--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/PrioG.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,2805 @@
+theory PrioG
+imports PrioGDef
+begin
+
+lemma runing_ready: "runing s \<subseteq> readys s"
+ by (auto simp only:runing_def readys_def)
+
+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 step 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 (waiting_queue (schs s) cs)"
+ and h3: "thread \<in> runing s"
+ show "False"
+ proof -
+ from h3 have "\<And> cs. thread \<in> set (waiting_queue (schs s) cs) \<Longrightarrow>
+ thread = hd ((waiting_queue (schs s) cs))"
+ by (simp add:runing_def readys_def s_waiting_def wq_def)
+ from this [OF h2] have "thread = hd (waiting_queue (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 ccs (e#s) \<Longrightarrow> vt ccs s"
+ by(ind_cases "vt ccs (e#s)", simp)
+
+lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e"
+ by(ind_cases "vt ccs (e#s)", simp)
+
+lemma block_pre:
+ fixes thread cs s
+ assumes vt_e: "vt step (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 (waiting_queue (schs s) cs)"
+ and h2: "q # qs = waiting_queue (schs s) cs"
+ and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+ and vt: "vt step (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 step ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
+ thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (depend s)^+"
+apply (ind_cases "vt step ((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 cs (a#s)"
+
+lemma abs2:
+ assumes vt: "vt step (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 step (V th cs # s)"
+ and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+ and eq_wq: "waiting_queue (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 cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
+proof(induct s, simp)
+ fix a s t
+ assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
+ and vt_a: "vt cs (a # s)"
+ and le_t: "t \<le> length (a # s)"
+ show "vt cs (moment t (a # s))"
+ proof(cases "t = 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
+ with le_t have le_t1: "t \<le> length s" by simp
+ from vt_a have "vt cs s"
+ by (erule_tac evt_cons, simp)
+ from h [OF this le_t1] have "vt cs (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 (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
+*)
+
+lemma waiting_unique_pre:
+ fixes cs1 cs2 s thread
+ assumes vt: "vt step 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 step (e#moment t2 s)"
+ proof -
+ from vt_moment [OF vt le_t3]
+ have "vt step (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 step ((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 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 step (e#moment t1 s)"
+ proof -
+ from vt_moment [OF vt le_t3]
+ have "vt step (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 step ((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 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 step (e#moment t1 s)"
+ proof -
+ from vt_moment [OF vt le_t3]
+ have "vt step (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 step (e#moment t2 s)" by simp
+ from abs2 [OF this True eq_th h2 h1]
+ show ?thesis .
+ next
+ case False
+ have vt_e: "vt step (e#moment t2 s)"
+ proof -
+ from vt_moment [OF vt le_t3] eqt12
+ have "vt step (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:
+ assumes "vt step s"
+ and "waiting s th cs1"
+ and "waiting s th cs2"
+ shows "cs1 = cs2"
+proof -
+ from waiting_unique_pre and prems
+ show ?thesis
+ by (auto simp add:s_waiting_def)
+qed
+
+lemma holded_unique:
+ assumes "vt step s"
+ and "holding s th1 cs"
+ and "holding s th2 cs"
+ shows "th1 = th2"
+proof -
+ from prems show ?thesis
+ unfolding s_holding_def
+ by auto
+qed
+
+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 step (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 step (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 step (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 step (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: "waiting_queue (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: "waiting_queue (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: "waiting_queue (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 step (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 step (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
+proof -
+ assume vt: "vt step (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]:
+ "waiting_queue (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 step (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 step (V th cs # s)"
+ and eq_wq[folded wq_def]: " waiting_queue (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 step (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 step (V th cs # s)" and eq_wq: "waiting_queue (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 step (V th cs # s)" and eq_wq: "waiting_queue (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 step (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 step ?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: "waiting_queue (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: "waiting_queue (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:
+assumes vt:
+ "vt step (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 step (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(unfold s_depend_def wq_def)
+ apply (auto split:list.splits prod.splits simp:Let_def cs_waiting_def cs_holding_def)
+ apply(case_tac "c = cs", auto)
+ apply(fold wq_def)
+ apply(drule_tac step_back_step)
+ by (ind_cases " step s (P (hd (wq s cs)) cs)",
+ auto simp:s_depend_def wq_def cs_holding_def)
+
+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 step 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 step 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 step (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)" by (unfold s_holding_def, 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 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, auto)
+ proof -
+ assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ and eq_wq: "wq 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" 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 "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 step (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 step 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 step 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 step (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 step (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 step 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 step 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 step 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)
+ by (fold wq_def, auto)
+ 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 (waiting_queue (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 = waiting_queue (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 "waiting_queue (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: "waiting_queue (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 (waiting_queue (schs s) cs')" by auto
+ from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+ next
+ assume th_in: "th \<in> set (waiting_queue (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 step 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 step 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 prems show ?thesis
+ apply (auto simp:readys_def)
+ apply (case_tac "cs = csa", simp add:s_waiting_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: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: "waiting_queue (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] 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 th_nin th_in show False by auto
+ qed
+qed
+
+lemma chain_building:
+ assumes vt: "vt step 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 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 step 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 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, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+ by (unfold s_holding_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 step 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 step 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 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 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 step (P th cs#s)"
+ and "wq s cs = []"
+ shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+ from prems show ?thesis
+ unfolding holdents_def step_depend_p[OF vt] by auto
+qed
+
+lemma step_holdents_p_eq:
+ fixes th cs s
+ assumes vt: "vt step (P th cs#s)"
+ and "wq s cs \<noteq> []"
+ shows "holdents (P th cs#s) th = holdents s th"
+proof -
+ from prems show ?thesis
+ unfolding holdents_def step_depend_p[OF vt] by auto
+qed
+
+
+lemma finite_holding:
+ fixes s th cs
+ assumes vt: "vt step 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_def, auto intro:finite_subset)
+qed
+
+lemma cntCS_v_dec:
+ fixes s thread cs
+ assumes vtv: "vt step (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_def s_depend_def, simp)
+ by (unfold cs_holding_def s_holding_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_def, 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 step 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 step 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_def
+ 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_def
+ 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 (subst (1 2) wq_def)
+ apply (simp add:Let_def)
+ apply (subst s_waiting_def, simp)
+ by (fold wq_def, simp)
+ 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 prems have vtp: "vt step (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 "waiting_queue (schs s) cs \<noteq> []", auto)
+ apply (erule_tac x = cs in allE, auto)
+ by (case_tac "(waiting_queue (schs s) cs)", auto)
+ moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
+ apply (simp add:cntCS_def holdents_def)
+ 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_def, 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_def)
+ 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)
+ 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_def 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 prems have vtv: "vt step (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: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: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: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_def 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)
+ 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_def 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: "waiting_queue (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: "waiting_queue (schs s) cs = thread # rest"
+ and t_in: "?t \<in> set rest"
+ and neq_cs: "csa \<noteq> cs"
+ and t_in': "?t \<in> set (waiting_queue (schs s) csa)"
+ show "?t = hd (waiting_queue (schs s) csa)"
+ proof -
+ { assume neq_hd': "?t \<noteq> hd (waiting_queue (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)
+ 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_def 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_def
+ 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_def s_depend_def wq_def cs_holding_def)
+ qed
+qed
+
+lemma not_thread_cncs:
+ fixes th s
+ assumes vt: "vt step 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 step 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_def)
+ 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_def)
+ 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 prems have vtp: "vt step (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_def 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 prems have vtv: "vt step (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: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_def 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_def)
+ by (simp add:depend_set_unchanged)
+ qed
+ next
+ case vt_nil
+ show ?case
+ by (unfold cntCS_def,
+ auto simp:count_def holdents_def 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)
+
+lemma dm_depend_threads:
+ fixes th s
+ assumes vt: "vt step 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 s (wq s) th"
+proof(unfold cp_def wq_def, induct s)
+ case (Cons e s')
+ show ?case
+ by (auto simp:Let_def)
+next
+ case Nil
+ show ?case by (auto simp:Let_def)
+qed
+
+
+lemma runing_unique:
+ fixes th1 th2 s
+ assumes vt: "vt step 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)^+)" using Domain_def [of "(depend s)^+"]
+ by auto
+ 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)^+)" using Domain_def [of "(depend s)^+"]
+ by auto
+ 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 step 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 step 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 step 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_def)
+ qed
+ ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
+ by (unfold cntCS_def holdents_def 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 step 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 step s"
+ shows "finite (threads s)"
+proof -
+ from vt show ?thesis
+ proof(induct)
+ case (vt_cons s e)
+ assume vt: "vt step s"
+ and step: "step s e"
+ and ih: "finite (threads s)"
+ from step
+ show ?case
+ proof(cases)
+ case (thread_create thread prio)
+ assume eq_e: "e = Create thread prio"
+ with ih
+ show ?thesis by (unfold eq_e, auto)
+ next
+ case (thread_exit thread)
+ assume eq_e: "e = Exit thread"
+ with ih show ?thesis
+ by (unfold eq_e, auto)
+ next
+ case (thread_P thread cs)
+ assume eq_e: "e = P thread cs"
+ with ih show ?thesis by (unfold eq_e, auto)
+ next
+ case (thread_V thread cs)
+ assume eq_e: "e = V thread cs"
+ with ih show ?thesis by (unfold eq_e, auto)
+ next
+ case (thread_set thread prio)
+ from vt_cons thread_set show ?thesis by simp
+ qed
+ next
+ case vt_nil
+ show ?case by (auto)
+ qed
+qed
+
+lemma Max_f_mono:
+ assumes seq: "A \<subseteq> B"
+ and np: "A \<noteq> {}"
+ and fnt: "finite B"
+ shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+ from seq show "f ` A \<subseteq> f ` B" by auto
+next
+ from np show "f ` A \<noteq> {}" by auto
+next
+ from fnt and seq show "finite (f ` B)" by auto
+qed
+
+lemma cp_le:
+ assumes vt: "vt step 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 step 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 step 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 step 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 step 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 readys_threads:
+ shows "readys s \<subseteq> threads s"
+proof
+ fix th
+ assume "th \<in> readys s"
+ thus "th \<in> threads s"
+ by (unfold readys_def, auto)
+qed
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+ apply (unfold s_holding_def cs_holding_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
+
+end
\ No newline at end of file