--- a/CpsG.thy~ Thu Jan 28 21:14:17 2016 +0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,4656 +0,0 @@
-theory CpsG
-imports PIPDefs
-begin
-
-lemma f_image_eq:
- assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
- shows "f ` A = g ` A"
-proof
- show "f ` A \<subseteq> g ` A"
- by(rule image_subsetI, auto intro:h)
-next
- show "g ` A \<subseteq> f ` A"
- by (rule image_subsetI, auto intro:h[symmetric])
-qed
-
-lemma Max_fg_mono:
- assumes "finite A"
- and "\<forall> a \<in> A. f a \<le> g a"
- shows "Max (f ` A) \<le> Max (g ` A)"
-proof(cases "A = {}")
- case True
- thus ?thesis by auto
-next
- case False
- show ?thesis
- proof(rule Max.boundedI)
- from assms show "finite (f ` A)" by auto
- next
- from False show "f ` A \<noteq> {}" by auto
- next
- fix fa
- assume "fa \<in> f ` A"
- then obtain a where h_fa: "a \<in> A" "fa = f a" by auto
- show "fa \<le> Max (g ` A)"
- proof(rule Max_ge_iff[THEN iffD2])
- from assms show "finite (g ` A)" by auto
- next
- from False show "g ` A \<noteq> {}" by auto
- next
- from h_fa have "g a \<in> g ` A" by auto
- moreover have "fa \<le> g a" using h_fa assms(2) by auto
- ultimately show "\<exists>a\<in>g ` A. fa \<le> a" by auto
- qed
- qed
-qed
-
-lemma Max_f_mono:
- assumes seq: "A \<subseteq> B"
- and np: "A \<noteq> {}"
- and fnt: "finite B"
- shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
- from seq show "f ` A \<subseteq> f ` B" by auto
-next
- from np show "f ` A \<noteq> {}" by auto
-next
- from fnt and seq show "finite (f ` B)" by auto
-qed
-
-lemma Max_UNION:
- assumes "finite A"
- and "A \<noteq> {}"
- and "\<forall> M \<in> f ` A. finite M"
- and "\<forall> M \<in> f ` A. M \<noteq> {}"
- shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
- using assms[simp]
-proof -
- have "?L = Max (\<Union>(f ` A))"
- by (fold Union_image_eq, simp)
- also have "... = ?R"
- by (subst Max_Union, simp+)
- finally show ?thesis .
-qed
-
-lemma max_Max_eq:
- assumes "finite A"
- and "A \<noteq> {}"
- and "x = y"
- shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
-proof -
- have "?R = Max (insert y A)" by simp
- also from assms have "... = ?L"
- by (subst Max.insert, simp+)
- finally show ?thesis by simp
-qed
-
-lemma birth_time_lt:
- assumes "s \<noteq> []"
- shows "last_set th s < length s"
- using assms
-proof(induct s)
- case (Cons a s)
- show ?case
- proof(cases "s \<noteq> []")
- case False
- thus ?thesis
- by (cases a, auto)
- next
- case True
- show ?thesis using Cons(1)[OF True]
- by (cases a, auto)
- qed
-qed simp
-
-lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
- by (induct s, auto)
-
-lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
- by (drule_tac th_in_ne, unfold preced_def, auto intro: birth_time_lt)
-
-lemma eq_RAG:
- "RAG (wq s) = RAG s"
- by (unfold cs_RAG_def s_RAG_def, auto)
-
-lemma waiting_holding:
- assumes "waiting (s::state) th cs"
- obtains th' where "holding s th' cs"
-proof -
- from assms[unfolded s_waiting_def, folded wq_def]
- obtain th' where "th' \<in> set (wq s cs)" "th' = hd (wq s cs)"
- by (metis empty_iff hd_in_set list.set(1))
- hence "holding s th' cs"
- by (unfold s_holding_def, fold wq_def, auto)
- from that[OF this] show ?thesis .
-qed
-
-lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
-unfolding cp_def wq_def
-apply(induct s rule: schs.induct)
-apply(simp add: Let_def cpreced_initial)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-done
-
-lemma cp_alt_def:
- "cp s th =
- Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
-proof -
- have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
- Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
- (is "Max (_ ` ?L) = Max (_ ` ?R)")
- proof -
- have "?L = ?R"
- by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
- thus ?thesis by simp
- qed
- thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
-qed
-
-(* ccc *)
-
-
-locale valid_trace =
- fixes s
- assumes vt : "vt s"
-
-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
-
-lemma vt_moment: "\<And> t. vt (moment t s)"
-proof(induct rule:ind)
- case Nil
- thus ?case by (simp add:vt_nil)
-next
- case (Cons s e t)
- show ?case
- proof(cases "t \<ge> length (e#s)")
- case True
- from True have "moment t (e#s) = e#s" by simp
- thus ?thesis using Cons
- by (simp add:valid_trace_def valid_trace_e_def, auto)
- next
- case False
- from Cons have "vt (moment t s)" by simp
- moreover have "moment t (e#s) = moment t s"
- proof -
- from False have "t \<le> length s" by simp
- from moment_app [OF this, of "[e]"]
- show ?thesis by simp
- qed
- ultimately show ?thesis by simp
- qed
-qed
-
-lemma finite_threads:
- shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-end
-
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
- by (unfold s_RAG_def, auto)
-
-locale valid_moment = valid_trace +
- fixes i :: nat
-
-sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
- by (unfold_locales, insert vt_moment, auto)
-
-lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
- by (unfold s_waiting_def cs_waiting_def wq_def, auto)
-
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
- by (unfold s_holding_def wq_def cs_holding_def, simp)
-
-lemma runing_ready:
- shows "runing s \<subseteq> readys s"
- unfolding runing_def readys_def
- by auto
-
-lemma readys_threads:
- shows "readys s \<subseteq> threads s"
- unfolding readys_def
- by auto
-
-lemma wq_v_neq [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_create
-begin
-
-lemma wq_neq_simp [simp]:
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_create wq_def
- by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
- using assms by simp
-end
-
-context valid_trace_exit
-begin
-
-lemma wq_neq_simp [simp]:
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_exit wq_def
- by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
- using assms by simp
-end
-
-context valid_trace_p
-begin
-
-lemma wq_neq_simp [simp]:
- assumes "cs' \<noteq> cs"
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_p wq_def
- by (auto simp:Let_def)
-
-lemma runing_th_s:
- shows "th \<in> runing s"
-proof -
- from pip_e[unfolded is_p]
- show ?thesis by (cases, simp)
-qed
-
-lemma ready_th_s: "th \<in> readys s"
- using runing_th_s
- by (unfold runing_def, auto)
-
-lemma live_th_s: "th \<in> threads s"
- using readys_threads ready_th_s by auto
-
-lemma live_th_es: "th \<in> threads (e#s)"
- using live_th_s
- by (unfold is_p, simp)
-
-lemma th_not_waiting:
- "\<not> waiting s th c"
-proof -
- have "th \<in> readys s"
- using runing_ready runing_th_s by blast
- thus ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma waiting_neq_th:
- assumes "waiting s t c"
- shows "t \<noteq> th"
- using assms using th_not_waiting by blast
-
-lemma th_not_in_wq:
- shows "th \<notin> set (wq s cs)"
-proof
- assume otherwise: "th \<in> set (wq s cs)"
- from runing_wqE[OF runing_th_s this]
- obtain rest where eq_wq: "wq s cs = th#rest" by blast
- with otherwise
- have "holding s th cs"
- by (unfold s_holding_def, fold wq_def, simp)
- hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
- by (unfold s_RAG_def, fold holding_eq, auto)
- from pip_e[unfolded is_p]
- show False
- proof(cases)
- case (thread_P)
- with cs_th_RAG show ?thesis by auto
- qed
-qed
-
-lemma wq_es_cs:
- "wq (e#s) cs = wq s cs @ [th]"
- by (unfold is_p wq_def, auto simp:Let_def)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
-proof(cases "cs' = cs")
- case True
- show ?thesis using True assms th_not_in_wq
- by (unfold True wq_es_cs, auto)
-qed (insert assms, simp)
-
-end
-
-context valid_trace_v
-begin
-
-lemma wq_neq_simp [simp]:
- assumes "cs' \<noteq> cs"
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_v wq_def
- by (auto simp:Let_def)
-
-lemma runing_th_s:
- shows "th \<in> runing s"
-proof -
- from pip_e[unfolded is_v]
- show ?thesis by (cases, simp)
-qed
-
-lemma th_not_waiting:
- "\<not> waiting s th c"
-proof -
- have "th \<in> readys s"
- using runing_ready runing_th_s by blast
- thus ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma waiting_neq_th:
- assumes "waiting s t c"
- shows "t \<noteq> th"
- using assms using th_not_waiting by blast
-
-lemma wq_s_cs:
- "wq s cs = th#rest"
-proof -
- from pip_e[unfolded is_v]
- show ?thesis
- proof(cases)
- case (thread_V)
- from this(2) show ?thesis
- by (unfold rest_def s_holding_def, fold wq_def,
- metis empty_iff list.collapse list.set(1))
- qed
-qed
-
-lemma wq_es_cs:
- "wq (e#s) cs = wq'"
- using wq_s_cs[unfolded wq_def]
- by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
-proof(cases "cs' = cs")
- case True
- show ?thesis
- proof(unfold True wq_es_cs wq'_def, rule someI2)
- show "distinct rest \<and> set rest = set rest"
- using assms[unfolded True wq_s_cs] by auto
- qed simp
-qed (insert assms, simp)
-
-end
-
-context valid_trace_set
-begin
-
-lemma wq_neq_simp [simp]:
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_set wq_def
- by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
- using assms by simp
-end
-
-context valid_trace
-begin
-
-lemma actor_inv:
- assumes "PIP s e"
- and "\<not> isCreate e"
- shows "actor e \<in> runing s"
- using assms
- by (induct, auto)
-
-lemma 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 (Create th prio)
- interpret vt_create: valid_trace_create s e th prio
- using Create by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept)
- next
- case (Exit th)
- interpret vt_exit: valid_trace_exit s e th
- using Exit by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept)
- next
- case (P th cs)
- interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept)
- next
- case (V th cs)
- interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept)
- next
- case (Set th prio)
- interpret vt_set: valid_trace_set s e th prio
- using Set by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_set.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
-
-text {*
- The following three lemmas show that @{text "RAG"} does not change
- by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
- events, respectively.
-*}
-
-lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma (in valid_trace_set)
- RAG_unchanged: "(RAG (e # s)) = RAG s"
- by (unfold is_set RAG_set_unchanged, simp)
-
-lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma (in valid_trace_create)
- RAG_unchanged: "(RAG (e # s)) = RAG s"
- by (unfold is_create RAG_create_unchanged, simp)
-
-lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma (in valid_trace_exit)
- RAG_unchanged: "(RAG (e # s)) = RAG s"
- by (unfold is_exit RAG_exit_unchanged, simp)
-
-context valid_trace_v
-begin
-
-lemma distinct_rest: "distinct rest"
- by (simp add: distinct_tl rest_def wq_distinct)
-
-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 distinct_wq': "distinct wq'"
- by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def)
-
-lemma set_wq': "set wq' = set rest"
- by (metis (mono_tags, lifting) distinct_rest rest_def
- some_eq_ex wq'_def)
-
-lemma th'_in_inv:
- assumes "th' \<in> set wq'"
- shows "th' \<in> set rest"
- using assms set_wq' by simp
-
-lemma neq_t_th:
- assumes "waiting (e#s) t c"
- shows "t \<noteq> th"
-proof
- assume otherwise: "t = th"
- show False
- proof(cases "c = cs")
- case True
- have "t \<in> set wq'"
- using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
- by simp
- from th'_in_inv[OF this] have "t \<in> set rest" .
- with wq_s_cs[folded otherwise] wq_distinct[of cs]
- show ?thesis by simp
- next
- case False
- 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_e
-begin
-
-lemma nil_wq': "wq' = []"
-proof (unfold wq'_def, rule someI2)
- show "distinct rest \<and> set rest = set rest"
- by (simp add: distinct_rest)
-next
- fix x
- assume " distinct x \<and> set x = set rest"
- thus "x = []" using rest_nil by auto
-qed
-
-lemma no_taker:
- assumes "next_th s th cs taker"
- shows "False"
-proof -
- from assms[unfolded next_th_def]
- obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
- by auto
- thus ?thesis using rest_def rest_nil by auto
-qed
-
-lemma waiting_set_eq:
- "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
- using no_taker by auto
-
-lemma holding_set_eq:
- "{(Cs cs, Th th') |th'. next_th s th cs th'} = {}"
- using no_taker by auto
-
-lemma no_holding:
- assumes "holding (e#s) taker cs"
- shows False
-proof -
- from wq_es_cs[unfolded nil_wq']
- have " wq (e # s) cs = []" .
- from assms[unfolded s_holding_def, folded wq_def, unfolded this]
- show ?thesis by auto
-qed
-
-lemma no_waiting:
- assumes "waiting (e#s) t cs"
- shows False
-proof -
- from wq_es_cs[unfolded nil_wq']
- have " wq (e # s) cs = []" .
- from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
- show ?thesis by auto
-qed
-
-lemma waiting_esI2:
- assumes "waiting s t c"
- shows "waiting (e#s) t c"
-proof -
- have "c \<noteq> cs" using assms
- using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto
- from waiting_esI1[OF assms this]
- show ?thesis .
-qed
-
-lemma waiting_esE:
- assumes "waiting (e#s) t c"
- obtains "c \<noteq> cs" "waiting s t c"
-proof(cases "c = cs")
- case False
- hence "wq (e#s) c = wq s c" using is_v by auto
- with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto
- from that(1)[OF False this] show ?thesis .
-next
- case True
- from no_waiting[OF assms[unfolded True]]
- show ?thesis by auto
-qed
-
-lemma holding_esE:
- assumes "holding (e#s) t c"
- obtains "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
- case True
- from no_holding[OF assms[unfolded True]]
- show ?thesis by auto
-next
- case False
- hence "wq (e#s) c = wq s c" using is_v by auto
- from assms[unfolded s_holding_def, folded wq_def,
- unfolded this, unfolded wq_def, folded s_holding_def]
- have "holding s t c" .
- from that[OF False this] show ?thesis .
-qed
-
-end
-
-lemma rel_eqI:
- assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
- and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
- shows "A = B"
- using assms by auto
-
-lemma in_RAG_E:
- assumes "(n1, n2) \<in> RAG (s::state)"
- obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
- | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
- using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
- by auto
-
-context valid_trace_v
-begin
-
-lemma RAG_es:
- "RAG (e # s) =
- RAG s - {(Cs cs, Th th)} -
- {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = ?R")
-proof(rule rel_eqI)
- fix n1 n2
- assume "(n1, n2) \<in> ?L"
- thus "(n1, n2) \<in> ?R"
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- show ?thesis
- proof(cases "rest = []")
- case False
- interpret h_n: valid_trace_v_n s e th cs
- by (unfold_locales, insert False, simp)
- from waiting(3)
- show ?thesis
- proof(cases rule:h_n.waiting_esE)
- case 1
- with waiting(1,2)
- show ?thesis
- by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
- next
- case 2
- with waiting(1,2)
- show ?thesis
- by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
- qed
- next
- case True
- interpret h_e: valid_trace_v_e s e th cs
- by (unfold_locales, insert True, simp)
- from waiting(3)
- show ?thesis
- proof(cases rule:h_e.waiting_esE)
- case 1
- with waiting(1,2)
- show ?thesis
- by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
- qed
- qed
- next
- case (holding th' cs')
- show ?thesis
- proof(cases "rest = []")
- case False
- interpret h_n: valid_trace_v_n s e th cs
- by (unfold_locales, insert False, simp)
- from holding(3)
- show ?thesis
- proof(cases rule:h_n.holding_esE)
- case 1
- with holding(1,2)
- show ?thesis
- by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
- next
- case 2
- with holding(1,2)
- show ?thesis
- by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold holding_eq, auto)
- qed
- next
- case True
- interpret h_e: valid_trace_v_e s e th cs
- by (unfold_locales, insert True, simp)
- from holding(3)
- show ?thesis
- proof(cases rule:h_e.holding_esE)
- case 1
- with holding(1,2)
- show ?thesis
- by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
- fold holding_eq, auto)
- qed
- qed
- qed
-next
- fix n1 n2
- assume h: "(n1, n2) \<in> ?R"
- show "(n1, n2) \<in> ?L"
- proof(cases "rest = []")
- case False
- interpret h_n: valid_trace_v_n s e th cs
- by (unfold_locales, insert False, simp)
- from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
- have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
- \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or>
- (n2 = Th h_n.taker \<and> n1 = Cs cs)"
- by auto
- thus ?thesis
- proof
- assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
- with h_n.holding_taker
- show ?thesis
- by (unfold s_RAG_def, fold holding_eq, auto)
- next
- assume h: "(n1, n2) \<in> RAG s \<and>
- (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
- hence "(n1, n2) \<in> RAG s" by simp
- thus ?thesis
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from h and this(1,2)
- have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
- hence "waiting (e#s) th' cs'"
- proof
- assume "cs' \<noteq> cs"
- from waiting_esI1[OF waiting(3) this]
- show ?thesis .
- next
- assume neq_th': "th' \<noteq> h_n.taker"
- show ?thesis
- proof(cases "cs' = cs")
- case False
- from waiting_esI1[OF waiting(3) this]
- show ?thesis .
- next
- 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
-
-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
-
-lemma (in valid_trace_create)
- th_not_in_threads: "th \<notin> threads s"
-proof -
- from pip_e[unfolded is_create]
- show ?thesis by (cases, simp)
-qed
-
-lemma (in valid_trace_create)
- threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
- by (unfold is_create, simp)
-
-lemma (in valid_trace_exit)
- threads_es [simp]: "threads (e#s) = threads s - {th}"
- by (unfold is_exit, simp)
-
-lemma (in valid_trace_p)
- threads_es [simp]: "threads (e#s) = threads s"
- by (unfold is_p, simp)
-
-lemma (in valid_trace_v)
- threads_es [simp]: "threads (e#s) = threads s"
- by (unfold is_v, simp)
-
-lemma (in valid_trace_v)
- th_not_in_rest[simp]: "th \<notin> set rest"
-proof
- assume otherwise: "th \<in> set rest"
- have "distinct (wq s cs)" by (simp add: wq_distinct)
- from this[unfolded wq_s_cs] and otherwise
- show False by auto
-qed
-
-lemma (in valid_trace_v)
- set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
-proof(unfold wq_es_cs wq'_def, rule someI2)
- show "distinct rest \<and> set rest = set rest"
- by (simp add: distinct_rest)
-next
- fix x
- assume "distinct x \<and> set x = set rest"
- thus "set x = set (wq s cs) - {th}"
- by (unfold wq_s_cs, simp)
-qed
-
-lemma (in valid_trace_exit)
- th_not_in_wq: "th \<notin> set (wq s cs)"
-proof -
- from pip_e[unfolded is_exit]
- show ?thesis
- by (cases, unfold holdents_def s_holding_def, fold wq_def,
- auto elim!:runing_wqE)
-qed
-
-lemma (in valid_trace) wq_threads:
- assumes "th \<in> set (wq s cs)"
- shows "th \<in> threads s"
- using assms
-proof(induct rule:ind)
- case (Nil)
- thus ?case by (auto simp:wq_def)
-next
- case (Cons s e)
- interpret vt_e: valid_trace_e s e using Cons by simp
- show ?case
- proof(cases e)
- case (Create th' prio')
- interpret vt: valid_trace_create s e th' prio'
- using Create by (unfold_locales, simp)
- show ?thesis
- using Cons.hyps(2) Cons.prems by auto
- next
- case (Exit th')
- interpret vt: valid_trace_exit s e th'
- using Exit by (unfold_locales, simp)
- show ?thesis
- using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto
- next
- case (P th' cs')
- interpret vt: valid_trace_p s e th' cs'
- using P by (unfold_locales, simp)
- show ?thesis
- using Cons.hyps(2) Cons.prems readys_threads
- runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv
- by fastforce
- next
- case (V th' cs')
- interpret vt: valid_trace_v s e th' cs'
- using V by (unfold_locales, simp)
- show ?thesis using Cons
- using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
- next
- case (Set th' prio)
- interpret vt: valid_trace_set s e th' prio
- using Set by (unfold_locales, simp)
- show ?thesis using Cons.hyps(2) Cons.prems vt.is_set
- by (auto simp:wq_def Let_def)
- qed
-qed
-
-context valid_trace
-begin
-
-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 rg_RAG_threads:
- assumes "(Th th) \<in> Range (RAG s)"
- shows "th \<in> threads s"
- using assms
- by (unfold s_RAG_def cs_waiting_def cs_holding_def,
- auto intro:wq_threads)
-
-lemma RAG_threads:
- assumes "(Th th) \<in> Field (RAG s)"
- shows "th \<in> threads s"
- using assms
- by (metis Field_def UnE dm_RAG_threads rg_RAG_threads)
-
-end
-
-lemma (in valid_trace_v)
- preced_es [simp]: "preced th (e#s) = preced th s"
- by (unfold is_v preced_def, simp)
-
-lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
-proof
- fix th'
- show "the_preced (V th cs # s) th' = the_preced s th'"
- by (unfold the_preced_def preced_def, simp)
-qed
-
-lemma (in valid_trace_v)
- the_preced_es: "the_preced (e#s) = the_preced s"
- by (unfold is_v preced_def, simp)
-
-context valid_trace_p
-begin
-
-lemma not_holding_s_th_cs: "\<not> holding s th cs"
-proof
- assume otherwise: "holding s th cs"
- from pip_e[unfolded is_p]
- show False
- proof(cases)
- case (thread_P)
- moreover have "(Cs cs, Th th) \<in> RAG s"
- using otherwise cs_holding_def
- holding_eq th_not_in_wq by auto
- ultimately show ?thesis by auto
- qed
-qed
-
-lemma waiting_kept:
- assumes "waiting s th' cs'"
- shows "waiting (e#s) th' cs'"
- using assms
- by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2)
- rotate1.simps(2) self_append_conv2 set_rotate1
- th_not_in_wq waiting_eq wq_es_cs wq_neq_simp)
-
-lemma holding_kept:
- assumes "holding s th' cs'"
- shows "holding (e#s) th' cs'"
-proof(cases "cs' = cs")
- case False
- hence "wq (e#s) cs' = wq s cs'" by simp
- with assms show ?thesis using cs_holding_def holding_eq by auto
-next
- case True
- from assms[unfolded s_holding_def, folded wq_def]
- obtain rest where eq_wq: "wq s cs' = th'#rest"
- by (metis empty_iff list.collapse list.set(1))
- hence "wq (e#s) cs' = th'#(rest@[th])"
- by (simp add: True wq_es_cs)
- thus ?thesis
- by (simp add: cs_holding_def holding_eq)
-qed
-
-end
-
-locale valid_trace_p_h = valid_trace_p +
- assumes we: "wq s cs = []"
-
-locale valid_trace_p_w = valid_trace_p +
- assumes wne: "wq s cs \<noteq> []"
-begin
-
-definition "holder = hd (wq s cs)"
-definition "waiters = tl (wq s cs)"
-definition "waiters' = waiters @ [th]"
-
-lemma wq_s_cs: "wq s cs = holder#waiters"
- by (simp add: holder_def waiters_def wne)
-
-lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
- by (simp add: wq_es_cs wq_s_cs)
-
-lemma waiting_es_th_cs: "waiting (e#s) th cs"
- using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
-
-lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
- by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
-
-lemma holding_esE:
- assumes "holding (e#s) th' cs'"
- obtains "holding s th' cs'"
- using assms
-proof(cases "cs' = cs")
- case False
- hence "wq (e#s) cs' = wq s cs'" by simp
- with assms show ?thesis
- using cs_holding_def holding_eq that by auto
-next
- case True
- with assms show ?thesis
- by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that
- wq_es_cs' wq_s_cs)
-qed
-
-lemma waiting_esE:
- assumes "waiting (e#s) th' cs'"
- obtains "th' \<noteq> th" "waiting s th' cs'"
- | "th' = th" "cs' = cs"
-proof(cases "waiting s th' cs'")
- case True
- have "th' \<noteq> th"
- proof
- assume otherwise: "th' = th"
- from True[unfolded this]
- show False by (simp add: th_not_waiting)
- qed
- from that(1)[OF this True] show ?thesis .
-next
- case False
- hence "th' = th \<and> cs' = cs"
- by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2)
- set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
- with that(2) show ?thesis by metis
-qed
-
-lemma RAG_es: "RAG (e # s) = RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
-proof(rule rel_eqI)
- fix n1 n2
- assume "(n1, n2) \<in> ?L"
- thus "(n1, n2) \<in> ?R"
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from this(3)
- show ?thesis
- proof(cases rule:waiting_esE)
- case 1
- thus ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- next
- case 2
- thus ?thesis using waiting(1,2) by auto
- qed
- next
- case (holding th' cs')
- from this(3)
- show ?thesis
- proof(cases rule:holding_esE)
- case 1
- with holding(1,2)
- show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
- qed
- qed
-next
- fix n1 n2
- assume "(n1, n2) \<in> ?R"
- hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
- thus "(n1, n2) \<in> ?L"
- proof
- assume "(n1, n2) \<in> RAG s"
- thus ?thesis
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from waiting_kept[OF this(3)]
- show ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- next
- case (holding th' cs')
- from holding_kept[OF this(3)]
- show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
- qed
- next
- assume "n1 = Th th \<and> n2 = Cs cs"
- thus ?thesis using RAG_edge by auto
- qed
-qed
-
-end
-
-context valid_trace_p_h
-begin
-
-lemma wq_es_cs': "wq (e#s) cs = [th]"
- using wq_es_cs[unfolded we] by simp
-
-lemma holding_es_th_cs:
- shows "holding (e#s) th cs"
-proof -
- from wq_es_cs'
- have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto
- thus ?thesis using cs_holding_def holding_eq by blast
-qed
-
-lemma RAG_edge: "(Cs cs, Th th) \<in> RAG (e#s)"
- by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto)
-
-lemma waiting_esE:
- assumes "waiting (e#s) th' cs'"
- obtains "waiting s th' cs'"
- using assms
- by (metis cs_waiting_def event.distinct(15) is_p list.sel(1)
- set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv)
-
-lemma holding_esE:
- assumes "holding (e#s) th' cs'"
- obtains "cs' \<noteq> cs" "holding s th' cs'"
- | "cs' = cs" "th' = th"
-proof(cases "cs' = cs")
- case True
- from held_unique[OF holding_es_th_cs assms[unfolded True]]
- have "th' = th" by simp
- from that(2)[OF True this] show ?thesis .
-next
- case False
- have "holding s th' cs'" using assms
- using False cs_holding_def holding_eq by auto
- from that(1)[OF False this] show ?thesis .
-qed
-
-lemma RAG_es: "RAG (e # s) = RAG s \<union> {(Cs cs, Th th)}" (is "?L = ?R")
-proof(rule rel_eqI)
- fix n1 n2
- assume "(n1, n2) \<in> ?L"
- thus "(n1, n2) \<in> ?R"
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from this(3)
- show ?thesis
- proof(cases rule:waiting_esE)
- case 1
- thus ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- qed
- next
- case (holding th' cs')
- from this(3)
- show ?thesis
- proof(cases rule:holding_esE)
- case 1
- with holding(1,2)
- show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
- next
- case 2
- with holding(1,2) show ?thesis by auto
- qed
- qed
-next
- fix n1 n2
- assume "(n1, n2) \<in> ?R"
- hence "(n1, n2) \<in> RAG s \<or> (n1 = Cs cs \<and> n2 = Th th)" by auto
- thus "(n1, n2) \<in> ?L"
- proof
- assume "(n1, n2) \<in> RAG s"
- thus ?thesis
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from waiting_kept[OF this(3)]
- show ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- next
- case (holding th' cs')
- from holding_kept[OF this(3)]
- show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
- qed
- next
- assume "n1 = Cs cs \<and> n2 = Th th"
- with holding_es_th_cs
- show ?thesis
- by (unfold s_RAG_def, fold holding_eq, auto)
- qed
-qed
-
-end
-
-context valid_trace_p
-begin
-
-lemma RAG_es': "RAG (e # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
- else RAG s \<union> {(Th th, Cs cs)})"
-proof(cases "wq s cs = []")
- case True
- interpret vt_p: valid_trace_p_h using True
- by (unfold_locales, simp)
- show ?thesis by (simp add: vt_p.RAG_es vt_p.we)
-next
- case False
- interpret vt_p: valid_trace_p_w using False
- by (unfold_locales, simp)
- show ?thesis by (simp add: vt_p.RAG_es vt_p.wne)
-qed
-
-end
-
-lemma (in valid_trace_v_n) finite_waiting_set:
- "finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
- by (simp add: waiting_set_eq)
-
-lemma (in valid_trace_v_n) finite_holding_set:
- "finite {(Cs cs, Th th') |th'. next_th s th cs th'}"
- by (simp add: holding_set_eq)
-
-lemma (in valid_trace_v_e) finite_waiting_set:
- "finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
- by (simp add: waiting_set_eq)
-
-lemma (in valid_trace_v_e) finite_holding_set:
- "finite {(Cs cs, Th th') |th'. next_th s th cs th'}"
- by (simp add: holding_set_eq)
-
-context valid_trace_v
-begin
-
-lemma
- finite_RAG_kept:
- assumes "finite (RAG s)"
- shows "finite (RAG (e#s))"
-proof(cases "rest = []")
- case True
- interpret vt: valid_trace_v_e using True
- by (unfold_locales, simp)
- show ?thesis using assms
- by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
-next
- case False
- interpret vt: valid_trace_v_n using False
- by (unfold_locales, simp)
- show ?thesis using assms
- by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
-qed
-
-end
-
-context valid_trace_v_e
-begin
-
-lemma
- acylic_RAG_kept:
- assumes "acyclic (RAG s)"
- shows "acyclic (RAG (e#s))"
-proof(rule acyclic_subset[OF assms])
- show "RAG (e # s) \<subseteq> RAG s"
- by (unfold RAG_es waiting_set_eq holding_set_eq, auto)
-qed
-
-end
-
-context valid_trace_v_n
-begin
-
-lemma waiting_taker: "waiting s taker cs"
- apply (unfold s_waiting_def, fold wq_def, unfold wq_s_cs taker_def)
- using eq_wq' th'_in_inv wq'_def by fastforce
-
-lemma
- acylic_RAG_kept:
- assumes "acyclic (RAG s)"
- shows "acyclic (RAG (e#s))"
-proof -
- have "acyclic ((RAG s - {(Cs cs, Th th)} - {(Th taker, Cs cs)}) \<union>
- {(Cs cs, Th taker)})" (is "acyclic (?A \<union> _)")
- proof -
- from assms
- have "acyclic ?A"
- by (rule acyclic_subset, auto)
- moreover have "(Th taker, Cs cs) \<notin> ?A^*"
- proof
- assume otherwise: "(Th taker, Cs cs) \<in> ?A^*"
- hence "(Th taker, Cs cs) \<in> ?A^+"
- by (unfold rtrancl_eq_or_trancl, auto)
- from tranclD[OF this]
- obtain cs' where h: "(Th taker, Cs cs') \<in> ?A"
- "(Th taker, Cs cs') \<in> RAG s"
- by (unfold s_RAG_def, auto)
- from this(2) have "waiting s taker cs'"
- by (unfold s_RAG_def, fold waiting_eq, auto)
- from waiting_unique[OF this waiting_taker]
- have "cs' = cs" .
- from h(1)[unfolded this] show False by auto
- qed
- ultimately show ?thesis by auto
- qed
- thus ?thesis
- by (unfold RAG_es waiting_set_eq holding_set_eq, simp)
-qed
-
-end
-
-context valid_trace_p_h
-begin
-
-lemma
- acylic_RAG_kept:
- assumes "acyclic (RAG s)"
- shows "acyclic (RAG (e#s))"
-proof -
- have "acyclic (RAG s \<union> {(Cs cs, Th th)})" (is "acyclic (?A \<union> _)")
- proof -
- from assms
- have "acyclic ?A"
- by (rule acyclic_subset, auto)
- moreover have "(Th th, Cs cs) \<notin> ?A^*"
- proof
- assume otherwise: "(Th th, Cs cs) \<in> ?A^*"
- hence "(Th th, Cs cs) \<in> ?A^+"
- by (unfold rtrancl_eq_or_trancl, auto)
- from tranclD[OF this]
- obtain cs' where h: "(Th th, Cs cs') \<in> RAG s"
- by (unfold s_RAG_def, auto)
- hence "waiting s th cs'"
- by (unfold s_RAG_def, fold waiting_eq, auto)
- with th_not_waiting show False by auto
- qed
- ultimately show ?thesis by auto
- qed
- thus ?thesis by (unfold RAG_es, simp)
-qed
-
-end
-
-context valid_trace_p_w
-begin
-
-lemma
- acylic_RAG_kept:
- assumes "acyclic (RAG s)"
- shows "acyclic (RAG (e#s))"
-proof -
- have "acyclic (RAG s \<union> {(Th th, Cs cs)})" (is "acyclic (?A \<union> _)")
- proof -
- from assms
- have "acyclic ?A"
- by (rule acyclic_subset, auto)
- moreover have "(Cs cs, Th th) \<notin> ?A^*"
- proof
- assume otherwise: "(Cs cs, Th th) \<in> ?A^*"
- from pip_e[unfolded is_p]
- show False
- proof(cases)
- case (thread_P)
- moreover from otherwise have "(Cs cs, Th th) \<in> ?A^+"
- by (unfold rtrancl_eq_or_trancl, auto)
- ultimately show ?thesis by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- thus ?thesis by (unfold RAG_es, simp)
-qed
-
-end
-
-context valid_trace
-begin
-
-lemma finite_RAG:
- shows "finite (RAG s)"
-proof(induct rule:ind)
- case Nil
- show ?case
- by (auto simp: s_RAG_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
-next
- case (Cons s e)
- interpret vt_e: valid_trace_e s e using Cons by simp
- show ?case
- proof(cases e)
- case (Create th prio)
- interpret vt: valid_trace_create s e th prio using Create
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt.RAG_unchanged)
- next
- case (Exit th)
- interpret vt: valid_trace_exit s e th using Exit
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt.RAG_unchanged)
- next
- case (P th cs)
- interpret vt: valid_trace_p s e th cs using P
- by (unfold_locales, simp)
- show ?thesis using Cons using vt.RAG_es' by auto
- next
- case (V th cs)
- interpret vt: valid_trace_v s e th cs using V
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt.finite_RAG_kept)
- next
- case (Set th prio)
- interpret vt: valid_trace_set s e th prio using Set
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt.RAG_unchanged)
- qed
-qed
-
-lemma acyclic_RAG:
- shows "acyclic (RAG s)"
-proof(induct rule:ind)
- case Nil
- show ?case
- by (auto simp: s_RAG_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
-next
- case (Cons s e)
- interpret vt_e: valid_trace_e s e using Cons by simp
- show ?case
- proof(cases e)
- case (Create th prio)
- interpret vt: valid_trace_create s e th prio using Create
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt.RAG_unchanged)
- next
- case (Exit th)
- interpret vt: valid_trace_exit s e th using Exit
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt.RAG_unchanged)
- next
- case (P th cs)
- interpret vt: valid_trace_p s e th cs using P
- by (unfold_locales, simp)
- show ?thesis
- proof(cases "wq s cs = []")
- case True
- then interpret vt_h: valid_trace_p_h s e th cs
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_h.acylic_RAG_kept)
- next
- case False
- then interpret vt_w: valid_trace_p_w s e th cs
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_w.acylic_RAG_kept)
- qed
- next
- case (V th cs)
- interpret vt: valid_trace_v s e th cs using V
- by (unfold_locales, simp)
- show ?thesis
- proof(cases "vt.rest = []")
- case True
- then interpret vt_e: valid_trace_v_e s e th cs
- by (unfold_locales, simp)
- show ?thesis by (simp add: Cons.hyps(2) vt_e.acylic_RAG_kept)
- next
- case False
- then interpret vt_n: valid_trace_v_n s e th cs
- by (unfold_locales, simp)
- show ?thesis by (simp add: Cons.hyps(2) vt_n.acylic_RAG_kept)
- qed
- next
- case (Set th prio)
- interpret vt: valid_trace_set s e th prio using Set
- by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt.RAG_unchanged)
- qed
-qed
-
-lemma wf_RAG: "wf (RAG s)"
-proof(rule finite_acyclic_wf)
- from finite_RAG show "finite (RAG s)" .
-next
- from acyclic_RAG show "acyclic (RAG s)" .
-qed
-
-lemma sgv_wRAG: "single_valued (wRAG s)"
- using waiting_unique
- by (unfold single_valued_def wRAG_def, auto)
-
-lemma sgv_hRAG: "single_valued (hRAG s)"
- using held_unique
- by (unfold single_valued_def hRAG_def, auto)
-
-lemma sgv_tRAG: "single_valued (tRAG s)"
- by (unfold tRAG_def, rule single_valued_relcomp,
- insert sgv_wRAG sgv_hRAG, auto)
-
-lemma acyclic_tRAG: "acyclic (tRAG s)"
-proof(unfold tRAG_def, rule acyclic_compose)
- show "acyclic (RAG s)" using acyclic_RAG .
-next
- show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-next
- show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-qed
-
-lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
- apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
- by(auto elim:waiting_unique held_unique)
-
-lemma sgv_RAG: "single_valued (RAG s)"
- using unique_RAG by (auto simp:single_valued_def)
-
-lemma rtree_RAG: "rtree (RAG s)"
- using sgv_RAG acyclic_RAG
- by (unfold rtree_def rtree_axioms_def sgv_def, auto)
-
-end
-
-sublocale valid_trace < rtree_RAG: rtree "RAG s"
-proof
- show "single_valued (RAG s)"
- apply (intro_locales)
- by (unfold single_valued_def,
- auto intro:unique_RAG)
-
- show "acyclic (RAG s)"
- by (rule acyclic_RAG)
-qed
-
-sublocale valid_trace < rtree_s: rtree "tRAG s"
-proof(unfold_locales)
- from sgv_tRAG show "single_valued (tRAG s)" .
-next
- from acyclic_tRAG show "acyclic (tRAG s)" .
-qed
-
-sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
-proof -
- show "fsubtree (RAG s)"
- proof(intro_locales)
- show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
- next
- show "fsubtree_axioms (RAG s)"
- proof(unfold fsubtree_axioms_def)
- from wf_RAG show "wf (RAG s)" .
- qed
- qed
-qed
-
-lemma tRAG_alt_def:
- "tRAG s = {(Th th1, Th th2) | th1 th2.
- \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
- by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
-
-sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
-proof -
- have "fsubtree (tRAG s)"
- proof -
- have "fbranch (tRAG s)"
- proof(unfold tRAG_def, rule fbranch_compose)
- show "fbranch (wRAG s)"
- proof(rule finite_fbranchI)
- from finite_RAG show "finite (wRAG s)"
- by (unfold RAG_split, auto)
- qed
- 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
-
-
-context valid_trace
-begin
-
-lemma finite_subtree_threads:
- "finite {th'. Th th' \<in> subtree (RAG s) (Th th)}" (is "finite ?A")
-proof -
- have "?A = the_thread ` {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}"
- by (auto, insert image_iff, fastforce)
- moreover have "finite {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}"
- (is "finite ?B")
- proof -
- have "?B = (subtree (RAG s) (Th th)) \<inter> {Th th' | th'. True}"
- by auto
- moreover have "... \<subseteq> (subtree (RAG s) (Th th))" by auto
- moreover have "finite ..." by (simp add: fsbtRAGs.finite_subtree)
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma le_cp:
- shows "preced th s \<le> cp s th"
- proof(unfold cp_alt_def, rule Max_ge)
- show "finite (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
- by (simp add: finite_subtree_threads)
- next
- show "preced th s \<in> the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}"
- by (simp add: subtree_def the_preced_def)
- qed
-
-lemma cp_le:
- assumes th_in: "th \<in> threads s"
- shows "cp s th \<le> Max (the_preced s ` threads s)"
-proof(unfold cp_alt_def, rule Max_f_mono)
- show "finite (threads s)" by (simp add: finite_threads)
-next
- show " {th'. Th th' \<in> subtree (RAG s) (Th th)} \<noteq> {}"
- using subtree_def by fastforce
-next
- show "{th'. Th th' \<in> subtree (RAG s) (Th th)} \<subseteq> threads s"
- using assms
- by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq
- node.inject(1) rtranclD subsetI subtree_def trancl_domain)
-qed
-
-lemma max_cp_eq:
- shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
- (is "?L = ?R")
-proof -
- have "?L \<le> ?R"
- proof(cases "threads s = {}")
- case False
- show ?thesis
- by (rule Max.boundedI,
- insert cp_le,
- auto simp:finite_threads False)
- qed auto
- moreover have "?R \<le> ?L"
- by (rule Max_fg_mono,
- simp add: finite_threads,
- simp add: le_cp the_preced_def)
- ultimately show ?thesis by auto
-qed
-
-lemma wf_RAG_converse:
- shows "wf ((RAG s)^-1)"
-proof(rule finite_acyclic_wf_converse)
- from finite_RAG
- show "finite (RAG s)" .
-next
- from acyclic_RAG
- show "acyclic (RAG s)" .
-qed
-
-lemma chain_building:
- assumes "node \<in> Domain (RAG s)"
- obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"
-proof -
- from assms have "node \<in> Range ((RAG s)^-1)" by auto
- from wf_base[OF wf_RAG_converse this]
- obtain b where h_b: "(b, node) \<in> ((RAG s)\<inverse>)\<^sup>+" "\<forall>c. (c, b) \<notin> (RAG s)\<inverse>" by auto
- obtain th' where eq_b: "b = Th th'"
- proof(cases b)
- case (Cs cs)
- from h_b(1)[unfolded trancl_converse]
- have "(node, b) \<in> ((RAG s)\<^sup>+)" by auto
- from tranclE[OF this]
- obtain n where "(n, b) \<in> RAG s" by auto
- from this[unfolded Cs]
- obtain th1 where "waiting s th1 cs"
- by (unfold s_RAG_def, fold waiting_eq, auto)
- from waiting_holding[OF this]
- obtain th2 where "holding s th2 cs" .
- hence "(Cs cs, Th th2) \<in> RAG s"
- by (unfold s_RAG_def, fold holding_eq, auto)
- with h_b(2)[unfolded Cs, rule_format]
- have False by auto
- thus ?thesis by auto
- qed auto
- have "th' \<in> readys s"
- proof -
- from h_b(2)[unfolded eq_b]
- have "\<forall>cs. \<not> waiting s th' cs"
- by (unfold s_RAG_def, fold waiting_eq, auto)
- moreover have "th' \<in> threads s"
- proof(rule rg_RAG_threads)
- from tranclD[OF h_b(1), unfolded eq_b]
- obtain z where "(z, Th th') \<in> (RAG s)" by auto
- thus "Th th' \<in> Range (RAG s)" by auto
- qed
- ultimately show ?thesis by (auto simp:readys_def)
- qed
- moreover have "(node, Th th') \<in> (RAG s)^+"
- using h_b(1)[unfolded trancl_converse] eq_b by auto
- ultimately show ?thesis using that by metis
-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 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_simp1[simp]:
- "cntP (P th cs'#s) th = cntP s th + 1"
- by (unfold cntP_def, simp)
-
-lemma cntP_simp2[simp]:
- assumes "th' \<noteq> th"
- shows "cntP (P th cs'#s) th' = cntP s th'"
- using assms
- by (unfold cntP_def, simp)
-
-lemma cntP_simp3[simp]:
- assumes "\<not> isP e"
- shows "cntP (e#s) th' = cntP s th'"
- using assms
- by (unfold cntP_def, cases e, simp+)
-
-lemma cntV_simp1[simp]:
- "cntV (V th cs'#s) th = cntV s th + 1"
- by (unfold cntV_def, simp)
-
-lemma cntV_simp2[simp]:
- assumes "th' \<noteq> th"
- shows "cntV (V th cs'#s) th' = cntV s th'"
- using assms
- by (unfold cntV_def, simp)
-
-lemma cntV_simp3[simp]:
- assumes "\<not> isV e"
- shows "cntV (e#s) th' = cntV s th'"
- using assms
- by (unfold cntV_def, cases e, simp+)
-
-lemma cntP_diff_inv:
- assumes "cntP (e#s) th \<noteq> cntP s th"
- shows "isP e \<and> actor e = th"
-proof(cases e)
- case (P th' pty)
- show ?thesis
- by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)",
- insert assms P, auto simp:cntP_def)
-qed (insert assms, auto simp:cntP_def)
-
-lemma cntV_diff_inv:
- assumes "cntV (e#s) th \<noteq> cntV s th"
- shows "isV e \<and> actor e = th"
-proof(cases e)
- case (V th' pty)
- show ?thesis
- by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)",
- insert assms V, auto simp:cntV_def)
-qed (insert assms, auto simp:cntV_def)
-
-lemma children_RAG_alt_def:
- "children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}"
- by (unfold s_RAG_def, auto simp:children_def holding_eq)
-
-lemma holdents_alt_def:
- "holdents s th = the_cs ` (children (RAG (s::state)) (Th th))"
- by (unfold children_RAG_alt_def holdents_def, simp add: image_image)
-
-lemma cntCS_alt_def:
- "cntCS s th = card (children (RAG s) (Th th))"
- apply (unfold children_RAG_alt_def cntCS_def holdents_def)
- by (rule card_image[symmetric], auto simp:inj_on_def)
-
-context valid_trace
-begin
-
-lemma finite_holdents: "finite (holdents s th)"
- by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)
-
-end
-
-context valid_trace_p_w
-begin
-
-lemma holding_s_holder: "holding s holder cs"
- by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
-
-lemma holding_es_holder: "holding (e#s) holder cs"
- by (unfold s_holding_def, fold wq_def, unfold wq_es_cs wq_s_cs, auto)
-
-lemma holdents_es:
- shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
-proof -
- { fix cs'
- assume "cs' \<in> ?L"
- hence h: "holding (e#s) th' cs'" by (auto simp:holdents_def)
- have "holding s th' cs'"
- proof(cases "cs' = cs")
- case True
- from held_unique[OF h[unfolded True] holding_es_holder]
- have "th' = holder" .
- thus ?thesis
- by (unfold True holdents_def, insert holding_s_holder, simp)
- next
- case False
- hence "wq (e#s) cs' = wq s cs'" by simp
- from h[unfolded s_holding_def, folded wq_def, unfolded this]
- show ?thesis
- by (unfold s_holding_def, fold wq_def, auto)
- qed
- hence "cs' \<in> ?R" by (auto simp:holdents_def)
- } moreover {
- fix cs'
- assume "cs' \<in> ?R"
- hence h: "holding s th' cs'" by (auto simp:holdents_def)
- have "holding (e#s) th' cs'"
- proof(cases "cs' = cs")
- case True
- from held_unique[OF h[unfolded True] holding_s_holder]
- have "th' = holder" .
- thus ?thesis
- by (unfold True holdents_def, insert holding_es_holder, simp)
- next
- case False
- hence "wq s cs' = wq (e#s) cs'" by simp
- from h[unfolded s_holding_def, folded wq_def, unfolded this]
- show ?thesis
- by (unfold s_holding_def, fold wq_def, auto)
- qed
- hence "cs' \<in> ?L" by (auto simp:holdents_def)
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_es_th[simp]: "cntCS (e#s) th' = cntCS s th'"
- by (unfold cntCS_def holdents_es, simp)
-
-lemma th_not_ready_es:
- shows "th \<notin> readys (e#s)"
- using waiting_es_th_cs
- by (unfold readys_def, auto)
-
-end
-
-context valid_trace_p_h
-begin
-
-lemma th_not_waiting':
- "\<not> waiting (e#s) th cs'"
-proof(cases "cs' = cs")
- case True
- show ?thesis
- by (unfold True s_waiting_def, fold wq_def, unfold wq_es_cs', auto)
-next
- case False
- from th_not_waiting[of cs', unfolded s_waiting_def, folded wq_def]
- show ?thesis
- by (unfold s_waiting_def, fold wq_def, insert False, simp)
-qed
-
-lemma ready_th_es:
- shows "th \<in> readys (e#s)"
- using th_not_waiting'
- by (unfold readys_def, insert live_th_es, auto)
-
-lemma holdents_es_th:
- "holdents (e#s) th = (holdents s th) \<union> {cs}" (is "?L = ?R")
-proof -
- { fix cs'
- assume "cs' \<in> ?L"
- hence "holding (e#s) th cs'"
- by (unfold holdents_def, auto)
- hence "cs' \<in> ?R"
- by (cases rule:holding_esE, auto simp:holdents_def)
- } moreover {
- fix cs'
- assume "cs' \<in> ?R"
- hence "holding s th cs' \<or> cs' = cs"
- by (auto simp:holdents_def)
- hence "cs' \<in> ?L"
- proof
- assume "holding s th cs'"
- from holding_kept[OF this]
- show ?thesis by (auto simp:holdents_def)
- next
- assume "cs' = cs"
- thus ?thesis using holding_es_th_cs
- by (unfold holdents_def, auto)
- qed
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_es_th: "cntCS (e#s) th = cntCS s th + 1"
-proof -
- have "card (holdents s th \<union> {cs}) = card (holdents s th) + 1"
- proof(subst card_Un_disjoint)
- show "holdents s th \<inter> {cs} = {}"
- using not_holding_s_th_cs by (auto simp:holdents_def)
- qed (auto simp:finite_holdents)
- thus ?thesis
- by (unfold cntCS_def holdents_es_th, simp)
-qed
-
-lemma no_holder:
- "\<not> holding s th' cs"
-proof
- assume otherwise: "holding s th' cs"
- from this[unfolded s_holding_def, folded wq_def, unfolded we]
- show False by auto
-qed
-
-lemma holdents_es_th':
- assumes "th' \<noteq> th"
- shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
-proof -
- { fix cs'
- assume "cs' \<in> ?L"
- hence h_e: "holding (e#s) th' cs'" by (auto simp:holdents_def)
- have "cs' \<noteq> cs"
- proof
- assume "cs' = cs"
- from held_unique[OF h_e[unfolded this] holding_es_th_cs]
- have "th' = th" .
- with assms show False by simp
- qed
- from h_e[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp[OF this]]
- have "th' \<in> set (wq s cs') \<and> th' = hd (wq s cs')" .
- hence "cs' \<in> ?R"
- by (unfold holdents_def s_holding_def, fold wq_def, auto)
- } moreover {
- fix cs'
- assume "cs' \<in> ?R"
- hence "holding s th' cs'" by (auto simp:holdents_def)
- from holding_kept[OF this]
- have "holding (e # s) th' cs'" .
- hence "cs' \<in> ?L"
- by (unfold holdents_def, auto)
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_es_th'[simp]:
- assumes "th' \<noteq> th"
- shows "cntCS (e#s) th' = cntCS s th'"
- by (unfold cntCS_def holdents_es_th'[OF assms], simp)
-
-end
-
-context valid_trace_p
-begin
-
-lemma readys_kept1:
- assumes "th' \<noteq> th"
- and "th' \<in> readys (e#s)"
- shows "th' \<in> readys s"
-proof -
- { fix cs'
- assume wait: "waiting s th' cs'"
- have n_wait: "\<not> waiting (e#s) th' cs'"
- using assms(2)[unfolded readys_def] by auto
- have False
- proof(cases "cs' = cs")
- case False
- with n_wait wait
- show ?thesis
- by (unfold s_waiting_def, fold wq_def, auto)
- next
- case True
- show ?thesis
- proof(cases "wq s cs = []")
- case True
- then interpret vt: valid_trace_p_h
- by (unfold_locales, simp)
- show ?thesis using n_wait wait waiting_kept by auto
- next
- case False
- then interpret vt: valid_trace_p_w by (unfold_locales, simp)
- show ?thesis using n_wait wait waiting_kept by blast
- qed
- qed
- } with assms(2) show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_kept2:
- assumes "th' \<noteq> th"
- and "th' \<in> readys s"
- shows "th' \<in> readys (e#s)"
-proof -
- { fix cs'
- assume wait: "waiting (e#s) th' cs'"
- have n_wait: "\<not> waiting s th' cs'"
- using assms(2)[unfolded readys_def] by auto
- have False
- proof(cases "cs' = cs")
- case False
- with n_wait wait
- show ?thesis
- by (unfold s_waiting_def, fold wq_def, auto)
- next
- case True
- show ?thesis
- proof(cases "wq s cs = []")
- case True
- then interpret vt: valid_trace_p_h
- by (unfold_locales, simp)
- show ?thesis using n_wait vt.waiting_esE wait by blast
- next
- case False
- then interpret vt: valid_trace_p_w by (unfold_locales, simp)
- show ?thesis using assms(1) n_wait vt.waiting_esE wait by auto
- qed
- qed
- } with assms(2) show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_simp [simp]:
- assumes "th' \<noteq> th"
- shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
- using readys_kept1[OF assms] readys_kept2[OF assms]
- by metis
-
-lemma cnp_cnv_cncs_kept: (* ddd *)
- assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
- shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
-proof(cases "th' = th")
- case True
- note eq_th' = this
- show ?thesis
- proof(cases "wq s cs = []")
- case True
- then interpret vt: valid_trace_p_h by (unfold_locales, simp)
- show ?thesis
- using assms eq_th' is_p ready_th_s vt.cntCS_es_th vt.ready_th_es pvD_def by auto
- next
- case False
- then interpret vt: valid_trace_p_w by (unfold_locales, simp)
- show ?thesis
- using add.commute add.left_commute assms eq_th' is_p live_th_s
- ready_th_s vt.th_not_ready_es pvD_def
- apply (auto)
- by (fold is_p, simp)
- qed
-next
- case False
- note h_False = False
- thus ?thesis
- proof(cases "wq s cs = []")
- case True
- then interpret vt: valid_trace_p_h by (unfold_locales, simp)
- show ?thesis using assms
- by (insert True h_False pvD_def, auto split:if_splits,unfold is_p, auto)
- next
- case False
- then interpret vt: valid_trace_p_w by (unfold_locales, simp)
- show ?thesis using assms
- by (insert False h_False pvD_def, auto split:if_splits,unfold is_p, auto)
- qed
-qed
-
-end
-
-
-context valid_trace_v (* ccc *)
-begin
-
-lemma holding_th_cs_s:
- "holding s th cs"
- by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
-
-lemma th_ready_s [simp]: "th \<in> readys s"
- using runing_th_s
- by (unfold runing_def readys_def, auto)
-
-lemma th_live_s [simp]: "th \<in> threads s"
- using th_ready_s by (unfold readys_def, auto)
-
-lemma th_ready_es [simp]: "th \<in> readys (e#s)"
- using runing_th_s neq_t_th
- by (unfold is_v runing_def readys_def, auto)
-
-lemma th_live_es [simp]: "th \<in> threads (e#s)"
- using th_ready_es by (unfold readys_def, auto)
-
-lemma pvD_th_s[simp]: "pvD s th = 0"
- by (unfold pvD_def, simp)
-
-lemma pvD_th_es[simp]: "pvD (e#s) th = 0"
- by (unfold pvD_def, simp)
-
-lemma cntCS_s_th [simp]: "cntCS s th > 0"
-proof -
- have "cs \<in> holdents s th" using holding_th_cs_s
- by (unfold holdents_def, simp)
- moreover have "finite (holdents s th)" using finite_holdents
- by simp
- ultimately show ?thesis
- by (unfold cntCS_def,
- auto intro!:card_gt_0_iff[symmetric, THEN iffD1])
-qed
-
-end
-
-context valid_trace_v_n
-begin
-
-lemma not_ready_taker_s[simp]:
- "taker \<notin> readys s"
- using waiting_taker
- by (unfold readys_def, auto)
-
-lemma taker_live_s [simp]: "taker \<in> threads s"
-proof -
- have "taker \<in> set wq'" by (simp add: eq_wq')
- from th'_in_inv[OF this]
- have "taker \<in> set rest" .
- hence "taker \<in> set (wq s cs)" by (simp add: wq_s_cs)
- thus ?thesis using wq_threads by auto
-qed
-
-lemma taker_live_es [simp]: "taker \<in> threads (e#s)"
- using taker_live_s threads_es by blast
-
-lemma taker_ready_es [simp]:
- shows "taker \<in> readys (e#s)"
-proof -
- { fix cs'
- assume "waiting (e#s) taker cs'"
- hence False
- proof(cases rule:waiting_esE)
- case 1
- thus ?thesis using waiting_taker waiting_unique by auto
- qed simp
- } thus ?thesis by (unfold readys_def, auto)
-qed
-
-lemma neq_taker_th: "taker \<noteq> th"
- using th_not_waiting waiting_taker by blast
-
-lemma not_holding_taker_s_cs:
- shows "\<not> holding s taker cs"
- using holding_cs_eq_th neq_taker_th by auto
-
-lemma holdents_es_taker:
- "holdents (e#s) taker = holdents s taker \<union> {cs}" (is "?L = ?R")
-proof -
- { fix cs'
- assume "cs' \<in> ?L"
- hence "holding (e#s) taker cs'" by (auto simp:holdents_def)
- hence "cs' \<in> ?R"
- proof(cases rule:holding_esE)
- case 2
- thus ?thesis by (auto simp:holdents_def)
- qed auto
- } moreover {
- fix cs'
- assume "cs' \<in> ?R"
- hence "holding s taker cs' \<or> cs' = cs" by (auto simp:holdents_def)
- hence "cs' \<in> ?L"
- proof
- assume "holding s taker cs'"
- hence "holding (e#s) taker cs'"
- using holding_esI2 holding_taker by fastforce
- thus ?thesis by (auto simp:holdents_def)
- next
- assume "cs' = cs"
- with holding_taker
- show ?thesis by (auto simp:holdents_def)
- qed
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_es_taker [simp]: "cntCS (e#s) taker = cntCS s taker + 1"
-proof -
- have "card (holdents s taker \<union> {cs}) = card (holdents s taker) + 1"
- proof(subst card_Un_disjoint)
- show "holdents s taker \<inter> {cs} = {}"
- using not_holding_taker_s_cs by (auto simp:holdents_def)
- qed (auto simp:finite_holdents)
- thus ?thesis
- by (unfold cntCS_def, insert holdents_es_taker, simp)
-qed
-
-lemma pvD_taker_s[simp]: "pvD s taker = 1"
- by (unfold pvD_def, simp)
-
-lemma pvD_taker_es[simp]: "pvD (e#s) taker = 0"
- by (unfold pvD_def, simp)
-
-lemma pvD_th_s[simp]: "pvD s th = 0"
- by (unfold pvD_def, simp)
-
-lemma pvD_th_es[simp]: "pvD (e#s) th = 0"
- by (unfold pvD_def, simp)
-
-lemma holdents_es_th:
- "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")
-proof -
- { fix cs'
- assume "cs' \<in> ?L"
- hence "holding (e#s) th cs'" by (auto simp:holdents_def)
- hence "cs' \<in> ?R"
- proof(cases rule:holding_esE)
- case 2
- thus ?thesis by (auto simp:holdents_def)
- qed (insert neq_taker_th, auto)
- } moreover {
- fix cs'
- assume "cs' \<in> ?R"
- hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def)
- from holding_esI2[OF this]
- have "cs' \<in> ?L" by (auto simp:holdents_def)
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1"
-proof -
- have "card (holdents s th - {cs}) = card (holdents s th) - 1"
- proof -
- have "cs \<in> holdents s th" using holding_th_cs_s
- by (auto simp:holdents_def)
- moreover have "finite (holdents s th)"
- by (simp add: finite_holdents)
- ultimately show ?thesis by auto
- qed
- thus ?thesis by (unfold cntCS_def holdents_es_th)
-qed
-
-lemma holdents_kept:
- assumes "th' \<noteq> taker"
- and "th' \<noteq> th"
- shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
-proof -
- { fix cs'
- assume h: "cs' \<in> ?L"
- have "cs' \<in> ?R"
- proof(cases "cs' = cs")
- case False
- hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
- from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
- from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
- show ?thesis
- by (unfold holdents_def s_holding_def, fold wq_def, auto)
- next
- case True
- from h[unfolded this]
- have "holding (e#s) th' cs" by (auto simp:holdents_def)
- from held_unique[OF this holding_taker]
- have "th' = taker" .
- with assms show ?thesis by auto
- qed
- } moreover {
- fix cs'
- assume h: "cs' \<in> ?R"
- have "cs' \<in> ?L"
- proof(cases "cs' = cs")
- case False
- hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
- from h have "holding s th' cs'" by (auto simp:holdents_def)
- from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
- show ?thesis
- by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
- next
- case True
- from h[unfolded this]
- have "holding s th' cs" by (auto simp:holdents_def)
- from held_unique[OF this holding_th_cs_s]
- have "th' = th" .
- with assms show ?thesis by auto
- qed
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_kept [simp]:
- assumes "th' \<noteq> taker"
- and "th' \<noteq> th"
- shows "cntCS (e#s) th' = cntCS s th'"
- by (unfold cntCS_def holdents_kept[OF assms], simp)
-
-lemma readys_kept1:
- assumes "th' \<noteq> taker"
- and "th' \<in> readys (e#s)"
- shows "th' \<in> readys s"
-proof -
- { fix cs'
- assume wait: "waiting s th' cs'"
- have n_wait: "\<not> waiting (e#s) th' cs'"
- using assms(2)[unfolded readys_def] by auto
- have False
- proof(cases "cs' = cs")
- case False
- with n_wait wait
- show ?thesis
- by (unfold s_waiting_def, fold wq_def, auto)
- next
- case True
- have "th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)"
- using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
- moreover have "\<not> (th' \<in> set rest \<and> th' \<noteq> hd (taker # rest'))"
- using n_wait[unfolded True s_waiting_def, folded wq_def,
- unfolded wq_es_cs set_wq', unfolded eq_wq'] .
- ultimately have "th' = taker" by auto
- with assms(1)
- show ?thesis by simp
- qed
- } with assms(2) show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_kept2:
- assumes "th' \<noteq> taker"
- and "th' \<in> readys s"
- shows "th' \<in> readys (e#s)"
-proof -
- { fix cs'
- assume wait: "waiting (e#s) th' cs'"
- have n_wait: "\<not> waiting s th' cs'"
- using assms(2)[unfolded readys_def] by auto
- have False
- proof(cases "cs' = cs")
- case False
- with n_wait wait
- show ?thesis
- by (unfold s_waiting_def, fold wq_def, auto)
- next
- case True
- have "th' \<in> set rest \<and> th' \<noteq> hd (taker # rest')"
- using wait [unfolded True s_waiting_def, folded wq_def,
- unfolded wq_es_cs set_wq', unfolded eq_wq'] .
- moreover have "\<not> (th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest))"
- using n_wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
- ultimately have "th' = taker" by auto
- with assms(1)
- show ?thesis by simp
- qed
- } with assms(2) show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_simp [simp]:
- assumes "th' \<noteq> taker"
- shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
- using readys_kept1[OF assms] readys_kept2[OF assms]
- by metis
-
-lemma cnp_cnv_cncs_kept:
- assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
- shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
-proof -
- { assume eq_th': "th' = taker"
- have ?thesis
- apply (unfold eq_th' pvD_taker_es cntCS_es_taker)
- by (insert neq_taker_th assms[unfolded eq_th'], unfold is_v, simp)
- } moreover {
- assume eq_th': "th' = th"
- have ?thesis
- apply (unfold eq_th' pvD_th_es cntCS_es_th)
- by (insert assms[unfolded eq_th'], unfold is_v, simp)
- } moreover {
- assume h: "th' \<noteq> taker" "th' \<noteq> th"
- have ?thesis using assms
- apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp)
- by (fold is_v, unfold pvD_def, simp)
- } ultimately show ?thesis by metis
-qed
-
-end
-
-context valid_trace_v_e
-begin
-
-lemma holdents_es_th:
- "holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")
-proof -
- { fix cs'
- assume "cs' \<in> ?L"
- hence "holding (e#s) th cs'" by (auto simp:holdents_def)
- hence "cs' \<in> ?R"
- proof(cases rule:holding_esE)
- case 1
- thus ?thesis by (auto simp:holdents_def)
- qed
- } moreover {
- fix cs'
- assume "cs' \<in> ?R"
- hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def)
- from holding_esI2[OF this]
- have "cs' \<in> ?L" by (auto simp:holdents_def)
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1"
-proof -
- have "card (holdents s th - {cs}) = card (holdents s th) - 1"
- proof -
- have "cs \<in> holdents s th" using holding_th_cs_s
- by (auto simp:holdents_def)
- moreover have "finite (holdents s th)"
- by (simp add: finite_holdents)
- ultimately show ?thesis by auto
- qed
- thus ?thesis by (unfold cntCS_def holdents_es_th)
-qed
-
-lemma holdents_kept:
- assumes "th' \<noteq> th"
- shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
-proof -
- { fix cs'
- assume h: "cs' \<in> ?L"
- have "cs' \<in> ?R"
- proof(cases "cs' = cs")
- case False
- hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
- from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
- from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
- show ?thesis
- by (unfold holdents_def s_holding_def, fold wq_def, auto)
- next
- case True
- from h[unfolded this]
- have "holding (e#s) th' cs" by (auto simp:holdents_def)
- from this[unfolded s_holding_def, folded wq_def,
- unfolded wq_es_cs nil_wq']
- show ?thesis by auto
- qed
- } moreover {
- fix cs'
- assume h: "cs' \<in> ?R"
- have "cs' \<in> ?L"
- proof(cases "cs' = cs")
- case False
- hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
- from h have "holding s th' cs'" by (auto simp:holdents_def)
- from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
- show ?thesis
- by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
- next
- case True
- from h[unfolded this]
- have "holding s th' cs" by (auto simp:holdents_def)
- from held_unique[OF this holding_th_cs_s]
- have "th' = th" .
- with assms show ?thesis by auto
- qed
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_kept [simp]:
- assumes "th' \<noteq> th"
- shows "cntCS (e#s) th' = cntCS s th'"
- by (unfold cntCS_def holdents_kept[OF assms], simp)
-
-lemma readys_kept1:
- assumes "th' \<in> readys (e#s)"
- shows "th' \<in> readys s"
-proof -
- { fix cs'
- assume wait: "waiting s th' cs'"
- have n_wait: "\<not> waiting (e#s) th' cs'"
- using assms(1)[unfolded readys_def] by auto
- have False
- proof(cases "cs' = cs")
- case False
- with n_wait wait
- show ?thesis
- by (unfold s_waiting_def, fold wq_def, auto)
- next
- case True
- have "th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)"
- using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
- hence "th' \<in> set rest" by auto
- with set_wq' have "th' \<in> set wq'" by metis
- with nil_wq' show ?thesis by simp
- qed
- } thus ?thesis using assms
- by (unfold readys_def, auto)
-qed
-
-lemma readys_kept2:
- assumes "th' \<in> readys s"
- shows "th' \<in> readys (e#s)"
-proof -
- { fix cs'
- assume wait: "waiting (e#s) th' cs'"
- have n_wait: "\<not> waiting s th' cs'"
- using assms[unfolded readys_def] by auto
- have False
- proof(cases "cs' = cs")
- case False
- with n_wait wait
- show ?thesis
- by (unfold s_waiting_def, fold wq_def, auto)
- next
- case True
- have "th' \<in> set [] \<and> th' \<noteq> hd []"
- using wait[unfolded True s_waiting_def, folded wq_def,
- unfolded wq_es_cs nil_wq'] .
- thus ?thesis by simp
- qed
- } with assms show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_simp [simp]:
- shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
- using readys_kept1[OF assms] readys_kept2[OF assms]
- by metis
-
-lemma cnp_cnv_cncs_kept:
- assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
- shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
-proof -
- {
- assume eq_th': "th' = th"
- have ?thesis
- apply (unfold eq_th' pvD_th_es cntCS_es_th)
- by (insert assms[unfolded eq_th'], unfold is_v, simp)
- } moreover {
- assume h: "th' \<noteq> th"
- have ?thesis using assms
- apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp)
- by (fold is_v, unfold pvD_def, simp)
- } ultimately show ?thesis by metis
-qed
-
-end
-
-context valid_trace_v
-begin
-
-lemma cnp_cnv_cncs_kept:
- assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
- shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
-proof(cases "rest = []")
- case True
- then interpret vt: valid_trace_v_e by (unfold_locales, simp)
- show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast
-next
- case False
- then interpret vt: valid_trace_v_n by (unfold_locales, simp)
- show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast
-qed
-
-end
-
-context valid_trace_create
-begin
-
-lemma th_not_live_s [simp]: "th \<notin> threads s"
-proof -
- from pip_e[unfolded is_create]
- show ?thesis by (cases, simp)
-qed
-
-lemma th_not_ready_s [simp]: "th \<notin> readys s"
- using th_not_live_s by (unfold readys_def, simp)
-
-lemma th_live_es [simp]: "th \<in> threads (e#s)"
- by (unfold is_create, simp)
-
-lemma not_waiting_th_s [simp]: "\<not> waiting s th cs'"
-proof
- assume "waiting s th cs'"
- from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
- have "th \<in> set (wq s cs')" by auto
- from wq_threads[OF this] have "th \<in> threads s" .
- with th_not_live_s show False by simp
-qed
-
-lemma not_holding_th_s [simp]: "\<not> holding s th cs'"
-proof
- assume "holding s th cs'"
- from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
- have "th \<in> set (wq s cs')" by auto
- from wq_threads[OF this] have "th \<in> threads s" .
- with th_not_live_s show False by simp
-qed
-
-lemma not_waiting_th_es [simp]: "\<not> waiting (e#s) th cs'"
-proof
- assume "waiting (e # s) th cs'"
- from this[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
- have "th \<in> set (wq s cs')" by auto
- from wq_threads[OF this] have "th \<in> threads s" .
- with th_not_live_s show False by simp
-qed
-
-lemma not_holding_th_es [simp]: "\<not> holding (e#s) th cs'"
-proof
- assume "holding (e # s) th cs'"
- from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
- have "th \<in> set (wq s cs')" by auto
- from wq_threads[OF this] have "th \<in> threads s" .
- with th_not_live_s show False by simp
-qed
-
-lemma ready_th_es [simp]: "th \<in> readys (e#s)"
- by (simp add:readys_def)
-
-lemma holdents_th_s: "holdents s th = {}"
- by (unfold holdents_def, auto)
-
-lemma holdents_th_es: "holdents (e#s) th = {}"
- by (unfold holdents_def, auto)
-
-lemma cntCS_th_s [simp]: "cntCS s th = 0"
- by (unfold cntCS_def, simp add:holdents_th_s)
-
-lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0"
- by (unfold cntCS_def, simp add:holdents_th_es)
-
-lemma pvD_th_s [simp]: "pvD s th = 0"
- by (unfold pvD_def, simp)
-
-lemma pvD_th_es [simp]: "pvD (e#s) th = 0"
- by (unfold pvD_def, simp)
-
-lemma holdents_kept:
- assumes "th' \<noteq> th"
- shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
-proof -
- { fix cs'
- assume h: "cs' \<in> ?L"
- hence "cs' \<in> ?R"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_neq_simp, auto)
- } moreover {
- fix cs'
- assume h: "cs' \<in> ?R"
- hence "cs' \<in> ?L"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_neq_simp, auto)
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_kept [simp]:
- assumes "th' \<noteq> th"
- shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
- using holdents_kept[OF assms]
- by (unfold cntCS_def, simp)
-
-lemma readys_kept1:
- assumes "th' \<noteq> th"
- and "th' \<in> readys (e#s)"
- shows "th' \<in> readys s"
-proof -
- { fix cs'
- assume wait: "waiting s th' cs'"
- have n_wait: "\<not> waiting (e#s) th' cs'"
- using assms by (auto simp:readys_def)
- from wait[unfolded s_waiting_def, folded wq_def]
- n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
- have False by auto
- } thus ?thesis using assms
- by (unfold readys_def, auto)
-qed
-
-lemma readys_kept2:
- assumes "th' \<noteq> th"
- and "th' \<in> readys s"
- shows "th' \<in> readys (e#s)"
-proof -
- { fix cs'
- assume wait: "waiting (e#s) th' cs'"
- have n_wait: "\<not> waiting s th' cs'"
- using assms(2) by (auto simp:readys_def)
- from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
- n_wait[unfolded s_waiting_def, folded wq_def]
- have False by auto
- } with assms show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_simp [simp]:
- assumes "th' \<noteq> th"
- shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
- using readys_kept1[OF assms] readys_kept2[OF assms]
- by metis
-
-lemma pvD_kept [simp]:
- assumes "th' \<noteq> th"
- shows "pvD (e#s) th' = pvD s th'"
- using assms
- by (unfold pvD_def, simp)
-
-lemma cnp_cnv_cncs_kept:
- assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
- shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
-proof -
- {
- assume eq_th': "th' = th"
- have ?thesis using assms
- by (unfold eq_th', simp, unfold is_create, simp)
- } moreover {
- assume h: "th' \<noteq> th"
- hence ?thesis using assms
- by (simp, simp add:is_create)
- } ultimately show ?thesis by metis
-qed
-
-end
-
-context valid_trace_exit
-begin
-
-lemma th_live_s [simp]: "th \<in> threads s"
-proof -
- from pip_e[unfolded is_exit]
- show ?thesis
- by (cases, unfold runing_def readys_def, simp)
-qed
-
-lemma th_ready_s [simp]: "th \<in> readys s"
-proof -
- from pip_e[unfolded is_exit]
- show ?thesis
- by (cases, unfold runing_def, simp)
-qed
-
-lemma th_not_live_es [simp]: "th \<notin> threads (e#s)"
- by (unfold is_exit, simp)
-
-lemma not_holding_th_s [simp]: "\<not> holding s th cs'"
-proof -
- from pip_e[unfolded is_exit]
- show ?thesis
- by (cases, unfold holdents_def, auto)
-qed
-
-lemma cntCS_th_s [simp]: "cntCS s th = 0"
-proof -
- from pip_e[unfolded is_exit]
- show ?thesis
- by (cases, unfold cntCS_def, simp)
-qed
-
-lemma not_holding_th_es [simp]: "\<not> holding (e#s) th cs'"
-proof
- assume "holding (e # s) th cs'"
- from this[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp]
- have "holding s th cs'"
- by (unfold s_holding_def, fold wq_def, auto)
- with not_holding_th_s
- show False by simp
-qed
-
-lemma ready_th_es [simp]: "th \<notin> readys (e#s)"
- by (simp add:readys_def)
-
-lemma holdents_th_s: "holdents s th = {}"
- by (unfold holdents_def, auto)
-
-lemma holdents_th_es: "holdents (e#s) th = {}"
- by (unfold holdents_def, auto)
-
-lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0"
- by (unfold cntCS_def, simp add:holdents_th_es)
-
-lemma pvD_th_s [simp]: "pvD s th = 0"
- by (unfold pvD_def, simp)
-
-lemma pvD_th_es [simp]: "pvD (e#s) th = 0"
- by (unfold pvD_def, simp)
-
-lemma holdents_kept:
- assumes "th' \<noteq> th"
- shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
-proof -
- { fix cs'
- assume h: "cs' \<in> ?L"
- hence "cs' \<in> ?R"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_neq_simp, auto)
- } moreover {
- fix cs'
- assume h: "cs' \<in> ?R"
- hence "cs' \<in> ?L"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_neq_simp, auto)
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_kept [simp]:
- assumes "th' \<noteq> th"
- shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
- using holdents_kept[OF assms]
- by (unfold cntCS_def, simp)
-
-lemma readys_kept1:
- assumes "th' \<noteq> th"
- and "th' \<in> readys (e#s)"
- shows "th' \<in> readys s"
-proof -
- { fix cs'
- assume wait: "waiting s th' cs'"
- have n_wait: "\<not> waiting (e#s) th' cs'"
- using assms by (auto simp:readys_def)
- from wait[unfolded s_waiting_def, folded wq_def]
- n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
- have False by auto
- } thus ?thesis using assms
- by (unfold readys_def, auto)
-qed
-
-lemma readys_kept2:
- assumes "th' \<noteq> th"
- and "th' \<in> readys s"
- shows "th' \<in> readys (e#s)"
-proof -
- { fix cs'
- assume wait: "waiting (e#s) th' cs'"
- have n_wait: "\<not> waiting s th' cs'"
- using assms(2) by (auto simp:readys_def)
- from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
- n_wait[unfolded s_waiting_def, folded wq_def]
- have False by auto
- } with assms show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_simp [simp]:
- assumes "th' \<noteq> th"
- shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
- using readys_kept1[OF assms] readys_kept2[OF assms]
- by metis
-
-lemma pvD_kept [simp]:
- assumes "th' \<noteq> th"
- shows "pvD (e#s) th' = pvD s th'"
- using assms
- by (unfold pvD_def, simp)
-
-lemma cnp_cnv_cncs_kept:
- assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
- shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
-proof -
- {
- assume eq_th': "th' = th"
- have ?thesis using assms
- by (unfold eq_th', simp, unfold is_exit, simp)
- } moreover {
- assume h: "th' \<noteq> th"
- hence ?thesis using assms
- by (simp, simp add:is_exit)
- } ultimately show ?thesis by metis
-qed
-
-end
-
-context valid_trace_set
-begin
-
-lemma th_live_s [simp]: "th \<in> threads s"
-proof -
- from pip_e[unfolded is_set]
- show ?thesis
- by (cases, unfold runing_def readys_def, simp)
-qed
-
-lemma th_ready_s [simp]: "th \<in> readys s"
-proof -
- from pip_e[unfolded is_set]
- show ?thesis
- by (cases, unfold runing_def, simp)
-qed
-
-lemma th_not_live_es [simp]: "th \<in> threads (e#s)"
- by (unfold is_set, simp)
-
-
-lemma holdents_kept:
- shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
-proof -
- { fix cs'
- assume h: "cs' \<in> ?L"
- hence "cs' \<in> ?R"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_neq_simp, auto)
- } moreover {
- fix cs'
- assume h: "cs' \<in> ?R"
- hence "cs' \<in> ?L"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_neq_simp, auto)
- } ultimately show ?thesis by auto
-qed
-
-lemma cntCS_kept [simp]:
- shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
- using holdents_kept
- by (unfold cntCS_def, simp)
-
-lemma threads_kept[simp]:
- "threads (e#s) = threads s"
- by (unfold is_set, simp)
-
-lemma readys_kept1:
- assumes "th' \<in> readys (e#s)"
- shows "th' \<in> readys s"
-proof -
- { fix cs'
- assume wait: "waiting s th' cs'"
- have n_wait: "\<not> waiting (e#s) th' cs'"
- using assms by (auto simp:readys_def)
- from wait[unfolded s_waiting_def, folded wq_def]
- n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
- have False by auto
- } moreover have "th' \<in> threads s"
- using assms[unfolded readys_def] by auto
- ultimately show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_kept2:
- assumes "th' \<in> readys s"
- shows "th' \<in> readys (e#s)"
-proof -
- { fix cs'
- assume wait: "waiting (e#s) th' cs'"
- have n_wait: "\<not> waiting s th' cs'"
- using assms by (auto simp:readys_def)
- from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_neq_simp]
- n_wait[unfolded s_waiting_def, folded wq_def]
- have False by auto
- } with assms show ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma readys_simp [simp]:
- shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
- using readys_kept1 readys_kept2
- by metis
-
-lemma pvD_kept [simp]:
- shows "pvD (e#s) th' = pvD s th'"
- by (unfold pvD_def, simp)
-
-lemma cnp_cnv_cncs_kept:
- assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
- shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
- using assms
- by (unfold is_set, simp, fold is_set, simp)
-
-end
-
-context valid_trace
-begin
-
-lemma cnp_cnv_cncs: "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
-proof(induct rule:ind)
- case Nil
- thus ?case
- by (unfold cntP_def cntV_def pvD_def cntCS_def holdents_def
- s_holding_def, simp)
-next
- case (Cons s e)
- interpret vt_e: valid_trace_e s e using Cons by simp
- show ?case
- proof(cases e)
- case (Create th prio)
- interpret vt_create: valid_trace_create s e th prio
- using Create by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_create.cnp_cnv_cncs_kept)
- next
- case (Exit th)
- interpret vt_exit: valid_trace_exit s e th
- using Exit by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_exit.cnp_cnv_cncs_kept)
- next
- case (P th cs)
- interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_p.cnp_cnv_cncs_kept)
- next
- case (V th cs)
- interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_v.cnp_cnv_cncs_kept)
- next
- case (Set th prio)
- interpret vt_set: valid_trace_set s e th prio
- using Set by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_set.cnp_cnv_cncs_kept)
- qed
-qed
-
-lemma not_thread_holdents:
- assumes not_in: "th \<notin> threads s"
- shows "holdents s th = {}"
-proof -
- { fix cs
- assume "cs \<in> holdents s th"
- hence "holding s th cs" by (auto simp:holdents_def)
- from this[unfolded s_holding_def, folded wq_def]
- have "th \<in> set (wq s cs)" by auto
- with wq_threads have "th \<in> threads s" by auto
- with assms
- have False by simp
- } thus ?thesis by auto
-qed
-
-lemma not_thread_cncs:
- assumes not_in: "th \<notin> threads s"
- shows "cntCS s th = 0"
- using not_thread_holdents[OF assms]
- by (simp add:cntCS_def)
-
-lemma cnp_cnv_eq:
- assumes "th \<notin> threads s"
- shows "cntP s th = cntV s th"
- using assms cnp_cnv_cncs not_thread_cncs pvD_def
- by (auto)
-
-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 by auto
- from this[unfolded cp_alt_def]
- have eq_max:
- "Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th1)}) =
- Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th2)})"
- (is "Max ?L = Max ?R") .
- have "Max ?L \<in> ?L"
- proof(rule Max_in)
- show "finite ?L" by (simp add: finite_subtree_threads)
- next
- show "?L \<noteq> {}" using subtree_def by fastforce
- qed
- then obtain th1' where
- h_1: "Th th1' \<in> subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L"
- by auto
- have "Max ?R \<in> ?R"
- proof(rule Max_in)
- show "finite ?R" by (simp add: finite_subtree_threads)
- next
- show "?R \<noteq> {}" using subtree_def by fastforce
- qed
- then obtain th2' where
- h_2: "Th th2' \<in> subtree (RAG s) (Th th2)" "the_preced s th2' = Max ?R"
- by auto
- have "th1' = th2'"
- proof(rule preced_unique)
- from h_1(1)
- show "th1' \<in> threads s"
- proof(cases rule:subtreeE)
- case 1
- hence "th1' = th1" by simp
- with runing_1 show ?thesis by (auto simp:runing_def readys_def)
- next
- case 2
- from this(2)
- have "(Th th1', Th th1) \<in> (RAG s)^+" by (auto simp:ancestors_def)
- from tranclD[OF this]
- have "(Th th1') \<in> Domain (RAG s)" by auto
- from dm_RAG_threads[OF this] show ?thesis .
- qed
- next
- from h_2(1)
- show "th2' \<in> threads s"
- proof(cases rule:subtreeE)
- case 1
- hence "th2' = th2" by simp
- with runing_2 show ?thesis by (auto simp:runing_def readys_def)
- next
- case 2
- from this(2)
- have "(Th th2', Th th2) \<in> (RAG s)^+" by (auto simp:ancestors_def)
- from tranclD[OF this]
- have "(Th th2') \<in> Domain (RAG s)" by auto
- from dm_RAG_threads[OF this] show ?thesis .
- qed
- next
- have "the_preced s th1' = the_preced s th2'"
- using eq_max h_1(2) h_2(2) by metis
- thus "preced th1' s = preced th2' s" by (simp add:the_preced_def)
- qed
- from h_1(1)[unfolded this]
- have star1: "(Th th2', Th th1) \<in> (RAG s)^*" by (auto simp:subtree_def)
- from h_2(1)[unfolded this]
- have star2: "(Th th2', Th th2) \<in> (RAG s)^*" by (auto simp:subtree_def)
- from star_rpath[OF star1] obtain xs1
- where rp1: "rpath (RAG s) (Th th2') xs1 (Th th1)"
- by auto
- from star_rpath[OF star2] obtain xs2
- where rp2: "rpath (RAG s) (Th th2') xs2 (Th th2)"
- by auto
- from rp1 rp2
- show ?thesis
- proof(cases)
- case (less_1 xs')
- moreover have "xs' = []"
- proof(rule ccontr)
- assume otherwise: "xs' \<noteq> []"
- from rpath_plus[OF less_1(3) this]
- have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+" .
- from tranclD[OF this]
- obtain cs where "waiting s th1 cs"
- by (unfold s_RAG_def, fold waiting_eq, auto)
- with runing_1 show False
- by (unfold runing_def readys_def, auto)
- qed
- ultimately have "xs2 = xs1" by simp
- from rpath_dest_eq[OF rp1 rp2[unfolded this]]
- show ?thesis by simp
- next
- case (less_2 xs')
- moreover have "xs' = []"
- proof(rule ccontr)
- assume otherwise: "xs' \<noteq> []"
- from rpath_plus[OF less_2(3) this]
- have "(Th th2, Th th1) \<in> (RAG s)\<^sup>+" .
- from tranclD[OF this]
- obtain cs where "waiting s th2 cs"
- by (unfold s_RAG_def, fold waiting_eq, auto)
- with runing_2 show False
- by (unfold runing_def readys_def, auto)
- qed
- ultimately have "xs2 = xs1" by simp
- from rpath_dest_eq[OF rp1 rp2[unfolded this]]
- show ?thesis by simp
- qed
-qed
-
-lemma card_runing: "card (runing s) \<le> 1"
-proof(cases "runing s = {}")
- case True
- thus ?thesis by auto
-next
- case False
- then obtain th where [simp]: "th \<in> runing s" by auto
- from runing_unique[OF this]
- have "runing s = {th}" by auto
- thus ?thesis by auto
-qed
-
-lemma create_pre:
- assumes stp: "step s e"
- and not_in: "th \<notin> threads s"
- and is_in: "th \<in> threads (e#s)"
- obtains prio where "e = Create th prio"
-proof -
- from assms
- show ?thesis
- proof(cases)
- case (thread_create thread prio)
- with is_in not_in have "e = Create th prio" by simp
- from that[OF this] show ?thesis .
- next
- case (thread_exit thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_P thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_V thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_set thread)
- with assms show ?thesis by (auto intro!:that)
- qed
-qed
-
-lemma eq_pv_children:
- assumes eq_pv: "cntP s th = cntV s th"
- shows "children (RAG s) (Th th) = {}"
-proof -
- from cnp_cnv_cncs and eq_pv
- have "cntCS s th = 0"
- by (auto split:if_splits)
- from this[unfolded cntCS_def holdents_alt_def]
- have card_0: "card (the_cs ` children (RAG s) (Th th)) = 0" .
- have "finite (the_cs ` children (RAG s) (Th th))"
- by (simp add: fsbtRAGs.finite_children)
- from card_0[unfolded card_0_eq[OF this]]
- show ?thesis by auto
-qed
-
-lemma eq_pv_holdents:
- assumes eq_pv: "cntP s th = cntV s th"
- shows "holdents s th = {}"
- by (unfold holdents_alt_def eq_pv_children[OF assms], simp)
-
-lemma eq_pv_subtree:
- assumes eq_pv: "cntP s th = cntV s th"
- shows "subtree (RAG s) (Th th) = {Th th}"
- using eq_pv_children[OF assms]
- by (unfold subtree_children, simp)
-
-end
-
-lemma 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
- 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 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, auto)
- then obtain cs where "x1 = Cs cs"
- by (unfold s_RAG_def, auto)
- from rpath_nnl_lastE[OF rp[unfolded this]]
- show ?thesis by auto
- next
- 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)
-
-lemma dependants_alt_def1:
- "dependants (s::state) th = {th'. (Th th', Th th) \<in> (RAG s)^+}"
- using dependants_alt_def tRAG_trancl_eq by auto
-
-context valid_trace
-begin
-lemma count_eq_RAG_plus:
- assumes "cntP s th = cntV s th"
- shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
-proof(rule ccontr)
- assume otherwise: "{th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} \<noteq> {}"
- then obtain th' where "(Th th', Th th) \<in> (RAG s)^+" by auto
- from tranclD2[OF this]
- obtain z where "z \<in> children (RAG s) (Th th)"
- by (auto simp:children_def)
- with eq_pv_children[OF assms]
- show False by simp
-qed
-
-lemma eq_pv_dependants:
- assumes eq_pv: "cntP s th = cntV s th"
- shows "dependants s th = {}"
-proof -
- from count_eq_RAG_plus[OF assms, folded dependants_alt_def1]
- show ?thesis .
-qed
-
-end
-
-lemma eq_dependants: "dependants (wq s) = dependants s"
- by (simp add: s_dependants_abv wq_def)
-
-context valid_trace
-begin
-
-lemma count_eq_tRAG_plus:
- assumes "cntP s th = cntV s th"
- shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
- using assms eq_pv_dependants dependants_alt_def eq_dependants by auto
-
-lemma count_eq_RAG_plus_Th:
- assumes "cntP s th = cntV s th"
- shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
- using count_eq_RAG_plus[OF assms] by auto
-
-lemma count_eq_tRAG_plus_Th:
- assumes "cntP s th = cntV s th"
- shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
- using count_eq_tRAG_plus[OF assms] by auto
-end
-
-lemma inj_the_preced:
- "inj_on (the_preced s) (threads s)"
- by (metis inj_onI preced_unique the_preced_def)
-
-lemma tRAG_Field:
- "Field (tRAG s) \<subseteq> Field (RAG s)"
- by (unfold tRAG_alt_def Field_def, auto)
-
-lemma tRAG_ancestorsE:
- assumes "x \<in> ancestors (tRAG s) u"
- obtains th where "x = Th th"
-proof -
- from assms have "(u, x) \<in> (tRAG s)^+"
- by (unfold ancestors_def, auto)
- from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
- then obtain th where "x = Th th"
- by (unfold tRAG_alt_def, auto)
- from that[OF this] show ?thesis .
-qed
-
-lemma tRAG_mono:
- assumes "RAG s' \<subseteq> RAG s"
- shows "tRAG s' \<subseteq> tRAG s"
- using assms
- by (unfold tRAG_alt_def, auto)
-
-lemma holding_next_thI:
- assumes "holding s th cs"
- and "length (wq s cs) > 1"
- obtains th' where "next_th s th cs th'"
-proof -
- from assms(1)[folded holding_eq, unfolded cs_holding_def]
- have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)"
- by (unfold s_holding_def, fold wq_def, auto)
- then obtain rest where h1: "wq s cs = th#rest"
- by (cases "wq s cs", auto)
- with assms(2) have h2: "rest \<noteq> []" by auto
- let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
- have "next_th s th cs ?th'" using h1(1) h2
- by (unfold next_th_def, auto)
- from that[OF this] show ?thesis .
-qed
-
-lemma RAG_tRAG_transfer:
- assumes "vt s'"
- assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
- and "(Cs cs, Th th'') \<in> RAG s'"
- shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
-proof -
- interpret vt_s': valid_trace "s'" using assms(1)
- by (unfold_locales, simp)
- { fix n1 n2
- assume "(n1, n2) \<in> ?L"
- from this[unfolded tRAG_alt_def]
- obtain th1 th2 cs' where
- h: "n1 = Th th1" "n2 = Th th2"
- "(Th th1, Cs cs') \<in> RAG s"
- "(Cs cs', Th th2) \<in> RAG s" by auto
- from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
- from h(3) and assms(2)
- have "(Th th1, Cs cs') = (Th th, Cs cs) \<or>
- (Th th1, Cs cs') \<in> RAG s'" by auto
- hence "(n1, n2) \<in> ?R"
- proof
- assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
- hence eq_th1: "th1 = th" by simp
- moreover have "th2 = th''"
- proof -
- from h1 have "cs' = cs" by simp
- from assms(3) cs_in[unfolded this]
- show ?thesis using vt_s'.unique_RAG by auto
- qed
- ultimately show ?thesis using h(1,2) by auto
- next
- assume "(Th th1, Cs cs') \<in> RAG s'"
- with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
- by (unfold tRAG_alt_def, auto)
- from this[folded h(1, 2)] show ?thesis by auto
- qed
- } moreover {
- fix n1 n2
- assume "(n1, n2) \<in> ?R"
- hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
- hence "(n1, n2) \<in> ?L"
- proof
- assume "(n1, n2) \<in> tRAG s'"
- moreover have "... \<subseteq> ?L"
- proof(rule tRAG_mono)
- show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
- qed
- ultimately show ?thesis by auto
- next
- assume eq_n: "(n1, n2) = (Th th, Th th'')"
- from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
- moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
- ultimately show ?thesis
- by (unfold eq_n tRAG_alt_def, auto)
- qed
- } ultimately show ?thesis by auto
-qed
-
-context valid_trace
-begin
-
-lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
-
-end
-
-lemma tRAG_subtree_eq:
- "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}"
- (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 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 rg_RAG_threads assms
- show False by (unfold Field_def, blast)
-qed
-
-end
-
-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 eq_pv cnp_cnv_cncs
- have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:pvD_def)
- thus ?thesis
- proof
- assume "th \<notin> threads s"
- with rg_RAG_threads dm_RAG_threads
- show ?thesis
- by (auto simp add: detached_def s_RAG_def s_waiting_abv
- s_holding_abv wq_def Domain_iff Range_iff)
- next
- assume "th \<in> readys s"
- moreover have "Th th \<notin> Range (RAG s)"
- proof -
- from eq_pv_children[OF assms]
- have "children (RAG s) (Th th) = {}" .
- thus ?thesis
- by (unfold children_def, auto)
- qed
- ultimately show ?thesis
- by (auto simp add: detached_def s_RAG_def s_waiting_abv
- s_holding_abv wq_def readys_def)
- qed
-qed
-
-lemma detached_elim:
- assumes dtc: "detached s th"
- shows "cntP s th = cntV s th"
-proof -
- have cncs_z: "cntCS s th = 0"
- proof -
- from dtc have "holdents s th = {}"
- unfolding detached_def holdents_test s_RAG_def
- by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
- thus ?thesis by (auto simp:cntCS_def)
- qed
- show ?thesis
- proof(cases "th \<in> threads s")
- case True
- with dtc
- have "th \<in> readys s"
- by (unfold readys_def detached_def Field_def Domain_def Range_def,
- auto simp:waiting_eq s_RAG_def)
- with cncs_z show ?thesis using cnp_cnv_cncs by (simp add:pvD_def)
- next
- case False
- with cncs_z and cnp_cnv_cncs show ?thesis by (simp add:pvD_def)
- qed
-qed
-
-lemma detached_eq:
- shows "(detached s th) = (cntP s th = cntV s th)"
- by (insert vt, auto intro:detached_intro detached_elim)
-
-end
-
-context valid_trace
-begin
-(* ddd *)
-lemma cp_gen_rec:
- assumes "x = Th th"
- shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
-proof(cases "children (tRAG s) x = {}")
- 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
-
-lemma next_th_holding:
- assumes 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
-
-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
-
-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)
-
-context valid_trace
-begin
-
-thm th_chain_to_ready
-
-find_theorems subtree Th RAG
-
-lemma "(threads s) = (\<Union> th \<in> readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)})"
- (is "?L = ?R")
-proof -
- { fix th1
- assume "th1 \<in> ?L"
- from th_chain_to_ready[OF this]
- have "th1 \<in> readys s \<or> (\<exists>th'a. th'a \<in> readys s \<and> (Th th1, Th th'a) \<in> (RAG s)\<^sup>+)" .
- hence "th1 \<in> ?R"
- proof
- assume "th1 \<in> readys s"
- thus ?thesis by (auto simp:subtree_def)
- next
- assume "\<exists>th'a. th'a \<in> readys s \<and> (Th th1, Th th'a) \<in> (RAG s)\<^sup>+"
- thus ?thesis
- qed
- } moreover
- { fix th'
- assume "th' \<in> ?R"
- have "th' \<in> ?L" sorry
- } ultimately show ?thesis by auto
-qed
-
-lemma max_cp_readys_threads_pre: (* ccc *)
- 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 (the_preced s ` threads s)"
- proof -
- let ?p = "Max (the_preced s ` threads s)"
- let ?f = "the_preced s"
- have "?p \<in> (?f ` 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
-
-end
-