pip: comparison PIPBasics.thy~

equal deleted inserted replaced

-:4805c6333fef
+:c7ba70dc49bd
-theory PIPBasics
-imports PIPDefs
-begin
-locale valid_trace =
-fixes s
-assumes vt : "vt s"
-locale valid_trace_e = valid_trace +
-fixes e
-assumes vt_e: "vt (e#s)"
-begin
-lemma pip_e: "PIP s e"
-using vt_e by (cases, simp)
-end
-lemma runing_ready:
-shows "runing s \<subseteq> readys s"
-unfolding runing_def readys_def
-by auto
-lemma readys_threads:
-shows "readys s \<subseteq> threads s"
-unfolding readys_def
-by auto
-lemma wq_v_neq:
-"cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
-by (auto simp:wq_def Let_def cp_def split:list.splits)
-context valid_trace
-begin
-lemma actor_inv:
-assumes "PIP s e"
-and "\<not> isCreate e"
-shows "actor e \<in> runing s"
-using assms
-by (induct, auto)
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
-assumes "PP []"
-and "(\<And>s e. valid_trace s \<Longrightarrow> valid_trace (e#s) \<Longrightarrow>
-PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
-shows "PP s"
-proof(rule vt.induct[OF vt])
-from assms(1) show "PP []" .
-next
-fix s e
-assume h: "vt s" "PP s" "PIP s e"
-show "PP (e # s)"
-proof(cases rule:assms(2))
-from h(1) show v1: "valid_trace s" by (unfold_locales, simp)
-next
-from h(1,3) have "vt (e#s)" by auto
-thus "valid_trace (e # s)" by (unfold_locales, simp)
-qed (insert h, auto)
-qed
-lemma wq_distinct: "distinct (wq s cs)"
-proof(rule ind, simp add:wq_def)
-fix s e
-assume h1: "step s e"
-and h2: "distinct (wq s cs)"
-thus "distinct (wq (e # s) cs)"
-proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
-fix thread s
-assume h1: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
-and h2: "thread \<in> set (wq_fun (schs s) cs)"
-and h3: "thread \<in> runing s"
-show "False"
-proof -
-from h3 have "\<And> cs. thread \<in>  set (wq_fun (schs s) cs) \<Longrightarrow>
-thread = hd ((wq_fun (schs s) cs))"
-by (simp add:runing_def readys_def s_waiting_def wq_def)
-from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" .
-with h2
-have "(Cs cs, Th thread) \<in> (RAG s)"
-by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
-with h1 show False by auto
-qed
-next
-fix thread s a list
-assume dst: "distinct list"
-show "distinct (SOME q. distinct q \<and> set q = set list)"
-proof(rule someI2)
-from dst show  "distinct list \<and> set list = set list" by auto
-next
-fix q assume "distinct q \<and> set q = set list"
-thus "distinct q" by auto
-qed
-qed
-qed
-end
-context valid_trace_e
-begin
-text {*
-The following lemma shows that only the @{text "P"}
-operation can add new thread into waiting queues.
-Such kind of lemmas are very obvious, but need to be checked formally.
-This is a kind of confirmation that our modelling is correct.
-*}
-lemma block_pre:
-assumes s_ni: "thread \<notin>  set (wq s cs)"
-and s_i: "thread \<in> set (wq (e#s) cs)"
-shows "e = P thread cs"
-proof -
-show ?thesis
-proof(cases e)
-case (P th cs)
-with assms
-show ?thesis
-by (auto simp:wq_def Let_def split:if_splits)
-next
-case (Create th prio)
-with assms show ?thesis
-by (auto simp:wq_def Let_def split:if_splits)
-next
-case (Exit th)
-with assms show ?thesis
-by (auto simp:wq_def Let_def split:if_splits)
-next
-case (Set th prio)
-with assms show ?thesis
-by (auto simp:wq_def Let_def split:if_splits)
-next
-case (V th cs)
-with vt_e assms show ?thesis
-apply (auto simp:wq_def Let_def split:if_splits)
-proof -
-fix q qs
-assume h1: "thread \<notin> set (wq_fun (schs s) cs)"
-and h2: "q # qs = wq_fun (schs s) cs"
-and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
-and vt: "vt (V th cs # s)"
-from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp
-moreover have "thread \<in> set qs"
-proof -
-have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
-proof(rule someI2)
-from wq_distinct [of cs]
-and h2[symmetric, folded wq_def]
-show "distinct qs \<and> set qs = set qs" by auto
-next
-fix x assume "distinct x \<and> set x = set qs"
-thus "set x = set qs" by auto
-qed
-with h3 show ?thesis by simp
-qed
-ultimately show "False" by auto
-qed
-qed
-qed
-end
-text {*
-The following lemmas is also obvious and shallow. It says
-that only running thread can request for a critical resource
-and that the requested resource must be one which is
-not current held by the thread.
-*}
-lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
-thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (RAG s)^+"
-apply (ind_cases "vt ((P thread cs)#s)")
-apply (ind_cases "step s (P thread cs)")
-by auto
-lemma abs1:
-assumes ein: "e \<in> set es"
-and neq: "hd es \<noteq> hd (es @ [x])"
-shows "False"
-proof -
-from ein have "es \<noteq> []" by auto
-then obtain e ess where "es = e # ess" by (cases es, auto)
-with neq show ?thesis by auto
-qed
-lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
-by (cases es, auto)
-inductive_cases evt_cons: "vt (a#s)"
-context valid_trace_e
-begin
-lemma abs2:
-assumes inq: "thread \<in> set (wq s cs)"
-and nh: "thread = hd (wq s cs)"
-and qt: "thread \<noteq> hd (wq (e#s) cs)"
-and inq': "thread \<in> set (wq (e#s) cs)"
-shows "False"
-proof -
-from vt_e assms show "False"
-apply (cases e)
-apply ((simp split:if_splits add:Let_def wq_def)[1])+
-apply (insert abs1, fast)[1]
-apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)
-proof -
-fix th qs
-assume vt: "vt (V th cs # s)"
-and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
-and eq_wq: "wq_fun (schs s) cs = thread # qs"
-show "False"
-proof -
-from wq_distinct[of cs]
-and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
-moreover have "thread \<in> set qs"
-proof -
-have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
-proof(rule someI2)
-from wq_distinct [of cs]
-and eq_wq [folded wq_def]
-show "distinct qs \<and> set qs = set qs" by auto
-next
-fix x assume "distinct x \<and> set x = set qs"
-thus "set x = set qs" by auto
-qed
-with th_in show ?thesis by auto
-qed
-ultimately show ?thesis by auto
-qed
-qed
-qed
-end
-context valid_trace
-begin
-lemma vt_moment: "\<And> t. vt (moment t s)"
-proof(induct rule:ind)
-case Nil
-thus ?case by (simp add:vt_nil)
-next
-case (Cons s e t)
-show ?case
-proof(cases "t \<ge> length (e#s)")
-case True
-from True have "moment t (e#s) = e#s" by simp
-thus ?thesis using Cons
-by (simp add:valid_trace_def)
-next
-case False
-from Cons have "vt (moment t s)" by simp
-moreover have "moment t (e#s) = moment t s"
-proof -
-from False have "t \<le> length s" by simp
-from moment_app [OF this, of "[e]"]
-show ?thesis by simp
-qed
-ultimately show ?thesis by simp
-qed
-qed
-(* Wrong:
-lemma \<lbrakk>thread \<in> set (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
-*)
-text {* (* ddd *)
-The nature of the work is like this: since it starts from a very simple and basic
-model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
-For instance, the fact
-that one thread can not be blocked by two critical resources at the same time
-is obvious, because only running threads can make new requests, if one is waiting for
-a critical resource and get blocked, it can not make another resource request and get
-blocked the second time (because it is not running).
-To derive this fact, one needs to prove by contraction and
-reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
-named @{text "p_split"}, which is about status changing along the time axis. It says if
-a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
-but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"}
-in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history
-of events leading to it), such that @{text "Q"} switched
-from being @{text "False"} to @{text "True"} and kept being @{text "True"}
-till the last moment of @{text "s"}.
-Suppose a thread @{text "th"} is blocked
-on @{text "cs1"} and @{text "cs2"} in some state @{text "s"},
-since no thread is blocked at the very beginning, by applying
-@{text "p_split"} to these two blocking facts, there exist
-two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that
-@{text "th"} got blocked on @{text "cs1"} and @{text "cs2"}
-and kept on blocked on them respectively ever since.
-Without lose of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
-However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
-in blocked state at moment @{text "t2"} and could not
-make any request and get blocked the second time: Contradiction.
-*}
-lemma waiting_unique_pre:
-assumes h11: "thread \<in> set (wq s cs1)"
-and h12: "thread \<noteq> hd (wq s cs1)"
-assumes h21: "thread \<in> set (wq s cs2)"
-and h22: "thread \<noteq> hd (wq s cs2)"
-and neq12: "cs1 \<noteq> cs2"
-shows "False"
-proof -
-let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
-from h11 and h12 have q1: "?Q cs1 s" by simp
-from h21 and h22 have q2: "?Q cs2 s" by simp
-have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
-have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
-from p_split [of "?Q cs1", OF q1 nq1]
-obtain t1 where lt1: "t1 < length s"
-and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
-thread \<noteq> hd (wq (moment t1 s) cs1))"
-and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
-thread \<noteq> hd (wq (moment i' s) cs1))" by auto
-from p_split [of "?Q cs2", OF q2 nq2]
-obtain t2 where lt2: "t2 < length s"
-and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
-thread \<noteq> hd (wq (moment t2 s) cs2))"
-and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
-thread \<noteq> hd (wq (moment i' s) cs2))" by auto
-show ?thesis
-proof -
-{
-assume lt12: "t1 < t2"
-let ?t3 = "Suc t2"
-from lt2 have le_t3: "?t3 \<le> length s" by auto
-from moment_plus [OF this]
-obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
-have "t2 < ?t3" by simp
-from nn2 [rule_format, OF this] and eq_m
-have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-have "vt (e#moment t2 s)"
-proof -
-from vt_moment
-have "vt (moment ?t3 s)" .
-with eq_m show ?thesis by simp
-qed
-then interpret vt_e: valid_trace_e "moment t2 s" "e"
-by (unfold_locales, auto, cases, simp)
-have ?thesis
-proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-case True
-from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
-by auto
-from vt_e.abs2 [OF True eq_th h2 h1]
-show ?thesis by auto
-next
-case False
-from vt_e.block_pre[OF False h1]
-have "e = P thread cs2" .
-with vt_e.vt_e have "vt ((P thread cs2)# moment t2 s)" by simp
-from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
-with runing_ready have "thread \<in> readys (moment t2 s)" by auto
-with nn1 [rule_format, OF lt12]
-show ?thesis  by (simp add:readys_def wq_def s_waiting_def, auto)
-qed
-} moreover {
-assume lt12: "t2 < t1"
-let ?t3 = "Suc t1"
-from lt1 have le_t3: "?t3 \<le> length s" by auto
-from moment_plus [OF this]
-obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
-have lt_t3: "t1 < ?t3" by simp
-from nn1 [rule_format, OF this] and eq_m
-have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
-h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
-have "vt  (e#moment t1 s)"
-proof -
-from vt_moment
-have "vt (moment ?t3 s)" .
-with eq_m show ?thesis by simp
-qed
-then interpret vt_e: valid_trace_e "moment t1 s" e
-by (unfold_locales, auto, cases, auto)
-have ?thesis
-proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
-case True
-from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
-by auto
-from vt_e.abs2 True eq_th h2 h1
-show ?thesis by auto
-next
-case False
-from vt_e.block_pre [OF False h1]
-have "e = P thread cs1" .
-with vt_e.vt_e have "vt ((P thread cs1)# moment t1 s)" by simp
-from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
-with runing_ready have "thread \<in> readys (moment t1 s)" by auto
-with nn2 [rule_format, OF lt12]
-show ?thesis  by (simp add:readys_def wq_def s_waiting_def, auto)
-qed
-} moreover {
-assume eqt12: "t1 = t2"
-let ?t3 = "Suc t1"
-from lt1 have le_t3: "?t3 \<le> length s" by auto
-from moment_plus [OF this]
-obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
-have lt_t3: "t1 < ?t3" by simp
-from nn1 [rule_format, OF this] and eq_m
-have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
-h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
-have vt_e: "vt (e#moment t1 s)"
-proof -
-from vt_moment
-have "vt (moment ?t3 s)" .
-with eq_m show ?thesis by simp
-qed
-then interpret vt_e: valid_trace_e "moment t1 s" e
-by (unfold_locales, auto, cases, auto)
-have ?thesis
-proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
-case True
-from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
-by auto
-from vt_e.abs2 [OF True eq_th h2 h1]
-show ?thesis by auto
-next
-case False
-from vt_e.block_pre [OF False h1]
-have eq_e1: "e = P thread cs1" .
-have lt_t3: "t1 < ?t3" by simp
-with eqt12 have "t2 < ?t3" by simp
-from nn2 [rule_format, OF this] and eq_m and eqt12
-have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
-h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
-show ?thesis
-proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
-case True
-from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
-by auto
-from vt_e and eqt12 have "vt (e#moment t2 s)" by simp
-then interpret vt_e2: valid_trace_e "moment t2 s" e
-by (unfold_locales, auto, cases, auto)
-from vt_e2.abs2 [OF True eq_th h2 h1]
-show ?thesis .
-next
-case False
-have "vt (e#moment t2 s)"
-proof -
-from vt_moment eqt12
-have "vt (moment (Suc t2) s)" by auto
-with eq_m eqt12 show ?thesis by simp
-qed
-then interpret vt_e2: valid_trace_e "moment t2 s" e
-by (unfold_locales, auto, cases, auto)
-from vt_e2.block_pre [OF False h1]
-have "e = P thread cs2" .
-with eq_e1 neq12 show ?thesis by auto
-qed
-qed
-} ultimately show ?thesis by arith
-qed
-qed
-text {*
-This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
-*}
-lemma waiting_unique:
-assumes "waiting s th cs1"
-and "waiting s th cs2"
-shows "cs1 = cs2"
-using waiting_unique_pre assms
-unfolding wq_def s_waiting_def
-by auto
-end
-(* not used *)
-text {*
-Every thread can only be blocked on one critical resource,
-symmetrically, every critical resource can only be held by one thread.
-This fact is much more easier according to our definition.
-*}
-lemma held_unique:
-assumes "holding (s::event list) th1 cs"
-and "holding s th2 cs"
-shows "th1 = th2"
-by (insert assms, unfold s_holding_def, auto)
-lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
-apply (induct s, auto)
-by (case_tac a, auto split:if_splits)
-lemma last_set_unique:
-"\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
-\<Longrightarrow> th1 = th2"
-apply (induct s, auto)
-by (case_tac a, auto split:if_splits dest:last_set_lt)
-lemma preced_unique :
-assumes pcd_eq: "preced th1 s = preced th2 s"
-and th_in1: "th1 \<in> threads s"
-and th_in2: " th2 \<in> threads s"
-shows "th1 = th2"
-proof -
-from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
-from last_set_unique [OF this th_in1 th_in2]
-show ?thesis .
-qed
-lemma preced_linorder:
-assumes neq_12: "th1 \<noteq> th2"
-and th_in1: "th1 \<in> threads s"
-and th_in2: " th2 \<in> threads s"
-shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
-from preced_unique [OF _ th_in1 th_in2] and neq_12
-have "preced th1 s \<noteq> preced th2 s" by auto
-thus ?thesis by auto
-qed
-(* An aux lemma used later *)
-lemma unique_minus:
-fixes x y z r
-assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-and xy: "(x, y) \<in> r"
-and xz: "(x, z) \<in> r^+"
-and neq: "y \<noteq> z"
-shows "(y, z) \<in> r^+"
-proof -
-from xz and neq show ?thesis
-proof(induct)
-case (base ya)
-have "(x, ya) \<in> r" by fact
-from unique [OF xy this] have "y = ya" .
-with base show ?case by auto
-next
-case (step ya z)
-show ?case
-proof(cases "y = ya")
-case True
-from step True show ?thesis by simp
-next
-case False
-from step False
-show ?thesis by auto
-qed
-qed
-qed
-lemma unique_base:
-fixes r x y z
-assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-and xy: "(x, y) \<in> r"
-and xz: "(x, z) \<in> r^+"
-and neq_yz: "y \<noteq> z"
-shows "(y, z) \<in> r^+"
-proof -
-from xz neq_yz show ?thesis
-proof(induct)
-case (base ya)
-from xy unique base show ?case by auto
-next
-case (step ya z)
-show ?case
-proof(cases "y = ya")
-case True
-from True step show ?thesis by auto
-next
-case False
-from False step
-have "(y, ya) \<in> r\<^sup>+" by auto
-with step show ?thesis by auto
-qed
-qed
-qed
-lemma unique_chain:
-fixes r x y z
-assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
-and xy: "(x, y) \<in> r^+"
-and xz: "(x, z) \<in> r^+"
-and neq_yz: "y \<noteq> z"
-shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
-proof -
-from xy xz neq_yz show ?thesis
-proof(induct)
-case (base y)
-have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
-from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
-next
-case (step y za)
-show ?case
-proof(cases "y = z")
-case True
-from True step show ?thesis by auto
-next
-case False
-from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
-thus ?thesis
-proof
-assume "(z, y) \<in> r\<^sup>+"
-with step have "(z, za) \<in> r\<^sup>+" by auto
-thus ?thesis by auto
-next
-assume h: "(y, z) \<in> r\<^sup>+"
-from step have yza: "(y, za) \<in> r" by simp
-from step have "za \<noteq> z" by simp
-from unique_minus [OF _ yza h this] and unique
-have "(za, z) \<in> r\<^sup>+" by auto
-thus ?thesis by auto
-qed
-qed
-qed
-qed
-text {*
-The following three lemmas show that @{text "RAG"} does not change
-by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
-events, respectively.
-*}
-lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-text {*
-The following lemmas are used in the proof of
-lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed
-by @{text "V"}-events.
-However, since our model is very concise, such  seemingly obvious lemmas need to be derived from scratch,
-starting from the model definitions.
-*}
-lemma step_v_hold_inv[elim_format]:
-"\<And>c t. \<lbrakk>vt (V th cs # s);
-\<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow>
-next_th s th cs t \<and> c = cs"
-proof -
-fix c t
-assume vt: "vt (V th cs # s)"
-and nhd: "\<not> holding (wq s) t c"
-and hd: "holding (wq (V th cs # s)) t c"
-show "next_th s th cs t \<and> c = cs"
-proof(cases "c = cs")
-case False
-with nhd hd show ?thesis
-by (unfold cs_holding_def wq_def, auto simp:Let_def)
-next
-case True
-with step_back_step [OF vt]
-have "step s (V th c)" by simp
-hence "next_th s th cs t"
-proof(cases)
-assume "holding s th c"
-with nhd hd show ?thesis
-apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
-auto simp:Let_def split:list.splits if_splits)
-proof -
-assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
-moreover have "\<dots> = set []"
-proof(rule someI2)
-show "distinct [] \<and> [] = []" by auto
-next
-fix x assume "distinct x \<and> x = []"
-thus "set x = set []" by auto
-qed
-ultimately show False by auto
-next
-assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
-moreover have "\<dots> = set []"
-proof(rule someI2)
-show "distinct [] \<and> [] = []" by auto
-next
-fix x assume "distinct x \<and> x = []"
-thus "set x = set []" by auto
-qed
-ultimately show False by auto
-qed
-qed
-with True show ?thesis by auto
-qed
-qed
-text {*
-The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be
-derived from scratch, which confirms the correctness of the definition of @{text "next_th"}.
-*}
-lemma step_v_wait_inv[elim_format]:
-"\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
-\<rbrakk>
-\<Longrightarrow> (next_th s th cs t \<and> cs = c)"
-proof -
-fix t c
-assume vt: "vt (V th cs # s)"
-and nw: "\<not> waiting (wq (V th cs # s)) t c"
-and wt: "waiting (wq s) t c"
-from vt interpret vt_v: valid_trace_e s "V th cs"
-by  (cases, unfold_locales, simp)
-show "next_th s th cs t \<and> cs = c"
-proof(cases "cs = c")
-case False
-with nw wt show ?thesis
-by (auto simp:cs_waiting_def wq_def Let_def)
-next
-case True
-from nw[folded True] wt[folded True]
-have "next_th s th cs t"
-apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
-proof -
-fix a list
-assume t_in: "t \<in> set list"
-and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
-and eq_wq: "wq_fun (schs s) cs = a # list"
-have " set (SOME q. distinct q \<and> set q = set list) = set list"
-proof(rule someI2)
-from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
-show "distinct list \<and> set list = set list" by auto
-next
-show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
-by auto
-qed
-with t_ni and t_in show "a = th" by auto
-next
-fix a list
-assume t_in: "t \<in> set list"
-and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
-and eq_wq: "wq_fun (schs s) cs = a # list"
-have " set (SOME q. distinct q \<and> set q = set list) = set list"
-proof(rule someI2)
-from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
-show "distinct list \<and> set list = set list" by auto
-next
-show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
-by auto
-qed
-with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto
-next
-fix a list
-assume eq_wq: "wq_fun (schs s) cs = a # list"
-from step_back_step[OF vt]
-show "a = th"
-proof(cases)
-assume "holding s th cs"
-with eq_wq show ?thesis
-by (unfold s_holding_def wq_def, auto)
-qed
-qed
-with True show ?thesis by simp
-qed
-qed
-lemma step_v_not_wait[consumes 3]:
-"\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
-by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
-lemma step_v_release:
-"\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
-proof -
-assume vt: "vt (V th cs # s)"
-and hd: "holding (wq (V th cs # s)) th cs"
-from vt interpret vt_v: valid_trace_e s "V th cs"
-by (cases, unfold_locales, simp+)
-from step_back_step [OF vt] and hd
-show "False"
-proof(cases)
-assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"
-thus ?thesis
-apply (unfold s_holding_def wq_def cs_holding_def)
-apply (auto simp:Let_def split:list.splits)
-proof -
-fix list
-assume eq_wq[folded wq_def]:
-"wq_fun (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
-and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
-\<in> set (SOME q. distinct q \<and> set q = set list)"
-have "set (SOME q. distinct q \<and> set q = set list) = set list"
-proof(rule someI2)
-from vt_v.wq_distinct[of cs] and eq_wq
-show "distinct list \<and> set list = set list" by auto
-next
-show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
-by auto
-qed
-moreover have "distinct  (hd (SOME q. distinct q \<and> set q = set list) # list)"
-proof -
-from vt_v.wq_distinct[of cs] and eq_wq
-show ?thesis by auto
-qed
-moreover note eq_wq and hd_in
-ultimately show "False" by auto
-qed
-qed
-qed
-lemma step_v_get_hold:
-"\<And>th'. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
-apply (unfold cs_holding_def next_th_def wq_def,
-auto simp:Let_def)
-proof -
-fix rest
-assume vt: "vt (V th cs # s)"
-and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest"
-and nrest: "rest \<noteq> []"
-and ni: "hd (SOME q. distinct q \<and> set q = set rest)
-\<notin> set (SOME q. distinct q \<and> set q = set rest)"
-from vt interpret vt_v: valid_trace_e s "V th cs"
-by (cases, unfold_locales, simp+)
-have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-proof(rule someI2)
-from vt_v.wq_distinct[of cs] and eq_wq
-show "distinct rest \<and> set rest = set rest" by auto
-next
-fix x assume "distinct x \<and> set x = set rest"
-hence "set x = set rest" by auto
-with nrest
-show "x \<noteq> []" by (case_tac x, auto)
-qed
-with ni show "False" by auto
-qed
-lemma step_v_release_inv[elim_format]:
-"\<And>c t. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow>
-c = cs \<and> t = th"
-apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
-proof -
-fix a list
-assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
-from step_back_step [OF vt] show "a = th"
-proof(cases)
-assume "holding s th cs" with eq_wq
-show ?thesis
-by (unfold s_holding_def wq_def, auto)
-qed
-next
-fix a list
-assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
-from step_back_step [OF vt] show "a = th"
-proof(cases)
-assume "holding s th cs" with eq_wq
-show ?thesis
-by (unfold s_holding_def wq_def, auto)
-qed
-qed
-lemma step_v_waiting_mono:
-"\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
-proof -
-fix t c
-let ?s' = "(V th cs # s)"
-assume vt: "vt ?s'"
-and wt: "waiting (wq ?s') t c"
-from vt interpret vt_v: valid_trace_e s "V th cs"
-by (cases, unfold_locales, simp+)
-show "waiting (wq s) t c"
-proof(cases "c = cs")
-case False
-assume neq_cs: "c \<noteq> cs"
-hence "waiting (wq ?s') t c = waiting (wq s) t c"
-by (unfold cs_waiting_def wq_def, auto simp:Let_def)
-with wt show ?thesis by simp
-next
-case True
-with wt show ?thesis
-apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
-proof -
-fix a list
-assume not_in: "t \<notin> set list"
-and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
-and eq_wq: "wq_fun (schs s) cs = a # list"
-have "set (SOME q. distinct q \<and> set q = set list) = set list"
-proof(rule someI2)
-from vt_v.wq_distinct [of cs]
-and eq_wq[folded wq_def]
-show "distinct list \<and> set list = set list" by auto
-next
-fix x assume "distinct x \<and> set x = set list"
-thus "set x = set list" by auto
-qed
-with not_in is_in show "t = a" by auto
-next
-fix list
-assume is_waiting: "waiting (wq (V th cs # s)) t cs"
-and eq_wq: "wq_fun (schs s) cs = t # list"
-hence "t \<in> set list"
-apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
-proof -
-assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
-moreover have "\<dots> = set list"
-proof(rule someI2)
-from vt_v.wq_distinct [of cs]
-and eq_wq[folded wq_def]
-show "distinct list \<and> set list = set list" by auto
-next
-fix x assume "distinct x \<and> set x = set list"
-thus "set x = set list" by auto
-qed
-ultimately show "t \<in> set list" by simp
-qed
-with eq_wq and vt_v.wq_distinct [of cs, unfolded wq_def]
-show False by auto
-qed
-qed
-qed
-text {* (* ddd *)
-The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
-with the happening of @{text "V"}-events:
-*}
-lemma step_RAG_v:
-fixes th::thread
-assumes vt:
-"vt (V th cs#s)"
-shows "
-RAG (V th cs # s) =
-RAG s - {(Cs cs, Th th)} -
-{(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-{(Cs cs, Th th') |th'.  next_th s th cs th'}"
-apply (insert vt, unfold s_RAG_def)
-apply (auto split:if_splits list.splits simp:Let_def)
-apply (auto elim: step_v_waiting_mono step_v_hold_inv
-step_v_release step_v_wait_inv
-step_v_get_hold step_v_release_inv)
-apply (erule_tac step_v_not_wait, auto)
-done
-text {*
-The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
-with the happening of @{text "P"}-events:
-*}
-lemma step_RAG_p:
-"vt (P th cs#s) \<Longrightarrow>
-RAG (P th cs # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
-else RAG s \<union> {(Th th, Cs cs)})"
-apply(simp only: s_RAG_def wq_def)
-apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
-apply(case_tac "csa = cs", auto)
-apply(fold wq_def)
-apply(drule_tac step_back_step)
-apply(ind_cases " step s (P (hd (wq s cs)) cs)")
-apply(simp add:s_RAG_def wq_def cs_holding_def)
-apply(auto)
-done
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
-by (unfold s_RAG_def, auto)
-context valid_trace
-begin
-text {*
-The following lemma shows that @{text "RAG"} is acyclic.
-The overall structure is by induction on the formation of @{text "vt s"}
-and then case analysis on event @{text "e"}, where the non-trivial cases
-for those for @{text "V"} and @{text "P"} events.
-*}
-lemma acyclic_RAG:
-shows "acyclic (RAG s)"
-using vt
-proof(induct)
-case (vt_cons s e)
-interpret vt_s: valid_trace s using vt_cons(1)
-by (unfold_locales, simp)
-assume ih: "acyclic (RAG s)"
-and stp: "step s e"
-and vt: "vt s"
-show ?case
-proof(cases e)
-case (Create th prio)
-with ih
-show ?thesis by (simp add:RAG_create_unchanged)
-next
-case (Exit th)
-with ih show ?thesis by (simp add:RAG_exit_unchanged)
-next
-case (V th cs)
-from V vt stp have vtt: "vt (V th cs#s)" by auto
-from step_RAG_v [OF this]
-have eq_de:
-"RAG (e # s) =
-RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-{(Cs cs, Th th') |th'. next_th s th cs th'}"
-(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
-from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
-from step_back_step [OF vtt]
-have "step s (V th cs)" .
-thus ?thesis
-proof(cases)
-assume "holding s th cs"
-hence th_in: "th \<in> set (wq s cs)" and
-eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
-then obtain rest where
-eq_wq: "wq s cs = th#rest"
-by (cases "wq s cs", auto)
-show ?thesis
-proof(cases "rest = []")
-case False
-let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
-from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
-by (unfold next_th_def, auto)
-let ?E = "(?A - ?B - ?C)"
-have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
-proof
-assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
-hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-from tranclD [OF this]
-obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
-hence th_d: "(Th ?th', x) \<in> ?A" by simp
-from RAG_target_th [OF this]
-obtain cs' where eq_x: "x = Cs cs'" by auto
-with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
-hence wt_th': "waiting s ?th' cs'"
-unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
-hence "cs' = cs"
-proof(rule vt_s.waiting_unique)
-from eq_wq vt_s.wq_distinct[of cs]
-show "waiting s ?th' cs"
-apply (unfold s_waiting_def wq_def, auto)
-proof -
-assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-and eq_wq: "wq_fun (schs s) cs = th # rest"
-have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-proof(rule someI2)
-from vt_s.wq_distinct[of cs] and eq_wq
-show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
-next
-fix x assume "distinct x \<and> set x = set rest"
-with False show "x \<noteq> []" by auto
-qed
-hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
-set (SOME q. distinct q \<and> set q = set rest)" by auto
-moreover have "\<dots> = set rest"
-proof(rule someI2)
-from vt_s.wq_distinct[of cs] and eq_wq
-show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
-next
-show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-qed
-moreover note hd_in
-ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
-next
-assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
-have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-proof(rule someI2)
-from vt_s.wq_distinct[of cs] and eq_wq
-show "distinct rest \<and> set rest = set rest" by auto
-next
-fix x assume "distinct x \<and> set x = set rest"
-with False show "x \<noteq> []" by auto
-qed
-hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
-set (SOME q. distinct q \<and> set q = set rest)" by auto
-moreover have "\<dots> = set rest"
-proof(rule someI2)
-from vt_s.wq_distinct[of cs] and eq_wq
-show "distinct rest \<and> set rest = set rest" by auto
-next
-show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-qed
-moreover note hd_in
-ultimately show False by auto
-qed
-qed
-with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
-with False
-show "False" by (auto simp: next_th_def eq_wq)
-qed
-with acyclic_insert[symmetric] and ac
-and eq_de eq_D show ?thesis by auto
-next
-case True
-with eq_wq
-have eq_D: "?D = {}"
-by (unfold next_th_def, auto)
-with eq_de ac
-show ?thesis by auto
-qed
-qed
-next
-case (P th cs)
-from P vt stp have vtt: "vt (P th cs#s)" by auto
-from step_RAG_p [OF this] P
-have "RAG (e # s) =
-(if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
-RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
-by simp
-moreover have "acyclic ?R"
-proof(cases "wq s cs = []")
-case True
-hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
-have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
-proof
-assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
-hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-from tranclD2 [OF this]
-obtain x where "(x, Cs cs) \<in> RAG s" by auto
-with True show False by (auto simp:s_RAG_def cs_waiting_def)
-qed
-with acyclic_insert ih eq_r show ?thesis by auto
-next
-case False
-hence eq_r: "?R =  RAG s \<union> {(Th th, Cs cs)}" by simp
-have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
-proof
-assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
-hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
-moreover from step_back_step [OF vtt] have "step s (P th cs)" .
-ultimately show False
-proof -
-show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
-by (ind_cases "step s (P th cs)", simp)
-qed
-qed
-with acyclic_insert ih eq_r show ?thesis by auto
-qed
-ultimately show ?thesis by simp
-next
-case (Set thread prio)
-with ih
-thm RAG_set_unchanged
-show ?thesis by (simp add:RAG_set_unchanged)
-qed
-next
-case vt_nil
-show "acyclic (RAG ([]::state))"
-by (auto simp: s_RAG_def cs_waiting_def
-cs_holding_def wq_def acyclic_def)
-qed
-lemma finite_RAG:
-shows "finite (RAG s)"
-proof -
-from vt show ?thesis
-proof(induct)
-case (vt_cons s e)
-interpret vt_s: valid_trace s using vt_cons(1)
-by (unfold_locales, simp)
-assume ih: "finite (RAG s)"
-and stp: "step s e"
-and vt: "vt s"
-show ?case
-proof(cases e)
-case (Create th prio)
-with ih
-show ?thesis by (simp add:RAG_create_unchanged)
-next
-case (Exit th)
-with ih show ?thesis by (simp add:RAG_exit_unchanged)
-next
-case (V th cs)
-from V vt stp have vtt: "vt (V th cs#s)" by auto
-from step_RAG_v [OF this]
-have eq_de: "RAG (e # s) =
-RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
-{(Cs cs, Th th') |th'. next_th s th cs th'}
-"
-(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
-moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
-moreover have "finite ?D"
-proof -
-have "?D = {} \<or> (\<exists> a. ?D = {a})"
-by (unfold next_th_def, auto)
-thus ?thesis
-proof
-assume h: "?D = {}"
-show ?thesis by (unfold h, simp)
-next
-assume "\<exists> a. ?D = {a}"
-thus ?thesis
-by (metis finite.simps)
-qed
-qed
-ultimately show ?thesis by simp
-next
-case (P th cs)
-from P vt stp have vtt: "vt (P th cs#s)" by auto
-from step_RAG_p [OF this] P
-have "RAG (e # s) =
-(if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
-RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
-by simp
-moreover have "finite ?R"
-proof(cases "wq s cs = []")
-case True
-hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
-with True and ih show ?thesis by auto
-next
-case False
-hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
-with False and ih show ?thesis by auto
-qed
-ultimately show ?thesis by auto
-next
-case (Set thread prio)
-with ih
-show ?thesis by (simp add:RAG_set_unchanged)
-qed
-next
-case vt_nil
-show "finite (RAG ([]::state))"
-by (auto simp: s_RAG_def cs_waiting_def
-cs_holding_def wq_def acyclic_def)
-qed
-qed
-text {* Several useful lemmas *}
-lemma wf_dep_converse:
-shows "wf ((RAG s)^-1)"
-proof(rule finite_acyclic_wf_converse)
-from finite_RAG
-show "finite (RAG s)" .
-next
-from acyclic_RAG
-show "acyclic (RAG s)" .
-qed
-end
-lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
-by (induct l, auto)
-lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
-by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-context valid_trace
-begin
-lemma wq_threads:
-assumes h: "th \<in> set (wq s cs)"
-shows "th \<in> threads s"
-proof -
-from vt and h show ?thesis
-proof(induct arbitrary: th cs)
-case (vt_cons s e)
-interpret vt_s: valid_trace s
-using vt_cons(1) by (unfold_locales, auto)
-assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
-and stp: "step s e"
-and vt: "vt s"
-and h: "th \<in> set (wq (e # s) cs)"
-show ?case
-proof(cases e)
-case (Create th' prio)
-with ih h show ?thesis
-by (auto simp:wq_def Let_def)
-next
-case (Exit th')
-with stp ih h show ?thesis
-apply (auto simp:wq_def Let_def)
-apply (ind_cases "step s (Exit th')")
-apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
-s_RAG_def s_holding_def cs_holding_def)
-done
-next
-case (V th' cs')
-show ?thesis
-proof(cases "cs' = cs")
-case False
-with h
-show ?thesis
-apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
-by (drule_tac ih, simp)
-next
-case True
-from h
-show ?thesis
-proof(unfold V wq_def)
-assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
-show "th \<in> threads (V th' cs' # s)"
-proof(cases "cs = cs'")
-case False
-hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
-with th_in have " th \<in> set (wq s cs)"
-by (fold wq_def, simp)
-from ih [OF this] show ?thesis by simp
-next
-case True
-show ?thesis
-proof(cases "wq_fun (schs s) cs'")
-case Nil
-with h V show ?thesis
-apply (auto simp:wq_def Let_def split:if_splits)
-by (fold wq_def, drule_tac ih, simp)
-next
-case (Cons a rest)
-assume eq_wq: "wq_fun (schs s) cs' = a # rest"
-with h V show ?thesis
-apply (auto simp:Let_def wq_def split:if_splits)
-proof -
-assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
-have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-proof(rule someI2)
-from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
-show "distinct rest \<and> set rest = set rest" by auto
-next
-show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
-by auto
-qed
-with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
-from ih[OF this[folded wq_def]] show "th \<in> threads s" .
-next
-assume th_in: "th \<in> set (wq_fun (schs s) cs)"
-from ih[OF this[folded wq_def]]
-show "th \<in> threads s" .
-qed
-qed
-qed
-qed
-qed
-next
-case (P th' cs')
-from h stp
-show ?thesis
-apply (unfold P wq_def)
-apply (auto simp:Let_def split:if_splits, fold wq_def)
-apply (auto intro:ih)
-apply(ind_cases "step s (P th' cs')")
-by (unfold runing_def readys_def, auto)
-next
-case (Set thread prio)
-with ih h show ?thesis
-by (auto simp:wq_def Let_def)
-qed
-next
-case vt_nil
-thus ?case by (auto simp:wq_def)
-qed
-qed
-lemma range_in: "\<lbrakk>(Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
-apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
-by (auto intro:wq_threads)
-lemma readys_v_eq:
-fixes th thread cs rest
-assumes neq_th: "th \<noteq> thread"
-and eq_wq: "wq s cs = thread#rest"
-and not_in: "th \<notin>  set rest"
-shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
-from assms show ?thesis
-apply (auto simp:readys_def)
-apply(simp add:s_waiting_def[folded wq_def])
-apply (erule_tac x = csa in allE)
-apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
-apply (case_tac "csa = cs", simp)
-apply (erule_tac x = cs in allE)
-apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
-apply(auto simp add: wq_def)
-apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
-proof -
-assume th_nin: "th \<notin> set rest"
-and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
-and eq_wq: "wq_fun (schs s) cs = thread # rest"
-have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-proof(rule someI2)
-from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
-show "distinct rest \<and> set rest = set rest" by auto
-next
-show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-qed
-with th_nin th_in show False by auto
-qed
-qed
-text {* \noindent
-The following lemmas shows that: starting from any node in @{text "RAG"},
-by chasing out-going edges, it is always possible to reach a node representing a ready
-thread. In this lemma, it is the @{text "th'"}.
-*}
-lemma chain_building:
-shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
-proof -
-from wf_dep_converse
-have h: "wf ((RAG s)\<inverse>)" .
-show ?thesis
-proof(induct rule:wf_induct [OF h])
-fix x
-assume ih [rule_format]:
-"\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow>
-y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
-show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
-proof
-assume x_d: "x \<in> Domain (RAG s)"
-show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
-proof(cases x)
-case (Th th)
-from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
-with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
-from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
-hence "Cs cs \<in> Domain (RAG s)" by auto
-from ih [OF x_in_r this] obtain th'
-where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
-have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
-with th'_ready show ?thesis by auto
-next
-case (Cs cs)
-from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
-show ?thesis
-proof(cases "th' \<in> readys s")
-case True
-from True and th'_d show ?thesis by auto
-next
-case False
-from th'_d and range_in  have "th' \<in> threads s" by auto
-with False have "Th th' \<in> Domain (RAG s)"
-by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
-from ih [OF th'_d this]
-obtain th'' where
-th''_r: "th'' \<in> readys s" and
-th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
-from th'_d and th''_in
-have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
-with th''_r show ?thesis by auto
-qed
-qed
-qed
-qed
-qed
-text {* \noindent
-The following is just an instance of @{text "chain_building"}.
-*}
-lemma th_chain_to_ready:
-assumes th_in: "th \<in> threads s"
-shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
-proof(cases "th \<in> readys s")
-case True
-thus ?thesis by auto
-next
-case False
-from False and th_in have "Th th \<in> Domain (RAG s)"
-by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
-from chain_building [rule_format, OF this]
-show ?thesis by auto
-qed
-end
-lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
-by  (unfold s_waiting_def cs_waiting_def wq_def, auto)
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
-by (unfold s_holding_def wq_def cs_holding_def, simp)
-lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
-by (unfold s_holding_def cs_holding_def, auto)
-context valid_trace
-begin
-lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
-apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
-by(auto elim:waiting_unique holding_unique)
-end
-lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
-by (induct rule:trancl_induct, auto)
-context valid_trace
-begin
-lemma dchain_unique:
-assumes th1_d: "(n, Th th1) \<in> (RAG s)^+"
-and th1_r: "th1 \<in> readys s"
-and th2_d: "(n, Th th2) \<in> (RAG s)^+"
-and th2_r: "th2 \<in> readys s"
-shows "th1 = th2"
-proof -
-{ assume neq: "th1 \<noteq> th2"
-hence "Th th1 \<noteq> Th th2" by simp
-from unique_chain [OF _ th1_d th2_d this] and unique_RAG
-have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
-hence "False"
-proof
-assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
-from trancl_split [OF this]
-obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
-then obtain cs where eq_n: "n = Cs cs"
-by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-from dd eq_n have "th1 \<notin> readys s"
-by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
-with th1_r show ?thesis by auto
-next
-assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
-from trancl_split [OF this]
-obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
-then obtain cs where eq_n: "n = Cs cs"
-by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-from dd eq_n have "th2 \<notin> readys s"
-by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
-with th2_r show ?thesis by auto
-qed
-} thus ?thesis by auto
-qed
-end
-lemma step_holdents_p_add:
-fixes th cs s
-assumes vt: "vt (P th cs#s)"
-and "wq s cs = []"
-shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
-proof -
-from assms show ?thesis
-unfolding  holdents_test step_RAG_p[OF vt] by (auto)
-qed
-lemma step_holdents_p_eq:
-fixes th cs s
-assumes vt: "vt (P th cs#s)"
-and "wq s cs \<noteq> []"
-shows "holdents (P th cs#s) th = holdents s th"
-proof -
-from assms show ?thesis
-unfolding  holdents_test step_RAG_p[OF vt] by auto
-qed
-lemma (in valid_trace) finite_holding :
-shows "finite (holdents s th)"
-proof -
-let ?F = "\<lambda> (x, y). the_cs x"
-from finite_RAG
-have "finite (RAG s)" .
-hence "finite (?F `(RAG s))" by simp
-moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>"
-proof -
-{ have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
-fix x assume "(Cs x, Th th) \<in> RAG s"
-hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
-moreover have "?F (Cs x, Th th) = x" by simp
-ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp
-} thus ?thesis by auto
-qed
-ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
-qed
-lemma cntCS_v_dec:
-fixes s thread cs
-assumes vtv: "vt (V thread cs#s)"
-shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
-proof -
-from vtv interpret vt_s: valid_trace s
-by (cases, unfold_locales, simp)
-from vtv interpret vt_v: valid_trace "V thread cs#s"
-by (unfold_locales, simp)
-from step_back_step[OF vtv]
-have cs_in: "cs \<in> holdents s thread"
-apply (cases, unfold holdents_test s_RAG_def, simp)
-by (unfold cs_holding_def s_holding_def wq_def, auto)
-moreover have cs_not_in:
-"(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
-apply (insert vt_s.wq_distinct[of cs])
-apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
-auto simp:next_th_def)
-proof -
-fix rest
-assume dst: "distinct (rest::thread list)"
-and ne: "rest \<noteq> []"
-and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-proof(rule someI2)
-from dst show "distinct rest \<and> set rest = set rest" by auto
-next
-show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-qed
-ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
-set (SOME q. distinct q \<and> set q = set rest)" by simp
-moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-proof(rule someI2)
-from dst show "distinct rest \<and> set rest = set rest" by auto
-next
-fix x assume " distinct x \<and> set x = set rest" with ne
-show "x \<noteq> []" by auto
-qed
-ultimately
-show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
-by auto
-next
-fix rest
-assume dst: "distinct (rest::thread list)"
-and ne: "rest \<noteq> []"
-and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-proof(rule someI2)
-from dst show "distinct rest \<and> set rest = set rest" by auto
-next
-show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
-qed
-ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
-set (SOME q. distinct q \<and> set q = set rest)" by simp
-moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-proof(rule someI2)
-from dst show "distinct rest \<and> set rest = set rest" by auto
-next
-fix x assume " distinct x \<and> set x = set rest" with ne
-show "x \<noteq> []" by auto
-qed
-ultimately show "False" by auto
-qed
-ultimately
-have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
-by auto
-moreover have "card \<dots> =
-Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
-proof(rule card_insert)
-from vt_v.finite_holding
-show " finite (holdents (V thread cs # s) thread)" .
-qed
-moreover from cs_not_in
-have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
-ultimately show ?thesis by (simp add:cntCS_def)
-qed
-lemma count_rec1 [simp]:
-assumes "Q e"
-shows "count Q (e#es) = Suc (count Q es)"
-using assms
-by (unfold count_def, auto)
-lemma count_rec2 [simp]:
-assumes "\<not>Q e"
-shows "count Q (e#es) = (count Q es)"
-using assms
-by (unfold count_def, auto)
-lemma count_rec3 [simp]:
-shows "count Q [] =  0"
-by (unfold count_def, auto)
-lemma cntP_diff_inv:
-assumes "cntP (e#s) th \<noteq> cntP s th"
-shows "isP e \<and> actor e = th"
-proof(cases e)
-case (P th' pty)
-show ?thesis
-by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)",
-insert assms P, auto simp:cntP_def)
-qed (insert assms, auto simp:cntP_def)
-lemma isP_E:
-assumes "isP e"
-obtains cs where "e = P (actor e) cs"
-using assms by (cases e, auto)
-lemma isV_E:
-assumes "isV e"
-obtains cs where "e = V (actor e) cs"
-using assms by (cases e, auto) (* ccc *)
-lemma cntV_diff_inv:
-assumes "cntV (e#s) th \<noteq> cntV s th"
-shows "isV e \<and> actor e = th"
-proof(cases e)
-case (V th' pty)
-show ?thesis
-by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)",
-insert assms V, auto simp:cntV_def)
-qed (insert assms, auto simp:cntV_def)
-context valid_trace
-begin
-text {* (* ddd *) \noindent
-The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"}
-of one particular thread.
-*}
-lemma cnp_cnv_cncs:
-shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
-then cntCS s th else cntCS s th + 1)"
-proof -
-from vt show ?thesis
-proof(induct arbitrary:th)
-case (vt_cons s e)
-interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
-assume vt: "vt s"
-and ih: "\<And>th. cntP s th  = cntV s th +
-(if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
-and stp: "step s e"
-from stp show ?case
-proof(cases)
-case (thread_create thread prio)
-assume eq_e: "e = Create thread prio"
-and not_in: "thread \<notin> threads s"
-show ?thesis
-proof -
-{ fix cs
-assume "thread \<in> set (wq s cs)"
-from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
-with not_in have "False" by simp
-} with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
-by (auto simp:readys_def threads.simps s_waiting_def
-wq_def cs_waiting_def Let_def)
-from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-have eq_cncs: "cntCS (e#s) th = cntCS s th"
-unfolding cntCS_def holdents_test
-by (simp add:RAG_create_unchanged eq_e)
-{ assume "th \<noteq> thread"
-with eq_readys eq_e
-have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
-(th \<in> readys (s) \<or> th \<notin> threads (s))"
-by (simp add:threads.simps)
-with eq_cnp eq_cnv eq_cncs ih not_in
-have ?thesis by simp
-} moreover {
-assume eq_th: "th = thread"
-with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp
-moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
-moreover note eq_cnp eq_cnv eq_cncs
-ultimately have ?thesis by auto
-} ultimately show ?thesis by blast
-qed
-next
-case (thread_exit thread)
-assume eq_e: "e = Exit thread"
-and is_runing: "thread \<in> runing s"
-and no_hold: "holdents s thread = {}"
-from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-have eq_cncs: "cntCS (e#s) th = cntCS s th"
-unfolding cntCS_def holdents_test
-by (simp add:RAG_exit_unchanged eq_e)
-{ assume "th \<noteq> thread"
-with eq_e
-have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
-(th \<in> readys (s) \<or> th \<notin> threads (s))"
-apply (simp add:threads.simps readys_def)
-apply (subst s_waiting_def)
-apply (simp add:Let_def)
-apply (subst s_waiting_def, simp)
-done
-with eq_cnp eq_cnv eq_cncs ih
-have ?thesis by simp
-} moreover {
-assume eq_th: "th = thread"
-with ih is_runing have " cntP s th = cntV s th + cntCS s th"
-by (simp add:runing_def)
-moreover from eq_th eq_e have "th \<notin> threads (e#s)"
-by simp
-moreover note eq_cnp eq_cnv eq_cncs
-ultimately have ?thesis by auto
-} ultimately show ?thesis by blast
-next
-case (thread_P thread cs)
-assume eq_e: "e = P thread cs"
-and is_runing: "thread \<in> runing s"
-and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
-from thread_P vt stp ih  have vtp: "vt (P thread cs#s)" by auto
-then interpret vt_p: valid_trace "(P thread cs#s)"
-by (unfold_locales, simp)
-show ?thesis
-proof -
-{ have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
-assume neq_th: "th \<noteq> thread"
-with eq_e
-have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
-apply (simp add:readys_def s_waiting_def wq_def Let_def)
-apply (rule_tac hh)
-apply (intro iffI allI, clarify)
-apply (erule_tac x = csa in allE, auto)
-apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
-apply (erule_tac x = cs in allE, auto)
-by (case_tac "(wq_fun (schs s) cs)", auto)
-moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
-apply (simp add:cntCS_def holdents_test)
-by (unfold  step_RAG_p [OF vtp], auto)
-moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
-by (simp add:cntP_def count_def)
-moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
-by (simp add:cntV_def count_def)
-moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
-moreover note ih [of th]
-ultimately have ?thesis by simp
-} moreover {
-assume eq_th: "th = thread"
-have ?thesis
-proof -
-from eq_e eq_th have eq_cnp: "cntP (e # s) th  = 1 + (cntP s th)"
-by (simp add:cntP_def count_def)
-from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
-by (simp add:cntV_def count_def)
-show ?thesis
-proof (cases "wq s cs = []")
-case True
-with is_runing
-have "th \<in> readys (e#s)"
-apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
-apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
-by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
-moreover have "cntCS (e # s) th = 1 + cntCS s th"
-proof -
-have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
-Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
-proof -
-have "?L = insert cs ?R" by auto
-moreover have "card \<dots> = Suc (card (?R - {cs}))"
-proof(rule card_insert)
-from vt_s.finite_holding [of thread]
-show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
-by (unfold holdents_test, simp)
-qed
-moreover have "?R - {cs} = ?R"
-proof -
-have "cs \<notin> ?R"
-proof
-assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
-with no_dep show False by auto
-qed
-thus ?thesis by auto
-qed
-ultimately show ?thesis by auto
-qed
-thus ?thesis
-apply (unfold eq_e eq_th cntCS_def)
-apply (simp add: holdents_test)
-by (unfold step_RAG_p [OF vtp], auto simp:True)
-qed
-moreover from is_runing have "th \<in> readys s"
-by (simp add:runing_def eq_th)
-moreover note eq_cnp eq_cnv ih [of th]
-ultimately show ?thesis by auto
-next
-case False
-have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
-by (unfold eq_th eq_e wq_def, auto simp:Let_def)
-have "th \<notin> readys (e#s)"
-proof
-assume "th \<in> readys (e#s)"
-hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
-from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
-hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
-by (simp add:s_waiting_def wq_def)
-moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
-ultimately have "th = hd (wq (e#s) cs)" by blast
-with eq_wq have "th = hd (wq s cs @ [th])" by simp
-hence "th = hd (wq s cs)" using False by auto
-with False eq_wq vt_p.wq_distinct [of cs]
-show False by (fold eq_e, auto)
-qed
-moreover from is_runing have "th \<in> threads (e#s)"
-by (unfold eq_e, auto simp:runing_def readys_def eq_th)
-moreover have "cntCS (e # s) th = cntCS s th"
-apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
-by (auto simp:False)
-moreover note eq_cnp eq_cnv ih[of th]
-moreover from is_runing have "th \<in> readys s"
-by (simp add:runing_def eq_th)
-ultimately show ?thesis by auto
-qed
-qed
-} ultimately show ?thesis by blast
-qed
-next
-case (thread_V thread cs)
-from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
-then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
-assume eq_e: "e = V thread cs"
-and is_runing: "thread \<in> runing s"
-and hold: "holding s thread cs"
-from hold obtain rest
-where eq_wq: "wq s cs = thread # rest"
-by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
-have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
-have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
-proof(rule someI2)
-from vt_v.wq_distinct[of cs] and eq_wq
-show "distinct rest \<and> set rest = set rest"
-by (metis distinct.simps(2) vt_s.wq_distinct)
-next
-show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
-by auto
-qed
-show ?thesis
-proof -
-{ assume eq_th: "th = thread"
-from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
-by (unfold eq_e, simp add:cntP_def count_def)
-moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
-by (unfold eq_e, simp add:cntV_def count_def)
-moreover from cntCS_v_dec [OF vtv]
-have "cntCS (e # s) thread + 1 = cntCS s thread"
-by (simp add:eq_e)
-moreover from is_runing have rd_before: "thread \<in> readys s"
-by (unfold runing_def, simp)
-moreover have "thread \<in> readys (e # s)"
-proof -
-from is_runing
-have "thread \<in> threads (e#s)"
-by (unfold eq_e, auto simp:runing_def readys_def)
-moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
-proof
-fix cs1
-{ assume eq_cs: "cs1 = cs"
-have "\<not> waiting (e # s) thread cs1"
-proof -
-from eq_wq
-have "thread \<notin> set (wq (e#s) cs1)"
-apply(unfold eq_e wq_def eq_cs s_holding_def)
-apply (auto simp:Let_def)
-proof -
-assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
-with eq_set have "thread \<in> set rest" by simp
-with vt_v.wq_distinct[of cs]
-and eq_wq show False
-by (metis distinct.simps(2) vt_s.wq_distinct)
-qed
-thus ?thesis by (simp add:wq_def s_waiting_def)
-qed
-} moreover {
-assume neq_cs: "cs1 \<noteq> cs"
-have "\<not> waiting (e # s) thread cs1"
-proof -
-from wq_v_neq [OF neq_cs[symmetric]]
-have "wq (V thread cs # s) cs1 = wq s cs1" .
-moreover have "\<not> waiting s thread cs1"
-proof -
-from runing_ready and is_runing
-have "thread \<in> readys s" by auto
-thus ?thesis by (simp add:readys_def)
-qed
-ultimately show ?thesis
-by (auto simp:wq_def s_waiting_def eq_e)
-qed
-} ultimately show "\<not> waiting (e # s) thread cs1" by blast
-qed
-ultimately show ?thesis by (simp add:readys_def)
-qed
-moreover note eq_th ih
-ultimately have ?thesis by auto
-} moreover {
-assume neq_th: "th \<noteq> thread"
-from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th"
-by (simp add:cntP_def count_def)
-from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th"
-by (simp add:cntV_def count_def)
-have ?thesis
-proof(cases "th \<in> set rest")
-case False
-have "(th \<in> readys (e # s)) = (th \<in> readys s)"
-apply (insert step_back_vt[OF vtv])
-by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
-moreover have "cntCS (e#s) th = cntCS s th"
-apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
-proof -
-have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
-{cs. (Cs cs, Th th) \<in> RAG s}"
-proof -
-from False eq_wq
-have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
-apply (unfold next_th_def, auto)
-proof -
-assume ne: "rest \<noteq> []"
-and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
-and eq_wq: "wq s cs = thread # rest"
-from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
-set (SOME q. distinct q \<and> set q = set rest)
-" by simp
-moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
-proof(rule someI2)
-from vt_s.wq_distinct[ of cs] and eq_wq
-show "distinct rest \<and> set rest = set rest" by auto
-next
-fix x assume "distinct x \<and> set x = set rest"
-with ne show "x \<noteq> []" by auto
-qed
-ultimately show
-"(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
-by auto
-qed
-thus ?thesis by auto
-qed
-thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
-card {cs. (Cs cs, Th th) \<in> RAG s}" by simp
-qed
-moreover note ih eq_cnp eq_cnv eq_threads
-ultimately show ?thesis by auto
-next
-case True
-assume th_in: "th \<in> set rest"
-show ?thesis
-proof(cases "next_th s thread cs th")
-case False
-with eq_wq and th_in have
-neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
-by (auto simp:next_th_def)
-have "(th \<in> readys (e # s)) = (th \<in> readys s)"
-proof -
-from eq_wq and th_in
-have "\<not> th \<in> readys s"
-apply (auto simp:readys_def s_waiting_def)
-apply (rule_tac x = cs in exI, auto)
-by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
-moreover
-from eq_wq and th_in and neq_hd
-have "\<not> (th \<in> readys (e # s))"
-apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
-by (rule_tac x = cs in exI, auto simp:eq_set)
-ultimately show ?thesis by auto
-qed
-moreover have "cntCS (e#s) th = cntCS s th"
-proof -
-from eq_wq and  th_in and neq_hd
-have "(holdents (e # s) th) = (holdents s th)"
-apply (unfold eq_e step_RAG_v[OF vtv],
-auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
-Let_def cs_holding_def)
-by (insert vt_s.wq_distinct[of cs], auto simp:wq_def)
-thus ?thesis by (simp add:cntCS_def)
-qed
-moreover note ih eq_cnp eq_cnv eq_threads
-ultimately show ?thesis by auto
-next
-case True
-let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
-let ?t = "hd ?rest"
-from True eq_wq th_in neq_th
-have "th \<in> readys (e # s)"
-apply (auto simp:eq_e readys_def s_waiting_def wq_def
-Let_def next_th_def)
-proof -
-assume eq_wq: "wq_fun (schs s) cs = thread # rest"
-and t_in: "?t \<in> set rest"
-show "?t \<in> threads s"
-proof(rule vt_s.wq_threads)
-from eq_wq and t_in
-show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
-qed
-next
-fix csa
-assume eq_wq: "wq_fun (schs s) cs = thread # rest"
-and t_in: "?t \<in> set rest"
-and neq_cs: "csa \<noteq> cs"
-and t_in': "?t \<in>  set (wq_fun (schs s) csa)"
-show "?t = hd (wq_fun (schs s) csa)"
-proof -
-{ assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
-from vt_s.wq_distinct[of cs] and
-eq_wq[folded wq_def] and t_in eq_wq
-have "?t \<noteq> thread" by auto
-with eq_wq and t_in
-have w1: "waiting s ?t cs"
-by (auto simp:s_waiting_def wq_def)
-from t_in' neq_hd'
-have w2: "waiting s ?t csa"
-by (auto simp:s_waiting_def wq_def)
-from vt_s.waiting_unique[OF w1 w2]
-and neq_cs have "False" by auto
-} thus ?thesis by auto
-qed
-qed
-moreover have "cntP s th = cntV s th + cntCS s th + 1"
-proof -
-have "th \<notin> readys s"
-proof -
-from True eq_wq neq_th th_in
-show ?thesis
-apply (unfold readys_def s_waiting_def, auto)
-by (rule_tac x = cs in exI, auto simp add: wq_def)
-qed
-moreover have "th \<in> threads s"
-proof -
-from th_in eq_wq
-have "th \<in> set (wq s cs)" by simp
-from vt_s.wq_threads [OF this]
-show ?thesis .
-qed
-ultimately show ?thesis using ih by auto
-qed
-moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
-apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
-proof -
-show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
-Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
-(is "card ?A = Suc (card ?B)")
-proof -
-have "?A = insert cs ?B" by auto
-hence "card ?A = card (insert cs ?B)" by simp
-also have "\<dots> = Suc (card ?B)"
-proof(rule card_insert_disjoint)
-have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)"
-apply (auto simp:image_def)
-by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
-with vt_s.finite_RAG
-show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
-next
-show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
-proof
-assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
-hence "(Cs cs, Th th) \<in> RAG s" by simp
-with True neq_th eq_wq show False
-by (auto simp:next_th_def s_RAG_def cs_holding_def)
-qed
-qed
-finally show ?thesis .
-qed
-qed
-moreover note eq_cnp eq_cnv
-ultimately show ?thesis by simp
-qed
-qed
-} ultimately show ?thesis by blast
-qed
-next
-case (thread_set thread prio)
-assume eq_e: "e = Set thread prio"
-and is_runing: "thread \<in> runing s"
-show ?thesis
-proof -
-from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
-from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
-have eq_cncs: "cntCS (e#s) th = cntCS s th"
-unfolding cntCS_def holdents_test
-by (simp add:RAG_set_unchanged eq_e)
-from eq_e have eq_readys: "readys (e#s) = readys s"
-by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
-auto simp:Let_def)
-{ assume "th \<noteq> thread"
-with eq_readys eq_e
-have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
-(th \<in> readys (s) \<or> th \<notin> threads (s))"
-by (simp add:threads.simps)
-with eq_cnp eq_cnv eq_cncs ih is_runing
-have ?thesis by simp
-} moreover {
-assume eq_th: "th = thread"
-with is_runing ih have " cntP s th  = cntV s th + cntCS s th"
-by (unfold runing_def, auto)
-moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
-by (simp add:runing_def)
-moreover note eq_cnp eq_cnv eq_cncs
-ultimately have ?thesis by auto
-} ultimately show ?thesis by blast
-qed
-qed
-next
-case vt_nil
-show ?case
-by (unfold cntP_def cntV_def cntCS_def,
-auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
-qed
-qed
-lemma not_thread_cncs:
-assumes not_in: "th \<notin> threads s"
-shows "cntCS s th = 0"
-proof -
-from vt not_in show ?thesis
-proof(induct arbitrary:th)
-case (vt_cons s e th)
-interpret vt_s: valid_trace s using vt_cons(1)
-by (unfold_locales, simp)
-assume vt: "vt s"
-and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
-and stp: "step s e"
-and not_in: "th \<notin> threads (e # s)"
-from stp show ?case
-proof(cases)
-case (thread_create thread prio)
-assume eq_e: "e = Create thread prio"
-and not_in': "thread \<notin> threads s"
-have "cntCS (e # s) th = cntCS s th"
-apply (unfold eq_e cntCS_def holdents_test)
-by (simp add:RAG_create_unchanged)
-moreover have "th \<notin> threads s"
-proof -
-from not_in eq_e show ?thesis by simp
-qed
-moreover note ih ultimately show ?thesis by auto
-next
-case (thread_exit thread)
-assume eq_e: "e = Exit thread"
-and nh: "holdents s thread = {}"
-have eq_cns: "cntCS (e # s) th = cntCS s th"
-apply (unfold eq_e cntCS_def holdents_test)
-by (simp add:RAG_exit_unchanged)
-show ?thesis
-proof(cases "th = thread")
-case True
-have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
-with eq_cns show ?thesis by simp
-next
-case False
-with not_in and eq_e
-have "th \<notin> threads s" by simp
-from ih[OF this] and eq_cns show ?thesis by simp
-qed
-next
-case (thread_P thread cs)
-assume eq_e: "e = P thread cs"
-and is_runing: "thread \<in> runing s"
-from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
-have neq_th: "th \<noteq> thread"
-proof -
-from not_in eq_e have "th \<notin> threads s" by simp
-moreover from is_runing have "thread \<in> threads s"
-by (simp add:runing_def readys_def)
-ultimately show ?thesis by auto
-qed
-hence "cntCS (e # s) th  = cntCS s th "
-apply (unfold cntCS_def holdents_test eq_e)
-by (unfold step_RAG_p[OF vtp], auto)
-moreover have "cntCS s th = 0"
-proof(rule ih)
-from not_in eq_e show "th \<notin> threads s" by simp
-qed
-ultimately show ?thesis by simp
-next
-case (thread_V thread cs)
-assume eq_e: "e = V thread cs"
-and is_runing: "thread \<in> runing s"
-and hold: "holding s thread cs"
-have neq_th: "th \<noteq> thread"
-proof -
-from not_in eq_e have "th \<notin> threads s" by simp
-moreover from is_runing have "thread \<in> threads s"
-by (simp add:runing_def readys_def)
-ultimately show ?thesis by auto
-qed
-from assms thread_V vt stp ih
-have vtv: "vt (V thread cs#s)" by auto
-then interpret vt_v: valid_trace "(V thread cs#s)"
-by (unfold_locales, simp)
-from hold obtain rest
-where eq_wq: "wq s cs = thread # rest"
-by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
-from not_in eq_e eq_wq
-have "\<not> next_th s thread cs th"
-apply (auto simp:next_th_def)
-proof -
-assume ne: "rest \<noteq> []"
-and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
-have "?t \<in> set rest"
-proof(rule someI2)
-from vt_v.wq_distinct[of cs] and eq_wq
-show "distinct rest \<and> set rest = set rest"
-by (metis distinct.simps(2) vt_s.wq_distinct)
-next
-fix x assume "distinct x \<and> set x = set rest" with ne
-show "hd x \<in> set rest" by (cases x, auto)
-qed
-with eq_wq have "?t \<in> set (wq s cs)" by simp
-from vt_s.wq_threads[OF this] and ni
-show False
-using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)`
-ni vt_s.wq_threads by blast
-qed
-moreover note neq_th eq_wq
-ultimately have "cntCS (e # s) th  = cntCS s th"
-by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
-moreover have "cntCS s th = 0"
-proof(rule ih)
-from not_in eq_e show "th \<notin> threads s" by simp
-qed
-ultimately show ?thesis by simp
-next
-case (thread_set thread prio)
-print_facts
-assume eq_e: "e = Set thread prio"
-and is_runing: "thread \<in> runing s"
-from not_in and eq_e have "th \<notin> threads s" by auto
-from ih [OF this] and eq_e
-show ?thesis
-apply (unfold eq_e cntCS_def holdents_test)
-by (simp add:RAG_set_unchanged)
-qed
-next
-case vt_nil
-show ?case
-by (unfold cntCS_def,
-auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
-qed
-qed
-end
-lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
-by (auto simp:s_waiting_def cs_waiting_def wq_def)
-context valid_trace
-begin
-lemma dm_RAG_threads:
-assumes in_dom: "(Th th) \<in> Domain (RAG s)"
-shows "th \<in> threads s"
-proof -
-from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
-moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
-ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
-hence "th \<in> set (wq s cs)"
-by (unfold s_RAG_def, auto simp:cs_waiting_def)
-from wq_threads [OF this] show ?thesis .
-qed
-end
-lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
-unfolding cp_def wq_def
-apply(induct s rule: schs.induct)
-thm cpreced_initial
-apply(simp add: Let_def cpreced_initial)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-done
-context valid_trace
-begin
-lemma runing_unique:
-assumes runing_1: "th1 \<in> runing s"
-and runing_2: "th2 \<in> runing s"
-shows "th1 = th2"
-proof -
-from runing_1 and runing_2 have "cp s th1 = cp s th2"
-unfolding runing_def
-apply(simp)
-done
-hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
-Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
-(is "Max (?f ` ?A) = Max (?f ` ?B)")
-unfolding cp_eq_cpreced
-unfolding cpreced_def .
-obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
-proof -
-have h1: "finite (?f ` ?A)"
-proof -
-have "finite ?A"
-proof -
-have "finite (dependants (wq s) th1)"
-proof-
-have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
-proof -
-let ?F = "\<lambda> (x, y). the_th x"
-have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
-apply (auto simp:image_def)
-by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
-moreover have "finite \<dots>"
-proof -
-from finite_RAG have "finite (RAG s)" .
-hence "finite ((RAG (wq s))\<^sup>+)"
-apply (unfold finite_trancl)
-by (auto simp: s_RAG_def cs_RAG_def wq_def)
-thus ?thesis by auto
-qed
-ultimately show ?thesis by (auto intro:finite_subset)
-qed
-thus ?thesis by (simp add:cs_dependants_def)
-qed
-thus ?thesis by simp
-qed
-thus ?thesis by auto
-qed
-moreover have h2: "(?f ` ?A) \<noteq> {}"
-proof -
-have "?A \<noteq> {}" by simp
-thus ?thesis by simp
-qed
-from Max_in [OF h1 h2]
-have "Max (?f ` ?A) \<in> (?f ` ?A)" .
-thus ?thesis
-thm cpreced_def
-unfolding cpreced_def[symmetric]
-unfolding cp_eq_cpreced[symmetric]
-unfolding cpreced_def
-using that[intro] by (auto)
-qed
-obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
-proof -
-have h1: "finite (?f ` ?B)"
-proof -
-have "finite ?B"
-proof -
-have "finite (dependants (wq s) th2)"
-proof-
-have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
-proof -
-let ?F = "\<lambda> (x, y). the_th x"
-have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
-apply (auto simp:image_def)
-by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
-moreover have "finite \<dots>"
-proof -
-from finite_RAG have "finite (RAG s)" .
-hence "finite ((RAG (wq s))\<^sup>+)"
-apply (unfold finite_trancl)
-by (auto simp: s_RAG_def cs_RAG_def wq_def)
-thus ?thesis by auto
-qed
-ultimately show ?thesis by (auto intro:finite_subset)
-qed
-thus ?thesis by (simp add:cs_dependants_def)
-qed
-thus ?thesis by simp
-qed
-thus ?thesis by auto
-qed
-moreover have h2: "(?f ` ?B) \<noteq> {}"
-proof -
-have "?B \<noteq> {}" by simp
-thus ?thesis by simp
-qed
-from Max_in [OF h1 h2]
-have "Max (?f ` ?B) \<in> (?f ` ?B)" .
-thus ?thesis by (auto intro:that)
-qed
-from eq_f_th1 eq_f_th2 eq_max
-have eq_preced: "preced th1' s = preced th2' s" by auto
-hence eq_th12: "th1' = th2'"
-proof (rule preced_unique)
-from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
-thus "th1' \<in> threads s"
-proof
-assume "th1' \<in> dependants (wq s) th1"
-hence "(Th th1') \<in> Domain ((RAG s)^+)"
-apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
-by (auto simp:Domain_def)
-hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
-from dm_RAG_threads[OF this] show ?thesis .
-next
-assume "th1' = th1"
-with runing_1 show ?thesis
-by (unfold runing_def readys_def, auto)
-qed
-next
-from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
-thus "th2' \<in> threads s"
-proof
-assume "th2' \<in> dependants (wq s) th2"
-hence "(Th th2') \<in> Domain ((RAG s)^+)"
-apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
-by (auto simp:Domain_def)
-hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
-from dm_RAG_threads[OF this] show ?thesis .
-next
-assume "th2' = th2"
-with runing_2 show ?thesis
-by (unfold runing_def readys_def, auto)
-qed
-qed
-from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
-thus ?thesis
-proof
-assume eq_th': "th1' = th1"
-from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
-thus ?thesis
-proof
-assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
-next
-assume "th2' \<in> dependants (wq s) th2"
-with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
-hence "(Th th1, Th th2) \<in> (RAG s)^+"
-by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
-hence "Th th1 \<in> Domain ((RAG s)^+)"
-apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
-by (auto simp:Domain_def)
-hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
-then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
-from RAG_target_th [OF this]
-obtain cs' where "n = Cs cs'" by auto
-with d have "(Th th1, Cs cs') \<in> RAG s" by simp
-with runing_1 have "False"
-apply (unfold runing_def readys_def s_RAG_def)
-by (auto simp:eq_waiting)
-thus ?thesis by simp
-qed
-next
-assume th1'_in: "th1' \<in> dependants (wq s) th1"
-from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
-thus ?thesis
-proof
-assume "th2' = th2"
-with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
-hence "(Th th2, Th th1) \<in> (RAG s)^+"
-by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
-hence "Th th2 \<in> Domain ((RAG s)^+)"
-apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
-by (auto simp:Domain_def)
-hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
-then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
-from RAG_target_th [OF this]
-obtain cs' where "n = Cs cs'" by auto
-with d have "(Th th2, Cs cs') \<in> RAG s" by simp
-with runing_2 have "False"
-apply (unfold runing_def readys_def s_RAG_def)
-by (auto simp:eq_waiting)
-thus ?thesis by simp
-next
-assume "th2' \<in> dependants (wq s) th2"
-with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
-hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
-by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
-from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
-by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
-show ?thesis
-proof(rule dchain_unique[OF h1 _ h2, symmetric])
-from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
-from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
-qed
-qed
-qed
-qed
-lemma "card (runing s) \<le> 1"
-apply(subgoal_tac "finite (runing s)")
-prefer 2
-apply (metis finite_nat_set_iff_bounded lessI runing_unique)
-apply(rule ccontr)
-apply(simp)
-apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
-apply(subst (asm) card_le_Suc_iff)
-apply(simp)
-apply(auto)[1]
-apply (metis insertCI runing_unique)
-apply(auto)
-done
-end
-lemma create_pre:
-assumes stp: "step s e"
-and not_in: "th \<notin> threads s"
-and is_in: "th \<in> threads (e#s)"
-obtains prio where "e = Create th prio"
-proof -
-from assms
-show ?thesis
-proof(cases)
-case (thread_create thread prio)
-with is_in not_in have "e = Create th prio" by simp
-from that[OF this] show ?thesis .
-next
-case (thread_exit thread)
-with assms show ?thesis by (auto intro!:that)
-next
-case (thread_P thread)
-with assms show ?thesis by (auto intro!:that)
-next
-case (thread_V thread)
-with assms show ?thesis by (auto intro!:that)
-next
-case (thread_set thread)
-with assms show ?thesis by (auto intro!:that)
-qed
-qed
-lemma length_down_to_in:
-assumes le_ij: "i \<le> j"
-and le_js: "j \<le> length s"
-shows "length (down_to j i s) = j - i"
-proof -
-have "length (down_to j i s) = length (from_to i j (rev s))"
-by (unfold down_to_def, auto)
-also have "\<dots> = j - i"
-proof(rule length_from_to_in[OF le_ij])
-from le_js show "j \<le> length (rev s)" by simp
-qed
-finally show ?thesis .
-qed
-lemma moment_head:
-assumes le_it: "Suc i \<le> length t"
-obtains e where "moment (Suc i) t = e#moment i t"
-proof -
-have "i \<le> Suc i" by simp
-from length_down_to_in [OF this le_it]
-have "length (down_to (Suc i) i t) = 1" by auto
-then obtain e where "down_to (Suc i) i t = [e]"
-apply (cases "(down_to (Suc i) i t)") by auto
-moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
-by (rule down_to_conc[symmetric], auto)
-ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
-by (auto simp:down_to_moment)
-from that [OF this] show ?thesis .
-qed
-context valid_trace
-begin
-lemma cnp_cnv_eq:
-assumes "th \<notin> threads s"
-shows "cntP s th = cntV s th"
-using assms
-using cnp_cnv_cncs not_thread_cncs by auto
-end
-lemma eq_RAG:
-"RAG (wq s) = RAG s"
-by (unfold cs_RAG_def s_RAG_def, auto)
-context valid_trace
-begin
-lemma count_eq_dependants:
-assumes eq_pv: "cntP s th = cntV s th"
-shows "dependants (wq s) th = {}"
-proof -
-from cnp_cnv_cncs and eq_pv
-have "cntCS s th = 0"
-by (auto split:if_splits)
-moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
-proof -
-from finite_holding[of th] show ?thesis
-by (simp add:holdents_test)
-qed
-ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
-by (unfold cntCS_def holdents_test cs_dependants_def, auto)
-show ?thesis
-proof(unfold cs_dependants_def)
-{ assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
-then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
-hence "False"
-proof(cases)
-assume "(Th th', Th th) \<in> RAG (wq s)"
-thus "False" by (auto simp:cs_RAG_def)
-next
-fix c
-assume "(c, Th th) \<in> RAG (wq s)"
-with h and eq_RAG show "False"
-by (cases c, auto simp:cs_RAG_def)
-qed
-} thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
-qed
-qed
-lemma dependants_threads:
-shows "dependants (wq s) th \<subseteq> threads s"
-proof
-{ fix th th'
-assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
-have "Th th \<in> Domain (RAG s)"
-proof -
-from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
-hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
-with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
-thus ?thesis using eq_RAG by simp
-qed
-from dm_RAG_threads[OF this]
-have "th \<in> threads s" .
-} note hh = this
-fix th1
-assume "th1 \<in> dependants (wq s) th"
-hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
-by (unfold cs_dependants_def, simp)
-from hh [OF this] show "th1 \<in> threads s" .
-qed
-lemma finite_threads:
-shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-end
-lemma Max_f_mono:
-assumes seq: "A \<subseteq> B"
-and np: "A \<noteq> {}"
-and fnt: "finite B"
-shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
-from seq show "f ` A \<subseteq> f ` B" by auto
-next
-from np show "f ` A \<noteq> {}" by auto
-next
-from fnt and seq show "finite (f ` B)" by auto
-qed
-context valid_trace
-begin
-lemma cp_le:
-assumes th_in: "th \<in> threads s"
-shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
-show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
-\<le> Max ((\<lambda>th. preced th s) ` threads s)"
-(is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
-proof(rule Max_f_mono)
-show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
-next
-from finite_threads
-show "finite (threads s)" .
-next
-from th_in
-show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
-apply (auto simp:Domain_def)
-apply (rule_tac dm_RAG_threads)
-apply (unfold trancl_domain [of "RAG s", symmetric])
-by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
-qed
-qed
-lemma le_cp:
-shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
-show "Prc (priority th s) (last_set th s)
-\<le> Max (insert (Prc (priority th s) (last_set th s))
-((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
-(is "?l \<le> Max (insert ?l ?A)")
-proof(cases "?A = {}")
-case False
-have "finite ?A" (is "finite (?f ` ?B)")
-proof -
-have "finite ?B"
-proof-
-have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
-proof -
-let ?F = "\<lambda> (x, y). the_th x"
-have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
-apply (auto simp:image_def)
-by (rule_tac x = "(Th x, Th th)" in bexI, auto)
-moreover have "finite \<dots>"
-proof -
-from finite_RAG have "finite (RAG s)" .
-hence "finite ((RAG (wq s))\<^sup>+)"
-apply (unfold finite_trancl)
-by (auto simp: s_RAG_def cs_RAG_def wq_def)
-thus ?thesis by auto
-qed
-ultimately show ?thesis by (auto intro:finite_subset)
-qed
-thus ?thesis by (simp add:cs_dependants_def)
-qed
-thus ?thesis by simp
-qed
-from Max_insert [OF this False, of ?l] show ?thesis by auto
-next
-case True
-thus ?thesis by auto
-qed
-qed
-lemma max_cp_eq:
-shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
-(is "?l = ?r")
-proof(cases "threads s = {}")
-case True
-thus ?thesis by auto
-next
-case False
-have "?l \<in> ((cp s) ` threads s)"
-proof(rule Max_in)
-from finite_threads
-show "finite (cp s ` threads s)" by auto
-next
-from False show "cp s ` threads s \<noteq> {}" by auto
-qed
-then obtain th
-where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
-have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
-moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
-proof -
-have "?r \<in> (?f ` ?A)"
-proof(rule Max_in)
-from finite_threads
-show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
-next
-from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
-qed
-then obtain th' where
-th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
-from le_cp [of th']  eq_r
-have "?r \<le> cp s th'" by auto
-moreover have "\<dots> \<le> cp s th"
-proof(fold eq_l)
-show " cp s th' \<le> Max (cp s ` threads s)"
-proof(rule Max_ge)
-from th_in' show "cp s th' \<in> cp s ` threads s"
-by auto
-next
-from finite_threads
-show "finite (cp s ` threads s)" by auto
-qed
-qed
-ultimately show ?thesis by auto
-qed
-ultimately show ?thesis using eq_l by auto
-qed
-lemma max_cp_readys_threads_pre:
-assumes np: "threads s \<noteq> {}"
-shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(unfold max_cp_eq)
-show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
-proof -
-let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
-let ?f = "(\<lambda>th. preced th s)"
-have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
-proof(rule Max_in)
-from finite_threads show "finite (?f ` threads s)" by simp
-next
-from np show "?f ` threads s \<noteq> {}" by simp
-qed
-then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
-by (auto simp:Image_def)
-from th_chain_to_ready [OF tm_in]
-have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
-thus ?thesis
-proof
-assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
-then obtain th' where th'_in: "th' \<in> readys s"
-and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
-have "cp s th' = ?f tm"
-proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
-from dependants_threads finite_threads
-show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))"
-by (auto intro:finite_subset)
-next
-fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
-from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
-moreover have "p \<le> \<dots>"
-proof(rule Max_ge)
-from finite_threads
-show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-next
-from p_in and th'_in and dependants_threads[of th']
-show "p \<in> (\<lambda>th. preced th s) ` threads s"
-by (auto simp:readys_def)
-qed
-ultimately show "p \<le> preced tm s" by auto
-next
-show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
-proof -
-from tm_chain
-have "tm \<in> dependants (wq s) th'"
-by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
-thus ?thesis by auto
-qed
-qed
-with tm_max
-have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
-show ?thesis
-proof (fold h, rule Max_eqI)
-fix q
-assume "q \<in> cp s ` readys s"
-then obtain th1 where th1_in: "th1 \<in> readys s"
-and eq_q: "q = cp s th1" by auto
-show "q \<le> cp s th'"
-apply (unfold h eq_q)
-apply (unfold cp_eq_cpreced cpreced_def)
-apply (rule Max_mono)
-proof -
-from dependants_threads [of th1] th1_in
-show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq>
-(\<lambda>th. preced th s) ` threads s"
-by (auto simp:readys_def)
-next
-show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
-next
-from finite_threads
-show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
-qed
-next
-from finite_threads
-show "finite (cp s ` readys s)" by (auto simp:readys_def)
-next
-from th'_in
-show "cp s th' \<in> cp s ` readys s" by simp
-qed
-next
-assume tm_ready: "tm \<in> readys s"
-show ?thesis
-proof(fold tm_max)
-have cp_eq_p: "cp s tm = preced tm s"
-proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
-fix y
-assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
-show "y \<le> preced tm s"
-proof -
-{ fix y'
-assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
-have "y' \<le> preced tm s"
-proof(unfold tm_max, rule Max_ge)
-from hy' dependants_threads[of tm]
-show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
-next
-from finite_threads
-show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-qed
-} with hy show ?thesis by auto
-qed
-next
-from dependants_threads[of tm] finite_threads
-show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
-by (auto intro:finite_subset)
-next
-show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
-by simp
-qed
-moreover have "Max (cp s ` readys s) = cp s tm"
-proof(rule Max_eqI)
-from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
-next
-from finite_threads
-show "finite (cp s ` readys s)" by (auto simp:readys_def)
-next
-fix y assume "y \<in> cp s ` readys s"
-then obtain th1 where th1_readys: "th1 \<in> readys s"
-and h: "y = cp s th1" by auto
-show "y \<le> cp s tm"
-apply(unfold cp_eq_p h)
-apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
-proof -
-from finite_threads
-show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
-next
-show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
-by simp
-next
-from dependants_threads[of th1] th1_readys
-show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)
-\<subseteq> (\<lambda>th. preced th s) ` threads s"
-by (auto simp:readys_def)
-qed
-qed
-ultimately show " Max (cp s ` readys s) = preced tm s" by simp
-qed
-qed
-qed
-qed
-text {* (* ccc *) \noindent
-Since the current precedence of the threads in ready queue will always be boosted,
-there must be one inside it has the maximum precedence of the whole system.
-*}
-lemma max_cp_readys_threads:
-shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(cases "threads s = {}")
-case True
-thus ?thesis
-by (auto simp:readys_def)
-next
-case False
-show ?thesis by (rule max_cp_readys_threads_pre[OF False])
-qed
-end
-lemma eq_holding: "holding (wq s) th cs = holding s th cs"
-apply (unfold s_holding_def cs_holding_def wq_def, simp)
-done
-lemma f_image_eq:
-assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
-shows "f ` A = g ` A"
-proof
-show "f ` A \<subseteq> g ` A"
-by(rule image_subsetI, auto intro:h)
-next
-show "g ` A \<subseteq> f ` A"
-by (rule image_subsetI, auto intro:h[symmetric])
-qed
-definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
-where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
-lemma detached_test:
-shows "detached s th = (Th th \<notin> Field (RAG s))"
-apply(simp add: detached_def Field_def)
-apply(simp add: s_RAG_def)
-apply(simp add: s_holding_abv s_waiting_abv)
-apply(simp add: Domain_iff Range_iff)
-apply(simp add: wq_def)
-apply(auto)
-done
-context valid_trace
-begin
-lemma detached_intro:
-assumes eq_pv: "cntP s th = cntV s th"
-shows "detached s th"
-proof -
-from cnp_cnv_cncs
-have eq_cnt: "cntP s th =
-cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
-hence cncs_zero: "cntCS s th = 0"
-by (auto simp:eq_pv split:if_splits)
-with eq_cnt
-have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
-thus ?thesis
-proof
-assume "th \<notin> threads s"
-with range_in dm_RAG_threads
-show ?thesis
-by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
-next
-assume "th \<in> readys s"
-moreover have "Th th \<notin> Range (RAG s)"
-proof -
-from card_0_eq [OF finite_holding] and cncs_zero
-have "holdents s th = {}"
-by (simp add:cntCS_def)
-thus ?thesis
-apply(auto simp:holdents_test)
-apply(case_tac a)
-apply(auto simp:holdents_test s_RAG_def)
-done
-qed
-ultimately show ?thesis
-by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
-qed
-qed
-lemma detached_elim:
-assumes dtc: "detached s th"
-shows "cntP s th = cntV s th"
-proof -
-from cnp_cnv_cncs
-have eq_pv: " cntP s th =
-cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
-have cncs_z: "cntCS s th = 0"
-proof -
-from dtc have "holdents s th = {}"
-unfolding detached_def holdents_test s_RAG_def
-by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
-thus ?thesis by (auto simp:cntCS_def)
-qed
-show ?thesis
-proof(cases "th \<in> threads s")
-case True
-with dtc
-have "th \<in> readys s"
-by (unfold readys_def detached_def Field_def Domain_def Range_def,
-auto simp:eq_waiting s_RAG_def)
-with cncs_z and eq_pv show ?thesis by simp
-next
-case False
-with cncs_z and eq_pv show ?thesis by simp
-qed
-qed
-lemma detached_eq:
-shows "(detached s th) = (cntP s th = cntV s th)"
-by (insert vt, auto intro:detached_intro detached_elim)
-end
-text {*
-The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived
-from the concise and miniature model of PIP given in PrioGDef.thy.
-*}
-lemma eq_dependants: "dependants (wq s) = dependants s"
-by (simp add: s_dependants_abv wq_def)
-lemma next_th_unique:
-assumes nt1: "next_th s th cs th1"
-and nt2: "next_th s th cs th2"
-shows "th1 = th2"
-using assms by (unfold next_th_def, auto)
-lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
-apply (induct s, simp)
-proof -
-fix a s
-assume ih: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
-and eq_as: "a # s \<noteq> []"
-show "last_set th (a # s) < length (a # s)"
-proof(cases "s \<noteq> []")
-case False
-from False show ?thesis
-by (cases a, auto simp:last_set.simps)
-next
-case True
-from ih [OF True] show ?thesis
-by (cases a, auto simp:last_set.simps)
-qed
-qed
-lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
-by (induct s, auto simp:threads.simps)
-lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
-apply (drule_tac th_in_ne)
-by (unfold preced_def, auto intro: birth_time_lt)
-lemma inj_the_preced:
-"inj_on (the_preced s) (threads s)"
-by (metis inj_onI preced_unique the_preced_def)
-lemma tRAG_alt_def:
-"tRAG s = {(Th th1, Th th2) | th1 th2.
-\<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
-by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
-lemma tRAG_Field:
-"Field (tRAG s) \<subseteq> Field (RAG s)"
-by (unfold tRAG_alt_def Field_def, auto)
-lemma tRAG_ancestorsE:
-assumes "x \<in> ancestors (tRAG s) u"
-obtains th where "x = Th th"
-proof -
-from assms have "(u, x) \<in> (tRAG s)^+"
-by (unfold ancestors_def, auto)
-from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
-then obtain th where "x = Th th"
-by (unfold tRAG_alt_def, auto)
-from that[OF this] show ?thesis .
-qed
-lemma tRAG_mono:
-assumes "RAG s' \<subseteq> RAG s"
-shows "tRAG s' \<subseteq> tRAG s"
-using assms
-by (unfold tRAG_alt_def, auto)
-lemma holding_next_thI:
-assumes "holding s th cs"
-and "length (wq s cs) > 1"
-obtains th' where "next_th s th cs th'"
-proof -
-from assms(1)[folded eq_holding, unfolded cs_holding_def]
-have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
-then obtain rest where h1: "wq s cs = th#rest"
-by (cases "wq s cs", auto)
-with assms(2) have h2: "rest \<noteq> []" by auto
-let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
-have "next_th s th cs ?th'" using  h1(1) h2
-by (unfold next_th_def, auto)
-from that[OF this] show ?thesis .
-qed
-lemma RAG_tRAG_transfer:
-assumes "vt s'"
-assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
-and "(Cs cs, Th th'') \<in> RAG s'"
-shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
-proof -
-interpret vt_s': valid_trace "s'" using assms(1)
-by (unfold_locales, simp)
-interpret rtree: rtree "RAG s'"
-proof
-show "single_valued (RAG s')"
-apply (intro_locales)
-by (unfold single_valued_def,
-auto intro:vt_s'.unique_RAG)
-show "acyclic (RAG s')"
-by (rule vt_s'.acyclic_RAG)
-qed
-{ fix n1 n2
-assume "(n1, n2) \<in> ?L"
-from this[unfolded tRAG_alt_def]
-obtain th1 th2 cs' where
-h: "n1 = Th th1" "n2 = Th th2"
-"(Th th1, Cs cs') \<in> RAG s"
-"(Cs cs', Th th2) \<in> RAG s" by auto
-from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
-from h(3) and assms(2)
-have "(Th th1, Cs cs') = (Th th, Cs cs) \<or>
-(Th th1, Cs cs') \<in> RAG s'" by auto
-hence "(n1, n2) \<in> ?R"
-proof
-assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
-hence eq_th1: "th1 = th" by simp
-moreover have "th2 = th''"
-proof -
-from h1 have "cs' = cs" by simp
-from assms(3) cs_in[unfolded this] rtree.sgv
-show ?thesis
-by (unfold single_valued_def, auto)
-qed
-ultimately show ?thesis using h(1,2) by auto
-next
-assume "(Th th1, Cs cs') \<in> RAG s'"
-with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
-by (unfold tRAG_alt_def, auto)
-from this[folded h(1, 2)] show ?thesis by auto
-qed
-} moreover {
-fix n1 n2
-assume "(n1, n2) \<in> ?R"
-hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
-hence "(n1, n2) \<in> ?L"
-proof
-assume "(n1, n2) \<in> tRAG s'"
-moreover have "... \<subseteq> ?L"
-proof(rule tRAG_mono)
-show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
-qed
-ultimately show ?thesis by auto
-next
-assume eq_n: "(n1, n2) = (Th th, Th th'')"
-from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
-moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
-ultimately show ?thesis
-by (unfold eq_n tRAG_alt_def, auto)
-qed
-} ultimately show ?thesis by auto
-qed
-context valid_trace
-begin
-lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
-end
-lemma cp_alt_def:
-"cp s th =
-Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
-proof -
-have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
-Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
-(is "Max (_ ` ?L) = Max (_ ` ?R)")
-proof -
-have "?L = ?R"
-by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
-thus ?thesis by simp
-qed
-thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
-qed
-lemma cp_gen_alt_def:
-"cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
-by (auto simp:cp_gen_def)
-lemma tRAG_nodeE:
-assumes "(n1, n2) \<in> tRAG s"
-obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
-using assms
-by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
-lemma subtree_nodeE:
-assumes "n \<in> subtree (tRAG s) (Th th)"
-obtains th1 where "n = Th th1"
-proof -
-show ?thesis
-proof(rule subtreeE[OF assms])
-assume "n = Th th"
-from that[OF this] show ?thesis .
-next
-assume "Th th \<in> ancestors (tRAG s) n"
-hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
-hence "\<exists> th1. n = Th th1"
-proof(induct)
-case (base y)
-from tRAG_nodeE[OF this] show ?case by metis
-next
-case (step y z)
-thus ?case by auto
-qed
-with that show ?thesis by auto
-qed
-qed
-lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
-proof -
-have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*"
-by (rule rtrancl_mono, auto simp:RAG_split)
-also have "... \<subseteq> ((RAG s)^*)^*"
-by (rule rtrancl_mono, auto)
-also have "... = (RAG s)^*" by simp
-finally show ?thesis by (unfold tRAG_def, simp)
-qed
-lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
-proof -
-{ fix a
-assume "a \<in> subtree (tRAG s) x"
-hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
-with tRAG_star_RAG[of s]
-have "(a, x) \<in> (RAG s)^*" by auto
-hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
-} thus ?thesis by auto
-qed
-lemma tRAG_trancl_eq:
-"{th'. (Th th', Th th)  \<in> (tRAG s)^+} =
-{th'. (Th th', Th th)  \<in> (RAG s)^+}"
-(is "?L = ?R")
-proof -
-{ fix th'
-assume "th' \<in> ?L"
-hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
-from tranclD[OF this]
-obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
-from tRAG_subtree_RAG[of s] and this(2)
-have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG)
-moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto
-ultimately have "th' \<in> ?R"  by auto
-} moreover
-{ fix th'
-assume "th' \<in> ?R"
-hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
-from plus_rpath[OF this]
-obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
-hence "(Th th', Th th) \<in> (tRAG s)^+"
-proof(induct xs arbitrary:th' th rule:length_induct)
-case (1 xs th' th)
-then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
-show ?case
-proof(cases "xs1")
-case Nil
-from 1(2)[unfolded Cons1 Nil]
-have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
-hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
-then obtain cs where "x1 = Cs cs"
-by (unfold s_RAG_def, auto)
-from rpath_nnl_lastE[OF rp[unfolded this]]
-show ?thesis by auto
-next
-case (Cons x2 xs2)
-from 1(2)[unfolded Cons1[unfolded this]]
-have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
-from rpath_edges_on[OF this]
-have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
-have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
-by (simp add: edges_on_unfold)
-with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
-then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
-have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
-by (simp add: edges_on_unfold)
-from this eds
-have rg2: "(x1, x2) \<in> RAG s" by auto
-from this[unfolded eq_x1]
-obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
-from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
-have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
-from rp have "rpath (RAG s) x2 xs2 (Th th)"
-by  (elim rpath_ConsE, simp)
-from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
-show ?thesis
-proof(cases "xs2 = []")
-case True
-from rpath_nilE[OF rp'[unfolded this]]
-have "th1 = th" by auto
-from rt1[unfolded this] show ?thesis by auto
-next
-case False
-from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
-have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
-with rt1 show ?thesis by auto
-qed
-qed
-qed
-hence "th' \<in> ?L" by auto
-} ultimately show ?thesis by blast
-qed
-lemma tRAG_trancl_eq_Th:
-"{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} =
-{Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}"
-using tRAG_trancl_eq by auto
-lemma dependants_alt_def:
-"dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
-by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
-context valid_trace
-begin
-lemma count_eq_tRAG_plus:
-assumes "cntP s th = cntV s th"
-shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
-using assms count_eq_dependants dependants_alt_def eq_dependants by auto
-lemma count_eq_RAG_plus:
-assumes "cntP s th = cntV s th"
-shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
-using assms count_eq_dependants cs_dependants_def eq_RAG by auto
-lemma count_eq_RAG_plus_Th:
-assumes "cntP s th = cntV s th"
-shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
-using count_eq_RAG_plus[OF assms] by auto
-lemma count_eq_tRAG_plus_Th:
-assumes "cntP s th = cntV s th"
-shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
-using count_eq_tRAG_plus[OF assms] by auto
-end
-lemma tRAG_subtree_eq:
-"(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th'  \<in> (subtree (RAG s) (Th th))}"
-(is "?L = ?R")
-proof -
-{ fix n
-assume h: "n \<in> ?L"
-hence "n \<in> ?R"
-by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG)
-} moreover {
-fix n
-assume "n \<in> ?R"
-then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
-by (auto simp:subtree_def)
-from rtranclD[OF this(2)]
-have "n \<in> ?L"
-proof
-assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
-with h have "n \<in> {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}" by auto
-thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
-qed (insert h, auto simp:subtree_def)
-} ultimately show ?thesis by auto
-qed
-lemma threads_set_eq:
-"the_thread ` (subtree (tRAG s) (Th th)) =
-{th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
-by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
-lemma cp_alt_def1:
-"cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
-proof -
-have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
-((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
-by auto
-thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
-qed
-lemma cp_gen_def_cond:
-assumes "x = Th th"
-shows "cp s th = cp_gen s (Th th)"
-by (unfold cp_alt_def1 cp_gen_def, simp)
-lemma cp_gen_over_set:
-assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
-shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
-proof(rule f_image_eq)
-fix a
-assume "a \<in> A"
-from assms[rule_format, OF this]
-obtain th where eq_a: "a = Th th" by auto
-show "cp_gen s a = (cp s \<circ> the_thread) a"
-by  (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
-qed
-context valid_trace
-begin
-lemma RAG_threads:
-assumes "(Th th) \<in> Field (RAG s)"
-shows "th \<in> threads s"
-using assms
-by (metis Field_def UnE dm_RAG_threads range_in vt)
-lemma subtree_tRAG_thread:
-assumes "th \<in> threads s"
-shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
-proof -
-have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
-by (unfold tRAG_subtree_eq, simp)
-also have "... \<subseteq> ?R"
-proof
-fix x
-assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
-then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
-from this(2)
-show "x \<in> ?R"
-proof(cases rule:subtreeE)
-case 1
-thus ?thesis by (simp add: assms h(1))
-next
-case 2
-thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI)
-qed
-qed
-finally show ?thesis .
-qed
-lemma readys_root:
-assumes "th \<in> readys s"
-shows "root (RAG s) (Th th)"
-proof -
-{ fix x
-assume "x \<in> ancestors (RAG s) (Th th)"
-hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
-from tranclD[OF this]
-obtain z where "(Th th, z) \<in> RAG s" by auto
-with assms(1) have False
-apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
-by (fold wq_def, blast)
-} thus ?thesis by (unfold root_def, auto)
-qed
-lemma readys_in_no_subtree:
-assumes "th \<in> readys s"
-and "th' \<noteq> th"
-shows "Th th \<notin> subtree (RAG s) (Th th')"
-proof
-assume "Th th \<in> subtree (RAG s) (Th th')"
-thus False
-proof(cases rule:subtreeE)
-case 1
-with assms show ?thesis by auto
-next
-case 2
-with readys_root[OF assms(1)]
-show ?thesis by (auto simp:root_def)
-qed
-qed
-lemma not_in_thread_isolated:
-assumes "th \<notin> threads s"
-shows "(Th th) \<notin> Field (RAG s)"
-proof
-assume "(Th th) \<in> Field (RAG s)"
-with dm_RAG_threads and range_in assms
-show False by (unfold Field_def, blast)
-qed
-lemma wf_RAG: "wf (RAG s)"
-proof(rule finite_acyclic_wf)
-from finite_RAG show "finite (RAG s)" .
-next
-from acyclic_RAG show "acyclic (RAG s)" .
-qed
-lemma sgv_wRAG: "single_valued (wRAG s)"
-using waiting_unique
-by (unfold single_valued_def wRAG_def, auto)
-lemma sgv_hRAG: "single_valued (hRAG s)"
-using holding_unique
-by (unfold single_valued_def hRAG_def, auto)
-lemma sgv_tRAG: "single_valued (tRAG s)"
-by (unfold tRAG_def, rule single_valued_relcomp,
-insert sgv_wRAG sgv_hRAG, auto)
-lemma acyclic_tRAG: "acyclic (tRAG s)"
-proof(unfold tRAG_def, rule acyclic_compose)
-show "acyclic (RAG s)" using acyclic_RAG .
-next
-show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-next
-show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
-qed
-lemma sgv_RAG: "single_valued (RAG s)"
-using unique_RAG by (auto simp:single_valued_def)
-lemma rtree_RAG: "rtree (RAG s)"
-using sgv_RAG acyclic_RAG
-by (unfold rtree_def rtree_axioms_def sgv_def, auto)
-end
-sublocale valid_trace < rtree_RAG: rtree "RAG s"
-proof
-show "single_valued (RAG s)"
-apply (intro_locales)
-by (unfold single_valued_def,
-auto intro:unique_RAG)
-show "acyclic (RAG s)"
-by (rule acyclic_RAG)
-qed
-sublocale valid_trace < rtree_s: rtree "tRAG s"
-proof(unfold_locales)
-from sgv_tRAG show "single_valued (tRAG s)" .
-next
-from acyclic_tRAG show "acyclic (tRAG s)" .
-qed
-sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
-proof -
-show "fsubtree (RAG s)"
-proof(intro_locales)
-show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
-next
-show "fsubtree_axioms (RAG s)"
-proof(unfold fsubtree_axioms_def)
-from wf_RAG show "wf (RAG s)" .
-qed
-qed
-qed
-sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
-proof -
-have "fsubtree (tRAG s)"
-proof -
-have "fbranch (tRAG s)"
-proof(unfold tRAG_def, rule fbranch_compose)
-show "fbranch (wRAG s)"
-proof(rule finite_fbranchI)
-from finite_RAG show "finite (wRAG s)"
-by (unfold RAG_split, auto)
-qed
-next
-show "fbranch (hRAG s)"
-proof(rule finite_fbranchI)
-from finite_RAG
-show "finite (hRAG s)" by (unfold RAG_split, auto)
-qed
-qed
-moreover have "wf (tRAG s)"
-proof(rule wf_subset)
-show "wf (RAG s O RAG s)" using wf_RAG
-by (fold wf_comp_self, simp)
-next
-show "tRAG s \<subseteq> (RAG s O RAG s)"
-by (unfold tRAG_alt_def, auto)
-qed
-ultimately show ?thesis
-by (unfold fsubtree_def fsubtree_axioms_def,auto)
-qed
-from this[folded tRAG_def] show "fsubtree (tRAG s)" .
-qed
-lemma Max_UNION:
-assumes "finite A"
-and "A \<noteq> {}"
-and "\<forall> M \<in> f ` A. finite M"
-and "\<forall> M \<in> f ` A. M \<noteq> {}"
-shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
-using assms[simp]
-proof -
-have "?L = Max (\<Union>(f ` A))"
-by (fold Union_image_eq, simp)
-also have "... = ?R"
-by (subst Max_Union, simp+)
-finally show ?thesis .
-qed
-lemma max_Max_eq:
-assumes "finite A"
-and "A \<noteq> {}"
-and "x = y"
-shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
-proof -
-have "?R = Max (insert y A)" by simp
-also from assms have "... = ?L"
-by (subst Max.insert, simp+)
-finally show ?thesis by simp
-qed
-context valid_trace
-begin
-(* ddd *)
-lemma cp_gen_rec:
-assumes "x = Th th"
-shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
-proof(cases "children (tRAG s) x = {}")
-case True
-show ?thesis
-by (unfold True cp_gen_def subtree_children, simp add:assms)
-next
-case False
-hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
-note fsbttRAGs.finite_subtree[simp]
-have [simp]: "finite (children (tRAG s) x)"
-by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree],
-rule children_subtree)
-{ fix r x
-have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
-} note this[simp]
-have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
-proof -
-from False obtain q where "q \<in> children (tRAG s) x" by blast
-moreover have "subtree (tRAG s) q \<noteq> {}" by simp
-ultimately show ?thesis by blast
-qed
-have h: "Max ((the_preced s \<circ> the_thread) `
-({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
-Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
-(is "?L = ?R")
-proof -
-let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
-let "Max (_ \<union> (?h ` ?B))" = ?R
-let ?L1 = "?f ` \<Union>(?g ` ?B)"
-have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
-proof -
-have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
-also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
-finally have "Max ?L1 = Max ..." by simp
-also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
-by (subst Max_UNION, simp+)
-also have "... = Max (cp_gen s ` children (tRAG s) x)"
-by (unfold image_comp cp_gen_alt_def, simp)
-finally show ?thesis .
-qed
-show ?thesis
-proof -
-have "?L = Max (?f ` ?A \<union> ?L1)" by simp
-also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
-by (subst Max_Un, simp+)
-also have "... = max (?f x) (Max (?h ` ?B))"
-by (unfold eq_Max_L1, simp)
-also have "... =?R"
-by (rule max_Max_eq, (simp)+, unfold assms, simp)
-finally show ?thesis .
-qed
-qed  thus ?thesis
-by (fold h subtree_children, unfold cp_gen_def, simp)
-qed
-lemma cp_rec:
-"cp s th = Max ({the_preced s th} \<union>
-(cp s o the_thread) ` children (tRAG s) (Th th))"
-proof -
-have "Th th = Th th" by simp
-note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
-show ?thesis
-proof -
-have "cp_gen s ` children (tRAG s) (Th th) =
-(cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
-proof(rule cp_gen_over_set)
-show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
-by (unfold tRAG_alt_def, auto simp:children_def)
-qed
-thus ?thesis by (subst (1) h(1), unfold h(2), simp)
-qed
-qed
-end
-(* keep *)
-lemma next_th_holding:
-assumes vt: "vt s"
-and nxt: "next_th s th cs th'"
-shows "holding (wq s) th cs"
-proof -
-from nxt[unfolded next_th_def]
-obtain rest where h: "wq s cs = th # rest"
-"rest \<noteq> []"
-"th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
-thus ?thesis
-by (unfold cs_holding_def, auto)
-qed
-context valid_trace
-begin
-lemma next_th_waiting:
-assumes nxt: "next_th s th cs th'"
-shows "waiting (wq s) th' cs"
-proof -
-from nxt[unfolded next_th_def]
-obtain rest where h: "wq s cs = th # rest"
-"rest \<noteq> []"
-"th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
-from wq_distinct[of cs, unfolded h]
-have dst: "distinct (th # rest)" .
-have in_rest: "th' \<in> set rest"
-proof(unfold h, rule someI2)
-show "distinct rest \<and> set rest = set rest" using dst by auto
-next
-fix x assume "distinct x \<and> set x = set rest"
-with h(2)
-show "hd x \<in> set (rest)" by (cases x, auto)
-qed
-hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
-moreover have "th' \<noteq> hd (wq s cs)"
-by (unfold h(1), insert in_rest dst, auto)
-ultimately show ?thesis by (auto simp:cs_waiting_def)
-qed
-lemma next_th_RAG:
-assumes nxt: "next_th (s::event list) th cs th'"
-shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
-using vt assms next_th_holding next_th_waiting
-by (unfold s_RAG_def, simp)
-end
--- {* A useless definition *}
-definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
-where "cps s = {(th, cp s th) | th . th \<in> threads s}"
-end

changeset 97	c7ba70dc49bd
parent 96	4805c6333fef
parent 93	524bd3caa6b6
child 98	382293d415f3