--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/CpsG_1.thy Wed Jan 27 19:28:42 2016 +0800
@@ -0,0 +1,4403 @@
+theory CpsG
+imports PIPDefs
+begin
+
+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
+
+(* I am going to use this file as a start point to retrofiting
+ PIPBasics.thy, which is originally called CpsG.ghy *)
+
+locale valid_trace =
+ fixes s
+ assumes vt : "vt s"
+
+locale valid_trace_e = valid_trace +
+ fixes e
+ assumes vt_e: "vt (e#s)"
+begin
+
+lemma pip_e: "PIP s e"
+ using vt_e by (cases, simp)
+
+end
+
+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
+
+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 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 [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"
+ and "th \<in> set (wq_fun (schs s) cs)"
+ shows "th = hd (wq_fun (schs s) cs)"
+ using assms
+ 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_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 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
+begin
+
+lemma actor_inv:
+ assumes "PIP s e"
+ and "\<not> isCreate e"
+ shows "actor e \<in> runing s"
+ using assms
+ by (induct, auto)
+
+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)
+
+lemma wq_distinct: "distinct (wq s cs)"
+proof(induct rule:ind)
+ case (Cons s e)
+ interpret vt_e: valid_trace_e s e using Cons by simp
+ show ?case
+ proof(cases e)
+ 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.wq_distinct_kept)
+ qed
+qed (unfold wq_def Let_def, simp)
+
+end
+
+context valid_trace_e
+begin
+
+text {*
+ The following lemma shows that only the @{text "P"}
+ operation can add new thread into waiting queues.
+ Such kind of lemmas are very obvious, but need to be checked formally.
+ This is a kind of confirmation that our modelling is correct.
+*}
+
+lemma 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: *}
+ case (V th cs1)
+ have False
+ proof(cases "cs1 = cs")
+ case True
+ show ?thesis
+ proof(cases "(wq s cs1)")
+ case (Cons w_hd w_tl)
+ have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
+ proof -
+ have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
+ using Cons V by (auto simp:wq_def Let_def True split:if_splits)
+ moreover have "set ... \<subseteq> set (wq s cs)"
+ proof(rule someI2)
+ show "distinct w_tl \<and> set w_tl = set w_tl"
+ by (metis distinct.simps(2) local.Cons wq_distinct)
+ qed (insert Cons True, auto)
+ ultimately show ?thesis by simp
+ qed
+ with assms show ?thesis by auto
+ 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
+
+
+
+context valid_trace
+begin
+
+
+text {* (* ddd *)
+ The nature of the work is like this: since it starts from a very simple and basic
+ model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
+ For instance, the fact
+ that one thread can not be blocked by two critical resources at the same time
+ is obvious, because only running threads can make new requests, if one is waiting for
+ a critical resource and get blocked, it can not make another resource request and get
+ blocked the second time (because it is not running).
+
+ To derive this fact, one needs to prove by contraction and
+ reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
+ named @{text "p_split"}, which is about status changing along the time axis. It says if
+ a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
+ but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"}
+ in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history
+ of events leading to it), such that @{text "Q"} switched
+ from being @{text "False"} to @{text "True"} and kept being @{text "True"}
+ till the last moment of @{text "s"}.
+
+ Suppose a thread @{text "th"} is blocked
+ on @{text "cs1"} and @{text "cs2"} in some state @{text "s"},
+ since no thread is blocked at the very beginning, by applying
+ @{text "p_split"} to these two blocking facts, there exist
+ two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that
+ @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"}
+ and kept on blocked on them respectively ever since.
+
+ Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
+ However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
+ in blocked state at moment @{text "t2"} and could not
+ make any request and get blocked the second time: Contradiction.
+*}
+
+lemma waiting_unique_pre: (* 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" = "\<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> ?Q cs1 (moment t1 s)"
+ and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
+ from p_split [of "?Q cs2", OF q2 nq2]
+ obtain t2 where lt2: "t2 < length s"
+ and np2: "\<not> ?Q cs2 (moment t2 s)"
+ and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
+ { fix s cs
+ 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"
+ 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 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 "e = V thread cs2 \<or> e = P thread cs2"
+ 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 by auto
+ next
+ case False
+ have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
+ thus ?thesis by auto
+ qed
+ moreover have "e = V thread cs1 \<or> e = P thread cs1"
+ proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+ case True
+ have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+ using True and np1 by auto
+ from vt_e.wq_out_inv[folded eq_12, OF True this g2]
+ have "e = V thread cs1" .
+ thus ?thesis by auto
+ next
+ case False
+ have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
+ thus ?thesis by auto
+ qed
+ ultimately have ?thesis using neq12 by auto
+ } ultimately show ?thesis using nat_neq_iff by blast
+ qed
+qed
+
+text {*
+ This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
+*}
+
+lemma waiting_unique:
+ assumes "waiting s th cs1"
+ and "waiting s th cs2"
+ shows "cs1 = cs2"
+ using waiting_unique_pre assms
+ unfolding wq_def s_waiting_def
+ by auto
+
+end
+
+(* not used *)
+text {*
+ Every thread can only be blocked on one critical resource,
+ symmetrically, every critical resource can only be held by one thread.
+ This fact is much more easier according to our definition.
+*}
+lemma held_unique:
+ assumes "holding (s::event list) th1 cs"
+ and "holding s th2 cs"
+ shows "th1 = th2"
+ by (insert assms, unfold s_holding_def, auto)
+
+
+lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+ apply (induct s, auto)
+ by (case_tac a, auto split:if_splits)
+
+lemma last_set_unique:
+ "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
+ \<Longrightarrow> th1 = th2"
+ apply (induct s, auto)
+ by (case_tac a, auto split:if_splits dest:last_set_lt)
+
+lemma preced_unique :
+ assumes pcd_eq: "preced th1 s = preced th2 s"
+ and th_in1: "th1 \<in> threads s"
+ and th_in2: " th2 \<in> threads s"
+ shows "th1 = th2"
+proof -
+ from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
+ from last_set_unique [OF this th_in1 th_in2]
+ show ?thesis .
+qed
+
+lemma preced_linorder:
+ assumes neq_12: "th1 \<noteq> th2"
+ and th_in1: "th1 \<in> threads s"
+ and th_in2: " th2 \<in> threads s"
+ shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
+proof -
+ from preced_unique [OF _ th_in1 th_in2] and neq_12
+ have "preced th1 s \<noteq> preced th2 s" by auto
+ thus ?thesis by auto
+qed
+
+(* An aux lemma used later *)
+lemma unique_minus:
+ 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 RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+
+context valid_trace_v
+begin
+
+
+lemma distinct_rest: "distinct rest"
+ by (simp add: distinct_tl rest_def wq_distinct)
+
+definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+
+lemma runing_th_s:
+ shows "th \<in> runing s"
+proof -
+ from pip_e[unfolded is_v]
+ show ?thesis by (cases, simp)
+qed
+
+lemma holding_cs_eq_th:
+ assumes "holding s t cs"
+ shows "t = th"
+proof -
+ from pip_e[unfolded is_v]
+ show ?thesis
+ proof(cases)
+ case (thread_V)
+ from held_unique[OF this(2) assms]
+ show ?thesis by simp
+ qed
+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 distinct_wq': "distinct wq'"
+ by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def)
+
+lemma th'_in_inv:
+ assumes "th' \<in> set wq'"
+ shows "th' \<in> set rest"
+ using assms
+ by (metis (mono_tags, lifting) distinct.simps(2)
+ rest_def some_eq_ex wq'_def wq_distinct wq_s_cs)
+
+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
+ have "wq (e#s) c = wq s c" using False
+ by (unfold is_v, simp)
+ hence "waiting s t c" using assms
+ by (simp add: cs_waiting_def waiting_eq)
+ hence "t \<notin> readys s" by (unfold readys_def, auto)
+ hence "t \<notin> runing s" using runing_ready by auto
+ with runing_th_s[folded otherwise] show ?thesis by auto
+ qed
+qed
+
+lemma waiting_esI1:
+ assumes "waiting s t c"
+ and "c \<noteq> cs"
+ shows "waiting (e#s) t c"
+proof -
+ have "wq (e#s) c = wq s c"
+ using assms(2) is_v by auto
+ with assms(1) show ?thesis
+ using cs_waiting_def waiting_eq by auto
+qed
+
+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_n
+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 (* ccc *)
+
+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
+ case True
+ from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
+ show ?thesis .
+ qed
+ qed
+ thus ?thesis using waiting(1,2)
+ by (unfold s_RAG_def, fold waiting_eq, auto)
+ next
+ case (holding th' cs')
+ from h this(1,2)
+ have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
+ hence "holding (e#s) th' cs'"
+ proof
+ assume "cs' \<noteq> cs"
+ from holding_esI2[OF this holding(3)]
+ show ?thesis .
+ next
+ assume "th' \<noteq> th"
+ from holding_esI1[OF holding(3) this]
+ show ?thesis .
+ qed
+ 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
+ assume "th' \<noteq> th"
+ from holding_esI1[OF holding(3) this]
+ show ?thesis using holding(1,2)
+ by (unfold s_RAG_def, fold holding_eq, auto)
+ qed
+ qed
+ qed
+qed
+
+end
+
+
+
+context valid_trace
+begin
+
+lemma finite_threads:
+ shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+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 RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+ by (unfold s_RAG_def, auto)
+
+lemma wq_threads:
+ assumes h: "th \<in> set (wq s cs)"
+ shows "th \<in> threads s"
+
+
+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 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
+
+
+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_eq_the_preced:
+ shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
+ using max_cp_eq using the_preced_def by presburger
+
+end
+
+lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
+ by (unfold 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 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'}" (is "?L = ?R")
+proof -
+ interpret vt_v: valid_trace_v s "V th cs"
+ using assms step_back_vt by (unfold_locales, auto)
+ show ?thesis using vt_v.RAG_es .
+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'}"
+ apply (insert vt, unfold s_RAG_def)
+ apply (auto split:if_splits list.splits simp:Let_def)
+ apply (auto elim: step_v_waiting_mono step_v_hold_inv
+ step_v_release step_v_wait_inv
+ step_v_get_hold step_v_release_inv)
+ apply (erule_tac step_v_not_wait, auto)
+ done
+
+text {*
+ The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
+ with the happening of @{text "P"}-events:
+*}
+lemma step_RAG_p:
+ "vt (P th cs#s) \<Longrightarrow>
+ RAG (P th cs # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+ else RAG s \<union> {(Th th, Cs cs)})"
+ apply(simp only: s_RAG_def wq_def)
+ apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
+ apply(case_tac "csa = cs", auto)
+ apply(fold wq_def)
+ apply(drule_tac step_back_step)
+ apply(ind_cases " step s (P (hd (wq s cs)) cs)")
+ apply(simp add:s_RAG_def wq_def cs_holding_def)
+ apply(auto)
+ done
+
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+ by (unfold s_RAG_def, auto)
+
+context valid_trace
+begin
+
+text {*
+ The following lemma shows that @{text "RAG"} is acyclic.
+ The overall structure is by induction on the formation of @{text "vt s"}
+ and then case analysis on event @{text "e"}, where the non-trivial cases
+ for those for @{text "V"} and @{text "P"} events.
+*}
+lemma acyclic_RAG:
+ shows "acyclic (RAG s)"
+using vt
+proof(induct)
+ case (vt_cons s e)
+ interpret vt_s: valid_trace s using vt_cons(1)
+ by (unfold_locales, simp)
+ assume ih: "acyclic (RAG s)"
+ and stp: "step s e"
+ and vt: "vt s"
+ show ?case
+ proof(cases e)
+ case (Create th prio)
+ with ih
+ show ?thesis by (simp add:RAG_create_unchanged)
+ next
+ case (Exit th)
+ with ih show ?thesis by (simp add:RAG_exit_unchanged)
+ next
+ case (V th cs)
+ from V vt stp have vtt: "vt (V th cs#s)" by auto
+ from step_RAG_v [OF this]
+ have eq_de:
+ "RAG (e # s) =
+ RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+ {(Cs cs, Th th') |th'. next_th s th cs th'}"
+ (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+ from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+ from step_back_step [OF vtt]
+ have "step s (V th cs)" .
+ thus ?thesis
+ proof(cases)
+ assume "holding s th cs"
+ hence th_in: "th \<in> set (wq s cs)" and
+ eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
+ then obtain rest where
+ eq_wq: "wq s cs = th#rest"
+ by (cases "wq s cs", auto)
+ show ?thesis
+ proof(cases "rest = []")
+ case False
+ let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+ from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
+ by (unfold next_th_def, auto)
+ let ?E = "(?A - ?B - ?C)"
+ have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
+ proof
+ assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
+ hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+ from tranclD [OF this]
+ obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
+ hence th_d: "(Th ?th', x) \<in> ?A" by simp
+ from RAG_target_th [OF this]
+ obtain cs' where eq_x: "x = Cs cs'" by auto
+ with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
+ hence wt_th': "waiting s ?th' cs'"
+ unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
+ hence "cs' = cs"
+ proof(rule vt_s.waiting_unique)
+ from eq_wq vt_s.wq_distinct[of cs]
+ show "waiting s ?th' cs"
+ apply (unfold s_waiting_def wq_def, auto)
+ proof -
+ assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ and eq_wq: "wq_fun (schs s) cs = th # rest"
+ have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ with False show "x \<noteq> []" by auto
+ qed
+ hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
+ set (SOME q. distinct q \<and> set q = set rest)" by auto
+ moreover have "\<dots> = set rest"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+ qed
+ moreover note hd_in
+ ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
+ next
+ assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+ have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ with False show "x \<noteq> []" by auto
+ qed
+ hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
+ set (SOME q. distinct q \<and> set q = set rest)" by auto
+ moreover have "\<dots> = set rest"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+ qed
+ moreover note hd_in
+ ultimately show False by auto
+ qed
+ qed
+ with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
+ with False
+ show "False" by (auto simp: next_th_def eq_wq)
+ qed
+ with acyclic_insert[symmetric] and ac
+ and eq_de eq_D show ?thesis by auto
+ next
+ case True
+ with eq_wq
+ have eq_D: "?D = {}"
+ by (unfold next_th_def, auto)
+ with eq_de ac
+ show ?thesis by auto
+ qed
+ qed
+ next
+ case (P th cs)
+ from P vt stp have vtt: "vt (P th cs#s)" by auto
+ from step_RAG_p [OF this] P
+ have "RAG (e # s) =
+ (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
+ RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+ by simp
+ moreover have "acyclic ?R"
+ proof(cases "wq s cs = []")
+ case True
+ hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
+ have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
+ proof
+ assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
+ hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+ from tranclD2 [OF this]
+ obtain x where "(x, Cs cs) \<in> RAG s" by auto
+ with True show False by (auto simp:s_RAG_def cs_waiting_def)
+ qed
+ with acyclic_insert ih eq_r show ?thesis by auto
+ next
+ case False
+ hence eq_r: "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
+ have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
+ proof
+ assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
+ hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+ moreover from step_back_step [OF vtt] have "step s (P th cs)" .
+ ultimately show False
+ proof -
+ show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+ by (ind_cases "step s (P th cs)", simp)
+ qed
+ qed
+ with acyclic_insert ih eq_r show ?thesis by auto
+ qed
+ ultimately show ?thesis by simp
+ next
+ case (Set thread prio)
+ with ih
+ thm RAG_set_unchanged
+ show ?thesis by (simp add:RAG_set_unchanged)
+ qed
+ next
+ case vt_nil
+ show "acyclic (RAG ([]::state))"
+ by (auto simp: s_RAG_def cs_waiting_def
+ cs_holding_def wq_def acyclic_def)
+qed
+
+
+lemma finite_RAG:
+ shows "finite (RAG s)"
+proof -
+ from vt show ?thesis
+ proof(induct)
+ case (vt_cons s e)
+ interpret vt_s: valid_trace s using vt_cons(1)
+ by (unfold_locales, simp)
+ assume ih: "finite (RAG s)"
+ and stp: "step s e"
+ and vt: "vt s"
+ show ?case
+ proof(cases e)
+ case (Create th prio)
+ with ih
+ show ?thesis by (simp add:RAG_create_unchanged)
+ next
+ case (Exit th)
+ with ih show ?thesis by (simp add:RAG_exit_unchanged)
+ next
+ case (V th cs)
+ from V vt stp have vtt: "vt (V th cs#s)" by auto
+ from step_RAG_v [OF this]
+ have eq_de: "RAG (e # s) =
+ RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+ {(Cs cs, Th th') |th'. next_th s th cs th'}
+"
+ (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+ moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
+ moreover have "finite ?D"
+ proof -
+ have "?D = {} \<or> (\<exists> a. ?D = {a})"
+ by (unfold next_th_def, auto)
+ thus ?thesis
+ proof
+ assume h: "?D = {}"
+ show ?thesis by (unfold h, simp)
+ next
+ assume "\<exists> a. ?D = {a}"
+ thus ?thesis
+ by (metis finite.simps)
+ qed
+ qed
+ ultimately show ?thesis by simp
+ next
+ case (P th cs)
+ from P vt stp have vtt: "vt (P th cs#s)" by auto
+ from step_RAG_p [OF this] P
+ have "RAG (e # s) =
+ (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
+ RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+ by simp
+ moreover have "finite ?R"
+ proof(cases "wq s cs = []")
+ case True
+ hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
+ with True and ih show ?thesis by auto
+ next
+ case False
+ hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
+ with False and ih show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ next
+ case (Set thread prio)
+ with ih
+ show ?thesis by (simp add:RAG_set_unchanged)
+ qed
+ next
+ case vt_nil
+ show "finite (RAG ([]::state))"
+ by (auto simp: s_RAG_def cs_waiting_def
+ cs_holding_def wq_def acyclic_def)
+ qed
+qed
+
+text {* Several useful lemmas *}
+
+lemma wf_dep_converse:
+ shows "wf ((RAG s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+ from finite_RAG
+ show "finite (RAG s)" .
+next
+ from acyclic_RAG
+ show "acyclic (RAG s)" .
+qed
+
+end
+
+lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
+ by (induct l, auto)
+
+lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
+ by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+
+context valid_trace
+begin
+
+lemma wq_threads:
+ assumes h: "th \<in> set (wq s cs)"
+ shows "th \<in> threads s"
+proof -
+ from vt and h show ?thesis
+ proof(induct arbitrary: th cs)
+ case (vt_cons s e)
+ interpret vt_s: valid_trace s
+ using vt_cons(1) by (unfold_locales, auto)
+ assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+ and stp: "step s e"
+ and vt: "vt s"
+ and h: "th \<in> set (wq (e # s) cs)"
+ show ?case
+ proof(cases e)
+ case (Create th' prio)
+ with ih h show ?thesis
+ by (auto simp:wq_def Let_def)
+ next
+ case (Exit th')
+ with stp ih h show ?thesis
+ apply (auto simp:wq_def Let_def)
+ apply (ind_cases "step s (Exit th')")
+ apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
+ s_RAG_def s_holding_def cs_holding_def)
+ done
+ next
+ case (V th' cs')
+ show ?thesis
+ proof(cases "cs' = cs")
+ case False
+ with h
+ show ?thesis
+ apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+ by (drule_tac ih, simp)
+ next
+ case True
+ from h
+ show ?thesis
+ proof(unfold V wq_def)
+ assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+ show "th \<in> threads (V th' cs' # s)"
+ proof(cases "cs = cs'")
+ case False
+ hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
+ with th_in have " th \<in> set (wq s cs)"
+ by (fold wq_def, simp)
+ from ih [OF this] show ?thesis by simp
+ next
+ case True
+ show ?thesis
+ proof(cases "wq_fun (schs s) cs'")
+ case Nil
+ with h V show ?thesis
+ apply (auto simp:wq_def Let_def split:if_splits)
+ by (fold wq_def, drule_tac ih, simp)
+ next
+ case (Cons a rest)
+ assume eq_wq: "wq_fun (schs s) cs' = a # rest"
+ with h V show ?thesis
+ apply (auto simp:Let_def wq_def split:if_splits)
+ proof -
+ assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+ have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+ by auto
+ qed
+ with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
+ from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+ next
+ assume th_in: "th \<in> set (wq_fun (schs s) cs)"
+ from ih[OF this[folded wq_def]]
+ show "th \<in> threads s" .
+ qed
+ qed
+ qed
+ qed
+ qed
+ next
+ case (P th' cs')
+ from h stp
+ show ?thesis
+ apply (unfold P wq_def)
+ apply (auto simp:Let_def split:if_splits, fold wq_def)
+ apply (auto intro:ih)
+ apply(ind_cases "step s (P th' cs')")
+ by (unfold runing_def readys_def, auto)
+ next
+ case (Set thread prio)
+ with ih h show ?thesis
+ by (auto simp:wq_def Let_def)
+ qed
+ next
+ case vt_nil
+ thus ?case by (auto simp:wq_def)
+ qed
+qed
+
+lemma range_in: "\<lbrakk>(Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+ apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
+ by (auto intro:wq_threads)
+
+lemma readys_v_eq:
+ assumes neq_th: "th \<noteq> thread"
+ and eq_wq: "wq s cs = thread#rest"
+ and not_in: "th \<notin> set rest"
+ shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
+proof -
+ from assms show ?thesis
+ apply (auto simp:readys_def)
+ apply(simp add:s_waiting_def[folded wq_def])
+ apply (erule_tac x = csa in allE)
+ apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
+ apply (case_tac "csa = cs", simp)
+ apply (erule_tac x = cs in allE)
+ apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
+ apply(auto simp add: wq_def)
+ apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
+ proof -
+ assume th_nin: "th \<notin> set rest"
+ and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+ and eq_wq: "wq_fun (schs s) cs = thread # rest"
+ have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+ qed
+ with th_nin th_in show False by auto
+ qed
+qed
+
+text {* \noindent
+ The following lemmas shows that: starting from any node in @{text "RAG"},
+ by chasing out-going edges, it is always possible to reach a node representing a ready
+ thread. In this lemma, it is the @{text "th'"}.
+*}
+
+lemma chain_building:
+ shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
+proof -
+ from wf_dep_converse
+ have h: "wf ((RAG s)\<inverse>)" .
+ show ?thesis
+ proof(induct rule:wf_induct [OF h])
+ fix x
+ assume ih [rule_format]:
+ "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow>
+ y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
+ show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
+ proof
+ assume x_d: "x \<in> Domain (RAG s)"
+ show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
+ proof(cases x)
+ case (Th th)
+ from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
+ with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
+ from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
+ hence "Cs cs \<in> Domain (RAG s)" by auto
+ from ih [OF x_in_r this] obtain th'
+ where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
+ have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
+ with th'_ready show ?thesis by auto
+ next
+ case (Cs cs)
+ from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
+ show ?thesis
+ proof(cases "th' \<in> readys s")
+ case True
+ from True and th'_d show ?thesis by auto
+ next
+ case False
+ from th'_d and range_in have "th' \<in> threads s" by auto
+ with False have "Th th' \<in> Domain (RAG s)"
+ by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
+ from ih [OF th'_d this]
+ obtain th'' where
+ th''_r: "th'' \<in> readys s" and
+ th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
+ from th'_d and th''_in
+ have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
+ with th''_r show ?thesis by auto
+ qed
+ qed
+ qed
+ qed
+qed
+
+text {* \noindent
+ The following is just an instance of @{text "chain_building"}.
+*}
+lemma th_chain_to_ready:
+ assumes th_in: "th \<in> threads s"
+ shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
+proof(cases "th \<in> readys s")
+ case True
+ thus ?thesis by auto
+next
+ case False
+ from False and th_in have "Th th \<in> Domain (RAG s)"
+ by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
+ from chain_building [rule_format, OF this]
+ show ?thesis by auto
+qed
+
+end
+
+
+
+lemma 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)
+
+lemma count_rec2 [simp]:
+ assumes "\<not>Q e"
+ shows "count Q (e#es) = (count Q es)"
+ using assms
+ by (unfold count_def, auto)
+
+lemma count_rec3 [simp]:
+ shows "count Q [] = 0"
+ by (unfold count_def, auto)
+
+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 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)
+
+context valid_trace
+begin
+
+text {* (* ddd *) \noindent
+ The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"}
+ of one particular thread. t
+*}
+
+lemma cnp_cnv_cncs:
+ shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
+ then cntCS s th else cntCS s th + 1)"
+proof -
+ from vt show ?thesis
+ proof(induct arbitrary:th)
+ case (vt_cons s e)
+ interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
+ assume vt: "vt s"
+ and ih: "\<And>th. cntP s th = cntV s th +
+ (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
+ and stp: "step s e"
+ from stp show ?case
+ proof(cases)
+ case (thread_create thread prio)
+ assume eq_e: "e = Create thread prio"
+ and not_in: "thread \<notin> threads s"
+ show ?thesis
+ proof -
+ { fix cs
+ assume "thread \<in> set (wq s cs)"
+ from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
+ with not_in have "False" by simp
+ } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
+ by (auto simp:readys_def threads.simps s_waiting_def
+ wq_def cs_waiting_def Let_def)
+ from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+ from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+ have eq_cncs: "cntCS (e#s) th = cntCS s th"
+ unfolding cntCS_def holdents_test
+ by (simp add:RAG_create_unchanged eq_e)
+ { assume "th \<noteq> thread"
+ with eq_readys eq_e
+ have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
+ (th \<in> readys (s) \<or> th \<notin> threads (s))"
+ by (simp add:threads.simps)
+ with eq_cnp eq_cnv eq_cncs ih not_in
+ have ?thesis by simp
+ } moreover {
+ assume eq_th: "th = thread"
+ with not_in ih have " cntP s th = cntV s th + cntCS s th" by simp
+ moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
+ moreover note eq_cnp eq_cnv eq_cncs
+ ultimately have ?thesis by auto
+ } ultimately show ?thesis by blast
+ qed
+ next
+ case (thread_exit thread)
+ assume eq_e: "e = Exit thread"
+ and is_runing: "thread \<in> runing s"
+ and no_hold: "holdents s thread = {}"
+ from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+ from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+ have eq_cncs: "cntCS (e#s) th = cntCS s th"
+ unfolding cntCS_def holdents_test
+ by (simp add:RAG_exit_unchanged eq_e)
+ { assume "th \<noteq> thread"
+ with eq_e
+ have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
+ (th \<in> readys (s) \<or> th \<notin> threads (s))"
+ apply (simp add:threads.simps readys_def)
+ apply (subst s_waiting_def)
+ apply (simp add:Let_def)
+ apply (subst s_waiting_def, simp)
+ done
+ with eq_cnp eq_cnv eq_cncs ih
+ have ?thesis by simp
+ } moreover {
+ assume eq_th: "th = thread"
+ with ih is_runing have " cntP s th = cntV s th + cntCS s th"
+ by (simp add:runing_def)
+ moreover from eq_th eq_e have "th \<notin> threads (e#s)"
+ by simp
+ moreover note eq_cnp eq_cnv eq_cncs
+ ultimately have ?thesis by auto
+ } ultimately show ?thesis by blast
+ next
+ case (thread_P thread cs)
+ assume eq_e: "e = P thread cs"
+ and is_runing: "thread \<in> runing s"
+ and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
+ from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto
+ then interpret vt_p: valid_trace "(P thread cs#s)"
+ by (unfold_locales, simp)
+ show ?thesis
+ proof -
+ { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
+ assume neq_th: "th \<noteq> thread"
+ with eq_e
+ have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
+ apply (simp add:readys_def s_waiting_def wq_def Let_def)
+ apply (rule_tac hh)
+ apply (intro iffI allI, clarify)
+ apply (erule_tac x = csa in allE, auto)
+ apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
+ apply (erule_tac x = cs in allE, auto)
+ by (case_tac "(wq_fun (schs s) cs)", auto)
+ moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+ apply (simp add:cntCS_def holdents_test)
+ by (unfold step_RAG_p [OF vtp], auto)
+ moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
+ by (simp add:cntP_def count_def)
+ moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
+ by (simp add:cntV_def count_def)
+ moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
+ moreover note ih [of th]
+ ultimately have ?thesis by simp
+ } moreover {
+ assume eq_th: "th = thread"
+ have ?thesis
+ proof -
+ from eq_e eq_th have eq_cnp: "cntP (e # s) th = 1 + (cntP s th)"
+ by (simp add:cntP_def count_def)
+ from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
+ by (simp add:cntV_def count_def)
+ show ?thesis
+ proof (cases "wq s cs = []")
+ case True
+ with is_runing
+ have "th \<in> readys (e#s)"
+ apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
+ apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
+ by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
+ moreover have "cntCS (e # s) th = 1 + cntCS s th"
+ proof -
+ have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
+ Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
+ proof -
+ have "?L = insert cs ?R" by auto
+ moreover have "card \<dots> = Suc (card (?R - {cs}))"
+ proof(rule card_insert)
+ from vt_s.finite_holding [of thread]
+ show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
+ by (unfold holdents_test, simp)
+ qed
+ moreover have "?R - {cs} = ?R"
+ proof -
+ have "cs \<notin> ?R"
+ proof
+ assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
+ with no_dep show False by auto
+ qed
+ thus ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ thus ?thesis
+ apply (unfold eq_e eq_th cntCS_def)
+ apply (simp add: holdents_test)
+ by (unfold step_RAG_p [OF vtp], auto simp:True)
+ qed
+ moreover from is_runing have "th \<in> readys s"
+ by (simp add:runing_def eq_th)
+ moreover note eq_cnp eq_cnv ih [of th]
+ ultimately show ?thesis by auto
+ next
+ case False
+ have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
+ by (unfold eq_th eq_e wq_def, auto simp:Let_def)
+ have "th \<notin> readys (e#s)"
+ proof
+ assume "th \<in> readys (e#s)"
+ hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
+ from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
+ hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
+ by (simp add:s_waiting_def wq_def)
+ moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
+ ultimately have "th = hd (wq (e#s) cs)" by blast
+ with eq_wq have "th = hd (wq s cs @ [th])" by simp
+ hence "th = hd (wq s cs)" using False by auto
+ with False eq_wq vt_p.wq_distinct [of cs]
+ show False by (fold eq_e, auto)
+ qed
+ moreover from is_runing have "th \<in> threads (e#s)"
+ by (unfold eq_e, auto simp:runing_def readys_def eq_th)
+ moreover have "cntCS (e # s) th = cntCS s th"
+ apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
+ by (auto simp:False)
+ moreover note eq_cnp eq_cnv ih[of th]
+ moreover from is_runing have "th \<in> readys s"
+ by (simp add:runing_def eq_th)
+ ultimately show ?thesis by auto
+ qed
+ qed
+ } ultimately show ?thesis by blast
+ qed
+ next
+ case (thread_V thread cs)
+ from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
+ then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
+ assume eq_e: "e = V thread cs"
+ and is_runing: "thread \<in> runing s"
+ and hold: "holding s thread cs"
+ from hold obtain rest
+ where eq_wq: "wq s cs = thread # rest"
+ by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+ have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+ have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from vt_v.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest"
+ by (metis distinct.simps(2) vt_s.wq_distinct)
+ next
+ show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+ by auto
+ qed
+ show ?thesis
+ proof -
+ { assume eq_th: "th = thread"
+ from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
+ by (unfold eq_e, simp add:cntP_def count_def)
+ moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
+ by (unfold eq_e, simp add:cntV_def count_def)
+ moreover from cntCS_v_dec [OF vtv]
+ have "cntCS (e # s) thread + 1 = cntCS s thread"
+ by (simp add:eq_e)
+ moreover from is_runing have rd_before: "thread \<in> readys s"
+ by (unfold runing_def, simp)
+ moreover have "thread \<in> readys (e # s)"
+ proof -
+ from is_runing
+ have "thread \<in> threads (e#s)"
+ by (unfold eq_e, auto simp:runing_def readys_def)
+ moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
+ proof
+ fix cs1
+ { assume eq_cs: "cs1 = cs"
+ have "\<not> waiting (e # s) thread cs1"
+ proof -
+ from eq_wq
+ have "thread \<notin> set (wq (e#s) cs1)"
+ apply(unfold eq_e wq_def eq_cs s_holding_def)
+ apply (auto simp:Let_def)
+ proof -
+ assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
+ with eq_set have "thread \<in> set rest" by simp
+ with vt_v.wq_distinct[of cs]
+ and eq_wq show False
+ by (metis distinct.simps(2) vt_s.wq_distinct)
+ qed
+ thus ?thesis by (simp add:wq_def s_waiting_def)
+ qed
+ } moreover {
+ assume neq_cs: "cs1 \<noteq> cs"
+ have "\<not> waiting (e # s) thread cs1"
+ proof -
+ from wq_v_neq [OF neq_cs[symmetric]]
+ have "wq (V thread cs # s) cs1 = wq s cs1" .
+ moreover have "\<not> waiting s thread cs1"
+ proof -
+ from runing_ready and is_runing
+ have "thread \<in> readys s" by auto
+ thus ?thesis by (simp add:readys_def)
+ qed
+ ultimately show ?thesis
+ by (auto simp:wq_def s_waiting_def eq_e)
+ qed
+ } ultimately show "\<not> waiting (e # s) thread cs1" by blast
+ qed
+ ultimately show ?thesis by (simp add:readys_def)
+ qed
+ moreover note eq_th ih
+ ultimately have ?thesis by auto
+ } moreover {
+ assume neq_th: "th \<noteq> thread"
+ from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th"
+ by (simp add:cntP_def count_def)
+ from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th"
+ by (simp add:cntV_def count_def)
+ have ?thesis
+ proof(cases "th \<in> set rest")
+ case False
+ have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+ apply (insert step_back_vt[OF vtv])
+ by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
+ moreover have "cntCS (e#s) th = cntCS s th"
+ apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
+ proof -
+ have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+ {cs. (Cs cs, Th th) \<in> RAG s}"
+ proof -
+ from False eq_wq
+ have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
+ apply (unfold next_th_def, auto)
+ proof -
+ assume ne: "rest \<noteq> []"
+ and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ and eq_wq: "wq s cs = thread # rest"
+ from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
+ set (SOME q. distinct q \<and> set q = set rest)
+ " by simp
+ moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from vt_s.wq_distinct[ of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ with ne show "x \<noteq> []" by auto
+ qed
+ ultimately show
+ "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
+ by auto
+ qed
+ thus ?thesis by auto
+ qed
+ thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+ card {cs. (Cs cs, Th th) \<in> RAG s}" by simp
+ qed
+ moreover note ih eq_cnp eq_cnv eq_threads
+ ultimately show ?thesis by auto
+ next
+ case True
+ assume th_in: "th \<in> set rest"
+ show ?thesis
+ proof(cases "next_th s thread cs th")
+ case False
+ with eq_wq and th_in have
+ neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
+ by (auto simp:next_th_def)
+ have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+ proof -
+ from eq_wq and th_in
+ have "\<not> th \<in> readys s"
+ apply (auto simp:readys_def s_waiting_def)
+ apply (rule_tac x = cs in exI, auto)
+ by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
+ moreover
+ from eq_wq and th_in and neq_hd
+ have "\<not> (th \<in> readys (e # s))"
+ apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
+ by (rule_tac x = cs in exI, auto simp:eq_set)
+ ultimately show ?thesis by auto
+ qed
+ moreover have "cntCS (e#s) th = cntCS s th"
+ proof -
+ from eq_wq and th_in and neq_hd
+ have "(holdents (e # s) th) = (holdents s th)"
+ apply (unfold eq_e step_RAG_v[OF vtv],
+ auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
+ Let_def cs_holding_def)
+ by (insert vt_s.wq_distinct[of cs], auto simp:wq_def)
+ thus ?thesis by (simp add:cntCS_def)
+ qed
+ moreover note ih eq_cnp eq_cnv eq_threads
+ ultimately show ?thesis by auto
+ next
+ case True
+ let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
+ let ?t = "hd ?rest"
+ from True eq_wq th_in neq_th
+ have "th \<in> readys (e # s)"
+ apply (auto simp:eq_e readys_def s_waiting_def wq_def
+ Let_def next_th_def)
+ proof -
+ assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+ and t_in: "?t \<in> set rest"
+ show "?t \<in> threads s"
+ proof(rule vt_s.wq_threads)
+ from eq_wq and t_in
+ show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
+ qed
+ next
+ fix csa
+ assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+ and t_in: "?t \<in> set rest"
+ and neq_cs: "csa \<noteq> cs"
+ and t_in': "?t \<in> set (wq_fun (schs s) csa)"
+ show "?t = hd (wq_fun (schs s) csa)"
+ proof -
+ { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
+ from vt_s.wq_distinct[of cs] and
+ eq_wq[folded wq_def] and t_in eq_wq
+ have "?t \<noteq> thread" by auto
+ with eq_wq and t_in
+ have w1: "waiting s ?t cs"
+ by (auto simp:s_waiting_def wq_def)
+ from t_in' neq_hd'
+ have w2: "waiting s ?t csa"
+ by (auto simp:s_waiting_def wq_def)
+ from vt_s.waiting_unique[OF w1 w2]
+ and neq_cs have "False" by auto
+ } thus ?thesis by auto
+ qed
+ qed
+ moreover have "cntP s th = cntV s th + cntCS s th + 1"
+ proof -
+ have "th \<notin> readys s"
+ proof -
+ from True eq_wq neq_th th_in
+ show ?thesis
+ apply (unfold readys_def s_waiting_def, auto)
+ by (rule_tac x = cs in exI, auto simp add: wq_def)
+ qed
+ moreover have "th \<in> threads s"
+ proof -
+ from th_in eq_wq
+ have "th \<in> set (wq s cs)" by simp
+ from vt_s.wq_threads [OF this]
+ show ?thesis .
+ qed
+ ultimately show ?thesis using ih by auto
+ qed
+ moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
+ apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
+ proof -
+ show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
+ Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
+ (is "card ?A = Suc (card ?B)")
+ proof -
+ have "?A = insert cs ?B" by auto
+ hence "card ?A = card (insert cs ?B)" by simp
+ also have "\<dots> = Suc (card ?B)"
+ proof(rule card_insert_disjoint)
+ have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
+ with vt_s.finite_RAG
+ show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
+ next
+ show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
+ proof
+ assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
+ hence "(Cs cs, Th th) \<in> RAG s" by simp
+ with True neq_th eq_wq show False
+ by (auto simp:next_th_def s_RAG_def cs_holding_def)
+ qed
+ qed
+ finally show ?thesis .
+ qed
+ qed
+ moreover note eq_cnp eq_cnv
+ ultimately show ?thesis by simp
+ qed
+ qed
+ } ultimately show ?thesis by blast
+ qed
+ next
+ case (thread_set thread prio)
+ assume eq_e: "e = Set thread prio"
+ and is_runing: "thread \<in> runing s"
+ show ?thesis
+ proof -
+ from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+ from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
+ have eq_cncs: "cntCS (e#s) th = cntCS s th"
+ unfolding cntCS_def holdents_test
+ by (simp add:RAG_set_unchanged eq_e)
+ from eq_e have eq_readys: "readys (e#s) = readys s"
+ by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
+ auto simp:Let_def)
+ { assume "th \<noteq> thread"
+ with eq_readys eq_e
+ have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
+ (th \<in> readys (s) \<or> th \<notin> threads (s))"
+ by (simp add:threads.simps)
+ with eq_cnp eq_cnv eq_cncs ih is_runing
+ have ?thesis by simp
+ } moreover {
+ assume eq_th: "th = thread"
+ with is_runing ih have " cntP s th = cntV s th + cntCS s th"
+ by (unfold runing_def, auto)
+ moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
+ by (simp add:runing_def)
+ moreover note eq_cnp eq_cnv eq_cncs
+ ultimately have ?thesis by auto
+ } ultimately show ?thesis by blast
+ qed
+ qed
+ next
+ case vt_nil
+ show ?case
+ by (unfold cntP_def cntV_def cntCS_def,
+ auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+ qed
+qed
+
+lemma not_thread_cncs:
+ assumes not_in: "th \<notin> threads s"
+ shows "cntCS s th = 0"
+proof -
+ from vt not_in show ?thesis
+ proof(induct arbitrary:th)
+ case (vt_cons s e th)
+ interpret vt_s: valid_trace s using vt_cons(1)
+ by (unfold_locales, simp)
+ assume vt: "vt s"
+ and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
+ and stp: "step s e"
+ and not_in: "th \<notin> threads (e # s)"
+ from stp show ?case
+ proof(cases)
+ case (thread_create thread prio)
+ assume eq_e: "e = Create thread prio"
+ and not_in': "thread \<notin> threads s"
+ have "cntCS (e # s) th = cntCS s th"
+ apply (unfold eq_e cntCS_def holdents_test)
+ by (simp add:RAG_create_unchanged)
+ moreover have "th \<notin> threads s"
+ proof -
+ from not_in eq_e show ?thesis by simp
+ qed
+ moreover note ih ultimately show ?thesis by auto
+ next
+ case (thread_exit thread)
+ assume eq_e: "e = Exit thread"
+ and nh: "holdents s thread = {}"
+ have eq_cns: "cntCS (e # s) th = cntCS s th"
+ apply (unfold eq_e cntCS_def holdents_test)
+ by (simp add:RAG_exit_unchanged)
+ show ?thesis
+ proof(cases "th = thread")
+ case True
+ have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
+ with eq_cns show ?thesis by simp
+ next
+ case False
+ with not_in and eq_e
+ have "th \<notin> threads s" by simp
+ from ih[OF this] and eq_cns show ?thesis by simp
+ qed
+ next
+ case (thread_P thread cs)
+ assume eq_e: "e = P thread cs"
+ and is_runing: "thread \<in> runing s"
+ from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
+ have neq_th: "th \<noteq> thread"
+ proof -
+ from not_in eq_e have "th \<notin> threads s" by simp
+ moreover from is_runing have "thread \<in> threads s"
+ by (simp add:runing_def readys_def)
+ ultimately show ?thesis by auto
+ qed
+ hence "cntCS (e # s) th = cntCS s th "
+ apply (unfold cntCS_def holdents_test eq_e)
+ by (unfold step_RAG_p[OF vtp], auto)
+ moreover have "cntCS s th = 0"
+ proof(rule ih)
+ from not_in eq_e show "th \<notin> threads s" by simp
+ qed
+ ultimately show ?thesis by simp
+ next
+ case (thread_V thread cs)
+ assume eq_e: "e = V thread cs"
+ and is_runing: "thread \<in> runing s"
+ and hold: "holding s thread cs"
+ have neq_th: "th \<noteq> thread"
+ proof -
+ from not_in eq_e have "th \<notin> threads s" by simp
+ moreover from is_runing have "thread \<in> threads s"
+ by (simp add:runing_def readys_def)
+ ultimately show ?thesis by auto
+ qed
+ from assms thread_V vt stp ih
+ have vtv: "vt (V thread cs#s)" by auto
+ then interpret vt_v: valid_trace "(V thread cs#s)"
+ by (unfold_locales, simp)
+ from hold obtain rest
+ where eq_wq: "wq s cs = thread # rest"
+ by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+ from not_in eq_e eq_wq
+ have "\<not> next_th s thread cs th"
+ apply (auto simp:next_th_def)
+ proof -
+ assume ne: "rest \<noteq> []"
+ and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
+ have "?t \<in> set rest"
+ proof(rule someI2)
+ from vt_v.wq_distinct[of cs] and eq_wq
+ show "distinct rest \<and> set rest = set rest"
+ by (metis distinct.simps(2) vt_s.wq_distinct)
+ next
+ fix x assume "distinct x \<and> set x = set rest" with ne
+ show "hd x \<in> set rest" by (cases x, auto)
+ qed
+ with eq_wq have "?t \<in> set (wq s cs)" by simp
+ from vt_s.wq_threads[OF this] and ni
+ show False
+ using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)`
+ ni vt_s.wq_threads by blast
+ qed
+ moreover note neq_th eq_wq
+ ultimately have "cntCS (e # s) th = cntCS s th"
+ by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
+ moreover have "cntCS s th = 0"
+ proof(rule ih)
+ from not_in eq_e show "th \<notin> threads s" by simp
+ qed
+ ultimately show ?thesis by simp
+ next
+ case (thread_set thread prio)
+ print_facts
+ assume eq_e: "e = Set thread prio"
+ and is_runing: "thread \<in> runing s"
+ from not_in and eq_e have "th \<notin> threads s" by auto
+ from ih [OF this] and eq_e
+ show ?thesis
+ apply (unfold eq_e cntCS_def holdents_test)
+ by (simp add:RAG_set_unchanged)
+ qed
+ next
+ case vt_nil
+ show ?case
+ by (unfold cntCS_def,
+ auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+ qed
+qed
+
+end
+
+
+context valid_trace
+begin
+
+lemma dm_RAG_threads:
+ assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+ shows "th \<in> threads s"
+proof -
+ from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
+ moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
+ ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
+ hence "th \<in> set (wq s cs)"
+ by (unfold s_RAG_def, auto simp:cs_waiting_def)
+ from wq_threads [OF this] show ?thesis .
+qed
+
+end
+
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+thm cpreced_initial
+apply(simp add: Let_def cpreced_initial)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+
+context valid_trace
+begin
+
+lemma runing_unique:
+ assumes runing_1: "th1 \<in> runing s"
+ and runing_2: "th2 \<in> runing s"
+ shows "th1 = th2"
+proof -
+ from runing_1 and runing_2 have "cp s th1 = cp s th2"
+ unfolding runing_def
+ apply(simp)
+ done
+ hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
+ Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
+ (is "Max (?f ` ?A) = Max (?f ` ?B)")
+ unfolding cp_eq_cpreced
+ unfolding cpreced_def .
+ obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
+ proof -
+ have h1: "finite (?f ` ?A)"
+ proof -
+ have "finite ?A"
+ proof -
+ have "finite (dependants (wq s) th1)"
+ proof-
+ have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
+ proof -
+ let ?F = "\<lambda> (x, y). the_th x"
+ have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
+ moreover have "finite \<dots>"
+ proof -
+ from finite_RAG have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
+ apply (unfold finite_trancl)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
+ thus ?thesis by auto
+ qed
+ ultimately show ?thesis by (auto intro:finite_subset)
+ qed
+ thus ?thesis by (simp add:cs_dependants_def)
+ qed
+ thus ?thesis by simp
+ qed
+ thus ?thesis by auto
+ qed
+ moreover have h2: "(?f ` ?A) \<noteq> {}"
+ proof -
+ have "?A \<noteq> {}" by simp
+ thus ?thesis by simp
+ qed
+ from Max_in [OF h1 h2]
+ have "Max (?f ` ?A) \<in> (?f ` ?A)" .
+ thus ?thesis
+ thm cpreced_def
+ unfolding cpreced_def[symmetric]
+ unfolding cp_eq_cpreced[symmetric]
+ unfolding cpreced_def
+ using that[intro] by (auto)
+ qed
+ obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
+ proof -
+ have h1: "finite (?f ` ?B)"
+ proof -
+ have "finite ?B"
+ proof -
+ have "finite (dependants (wq s) th2)"
+ proof-
+ have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
+ proof -
+ let ?F = "\<lambda> (x, y). the_th x"
+ have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
+ moreover have "finite \<dots>"
+ proof -
+ from finite_RAG have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
+ apply (unfold finite_trancl)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
+ thus ?thesis by auto
+ qed
+ ultimately show ?thesis by (auto intro:finite_subset)
+ qed
+ thus ?thesis by (simp add:cs_dependants_def)
+ qed
+ thus ?thesis by simp
+ qed
+ thus ?thesis by auto
+ qed
+ moreover have h2: "(?f ` ?B) \<noteq> {}"
+ proof -
+ have "?B \<noteq> {}" by simp
+ thus ?thesis by simp
+ qed
+ from Max_in [OF h1 h2]
+ have "Max (?f ` ?B) \<in> (?f ` ?B)" .
+ thus ?thesis by (auto intro:that)
+ qed
+ from eq_f_th1 eq_f_th2 eq_max
+ have eq_preced: "preced th1' s = preced th2' s" by auto
+ hence eq_th12: "th1' = th2'"
+ proof (rule preced_unique)
+ from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
+ thus "th1' \<in> threads s"
+ proof
+ assume "th1' \<in> dependants (wq s) th1"
+ hence "(Th th1') \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+ by (auto simp:Domain_def)
+ hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ from dm_RAG_threads[OF this] show ?thesis .
+ next
+ assume "th1' = th1"
+ with runing_1 show ?thesis
+ by (unfold runing_def readys_def, auto)
+ qed
+ next
+ from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
+ thus "th2' \<in> threads s"
+ proof
+ assume "th2' \<in> dependants (wq s) th2"
+ hence "(Th th2') \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+ by (auto simp:Domain_def)
+ hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ from dm_RAG_threads[OF this] show ?thesis .
+ next
+ assume "th2' = th2"
+ with runing_2 show ?thesis
+ by (unfold runing_def readys_def, auto)
+ qed
+ qed
+ from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
+ thus ?thesis
+ proof
+ assume eq_th': "th1' = th1"
+ from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+ thus ?thesis
+ proof
+ assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+ next
+ assume "th2' \<in> dependants (wq s) th2"
+ with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
+ hence "(Th th1, Th th2) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ hence "Th th1 \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
+ by (auto simp:Domain_def)
+ hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
+ from RAG_target_th [OF this]
+ obtain cs' where "n = Cs cs'" by auto
+ with d have "(Th th1, Cs cs') \<in> RAG s" by simp
+ with runing_1 have "False"
+ apply (unfold runing_def readys_def s_RAG_def)
+ by (auto simp:waiting_eq)
+ 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:waiting_eq)
+ thus ?thesis by simp
+ next
+ assume "th2' \<in> dependants (wq s) th2"
+ with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
+ hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ show ?thesis
+ proof(rule dchain_unique[OF h1 _ h2, symmetric])
+ from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
+ from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
+ qed
+ qed
+ qed
+qed
+
+
+lemma "card (runing s) \<le> 1"
+apply(subgoal_tac "finite (runing s)")
+prefer 2
+apply (metis finite_nat_set_iff_bounded lessI runing_unique)
+apply(rule ccontr)
+apply(simp)
+apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
+apply(subst (asm) card_le_Suc_iff)
+apply(simp)
+apply(auto)[1]
+apply (metis insertCI runing_unique)
+apply(auto)
+done
+
+end
+
+
+lemma create_pre:
+ assumes stp: "step s e"
+ and not_in: "th \<notin> threads s"
+ and is_in: "th \<in> threads (e#s)"
+ obtains prio where "e = Create th prio"
+proof -
+ from assms
+ show ?thesis
+ proof(cases)
+ case (thread_create thread prio)
+ with is_in not_in have "e = Create th prio" by simp
+ from that[OF this] show ?thesis .
+ next
+ case (thread_exit thread)
+ with assms show ?thesis by (auto intro!:that)
+ next
+ case (thread_P thread)
+ with assms show ?thesis by (auto intro!:that)
+ next
+ case (thread_V thread)
+ with assms show ?thesis by (auto intro!:that)
+ next
+ case (thread_set thread)
+ with assms show ?thesis by (auto intro!:that)
+ qed
+qed
+
+context valid_trace
+begin
+
+lemma cnp_cnv_eq:
+ assumes "th \<notin> threads s"
+ shows "cntP s th = cntV s th"
+ using assms
+ using cnp_cnv_cncs not_thread_cncs by auto
+
+end
+
+
+lemma eq_RAG:
+ "RAG (wq s) = RAG s"
+by (unfold cs_RAG_def s_RAG_def, auto)
+
+context valid_trace
+begin
+
+lemma count_eq_dependants:
+ assumes eq_pv: "cntP s th = cntV s th"
+ shows "dependants (wq s) th = {}"
+proof -
+ from cnp_cnv_cncs and eq_pv
+ have "cntCS s th = 0"
+ by (auto split:if_splits)
+ moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
+ proof -
+ from finite_holding[of th] show ?thesis
+ by (simp add:holdents_test)
+ qed
+ ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
+ by (unfold cntCS_def holdents_test cs_dependants_def, auto)
+ show ?thesis
+ proof(unfold cs_dependants_def)
+ { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
+ then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
+ hence "False"
+ proof(cases)
+ assume "(Th th', Th th) \<in> RAG (wq s)"
+ thus "False" by (auto simp:cs_RAG_def)
+ next
+ fix c
+ assume "(c, Th th) \<in> RAG (wq s)"
+ with h and eq_RAG show "False"
+ by (cases c, auto simp:cs_RAG_def)
+ qed
+ } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
+ qed
+qed
+
+lemma dependants_threads:
+ shows "dependants (wq s) th \<subseteq> threads s"
+proof
+ { fix th th'
+ assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
+ have "Th th \<in> Domain (RAG s)"
+ proof -
+ from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
+ hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
+ with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
+ thus ?thesis using eq_RAG by simp
+ qed
+ from dm_RAG_threads[OF this]
+ have "th \<in> threads s" .
+ } note hh = this
+ fix th1
+ assume "th1 \<in> dependants (wq s) th"
+ hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
+ by (unfold cs_dependants_def, simp)
+ from hh [OF this] show "th1 \<in> threads s" .
+qed
+
+lemma finite_threads:
+ shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+lemma Max_f_mono:
+ assumes seq: "A \<subseteq> B"
+ and np: "A \<noteq> {}"
+ and fnt: "finite B"
+ shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+ from seq show "f ` A \<subseteq> f ` B" by auto
+next
+ from np show "f ` A \<noteq> {}" by auto
+next
+ from fnt and seq show "finite (f ` B)" by auto
+qed
+
+context valid_trace
+begin
+
+lemma cp_le:
+ assumes th_in: "th \<in> threads s"
+ shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
+ show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
+ \<le> Max ((\<lambda>th. preced th s) ` threads s)"
+ (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
+ proof(rule Max_f_mono)
+ show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
+ next
+ from finite_threads
+ show "finite (threads s)" .
+ next
+ from th_in
+ show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
+ apply (auto simp:Domain_def)
+ apply (rule_tac dm_RAG_threads)
+ apply (unfold trancl_domain [of "RAG s", symmetric])
+ by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
+ qed
+qed
+
+lemma le_cp:
+ shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+ show "Prc (priority th s) (last_set th s)
+ \<le> Max (insert (Prc (priority th s) (last_set th s))
+ ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
+ (is "?l \<le> Max (insert ?l ?A)")
+ proof(cases "?A = {}")
+ case False
+ have "finite ?A" (is "finite (?f ` ?B)")
+ proof -
+ have "finite ?B"
+ proof-
+ have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
+ proof -
+ let ?F = "\<lambda> (x, y). the_th x"
+ have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Th x, Th th)" in bexI, auto)
+ moreover have "finite \<dots>"
+ proof -
+ from finite_RAG have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
+ apply (unfold finite_trancl)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
+ thus ?thesis by auto
+ qed
+ ultimately show ?thesis by (auto intro:finite_subset)
+ qed
+ thus ?thesis by (simp add:cs_dependants_def)
+ qed
+ thus ?thesis by simp
+ qed
+ from Max_insert [OF this False, of ?l] show ?thesis by auto
+ next
+ case True
+ thus ?thesis by auto
+ qed
+qed
+
+lemma max_cp_eq:
+ shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
+ (is "?l = ?r")
+proof(cases "threads s = {}")
+ case True
+ thus ?thesis by auto
+next
+ case False
+ have "?l \<in> ((cp s) ` threads s)"
+ proof(rule Max_in)
+ from finite_threads
+ show "finite (cp s ` threads s)" by auto
+ next
+ from False show "cp s ` threads s \<noteq> {}" by auto
+ qed
+ then obtain th
+ where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
+ have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
+ moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
+ proof -
+ have "?r \<in> (?f ` ?A)"
+ proof(rule Max_in)
+ from finite_threads
+ show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
+ next
+ from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
+ qed
+ then obtain th' where
+ th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
+ from le_cp [of th'] eq_r
+ have "?r \<le> cp s th'" by auto
+ moreover have "\<dots> \<le> cp s th"
+ proof(fold eq_l)
+ show " cp s th' \<le> Max (cp s ` threads s)"
+ proof(rule Max_ge)
+ from th_in' show "cp s th' \<in> cp s ` threads s"
+ by auto
+ next
+ from finite_threads
+ show "finite (cp s ` threads s)" by auto
+ qed
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis using eq_l by auto
+qed
+
+lemma max_cp_readys_threads_pre:
+ assumes np: "threads s \<noteq> {}"
+ shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq)
+ show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
+ proof -
+ let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
+ let ?f = "(\<lambda>th. preced th s)"
+ have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
+ proof(rule Max_in)
+ from finite_threads show "finite (?f ` threads s)" by simp
+ next
+ from np show "?f ` threads s \<noteq> {}" by simp
+ qed
+ then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
+ by (auto simp:Image_def)
+ from th_chain_to_ready [OF tm_in]
+ have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
+ thus ?thesis
+ proof
+ assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
+ then obtain th' where th'_in: "th' \<in> readys s"
+ and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
+ have "cp s th' = ?f tm"
+ proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
+ from dependants_threads finite_threads
+ show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))"
+ by (auto intro:finite_subset)
+ next
+ fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+ from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
+ moreover have "p \<le> \<dots>"
+ proof(rule Max_ge)
+ from finite_threads
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ next
+ from p_in and th'_in and dependants_threads[of th']
+ show "p \<in> (\<lambda>th. preced th s) ` threads s"
+ by (auto simp:readys_def)
+ qed
+ ultimately show "p \<le> preced tm s" by auto
+ next
+ show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+ proof -
+ from tm_chain
+ have "tm \<in> dependants (wq s) th'"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
+ thus ?thesis by auto
+ qed
+ qed
+ with tm_max
+ have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+ show ?thesis
+ proof (fold h, rule Max_eqI)
+ fix q
+ assume "q \<in> cp s ` readys s"
+ then obtain th1 where th1_in: "th1 \<in> readys s"
+ and eq_q: "q = cp s th1" by auto
+ show "q \<le> cp s th'"
+ apply (unfold h eq_q)
+ apply (unfold cp_eq_cpreced cpreced_def)
+ apply (rule Max_mono)
+ proof -
+ from dependants_threads [of th1] th1_in
+ show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq>
+ (\<lambda>th. preced th s) ` threads s"
+ by (auto simp:readys_def)
+ next
+ show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
+ next
+ from finite_threads
+ show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ qed
+ next
+ from finite_threads
+ show "finite (cp s ` readys s)" by (auto simp:readys_def)
+ next
+ from th'_in
+ show "cp s th' \<in> cp s ` readys s" by simp
+ qed
+ next
+ assume tm_ready: "tm \<in> readys s"
+ show ?thesis
+ proof(fold tm_max)
+ have cp_eq_p: "cp s tm = preced tm s"
+ proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+ fix y
+ assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+ show "y \<le> preced tm s"
+ proof -
+ { fix y'
+ assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
+ have "y' \<le> preced tm s"
+ proof(unfold tm_max, rule Max_ge)
+ from hy' dependants_threads[of tm]
+ show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+ next
+ from finite_threads
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ qed
+ } with hy show ?thesis by auto
+ qed
+ next
+ from dependants_threads[of tm] finite_threads
+ show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
+ by (auto intro:finite_subset)
+ next
+ show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+ by simp
+ qed
+ moreover have "Max (cp s ` readys s) = cp s tm"
+ proof(rule Max_eqI)
+ from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
+ next
+ from finite_threads
+ show "finite (cp s ` readys s)" by (auto simp:readys_def)
+ next
+ fix y assume "y \<in> cp s ` readys s"
+ then obtain th1 where th1_readys: "th1 \<in> readys s"
+ and h: "y = cp s th1" by auto
+ show "y \<le> cp s tm"
+ apply(unfold cp_eq_p h)
+ apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
+ proof -
+ from finite_threads
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ next
+ show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
+ by simp
+ next
+ from dependants_threads[of th1] th1_readys
+ show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)
+ \<subseteq> (\<lambda>th. preced th s) ` threads s"
+ by (auto simp:readys_def)
+ qed
+ qed
+ ultimately show " Max (cp s ` readys s) = preced tm s" by simp
+ qed
+ qed
+ qed
+qed
+
+text {* (* ccc *) \noindent
+ Since the current precedence of the threads in ready queue will always be boosted,
+ there must be one inside it has the maximum precedence of the whole system.
+*}
+lemma max_cp_readys_threads:
+ shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(cases "threads s = {}")
+ case True
+ thus ?thesis
+ by (auto simp:readys_def)
+next
+ case False
+ show ?thesis by (rule max_cp_readys_threads_pre[OF False])
+qed
+
+end
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+ apply (unfold s_holding_def cs_holding_def wq_def, simp)
+ done
+
+lemma f_image_eq:
+ assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
+ shows "f ` A = g ` A"
+proof
+ show "f ` A \<subseteq> g ` A"
+ by(rule image_subsetI, auto intro:h)
+next
+ show "g ` A \<subseteq> f ` A"
+ by (rule image_subsetI, auto intro:h[symmetric])
+qed
+
+
+definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
+ where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
+
+lemma detached_test:
+ shows "detached s th = (Th th \<notin> Field (RAG s))"
+apply(simp add: detached_def Field_def)
+apply(simp add: s_RAG_def)
+apply(simp add: s_holding_abv s_waiting_abv)
+apply(simp add: Domain_iff Range_iff)
+apply(simp add: wq_def)
+apply(auto)
+done
+
+context valid_trace
+begin
+
+lemma detached_intro:
+ assumes eq_pv: "cntP s th = cntV s th"
+ shows "detached s th"
+proof -
+ from cnp_cnv_cncs
+ have eq_cnt: "cntP s th =
+ cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+ hence cncs_zero: "cntCS s th = 0"
+ by (auto simp:eq_pv split:if_splits)
+ with eq_cnt
+ have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
+ thus ?thesis
+ proof
+ assume "th \<notin> threads s"
+ with range_in dm_RAG_threads
+ show ?thesis
+ by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
+ next
+ assume "th \<in> readys s"
+ moreover have "Th th \<notin> Range (RAG s)"
+ proof -
+ from card_0_eq [OF finite_holding] and cncs_zero
+ have "holdents s th = {}"
+ by (simp add:cntCS_def)
+ thus ?thesis
+ apply(auto simp:holdents_test)
+ apply(case_tac a)
+ apply(auto simp:holdents_test s_RAG_def)
+ done
+ qed
+ ultimately show ?thesis
+ by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
+ qed
+qed
+
+lemma detached_elim:
+ assumes dtc: "detached s th"
+ shows "cntP s th = cntV s th"
+proof -
+ from cnp_cnv_cncs
+ have eq_pv: " cntP s th =
+ cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+ have cncs_z: "cntCS s th = 0"
+ proof -
+ from dtc have "holdents s th = {}"
+ unfolding detached_def holdents_test s_RAG_def
+ by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
+ thus ?thesis by (auto simp:cntCS_def)
+ qed
+ show ?thesis
+ proof(cases "th \<in> threads s")
+ case True
+ with dtc
+ have "th \<in> readys s"
+ by (unfold readys_def detached_def Field_def Domain_def Range_def,
+ auto simp:waiting_eq 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_nodeE:
+ assumes "(n1, n2) \<in> tRAG s"
+ obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
+ using assms
+ by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
+
+lemma subtree_nodeE:
+ assumes "n \<in> subtree (tRAG s) (Th th)"
+ obtains th1 where "n = Th th1"
+proof -
+ show ?thesis
+ proof(rule subtreeE[OF assms])
+ assume "n = Th th"
+ from that[OF this] show ?thesis .
+ next
+ assume "Th th \<in> ancestors (tRAG s) n"
+ hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+ hence "\<exists> th1. n = Th th1"
+ proof(induct)
+ case (base y)
+ from tRAG_nodeE[OF this] show ?case by metis
+ next
+ case (step y z)
+ thus ?case by auto
+ qed
+ with that show ?thesis by auto
+ qed
+qed
+
+lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
+proof -
+ have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*"
+ by (rule rtrancl_mono, auto simp:RAG_split)
+ also have "... \<subseteq> ((RAG s)^*)^*"
+ by (rule rtrancl_mono, auto)
+ also have "... = (RAG s)^*" by simp
+ finally show ?thesis by (unfold tRAG_def, simp)
+qed
+
+lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
+proof -
+ { fix a
+ assume "a \<in> subtree (tRAG s) x"
+ hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
+ with tRAG_star_RAG[of s]
+ have "(a, x) \<in> (RAG s)^*" by auto
+ hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
+ } thus ?thesis by auto
+qed
+
+lemma tRAG_trancl_eq:
+ "{th'. (Th th', Th th) \<in> (tRAG s)^+} =
+ {th'. (Th th', Th th) \<in> (RAG s)^+}"
+ (is "?L = ?R")
+proof -
+ { fix th'
+ assume "th' \<in> ?L"
+ hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
+ from tranclD[OF this]
+ obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
+ from tRAG_subtree_RAG[of s] and this(2)
+ have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG)
+ moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto
+ ultimately have "th' \<in> ?R" by auto
+ } moreover
+ { fix th'
+ assume "th' \<in> ?R"
+ hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
+ from plus_rpath[OF this]
+ obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
+ hence "(Th th', Th th) \<in> (tRAG s)^+"
+ proof(induct xs arbitrary:th' th rule:length_induct)
+ case (1 xs th' th)
+ then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
+ show ?case
+ proof(cases "xs1")
+ case Nil
+ from 1(2)[unfolded Cons1 Nil]
+ have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
+ hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
+ then obtain cs where "x1 = Cs cs"
+ by (unfold s_RAG_def, auto)
+ from rpath_nnl_lastE[OF rp[unfolded this]]
+ show ?thesis by auto
+ next
+ case (Cons x2 xs2)
+ from 1(2)[unfolded Cons1[unfolded this]]
+ have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
+ from rpath_edges_on[OF this]
+ have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
+ have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+ by (simp add: edges_on_unfold)
+ with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
+ then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
+ have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+ by (simp add: edges_on_unfold)
+ from this eds
+ have rg2: "(x1, x2) \<in> RAG s" by auto
+ from this[unfolded eq_x1]
+ obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
+ from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
+ have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
+ from rp have "rpath (RAG s) x2 xs2 (Th th)"
+ by (elim rpath_ConsE, simp)
+ from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
+ show ?thesis
+ proof(cases "xs2 = []")
+ case True
+ from rpath_nilE[OF rp'[unfolded this]]
+ have "th1 = th" by auto
+ from rt1[unfolded this] show ?thesis by auto
+ next
+ case False
+ from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
+ have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
+ with rt1 show ?thesis by auto
+ qed
+ qed
+ qed
+ hence "th' \<in> ?L" by auto
+ } ultimately show ?thesis by blast
+qed
+
+lemma tRAG_trancl_eq_Th:
+ "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} =
+ {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}"
+ using tRAG_trancl_eq by auto
+
+lemma dependants_alt_def:
+ "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
+ by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
+
+context valid_trace
+begin
+
+lemma count_eq_tRAG_plus:
+ assumes "cntP s th = cntV s th"
+ shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+ using assms count_eq_dependants dependants_alt_def eq_dependants by auto
+
+lemma count_eq_RAG_plus:
+ assumes "cntP s th = cntV s th"
+ shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+ using assms count_eq_dependants cs_dependants_def eq_RAG by auto
+
+lemma count_eq_RAG_plus_Th:
+ assumes "cntP s th = cntV s th"
+ shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
+ using count_eq_RAG_plus[OF assms] by auto
+
+lemma count_eq_tRAG_plus_Th:
+ assumes "cntP s th = cntV s th"
+ shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
+ using count_eq_tRAG_plus[OF assms] by auto
+
+end
+
+lemma tRAG_subtree_eq:
+ "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}"
+ (is "?L = ?R")
+proof -
+ { fix n
+ assume h: "n \<in> ?L"
+ hence "n \<in> ?R"
+ by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG)
+ } moreover {
+ fix n
+ assume "n \<in> ?R"
+ then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
+ by (auto simp:subtree_def)
+ from rtranclD[OF this(2)]
+ have "n \<in> ?L"
+ proof
+ assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
+ with h have "n \<in> {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}" by auto
+ thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
+ qed (insert h, auto simp:subtree_def)
+ } ultimately show ?thesis by auto
+qed
+
+lemma threads_set_eq:
+ "the_thread ` (subtree (tRAG s) (Th th)) =
+ {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
+ by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
+
+lemma cp_alt_def1:
+ "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
+proof -
+ have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
+ ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
+ by auto
+ thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
+qed
+
+lemma cp_gen_def_cond:
+ assumes "x = Th th"
+ shows "cp s th = cp_gen s (Th th)"
+by (unfold cp_alt_def1 cp_gen_def, simp)
+
+lemma cp_gen_over_set:
+ assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
+ shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
+proof(rule f_image_eq)
+ fix a
+ assume "a \<in> A"
+ from assms[rule_format, OF this]
+ obtain th where eq_a: "a = Th th" by auto
+ show "cp_gen s a = (cp s \<circ> the_thread) a"
+ by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
+qed
+
+
+context valid_trace
+begin
+
+lemma RAG_threads:
+ assumes "(Th th) \<in> Field (RAG s)"
+ shows "th \<in> threads s"
+ using assms
+ by (metis Field_def UnE dm_RAG_threads range_in vt)
+
+lemma subtree_tRAG_thread:
+ assumes "th \<in> threads s"
+ shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
+proof -
+ have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+ by (unfold tRAG_subtree_eq, simp)
+ also have "... \<subseteq> ?R"
+ proof
+ fix x
+ assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+ then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
+ from this(2)
+ show "x \<in> ?R"
+ proof(cases rule:subtreeE)
+ case 1
+ thus ?thesis by (simp add: assms h(1))
+ next
+ case 2
+ thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI)
+ qed
+ qed
+ finally show ?thesis .
+qed
+
+lemma readys_root:
+ assumes "th \<in> readys s"
+ shows "root (RAG s) (Th th)"
+proof -
+ { fix x
+ assume "x \<in> ancestors (RAG s) (Th th)"
+ hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+ from tranclD[OF this]
+ obtain z where "(Th th, z) \<in> RAG s" by auto
+ with assms(1) have False
+ apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
+ by (fold wq_def, blast)
+ } thus ?thesis by (unfold root_def, auto)
+qed
+
+lemma readys_in_no_subtree:
+ assumes "th \<in> readys s"
+ and "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s) (Th th')"
+proof
+ assume "Th th \<in> subtree (RAG s) (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 1
+ with assms show ?thesis by auto
+ next
+ case 2
+ with readys_root[OF assms(1)]
+ show ?thesis by (auto simp:root_def)
+ qed
+qed
+
+lemma not_in_thread_isolated:
+ assumes "th \<notin> threads s"
+ shows "(Th th) \<notin> Field (RAG s)"
+proof
+ assume "(Th th) \<in> Field (RAG s)"
+ with dm_RAG_threads and range_in assms
+ show False by (unfold Field_def, blast)
+qed
+
+lemma wf_RAG: "wf (RAG s)"
+proof(rule finite_acyclic_wf)
+ from finite_RAG show "finite (RAG s)" .
+next
+ from acyclic_RAG show "acyclic (RAG s)" .
+qed
+
+lemma sgv_wRAG: "single_valued (wRAG s)"
+ using waiting_unique
+ by (unfold single_valued_def wRAG_def, auto)
+
+lemma sgv_hRAG: "single_valued (hRAG s)"
+ using holding_unique
+ by (unfold single_valued_def hRAG_def, auto)
+
+lemma sgv_tRAG: "single_valued (tRAG s)"
+ by (unfold tRAG_def, rule single_valued_relcomp,
+ insert sgv_wRAG sgv_hRAG, auto)
+
+lemma acyclic_tRAG: "acyclic (tRAG s)"
+proof(unfold tRAG_def, rule acyclic_compose)
+ show "acyclic (RAG s)" using acyclic_RAG .
+next
+ show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+next
+ show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
+qed
+
+lemma sgv_RAG: "single_valued (RAG s)"
+ using unique_RAG by (auto simp:single_valued_def)
+
+lemma rtree_RAG: "rtree (RAG s)"
+ using sgv_RAG acyclic_RAG
+ by (unfold rtree_def rtree_axioms_def sgv_def, auto)
+
+end
+
+sublocale valid_trace < rtree_RAG: rtree "RAG s"
+proof
+ show "single_valued (RAG s)"
+ apply (intro_locales)
+ by (unfold single_valued_def,
+ auto intro:unique_RAG)
+
+ show "acyclic (RAG s)"
+ by (rule acyclic_RAG)
+qed
+
+sublocale valid_trace < rtree_s: rtree "tRAG s"
+proof(unfold_locales)
+ from sgv_tRAG show "single_valued (tRAG s)" .
+next
+ from acyclic_tRAG show "acyclic (tRAG s)" .
+qed
+
+sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
+proof -
+ show "fsubtree (RAG s)"
+ proof(intro_locales)
+ show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
+ next
+ show "fsubtree_axioms (RAG s)"
+ proof(unfold fsubtree_axioms_def)
+ from wf_RAG show "wf (RAG s)" .
+ qed
+ qed
+qed
+
+sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
+proof -
+ have "fsubtree (tRAG s)"
+ proof -
+ have "fbranch (tRAG s)"
+ proof(unfold tRAG_def, rule fbranch_compose)
+ show "fbranch (wRAG s)"
+ proof(rule finite_fbranchI)
+ from finite_RAG show "finite (wRAG s)"
+ by (unfold RAG_split, auto)
+ qed
+ next
+ show "fbranch (hRAG s)"
+ proof(rule finite_fbranchI)
+ from finite_RAG
+ show "finite (hRAG s)" by (unfold RAG_split, auto)
+ qed
+ qed
+ moreover have "wf (tRAG s)"
+ proof(rule wf_subset)
+ show "wf (RAG s O RAG s)" using wf_RAG
+ by (fold wf_comp_self, simp)
+ next
+ show "tRAG s \<subseteq> (RAG s O RAG s)"
+ by (unfold tRAG_alt_def, auto)
+ qed
+ ultimately show ?thesis
+ by (unfold fsubtree_def fsubtree_axioms_def,auto)
+ qed
+ from this[folded tRAG_def] show "fsubtree (tRAG s)" .
+qed
+
+lemma Max_UNION:
+ assumes "finite A"
+ and "A \<noteq> {}"
+ and "\<forall> M \<in> f ` A. finite M"
+ and "\<forall> M \<in> f ` A. M \<noteq> {}"
+ shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
+ using assms[simp]
+proof -
+ have "?L = Max (\<Union>(f ` A))"
+ by (fold Union_image_eq, simp)
+ also have "... = ?R"
+ by (subst Max_Union, simp+)
+ finally show ?thesis .
+qed
+
+lemma max_Max_eq:
+ assumes "finite A"
+ and "A \<noteq> {}"
+ and "x = y"
+ shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
+proof -
+ have "?R = Max (insert y A)" by simp
+ also from assms have "... = ?L"
+ by (subst Max.insert, simp+)
+ finally show ?thesis by simp
+qed
+
+context valid_trace
+begin
+
+(* ddd *)
+lemma cp_gen_rec:
+ assumes "x = Th th"
+ shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
+proof(cases "children (tRAG s) x = {}")
+ case True
+ show ?thesis
+ by (unfold True cp_gen_def subtree_children, simp add:assms)
+next
+ case False
+ hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
+ note fsbttRAGs.finite_subtree[simp]
+ have [simp]: "finite (children (tRAG s) x)"
+ by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree],
+ rule children_subtree)
+ { fix r x
+ have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
+ } note this[simp]
+ have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
+ proof -
+ from False obtain q where "q \<in> children (tRAG s) x" by blast
+ moreover have "subtree (tRAG s) q \<noteq> {}" by simp
+ ultimately show ?thesis by blast
+ qed
+ have h: "Max ((the_preced s \<circ> the_thread) `
+ ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
+ Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
+ (is "?L = ?R")
+ proof -
+ let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
+ let "Max (_ \<union> (?h ` ?B))" = ?R
+ let ?L1 = "?f ` \<Union>(?g ` ?B)"
+ have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
+ proof -
+ have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
+ also have "... = (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
+ finally have "Max ?L1 = Max ..." by simp
+ also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
+ by (subst Max_UNION, simp+)
+ also have "... = Max (cp_gen s ` children (tRAG s) x)"
+ by (unfold image_comp cp_gen_alt_def, simp)
+ finally show ?thesis .
+ qed
+ show ?thesis
+ proof -
+ have "?L = Max (?f ` ?A \<union> ?L1)" by simp
+ also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
+ by (subst Max_Un, simp+)
+ also have "... = max (?f x) (Max (?h ` ?B))"
+ by (unfold eq_Max_L1, simp)
+ also have "... =?R"
+ by (rule max_Max_eq, (simp)+, unfold assms, simp)
+ finally show ?thesis .
+ qed
+ qed thus ?thesis
+ by (fold h subtree_children, unfold cp_gen_def, simp)
+qed
+
+lemma cp_rec:
+ "cp s th = Max ({the_preced s th} \<union>
+ (cp s o the_thread) ` children (tRAG s) (Th th))"
+proof -
+ have "Th th = Th th" by simp
+ note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this]
+ show ?thesis
+ proof -
+ have "cp_gen s ` children (tRAG s) (Th th) =
+ (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
+ proof(rule cp_gen_over_set)
+ show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
+ by (unfold tRAG_alt_def, auto simp:children_def)
+ qed
+ thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+ qed
+qed
+
+end
+
+(* keep *)
+lemma next_th_holding:
+ assumes vt: "vt s"
+ and nxt: "next_th s th cs th'"
+ shows "holding (wq s) th cs"
+proof -
+ from nxt[unfolded next_th_def]
+ obtain rest where h: "wq s cs = th # rest"
+ "rest \<noteq> []"
+ "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+ thus ?thesis
+ by (unfold cs_holding_def, auto)
+qed
+
+context valid_trace
+begin
+
+lemma next_th_waiting:
+ assumes nxt: "next_th s th cs th'"
+ shows "waiting (wq s) th' cs"
+proof -
+ from nxt[unfolded next_th_def]
+ obtain rest where h: "wq s cs = th # rest"
+ "rest \<noteq> []"
+ "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+ from wq_distinct[of cs, unfolded h]
+ have dst: "distinct (th # rest)" .
+ have in_rest: "th' \<in> set rest"
+ proof(unfold h, rule someI2)
+ show "distinct rest \<and> set rest = set rest" using dst by auto
+ next
+ fix x assume "distinct x \<and> set x = set rest"
+ with h(2)
+ show "hd x \<in> set (rest)" by (cases x, auto)
+ qed
+ hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
+ moreover have "th' \<noteq> hd (wq s cs)"
+ by (unfold h(1), insert in_rest dst, auto)
+ ultimately show ?thesis by (auto simp:cs_waiting_def)
+qed
+
+lemma next_th_RAG:
+ assumes nxt: "next_th (s::event list) th cs th'"
+ shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
+ using vt assms next_th_holding next_th_waiting
+ by (unfold s_RAG_def, simp)
+
+end
+
+-- {* A useless definition *}
+definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+
+lemma "wq (V th cs # s) cs1 = ttt"
+ apply (unfold wq_def, auto simp:Let_def)
+
+end
+