pip: comparison PrioG.thy

equal deleted inserted replaced

--1:000000000000
+:110247f9d47e
+theory PrioG
+imports PrioGDef
+begin
+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)
+lemma wq_distinct: "vt s \<Longrightarrow> distinct (wq s cs)"
+proof(erule_tac vt.induct, 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> (depend 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> (depend s)"
+by (simp add:s_depend_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
+lemma step_back_vt: "vt (e#s) \<Longrightarrow> vt s"
+by(ind_cases "vt (e#s)", simp)
+lemma step_back_step: "vt (e#s) \<Longrightarrow> step s e"
+by(ind_cases "vt (e#s)", simp)
+lemma block_pre:
+fixes thread cs s
+assumes vt_e: "vt (e#s)"
+and 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 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 step_back_vt[OF vt], 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
+lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
+thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (depend s)^+"
+apply (ind_cases "vt ((P thread cs)#s)")
+apply (ind_cases "step s (P thread cs)")
+by auto
+lemma abs1:
+fixes e es
+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)"
+lemma abs2:
+assumes vt: "vt (e#s)"
+and 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 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 step_back_vt[OF vt], 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 step_back_vt[OF vt], 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
+lemma vt_moment: "\<And> t. \<lbrakk>vt s\<rbrakk> \<Longrightarrow> vt (moment t s)"
+proof(induct s, simp)
+fix a s t
+assume h: "\<And>t.\<lbrakk>vt s\<rbrakk> \<Longrightarrow> vt (moment t s)"
+and vt_a: "vt (a # s)"
+show "vt (moment t (a # s))"
+proof(cases "t \<ge> length (a#s)")
+case True
+from True have "moment t (a#s) = a#s" by simp
+with vt_a show ?thesis by simp
+next
+case False
+hence le_t1: "t \<le> length s" by simp
+from vt_a have "vt s"
+by (erule_tac evt_cons, simp)
+from h [OF this] have "vt (moment t s)" .
+moreover have "moment t (a#s) = moment t s"
+proof -
+from moment_app [OF le_t1, of "[a]"]
+show ?thesis by simp
+qed
+ultimately show ?thesis by auto
+qed
+qed
+(* Wrong:
+lemma \<lbrakk>thread \<in> set (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
+*)
+lemma waiting_unique_pre:
+fixes cs1 cs2 s thread
+assumes vt: "vt s"
+and 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: "vt (e#moment t2 s)"
+proof -
+from vt_moment [OF vt]
+have "vt (moment ?t3 s)" .
+with eq_m show ?thesis by simp
+qed
+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 abs2 [OF vt_e True eq_th h2 h1]
+show ?thesis by auto
+next
+case False
+from block_pre [OF vt_e False h1]
+have "e = P thread cs2" .
+with 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: "vt  (e#moment t1 s)"
+proof -
+from vt_moment [OF vt]
+have "vt (moment ?t3 s)" .
+with eq_m show ?thesis by simp
+qed
+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 abs2 [OF vt_e True eq_th h2 h1]
+show ?thesis by auto
+next
+case False
+from block_pre [OF vt_e False h1]
+have "e = P thread cs1" .
+with 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 [OF vt]
+have "vt (moment ?t3 s)" .
+with eq_m show ?thesis by simp
+qed
+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 abs2 [OF vt_e True eq_th h2 h1]
+show ?thesis by auto
+next
+case False
+from block_pre [OF vt_e 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
+from abs2 [OF this True eq_th h2 h1]
+show ?thesis .
+next
+case False
+have vt_e: "vt (e#moment t2 s)"
+proof -
+from vt_moment [OF vt] eqt12
+have "vt (moment (Suc t2) s)" by auto
+with eq_m eqt12 show ?thesis by simp
+qed
+from block_pre [OF vt_e False h1]
+have "e = P thread cs2" .
+with eq_e1 neq12 show ?thesis by auto
+qed
+qed
+} ultimately show ?thesis by arith
+qed
+qed
+lemma waiting_unique:
+fixes s cs1 cs2
+assumes "vt s"
+and "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
+(* not used *)
+lemma held_unique:
+fixes s::"state"
+assumes "holding s th1 cs"
+and "holding s th2 cs"
+shows "th1 = th2"
+using assms
+unfolding s_holding_def
+by auto
+lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
+apply (induct s, auto)
+by (case_tac a, auto split:if_splits)
+lemma birthtime_unique:
+"\<lbrakk>birthtime th1 s = birthtime 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:birthtime_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 "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
+from birthtime_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
+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
+lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
+apply (unfold s_depend_def s_waiting_def wq_def)
+by (simp add:Let_def)
+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
+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"
+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 wq_distinct[OF step_back_vt[OF vt], 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 wq_distinct[OF step_back_vt[OF vt], 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 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 wq_distinct[OF step_back_vt[OF vt], 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 wq_distinct[OF step_back_vt[OF vt], 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)"
+have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+proof(rule someI2)
+from wq_distinct[OF step_back_vt[OF vt], 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"
+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 wq_distinct [OF step_back_vt[OF vt], 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 wq_distinct [OF step_back_vt[OF vt], 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 wq_distinct [OF step_back_vt[OF vt], of cs, unfolded wq_def]
+show False by auto
+qed
+qed
+qed
+lemma step_depend_v:
+fixes th::thread
+assumes vt:
+"vt (V th cs#s)"
+shows "
+depend (V th cs # s) =
+depend 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_depend_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
+lemma step_depend_p:
+"vt (P th cs#s) \<Longrightarrow>
+depend (P th cs # s) =  (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
+else depend s \<union> {(Th th, Cs cs)})"
+apply(simp only: s_depend_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(auto simp:s_depend_def wq_def cs_holding_def)
+done
+lemma simple_A:
+fixes A
+assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
+shows "A = {} \<or> (\<exists> a. A = {a})"
+proof(cases "A = {}")
+case True thus ?thesis by simp
+next
+case False then obtain a where "a \<in> A" by auto
+with h have "A = {a}" by auto
+thus ?thesis by simp
+qed
+lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+by (unfold s_depend_def, auto)
+lemma acyclic_depend:
+fixes s
+assumes vt: "vt s"
+shows "acyclic (depend s)"
+proof -
+from vt show ?thesis
+proof(induct)
+case (vt_cons s e)
+assume ih: "acyclic (depend 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:depend_create_unchanged)
+next
+case (Exit th)
+with ih show ?thesis by (simp add:depend_exit_unchanged)
+next
+case (V th cs)
+from V vt stp have vtt: "vt (V th cs#s)" by auto
+from step_depend_v [OF this]
+have eq_de:
+"depend (e # s) =
+depend 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 depend_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_depend_def s_waiting_def cs_waiting_def wq_def by simp
+hence "cs' = cs"
+proof(rule waiting_unique [OF vt])
+from eq_wq wq_distinct[OF vt, 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 wq_distinct[OF vt, 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 wq_distinct[OF vt, 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 wq_distinct[OF vt, 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 wq_distinct[OF vt, 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_depend_p [OF this] P
+have "depend (e # s) =
+(if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
+depend 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 =  depend s \<union> {(Cs cs, Th th)}" by simp
+have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
+proof
+assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
+hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+from tranclD2 [OF this]
+obtain x where "(x, Cs cs) \<in> depend s" by auto
+with True show False by (auto simp:s_depend_def cs_waiting_def)
+qed
+with acyclic_insert ih eq_r show ?thesis by auto
+next
+case False
+hence eq_r: "?R =  depend s \<union> {(Th th, Cs cs)}" by simp
+have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
+proof
+assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
+hence "(Cs cs, Th th) \<in> (depend 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> (depend 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 depend_set_unchanged
+show ?thesis by (simp add:depend_set_unchanged)
+qed
+next
+case vt_nil
+show "acyclic (depend ([]::state))"
+by (auto simp: s_depend_def cs_waiting_def
+cs_holding_def wq_def acyclic_def)
+qed
+qed
+lemma finite_depend:
+fixes s
+assumes vt: "vt s"
+shows "finite (depend s)"
+proof -
+from vt show ?thesis
+proof(induct)
+case (vt_cons s e)
+assume ih: "finite (depend 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:depend_create_unchanged)
+next
+case (Exit th)
+with ih show ?thesis by (simp add:depend_exit_unchanged)
+next
+case (V th cs)
+from V vt stp have vtt: "vt (V th cs#s)" by auto
+from step_depend_v [OF this]
+have eq_de: "depend (e # s) =
+depend 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 auto
+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_depend_p [OF this] P
+have "depend (e # s) =
+(if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
+depend 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 =  depend s \<union> {(Cs cs, Th th)}" by simp
+with True and ih show ?thesis by auto
+next
+case False
+hence "?R = depend 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:depend_set_unchanged)
+qed
+next
+case vt_nil
+show "finite (depend ([]::state))"
+by (auto simp: s_depend_def cs_waiting_def
+cs_holding_def wq_def acyclic_def)
+qed
+qed
+text {* Several useful lemmas *}
+lemma wf_dep_converse:
+fixes s
+assumes vt: "vt s"
+shows "wf ((depend s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+from finite_depend [OF vt]
+show "finite (depend s)" .
+next
+from acyclic_depend[OF vt]
+show "acyclic (depend s)" .
+qed
+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> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
+by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+lemma wq_threads:
+fixes s cs
+assumes vt: "vt s"
+and 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)
+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_depend_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 wq_distinct[OF vt, 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>vt s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+apply(unfold s_depend_def cs_waiting_def cs_holding_def)
+by (auto intro:wq_threads)
+lemma readys_v_eq:
+fixes th thread cs rest
+assumes vt: "vt s"
+and 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 vt, 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
+lemma chain_building:
+assumes vt: "vt s"
+shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
+proof -
+from wf_dep_converse [OF vt]
+have h: "wf ((depend s)\<inverse>)" .
+show ?thesis
+proof(induct rule:wf_induct [OF h])
+fix x
+assume ih [rule_format]:
+"\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow>
+y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
+show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
+proof
+assume x_d: "x \<in> Domain (depend s)"
+show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
+proof(cases x)
+case (Th th)
+from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
+with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
+from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
+hence "Cs cs \<in> Domain (depend 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> (depend s)\<^sup>+" by auto
+have "(x, Th th') \<in> (depend 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> (depend s)^-1" by (auto simp:s_depend_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 [OF vt] have "th' \<in> threads s" by auto
+with False have "Th th' \<in> Domain (depend s)"
+by (auto simp:readys_def wq_def s_waiting_def s_depend_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> (depend s)\<^sup>+" by auto
+from th'_d and th''_in
+have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
+with th''_r show ?thesis by auto
+qed
+qed
+qed
+qed
+qed
+lemma th_chain_to_ready:
+fixes s th
+assumes vt: "vt s"
+and th_in: "th \<in> threads s"
+shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend 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 (depend s)"
+by (auto simp:readys_def s_waiting_def s_depend_def wq_def cs_waiting_def Domain_def)
+from chain_building [rule_format, OF vt this]
+show ?thesis by auto
+qed
+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)
+lemma unique_depend: "\<lbrakk>vt s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
+apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
+by(auto elim:waiting_unique holding_unique)
+lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
+by (induct rule:trancl_induct, auto)
+lemma dchain_unique:
+assumes vt: "vt s"
+and th1_d: "(n, Th th1) \<in> (depend s)^+"
+and th1_r: "th1 \<in> readys s"
+and th2_d: "(n, Th th2) \<in> (depend 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_depend [OF vt]
+have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
+hence "False"
+proof
+assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
+from trancl_split [OF this]
+obtain n where dd: "(Th th1, n) \<in> depend s" by auto
+then obtain cs where eq_n: "n = Cs cs"
+by (auto simp:s_depend_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_depend_def wq_def s_waiting_def cs_waiting_def)
+with th1_r show ?thesis by auto
+next
+assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
+from trancl_split [OF this]
+obtain n where dd: "(Th th2, n) \<in> depend s" by auto
+then obtain cs where eq_n: "n = Cs cs"
+by (auto simp:s_depend_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_depend_def s_waiting_def cs_waiting_def)
+with th2_r show ?thesis by auto
+qed
+} thus ?thesis by auto
+qed
+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_depend_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_depend_p[OF vt] by auto
+qed
+lemma finite_holding:
+fixes s th cs
+assumes vt: "vt s"
+shows "finite (holdents s th)"
+proof -
+let ?F = "\<lambda> (x, y). the_cs x"
+from finite_depend [OF vt]
+have "finite (depend s)" .
+hence "finite (?F `(depend s))" by simp
+moreover have "{cs . (Cs cs, Th th) \<in> depend 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> depend s"
+hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
+moreover have "?F (Cs x, Th th) = x" by simp
+ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend 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 step_back_step[OF vtv]
+have cs_in: "cs \<in> holdents s thread"
+apply (cases, unfold holdents_test s_depend_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 wq_distinct[OF step_back_vt[OF vtv], of cs])
+apply (unfold holdents_test, unfold step_depend_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> depend 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 finite_holding [OF vtv]
+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 cnp_cnv_cncs:
+fixes s th
+assumes vt: "vt s"
+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)
+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 wq_threads [OF vt 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:depend_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:depend_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> (depend s)\<^sup>+"
+from thread_P vt stp ih  have vtp: "vt (P thread cs#s)" by auto
+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, clarify)
+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_depend_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_depend_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> depend s} =
+Suc (card {cs. (Cs cs, Th thread) \<in> depend 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 finite_holding [OF vt, of thread]
+show " finite {cs. (Cs cs, Th thread) \<in> depend 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> depend 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_depend_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 wq_distinct [OF vtp, 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_depend_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
+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 wq_distinct[OF step_back_vt[OF vtv], 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
+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 wq_distinct[OF step_back_vt[OF vtv], of cs]
+and eq_wq show False by auto
+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 (unfold eq_e, rule readys_v_eq [OF _ neq_th eq_wq False], auto)
+moreover have "cntCS (e#s) th = cntCS s th"
+apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
+proof -
+have "{csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
+{cs. (Cs cs, Th th) \<in> depend s}"
+proof -
+from False eq_wq
+have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> depend 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 wq_distinct[OF step_back_vt[OF vtv], 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> depend s"
+by auto
+qed
+thus ?thesis by auto
+qed
+thus "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
+card {cs. (Cs cs, Th th) \<in> depend 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 wq_distinct[OF step_back_vt[OF vtv], 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_depend_v[OF vtv],
+auto simp:next_th_def eq_set s_depend_def holdents_test wq_def
+Let_def cs_holding_def)
+by (insert wq_distinct[OF step_back_vt[OF vtv], 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 wq_threads[OF step_back_vt[OF vtv]])
+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 wq_distinct[OF step_back_vt[OF vtv], 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 waiting_unique[OF step_back_vt[OF vtv] 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 wq_threads [OF step_back_vt[OF vtv] 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_depend_v[OF vtv], auto)
+proof -
+show "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs} =
+Suc (card {cs. (Cs cs, Th th) \<in> depend 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) ` depend s)"
+apply (auto simp:image_def)
+by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
+with finite_depend[OF step_back_vt[OF vtv]]
+show "finite {cs. (Cs cs, Th th) \<in> depend s}" by (auto intro:finite_subset)
+next
+show "cs \<notin> {cs. (Cs cs, Th th) \<in> depend s}"
+proof
+assume "cs \<in> {cs. (Cs cs, Th th) \<in> depend s}"
+hence "(Cs cs, Th th) \<in> depend s" by simp
+with True neq_th eq_wq show False
+by (auto simp:next_th_def s_depend_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:depend_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_depend_def wq_def cs_holding_def)
+qed
+qed
+lemma not_thread_cncs:
+fixes th s
+assumes vt: "vt s"
+and 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)
+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:depend_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:depend_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_depend_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
+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 wq_distinct[OF step_back_vt[OF vtv], 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 "hd x \<in> set rest" by (cases x, auto)
+qed
+with eq_wq have "?t \<in> set (wq s cs)" by simp
+from wq_threads[OF step_back_vt[OF vtv], OF this] and ni
+show False by auto
+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_depend_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:depend_set_unchanged)
+qed
+next
+case vt_nil
+show ?case
+by (unfold cntCS_def,
+auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
+qed
+qed
+lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
+by (auto simp:s_waiting_def cs_waiting_def wq_def)
+lemma dm_depend_threads:
+fixes th s
+assumes vt: "vt s"
+and in_dom: "(Th th) \<in> Domain (depend s)"
+shows "th \<in> threads s"
+proof -
+from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
+moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
+ultimately have "(Th th, Cs cs) \<in> depend s" by simp
+hence "th \<in> set (wq s cs)"
+by (unfold s_depend_def, auto simp:cs_waiting_def)
+from wq_threads [OF vt this] show ?thesis .
+qed
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+apply(simp add: Let_def cpreced_initial)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+lemma runing_unique:
+fixes th1 th2 s
+assumes vt: "vt s"
+and 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"
+by (unfold runing_def, simp)
+hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
+Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
+(is "Max (?f ` ?A) = Max (?f ` ?B)")
+by (unfold cp_eq_cpreced 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 (dependents (wq s) th1)"
+proof-
+have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
+proof -
+let ?F = "\<lambda> (x, y). the_th x"
+have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+hence "finite ((depend (wq s))\<^sup>+)"
+apply (unfold finite_trancl)
+by (auto simp: s_depend_def cs_depend_def wq_def)
+thus ?thesis by auto
+qed
+ultimately show ?thesis by (auto intro:finite_subset)
+qed
+thus ?thesis by (simp add:cs_dependents_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 by (auto intro:that)
+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 (dependents (wq s) th2)"
+proof-
+have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
+proof -
+let ?F = "\<lambda> (x, y). the_th x"
+have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+hence "finite ((depend (wq s))\<^sup>+)"
+apply (unfold finite_trancl)
+by (auto simp: s_depend_def cs_depend_def wq_def)
+thus ?thesis by auto
+qed
+ultimately show ?thesis by (auto intro:finite_subset)
+qed
+thus ?thesis by (simp add:cs_dependents_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> dependents (wq s) th1)" by simp
+thus "th1' \<in> threads s"
+proof
+assume "th1' \<in> dependents (wq s) th1"
+hence "(Th th1') \<in> Domain ((depend s)^+)"
+apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+by (auto simp:Domain_def)
+hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
+from dm_depend_threads[OF vt 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> dependents (wq s) th2)" by simp
+thus "th2' \<in> threads s"
+proof
+assume "th2' \<in> dependents (wq s) th2"
+hence "(Th th2') \<in> Domain ((depend s)^+)"
+apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+by (auto simp:Domain_def)
+hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
+from dm_depend_threads[OF vt 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> dependents (wq s) th1" by simp
+thus ?thesis
+proof
+assume eq_th': "th1' = th1"
+from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
+thus ?thesis
+proof
+assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+next
+assume "th2' \<in> dependents (wq s) th2"
+with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
+hence "(Th th1, Th th2) \<in> (depend s)^+"
+by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+hence "Th th1 \<in> Domain ((depend s)^+)"
+apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+by (auto simp:Domain_def)
+hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
+then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
+from depend_target_th [OF this]
+obtain cs' where "n = Cs cs'" by auto
+with d have "(Th th1, Cs cs') \<in> depend s" by simp
+with runing_1 have "False"
+apply (unfold runing_def readys_def s_depend_def)
+by (auto simp:eq_waiting)
+thus ?thesis by simp
+qed
+next
+assume th1'_in: "th1' \<in> dependents (wq s) th1"
+from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
+thus ?thesis
+proof
+assume "th2' = th2"
+with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
+hence "(Th th2, Th th1) \<in> (depend s)^+"
+by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+hence "Th th2 \<in> Domain ((depend s)^+)"
+apply (unfold cs_dependents_def cs_depend_def s_depend_def)
+by (auto simp:Domain_def)
+hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
+then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
+from depend_target_th [OF this]
+obtain cs' where "n = Cs cs'" by auto
+with d have "(Th th2, Cs cs') \<in> depend s" by simp
+with runing_2 have "False"
+apply (unfold runing_def readys_def s_depend_def)
+by (auto simp:eq_waiting)
+thus ?thesis by simp
+next
+assume "th2' \<in> dependents (wq s) th2"
+with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
+hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
+by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
+by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
+show ?thesis
+proof(rule dchain_unique[OF vt 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 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
+lemma cnp_cnv_eq:
+fixes th s
+assumes "vt s"
+and "th \<notin> threads s"
+shows "cntP s th = cntV s th"
+proof -
+from assms show ?thesis
+proof(induct)
+case (vt_cons s e)
+have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
+have not_in: "th \<notin> threads (e # s)" by fact
+have "step s e" by fact
+thus ?case proof(cases)
+case (thread_create thread prio)
+assume eq_e: "e = Create thread prio"
+hence "thread \<in> threads (e#s)" by simp
+with not_in and eq_e have "th \<notin> threads s" by auto
+from ih [OF this] show ?thesis using eq_e
+by (auto simp:cntP_def cntV_def count_def)
+next
+case (thread_exit thread)
+assume eq_e: "e = Exit thread"
+and not_holding: "holdents s thread = {}"
+have vt_s: "vt s" by fact
+from finite_holding[OF vt_s] have "finite (holdents s thread)" .
+with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
+moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
+moreover note cnp_cnv_cncs[OF vt_s, of thread]
+ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
+show ?thesis
+proof(cases "th = thread")
+case True
+with eq_thread eq_e show ?thesis
+by (auto simp:cntP_def cntV_def count_def)
+next
+case False
+with not_in and eq_e have "th \<notin> threads s" by simp
+from ih[OF this] and eq_e show ?thesis
+by (auto simp:cntP_def cntV_def count_def)
+qed
+next
+case (thread_P thread cs)
+assume eq_e: "e = P thread cs"
+have "thread \<in> runing s" by fact
+with not_in eq_e have neq_th: "thread \<noteq> th"
+by (auto simp:runing_def readys_def)
+from not_in eq_e have "th \<notin> threads s" by simp
+from ih[OF this] and neq_th and eq_e show ?thesis
+by (auto simp:cntP_def cntV_def count_def)
+next
+case (thread_V thread cs)
+assume eq_e: "e = V thread cs"
+have "thread \<in> runing s" by fact
+with not_in eq_e have neq_th: "thread \<noteq> th"
+by (auto simp:runing_def readys_def)
+from not_in eq_e have "th \<notin> threads s" by simp
+from ih[OF this] and neq_th and eq_e show ?thesis
+by (auto simp:cntP_def cntV_def count_def)
+next
+case (thread_set thread prio)
+assume eq_e: "e = Set thread prio"
+and "thread \<in> runing s"
+hence "thread \<in> threads (e#s)"
+by (simp add:runing_def readys_def)
+with not_in and eq_e have "th \<notin> threads s" by auto
+from ih [OF this] show ?thesis using eq_e
+by (auto simp:cntP_def cntV_def count_def)
+qed
+next
+case vt_nil
+show ?case by (auto simp:cntP_def cntV_def count_def)
+qed
+qed
+lemma eq_depend:
+"depend (wq s) = depend s"
+by (unfold cs_depend_def s_depend_def, auto)
+lemma count_eq_dependents:
+assumes vt: "vt s"
+and eq_pv: "cntP s th = cntV s th"
+shows "dependents (wq s) th = {}"
+proof -
+from cnp_cnv_cncs[OF vt] and eq_pv
+have "cntCS s th = 0"
+by (auto split:if_splits)
+moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
+proof -
+from finite_holding[OF vt, of th] show ?thesis
+by (simp add:holdents_test)
+qed
+ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
+by (unfold cntCS_def holdents_test cs_dependents_def, auto)
+show ?thesis
+proof(unfold cs_dependents_def)
+{ assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
+then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
+hence "False"
+proof(cases)
+assume "(Th th', Th th) \<in> depend (wq s)"
+thus "False" by (auto simp:cs_depend_def)
+next
+fix c
+assume "(c, Th th) \<in> depend (wq s)"
+with h and eq_depend show "False"
+by (cases c, auto simp:cs_depend_def)
+qed
+} thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
+qed
+qed
+lemma dependents_threads:
+fixes s th
+assumes vt: "vt s"
+shows "dependents (wq s) th \<subseteq> threads s"
+proof
+{ fix th th'
+assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
+have "Th th \<in> Domain (depend s)"
+proof -
+from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
+hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
+with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
+thus ?thesis using eq_depend by simp
+qed
+from dm_depend_threads[OF vt this]
+have "th \<in> threads s" .
+} note hh = this
+fix th1
+assume "th1 \<in> dependents (wq s) th"
+hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
+by (unfold cs_dependents_def, simp)
+from hh [OF this] show "th1 \<in> threads s" .
+qed
+lemma finite_threads:
+assumes vt: "vt s"
+shows "finite (threads s)"
+using vt
+by (induct) (auto elim: step.cases)
+lemma Max_f_mono:
+assumes seq: "A \<subseteq> B"
+and np: "A \<noteq> {}"
+and fnt: "finite B"
+shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+from seq show "f ` A \<subseteq> f ` B" by auto
+next
+from np show "f ` A \<noteq> {}" by auto
+next
+from fnt and seq show "finite (f ` B)" by auto
+qed
+lemma cp_le:
+assumes vt: "vt s"
+and 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_dependents_def)
+show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (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> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
+next
+from finite_threads [OF vt]
+show "finite (threads s)" .
+next
+from th_in
+show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
+apply (auto simp:Domain_def)
+apply (rule_tac dm_depend_threads[OF vt])
+apply (unfold trancl_domain [of "depend s", symmetric])
+by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
+qed
+qed
+lemma le_cp:
+assumes vt: "vt s"
+shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+show "Prc (original_priority th s) (birthtime th s)
+\<le> Max (insert (Prc (original_priority th s) (birthtime th s))
+((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (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> (depend (wq s))\<^sup>+}"
+proof -
+let ?F = "\<lambda> (x, y). the_th x"
+have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (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_depend[OF vt] have "finite (depend s)" .
+hence "finite ((depend (wq s))\<^sup>+)"
+apply (unfold finite_trancl)
+by (auto simp: s_depend_def cs_depend_def wq_def)
+thus ?thesis by auto
+qed
+ultimately show ?thesis by (auto intro:finite_subset)
+qed
+thus ?thesis by (simp add:cs_dependents_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:
+assumes vt: "vt s"
+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[OF vt]
+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 vt 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[OF vt]
+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 vt, 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[OF vt]
+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 vt: "vt s"
+and np: "threads s \<noteq> {}"
+shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq[OF vt])
+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[OF vt] 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 vt tm_in]
+have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
+thus ?thesis
+proof
+assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
+then obtain th' where th'_in: "th' \<in> readys s"
+and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
+have "cp s th' = ?f tm"
+proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
+from dependents_threads[OF vt] finite_threads[OF vt]
+show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))"
+by (auto intro:finite_subset)
+next
+fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (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[OF vt]
+show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+next
+from p_in and th'_in and dependents_threads[OF vt, 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> dependents (wq s) th')"
+proof -
+from tm_chain
+have "tm \<in> dependents (wq s) th'"
+by (unfold cs_dependents_def s_depend_def cs_depend_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 dependents_threads [OF vt, of th1] th1_in
+show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (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> dependents (wq s) th1) \<noteq> {}" by simp
+next
+from finite_threads[OF vt]
+show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+qed
+next
+from finite_threads[OF vt]
+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> dependents (wq s) tm)"
+show "y \<le> preced tm s"
+proof -
+{ fix y'
+assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
+have "y' \<le> preced tm s"
+proof(unfold tm_max, rule Max_ge)
+from hy' dependents_threads[OF vt, of tm]
+show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+next
+from finite_threads[OF vt]
+show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+qed
+} with hy show ?thesis by auto
+qed
+next
+from dependents_threads[OF vt, of tm] finite_threads[OF vt]
+show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
+by (auto intro:finite_subset)
+next
+show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (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[OF vt]
+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[OF vt]
+show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+next
+show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
+by simp
+next
+from dependents_threads[OF vt, of th1] th1_readys
+show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (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
+lemma max_cp_readys_threads:
+assumes vt: "vt s"
+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 vt False])
+qed
+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 (depend s))"
+apply(simp add: detached_def Field_def)
+apply(simp add: s_depend_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
+lemma detached_intro:
+fixes s th
+assumes vt: "vt s"
+and eq_pv: "cntP s th = cntV s th"
+shows "detached s th"
+proof -
+from cnp_cnv_cncs[OF vt]
+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[OF vt] dm_depend_threads[OF vt]
+show ?thesis
+by (auto simp add: detached_def s_depend_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 (depend s)"
+proof -
+from card_0_eq [OF finite_holding [OF vt]] 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_depend_def)
+done
+qed
+ultimately show ?thesis
+by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def readys_def)
+qed
+qed
+lemma detached_elim:
+fixes s th
+assumes vt: "vt s"
+and dtc: "detached s th"
+shows "cntP s th = cntV s th"
+proof -
+from cnp_cnv_cncs[OF vt]
+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_depend_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_depend_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:
+fixes s th
+assumes vt: "vt s"
+shows "(detached s th) = (cntP s th = cntV s th)"
+by (insert vt, auto intro:detached_intro detached_elim)
+end

changeset 0	110247f9d47e
child 3	51019d035a79