thread cs)")
-by auto
-
-lemma abs1:
- assumes ein: "e \<in> set es"
- and neq: "hd es \<noteq> hd (es @ [x])"
- shows "False"
-proof -
- from ein have "es \<noteq> []" by auto
- then obtain e ess where "es = e # ess" by (cases es, auto)
- with neq show ?thesis by auto
-qed
-
-lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
- by (cases es, auto)
-
-inductive_cases evt_cons: "vt (a#s)"
-
-context valid_trace_e
-begin
-
-lemma abs2:
- assumes inq: "thread \<in> set (wq s cs)"
- and nh: "thread = hd (wq s cs)"
- and qt: "thread \<noteq> hd (wq (e#s) cs)"
- and inq': "thread \<in> set (wq (e#s) cs)"
- shows "False"
-proof -
- from vt_e assms show "False"
- apply (cases e)
- apply ((simp split:if_splits add:Let_def wq_def)[1])+
- apply (insert abs1, fast)[1]
- apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)
- proof -
- fix th qs
- assume vt: "vt (V th cs # s)"
- and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
- and eq_wq: "wq_fun (schs s) cs = thread # qs"
- show "False"
- proof -
- from wq_distinct[of cs]
- and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
- moreover have "thread \<in> set qs"
- proof -
- have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
- proof(rule someI2)
- from wq_distinct [of cs]
- and eq_wq [folded wq_def]
- show "distinct qs \<and> set qs = set qs" by auto
- next
- fix x assume "distinct x \<and> set x = set qs"
- thus "set x = set qs" by auto
- qed
- with th_in show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- qed
-qed
-
-end
-
-context valid_trace
-begin
-
-lemma vt_moment: "\<And> t. vt (moment t s)"
-proof(induct rule:ind)
- case Nil
- thus ?case by (simp add:vt_nil)
-next
- case (Cons s e t)
- show ?case
- proof(cases "t \<ge> length (e#s)")
- case True
- from True have "moment t (e#s) = e#s" by simp
- thus ?thesis using Cons
- by (simp add:valid_trace_def)
- next
- case False
- from Cons have "vt (moment t s)" by simp
- moreover have "moment t (e#s) = moment t s"
- proof -
- from False have "t \<le> length s" by simp
- from moment_app [OF this, of "[e]"]
- show ?thesis by simp
- qed
- ultimately show ?thesis by simp
- qed
-qed
-
-(* Wrong:
- lemma \<lbrakk>thread \<in> set (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
-*)
-
-text {* (* ddd *)
- The nature of the work is like this: since it starts from a very simple and basic
- model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
- For instance, the fact
- that one thread can not be blocked by two critical resources at the same time
- is obvious, because only running threads can make new requests, if one is waiting for
- a critical resource and get blocked, it can not make another resource request and get
- blocked the second time (because it is not running).
-
- To derive this fact, one needs to prove by contraction and
- reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
- named @{text "p_split"}, which is about status changing along the time axis. It says if
- a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
- but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"}
- in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history
- of events leading to it), such that @{text "Q"} switched
- from being @{text "False"} to @{text "True"} and kept being @{text "True"}
- till the last moment of @{text "s"}.
-
- Suppose a thread @{text "th"} is blocked
- on @{text "cs1"} and @{text "cs2"} in some state @{text "s"},
- since no thread is blocked at the very beginning, by applying
- @{text "p_split"} to these two blocking facts, there exist
- two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that
- @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"}
- and kept on blocked on them respectively ever since.
-
- Without lose of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
- However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
- in blocked state at moment @{text "t2"} and could not
- make any request and get blocked the second time: Contradiction.
-*}
-
-lemma waiting_unique_pre:
- assumes h11: "thread \<in> set (wq s cs1)"
- and h12: "thread \<noteq> hd (wq s cs1)"
- assumes h21: "thread \<in> set (wq s cs2)"
- and h22: "thread \<noteq> hd (wq s cs2)"
- and neq12: "cs1 \<noteq> cs2"
- shows "False"
-proof -
- let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
- from h11 and h12 have q1: "?Q cs1 s" by simp
- from h21 and h22 have q2: "?Q cs2 s" by simp
- have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
- have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
- from p_split [of "?Q cs1", OF q1 nq1]
- obtain t1 where lt1: "t1 < length s"
- and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
- thread \<noteq> hd (wq (moment t1 s) cs1))"
- and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
- thread \<noteq> hd (wq (moment i' s) cs1))" by auto
- from p_split [of "?Q cs2", OF q2 nq2]
- obtain t2 where lt2: "t2 < length s"
- and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
- thread \<noteq> hd (wq (moment t2 s) cs2))"
- and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
- thread \<noteq> hd (wq (moment i' s) cs2))" by auto
- show ?thesis
- proof -
- {
- assume lt12: "t1 < t2"
- let ?t3 = "Suc t2"
- from lt2 have le_t3: "?t3 \<le> length s" by auto
- from moment_plus [OF this]
- obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
- have "t2 < ?t3" by simp
- from nn2 [rule_format, OF this] and eq_m
- have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- have "vt (e#moment t2 s)"
- proof -
- from vt_moment
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- then interpret vt_e: valid_trace_e "moment t2 s" "e"
- by (unfold_locales, auto, cases, simp)
- have ?thesis
- proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
- case True
- from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
- by auto
- from vt_e.abs2 [OF True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from vt_e.block_pre[OF False h1]
- have "e = P thread cs2" .
- with vt_e.vt_e have "vt ((P thread cs2)# moment t2 s)" by simp
- from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
- with runing_ready have "thread \<in> readys (moment t2 s)" by auto
- with nn1 [rule_format, OF lt12]
- show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto)
- qed
- } moreover {
- assume lt12: "t2 < t1"
- let ?t3 = "Suc t1"
- from lt1 have le_t3: "?t3 \<le> length s" by auto
- from moment_plus [OF this]
- obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
- have lt_t3: "t1 < ?t3" by simp
- from nn1 [rule_format, OF this] and eq_m
- have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
- h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have "vt (e#moment t1 s)"
- proof -
- from vt_moment
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- then interpret vt_e: valid_trace_e "moment t1 s" e
- by (unfold_locales, auto, cases, auto)
- have ?thesis
- proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
- case True
- from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
- by auto
- from vt_e.abs2 True eq_th h2 h1
- show ?thesis by auto
- next
- case False
- from vt_e.block_pre [OF False h1]
- have "e = P thread cs1" .
- with vt_e.vt_e have "vt ((P thread cs1)# moment t1 s)" by simp
- from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
- with runing_ready have "thread \<in> readys (moment t1 s)" by auto
- with nn2 [rule_format, OF lt12]
- show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto)
- qed
- } moreover {
- assume eqt12: "t1 = t2"
- let ?t3 = "Suc t1"
- from lt1 have le_t3: "?t3 \<le> length s" by auto
- from moment_plus [OF this]
- obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
- have lt_t3: "t1 < ?t3" by simp
- from nn1 [rule_format, OF this] and eq_m
- have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
- h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have vt_e: "vt (e#moment t1 s)"
- proof -
- from vt_moment
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- then interpret vt_e: valid_trace_e "moment t1 s" e
- by (unfold_locales, auto, cases, auto)
- have ?thesis
- proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
- case True
- from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
- by auto
- from vt_e.abs2 [OF True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from vt_e.block_pre [OF False h1]
- have eq_e1: "e = P thread cs1" .
- have lt_t3: "t1 < ?t3" by simp
- with eqt12 have "t2 < ?t3" by simp
- from nn2 [rule_format, OF this] and eq_m and eqt12
- have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- show ?thesis
- proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
- case True
- from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
- by auto
- from vt_e and eqt12 have "vt (e#moment t2 s)" by simp
- then interpret vt_e2: valid_trace_e "moment t2 s" e
- by (unfold_locales, auto, cases, auto)
- from vt_e2.abs2 [OF True eq_th h2 h1]
- show ?thesis .
- next
- case False
- have "vt (e#moment t2 s)"
- proof -
- from vt_moment eqt12
- have "vt (moment (Suc t2) s)" by auto
- with eq_m eqt12 show ?thesis by simp
- qed
- then interpret vt_e2: valid_trace_e "moment t2 s" e
- by (unfold_locales, auto, cases, auto)
- from vt_e2.block_pre [OF False h1]
- have "e = P thread cs2" .
- with eq_e1 neq12 show ?thesis by auto
- qed
- qed
- } ultimately show ?thesis by arith
- qed
-qed
-
-text {*
- This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
-*}
-
-lemma waiting_unique:
- assumes "waiting s th cs1"
- and "waiting s th cs2"
- shows "cs1 = cs2"
-using waiting_unique_pre assms
-unfolding wq_def s_waiting_def
-by auto
-
-end
-
-(* not used *)
-text {*
- Every thread can only be blocked on one critical resource,
- symmetrically, every critical resource can only be held by one thread.
- This fact is much more easier according to our definition.
-*}
-lemma held_unique:
- assumes "holding (s::event list) th1 cs"
- and "holding s th2 cs"
- shows "th1 = th2"
- by (insert assms, unfold s_holding_def, auto)
-
-
-lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits)
-
-lemma last_set_unique:
- "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
- \<Longrightarrow> th1 = th2"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits dest:last_set_lt)
-
-lemma preced_unique :
- assumes pcd_eq: "preced th1 s = preced th2 s"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "th1 = th2"
-proof -
- from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
- from last_set_unique [OF this th_in1 th_in2]
- show ?thesis .
-qed
-
-lemma preced_linorder:
- assumes neq_12: "th1 \<noteq> th2"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
- from preced_unique [OF _ th_in1 th_in2] and neq_12
- have "preced th1 s \<noteq> preced th2 s" by auto
- thus ?thesis by auto
-qed
-
-(* An aux lemma used later *)
-lemma unique_minus:
- fixes x y z r
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq: "y \<noteq> z"
- shows "(y, z) \<in> r^+"
-proof -
- from xz and neq show ?thesis
- proof(induct)
- case (base ya)
- have "(x, ya) \<in> r" by fact
- from unique [OF xy this] have "y = ya" .
- with base show ?case by auto
- next
- case (step ya z)
- show ?case
- proof(cases "y = ya")
- case True
- from step True show ?thesis by simp
- next
- case False
- from step False
- show ?thesis by auto
- qed
- qed
-qed
-
-lemma unique_base:
- fixes r x y z
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> r^+"
-proof -
- from xz neq_yz show ?thesis
- proof(induct)
- case (base ya)
- from xy unique base show ?case by auto
- next
- case (step ya z)
- show ?case
- proof(cases "y = ya")
- case True
- from True step show ?thesis by auto
- next
- case False
- from False step
- have "(y, ya) \<in> r\<^sup>+" by auto
- with step show ?thesis by auto
- qed
- qed
-qed
-
-lemma unique_chain:
- fixes r x y z
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r^+"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
-proof -
- from xy xz neq_yz show ?thesis
- proof(induct)
- case (base y)
- have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
- from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
- next
- case (step y za)
- show ?case
- proof(cases "y = z")
- case True
- from True step show ?thesis by auto
- next
- case False
- from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
- thus ?thesis
- proof
- assume "(z, y) \<in> r\<^sup>+"
- with step have "(z, za) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- next
- assume h: "(y, z) \<in> r\<^sup>+"
- from step have yza: "(y, za) \<in> r" by simp
- from step have "za \<noteq> z" by simp
- from unique_minus [OF _ yza h this] and unique
- have "(za, z) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- qed
- qed
- qed
-qed
-
-text {*
- The following three lemmas show that @{text "RAG"} does not change
- by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
- events, respectively.
-*}
-
-lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-
-text {*
- The following lemmas are used in the proof of
- lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed
- by @{text "V"}-events.
- However, since our model is very concise, such seemingly obvious lemmas need to be derived from scratch,
- starting from the model definitions.
-*}
-lemma step_v_hold_inv[elim_format]:
- "\<And>c t. \<lbrakk>vt (V th cs # s);
- \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow>
- next_th s th cs t \<and> c = cs"
-proof -
- fix c t
- assume vt: "vt (V th cs # s)"
- and nhd: "\<not> holding (wq s) t c"
- and hd: "holding (wq (V th cs # s)) t c"
- show "next_th s th cs t \<and> c = cs"
- proof(cases "c = cs")
- case False
- with nhd hd show ?thesis
- by (unfold cs_holding_def wq_def, auto simp:Let_def)
- next
- case True
- with step_back_step [OF vt]
- have "step s (V th c)" by simp
- hence "next_th s th cs t"
- proof(cases)
- assume "holding s th c"
- with nhd hd show ?thesis
- apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
- auto simp:Let_def split:list.splits if_splits)
- proof -
- assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
- moreover have "\<dots> = set []"
- proof(rule someI2)
- show "distinct [] \<and> [] = []" by auto
- next
- fix x assume "distinct x \<and> x = []"
- thus "set x = set []" by auto
- qed
- ultimately show False by auto
- next
- assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
- moreover have "\<dots> = set []"
- proof(rule someI2)
- show "distinct [] \<and> [] = []" by auto
- next
- fix x assume "distinct x \<and> x = []"
- thus "set x = set []" by auto
- qed
- ultimately show False by auto
- qed
- qed
- with True show ?thesis by auto
- qed
-qed
-
-text {*
- The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be
- derived from scratch, which confirms the correctness of the definition of @{text "next_th"}.
-*}
-lemma step_v_wait_inv[elim_format]:
- "\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
- \<rbrakk>
- \<Longrightarrow> (next_th s th cs t \<and> cs = c)"
-proof -
- fix t c
- assume vt: "vt (V th cs # s)"
- and nw: "\<not> waiting (wq (V th cs # s)) t c"
- and wt: "waiting (wq s) t c"
- from vt interpret vt_v: valid_trace_e s "V th cs"
- by (cases, unfold_locales, simp)
- show "next_th s th cs t \<and> cs = c"
- proof(cases "cs = c")
- case False
- with nw wt show ?thesis
- by (auto simp:cs_waiting_def wq_def Let_def)
- next
- case True
- from nw[folded True] wt[folded True]
- have "next_th s th cs t"
- apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
- proof -
- fix a list
- assume t_in: "t \<in> set list"
- and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have " set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
- by auto
- qed
- with t_ni and t_in show "a = th" by auto
- next
- fix a list
- assume t_in: "t \<in> set list"
- and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have " set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
- by auto
- qed
- with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto
- next
- fix a list
- assume eq_wq: "wq_fun (schs s) cs = a # list"
- from step_back_step[OF vt]
- show "a = th"
- proof(cases)
- assume "holding s th cs"
- with eq_wq show ?thesis
- by (unfold s_holding_def wq_def, auto)
- qed
- qed
- with True show ?thesis by simp
- qed
-qed
-
-lemma step_v_not_wait[consumes 3]:
- "\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
- by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
-
-lemma step_v_release:
- "\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
-proof -
- assume vt: "vt (V th cs # s)"
- and hd: "holding (wq (V th cs # s)) th cs"
- from vt interpret vt_v: valid_trace_e s "V th cs"
- by (cases, unfold_locales, simp+)
- from step_back_step [OF vt] and hd
- show "False"
- proof(cases)
- assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"
- thus ?thesis
- apply (unfold s_holding_def wq_def cs_holding_def)
- apply (auto simp:Let_def split:list.splits)
- proof -
- fix list
- assume eq_wq[folded wq_def]:
- "wq_fun (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
- and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
- \<in> set (SOME q. distinct q \<and> set q = set list)"
- have "set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
- by auto
- qed
- moreover have "distinct (hd (SOME q. distinct q \<and> set q = set list) # list)"
- proof -
- from vt_v.wq_distinct[of cs] and eq_wq
- show ?thesis by auto
- qed
- moreover note eq_wq and hd_in
- ultimately show "False" by auto
- qed
- qed
-qed
-
-lemma step_v_get_hold:
- "\<And>th'. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
- apply (unfold cs_holding_def next_th_def wq_def,
- auto simp:Let_def)
-proof -
- fix rest
- assume vt: "vt (V th cs # s)"
- and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest"
- and nrest: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest)
- \<notin> set (SOME q. distinct q \<and> set q = set rest)"
- from vt interpret vt_v: valid_trace_e s "V th cs"
- by (cases, unfold_locales, simp+)
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- hence "set x = set rest" by auto
- with nrest
- show "x \<noteq> []" by (case_tac x, auto)
- qed
- with ni show "False" by auto
-qed
-
-lemma step_v_release_inv[elim_format]:
-"\<And>c t. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow>
- c = cs \<and> t = th"
- apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
- proof -
- fix a list
- assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
- from step_back_step [OF vt] show "a = th"
- proof(cases)
- assume "holding s th cs" with eq_wq
- show ?thesis
- by (unfold s_holding_def wq_def, auto)
- qed
- next
- fix a list
- assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
- from step_back_step [OF vt] show "a = th"
- proof(cases)
- assume "holding s th cs" with eq_wq
- show ?thesis
- by (unfold s_holding_def wq_def, auto)
- qed
- qed
-
-lemma step_v_waiting_mono:
- "\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
-proof -
- fix t c
- let ?s' = "(V th cs # s)"
- assume vt: "vt ?s'"
- and wt: "waiting (wq ?s') t c"
- from vt interpret vt_v: valid_trace_e s "V th cs"
- by (cases, unfold_locales, simp+)
- show "waiting (wq s) t c"
- proof(cases "c = cs")
- case False
- assume neq_cs: "c \<noteq> cs"
- hence "waiting (wq ?s') t c = waiting (wq s) t c"
- by (unfold cs_waiting_def wq_def, auto simp:Let_def)
- with wt show ?thesis by simp
- next
- case True
- with wt show ?thesis
- apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
- proof -
- fix a list
- assume not_in: "t \<notin> set list"
- and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have "set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from vt_v.wq_distinct [of cs]
- and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- fix x assume "distinct x \<and> set x = set list"
- thus "set x = set list" by auto
- qed
- with not_in is_in show "t = a" by auto
- next
- fix list
- assume is_waiting: "waiting (wq (V th cs # s)) t cs"
- and eq_wq: "wq_fun (schs s) cs = t # list"
- hence "t \<in> set list"
- apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
- proof -
- assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
- moreover have "\<dots> = set list"
- proof(rule someI2)
- from vt_v.wq_distinct [of cs]
- and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- fix x assume "distinct x \<and> set x = set list"
- thus "set x = set list" by auto
- qed
- ultimately show "t \<in> set list" by simp
- qed
- with eq_wq and vt_v.wq_distinct [of cs, unfolded wq_def]
- show False by auto
- qed
- qed
-qed
-
-text {* (* ddd *)
- The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
- with the happening of @{text "V"}-events:
-*}
-lemma step_RAG_v:
-fixes th::thread
-assumes vt:
- "vt (V th cs#s)"
-shows "
- RAG (V th cs # s) =
- RAG s - {(Cs cs, Th th)} -
- {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}"
- apply (insert vt, unfold s_RAG_def)
- apply (auto split:if_splits list.splits simp:Let_def)
- apply (auto elim: step_v_waiting_mono step_v_hold_inv
- step_v_release step_v_wait_inv
- step_v_get_hold step_v_release_inv)
- apply (erule_tac step_v_not_wait, auto)
- done
-
-text {*
- The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
- with the happening of @{text "P"}-events:
-*}
-lemma step_RAG_p:
- "vt (P th cs#s) \<Longrightarrow>
- RAG (P th cs # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
- else RAG s \<union> {(Th th, Cs cs)})"
- apply(simp only: s_RAG_def wq_def)
- apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
- apply(case_tac "csa = cs", auto)
- apply(fold wq_def)
- apply(drule_tac step_back_step)
- apply(ind_cases " step s (P (hd (wq s cs)) cs)")
- apply(simp add:s_RAG_def wq_def cs_holding_def)
- apply(auto)
- done
-
-
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
- by (unfold s_RAG_def, auto)
-
-context valid_trace
-begin
-
-text {*
- The following lemma shows that @{text "RAG"} is acyclic.
- The overall structure is by induction on the formation of @{text "vt s"}
- and then case analysis on event @{text "e"}, where the non-trivial cases
- for those for @{text "V"} and @{text "P"} events.
-*}
-lemma acyclic_RAG:
- shows "acyclic (RAG s)"
-using vt
-proof(induct)
- case (vt_cons s e)
- interpret vt_s: valid_trace s using vt_cons(1)
- by (unfold_locales, simp)
- assume ih: "acyclic (RAG s)"
- and stp: "step s e"
- and vt: "vt s"
- show ?case
- proof(cases e)
- case (Create th prio)
- with ih
- show ?thesis by (simp add:RAG_create_unchanged)
- next
- case (Exit th)
- with ih show ?thesis by (simp add:RAG_exit_unchanged)
- next
- case (V th cs)
- from V vt stp have vtt: "vt (V th cs#s)" by auto
- from step_RAG_v [OF this]
- have eq_de:
- "RAG (e # s) =
- RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}"
- (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
- from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
- from step_back_step [OF vtt]
- have "step s (V th cs)" .
- thus ?thesis
- proof(cases)
- assume "holding s th cs"
- hence th_in: "th \<in> set (wq s cs)" and
- eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
- then obtain rest where
- eq_wq: "wq s cs = th#rest"
- by (cases "wq s cs", auto)
- show ?thesis
- proof(cases "rest = []")
- case False
- let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
- from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
- by (unfold next_th_def, auto)
- let ?E = "(?A - ?B - ?C)"
- have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
- proof
- assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
- hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- from tranclD [OF this]
- obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
- hence th_d: "(Th ?th', x) \<in> ?A" by simp
- from RAG_target_th [OF this]
- obtain cs' where eq_x: "x = Cs cs'" by auto
- with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
- hence wt_th': "waiting s ?th' cs'"
- unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
- hence "cs' = cs"
- proof(rule vt_s.waiting_unique)
- from eq_wq vt_s.wq_distinct[of cs]
- show "waiting s ?th' cs"
- apply (unfold s_waiting_def wq_def, auto)
- proof -
- assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq_fun (schs s) cs = th # rest"
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with False show "x \<noteq> []" by auto
- qed
- hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
- set (SOME q. distinct q \<and> set q = set rest)" by auto
- moreover have "\<dots> = set rest"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- moreover note hd_in
- ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
- next
- assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with False show "x \<noteq> []" by auto
- qed
- hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
- set (SOME q. distinct q \<and> set q = set rest)" by auto
- moreover have "\<dots> = set rest"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- moreover note hd_in
- ultimately show False by auto
- qed
- qed
- with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
- with False
- show "False" by (auto simp: next_th_def eq_wq)
- qed
- with acyclic_insert[symmetric] and ac
- and eq_de eq_D show ?thesis by auto
- next
- case True
- with eq_wq
- have eq_D: "?D = {}"
- by (unfold next_th_def, auto)
- with eq_de ac
- show ?thesis by auto
- qed
- qed
- next
- case (P th cs)
- from P vt stp have vtt: "vt (P th cs#s)" by auto
- from step_RAG_p [OF this] P
- have "RAG (e # s) =
- (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
- RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
- by simp
- moreover have "acyclic ?R"
- proof(cases "wq s cs = []")
- case True
- hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
- have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
- proof
- assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
- hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- from tranclD2 [OF this]
- obtain x where "(x, Cs cs) \<in> RAG s" by auto
- with True show False by (auto simp:s_RAG_def cs_waiting_def)
- qed
- with acyclic_insert ih eq_r show ?thesis by auto
- next
- case False
- hence eq_r: "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
- have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
- proof
- assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
- hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- moreover from step_back_step [OF vtt] have "step s (P th cs)" .
- ultimately show False
- proof -
- show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
- by (ind_cases "step s (P th cs)", simp)
- qed
- qed
- with acyclic_insert ih eq_r show ?thesis by auto
- qed
- ultimately show ?thesis by simp
- next
- case (Set thread prio)
- with ih
- thm RAG_set_unchanged
- show ?thesis by (simp add:RAG_set_unchanged)
- qed
- next
- case vt_nil
- show "acyclic (RAG ([]::state))"
- by (auto simp: s_RAG_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
-qed
-
-
-lemma finite_RAG:
- shows "finite (RAG s)"
-proof -
- from vt show ?thesis
- proof(induct)
- case (vt_cons s e)
- interpret vt_s: valid_trace s using vt_cons(1)
- by (unfold_locales, simp)
- assume ih: "finite (RAG s)"
- and stp: "step s e"
- and vt: "vt s"
- show ?case
- proof(cases e)
- case (Create th prio)
- with ih
- show ?thesis by (simp add:RAG_create_unchanged)
- next
- case (Exit th)
- with ih show ?thesis by (simp add:RAG_exit_unchanged)
- next
- case (V th cs)
- from V vt stp have vtt: "vt (V th cs#s)" by auto
- from step_RAG_v [OF this]
- have eq_de: "RAG (e # s) =
- RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}
-"
- (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
- moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
- moreover have "finite ?D"
- proof -
- have "?D = {} \<or> (\<exists> a. ?D = {a})"
- by (unfold next_th_def, auto)
- thus ?thesis
- proof
- assume h: "?D = {}"
- show ?thesis by (unfold h, simp)
- next
- assume "\<exists> a. ?D = {a}"
- thus ?thesis
- by (metis finite.simps)
- qed
- qed
- ultimately show ?thesis by simp
- next
- case (P th cs)
- from P vt stp have vtt: "vt (P th cs#s)" by auto
- from step_RAG_p [OF this] P
- have "RAG (e # s) =
- (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
- RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
- by simp
- moreover have "finite ?R"
- proof(cases "wq s cs = []")
- case True
- hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
- with True and ih show ?thesis by auto
- next
- case False
- hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
- with False and ih show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- next
- case (Set thread prio)
- with ih
- show ?thesis by (simp add:RAG_set_unchanged)
- qed
- next
- case vt_nil
- show "finite (RAG ([]::state))"
- by (auto simp: s_RAG_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
- qed
-qed
-
-text {* Several useful lemmas *}
-
-lemma wf_dep_converse:
- shows "wf ((RAG s)^-1)"
-proof(rule finite_acyclic_wf_converse)
- from finite_RAG
- show "finite (RAG s)" .
-next
- from acyclic_RAG
- show "acyclic (RAG s)" .
-qed
-
-end
-
-lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
- by (induct l, auto)
-
-lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
- by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-
-context valid_trace
-begin
-
-lemma wq_threads:
- assumes h: "th \<in> set (wq s cs)"
- shows "th \<in> threads s"
-proof -
- from vt and h show ?thesis
- proof(induct arbitrary: th cs)
- case (vt_cons s e)
- interpret vt_s: valid_trace s
- using vt_cons(1) by (unfold_locales, auto)
- assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
- and stp: "step s e"
- and vt: "vt s"
- and h: "th \<in> set (wq (e # s) cs)"
- show ?case
- proof(cases e)
- case (Create th' prio)
- with ih h show ?thesis
- by (auto simp:wq_def Let_def)
- next
- case (Exit th')
- with stp ih h show ?thesis
- apply (auto simp:wq_def Let_def)
- apply (ind_cases "step s (Exit th')")
- apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
- s_RAG_def s_holding_def cs_holding_def)
- done
- next
- case (V th' cs')
- show ?thesis
- proof(cases "cs' = cs")
- case False
- with h
- show ?thesis
- apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
- by (drule_tac ih, simp)
- next
- case True
- from h
- show ?thesis
- proof(unfold V wq_def)
- assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
- show "th \<in> threads (V th' cs' # s)"
- proof(cases "cs = cs'")
- case False
- hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
- with th_in have " th \<in> set (wq s cs)"
- by (fold wq_def, simp)
- from ih [OF this] show ?thesis by simp
- next
- case True
- show ?thesis
- proof(cases "wq_fun (schs s) cs'")
- case Nil
- with h V show ?thesis
- apply (auto simp:wq_def Let_def split:if_splits)
- by (fold wq_def, drule_tac ih, simp)
- next
- case (Cons a rest)
- assume eq_wq: "wq_fun (schs s) cs' = a # rest"
- with h V show ?thesis
- apply (auto simp:Let_def wq_def split:if_splits)
- proof -
- assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
- have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
- by auto
- qed
- with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
- from ih[OF this[folded wq_def]] show "th \<in> threads s" .
- next
- assume th_in: "th \<in> set (wq_fun (schs s) cs)"
- from ih[OF this[folded wq_def]]
- show "th \<in> threads s" .
- qed
- qed
- qed
- qed
- qed
- next
- case (P th' cs')
- from h stp
- show ?thesis
- apply (unfold P wq_def)
- apply (auto simp:Let_def split:if_splits, fold wq_def)
- apply (auto intro:ih)
- apply(ind_cases "step s (P th' cs')")
- by (unfold runing_def readys_def, auto)
- next
- case (Set thread prio)
- with ih h show ?thesis
- by (auto simp:wq_def Let_def)
- qed
- next
- case vt_nil
- thus ?case by (auto simp:wq_def)
- qed
-qed
-
-lemma range_in: "\<lbrakk>(Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
- apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
- by (auto intro:wq_threads)
-
-lemma readys_v_eq:
- fixes th thread cs rest
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and not_in: "th \<notin> set rest"
- shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
- from assms show ?thesis
- apply (auto simp:readys_def)
- apply(simp add:s_waiting_def[folded wq_def])
- apply (erule_tac x = csa in allE)
- apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
- apply (case_tac "csa = cs", simp)
- apply (erule_tac x = cs in allE)
- apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
- apply(auto simp add: wq_def)
- apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
- proof -
- assume th_nin: "th \<notin> set rest"
- and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
- and eq_wq: "wq_fun (schs s) cs = thread # rest"
- have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- with th_nin th_in show False by auto
- qed
-qed
-
-text {* \noindent
- The following lemmas shows that: starting from any node in @{text "RAG"},
- by chasing out-going edges, it is always possible to reach a node representing a ready
- thread. In this lemma, it is the @{text "th'"}.
-*}
-
-lemma chain_building:
- shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
-proof -
- from wf_dep_converse
- have h: "wf ((RAG s)\<inverse>)" .
- show ?thesis
- proof(induct rule:wf_induct [OF h])
- fix x
- assume ih [rule_format]:
- "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow>
- y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
- show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
- proof
- assume x_d: "x \<in> Domain (RAG s)"
- show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
- proof(cases x)
- case (Th th)
- from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
- with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
- from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
- hence "Cs cs \<in> Domain (RAG s)" by auto
- from ih [OF x_in_r this] obtain th'
- where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
- have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
- with th'_ready show ?thesis by auto
- next
- case (Cs cs)
- from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
- show ?thesis
- proof(cases "th' \<in> readys s")
- case True
- from True and th'_d show ?thesis by auto
- next
- case False
- from th'_d and range_in have "th' \<in> threads s" by auto
- with False have "Th th' \<in> Domain (RAG s)"
- by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
- from ih [OF th'_d this]
- obtain th'' where
- th''_r: "th'' \<in> readys s" and
- th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
- from th'_d and th''_in
- have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
- with th''_r show ?thesis by auto
- qed
- qed
- qed
- qed
-qed
-
-text {* \noindent
- The following is just an instance of @{text "chain_building"}.
-*}
-lemma th_chain_to_ready:
- assumes th_in: "th \<in> threads s"
- shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
-proof(cases "th \<in> readys s")
- case True
- thus ?thesis by auto
-next
- case False
- from False and th_in have "Th th \<in> Domain (RAG s)"
- by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
- from chain_building [rule_format, OF this]
- show ?thesis by auto
-qed
-
-end
-
-lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
- by (unfold s_waiting_def cs_waiting_def wq_def, auto)
-
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
- by (unfold s_holding_def wq_def cs_holding_def, simp)
-
-lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
- by (unfold s_holding_def cs_holding_def, auto)
-
-context valid_trace
-begin
-
-lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
- apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
- by(auto elim:waiting_unique holding_unique)
-
-end
-
-
-lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
-by (induct rule:trancl_induct, auto)
-
-context valid_trace
-begin
-
-lemma dchain_unique:
- assumes th1_d: "(n, Th th1) \<in> (RAG s)^+"
- and th1_r: "th1 \<in> readys s"
- and th2_d: "(n, Th th2) \<in> (RAG s)^+"
- and th2_r: "th2 \<in> readys s"
- shows "th1 = th2"
-proof -
- { assume neq: "th1 \<noteq> th2"
- hence "Th th1 \<noteq> Th th2" by simp
- from unique_chain [OF _ th1_d th2_d this] and unique_RAG
- have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
- hence "False"
- proof
- assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
- from trancl_split [OF this]
- obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
- then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
- from dd eq_n have "th1 \<notin> readys s"
- by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
- with th1_r show ?thesis by auto
- next
- assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
- from trancl_split [OF this]
- obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
- then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
- from dd eq_n have "th2 \<notin> readys s"
- by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
- with th2_r show ?thesis by auto
- qed
- } thus ?thesis by auto
-qed
-
-end
-
-
-lemma step_holdents_p_add:
- fixes th cs s
- assumes vt: "vt (P th cs#s)"
- and "wq s cs = []"
- shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
-proof -
- from assms show ?thesis
- unfolding holdents_test step_RAG_p[OF vt] by (auto)
-qed
-
-lemma step_holdents_p_eq:
- fixes th cs s
- assumes vt: "vt (P th cs#s)"
- and "wq s cs \<noteq> []"
- shows "holdents (P th cs#s) th = holdents s th"
-proof -
- from assms show ?thesis
- unfolding holdents_test step_RAG_p[OF vt] by auto
-qed
-
-
-lemma (in valid_trace) finite_holding :
- shows "finite (holdents s th)"
-proof -
- let ?F = "\<lambda> (x, y). the_cs x"
- from finite_RAG
- have "finite (RAG s)" .
- hence "finite (?F `(RAG s))" by simp
- moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>"
- proof -
- { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
- fix x assume "(Cs x, Th th) \<in> RAG s"
- hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
- moreover have "?F (Cs x, Th th) = x" by simp
- ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp
- } thus ?thesis by auto
- qed
- ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
-qed
-
-lemma cntCS_v_dec:
- fixes s thread cs
- assumes vtv: "vt (V thread cs#s)"
- shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
-proof -
- from vtv interpret vt_s: valid_trace s
- by (cases, unfold_locales, simp)
- from vtv interpret vt_v: valid_trace "V thread cs#s"
- by (unfold_locales, simp)
- from step_back_step[OF vtv]
- have cs_in: "cs \<in> holdents s thread"
- apply (cases, unfold holdents_test s_RAG_def, simp)
- by (unfold cs_holding_def s_holding_def wq_def, auto)
- moreover have cs_not_in:
- "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
- apply (insert vt_s.wq_distinct[of cs])
- apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
- auto simp:next_th_def)
- proof -
- fix rest
- assume dst: "distinct (rest::thread list)"
- and ne: "rest \<noteq> []"
- and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)" by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume " distinct x \<and> set x = set rest" with ne
- show "x \<noteq> []" by auto
- qed
- ultimately
- show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
- by auto
- next
- fix rest
- assume dst: "distinct (rest::thread list)"
- and ne: "rest \<noteq> []"
- and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)" by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume " distinct x \<and> set x = set rest" with ne
- show "x \<noteq> []" by auto
- qed
- ultimately show "False" by auto
- qed
- ultimately
- have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
- by auto
- moreover have "card \<dots> =
- Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
- proof(rule card_insert)
- from vt_v.finite_holding
- show " finite (holdents (V thread cs # s) thread)" .
- qed
- moreover from cs_not_in
- have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
- ultimately show ?thesis by (simp add:cntCS_def)
-qed
-
-context valid_trace
-begin
-
-text {* (* ddd *) \noindent
- The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"}
- of one particular thread.
-*}
-
-lemma cnp_cnv_cncs:
- shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
- then cntCS s th else cntCS s th + 1)"
-proof -
- from vt show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e)
- interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
- assume vt: "vt s"
- and ih: "\<And>th. cntP s th = cntV s th +
- (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
- and stp: "step s e"
- from stp show ?case
- proof(cases)
- case (thread_create thread prio)
- assume eq_e: "e = Create thread prio"
- and not_in: "thread \<notin> threads s"
- show ?thesis
- proof -
- { fix cs
- assume "thread \<in> set (wq s cs)"
- from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
- with not_in have "False" by simp
- } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
- by (auto simp:readys_def threads.simps s_waiting_def
- wq_def cs_waiting_def Let_def)
- from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
- from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
- have eq_cncs: "cntCS (e#s) th = cntCS s th"
- unfolding cntCS_def holdents_test
- by (simp add:RAG_create_unchanged eq_e)
- { assume "th \<noteq> thread"
- with eq_readys eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> threads (s))"
- by (simp add:threads.simps)
- with eq_cnp eq_cnv eq_cncs ih not_in
- have ?thesis by simp
- } moreover {
- assume eq_th: "th = thread"
- with not_in ih have " cntP s th = cntV s th + cntCS s th" by simp
- moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
- moreover note eq_cnp eq_cnv eq_cncs
- ultimately have ?thesis by auto
- } ultimately show ?thesis by blast
- qed
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- and is_runing: "thread \<in> runing s"
- and no_hold: "holdents s thread = {}"
- from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
- from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
- have eq_cncs: "cntCS (e#s) th = cntCS s th"
- unfolding cntCS_def holdents_test
- by (simp add:RAG_exit_unchanged eq_e)
- { assume "th \<noteq> thread"
- with eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> threads (s))"
- apply (simp add:threads.simps readys_def)
- apply (subst s_waiting_def)
- apply (simp add:Let_def)
- apply (subst s_waiting_def, simp)
- done
- with eq_cnp eq_cnv eq_cncs ih
- have ?thesis by simp
- } moreover {
- assume eq_th: "th = thread"
- with ih is_runing have " cntP s th = cntV s th + cntCS s th"
- by (simp add:runing_def)
- moreover from eq_th eq_e have "th \<notin> threads (e#s)"
- by simp
- moreover note eq_cnp eq_cnv eq_cncs
- ultimately have ?thesis by auto
- } ultimately show ?thesis by blast
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- and is_runing: "thread \<in> runing s"
- and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
- from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto
- then interpret vt_p: valid_trace "(P thread cs#s)"
- by (unfold_locales, simp)
- show ?thesis
- proof -
- { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
- assume neq_th: "th \<noteq> thread"
- with eq_e
- have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
- apply (simp add:readys_def s_waiting_def wq_def Let_def)
- apply (rule_tac hh)
- apply (intro iffI allI, clarify)
- apply (erule_tac x = csa in allE, auto)
- apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
- apply (erule_tac x = cs in allE, auto)
- by (case_tac "(wq_fun (schs s) cs)", auto)
- moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
- apply (simp add:cntCS_def holdents_test)
- by (unfold step_RAG_p [OF vtp], auto)
- moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
- by (simp add:cntP_def count_def)
- moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
- by (simp add:cntV_def count_def)
- moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
- moreover note ih [of th]
- ultimately have ?thesis by simp
- } moreover {
- assume eq_th: "th = thread"
- have ?thesis
- proof -
- from eq_e eq_th have eq_cnp: "cntP (e # s) th = 1 + (cntP s th)"
- by (simp add:cntP_def count_def)
- from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
- by (simp add:cntV_def count_def)
- show ?thesis
- proof (cases "wq s cs = []")
- case True
- with is_runing
- have "th \<in> readys (e#s)"
- apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
- apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
- by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
- moreover have "cntCS (e # s) th = 1 + cntCS s th"
- proof -
- have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
- Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
- proof -
- have "?L = insert cs ?R" by auto
- moreover have "card \<dots> = Suc (card (?R - {cs}))"
- proof(rule card_insert)
- from vt_s.finite_holding [of thread]
- show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
- by (unfold holdents_test, simp)
- qed
- moreover have "?R - {cs} = ?R"
- proof -
- have "cs \<notin> ?R"
- proof
- assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
- with no_dep show False by auto
- qed
- thus ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- thus ?thesis
- apply (unfold eq_e eq_th cntCS_def)
- apply (simp add: holdents_test)
- by (unfold step_RAG_p [OF vtp], auto simp:True)
- qed
- moreover from is_runing have "th \<in> readys s"
- by (simp add:runing_def eq_th)
- moreover note eq_cnp eq_cnv ih [of th]
- ultimately show ?thesis by auto
- next
- case False
- have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
- by (unfold eq_th eq_e wq_def, auto simp:Let_def)
- have "th \<notin> readys (e#s)"
- proof
- assume "th \<in> readys (e#s)"
- hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
- from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
- hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
- by (simp add:s_waiting_def wq_def)
- moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
- ultimately have "th = hd (wq (e#s) cs)" by blast
- with eq_wq have "th = hd (wq s cs @ [th])" by simp
- hence "th = hd (wq s cs)" using False by auto
- with False eq_wq vt_p.wq_distinct [of cs]
- show False by (fold eq_e, auto)
- qed
- moreover from is_runing have "th \<in> threads (e#s)"
- by (unfold eq_e, auto simp:runing_def readys_def eq_th)
- moreover have "cntCS (e # s) th = cntCS s th"
- apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
- by (auto simp:False)
- moreover note eq_cnp eq_cnv ih[of th]
- moreover from is_runing have "th \<in> readys s"
- by (simp add:runing_def eq_th)
- ultimately show ?thesis by auto
- qed
- qed
- } ultimately show ?thesis by blast
- qed
- next
- case (thread_V thread cs)
- from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
- then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
- assume eq_e: "e = V thread cs"
- and is_runing: "thread \<in> runing s"
- and hold: "holding s thread cs"
- from hold obtain rest
- where eq_wq: "wq s cs = thread # rest"
- by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
- have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
- have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest"
- by (metis distinct.simps(2) vt_s.wq_distinct)
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
- by auto
- qed
- show ?thesis
- proof -
- { assume eq_th: "th = thread"
- from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
- by (unfold eq_e, simp add:cntP_def count_def)
- moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
- by (unfold eq_e, simp add:cntV_def count_def)
- moreover from cntCS_v_dec [OF vtv]
- have "cntCS (e # s) thread + 1 = cntCS s thread"
- by (simp add:eq_e)
- moreover from is_runing have rd_before: "thread \<in> readys s"
- by (unfold runing_def, simp)
- moreover have "thread \<in> readys (e # s)"
- proof -
- from is_runing
- have "thread \<in> threads (e#s)"
- by (unfold eq_e, auto simp:runing_def readys_def)
- moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
- proof
- fix cs1
- { assume eq_cs: "cs1 = cs"
- have "\<not> waiting (e # s) thread cs1"
- proof -
- from eq_wq
- have "thread \<notin> set (wq (e#s) cs1)"
- apply(unfold eq_e wq_def eq_cs s_holding_def)
- apply (auto simp:Let_def)
- proof -
- assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
- with eq_set have "thread \<in> set rest" by simp
- with vt_v.wq_distinct[of cs]
- and eq_wq show False
- by (metis distinct.simps(2) vt_s.wq_distinct)
- qed
- thus ?thesis by (simp add:wq_def s_waiting_def)
- qed
- } moreover {
- assume neq_cs: "cs1 \<noteq> cs"
- have "\<not> waiting (e # s) thread cs1"
- proof -
- from wq_v_neq [OF neq_cs[symmetric]]
- have "wq (V thread cs # s) cs1 = wq s cs1" .
- moreover have "\<not> waiting s thread cs1"
- proof -
- from runing_ready and is_runing
- have "thread \<in> readys s" by auto
- thus ?thesis by (simp add:readys_def)
- qed
- ultimately show ?thesis
- by (auto simp:wq_def s_waiting_def eq_e)
- qed
- } ultimately show "\<not> waiting (e # s) thread cs1" by blast
- qed
- ultimately show ?thesis by (simp add:readys_def)
- qed
- moreover note eq_th ih
- ultimately have ?thesis by auto
- } moreover {
- assume neq_th: "th \<noteq> thread"
- from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th"
- by (simp add:cntP_def count_def)
- from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th"
- by (simp add:cntV_def count_def)
- have ?thesis
- proof(cases "th \<in> set rest")
- case False
- have "(th \<in> readys (e # s)) = (th \<in> readys s)"
- apply (insert step_back_vt[OF vtv])
- by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
- moreover have "cntCS (e#s) th = cntCS s th"
- apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
- proof -
- have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
- {cs. (Cs cs, Th th) \<in> RAG s}"
- proof -
- from False eq_wq
- have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
- apply (unfold next_th_def, auto)
- proof -
- assume ne: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq s cs = thread # rest"
- from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)
- " by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from vt_s.wq_distinct[ of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with ne show "x \<noteq> []" by auto
- qed
- ultimately show
- "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
- by auto
- qed
- thus ?thesis by auto
- qed
- thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
- card {cs. (Cs cs, Th th) \<in> RAG s}" by simp
- qed
- moreover note ih eq_cnp eq_cnv eq_threads
- ultimately show ?thesis by auto
- next
- case True
- assume th_in: "th \<in> set rest"
- show ?thesis
- proof(cases "next_th s thread cs th")
- case False
- with eq_wq and th_in have
- neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
- by (auto simp:next_th_def)
- have "(th \<in> readys (e # s)) = (th \<in> readys s)"
- proof -
- from eq_wq and th_in
- have "\<not> th \<in> readys s"
- apply (auto simp:readys_def s_waiting_def)
- apply (rule_tac x = cs in exI, auto)
- by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
- moreover
- from eq_wq and th_in and neq_hd
- have "\<not> (th \<in> readys (e # s))"
- apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
- by (rule_tac x = cs in exI, auto simp:eq_set)
- ultimately show ?thesis by auto
- qed
- moreover have "cntCS (e#s) th = cntCS s th"
- proof -
- from eq_wq and th_in and neq_hd
- have "(holdents (e # s) th) = (holdents s th)"
- apply (unfold eq_e step_RAG_v[OF vtv],
- auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
- Let_def cs_holding_def)
- by (insert vt_s.wq_distinct[of cs], auto simp:wq_def)
- thus ?thesis by (simp add:cntCS_def)
- qed
- moreover note ih eq_cnp eq_cnv eq_threads
- ultimately show ?thesis by auto
- next
- case True
- let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
- let ?t = "hd ?rest"
- from True eq_wq th_in neq_th
- have "th \<in> readys (e # s)"
- apply (auto simp:eq_e readys_def s_waiting_def wq_def
- Let_def next_th_def)
- proof -
- assume eq_wq: "wq_fun (schs s) cs = thread # rest"
- and t_in: "?t \<in> set rest"
- show "?t \<in> threads s"
- proof(rule vt_s.wq_threads)
- from eq_wq and t_in
- show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
- qed
- next
- fix csa
- assume eq_wq: "wq_fun (schs s) cs = thread # rest"
- and t_in: "?t \<in> set rest"
- and neq_cs: "csa \<noteq> cs"
- and t_in': "?t \<in> set (wq_fun (schs s) csa)"
- show "?t = hd (wq_fun (schs s) csa)"
- proof -
- { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
- from vt_s.wq_distinct[of cs] and
- eq_wq[folded wq_def] and t_in eq_wq
- have "?t \<noteq> thread" by auto
- with eq_wq and t_in
- have w1: "waiting s ?t cs"
- by (auto simp:s_waiting_def wq_def)
- from t_in' neq_hd'
- have w2: "waiting s ?t csa"
- by (auto simp:s_waiting_def wq_def)
- from vt_s.waiting_unique[OF w1 w2]
- and neq_cs have "False" by auto
- } thus ?thesis by auto
- qed
- qed
- moreover have "cntP s th = cntV s th + cntCS s th + 1"
- proof -
- have "th \<notin> readys s"
- proof -
- from True eq_wq neq_th th_in
- show ?thesis
- apply (unfold readys_def s_waiting_def, auto)
- by (rule_tac x = cs in exI, auto simp add: wq_def)
- qed
- moreover have "th \<in> threads s"
- proof -
- from th_in eq_wq
- have "th \<in> set (wq s cs)" by simp
- from vt_s.wq_threads [OF this]
- show ?thesis .
- qed
- ultimately show ?thesis using ih by auto
- qed
- moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
- apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
- proof -
- show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
- Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
- (is "card ?A = Suc (card ?B)")
- proof -
- have "?A = insert cs ?B" by auto
- hence "card ?A = card (insert cs ?B)" by simp
- also have "\<dots> = Suc (card ?B)"
- proof(rule card_insert_disjoint)
- have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
- with vt_s.finite_RAG
- show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
- next
- show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
- proof
- assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
- hence "(Cs cs, Th th) \<in> RAG s" by simp
- with True neq_th eq_wq show False
- by (auto simp:next_th_def s_RAG_def cs_holding_def)
- qed
- qed
- finally show ?thesis .
- qed
- qed
- moreover note eq_cnp eq_cnv
- ultimately show ?thesis by simp
- qed
- qed
- } ultimately show ?thesis by blast
- qed
- next
- case (thread_set thread prio)
- assume eq_e: "e = Set thread prio"
- and is_runing: "thread \<in> runing s"
- show ?thesis
- proof -
- from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
- from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
- have eq_cncs: "cntCS (e#s) th = cntCS s th"
- unfolding cntCS_def holdents_test
- by (simp add:RAG_set_unchanged eq_e)
- from eq_e have eq_readys: "readys (e#s) = readys s"
- by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
- auto simp:Let_def)
- { assume "th \<noteq> thread"
- with eq_readys eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> threads (s))"
- by (simp add:threads.simps)
- with eq_cnp eq_cnv eq_cncs ih is_runing
- have ?thesis by simp
- } moreover {
- assume eq_th: "th = thread"
- with is_runing ih have " cntP s th = cntV s th + cntCS s th"
- by (unfold runing_def, auto)
- moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
- by (simp add:runing_def)
- moreover note eq_cnp eq_cnv eq_cncs
- ultimately have ?thesis by auto
- } ultimately show ?thesis by blast
- qed
- qed
- next
- case vt_nil
- show ?case
- by (unfold cntP_def cntV_def cntCS_def,
- auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
- qed
-qed
-
-lemma not_thread_cncs:
- assumes not_in: "th \<notin> threads s"
- shows "cntCS s th = 0"
-proof -
- from vt not_in show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e th)
- interpret vt_s: valid_trace s using vt_cons(1)
- by (unfold_locales, simp)
- assume vt: "vt s"
- and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
- and stp: "step s e"
- and not_in: "th \<notin> threads (e # s)"
- from stp show ?case
- proof(cases)
- case (thread_create thread prio)
- assume eq_e: "e = Create thread prio"
- and not_in': "thread \<notin> threads s"
- have "cntCS (e # s) th = cntCS s th"
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:RAG_create_unchanged)
- moreover have "th \<notin> threads s"
- proof -
- from not_in eq_e show ?thesis by simp
- qed
- moreover note ih ultimately show ?thesis by auto
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- and nh: "holdents s thread = {}"
- have eq_cns: "cntCS (e # s) th = cntCS s th"
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:RAG_exit_unchanged)
- show ?thesis
- proof(cases "th = thread")
- case True
- have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
- with eq_cns show ?thesis by simp
- next
- case False
- with not_in and eq_e
- have "th \<notin> threads s" by simp
- from ih[OF this] and eq_cns show ?thesis by simp
- qed
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- and is_runing: "thread \<in> runing s"
- from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- hence "cntCS (e # s) th = cntCS s th "
- apply (unfold cntCS_def holdents_test eq_e)
- by (unfold step_RAG_p[OF vtp], auto)
- moreover have "cntCS s th = 0"
- proof(rule ih)
- from not_in eq_e show "th \<notin> threads s" by simp
- qed
- ultimately show ?thesis by simp
- next
- case (thread_V thread cs)
- assume eq_e: "e = V thread cs"
- and is_runing: "thread \<in> runing s"
- and hold: "holding s thread cs"
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- from assms thread_V vt stp ih
- have vtv: "vt (V thread cs#s)" by auto
- then interpret vt_v: valid_trace "(V thread cs#s)"
- by (unfold_locales, simp)
- from hold obtain rest
- where eq_wq: "wq s cs = thread # rest"
- by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
- from not_in eq_e eq_wq
- have "\<not> next_th s thread cs th"
- apply (auto simp:next_th_def)
- proof -
- assume ne: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
- have "?t \<in> set rest"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest"
- by (metis distinct.simps(2) vt_s.wq_distinct)
- next
- fix x assume "distinct x \<and> set x = set rest" with ne
- show "hd x \<in> set rest" by (cases x, auto)
- qed
- with eq_wq have "?t \<in> set (wq s cs)" by simp
- from vt_s.wq_threads[OF this] and ni
- show False
- using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)`
- ni vt_s.wq_threads by blast
- qed
- moreover note neq_th eq_wq
- ultimately have "cntCS (e # s) th = cntCS s th"
- by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
- moreover have "cntCS s th = 0"
- proof(rule ih)
- from not_in eq_e show "th \<notin> threads s" by simp
- qed
- ultimately show ?thesis by simp
- next
- case (thread_set thread prio)
- print_facts
- assume eq_e: "e = Set thread prio"
- and is_runing: "thread \<in> runing s"
- from not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] and eq_e
- show ?thesis
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:RAG_set_unchanged)
- qed
- next
- case vt_nil
- show ?case
- by (unfold cntCS_def,
- auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
- qed
-qed
-
-end
-
-lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
- by (auto simp:s_waiting_def cs_waiting_def wq_def)
-
-context valid_trace
-begin
-
-lemma dm_RAG_threads:
- assumes in_dom: "(Th th) \<in> Domain (RAG s)"
- shows "th \<in> threads s"
-proof -
- from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
- moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
- ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
- hence "th \<in> set (wq s cs)"
- by (unfold s_RAG_def, auto simp:cs_waiting_def)
- from wq_threads [OF this] show ?thesis .
-qed
-
-end
-
-lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
-unfolding cp_def wq_def
-apply(induct s rule: schs.induct)
-thm cpreced_initial
-apply(simp add: Let_def cpreced_initial)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-done
-
-context valid_trace
-begin
-
-lemma runing_unique:
- assumes runing_1: "th1 \<in> runing s"
- and runing_2: "th2 \<in> runing s"
- shows "th1 = th2"
-proof -
- from runing_1 and runing_2 have "cp s th1 = cp s th2"
- unfolding runing_def
- apply(simp)
- done
- hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
- Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
- (is "Max (?f ` ?A) = Max (?f ` ?B)")
- unfolding cp_eq_cpreced
- unfolding cpreced_def .
- obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
- proof -
- have h1: "finite (?f ` ?A)"
- proof -
- have "finite ?A"
- proof -
- have "finite (dependants (wq s) th1)"
- proof-
- have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_RAG have "finite (RAG s)" .
- hence "finite ((RAG (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_RAG_def cs_RAG_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependants_def)
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by auto
- qed
- moreover have h2: "(?f ` ?A) \<noteq> {}"
- proof -
- have "?A \<noteq> {}" by simp
- thus ?thesis by simp
- qed
- from Max_in [OF h1 h2]
- have "Max (?f ` ?A) \<in> (?f ` ?A)" .
- thus ?thesis
- thm cpreced_def
- unfolding cpreced_def[symmetric]
- unfolding cp_eq_cpreced[symmetric]
- unfolding cpreced_def
- using that[intro] by (auto)
- qed
- obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
- proof -
- have h1: "finite (?f ` ?B)"
- proof -
- have "finite ?B"
- proof -
- have "finite (dependants (wq s) th2)"
- proof-
- have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_RAG have "finite (RAG s)" .
- hence "finite ((RAG (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_RAG_def cs_RAG_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependants_def)
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by auto
- qed
- moreover have h2: "(?f ` ?B) \<noteq> {}"
- proof -
- have "?B \<noteq> {}" by simp
- thus ?thesis by simp
- qed
- from Max_in [OF h1 h2]
- have "Max (?f ` ?B) \<in> (?f ` ?B)" .
- thus ?thesis by (auto intro:that)
- qed
- from eq_f_th1 eq_f_th2 eq_max
- have eq_preced: "preced th1' s = preced th2' s" by auto
- hence eq_th12: "th1' = th2'"
- proof (rule preced_unique)
- from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
- thus "th1' \<in> threads s"
- proof
- assume "th1' \<in> dependants (wq s) th1"
- hence "(Th th1') \<in> Domain ((RAG s)^+)"
- apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
- by (auto simp:Domain_def)
- hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
- from dm_RAG_threads[OF this] show ?thesis .
- next
- assume "th1' = th1"
- with runing_1 show ?thesis
- by (unfold runing_def readys_def, auto)
- qed
- next
- from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
- thus "th2' \<in> threads s"
- proof
- assume "th2' \<in> dependants (wq s) th2"
- hence "(Th th2') \<in> Domain ((RAG s)^+)"
- apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
- by (auto simp:Domain_def)
- hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
- from dm_RAG_threads[OF this] show ?thesis .
- next
- assume "th2' = th2"
- with runing_2 show ?thesis
- by (unfold runing_def readys_def, auto)
- qed
- qed
- from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
- thus ?thesis
- proof
- assume eq_th': "th1' = th1"
- from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
- thus ?thesis
- proof
- assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
- next
- assume "th2' \<in> dependants (wq s) th2"
- with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
- hence "(Th th1, Th th2) \<in> (RAG s)^+"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
- hence "Th th1 \<in> Domain ((RAG s)^+)"
- apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
- by (auto simp:Domain_def)
- hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
- from RAG_target_th [OF this]
- obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th1, Cs cs') \<in> RAG s" by simp
- with runing_1 have "False"
- apply (unfold runing_def readys_def s_RAG_def)
- by (auto simp:eq_waiting)
- thus ?thesis by simp
- qed
- next
- assume th1'_in: "th1' \<in> dependants (wq s) th1"
- from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
- thus ?thesis
- proof
- assume "th2' = th2"
- with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
- hence "(Th th2, Th th1) \<in> (RAG s)^+"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
- hence "Th th2 \<in> Domain ((RAG s)^+)"
- apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
- by (auto simp:Domain_def)
- hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
- from RAG_target_th [OF this]
- obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th2, Cs cs') \<in> RAG s" by simp
- with runing_2 have "False"
- apply (unfold runing_def readys_def s_RAG_def)
- by (auto simp:eq_waiting)
- thus ?thesis by simp
- next
- assume "th2' \<in> dependants (wq s) th2"
- with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
- hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
- from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
- show ?thesis
- proof(rule dchain_unique[OF h1 _ h2, symmetric])
- from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
- from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
- qed
- qed
- qed
-qed
-
-
-lemma "card (runing s) \<le> 1"
-apply(subgoal_tac "finite (runing s)")
-prefer 2
-apply (metis finite_nat_set_iff_bounded lessI runing_unique)
-apply(rule ccontr)
-apply(simp)
-apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
-apply(subst (asm) card_le_Suc_iff)
-apply(simp)
-apply(auto)[1]
-apply (metis insertCI runing_unique)
-apply(auto)
-done
-
-end
-
-
-lemma create_pre:
- assumes stp: "step s e"
- and not_in: "th \<notin> threads s"
- and is_in: "th \<in> threads (e#s)"
- obtains prio where "e = Create th prio"
-proof -
- from assms
- show ?thesis
- proof(cases)
- case (thread_create thread prio)
- with is_in not_in have "e = Create th prio" by simp
- from that[OF this] show ?thesis .
- next
- case (thread_exit thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_P thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_V thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_set thread)
- with assms show ?thesis by (auto intro!:that)
- qed
-qed
-
-lemma length_down_to_in:
- assumes le_ij: "i \<le> j"
- and le_js: "j \<le> length s"
- shows "length (down_to j i s) = j - i"
-proof -
- have "length (down_to j i s) = length (from_to i j (rev s))"
- by (unfold down_to_def, auto)
- also have "\<dots> = j - i"
- proof(rule length_from_to_in[OF le_ij])
- from le_js show "j \<le> length (rev s)" by simp
- qed
- finally show ?thesis .
-qed
-
-
-lemma moment_head:
- assumes le_it: "Suc i \<le> length t"
- obtains e where "moment (Suc i) t = e#moment i t"
-proof -
- have "i \<le> Suc i" by simp
- from length_down_to_in [OF this le_it]
- have "length (down_to (Suc i) i t) = 1" by auto
- then obtain e where "down_to (Suc i) i t = [e]"
- apply (cases "(down_to (Suc i) i t)") by auto
- moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
- by (rule down_to_conc[symmetric], auto)
- ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
- by (auto simp:down_to_moment)
- from that [OF this] show ?thesis .
-qed
-
-context valid_trace
-begin
-
-lemma cnp_cnv_eq:
- assumes "th \<notin> threads s"
- shows "cntP s th = cntV s th"
- using assms
- using cnp_cnv_cncs not_thread_cncs by auto
-
-end
-
-
-lemma eq_RAG:
- "RAG (wq s) = RAG s"
-by (unfold cs_RAG_def s_RAG_def, auto)
-
-context valid_trace
-begin
-
-lemma count_eq_dependants:
- assumes eq_pv: "cntP s th = cntV s th"
- shows "dependants (wq s) th = {}"
-proof -
- from cnp_cnv_cncs and eq_pv
- have "cntCS s th = 0"
- by (auto split:if_splits)
- moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
- proof -
- from finite_holding[of th] show ?thesis
- by (simp add:holdents_test)
- qed
- ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
- by (unfold cntCS_def holdents_test cs_dependants_def, auto)
- show ?thesis
- proof(unfold cs_dependants_def)
- { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
- then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
- hence "False"
- proof(cases)
- assume "(Th th', Th th) \<in> RAG (wq s)"
- thus "False" by (auto simp:cs_RAG_def)
- next
- fix c
- assume "(c, Th th) \<in> RAG (wq s)"
- with h and eq_RAG show "False"
- by (cases c, auto simp:cs_RAG_def)
- qed
- } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
- qed
-qed
-
-lemma dependants_threads:
- shows "dependants (wq s) th \<subseteq> threads s"
-proof
- { fix th th'
- assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
- have "Th th \<in> Domain (RAG s)"
- proof -
- from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
- hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
- with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
- thus ?thesis using eq_RAG by simp
- qed
- from dm_RAG_threads[OF this]
- have "th \<in> threads s" .
- } note hh = this
- fix th1
- assume "th1 \<in> dependants (wq s) th"
- hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
- by (unfold cs_dependants_def, simp)
- from hh [OF this] show "th1 \<in> threads s" .
-qed
-
-lemma finite_threads:
- shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-end
-
-lemma Max_f_mono:
- assumes seq: "A \<subseteq> B"
- and np: "A \<noteq> {}"
- and fnt: "finite B"
- shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
- from seq show "f ` A \<subseteq> f ` B" by auto
-next
- from np show "f ` A \<noteq> {}" by auto
-next
- from fnt and seq show "finite (f ` B)" by auto
-qed
-
-context valid_trace
-begin
-
-lemma cp_le:
- assumes th_in: "th \<in> threads s"
- shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
- show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
- \<le> Max ((\<lambda>th. preced th s) ` threads s)"
- (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
- proof(rule Max_f_mono)
- show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
- next
- from finite_threads
- show "finite (threads s)" .
- next
- from th_in
- show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
- apply (auto simp:Domain_def)
- apply (rule_tac dm_RAG_threads)
- apply (unfold trancl_domain [of "RAG s", symmetric])
- by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
- qed
-qed
-
-lemma le_cp:
- shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
- show "Prc (priority th s) (last_set th s)
- \<le> Max (insert (Prc (priority th s) (last_set th s))
- ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
- (is "?l \<le> Max (insert ?l ?A)")
- proof(cases "?A = {}")
- case False
- have "finite ?A" (is "finite (?f ` ?B)")
- proof -
- have "finite ?B"
- proof-
- have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_RAG have "finite (RAG s)" .
- hence "finite ((RAG (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_RAG_def cs_RAG_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependants_def)
- qed
- thus ?thesis by simp
- qed
- from Max_insert [OF this False, of ?l] show ?thesis by auto
- next
- case True
- thus ?thesis by auto
- qed
-qed
-
-lemma max_cp_eq:
- shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
- (is "?l = ?r")
-proof(cases "threads s = {}")
- case True
- thus ?thesis by auto
-next
- case False
- have "?l \<in> ((cp s) ` threads s)"
- proof(rule Max_in)
- from finite_threads
- show "finite (cp s ` threads s)" by auto
- next
- from False show "cp s ` threads s \<noteq> {}" by auto
- qed
- then obtain th
- where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
- have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
- moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
- proof -
- have "?r \<in> (?f ` ?A)"
- proof(rule Max_in)
- from finite_threads
- show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
- next
- from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
- qed
- then obtain th' where
- th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
- from le_cp [of th'] eq_r
- have "?r \<le> cp s th'" by auto
- moreover have "\<dots> \<le> cp s th"
- proof(fold eq_l)
- show " cp s th' \<le> Max (cp s ` threads s)"
- proof(rule Max_ge)
- from th_in' show "cp s th' \<in> cp s ` threads s"
- by auto
- next
- from finite_threads
- show "finite (cp s ` threads s)" by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis using eq_l by auto
-qed
-
-lemma max_cp_readys_threads_pre:
- assumes np: "threads s \<noteq> {}"
- shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(unfold max_cp_eq)
- show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
- proof -
- let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
- let ?f = "(\<lambda>th. preced th s)"
- have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
- proof(rule Max_in)
- from finite_threads show "finite (?f ` threads s)" by simp
- next
- from np show "?f ` threads s \<noteq> {}" by simp
- qed
- then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
- by (auto simp:Image_def)
- from th_chain_to_ready [OF tm_in]
- have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
- thus ?thesis
- proof
- assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
- then obtain th' where th'_in: "th' \<in> readys s"
- and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
- have "cp s th' = ?f tm"
- proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
- from dependants_threads finite_threads
- show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))"
- by (auto intro:finite_subset)
- next
- fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
- from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
- moreover have "p \<le> \<dots>"
- proof(rule Max_ge)
- from finite_threads
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- from p_in and th'_in and dependants_threads[of th']
- show "p \<in> (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- qed
- ultimately show "p \<le> preced tm s" by auto
- next
- show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
- proof -
- from tm_chain
- have "tm \<in> dependants (wq s) th'"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
- thus ?thesis by auto
- qed
- qed
- with tm_max
- have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
- show ?thesis
- proof (fold h, rule Max_eqI)
- fix q
- assume "q \<in> cp s ` readys s"
- then obtain th1 where th1_in: "th1 \<in> readys s"
- and eq_q: "q = cp s th1" by auto
- show "q \<le> cp s th'"
- apply (unfold h eq_q)
- apply (unfold cp_eq_cpreced cpreced_def)
- apply (rule Max_mono)
- proof -
- from dependants_threads [of th1] th1_in
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq>
- (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- next
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
- next
- from finite_threads
- show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
- qed
- next
- from finite_threads
- show "finite (cp s ` readys s)" by (auto simp:readys_def)
- next
- from th'_in
- show "cp s th' \<in> cp s ` readys s" by simp
- qed
- next
- assume tm_ready: "tm \<in> readys s"
- show ?thesis
- proof(fold tm_max)
- have cp_eq_p: "cp s tm = preced tm s"
- proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
- fix y
- assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
- show "y \<le> preced tm s"
- proof -
- { fix y'
- assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
- have "y' \<le> preced tm s"
- proof(unfold tm_max, rule Max_ge)
- from hy' dependants_threads[of tm]
- show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
- next
- from finite_threads
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- qed
- } with hy show ?thesis by auto
- qed
- next
- from dependants_threads[of tm] finite_threads
- show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
- by (auto intro:finite_subset)
- next
- show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
- by simp
- qed
- moreover have "Max (cp s ` readys s) = cp s tm"
- proof(rule Max_eqI)
- from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
- next
- from finite_threads
- show "finite (cp s ` readys s)" by (auto simp:readys_def)
- next
- fix y assume "y \<in> cp s ` readys s"
- then obtain th1 where th1_readys: "th1 \<in> readys s"
- and h: "y = cp s th1" by auto
- show "y \<le> cp s tm"
- apply(unfold cp_eq_p h)
- apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
- proof -
- from finite_threads
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
- by simp
- next
- from dependants_threads[of th1] th1_readys
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)
- \<subseteq> (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- qed
- qed
- ultimately show " Max (cp s ` readys s) = preced tm s" by simp
- qed
- qed
- qed
-qed
-
-text {* (* ccc *) \noindent
- Since the current precedence of the threads in ready queue will always be boosted,
- there must be one inside it has the maximum precedence of the whole system.
-*}
-lemma max_cp_readys_threads:
- shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(cases "threads s = {}")
- case True
- thus ?thesis
- by (auto simp:readys_def)
-next
- case False
- show ?thesis by (rule max_cp_readys_threads_pre[OF False])
-qed
-
-end
-
-lemma eq_holding: "holding (wq s) th cs = holding s th cs"
- apply (unfold s_holding_def cs_holding_def wq_def, simp)
- done
-
-lemma f_image_eq:
- assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
- shows "f ` A = g ` A"
-proof
- show "f ` A \<subseteq> g ` A"
- by(rule image_subsetI, auto intro:h)
-next
- show "g ` A \<subseteq> f ` A"
- by (rule image_subsetI, auto intro:h[symmetric])
-qed
-
-
-definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
- where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
-
-
-lemma detached_test:
- shows "detached s th = (Th th \<notin> Field (RAG s))"
-apply(simp add: detached_def Field_def)
-apply(simp add: s_RAG_def)
-apply(simp add: s_holding_abv s_waiting_abv)
-apply(simp add: Domain_iff Range_iff)
-apply(simp add: wq_def)
-apply(auto)
-done
-
-context valid_trace
-begin
-
-lemma detached_intro:
- assumes eq_pv: "cntP s th = cntV s th"
- shows "detached s th"
-proof -
- from cnp_cnv_cncs
- have eq_cnt: "cntP s th =
- cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
- hence cncs_zero: "cntCS s th = 0"
- by (auto simp:eq_pv split:if_splits)
- with eq_cnt
- have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
- thus ?thesis
- proof
- assume "th \<notin> threads s"
- with range_in dm_RAG_threads
- show ?thesis
- by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
- next
- assume "th \<in> readys s"
- moreover have "Th th \<notin> Range (RAG s)"
- proof -
- from card_0_eq [OF finite_holding] and cncs_zero
- have "holdents s th = {}"
- by (simp add:cntCS_def)
- thus ?thesis
- apply(auto simp:holdents_test)
- apply(case_tac a)
- apply(auto simp:holdents_test s_RAG_def)
- done
- qed
- ultimately show ?thesis
- by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
- qed
-qed
-
-lemma detached_elim:
- assumes dtc: "detached s th"
- shows "cntP s th = cntV s th"
-proof -
- from cnp_cnv_cncs
- have eq_pv: " cntP s th =
- cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
- have cncs_z: "cntCS s th = 0"
- proof -
- from dtc have "holdents s th = {}"
- unfolding detached_def holdents_test s_RAG_def
- by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
- thus ?thesis by (auto simp:cntCS_def)
- qed
- show ?thesis
- proof(cases "th \<in> threads s")
- case True
- with dtc
- have "th \<in> readys s"
- by (unfold readys_def detached_def Field_def Domain_def Range_def,
- auto simp:eq_waiting s_RAG_def)
- with cncs_z and eq_pv show ?thesis by simp
- next
- case False
- with cncs_z and eq_pv show ?thesis by simp
- qed
-qed
-
-lemma detached_eq:
- shows "(detached s th) = (cntP s th = cntV s th)"
- by (insert vt, auto intro:detached_intro detached_elim)
-
-end
-
-text {*
- The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived
- from the concise and miniature model of PIP given in PrioGDef.thy.
-*}
-
-lemma eq_dependants: "dependants (wq s) = dependants s"
- by (simp add: s_dependants_abv wq_def)
-
-lemma next_th_unique:
- assumes nt1: "next_th s th cs th1"
- and nt2: "next_th s th cs th2"
- shows "th1 = th2"
-using assms by (unfold next_th_def, auto)
-
-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
-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)
-
-end
--- a/PrioG.thy~ Wed Jan 06 20:46:14 2016 +0800
+++ b/PrioG.thy~ Wed Jan 06 16:34:26 2016 +0000
@@ -1,5 +1,5 @@
theory PrioG
-imports PrioGDef
+imports PrioGDef RTree
begin
locale valid_trace =
@@ -3023,4 +3023,606 @@
shows "th1 = th2"
using assms by (unfold next_th_def, auto)
+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
+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 {* @{text "the_preced"} is also the same as @{text "preced"}, the only
+ difference is the order of arguemts. *}
+definition "the_preced s th = preced th s"
+
+lemma inj_the_preced:
+ "inj_on (the_preced s) (threads s)"
+ by (metis inj_onI preced_unique the_preced_def)
+
+text {* @{term "the_thread"} extracts thread out of RAG node. *}
+fun the_thread :: "node \<Rightarrow> thread" where
+ "the_thread (Th th) = th"
+
+text {* The following @{text "wRAG"} is the waiting sub-graph of @{text "RAG"}. *}
+definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"
+
+text {* The following @{text "hRAG"} is the holding sub-graph of @{text "RAG"}. *}
+definition "hRAG (s::state) = {(Cs cs, Th th) | th cs. holding s th cs}"
+
+text {* The following lemma splits @{term "RAG"} graph into the above two sub-graphs. *}
+lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"
+ by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv
+ s_holding_abv cs_RAG_def, auto)
+
+text {*
+ The following @{text "tRAG"} is the thread-graph derived from @{term "RAG"}.
+ It characterizes the dependency between threads when calculating current
+ precedences. It is defined as the composition of the above two sub-graphs,
+ names @{term "wRAG"} and @{term "hRAG"}.
+ *}
+definition "tRAG s = wRAG s O hRAG s"
+
+(* ccc *)
+
+definition "cp_gen s x =
+ Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"
+
+lemma tRAG_alt_def:
+ "tRAG s = {(Th th1, Th th2) | th1 th2.
+ \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
+ by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
+
+lemma tRAG_Field:
+ "Field (tRAG s) \<subseteq> Field (RAG s)"
+ by (unfold tRAG_alt_def Field_def, auto)
+
+lemma tRAG_ancestorsE:
+ assumes "x \<in> ancestors (tRAG s) u"
+ obtains th where "x = Th th"
+proof -
+ from assms have "(u, x) \<in> (tRAG s)^+"
+ by (unfold ancestors_def, auto)
+ from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
+ then obtain th where "x = Th th"
+ by (unfold tRAG_alt_def, auto)
+ from that[OF this] show ?thesis .
+qed
+
+lemma tRAG_mono:
+ assumes "RAG s' \<subseteq> RAG s"
+ shows "tRAG s' \<subseteq> tRAG s"
+ using assms
+ by (unfold tRAG_alt_def, auto)
+
+lemma holding_next_thI:
+ assumes "holding s th cs"
+ and "length (wq s cs) > 1"
+ obtains th' where "next_th s th cs th'"
+proof -
+ from assms(1)[folded eq_holding, unfolded cs_holding_def]
+ have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
+ then obtain rest where h1: "wq s cs = th#rest"
+ by (cases "wq s cs", auto)
+ with assms(2) have h2: "rest \<noteq> []" by auto
+ let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+ have "next_th s th cs ?th'" using h1(1) h2
+ by (unfold next_th_def, auto)
+ from that[OF this] show ?thesis .
+qed
+
+lemma RAG_tRAG_transfer:
+ assumes "vt s'"
+ assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+ and "(Cs cs, Th th'') \<in> RAG s'"
+ shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
+proof -
+ interpret vt_s': valid_trace "s'" using assms(1)
+ by (unfold_locales, simp)
+ interpret rtree: rtree "RAG s'"
+ proof
+ show "single_valued (RAG s')"
+ apply (intro_locales)
+ by (unfold single_valued_def,
+ auto intro:vt_s'.unique_RAG)
+
+ show "acyclic (RAG s')"
+ by (rule vt_s'.acyclic_RAG)
+ qed
+ { fix n1 n2
+ assume "(n1, n2) \<in> ?L"
+ from this[unfolded tRAG_alt_def]
+ obtain th1 th2 cs' where
+ h: "n1 = Th th1" "n2 = Th th2"
+ "(Th th1, Cs cs') \<in> RAG s"
+ "(Cs cs', Th th2) \<in> RAG s" by auto
+ from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
+ from h(3) and assms(2)
+ have "(Th th1, Cs cs') = (Th th, Cs cs) \<or>
+ (Th th1, Cs cs') \<in> RAG s'" by auto
+ hence "(n1, n2) \<in> ?R"
+ proof
+ assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
+ hence eq_th1: "th1 = th" by simp
+ moreover have "th2 = th''"
+ proof -
+ from h1 have "cs' = cs" by simp
+ from assms(3) cs_in[unfolded this] rtree.sgv
+ show ?thesis
+ by (unfold single_valued_def, auto)
+ qed
+ ultimately show ?thesis using h(1,2) by auto
+ next
+ assume "(Th th1, Cs cs') \<in> RAG s'"
+ with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
+ by (unfold tRAG_alt_def, auto)
+ from this[folded h(1, 2)] show ?thesis by auto
+ qed
+ } moreover {
+ fix n1 n2
+ assume "(n1, n2) \<in> ?R"
+ hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
+ hence "(n1, n2) \<in> ?L"
+ proof
+ assume "(n1, n2) \<in> tRAG s'"
+ moreover have "... \<subseteq> ?L"
+ proof(rule tRAG_mono)
+ show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
+ qed
+ ultimately show ?thesis by auto
+ next
+ assume eq_n: "(n1, n2) = (Th th, Th th'')"
+ from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
+ moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
+ ultimately show ?thesis
+ by (unfold eq_n tRAG_alt_def, auto)
+ qed
+ } ultimately show ?thesis by auto
+qed
+
+context valid_trace
+begin
+
+lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
+
end
+
+lemma cp_alt_def:
+ "cp s th =
+ Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
+proof -
+ have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
+ Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
+ (is "Max (_ ` ?L) = Max (_ ` ?R)")
+ proof -
+ have "?L = ?R"
+ by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
+ thus ?thesis by simp
+ qed
+ thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
+qed
+
+lemma cp_gen_alt_def:
+ "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
+ by (auto simp:cp_gen_def)
+
+lemma tRAG_nodeE:
+ assumes "(n1, n2) \<in> tRAG s"
+ obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
+ using assms
+ by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
+
+lemma subtree_nodeE:
+ assumes "n \<in> subtree (tRAG s) (Th th)"
+ obtains th1 where "n = Th th1"
+proof -
+ show ?thesis
+ proof(rule subtreeE[OF assms])
+ assume "n = Th th"
+ from that[OF this] show ?thesis .
+ next
+ assume "Th th \<in> ancestors (tRAG s) n"
+ hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+ hence "\<exists> th1. n = Th th1"
+ proof(induct)
+ case (base y)
+ from tRAG_nodeE[OF this] show ?case by metis
+ next
+ case (step y z)
+ thus ?case by auto
+ qed
+ with that show ?thesis by auto
+ qed
+qed
+
+lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
+proof -
+ have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*"
+ by (rule rtrancl_mono, auto simp:RAG_split)
+ also have "... \<subseteq> ((RAG s)^*)^*"
+ by (rule rtrancl_mono, auto)
+ also have "... = (RAG s)^*" by simp
+ finally show ?thesis by (unfold tRAG_def, simp)
+qed
+
+lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
+proof -
+ { fix a
+ assume "a \<in> subtree (tRAG s) x"
+ hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
+ with tRAG_star_RAG[of s]
+ have "(a, x) \<in> (RAG s)^*" by auto
+ hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
+ } thus ?thesis by auto
+qed
+
+lemma tRAG_trancl_eq:
+ "{th'. (Th th', Th th) \<in> (tRAG s)^+} =
+ {th'. (Th th', Th th) \<in> (RAG s)^+}"
+ (is "?L = ?R")
+proof -
+ { fix th'
+ assume "th' \<in> ?L"
+ hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
+ from tranclD[OF this]
+ obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
+ from tRAG_subtree_RAG[of s] and this(2)
+ have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG)
+ moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto
+ ultimately have "th' \<in> ?R" by auto
+ } moreover
+ { fix th'
+ assume "th' \<in> ?R"
+ hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
+ from plus_rpath[OF this]
+ obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
+ hence "(Th th', Th th) \<in> (tRAG s)^+"
+ proof(induct xs arbitrary:th' th rule:length_induct)
+ case (1 xs th' th)
+ then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
+ show ?case
+ proof(cases "xs1")
+ case Nil
+ from 1(2)[unfolded Cons1 Nil]
+ have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
+ hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
+ then obtain cs where "x1 = Cs cs"
+ by (unfold s_RAG_def, auto)
+ from rpath_nnl_lastE[OF rp[unfolded this]]
+ show ?thesis by auto
+ next
+ case (Cons x2 xs2)
+ from 1(2)[unfolded Cons1[unfolded this]]
+ have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
+ from rpath_edges_on[OF this]
+ have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
+ have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+ by (simp add: edges_on_unfold)
+ with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
+ then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
+ have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+ by (simp add: edges_on_unfold)
+ from this eds
+ have rg2: "(x1, x2) \<in> RAG s" by auto
+ from this[unfolded eq_x1]
+ obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
+ from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
+ have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
+ from rp have "rpath (RAG s) x2 xs2 (Th th)"
+ by (elim rpath_ConsE, simp)
+ from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
+ show ?thesis
+ proof(cases "xs2 = []")
+ case True
+ from rpath_nilE[OF rp'[unfolded this]]
+ have "th1 = th" by auto
+ from rt1[unfolded this] show ?thesis by auto
+ next
+ case False
+ from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
+ have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
+ with rt1 show ?thesis by auto
+ qed
+ qed
+ qed
+ hence "th' \<in> ?L" by auto
+ } ultimately show ?thesis by blast
+qed
+
+lemma tRAG_trancl_eq_Th:
+ "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} =
+ {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}"
+ using tRAG_trancl_eq by auto
+
+lemma dependants_alt_def:
+ "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
+ by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
+
+context valid_trace
+begin
+
+lemma count_eq_tRAG_plus:
+ assumes "cntP s th = cntV s th"
+ shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+ using assms count_eq_dependants dependants_alt_def eq_dependants by auto
+
+lemma count_eq_RAG_plus:
+ assumes "cntP s th = cntV s th"
+ shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+ using assms count_eq_dependants cs_dependants_def eq_RAG by auto
+
+lemma count_eq_RAG_plus_Th:
+ assumes "cntP s th = cntV s th"
+ shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+ using count_eq_RAG_plus[OF assms] by auto
+
+lemma count_eq_tRAG_plus_Th:
+ assumes "cntP s th = cntV s th"
+ shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+ using count_eq_tRAG_plus[OF assms] by auto
+
+end
+
+lemma tRAG_subtree_eq:
+ "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}"
+ (is "?L = ?R")
+proof -
+ { fix n
+ assume h: "n \<in> ?L"
+ hence "n \<in> ?R"
+ by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG)
+ } moreover {
+ fix n
+ assume "n \<in> ?R"
+ then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
+ by (auto simp:subtree_def)
+ from rtranclD[OF this(2)]
+ have "n \<in> ?L"
+ proof
+ assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
+ with h have "n \<in> {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}" by auto
+ thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
+ qed (insert h, auto simp:subtree_def)
+ } ultimately show ?thesis by auto
+qed
+
+lemma threads_set_eq:
+ "the_thread ` (subtree (tRAG s) (Th th)) =
+ {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
+ by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
+
+lemma cp_alt_def1:
+ "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
+proof -
+ have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
+ ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
+ by auto
+ thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
+qed
+
+lemma cp_gen_def_cond:
+ assumes "x = Th th"
+ shows "cp s th = cp_gen s (Th th)"
+by (unfold cp_alt_def1 cp_gen_def, simp)
+
+lemma cp_gen_over_set:
+ assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
+ shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
+proof(rule f_image_eq)
+ fix a
+ assume "a \<in> A"
+ from assms[rule_format, OF this]
+ obtain th where eq_a: "a = Th th" by auto
+ show "cp_gen s a = (cp s \<circ> the_thread) a"
+ by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
+qed
+
+
+context valid_trace
+begin
+
+lemma RAG_threads:
+ assumes "(Th th) \<in> Field (RAG s)"
+ shows "th \<in> threads s"
+ using assms
+ by (metis Field_def UnE dm_RAG_threads range_in vt)
+
+lemma subtree_tRAG_thread:
+ assumes "th \<in> threads s"
+ shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
+proof -
+ have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+ by (unfold tRAG_subtree_eq, simp)
+ also have "... \<subseteq> ?R"
+ proof
+ fix x
+ assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+ then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
+ from this(2)
+ show "x \<in> ?R"
+ proof(cases rule:subtreeE)
+ case 1
+ thus ?thesis by (simp add: assms h(1))
+ next
+ case 2
+ thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI)
+ qed
+ qed
+ finally show ?thesis .
+qed
+
+lemma readys_root:
+ assumes "th \<in> readys s"
+ shows "root (RAG s) (Th th)"
+proof -
+ { fix x
+ assume "x \<in> ancestors (RAG s) (Th th)"
+ hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+ from tranclD[OF this]
+ obtain z where "(Th th, z) \<in> RAG s" by auto
+ with assms(1) have False
+ apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
+ by (fold wq_def, blast)
+ } thus ?thesis by (unfold root_def, auto)
+qed
+
+lemma readys_in_no_subtree:
+ assumes "th \<in> readys s"
+ and "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s) (Th th')"
+proof
+ assume "Th th \<in> subtree (RAG s) (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 1
+ with assms show ?thesis by auto
+ next
+ case 2
+ with readys_root[OF assms(1)]
+ show ?thesis by (auto simp:root_def)
+ qed
+qed
+
+lemma not_in_thread_isolated:
+ assumes "th \<notin> threads s"
+ shows "(Th th) \<notin> Field (RAG s)"
+proof
+ assume "(Th th) \<in> Field (RAG s)"
+ with dm_RAG_threads and range_in assms
+ show False by (unfold Field_def, blast)
+qed
+
+lemma wf_RAG: "wf (RAG s)"
+proof(rule finite_acyclic_wf)
+ from finite_RAG show "finite (RAG s)" .
+next
+ from acyclic_RAG show "acyclic (RAG s)" .
+qed
+
+lemma sgv_wRAG: "single_valued (wRAG s)"
+ using waiting_unique
+ by (unfold single_valued_def wRAG_def, auto)
+
+lemma sgv_hRAG: "single_valued (hRAG s)"
+ using holding_unique
+ by (unfold single_valued_def hRAG_def, auto)
+
+lemma sgv_tRAG: "single_valued (tRAG s)"
+ by (unfold tRAG_def, rule single_valued_relcomp,
+ insert sgv_wRAG sgv_hRAG, auto)
+
+lemma acyclic_tRAG: "acyclic (tRAG s)"
+proof(unfold tRAG_def, rule acyclic_compose)
+ show "acyclic (RAG s)" using acyclic_RAG .
+next
+ show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+next
+ show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+qed
+
+lemma sgv_RAG: "single_valued (RAG s)"
+ using unique_RAG by (auto simp:single_valued_def)
+
+lemma rtree_RAG: "rtree (RAG s)"
+ using sgv_RAG acyclic_RAG
+ by (unfold rtree_def rtree_axioms_def sgv_def, auto)
+
+end
+context valid_trace
+begin
+
+(* ddd *)
+lemma cp_gen_rec:
+ assumes "x = Th th"
+ shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
+proof(cases "children (tRAG s) x = {}")
+ case True
+ show ?thesis
+ by (unfold True cp_gen_def subtree_children, simp add:assms)
+next
+ case False
+ hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
+ note fsbttRAGs.finite_subtree[simp]
+ have [simp]: "finite (children (tRAG s) x)"
+ by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree],
+ rule children_subtree)
+ { fix r x
+ have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
+ } note this[simp]
+ have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
+ proof -
+ from False obtain q where "q \<in> children (tRAG s) x" by blast
+ moreover have "subtree (tRAG s) q \<noteq> {}" by simp
+ ultimately show ?thesis by blast
+ qed
+ have h: "Max ((the_preced s \<circ> the_thread) `
+ ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
+ Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
+ (is "?L = ?R")
+ proof -
+ let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
+ let "Max (_ \<union> (?h ` ?B))" = ?R
+ let ?L1 = "?f ` \<Union>(?g ` ?B)"
+ have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
+ proof -
+ have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
+ also have "... = (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
+ finally have "Max ?L1 = Max ..." by simp
+ also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
+ by (subst Max_UNION, simp+)
+ also have "... = Max (cp_gen s ` children (tRAG s) x)"
+ by (unfold image_comp cp_gen_alt_def, simp)
+ finally show ?thesis .
+ qed
+ show ?thesis
+ proof -
+ have "?L = Max (?f ` ?A \<union> ?L1)" by simp
+ also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
+ by (subst Max_Un, simp+)
+ also have "... = max (?f x) (Max (?h ` ?B))"
+ by (unfold eq_Max_L1, simp)
+ also have "... =?R"
+ by (rule max_Max_eq, (simp)+, unfold assms, simp)
+ finally show ?thesis .
+ qed
+ qed thus ?thesis
+ by (fold h subtree_children, unfold cp_gen_def, simp)
+qed
+
+lemma cp_rec:
+ "cp s th = Max ({the_preced s th} \<union>
+ (cp s o the_thread) ` children (tRAG s) (Th th))"
+proof -
+ have "Th th = Th th" by simp
+ note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this]
+ show ?thesis
+ proof -
+ have "cp_gen s ` children (tRAG s) (Th th) =
+ (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
+ proof(rule cp_gen_over_set)
+ show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
+ by (unfold tRAG_alt_def, auto simp:children_def)
+ qed
+ thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+ qed
+qed
+
+end
+
+end
--- a/PrioGDef.thy Wed Jan 06 20:46:14 2016 +0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,614 +0,0 @@
-chapter {* Definitions *}
-(*<*)
-theory PrioGDef
-imports Precedence_ord Moment
-begin
-(*>*)
-
-text {*
- In this section, the formal model of Priority Inheritance Protocol (PIP) is presented.
- The model is based on Paulson's inductive protocol verification method, where
- the state of the system is modelled as a list of events happened so far with the latest
- event put at the head.
-*}
-
-text {*
- To define events, the identifiers of {\em threads},
- {\em priority} and {\em critical resources } (abbreviated as @{text "cs"})
- need to be represented. All three are represetned using standard
- Isabelle/HOL type @{typ "nat"}:
-*}
-
-type_synonym thread = nat -- {* Type for thread identifiers. *}
-type_synonym priority = nat -- {* Type for priorities. *}
-type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
-
-text {*
- \noindent
- The abstraction of Priority Inheritance Protocol (PIP) is set at the system call level.
- Every system call is represented as an event. The format of events is defined
- defined as follows:
- *}
-
-datatype event =
- Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *}
- Exit thread | -- {* Thread @{text "thread"} finishing its execution. *}
- P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *}
- V thread cs | -- {* Thread @{text "thread"} releasing critical resource @{text "cs"}. *}
- Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
-
-
-text {*
- As mentioned earlier, in Paulson's inductive method, the states of system are represented as lists of events,
- which is defined by the following type @{text "state"}:
- *}
-type_synonym state = "event list"
-
-
-text {*
-\noindent
- Resource Allocation Graph (RAG for short) is used extensively in our formal analysis.
- The following type @{text "node"} is used to represent nodes in RAG.
- *}
-datatype node =
- Th "thread" | -- {* Node for thread. *}
- Cs "cs" -- {* Node for critical resource. *}
-
-text {*
- \noindent
- The following function
- @{text "threads"} is used to calculate the set of live threads (@{text "threads s"})
- in state @{text "s"}.
- *}
-fun threads :: "state \<Rightarrow> thread set"
- where
- -- {* At the start of the system, the set of threads is empty: *}
- "threads [] = {}" |
- -- {* New thread is added to the @{text "threads"}: *}
- "threads (Create thread prio#s) = {thread} \<union> threads s" |
- -- {* Finished thread is removed: *}
- "threads (Exit thread # s) = (threads s) - {thread}" |
- -- {* Other kind of events does not affect the value of @{text "threads"}: *}
- "threads (e#s) = threads s"
-
-text {*
- \noindent
- The function @{text "threads"} defined above is one of
- the so called {\em observation function}s which forms
- the very basis of Paulson's inductive protocol verification method.
- Each observation function {\em observes} one particular aspect (or attribute)
- of the system. For example, the attribute observed by @{text "threads s"}
- is the set of threads living in state @{text "s"}.
- The protocol being modelled
- The decision made the protocol being modelled is based on the {\em observation}s
- returned by {\em observation function}s. Since {\observation function}s forms
- the very basis on which Paulson's inductive method is based, there will be
- a lot of such observation functions introduced in the following. In fact, any function
- which takes event list as argument is a {\em observation function}.
- *}
-
-text {* \noindent
- Observation @{text "priority th s"} is
- the {\em original priority} of thread @{text "th"} in state @{text "s"}.
- The {\em original priority} is the priority
- assigned to a thread when it is created or when it is reset by system call
- (represented by event @{text "Set thread priority"}).
-*}
-
-fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
- where
- -- {* @{text "0"} is assigned to threads which have never been created: *}
- "priority thread [] = 0" |
- "priority thread (Create thread' prio#s) =
- (if thread' = thread then prio else priority thread s)" |
- "priority thread (Set thread' prio#s) =
- (if thread' = thread then prio else priority thread s)" |
- "priority thread (e#s) = priority thread s"
-
-text {*
- \noindent
- Observation @{text "last_set th s"} is the last time when the priority of thread @{text "th"} is set,
- observed from state @{text "s"}.
- The time in the system is measured by the number of events happened so far since the very beginning.
-*}
-fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
- where
- "last_set thread [] = 0" |
- "last_set thread ((Create thread' prio)#s) =
- (if (thread = thread') then length s else last_set thread s)" |
- "last_set thread ((Set thread' prio)#s) =
- (if (thread = thread') then length s else last_set thread s)" |
- "last_set thread (_#s) = last_set thread s"
-
-text {*
- \noindent
- The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of
- a thread is the combination of its {\em original priority} and {\em time} the priority is set.
- The intention is to discriminate threads with the same priority by giving threads whose priority
- is assigned earlier higher precedences, becasue such threads are more urgent to finish.
- This explains the following definition:
- *}
-definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
- where "preced thread s \<equiv> Prc (priority thread s) (last_set thread s)"
-
-
-text {*
- \noindent
- A number of important notions in PIP are represented as the following functions,
- defined in terms of the waiting queues of the system, where the waiting queues
- , as a whole, is represented by the @{text "wq"} argument of every notion function.
- The @{text "wq"} argument is itself a functions which maps every critical resource
- @{text "cs"} to the list of threads which are holding or waiting for it.
- The thread at the head of this list is designated as the thread which is current
- holding the resrouce, which is slightly different from tradition where
- all threads in the waiting queue are considered as waiting for the resource.
- *}
-
-consts
- holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
- waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
- RAG :: "'b \<Rightarrow> (node \<times> node) set"
- dependants :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
-
-defs (overloaded)
- -- {*
- \begin{minipage}{0.9\textwidth}
- This meaning of @{text "wq"} is reflected in the following definition of @{text "holding wq th cs"},
- where @{text "holding wq th cs"} means thread @{text "th"} is holding the critical
- resource @{text "cs"}. This decision is based on @{text "wq"}.
- \end{minipage}
- *}
-
- cs_holding_def:
- "holding wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
- -- {*
- \begin{minipage}{0.9\textwidth}
- In accordance with the definition of @{text "holding wq th cs"},
- a thread @{text "th"} is considered waiting for @{text "cs"} if
- it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head.
- This is reflected in the definition of @{text "waiting wq th cs"} as follows:
- \end{minipage}
- *}
- cs_waiting_def:
- "waiting wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
- -- {*
- \begin{minipage}{0.9\textwidth}
- @{text "RAG wq"} generates RAG (a binary relations on @{text "node"})
- out of waiting queues of the system (represented by the @{text "wq"} argument):
- \end{minipage}
- *}
- cs_RAG_def:
- "RAG (wq::cs \<Rightarrow> thread list) \<equiv>
- {(Th th, Cs cs) | th cs. waiting wq th cs} \<union> {(Cs cs, Th th) | cs th. holding wq th cs}"
- -- {*
- \begin{minipage}{0.9\textwidth}
- The following @{text "dependants wq th"} represents the set of threads which are RAGing on
- thread @{text "th"} in Resource Allocation Graph @{text "RAG wq"}.
- Here, "RAGing" means waiting directly or indirectly on the critical resource.
- \end{minipage}
- *}
- cs_dependants_def:
- "dependants (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
-
-
-text {* \noindent
- The following
- @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under
- state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of
- Priority Inheritance that the {\em current precedence} of a thread is the precedence
- inherited from the maximum of all its dependants, i.e. the threads which are waiting
- directly or indirectly waiting for some resources from it. If no such thread exits,
- @{text "th"}'s {\em current precedence} equals its original precedence, i.e.
- @{text "preced th s"}.
- *}
-
-definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
- where "cpreced wq s = (\<lambda>th. Max ((\<lambda>th'. preced th' s) ` ({th} \<union> dependants wq th)))"
-
-text {*
- Notice that the current precedence (@{text "cpreced"}) of one thread @{text "th"} can be boosted
- (becoming larger than its own precedence) by those threads in
- the @{text "dependants wq th"}-set. If one thread get boosted, we say
- it inherits the priority (or, more precisely, the precedence) of
- its dependants. This is how the word "Inheritance" in
- Priority Inheritance Protocol comes.
-*}
-
-(*<*)
-lemma
- cpreced_def2:
- "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
- unfolding cpreced_def image_def
- apply(rule eq_reflection)
- apply(rule_tac f="Max" in arg_cong)
- by (auto)
-(*>*)
-
-
-text {* \noindent
- Assuming @{text "qs"} be the waiting queue of a critical resource,
- the following abbreviation "release qs" is the waiting queue after the thread
- holding the resource (which is thread at the head of @{text "qs"}) released
- the resource:
-*}
-abbreviation
- "release qs \<equiv> case qs of
- [] => []
- | (_#qs') => (SOME q. distinct q \<and> set q = set qs')"
-text {* \noindent
- It can be seen from the definition that the thread at the head of @{text "qs"} is removed
- from the return value, and the value @{term "q"} is an reordering of @{text "qs'"}, the
- tail of @{text "qs"}. Through this reordering, one of the waiting threads (those in @{text "qs'"} }
- is chosen nondeterministically to be the head of the new queue @{text "q"}.
- Therefore, this thread is the one who takes over the resource. This is a little better different
- from common sense that the thread who comes the earliest should take over.
- The intention of this definition is to show that the choice of which thread to take over the
- release resource does not affect the correctness of the PIP protocol.
-*}
-
-text {*
- The data structure used by the operating system for scheduling is referred to as
- {\em schedule state}. It is represented as a record consisting of
- a function assigning waiting queue to resources
- (to be used as the @{text "wq"} argument in @{text "holding"}, @{text "waiting"}
- and @{text "RAG"}, etc) and a function assigning precedence to threads:
- *}
-
-record schedule_state =
- wq_fun :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
- cprec_fun :: "thread \<Rightarrow> precedence" -- {* The function assigning precedence. *}
-
-text {* \noindent
- The following two abbreviations (@{text "all_unlocked"} and @{text "initial_cprec"})
- are used to set the initial values of the @{text "wq_fun"} @{text "cprec_fun"} fields
- respectively of the @{text "schedule_state"} record by the following function @{text "sch"},
- which is used to calculate the system's {\em schedule state}.
-
- Since there is no thread at the very beginning to make request, all critical resources
- are free (or unlocked). This status is represented by the abbreviation
- @{text "all_unlocked"}.
- *}
-abbreviation
- "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
-
-
-text {* \noindent
- The initial current precedence for a thread can be anything, because there is no thread then.
- We simply assume every thread has precedence @{text "Prc 0 0"}.
- *}
-
-abbreviation
- "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
-
-
-text {* \noindent
- The following function @{text "schs"} is used to calculate the system's schedule state @{text "schs s"}
- out of the current system state @{text "s"}. It is the central function to model Priority Inheritance:
- *}
-fun schs :: "state \<Rightarrow> schedule_state"
- where
- -- {*
- \begin{minipage}{0.9\textwidth}
- Setting the initial value of the @{text "schedule_state"} record (see the explanations above).
- \end{minipage}
- *}
- "schs [] = (| wq_fun = all_unlocked, cprec_fun = initial_cprec |)" |
-
- -- {*
- \begin{minipage}{0.9\textwidth}
- \begin{enumerate}
- \item @{text "ps"} is the schedule state of last moment.
- \item @{text "pwq"} is the waiting queue function of last moment.
- \item @{text "pcp"} is the precedence function of last moment (NOT USED).
- \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement:
- \begin{enumerate}
- \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to
- the end of @{text "cs"}'s waiting queue.
- \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state,
- @{text "th'"} must equal to @{text "thread"},
- because @{text "thread"} is the one currently holding @{text "cs"}.
- The case @{text "[] \<Longrightarrow> []"} may never be executed in a legal state.
- the @{text "(SOME q. distinct q \<and> set q = set qs)"} is used to choose arbitrarily one
- thread in waiting to take over the released resource @{text "cs"}. In our representation,
- this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
- \item For other happening event, the schedule state just does not change.
- \end{enumerate}
- \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue
- function. The RAGency of precedence function on waiting queue function is the reason to
- put them in the same record so that they can evolve together.
- \end{enumerate}
-
-
- The calculation of @{text "cprec_fun"} depends on the value of @{text "wq_fun"}.
- Therefore, in the following cases, @{text "wq_fun"} is always calculated first, in
- the name of @{text "wq"} (if @{text "wq_fun"} is not changed
- by the happening event) or @{text "new_wq"} (if the value of @{text "wq_fun"} is changed).
- \end{minipage}
- *}
- "schs (Create th prio # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"
-| "schs (Exit th # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"
-| "schs (Set th prio # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"
- -- {*
- \begin{minipage}{0.9\textwidth}
- Different from the forth coming cases, the @{text "wq_fun"} field of the schedule state
- is changed. So, the new value is calculated first, in the name of @{text "new_wq"}.
- \end{minipage}
- *}
-| "schs (P th cs # s) =
- (let wq = wq_fun (schs s) in
- let new_wq = wq(cs := (wq cs @ [th])) in
- (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"
-| "schs (V th cs # s) =
- (let wq = wq_fun (schs s) in
- let new_wq = wq(cs := release (wq cs)) in
- (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"
-
-lemma cpreced_initial:
- "cpreced (\<lambda> cs. []) [] = (\<lambda>_. (Prc 0 0))"
-apply(simp add: cpreced_def)
-apply(simp add: cs_dependants_def cs_RAG_def cs_waiting_def cs_holding_def)
-apply(simp add: preced_def)
-done
-
-lemma sch_old_def:
- "schs (e#s) = (let ps = schs s in
- let pwq = wq_fun ps in
- let nwq = case e of
- P th cs \<Rightarrow> pwq(cs:=(pwq cs @ [th])) |
- V th cs \<Rightarrow> let nq = case (pwq cs) of
- [] \<Rightarrow> [] |
- (_#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
- in pwq(cs:=nq) |
- _ \<Rightarrow> pwq
- in let ncp = cpreced nwq (e#s) in
- \<lparr>wq_fun = nwq, cprec_fun = ncp\<rparr>
- )"
-apply(cases e)
-apply(simp_all)
-done
-
-
-text {*
- \noindent
- The following @{text "wq"} is a shorthand for @{text "wq_fun"}.
- *}
-definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
- where "wq s = wq_fun (schs s)"
-
-text {* \noindent
- The following @{text "cp"} is a shorthand for @{text "cprec_fun"}.
- *}
-definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
- where "cp s \<equiv> cprec_fun (schs s)"
-
-text {* \noindent
- Functions @{text "holding"}, @{text "waiting"}, @{text "RAG"} and
- @{text "dependants"} still have the
- same meaning, but redefined so that they no longer RAG on the
- fictitious {\em waiting queue function}
- @{text "wq"}, but on system state @{text "s"}.
- *}
-defs (overloaded)
- s_holding_abv:
- "holding (s::state) \<equiv> holding (wq_fun (schs s))"
- s_waiting_abv:
- "waiting (s::state) \<equiv> waiting (wq_fun (schs s))"
- s_RAG_abv:
- "RAG (s::state) \<equiv> RAG (wq_fun (schs s))"
- s_dependants_abv:
- "dependants (s::state) \<equiv> dependants (wq_fun (schs s))"
-
-
-text {*
- The following lemma can be proved easily, and the meaning is obvious.
- *}
-lemma
- s_holding_def:
- "holding (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th = hd (wq_fun (schs s) cs))"
- by (auto simp:s_holding_abv wq_def cs_holding_def)
-
-lemma s_waiting_def:
- "waiting (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th \<noteq> hd (wq_fun (schs s) cs))"
- by (auto simp:s_waiting_abv wq_def cs_waiting_def)
-
-lemma s_RAG_def:
- "RAG (s::state) =
- {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \<union> {(Cs cs, Th th) | cs th. holding (wq s) th cs}"
- by (auto simp:s_RAG_abv wq_def cs_RAG_def)
-
-lemma
- s_dependants_def:
- "dependants (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (RAG (wq s))^+}"
- by (auto simp:s_dependants_abv wq_def cs_dependants_def)
-
-text {*
- The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready}
- for running if it is a live thread and it is not waiting for any critical resource.
- *}
-definition readys :: "state \<Rightarrow> thread set"
- where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waiting s th cs)}"
-
-text {* \noindent
- The following function @{text "runing"} calculates the set of running thread, which is the ready
- thread with the highest precedence.
- *}
-definition runing :: "state \<Rightarrow> thread set"
- where "runing s \<equiv> {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
-
-text {* \noindent
- Notice that the definition of @{text "running"} reflects the preemptive scheduling strategy,
- because, if the @{text "running"}-thread (the one in @{text "runing"} set)
- lowered its precedence by resetting its own priority to a lower
- one, it will lose its status of being the max in @{text "ready"}-set and be superseded.
-*}
-
-text {* \noindent
- The following function @{text "holdents s th"} returns the set of resources held by thread
- @{text "th"} in state @{text "s"}.
- *}
-definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
- where "holdents s th \<equiv> {cs . holding s th cs}"
-
-lemma holdents_test:
- "holdents s th = {cs . (Cs cs, Th th) \<in> RAG s}"
-unfolding holdents_def
-unfolding s_RAG_def
-unfolding s_holding_abv
-unfolding wq_def
-by (simp)
-
-text {* \noindent
- Observation @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in
- state @{text "s"}:
- *}
-definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
- where "cntCS s th = card (holdents s th)"
-
-text {* \noindent
- According to the convention of Paulson's inductive method,
- the decision made by a protocol that event @{text "e"} is eligible to happen next under state @{text "s"}
- is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as
- follows (notice how the decision is based on the {\em observation function}s
- defined above, and also notice how a complicated protocol is modeled by a few simple
- observations, and how such a kind of simplicity gives rise to improved trust on
- faithfulness):
- *}
-inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
- where
- -- {*
- A thread can be created if it is not a live thread:
- *}
- thread_create: "\<lbrakk>thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
- -- {*
- A thread can exit if it no longer hold any resource:
- *}
- thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
- -- {*
- \begin{minipage}{0.9\textwidth}
- A thread can request for an critical resource @{text "cs"}, if it is running and
- the request does not form a loop in the current RAG. The latter condition
- is set up to avoid deadlock. The condition also reflects our assumption all threads are
- carefully programmed so that deadlock can not happen:
- \end{minipage}
- *}
- thread_P: "\<lbrakk>thread \<in> runing s; (Cs cs, Th thread) \<notin> (RAG s)^+\<rbrakk> \<Longrightarrow>
- step s (P thread cs)" |
- -- {*
- \begin{minipage}{0.9\textwidth}
- A thread can release a critical resource @{text "cs"}
- if it is running and holding that resource:
- \end{minipage}
- *}
- thread_V: "\<lbrakk>thread \<in> runing s; holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
- -- {*
- \begin{minipage}{0.9\textwidth}
- A thread can adjust its own priority as long as it is current running.
- With the resetting of one thread's priority, its precedence may change.
- If this change lowered the precedence, according to the definition of @{text "running"}
- function,
- \end{minipage}
- *}
- thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
-
-text {*
- In Paulson's inductive method, every protocol is defined by such a @{text "step"}
- predicate. For instance, the predicate @{text "step"} given above
- defines the PIP protocol. So, it can also be called "PIP".
-*}
-
-abbreviation
- "PIP \<equiv> step"
-
-
-text {* \noindent
- For any protocol defined by a @{text "step"} predicate,
- the fact that @{text "s"} is a legal state in
- the protocol is expressed as: @{text "vt step s"}, where
- the predicate @{text "vt"} can be defined as the following:
- *}
-inductive vt :: "state \<Rightarrow> bool"
- where
- -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
- vt_nil[intro]: "vt []" |
- -- {*
- \begin{minipage}{0.9\textwidth}
- If @{text "s"} a legal state of the protocol defined by predicate @{text "step"},
- and event @{text "e"} is allowed to happen under state @{text "s"} by the protocol
- predicate @{text "step"}, then @{text "e#s"} is a new legal state rendered by the
- happening of @{text "e"}:
- \end{minipage}
- *}
- vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
-
-text {* \noindent
- It is easy to see that the definition of @{text "vt"} is generic. It can be applied to
- any specific protocol specified by a @{text "step"}-predicate to get the set of
- legal states of that particular protocol.
- *}
-
-text {*
- The following are two very basic properties of @{text "vt"}.
-*}
-
-lemma step_back_vt: "vt (e#s) \<Longrightarrow> vt s"
- by(ind_cases "vt (e#s)", simp)
-
-lemma step_back_step: "vt (e#s) \<Longrightarrow> step s e"
- by(ind_cases "vt (e#s)", simp)
-
-text {* \noindent
- The following two auxiliary functions @{text "the_cs"} and @{text "the_th"} are used to extract
- critical resource and thread respectively out of RAG nodes.
- *}
-fun the_cs :: "node \<Rightarrow> cs"
- where "the_cs (Cs cs) = cs"
-
-fun the_th :: "node \<Rightarrow> thread"
- where "the_th (Th th) = th"
-
-text {* \noindent
- The following predicate @{text "next_th"} describe the next thread to
- take over when a critical resource is released. In @{text "next_th s th cs t"},
- @{text "th"} is the thread to release, @{text "t"} is the one to take over.
- Notice how this definition is backed up by the @{text "release"} function and its use
- in the @{text "V"}-branch of @{text "schs"} function. This @{text "next_th"} function
- is not needed for the execution of PIP. It is introduced as an auxiliary function
- to state lemmas. The correctness of this definition will be confirmed by
- lemmas @{text "step_v_hold_inv"}, @{text " step_v_wait_inv"},
- @{text "step_v_get_hold"} and @{text "step_v_not_wait"}.
- *}
-definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
- where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and>
- t = hd (SOME q. distinct q \<and> set q = set rest))"
-
-text {* \noindent
- The aux function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
- in list @{text "l"}:
- *}
-definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
- where "count Q l = length (filter Q l)"
-
-text {* \noindent
- The following observation @{text "cntP s"} returns the number of operation @{text "P"} happened
- before reaching state @{text "s"}.
- *}
-definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
- where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
-
-text {* \noindent
- The following observation @{text "cntV s"} returns the number of operation @{text "V"} happened
- before reaching state @{text "s"}.
- *}
-definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
- where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
-(*<*)
-
-end
-(*>*)
-
--- a/PrioGDef.thy~ Wed Jan 06 20:46:14 2016 +0800
+++ b/PrioGDef.thy~ Wed Jan 06 16:34:26 2016 +0000
@@ -1,4 +1,4 @@
- {* Definitions *}
+chapter {* Definitions *}
(*<*)
theory PrioGDef
imports Precedence_ord Moment
@@ -387,6 +387,8 @@
definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
where "cp s \<equiv> cprec_fun (schs s)"
+definition "cp_gen s x = Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"
+
text {* \noindent
Functions @{text "holding"}, @{text "waiting"}, @{text "RAG"} and
@{text "dependants"} still have the
@@ -585,6 +587,7 @@
*}
definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and>
+ t = hd (SOME q. distinct q \<and> set q = set rest))"
text {* \noindent
The aux function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
--- a/README Wed Jan 06 20:46:14 2016 +0800
+++ b/README Wed Jan 06 16:34:26 2016 +0000
@@ -4,10 +4,10 @@
Max.thy Some generic facts about Max.
Precedence_ord.thy A theory of precedences.
Moment.thy The notion of moment.
- PrioGDef.thy The formal definition of the PIP-model.
- PrioG.thy Basic properties of the PIP-model.
- ExtGG.thy The correctness proof of the PIP-model.
- CpsG.thy Properties interesting for an implementation.
+ PIPDefs.thy The formal definition of the PIP-model.
+ PIPBasics.thy Basic properties of the PIP-model.
+ Correctness.thy The correctness proof of the PIP-model.
+ Implementation.thy Properties interesting for an implementation.
The repository can be checked using Isabelle 2013-2.
--- a/ROOT Wed Jan 06 20:46:14 2016 +0800
+++ b/ROOT Wed Jan 06 16:34:26 2016 +0000
@@ -1,7 +1,7 @@
session "PIP" = HOL +
theories [document = false, quick_and_dirty]
- "CpsG"
- "ExtGG"
+ "Implementation"
+ "Correctness"
"Test"
session "Slides2" in "Slides" = PIP +
--- a/RTree.thy Wed Jan 06 20:46:14 2016 +0800
+++ b/RTree.thy Wed Jan 06 16:34:26 2016 +0000
@@ -154,6 +154,7 @@
lemma index_minimize:
assumes "P (i::nat)"
obtains j where "P j" and "\<forall> k < j. \<not> P k"
+using assms
proof -
have "\<exists> j. P j \<and> (\<forall> k < j. \<not> P k)"
using assms
Binary file journal.pdf has changed