pip: comparison PIPBasics.thy

equal deleted inserted replaced

-:0525670d8e6a
+:4763aa246dbd
-theory PIPBasics
+theory CpsG
 imports PIPDefs
 begin
+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
+lemma Max_fg_mono:
+assumes "finite A"
+and "\<forall> a \<in> A. f a \<le> g a"
+shows "Max (f ` A) \<le> Max (g ` A)"
+proof(cases "A = {}")
+case True
+thus ?thesis by auto
+next
+case False
+show ?thesis
+proof(rule Max.boundedI)
+from assms show "finite (f ` A)" by auto
+next
+from False show "f ` A \<noteq> {}" by auto
+next
+fix fa
+assume "fa \<in> f ` A"
+then obtain a where h_fa: "a \<in> A" "fa = f a" by auto
+show "fa \<le> Max (g ` A)"
+proof(rule Max_ge_iff[THEN iffD2])
+from assms show "finite (g ` A)" by auto
+next
+from False show "g ` A \<noteq> {}" by auto
+next
+from h_fa have "g a \<in> g ` A" by auto
+moreover have "fa \<le> g a" using h_fa assms(2) by auto
+ultimately show "\<exists>a\<in>g ` A. fa \<le> a" by auto
+qed
+qed
+qed
+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
+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
+lemma birth_time_lt:
+assumes "s \<noteq> []"
+shows "last_set th s < length s"
+using assms
+proof(induct s)
+case (Cons a s)
+show ?case
+proof(cases "s \<noteq> []")
+case False
+thus ?thesis
+by (cases a, auto)
+next
+case True
+show ?thesis using Cons(1)[OF True]
+by (cases a, auto)
+qed
+qed simp
+lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
+by (induct s, auto)
+lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
+by (drule_tac th_in_ne, unfold preced_def, auto intro: birth_time_lt)
+lemma eq_RAG:
+"RAG (wq s) = RAG s"
+by (unfold cs_RAG_def s_RAG_def, auto)
+lemma waiting_holding:
+assumes "waiting (s::state) th cs"
+obtains th' where "holding s th' cs"
+proof -
+from assms[unfolded s_waiting_def, folded wq_def]
+obtain th' where "th' \<in> set (wq s cs)" "th' = hd (wq s cs)"
+by (metis empty_iff hd_in_set list.set(1))
+hence "holding s th' cs"
+by (unfold s_holding_def, fold wq_def, auto)
+from that[OF this] show ?thesis .
+qed
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+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
+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
+(* ccc *)
 locale valid_trace =
 fixes s
 assumes vt : "vt s"
 lemma pip_e: "PIP s e"
 using vt_e by (cases, simp)
 end
+locale valid_trace_create = valid_trace_e +
+fixes th prio
+assumes is_create: "e = Create th prio"
+locale valid_trace_exit = valid_trace_e +
+fixes th
+assumes is_exit: "e = Exit th"
+locale valid_trace_p = valid_trace_e +
+fixes th cs
+assumes is_p: "e = P th cs"
+locale valid_trace_v = valid_trace_e +
+fixes th cs
+assumes is_v: "e = V th cs"
+begin
+definition "rest = tl (wq s cs)"
+definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+end
+locale valid_trace_v_n = valid_trace_v +
+assumes rest_nnl: "rest \<noteq> []"
+locale valid_trace_v_e = valid_trace_v +
+assumes rest_nil: "rest = []"
+locale valid_trace_set= valid_trace_e +
+fixes th prio
+assumes is_set: "e = Set th prio"
+context valid_trace
+begin
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+assumes "PP []"
+and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
+PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
+shows "PP s"
+proof(induct rule:vt.induct[OF vt, case_names Init Step])
+case Init
+from assms(1) show ?case .
+next
+case (Step s e)
+show ?case
+proof(rule assms(2))
+show "valid_trace_e s e" using Step by (unfold_locales, auto)
+next
+show "PP s" using Step by simp
+next
+show "PIP s e" using Step by simp
+qed
+qed
+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 valid_trace_e_def, auto)
+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
+lemma finite_threads:
+shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+end
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+by (unfold s_RAG_def, auto)
+locale valid_moment = valid_trace +
+fixes i :: nat
+sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
+by (unfold_locales, insert vt_moment, auto)
+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 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:
+lemma wq_v_neq [simp]:
 "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)
 lemma runing_head:
 assumes "th \<in> runing s"
 by (simp add:runing_def readys_def s_waiting_def wq_def)
 context valid_trace
 begin
+lemma runing_wqE:
+assumes "th \<in> runing s"
+and "th \<in> set (wq s cs)"
+obtains rest where "wq s cs = th#rest"
+proof -
+from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
+by (meson list.set_cases)
+have "th' = th"
+proof(rule ccontr)
+assume "th' \<noteq> th"
+hence "th \<noteq> hd (wq s cs)" using eq_wq by auto
+with assms(2)
+have "waiting s th cs"
+by (unfold s_waiting_def, fold wq_def, auto)
+with assms show False
+by (unfold runing_def readys_def, auto)
+qed
+with eq_wq that show ?thesis by metis
+qed
+end
+context valid_trace_create
+begin
+lemma wq_neq_simp [simp]:
+shows "wq (e#s) cs' = wq s cs'"
+using assms unfolding is_create wq_def
+by (auto simp:Let_def)
+lemma wq_distinct_kept:
+assumes "distinct (wq s cs')"
+shows "distinct (wq (e#s) cs')"
+using assms by simp
+end
+context valid_trace_exit
+begin
+lemma wq_neq_simp [simp]:
+shows "wq (e#s) cs' = wq s cs'"
+using assms unfolding is_exit wq_def
+by (auto simp:Let_def)
+lemma wq_distinct_kept:
+assumes "distinct (wq s cs')"
+shows "distinct (wq (e#s) cs')"
+using assms by simp
+end
+context valid_trace_p
+begin
+lemma wq_neq_simp [simp]:
+assumes "cs' \<noteq> cs"
+shows "wq (e#s) cs' = wq s cs'"
+using assms unfolding is_p wq_def
+by (auto simp:Let_def)
+lemma runing_th_s:
+shows "th \<in> runing s"
+proof -
+from pip_e[unfolded is_p]
+show ?thesis by (cases, simp)
+qed
+lemma ready_th_s: "th \<in> readys s"
+using runing_th_s
+by (unfold runing_def, auto)
+lemma live_th_s: "th \<in> threads s"
+using readys_threads ready_th_s by auto
+lemma live_th_es: "th \<in> threads (e#s)"
+using live_th_s
+by (unfold is_p, simp)
+lemma th_not_waiting:
+"\<not> waiting s th c"
+proof -
+have "th \<in> readys s"
+using runing_ready runing_th_s by blast
+thus ?thesis
+by (unfold readys_def, auto)
+qed
+lemma waiting_neq_th:
+assumes "waiting s t c"
+shows "t \<noteq> th"
+using assms using th_not_waiting by blast
+lemma th_not_in_wq:
+shows "th \<notin> set (wq s cs)"
+proof
+assume otherwise: "th \<in> set (wq s cs)"
+from runing_wqE[OF runing_th_s this]
+obtain rest where eq_wq: "wq s cs = th#rest" by blast
+with otherwise
+have "holding s th cs"
+by (unfold s_holding_def, fold wq_def, simp)
+hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
+by (unfold s_RAG_def, fold holding_eq, auto)
+from pip_e[unfolded is_p]
+show False
+proof(cases)
+case (thread_P)
+with cs_th_RAG show ?thesis by auto
+qed
+qed
+lemma wq_es_cs:
+"wq (e#s) cs =  wq s cs @ [th]"
+by (unfold is_p wq_def, auto simp:Let_def)
+lemma wq_distinct_kept:
+assumes "distinct (wq s cs')"
+shows "distinct (wq (e#s) cs')"
+proof(cases "cs' = cs")
+case True
+show ?thesis using True assms th_not_in_wq
+by (unfold True wq_es_cs, auto)
+qed (insert assms, simp)
+end
+context valid_trace_v
+begin
+lemma wq_neq_simp [simp]:
+assumes "cs' \<noteq> cs"
+shows "wq (e#s) cs' = wq s cs'"
+using assms unfolding is_v wq_def
+by (auto simp:Let_def)
+lemma runing_th_s:
+shows "th \<in> runing s"
+proof -
+from pip_e[unfolded is_v]
+show ?thesis by (cases, simp)
+qed
+lemma th_not_waiting:
+"\<not> waiting s th c"
+proof -
+have "th \<in> readys s"
+using runing_ready runing_th_s by blast
+thus ?thesis
+by (unfold readys_def, auto)
+qed
+lemma waiting_neq_th:
+assumes "waiting s t c"
+shows "t \<noteq> th"
+using assms using th_not_waiting by blast
+lemma wq_s_cs:
+"wq s cs = th#rest"
+proof -
+from pip_e[unfolded is_v]
+show ?thesis
+proof(cases)
+case (thread_V)
+from this(2) show ?thesis
+by (unfold rest_def s_holding_def, fold wq_def,
+metis empty_iff list.collapse list.set(1))
+qed
+qed
+lemma wq_es_cs:
+"wq (e#s) cs = wq'"
+using wq_s_cs[unfolded wq_def]
+by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp)
+lemma wq_distinct_kept:
+assumes "distinct (wq s cs')"
+shows "distinct (wq (e#s) cs')"
+proof(cases "cs' = cs")
+case True
+show ?thesis
+proof(unfold True wq_es_cs wq'_def, rule someI2)
+show "distinct rest \<and> set rest = set rest"
+using assms[unfolded True wq_s_cs] by auto
+qed simp
+qed (insert assms, simp)
+end
+context valid_trace_set
+begin
+lemma wq_neq_simp [simp]:
+shows "wq (e#s) cs' = wq s cs'"
+using assms unfolding is_set wq_def
+by (auto simp:Let_def)
+lemma wq_distinct_kept:
+assumes "distinct (wq s cs')"
+shows "distinct (wq (e#s) cs')"
+using assms by simp
+end
+context valid_trace
+begin
 lemma actor_inv:
 assumes "PIP s e"
 and "\<not> isCreate e"
 shows "actor e \<in> runing s"
 using assms
 by (induct, auto)
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
+lemma isP_E:
-assumes "PP []"
+assumes "isP e"
-and "(\<And>s e. valid_trace s \<Longrightarrow> valid_trace (e#s) \<Longrightarrow>
+obtains cs where "e = P (actor e) cs"
-PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
+using assms by (cases e, auto)
-shows "PP s"
-proof(rule vt.induct[OF vt])
+lemma isV_E:
-from assms(1) show "PP []" .
+assumes "isV e"
-next
+obtains cs where "e = V (actor e) cs"
-fix s e
+using assms by (cases e, auto)
-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(induct rule:ind)
 case (Cons s e)
-from Cons(4,3)
+interpret vt_e: valid_trace_e s e using Cons by simp
 show ?case
-proof(induct)
+proof(cases e)
-case (thread_P th s cs1)
+case (Create th prio)
-show ?case
+interpret vt_create: valid_trace_create s e th prio
-proof(cases "cs = cs1")
+using Create by (unfold_locales, simp)
-case True
+show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept)
-thus ?thesis (is "distinct ?L")
+next
-proof -
+case (Exit th)
-have "?L = wq_fun (schs s) cs1 @ [th]" using True
+interpret vt_exit: valid_trace_exit s e th
-by (simp add:wq_def wf_def Let_def split:list.splits)
+using Exit by (unfold_locales, simp)
-moreover have "distinct ..."
+show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept)
-proof -
+next
-have "th \<notin> set (wq_fun (schs s) cs1)"
+case (P th cs)
-proof
+interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
-assume otherwise: "th \<in> set (wq_fun (schs s) cs1)"
+show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept)
-from runing_head[OF thread_P(1) this]
+next
-have "th = hd (wq_fun (schs s) cs1)" .
+case (V th cs)
-hence "(Cs cs1, Th th) \<in> (RAG s)" using otherwise
+interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
-by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
+show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept)
-with thread_P(2) show False by auto
+next
-qed
+case (Set th prio)
-moreover have "distinct (wq_fun (schs s) cs1)"
+interpret vt_set: valid_trace_set s e th prio
-using True thread_P wq_def by auto
+using Set by (unfold_locales, simp)
-ultimately show ?thesis by auto
+show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept)
 qed
-ultimately show ?thesis by simp
-qed
-next
-case False
-with thread_P(3)
-show ?thesis
-by (auto simp:wq_def wf_def Let_def split:list.splits)
-qed
-next
-case (thread_V th s cs1)
-thus ?case
-proof(cases "cs = cs1")
-case True
-show ?thesis (is "distinct ?L")
-proof(cases "(wq s cs)")
-case Nil
-thus ?thesis
-by (auto simp:wq_def wf_def Let_def split:list.splits)
-next
-case (Cons w_hd w_tl)
-moreover have "distinct (SOME q. distinct q \<and> set q = set w_tl)"
-proof(rule someI2)
-from thread_V(3)[unfolded Cons]
-show  "distinct w_tl \<and> set w_tl = set w_tl" by auto
-qed auto
-ultimately show ?thesis
-by (auto simp:wq_def wf_def Let_def True split:list.splits)
-qed
-next
-case False
-with thread_V(3)
-show ?thesis
-by (auto simp:wq_def wf_def Let_def split:list.splits)
-qed
-qed (insert Cons, auto simp: wq_def Let_def split:list.splits)
 qed (unfold wq_def Let_def, simp)
 end
 context valid_trace_e
 begin
 text {*
 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:
+lemma wq_in_inv:
 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(cases e)
 -- {* This is the only non-trivial case: *}
 qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
 qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
 thus ?thesis by auto
 qed (insert assms, auto simp:wq_def Let_def split:if_splits)
+lemma wq_out_inv:
+assumes s_in: "thread \<in> set (wq s cs)"
+and s_hd: "thread = hd (wq s cs)"
+and s_i: "thread \<noteq> hd (wq (e#s) cs)"
+shows "e = V thread cs"
+proof(cases e)
+-- {* There are only two non-trivial cases: *}
+case (V th cs1)
+show ?thesis
+proof(cases "cs1 = cs")
+case True
+have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
+thus ?thesis
+proof(cases)
+case (thread_V)
+moreover have "th = thread" using thread_V(2) s_hd
+by (unfold s_holding_def wq_def, simp)
+ultimately show ?thesis using V True by simp
+qed
+qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
+next
+case (P th cs1)
+show ?thesis
+proof(cases "cs1 = cs")
+case True
+with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
+by (auto simp:wq_def Let_def split:if_splits)
+with s_i s_hd s_in have False
+by (metis empty_iff hd_append2 list.set(1) wq_def)
+thus ?thesis by simp
+qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
+qed (insert assms, auto simp:wq_def Let_def split:if_splits)
 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
-end
-locale valid_moment = valid_trace +
-fixes i :: nat
-sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
-by (unfold_locales, insert vt_moment, auto)
 context valid_trace
 begin
 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: (* ccc *)
+lemma waiting_unique_pre: (* ddd *)
 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)"
+let "?Q" = "\<lambda> 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>
+and np1: "\<not> ?Q cs1 (moment t1 s)"
-thread \<noteq> hd (wq (moment t1 s) cs1))"
+and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
-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>
+and np2: "\<not> ?Q cs2 (moment t2 s)"
-thread \<noteq> hd (wq (moment t2 s) cs2))"
+and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
-and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
+{ fix s cs
-thread \<noteq> hd (wq (moment i' s) cs2))" by auto
+assume q: "?Q cs s"
+have "thread \<notin> runing s"
+proof
+assume "thread \<in> runing s"
+hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and>
+thread \<noteq> hd (wq_fun (schs s) cs))"
+by (unfold runing_def s_waiting_def readys_def, auto)
+from this[rule_format, of cs] q
+show False by (simp add: wq_def)
+qed
+} note q_not_runing = this
+{ fix t1 t2 cs1 cs2
+assume  lt1: "t1 < length s"
+and np1: "\<not> ?Q cs1 (moment t1 s)"
+and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
+and lt2: "t2 < length s"
+and np2: "\<not> ?Q cs2 (moment t2 s)"
+and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
+and 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 -
+have "thread \<in> runing (moment t2 s)"
+proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+case True
+have "e = V thread cs2"
+proof -
+have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+using True and np2  by auto
+from vt_e.wq_out_inv[OF True this h2]
+show ?thesis .
+qed
+thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto
+next
+case False
+have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+with vt_e.actor_inv[OF vt_e.pip_e]
+show ?thesis by auto
+qed
+moreover have "thread \<notin> runing (moment t2 s)"
+by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
+ultimately show ?thesis by simp
+qed
+} note lt_case = this
 show ?thesis
 proof -
-{
+{ assume "t1 < t2"
-assume lt12: "t1 < t2"
+from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
+have ?thesis .
+} moreover {
+assume "t2 < t1"
+from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
+have ?thesis .
+} moreover {
+assume eq_12: "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
+have lt_2: "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
+from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12]
+have g1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+g2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" 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
+have "e = V thread cs2 \<or> e = P thread cs2"
 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)"
+have "e = V thread cs2"
-by auto
+proof -
-from vt_e.abs2 [OF True eq_th h2 h1]
+have eq_th: "thread = hd (wq (moment t2 s) cs2)"
-show ?thesis by auto
+using True and np2  by auto
+from vt_e.wq_out_inv[OF True this h2]
+show ?thesis .
+qed
+thus ?thesis by auto
 next
 case False
-from vt_e.block_pre[OF False h1]
+have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
-have "e = P thread cs2" .
+thus ?thesis by auto
-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 {
+moreover have "e = V thread cs1 \<or> e = P thread cs1"
-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)"
+have eq_th: "thread = hd (wq (moment t1 s) cs1)"
-by auto
+using True and np1  by auto
-from vt_e.abs2 True eq_th h2 h1
+from vt_e.wq_out_inv[folded eq_12, OF True this g2]
-show ?thesis by auto
+have "e = V thread cs1" .
+thus ?thesis by auto
 next
 case False
-from vt_e.block_pre [OF False h1]
+have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
-have "e = P thread cs1" .
+thus ?thesis by auto
-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 {
+ultimately have ?thesis using neq12 by auto
-assume eqt12: "t1 = t2"
+} ultimately show ?thesis using nat_neq_iff by blast
-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 {*
 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:
 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"
 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:
-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:
-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:
-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 (in valid_trace_set)
+RAG_unchanged: "(RAG (e # s)) = RAG s"
+by (unfold is_set RAG_set_unchanged, simp)
 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 (in valid_trace_create)
+RAG_unchanged: "(RAG (e # s)) = RAG s"
+by (unfold is_create RAG_create_unchanged, simp)
 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)
+lemma (in valid_trace_exit)
-text {*
+RAG_unchanged: "(RAG (e # s)) = RAG s"
-The following lemmas are used in the proof of
+by (unfold is_exit RAG_exit_unchanged, simp)
-lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed
-by @{text "V"}-events.
+context valid_trace_v
-However, since our model is very concise, such  seemingly obvious lemmas need to be derived from scratch,
+begin
-starting from the model definitions.
-*}
+lemma distinct_rest: "distinct rest"
-lemma step_v_hold_inv[elim_format]:
+by (simp add: distinct_tl rest_def wq_distinct)
-"\<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>
+lemma holding_cs_eq_th:
-next_th s th cs t \<and> c = cs"
+assumes "holding s t cs"
-proof -
+shows "t = th"
-fix c t
+proof -
-assume vt: "vt (V th cs # s)"
+from pip_e[unfolded is_v]
-and nhd: "\<not> holding (wq s) t c"
+show ?thesis
-and hd: "holding (wq (V th cs # s)) t c"
+proof(cases)
-show "next_th s th cs t \<and> c = cs"
+case (thread_V)
+from held_unique[OF this(2) assms]
+show ?thesis by simp
+qed
+qed
+lemma distinct_wq': "distinct wq'"
+by (metis (mono_tags, lifting) distinct_rest  some_eq_ex wq'_def)
+lemma set_wq': "set wq' = set rest"
+by (metis (mono_tags, lifting) distinct_rest rest_def
+some_eq_ex wq'_def)
+lemma th'_in_inv:
+assumes "th' \<in> set wq'"
+shows "th' \<in> set rest"
+using assms set_wq' by simp
+lemma neq_t_th:
+assumes "waiting (e#s) t c"
+shows "t \<noteq> th"
+proof
+assume otherwise: "t = th"
+show False
 proof(cases "c = cs")
+case True
+have "t \<in> set wq'"
+using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
+by simp
+from th'_in_inv[OF this] have "t \<in> set rest" .
+with wq_s_cs[folded otherwise] wq_distinct[of cs]
+show ?thesis by simp
+next
 case False
-with nhd hd show ?thesis
+have "wq (e#s) c = wq s c" using False
-by (unfold cs_holding_def wq_def, auto simp:Let_def)
+by (unfold is_v, simp)
-next
+hence "waiting s t c" using assms
-case True
+by (simp add: cs_waiting_def waiting_eq)
-with step_back_step [OF vt]
+hence "t \<notin> readys s" by (unfold readys_def, auto)
-have "step s (V th c)" by simp
+hence "t \<notin> runing s" using runing_ready by auto
-hence "next_th s th cs t"
+with runing_th_s[folded otherwise] show ?thesis by auto
-proof(cases)
+qed
-assume "holding s th c"
+qed
-with nhd hd show ?thesis
-apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
+lemma waiting_esI1:
-auto simp:Let_def split:list.splits if_splits)
+assumes "waiting s t c"
-proof -
+and "c \<noteq> cs"
-assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
+shows "waiting (e#s) t c"
-moreover have "\<dots> = set []"
+proof -
-proof(rule someI2)
+have "wq (e#s) c = wq s c"
-show "distinct [] \<and> [] = []" by auto
+using assms(2) is_v by auto
-next
+with assms(1) show ?thesis
-fix x assume "distinct x \<and> x = []"
+using cs_waiting_def waiting_eq by auto
-thus "set x = set []" by auto
+qed
-qed
-ultimately show False by auto
+lemma holding_esI2:
+assumes "c \<noteq> cs"
+and "holding s t c"
+shows "holding (e#s) t c"
+proof -
+from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
+from assms(2)[unfolded s_holding_def, folded wq_def,
+folded this, unfolded wq_def, folded s_holding_def]
+show ?thesis .
+qed
+lemma holding_esI1:
+assumes "holding s t c"
+and "t \<noteq> th"
+shows "holding (e#s) t c"
+proof -
+have "c \<noteq> cs" using assms using holding_cs_eq_th by blast
+from holding_esI2[OF this assms(1)]
+show ?thesis .
+qed
+end
+context valid_trace_v_n
+begin
+lemma neq_wq': "wq' \<noteq> []"
+proof (unfold wq'_def, rule someI2)
+show "distinct rest \<and> set rest = set rest"
+by (simp add: distinct_rest)
+next
+fix x
+assume " distinct x \<and> set x = set rest"
+thus "x \<noteq> []" using rest_nnl by auto
+qed
+definition "taker = hd wq'"
+definition "rest' = tl wq'"
+lemma eq_wq': "wq' = taker # rest'"
+by (simp add: neq_wq' rest'_def taker_def)
+lemma next_th_taker:
+shows "next_th s th cs taker"
+using rest_nnl taker_def wq'_def wq_s_cs
+by (auto simp:next_th_def)
+lemma taker_unique:
+assumes "next_th s th cs taker'"
+shows "taker' = taker"
+proof -
+from assms
+obtain rest' where
+h: "wq s cs = th # rest'"
+"taker' = hd (SOME q. distinct q \<and> set q = set rest')"
+by (unfold next_th_def, auto)
+with wq_s_cs have "rest' = rest" by auto
+thus ?thesis using h(2) taker_def wq'_def by auto
+qed
+lemma waiting_set_eq:
+"{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
+by (smt all_not_in_conv bot.extremum insertI1 insert_subset
+mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
+lemma holding_set_eq:
+"{(Cs cs, Th th') |th'.  next_th s th cs th'} = {(Cs cs, Th taker)}"
+using next_th_taker taker_def waiting_set_eq
+by fastforce
+lemma holding_taker:
+shows "holding (e#s) taker cs"
+by (unfold s_holding_def, fold wq_def, unfold wq_es_cs,
+auto simp:neq_wq' taker_def)
+lemma waiting_esI2:
+assumes "waiting s t cs"
+and "t \<noteq> taker"
+shows "waiting (e#s) t cs"
+proof -
+have "t \<in> set wq'"
+proof(unfold wq'_def, rule someI2)
+show "distinct rest \<and> set rest = set rest"
+by (simp add: distinct_rest)
+next
+fix x
+assume "distinct x \<and> set x = set rest"
+moreover have "t \<in> set rest"
+using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto
+ultimately show "t \<in> set x" by simp
+qed
+moreover have "t \<noteq> hd wq'"
+using assms(2) taker_def by auto
+ultimately show ?thesis
+by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
+qed
+lemma waiting_esE:
+assumes "waiting (e#s) t c"
+obtains "c \<noteq> cs" "waiting s t c"
+|    "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
+proof(cases "c = cs")
+case False
+hence "wq (e#s) c = wq s c" using is_v by auto
+with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto
+from that(1)[OF False this] show ?thesis .
+next
+case True
+from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
+have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
+hence "t \<noteq> taker" by (simp add: taker_def)
+moreover hence "t \<noteq> th" using assms neq_t_th by blast
+moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv)
+ultimately have "waiting s t cs"
+by (metis cs_waiting_def list.distinct(2) list.sel(1)
+list.set_sel(2) rest_def waiting_eq wq_s_cs)
+show ?thesis using that(2)
+using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto
+qed
+lemma holding_esI1:
+assumes "c = cs"
+and "t = taker"
+shows "holding (e#s) t c"
+by (unfold assms, simp add: holding_taker)
+lemma holding_esE:
+assumes "holding (e#s) t c"
+obtains "c = cs" "t = taker"
+| "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+case True
+from assms[unfolded True, unfolded s_holding_def,
+folded wq_def, unfolded wq_es_cs]
+have "t = taker" by (simp add: taker_def)
+from that(1)[OF True this] show ?thesis .
+next
+case False
+hence "wq (e#s) c = wq s c" using is_v by auto
+from assms[unfolded s_holding_def, folded wq_def,
+unfolded this, unfolded wq_def, folded s_holding_def]
+have "holding s t c"  .
+from that(2)[OF False this] show ?thesis .
+qed
+end
+context valid_trace_v_e
+begin
+lemma nil_wq': "wq' = []"
+proof (unfold wq'_def, rule someI2)
+show "distinct rest \<and> set rest = set rest"
+by (simp add: distinct_rest)
+next
+fix x
+assume " distinct x \<and> set x = set rest"
+thus "x = []" using rest_nil by auto
+qed
+lemma no_taker:
+assumes "next_th s th cs taker"
+shows "False"
+proof -
+from assms[unfolded next_th_def]
+obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
+by auto
+thus ?thesis using rest_def rest_nil by auto
+qed
+lemma waiting_set_eq:
+"{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
+using no_taker by auto
+lemma holding_set_eq:
+"{(Cs cs, Th th') |th'.  next_th s th cs th'} = {}"
+using no_taker by auto
+lemma no_holding:
+assumes "holding (e#s) taker cs"
+shows False
+proof -
+from wq_es_cs[unfolded nil_wq']
+have " wq (e # s) cs = []" .
+from assms[unfolded s_holding_def, folded wq_def, unfolded this]
+show ?thesis by auto
+qed
+lemma no_waiting:
+assumes "waiting (e#s) t cs"
+shows False
+proof -
+from wq_es_cs[unfolded nil_wq']
+have " wq (e # s) cs = []" .
+from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
+show ?thesis by auto
+qed
+lemma waiting_esI2:
+assumes "waiting s t c"
+shows "waiting (e#s) t c"
+proof -
+have "c \<noteq> cs" using assms
+using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto
+from waiting_esI1[OF assms this]
+show ?thesis .
+qed
+lemma waiting_esE:
+assumes "waiting (e#s) t c"
+obtains "c \<noteq> cs" "waiting s t c"
+proof(cases "c = cs")
+case False
+hence "wq (e#s) c = wq s c" using is_v by auto
+with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto
+from that(1)[OF False this] show ?thesis .
+next
+case True
+from no_waiting[OF assms[unfolded True]]
+show ?thesis by auto
+qed
+lemma holding_esE:
+assumes "holding (e#s) t c"
+obtains "c \<noteq> cs" "holding s t c"
+proof(cases "c = cs")
+case True
+from no_holding[OF assms[unfolded True]]
+show ?thesis by auto
+next
+case False
+hence "wq (e#s) c = wq s c" using is_v by auto
+from assms[unfolded s_holding_def, folded wq_def,
+unfolded this, unfolded wq_def, folded s_holding_def]
+have "holding s t c"  .
+from that[OF False this] show ?thesis .
+qed
+end
+lemma rel_eqI:
+assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
+and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
+shows "A = B"
+using assms by auto
+lemma in_RAG_E:
+assumes "(n1, n2) \<in> RAG (s::state)"
+obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
+| (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
+using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
+by auto
+context valid_trace_v
+begin
+lemma RAG_es:
+"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 = ?R")
+proof(rule rel_eqI)
+fix n1 n2
+assume "(n1, n2) \<in> ?L"
+thus "(n1, n2) \<in> ?R"
+proof(cases rule:in_RAG_E)
+case (waiting th' cs')
+show ?thesis
+proof(cases "rest = []")
+case False
+interpret h_n: valid_trace_v_n s e th cs
+by (unfold_locales, insert False, simp)
+from waiting(3)
+show ?thesis
+proof(cases rule:h_n.waiting_esE)
+case 1
+with waiting(1,2)
+show ?thesis
+by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
+fold waiting_eq, auto)
+next
+case 2
+with waiting(1,2)
+show ?thesis
+by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
+fold waiting_eq, auto)
+qed
+next
+case True
+interpret h_e: valid_trace_v_e s e th cs
+by (unfold_locales, insert True, simp)
+from waiting(3)
+show ?thesis
+proof(cases rule:h_e.waiting_esE)
+case 1
+with waiting(1,2)
+show ?thesis
+by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
+fold waiting_eq, auto)
+qed
+qed
+next
+case (holding th' cs')
+show ?thesis
+proof(cases "rest = []")
+case False
+interpret h_n: valid_trace_v_n s e th cs
+by (unfold_locales, insert False, simp)
+from holding(3)
+show ?thesis
+proof(cases rule:h_n.holding_esE)
+case 1
+with holding(1,2)
+show ?thesis
+by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
+fold waiting_eq, auto)
+next
+case 2
+with holding(1,2)
+show ?thesis
+by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
+fold holding_eq, auto)
+qed
+next
+case True
+interpret h_e: valid_trace_v_e s e th cs
+by (unfold_locales, insert True, simp)
+from holding(3)
+show ?thesis
+proof(cases rule:h_e.holding_esE)
+case 1
+with holding(1,2)
+show ?thesis
+by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
+fold holding_eq, auto)
+qed
+qed
+qed
+next
+fix n1 n2
+assume h: "(n1, n2) \<in> ?R"
+show "(n1, n2) \<in> ?L"
+proof(cases "rest = []")
+case False
+interpret h_n: valid_trace_v_n s e th cs
+by (unfold_locales, insert False, simp)
+from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
+have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
+\<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or>
+(n2 = Th h_n.taker \<and> n1 = Cs cs)"
+by auto
+thus ?thesis
+proof
+assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
+with h_n.holding_taker
+show ?thesis
+by (unfold s_RAG_def, fold holding_eq, auto)
+next
+assume h: "(n1, n2) \<in> RAG s \<and>
+(n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
+hence "(n1, n2) \<in> RAG s" by simp
+thus ?thesis
+proof(cases rule:in_RAG_E)
+case (waiting th' cs')
+from h and this(1,2)
+have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
+hence "waiting (e#s) th' cs'"
+proof
+assume "cs' \<noteq> cs"
+from waiting_esI1[OF waiting(3) this]
+show ?thesis .
+next
+assume neq_th': "th' \<noteq> h_n.taker"
+show ?thesis
+proof(cases "cs' = cs")
+case False
+from waiting_esI1[OF waiting(3) this]
+show ?thesis .
 next
-assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
+case True
-moreover have "\<dots> = set []"
+from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
-proof(rule someI2)
+show ?thesis .
-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
+thus ?thesis using waiting(1,2)
-qed
+by (unfold s_RAG_def, fold waiting_eq, auto)
-qed
+next
+case (holding th' cs')
-text {*
+from h this(1,2)
-The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be
+have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
-derived from scratch, which confirms the correctness of the definition of @{text "next_th"}.
+hence "holding (e#s) th' cs'"
-*}
+proof
-lemma step_v_wait_inv[elim_format]:
+assume "cs' \<noteq> cs"
-"\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
+from holding_esI2[OF this holding(3)]
-\<rbrakk>
+show ?thesis .
-\<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"
+assume "th' \<noteq> th"
-by auto
+from holding_esI1[OF holding(3) this]
+show ?thesis .
 qed
-with t_ni and t_in show "a = th" by auto
+thus ?thesis using holding(1,2)
+by (unfold s_RAG_def, fold holding_eq, auto)
+qed
+qed
+next
+case True
+interpret h_e: valid_trace_v_e s e th cs
+by (unfold_locales, insert True, simp)
+from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
+have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)"
+by auto
+from h_s(1)
+show ?thesis
+proof(cases rule:in_RAG_E)
+case (waiting th' cs')
+from h_e.waiting_esI2[OF this(3)]
+show ?thesis using waiting(1,2)
+by (unfold s_RAG_def, fold waiting_eq, auto)
+next
+case (holding th' cs')
+with h_s(2)
+have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+thus ?thesis
+proof
+assume neq_cs: "cs' \<noteq> cs"
+from holding_esI2[OF this holding(3)]
+show ?thesis using holding(1,2)
+by (unfold s_RAG_def, fold holding_eq, auto)
 next
-fix a list
+assume "th' \<noteq> th"
-assume t_in: "t \<in> set list"
+from holding_esI1[OF holding(3) this]
-and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
+show ?thesis using holding(1,2)
-and eq_wq: "wq_fun (schs s) cs = a # list"
+by (unfold s_RAG_def, fold holding_eq, auto)
-have " set (SOME q. distinct q \<and> set q = set list) = set list"
+qed
-proof(rule someI2)
+qed
-from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
+qed
-show "distinct list \<and> set list = set list" by auto
+qed
-next
-show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+end
-by auto
-qed
+lemma step_RAG_v:
-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:
 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'}"
+{(Cs cs, Th th') |th'.  next_th s th cs th'}" (is "?L = ?R")
-apply (insert vt, unfold s_RAG_def)
+proof -
-apply (auto split:if_splits list.splits simp:Let_def)
+interpret vt_v: valid_trace_v s "V th cs"
-apply (auto elim: step_v_waiting_mono step_v_hold_inv
+using assms step_back_vt by (unfold_locales, auto)
-step_v_release step_v_wait_inv
+show ?thesis using vt_v.RAG_es .
-step_v_get_hold step_v_release_inv)
+qed
-apply (erule_tac step_v_not_wait, auto)
-done
+lemma (in valid_trace_create)
+th_not_in_threads: "th \<notin> threads s"
-text {*
+proof -
-The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
+from pip_e[unfolded is_create]
-with the happening of @{text "P"}-events:
+show ?thesis by (cases, simp)
-*}
+qed
-lemma step_RAG_p:
-"vt (P th cs#s) \<Longrightarrow>
+lemma (in valid_trace_create)
-RAG (P th cs # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
-else RAG s \<union> {(Th th, Cs cs)})"
+by (unfold is_create, simp)
-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)
+lemma (in valid_trace_exit)
-apply(case_tac "csa = cs", auto)
+threads_es [simp]: "threads (e#s) = threads s - {th}"
-apply(fold wq_def)
+by (unfold is_exit, simp)
-apply(drule_tac step_back_step)
-apply(ind_cases " step s (P (hd (wq s cs)) cs)")
+lemma (in valid_trace_p)
-apply(simp add:s_RAG_def wq_def cs_holding_def)
+threads_es [simp]: "threads (e#s) = threads s"
-apply(auto)
+by (unfold is_p, simp)
-done
+lemma (in valid_trace_v)
+threads_es [simp]: "threads (e#s) = threads s"
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+by (unfold is_v, simp)
-by (unfold s_RAG_def, auto)
+lemma (in valid_trace_v)
+th_not_in_rest[simp]: "th \<notin> set rest"
+proof
+assume otherwise: "th \<in> set rest"
+have "distinct (wq s cs)" by (simp add: wq_distinct)
+from this[unfolded wq_s_cs] and otherwise
+show False by auto
+qed
+lemma (in valid_trace_v)
+set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
+proof(unfold wq_es_cs wq'_def, rule someI2)
+show "distinct rest \<and> set rest = set rest"
+by (simp add: distinct_rest)
+next
+fix x
+assume "distinct x \<and> set x = set rest"
+thus "set x = set (wq s cs) - {th}"
+by (unfold wq_s_cs, simp)
+qed
+lemma (in valid_trace_exit)
+th_not_in_wq: "th \<notin> set (wq s cs)"
+proof -
+from pip_e[unfolded is_exit]
+show ?thesis
+by (cases, unfold holdents_def s_holding_def, fold wq_def,
+auto elim!:runing_wqE)
+qed
+lemma (in valid_trace) wq_threads:
+assumes "th \<in> set (wq s cs)"
+shows "th \<in> threads s"
+using assms
+proof(induct rule:ind)
+case (Nil)
+thus ?case by (auto simp:wq_def)
+next
+case (Cons s e)
+interpret vt_e: valid_trace_e s e using Cons by simp
+show ?case
+proof(cases e)
+case (Create th' prio')
+interpret vt: valid_trace_create s e th' prio'
+using Create by (unfold_locales, simp)
+show ?thesis
+using Cons.hyps(2) Cons.prems by auto
+next
+case (Exit th')
+interpret vt: valid_trace_exit s e th'
+using Exit by (unfold_locales, simp)
+show ?thesis
+using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto
+next
+case (P th' cs')
+interpret vt: valid_trace_p s e th' cs'
+using P by (unfold_locales, simp)
+show ?thesis
+using Cons.hyps(2) Cons.prems readys_threads
+runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv
+by fastforce
+next
+case (V th' cs')
+interpret vt: valid_trace_v s e th' cs'
+using V by (unfold_locales, simp)
+show ?thesis using Cons
+using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
+next
+case (Set th' prio)
+interpret vt: valid_trace_set s e th' prio
+using Set by (unfold_locales, simp)
+show ?thesis using Cons.hyps(2) Cons.prems vt.is_set
+by (auto simp:wq_def Let_def)
+qed
+qed
 context valid_trace
 begin
-text {*
+lemma  dm_RAG_threads:
-The following lemma shows that @{text "RAG"} is acyclic.
+assumes in_dom: "(Th th) \<in> Domain (RAG s)"
-The overall structure is by induction on the formation of @{text "vt s"}
+shows "th \<in> threads s"
-and then case analysis on event @{text "e"}, where the non-trivial cases
+proof -
-for those for @{text "V"} and @{text "P"} events.
+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
-lemma acyclic_RAG:
+ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
-shows "acyclic (RAG s)"
+hence "th \<in> set (wq s cs)"
-using vt
+by (unfold s_RAG_def, auto simp:cs_waiting_def)
-proof(induct)
+from wq_threads [OF this] show ?thesis .
-case (vt_cons s e)
+qed
-interpret vt_s: valid_trace s using vt_cons(1)
+lemma rg_RAG_threads:
+assumes "(Th th) \<in> Range (RAG s)"
+shows "th \<in> threads s"
+using assms
+by (unfold s_RAG_def cs_waiting_def cs_holding_def,
+auto intro:wq_threads)
+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 rg_RAG_threads)
+end
+lemma (in valid_trace_v)
+preced_es [simp]: "preced th (e#s) = preced th s"
+by (unfold is_v preced_def, simp)
+lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
+proof
+fix th'
+show "the_preced (V th cs # s) th' = the_preced s th'"
+by (unfold the_preced_def preced_def, simp)
+qed
+lemma (in valid_trace_v)
+the_preced_es: "the_preced (e#s) = the_preced s"
+by (unfold is_v preced_def, simp)
+context valid_trace_p
+begin
+lemma not_holding_s_th_cs: "\<not> holding s th cs"
+proof
+assume otherwise: "holding s th cs"
+from pip_e[unfolded is_p]
+show False
+proof(cases)
+case (thread_P)
+moreover have "(Cs cs, Th th) \<in> RAG s"
+using otherwise cs_holding_def
+holding_eq th_not_in_wq by auto
+ultimately show ?thesis by auto
+qed
+qed
+lemma waiting_kept:
+assumes "waiting s th' cs'"
+shows "waiting (e#s) th' cs'"
+using assms
+by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2)
+rotate1.simps(2) self_append_conv2 set_rotate1
+th_not_in_wq waiting_eq wq_es_cs wq_neq_simp)
+lemma holding_kept:
+assumes "holding s th' cs'"
+shows "holding (e#s) th' cs'"
+proof(cases "cs' = cs")
+case False
+hence "wq (e#s) cs' = wq s cs'" by simp
+with assms show ?thesis using cs_holding_def holding_eq by auto
+next
+case True
+from assms[unfolded s_holding_def, folded wq_def]
+obtain rest where eq_wq: "wq s cs' = th'#rest"
+by (metis empty_iff list.collapse list.set(1))
+hence "wq (e#s) cs' = th'#(rest@[th])"
+by (simp add: True wq_es_cs)
+thus ?thesis
+by (simp add: cs_holding_def holding_eq)
+qed
+end
+locale valid_trace_p_h = valid_trace_p +
+assumes we: "wq s cs = []"
+locale valid_trace_p_w = valid_trace_p +
+assumes wne: "wq s cs \<noteq> []"
+begin
+definition "holder = hd (wq s cs)"
+definition "waiters = tl (wq s cs)"
+definition "waiters' = waiters @ [th]"
+lemma wq_s_cs: "wq s cs = holder#waiters"
+by (simp add: holder_def waiters_def wne)
+lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
+by (simp add: wq_es_cs wq_s_cs)
+lemma waiting_es_th_cs: "waiting (e#s) th cs"
+using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
+lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
+by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
+lemma holding_esE:
+assumes "holding (e#s) th' cs'"
+obtains "holding s th' cs'"
+using assms
+proof(cases "cs' = cs")
+case False
+hence "wq (e#s) cs' = wq s cs'" by simp
+with assms show ?thesis
+using cs_holding_def holding_eq that by auto
+next
+case True
+with assms show ?thesis
+by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that
+wq_es_cs' wq_s_cs)
+qed
+lemma waiting_esE:
+assumes "waiting (e#s) th' cs'"
+obtains "th' \<noteq> th" "waiting s th' cs'"
+|  "th' = th" "cs' = cs"
+proof(cases "waiting s th' cs'")
+case True
+have "th' \<noteq> th"
+proof
+assume otherwise: "th' = th"
+from True[unfolded this]
+show False by (simp add: th_not_waiting)
+qed
+from that(1)[OF this True] show ?thesis .
+next
+case False
+hence "th' = th \<and> cs' = cs"
+by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2)
+set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
+with that(2) show ?thesis by metis
+qed
+lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
+proof(rule rel_eqI)
+fix n1 n2
+assume "(n1, n2) \<in> ?L"
+thus "(n1, n2) \<in> ?R"
+proof(cases rule:in_RAG_E)
+case (waiting th' cs')
+from this(3)
+show ?thesis
+proof(cases rule:waiting_esE)
+case 1
+thus ?thesis using waiting(1,2)
+by (unfold s_RAG_def, fold waiting_eq, auto)
+next
+case 2
+thus ?thesis using waiting(1,2) by auto
+qed
+next
+case (holding th' cs')
+from this(3)
+show ?thesis
+proof(cases rule:holding_esE)
+case 1
+with holding(1,2)
+show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+qed
+qed
+next
+fix n1 n2
+assume "(n1, n2) \<in> ?R"
+hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
+thus "(n1, n2) \<in> ?L"
+proof
+assume "(n1, n2) \<in> RAG s"
+thus ?thesis
+proof(cases rule:in_RAG_E)
+case (waiting th' cs')
+from waiting_kept[OF this(3)]
+show ?thesis using waiting(1,2)
+by (unfold s_RAG_def, fold waiting_eq, auto)
+next
+case (holding th' cs')
+from holding_kept[OF this(3)]
+show ?thesis using holding(1,2)
+by (unfold s_RAG_def, fold holding_eq, auto)
+qed
+next
+assume "n1 = Th th \<and> n2 = Cs cs"
+thus ?thesis using RAG_edge by auto
+qed
+qed
+end
+context valid_trace_p_h
+begin
+lemma wq_es_cs': "wq (e#s) cs = [th]"
+using wq_es_cs[unfolded we] by simp
+lemma holding_es_th_cs:
+shows "holding (e#s) th cs"
+proof -
+from wq_es_cs'
+have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto
+thus ?thesis using cs_holding_def holding_eq by blast
+qed
+lemma RAG_edge: "(Cs cs, Th th) \<in> RAG (e#s)"
+by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto)
+lemma waiting_esE:
+assumes "waiting (e#s) th' cs'"
+obtains "waiting s th' cs'"
+using assms
+by (metis cs_waiting_def event.distinct(15) is_p list.sel(1)
+set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv)
+lemma holding_esE:
+assumes "holding (e#s) th' cs'"
+obtains "cs' \<noteq> cs" "holding s th' cs'"
+| "cs' = cs" "th' = th"
+proof(cases "cs' = cs")
+case True
+from held_unique[OF holding_es_th_cs assms[unfolded True]]
+have "th' = th" by simp
+from that(2)[OF True this] show ?thesis .
+next
+case False
+have "holding s th' cs'" using assms
+using False cs_holding_def holding_eq by auto
+from that(1)[OF False this] show ?thesis .
+qed
+lemma RAG_es: "RAG (e # s) =  RAG s \<union> {(Cs cs, Th th)}" (is "?L = ?R")
+proof(rule rel_eqI)
+fix n1 n2
+assume "(n1, n2) \<in> ?L"
+thus "(n1, n2) \<in> ?R"
+proof(cases rule:in_RAG_E)
+case (waiting th' cs')
+from this(3)
+show ?thesis
+proof(cases rule:waiting_esE)
+case 1
+thus ?thesis using waiting(1,2)
+by (unfold s_RAG_def, fold waiting_eq, auto)
+qed
+next
+case (holding th' cs')
+from this(3)
+show ?thesis
+proof(cases rule:holding_esE)
+case 1
+with holding(1,2)
+show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+next
+case 2
+with holding(1,2) show ?thesis by auto
+qed
+qed
+next
+fix n1 n2
+assume "(n1, n2) \<in> ?R"
+hence "(n1, n2) \<in> RAG s \<or> (n1 = Cs cs \<and> n2 = Th th)" by auto
+thus "(n1, n2) \<in> ?L"
+proof
+assume "(n1, n2) \<in> RAG s"
+thus ?thesis
+proof(cases rule:in_RAG_E)
+case (waiting th' cs')
+from waiting_kept[OF this(3)]
+show ?thesis using waiting(1,2)
+by (unfold s_RAG_def, fold waiting_eq, auto)
+next
+case (holding th' cs')
+from holding_kept[OF this(3)]
+show ?thesis using holding(1,2)
+by (unfold s_RAG_def, fold holding_eq, auto)
+qed
+next
+assume "n1 = Cs cs \<and> n2 = Th th"
+with holding_es_th_cs
+show ?thesis
+by (unfold s_RAG_def, fold holding_eq, auto)
+qed
+qed
+end
+context valid_trace_p
+begin
+lemma RAG_es': "RAG (e # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+else RAG s \<union> {(Th th, Cs cs)})"
+proof(cases "wq s cs = []")
+case True
+interpret vt_p: valid_trace_p_h using True
 by (unfold_locales, simp)
-assume ih: "acyclic (RAG s)"
+show ?thesis by (simp add: vt_p.RAG_es vt_p.we)
-and stp: "step s e"
+next
-and vt: "vt s"
+case False
+interpret vt_p: valid_trace_p_w using False
+by (unfold_locales, simp)
+show ?thesis by (simp add: vt_p.RAG_es vt_p.wne)
+qed
+end
+lemma (in valid_trace_v_n) finite_waiting_set:
+"finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
+by (simp add: waiting_set_eq)
+lemma (in valid_trace_v_n) finite_holding_set:
+"finite {(Cs cs, Th th') |th'. next_th s th cs th'}"
+by (simp add: holding_set_eq)
+lemma (in valid_trace_v_e) finite_waiting_set:
+"finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
+by (simp add: waiting_set_eq)
+lemma (in valid_trace_v_e) finite_holding_set:
+"finite {(Cs cs, Th th') |th'. next_th s th cs th'}"
+by (simp add: holding_set_eq)
+context valid_trace_v
+begin
+lemma
+finite_RAG_kept:
+assumes "finite (RAG s)"
+shows "finite (RAG (e#s))"
+proof(cases "rest = []")
+case True
+interpret vt: valid_trace_v_e using True
+by (unfold_locales, simp)
+show ?thesis using assms
+by  (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
+next
+case False
+interpret vt: valid_trace_v_n using False
+by (unfold_locales, simp)
+show ?thesis using assms
+by  (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
+qed
+end
+context valid_trace_v_e
+begin
+lemma
+acylic_RAG_kept:
+assumes "acyclic (RAG s)"
+shows "acyclic (RAG (e#s))"
+proof(rule acyclic_subset[OF assms])
+show "RAG (e # s) \<subseteq> RAG s"
+by (unfold RAG_es waiting_set_eq holding_set_eq, auto)
+qed
+end
+context valid_trace_v_n
+begin
+lemma waiting_taker: "waiting s taker cs"
+apply (unfold s_waiting_def, fold wq_def, unfold wq_s_cs taker_def)
+using eq_wq' th'_in_inv wq'_def by fastforce
+lemma
+acylic_RAG_kept:
+assumes "acyclic (RAG s)"
+shows "acyclic (RAG (e#s))"
+proof -
+have "acyclic ((RAG s - {(Cs cs, Th th)} - {(Th taker, Cs cs)}) \<union>
+{(Cs cs, Th taker)})" (is "acyclic (?A \<union> _)")
+proof -
+from assms
+have "acyclic ?A"
+by (rule acyclic_subset, auto)
+moreover have "(Th taker, Cs cs) \<notin> ?A^*"
+proof
+assume otherwise: "(Th taker, Cs cs) \<in> ?A^*"
+hence "(Th taker, Cs cs) \<in> ?A^+"
+by (unfold rtrancl_eq_or_trancl, auto)
+from tranclD[OF this]
+obtain cs' where h: "(Th taker, Cs cs') \<in> ?A"
+"(Th taker, Cs cs') \<in> RAG s"
+by (unfold s_RAG_def, auto)
+from this(2) have "waiting s taker cs'"
+by (unfold s_RAG_def, fold waiting_eq, auto)
+from waiting_unique[OF this waiting_taker]
+have "cs' = cs" .
+from h(1)[unfolded this] show False by auto
+qed
+ultimately show ?thesis by auto
+qed
+thus ?thesis
+by (unfold RAG_es waiting_set_eq holding_set_eq, simp)
+qed
+end
+context valid_trace_p_h
+begin
+lemma
+acylic_RAG_kept:
+assumes "acyclic (RAG s)"
+shows "acyclic (RAG (e#s))"
+proof -
+have "acyclic (RAG s \<union> {(Cs cs, Th th)})" (is "acyclic (?A \<union> _)")
+proof -
+from assms
+have "acyclic ?A"
+by (rule acyclic_subset, auto)
+moreover have "(Th th, Cs cs) \<notin> ?A^*"
+proof
+assume otherwise: "(Th th, Cs cs) \<in> ?A^*"
+hence "(Th th, Cs cs) \<in> ?A^+"
+by (unfold rtrancl_eq_or_trancl, auto)
+from tranclD[OF this]
+obtain cs' where h: "(Th th, Cs cs') \<in> RAG s"
+by (unfold s_RAG_def, auto)
+hence "waiting s th cs'"
+by (unfold s_RAG_def, fold waiting_eq, auto)
+with th_not_waiting show False by auto
+qed
+ultimately show ?thesis by auto
+qed
+thus ?thesis by (unfold RAG_es, simp)
+qed
+end
+context valid_trace_p_w
+begin
+lemma
+acylic_RAG_kept:
+assumes "acyclic (RAG s)"
+shows "acyclic (RAG (e#s))"
+proof -
+have "acyclic (RAG s \<union> {(Th th, Cs cs)})" (is "acyclic (?A \<union> _)")
+proof -
+from assms
+have "acyclic ?A"
+by (rule acyclic_subset, auto)
+moreover have "(Cs cs, Th th) \<notin> ?A^*"
+proof
+assume otherwise: "(Cs cs, Th th) \<in> ?A^*"
+from pip_e[unfolded is_p]
+show False
+proof(cases)
+case (thread_P)
+moreover from otherwise have "(Cs cs, Th th) \<in> ?A^+"
+by (unfold rtrancl_eq_or_trancl, auto)
+ultimately show ?thesis by auto
+qed
+qed
+ultimately show ?thesis by auto
+qed
+thus ?thesis by (unfold RAG_es, simp)
+qed
+end
+context valid_trace
+begin
+lemma finite_RAG:
+shows "finite (RAG s)"
+proof(induct rule:ind)
+case Nil
+show ?case
+by (auto simp: s_RAG_def cs_waiting_def
+cs_holding_def wq_def acyclic_def)
+next
+case (Cons s e)
+interpret vt_e: valid_trace_e s e using Cons by simp
 show ?case
 proof(cases e)
 case (Create th prio)
-with ih
+interpret vt: valid_trace_create s e th prio using Create
-show ?thesis by (simp add:RAG_create_unchanged)
+by (unfold_locales, simp)
+show ?thesis using Cons by (simp add: vt.RAG_unchanged)
 next
 case (Exit th)
-with ih show ?thesis by (simp add:RAG_exit_unchanged)
+interpret vt: valid_trace_exit s e th using Exit
+by (unfold_locales, simp)
+show ?thesis using Cons by (simp add: vt.RAG_unchanged)
+next
+case (P th cs)
+interpret vt: valid_trace_p s e th cs using P
+by (unfold_locales, simp)
+show ?thesis using Cons using vt.RAG_es' by auto
 next
 case (V th cs)
-from V vt stp have vtt: "vt (V th cs#s)" by auto
+interpret vt: valid_trace_v s e th cs using V
-from step_RAG_v [OF this]
+by (unfold_locales, simp)
-have eq_de:
+show ?thesis using Cons by (simp add: vt.finite_RAG_kept)
-"RAG (e # s) =
+next
-RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+case (Set th prio)
-{(Cs cs, Th th') |th'. next_th s th cs th'}"
+interpret vt: valid_trace_set s e th prio using Set
-(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+by (unfold_locales, simp)
-from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+show ?thesis using Cons by (simp add: vt.RAG_unchanged)
-from step_back_step [OF vtt]
+qed
-have "step s (V th cs)" .
+qed
-thus ?thesis
-proof(cases)
+lemma acyclic_RAG:
-assume "holding s th cs"
+shows "acyclic (RAG s)"
-hence th_in: "th \<in> set (wq s cs)" and
+proof(induct rule:ind)
-eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
+case Nil
-then obtain rest where
+show ?case
-eq_wq: "wq s cs = th#rest"
+by (auto simp: s_RAG_def cs_waiting_def
-by (cases "wq s cs", auto)
+cs_holding_def wq_def acyclic_def)
-show ?thesis
+next
-proof(cases "rest = []")
+case (Cons s e)
-case False
+interpret vt_e: valid_trace_e s e using Cons by simp
-let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+show ?case
-from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
+proof(cases e)
-by (unfold next_th_def, auto)
+case (Create th prio)
-let ?E = "(?A - ?B - ?C)"
+interpret vt: valid_trace_create s e th prio using Create
-have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
+by (unfold_locales, simp)
-proof
+show ?thesis using Cons by (simp add: vt.RAG_unchanged)
-assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
+next
-hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+case (Exit th)
-from tranclD [OF this]
+interpret vt: valid_trace_exit s e th using Exit
-obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
+by (unfold_locales, simp)
-hence th_d: "(Th ?th', x) \<in> ?A" by simp
+show ?thesis using Cons by (simp add: vt.RAG_unchanged)
-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
+interpret vt: valid_trace_p s e th cs using P
-from step_RAG_p [OF this] P
+by (unfold_locales, simp)
-have "RAG (e # s) =
+show ?thesis
-(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
+then interpret vt_h: valid_trace_p_h s e th cs
-have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
+by (unfold_locales, simp)
-proof
+show ?thesis using Cons by (simp add: vt_h.acylic_RAG_kept)
-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
+then interpret vt_w: valid_trace_p_w s e th cs
-have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
+by (unfold_locales, simp)
-proof
+show ?thesis using Cons by (simp add: vt_w.acylic_RAG_kept)
-assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
+qed
-hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+next
-moreover from step_back_step [OF vtt] have "step s (P th cs)" .
+case (V th cs)
-ultimately show False
+interpret vt: valid_trace_v s e th cs using V
-proof -
+by (unfold_locales, simp)
-show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+show ?thesis
-by (ind_cases "step s (P th cs)", simp)
+proof(cases "vt.rest = []")
+case True
+then interpret vt_e: valid_trace_v_e s e th cs
+by (unfold_locales, simp)
+show ?thesis by (simp add: Cons.hyps(2) vt_e.acylic_RAG_kept)
+next
+case False
+then interpret vt_n: valid_trace_v_n s e th cs
+by (unfold_locales, simp)
+show ?thesis by (simp add: Cons.hyps(2) vt_n.acylic_RAG_kept)
+qed
+next
+case (Set th prio)
+interpret vt: valid_trace_set s e th prio using Set
+by (unfold_locales, simp)
+show ?thesis using Cons by (simp add: vt.RAG_unchanged)
+qed
+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 held_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 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 held_unique)
+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)
+from wf_RAG show "wf (RAG s)" .
+qed
+qed
+qed
+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)
+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
-qed
-with acyclic_insert ih eq_r show ?thesis by auto
-qed
-ultimately show ?thesis by simp
 next
-case (Set thread prio)
+show "fbranch (hRAG s)"
-with ih
+proof(rule finite_fbranchI)
-thm RAG_set_unchanged
+from finite_RAG
-show ?thesis by (simp add:RAG_set_unchanged)
+show "finite (hRAG s)" by (unfold RAG_split, auto)
 qed
-next
+qed
-case vt_nil
+moreover have "wf (tRAG s)"
-show "acyclic (RAG ([]::state))"
+proof(rule wf_subset)
-by (auto simp: s_RAG_def cs_waiting_def
+show "wf (RAG s O RAG s)" using wf_RAG
-cs_holding_def wq_def acyclic_def)
+by (fold wf_comp_self, simp)
-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)
+show "tRAG s \<subseteq> (RAG s O RAG s)"
-with ih show ?thesis by (simp add:RAG_exit_unchanged)
+by (unfold tRAG_alt_def, auto)
-next
+qed
-case (V th cs)
+ultimately show ?thesis
-from V vt stp have vtt: "vt (V th cs#s)" by auto
+by (unfold fsubtree_def fsubtree_axioms_def,auto)
-from step_RAG_v [OF this]
+qed
-have eq_de: "RAG (e # s) =
+from this[folded tRAG_def] show "fsubtree (tRAG s)" .
-RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+qed
-{(Cs cs, Th th') |th'. next_th s th cs th'}
-"
-(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+context valid_trace
-moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
+begin
-moreover have "finite ?D"
-proof -
+lemma finite_subtree_threads:
-have "?D = {} \<or> (\<exists> a. ?D = {a})"
+"finite {th'. Th th' \<in> subtree (RAG s) (Th th)}" (is "finite ?A")
-by (unfold next_th_def, auto)
+proof -
-thus ?thesis
+have "?A = the_thread ` {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}"
-proof
+by (auto, insert image_iff, fastforce)
-assume h: "?D = {}"
+moreover have "finite {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}"
-show ?thesis by (unfold h, simp)
+(is "finite ?B")
-next
+proof -
-assume "\<exists> a. ?D = {a}"
+have "?B = (subtree (RAG s) (Th th)) \<inter> {Th th' | th'. True}"
-thus ?thesis
+by auto
-by (metis finite.simps)
+moreover have "... \<subseteq> (subtree (RAG s) (Th th))" by auto
-qed
+moreover have "finite ..." by (simp add: fsbtRAGs.finite_subtree)
-qed
+ultimately show ?thesis by auto
-ultimately show ?thesis by simp
+qed
-next
+ultimately show ?thesis by auto
-case (P th cs)
+qed
-from P vt stp have vtt: "vt (P th cs#s)" by auto
-from step_RAG_p [OF this] P
+lemma le_cp:
-have "RAG (e # s) =
+shows "preced th s \<le> cp s th"
-(if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
+proof(unfold cp_alt_def, rule Max_ge)
-RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+show "finite (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
-by simp
+by (simp add: finite_subtree_threads)
-moreover have "finite ?R"
+next
-proof(cases "wq s cs = []")
+show "preced th s \<in> the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}"
-case True
+by (simp add: subtree_def the_preced_def)
-hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
+qed
-with True and ih show ?thesis by auto
-next
+lemma cp_le:
-case False
+assumes th_in: "th \<in> threads s"
-hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
+shows "cp s th \<le> Max (the_preced s ` threads s)"
-with False and ih show ?thesis by auto
+proof(unfold cp_alt_def, rule Max_f_mono)
-qed
+show "finite (threads s)" by (simp add: finite_threads)
-ultimately show ?thesis by auto
+next
-next
+show " {th'. Th th' \<in> subtree (RAG s) (Th th)} \<noteq> {}"
-case (Set thread prio)
+using subtree_def by fastforce
-with ih
+next
-show ?thesis by (simp add:RAG_set_unchanged)
+show "{th'. Th th' \<in> subtree (RAG s) (Th th)} \<subseteq> threads s"
-qed
+using assms
-next
+by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq
-case vt_nil
+node.inject(1) rtranclD subsetI subtree_def trancl_domain)
-show "finite (RAG ([]::state))"
+qed
-by (auto simp: s_RAG_def cs_waiting_def
-cs_holding_def wq_def acyclic_def)
+lemma max_cp_eq:
-qed
+shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
-qed
+(is "?L = ?R")
+proof -
-text {* Several useful lemmas *}
+have "?L \<le> ?R"
+proof(cases "threads s = {}")
-lemma wf_dep_converse:
+case False
+show ?thesis
+by (rule Max.boundedI,
+insert cp_le,
+auto simp:finite_threads False)
+qed auto
+moreover have "?R \<le> ?L"
+by (rule Max_fg_mono,
+simp add: finite_threads,
+simp add: le_cp the_preced_def)
+ultimately show ?thesis by auto
+qed
+lemma wf_RAG_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:
-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)^+)"
+assumes "node \<in> Domain (RAG s)"
-proof -
+obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"
-from wf_dep_converse
+proof -
-have h: "wf ((RAG s)\<inverse>)" .
+from assms have "node \<in> Range ((RAG s)^-1)" by auto
-show ?thesis
+from wf_base[OF wf_RAG_converse this]
-proof(induct rule:wf_induct [OF h])
+obtain b where h_b: "(b, node) \<in> ((RAG s)\<inverse>)\<^sup>+" "\<forall>c. (c, b) \<notin> (RAG s)\<inverse>" by auto
-fix x
+obtain th' where eq_b: "b = Th th'"
-assume ih [rule_format]:
+proof(cases b)
-"\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow>
+case (Cs cs)
-y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
+from h_b(1)[unfolded trancl_converse]
-show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
+have "(node, b) \<in> ((RAG s)\<^sup>+)" by auto
-proof
+from tranclE[OF this]
-assume x_d: "x \<in> Domain (RAG s)"
+obtain n where "(n, b) \<in> RAG s" by auto
-show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
+from this[unfolded Cs]
-proof(cases x)
+obtain th1 where "waiting s th1 cs"
-case (Th th)
+by (unfold s_RAG_def, fold waiting_eq, auto)
-from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
+from waiting_holding[OF this]
-with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
+obtain th2 where "holding s th2 cs" .
-from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
+hence "(Cs cs, Th th2) \<in> RAG s"
-hence "Cs cs \<in> Domain (RAG s)" by auto
+by (unfold s_RAG_def, fold holding_eq, auto)
-from ih [OF x_in_r this] obtain th'
+with h_b(2)[unfolded Cs, rule_format]
-where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
+have False by auto
-have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
+thus ?thesis by auto
-with th'_ready show ?thesis by auto
+qed auto
-next
+have "th' \<in> readys s"
-case (Cs cs)
+proof -
-from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
+from h_b(2)[unfolded eq_b]
-show ?thesis
+have "\<forall>cs. \<not> waiting s th' cs"
-proof(cases "th' \<in> readys s")
+by (unfold s_RAG_def, fold waiting_eq, auto)
-case True
+moreover have "th' \<in> threads s"
-from True and th'_d show ?thesis by auto
+proof(rule rg_RAG_threads)
-next
+from tranclD[OF h_b(1), unfolded eq_b]
-case False
+obtain z where "(z, Th th') \<in> (RAG s)" by auto
-from th'_d and range_in  have "th' \<in> threads s" by auto
+thus "Th th' \<in> Range (RAG s)" by auto
-with False have "Th th' \<in> Domain (RAG s)"
+qed
-by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
+ultimately show ?thesis by (auto simp:readys_def)
-from ih [OF th'_d this]
+qed
-obtain th'' where
+moreover have "(node, Th th') \<in> (RAG s)^+"
-th''_r: "th'' \<in> readys s" and
+using h_b(1)[unfolded trancl_converse] eq_b by auto
-th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
+ultimately show ?thesis using that by metis
-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"}.
 *}
 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:
-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:
-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:
-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
 lemma count_rec1 [simp]:
 assumes "Q e"
 shows "count Q (e#es) = Suc (count Q es)"
 using assms
 by (unfold count_def, auto)
 by (unfold count_def, auto)
 lemma count_rec3 [simp]:
 shows "count Q [] =  0"
 by (unfold count_def, auto)
+lemma cntP_simp1[simp]:
+"cntP (P th cs'#s) th = cntP s th + 1"
+by (unfold cntP_def, simp)
+lemma cntP_simp2[simp]:
+assumes "th' \<noteq> th"
+shows "cntP (P th cs'#s) th' = cntP s th'"
+using assms
+by (unfold cntP_def, simp)
+lemma cntP_simp3[simp]:
+assumes "\<not> isP e"
+shows "cntP (e#s) th' = cntP s th'"
+using assms
+by (unfold cntP_def, cases e, simp+)
+lemma cntV_simp1[simp]:
+"cntV (V th cs'#s) th = cntV s th + 1"
+by (unfold cntV_def, simp)
+lemma cntV_simp2[simp]:
+assumes "th' \<noteq> th"
+shows "cntV (V th cs'#s) th' = cntV s th'"
+using assms
+by (unfold cntV_def, simp)
+lemma cntV_simp3[simp]:
+assumes "\<not> isV e"
+shows "cntV (e#s) th' = cntV s th'"
+using assms
+by (unfold cntV_def, cases e, simp+)
 lemma cntP_diff_inv:
 assumes "cntP (e#s) th \<noteq> cntP s th"
 shows "isP e \<and> actor e = th"
 proof(cases e)
 case (P th' pty)
 show ?thesis
 by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)",
 insert assms P, auto simp:cntP_def)
 qed (insert assms, auto simp:cntP_def)
-lemma isP_E:
-assumes "isP e"
-obtains cs where "e = P (actor e) cs"
-using assms by (cases e, auto)
-lemma isV_E:
-assumes "isV e"
-obtains cs where "e = V (actor e) cs"
-using assms by (cases e, auto) (* ccc *)
 lemma cntV_diff_inv:
 assumes "cntV (e#s) th \<noteq> cntV s th"
 shows "isV e \<and> actor e = th"
 proof(cases e)
 case (V th' pty)
 show ?thesis
 by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)",
 insert assms V, auto simp:cntV_def)
 qed (insert assms, auto simp:cntV_def)
+lemma children_RAG_alt_def:
+"children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}"
+by (unfold s_RAG_def, auto simp:children_def holding_eq)
+lemma holdents_alt_def:
+"holdents s th = the_cs ` (children (RAG (s::state)) (Th th))"
+by (unfold children_RAG_alt_def holdents_def, simp add: image_image)
+lemma cntCS_alt_def:
+"cntCS s th = card (children (RAG s) (Th th))"
+apply (unfold children_RAG_alt_def cntCS_def holdents_def)
+by (rule card_image[symmetric], auto simp:inj_on_def)
 context valid_trace
 begin
-text {* (* ddd *) \noindent
+lemma finite_holdents: "finite (holdents s th)"
-The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"}
+by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)
-of one particular thread.
-*}
+end
-lemma cnp_cnv_cncs:
+context valid_trace_p_w
-shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
+begin
-then cntCS s th else cntCS s th + 1)"
-proof -
+lemma holding_s_holder: "holding s holder cs"
-from vt show ?thesis
+by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
-proof(induct arbitrary:th)
-case (vt_cons s e)
+lemma holding_es_holder: "holding (e#s) holder cs"
-interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
+by (unfold s_holding_def, fold wq_def, unfold wq_es_cs wq_s_cs, auto)
-assume vt: "vt s"
-and ih: "\<And>th. cntP s th  = cntV s th +
+lemma holdents_es:
-(if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
+shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
-and stp: "step s e"
+proof -
-from stp show ?case
+{ fix cs'
-proof(cases)
+assume "cs' \<in> ?L"
-case (thread_create thread prio)
+hence h: "holding (e#s) th' cs'" by (auto simp:holdents_def)
-assume eq_e: "e = Create thread prio"
+have "holding s th' cs'"
-and not_in: "thread \<notin> threads s"
+proof(cases "cs' = cs")
+case True
+from held_unique[OF h[unfolded True] holding_es_holder]
+have "th' = holder" .
+thus ?thesis
+by (unfold True holdents_def, insert holding_s_holder, simp)
+next
+case False
+hence "wq (e#s) cs' = wq s cs'" by simp
+from h[unfolded s_holding_def, folded wq_def, unfolded this]
 show ?thesis
-proof -
+by (unfold s_holding_def, fold wq_def, auto)
-{ fix cs
+qed
-assume "thread \<in> set (wq s cs)"
+hence "cs' \<in> ?R" by (auto simp:holdents_def)
-from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
+} moreover {
-with not_in have "False" by simp
+fix cs'
-} with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
+assume "cs' \<in> ?R"
-by (auto simp:readys_def threads.simps s_waiting_def
+hence h: "holding s th' cs'" by (auto simp:holdents_def)
-wq_def cs_waiting_def Let_def)
+have "holding (e#s) th' cs'"
-from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+proof(cases "cs' = cs")
-from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+case True
-have eq_cncs: "cntCS (e#s) th = cntCS s th"
+from held_unique[OF h[unfolded True] holding_s_holder]
-unfolding cntCS_def holdents_test
+have "th' = holder" .
-by (simp add:RAG_create_unchanged eq_e)
+thus ?thesis
-{ assume "th \<noteq> thread"
+by (unfold True holdents_def, insert holding_es_holder, simp)
-with eq_readys eq_e
+next
-have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
+case False
-(th \<in> readys (s) \<or> th \<notin> threads (s))"
+hence "wq s cs' = wq (e#s) cs'" by simp
-by (simp add:threads.simps)
+from h[unfolded s_holding_def, folded wq_def, unfolded this]
-with eq_cnp eq_cnv eq_cncs ih not_in
+show ?thesis
-have ?thesis by simp
+by (unfold s_holding_def, fold wq_def, auto)
-} moreover {
+qed
-assume eq_th: "th = thread"
+hence "cs' \<in> ?L" by (auto simp:holdents_def)
-with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp
+} ultimately show ?thesis by auto
-moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
+qed
-moreover note eq_cnp eq_cnv eq_cncs
-ultimately have ?thesis by auto
+lemma cntCS_es_th[simp]: "cntCS (e#s) th' = cntCS s th'"
-} ultimately show ?thesis by blast
+by (unfold cntCS_def holdents_es, simp)
+lemma th_not_ready_es:
+shows "th \<notin> readys (e#s)"
+using waiting_es_th_cs
+by (unfold readys_def, auto)
+end
+context valid_trace_p_h
+begin
+lemma th_not_waiting':
+"\<not> waiting (e#s) th cs'"
+proof(cases "cs' = cs")
+case True
+show ?thesis
+by (unfold True s_waiting_def, fold wq_def, unfold wq_es_cs', auto)
+next
+case False
+from th_not_waiting[of cs', unfolded s_waiting_def, folded wq_def]
+show ?thesis
+by (unfold s_waiting_def, fold wq_def, insert False, simp)
+qed
+lemma ready_th_es:
+shows "th \<in> readys (e#s)"
+using th_not_waiting'
+by (unfold readys_def, insert live_th_es, auto)
+lemma holdents_es_th:
+"holdents (e#s) th = (holdents s th) \<union> {cs}" (is "?L = ?R")
+proof -
+{ fix cs'
+assume "cs' \<in> ?L"
+hence "holding (e#s) th cs'"
+by (unfold holdents_def, auto)
+hence "cs' \<in> ?R"
+by (cases rule:holding_esE, auto simp:holdents_def)
+} moreover {
+fix cs'
+assume "cs' \<in> ?R"
+hence "holding s th cs' \<or> cs' = cs"
+by (auto simp:holdents_def)
+hence "cs' \<in> ?L"
+proof
+assume "holding s th cs'"
+from holding_kept[OF this]
+show ?thesis by (auto simp:holdents_def)
+next
+assume "cs' = cs"
+thus ?thesis using holding_es_th_cs
+by (unfold holdents_def, auto)
+qed
+} ultimately show ?thesis by auto
+qed
+lemma cntCS_es_th: "cntCS (e#s) th = cntCS s th + 1"
+proof -
+have "card (holdents s th \<union> {cs}) = card (holdents s th) + 1"
+proof(subst card_Un_disjoint)
+show "holdents s th \<inter> {cs} = {}"
+using not_holding_s_th_cs by (auto simp:holdents_def)
+qed (auto simp:finite_holdents)
+thus ?thesis
+by (unfold cntCS_def holdents_es_th, simp)
+qed
+lemma no_holder:
+"\<not> holding s th' cs"
+proof
+assume otherwise: "holding s th' cs"
+from this[unfolded s_holding_def, folded wq_def, unfolded we]
+show False by auto
+qed
+lemma holdents_es_th':
+assumes "th' \<noteq> th"
+shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+{ fix cs'
+assume "cs' \<in> ?L"
+hence h_e: "holding (e#s) th' cs'" by (auto simp:holdents_def)
+have "cs' \<noteq> cs"
+proof
+assume "cs' = cs"
+from held_unique[OF h_e[unfolded this] holding_es_th_cs]
+have "th' = th" .
+with assms show False by simp
+qed
+from h_e[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp[OF this]]
+have "th' \<in> set (wq s cs') \<and> th' = hd (wq s cs')" .
+hence "cs' \<in> ?R"
+by (unfold holdents_def s_holding_def, fold wq_def, auto)
+} moreover {
+fix cs'
+assume "cs' \<in> ?R"
+hence "holding s th' cs'" by (auto simp:holdents_def)
+from holding_kept[OF this]
+have "holding (e # s) th' cs'" .
+hence "cs' \<in> ?L"
+by (unfold holdents_def, auto)
+} ultimately show ?thesis by auto
+qed
+lemma cntCS_es_th'[simp]:
+assumes "th' \<noteq> th"
+shows "cntCS (e#s) th' = cntCS s th'"
+by (unfold cntCS_def holdents_es_th'[OF assms], simp)
+end
+context valid_trace_p
+begin
+lemma readys_kept1:
+assumes "th' \<noteq> th"
+and "th' \<in> readys (e#s)"
+shows "th' \<in> readys s"
+proof -
+{ fix cs'
+assume wait: "waiting s th' cs'"
+have n_wait: "\<not> waiting (e#s) th' cs'"
+using assms(2)[unfolded readys_def] by auto
+have False
+proof(cases "cs' = cs")
+case False
+with n_wait wait
+show ?thesis
+by (unfold s_waiting_def, fold wq_def, auto)
+next
+case True
+show ?thesis
+proof(cases "wq s cs = []")
+case True
+then interpret vt: valid_trace_p_h
+by (unfold_locales, simp)
+show ?thesis using n_wait wait waiting_kept by auto
+next
+case False
+then interpret vt: valid_trace_p_w by (unfold_locales, simp)
+show ?thesis using n_wait wait waiting_kept by blast
 qed
+qed
+} with assms(2) show ?thesis
+by (unfold readys_def, auto)
+qed
+lemma readys_kept2:
+assumes "th' \<noteq> th"
+and "th' \<in> readys s"
+shows "th' \<in> readys (e#s)"
+proof -
+{ fix cs'
+assume wait: "waiting (e#s) th' cs'"
+have n_wait: "\<not> waiting s th' cs'"
+using assms(2)[unfolded readys_def] by auto
+have False
+proof(cases "cs' = cs")
+case False
+with n_wait wait
+show ?thesis
+by (unfold s_waiting_def, fold wq_def, auto)
 next
-case (thread_exit thread)
+case True
-assume eq_e: "e = Exit thread"
+show ?thesis
-and is_runing: "thread \<in> runing s"
+proof(cases "wq s cs = []")
-and no_hold: "holdents s thread = {}"
+case True
-from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+then interpret vt: valid_trace_p_h
-from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+by (unfold_locales, simp)
-have eq_cncs: "cntCS (e#s) th = cntCS s th"
+show ?thesis using n_wait vt.waiting_esE wait by blast
-unfolding cntCS_def holdents_test
+next
-by (simp add:RAG_exit_unchanged eq_e)
+case False
-{ assume "th \<noteq> thread"
+then interpret vt: valid_trace_p_w by (unfold_locales, simp)
-with eq_e
+show ?thesis using assms(1) n_wait vt.waiting_esE wait by auto
-have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
+qed
-(th \<in> readys (s) \<or> th \<notin> threads (s))"
+qed
-apply (simp add:threads.simps readys_def)
+} with assms(2) show ?thesis
-apply (subst s_waiting_def)
+by (unfold readys_def, auto)
-apply (simp add:Let_def)
+qed
-apply (subst s_waiting_def, simp)
-done
+lemma readys_simp [simp]:
-with eq_cnp eq_cnv eq_cncs ih
+assumes "th' \<noteq> th"
-have ?thesis by simp
+shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
-} moreover {
+using readys_kept1[OF assms] readys_kept2[OF assms]
-assume eq_th: "th = thread"
+by metis
-with ih is_runing have " cntP s th = cntV s th + cntCS s th"
-by (simp add:runing_def)
+lemma cnp_cnv_cncs_kept: (* ddd *)
-moreover from eq_th eq_e have "th \<notin> threads (e#s)"
+assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
-by simp
+shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
-moreover note eq_cnp eq_cnv eq_cncs
+proof(cases "th' = th")
-ultimately have ?thesis by auto
+case True
-} ultimately show ?thesis by blast
+note eq_th' = this
+show ?thesis
+proof(cases "wq s cs = []")
+case True
+then interpret vt: valid_trace_p_h by (unfold_locales, simp)
+show ?thesis
+using assms eq_th' is_p ready_th_s vt.cntCS_es_th vt.ready_th_es pvD_def by auto
+next
+case False
+then interpret vt: valid_trace_p_w by (unfold_locales, simp)
+show ?thesis
+using add.commute add.left_commute assms eq_th' is_p live_th_s
+ready_th_s vt.th_not_ready_es pvD_def
+apply (auto)
+by (fold is_p, simp)
+qed
+next
+case False
+note h_False = False
+thus ?thesis
+proof(cases "wq s cs = []")
+case True
+then interpret vt: valid_trace_p_h by (unfold_locales, simp)
+show ?thesis using assms
+by (insert True h_False pvD_def, auto split:if_splits,unfold is_p, auto)
+next
+case False
+then interpret vt: valid_trace_p_w by (unfold_locales, simp)
+show ?thesis using assms
+by (insert False h_False pvD_def, auto split:if_splits,unfold is_p, auto)
+qed
+qed
+end
+context valid_trace_v (* ccc *)
+begin
+lemma holding_th_cs_s:
+"holding s th cs"
+by  (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
+lemma th_ready_s [simp]: "th \<in> readys s"
+using runing_th_s
+by (unfold runing_def readys_def, auto)
+lemma th_live_s [simp]: "th \<in> threads s"
+using th_ready_s by (unfold readys_def, auto)
+lemma th_ready_es [simp]: "th \<in> readys (e#s)"
+using runing_th_s neq_t_th
+by (unfold is_v runing_def readys_def, auto)
+lemma th_live_es [simp]: "th \<in> threads (e#s)"
+using th_ready_es by (unfold readys_def, auto)
+lemma pvD_th_s[simp]: "pvD s th = 0"
+by (unfold pvD_def, simp)
+lemma pvD_th_es[simp]: "pvD (e#s) th = 0"
+by (unfold pvD_def, simp)
+lemma cntCS_s_th [simp]: "cntCS s th > 0"
+proof -
+have "cs \<in> holdents s th" using holding_th_cs_s
+by (unfold holdents_def, simp)
+moreover have "finite (holdents s th)" using finite_holdents
+by simp
+ultimately show ?thesis
+by (unfold cntCS_def,
+auto intro!:card_gt_0_iff[symmetric, THEN iffD1])
+qed
+end
+context valid_trace_v_n
+begin
+lemma not_ready_taker_s[simp]:
+"taker \<notin> readys s"
+using waiting_taker
+by (unfold readys_def, auto)
+lemma taker_live_s [simp]: "taker \<in> threads s"
+proof -
+have "taker \<in> set wq'" by (simp add: eq_wq')
+from th'_in_inv[OF this]
+have "taker \<in> set rest" .
+hence "taker \<in> set (wq s cs)" by (simp add: wq_s_cs)
+thus ?thesis using wq_threads by auto
+qed
+lemma taker_live_es [simp]: "taker \<in> threads (e#s)"
+using taker_live_s threads_es by blast
+lemma taker_ready_es [simp]:
+shows "taker \<in> readys (e#s)"
+proof -
+{ fix cs'
+assume "waiting (e#s) taker cs'"
+hence False
+proof(cases rule:waiting_esE)
+case 1
+thus ?thesis using waiting_taker waiting_unique by auto
+qed simp
+} thus ?thesis by (unfold readys_def, auto)
+qed
+lemma neq_taker_th: "taker \<noteq> th"
+using th_not_waiting waiting_taker by blast
+lemma not_holding_taker_s_cs:
+shows "\<not> holding s taker cs"
+using holding_cs_eq_th neq_taker_th by auto
+lemma holdents_es_taker:
+"holdents (e#s) taker = holdents s taker \<union> {cs}" (is "?L = ?R")
+proof -
+{ fix cs'
+assume "cs' \<in> ?L"
+hence "holding (e#s) taker cs'" by (auto simp:holdents_def)
+hence "cs' \<in> ?R"
+proof(cases rule:holding_esE)
+case 2
+thus ?thesis by (auto simp:holdents_def)
+qed auto
+} moreover {
+fix cs'
+assume "cs' \<in> ?R"
+hence "holding s taker cs' \<or> cs' = cs" by (auto simp:holdents_def)
+hence "cs' \<in> ?L"
+proof
+assume "holding s taker cs'"
+hence "holding (e#s) taker cs'"
+using holding_esI2 holding_taker by fastforce
+thus ?thesis by (auto simp:holdents_def)
 next
-case (thread_P thread cs)
+assume "cs' = cs"
-assume eq_e: "e = P thread cs"
+with holding_taker
-and is_runing: "thread \<in> runing s"
+show ?thesis by (auto simp:holdents_def)
-and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
+qed
-from thread_P vt stp ih  have vtp: "vt (P thread cs#s)" by auto
+} ultimately show ?thesis by auto
-then interpret vt_p: valid_trace "(P thread cs#s)"
+qed
-by (unfold_locales, simp)
+lemma cntCS_es_taker [simp]: "cntCS (e#s) taker = cntCS s taker + 1"
+proof -
+have "card (holdents s taker \<union> {cs}) = card (holdents s taker) + 1"
+proof(subst card_Un_disjoint)
+show "holdents s taker \<inter> {cs} = {}"
+using not_holding_taker_s_cs by (auto simp:holdents_def)
+qed (auto simp:finite_holdents)
+thus ?thesis
+by (unfold cntCS_def, insert holdents_es_taker, simp)
+qed
+lemma pvD_taker_s[simp]: "pvD s taker = 1"
+by (unfold pvD_def, simp)
+lemma pvD_taker_es[simp]: "pvD (e#s) taker = 0"
+by (unfold pvD_def, simp)
+lemma pvD_th_s[simp]: "pvD s th = 0"
+by (unfold pvD_def, simp)
+lemma pvD_th_es[simp]: "pvD (e#s) th = 0"
+by (unfold pvD_def, simp)
+lemma holdents_es_th:
+"holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")
+proof -
+{ fix cs'
+assume "cs' \<in> ?L"
+hence "holding (e#s) th cs'" by (auto simp:holdents_def)
+hence "cs' \<in> ?R"
+proof(cases rule:holding_esE)
+case 2
+thus ?thesis by (auto simp:holdents_def)
+qed (insert neq_taker_th, auto)
+} moreover {
+fix cs'
+assume "cs' \<in> ?R"
+hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def)
+from holding_esI2[OF this]
+have "cs' \<in> ?L" by (auto simp:holdents_def)
+} ultimately show ?thesis by auto
+qed
+lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1"
+proof -
+have "card (holdents s th - {cs}) = card (holdents s th) - 1"
+proof -
+have "cs \<in> holdents s th" using holding_th_cs_s
+by (auto simp:holdents_def)
+moreover have "finite (holdents s th)"
+by (simp add: finite_holdents)
+ultimately show ?thesis by auto
+qed
+thus ?thesis by (unfold cntCS_def holdents_es_th)
+qed
+lemma holdents_kept:
+assumes "th' \<noteq> taker"
+and "th' \<noteq> th"
+shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+{ fix cs'
+assume h: "cs' \<in> ?L"
+have "cs' \<in> ?R"
+proof(cases "cs' = cs")
+case False
+hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
+from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
+from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
+show ?thesis
+by (unfold holdents_def s_holding_def, fold wq_def, auto)
+next
+case True
+from h[unfolded this]
+have "holding (e#s) th' cs" by (auto simp:holdents_def)
+from held_unique[OF this holding_taker]
+have "th' = taker" .
+with assms show ?thesis by auto
+qed
+} moreover {
+fix cs'
+assume h: "cs' \<in> ?R"
+have "cs' \<in> ?L"
+proof(cases "cs' = cs")
+case False
+hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
+from h have "holding s th' cs'" by (auto simp:holdents_def)
+from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
+show ?thesis
+by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
+next
+case True
+from h[unfolded this]
+have "holding s th' cs" by (auto simp:holdents_def)
+from held_unique[OF this holding_th_cs_s]
+have "th' = th" .
+with assms show ?thesis by auto
+qed
+} ultimately show ?thesis by auto
+qed
+lemma cntCS_kept [simp]:
+assumes "th' \<noteq> taker"
+and "th' \<noteq> th"
+shows "cntCS (e#s) th' = cntCS s th'"
+by (unfold cntCS_def holdents_kept[OF assms], simp)
+lemma readys_kept1:
+assumes "th' \<noteq> taker"
+and "th' \<in> readys (e#s)"
+shows "th' \<in> readys s"
+proof -
+{ fix cs'
+assume wait: "waiting s th' cs'"
+have n_wait: "\<not> waiting (e#s) th' cs'"
+using assms(2)[unfolded readys_def] by auto
+have False
+proof(cases "cs' = cs")
+case False
+with n_wait wait
 show ?thesis
-proof -
+by (unfold s_waiting_def, fold wq_def, auto)
-{ 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)
+case True
-from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
+have "th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)"
-then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
+using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
-assume eq_e: "e = V thread cs"
+moreover have "\<not> (th' \<in> set rest \<and> th' \<noteq> hd (taker # rest'))"
-and is_runing: "thread \<in> runing s"
+using n_wait[unfolded True s_waiting_def, folded wq_def,
-and hold: "holding s thread cs"
+unfolded wq_es_cs set_wq', unfolded eq_wq'] .
-from hold obtain rest
+ultimately have "th' = taker" by auto
-where eq_wq: "wq s cs = thread # rest"
+with assms(1)
-by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+show ?thesis by simp
-have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+qed
-have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+} with assms(2) show ?thesis
-proof(rule someI2)
+by (unfold readys_def, auto)
-from vt_v.wq_distinct[of cs] and eq_wq
+qed
-show "distinct rest \<and> set rest = set rest"
-by (metis distinct.simps(2) vt_s.wq_distinct)
+lemma readys_kept2:
-next
+assumes "th' \<noteq> taker"
-show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+and "th' \<in> readys s"
-by auto
+shows "th' \<in> readys (e#s)"
-qed
+proof -
+{ fix cs'
+assume wait: "waiting (e#s) th' cs'"
+have n_wait: "\<not> waiting s th' cs'"
+using assms(2)[unfolded readys_def] by auto
+have False
+proof(cases "cs' = cs")
+case False
+with n_wait wait
+show ?thesis
+by (unfold s_waiting_def, fold wq_def, auto)
+next
+case True
+have "th' \<in> set rest \<and> th' \<noteq> hd (taker # rest')"
+using  wait [unfolded True s_waiting_def, folded wq_def,
+unfolded wq_es_cs set_wq', unfolded eq_wq']  .
+moreover have "\<not> (th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest))"
+using n_wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
+ultimately have "th' = taker" by auto
+with assms(1)
+show ?thesis by simp
+qed
+} with assms(2) show ?thesis
+by (unfold readys_def, auto)
+qed
+lemma readys_simp [simp]:
+assumes "th' \<noteq> taker"
+shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+using readys_kept1[OF assms] readys_kept2[OF assms]
+by metis
+lemma cnp_cnv_cncs_kept:
+assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof -
+{ assume eq_th': "th' = taker"
+have ?thesis
+apply (unfold eq_th' pvD_taker_es cntCS_es_taker)
+by (insert neq_taker_th assms[unfolded eq_th'], unfold is_v, simp)
+} moreover {
+assume eq_th': "th' = th"
+have ?thesis
+apply (unfold eq_th' pvD_th_es cntCS_es_th)
+by (insert assms[unfolded eq_th'], unfold is_v, simp)
+} moreover {
+assume h: "th' \<noteq> taker" "th' \<noteq> th"
+have ?thesis using assms
+apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp)
+by (fold is_v, unfold pvD_def, simp)
+} ultimately show ?thesis by metis
+qed
+end
+context valid_trace_v_e
+begin
+lemma holdents_es_th:
+"holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")
+proof -
+{ fix cs'
+assume "cs' \<in> ?L"
+hence "holding (e#s) th cs'" by (auto simp:holdents_def)
+hence "cs' \<in> ?R"
+proof(cases rule:holding_esE)
+case 1
+thus ?thesis by (auto simp:holdents_def)
+qed
+} moreover {
+fix cs'
+assume "cs' \<in> ?R"
+hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def)
+from holding_esI2[OF this]
+have "cs' \<in> ?L" by (auto simp:holdents_def)
+} ultimately show ?thesis by auto
+qed
+lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1"
+proof -
+have "card (holdents s th - {cs}) = card (holdents s th) - 1"
+proof -
+have "cs \<in> holdents s th" using holding_th_cs_s
+by (auto simp:holdents_def)
+moreover have "finite (holdents s th)"
+by (simp add: finite_holdents)
+ultimately show ?thesis by auto
+qed
+thus ?thesis by (unfold cntCS_def holdents_es_th)
+qed
+lemma holdents_kept:
+assumes "th' \<noteq> th"
+shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+{ fix cs'
+assume h: "cs' \<in> ?L"
+have "cs' \<in> ?R"
+proof(cases "cs' = cs")
+case False
+hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
+from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
+from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
 show ?thesis
-proof -
+by (unfold holdents_def s_holding_def, fold wq_def, auto)
-{ 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)
+case True
-assume eq_e: "e = Set thread prio"
+from h[unfolded this]
-and is_runing: "thread \<in> runing s"
+have "holding (e#s) th' cs" by (auto simp:holdents_def)
+from this[unfolded s_holding_def, folded wq_def,
+unfolded wq_es_cs nil_wq']
+show ?thesis by auto
+qed
+} moreover {
+fix cs'
+assume h: "cs' \<in> ?R"
+have "cs' \<in> ?L"
+proof(cases "cs' = cs")
+case False
+hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
+from h have "holding s th' cs'" by (auto simp:holdents_def)
+from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
 show ?thesis
-proof -
+by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
-from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+next
-from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+case True
-have eq_cncs: "cntCS (e#s) th = cntCS s th"
+from h[unfolded this]
-unfolding cntCS_def holdents_test
+have "holding s th' cs" by (auto simp:holdents_def)
-by (simp add:RAG_set_unchanged eq_e)
+from held_unique[OF this holding_th_cs_s]
-from eq_e have eq_readys: "readys (e#s) = readys s"
+have "th' = th" .
-by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
+with assms show ?thesis by auto
-auto simp:Let_def)
+qed
-{ assume "th \<noteq> thread"
+} ultimately show ?thesis by auto
-with eq_readys eq_e
+qed
-have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
-(th \<in> readys (s) \<or> th \<notin> threads (s))"
+lemma cntCS_kept [simp]:
-by (simp add:threads.simps)
+assumes "th' \<noteq> th"
-with eq_cnp eq_cnv eq_cncs ih is_runing
+shows "cntCS (e#s) th' = cntCS s th'"
-have ?thesis by simp
+by (unfold cntCS_def holdents_kept[OF assms], simp)
-} moreover {
-assume eq_th: "th = thread"
+lemma readys_kept1:
-with is_runing ih have " cntP s th  = cntV s th + cntCS s th"
+assumes "th' \<in> readys (e#s)"
-by (unfold runing_def, auto)
+shows "th' \<in> readys s"
-moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
+proof -
-by (simp add:runing_def)
+{ fix cs'
-moreover note eq_cnp eq_cnv eq_cncs
+assume wait: "waiting s th' cs'"
-ultimately have ?thesis by auto
+have n_wait: "\<not> waiting (e#s) th' cs'"
-} ultimately show ?thesis by blast
+using assms(1)[unfolded readys_def] by auto
-qed
+have False
-qed
+proof(cases "cs' = cs")
-next
+case False
-case vt_nil
+with n_wait wait
-show ?case
+show ?thesis
-by (unfold cntP_def cntV_def cntCS_def,
+by (unfold s_waiting_def, fold wq_def, auto)
-auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+next
-qed
+case True
+have "th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)"
+using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
+hence "th' \<in> set rest" by auto
+with set_wq' have "th' \<in> set wq'" by metis
+with nil_wq' show ?thesis by simp
+qed
+} thus ?thesis using assms
+by (unfold readys_def, auto)
+qed
+lemma readys_kept2:
+assumes "th' \<in> readys s"
+shows "th' \<in> readys (e#s)"
+proof -
+{ fix cs'
+assume wait: "waiting (e#s) th' cs'"
+have n_wait: "\<not> waiting s th' cs'"
+using assms[unfolded readys_def] by auto
+have False
+proof(cases "cs' = cs")
+case False
+with n_wait wait
+show ?thesis
+by (unfold s_waiting_def, fold wq_def, auto)
+next
+case True
+have "th' \<in> set [] \<and> th' \<noteq> hd []"
+using wait[unfolded True s_waiting_def, folded wq_def,
+unfolded wq_es_cs nil_wq'] .
+thus ?thesis by simp
+qed
+} with assms show ?thesis
+by (unfold readys_def, auto)
+qed
+lemma readys_simp [simp]:
+shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+using readys_kept1[OF assms] readys_kept2[OF assms]
+by metis
+lemma cnp_cnv_cncs_kept:
+assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof -
+{
+assume eq_th': "th' = th"
+have ?thesis
+apply (unfold eq_th' pvD_th_es cntCS_es_th)
+by (insert assms[unfolded eq_th'], unfold is_v, simp)
+} moreover {
+assume h: "th' \<noteq> th"
+have ?thesis using assms
+apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp)
+by (fold is_v, unfold pvD_def, simp)
+} ultimately show ?thesis by metis
+qed
+end
+context valid_trace_v
+begin
+lemma cnp_cnv_cncs_kept:
+assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof(cases "rest = []")
+case True
+then interpret vt: valid_trace_v_e by (unfold_locales, simp)
+show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast
+next
+case False
+then interpret vt: valid_trace_v_n by (unfold_locales, simp)
+show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast
+qed
+end
+context valid_trace_create
+begin
+lemma th_not_live_s [simp]: "th \<notin> threads s"
+proof -
+from pip_e[unfolded is_create]
+show ?thesis by (cases, simp)
+qed
+lemma th_not_ready_s [simp]: "th \<notin> readys s"
+using th_not_live_s by (unfold readys_def, simp)
+lemma th_live_es [simp]: "th \<in> threads (e#s)"
+by (unfold is_create, simp)
+lemma not_waiting_th_s [simp]: "\<not> waiting s th cs'"
+proof
+assume "waiting s th cs'"
+from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+have "th \<in> set (wq s cs')" by auto
+from wq_threads[OF this] have "th \<in> threads s" .
+with th_not_live_s show False by simp
+qed
+lemma not_holding_th_s [simp]: "\<not> holding s th cs'"
+proof
+assume "holding s th cs'"
+from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
+have "th \<in> set (wq s cs')" by auto
+from wq_threads[OF this] have "th \<in> threads s" .
+with th_not_live_s show False by simp
+qed
+lemma not_waiting_th_es [simp]: "\<not> waiting (e#s) th cs'"
+proof
+assume "waiting (e # s) th cs'"
+from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+have "th \<in> set (wq s cs')" by auto
+from wq_threads[OF this] have "th \<in> threads s" .
+with th_not_live_s show False by simp
+qed
+lemma not_holding_th_es [simp]: "\<not> holding (e#s) th cs'"
+proof
+assume "holding (e # s) th cs'"
+from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
+have "th \<in> set (wq s cs')" by auto
+from wq_threads[OF this] have "th \<in> threads s" .
+with th_not_live_s show False by simp
+qed
+lemma ready_th_es [simp]: "th \<in> readys (e#s)"
+by (simp add:readys_def)
+lemma holdents_th_s: "holdents s th = {}"
+by (unfold holdents_def, auto)
+lemma holdents_th_es: "holdents (e#s) th = {}"
+by (unfold holdents_def, auto)
+lemma cntCS_th_s [simp]: "cntCS s th = 0"
+by (unfold cntCS_def, simp add:holdents_th_s)
+lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0"
+by (unfold cntCS_def, simp add:holdents_th_es)
+lemma pvD_th_s [simp]: "pvD s th = 0"
+by (unfold pvD_def, simp)
+lemma pvD_th_es [simp]: "pvD (e#s) th = 0"
+by (unfold pvD_def, simp)
+lemma holdents_kept:
+assumes "th' \<noteq> th"
+shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+{ fix cs'
+assume h: "cs' \<in> ?L"
+hence "cs' \<in> ?R"
+by (unfold holdents_def s_holding_def, fold wq_def,
+unfold wq_neq_simp, auto)
+} moreover {
+fix cs'
+assume h: "cs' \<in> ?R"
+hence "cs' \<in> ?L"
+by (unfold holdents_def s_holding_def, fold wq_def,
+unfold wq_neq_simp, auto)
+} ultimately show ?thesis by auto
+qed
+lemma cntCS_kept [simp]:
+assumes "th' \<noteq> th"
+shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
+using holdents_kept[OF assms]
+by (unfold cntCS_def, simp)
+lemma readys_kept1:
+assumes "th' \<noteq> th"
+and "th' \<in> readys (e#s)"
+shows "th' \<in> readys s"
+proof -
+{ fix cs'
+assume wait: "waiting s th' cs'"
+have n_wait: "\<not> waiting (e#s) th' cs'"
+using assms by (auto simp:readys_def)
+from wait[unfolded s_waiting_def, folded wq_def]
+n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+have False by auto
+} thus ?thesis using assms
+by (unfold readys_def, auto)
+qed
+lemma readys_kept2:
+assumes "th' \<noteq> th"
+and "th' \<in> readys s"
+shows "th' \<in> readys (e#s)"
+proof -
+{ fix cs'
+assume wait: "waiting (e#s) th' cs'"
+have n_wait: "\<not> waiting s th' cs'"
+using assms(2) by (auto simp:readys_def)
+from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+n_wait[unfolded s_waiting_def, folded wq_def]
+have False by auto
+} with assms show ?thesis
+by (unfold readys_def, auto)
+qed
+lemma readys_simp [simp]:
+assumes "th' \<noteq> th"
+shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+using readys_kept1[OF assms] readys_kept2[OF assms]
+by metis
+lemma pvD_kept [simp]:
+assumes "th' \<noteq> th"
+shows "pvD (e#s) th' = pvD s th'"
+using assms
+by (unfold pvD_def, simp)
+lemma cnp_cnv_cncs_kept:
+assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof -
+{
+assume eq_th': "th' = th"
+have ?thesis using assms
+by (unfold eq_th', simp, unfold is_create, simp)
+} moreover {
+assume h: "th' \<noteq> th"
+hence ?thesis using assms
+by (simp, simp add:is_create)
+} ultimately show ?thesis by metis
+qed
+end
+context valid_trace_exit
+begin
+lemma th_live_s [simp]: "th \<in> threads s"
+proof -
+from pip_e[unfolded is_exit]
+show ?thesis
+by (cases, unfold runing_def readys_def, simp)
+qed
+lemma th_ready_s [simp]: "th \<in> readys s"
+proof -
+from pip_e[unfolded is_exit]
+show ?thesis
+by (cases, unfold runing_def, simp)
+qed
+lemma th_not_live_es [simp]: "th \<notin> threads (e#s)"
+by (unfold is_exit, simp)
+lemma not_holding_th_s [simp]: "\<not> holding s th cs'"
+proof -
+from pip_e[unfolded is_exit]
+show ?thesis
+by (cases, unfold holdents_def, auto)
+qed
+lemma cntCS_th_s [simp]: "cntCS s th = 0"
+proof -
+from pip_e[unfolded is_exit]
+show ?thesis
+by (cases, unfold cntCS_def, simp)
+qed
+lemma not_holding_th_es [simp]: "\<not> holding (e#s) th cs'"
+proof
+assume "holding (e # s) th cs'"
+from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
+have "holding s th cs'"
+by (unfold s_holding_def, fold wq_def, auto)
+with not_holding_th_s
+show False by simp
+qed
+lemma ready_th_es [simp]: "th \<notin> readys (e#s)"
+by (simp add:readys_def)
+lemma holdents_th_s: "holdents s th = {}"
+by (unfold holdents_def, auto)
+lemma holdents_th_es: "holdents (e#s) th = {}"
+by (unfold holdents_def, auto)
+lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0"
+by (unfold cntCS_def, simp add:holdents_th_es)
+lemma pvD_th_s [simp]: "pvD s th = 0"
+by (unfold pvD_def, simp)
+lemma pvD_th_es [simp]: "pvD (e#s) th = 0"
+by (unfold pvD_def, simp)
+lemma holdents_kept:
+assumes "th' \<noteq> th"
+shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+{ fix cs'
+assume h: "cs' \<in> ?L"
+hence "cs' \<in> ?R"
+by (unfold holdents_def s_holding_def, fold wq_def,
+unfold wq_neq_simp, auto)
+} moreover {
+fix cs'
+assume h: "cs' \<in> ?R"
+hence "cs' \<in> ?L"
+by (unfold holdents_def s_holding_def, fold wq_def,
+unfold wq_neq_simp, auto)
+} ultimately show ?thesis by auto
+qed
+lemma cntCS_kept [simp]:
+assumes "th' \<noteq> th"
+shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
+using holdents_kept[OF assms]
+by (unfold cntCS_def, simp)
+lemma readys_kept1:
+assumes "th' \<noteq> th"
+and "th' \<in> readys (e#s)"
+shows "th' \<in> readys s"
+proof -
+{ fix cs'
+assume wait: "waiting s th' cs'"
+have n_wait: "\<not> waiting (e#s) th' cs'"
+using assms by (auto simp:readys_def)
+from wait[unfolded s_waiting_def, folded wq_def]
+n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+have False by auto
+} thus ?thesis using assms
+by (unfold readys_def, auto)
+qed
+lemma readys_kept2:
+assumes "th' \<noteq> th"
+and "th' \<in> readys s"
+shows "th' \<in> readys (e#s)"
+proof -
+{ fix cs'
+assume wait: "waiting (e#s) th' cs'"
+have n_wait: "\<not> waiting s th' cs'"
+using assms(2) by (auto simp:readys_def)
+from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+n_wait[unfolded s_waiting_def, folded wq_def]
+have False by auto
+} with assms show ?thesis
+by (unfold readys_def, auto)
+qed
+lemma readys_simp [simp]:
+assumes "th' \<noteq> th"
+shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+using readys_kept1[OF assms] readys_kept2[OF assms]
+by metis
+lemma pvD_kept [simp]:
+assumes "th' \<noteq> th"
+shows "pvD (e#s) th' = pvD s th'"
+using assms
+by (unfold pvD_def, simp)
+lemma cnp_cnv_cncs_kept:
+assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+proof -
+{
+assume eq_th': "th' = th"
+have ?thesis using assms
+by (unfold eq_th', simp, unfold is_exit, simp)
+} moreover {
+assume h: "th' \<noteq> th"
+hence ?thesis using assms
+by (simp, simp add:is_exit)
+} ultimately show ?thesis by metis
+qed
+end
+context valid_trace_set
+begin
+lemma th_live_s [simp]: "th \<in> threads s"
+proof -
+from pip_e[unfolded is_set]
+show ?thesis
+by (cases, unfold runing_def readys_def, simp)
+qed
+lemma th_ready_s [simp]: "th \<in> readys s"
+proof -
+from pip_e[unfolded is_set]
+show ?thesis
+by (cases, unfold runing_def, simp)
+qed
+lemma th_not_live_es [simp]: "th \<in> threads (e#s)"
+by (unfold is_set, simp)
+lemma holdents_kept:
+shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
+proof -
+{ fix cs'
+assume h: "cs' \<in> ?L"
+hence "cs' \<in> ?R"
+by (unfold holdents_def s_holding_def, fold wq_def,
+unfold wq_neq_simp, auto)
+} moreover {
+fix cs'
+assume h: "cs' \<in> ?R"
+hence "cs' \<in> ?L"
+by (unfold holdents_def s_holding_def, fold wq_def,
+unfold wq_neq_simp, auto)
+} ultimately show ?thesis by auto
+qed
+lemma cntCS_kept [simp]:
+shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
+using holdents_kept
+by (unfold cntCS_def, simp)
+lemma threads_kept[simp]:
+"threads (e#s) = threads s"
+by (unfold is_set, simp)
+lemma readys_kept1:
+assumes "th' \<in> readys (e#s)"
+shows "th' \<in> readys s"
+proof -
+{ fix cs'
+assume wait: "waiting s th' cs'"
+have n_wait: "\<not> waiting (e#s) th' cs'"
+using assms by (auto simp:readys_def)
+from wait[unfolded s_waiting_def, folded wq_def]
+n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+have False by auto
+} moreover have "th' \<in> threads s"
+using assms[unfolded readys_def] by auto
+ultimately show ?thesis
+by (unfold readys_def, auto)
+qed
+lemma readys_kept2:
+assumes "th' \<in> readys s"
+shows "th' \<in> readys (e#s)"
+proof -
+{ fix cs'
+assume wait: "waiting (e#s) th' cs'"
+have n_wait: "\<not> waiting s th' cs'"
+using assms by (auto simp:readys_def)
+from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
+n_wait[unfolded s_waiting_def, folded wq_def]
+have False by auto
+} with assms show ?thesis
+by (unfold readys_def, auto)
+qed
+lemma readys_simp [simp]:
+shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
+using readys_kept1 readys_kept2
+by metis
+lemma pvD_kept [simp]:
+shows "pvD (e#s) th' = pvD s th'"
+by (unfold pvD_def, simp)
+lemma cnp_cnv_cncs_kept:
+assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+shows "cntP (e#s) th' = cntV (e#s) th' +  cntCS (e#s) th' + pvD (e#s) th'"
+using assms
+by (unfold is_set, simp, fold is_set, simp)
+end
+context valid_trace
+begin
+lemma cnp_cnv_cncs: "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
+proof(induct rule:ind)
+case Nil
+thus ?case
+by (unfold cntP_def cntV_def pvD_def cntCS_def holdents_def
+s_holding_def, simp)
+next
+case (Cons s e)
+interpret vt_e: valid_trace_e s e using Cons by simp
+show ?case
+proof(cases e)
+case (Create th prio)
+interpret vt_create: valid_trace_create s e th prio
+using Create by (unfold_locales, simp)
+show ?thesis using Cons by (simp add: vt_create.cnp_cnv_cncs_kept)
+next
+case (Exit th)
+interpret vt_exit: valid_trace_exit s e th
+using Exit by (unfold_locales, simp)
+show ?thesis using Cons by (simp add: vt_exit.cnp_cnv_cncs_kept)
+next
+case (P th cs)
+interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
+show ?thesis using Cons by (simp add: vt_p.cnp_cnv_cncs_kept)
+next
+case (V th cs)
+interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
+show ?thesis using Cons by (simp add: vt_v.cnp_cnv_cncs_kept)
+next
+case (Set th prio)
+interpret vt_set: valid_trace_set s e th prio
+using Set by (unfold_locales, simp)
+show ?thesis using Cons by (simp add: vt_set.cnp_cnv_cncs_kept)
+qed
+qed
+lemma not_thread_holdents:
+assumes not_in: "th \<notin> threads s"
+shows "holdents s th = {}"
+proof -
+{ fix cs
+assume "cs \<in> holdents s th"
+hence "holding s th cs" by (auto simp:holdents_def)
+from this[unfolded s_holding_def, folded wq_def]
+have "th \<in> set (wq s cs)" by auto
+with wq_threads have "th \<in> threads s" by auto
+with assms
+have False by simp
+} thus ?thesis by auto
 qed
 lemma not_thread_cncs:
 assumes not_in: "th \<notin> threads s"
 shows "cntCS s th = 0"
-proof -
+using not_thread_holdents[OF assms]
-from vt not_in show ?thesis
+by (simp add:cntCS_def)
-proof(induct arbitrary:th)
-case (vt_cons s e th)
+lemma cnp_cnv_eq:
-interpret vt_s: valid_trace s using vt_cons(1)
+assumes "th \<notin> threads s"
-by (unfold_locales, simp)
+shows "cntP s th = cntV s th"
-assume vt: "vt s"
+using assms cnp_cnv_cncs not_thread_cncs pvD_def
-and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
+by (auto)
-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
+unfolding runing_def by auto
-apply(simp)
+from this[unfolded cp_alt_def]
-done
+have eq_max:
-hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
+"Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th1)}) =
-Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
+Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th2)})"
-(is "Max (?f ` ?A) = Max (?f ` ?B)")
+(is "Max ?L = Max ?R") .
-unfolding cp_eq_cpreced
+have "Max ?L \<in> ?L"
-unfolding cpreced_def .
+proof(rule Max_in)
-obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
+show "finite ?L" by (simp add: finite_subtree_threads)
-proof -
+next
-have h1: "finite (?f ` ?A)"
+show "?L \<noteq> {}" using subtree_def by fastforce
-proof -
+qed
-have "finite ?A"
+then obtain th1' where
-proof -
+h_1: "Th th1' \<in> subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L"
-have "finite (dependants (wq s) th1)"
+by auto
-proof-
+have "Max ?R \<in> ?R"
-have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
+proof(rule Max_in)
-proof -
+show "finite ?R" by (simp add: finite_subtree_threads)
-let ?F = "\<lambda> (x, y). the_th x"
+next
-have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+show "?R \<noteq> {}" using subtree_def by fastforce
-apply (auto simp:image_def)
+qed
-by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
+then obtain th2' where
-moreover have "finite \<dots>"
+h_2: "Th th2' \<in> subtree (RAG s) (Th th2)" "the_preced s th2' = Max ?R"
-proof -
+by auto
-from finite_RAG have "finite (RAG s)" .
+have "th1' = th2'"
-hence "finite ((RAG (wq s))\<^sup>+)"
+proof(rule preced_unique)
-apply (unfold finite_trancl)
+from h_1(1)
-by (auto simp: s_RAG_def cs_RAG_def wq_def)
+show "th1' \<in> threads s"
-thus ?thesis by auto
+proof(cases rule:subtreeE)
-qed
+case 1
-ultimately show ?thesis by (auto intro:finite_subset)
+hence "th1' = th1" by simp
-qed
+with runing_1 show ?thesis by (auto simp:runing_def readys_def)
-thus ?thesis by (simp add:cs_dependants_def)
+next
-qed
+case 2
-thus ?thesis by simp
+from this(2)
-qed
+have "(Th th1', Th th1) \<in> (RAG s)^+" by (auto simp:ancestors_def)
-thus ?thesis by auto
+from tranclD[OF this]
-qed
+have "(Th th1') \<in> Domain (RAG s)" by auto
-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 .
+qed
+next
+from h_2(1)
+show "th2' \<in> threads s"
+proof(cases rule:subtreeE)
+case 1
+hence "th2' = th2" by simp
+with runing_2 show ?thesis by (auto simp:runing_def readys_def)
 next
-assume "th1' = th1"
+case 2
-with runing_1 show ?thesis
+from this(2)
+have "(Th th2', Th th2) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+from tranclD[OF this]
+have "(Th th2') \<in> Domain (RAG s)" by auto
+from dm_RAG_threads[OF this] show ?thesis .
+qed
+next
+have "the_preced s th1' = the_preced s th2'"
+using eq_max h_1(2) h_2(2) by metis
+thus "preced th1' s = preced th2' s" by (simp add:the_preced_def)
+qed
+from h_1(1)[unfolded this]
+have star1: "(Th th2', Th th1) \<in> (RAG s)^*" by (auto simp:subtree_def)
+from h_2(1)[unfolded this]
+have star2: "(Th th2', Th th2) \<in> (RAG s)^*" by (auto simp:subtree_def)
+from star_rpath[OF star1] obtain xs1
+where rp1: "rpath (RAG s) (Th th2') xs1 (Th th1)"
+by auto
+from star_rpath[OF star2] obtain xs2
+where rp2: "rpath (RAG s) (Th th2') xs2 (Th th2)"
+by auto
+from rp1 rp2
+show ?thesis
+proof(cases)
+case (less_1 xs')
+moreover have "xs' = []"
+proof(rule ccontr)
+assume otherwise: "xs' \<noteq> []"
+from rpath_plus[OF less_1(3) this]
+have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+" .
+from tranclD[OF this]
+obtain cs where "waiting s th1 cs"
+by (unfold s_RAG_def, fold waiting_eq, auto)
+with runing_1 show False
 by (unfold runing_def readys_def, auto)
 qed
-next
+ultimately have "xs2 = xs1" by simp
-from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
+from rpath_dest_eq[OF rp1 rp2[unfolded this]]
-thus "th2' \<in> threads s"
+show ?thesis by simp
-proof
+next
-assume "th2' \<in> dependants (wq s) th2"
+case (less_2 xs')
-hence "(Th th2') \<in> Domain ((RAG s)^+)"
+moreover have "xs' = []"
-apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+proof(rule ccontr)
-by (auto simp:Domain_def)
+assume otherwise: "xs' \<noteq> []"
-hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+from rpath_plus[OF less_2(3) this]
-from dm_RAG_threads[OF this] show ?thesis .
+have "(Th th2, Th th1) \<in> (RAG s)\<^sup>+" .
-next
+from tranclD[OF this]
-assume "th2' = th2"
+obtain cs where "waiting s th2 cs"
-with runing_2 show ?thesis
+by (unfold s_RAG_def, fold waiting_eq, auto)
+with runing_2 show False
 by (unfold runing_def readys_def, auto)
 qed
-qed
+ultimately have "xs2 = xs1" by simp
-from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
+from rpath_dest_eq[OF rp1 rp2[unfolded this]]
-thus ?thesis
+show ?thesis by simp
-proof
+qed
-assume eq_th': "th1' = th1"
+qed
-from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
-thus ?thesis
+lemma card_runing: "card (runing s) \<le> 1"
-proof
+proof(cases "runing s = {}")
-assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+case True
-next
+thus ?thesis by auto
-assume "th2' \<in> dependants (wq s) th2"
+next
-with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
+case False
-hence "(Th th1, Th th2) \<in> (RAG s)^+"
+then obtain th where [simp]: "th \<in> runing s" by auto
-by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+from runing_unique[OF this]
-hence "Th th1 \<in> Domain ((RAG s)^+)"
+have "runing s = {th}" by auto
-apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+thus ?thesis by auto
-by (auto simp:Domain_def)
+qed
-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)"
 case (thread_set thread)
 with assms show ?thesis by (auto intro!:that)
 qed
 qed
+lemma eq_pv_children:
-context valid_trace
+assumes eq_pv: "cntP s th = cntV s th"
-begin
+shows "children (RAG s) (Th th) = {}"
+proof -
-lemma cnp_cnv_eq:
+from cnp_cnv_cncs and eq_pv
-assumes "th \<notin> threads s"
+have "cntCS s th = 0"
-shows "cntP s th = cntV s th"
+by (auto split:if_splits)
-using assms
+from this[unfolded cntCS_def holdents_alt_def]
-using cnp_cnv_cncs not_thread_cncs by auto
+have card_0: "card (the_cs ` children (RAG s) (Th th)) = 0" .
+have "finite (the_cs ` children (RAG s) (Th th))"
+by (simp add: fsbtRAGs.finite_children)
+from card_0[unfolded card_0_eq[OF this]]
+show ?thesis by auto
+qed
+lemma eq_pv_holdents:
+assumes eq_pv: "cntP s th = cntV s th"
+shows "holdents s th = {}"
+by (unfold holdents_alt_def eq_pv_children[OF assms], simp)
+lemma eq_pv_subtree:
+assumes eq_pv: "cntP s th = cntV s th"
+shows "subtree (RAG s) (Th th) = {Th th}"
+using eq_pv_children[OF assms]
+by (unfold subtree_children, simp)
 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)
-lemma inj_the_preced:
-"inj_on (the_preced s) (threads s)"
-by (metis inj_onI preced_unique the_preced_def)
-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_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]
+with tRAG_star_RAG
 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
 { 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)
+from tRAG_subtree_RAG 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'
 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)
+hence "(Th th', x1) \<in> (RAG s)"
+by (cases, auto)
 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
 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)
+lemma dependants_alt_def1:
+"dependants (s::state) th = {th'. (Th th', Th th) \<in> (RAG s)^+}"
+using dependants_alt_def tRAG_trancl_eq by auto
+context valid_trace
+begin
+lemma count_eq_RAG_plus:
+assumes "cntP s th = cntV s th"
+shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+proof(rule ccontr)
+assume otherwise: "{th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} \<noteq> {}"
+then obtain th' where "(Th th', Th th) \<in> (RAG s)^+" by auto
+from tranclD2[OF this]
+obtain z where "z \<in> children (RAG s) (Th th)"
+by (auto simp:children_def)
+with eq_pv_children[OF assms]
+show False by simp
+qed
+lemma eq_pv_dependants:
+assumes eq_pv: "cntP s th = cntV s th"
+shows "dependants s th = {}"
+proof -
+from count_eq_RAG_plus[OF assms, folded dependants_alt_def1]
+show ?thesis .
+qed
+end
+lemma eq_dependants: "dependants (wq s) = dependants s"
+by (simp add: s_dependants_abv wq_def)
 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
+using assms eq_pv_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 inj_the_preced:
+"inj_on (the_preced s) (threads s)"
+by (metis inj_onI preced_unique the_preced_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 holding_eq, unfolded cs_holding_def]
+have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)"
+by (unfold s_holding_def, fold wq_def, auto)
+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)
+{ 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]
+show ?thesis using vt_s'.unique_RAG by 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 tRAG_subtree_eq:
 "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th'  \<in> (subtree (RAG s) (Th th))}"
 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 -
 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
+with dm_RAG_threads and rg_RAG_threads 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"
+definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
-proof
+where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
-show "single_valued (RAG s)"
-apply (intro_locales)
-by (unfold single_valued_def,
+lemma detached_test:
-auto intro:unique_RAG)
+shows "detached s th = (Th th \<notin> Field (RAG s))"
+apply(simp add: detached_def Field_def)
-show "acyclic (RAG s)"
+apply(simp add: s_RAG_def)
-by (rule acyclic_RAG)
+apply(simp add: s_holding_abv s_waiting_abv)
-qed
+apply(simp add: Domain_iff Range_iff)
+apply(simp add: wq_def)
-sublocale valid_trace < rtree_s: rtree "tRAG s"
+apply(auto)
-proof(unfold_locales)
+done
-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)
-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
+lemma detached_intro:
+assumes eq_pv: "cntP s th = cntV s th"
+shows "detached s th"
+proof -
+from eq_pv cnp_cnv_cncs
+have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:pvD_def)
+thus ?thesis
+proof
+assume "th \<notin> threads s"
+with rg_RAG_threads 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 eq_pv_children[OF assms]
+have "children (RAG s) (Th th) = {}" .
+thus ?thesis
+by (unfold children_def, auto)
+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 -
+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:waiting_eq s_RAG_def)
+with cncs_z  show ?thesis using cnp_cnv_cncs by (simp add:pvD_def)
+next
+case False
+with cncs_z and cnp_cnv_cncs show ?thesis by (simp add:pvD_def)
+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
+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 = {}")
 qed
 thus ?thesis by (subst (1) h(1), unfold h(2), simp)
 qed
 qed
-end
-(* keep *)
 lemma next_th_holding:
-assumes vt: "vt s"
+assumes nxt: "next_th s th cs th'"
-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 -
 using vt assms next_th_holding next_th_waiting
 by (unfold s_RAG_def, simp)
 end
--- {* A useless definition *}
+lemma next_th_unique:
-definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+assumes nt1: "next_th s th cs th1"
-where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+and nt2: "next_th s th cs th2"
+shows "th1 = th2"
+using assms by (unfold next_th_def, auto)
-find_theorems "waiting" holding
 context valid_trace
 begin
-find_theorems "waiting" holding
+thm th_chain_to_ready
+find_theorems subtree Th RAG
+lemma threads_alt_def:
+"(threads s) = (\<Union> th \<in> readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)})"
+(is "?L = ?R")
+proof -
+{ fix th1
+assume "th1 \<in> ?L"
+from th_chain_to_ready[OF this]
+have "th1 \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th th1, Th th') \<in> (RAG s)\<^sup>+)" .
+hence "th1 \<in> ?R" by (auto simp:subtree_def)
+} moreover
+{ fix th'
+assume "th' \<in> ?R"
+then obtain th where h: "th \<in> readys s" " Th th' \<in> subtree (RAG s) (Th th)"
+by auto
+from this(2)
+have "th' \<in> ?L"
+proof(cases rule:subtreeE)
+case 1
+with h(1) show ?thesis by (auto simp:readys_def)
+next
+case 2
+from tranclD[OF this(2)[unfolded ancestors_def, simplified]]
+have "Th th' \<in> Domain (RAG s)" by auto
+from dm_RAG_threads[OF this]
+show ?thesis .
+qed
+} ultimately show ?thesis by auto
+qed
+lemma finite_readys [simp]: "finite (readys s)"
+using finite_threads readys_threads rev_finite_subset by blast
+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)" (is "?L = ?R")
+proof(cases "readys s = {}")
+case False
+have "?R = Max (the_preced s ` threads s)" by (unfold max_cp_eq, simp)
+also have "... =
+Max (the_preced s ` (\<Union>th\<in>readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)}))"
+by (unfold threads_alt_def, simp)
+also have "... =
+Max ((\<Union>th\<in>readys s. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}))"
+by (unfold image_UN, simp)
+also have "... =
+Max (Max ` (\<lambda>th. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}) ` readys s)"
+proof(rule Max_UNION)
+show "\<forall>M\<in>(\<lambda>x. the_preced s `
+{th'. Th th' \<in> subtree (RAG s) (Th x)}) ` readys s. finite M"
+using finite_subtree_threads by auto
+qed (auto simp:False subtree_def)
+also have "... =
+Max ((Max \<circ> (\<lambda>th. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})) ` readys s)"
+by (unfold image_comp, simp)
+also have "... = ?L" (is "Max (?f ` ?A) = Max (?g ` ?A)")
+proof -
+have "(?f ` ?A) = (?g ` ?A)"
+proof(rule f_image_eq)
+fix th1
+assume "th1 \<in> ?A"
+thus "?f th1 = ?g th1"
+by (unfold cp_alt_def, simp)
+qed
+thus ?thesis by simp
+qed
+finally show ?thesis by simp
+qed (auto simp:threads_alt_def)
 end
 end

changeset 92	4763aa246dbd
parent 90	ed938e2246b9
child 93	524bd3caa6b6