--- a/Matcher.thy Mon Dec 26 08:21:00 2011 +0000
+++ b/Matcher.thy Tue Jan 24 00:20:09 2012 +0000
@@ -171,6 +171,17 @@
section {* Examples *}
+definition
+ "CHRA \<equiv> CHAR (CHR ''a'')"
+
+definition
+ "ALT1 \<equiv> ALT CHRA EMPTY"
+
+definition
+ "SEQ3 \<equiv> SEQ (SEQ ALT1 ALT1) ALT1"
+
+value "matcher SEQ3 ''aaa''"
+
value "matcher NULL []"
value "matcher (CHAR (CHR ''a'')) [CHR ''a'']"
value "matcher (CHAR a) [a,a]"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/CpsG.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1826 @@
+theory CpsG
+imports PrioG
+begin
+
+lemma not_thread_holdents:
+ fixes th s
+ assumes vt: "vt step s"
+ and not_in: "th \<notin> threads s"
+ shows "holdents s th = {}"
+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> holdents s th = {}"
+ 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 "holdents (e # s) th = holdents s th"
+ apply (unfold eq_e 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 = {}"
+ show ?thesis
+ proof(cases "th = thread")
+ case True
+ with nh eq_e
+ show ?thesis
+ by (auto simp:holdents_def depend_exit_unchanged)
+ next
+ case False
+ with not_in and eq_e
+ have "th \<notin> threads s" by simp
+ from ih[OF this] False eq_e show ?thesis
+ by (auto simp:holdents_def depend_exit_unchanged)
+ 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 "holdents (e # s) th = holdents s th "
+ apply (unfold cntCS_def holdents_def eq_e)
+ by (unfold step_depend_p[OF vtp], auto)
+ moreover have "holdents s th = {}"
+ 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 "holdents (e # s) th = holdents s th"
+ by (unfold eq_e cntCS_def holdents_def step_depend_v[OF vtv], auto)
+ moreover have "holdents s th = {}"
+ 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 (auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
+ qed
+qed
+
+
+
+lemma next_th_neq:
+ assumes vt: "vt step s"
+ and nt: "next_th s th cs th'"
+ shows "th' \<noteq> th"
+proof -
+ from nt show ?thesis
+ apply (auto simp:next_th_def)
+ proof -
+ fix rest
+ assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+ and ne: "rest \<noteq> []"
+ have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest"
+ proof(rule someI2)
+ from wq_distinct[OF vt, of cs] eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ fix x
+ assume "distinct x \<and> set x = set rest"
+ hence eq_set: "set x = set rest" by auto
+ with ne have "x \<noteq> []" by auto
+ hence "hd x \<in> set x" by auto
+ with eq_set show "hd x \<in> set rest" by auto
+ qed
+ with wq_distinct[OF vt, of cs] eq_wq show False by auto
+ qed
+qed
+
+lemma next_th_unique:
+ assumes nt1: "next_th s th cs th1"
+ and nt2: "next_th s th cs th2"
+ shows "th1 = th2"
+proof -
+ from assms show ?thesis
+ by (unfold next_th_def, auto)
+qed
+
+lemma pp_sub: "(r^+)^+ \<subseteq> r^+"
+ by auto
+
+lemma wf_depend:
+ assumes vt: "vt step s"
+ shows "wf (depend s)"
+proof(rule finite_acyclic_wf)
+ from finite_depend[OF vt] show "finite (depend s)" .
+next
+ from acyclic_depend[OF vt] show "acyclic (depend s)" .
+qed
+
+lemma Max_Union:
+ assumes fc: "finite C"
+ and ne: "C \<noteq> {}"
+ and fa: "\<And> A. A \<in> C \<Longrightarrow> finite A \<and> A \<noteq> {}"
+ shows "Max (\<Union> C) = Max (Max ` C)"
+proof -
+ from fc ne fa show ?thesis
+ proof(induct)
+ case (insert x F)
+ assume ih: "\<lbrakk>F \<noteq> {}; \<And>A. A \<in> F \<Longrightarrow> finite A \<and> A \<noteq> {}\<rbrakk> \<Longrightarrow> Max (\<Union>F) = Max (Max ` F)"
+ and h: "\<And> A. A \<in> insert x F \<Longrightarrow> finite A \<and> A \<noteq> {}"
+ show ?case (is "?L = ?R")
+ proof(cases "F = {}")
+ case False
+ from Union_insert have "?L = Max (x \<union> (\<Union> F))" by simp
+ also have "\<dots> = max (Max x) (Max(\<Union> F))"
+ proof(rule Max_Un)
+ from h[of x] show "finite x" by auto
+ next
+ from h[of x] show "x \<noteq> {}" by auto
+ next
+ show "finite (\<Union>F)"
+ proof(rule finite_Union)
+ show "finite F" by fact
+ next
+ from h show "\<And>M. M \<in> F \<Longrightarrow> finite M" by auto
+ qed
+ next
+ from False and h show "\<Union>F \<noteq> {}" by auto
+ qed
+ also have "\<dots> = ?R"
+ proof -
+ have "?R = Max (Max ` ({x} \<union> F))" by simp
+ also have "\<dots> = Max ({Max x} \<union> (Max ` F))" by simp
+ also have "\<dots> = max (Max x) (Max (\<Union>F))"
+ proof -
+ have "Max ({Max x} \<union> Max ` F) = max (Max {Max x}) (Max (Max ` F))"
+ proof(rule Max_Un)
+ show "finite {Max x}" by simp
+ next
+ show "{Max x} \<noteq> {}" by simp
+ next
+ from insert show "finite (Max ` F)" by auto
+ next
+ from False show "Max ` F \<noteq> {}" by auto
+ qed
+ moreover have "Max {Max x} = Max x" by simp
+ moreover have "Max (\<Union>F) = Max (Max ` F)"
+ proof(rule ih)
+ show "F \<noteq> {}" by fact
+ next
+ from h show "\<And>A. A \<in> F \<Longrightarrow> finite A \<and> A \<noteq> {}"
+ by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ finally show ?thesis by simp
+ qed
+ finally show ?thesis by simp
+ next
+ case True
+ thus ?thesis by auto
+ qed
+ next
+ case empty
+ assume "{} \<noteq> {}" show ?case by auto
+ qed
+qed
+
+definition child :: "state \<Rightarrow> (node \<times> node) set"
+ where "child s =
+ {(Th th', Th th) | th th'. \<exists> cs. (Th th', Cs cs) \<in> depend s \<and> (Cs cs, Th th) \<in> depend s}"
+
+definition children :: "state \<Rightarrow> thread \<Rightarrow> thread set"
+ where "children s th = {th'. (Th th', Th th) \<in> child s}"
+
+
+lemma children_dependents: "children s th \<subseteq> dependents (wq s) th"
+ by (unfold children_def child_def cs_dependents_def, auto simp:eq_depend)
+
+lemma child_unique:
+ assumes vt: "vt step s"
+ and ch1: "(Th th, Th th1) \<in> child s"
+ and ch2: "(Th th, Th th2) \<in> child s"
+ shows "th1 = th2"
+proof -
+ from ch1 ch2 show ?thesis
+ proof(unfold child_def, clarsimp)
+ fix cs csa
+ assume h1: "(Th th, Cs cs) \<in> depend s"
+ and h2: "(Cs cs, Th th1) \<in> depend s"
+ and h3: "(Th th, Cs csa) \<in> depend s"
+ and h4: "(Cs csa, Th th2) \<in> depend s"
+ from unique_depend[OF vt h1 h3] have "cs = csa" by simp
+ with h4 have "(Cs cs, Th th2) \<in> depend s" by simp
+ from unique_depend[OF vt h2 this]
+ show "th1 = th2" by simp
+ qed
+qed
+
+
+lemma cp_eq_cpreced_f: "cp s = cpreced s (wq s)"
+proof -
+ from fun_eq_iff
+ have h:"\<And>f g. (\<forall> x. f x = g x) \<Longrightarrow> f = g" by auto
+ show ?thesis
+ proof(rule h)
+ from cp_eq_cpreced show "\<forall>x. cp s x = cpreced s (wq s) x" by auto
+ qed
+qed
+
+lemma depend_children:
+ assumes h: "(Th th1, Th th2) \<in> (depend s)^+"
+ shows "th1 \<in> children s th2 \<or> (\<exists> th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)^+)"
+proof -
+ from h show ?thesis
+ proof(induct rule: tranclE)
+ fix c th2
+ assume h1: "(Th th1, c) \<in> (depend s)\<^sup>+"
+ and h2: "(c, Th th2) \<in> depend s"
+ from h2 obtain cs where eq_c: "c = Cs cs"
+ by (case_tac c, auto simp:s_depend_def)
+ show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)\<^sup>+)"
+ proof(rule tranclE[OF h1])
+ fix ca
+ assume h3: "(Th th1, ca) \<in> (depend s)\<^sup>+"
+ and h4: "(ca, c) \<in> depend s"
+ show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)\<^sup>+)"
+ proof -
+ from eq_c and h4 obtain th3 where eq_ca: "ca = Th th3"
+ by (case_tac ca, auto simp:s_depend_def)
+ from eq_ca h4 h2 eq_c
+ have "th3 \<in> children s th2" by (auto simp:children_def child_def)
+ moreover from h3 eq_ca have "(Th th1, Th th3) \<in> (depend s)\<^sup>+" by simp
+ ultimately show ?thesis by auto
+ qed
+ next
+ assume "(Th th1, c) \<in> depend s"
+ with h2 eq_c
+ have "th1 \<in> children s th2"
+ by (auto simp:children_def child_def)
+ thus ?thesis by auto
+ qed
+ next
+ assume "(Th th1, Th th2) \<in> depend s"
+ thus ?thesis
+ by (auto simp:s_depend_def)
+ qed
+qed
+
+lemma sub_child: "child s \<subseteq> (depend s)^+"
+ by (unfold child_def, auto)
+
+lemma wf_child:
+ assumes vt: "vt step s"
+ shows "wf (child s)"
+proof(rule wf_subset)
+ from wf_trancl[OF wf_depend[OF vt]]
+ show "wf ((depend s)\<^sup>+)" .
+next
+ from sub_child show "child s \<subseteq> (depend s)\<^sup>+" .
+qed
+
+lemma depend_child_pre:
+ assumes vt: "vt step s"
+ shows
+ "(Th th, n) \<in> (depend s)^+ \<longrightarrow> (\<forall> th'. n = (Th th') \<longrightarrow> (Th th, Th th') \<in> (child s)^+)" (is "?P n")
+proof -
+ from wf_trancl[OF wf_depend[OF vt]]
+ have wf: "wf ((depend s)^+)" .
+ show ?thesis
+ proof(rule wf_induct[OF wf, of ?P], clarsimp)
+ fix th'
+ assume ih[rule_format]: "\<forall>y. (y, Th th') \<in> (depend s)\<^sup>+ \<longrightarrow>
+ (Th th, y) \<in> (depend s)\<^sup>+ \<longrightarrow> (\<forall>th'. y = Th th' \<longrightarrow> (Th th, Th th') \<in> (child s)\<^sup>+)"
+ and h: "(Th th, Th th') \<in> (depend s)\<^sup>+"
+ show "(Th th, Th th') \<in> (child s)\<^sup>+"
+ proof -
+ from depend_children[OF h]
+ have "th \<in> children s th' \<or> (\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (depend s)\<^sup>+)" .
+ thus ?thesis
+ proof
+ assume "th \<in> children s th'"
+ thus "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def)
+ next
+ assume "\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (depend s)\<^sup>+"
+ then obtain th3 where th3_in: "th3 \<in> children s th'"
+ and th_dp: "(Th th, Th th3) \<in> (depend s)\<^sup>+" by auto
+ from th3_in have "(Th th3, Th th') \<in> (depend s)^+" by (auto simp:children_def child_def)
+ from ih[OF this th_dp, of th3] have "(Th th, Th th3) \<in> (child s)\<^sup>+" by simp
+ with th3_in show "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def)
+ qed
+ qed
+ qed
+qed
+
+lemma depend_child: "\<lbrakk>vt step s; (Th th, Th th') \<in> (depend s)^+\<rbrakk> \<Longrightarrow> (Th th, Th th') \<in> (child s)^+"
+ by (insert depend_child_pre, auto)
+
+lemma child_depend_p:
+ assumes "(n1, n2) \<in> (child s)^+"
+ shows "(n1, n2) \<in> (depend s)^+"
+proof -
+ from assms show ?thesis
+ proof(induct)
+ case (base y)
+ with sub_child show ?case by auto
+ next
+ case (step y z)
+ assume "(y, z) \<in> child s"
+ with sub_child have "(y, z) \<in> (depend s)^+" by auto
+ moreover have "(n1, y) \<in> (depend s)^+" by fact
+ ultimately show ?case by auto
+ qed
+qed
+
+lemma child_depend_eq:
+ assumes vt: "vt step s"
+ shows
+ "((Th th1, Th th2) \<in> (child s)^+) =
+ ((Th th1, Th th2) \<in> (depend s)^+)"
+ by (auto intro: depend_child[OF vt] child_depend_p)
+
+lemma children_no_dep:
+ fixes s th th1 th2 th3
+ assumes vt: "vt step s"
+ and ch1: "(Th th1, Th th) \<in> child s"
+ and ch2: "(Th th2, Th th) \<in> child s"
+ and ch3: "(Th th1, Th th2) \<in> (depend s)^+"
+ shows "False"
+proof -
+ from depend_child[OF vt ch3]
+ have "(Th th1, Th th2) \<in> (child s)\<^sup>+" .
+ thus ?thesis
+ proof(rule converse_tranclE)
+ thm tranclD
+ assume "(Th th1, Th th2) \<in> child s"
+ from child_unique[OF vt ch1 this] have "th = th2" by simp
+ with ch2 have "(Th th2, Th th2) \<in> child s" by simp
+ with wf_child[OF vt] show ?thesis by auto
+ next
+ fix c
+ assume h1: "(Th th1, c) \<in> child s"
+ and h2: "(c, Th th2) \<in> (child s)\<^sup>+"
+ from h1 obtain th3 where eq_c: "c = Th th3" by (unfold child_def, auto)
+ with h1 have "(Th th1, Th th3) \<in> child s" by simp
+ from child_unique[OF vt ch1 this] have eq_th3: "th3 = th" by simp
+ with eq_c and h2 have "(Th th, Th th2) \<in> (child s)\<^sup>+" by simp
+ with ch2 have "(Th th, Th th) \<in> (child s)\<^sup>+" by auto
+ moreover have "wf ((child s)\<^sup>+)"
+ proof(rule wf_trancl)
+ from wf_child[OF vt] show "wf (child s)" .
+ qed
+ ultimately show False by auto
+ qed
+qed
+
+lemma unique_depend_p:
+ assumes vt: "vt step s"
+ and dp1: "(n, n1) \<in> (depend s)^+"
+ and dp2: "(n, n2) \<in> (depend s)^+"
+ and neq: "n1 \<noteq> n2"
+ shows "(n1, n2) \<in> (depend s)^+ \<or> (n2, n1) \<in> (depend s)^+"
+proof(rule unique_chain [OF _ dp1 dp2 neq])
+ from unique_depend[OF vt]
+ show "\<And>a b c. \<lbrakk>(a, b) \<in> depend s; (a, c) \<in> depend s\<rbrakk> \<Longrightarrow> b = c" by auto
+qed
+
+lemma dependents_child_unique:
+ fixes s th th1 th2 th3
+ assumes vt: "vt step s"
+ and ch1: "(Th th1, Th th) \<in> child s"
+ and ch2: "(Th th2, Th th) \<in> child s"
+ and dp1: "th3 \<in> dependents s th1"
+ and dp2: "th3 \<in> dependents s th2"
+shows "th1 = th2"
+proof -
+ { assume neq: "th1 \<noteq> th2"
+ from dp1 have dp1: "(Th th3, Th th1) \<in> (depend s)^+"
+ by (simp add:s_dependents_def eq_depend)
+ from dp2 have dp2: "(Th th3, Th th2) \<in> (depend s)^+"
+ by (simp add:s_dependents_def eq_depend)
+ from unique_depend_p[OF vt dp1 dp2] and neq
+ 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 children_no_dep[OF vt ch1 ch2 this] show ?thesis .
+ next
+ assume " (Th th2, Th th1) \<in> (depend s)\<^sup>+"
+ from children_no_dep[OF vt ch2 ch1 this] show ?thesis .
+ qed
+ } thus ?thesis by auto
+qed
+
+lemma cp_rec:
+ fixes s th
+ assumes vt: "vt step s"
+ shows "cp s th = Max ({preced th s} \<union> (cp s ` children s th))"
+proof(unfold cp_eq_cpreced_f cpreced_def)
+ let ?f = "(\<lambda>th. preced th s)"
+ show "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) =
+ Max ({preced th s} \<union> (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th)"
+ proof(cases " children s th = {}")
+ case False
+ have "(\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th =
+ {Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) | th' . th' \<in> children s th}"
+ (is "?L = ?R")
+ by auto
+ also have "\<dots> =
+ Max ` {((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) | th' . th' \<in> children s th}"
+ (is "_ = Max ` ?C")
+ by auto
+ finally have "Max ?L = Max (Max ` ?C)" by auto
+ also have "\<dots> = Max (\<Union> ?C)"
+ proof(rule Max_Union[symmetric])
+ from children_dependents[of s th] finite_threads[OF vt] and dependents_threads[OF vt, of th]
+ show "finite {(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+ by (auto simp:finite_subset)
+ next
+ from False
+ show "{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th} \<noteq> {}"
+ by simp
+ next
+ show "\<And>A. A \<in> {(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th} \<Longrightarrow>
+ finite A \<and> A \<noteq> {}"
+ apply (auto simp:finite_subset)
+ proof -
+ fix th'
+ from finite_threads[OF vt] and dependents_threads[OF vt, of th']
+ show "finite ((\<lambda>th. preced th s) ` dependents (wq s) th')" by (auto simp:finite_subset)
+ qed
+ qed
+ also have "\<dots> = Max ((\<lambda>th. preced th s) ` dependents (wq s) th)"
+ (is "Max ?A = Max ?B")
+ proof -
+ have "?A = ?B"
+ proof
+ show "\<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}
+ \<subseteq> (\<lambda>th. preced th s) ` dependents (wq s) th"
+ proof
+ fix x
+ assume "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+ then obtain th' where
+ th'_in: "th' \<in> children s th"
+ and x_in: "x \<in> ?f ` ({th'} \<union> dependents (wq s) th')" by auto
+ hence "x = ?f th' \<or> x \<in> (?f ` dependents (wq s) th')" by auto
+ thus "x \<in> ?f ` dependents (wq s) th"
+ proof
+ assume "x = preced th' s"
+ with th'_in and children_dependents
+ show "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th" by auto
+ next
+ assume "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th'"
+ moreover note th'_in
+ ultimately show " x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th"
+ by (unfold cs_dependents_def children_def child_def, auto simp:eq_depend)
+ qed
+ qed
+ next
+ show "?f ` dependents (wq s) th
+ \<subseteq> \<Union>{?f ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+ proof
+ fix x
+ assume x_in: "x \<in> (\<lambda>th. preced th s) ` dependents (wq s) th"
+ then obtain th' where
+ eq_x: "x = ?f th'" and dp: "(Th th', Th th) \<in> (depend s)^+"
+ by (auto simp:cs_dependents_def eq_depend)
+ from depend_children[OF dp]
+ have "th' \<in> children s th \<or> (\<exists>th3. th3 \<in> children s th \<and> (Th th', Th th3) \<in> (depend s)\<^sup>+)" .
+ thus "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+ proof
+ assume "th' \<in> children s th"
+ with eq_x
+ show "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+ by auto
+ next
+ assume "\<exists>th3. th3 \<in> children s th \<and> (Th th', Th th3) \<in> (depend s)\<^sup>+"
+ then obtain th3 where th3_in: "th3 \<in> children s th"
+ and dp3: "(Th th', Th th3) \<in> (depend s)\<^sup>+" by auto
+ show "x \<in> \<Union>{(\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th') |th'. th' \<in> children s th}"
+ proof -
+ from dp3
+ have "th' \<in> dependents (wq s) th3"
+ by (auto simp:cs_dependents_def eq_depend)
+ with eq_x th3_in show ?thesis by auto
+ qed
+ qed
+ qed
+ qed
+ thus ?thesis by simp
+ qed
+ finally have "Max ((\<lambda>th. preced th s) ` dependents (wq s) th) = Max (?L)"
+ (is "?X = ?Y") by auto
+ moreover have "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) =
+ max (?f th) ?X"
+ proof -
+ have "Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)) =
+ Max ({?f th} \<union> ?f ` (dependents (wq s) th))" by simp
+ also have "\<dots> = max (Max {?f th}) (Max (?f ` (dependents (wq s) th)))"
+ proof(rule Max_Un, auto)
+ from finite_threads[OF vt] and dependents_threads[OF vt, of th]
+ show "finite ((\<lambda>th. preced th s) ` dependents (wq s) th)" by (auto simp:finite_subset)
+ next
+ assume "dependents (wq s) th = {}"
+ with False and children_dependents show False by auto
+ qed
+ also have "\<dots> = max (?f th) ?X" by simp
+ finally show ?thesis .
+ qed
+ moreover have "Max ({preced th s} \<union>
+ (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th) =
+ max (?f th) ?Y"
+ proof -
+ have "Max ({preced th s} \<union>
+ (\<lambda>th. Max ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))) ` children s th) =
+ max (Max {preced th s}) ?Y"
+ proof(rule Max_Un, auto)
+ from finite_threads[OF vt] dependents_threads[OF vt, of th] children_dependents [of s th]
+ show "finite ((\<lambda>th. Max (insert (preced th s) ((\<lambda>th. preced th s) ` dependents (wq s) th))) `
+ children s th)"
+ by (auto simp:finite_subset)
+ next
+ assume "children s th = {}"
+ with False show False by auto
+ qed
+ thus ?thesis by simp
+ qed
+ ultimately show ?thesis by auto
+ next
+ case True
+ moreover have "dependents (wq s) th = {}"
+ proof -
+ { fix th'
+ assume "th' \<in> dependents (wq s) th"
+ hence " (Th th', Th th) \<in> (depend s)\<^sup>+" by (simp add:cs_dependents_def eq_depend)
+ from depend_children[OF this] and True
+ have "False" by auto
+ } thus ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+qed
+
+definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+
+locale step_set_cps =
+ fixes s' th prio s
+ defines s_def : "s \<equiv> (Set th prio#s')"
+ assumes vt_s: "vt step s"
+
+context step_set_cps
+begin
+
+lemma eq_preced:
+ fixes th'
+ assumes "th' \<noteq> th"
+ shows "preced th' s = preced th' s'"
+proof -
+ from assms show ?thesis
+ by (unfold s_def, auto simp:preced_def)
+qed
+
+lemma eq_dep: "depend s = depend s'"
+ by (unfold s_def depend_set_unchanged, auto)
+
+lemma eq_cp:
+ fixes th'
+ assumes neq_th: "th' \<noteq> th"
+ and nd: "th \<notin> dependents s th'"
+ shows "cp s th' = cp s' th'"
+ apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+ have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
+ by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
+ moreover {
+ fix th1
+ assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
+ hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
+ hence "preced th1 s = preced th1 s'"
+ proof
+ assume "th1 = th'"
+ with eq_preced[OF neq_th]
+ show "preced th1 s = preced th1 s'" by simp
+ next
+ assume "th1 \<in> dependents (wq s') th'"
+ with nd and eq_dp have "th1 \<noteq> th"
+ by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
+ from eq_preced[OF this] show "preced th1 s = preced th1 s'" by simp
+ qed
+ } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
+ by (auto simp:image_def)
+ thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+lemma eq_up:
+ fixes th' th''
+ assumes dp1: "th \<in> dependents s th'"
+ and dp2: "th' \<in> dependents s th''"
+ and eq_cps: "cp s th' = cp s' th'"
+ shows "cp s th'' = cp s' th''"
+proof -
+ from dp2
+ have "(Th th', Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
+ from depend_child[OF vt_s this[unfolded eq_depend]]
+ have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" .
+ moreover { fix n th''
+ have "\<lbrakk>(Th th', n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
+ (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
+ proof(erule trancl_induct, auto)
+ fix y th''
+ assume y_ch: "(y, Th th'') \<in> child s"
+ and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
+ and ch': "(Th th', y) \<in> (child s)\<^sup>+"
+ from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
+ with ih have eq_cpy:"cp s thy = cp s' thy" by blast
+ from dp1 have "(Th th, Th th') \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
+ moreover from child_depend_p[OF ch'] and eq_y
+ have "(Th th', Th thy) \<in> (depend s)^+" by simp
+ ultimately have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by auto
+ show "cp s th'' = cp s' th''"
+ apply (subst cp_rec[OF vt_s])
+ proof -
+ have "preced th'' s = preced th'' s'"
+ proof(rule eq_preced)
+ show "th'' \<noteq> th"
+ proof
+ assume "th'' = th"
+ with dp_thy y_ch[unfolded eq_y]
+ have "(Th th, Th th) \<in> (depend s)^+"
+ by (auto simp:child_def)
+ with wf_trancl[OF wf_depend[OF vt_s]]
+ show False by auto
+ qed
+ qed
+ moreover {
+ fix th1
+ assume th1_in: "th1 \<in> children s th''"
+ have "cp s th1 = cp s' th1"
+ proof(cases "th1 = thy")
+ case True
+ with eq_cpy show ?thesis by simp
+ next
+ case False
+ have neq_th1: "th1 \<noteq> th"
+ proof
+ assume eq_th1: "th1 = th"
+ with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+ from children_no_dep[OF vt_s _ _ this] and
+ th1_in y_ch eq_y show False by (auto simp:children_def)
+ qed
+ have "th \<notin> dependents s th1"
+ proof
+ assume h:"th \<in> dependents s th1"
+ from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
+ from dependents_child_unique[OF vt_s _ _ h this]
+ th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
+ with False show False by auto
+ qed
+ from eq_cp[OF neq_th1 this]
+ show ?thesis .
+ qed
+ }
+ ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
+ {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+ moreover have "children s th'' = children s' th''"
+ by (unfold children_def child_def s_def depend_set_unchanged, simp)
+ ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+ by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+ qed
+ next
+ fix th''
+ assume dp': "(Th th', Th th'') \<in> child s"
+ show "cp s th'' = cp s' th''"
+ apply (subst cp_rec[OF vt_s])
+ proof -
+ have "preced th'' s = preced th'' s'"
+ proof(rule eq_preced)
+ show "th'' \<noteq> th"
+ proof
+ assume "th'' = th"
+ with dp1 dp'
+ have "(Th th, Th th) \<in> (depend s)^+"
+ by (auto simp:child_def s_dependents_def eq_depend)
+ with wf_trancl[OF wf_depend[OF vt_s]]
+ show False by auto
+ qed
+ qed
+ moreover {
+ fix th1
+ assume th1_in: "th1 \<in> children s th''"
+ have "cp s th1 = cp s' th1"
+ proof(cases "th1 = th'")
+ case True
+ with eq_cps show ?thesis by simp
+ next
+ case False
+ have neq_th1: "th1 \<noteq> th"
+ proof
+ assume eq_th1: "th1 = th"
+ with dp1 have "(Th th1, Th th') \<in> (depend s)^+"
+ by (auto simp:s_dependents_def eq_depend)
+ from children_no_dep[OF vt_s _ _ this]
+ th1_in dp'
+ show False by (auto simp:children_def)
+ qed
+ thus ?thesis
+ proof(rule eq_cp)
+ show "th \<notin> dependents s th1"
+ proof
+ assume "th \<in> dependents s th1"
+ from dependents_child_unique[OF vt_s _ _ this dp1]
+ th1_in dp' have "th1 = th'"
+ by (auto simp:children_def)
+ with False show False by auto
+ qed
+ qed
+ qed
+ }
+ ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
+ {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+ moreover have "children s th'' = children s' th''"
+ by (unfold children_def child_def s_def depend_set_unchanged, simp)
+ ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+ by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+ qed
+ qed
+ }
+ ultimately show ?thesis by auto
+qed
+
+lemma eq_up_self:
+ fixes th' th''
+ assumes dp: "th \<in> dependents s th''"
+ and eq_cps: "cp s th = cp s' th"
+ shows "cp s th'' = cp s' th''"
+proof -
+ from dp
+ have "(Th th, Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
+ from depend_child[OF vt_s this[unfolded eq_depend]]
+ have ch_th': "(Th th, Th th'') \<in> (child s)\<^sup>+" .
+ moreover { fix n th''
+ have "\<lbrakk>(Th th, n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
+ (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
+ proof(erule trancl_induct, auto)
+ fix y th''
+ assume y_ch: "(y, Th th'') \<in> child s"
+ and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
+ and ch': "(Th th, y) \<in> (child s)\<^sup>+"
+ from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
+ with ih have eq_cpy:"cp s thy = cp s' thy" by blast
+ from child_depend_p[OF ch'] and eq_y
+ have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by simp
+ show "cp s th'' = cp s' th''"
+ apply (subst cp_rec[OF vt_s])
+ proof -
+ have "preced th'' s = preced th'' s'"
+ proof(rule eq_preced)
+ show "th'' \<noteq> th"
+ proof
+ assume "th'' = th"
+ with dp_thy y_ch[unfolded eq_y]
+ have "(Th th, Th th) \<in> (depend s)^+"
+ by (auto simp:child_def)
+ with wf_trancl[OF wf_depend[OF vt_s]]
+ show False by auto
+ qed
+ qed
+ moreover {
+ fix th1
+ assume th1_in: "th1 \<in> children s th''"
+ have "cp s th1 = cp s' th1"
+ proof(cases "th1 = thy")
+ case True
+ with eq_cpy show ?thesis by simp
+ next
+ case False
+ have neq_th1: "th1 \<noteq> th"
+ proof
+ assume eq_th1: "th1 = th"
+ with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+ from children_no_dep[OF vt_s _ _ this] and
+ th1_in y_ch eq_y show False by (auto simp:children_def)
+ qed
+ have "th \<notin> dependents s th1"
+ proof
+ assume h:"th \<in> dependents s th1"
+ from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
+ from dependents_child_unique[OF vt_s _ _ h this]
+ th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
+ with False show False by auto
+ qed
+ from eq_cp[OF neq_th1 this]
+ show ?thesis .
+ qed
+ }
+ ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
+ {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+ moreover have "children s th'' = children s' th''"
+ by (unfold children_def child_def s_def depend_set_unchanged, simp)
+ ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+ by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+ qed
+ next
+ fix th''
+ assume dp': "(Th th, Th th'') \<in> child s"
+ show "cp s th'' = cp s' th''"
+ apply (subst cp_rec[OF vt_s])
+ proof -
+ have "preced th'' s = preced th'' s'"
+ proof(rule eq_preced)
+ show "th'' \<noteq> th"
+ proof
+ assume "th'' = th"
+ with dp dp'
+ have "(Th th, Th th) \<in> (depend s)^+"
+ by (auto simp:child_def s_dependents_def eq_depend)
+ with wf_trancl[OF wf_depend[OF vt_s]]
+ show False by auto
+ qed
+ qed
+ moreover {
+ fix th1
+ assume th1_in: "th1 \<in> children s th''"
+ have "cp s th1 = cp s' th1"
+ proof(cases "th1 = th")
+ case True
+ with eq_cps show ?thesis by simp
+ next
+ case False
+ assume neq_th1: "th1 \<noteq> th"
+ thus ?thesis
+ proof(rule eq_cp)
+ show "th \<notin> dependents s th1"
+ proof
+ assume "th \<in> dependents s th1"
+ hence "(Th th, Th th1) \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
+ from children_no_dep[OF vt_s _ _ this]
+ and th1_in dp' show False
+ by (auto simp:children_def)
+ qed
+ qed
+ qed
+ }
+ ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
+ {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+ moreover have "children s th'' = children s' th''"
+ by (unfold children_def child_def s_def depend_set_unchanged, simp)
+ ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+ by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+ qed
+ qed
+ }
+ ultimately show ?thesis by auto
+qed
+end
+
+lemma next_waiting:
+ assumes vt: "vt step s"
+ and nxt: "next_th s th cs th'"
+ shows "waiting s th' cs"
+proof -
+ from assms show ?thesis
+ apply (auto simp:next_th_def s_waiting_def)
+ proof -
+ fix rest
+ assume ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+ and eq_wq: "wq s cs = th # rest"
+ and ne: "rest \<noteq> []"
+ have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from wq_distinct[OF vt, of cs] 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
+ with 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 vt, of cs] eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ from ne show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> x \<noteq> []" by auto
+ qed
+ ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
+ next
+ fix rest
+ assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+ and ne: "rest \<noteq> []"
+ have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+ proof(rule someI2)
+ from wq_distinct[OF vt, of cs] eq_wq
+ show "distinct rest \<and> set rest = set rest" by auto
+ next
+ from ne show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> 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 "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+ proof(rule someI2)
+ from wq_distinct[OF vt, of cs] 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
+ ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest" by simp
+ with eq_wq and wq_distinct[OF vt, of cs]
+ show False by auto
+ qed
+qed
+
+locale step_v_cps =
+ fixes s' th cs s
+ defines s_def : "s \<equiv> (V th cs#s')"
+ assumes vt_s: "vt step s"
+
+locale step_v_cps_nt = step_v_cps +
+ fixes th'
+ assumes nt: "next_th s' th cs th'"
+
+context step_v_cps_nt
+begin
+
+lemma depend_s:
+ "depend s = (depend s' - {(Cs cs, Th th)} - {(Th th', Cs cs)}) \<union>
+ {(Cs cs, Th th')}"
+proof -
+ from step_depend_v[OF vt_s[unfolded s_def], folded s_def]
+ and nt show ?thesis by (auto intro:next_th_unique)
+qed
+
+lemma dependents_kept:
+ fixes th''
+ assumes neq1: "th'' \<noteq> th"
+ and neq2: "th'' \<noteq> th'"
+ shows "dependents (wq s) th'' = dependents (wq s') th''"
+proof(auto)
+ fix x
+ assume "x \<in> dependents (wq s) th''"
+ hence dp: "(Th x, Th th'') \<in> (depend s)^+"
+ by (auto simp:cs_dependents_def eq_depend)
+ { fix n
+ have "(n, Th th'') \<in> (depend s)^+ \<Longrightarrow> (n, Th th'') \<in> (depend s')^+"
+ proof(induct rule:converse_trancl_induct)
+ fix y
+ assume "(y, Th th'') \<in> depend s"
+ with depend_s neq1 neq2
+ have "(y, Th th'') \<in> depend s'" by auto
+ thus "(y, Th th'') \<in> (depend s')\<^sup>+" by auto
+ next
+ fix y z
+ assume yz: "(y, z) \<in> depend s"
+ and ztp: "(z, Th th'') \<in> (depend s)\<^sup>+"
+ and ztp': "(z, Th th'') \<in> (depend s')\<^sup>+"
+ have "y \<noteq> Cs cs \<and> y \<noteq> Th th'"
+ proof
+ show "y \<noteq> Cs cs"
+ proof
+ assume eq_y: "y = Cs cs"
+ with yz have dp_yz: "(Cs cs, z) \<in> depend s" by simp
+ from depend_s
+ have cst': "(Cs cs, Th th') \<in> depend s" by simp
+ from unique_depend[OF vt_s this dp_yz]
+ have eq_z: "z = Th th'" by simp
+ with ztp have "(Th th', Th th'') \<in> (depend s)^+" by simp
+ from converse_tranclE[OF this]
+ obtain cs' where dp'': "(Th th', Cs cs') \<in> depend s"
+ by (auto simp:s_depend_def)
+ with depend_s have dp': "(Th th', Cs cs') \<in> depend s'" by auto
+ from dp'' eq_y yz eq_z have "(Cs cs, Cs cs') \<in> (depend s)^+" by auto
+ moreover have "cs' = cs"
+ proof -
+ from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
+ have "(Th th', Cs cs) \<in> depend s'"
+ by (auto simp:s_waiting_def s_depend_def cs_waiting_def)
+ from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this dp']
+ show ?thesis by simp
+ qed
+ ultimately have "(Cs cs, Cs cs) \<in> (depend s)^+" by simp
+ moreover note wf_trancl[OF wf_depend[OF vt_s]]
+ ultimately show False by auto
+ qed
+ next
+ show "y \<noteq> Th th'"
+ proof
+ assume eq_y: "y = Th th'"
+ with yz have dps: "(Th th', z) \<in> depend s" by simp
+ with depend_s have dps': "(Th th', z) \<in> depend s'" by auto
+ have "z = Cs cs"
+ proof -
+ from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
+ have "(Th th', Cs cs) \<in> depend s'"
+ by (auto simp:s_waiting_def s_depend_def cs_waiting_def)
+ from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] dps' this]
+ show ?thesis .
+ qed
+ with dps depend_s show False by auto
+ qed
+ qed
+ with depend_s yz have "(y, z) \<in> depend s'" by auto
+ with ztp'
+ show "(y, Th th'') \<in> (depend s')\<^sup>+" by auto
+ qed
+ }
+ from this[OF dp]
+ show "x \<in> dependents (wq s') th''"
+ by (auto simp:cs_dependents_def eq_depend)
+next
+ fix x
+ assume "x \<in> dependents (wq s') th''"
+ hence dp: "(Th x, Th th'') \<in> (depend s')^+"
+ by (auto simp:cs_dependents_def eq_depend)
+ { fix n
+ have "(n, Th th'') \<in> (depend s')^+ \<Longrightarrow> (n, Th th'') \<in> (depend s)^+"
+ proof(induct rule:converse_trancl_induct)
+ fix y
+ assume "(y, Th th'') \<in> depend s'"
+ with depend_s neq1 neq2
+ have "(y, Th th'') \<in> depend s" by auto
+ thus "(y, Th th'') \<in> (depend s)\<^sup>+" by auto
+ next
+ fix y z
+ assume yz: "(y, z) \<in> depend s'"
+ and ztp: "(z, Th th'') \<in> (depend s')\<^sup>+"
+ and ztp': "(z, Th th'') \<in> (depend s)\<^sup>+"
+ have "y \<noteq> Cs cs \<and> y \<noteq> Th th'"
+ proof
+ show "y \<noteq> Cs cs"
+ proof
+ assume eq_y: "y = Cs cs"
+ with yz have dp_yz: "(Cs cs, z) \<in> depend s'" by simp
+ from this have eq_z: "z = Th th"
+ proof -
+ from step_back_step[OF vt_s[unfolded s_def]]
+ have "(Cs cs, Th th) \<in> depend s'"
+ by(cases, auto simp: s_depend_def cs_holding_def s_holding_def)
+ from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this dp_yz]
+ show ?thesis by simp
+ qed
+ from converse_tranclE[OF ztp]
+ obtain u where "(z, u) \<in> depend s'" by auto
+ moreover
+ from step_back_step[OF vt_s[unfolded s_def]]
+ have "th \<in> readys s'" by (cases, simp add:runing_def)
+ moreover note eq_z
+ ultimately show False
+ by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
+ qed
+ next
+ show "y \<noteq> Th th'"
+ proof
+ assume eq_y: "y = Th th'"
+ with yz have dps: "(Th th', z) \<in> depend s'" by simp
+ have "z = Cs cs"
+ proof -
+ from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
+ have "(Th th', Cs cs) \<in> depend s'"
+ by (auto simp:s_waiting_def s_depend_def cs_waiting_def)
+ from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] dps this]
+ show ?thesis .
+ qed
+ with ztp have cs_i: "(Cs cs, Th th'') \<in> (depend s')\<^sup>+" by simp
+ from step_back_step[OF vt_s[unfolded s_def]]
+ have cs_th: "(Cs cs, Th th) \<in> depend s'"
+ by(cases, auto simp: s_depend_def cs_holding_def s_holding_def)
+ have "(Cs cs, Th th'') \<notin> depend s'"
+ proof
+ assume "(Cs cs, Th th'') \<in> depend s'"
+ from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this cs_th]
+ and neq1 show "False" by simp
+ qed
+ with converse_tranclE[OF cs_i]
+ obtain u where cu: "(Cs cs, u) \<in> depend s'"
+ and u_t: "(u, Th th'') \<in> (depend s')\<^sup>+" by auto
+ have "u = Th th"
+ proof -
+ from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] cu cs_th]
+ show ?thesis .
+ qed
+ with u_t have "(Th th, Th th'') \<in> (depend s')\<^sup>+" by simp
+ from converse_tranclE[OF this]
+ obtain v where "(Th th, v) \<in> (depend s')" by auto
+ moreover from step_back_step[OF vt_s[unfolded s_def]]
+ have "th \<in> readys s'" by (cases, simp add:runing_def)
+ ultimately show False
+ by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
+ qed
+ qed
+ with depend_s yz have "(y, z) \<in> depend s" by auto
+ with ztp'
+ show "(y, Th th'') \<in> (depend s)\<^sup>+" by auto
+ qed
+ }
+ from this[OF dp]
+ show "x \<in> dependents (wq s) th''"
+ by (auto simp:cs_dependents_def eq_depend)
+qed
+
+lemma cp_kept:
+ fixes th''
+ assumes neq1: "th'' \<noteq> th"
+ and neq2: "th'' \<noteq> th'"
+ shows "cp s th'' = cp s' th''"
+proof -
+ from dependents_kept[OF neq1 neq2]
+ have "dependents (wq s) th'' = dependents (wq s') th''" .
+ moreover {
+ fix th1
+ assume "th1 \<in> dependents (wq s) th''"
+ have "preced th1 s = preced th1 s'"
+ by (unfold s_def, auto simp:preced_def)
+ }
+ moreover have "preced th'' s = preced th'' s'"
+ by (unfold s_def, auto simp:preced_def)
+ ultimately have "((\<lambda>th. preced th s) ` ({th''} \<union> dependents (wq s) th'')) =
+ ((\<lambda>th. preced th s') ` ({th''} \<union> dependents (wq s') th''))"
+ by (auto simp:image_def)
+ thus ?thesis
+ by (unfold cp_eq_cpreced cpreced_def, simp)
+qed
+
+end
+
+locale step_v_cps_nnt = step_v_cps +
+ assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
+
+context step_v_cps_nnt
+begin
+
+lemma nw_cs: "(Th th1, Cs cs) \<notin> depend s'"
+proof
+ assume "(Th th1, Cs cs) \<in> depend s'"
+ thus "False"
+ apply (auto simp:s_depend_def cs_waiting_def)
+ proof -
+ assume h1: "th1 \<in> set (wq s' cs)"
+ and h2: "th1 \<noteq> hd (wq s' cs)"
+ from step_back_step[OF vt_s[unfolded s_def]]
+ show "False"
+ proof(cases)
+ assume "holding s' th cs"
+ then obtain rest where
+ eq_wq: "wq s' cs = th#rest"
+ apply (unfold s_holding_def)
+ by (case_tac "(wq s' cs)", auto)
+ with h1 h2 have ne: "rest \<noteq> []" by auto
+ with eq_wq
+ have "next_th s' th cs (hd (SOME q. distinct q \<and> set q = set rest))"
+ by(unfold next_th_def, rule_tac x = "rest" in exI, auto)
+ with nnt show ?thesis by auto
+ qed
+ qed
+qed
+
+lemma depend_s: "depend s = depend s' - {(Cs cs, Th th)}"
+proof -
+ from nnt and step_depend_v[OF vt_s[unfolded s_def], folded s_def]
+ show ?thesis by auto
+qed
+
+lemma child_kept_left:
+ assumes
+ "(n1, n2) \<in> (child s')^+"
+ shows "(n1, n2) \<in> (child s)^+"
+proof -
+ from assms show ?thesis
+ proof(induct rule: converse_trancl_induct)
+ case (base y)
+ from base obtain th1 cs1 th2
+ where h1: "(Th th1, Cs cs1) \<in> depend s'"
+ and h2: "(Cs cs1, Th th2) \<in> depend s'"
+ and eq_y: "y = Th th1" and eq_n2: "n2 = Th th2" by (auto simp:child_def)
+ have "cs1 \<noteq> cs"
+ proof
+ assume eq_cs: "cs1 = cs"
+ with h1 have "(Th th1, Cs cs1) \<in> depend s'" by simp
+ with nw_cs eq_cs show False by auto
+ qed
+ with h1 h2 depend_s have
+ h1': "(Th th1, Cs cs1) \<in> depend s" and
+ h2': "(Cs cs1, Th th2) \<in> depend s" by auto
+ hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
+ with eq_y eq_n2 have "(y, n2) \<in> child s" by simp
+ thus ?case by auto
+ next
+ case (step y z)
+ have "(y, z) \<in> child s'" by fact
+ then obtain th1 cs1 th2
+ where h1: "(Th th1, Cs cs1) \<in> depend s'"
+ and h2: "(Cs cs1, Th th2) \<in> depend s'"
+ and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def)
+ have "cs1 \<noteq> cs"
+ proof
+ assume eq_cs: "cs1 = cs"
+ with h1 have "(Th th1, Cs cs1) \<in> depend s'" by simp
+ with nw_cs eq_cs show False by auto
+ qed
+ with h1 h2 depend_s have
+ h1': "(Th th1, Cs cs1) \<in> depend s" and
+ h2': "(Cs cs1, Th th2) \<in> depend s" by auto
+ hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
+ with eq_y eq_z have "(y, z) \<in> child s" by simp
+ moreover have "(z, n2) \<in> (child s)^+" by fact
+ ultimately show ?case by auto
+ qed
+qed
+
+lemma child_kept_right:
+ assumes
+ "(n1, n2) \<in> (child s)^+"
+ shows "(n1, n2) \<in> (child s')^+"
+proof -
+ from assms show ?thesis
+ proof(induct)
+ case (base y)
+ from base and depend_s
+ have "(n1, y) \<in> child s'"
+ by (auto simp:child_def)
+ thus ?case by auto
+ next
+ case (step y z)
+ have "(y, z) \<in> child s" by fact
+ with depend_s have "(y, z) \<in> child s'"
+ by (auto simp:child_def)
+ moreover have "(n1, y) \<in> (child s')\<^sup>+" by fact
+ ultimately show ?case by auto
+ qed
+qed
+
+lemma eq_child: "(child s)^+ = (child s')^+"
+ by (insert child_kept_left child_kept_right, auto)
+
+lemma eq_cp:
+ fixes th'
+ shows "cp s th' = cp s' th'"
+ apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+ have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
+ apply (unfold cs_dependents_def, unfold eq_depend)
+ proof -
+ from eq_child
+ have "\<And>th. {th'. (Th th', Th th) \<in> (child s)\<^sup>+} = {th'. (Th th', Th th) \<in> (child s')\<^sup>+}"
+ by simp
+ with child_depend_eq[OF vt_s] child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ show "\<And>th. {th'. (Th th', Th th) \<in> (depend s)\<^sup>+} = {th'. (Th th', Th th) \<in> (depend s')\<^sup>+}"
+ by simp
+ qed
+ moreover {
+ fix th1
+ assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
+ hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
+ hence "preced th1 s = preced th1 s'"
+ proof
+ assume "th1 = th'"
+ show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
+ next
+ assume "th1 \<in> dependents (wq s') th'"
+ show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
+ qed
+ } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
+ by (auto simp:image_def)
+ thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+end
+
+locale step_P_cps =
+ fixes s' th cs s
+ defines s_def : "s \<equiv> (P th cs#s')"
+ assumes vt_s: "vt step s"
+
+locale step_P_cps_ne =step_P_cps +
+ assumes ne: "wq s' cs \<noteq> []"
+
+context step_P_cps_ne
+begin
+
+lemma depend_s: "depend s = depend s' \<union> {(Th th, Cs cs)}"
+proof -
+ from step_depend_p[OF vt_s[unfolded s_def]] and ne
+ show ?thesis by (simp add:s_def)
+qed
+
+lemma eq_child_left:
+ assumes nd: "(Th th, Th th') \<notin> (child s)^+"
+ shows "(n1, Th th') \<in> (child s)^+ \<Longrightarrow> (n1, Th th') \<in> (child s')^+"
+proof(induct rule:converse_trancl_induct)
+ case (base y)
+ from base obtain th1 cs1
+ where h1: "(Th th1, Cs cs1) \<in> depend s"
+ and h2: "(Cs cs1, Th th') \<in> depend s"
+ and eq_y: "y = Th th1" by (auto simp:child_def)
+ have "th1 \<noteq> th"
+ proof
+ assume "th1 = th"
+ with base eq_y have "(Th th, Th th') \<in> child s" by simp
+ with nd show False by auto
+ qed
+ with h1 h2 depend_s
+ have h1': "(Th th1, Cs cs1) \<in> depend s'" and
+ h2': "(Cs cs1, Th th') \<in> depend s'" by auto
+ with eq_y show ?case by (auto simp:child_def)
+next
+ case (step y z)
+ have yz: "(y, z) \<in> child s" by fact
+ then obtain th1 cs1 th2
+ where h1: "(Th th1, Cs cs1) \<in> depend s"
+ and h2: "(Cs cs1, Th th2) \<in> depend s"
+ and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def)
+ have "th1 \<noteq> th"
+ proof
+ assume "th1 = th"
+ with yz eq_y have "(Th th, z) \<in> child s" by simp
+ moreover have "(z, Th th') \<in> (child s)^+" by fact
+ ultimately have "(Th th, Th th') \<in> (child s)^+" by auto
+ with nd show False by auto
+ qed
+ with h1 h2 depend_s have h1': "(Th th1, Cs cs1) \<in> depend s'"
+ and h2': "(Cs cs1, Th th2) \<in> depend s'" by auto
+ with eq_y eq_z have "(y, z) \<in> child s'" by (auto simp:child_def)
+ moreover have "(z, Th th') \<in> (child s')^+" by fact
+ ultimately show ?case by auto
+qed
+
+lemma eq_child_right:
+ shows "(n1, Th th') \<in> (child s')^+ \<Longrightarrow> (n1, Th th') \<in> (child s)^+"
+proof(induct rule:converse_trancl_induct)
+ case (base y)
+ with depend_s show ?case by (auto simp:child_def)
+next
+ case (step y z)
+ have "(y, z) \<in> child s'" by fact
+ with depend_s have "(y, z) \<in> child s" by (auto simp:child_def)
+ moreover have "(z, Th th') \<in> (child s)^+" by fact
+ ultimately show ?case by auto
+qed
+
+lemma eq_child:
+ assumes nd: "(Th th, Th th') \<notin> (child s)^+"
+ shows "((n1, Th th') \<in> (child s)^+) = ((n1, Th th') \<in> (child s')^+)"
+ by (insert eq_child_left[OF nd] eq_child_right, auto)
+
+lemma eq_cp:
+ fixes th'
+ assumes nd: "th \<notin> dependents s th'"
+ shows "cp s th' = cp s' th'"
+ apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+ have nd': "(Th th, Th th') \<notin> (child s)^+"
+ proof
+ assume "(Th th, Th th') \<in> (child s)\<^sup>+"
+ with child_depend_eq[OF vt_s]
+ have "(Th th, Th th') \<in> (depend s)\<^sup>+" by simp
+ with nd show False
+ by (simp add:s_dependents_def eq_depend)
+ qed
+ have eq_dp: "dependents (wq s) th' = dependents (wq s') th'"
+ proof(auto)
+ fix x assume " x \<in> dependents (wq s) th'"
+ thus "x \<in> dependents (wq s') th'"
+ apply (auto simp:cs_dependents_def eq_depend)
+ proof -
+ assume "(Th x, Th th') \<in> (depend s)\<^sup>+"
+ with child_depend_eq[OF vt_s] have "(Th x, Th th') \<in> (child s)\<^sup>+" by simp
+ with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s')^+" by simp
+ with child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ show "(Th x, Th th') \<in> (depend s')\<^sup>+" by simp
+ qed
+ next
+ fix x assume "x \<in> dependents (wq s') th'"
+ thus "x \<in> dependents (wq s) th'"
+ apply (auto simp:cs_dependents_def eq_depend)
+ proof -
+ assume "(Th x, Th th') \<in> (depend s')\<^sup>+"
+ with child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ have "(Th x, Th th') \<in> (child s')\<^sup>+" by simp
+ with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s)^+" by simp
+ with child_depend_eq[OF vt_s]
+ show "(Th x, Th th') \<in> (depend s)\<^sup>+" by simp
+ qed
+ qed
+ moreover {
+ fix th1 have "preced th1 s = preced th1 s'" by (simp add:s_def preced_def)
+ } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
+ by (auto simp:image_def)
+ thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+lemma eq_up:
+ fixes th' th''
+ assumes dp1: "th \<in> dependents s th'"
+ and dp2: "th' \<in> dependents s th''"
+ and eq_cps: "cp s th' = cp s' th'"
+ shows "cp s th'' = cp s' th''"
+proof -
+ from dp2
+ have "(Th th', Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependents_def)
+ from depend_child[OF vt_s this[unfolded eq_depend]]
+ have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" .
+ moreover {
+ fix n th''
+ have "\<lbrakk>(Th th', n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
+ (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')"
+ proof(erule trancl_induct, auto)
+ fix y th''
+ assume y_ch: "(y, Th th'') \<in> child s"
+ and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''"
+ and ch': "(Th th', y) \<in> (child s)\<^sup>+"
+ from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
+ with ih have eq_cpy:"cp s thy = cp s' thy" by blast
+ from dp1 have "(Th th, Th th') \<in> (depend s)^+" by (auto simp:s_dependents_def eq_depend)
+ moreover from child_depend_p[OF ch'] and eq_y
+ have "(Th th', Th thy) \<in> (depend s)^+" by simp
+ ultimately have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by auto
+ show "cp s th'' = cp s' th''"
+ apply (subst cp_rec[OF vt_s])
+ proof -
+ have "preced th'' s = preced th'' s'"
+ by (simp add:s_def preced_def)
+ moreover {
+ fix th1
+ assume th1_in: "th1 \<in> children s th''"
+ have "cp s th1 = cp s' th1"
+ proof(cases "th1 = thy")
+ case True
+ with eq_cpy show ?thesis by simp
+ next
+ case False
+ have neq_th1: "th1 \<noteq> th"
+ proof
+ assume eq_th1: "th1 = th"
+ with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+ from children_no_dep[OF vt_s _ _ this] and
+ th1_in y_ch eq_y show False by (auto simp:children_def)
+ qed
+ have "th \<notin> dependents s th1"
+ proof
+ assume h:"th \<in> dependents s th1"
+ from eq_y dp_thy have "th \<in> dependents s thy" by (auto simp:s_dependents_def eq_depend)
+ from dependents_child_unique[OF vt_s _ _ h this]
+ th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
+ with False show False by auto
+ qed
+ from eq_cp[OF this]
+ show ?thesis .
+ qed
+ }
+ ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
+ {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+ moreover have "children s th'' = children s' th''"
+ apply (unfold children_def child_def s_def depend_set_unchanged, simp)
+ apply (fold s_def, auto simp:depend_s)
+ proof -
+ assume "(Cs cs, Th th'') \<in> depend s'"
+ with depend_s have cs_th': "(Cs cs, Th th'') \<in> depend s" by auto
+ from dp1 have "(Th th, Th th') \<in> (depend s)^+"
+ by (auto simp:s_dependents_def eq_depend)
+ from converse_tranclE[OF this]
+ obtain cs1 where h1: "(Th th, Cs cs1) \<in> depend s"
+ and h2: "(Cs cs1 , Th th') \<in> (depend s)\<^sup>+"
+ by (auto simp:s_depend_def)
+ have eq_cs: "cs1 = cs"
+ proof -
+ from depend_s have "(Th th, Cs cs) \<in> depend s" by simp
+ from unique_depend[OF vt_s this h1]
+ show ?thesis by simp
+ qed
+ have False
+ proof(rule converse_tranclE[OF h2])
+ assume "(Cs cs1, Th th') \<in> depend s"
+ with eq_cs have "(Cs cs, Th th') \<in> depend s" by simp
+ from unique_depend[OF vt_s this cs_th']
+ have "th' = th''" by simp
+ with ch' y_ch have "(Th th'', Th th'') \<in> (child s)^+" by auto
+ with wf_trancl[OF wf_child[OF vt_s]]
+ show False by auto
+ next
+ fix y
+ assume "(Cs cs1, y) \<in> depend s"
+ and ytd: " (y, Th th') \<in> (depend s)\<^sup>+"
+ with eq_cs have csy: "(Cs cs, y) \<in> depend s" by simp
+ from unique_depend[OF vt_s this cs_th']
+ have "y = Th th''" .
+ with ytd have "(Th th'', Th th') \<in> (depend s)^+" by simp
+ from depend_child[OF vt_s this]
+ have "(Th th'', Th th') \<in> (child s)\<^sup>+" .
+ moreover from ch' y_ch have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto
+ ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto
+ with wf_trancl[OF wf_child[OF vt_s]]
+ show False by auto
+ qed
+ thus "\<exists>cs. (Th th, Cs cs) \<in> depend s' \<and> (Cs cs, Th th'') \<in> depend s'" by auto
+ qed
+ ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+ by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+ qed
+ next
+ fix th''
+ assume dp': "(Th th', Th th'') \<in> child s"
+ show "cp s th'' = cp s' th''"
+ apply (subst cp_rec[OF vt_s])
+ proof -
+ have "preced th'' s = preced th'' s'"
+ by (simp add:s_def preced_def)
+ moreover {
+ fix th1
+ assume th1_in: "th1 \<in> children s th''"
+ have "cp s th1 = cp s' th1"
+ proof(cases "th1 = th'")
+ case True
+ with eq_cps show ?thesis by simp
+ next
+ case False
+ have neq_th1: "th1 \<noteq> th"
+ proof
+ assume eq_th1: "th1 = th"
+ with dp1 have "(Th th1, Th th') \<in> (depend s)^+"
+ by (auto simp:s_dependents_def eq_depend)
+ from children_no_dep[OF vt_s _ _ this]
+ th1_in dp'
+ show False by (auto simp:children_def)
+ qed
+ show ?thesis
+ proof(rule eq_cp)
+ show "th \<notin> dependents s th1"
+ proof
+ assume "th \<in> dependents s th1"
+ from dependents_child_unique[OF vt_s _ _ this dp1]
+ th1_in dp' have "th1 = th'"
+ by (auto simp:children_def)
+ with False show False by auto
+ qed
+ qed
+ qed
+ }
+ ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
+ {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
+ moreover have "children s th'' = children s' th''"
+ apply (unfold children_def child_def s_def depend_set_unchanged, simp)
+ apply (fold s_def, auto simp:depend_s)
+ proof -
+ assume "(Cs cs, Th th'') \<in> depend s'"
+ with depend_s have cs_th': "(Cs cs, Th th'') \<in> depend s" by auto
+ from dp1 have "(Th th, Th th') \<in> (depend s)^+"
+ by (auto simp:s_dependents_def eq_depend)
+ from converse_tranclE[OF this]
+ obtain cs1 where h1: "(Th th, Cs cs1) \<in> depend s"
+ and h2: "(Cs cs1 , Th th') \<in> (depend s)\<^sup>+"
+ by (auto simp:s_depend_def)
+ have eq_cs: "cs1 = cs"
+ proof -
+ from depend_s have "(Th th, Cs cs) \<in> depend s" by simp
+ from unique_depend[OF vt_s this h1]
+ show ?thesis by simp
+ qed
+ have False
+ proof(rule converse_tranclE[OF h2])
+ assume "(Cs cs1, Th th') \<in> depend s"
+ with eq_cs have "(Cs cs, Th th') \<in> depend s" by simp
+ from unique_depend[OF vt_s this cs_th']
+ have "th' = th''" by simp
+ with dp' have "(Th th'', Th th'') \<in> (child s)^+" by auto
+ with wf_trancl[OF wf_child[OF vt_s]]
+ show False by auto
+ next
+ fix y
+ assume "(Cs cs1, y) \<in> depend s"
+ and ytd: " (y, Th th') \<in> (depend s)\<^sup>+"
+ with eq_cs have csy: "(Cs cs, y) \<in> depend s" by simp
+ from unique_depend[OF vt_s this cs_th']
+ have "y = Th th''" .
+ with ytd have "(Th th'', Th th') \<in> (depend s)^+" by simp
+ from depend_child[OF vt_s this]
+ have "(Th th'', Th th') \<in> (child s)\<^sup>+" .
+ moreover from dp' have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto
+ ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto
+ with wf_trancl[OF wf_child[OF vt_s]]
+ show False by auto
+ qed
+ thus "\<exists>cs. (Th th, Cs cs) \<in> depend s' \<and> (Cs cs, Th th'') \<in> depend s'" by auto
+ qed
+ ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
+ by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
+ qed
+ qed
+ }
+ ultimately show ?thesis by auto
+qed
+
+end
+
+locale step_create_cps =
+ fixes s' th prio s
+ defines s_def : "s \<equiv> (Create th prio#s')"
+ assumes vt_s: "vt step s"
+
+context step_create_cps
+begin
+
+lemma eq_dep: "depend s = depend s'"
+ by (unfold s_def depend_create_unchanged, auto)
+
+lemma eq_cp:
+ fixes th'
+ assumes neq_th: "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+ apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+ have nd: "th \<notin> dependents s th'"
+ proof
+ assume "th \<in> dependents s th'"
+ hence "(Th th, Th th') \<in> (depend s)^+" by (simp add:s_dependents_def eq_depend)
+ with eq_dep have "(Th th, Th th') \<in> (depend s')^+" by simp
+ from converse_tranclE[OF this]
+ obtain y where "(Th th, y) \<in> depend s'" by auto
+ with dm_depend_threads[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ have in_th: "th \<in> threads s'" by auto
+ from step_back_step[OF vt_s[unfolded s_def]]
+ show False
+ proof(cases)
+ assume "th \<notin> threads s'"
+ with in_th show ?thesis by simp
+ qed
+ qed
+ have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
+ by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
+ moreover {
+ fix th1
+ assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
+ hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
+ hence "preced th1 s = preced th1 s'"
+ proof
+ assume "th1 = th'"
+ with neq_th
+ show "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
+ next
+ assume "th1 \<in> dependents (wq s') th'"
+ with nd and eq_dp have "th1 \<noteq> th"
+ by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
+ thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
+ qed
+ } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
+ by (auto simp:image_def)
+ thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+lemma nil_dependents: "dependents s th = {}"
+proof -
+ from step_back_step[OF vt_s[unfolded s_def]]
+ show ?thesis
+ proof(cases)
+ assume "th \<notin> threads s'"
+ from not_thread_holdents[OF step_back_vt[OF vt_s[unfolded s_def]] this]
+ have hdn: " holdents s' th = {}" .
+ have "dependents s' th = {}"
+ proof -
+ { assume "dependents s' th \<noteq> {}"
+ then obtain th' where dp: "(Th th', Th th) \<in> (depend s')^+"
+ by (auto simp:s_dependents_def eq_depend)
+ from tranclE[OF this] obtain cs' where
+ "(Cs cs', Th th) \<in> depend s'" by (auto simp:s_depend_def)
+ with hdn
+ have False by (auto simp:holdents_def)
+ } thus ?thesis by auto
+ qed
+ thus ?thesis
+ by (unfold s_def s_dependents_def eq_depend depend_create_unchanged, simp)
+ qed
+qed
+
+lemma eq_cp_th: "cp s th = preced th s"
+ apply (unfold cp_eq_cpreced cpreced_def)
+ by (insert nil_dependents, unfold s_dependents_def cs_dependents_def, auto)
+
+end
+
+
+locale step_exit_cps =
+ fixes s' th prio s
+ defines s_def : "s \<equiv> (Exit th#s')"
+ assumes vt_s: "vt step s"
+
+context step_exit_cps
+begin
+
+lemma eq_dep: "depend s = depend s'"
+ by (unfold s_def depend_exit_unchanged, auto)
+
+lemma eq_cp:
+ fixes th'
+ assumes neq_th: "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+ apply (unfold cp_eq_cpreced cpreced_def)
+proof -
+ have nd: "th \<notin> dependents s th'"
+ proof
+ assume "th \<in> dependents s th'"
+ hence "(Th th, Th th') \<in> (depend s)^+" by (simp add:s_dependents_def eq_depend)
+ with eq_dep have "(Th th, Th th') \<in> (depend s')^+" by simp
+ from converse_tranclE[OF this]
+ obtain cs' where bk: "(Th th, Cs cs') \<in> depend s'"
+ by (auto simp:s_depend_def)
+ from step_back_step[OF vt_s[unfolded s_def]]
+ show False
+ proof(cases)
+ assume "th \<in> runing s'"
+ with bk show ?thesis
+ apply (unfold runing_def readys_def s_waiting_def s_depend_def)
+ by (auto simp:cs_waiting_def)
+ qed
+ qed
+ have eq_dp: "\<And> th. dependents (wq s) th = dependents (wq s') th"
+ by (unfold cs_dependents_def, auto simp:eq_dep eq_depend)
+ moreover {
+ fix th1
+ assume "th1 \<in> {th'} \<union> dependents (wq s') th'"
+ hence "th1 = th' \<or> th1 \<in> dependents (wq s') th'" by auto
+ hence "preced th1 s = preced th1 s'"
+ proof
+ assume "th1 = th'"
+ with neq_th
+ show "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
+ next
+ assume "th1 \<in> dependents (wq s') th'"
+ with nd and eq_dp have "th1 \<noteq> th"
+ by (auto simp:eq_depend cs_dependents_def s_dependents_def eq_dep)
+ thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
+ qed
+ } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))"
+ by (auto simp:image_def)
+ thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')) =
+ Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependents (wq s') th'))" by simp
+qed
+
+end
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Ext.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1057 @@
+theory Ext
+imports Prio
+begin
+
+locale highest_create =
+ fixes s' th prio fixes s
+ defines s_def : "s \<equiv> (Create th prio#s')"
+ assumes vt_s: "vt step s"
+ and highest: "cp s th = Max ((cp s)`threads s)"
+
+context highest_create
+begin
+
+lemma threads_s: "threads s = threads s' \<union> {th}"
+ by (unfold s_def, simp)
+
+lemma vt_s': "vt step s'"
+ by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_create: "step s' (Create th prio)"
+ by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_create_elim:
+ "\<lbrakk>\<And>max_prio. \<lbrakk>prio \<le> max_prio; th \<notin> threads s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+ by (insert step_create, ind_cases "step s' (Create th prio)", auto)
+
+lemma eq_cp_s:
+ assumes th'_in: "th' \<in> threads s'"
+ shows "cp s th' = cp s' th'"
+proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def s_def
+ eq_depend depend_create_unchanged)
+ show "Max ((\<lambda>tha. preced tha (Create th prio # s')) `
+ ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+})) =
+ Max ((\<lambda>th. preced th s') ` ({th'} \<union> {th'a. (Th th'a, Th th') \<in> (depend s')\<^sup>+}))"
+ (is "Max (?f ` ?A) = Max (?g ` ?A)")
+ proof -
+ have "?f ` ?A = ?g ` ?A"
+ proof(rule f_image_eq)
+ fix a
+ assume a_in: "a \<in> ?A"
+ thus "?f a = ?g a"
+ proof -
+ from a_in
+ have "a = th' \<or> (Th a, Th th') \<in> (depend s')\<^sup>+" by auto
+ hence "a \<noteq> th"
+ proof
+ assume "a = th'"
+ moreover have "th' \<noteq> th"
+ proof(rule step_create_elim)
+ assume th_not_in: "th \<notin> threads s'" with th'_in
+ show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ next
+ assume "(Th a, Th th') \<in> (depend s')\<^sup>+"
+ hence "Th a \<in> Domain \<dots>"
+ by (auto simp:Domain_def)
+ hence "Th a \<in> Domain (depend s')"
+ by (simp add:trancl_domain)
+ from dm_depend_threads[OF vt_s' this]
+ have h: "a \<in> threads s'" .
+ show ?thesis
+ proof(rule step_create_elim)
+ assume "th \<notin> threads s'" with h
+ show ?thesis by auto
+ qed
+ qed
+ thus ?thesis
+ by (unfold preced_def, auto)
+ qed
+ qed
+ thus ?thesis by auto
+ qed
+qed
+
+lemma same_depend: "depend s = depend s'"
+ by (insert depend_create_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+ "dependents (wq s) th = dependents (wq s') th"
+ apply (unfold cs_dependents_def)
+ by (unfold eq_depend same_depend, simp)
+
+lemma nil_dependents_s': "dependents (wq s') th = {}"
+proof -
+ { assume ne: "dependents (wq s') th \<noteq> {}"
+ then obtain th' where "th' \<in> dependents (wq s') th"
+ by (unfold cs_dependents_def, auto)
+ hence "(Th th', Th th) \<in> (depend (wq s'))^+"
+ by (unfold cs_dependents_def, auto)
+ hence "(Th th', Th th) \<in> (depend s')^+"
+ by (simp add:eq_depend)
+ hence "Th th \<in> Range ((depend s')^+)" by (auto simp:Range_def Domain_def)
+ hence "Th th \<in> Range (depend s')" by (simp add:trancl_range)
+ from range_in [OF vt_s' this]
+ have h: "th \<in> threads s'" .
+ have "False"
+ proof(rule step_create_elim)
+ assume "th \<notin> threads s'" with h show ?thesis by auto
+ qed
+ } thus ?thesis by auto
+qed
+
+lemma nil_dependents: "dependents (wq s) th = {}"
+proof -
+ have "wq s' = wq s"
+ by (unfold wq_def s_def, auto simp:Let_def)
+ with nil_dependents_s' show ?thesis by auto
+qed
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+ by (unfold cp_eq_cpreced cpreced_def nil_dependents, auto)
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold max_cp_eq[OF vt_s], unfold highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+ { assume "th \<notin> readys s"
+ with threads_s obtain cs where
+ "waiting s th cs"
+ by (unfold readys_def, auto)
+ hence "(Th th, Cs cs) \<in> depend s"
+ by (unfold s_depend_def, unfold eq_waiting, simp)
+ hence "Th th \<in> Domain (depend s')"
+ by (unfold same_depend, auto simp:Domain_def)
+ from dm_depend_threads [OF vt_s' this]
+ have h: "th \<in> threads s'" .
+ have "False"
+ proof (rule_tac step_create_elim)
+ assume "th \<notin> threads s'" with h show ?thesis by simp
+ qed
+ } thus ?thesis by auto
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+ have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+ proof -
+ have " Max (cp s ` readys s) = cp s th"
+ proof(rule Max_eqI)
+ from finite_threads[OF vt_s] readys_threads finite_subset
+ have "finite (readys s)" by blast
+ thus "finite (cp s ` readys s)" by auto
+ next
+ from is_ready show "cp s th \<in> cp s ` readys s" by auto
+ next
+ fix y
+ assume h: "y \<in> cp s ` readys s"
+ have "y \<le> Max (cp s ` readys s)"
+ proof(rule Max_ge [OF _ h])
+ from finite_threads[OF vt_s] readys_threads finite_subset
+ have "finite (readys s)" by blast
+ thus "finite (cp s ` readys s)" by auto
+ qed
+ moreover have "\<dots> \<le> Max (cp s ` threads s)"
+ proof(rule Max_mono)
+ from readys_threads
+ show "cp s ` readys s \<subseteq> cp s ` threads s" by auto
+ next
+ from is_ready show "cp s ` readys s \<noteq> {}" by auto
+ next
+ from finite_threads [OF vt_s]
+ show "finite (cp s ` threads s)" by auto
+ qed
+ moreover note highest
+ ultimately show "y \<le> cp s th" by auto
+ qed
+ with highest show ?thesis by auto
+ qed
+ thus ?thesis
+ by (unfold runing_def, insert highest is_ready, auto)
+qed
+
+end
+
+locale extend_highest = highest_create +
+ fixes t
+ assumes vt_t: "vt step (t@s)"
+ and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+ and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+ and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app:
+ assumes vt_ts: "vt cs (t@s)"
+ shows "vt cs s"
+proof -
+ from vt_ts show ?thesis
+ proof(induct t)
+ case Nil
+ from Nil show ?case by auto
+ next
+ case (Cons e t)
+ assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+ and vt_et: "vt cs ((e # t) @ s)"
+ show ?case
+ proof(rule ih)
+ show "vt cs (t @ s)"
+ proof(rule step_back_vt)
+ from vt_et show "vt cs (e # t @ s)" by simp
+ qed
+ qed
+ qed
+qed
+
+context extend_highest
+begin
+
+lemma red_moment:
+ "extend_highest s' th prio (moment i t)"
+ apply (insert extend_highest_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+ apply (unfold extend_highest_def extend_highest_axioms_def, clarsimp)
+ by (unfold highest_create_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes
+ h0: "R []"
+ and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
+ extend_highest s' th prio t;
+ extend_highest s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+ shows "R t"
+proof -
+ from vt_t extend_highest_axioms show ?thesis
+ proof(induct t)
+ from h0 show "R []" .
+ next
+ case (Cons e t')
+ assume ih: "\<lbrakk>vt step (t' @ s); extend_highest s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+ and vt_e: "vt step ((e # t') @ s)"
+ and et: "extend_highest s' th prio (e # t')"
+ from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+ from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+ show ?case
+ proof(rule h2 [OF vt_ts stp _ _ _ ])
+ show "R t'"
+ proof(rule ih)
+ from et show ext': "extend_highest s' th prio t'"
+ by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
+ next
+ from vt_ts show "vt step (t' @ s)" .
+ qed
+ next
+ from et show "extend_highest s' th prio (e # t')" .
+ next
+ from et show ext': "extend_highest s' th prio t'"
+ by (unfold extend_highest_def extend_highest_axioms_def, auto dest:step_back_vt)
+ qed
+ qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and>
+ preced th (t@s) = preced th s" (is "?Q t")
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case Nil
+ from threads_s
+ show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+ by auto
+ next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio)
+ assume eq_e: " e = Create thread prio"
+ show ?thesis
+ proof -
+ from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+ hence "th \<noteq> thread"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ with Cons show ?thesis by auto
+ qed
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have "extend_highest s' th prio (e # t)" by auto
+ from extend_highest.exit_diff [OF this] and eq_e
+ have neq_th: "thread \<noteq> th" by auto
+ with Cons
+ show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ next
+ case (P thread cs)
+ assume eq_e: "e = P thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (V thread cs)
+ assume eq_e: "e = V thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (Set thread prio')
+ assume eq_e: " e = Set thread prio'"
+ show ?thesis
+ proof -
+ from Cons have "extend_highest s' th prio (e # t)" by auto
+ from extend_highest.set_diff_low[OF this] and eq_e
+ have "th \<noteq> thread" by auto
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ qed
+ qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+ case Nil
+ from highest_preced_thread
+ show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+ by simp
+next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio')
+ assume eq_e: " e = Create thread prio'"
+ from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+ hence neq_thread: "thread \<noteq> th"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th \<in> threads (t@s)"
+ proof -
+ from Cons have "extend_highest s' th prio t" by auto
+ from extend_highest.th_kept[OF this] show ?thesis by (simp add:s_def)
+ qed
+ ultimately show ?thesis by auto
+ qed
+ from Cons have "extend_highest s' th prio t" by auto
+ from extend_highest.th_kept[OF this]
+ have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
+ by (auto simp:s_def)
+ from stp
+ have thread_ts: "thread \<notin> threads (t @ s)"
+ by (cases, auto)
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+ by (unfold eq_e, simp)
+ moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+ proof(rule Max_insert)
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this]
+ show "finite (?f ` (threads (t@s)))" by simp
+ next
+ from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+ qed
+ moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+ proof -
+ have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
+ (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+ proof -
+ { fix th'
+ assume "th' \<in> ?B"
+ with thread_ts eq_e
+ have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+ } thus ?thesis
+ apply (auto simp:Image_def)
+ proof -
+ fix th'
+ assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
+ preced th' (e # t @ s) = preced th' (t @ s)"
+ and h1: "th' \<in> threads (t @ s)"
+ show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+ proof -
+ from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+ moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+ ultimately show ?thesis by simp
+ qed
+ qed
+ qed
+ with Cons show ?thesis by auto
+ qed
+ moreover have "?f thread < ?t"
+ proof -
+ from Cons have " extend_highest s' th prio (e # t)" by auto
+ from extend_highest.create_low[OF this] and eq_e
+ have "prio' \<le> prio" by auto
+ thus ?thesis
+ by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+ qed
+ ultimately show ?thesis by (auto simp:max_def)
+ qed
+next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have vt_e: "vt step (e#(t @ s))" by auto
+ from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+ from stp have thread_ts: "thread \<in> threads (t @ s)"
+ by(cases, unfold runing_def readys_def, auto)
+ from Cons have "extend_highest s' th prio (e # t)" by auto
+ from extend_highest.exit_diff[OF this] and eq_e
+ have neq_thread: "thread \<noteq> th" by auto
+ from Cons have "extend_highest s' th prio t" by auto
+ from extend_highest.th_kept[OF this, folded s_def]
+ have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "threads (t@s) = insert thread ?A"
+ by (insert stp thread_ts, unfold eq_e, auto)
+ hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+ also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+ also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ from finite_threads [OF vt_e]
+ show "finite (?f ` ?A)" by simp
+ next
+ from Cons have "extend_highest s' th prio (e # t)" by auto
+ from extend_highest.th_kept[OF this]
+ show "?f ` ?A \<noteq> {}" by (auto simp:s_def)
+ qed
+ finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+ moreover have "Max (?f ` (threads (t@s))) = ?t"
+ proof -
+ from Cons show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ qed
+ ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+ moreover have "?f thread < ?t"
+ proof(unfold eq_e, simp add:preced_def, fold preced_def)
+ show "preced thread (t @ s) < ?t"
+ proof -
+ have "preced thread (t @ s) \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "?t = Max (?g ` ?B)") by simp
+ moreover have "?g thread \<le> \<dots>"
+ proof(rule Max_ge)
+ have "vt step (t@s)" by fact
+ from finite_threads [OF this]
+ show "finite (?g ` ?B)" by simp
+ next
+ from thread_ts
+ show "?g thread \<in> (?g ` ?B)" by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ moreover have "preced thread (t @ s) \<noteq> ?t"
+ proof
+ assume "preced thread (t @ s) = preced th s"
+ with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+ from preced_unique [OF this] have "thread = th"
+ proof
+ from h' show "th \<in> threads (t @ s)" by simp
+ next
+ from thread_ts show "thread \<in> threads (t @ s)" .
+ qed(simp)
+ with neq_thread show "False" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def split:if_splits)
+ qed
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ let ?B = "threads (t@s)"
+ from Cons have "extend_highest s' th prio (e # t)" by auto
+ from extend_highest.set_diff_low[OF this] and Set
+ have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+ from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+ also have "\<dots> = ?t"
+ proof(rule Max_eqI)
+ fix y
+ assume y_in: "y \<in> ?f ` ?B"
+ then obtain th1 where
+ th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+ show "y \<le> ?t"
+ proof(cases "th1 = thread")
+ case True
+ with neq_thread le_p eq_y s_def Set
+ show ?thesis
+ by (auto simp:preced_def precedence_le_def)
+ next
+ case False
+ with Set eq_y
+ have "y = preced th1 (t@s)"
+ by (simp add:preced_def)
+ moreover have "\<dots> \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+ by auto
+ moreover have "preced th1 (t@s) \<le> \<dots>"
+ proof(rule Max_ge)
+ from th1_in
+ show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+ by simp
+ next
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ proof -
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this] show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from Cons and finite_threads
+ show "finite (?f ` ?B)" by auto
+ next
+ from Cons have "extend_highest s' th prio t" by auto
+ from extend_highest.th_kept [OF this, folded s_def]
+ have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show "?t \<in> (?f ` ?B)"
+ proof -
+ from neq_thread Set h
+ have "?t = ?f th" by (auto simp:preced_def)
+ with h show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ (is "?L = ?R")
+proof -
+ have "?L = cpreced (t@s) (wq (t@s)) th"
+ by (unfold cp_eq_cpreced, simp)
+ also have "\<dots> = ?R"
+ proof(unfold cpreced_def)
+ show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+ Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+ proof(cases "?A = {}")
+ case False
+ have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+ moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ show "finite (?f ` ?A)"
+ proof -
+ from dependents_threads[OF vt_t]
+ have "?A \<subseteq> threads (t@s)" .
+ moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+ ultimately show ?thesis
+ by (auto simp:finite_subset)
+ qed
+ next
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ qed
+ moreover have "\<dots> = Max (?f ` ?B)"
+ proof -
+ from max_preced have "?f th = Max (?f ` ?B)" .
+ moreover have "Max (?f ` ?A) \<le> \<dots>"
+ proof(rule Max_mono)
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ next
+ show "?f ` ?A \<subseteq> ?f ` ?B"
+ proof -
+ have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+ thus ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t]
+ show "finite (?f ` ?B)" by simp
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def)
+ qed
+ ultimately show ?thesis by auto
+ next
+ case True
+ with max_preced show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+ by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+ by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ shows "preced th' s < preced th s"
+proof -
+ have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+ proof(rule Max_ge)
+ from finite_threads [OF vt_s]
+ show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+ next
+ from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+ by simp
+ qed
+ moreover have "preced th' s \<noteq> preced th s"
+ proof
+ assume "preced th' s = preced th s"
+ from preced_unique[OF this th'_in] neq_th' is_ready
+ show "False" by (auto simp:readys_def)
+ qed
+ ultimately show ?thesis using highest_preced_thread
+ by auto
+qed
+
+lemma pv_blocked:
+ fixes th'
+ assumes th'_in: "th' \<in> threads (t@s)"
+ and neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+ shows "th' \<notin> runing (t@s)"
+proof
+ assume "th' \<in> runing (t@s)"
+ hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
+ by (auto simp:runing_def)
+ with max_cp_readys_threads [OF vt_t]
+ have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+ by auto
+ moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+ ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+ moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+ by simp
+ finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+ show False
+ proof -
+ have "dependents (wq (t @ s)) th' = {}"
+ by (rule count_eq_dependents [OF vt_t eq_pv])
+ moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+ proof
+ assume "preced th' (t @ s) = preced th (t @ s)"
+ hence "th' = th"
+ proof(rule preced_unique)
+ from th_kept show "th \<in> threads (t @ s)" by simp
+ next
+ from th'_in show "th' \<in> threads (t @ s)" by simp
+ qed
+ with assms show False by simp
+ qed
+ ultimately show ?thesis
+ by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+qed
+
+lemma runing_precond_pre:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads (t@s) \<and>
+ cntP (t@s) th' = cntV (t@s) th'"
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case (Cons e t)
+ from Cons
+ have in_thread: "th' \<in> threads (t @ s)"
+ and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ have "extend_highest s' th prio t" by fact
+ from extend_highest.pv_blocked
+ [OF this, folded s_def, OF in_thread neq_th' not_holding]
+ have not_runing: "th' \<notin> runing (t @ s)" .
+ show ?case
+ proof(cases e)
+ case (V thread cs)
+ from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+ show ?thesis
+ proof -
+ from Cons and V have "step (t@s) (V thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (unfold V, simp add:cntP_def cntV_def count_def)
+ moreover from in_thread
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (P thread cs)
+ from Cons and P have "step (t@s) (P thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Create thread prio')
+ from Cons and Create have "step (t@s) (Create thread prio')" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th' \<in> threads (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Create
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Create
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+ ultimately show ?thesis by auto
+ next
+ case (Exit thread)
+ from Cons and Exit have "step (t@s) (Exit thread)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t @ s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Exit
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Exit and neq_th'
+ have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Set thread prio')
+ with Cons
+ show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ next
+ case Nil
+ with assms
+ show ?case by auto
+ qed
+qed
+
+(*
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<notin> runing (t@s)"
+proof -
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2]
+ show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ and is_runing: "th' \<in> runing (t@s)"
+ shows "cntP s th' > cntV s th'"
+proof -
+ have "cntP s th' \<noteq> cntV s th'"
+ proof
+ assume eq_pv: "cntP s th' = cntV s th'"
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)"
+ and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+ with is_runing show "False" by simp
+ qed
+ moreover from cnp_cnv_cncs[OF vt_s, of th']
+ have "cntV s th' \<le> cntP s th'" by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+ th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+ case (Suc k)
+ show ?case
+ proof -
+ { assume True: "Suc (i+k) \<le> length t"
+ from moment_head [OF this]
+ obtain e where
+ eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+ by blast
+ from red_moment[of "Suc(i+k)"]
+ and eq_me have "extend_highest s' th prio (e # moment (i + k) t)" by simp
+ hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+ by (unfold extend_highest_def extend_highest_axioms_def
+ highest_create_def s_def, auto)
+ have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
+ proof(unfold s_def)
+ show "th' \<notin> runing (moment (i + k) t @ Create th prio # s')"
+ proof(rule extend_highest.pv_blocked)
+ from Suc show "th' \<in> threads (moment (i + k) t @ Create th prio # s')"
+ by (simp add:s_def)
+ next
+ from neq_th' show "th' \<noteq> th" .
+ next
+ from red_moment show "extend_highest s' th prio (moment (i + k) t)" .
+ next
+ from Suc show "cntP (moment (i + k) t @ Create th prio # s') th' =
+ cntV (moment (i + k) t @ Create th prio # s') th'"
+ by (auto simp:s_def)
+ qed
+ qed
+ from step_back_step[OF vt_e]
+ have "step ((moment (i + k) t)@s) e" .
+ hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+ th' \<in> threads (e#(moment (i + k) t)@s)
+ "
+ proof(cases)
+ case (thread_create thread prio)
+ with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_exit thread)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_P thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_V thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_set thread prio')
+ with Suc show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ with eq_me have ?thesis using eq_me by auto
+ } note h = this
+ show ?thesis
+ proof(cases "Suc (i+k) \<le> length t")
+ case True
+ from h [OF this] show ?thesis .
+ next
+ case False
+ with moment_ge
+ have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+ with Suc show ?thesis by auto
+ qed
+ qed
+next
+ case 0
+ from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ and le_ij: "i \<le> j"
+ shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+ have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+ and h2: "th' \<in> threads ((moment j t)@s)" by auto
+ with extend_highest.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1]
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+ case True
+ with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+ case False
+ let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+ let ?q = "moment 0 t"
+ from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+ from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+ from p_split_gen [of ?Q, OF this not_thread]
+ obtain i where lt_its: "i < length t"
+ and le_i: "0 \<le> i"
+ and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+ and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+ from lt_its have "Suc i \<le> length t" by auto
+ from moment_head[OF this] obtain e where
+ eq_me: "moment (Suc i) t = e # moment i t" by blast
+ from red_moment[of "Suc i"] and eq_me
+ have "extend_highest s' th prio (e # moment i t)" by simp
+ hence vt_e: "vt step (e#(moment i t)@s)"
+ by (unfold extend_highest_def extend_highest_axioms_def
+ highest_create_def s_def, auto)
+ from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+ from post[rule_format, of "Suc i"] and eq_me
+ have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+ from create_pre[OF stp_i pre this]
+ obtain prio where eq_e: "e = Create th' prio" .
+ have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+ proof(rule cnp_cnv_eq)
+ from step_back_vt [OF vt_e]
+ show "vt step (moment i t @ s)" .
+ next
+ from eq_e and stp_i
+ have "step (moment i t @ s) (Create th' prio)" by simp
+ thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+ qed
+ with eq_e
+ have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
+ by (simp add:cntP_def cntV_def count_def)
+ with eq_me[symmetric]
+ have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+ by simp
+ from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+ with eq_me [symmetric]
+ have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+ from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+ and moment_ge
+ have "th' \<notin> runing (t @ s)" by auto
+ with runing'
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+ from runing_inversion_1[OF _ runing']
+ show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+ case True thus ?thesis by auto
+next
+ case False
+ then have not_ready: "th \<notin> readys (t@s)"
+ apply (unfold runing_def,
+ insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+ by auto
+ from th_kept have "th \<in> threads (t@s)" by auto
+ from th_chain_to_ready[OF vt_t this] and not_ready
+ obtain th' where th'_in: "th' \<in> readys (t@s)"
+ and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+ have "th' \<in> runing (t@s)"
+ proof -
+ have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
+ preced th (t@s)"
+ proof(rule Max_eqI)
+ fix y
+ assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ then obtain th1 where
+ h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
+ and eq_y: "y = preced th1 (t@s)" by auto
+ show "y \<le> preced th (t @ s)"
+ proof -
+ from max_preced
+ have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+ moreover have "y \<le> \<dots>"
+ proof(rule Max_ge)
+ from h1
+ have "th1 \<in> threads (t@s)"
+ proof
+ assume "th1 = th'"
+ with th'_in show ?thesis by (simp add:readys_def)
+ next
+ assume "th1 \<in> dependents (wq (t @ s)) th'"
+ with dependents_threads [OF vt_t]
+ show "th1 \<in> threads (t @ s)" by auto
+ qed
+ with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+ next
+ from finite_threads[OF vt_t]
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+ show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+ by (auto intro:finite_subset)
+ next
+ from dp
+ have "th \<in> dependents (wq (t @ s)) th'"
+ by (unfold cs_dependents_def, auto simp:eq_depend)
+ thus "preced th (t @ s) \<in>
+ (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ by auto
+ qed
+ moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ from max_preced and max_cp_eq[OF vt_t, symmetric]
+ have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+ with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+ qed
+ ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+ with th'_in show ?thesis by (auto simp:runing_def)
+ qed
+ thus ?thesis by auto
+qed
+
+end
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtGG.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,970 @@
+theory ExtGG
+imports PrioG
+begin
+
+lemma birth_time_lt: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+ apply (induct s, simp)
+proof -
+ fix a s
+ assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+ and eq_as: "a # s \<noteq> []"
+ show "birthtime th (a # s) < length (a # s)"
+ proof(cases "s \<noteq> []")
+ case False
+ from False show ?thesis
+ by (cases a, auto simp:birthtime.simps)
+ next
+ case True
+ from ih [OF True] show ?thesis
+ by (cases a, auto simp:birthtime.simps)
+ qed
+qed
+
+lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
+ by (induct s, auto simp:threads.simps)
+
+lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
+ apply (drule_tac th_in_ne)
+ by (unfold preced_def, auto intro: birth_time_lt)
+
+locale highest_gen =
+ fixes s th prio tm
+ assumes vt_s: "vt step s"
+ and threads_s: "th \<in> threads s"
+ and highest: "preced th s = Max ((cp s)`threads s)"
+ and nh: "preced th s' \<noteq> Max ((cp s)`threads s')"
+ and preced_th: "preced th s = Prc prio tm"
+
+context highest_gen
+begin
+
+lemma lt_tm: "tm < length s"
+ by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+ from highest and max_cp_eq[OF vt_s]
+ have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+ have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+ proof -
+ from threads_s and dependents_threads[OF vt_s, of th]
+ show ?thesis by auto
+ qed
+ show ?thesis
+ proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+ show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+ next
+ fix y
+ assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+ then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+ and eq_y: "y = preced th1 s" by auto
+ show "y \<le> preced th s"
+ proof(unfold is_max, rule Max_ge)
+ from finite_threads[OF vt_s]
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ next
+ from sbs th1_in and eq_y
+ show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+ qed
+ next
+ from sbs and finite_threads[OF vt_s]
+ show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+ by (auto intro:finite_subset)
+ qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+ from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+ show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen +
+ fixes t
+ assumes vt_t: "vt step (t@s)"
+ and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+ and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+ and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app:
+ assumes vt_ts: "vt cs (t@s)"
+ shows "vt cs s"
+proof -
+ from vt_ts show ?thesis
+ proof(induct t)
+ case Nil
+ from Nil show ?case by auto
+ next
+ case (Cons e t)
+ assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+ and vt_et: "vt cs ((e # t) @ s)"
+ show ?case
+ proof(rule ih)
+ show "vt cs (t @ s)"
+ proof(rule step_back_vt)
+ from vt_et show "vt cs (e # t @ s)" by simp
+ qed
+ qed
+ qed
+qed
+
+context extend_highest_gen
+begin
+
+thm extend_highest_gen.axioms
+
+lemma red_moment:
+ "extend_highest_gen s th prio tm (moment i t)"
+ apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+ apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+ by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes
+ h0: "R []"
+ and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
+ extend_highest_gen s th prio tm t;
+ extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+ shows "R t"
+proof -
+ from vt_t extend_highest_gen_axioms show ?thesis
+ proof(induct t)
+ from h0 show "R []" .
+ next
+ case (Cons e t')
+ assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+ and vt_e: "vt step ((e # t') @ s)"
+ and et: "extend_highest_gen s th prio tm (e # t')"
+ from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+ from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+ show ?case
+ proof(rule h2 [OF vt_ts stp _ _ _ ])
+ show "R t'"
+ proof(rule ih)
+ from et show ext': "extend_highest_gen s th prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ next
+ from vt_ts show "vt step (t' @ s)" .
+ qed
+ next
+ from et show "extend_highest_gen s th prio tm (e # t')" .
+ next
+ from et show ext': "extend_highest_gen s th prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ qed
+ qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and>
+ preced th (t@s) = preced th s" (is "?Q t")
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case Nil
+ from threads_s
+ show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+ by auto
+ next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio)
+ assume eq_e: " e = Create thread prio"
+ show ?thesis
+ proof -
+ from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+ hence "th \<noteq> thread"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ with Cons show ?thesis by auto
+ qed
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+ from extend_highest_gen.exit_diff [OF this] and eq_e
+ have neq_th: "thread \<noteq> th" by auto
+ with Cons
+ show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ next
+ case (P thread cs)
+ assume eq_e: "e = P thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (V thread cs)
+ assume eq_e: "e = V thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (Set thread prio')
+ assume eq_e: " e = Set thread prio'"
+ show ?thesis
+ proof -
+ from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+ from extend_highest_gen.set_diff_low[OF this] and eq_e
+ have "th \<noteq> thread" by auto
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ qed
+ qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+ case Nil
+ from highest_preced_thread
+ show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+ by simp
+next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio')
+ assume eq_e: " e = Create thread prio'"
+ from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+ hence neq_thread: "thread \<noteq> th"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th \<in> threads (t@s)"
+ proof -
+ from Cons have "extend_highest_gen s th prio tm t" by auto
+ from extend_highest_gen.th_kept[OF this] show ?thesis by (simp)
+ qed
+ ultimately show ?thesis by auto
+ qed
+ from Cons have "extend_highest_gen s th prio tm t" by auto
+ from extend_highest_gen.th_kept[OF this]
+ have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
+ by (auto)
+ from stp
+ have thread_ts: "thread \<notin> threads (t @ s)"
+ by (cases, auto)
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+ by (unfold eq_e, simp)
+ moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+ proof(rule Max_insert)
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this]
+ show "finite (?f ` (threads (t@s)))" by simp
+ next
+ from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+ qed
+ moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+ proof -
+ have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
+ (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+ proof -
+ { fix th'
+ assume "th' \<in> ?B"
+ with thread_ts eq_e
+ have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+ } thus ?thesis
+ apply (auto simp:Image_def)
+ proof -
+ fix th'
+ assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
+ preced th' (e # t @ s) = preced th' (t @ s)"
+ and h1: "th' \<in> threads (t @ s)"
+ show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+ proof -
+ from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+ moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+ ultimately show ?thesis by simp
+ qed
+ qed
+ qed
+ with Cons show ?thesis by auto
+ qed
+ moreover have "?f thread < ?t"
+ proof -
+ from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+ from extend_highest_gen.create_low[OF this] and eq_e
+ have "prio' \<le> prio" by auto
+ thus ?thesis
+ by (unfold preced_th, unfold eq_e, insert lt_tm,
+ auto simp:preced_def precedence_less_def preced_th)
+ qed
+ ultimately show ?thesis by (auto simp:max_def)
+ qed
+next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have vt_e: "vt step (e#(t @ s))" by auto
+ from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+ from stp have thread_ts: "thread \<in> threads (t @ s)"
+ by(cases, unfold runing_def readys_def, auto)
+ from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+ from extend_highest_gen.exit_diff[OF this] and eq_e
+ have neq_thread: "thread \<noteq> th" by auto
+ from Cons have "extend_highest_gen s th prio tm t" by auto
+ from extend_highest_gen.th_kept[OF this]
+ have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "threads (t@s) = insert thread ?A"
+ by (insert stp thread_ts, unfold eq_e, auto)
+ hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+ also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+ also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ from finite_threads [OF vt_e]
+ show "finite (?f ` ?A)" by simp
+ next
+ from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+ from extend_highest_gen.th_kept[OF this]
+ show "?f ` ?A \<noteq> {}" by auto
+ qed
+ finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+ moreover have "Max (?f ` (threads (t@s))) = ?t"
+ proof -
+ from Cons show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ qed
+ ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+ moreover have "?f thread < ?t"
+ proof(unfold eq_e, simp add:preced_def, fold preced_def)
+ show "preced thread (t @ s) < ?t"
+ proof -
+ have "preced thread (t @ s) \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "?t = Max (?g ` ?B)") by simp
+ moreover have "?g thread \<le> \<dots>"
+ proof(rule Max_ge)
+ have "vt step (t@s)" by fact
+ from finite_threads [OF this]
+ show "finite (?g ` ?B)" by simp
+ next
+ from thread_ts
+ show "?g thread \<in> (?g ` ?B)" by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ moreover have "preced thread (t @ s) \<noteq> ?t"
+ proof
+ assume "preced thread (t @ s) = preced th s"
+ with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+ from preced_unique [OF this] have "thread = th"
+ proof
+ from h' show "th \<in> threads (t @ s)" by simp
+ next
+ from thread_ts show "thread \<in> threads (t @ s)" .
+ qed(simp)
+ with neq_thread show "False" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def split:if_splits)
+ qed
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ let ?B = "threads (t@s)"
+ from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
+ from extend_highest_gen.set_diff_low[OF this] and Set
+ have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+ from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+ also have "\<dots> = ?t"
+ proof(rule Max_eqI)
+ fix y
+ assume y_in: "y \<in> ?f ` ?B"
+ then obtain th1 where
+ th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+ show "y \<le> ?t"
+ proof(cases "th1 = thread")
+ case True
+ with neq_thread le_p eq_y Set
+ show ?thesis
+ apply (subst preced_th, insert lt_tm)
+ by (auto simp:preced_def precedence_le_def)
+ next
+ case False
+ with Set eq_y
+ have "y = preced th1 (t@s)"
+ by (simp add:preced_def)
+ moreover have "\<dots> \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+ by auto
+ moreover have "preced th1 (t@s) \<le> \<dots>"
+ proof(rule Max_ge)
+ from th1_in
+ show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+ by simp
+ next
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ proof -
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this] show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from Cons and finite_threads
+ show "finite (?f ` ?B)" by auto
+ next
+ from Cons have "extend_highest_gen s th prio tm t" by auto
+ from extend_highest_gen.th_kept [OF this]
+ have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show "?t \<in> (?f ` ?B)"
+ proof -
+ from neq_thread Set h
+ have "?t = ?f th" by (auto simp:preced_def)
+ with h show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ (is "?L = ?R")
+proof -
+ have "?L = cpreced (t@s) (wq (t@s)) th"
+ by (unfold cp_eq_cpreced, simp)
+ also have "\<dots> = ?R"
+ proof(unfold cpreced_def)
+ show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+ Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+ proof(cases "?A = {}")
+ case False
+ have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+ moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ show "finite (?f ` ?A)"
+ proof -
+ from dependents_threads[OF vt_t]
+ have "?A \<subseteq> threads (t@s)" .
+ moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+ ultimately show ?thesis
+ by (auto simp:finite_subset)
+ qed
+ next
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ qed
+ moreover have "\<dots> = Max (?f ` ?B)"
+ proof -
+ from max_preced have "?f th = Max (?f ` ?B)" .
+ moreover have "Max (?f ` ?A) \<le> \<dots>"
+ proof(rule Max_mono)
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ next
+ show "?f ` ?A \<subseteq> ?f ` ?B"
+ proof -
+ have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+ thus ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t]
+ show "finite (?f ` ?B)" by simp
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def)
+ qed
+ ultimately show ?thesis by auto
+ next
+ case True
+ with max_preced show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+ by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+ by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ shows "preced th' s < preced th s"
+proof -
+ have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+ proof(rule Max_ge)
+ from finite_threads [OF vt_s]
+ show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+ next
+ from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+ by simp
+ qed
+ moreover have "preced th' s \<noteq> preced th s"
+ proof
+ assume "preced th' s = preced th s"
+ from preced_unique[OF this th'_in] neq_th' threads_s
+ show "False" by (auto simp:readys_def)
+ qed
+ ultimately show ?thesis using highest_preced_thread
+ by auto
+qed
+
+lemma pv_blocked:
+ fixes th'
+ assumes th'_in: "th' \<in> threads (t@s)"
+ and neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+ shows "th' \<notin> runing (t@s)"
+proof
+ assume "th' \<in> runing (t@s)"
+ hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
+ by (auto simp:runing_def)
+ with max_cp_readys_threads [OF vt_t]
+ have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+ by auto
+ moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+ ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+ moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+ by simp
+ finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+ show False
+ proof -
+ have "dependents (wq (t @ s)) th' = {}"
+ by (rule count_eq_dependents [OF vt_t eq_pv])
+ moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+ proof
+ assume "preced th' (t @ s) = preced th (t @ s)"
+ hence "th' = th"
+ proof(rule preced_unique)
+ from th_kept show "th \<in> threads (t @ s)" by simp
+ next
+ from th'_in show "th' \<in> threads (t @ s)" by simp
+ qed
+ with assms show False by simp
+ qed
+ ultimately show ?thesis
+ by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+qed
+
+lemma runing_precond_pre:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads (t@s) \<and>
+ cntP (t@s) th' = cntV (t@s) th'"
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case (Cons e t)
+ from Cons
+ have in_thread: "th' \<in> threads (t @ s)"
+ and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from Cons have "extend_highest_gen s th prio tm t" by auto
+ from extend_highest_gen.pv_blocked
+ [OF this, OF in_thread neq_th' not_holding]
+ have not_runing: "th' \<notin> runing (t @ s)" .
+ show ?case
+ proof(cases e)
+ case (V thread cs)
+ from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+ show ?thesis
+ proof -
+ from Cons and V have "step (t@s) (V thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (unfold V, simp add:cntP_def cntV_def count_def)
+ moreover from in_thread
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (P thread cs)
+ from Cons and P have "step (t@s) (P thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Create thread prio')
+ from Cons and Create have "step (t@s) (Create thread prio')" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th' \<in> threads (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Create
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Create
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+ ultimately show ?thesis by auto
+ next
+ case (Exit thread)
+ from Cons and Exit have "step (t@s) (Exit thread)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t @ s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Exit
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Exit and neq_th'
+ have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Set thread prio')
+ with Cons
+ show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ next
+ case Nil
+ with assms
+ show ?case by auto
+ qed
+qed
+
+(*
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<notin> runing (t@s)"
+proof -
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2]
+ show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ and is_runing: "th' \<in> runing (t@s)"
+ shows "cntP s th' > cntV s th'"
+proof -
+ have "cntP s th' \<noteq> cntV s th'"
+ proof
+ assume eq_pv: "cntP s th' = cntV s th'"
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)"
+ and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+ with is_runing show "False" by simp
+ qed
+ moreover from cnp_cnv_cncs[OF vt_s, of th']
+ have "cntV s th' \<le> cntP s th'" by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+ th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+ case (Suc k)
+ show ?case
+ proof -
+ { assume True: "Suc (i+k) \<le> length t"
+ from moment_head [OF this]
+ obtain e where
+ eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+ by blast
+ from red_moment[of "Suc(i+k)"]
+ and eq_me have "extend_highest_gen s th prio tm (e # moment (i + k) t)" by simp
+ hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
+ highest_gen_def, auto)
+ have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
+ proof -
+ show "th' \<notin> runing (moment (i + k) t @ s)"
+ proof(rule extend_highest_gen.pv_blocked)
+ from Suc show "th' \<in> threads (moment (i + k) t @ s)"
+ by simp
+ next
+ from neq_th' show "th' \<noteq> th" .
+ next
+ from red_moment show "extend_highest_gen s th prio tm (moment (i + k) t)" .
+ next
+ from Suc show "cntP (moment (i + k) t @ s) th' = cntV (moment (i + k) t @ s) th'"
+ by (auto)
+ qed
+ qed
+ from step_back_step[OF vt_e]
+ have "step ((moment (i + k) t)@s) e" .
+ hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+ th' \<in> threads (e#(moment (i + k) t)@s)
+ "
+ proof(cases)
+ case (thread_create thread prio)
+ with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_exit thread)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_P thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_V thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_set thread prio')
+ with Suc show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ with eq_me have ?thesis using eq_me by auto
+ } note h = this
+ show ?thesis
+ proof(cases "Suc (i+k) \<le> length t")
+ case True
+ from h [OF this] show ?thesis .
+ next
+ case False
+ with moment_ge
+ have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+ with Suc show ?thesis by auto
+ qed
+ qed
+next
+ case 0
+ from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ and le_ij: "i \<le> j"
+ shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+ have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+ and h2: "th' \<in> threads ((moment j t)@s)" by auto
+ with extend_highest_gen.pv_blocked [OF red_moment [of j], OF h2 neq_th' h1]
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+ case True
+ with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+ case False
+ let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+ let ?q = "moment 0 t"
+ from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+ from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+ from p_split_gen [of ?Q, OF this not_thread]
+ obtain i where lt_its: "i < length t"
+ and le_i: "0 \<le> i"
+ and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+ and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+ from lt_its have "Suc i \<le> length t" by auto
+ from moment_head[OF this] obtain e where
+ eq_me: "moment (Suc i) t = e # moment i t" by blast
+ from red_moment[of "Suc i"] and eq_me
+ have "extend_highest_gen s th prio tm (e # moment i t)" by simp
+ hence vt_e: "vt step (e#(moment i t)@s)"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
+ highest_gen_def, auto)
+ from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+ from post[rule_format, of "Suc i"] and eq_me
+ have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+ from create_pre[OF stp_i pre this]
+ obtain prio where eq_e: "e = Create th' prio" .
+ have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+ proof(rule cnp_cnv_eq)
+ from step_back_vt [OF vt_e]
+ show "vt step (moment i t @ s)" .
+ next
+ from eq_e and stp_i
+ have "step (moment i t @ s) (Create th' prio)" by simp
+ thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+ qed
+ with eq_e
+ have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
+ by (simp add:cntP_def cntV_def count_def)
+ with eq_me[symmetric]
+ have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+ by simp
+ from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+ with eq_me [symmetric]
+ have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+ from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+ and moment_ge
+ have "th' \<notin> runing (t @ s)" by auto
+ with runing'
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+ from runing_inversion_1[OF _ runing']
+ show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+ case True thus ?thesis by auto
+next
+ case False
+ then have not_ready: "th \<notin> readys (t@s)"
+ apply (unfold runing_def,
+ insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+ by auto
+ from th_kept have "th \<in> threads (t@s)" by auto
+ from th_chain_to_ready[OF vt_t this] and not_ready
+ obtain th' where th'_in: "th' \<in> readys (t@s)"
+ and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+ have "th' \<in> runing (t@s)"
+ proof -
+ have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
+ preced th (t@s)"
+ proof(rule Max_eqI)
+ fix y
+ assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ then obtain th1 where
+ h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
+ and eq_y: "y = preced th1 (t@s)" by auto
+ show "y \<le> preced th (t @ s)"
+ proof -
+ from max_preced
+ have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+ moreover have "y \<le> \<dots>"
+ proof(rule Max_ge)
+ from h1
+ have "th1 \<in> threads (t@s)"
+ proof
+ assume "th1 = th'"
+ with th'_in show ?thesis by (simp add:readys_def)
+ next
+ assume "th1 \<in> dependents (wq (t @ s)) th'"
+ with dependents_threads [OF vt_t]
+ show "th1 \<in> threads (t @ s)" by auto
+ qed
+ with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+ next
+ from finite_threads[OF vt_t]
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+ show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+ by (auto intro:finite_subset)
+ next
+ from dp
+ have "th \<in> dependents (wq (t @ s)) th'"
+ by (unfold cs_dependents_def, auto simp:eq_depend)
+ thus "preced th (t @ s) \<in>
+ (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ by auto
+ qed
+ moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ from max_preced and max_cp_eq[OF vt_t, symmetric]
+ have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+ with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+ qed
+ ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+ with th'_in show ?thesis by (auto simp:runing_def)
+ qed
+ thus ?thesis by auto
+qed
+
+end
+end
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtGG_1.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,973 @@
+theory ExtGG
+imports PrioG
+begin
+
+lemma birth_time_lt: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+ apply (induct s, simp)
+proof -
+ fix a s
+ assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
+ and eq_as: "a # s \<noteq> []"
+ show "birthtime th (a # s) < length (a # s)"
+ proof(cases "s \<noteq> []")
+ case False
+ from False show ?thesis
+ by (cases a, auto simp:birthtime.simps)
+ next
+ case True
+ from ih [OF True] show ?thesis
+ by (cases a, auto simp:birthtime.simps)
+ qed
+qed
+
+lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
+ by (induct s, auto simp:threads.simps)
+
+lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
+ apply (drule_tac th_in_ne)
+ by (unfold preced_def, auto intro: birth_time_lt)
+
+locale highest_gen =
+ fixes s' th s e' prio tm
+ defines s_def : "s \<equiv> (e'#s')"
+ assumes vt_s: "vt step s"
+ and threads_s: "th \<in> threads s"
+ and highest: "preced th s = Max ((cp s)`threads s)"
+ and nh: "preced th s' \<noteq> Max ((cp s)`threads s')"
+ and preced_th: "preced th s = Prc prio tm"
+
+context highest_gen
+begin
+
+lemma lt_tm: "tm < length s"
+ by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+lemma vt_s': "vt step s'"
+ by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+ from highest and max_cp_eq[OF vt_s]
+ have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+ have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+ proof -
+ from threads_s and dependents_threads[OF vt_s, of th]
+ show ?thesis by auto
+ qed
+ show ?thesis
+ proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+ show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+ next
+ fix y
+ assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+ then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+ and eq_y: "y = preced th1 s" by auto
+ show "y \<le> preced th s"
+ proof(unfold is_max, rule Max_ge)
+ from finite_threads[OF vt_s]
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ next
+ from sbs th1_in and eq_y
+ show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+ qed
+ next
+ from sbs and finite_threads[OF vt_s]
+ show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+ by (auto intro:finite_subset)
+ qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+ from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+ show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen +
+ fixes t
+ assumes vt_t: "vt step (t@s)"
+ and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+ and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+ and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app:
+ assumes vt_ts: "vt cs (t@s)"
+ shows "vt cs s"
+proof -
+ from vt_ts show ?thesis
+ proof(induct t)
+ case Nil
+ from Nil show ?case by auto
+ next
+ case (Cons e t)
+ assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+ and vt_et: "vt cs ((e # t) @ s)"
+ show ?case
+ proof(rule ih)
+ show "vt cs (t @ s)"
+ proof(rule step_back_vt)
+ from vt_et show "vt cs (e # t @ s)" by simp
+ qed
+ qed
+ qed
+qed
+
+context extend_highest_gen
+begin
+
+lemma red_moment:
+ "extend_highest_gen s' th e' prio tm (moment i t)"
+ apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+ apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+ by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes
+ h0: "R []"
+ and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
+ extend_highest_gen s' th e' prio tm t;
+ extend_highest_gen s' th e' prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+ shows "R t"
+proof -
+ from vt_t extend_highest_gen_axioms show ?thesis
+ proof(induct t)
+ from h0 show "R []" .
+ next
+ case (Cons e t')
+ assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_gen s' th e' prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+ and vt_e: "vt step ((e # t') @ s)"
+ and et: "extend_highest_gen s' th e' prio tm (e # t')"
+ from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+ from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+ show ?case
+ proof(rule h2 [OF vt_ts stp _ _ _ ])
+ show "R t'"
+ proof(rule ih)
+ from et show ext': "extend_highest_gen s' th e' prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ next
+ from vt_ts show "vt step (t' @ s)" .
+ qed
+ next
+ from et show "extend_highest_gen s' th e' prio tm (e # t')" .
+ next
+ from et show ext': "extend_highest_gen s' th e' prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ qed
+ qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and>
+ preced th (t@s) = preced th s" (is "?Q t")
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case Nil
+ from threads_s
+ show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+ by auto
+ next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio)
+ assume eq_e: " e = Create thread prio"
+ show ?thesis
+ proof -
+ from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+ hence "th \<noteq> thread"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ with Cons show ?thesis by auto
+ qed
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+ from extend_highest_gen.exit_diff [OF this] and eq_e
+ have neq_th: "thread \<noteq> th" by auto
+ with Cons
+ show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ next
+ case (P thread cs)
+ assume eq_e: "e = P thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (V thread cs)
+ assume eq_e: "e = V thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (Set thread prio')
+ assume eq_e: " e = Set thread prio'"
+ show ?thesis
+ proof -
+ from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+ from extend_highest_gen.set_diff_low[OF this] and eq_e
+ have "th \<noteq> thread" by auto
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ qed
+ qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+ case Nil
+ from highest_preced_thread
+ show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+ by simp
+next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio')
+ assume eq_e: " e = Create thread prio'"
+ from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+ hence neq_thread: "thread \<noteq> th"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th \<in> threads (t@s)"
+ proof -
+ from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+ from extend_highest_gen.th_kept[OF this] show ?thesis by (simp add:s_def)
+ qed
+ ultimately show ?thesis by auto
+ qed
+ from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+ from extend_highest_gen.th_kept[OF this]
+ have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
+ by (auto simp:s_def)
+ from stp
+ have thread_ts: "thread \<notin> threads (t @ s)"
+ by (cases, auto)
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+ by (unfold eq_e, simp)
+ moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+ proof(rule Max_insert)
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this]
+ show "finite (?f ` (threads (t@s)))" by simp
+ next
+ from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+ qed
+ moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+ proof -
+ have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
+ (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+ proof -
+ { fix th'
+ assume "th' \<in> ?B"
+ with thread_ts eq_e
+ have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+ } thus ?thesis
+ apply (auto simp:Image_def)
+ proof -
+ fix th'
+ assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
+ preced th' (e # t @ s) = preced th' (t @ s)"
+ and h1: "th' \<in> threads (t @ s)"
+ show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+ proof -
+ from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+ moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+ ultimately show ?thesis by simp
+ qed
+ qed
+ qed
+ with Cons show ?thesis by auto
+ qed
+ moreover have "?f thread < ?t"
+ proof -
+ from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+ from extend_highest_gen.create_low[OF this] and eq_e
+ have "prio' \<le> prio" by auto
+ thus ?thesis
+ by (unfold preced_th, unfold eq_e, insert lt_tm,
+ auto simp:preced_def s_def precedence_less_def preced_th)
+ qed
+ ultimately show ?thesis by (auto simp:max_def)
+ qed
+next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have vt_e: "vt step (e#(t @ s))" by auto
+ from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+ from stp have thread_ts: "thread \<in> threads (t @ s)"
+ by(cases, unfold runing_def readys_def, auto)
+ from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+ from extend_highest_gen.exit_diff[OF this] and eq_e
+ have neq_thread: "thread \<noteq> th" by auto
+ from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+ from extend_highest_gen.th_kept[OF this, folded s_def]
+ have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "threads (t@s) = insert thread ?A"
+ by (insert stp thread_ts, unfold eq_e, auto)
+ hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+ also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+ also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ from finite_threads [OF vt_e]
+ show "finite (?f ` ?A)" by simp
+ next
+ from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+ from extend_highest_gen.th_kept[OF this]
+ show "?f ` ?A \<noteq> {}" by (auto simp:s_def)
+ qed
+ finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+ moreover have "Max (?f ` (threads (t@s))) = ?t"
+ proof -
+ from Cons show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ qed
+ ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+ moreover have "?f thread < ?t"
+ proof(unfold eq_e, simp add:preced_def, fold preced_def)
+ show "preced thread (t @ s) < ?t"
+ proof -
+ have "preced thread (t @ s) \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "?t = Max (?g ` ?B)") by simp
+ moreover have "?g thread \<le> \<dots>"
+ proof(rule Max_ge)
+ have "vt step (t@s)" by fact
+ from finite_threads [OF this]
+ show "finite (?g ` ?B)" by simp
+ next
+ from thread_ts
+ show "?g thread \<in> (?g ` ?B)" by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ moreover have "preced thread (t @ s) \<noteq> ?t"
+ proof
+ assume "preced thread (t @ s) = preced th s"
+ with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+ from preced_unique [OF this] have "thread = th"
+ proof
+ from h' show "th \<in> threads (t @ s)" by simp
+ next
+ from thread_ts show "thread \<in> threads (t @ s)" .
+ qed(simp)
+ with neq_thread show "False" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def split:if_splits)
+ qed
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ let ?B = "threads (t@s)"
+ from Cons have "extend_highest_gen s' th e' prio tm (e # t)" by auto
+ from extend_highest_gen.set_diff_low[OF this] and Set
+ have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+ from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+ also have "\<dots> = ?t"
+ proof(rule Max_eqI)
+ fix y
+ assume y_in: "y \<in> ?f ` ?B"
+ then obtain th1 where
+ th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+ show "y \<le> ?t"
+ proof(cases "th1 = thread")
+ case True
+ with neq_thread le_p eq_y s_def Set
+ show ?thesis
+ apply (subst preced_th, insert lt_tm)
+ by (auto simp:preced_def precedence_le_def)
+ next
+ case False
+ with Set eq_y
+ have "y = preced th1 (t@s)"
+ by (simp add:preced_def)
+ moreover have "\<dots> \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+ by auto
+ moreover have "preced th1 (t@s) \<le> \<dots>"
+ proof(rule Max_ge)
+ from th1_in
+ show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+ by simp
+ next
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ proof -
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this] show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from Cons and finite_threads
+ show "finite (?f ` ?B)" by auto
+ next
+ from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+ from extend_highest_gen.th_kept [OF this, folded s_def]
+ have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show "?t \<in> (?f ` ?B)"
+ proof -
+ from neq_thread Set h
+ have "?t = ?f th" by (auto simp:preced_def)
+ with h show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ (is "?L = ?R")
+proof -
+ have "?L = cpreced (t@s) (wq (t@s)) th"
+ by (unfold cp_eq_cpreced, simp)
+ also have "\<dots> = ?R"
+ proof(unfold cpreced_def)
+ show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+ Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+ proof(cases "?A = {}")
+ case False
+ have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+ moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ show "finite (?f ` ?A)"
+ proof -
+ from dependents_threads[OF vt_t]
+ have "?A \<subseteq> threads (t@s)" .
+ moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+ ultimately show ?thesis
+ by (auto simp:finite_subset)
+ qed
+ next
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ qed
+ moreover have "\<dots> = Max (?f ` ?B)"
+ proof -
+ from max_preced have "?f th = Max (?f ` ?B)" .
+ moreover have "Max (?f ` ?A) \<le> \<dots>"
+ proof(rule Max_mono)
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ next
+ show "?f ` ?A \<subseteq> ?f ` ?B"
+ proof -
+ have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+ thus ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t]
+ show "finite (?f ` ?B)" by simp
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def)
+ qed
+ ultimately show ?thesis by auto
+ next
+ case True
+ with max_preced show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+ by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+ by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ shows "preced th' s < preced th s"
+proof -
+ have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+ proof(rule Max_ge)
+ from finite_threads [OF vt_s]
+ show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+ next
+ from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+ by simp
+ qed
+ moreover have "preced th' s \<noteq> preced th s"
+ proof
+ assume "preced th' s = preced th s"
+ from preced_unique[OF this th'_in] neq_th' threads_s
+ show "False" by (auto simp:readys_def)
+ qed
+ ultimately show ?thesis using highest_preced_thread
+ by auto
+qed
+
+lemma pv_blocked:
+ fixes th'
+ assumes th'_in: "th' \<in> threads (t@s)"
+ and neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+ shows "th' \<notin> runing (t@s)"
+proof
+ assume "th' \<in> runing (t@s)"
+ hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
+ by (auto simp:runing_def)
+ with max_cp_readys_threads [OF vt_t]
+ have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+ by auto
+ moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+ ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+ moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+ by simp
+ finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+ show False
+ proof -
+ have "dependents (wq (t @ s)) th' = {}"
+ by (rule count_eq_dependents [OF vt_t eq_pv])
+ moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+ proof
+ assume "preced th' (t @ s) = preced th (t @ s)"
+ hence "th' = th"
+ proof(rule preced_unique)
+ from th_kept show "th \<in> threads (t @ s)" by simp
+ next
+ from th'_in show "th' \<in> threads (t @ s)" by simp
+ qed
+ with assms show False by simp
+ qed
+ ultimately show ?thesis
+ by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+qed
+
+lemma runing_precond_pre:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads (t@s) \<and>
+ cntP (t@s) th' = cntV (t@s) th'"
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case (Cons e t)
+ from Cons
+ have in_thread: "th' \<in> threads (t @ s)"
+ and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from Cons have "extend_highest_gen s' th e' prio tm t" by auto
+ from extend_highest_gen.pv_blocked
+ [OF this, folded s_def, OF in_thread neq_th' not_holding]
+ have not_runing: "th' \<notin> runing (t @ s)" .
+ show ?case
+ proof(cases e)
+ case (V thread cs)
+ from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+ show ?thesis
+ proof -
+ from Cons and V have "step (t@s) (V thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (unfold V, simp add:cntP_def cntV_def count_def)
+ moreover from in_thread
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (P thread cs)
+ from Cons and P have "step (t@s) (P thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Create thread prio')
+ from Cons and Create have "step (t@s) (Create thread prio')" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th' \<in> threads (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Create
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Create
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+ ultimately show ?thesis by auto
+ next
+ case (Exit thread)
+ from Cons and Exit have "step (t@s) (Exit thread)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t @ s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Exit
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Exit and neq_th'
+ have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Set thread prio')
+ with Cons
+ show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ next
+ case Nil
+ with assms
+ show ?case by auto
+ qed
+qed
+
+(*
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<notin> runing (t@s)"
+proof -
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2]
+ show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ and is_runing: "th' \<in> runing (t@s)"
+ shows "cntP s th' > cntV s th'"
+proof -
+ have "cntP s th' \<noteq> cntV s th'"
+ proof
+ assume eq_pv: "cntP s th' = cntV s th'"
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)"
+ and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+ with is_runing show "False" by simp
+ qed
+ moreover from cnp_cnv_cncs[OF vt_s, of th']
+ have "cntV s th' \<le> cntP s th'" by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+ th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+ case (Suc k)
+ show ?case
+ proof -
+ { assume True: "Suc (i+k) \<le> length t"
+ from moment_head [OF this]
+ obtain e where
+ eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+ by blast
+ from red_moment[of "Suc(i+k)"]
+ and eq_me have "extend_highest_gen s' th e' prio tm (e # moment (i + k) t)" by simp
+ hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
+ highest_gen_def s_def, auto)
+ have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
+ proof(unfold s_def)
+ show "th' \<notin> runing (moment (i + k) t @ e' # s')"
+ proof(rule extend_highest_gen.pv_blocked)
+ from Suc show "th' \<in> threads (moment (i + k) t @ e' # s')"
+ by (simp add:s_def)
+ next
+ from neq_th' show "th' \<noteq> th" .
+ next
+ from red_moment show "extend_highest_gen s' th e' prio tm (moment (i + k) t)" .
+ next
+ from Suc show "cntP (moment (i + k) t @ e' # s') th' = cntV (moment (i + k) t @ e' # s') th'"
+ by (auto simp:s_def)
+ qed
+ qed
+ from step_back_step[OF vt_e]
+ have "step ((moment (i + k) t)@s) e" .
+ hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+ th' \<in> threads (e#(moment (i + k) t)@s)
+ "
+ proof(cases)
+ case (thread_create thread prio)
+ with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_exit thread)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_P thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_V thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_set thread prio')
+ with Suc show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ with eq_me have ?thesis using eq_me by auto
+ } note h = this
+ show ?thesis
+ proof(cases "Suc (i+k) \<le> length t")
+ case True
+ from h [OF this] show ?thesis .
+ next
+ case False
+ with moment_ge
+ have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+ with Suc show ?thesis by auto
+ qed
+ qed
+next
+ case 0
+ from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ and le_ij: "i \<le> j"
+ shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+ have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+ and h2: "th' \<in> threads ((moment j t)@s)" by auto
+ with extend_highest_gen.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1]
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+ case True
+ with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+ case False
+ let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+ let ?q = "moment 0 t"
+ from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+ from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+ from p_split_gen [of ?Q, OF this not_thread]
+ obtain i where lt_its: "i < length t"
+ and le_i: "0 \<le> i"
+ and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+ and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+ from lt_its have "Suc i \<le> length t" by auto
+ from moment_head[OF this] obtain e where
+ eq_me: "moment (Suc i) t = e # moment i t" by blast
+ from red_moment[of "Suc i"] and eq_me
+ have "extend_highest_gen s' th e' prio tm (e # moment i t)" by simp
+ hence vt_e: "vt step (e#(moment i t)@s)"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
+ highest_gen_def s_def, auto)
+ from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+ from post[rule_format, of "Suc i"] and eq_me
+ have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+ from create_pre[OF stp_i pre this]
+ obtain prio where eq_e: "e = Create th' prio" .
+ have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+ proof(rule cnp_cnv_eq)
+ from step_back_vt [OF vt_e]
+ show "vt step (moment i t @ s)" .
+ next
+ from eq_e and stp_i
+ have "step (moment i t @ s) (Create th' prio)" by simp
+ thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+ qed
+ with eq_e
+ have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
+ by (simp add:cntP_def cntV_def count_def)
+ with eq_me[symmetric]
+ have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+ by simp
+ from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+ with eq_me [symmetric]
+ have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+ from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+ and moment_ge
+ have "th' \<notin> runing (t @ s)" by auto
+ with runing'
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+ from runing_inversion_1[OF _ runing']
+ show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+ case True thus ?thesis by auto
+next
+ case False
+ then have not_ready: "th \<notin> readys (t@s)"
+ apply (unfold runing_def,
+ insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+ by auto
+ from th_kept have "th \<in> threads (t@s)" by auto
+ from th_chain_to_ready[OF vt_t this] and not_ready
+ obtain th' where th'_in: "th' \<in> readys (t@s)"
+ and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+ have "th' \<in> runing (t@s)"
+ proof -
+ have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
+ preced th (t@s)"
+ proof(rule Max_eqI)
+ fix y
+ assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ then obtain th1 where
+ h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
+ and eq_y: "y = preced th1 (t@s)" by auto
+ show "y \<le> preced th (t @ s)"
+ proof -
+ from max_preced
+ have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+ moreover have "y \<le> \<dots>"
+ proof(rule Max_ge)
+ from h1
+ have "th1 \<in> threads (t@s)"
+ proof
+ assume "th1 = th'"
+ with th'_in show ?thesis by (simp add:readys_def)
+ next
+ assume "th1 \<in> dependents (wq (t @ s)) th'"
+ with dependents_threads [OF vt_t]
+ show "th1 \<in> threads (t @ s)" by auto
+ qed
+ with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+ next
+ from finite_threads[OF vt_t]
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+ show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+ by (auto intro:finite_subset)
+ next
+ from dp
+ have "th \<in> dependents (wq (t @ s)) th'"
+ by (unfold cs_dependents_def, auto simp:eq_depend)
+ thus "preced th (t @ s) \<in>
+ (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ by auto
+ qed
+ moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ from max_preced and max_cp_eq[OF vt_t, symmetric]
+ have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+ with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+ qed
+ ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+ with th'_in show ?thesis by (auto simp:runing_def)
+ qed
+ thus ?thesis by auto
+qed
+
+end
+
+end
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtS.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1019 @@
+theory ExtS
+imports Prio
+begin
+
+locale highest_set =
+ fixes s' th prio fixes s
+ defines s_def : "s \<equiv> (Set th prio#s')"
+ assumes vt_s: "vt step s"
+ and highest: "preced th s = Max ((cp s)`threads s)"
+
+context highest_set
+begin
+
+
+lemma vt_s': "vt step s'"
+ by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_set: "step s' (Set th prio)"
+ by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_set_elim:
+ "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+ by (insert step_set, ind_cases "step s' (Set th prio)", auto)
+
+
+lemma threads_s: "th \<in> threads s"
+ by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
+
+lemma same_depend: "depend s = depend s'"
+ by (insert depend_set_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+ "dependents (wq s) th = dependents (wq s') th"
+ apply (unfold cs_dependents_def)
+ by (unfold eq_depend same_depend, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+ from highest and max_cp_eq[OF vt_s]
+ have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+ have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+ proof -
+ from threads_s and dependents_threads[OF vt_s, of th]
+ show ?thesis by auto
+ qed
+ show ?thesis
+ proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+ show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+ next
+ fix y
+ assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+ then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+ and eq_y: "y = preced th1 s" by auto
+ show "y \<le> preced th s"
+ proof(unfold is_max, rule Max_ge)
+ from finite_threads[OF vt_s]
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ next
+ from sbs th1_in and eq_y
+ show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+ qed
+ next
+ from sbs and finite_threads[OF vt_s]
+ show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+ by (auto intro:finite_subset)
+ qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+ have "\<forall>cs. \<not> waiting s th cs"
+ apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
+ apply (unfold s_depend_def, unfold runing_def readys_def)
+ apply (auto, fold s_def)
+ apply (erule_tac x = cs in allE, auto simp:waiting_eq)
+ proof -
+ fix cs
+ assume h:
+ "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
+ {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
+ (is "?L = ?R")
+ and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
+ from wt have "(Th th, Cs cs) \<in> ?L" by auto
+ with h have "(Th th, Cs cs) \<in> ?R" by simp
+ hence "waiting (wq s') th cs" by auto with nwt
+ show False by auto
+ qed
+ with threads_s show ?thesis
+ by (unfold readys_def, auto)
+qed
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+ from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+ show ?thesis by simp
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+ have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+ proof -
+ have " Max (cp s ` readys s) = cp s th"
+ proof(rule Max_eqI)
+ from finite_threads[OF vt_s] readys_threads finite_subset
+ have "finite (readys s)" by blast
+ thus "finite (cp s ` readys s)" by auto
+ next
+ from is_ready show "cp s th \<in> cp s ` readys s" by auto
+ next
+ fix y
+ assume "y \<in> cp s ` readys s"
+ then obtain th1 where
+ eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
+ show "y \<le> cp s th"
+ proof -
+ have "y \<le> Max (cp s ` threads s)"
+ proof(rule Max_ge)
+ from eq_y and th1_in
+ show "y \<in> cp s ` threads s"
+ by (auto simp:readys_def)
+ next
+ from finite_threads[OF vt_s]
+ show "finite (cp s ` threads s)" by auto
+ qed
+ with highest' show ?thesis by auto
+ qed
+ qed
+ with highest' show ?thesis by auto
+ qed
+ thus ?thesis
+ by (unfold runing_def, insert highest' is_ready, auto)
+qed
+
+end
+
+locale extend_highest_set = highest_set +
+ fixes t
+ assumes vt_t: "vt step (t@s)"
+ and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+ and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+ and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app:
+ assumes vt_ts: "vt cs (t@s)"
+ shows "vt cs s"
+proof -
+ from vt_ts show ?thesis
+ proof(induct t)
+ case Nil
+ from Nil show ?case by auto
+ next
+ case (Cons e t)
+ assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+ and vt_et: "vt cs ((e # t) @ s)"
+ show ?case
+ proof(rule ih)
+ show "vt cs (t @ s)"
+ proof(rule step_back_vt)
+ from vt_et show "vt cs (e # t @ s)" by simp
+ qed
+ qed
+ qed
+qed
+
+context extend_highest_set
+begin
+
+lemma red_moment:
+ "extend_highest_set s' th prio (moment i t)"
+ apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+ apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
+ by (unfold highest_set_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes
+ h0: "R []"
+ and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
+ extend_highest_set s' th prio t;
+ extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+ shows "R t"
+proof -
+ from vt_t extend_highest_set_axioms show ?thesis
+ proof(induct t)
+ from h0 show "R []" .
+ next
+ case (Cons e t')
+ assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+ and vt_e: "vt step ((e # t') @ s)"
+ and et: "extend_highest_set s' th prio (e # t')"
+ from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+ from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+ show ?case
+ proof(rule h2 [OF vt_ts stp _ _ _ ])
+ show "R t'"
+ proof(rule ih)
+ from et show ext': "extend_highest_set s' th prio t'"
+ by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+ next
+ from vt_ts show "vt step (t' @ s)" .
+ qed
+ next
+ from et show "extend_highest_set s' th prio (e # t')" .
+ next
+ from et show ext': "extend_highest_set s' th prio t'"
+ by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+ qed
+ qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and>
+ preced th (t@s) = preced th s" (is "?Q t")
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case Nil
+ from threads_s
+ show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+ by auto
+ next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio)
+ assume eq_e: " e = Create thread prio"
+ show ?thesis
+ proof -
+ from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+ hence "th \<noteq> thread"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ with Cons show ?thesis by auto
+ qed
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.exit_diff [OF this] and eq_e
+ have neq_th: "thread \<noteq> th" by auto
+ with Cons
+ show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ next
+ case (P thread cs)
+ assume eq_e: "e = P thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (V thread cs)
+ assume eq_e: "e = V thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (Set thread prio')
+ assume eq_e: " e = Set thread prio'"
+ show ?thesis
+ proof -
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.set_diff_low[OF this] and eq_e
+ have "th \<noteq> thread" by auto
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ qed
+ qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+ case Nil
+ from highest_preced_thread
+ show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+ by simp
+next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio')
+ assume eq_e: " e = Create thread prio'"
+ from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+ hence neq_thread: "thread \<noteq> th"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th \<in> threads (t@s)"
+ proof -
+ from Cons have "extend_highest_set s' th prio t" by auto
+ from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
+ qed
+ ultimately show ?thesis by auto
+ qed
+ from Cons have "extend_highest_set s' th prio t" by auto
+ from extend_highest_set.th_kept[OF this]
+ have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
+ by (auto simp:s_def)
+ from stp
+ have thread_ts: "thread \<notin> threads (t @ s)"
+ by (cases, auto)
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+ by (unfold eq_e, simp)
+ moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+ proof(rule Max_insert)
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this]
+ show "finite (?f ` (threads (t@s)))" by simp
+ next
+ from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+ qed
+ moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+ proof -
+ have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
+ (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+ proof -
+ { fix th'
+ assume "th' \<in> ?B"
+ with thread_ts eq_e
+ have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+ } thus ?thesis
+ apply (auto simp:Image_def)
+ proof -
+ fix th'
+ assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
+ preced th' (e # t @ s) = preced th' (t @ s)"
+ and h1: "th' \<in> threads (t @ s)"
+ show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+ proof -
+ from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+ moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+ ultimately show ?thesis by simp
+ qed
+ qed
+ qed
+ with Cons show ?thesis by auto
+ qed
+ moreover have "?f thread < ?t"
+ proof -
+ from Cons have " extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.create_low[OF this] and eq_e
+ have "prio' \<le> prio" by auto
+ thus ?thesis
+ by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+ qed
+ ultimately show ?thesis by (auto simp:max_def)
+ qed
+next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have vt_e: "vt step (e#(t @ s))" by auto
+ from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+ from stp have thread_ts: "thread \<in> threads (t @ s)"
+ by(cases, unfold runing_def readys_def, auto)
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.exit_diff[OF this] and eq_e
+ have neq_thread: "thread \<noteq> th" by auto
+ from Cons have "extend_highest_set s' th prio t" by auto
+ from extend_highest_set.th_kept[OF this, folded s_def]
+ have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "threads (t@s) = insert thread ?A"
+ by (insert stp thread_ts, unfold eq_e, auto)
+ hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+ also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+ also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ from finite_threads [OF vt_e]
+ show "finite (?f ` ?A)" by simp
+ next
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.th_kept[OF this]
+ show "?f ` ?A \<noteq> {}" by (auto simp:s_def)
+ qed
+ finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+ moreover have "Max (?f ` (threads (t@s))) = ?t"
+ proof -
+ from Cons show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ qed
+ ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+ moreover have "?f thread < ?t"
+ proof(unfold eq_e, simp add:preced_def, fold preced_def)
+ show "preced thread (t @ s) < ?t"
+ proof -
+ have "preced thread (t @ s) \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "?t = Max (?g ` ?B)") by simp
+ moreover have "?g thread \<le> \<dots>"
+ proof(rule Max_ge)
+ have "vt step (t@s)" by fact
+ from finite_threads [OF this]
+ show "finite (?g ` ?B)" by simp
+ next
+ from thread_ts
+ show "?g thread \<in> (?g ` ?B)" by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ moreover have "preced thread (t @ s) \<noteq> ?t"
+ proof
+ assume "preced thread (t @ s) = preced th s"
+ with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+ from preced_unique [OF this] have "thread = th"
+ proof
+ from h' show "th \<in> threads (t @ s)" by simp
+ next
+ from thread_ts show "thread \<in> threads (t @ s)" .
+ qed(simp)
+ with neq_thread show "False" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def split:if_splits)
+ qed
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ let ?B = "threads (t@s)"
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.set_diff_low[OF this] and Set
+ have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+ from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+ also have "\<dots> = ?t"
+ proof(rule Max_eqI)
+ fix y
+ assume y_in: "y \<in> ?f ` ?B"
+ then obtain th1 where
+ th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+ show "y \<le> ?t"
+ proof(cases "th1 = thread")
+ case True
+ with neq_thread le_p eq_y s_def Set
+ show ?thesis
+ by (auto simp:preced_def precedence_le_def)
+ next
+ case False
+ with Set eq_y
+ have "y = preced th1 (t@s)"
+ by (simp add:preced_def)
+ moreover have "\<dots> \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+ by auto
+ moreover have "preced th1 (t@s) \<le> \<dots>"
+ proof(rule Max_ge)
+ from th1_in
+ show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+ by simp
+ next
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ proof -
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this] show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from Cons and finite_threads
+ show "finite (?f ` ?B)" by auto
+ next
+ from Cons have "extend_highest_set s' th prio t" by auto
+ from extend_highest_set.th_kept [OF this, folded s_def]
+ have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show "?t \<in> (?f ` ?B)"
+ proof -
+ from neq_thread Set h
+ have "?t = ?f th" by (auto simp:preced_def)
+ with h show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ (is "?L = ?R")
+proof -
+ have "?L = cpreced (t@s) (wq (t@s)) th"
+ by (unfold cp_eq_cpreced, simp)
+ also have "\<dots> = ?R"
+ proof(unfold cpreced_def)
+ show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+ Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+ proof(cases "?A = {}")
+ case False
+ have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+ moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ show "finite (?f ` ?A)"
+ proof -
+ from dependents_threads[OF vt_t]
+ have "?A \<subseteq> threads (t@s)" .
+ moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+ ultimately show ?thesis
+ by (auto simp:finite_subset)
+ qed
+ next
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ qed
+ moreover have "\<dots> = Max (?f ` ?B)"
+ proof -
+ from max_preced have "?f th = Max (?f ` ?B)" .
+ moreover have "Max (?f ` ?A) \<le> \<dots>"
+ proof(rule Max_mono)
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ next
+ show "?f ` ?A \<subseteq> ?f ` ?B"
+ proof -
+ have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+ thus ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t]
+ show "finite (?f ` ?B)" by simp
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def)
+ qed
+ ultimately show ?thesis by auto
+ next
+ case True
+ with max_preced show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+ by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+ by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ shows "preced th' s < preced th s"
+proof -
+ have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+ proof(rule Max_ge)
+ from finite_threads [OF vt_s]
+ show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+ next
+ from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+ by simp
+ qed
+ moreover have "preced th' s \<noteq> preced th s"
+ proof
+ assume "preced th' s = preced th s"
+ from preced_unique[OF this th'_in] neq_th' is_ready
+ show "False" by (auto simp:readys_def)
+ qed
+ ultimately show ?thesis using highest_preced_thread
+ by auto
+qed
+
+lemma pv_blocked:
+ fixes th'
+ assumes th'_in: "th' \<in> threads (t@s)"
+ and neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+ shows "th' \<notin> runing (t@s)"
+proof
+ assume "th' \<in> runing (t@s)"
+ hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
+ by (auto simp:runing_def)
+ with max_cp_readys_threads [OF vt_t]
+ have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+ by auto
+ moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+ ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+ moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+ by simp
+ finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+ show False
+ proof -
+ have "dependents (wq (t @ s)) th' = {}"
+ by (rule count_eq_dependents [OF vt_t eq_pv])
+ moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+ proof
+ assume "preced th' (t @ s) = preced th (t @ s)"
+ hence "th' = th"
+ proof(rule preced_unique)
+ from th_kept show "th \<in> threads (t @ s)" by simp
+ next
+ from th'_in show "th' \<in> threads (t @ s)" by simp
+ qed
+ with assms show False by simp
+ qed
+ ultimately show ?thesis
+ by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+qed
+
+lemma runing_precond_pre:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads (t@s) \<and>
+ cntP (t@s) th' = cntV (t@s) th'"
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case (Cons e t)
+ from Cons
+ have in_thread: "th' \<in> threads (t @ s)"
+ and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ have "extend_highest_set s' th prio t" by fact
+ from extend_highest_set.pv_blocked
+ [OF this, folded s_def, OF in_thread neq_th' not_holding]
+ have not_runing: "th' \<notin> runing (t @ s)" .
+ show ?case
+ proof(cases e)
+ case (V thread cs)
+ from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+ show ?thesis
+ proof -
+ from Cons and V have "step (t@s) (V thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (unfold V, simp add:cntP_def cntV_def count_def)
+ moreover from in_thread
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (P thread cs)
+ from Cons and P have "step (t@s) (P thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Create thread prio')
+ from Cons and Create have "step (t@s) (Create thread prio')" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th' \<in> threads (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Create
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Create
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+ ultimately show ?thesis by auto
+ next
+ case (Exit thread)
+ from Cons and Exit have "step (t@s) (Exit thread)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t @ s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Exit
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Exit and neq_th'
+ have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Set thread prio')
+ with Cons
+ show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ next
+ case Nil
+ with assms
+ show ?case by auto
+ qed
+qed
+
+(*
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<notin> runing (t@s)"
+proof -
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2]
+ show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ and is_runing: "th' \<in> runing (t@s)"
+ shows "cntP s th' > cntV s th'"
+proof -
+ have "cntP s th' \<noteq> cntV s th'"
+ proof
+ assume eq_pv: "cntP s th' = cntV s th'"
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)"
+ and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+ with is_runing show "False" by simp
+ qed
+ moreover from cnp_cnv_cncs[OF vt_s, of th']
+ have "cntV s th' \<le> cntP s th'" by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+ th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+ case (Suc k)
+ show ?case
+ proof -
+ { assume True: "Suc (i+k) \<le> length t"
+ from moment_head [OF this]
+ obtain e where
+ eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+ by blast
+ from red_moment[of "Suc(i+k)"]
+ and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
+ hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+ by (unfold extend_highest_set_def extend_highest_set_axioms_def
+ highest_set_def s_def, auto)
+ have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
+ proof(unfold s_def)
+ show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
+ proof(rule extend_highest_set.pv_blocked)
+ from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
+ by (simp add:s_def)
+ next
+ from neq_th' show "th' \<noteq> th" .
+ next
+ from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
+ next
+ from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
+ cntV (moment (i + k) t @ Set th prio # s') th'"
+ by (auto simp:s_def)
+ qed
+ qed
+ from step_back_step[OF vt_e]
+ have "step ((moment (i + k) t)@s) e" .
+ hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+ th' \<in> threads (e#(moment (i + k) t)@s)
+ "
+ proof(cases)
+ case (thread_create thread prio)
+ with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_exit thread)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_P thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_V thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_set thread prio')
+ with Suc show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ with eq_me have ?thesis using eq_me by auto
+ } note h = this
+ show ?thesis
+ proof(cases "Suc (i+k) \<le> length t")
+ case True
+ from h [OF this] show ?thesis .
+ next
+ case False
+ with moment_ge
+ have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+ with Suc show ?thesis by auto
+ qed
+ qed
+next
+ case 0
+ from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ and le_ij: "i \<le> j"
+ shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+ have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+ and h2: "th' \<in> threads ((moment j t)@s)" by auto
+ with extend_highest_set.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1]
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+ case True
+ with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+ case False
+ let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+ let ?q = "moment 0 t"
+ from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+ from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+ from p_split_gen [of ?Q, OF this not_thread]
+ obtain i where lt_its: "i < length t"
+ and le_i: "0 \<le> i"
+ and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+ and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+ from lt_its have "Suc i \<le> length t" by auto
+ from moment_head[OF this] obtain e where
+ eq_me: "moment (Suc i) t = e # moment i t" by blast
+ from red_moment[of "Suc i"] and eq_me
+ have "extend_highest_set s' th prio (e # moment i t)" by simp
+ hence vt_e: "vt step (e#(moment i t)@s)"
+ by (unfold extend_highest_set_def extend_highest_set_axioms_def
+ highest_set_def s_def, auto)
+ from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+ from post[rule_format, of "Suc i"] and eq_me
+ have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+ from create_pre[OF stp_i pre this]
+ obtain prio where eq_e: "e = Create th' prio" .
+ have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+ proof(rule cnp_cnv_eq)
+ from step_back_vt [OF vt_e]
+ show "vt step (moment i t @ s)" .
+ next
+ from eq_e and stp_i
+ have "step (moment i t @ s) (Create th' prio)" by simp
+ thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+ qed
+ with eq_e
+ have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
+ by (simp add:cntP_def cntV_def count_def)
+ with eq_me[symmetric]
+ have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+ by simp
+ from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+ with eq_me [symmetric]
+ have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+ from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+ and moment_ge
+ have "th' \<notin> runing (t @ s)" by auto
+ with runing'
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+ from runing_inversion_1[OF _ runing']
+ show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+ case True thus ?thesis by auto
+next
+ case False
+ then have not_ready: "th \<notin> readys (t@s)"
+ apply (unfold runing_def,
+ insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+ by auto
+ from th_kept have "th \<in> threads (t@s)" by auto
+ from th_chain_to_ready[OF vt_t this] and not_ready
+ obtain th' where th'_in: "th' \<in> readys (t@s)"
+ and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+ have "th' \<in> runing (t@s)"
+ proof -
+ have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
+ preced th (t@s)"
+ proof(rule Max_eqI)
+ fix y
+ assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ then obtain th1 where
+ h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
+ and eq_y: "y = preced th1 (t@s)" by auto
+ show "y \<le> preced th (t @ s)"
+ proof -
+ from max_preced
+ have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+ moreover have "y \<le> \<dots>"
+ proof(rule Max_ge)
+ from h1
+ have "th1 \<in> threads (t@s)"
+ proof
+ assume "th1 = th'"
+ with th'_in show ?thesis by (simp add:readys_def)
+ next
+ assume "th1 \<in> dependents (wq (t @ s)) th'"
+ with dependents_threads [OF vt_t]
+ show "th1 \<in> threads (t @ s)" by auto
+ qed
+ with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+ next
+ from finite_threads[OF vt_t]
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+ show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+ by (auto intro:finite_subset)
+ next
+ from dp
+ have "th \<in> dependents (wq (t @ s)) th'"
+ by (unfold cs_dependents_def, auto simp:eq_depend)
+ thus "preced th (t @ s) \<in>
+ (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ by auto
+ qed
+ moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ from max_preced and max_cp_eq[OF vt_t, symmetric]
+ have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+ with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+ qed
+ ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+ with th'_in show ?thesis by (auto simp:runing_def)
+ qed
+ thus ?thesis by auto
+qed
+
+end
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ExtSG.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1019 @@
+theory ExtSG
+imports PrioG
+begin
+
+locale highest_set =
+ fixes s' th prio fixes s
+ defines s_def : "s \<equiv> (Set th prio#s')"
+ assumes vt_s: "vt step s"
+ and highest: "preced th s = Max ((cp s)`threads s)"
+
+context highest_set
+begin
+
+lemma vt_s': "vt step s'"
+ by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)
+
+lemma step_set: "step s' (Set th prio)"
+ by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)
+
+lemma step_set_elim:
+ "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+ by (insert step_set, ind_cases "step s' (Set th prio)", auto)
+
+
+lemma threads_s: "th \<in> threads s"
+ by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)
+
+lemma same_depend: "depend s = depend s'"
+ by (insert depend_set_unchanged, unfold s_def, simp)
+
+lemma same_dependents:
+ "dependents (wq s) th = dependents (wq s') th"
+ apply (unfold cs_dependents_def)
+ by (unfold eq_depend same_depend, simp)
+
+lemma eq_cp_s_th: "cp s th = preced th s"
+proof -
+ from highest and max_cp_eq[OF vt_s]
+ have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+ have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
+ proof -
+ from threads_s and dependents_threads[OF vt_s, of th]
+ show ?thesis by auto
+ qed
+ show ?thesis
+ proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+ show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
+ next
+ fix y
+ assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
+ then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
+ and eq_y: "y = preced th1 s" by auto
+ show "y \<le> preced th s"
+ proof(unfold is_max, rule Max_ge)
+ from finite_threads[OF vt_s]
+ show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+ next
+ from sbs th1_in and eq_y
+ show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
+ qed
+ next
+ from sbs and finite_threads[OF vt_s]
+ show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
+ by (auto intro:finite_subset)
+ qed
+qed
+
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma is_ready: "th \<in> readys s"
+proof -
+ have "\<forall>cs. \<not> waiting s th cs"
+ apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])
+ apply (unfold s_depend_def, unfold runing_def readys_def)
+ apply (auto, fold s_def)
+ apply (erule_tac x = cs in allE, auto simp:waiting_eq)
+ proof -
+ fix cs
+ assume h:
+ "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =
+ {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"
+ (is "?L = ?R")
+ and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"
+ from wt have "(Th th, Cs cs) \<in> ?L" by auto
+ with h have "(Th th, Cs cs) \<in> ?R" by simp
+ hence "waiting (wq s') th cs" by auto with nwt
+ show False by auto
+ qed
+ with threads_s show ?thesis
+ by (unfold readys_def, auto)
+qed
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+ from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
+ show ?thesis by simp
+qed
+
+lemma is_runing: "th \<in> runing s"
+proof -
+ have "Max (cp s ` threads s) = Max (cp s ` readys s)"
+ proof -
+ have " Max (cp s ` readys s) = cp s th"
+ proof(rule Max_eqI)
+ from finite_threads[OF vt_s] readys_threads finite_subset
+ have "finite (readys s)" by blast
+ thus "finite (cp s ` readys s)" by auto
+ next
+ from is_ready show "cp s th \<in> cp s ` readys s" by auto
+ next
+ fix y
+ assume "y \<in> cp s ` readys s"
+ then obtain th1 where
+ eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto
+ show "y \<le> cp s th"
+ proof -
+ have "y \<le> Max (cp s ` threads s)"
+ proof(rule Max_ge)
+ from eq_y and th1_in
+ show "y \<in> cp s ` threads s"
+ by (auto simp:readys_def)
+ next
+ from finite_threads[OF vt_s]
+ show "finite (cp s ` threads s)" by auto
+ qed
+ with highest' show ?thesis by auto
+ qed
+ qed
+ with highest' show ?thesis by auto
+ qed
+ thus ?thesis
+ by (unfold runing_def, insert highest' is_ready, auto)
+qed
+
+end
+
+locale extend_highest_set = highest_set +
+ fixes t
+ assumes vt_t: "vt step (t@s)"
+ and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+ and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+ and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+lemma step_back_vt_app:
+ assumes vt_ts: "vt cs (t@s)"
+ shows "vt cs s"
+proof -
+ from vt_ts show ?thesis
+ proof(induct t)
+ case Nil
+ from Nil show ?case by auto
+ next
+ case (Cons e t)
+ assume ih: " vt cs (t @ s) \<Longrightarrow> vt cs s"
+ and vt_et: "vt cs ((e # t) @ s)"
+ show ?case
+ proof(rule ih)
+ show "vt cs (t @ s)"
+ proof(rule step_back_vt)
+ from vt_et show "vt cs (e # t @ s)" by simp
+ qed
+ qed
+ qed
+qed
+
+context extend_highest_set
+begin
+
+lemma red_moment:
+ "extend_highest_set s' th prio (moment i t)"
+ apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+ apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)
+ by (unfold highest_set_def, auto dest:step_back_vt_app)
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes
+ h0: "R []"
+ and h2: "\<And> e t. \<lbrakk>vt step (t@s); step (t@s) e;
+ extend_highest_set s' th prio t;
+ extend_highest_set s' th prio (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+ shows "R t"
+proof -
+ from vt_t extend_highest_set_axioms show ?thesis
+ proof(induct t)
+ from h0 show "R []" .
+ next
+ case (Cons e t')
+ assume ih: "\<lbrakk>vt step (t' @ s); extend_highest_set s' th prio t'\<rbrakk> \<Longrightarrow> R t'"
+ and vt_e: "vt step ((e # t') @ s)"
+ and et: "extend_highest_set s' th prio (e # t')"
+ from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+ from vt_e and step_back_vt have vt_ts: "vt step (t'@s)" by auto
+ show ?case
+ proof(rule h2 [OF vt_ts stp _ _ _ ])
+ show "R t'"
+ proof(rule ih)
+ from et show ext': "extend_highest_set s' th prio t'"
+ by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+ next
+ from vt_ts show "vt step (t' @ s)" .
+ qed
+ next
+ from et show "extend_highest_set s' th prio (e # t')" .
+ next
+ from et show ext': "extend_highest_set s' th prio t'"
+ by (unfold extend_highest_set_def extend_highest_set_axioms_def, auto dest:step_back_vt)
+ qed
+ qed
+qed
+
+lemma th_kept: "th \<in> threads (t @ s) \<and>
+ preced th (t@s) = preced th s" (is "?Q t")
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case Nil
+ from threads_s
+ show "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
+ by auto
+ next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio)
+ assume eq_e: " e = Create thread prio"
+ show ?thesis
+ proof -
+ from Cons and eq_e have "step (t@s) (Create thread prio)" by auto
+ hence "th \<noteq> thread"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ with Cons show ?thesis by auto
+ qed
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.exit_diff [OF this] and eq_e
+ have neq_th: "thread \<noteq> th" by auto
+ with Cons
+ show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ next
+ case (P thread cs)
+ assume eq_e: "e = P thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (V thread cs)
+ assume eq_e: "e = V thread cs"
+ with Cons
+ show ?thesis
+ by (auto simp:eq_e preced_def)
+ next
+ case (Set thread prio')
+ assume eq_e: " e = Set thread prio'"
+ show ?thesis
+ proof -
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.set_diff_low[OF this] and eq_e
+ have "th \<noteq> thread" by auto
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold eq_e, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:eq_e)
+ qed
+ qed
+ qed
+qed
+
+lemma max_kept: "Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+ case Nil
+ from highest_preced_thread
+ show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
+ by simp
+next
+ case (Cons e t)
+ show ?case
+ proof(cases e)
+ case (Create thread prio')
+ assume eq_e: " e = Create thread prio'"
+ from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
+ hence neq_thread: "thread \<noteq> th"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th \<in> threads (t@s)"
+ proof -
+ from Cons have "extend_highest_set s' th prio t" by auto
+ from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)
+ qed
+ ultimately show ?thesis by auto
+ qed
+ from Cons have "extend_highest_set s' th prio t" by auto
+ from extend_highest_set.th_kept[OF this]
+ have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
+ by (auto simp:s_def)
+ from stp
+ have thread_ts: "thread \<notin> threads (t @ s)"
+ by (cases, auto)
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
+ by (unfold eq_e, simp)
+ moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
+ proof(rule Max_insert)
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this]
+ show "finite (?f ` (threads (t@s)))" by simp
+ next
+ from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
+ qed
+ moreover have "(Max (?f ` (threads (t@s)))) = ?t"
+ proof -
+ have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
+ (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
+ proof -
+ { fix th'
+ assume "th' \<in> ?B"
+ with thread_ts eq_e
+ have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
+ } thus ?thesis
+ apply (auto simp:Image_def)
+ proof -
+ fix th'
+ assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
+ preced th' (e # t @ s) = preced th' (t @ s)"
+ and h1: "th' \<in> threads (t @ s)"
+ show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
+ proof -
+ from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
+ moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
+ ultimately show ?thesis by simp
+ qed
+ qed
+ qed
+ with Cons show ?thesis by auto
+ qed
+ moreover have "?f thread < ?t"
+ proof -
+ from Cons have " extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.create_low[OF this] and eq_e
+ have "prio' \<le> prio" by auto
+ thus ?thesis
+ by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)
+ qed
+ ultimately show ?thesis by (auto simp:max_def)
+ qed
+next
+ case (Exit thread)
+ assume eq_e: "e = Exit thread"
+ from Cons have vt_e: "vt step (e#(t @ s))" by auto
+ from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
+ from stp have thread_ts: "thread \<in> threads (t @ s)"
+ by(cases, unfold runing_def readys_def, auto)
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.exit_diff[OF this] and eq_e
+ have neq_thread: "thread \<noteq> th" by auto
+ from Cons have "extend_highest_set s' th prio t" by auto
+ from extend_highest_set.th_kept[OF this, folded s_def]
+ have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "threads (t@s) = insert thread ?A"
+ by (insert stp thread_ts, unfold eq_e, auto)
+ hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
+ also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
+ also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ from finite_threads [OF vt_e]
+ show "finite (?f ` ?A)" by simp
+ next
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.th_kept[OF this]
+ show "?f ` ?A \<noteq> {}" by (auto simp:s_def)
+ qed
+ finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
+ moreover have "Max (?f ` (threads (t@s))) = ?t"
+ proof -
+ from Cons show ?thesis
+ by (unfold eq_e, auto simp:preced_def)
+ qed
+ ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
+ moreover have "?f thread < ?t"
+ proof(unfold eq_e, simp add:preced_def, fold preced_def)
+ show "preced thread (t @ s) < ?t"
+ proof -
+ have "preced thread (t @ s) \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "?t = Max (?g ` ?B)") by simp
+ moreover have "?g thread \<le> \<dots>"
+ proof(rule Max_ge)
+ have "vt step (t@s)" by fact
+ from finite_threads [OF this]
+ show "finite (?g ` ?B)" by simp
+ next
+ from thread_ts
+ show "?g thread \<in> (?g ` ?B)" by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ moreover have "preced thread (t @ s) \<noteq> ?t"
+ proof
+ assume "preced thread (t @ s) = preced th s"
+ with h' have "preced thread (t @ s) = preced th (t@s)" by simp
+ from preced_unique [OF this] have "thread = th"
+ proof
+ from h' show "th \<in> threads (t @ s)" by simp
+ next
+ from thread_ts show "thread \<in> threads (t @ s)" .
+ qed(simp)
+ with neq_thread show "False" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def split:if_splits)
+ qed
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ let ?B = "threads (t@s)"
+ from Cons have "extend_highest_set s' th prio (e # t)" by auto
+ from extend_highest_set.set_diff_low[OF this] and Set
+ have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
+ from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
+ also have "\<dots> = ?t"
+ proof(rule Max_eqI)
+ fix y
+ assume y_in: "y \<in> ?f ` ?B"
+ then obtain th1 where
+ th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
+ show "y \<le> ?t"
+ proof(cases "th1 = thread")
+ case True
+ with neq_thread le_p eq_y s_def Set
+ show ?thesis
+ by (auto simp:preced_def precedence_le_def)
+ next
+ case False
+ with Set eq_y
+ have "y = preced th1 (t@s)"
+ by (simp add:preced_def)
+ moreover have "\<dots> \<le> ?t"
+ proof -
+ from Cons
+ have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
+ by auto
+ moreover have "preced th1 (t@s) \<le> \<dots>"
+ proof(rule Max_ge)
+ from th1_in
+ show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
+ by simp
+ next
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ proof -
+ from Cons have "vt step (t @ s)" by auto
+ from finite_threads[OF this] show ?thesis by auto
+ qed
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from Cons and finite_threads
+ show "finite (?f ` ?B)" by auto
+ next
+ from Cons have "extend_highest_set s' th prio t" by auto
+ from extend_highest_set.th_kept [OF this, folded s_def]
+ have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
+ show "?t \<in> (?f ` ?B)"
+ proof -
+ from neq_thread Set h
+ have "?t = ?f th" by (auto simp:preced_def)
+ with h show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ by (insert th_kept max_kept, auto)
+
+lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
+ (is "?L = ?R")
+proof -
+ have "?L = cpreced (t@s) (wq (t@s)) th"
+ by (unfold cp_eq_cpreced, simp)
+ also have "\<dots> = ?R"
+ proof(unfold cpreced_def)
+ show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
+ Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
+ (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
+ proof(cases "?A = {}")
+ case False
+ have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
+ moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
+ proof(rule Max_insert)
+ show "finite (?f ` ?A)"
+ proof -
+ from dependents_threads[OF vt_t]
+ have "?A \<subseteq> threads (t@s)" .
+ moreover from finite_threads[OF vt_t] have "finite \<dots>" .
+ ultimately show ?thesis
+ by (auto simp:finite_subset)
+ qed
+ next
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ qed
+ moreover have "\<dots> = Max (?f ` ?B)"
+ proof -
+ from max_preced have "?f th = Max (?f ` ?B)" .
+ moreover have "Max (?f ` ?A) \<le> \<dots>"
+ proof(rule Max_mono)
+ from False show "(?f ` ?A) \<noteq> {}" by simp
+ next
+ show "?f ` ?A \<subseteq> ?f ` ?B"
+ proof -
+ have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
+ thus ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t]
+ show "finite (?f ` ?B)" by simp
+ qed
+ ultimately show ?thesis
+ by (auto simp:max_def)
+ qed
+ ultimately show ?thesis by auto
+ next
+ case True
+ with max_preced show ?thesis by auto
+ qed
+ qed
+ finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+ by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+ by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less':
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ shows "preced th' s < preced th s"
+proof -
+ have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
+ proof(rule Max_ge)
+ from finite_threads [OF vt_s]
+ show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
+ next
+ from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
+ by simp
+ qed
+ moreover have "preced th' s \<noteq> preced th s"
+ proof
+ assume "preced th' s = preced th s"
+ from preced_unique[OF this th'_in] neq_th' is_ready
+ show "False" by (auto simp:readys_def)
+ qed
+ ultimately show ?thesis using highest_preced_thread
+ by auto
+qed
+
+lemma pv_blocked:
+ fixes th'
+ assumes th'_in: "th' \<in> threads (t@s)"
+ and neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+ shows "th' \<notin> runing (t@s)"
+proof
+ assume "th' \<in> runing (t@s)"
+ hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
+ by (auto simp:runing_def)
+ with max_cp_readys_threads [OF vt_t]
+ have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
+ by auto
+ moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
+ ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
+ moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
+ by simp
+ finally have h: "cp (t @ s) th' = preced th (t @ s)" .
+ show False
+ proof -
+ have "dependents (wq (t @ s)) th' = {}"
+ by (rule count_eq_dependents [OF vt_t eq_pv])
+ moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
+ proof
+ assume "preced th' (t @ s) = preced th (t @ s)"
+ hence "th' = th"
+ proof(rule preced_unique)
+ from th_kept show "th \<in> threads (t @ s)" by simp
+ next
+ from th'_in show "th' \<in> threads (t @ s)" by simp
+ qed
+ with assms show False by simp
+ qed
+ ultimately show ?thesis
+ by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+qed
+
+lemma runing_precond_pre:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads (t@s) \<and>
+ cntP (t@s) th' = cntV (t@s) th'"
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case (Cons e t)
+ from Cons
+ have in_thread: "th' \<in> threads (t @ s)"
+ and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ have "extend_highest_set s' th prio t" by fact
+ from extend_highest_set.pv_blocked
+ [OF this, folded s_def, OF in_thread neq_th' not_holding]
+ have not_runing: "th' \<notin> runing (t @ s)" .
+ show ?case
+ proof(cases e)
+ case (V thread cs)
+ from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
+
+ show ?thesis
+ proof -
+ from Cons and V have "step (t@s) (V thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (unfold V, simp add:cntP_def cntV_def count_def)
+ moreover from in_thread
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (P thread cs)
+ from Cons and P have "step (t@s) (P thread cs)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Create thread prio')
+ from Cons and Create have "step (t@s) (Create thread prio')" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<notin> threads (t @ s)"
+ moreover have "th' \<in> threads (t@s)" by fact
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Create
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Create
+ have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
+ ultimately show ?thesis by auto
+ next
+ case (Exit thread)
+ from Cons and Exit have "step (t@s) (Exit thread)" by auto
+ hence neq_th': "thread \<noteq> th'"
+ proof(cases)
+ assume "thread \<in> runing (t @ s)"
+ moreover note not_runing
+ ultimately show ?thesis by auto
+ qed
+ with Cons and Exit
+ have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ by (auto simp:cntP_def cntV_def count_def)
+ moreover from Cons and Exit and neq_th'
+ have in_thread': "th' \<in> threads ((e # t) @ s)"
+ by auto
+ ultimately show ?thesis by auto
+ next
+ case (Set thread prio')
+ with Cons
+ show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ next
+ case Nil
+ with assms
+ show ?case by auto
+ qed
+qed
+
+(*
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<notin> runing (t@s)"
+proof -
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2]
+ show ?thesis .
+qed
+*)
+
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ and is_runing: "th' \<in> runing (t@s)"
+ shows "cntP s th' > cntV s th'"
+proof -
+ have "cntP s th' \<noteq> cntV s th'"
+ proof
+ assume eq_pv: "cntP s th' = cntV s th'"
+ from runing_precond_pre[OF th'_in eq_pv neq_th']
+ have h1: "th' \<in> threads (t @ s)"
+ and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
+ from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
+ with is_runing show "False" by simp
+ qed
+ moreover from cnp_cnv_cncs[OF vt_s, of th']
+ have "cntV s th' \<le> cntP s th'" by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+ th' \<in> threads ((moment (i+j) t)@s)"
+proof(induct j)
+ case (Suc k)
+ show ?case
+ proof -
+ { assume True: "Suc (i+k) \<le> length t"
+ from moment_head [OF this]
+ obtain e where
+ eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
+ by blast
+ from red_moment[of "Suc(i+k)"]
+ and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp
+ hence vt_e: "vt step (e#(moment (i + k) t)@s)"
+ by (unfold extend_highest_set_def extend_highest_set_axioms_def
+ highest_set_def s_def, auto)
+ have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
+ proof(unfold s_def)
+ show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"
+ proof(rule extend_highest_set.pv_blocked)
+ from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"
+ by (simp add:s_def)
+ next
+ from neq_th' show "th' \<noteq> th" .
+ next
+ from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .
+ next
+ from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =
+ cntV (moment (i + k) t @ Set th prio # s') th'"
+ by (auto simp:s_def)
+ qed
+ qed
+ from step_back_step[OF vt_e]
+ have "step ((moment (i + k) t)@s) e" .
+ hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
+ th' \<in> threads (e#(moment (i + k) t)@s)
+ "
+ proof(cases)
+ case (thread_create thread prio)
+ with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_exit thread)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_P thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_V thread cs)
+ moreover have "thread \<noteq> th'"
+ proof -
+ have "thread \<in> runing (moment (i + k) t @ s)" by fact
+ moreover note not_runing'
+ ultimately show ?thesis by auto
+ qed
+ moreover note Suc
+ ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
+ next
+ case (thread_set thread prio')
+ with Suc show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+ with eq_me have ?thesis using eq_me by auto
+ } note h = this
+ show ?thesis
+ proof(cases "Suc (i+k) \<le> length t")
+ case True
+ from h [OF this] show ?thesis .
+ next
+ case False
+ with moment_ge
+ have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
+ with Suc show ?thesis by auto
+ qed
+ qed
+next
+ case 0
+ from assms show ?case by auto
+qed
+
+lemma moment_blocked:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ and le_ij: "i \<le> j"
+ shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+ have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+ and h2: "th' \<in> threads ((moment j t)@s)" by auto
+ with extend_highest_set.pv_blocked [OF red_moment [of j], folded s_def, OF h2 neq_th' h1]
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_1:
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof(cases "th' \<in> threads s")
+ case True
+ with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+next
+ case False
+ let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
+ let ?q = "moment 0 t"
+ from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
+ from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
+ from p_split_gen [of ?Q, OF this not_thread]
+ obtain i where lt_its: "i < length t"
+ and le_i: "0 \<le> i"
+ and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+ and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
+ from lt_its have "Suc i \<le> length t" by auto
+ from moment_head[OF this] obtain e where
+ eq_me: "moment (Suc i) t = e # moment i t" by blast
+ from red_moment[of "Suc i"] and eq_me
+ have "extend_highest_set s' th prio (e # moment i t)" by simp
+ hence vt_e: "vt step (e#(moment i t)@s)"
+ by (unfold extend_highest_set_def extend_highest_set_axioms_def
+ highest_set_def s_def, auto)
+ from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
+ from post[rule_format, of "Suc i"] and eq_me
+ have not_in': "th' \<in> threads (e # moment i t@s)" by auto
+ from create_pre[OF stp_i pre this]
+ obtain prio where eq_e: "e = Create th' prio" .
+ have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+ proof(rule cnp_cnv_eq)
+ from step_back_vt [OF vt_e]
+ show "vt step (moment i t @ s)" .
+ next
+ from eq_e and stp_i
+ have "step (moment i t @ s) (Create th' prio)" by simp
+ thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
+ qed
+ with eq_e
+ have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
+ by (simp add:cntP_def cntV_def count_def)
+ with eq_me[symmetric]
+ have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+ by simp
+ from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
+ with eq_me [symmetric]
+ have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
+ from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
+ and moment_ge
+ have "th' \<notin> runing (t @ s)" by auto
+ with runing'
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_2:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+ from runing_inversion_1[OF _ runing']
+ show ?thesis by auto
+qed
+
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+ case True thus ?thesis by auto
+next
+ case False
+ then have not_ready: "th \<notin> readys (t@s)"
+ apply (unfold runing_def,
+ insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
+ by auto
+ from th_kept have "th \<in> threads (t@s)" by auto
+ from th_chain_to_ready[OF vt_t this] and not_ready
+ obtain th' where th'_in: "th' \<in> readys (t@s)"
+ and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
+ have "th' \<in> runing (t@s)"
+ proof -
+ have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
+ preced th (t@s)"
+ proof(rule Max_eqI)
+ fix y
+ assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ then obtain th1 where
+ h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
+ and eq_y: "y = preced th1 (t@s)" by auto
+ show "y \<le> preced th (t @ s)"
+ proof -
+ from max_preced
+ have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
+ moreover have "y \<le> \<dots>"
+ proof(rule Max_ge)
+ from h1
+ have "th1 \<in> threads (t@s)"
+ proof
+ assume "th1 = th'"
+ with th'_in show ?thesis by (simp add:readys_def)
+ next
+ assume "th1 \<in> dependents (wq (t @ s)) th'"
+ with dependents_threads [OF vt_t]
+ show "th1 \<in> threads (t @ s)" by auto
+ qed
+ with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
+ next
+ from finite_threads[OF vt_t]
+ show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
+ qed
+ ultimately show ?thesis by auto
+ qed
+ next
+ from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
+ show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
+ by (auto intro:finite_subset)
+ next
+ from dp
+ have "th \<in> dependents (wq (t @ s)) th'"
+ by (unfold cs_dependents_def, auto simp:eq_depend)
+ thus "preced th (t @ s) \<in>
+ (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
+ by auto
+ qed
+ moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
+ proof -
+ from max_preced and max_cp_eq[OF vt_t, symmetric]
+ have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
+ with max_cp_readys_threads[OF vt_t] show ?thesis by simp
+ qed
+ ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
+ qed
+ with th'_in show ?thesis by (auto simp:runing_def)
+ qed
+ thus ?thesis by auto
+qed
+
+end
+
+end
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Happen_within.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,126 @@
+theory Happen_within
+imports Main Moment
+begin
+
+(*
+ lemma
+ fixes P :: "('a list) \<Rightarrow> bool"
+ and Q :: "('a list) \<Rightarrow> bool"
+ and k :: nat
+ and f :: "('a list) \<Rightarrow> nat"
+ assumes "\<And> s t. \<lbrakk>P s; \<not> Q s; P (t@s); k < length t\<rbrakk> \<Longrightarrow> f (t@s) < f s"
+ shows "\<And> s t. \<lbrakk> P s; P(t @ s); f(s) * k < length t\<rbrakk> \<Longrightarrow> Q (t@s)"
+ sorry
+*)
+
+text {*
+ The following two notions are introduced to improve the situation.
+ *}
+
+definition all_future :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> ('a list) \<Rightarrow> bool"
+where "all_future G R s = (\<forall> t. G (t@s) \<longrightarrow> R t)"
+
+definition happen_within :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> ('a list) \<Rightarrow> bool"
+where "happen_within G R k s = all_future G (\<lambda> t. k < length t \<longrightarrow>
+ (\<exists> i \<le> k. R (moment i t @ s) \<and> G (moment i t @ s))) s"
+
+lemma happen_within_intro:
+ fixes P :: "('a list) \<Rightarrow> bool"
+ and Q :: "('a list) \<Rightarrow> bool"
+ and k :: nat
+ and f :: "('a list) \<Rightarrow> nat"
+ assumes
+ lt_k: "0 < k"
+ and step: "\<And> s. \<lbrakk>P s; \<not> Q s\<rbrakk> \<Longrightarrow> happen_within P (\<lambda> s'. f s' < f s) k s"
+ shows "\<And> s. P s \<Longrightarrow> happen_within P Q ((f s + 1) * k) s"
+proof -
+ fix s
+ assume "P s"
+ thus "happen_within P Q ((f s + 1) * k) s"
+ proof(induct n == "f s + 1" arbitrary:s rule:nat_less_induct)
+ fix s
+ assume ih [rule_format]: "\<forall>m<f s + 1. \<forall>x. m = f x + 1 \<longrightarrow> P x
+ \<longrightarrow> happen_within P Q ((f x + 1) * k) x"
+ and ps: "P s"
+ show "happen_within P Q ((f s + 1) * k) s"
+ proof(cases "Q s")
+ case True
+ show ?thesis
+ proof -
+ { fix t
+ from True and ps have "0 \<le> ((f s + 1)*k) \<and> Q (moment 0 t @ s) \<and> P (moment 0 t @ s)" by auto
+ hence "\<exists>i\<le>(f s + 1) * k. Q (moment i t @ s) \<and> P (moment i t @ s)" by auto
+ } thus ?thesis by (auto simp: happen_within_def all_future_def)
+ qed
+ next
+ case False
+ from step [OF ps False] have kk: "happen_within P (\<lambda>s'. f s' < f s) k s" .
+ show ?thesis
+ proof -
+ { fix t
+ assume pts: "P (t @ s)" and ltk: "(f s + 1) * k < length t"
+ from ltk have lt_k_lt: "k < length t" by auto
+ with kk pts obtain i
+ where le_ik: "i \<le> k"
+ and lt_f: "f (moment i t @ s) < f s"
+ and p_m: "P (moment i t @ s)"
+ by (auto simp:happen_within_def all_future_def)
+ from ih [of "f (moment i t @ s) + 1" "(moment i t @ s)", OF _ _ p_m] and lt_f
+ have hw: "happen_within P Q ((f (moment i t @ s) + 1) * k) (moment i t @ s)" by auto
+ have "(\<exists>j\<le>(f s + 1) * k. Q (moment j t @ s) \<and> P (moment j t @ s))" (is "\<exists> j. ?T j")
+ proof -
+ let ?t = "restm i t"
+ have eq_t: "t = ?t @ moment i t" by (simp add:moment_restm_s)
+ have h1: "P (restm i t @ moment i t @ s)"
+ proof -
+ from pts and eq_t have "P ((restm i t @ moment i t) @ s)" by simp
+ thus ?thesis by simp
+ qed
+ moreover have h2: "(f (moment i t @ s) + 1) * k < length (restm i t)"
+ proof -
+ have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
+ from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
+ from h [OF this, of k]
+ have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
+ moreover from le_ik have "\<dots> \<le> ((f s) * k + k - i)" by simp
+ moreover from le_ik lt_k_lt and ltk have "(f s) * k + k - i < length t - i" by simp
+ moreover have "length (restm i t) = length t - i" using length_restm by metis
+ ultimately show ?thesis by simp
+ qed
+ from hw [unfolded happen_within_def all_future_def, rule_format, OF h1 h2]
+ obtain m where le_m: "m \<le> (f (moment i t @ s) + 1) * k"
+ and q_m: "Q (moment m ?t @ moment i t @ s)"
+ and p_m: "P (moment m ?t @ moment i t @ s)" by auto
+ have eq_mm: "moment m ?t @ moment i t @ s = (moment (m+i) t)@s"
+ proof -
+ have "moment m (restm i t) @ moment i t = moment (m + i) t"
+ using moment_plus_split by metis
+ thus ?thesis by simp
+ qed
+ let ?j = "m + i"
+ have "?T ?j"
+ proof -
+ have "m + i \<le> (f s + 1) * k"
+ proof -
+ have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp
+ from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp
+ from h [OF this, of k]
+ have "(f (moment i t @ s) + 1) * k \<le> f s * k" .
+ with le_m have "m \<le> f s * k" by simp
+ hence "m + i \<le> f s * k + i" by simp
+ with le_ik show ?thesis by simp
+ qed
+ moreover from eq_mm q_m have " Q (moment (m + i) t @ s)" by metis
+ moreover from eq_mm p_m have " P (moment (m + i) t @ s)" by metis
+ ultimately show ?thesis by blast
+ qed
+ thus ?thesis by blast
+ qed
+ } thus ?thesis by (simp add:happen_within_def all_future_def firstn.simps)
+ qed
+ qed
+ qed
+qed
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/IsaMakefile Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,29 @@
+
+## targets
+
+default: paper
+all: session paper
+
+## global settings
+
+SRC = $(ISABELLE_HOME)/src
+OUT = $(ISABELLE_OUTPUT)
+LOG = $(OUT)/log
+
+
+USEDIR = $(ISABELLE_TOOL) usedir -v true -t true
+
+
+## Slides
+
+session: ./ROOT.ML ./*.thy
+ @$(USEDIR) -b -D generated -f ROOT.ML HOL Prio
+
+paper: Paper/ROOT.ML \
+ Paper/*.thy
+ @$(USEDIR) -D generated -f ROOT.ML Prio Paper
+ rm -f Paper/generated/*.aux # otherwise latex will fall over
+ cd Paper/generated ; $(ISABELLE_TOOL) latex -o pdf root.tex
+ cd Paper/generated ; bibtex root
+ cd Paper/generated ; $(ISABELLE_TOOL) latex -o pdf root.tex
+ cp Paper/generated/root.pdf paper.pdf
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Lsp.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,323 @@
+theory Lsp
+imports Main
+begin
+
+fun lsp :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list)"
+where
+ "lsp f [] = ([], [], [])" |
+ "lsp f [x] = ([], [x], [])" |
+ "lsp f (x#xs) = (case (lsp f xs) of
+ (l, [], r) \<Rightarrow> ([], [x], []) |
+ (l, y#ys, r) \<Rightarrow> if f x \<ge> f y then ([], [x], xs) else (x#l, y#ys, r))"
+
+inductive lsp_p :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list) \<Rightarrow> bool"
+for f :: "('a \<Rightarrow> ('b::linorder))"
+where
+ lsp_nil [intro]: "lsp_p f [] ([], [], [])" |
+ lsp_single [intro]: "lsp_p f [x] ([], [x], [])" |
+ lsp_cons_1 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x \<ge> f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) ([], [x], xs)" |
+ lsp_cons_2 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) (x#l, [m], r)"
+
+lemma lsp_p_lsp_1: "lsp_p f x y \<Longrightarrow> y = lsp f x"
+proof (induct rule:lsp_p.induct)
+ case (lsp_cons_1 xs l m r x)
+ assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
+ and le_mx: "f m \<le> f x"
+ show ?case (is "?L = ?R")
+ proof(cases xs, simp)
+ case (Cons v vs)
+ show ?thesis
+ apply (simp add:Cons)
+ apply (fold Cons)
+ by (simp add:lsp_xs le_mx)
+ qed
+next
+ case (lsp_cons_2 xs l m r x)
+ assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
+ and lt_xm: "f x < f m"
+ show ?case (is "?L = ?R")
+ proof(cases xs)
+ case (Cons v vs)
+ show ?thesis
+ apply (simp add:Cons)
+ apply (fold Cons)
+ apply (simp add:lsp_xs)
+ by (insert lt_xm, auto)
+ next
+ case Nil
+ from prems show ?thesis by simp
+ qed
+qed auto
+
+lemma lsp_mid_nil: "lsp f xs = (a, [], c) \<Longrightarrow> xs = []"
+ apply (induct xs arbitrary:a c, auto)
+ apply (case_tac xs, auto)
+ by (case_tac "(lsp f (ab # list))", auto split:if_splits list.splits)
+
+
+lemma lsp_mid_length: "lsp f x = (u, v, w) \<Longrightarrow> length v \<le> 1"
+proof(induct x arbitrary:u v w, simp)
+ case (Cons x xs)
+ assume ih: "\<And> u v w. lsp f xs = (u, v, w) \<Longrightarrow> length v \<le> 1"
+ and h: "lsp f (x # xs) = (u, v, w)"
+ show "length v \<le> 1" using h
+ proof(cases xs, simp add:h)
+ case (Cons z zs)
+ assume eq_xs: "xs = z # zs"
+ show ?thesis
+ proof(cases "lsp f xs")
+ fix l m r
+ assume eq_lsp: "lsp f xs = (l, m, r)"
+ show ?thesis
+ proof(cases m)
+ case Nil
+ from Nil and eq_lsp have "lsp f xs = (l, [], r)" by simp
+ from lsp_mid_nil [OF this] have "xs = []" .
+ with h show ?thesis by auto
+ next
+ case (Cons y ys)
+ assume eq_m: "m = y # ys"
+ from ih [OF eq_lsp] have eq_xs_1: "length m \<le> 1" .
+ show ?thesis
+ proof(cases "f x \<ge> f y")
+ case True
+ from eq_xs eq_xs_1 True h eq_lsp show ?thesis
+ by (auto split:list.splits if_splits)
+ next
+ case False
+ from eq_xs eq_xs_1 False h eq_lsp show ?thesis
+ by (auto split:list.splits if_splits)
+ qed
+ qed
+ qed
+ next
+ assume "[] = u \<and> [x] = v \<and> [] = w"
+ hence "v = [x]" by simp
+ thus "length v \<le> Suc 0" by simp
+ qed
+qed
+
+lemma lsp_p_lsp_2: "lsp_p f x (lsp f x)"
+proof(induct x, auto)
+ case (Cons x xs)
+ assume ih: "lsp_p f xs (lsp f xs)"
+ show ?case
+ proof(cases xs)
+ case Nil
+ thus ?thesis by auto
+ next
+ case (Cons v vs)
+ show ?thesis
+ proof(cases "xs")
+ case Nil
+ thus ?thesis by auto
+ next
+ case (Cons v vs)
+ assume eq_xs: "xs = v # vs"
+ show ?thesis
+ proof(cases "lsp f xs")
+ fix l m r
+ assume eq_lsp_xs: "lsp f xs = (l, m, r)"
+ show ?thesis
+ proof(cases m)
+ case Nil
+ from eq_lsp_xs and Nil have "lsp f xs = (l, [], r)" by simp
+ from lsp_mid_nil [OF this] have eq_xs: "xs = []" .
+ hence "lsp f (x#xs) = ([], [x], [])" by simp
+ with eq_xs show ?thesis by auto
+ next
+ case (Cons y ys)
+ assume eq_m: "m = y # ys"
+ show ?thesis
+ proof(cases "f x \<ge> f y")
+ case True
+ from eq_xs eq_lsp_xs Cons True
+ have eq_lsp: "lsp f (x#xs) = ([], [x], v # vs)" by simp
+ show ?thesis
+ proof (simp add:eq_lsp)
+ show "lsp_p f (x # xs) ([], [x], v # vs)"
+ proof(fold eq_xs, rule lsp_cons_1 [OF _])
+ from eq_xs show "xs \<noteq> []" by simp
+ next
+ from lsp_mid_length [OF eq_lsp_xs] and Cons
+ have "m = [y]" by simp
+ with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
+ with ih show "lsp_p f xs (l, [y], r)" by simp
+ next
+ from True show "f y \<le> f x" by simp
+ qed
+ qed
+ next
+ case False
+ from eq_xs eq_lsp_xs Cons False
+ have eq_lsp: "lsp f (x#xs) = (x # l, y # ys, r) " by simp
+ show ?thesis
+ proof (simp add:eq_lsp)
+ from lsp_mid_length [OF eq_lsp_xs] and eq_m
+ have "ys = []" by simp
+ moreover have "lsp_p f (x # xs) (x # l, [y], r)"
+ proof(rule lsp_cons_2)
+ from eq_xs show "xs \<noteq> []" by simp
+ next
+ from lsp_mid_length [OF eq_lsp_xs] and Cons
+ have "m = [y]" by simp
+ with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
+ with ih show "lsp_p f xs (l, [y], r)" by simp
+ next
+ from False show "f x < f y" by simp
+ qed
+ ultimately show "lsp_p f (x # xs) (x # l, y # ys, r)" by simp
+ qed
+ qed
+ qed
+ qed
+ qed
+ qed
+qed
+
+lemma lsp_induct:
+ fixes f x1 x2 P
+ assumes h: "lsp f x1 = x2"
+ and p1: "P [] ([], [], [])"
+ and p2: "\<And>x. P [x] ([], [x], [])"
+ and p3: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f m \<le> f x\<rbrakk> \<Longrightarrow> P (x # xs) ([], [x], xs)"
+ and p4: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> P (x # xs) (x # l, [m], r)"
+ shows "P x1 x2"
+proof(rule lsp_p.induct)
+ from lsp_p_lsp_2 and h
+ show "lsp_p f x1 x2" by metis
+next
+ from p1 show "P [] ([], [], [])" by metis
+next
+ from p2 show "\<And>x. P [x] ([], [x], [])" by metis
+next
+ fix xs l m r x
+ assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f m \<le> f x"
+ show "P (x # xs) ([], [x], xs)"
+ proof(rule p3 [OF h1 _ h3 h4])
+ from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
+ qed
+next
+ fix xs l m r x
+ assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f x < f m"
+ show "P (x # xs) (x # l, [m], r)"
+ proof(rule p4 [OF h1 _ h3 h4])
+ from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
+ qed
+qed
+
+lemma lsp_set_eq:
+ fixes f x u v w
+ assumes h: "lsp f x = (u, v, w)"
+ shows "x = u@v@w"
+proof -
+ have "\<And> f x r. lsp f x = r \<Longrightarrow> \<forall> u v w. (r = (u, v, w) \<longrightarrow> x = u@v@w)"
+ by (erule lsp_induct, simp+)
+ from this [rule_format, OF h] show ?thesis by simp
+qed
+
+lemma lsp_set:
+ assumes h: "(u, v, w) = lsp f x"
+ shows "set (u@v@w) = set x"
+proof -
+ from lsp_set_eq [OF h[symmetric]]
+ show ?thesis by simp
+qed
+
+lemma max_insert_gt:
+ fixes S fx
+ assumes h: "fx < Max S"
+ and np: "S \<noteq> {}"
+ and fn: "finite S"
+ shows "Max S = Max (insert fx S)"
+proof -
+ from Max_insert [OF fn np]
+ have "Max (insert fx S) = max fx (Max S)" .
+ moreover have "\<dots> = Max S"
+ proof(cases "fx \<le> Max S")
+ case False
+ with h
+ show ?thesis by (simp add:max_def)
+ next
+ case True
+ thus ?thesis by (simp add:max_def)
+ qed
+ ultimately show ?thesis by simp
+qed
+
+lemma max_insert_le:
+ fixes S fx
+ assumes h: "Max S \<le> fx"
+ and fn: "finite S"
+ shows "fx = Max (insert fx S)"
+proof(cases "S = {}")
+ case True
+ thus ?thesis by simp
+next
+ case False
+ from Max_insert [OF fn False]
+ have "Max (insert fx S) = max fx (Max S)" .
+ moreover have "\<dots> = fx"
+ proof(cases "fx \<le> Max S")
+ case False
+ thus ?thesis by (simp add:max_def)
+ next
+ case True
+ have hh: "\<And> x y. \<lbrakk> x \<le> (y::('a::linorder)); y \<le> x\<rbrakk> \<Longrightarrow> x = y" by auto
+ from hh [OF True h]
+ have "fx = Max S" .
+ thus ?thesis by simp
+ qed
+ ultimately show ?thesis by simp
+qed
+
+lemma lsp_max:
+ fixes f x u m w
+ assumes h: "lsp f x = (u, [m], w)"
+ shows "f m = Max (f ` (set x))"
+proof -
+ { fix y
+ have "lsp f x = y \<Longrightarrow> \<forall> u m w. y = (u, [m], w) \<longrightarrow> f m = Max (f ` (set x))"
+ proof(erule lsp_induct, simp)
+ { fix x u m w
+ assume "(([]::'a list), ([x]::'a list), ([]::'a list)) = (u, [m], w)"
+ hence "f m = Max (f ` set [x])" by simp
+ } thus "\<And>x. \<forall>u m w. ([], [x], []) = (u, [m], w) \<longrightarrow> f m = Max (f ` set [x])" by simp
+ next
+ fix xs l m r x
+ assume h1: "xs \<noteq> []"
+ and h2: " lsp f xs = (l, [m], r)"
+ and h3: "\<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
+ and h4: "f m \<le> f x"
+ show " \<forall>u m w. ([], [x], xs) = (u, [m], w) \<longrightarrow> f m = Max (f ` set (x # xs))"
+ proof -
+ have "f x = Max (f ` set (x # xs))"
+ proof -
+ from h2 h3 have "f m = Max (f ` set xs)" by simp
+ with h4 show ?thesis
+ apply auto
+ by (rule_tac max_insert_le, auto)
+ qed
+ thus ?thesis by simp
+ qed
+ next
+ fix xs l m r x
+ assume h1: "xs \<noteq> []"
+ and h2: " lsp f xs = (l, [m], r)"
+ and h3: " \<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
+ and h4: "f x < f m"
+ show "\<forall>u ma w. (x # l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set (x # xs))"
+ proof -
+ from h2 h3 have "f m = Max (f ` set xs)" by simp
+ with h4
+ have "f m = Max (f ` set (x # xs))"
+ apply auto
+ apply (rule_tac max_insert_gt, simp+)
+ by (insert h1, simp+)
+ thus ?thesis by auto
+ qed
+ qed
+ } with h show ?thesis by metis
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Moment.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,773 @@
+theory Moment
+imports Main
+begin
+
+fun firstn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ "firstn 0 s = []" |
+ "firstn (Suc n) [] = []" |
+ "firstn (Suc n) (e#s) = e#(firstn n s)"
+
+fun restn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where "restn n s = rev (firstn (length s - n) (rev s))"
+
+definition moment :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where "moment n s = rev (firstn n (rev s))"
+
+definition restm :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where "restm n s = rev (restn n (rev s))"
+
+definition from_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+ where "from_to i j s = firstn (j - i) (restn i s)"
+
+definition down_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where "down_to j i s = rev (from_to i j (rev s))"
+
+(*
+value "down_to 6 2 [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]"
+value "from_to 2 6 [0, 1, 2, 3, 4, 5, 6, 7]"
+*)
+
+lemma length_eq_elim_l: "\<lbrakk>length xs = length ys; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"
+ by auto
+
+lemma length_eq_elim_r: "\<lbrakk>length us = length vs; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"
+ by simp
+
+lemma firstn_nil [simp]: "firstn n [] = []"
+ by (cases n, simp+)
+
+(*
+value "from_to 0 2 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] @
+ from_to 2 5 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"
+*)
+
+lemma firstn_le: "\<And> n s'. n \<le> length s \<Longrightarrow> firstn n (s@s') = firstn n s"
+proof (induct s, simp)
+ fix a s n s'
+ assume ih: "\<And>n s'. n \<le> length s \<Longrightarrow> firstn n (s @ s') = firstn n s"
+ and le_n: " n \<le> length (a # s)"
+ show "firstn n ((a # s) @ s') = firstn n (a # s)"
+ proof(cases n, simp)
+ fix k
+ assume eq_n: "n = Suc k"
+ with le_n have "k \<le> length s" by auto
+ from ih [OF this] and eq_n
+ show "firstn n ((a # s) @ s') = firstn n (a # s)" by auto
+ qed
+qed
+
+lemma firstn_ge [simp]: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"
+proof(induct s, simp)
+ fix a s n
+ assume ih: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"
+ and le: "length (a # s) \<le> n"
+ show "firstn n (a # s) = a # s"
+ proof(cases n)
+ assume eq_n: "n = 0" with le show ?thesis by simp
+ next
+ fix k
+ assume eq_n: "n = Suc k"
+ with le have le_k: "length s \<le> k" by simp
+ from ih [OF this] have "firstn k s = s" .
+ from eq_n and this
+ show ?thesis by simp
+ qed
+qed
+
+lemma firstn_eq [simp]: "firstn (length s) s = s"
+ by simp
+
+lemma firstn_restn_s: "(firstn n (s::'a list)) @ (restn n s) = s"
+proof(induct n arbitrary:s, simp)
+ fix n s
+ assume ih: "\<And>t. firstn n (t::'a list) @ restn n t = t"
+ show "firstn (Suc n) (s::'a list) @ restn (Suc n) s = s"
+ proof(cases s, simp)
+ fix x xs
+ assume eq_s: "s = x#xs"
+ show "firstn (Suc n) s @ restn (Suc n) s = s"
+ proof -
+ have "firstn (Suc n) s @ restn (Suc n) s = x # (firstn n xs @ restn n xs)"
+ proof -
+ from eq_s have "firstn (Suc n) s = x # firstn n xs" by simp
+ moreover have "restn (Suc n) s = restn n xs"
+ proof -
+ from eq_s have "restn (Suc n) s = rev (firstn (length xs - n) (rev xs @ [x]))" by simp
+ also have "\<dots> = restn n xs"
+ proof -
+ have "(firstn (length xs - n) (rev xs @ [x])) = (firstn (length xs - n) (rev xs))"
+ by(rule firstn_le, simp)
+ hence "rev (firstn (length xs - n) (rev xs @ [x])) =
+ rev (firstn (length xs - n) (rev xs))" by simp
+ also have "\<dots> = rev (firstn (length (rev xs) - n) (rev xs))" by simp
+ finally show ?thesis by simp
+ qed
+ finally show ?thesis by simp
+ qed
+ ultimately show ?thesis by simp
+ qed with ih eq_s show ?thesis by simp
+ qed
+ qed
+qed
+
+lemma moment_restm_s: "(restm n s)@(moment n s) = s"
+proof -
+ have " rev ((firstn n (rev s)) @ (restn n (rev s))) = s" (is "rev ?x = s")
+ proof -
+ have "?x = rev s" by (simp only:firstn_restn_s)
+ thus ?thesis by auto
+ qed
+ thus ?thesis
+ by (auto simp:restm_def moment_def)
+qed
+
+declare restn.simps [simp del] firstn.simps[simp del]
+
+lemma length_firstn_ge: "length s \<le> n \<Longrightarrow> length (firstn n s) = length s"
+proof(induct n arbitrary:s, simp add:firstn.simps)
+ case (Suc k)
+ assume ih: "\<And> s. length (s::'a list) \<le> k \<Longrightarrow> length (firstn k s) = length s"
+ and le: "length s \<le> Suc k"
+ show ?case
+ proof(cases s)
+ case Nil
+ from Nil show ?thesis by simp
+ next
+ case (Cons x xs)
+ from le and Cons have "length xs \<le> k" by simp
+ from ih [OF this] have "length (firstn k xs) = length xs" .
+ moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))"
+ by (simp add:firstn.simps)
+ moreover note Cons
+ ultimately show ?thesis by simp
+ qed
+qed
+
+lemma length_firstn_le: "n \<le> length s \<Longrightarrow> length (firstn n s) = n"
+proof(induct n arbitrary:s, simp add:firstn.simps)
+ case (Suc k)
+ assume ih: "\<And>s. k \<le> length (s::'a list) \<Longrightarrow> length (firstn k s) = k"
+ and le: "Suc k \<le> length s"
+ show ?case
+ proof(cases s)
+ case Nil
+ from Nil and le show ?thesis by auto
+ next
+ case (Cons x xs)
+ from le and Cons have "k \<le> length xs" by simp
+ from ih [OF this] have "length (firstn k xs) = k" .
+ moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))"
+ by (simp add:firstn.simps)
+ ultimately show ?thesis by simp
+ qed
+qed
+
+lemma app_firstn_restn:
+ fixes s1 s2
+ shows "s1 = firstn (length s1) (s1 @ s2) \<and> s2 = restn (length s1) (s1 @ s2)"
+proof(rule length_eq_elim_l)
+ have "length s1 \<le> length (s1 @ s2)" by simp
+ from length_firstn_le [OF this]
+ show "length s1 = length (firstn (length s1) (s1 @ s2))" by simp
+next
+ from firstn_restn_s
+ show "s1 @ s2 = firstn (length s1) (s1 @ s2) @ restn (length s1) (s1 @ s2)"
+ by metis
+qed
+
+
+lemma length_moment_le:
+ fixes k s
+ assumes le_k: "k \<le> length s"
+ shows "length (moment k s) = k"
+proof -
+ have "length (rev (firstn k (rev s))) = k"
+ proof -
+ have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
+ also have "\<dots> = k"
+ proof(rule length_firstn_le)
+ from le_k show "k \<le> length (rev s)" by simp
+ qed
+ finally show ?thesis .
+ qed
+ thus ?thesis by (simp add:moment_def)
+qed
+
+lemma app_moment_restm:
+ fixes s1 s2
+ shows "s1 = restm (length s2) (s1 @ s2) \<and> s2 = moment (length s2) (s1 @ s2)"
+proof(rule length_eq_elim_r)
+ have "length s2 \<le> length (s1 @ s2)" by simp
+ from length_moment_le [OF this]
+ show "length s2 = length (moment (length s2) (s1 @ s2))" by simp
+next
+ from moment_restm_s
+ show "s1 @ s2 = restm (length s2) (s1 @ s2) @ moment (length s2) (s1 @ s2)"
+ by metis
+qed
+
+lemma length_moment_ge:
+ fixes k s
+ assumes le_k: "length s \<le> k"
+ shows "length (moment k s) = (length s)"
+proof -
+ have "length (rev (firstn k (rev s))) = length s"
+ proof -
+ have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
+ also have "\<dots> = length s"
+ proof -
+ have "\<dots> = length (rev s)"
+ proof(rule length_firstn_ge)
+ from le_k show "length (rev s) \<le> k" by simp
+ qed
+ also have "\<dots> = length s" by simp
+ finally show ?thesis .
+ qed
+ finally show ?thesis .
+ qed
+ thus ?thesis by (simp add:moment_def)
+qed
+
+lemma length_firstn: "(length (firstn n s) = length s) \<or> (length (firstn n s) = n)"
+proof(cases "n \<le> length s")
+ case True
+ from length_firstn_le [OF True] show ?thesis by auto
+next
+ case False
+ from False have "length s \<le> n" by simp
+ from firstn_ge [OF this] show ?thesis by auto
+qed
+
+lemma firstn_conc:
+ fixes m n
+ assumes le_mn: "m \<le> n"
+ shows "firstn m s = firstn m (firstn n s)"
+proof(cases "m \<le> length s")
+ case True
+ have "s = (firstn n s) @ (restn n s)" by (simp add:firstn_restn_s)
+ hence "firstn m s = firstn m \<dots>" by simp
+ also have "\<dots> = firstn m (firstn n s)"
+ proof -
+ from length_firstn [of n s]
+ have "m \<le> length (firstn n s)"
+ proof
+ assume "length (firstn n s) = length s" with True show ?thesis by simp
+ next
+ assume "length (firstn n s) = n " with le_mn show ?thesis by simp
+ qed
+ from firstn_le [OF this, of "restn n s"]
+ show ?thesis .
+ qed
+ finally show ?thesis by simp
+next
+ case False
+ from False and le_mn have "length s \<le> n" by simp
+ from firstn_ge [OF this] show ?thesis by simp
+qed
+
+lemma restn_conc:
+ fixes i j k s
+ assumes eq_k: "j + i = k"
+ shows "restn k s = restn j (restn i s)"
+proof -
+ have "(firstn (length s - k) (rev s)) =
+ (firstn (length (rev (firstn (length s - i) (rev s))) - j)
+ (rev (rev (firstn (length s - i) (rev s)))))"
+ proof -
+ have "(firstn (length s - k) (rev s)) =
+ (firstn (length (rev (firstn (length s - i) (rev s))) - j)
+ (firstn (length s - i) (rev s)))"
+ proof -
+ have " (length (rev (firstn (length s - i) (rev s))) - j) = length s - k"
+ proof -
+ have "(length (rev (firstn (length s - i) (rev s))) - j) = (length s - i) - j"
+ proof -
+ have "(length (rev (firstn (length s - i) (rev s))) - j) =
+ length ((firstn (length s - i) (rev s))) - j"
+ by simp
+ also have "\<dots> = length ((firstn (length (rev s) - i) (rev s))) - j" by simp
+ also have "\<dots> = (length (rev s) - i) - j"
+ proof -
+ have "length ((firstn (length (rev s) - i) (rev s))) = (length (rev s) - i)"
+ by (rule length_firstn_le, simp)
+ thus ?thesis by simp
+ qed
+ also have "\<dots> = (length s - i) - j" by simp
+ finally show ?thesis .
+ qed
+ with eq_k show ?thesis by auto
+ qed
+ moreover have "(firstn (length s - k) (rev s)) =
+ (firstn (length s - k) (firstn (length s - i) (rev s)))"
+ proof(rule firstn_conc)
+ from eq_k show "length s - k \<le> length s - i" by simp
+ qed
+ ultimately show ?thesis by simp
+ qed
+ thus ?thesis by simp
+ qed
+ thus ?thesis by (simp only:restn.simps)
+qed
+
+(*
+value "down_to 2 0 [5, 4, 3, 2, 1, 0]"
+value "moment 2 [5, 4, 3, 2, 1, 0]"
+*)
+
+lemma from_to_firstn: "from_to 0 k s = firstn k s"
+by (simp add:from_to_def restn.simps)
+
+lemma moment_app [simp]:
+ assumes
+ ile: "i \<le> length s"
+ shows "moment i (s'@s) = moment i s"
+proof -
+ have "moment i (s'@s) = rev (firstn i (rev (s'@s)))" by (simp add:moment_def)
+ moreover have "firstn i (rev (s'@s)) = firstn i (rev s @ rev s')" by simp
+ moreover have "\<dots> = firstn i (rev s)"
+ proof(rule firstn_le)
+ have "length (rev s) = length s" by simp
+ with ile show "i \<le> length (rev s)" by simp
+ qed
+ ultimately show ?thesis by (simp add:moment_def)
+qed
+
+lemma moment_eq [simp]: "moment (length s) (s'@s) = s"
+proof -
+ have "length s \<le> length s" by simp
+ from moment_app [OF this, of s']
+ have " moment (length s) (s' @ s) = moment (length s) s" .
+ moreover have "\<dots> = s" by (simp add:moment_def)
+ ultimately show ?thesis by simp
+qed
+
+lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"
+ by (unfold moment_def, simp)
+
+lemma moment_zero [simp]: "moment 0 s = []"
+ by (simp add:moment_def firstn.simps)
+
+lemma p_split_gen:
+ "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk> \<Longrightarrow>
+ (\<exists> i. i < length s \<and> k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
+proof (induct s, simp)
+ fix a s
+ assume ih: "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk>
+ \<Longrightarrow> \<exists>i<length s. k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall>i'>i. Q (moment i' s))"
+ and nq: "\<not> Q (moment k (a # s))" and qa: "Q (a # s)"
+ have le_k: "k \<le> length s"
+ proof -
+ { assume "length s < k"
+ hence "length (a#s) \<le> k" by simp
+ from moment_ge [OF this] and nq and qa
+ have "False" by auto
+ } thus ?thesis by arith
+ qed
+ have nq_k: "\<not> Q (moment k s)"
+ proof -
+ have "moment k (a#s) = moment k s"
+ proof -
+ from moment_app [OF le_k, of "[a]"] show ?thesis by simp
+ qed
+ with nq show ?thesis by simp
+ qed
+ show "\<exists>i<length (a # s). k \<le> i \<and> \<not> Q (moment i (a # s)) \<and> (\<forall>i'>i. Q (moment i' (a # s)))"
+ proof -
+ { assume "Q s"
+ from ih [OF this nq_k]
+ obtain i where lti: "i < length s"
+ and nq: "\<not> Q (moment i s)"
+ and rst: "\<forall>i'>i. Q (moment i' s)"
+ and lki: "k \<le> i" by auto
+ have ?thesis
+ proof -
+ from lti have "i < length (a # s)" by auto
+ moreover have " \<not> Q (moment i (a # s))"
+ proof -
+ from lti have "i \<le> (length s)" by simp
+ from moment_app [OF this, of "[a]"]
+ have "moment i (a # s) = moment i s" by simp
+ with nq show ?thesis by auto
+ qed
+ moreover have " (\<forall>i'>i. Q (moment i' (a # s)))"
+ proof -
+ {
+ fix i'
+ assume lti': "i < i'"
+ have "Q (moment i' (a # s))"
+ proof(cases "length (a#s) \<le> i'")
+ case True
+ from True have "moment i' (a#s) = a#s" by simp
+ with qa show ?thesis by simp
+ next
+ case False
+ from False have "i' \<le> length s" by simp
+ from moment_app [OF this, of "[a]"]
+ have "moment i' (a#s) = moment i' s" by simp
+ with rst lti' show ?thesis by auto
+ qed
+ } thus ?thesis by auto
+ qed
+ moreover note lki
+ ultimately show ?thesis by auto
+ qed
+ } moreover {
+ assume ns: "\<not> Q s"
+ have ?thesis
+ proof -
+ let ?i = "length s"
+ have "\<not> Q (moment ?i (a#s))"
+ proof -
+ have "?i \<le> length s" by simp
+ from moment_app [OF this, of "[a]"]
+ have "moment ?i (a#s) = moment ?i s" by simp
+ moreover have "\<dots> = s" by simp
+ ultimately show ?thesis using ns by auto
+ qed
+ moreover have "\<forall> i' > ?i. Q (moment i' (a#s))"
+ proof -
+ { fix i'
+ assume "i' > ?i"
+ hence "length (a#s) \<le> i'" by simp
+ from moment_ge [OF this]
+ have " moment i' (a # s) = a # s" .
+ with qa have "Q (moment i' (a#s))" by simp
+ } thus ?thesis by auto
+ qed
+ moreover have "?i < length (a#s)" by simp
+ moreover note le_k
+ ultimately show ?thesis by auto
+ qed
+ } ultimately show ?thesis by auto
+ qed
+qed
+
+lemma p_split:
+ "\<And> s Q. \<lbrakk>Q s; \<not> Q []\<rbrakk> \<Longrightarrow>
+ (\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
+proof -
+ fix s Q
+ assume qs: "Q s" and nq: "\<not> Q []"
+ from nq have "\<not> Q (moment 0 s)" by simp
+ from p_split_gen [of Q s 0, OF qs this]
+ show "(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
+ by auto
+qed
+
+lemma moment_plus:
+ "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = (hd (moment (Suc i) s)) # (moment i s)"
+proof(induct s, simp+)
+ fix a s
+ assume ih: "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = hd (moment (Suc i) s) # moment i s"
+ and le_i: "i \<le> length s"
+ show "moment (Suc i) (a # s) = hd (moment (Suc i) (a # s)) # moment i (a # s)"
+ proof(cases "i= length s")
+ case True
+ hence "Suc i = length (a#s)" by simp
+ with moment_eq have "moment (Suc i) (a#s) = a#s" by auto
+ moreover have "moment i (a#s) = s"
+ proof -
+ from moment_app [OF le_i, of "[a]"]
+ and True show ?thesis by simp
+ qed
+ ultimately show ?thesis by auto
+ next
+ case False
+ from False and le_i have lti: "i < length s" by arith
+ hence les_i: "Suc i \<le> length s" by arith
+ show ?thesis
+ proof -
+ from moment_app [OF les_i, of "[a]"]
+ have "moment (Suc i) (a # s) = moment (Suc i) s" by simp
+ moreover have "moment i (a#s) = moment i s"
+ proof -
+ from lti have "i \<le> length s" by simp
+ from moment_app [OF this, of "[a]"] show ?thesis by simp
+ qed
+ moreover note ih [OF les_i]
+ ultimately show ?thesis by auto
+ qed
+ qed
+qed
+
+lemma from_to_conc:
+ fixes i j k s
+ assumes le_ij: "i \<le> j"
+ and le_jk: "j \<le> k"
+ shows "from_to i j s @ from_to j k s = from_to i k s"
+proof -
+ let ?ris = "restn i s"
+ have "firstn (j - i) (restn i s) @ firstn (k - j) (restn j s) =
+ firstn (k - i) (restn i s)" (is "?x @ ?y = ?z")
+ proof -
+ let "firstn (k-j) ?u" = "?y"
+ let ?rst = " restn (k - j) (restn (j - i) ?ris)"
+ let ?rst' = "restn (k - i) ?ris"
+ have "?u = restn (j-i) ?ris"
+ proof(rule restn_conc)
+ from le_ij show "j - i + i = j" by simp
+ qed
+ hence "?x @ ?y = ?x @ firstn (k-j) (restn (j-i) ?ris)" by simp
+ moreover have "firstn (k - j) (restn (j - i) (restn i s)) @ ?rst =
+ restn (j-i) ?ris" by (simp add:firstn_restn_s)
+ ultimately have "?x @ ?y @ ?rst = ?x @ (restn (j-i) ?ris)" by simp
+ also have "\<dots> = ?ris" by (simp add:firstn_restn_s)
+ finally have "?x @ ?y @ ?rst = ?ris" .
+ moreover have "?z @ ?rst = ?ris"
+ proof -
+ have "?z @ ?rst' = ?ris" by (simp add:firstn_restn_s)
+ moreover have "?rst' = ?rst"
+ proof(rule restn_conc)
+ from le_ij le_jk show "k - j + (j - i) = k - i" by auto
+ qed
+ ultimately show ?thesis by simp
+ qed
+ ultimately have "?x @ ?y @ ?rst = ?z @ ?rst" by simp
+ thus ?thesis by auto
+ qed
+ thus ?thesis by (simp only:from_to_def)
+qed
+
+lemma down_to_conc:
+ fixes i j k s
+ assumes le_ij: "i \<le> j"
+ and le_jk: "j \<le> k"
+ shows "down_to k j s @ down_to j i s = down_to k i s"
+proof -
+ have "rev (from_to j k (rev s)) @ rev (from_to i j (rev s)) = rev (from_to i k (rev s))"
+ (is "?L = ?R")
+ proof -
+ have "?L = rev (from_to i j (rev s) @ from_to j k (rev s))" by simp
+ also have "\<dots> = ?R" (is "rev ?x = rev ?y")
+ proof -
+ have "?x = ?y" by (rule from_to_conc[OF le_ij le_jk])
+ thus ?thesis by simp
+ qed
+ finally show ?thesis .
+ qed
+ thus ?thesis by (simp add:down_to_def)
+qed
+
+lemma restn_ge:
+ fixes s k
+ assumes le_k: "length s \<le> k"
+ shows "restn k s = []"
+proof -
+ from firstn_restn_s [of k s, symmetric] have "s = (firstn k s) @ (restn k s)" .
+ hence "length s = length \<dots>" by simp
+ also have "\<dots> = length (firstn k s) + length (restn k s)" by simp
+ finally have "length s = ..." by simp
+ moreover from length_firstn_ge and le_k
+ have "length (firstn k s) = length s" by simp
+ ultimately have "length (restn k s) = 0" by auto
+ thus ?thesis by auto
+qed
+
+lemma from_to_ge: "length s \<le> k \<Longrightarrow> from_to k j s = []"
+proof(simp only:from_to_def)
+ assume "length s \<le> k"
+ from restn_ge [OF this]
+ show "firstn (j - k) (restn k s) = []" by simp
+qed
+
+(*
+value "from_to 2 5 [0, 1, 2, 3, 4]"
+value "restn 2 [0, 1, 2, 3, 4]"
+*)
+
+lemma from_to_restn:
+ fixes k j s
+ assumes le_j: "length s \<le> j"
+ shows "from_to k j s = restn k s"
+proof -
+ have "from_to 0 k s @ from_to k j s = from_to 0 j s"
+ proof(cases "k \<le> j")
+ case True
+ from from_to_conc True show ?thesis by auto
+ next
+ case False
+ from False le_j have lek: "length s \<le> k" by auto
+ from from_to_ge [OF this] have "from_to k j s = []" .
+ hence "from_to 0 k s @ from_to k j s = from_to 0 k s" by simp
+ also have "\<dots> = s"
+ proof -
+ from from_to_firstn [of k s]
+ have "\<dots> = firstn k s" .
+ also have "\<dots> = s" by (rule firstn_ge [OF lek])
+ finally show ?thesis .
+ qed
+ finally have "from_to 0 k s @ from_to k j s = s" .
+ moreover have "from_to 0 j s = s"
+ proof -
+ have "from_to 0 j s = firstn j s" by (simp add:from_to_firstn)
+ also have "\<dots> = s"
+ proof(rule firstn_ge)
+ from le_j show "length s \<le> j " by simp
+ qed
+ finally show ?thesis .
+ qed
+ ultimately show ?thesis by auto
+ qed
+ also have "\<dots> = s"
+ proof -
+ from from_to_firstn have "\<dots> = firstn j s" .
+ also have "\<dots> = s"
+ proof(rule firstn_ge)
+ from le_j show "length s \<le> j" by simp
+ qed
+ finally show ?thesis .
+ qed
+ finally have "from_to 0 k s @ from_to k j s = s" .
+ moreover have "from_to 0 k s @ restn k s = s"
+ proof -
+ from from_to_firstn [of k s]
+ have "from_to 0 k s = firstn k s" .
+ thus ?thesis by (simp add:firstn_restn_s)
+ qed
+ ultimately have "from_to 0 k s @ from_to k j s =
+ from_to 0 k s @ restn k s" by simp
+ thus ?thesis by auto
+qed
+
+lemma down_to_moment: "down_to k 0 s = moment k s"
+proof -
+ have "rev (from_to 0 k (rev s)) = rev (firstn k (rev s))"
+ using from_to_firstn by metis
+ thus ?thesis by (simp add:down_to_def moment_def)
+qed
+
+lemma down_to_restm:
+ assumes le_s: "length s \<le> j"
+ shows "down_to j k s = restm k s"
+proof -
+ have "rev (from_to k j (rev s)) = rev (restn k (rev s))" (is "?L = ?R")
+ proof -
+ from le_s have "length (rev s) \<le> j" by simp
+ from from_to_restn [OF this, of k] show ?thesis by simp
+ qed
+ thus ?thesis by (simp add:down_to_def restm_def)
+qed
+
+lemma moment_split: "moment (m+i) s = down_to (m+i) i s @down_to i 0 s"
+proof -
+ have "moment (m + i) s = down_to (m+i) 0 s" using down_to_moment by metis
+ also have "\<dots> = (down_to (m+i) i s) @ (down_to i 0 s)"
+ by(rule down_to_conc[symmetric], auto)
+ finally show ?thesis .
+qed
+
+lemma length_restn: "length (restn i s) = length s - i"
+proof(cases "i \<le> length s")
+ case True
+ from length_firstn_le [OF this] have "length (firstn i s) = i" .
+ moreover have "length s = length (firstn i s) + length (restn i s)"
+ proof -
+ have "s = firstn i s @ restn i s" using firstn_restn_s by metis
+ hence "length s = length \<dots>" by simp
+ thus ?thesis by simp
+ qed
+ ultimately show ?thesis by simp
+next
+ case False
+ hence "length s \<le> i" by simp
+ from restn_ge [OF this] have "restn i s = []" .
+ with False show ?thesis by simp
+qed
+
+lemma length_from_to_in:
+ fixes i j s
+ assumes le_ij: "i \<le> j"
+ and le_j: "j \<le> length s"
+ shows "length (from_to i j s) = j - i"
+proof -
+ have "from_to 0 j s = from_to 0 i s @ from_to i j s"
+ by (rule from_to_conc[symmetric, OF _ le_ij], simp)
+ moreover have "length (from_to 0 j s) = j"
+ proof -
+ have "from_to 0 j s = firstn j s" using from_to_firstn by metis
+ moreover have "length \<dots> = j" by (rule length_firstn_le [OF le_j])
+ ultimately show ?thesis by simp
+ qed
+ moreover have "length (from_to 0 i s) = i"
+ proof -
+ have "from_to 0 i s = firstn i s" using from_to_firstn by metis
+ moreover have "length \<dots> = i"
+ proof (rule length_firstn_le)
+ from le_ij le_j show "i \<le> length s" by simp
+ qed
+ ultimately show ?thesis by simp
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma firstn_restn_from_to: "from_to i (m + i) s = firstn m (restn i s)"
+proof(cases "m+i \<le> length s")
+ case True
+ have "restn i s = from_to i (m+i) s @ from_to (m+i) (length s) s"
+ proof -
+ have "restn i s = from_to i (length s) s"
+ by(rule from_to_restn[symmetric], simp)
+ also have "\<dots> = from_to i (m+i) s @ from_to (m+i) (length s) s"
+ by(rule from_to_conc[symmetric, OF _ True], simp)
+ finally show ?thesis .
+ qed
+ hence "firstn m (restn i s) = firstn m \<dots>" by simp
+ moreover have "\<dots> = firstn (length (from_to i (m+i) s))
+ (from_to i (m+i) s @ from_to (m+i) (length s) s)"
+ proof -
+ have "length (from_to i (m+i) s) = m"
+ proof -
+ have "length (from_to i (m+i) s) = (m+i) - i"
+ by(rule length_from_to_in [OF _ True], simp)
+ thus ?thesis by simp
+ qed
+ thus ?thesis by simp
+ qed
+ ultimately show ?thesis using app_firstn_restn by metis
+next
+ case False
+ hence "length s \<le> m + i" by simp
+ from from_to_restn [OF this]
+ have "from_to i (m + i) s = restn i s" .
+ moreover have "firstn m (restn i s) = restn i s"
+ proof(rule firstn_ge)
+ show "length (restn i s) \<le> m"
+ proof -
+ have "length (restn i s) = length s - i" using length_restn by metis
+ with False show ?thesis by simp
+ qed
+ qed
+ ultimately show ?thesis by simp
+qed
+
+lemma down_to_moment_restm:
+ fixes m i s
+ shows "down_to (m + i) i s = moment m (restm i s)"
+ by (simp add:firstn_restn_from_to down_to_def moment_def restm_def)
+
+lemma moment_plus_split:
+ fixes m i s
+ shows "moment (m + i) s = moment m (restm i s) @ moment i s"
+proof -
+ from moment_split [of m i s]
+ have "moment (m + i) s = down_to (m + i) i s @ down_to i 0 s" .
+ also have "\<dots> = down_to (m+i) i s @ moment i s" using down_to_moment by simp
+ also from down_to_moment_restm have "\<dots> = moment m (restm i s) @ moment i s"
+ by simp
+ finally show ?thesis .
+qed
+
+lemma length_restm: "length (restm i s) = length s - i"
+proof -
+ have "length (rev (restn i (rev s))) = length s - i" (is "?L = ?R")
+ proof -
+ have "?L = length (restn i (rev s))" by simp
+ also have "\<dots> = length (rev s) - i" using length_restn by metis
+ also have "\<dots> = ?R" by simp
+ finally show ?thesis .
+ qed
+ thus ?thesis by (simp add:restm_def)
+qed
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/Paper.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,168 @@
+(*<*)
+theory Paper
+imports CpsG ExtGG
+begin
+(*>*)
+
+section {* Introduction *}
+
+text {*
+
+ Priority inversion referrers to the phenomena where tasks with higher
+ priority are blocked by ones with lower priority. If priority inversion
+ is not controlled, there will be no guarantee the urgent tasks will be
+ processed in time. As reported in \cite{Reeves-Glenn-1998},
+ priority inversion used to cause software system resets and data lose in
+ JPL's Mars pathfinder project. Therefore, the avoiding, detecting and controlling
+ of priority inversion is a key issue to attain predictability in priority
+ based real-time systems.
+
+ The priority inversion phenomenon was first published in \cite{Lampson:Redell:cacm:1980}.
+ The two protocols widely used to eliminate priority inversion, namely
+ PI (Priority Inheritance) and PCE (Priority Ceiling Emulation), were proposed
+ in \cite{journals/tc/ShaRL90}. PCE is less convenient to use because it requires
+ static analysis of programs. Therefore, PI is more commonly used in
+ practice\cite{locke-july02}. However, as pointed out in the literature,
+ the analysis of priority inheritance protocol is quite subtle\cite{yodaiken-july02}.
+ A formal analysis will certainly be helpful for us to understand and correctly
+ implement PI. All existing formal analysis of PI
+ \cite{conf/fase/JahierHR09,WellingsBSB07,Faria08} are based on the model checking
+ technology. Because of the state explosion problem, model check
+ is much like an exhaustive testing of finite models with limited size.
+ The results obtained can not be safely generalized to models with arbitrarily
+ large size. Worse still, since model checking is fully automatic, it give little
+ insight on why the formal model is correct. It is therefore
+ definitely desirable to analyze PI using theorem proving, which gives
+ more general results as well as deeper insight. And this is the purpose
+ of this paper which gives a formal analysis of PI in the interactive
+ theorem prover Isabelle using Higher Order Logic (HOL). The formalization
+ focuses on on two issues:
+
+ \begin{enumerate}
+ \item The correctness of the protocol model itself. A series of desirable properties is
+ derived until we are fully convinced that the formal model of PI does
+ eliminate priority inversion. And a better understanding of PI is so obtained
+ in due course. For example, we find through formalization that the choice of
+ next thread to take hold when a
+ resource is released is irrelevant for the very basic property of PI to hold.
+ A point never mentioned in literature.
+ \item The correctness of the implementation. A series of properties is derived the meaning
+ of which can be used as guidelines on how PI can be implemented efficiently and correctly.
+ \end{enumerate}
+
+ The rest of the paper is organized as follows: Section \ref{overview} gives an overview
+ of PI. Section \ref{model} introduces the formal model of PI. Section \ref{general}
+ discusses a series of basic properties of PI. Section \ref{extension} shows formally
+ how priority inversion is controlled by PI. Section \ref{implement} gives properties
+ which can be used for guidelines of implementation. Section \ref{related} discusses
+ related works. Section \ref{conclusion} concludes the whole paper.
+*}
+
+section {* An overview of priority inversion and priority inheritance \label{overview} *}
+
+text {*
+
+ Priority inversion refers to the phenomenon when a thread with high priority is blocked
+ by a thread with low priority. Priority happens when the high priority thread requests
+ for some critical resource already taken by the low priority thread. Since the high
+ priority thread has to wait for the low priority thread to complete, it is said to be
+ blocked by the low priority thread. Priority inversion might prevent high priority
+ thread from fulfill its task in time if the duration of priority inversion is indefinite
+ and unpredictable. Indefinite priority inversion happens when indefinite number
+ of threads with medium priorities is activated during the period when the high
+ priority thread is blocked by the low priority thread. Although these medium
+ priority threads can not preempt the high priority thread directly, they are able
+ to preempt the low priority threads and cause it to stay in critical section for
+ an indefinite long duration. In this way, the high priority thread may be blocked indefinitely.
+
+ Priority inheritance is one protocol proposed to avoid indefinite priority inversion.
+ The basic idea is to let the high priority thread donate its priority to the low priority
+ thread holding the critical resource, so that it will not be preempted by medium priority
+ threads. The thread with highest priority will not be blocked unless it is requesting
+ some critical resource already taken by other threads. Viewed from a different angle,
+ any thread which is able to block the highest priority threads must already hold some
+ critical resource. Further more, it must have hold some critical resource at the
+ moment the highest priority is created, otherwise, it may never get change to run and
+ get hold. Since the number of such resource holding lower priority threads is finite,
+ if every one of them finishes with its own critical section in a definite duration,
+ the duration the highest priority thread is blocked is definite as well. The key to
+ guarantee lower priority threads to finish in definite is to donate them the highest
+ priority. In such cases, the lower priority threads is said to have inherited the
+ highest priority. And this explains the name of the protocol:
+ {\em Priority Inheritance} and how Priority Inheritance prevents indefinite delay.
+
+ The objectives of this paper are:
+ \begin{enumerate}
+ \item Build the above mentioned idea into formal model and prove a series of properties
+ until we are convinced that the formal model does fulfill the original idea.
+ \item Show how formally derived properties can be used as guidelines for correct
+ and efficient implementation.
+ \end{enumerate}
+ The proof is totally formal in the sense that every detail is reduced to the
+ very first principles of Higher Order Logic. The nature of interactive theorem
+ proving is for the human user to persuade computer program to accept its arguments.
+ A clear and simple understanding of the problem at hand is both a prerequisite and a
+ byproduct of such an effort, because everything has finally be reduced to the very
+ first principle to be checked mechanically. The former intuitive explanation of
+ Priority Inheritance is just such a byproduct.
+ *}
+
+section {* Formal model of Priority Inheritance \label{model} *}
+text {*
+ \input{../../generated/PrioGDef}
+*}
+
+section {* General properties of Priority Inheritance \label{general} *}
+
+section {* Key properties \label{extension} *}
+
+section {* Properties to guide implementation \label{implement} *}
+
+section {* Related works \label{related} *}
+
+text {*
+ \begin{enumerate}
+ \item {\em Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java}
+ \cite{WellingsBSB07} models and verifies the combination of Priority Inheritance (PI) and
+ Priority Ceiling Emulation (PCE) protocols in the setting of Java virtual machine
+ using extended Timed Automata(TA) formalism of the UPPAAL tool. Although a detailed
+ formal model of combined PI and PCE is given, the number of properties is quite
+ small and the focus is put on the harmonious working of PI and PCE. Most key features of PI
+ (as well as PCE) are not shown. Because of the limitation of the model checking technique
+ used there, properties are shown only for a small number of scenarios. Therefore,
+ the verification does not show the correctness of the formal model itself in a
+ convincing way.
+ \item {\em Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC}
+ \cite{Faria08}. A formal model of PI is given in TLA+. Only 3 properties are shown
+ for PI using model checking. The limitation of model checking is intrinsic to the work.
+ \item {\em Synchronous modeling and validation of priority inheritance schedulers}
+ \cite{conf/fase/JahierHR09}. Gives a formal model
+ of PI and PCE in AADL (Architecture Analysis \& Design Language) and checked
+ several properties using model checking. The number of properties shown there is
+ less than here and the scale is also limited by the model checking technique.
+ \item {\em The Priority Ceiling Protocol: Formalization and Analysis Using PVS}
+ \cite{dutertre99b}. Formalized another protocol for Priority Inversion in the
+ interactive theorem proving system PVS.
+\end{enumerate}
+
+
+ There are several works on inversion avoidance:
+ \begin{enumerate}
+ \item {\em Solving the group priority inversion problem in a timed asynchronous system}
+ \cite{Wang:2002:SGP}. The notion of Group Priority Inversion is introduced. The main
+ strategy is still inversion avoidance. The method is by reordering requests
+ in the setting of Client-Server.
+ \item {\em A Formalization of Priority Inversion} \cite{journals/rts/BabaogluMS93}.
+ Formalized the notion of Priority
+ Inversion and proposes methods to avoid it.
+ \end{enumerate}
+
+ {\em Examples of inaccurate specification of the protocol ???}.
+
+*}
+
+section {* Conclusions \label{conclusion} *}
+
+(*<*)
+end
+(*>*)
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/PrioGDef.tex Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,488 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{PrioGDef}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\begin{isamarkuptext}%
+In this section, the formal model of Priority Inheritance is presented. First, the identifiers of {\em threads},
+ {\em priority} and {\em critical resources } (abbreviated as \isa{cs}) are all represented as natural numbers,
+ i.e. standard Isabelle/HOL type \isa{nat}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ thread\ {\isaliteral{3D}{\isacharequal}}\ nat\ %
+\isamarkupcmt{Type for thread identifiers.%
+}
+\isanewline
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ priority\ {\isaliteral{3D}{\isacharequal}}\ nat\ \ %
+\isamarkupcmt{Type for priorities.%
+}
+\isanewline
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ cs\ {\isaliteral{3D}{\isacharequal}}\ nat\ %
+\isamarkupcmt{Type for critical sections (or critical resources).%
+}
+%
+\begin{isamarkuptext}%
+Priority Inheritance protocol is modeled as an event driven system, where every event represents an
+ system call. Event format is given by the following type definition:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ event\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ Create\ thread\ priority\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Thread \isa{thread} is created with priority \isa{priority}.%
+}
+\isanewline
+\ \ Exit\ thread\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Thread \isa{thread} finishing its execution.%
+}
+\isanewline
+\ \ P\ thread\ cs\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Thread \isa{thread} requesting critical resource \isa{cs}.%
+}
+\isanewline
+\ \ V\ thread\ cs\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Thread \isa{thread} releasing critical resource \isa{cs}.%
+}
+\isanewline
+\ \ Set\ thread\ priority\ %
+\isamarkupcmt{Thread \isa{thread} resets its priority to \isa{priority}.%
+}
+%
+\begin{isamarkuptext}%
+Resource Allocation Graph (RAG for short) is used extensively in the analysis of Priority Inheritance.
+ The following type \isa{node} is used to model nodes in RAG.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ node\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ Th\ {\isaliteral{22}{\isachardoublequoteopen}}thread{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Node for thread.%
+}
+\isanewline
+\ \ \ Cs\ {\isaliteral{22}{\isachardoublequoteopen}}cs{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\isamarkupcmt{Node for critical resource.%
+}
+%
+\begin{isamarkuptext}%
+The protocol is analyzed using Paulson's inductive protocol verification method, where
+ the state of the system is modelled as the list of events happened so far with the latest
+ event at the head. Therefore, the state of the system is represented by the following
+ type \isa{state}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ state\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}event\ list{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following \isa{threads} is used to calculate the set of live threads (\isa{threads\ s})
+ in state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ threads\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ \isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{At the start of the system, the set of threads is empty.%
+}
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}Create\ thread\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}thread{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ threads\ s{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{New thread is added to the \isa{threads}.%
+}
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}Exit\ thread\ {\isaliteral{23}{\isacharhash}}\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}threads\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{7B}{\isacharbraceleft}}thread{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{Finished thread is removed.%
+}
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}threads\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ threads\ s{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\isamarkupcmt{other kind of events does not affect the value of \isa{threads}.%
+}
+%
+\begin{isamarkuptext}%
+Functions such as \isa{threads}, which extract information out of system states, are called
+ {\em observing functions}. A series of observing functions will be defined in the sequel in order to
+ model the protocol.
+ Observing function \isa{original{\isaliteral{5F}{\isacharunderscore}}priority} calculates
+ the {\em original priority} of thread \isa{th} in state \isa{s}, expressed as
+ : \isa{original{\isaliteral{5F}{\isacharunderscore}}priority\ th\ s}. The {\em original priority} is the priority
+ assigned to a thread when it is created or when it is reset by system call \isa{Set\ thread\ priority}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ original{\isaliteral{5F}{\isacharunderscore}}priority\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ priority{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ %
+\isamarkupcmt{\isa{{\isadigit{0}}} is assigned to threads which have never been created.%
+}
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}Create\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ thread{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ thread\ then\ prio\ else\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}Set\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ thread{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ thread\ then\ prio\ else\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{birthtime\ th\ s} is the time when thread \isa{th} is created, observed from state \isa{s}.
+ The time in the system is measured by the number of events happened so far since the very beginning.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ birthtime\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Create\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{3D}{\isacharequal}}\ thread{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ then\ length\ s\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Set\ thread{\isaliteral{27}{\isacharprime}}\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{3D}{\isacharequal}}\ thread{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ then\ length\ s\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}birthtime\ thread\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ birthtime\ thread\ s{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of
+ a thread is the combination of its {\em original priority} and {\em birth time}. The intention is
+ to discriminate threads with the same priority by giving threads with the earlier assigned priority
+ higher precedence in scheduling. This explains the following definition:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ preced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}preced\ thread\ s\ {\isaliteral{3D}{\isacharequal}}\ Prc\ {\isaliteral{28}{\isacharparenleft}}original{\isaliteral{5F}{\isacharunderscore}}priority\ thread\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}birthtime\ thread\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+A number of important notions are defined here:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{consts}\isamarkupfalse%
+\ \isanewline
+\ \ holding\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isanewline
+\ \ \ \ \ \ \ waiting\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \ \ \ depend\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}node\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ node{\isaliteral{29}{\isacharparenright}}\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \ \ \ dependents\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The definition of the following several functions, it is supposed that
+ the waiting queue of every critical resource is given by a waiting queue
+ function \isa{wq}, which servers as arguments of these functions.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{defs}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{overloaded}{\isaliteral{29}{\isacharparenright}}\ \isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+ We define that the thread which is at the head of waiting queue of resource \isa{cs}
+ is holding the resource. This definition is slightly different from tradition where
+ all threads in the waiting queue are considered as waiting for the resource.
+ This notion is reflected in the definition of \isa{holding\ wq\ th\ cs} as follows:
+ \end{minipage}%
+}
+\isanewline
+\ \ cs{\isaliteral{5F}{\isacharunderscore}}holding{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}holding\ wq\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+ In accordance with the definition of \isa{holding\ wq\ th\ cs},
+ a thread \isa{th} is considered waiting for \isa{cs} if
+ it is in the {\em waiting queue} of critical resource \isa{cs}, but not at the head.
+ This is reflected in the definition of \isa{waiting\ wq\ th\ cs} as follows:
+ \end{minipage}%
+}
+\isanewline
+\ \ cs{\isaliteral{5F}{\isacharunderscore}}waiting{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}waiting\ wq\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+ \isa{depend\ wq} represents the Resource Allocation Graph of the system under the waiting
+ queue function \isa{wq}.
+ \end{minipage}%
+}
+\isanewline
+\ \ cs{\isaliteral{5F}{\isacharunderscore}}depend{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}depend\ {\isaliteral{28}{\isacharparenleft}}wq{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Th\ t{\isaliteral{2C}{\isacharcomma}}\ Cs\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ t\ c{\isaliteral{2E}{\isachardot}}\ waiting\ wq\ t\ c{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Cs\ c{\isaliteral{2C}{\isacharcomma}}\ Th\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ c\ t{\isaliteral{2E}{\isachardot}}\ holding\ wq\ t\ c{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+ \isa{dependents\ wq\ th} represents the set of threads which are depending on
+ thread \isa{th} in Resource Allocation Graph \isa{depend\ wq}:
+ \end{minipage}%
+}
+\isanewline
+\ \ cs{\isaliteral{5F}{\isacharunderscore}}dependents{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dependents\ {\isaliteral{28}{\isacharparenleft}}wq{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ th\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{27}{\isacharprime}}\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ wq{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The data structure used by the operating system for scheduling is referred to as
+ {\em schedule state}. It is represented as a record consisting of
+ a function assigning waiting queue to resources and a function assigning precedence to
+ threads:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{record}\isamarkupfalse%
+\ schedule{\isaliteral{5F}{\isacharunderscore}}state\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ \ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\isamarkupcmt{The function assigning waiting queue.%
+}
+\isanewline
+\ \ \ \ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\isamarkupcmt{The function assigning precedence.%
+}
+%
+\begin{isamarkuptext}%
+\isa{cpreced\ s\ th} gives the {\em current precedence} of thread \isa{th} under
+ state \isa{s}. The definition of \isa{cpreced} reflects the basic idea of
+ Priority Inheritance that the {\em current precedence} of a thread is the precedence
+ inherited from the maximum of all its dependents, i.e. the threads which are waiting
+ directly or indirectly waiting for some resources from it. If no such thread exits,
+ \isa{th}'s {\em current precedence} equals its original precedence, i.e.
+ \isa{preced\ th\ s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cpreced\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cpreced\ s\ wq\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ th{\isaliteral{2E}{\isachardot}}\ Max\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ th{\isaliteral{2E}{\isachardot}}\ preced\ th\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ dependents\ wq\ th{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following function \isa{schs} is used to calculate the schedule state \isa{schs\ s}.
+ It is the key function to model Priority Inheritance:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ schs\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ schedule{\isaliteral{5F}{\isacharunderscore}}state{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ \ {\isaliteral{22}{\isachardoublequoteopen}}schs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{2C}{\isacharcomma}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3D}{\isacharequal}}\ cpreced\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{\begin{minipage}{0.8\textwidth}
+ \begin{enumerate}
+ \item \isa{ps} is the schedule state of last moment.
+ \item \isa{pwq} is the waiting queue function of last moment.
+ \item \isa{pcp} is the precedence function of last moment.
+ \item \isa{nwq} is the new waiting queue function. It is calculated using a \isa{case} statement:
+ \begin{enumerate}
+ \item If the happening event is \isa{P\ thread\ cs}, \isa{thread} is added to
+ the end of \isa{cs}'s waiting queue.
+ \item If the happening event is \isa{V\ thread\ cs} and \isa{s} is a legal state,
+ \isa{th{\isaliteral{27}{\isacharprime}}} must equal to \isa{thread},
+ because \isa{thread} is the one currently holding \isa{cs}.
+ The case \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} may never be executed in a legal state.
+ the \isa{{\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ qs{\isaliteral{29}{\isacharparenright}}} is used to choose arbitrarily one
+ thread in waiting to take over the released resource \isa{cs}. In our representation,
+ this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
+ \item For other happening event, the schedule state just does not change.
+ \end{enumerate}
+ \item \isa{ncp} is new precedence function, it is calculated from the newly updated waiting queue
+ function. The dependency of precedence function on waiting queue function is the reason to
+ put them in the same record so that they can evolve together.
+ \end{enumerate}
+ \end{minipage}%
+}
+\isanewline
+\ \ \ {\isaliteral{22}{\isachardoublequoteopen}}schs\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}let\ ps\ {\isaliteral{3D}{\isacharequal}}\ schs\ s\ in\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ pwq\ {\isaliteral{3D}{\isacharequal}}\ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ ps\ in\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ pcp\ {\isaliteral{3D}{\isacharequal}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ ps\ in\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ nwq\ {\isaliteral{3D}{\isacharequal}}\ case\ e\ of\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P\ thread\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ \ pwq{\isaliteral{28}{\isacharparenleft}}cs{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{28}{\isacharparenleft}}pwq\ cs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}thread{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ V\ thread\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ let\ nq\ {\isaliteral{3D}{\isacharequal}}\ case\ {\isaliteral{28}{\isacharparenleft}}pwq\ cs{\isaliteral{29}{\isacharparenright}}\ of\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{7C}{\isacharbar}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}th{\isaliteral{27}{\isacharprime}}{\isaliteral{23}{\isacharhash}}qs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ qs{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ pwq{\isaliteral{28}{\isacharparenleft}}cs{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}nq{\isaliteral{29}{\isacharparenright}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ pwq\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ let\ ncp\ {\isaliteral{3D}{\isacharequal}}\ cpreced\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}\ nwq\ in\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{3D}{\isacharequal}}\ nwq{\isaliteral{2C}{\isacharcomma}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{3D}{\isacharequal}}\ ncp{\isaliteral{5C3C72706172723E}{\isasymrparr}}\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{wq} is a shorthand for \isa{waiting{\isaliteral{5F}{\isacharunderscore}}queue}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ wq\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ list{\isaliteral{22}{\isachardoublequoteclose}}\ \isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}wq\ s\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ waiting{\isaliteral{5F}{\isacharunderscore}}queue\ {\isaliteral{28}{\isacharparenleft}}schs\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{cp} is a shorthand for \isa{cur{\isaliteral{5F}{\isacharunderscore}}preced}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cp\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ precedence{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cp\ s\ {\isaliteral{3D}{\isacharequal}}\ cur{\isaliteral{5F}{\isacharunderscore}}preced\ {\isaliteral{28}{\isacharparenleft}}schs\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+Functions \isa{holding}, \isa{waiting}, \isa{depend} and \isa{dependents} still have the
+ same meaning, but redefined so that they no longer depend on the fictitious {\em waiting queue function}
+ \isa{wq}, but on system state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{defs}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{overloaded}{\isaliteral{29}{\isacharparenright}}\ \isanewline
+\ \ s{\isaliteral{5F}{\isacharunderscore}}holding{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}holding\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ s{\isaliteral{5F}{\isacharunderscore}}waiting{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}waiting\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ thread\ cs\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ thread\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ hd\ {\isaliteral{28}{\isacharparenleft}}wq\ s\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ s{\isaliteral{5F}{\isacharunderscore}}depend{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}depend\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Th\ t{\isaliteral{2C}{\isacharcomma}}\ Cs\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ t\ c{\isaliteral{2E}{\isachardot}}\ waiting\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}\ t\ c{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}Cs\ c{\isaliteral{2C}{\isacharcomma}}\ Th\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ c\ t{\isaliteral{2E}{\isachardot}}\ holding\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}\ t\ c{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ s{\isaliteral{5F}{\isacharunderscore}}dependents{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dependents\ {\isaliteral{28}{\isacharparenleft}}s{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}state{\isaliteral{29}{\isacharparenright}}\ th\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th{\isaliteral{27}{\isacharprime}}\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ {\isaliteral{28}{\isacharparenleft}}wq\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following function \isa{readys} calculates the set of ready threads. A thread is {\em ready}
+ for running if it is a live thread and it is not waiting for any critical resource.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ readys\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}readys\ s\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
+\ \ \ \ \ {\isaliteral{7B}{\isacharbraceleft}}thread\ {\isaliteral{2E}{\isachardot}}\ thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ threads\ s\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}\ cs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ waiting\ s\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following function \isa{runing} calculates the set of running thread, which is the ready
+ thread with the highest precedence.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ runing\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}runing\ s\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}th\ {\isaliteral{2E}{\isachardot}}\ th\ {\isaliteral{5C3C696E3E}{\isasymin}}\ readys\ s\ {\isaliteral{5C3C616E643E}{\isasymand}}\ cp\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ Max\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}cp\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{60}{\isacharbackquote}}\ {\isaliteral{28}{\isacharparenleft}}readys\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following function \isa{holdents\ s\ th} returns the set of resources held by thread
+ \isa{th} in state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ holdents\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}holdents\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}cs\ {\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{2C}{\isacharcomma}}\ Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ depend\ s{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{cntCS\ s\ th} returns the number of resources held by thread \isa{th} in
+ state \isa{s}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cntCS\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntCS\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ card\ {\isaliteral{28}{\isacharparenleft}}holdents\ s\ th{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The fact that event \isa{e} is eligible to happen next in state \isa{s}
+ is expressed as \isa{step\ s\ e}. The predicate \isa{step} is inductively defined as
+ follows:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{inductive}\isamarkupfalse%
+\ step\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ event\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\isanewline
+\ \ %
+\isamarkupcmt{A thread can be created if it is not a live thread:%
+}
+\isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}create{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ threads\ s{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Create\ thread\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{A thread can exit if it no longer hold any resource:%
+}
+\isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}exit{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ holdents\ s\ thread\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Exit\ thread{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{A thread can request for an critical resource \isa{cs}, if it is running and
+ the request does not form a loop in the current RAG. The latter condition
+ is set up to avoid deadlock. The condition also reflects our assumption all threads are
+ carefully programmed so that deadlock can not happen.%
+}
+\isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}P{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ \ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{2C}{\isacharcomma}}\ Th\ thread{\isaliteral{29}{\isacharparenright}}\ \ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ {\isaliteral{28}{\isacharparenleft}}depend\ s{\isaliteral{29}{\isacharparenright}}{\isaliteral{5E}{\isacharcircum}}{\isaliteral{2B}{\isacharplus}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}P\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{A thread can release a critical resource \isa{cs} if it is running and holding that resource.%
+}
+\isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}V{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{3B}{\isacharsemicolon}}\ \ holding\ s\ thread\ cs{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}V\ thread\ cs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{A thread can adjust its own priority as long as it is current running.%
+}
+\ \ \isanewline
+\ \ thread{\isaliteral{5F}{\isacharunderscore}}set{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}thread\ {\isaliteral{5C3C696E3E}{\isasymin}}\ runing\ s{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ step\ s\ {\isaliteral{28}{\isacharparenleft}}Set\ thread\ prio{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+With predicate \isa{step}, the fact that \isa{s} is a legal state in
+ Priority Inheritance protocol can be expressed as: \isa{vt\ step\ s}, where
+ the predicate \isa{vt} can be defined as the following:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{inductive}\isamarkupfalse%
+\ vt\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ event\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \isakeyword{for}\ cs\ %
+\isamarkupcmt{\isa{cs} is an argument representing any step predicate.%
+}
+\isanewline
+\isakeyword{where}\isanewline
+\ \ %
+\isamarkupcmt{Empty list \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} is a legal state in any protocol:%
+}
+\isanewline
+\ \ vt{\isaliteral{5F}{\isacharunderscore}}nil{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}vt\ cs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+\ \ %
+\isamarkupcmt{If \isa{s} a legal state, and event \isa{e} is eligible to happen
+ in state \isa{s}, then \isa{e{\isaliteral{23}{\isacharhash}}{\isaliteral{23}{\isacharhash}}s} is a legal state as well:%
+}
+\isanewline
+\ \ vt{\isaliteral{5F}{\isacharunderscore}}cons{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}vt\ cs\ s{\isaliteral{3B}{\isacharsemicolon}}\ cs\ s\ e{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ vt\ cs\ {\isaliteral{28}{\isacharparenleft}}e{\isaliteral{23}{\isacharhash}}s{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+It is easy to see that the definition of \isa{vt} is generic. It can be applied to
+ any step predicate to get the set of legal states.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The following two functions \isa{the{\isaliteral{5F}{\isacharunderscore}}cs} and \isa{the{\isaliteral{5F}{\isacharunderscore}}th} are used to extract
+ critical resource and thread respectively out of RAG nodes.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ the{\isaliteral{5F}{\isacharunderscore}}cs\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}node\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}the{\isaliteral{5F}{\isacharunderscore}}cs\ {\isaliteral{28}{\isacharparenleft}}Cs\ cs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ cs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\isacommand{fun}\isamarkupfalse%
+\ the{\isaliteral{5F}{\isacharunderscore}}th\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}node\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}the{\isaliteral{5F}{\isacharunderscore}}th\ {\isaliteral{28}{\isacharparenleft}}Th\ th{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ th{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following predicate \isa{next{\isaliteral{5F}{\isacharunderscore}}th} describe the next thread to
+ take over when a critical resource is released. In \isa{next{\isaliteral{5F}{\isacharunderscore}}th\ s\ th\ cs\ t},
+ \isa{th} is the thread to release, \isa{t} is the one to take over.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ next{\isaliteral{5F}{\isacharunderscore}}th{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ cs\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}next{\isaliteral{5F}{\isacharunderscore}}th\ s\ th\ cs\ t\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}\ rest{\isaliteral{2E}{\isachardot}}\ wq\ s\ cs\ {\isaliteral{3D}{\isacharequal}}\ th{\isaliteral{23}{\isacharhash}}rest\ {\isaliteral{5C3C616E643E}{\isasymand}}\ rest\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t\ {\isaliteral{3D}{\isacharequal}}\ hd\ {\isaliteral{28}{\isacharparenleft}}SOME\ q{\isaliteral{2E}{\isachardot}}\ distinct\ q\ {\isaliteral{5C3C616E643E}{\isasymand}}\ set\ q\ {\isaliteral{3D}{\isacharequal}}\ set\ rest{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The function \isa{count\ Q\ l} is used to count the occurrence of situation \isa{Q}
+ in list \isa{l}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ count\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}count\ Q\ l\ {\isaliteral{3D}{\isacharequal}}\ length\ {\isaliteral{28}{\isacharparenleft}}filter\ Q\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{cntP\ s} returns the number of operation \isa{P} happened
+ before reaching state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cntP\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntP\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ count\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ e{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}\ cs{\isaliteral{2E}{\isachardot}}\ e\ {\isaliteral{3D}{\isacharequal}}\ P\ th\ cs{\isaliteral{29}{\isacharparenright}}\ s{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+\isa{cntV\ s} returns the number of operation \isa{V} happened
+ before reaching state \isa{s}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ cntV\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}state\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ thread\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}cntV\ s\ th\ {\isaliteral{3D}{\isacharequal}}\ count\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\ e{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}\ cs{\isaliteral{2E}{\isachardot}}\ e\ {\isaliteral{3D}{\isacharequal}}\ V\ th\ cs{\isaliteral{29}{\isacharparenright}}\ s{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+\isacommand{end}\isamarkupfalse%
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+\isanewline
+%
+\endisadelimtheory
+\isanewline
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/ROOT.ML Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1 @@
+use_thy "Paper";
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/document/llncs.cls Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1189 @@
+% LLNCS DOCUMENT CLASS -- version 2.13 (28-Jan-2002)
+% Springer Verlag LaTeX2e support for Lecture Notes in Computer Science
+%
+%%
+%% \CharacterTable
+%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%% Digits \0\1\2\3\4\5\6\7\8\9
+%% Exclamation \! Double quote \" Hash (number) \#
+%% Dollar \$ Percent \% Ampersand \&
+%% Acute accent \' Left paren \( Right paren \)
+%% Asterisk \* Plus \+ Comma \,
+%% Minus \- Point \. Solidus \/
+%% Colon \: Semicolon \; Less than \<
+%% Equals \= Greater than \> Question mark \?
+%% Commercial at \@ Left bracket \[ Backslash \\
+%% Right bracket \] Circumflex \^ Underscore \_
+%% Grave accent \` Left brace \{ Vertical bar \|
+%% Right brace \} Tilde \~}
+%%
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesClass{llncs}[2002/01/28 v2.13
+^^J LaTeX document class for Lecture Notes in Computer Science]
+% Options
+\let\if@envcntreset\iffalse
+\DeclareOption{envcountreset}{\let\if@envcntreset\iftrue}
+\DeclareOption{citeauthoryear}{\let\citeauthoryear=Y}
+\DeclareOption{oribibl}{\let\oribibl=Y}
+\let\if@custvec\iftrue
+\DeclareOption{orivec}{\let\if@custvec\iffalse}
+\let\if@envcntsame\iffalse
+\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue}
+\let\if@envcntsect\iffalse
+\DeclareOption{envcountsect}{\let\if@envcntsect\iftrue}
+\let\if@runhead\iffalse
+\DeclareOption{runningheads}{\let\if@runhead\iftrue}
+
+\let\if@openbib\iffalse
+\DeclareOption{openbib}{\let\if@openbib\iftrue}
+
+% languages
+\let\switcht@@therlang\relax
+\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}}
+\def\ds@francais{\def\switcht@@therlang{\switcht@francais}}
+
+\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
+
+\ProcessOptions
+
+\LoadClass[twoside]{article}
+\RequirePackage{multicol} % needed for the list of participants, index
+
+\setlength{\textwidth}{12.2cm}
+\setlength{\textheight}{19.3cm}
+\renewcommand\@pnumwidth{2em}
+\renewcommand\@tocrmarg{3.5em}
+%
+\def\@dottedtocline#1#2#3#4#5{%
+ \ifnum #1>\c@tocdepth \else
+ \vskip \z@ \@plus.2\p@
+ {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \parindent #2\relax\@afterindenttrue
+ \interlinepenalty\@M
+ \leavevmode
+ \@tempdima #3\relax
+ \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
+ {#4}\nobreak
+ \leaders\hbox{$\m@th
+ \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
+ mu$}\hfill
+ \nobreak
+ \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}%
+ \par}%
+ \fi}
+%
+\def\switcht@albion{%
+\def\abstractname{Abstract.}
+\def\ackname{Acknowledgement.}
+\def\andname{and}
+\def\lastandname{\unskip, and}
+\def\appendixname{Appendix}
+\def\chaptername{Chapter}
+\def\claimname{Claim}
+\def\conjecturename{Conjecture}
+\def\contentsname{Table of Contents}
+\def\corollaryname{Corollary}
+\def\definitionname{Definition}
+\def\examplename{Example}
+\def\exercisename{Exercise}
+\def\figurename{Fig.}
+\def\keywordname{{\bf Key words:}}
+\def\indexname{Index}
+\def\lemmaname{Lemma}
+\def\contriblistname{List of Contributors}
+\def\listfigurename{List of Figures}
+\def\listtablename{List of Tables}
+\def\mailname{{\it Correspondence to\/}:}
+\def\noteaddname{Note added in proof}
+\def\notename{Note}
+\def\partname{Part}
+\def\problemname{Problem}
+\def\proofname{Proof}
+\def\propertyname{Property}
+\def\propositionname{Proposition}
+\def\questionname{Question}
+\def\remarkname{Remark}
+\def\seename{see}
+\def\solutionname{Solution}
+\def\subclassname{{\it Subject Classifications\/}:}
+\def\tablename{Table}
+\def\theoremname{Theorem}}
+\switcht@albion
+% Names of theorem like environments are already defined
+% but must be translated if another language is chosen
+%
+% French section
+\def\switcht@francais{%\typeout{On parle francais.}%
+ \def\abstractname{R\'esum\'e.}%
+ \def\ackname{Remerciements.}%
+ \def\andname{et}%
+ \def\lastandname{ et}%
+ \def\appendixname{Appendice}
+ \def\chaptername{Chapitre}%
+ \def\claimname{Pr\'etention}%
+ \def\conjecturename{Hypoth\`ese}%
+ \def\contentsname{Table des mati\`eres}%
+ \def\corollaryname{Corollaire}%
+ \def\definitionname{D\'efinition}%
+ \def\examplename{Exemple}%
+ \def\exercisename{Exercice}%
+ \def\figurename{Fig.}%
+ \def\keywordname{{\bf Mots-cl\'e:}}
+ \def\indexname{Index}
+ \def\lemmaname{Lemme}%
+ \def\contriblistname{Liste des contributeurs}
+ \def\listfigurename{Liste des figures}%
+ \def\listtablename{Liste des tables}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}%
+ \def\notename{Remarque}%
+ \def\partname{Partie}%
+ \def\problemname{Probl\`eme}%
+ \def\proofname{Preuve}%
+ \def\propertyname{Caract\'eristique}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Question}%
+ \def\remarkname{Remarque}%
+ \def\seename{voir}
+ \def\solutionname{Solution}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tableau}%
+ \def\theoremname{Th\'eor\`eme}%
+}
+%
+% German section
+\def\switcht@deutsch{%\typeout{Man spricht deutsch.}%
+ \def\abstractname{Zusammenfassung.}%
+ \def\ackname{Danksagung.}%
+ \def\andname{und}%
+ \def\lastandname{ und}%
+ \def\appendixname{Anhang}%
+ \def\chaptername{Kapitel}%
+ \def\claimname{Behauptung}%
+ \def\conjecturename{Hypothese}%
+ \def\contentsname{Inhaltsverzeichnis}%
+ \def\corollaryname{Korollar}%
+%\def\definitionname{Definition}%
+ \def\examplename{Beispiel}%
+ \def\exercisename{\"Ubung}%
+ \def\figurename{Abb.}%
+ \def\keywordname{{\bf Schl\"usselw\"orter:}}
+ \def\indexname{Index}
+%\def\lemmaname{Lemma}%
+ \def\contriblistname{Mitarbeiter}
+ \def\listfigurename{Abbildungsverzeichnis}%
+ \def\listtablename{Tabellenverzeichnis}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Nachtrag}%
+ \def\notename{Anmerkung}%
+ \def\partname{Teil}%
+%\def\problemname{Problem}%
+ \def\proofname{Beweis}%
+ \def\propertyname{Eigenschaft}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Frage}%
+ \def\remarkname{Anmerkung}%
+ \def\seename{siehe}
+ \def\solutionname{L\"osung}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tabelle}%
+%\def\theoremname{Theorem}%
+}
+
+% Ragged bottom for the actual page
+\def\thisbottomragged{\def\@textbottom{\vskip\z@ plus.0001fil
+\global\let\@textbottom\relax}}
+
+\renewcommand\small{%
+ \@setfontsize\small\@ixpt{11}%
+ \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@
+ \abovedisplayshortskip \z@ \@plus2\p@
+ \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@
+ \def\@listi{\leftmargin\leftmargini
+ \parsep 0\p@ \@plus1\p@ \@minus\p@
+ \topsep 8\p@ \@plus2\p@ \@minus4\p@
+ \itemsep0\p@}%
+ \belowdisplayskip \abovedisplayskip
+}
+
+\frenchspacing
+\widowpenalty=10000
+\clubpenalty=10000
+
+\setlength\oddsidemargin {63\p@}
+\setlength\evensidemargin {63\p@}
+\setlength\marginparwidth {90\p@}
+
+\setlength\headsep {16\p@}
+
+\setlength\footnotesep{7.7\p@}
+\setlength\textfloatsep{8mm\@plus 2\p@ \@minus 4\p@}
+\setlength\intextsep {8mm\@plus 2\p@ \@minus 2\p@}
+
+\setcounter{secnumdepth}{2}
+
+\newcounter {chapter}
+\renewcommand\thechapter {\@arabic\c@chapter}
+
+\newif\if@mainmatter \@mainmattertrue
+\newcommand\frontmatter{\cleardoublepage
+ \@mainmatterfalse\pagenumbering{Roman}}
+\newcommand\mainmatter{\cleardoublepage
+ \@mainmattertrue\pagenumbering{arabic}}
+\newcommand\backmatter{\if@openright\cleardoublepage\else\clearpage\fi
+ \@mainmatterfalse}
+
+\renewcommand\part{\cleardoublepage
+ \thispagestyle{empty}%
+ \if@twocolumn
+ \onecolumn
+ \@tempswatrue
+ \else
+ \@tempswafalse
+ \fi
+ \null\vfil
+ \secdef\@part\@spart}
+
+\def\@part[#1]#2{%
+ \ifnum \c@secnumdepth >-2\relax
+ \refstepcounter{part}%
+ \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
+ \else
+ \addcontentsline{toc}{part}{#1}%
+ \fi
+ \markboth{}{}%
+ {\centering
+ \interlinepenalty \@M
+ \normalfont
+ \ifnum \c@secnumdepth >-2\relax
+ \huge\bfseries \partname~\thepart
+ \par
+ \vskip 20\p@
+ \fi
+ \Huge \bfseries #2\par}%
+ \@endpart}
+\def\@spart#1{%
+ {\centering
+ \interlinepenalty \@M
+ \normalfont
+ \Huge \bfseries #1\par}%
+ \@endpart}
+\def\@endpart{\vfil\newpage
+ \if@twoside
+ \null
+ \thispagestyle{empty}%
+ \newpage
+ \fi
+ \if@tempswa
+ \twocolumn
+ \fi}
+
+\newcommand\chapter{\clearpage
+ \thispagestyle{empty}%
+ \global\@topnum\z@
+ \@afterindentfalse
+ \secdef\@chapter\@schapter}
+\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
+ \if@mainmatter
+ \refstepcounter{chapter}%
+ \typeout{\@chapapp\space\thechapter.}%
+ \addcontentsline{toc}{chapter}%
+ {\protect\numberline{\thechapter}#1}%
+ \else
+ \addcontentsline{toc}{chapter}{#1}%
+ \fi
+ \else
+ \addcontentsline{toc}{chapter}{#1}%
+ \fi
+ \chaptermark{#1}%
+ \addtocontents{lof}{\protect\addvspace{10\p@}}%
+ \addtocontents{lot}{\protect\addvspace{10\p@}}%
+ \if@twocolumn
+ \@topnewpage[\@makechapterhead{#2}]%
+ \else
+ \@makechapterhead{#2}%
+ \@afterheading
+ \fi}
+\def\@makechapterhead#1{%
+% \vspace*{50\p@}%
+ {\centering
+ \ifnum \c@secnumdepth >\m@ne
+ \if@mainmatter
+ \large\bfseries \@chapapp{} \thechapter
+ \par\nobreak
+ \vskip 20\p@
+ \fi
+ \fi
+ \interlinepenalty\@M
+ \Large \bfseries #1\par\nobreak
+ \vskip 40\p@
+ }}
+\def\@schapter#1{\if@twocolumn
+ \@topnewpage[\@makeschapterhead{#1}]%
+ \else
+ \@makeschapterhead{#1}%
+ \@afterheading
+ \fi}
+\def\@makeschapterhead#1{%
+% \vspace*{50\p@}%
+ {\centering
+ \normalfont
+ \interlinepenalty\@M
+ \Large \bfseries #1\par\nobreak
+ \vskip 40\p@
+ }}
+
+\renewcommand\section{\@startsection{section}{1}{\z@}%
+ {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+ {12\p@ \@plus 4\p@ \@minus 4\p@}%
+ {\normalfont\large\bfseries\boldmath
+ \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+ {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+ {8\p@ \@plus 4\p@ \@minus 4\p@}%
+ {\normalfont\normalsize\bfseries\boldmath
+ \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+ {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+ {-0.5em \@plus -0.22em \@minus -0.1em}%
+ {\normalfont\normalsize\bfseries\boldmath}}
+\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
+ {-12\p@ \@plus -4\p@ \@minus -4\p@}%
+ {-0.5em \@plus -0.22em \@minus -0.1em}%
+ {\normalfont\normalsize\itshape}}
+\renewcommand\subparagraph[1]{\typeout{LLNCS warning: You should not use
+ \string\subparagraph\space with this class}\vskip0.5cm
+You should not use \verb|\subparagraph| with this class.\vskip0.5cm}
+
+\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
+\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
+\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
+\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
+\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
+\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
+\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
+\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
+\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
+\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
+\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}
+
+\let\footnotesize\small
+
+\if@custvec
+\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}}
+{\mbox{\boldmath$\textstyle#1$}}
+{\mbox{\boldmath$\scriptstyle#1$}}
+{\mbox{\boldmath$\scriptscriptstyle#1$}}}
+\fi
+
+\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}}
+\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil
+\penalty50\hskip1em\null\nobreak\hfil\squareforqed
+\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
+
+\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+\gets\cr\to\cr}}}}}
+\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+<\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
+>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.8pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.3pt}<\cr}}}}}
+\def\bbbr{{\rm I\!R}} %reelle Zahlen
+\def\bbbm{{\rm I\!M}}
+\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
+\def\bbbf{{\rm I\!F}}
+\def\bbbh{{\rm I\!H}}
+\def\bbbk{{\rm I\!K}}
+\def\bbbp{{\rm I\!P}}
+\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
+{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
+\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
+\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbs{{\mathchoice
+{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
+\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
+{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
+
+\let\ts\,
+
+\setlength\leftmargini {17\p@}
+\setlength\leftmargin {\leftmargini}
+\setlength\leftmarginii {\leftmargini}
+\setlength\leftmarginiii {\leftmargini}
+\setlength\leftmarginiv {\leftmargini}
+\setlength \labelsep {.5em}
+\setlength \labelwidth{\leftmargini}
+\addtolength\labelwidth{-\labelsep}
+
+\def\@listI{\leftmargin\leftmargini
+ \parsep 0\p@ \@plus1\p@ \@minus\p@
+ \topsep 8\p@ \@plus2\p@ \@minus4\p@
+ \itemsep0\p@}
+\let\@listi\@listI
+\@listi
+\def\@listii {\leftmargin\leftmarginii
+ \labelwidth\leftmarginii
+ \advance\labelwidth-\labelsep
+ \topsep 0\p@ \@plus2\p@ \@minus\p@}
+\def\@listiii{\leftmargin\leftmarginiii
+ \labelwidth\leftmarginiii
+ \advance\labelwidth-\labelsep
+ \topsep 0\p@ \@plus\p@\@minus\p@
+ \parsep \z@
+ \partopsep \p@ \@plus\z@ \@minus\p@}
+
+\renewcommand\labelitemi{\normalfont\bfseries --}
+\renewcommand\labelitemii{$\m@th\bullet$}
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
+ {{\contentsname}}}
+ \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
+ \def\lastand{\ifnum\value{auco}=2\relax
+ \unskip{} \andname\
+ \else
+ \unskip \lastandname\
+ \fi}%
+ \def\and{\stepcounter{@auth}\relax
+ \ifnum\value{@auth}=\value{auco}%
+ \lastand
+ \else
+ \unskip,
+ \fi}%
+ \@starttoc{toc}\if@restonecol\twocolumn\fi}
+
+\def\l@part#1#2{\addpenalty{\@secpenalty}%
+ \addvspace{2em plus\p@}% % space above part line
+ \begingroup
+ \parindent \z@
+ \rightskip \z@ plus 5em
+ \hrule\vskip5pt
+ \large % same size as for a contribution heading
+ \bfseries\boldmath % set line in boldface
+ \leavevmode % TeX command to enter horizontal mode.
+ #1\par
+ \vskip5pt
+ \hrule
+ \vskip1pt
+ \nobreak % Never break after part entry
+ \endgroup}
+
+\def\@dotsep{2}
+
+\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
+{chapter.\thechapter}\fi}
+
+\def\addnumcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
+ {\thechapter}#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmarkwop#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}
+
+\def\@adcmk[#1]{\ifcase #1 \or
+\def\@gtempa{\addnumcontentsmark}%
+ \or \def\@gtempa{\addcontentsmark}%
+ \or \def\@gtempa{\addcontentsmarkwop}%
+ \fi\@gtempa{toc}{chapter}}
+\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
+
+\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
+ \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ {\large\bfseries\boldmath#1}\ifx0#2\hfil\null
+ \else
+ \nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hss #2}%
+ \fi\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@title#1#2{\addpenalty{-\@highpenalty}
+ \addvspace{8pt plus 1pt}
+ \@tempdima \z@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ #1\nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hss #2}\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@author#1#2{\addpenalty{\@highpenalty}
+ \@tempdima=\z@ %15\p@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
+ \textit{#1}\par
+ \penalty\@highpenalty \endgroup}
+
+%\setcounter{tocdepth}{0}
+\newdimen\tocchpnum
+\newdimen\tocsecnum
+\newdimen\tocsectotal
+\newdimen\tocsubsecnum
+\newdimen\tocsubsectotal
+\newdimen\tocsubsubsecnum
+\newdimen\tocsubsubsectotal
+\newdimen\tocparanum
+\newdimen\tocparatotal
+\newdimen\tocsubparanum
+\tocchpnum=\z@ % no chapter numbers
+\tocsecnum=15\p@ % section 88. plus 2.222pt
+\tocsubsecnum=23\p@ % subsection 88.8 plus 2.222pt
+\tocsubsubsecnum=27\p@ % subsubsection 88.8.8 plus 1.444pt
+\tocparanum=35\p@ % paragraph 88.8.8.8 plus 1.666pt
+\tocsubparanum=43\p@ % subparagraph 88.8.8.8.8 plus 1.888pt
+\def\calctocindent{%
+\tocsectotal=\tocchpnum
+\advance\tocsectotal by\tocsecnum
+\tocsubsectotal=\tocsectotal
+\advance\tocsubsectotal by\tocsubsecnum
+\tocsubsubsectotal=\tocsubsectotal
+\advance\tocsubsubsectotal by\tocsubsubsecnum
+\tocparatotal=\tocsubsubsectotal
+\advance\tocparatotal by\tocparanum}
+\calctocindent
+
+\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
+\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
+\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
+\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
+\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
+
+\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
+ \@starttoc{lof}\if@restonecol\twocolumn\fi}
+\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
+
+\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
+ \@starttoc{lot}\if@restonecol\twocolumn\fi}
+\let\l@table\l@figure
+
+\renewcommand\listoffigures{%
+ \section*{\listfigurename
+ \@mkboth{\listfigurename}{\listfigurename}}%
+ \@starttoc{lof}%
+ }
+
+\renewcommand\listoftables{%
+ \section*{\listtablename
+ \@mkboth{\listtablename}{\listtablename}}%
+ \@starttoc{lot}%
+ }
+
+\ifx\oribibl\undefined
+\ifx\citeauthoryear\undefined
+\renewenvironment{thebibliography}[1]
+ {\section*{\refname}
+ \def\@biblabel##1{##1.}
+ \small
+ \list{\@biblabel{\@arabic\c@enumiv}}%
+ {\settowidth\labelwidth{\@biblabel{#1}}%
+ \leftmargin\labelwidth
+ \advance\leftmargin\labelsep
+ \if@openbib
+ \advance\leftmargin\bibindent
+ \itemindent -\bibindent
+ \listparindent \itemindent
+ \parsep \z@
+ \fi
+ \usecounter{enumiv}%
+ \let\p@enumiv\@empty
+ \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+ \if@openbib
+ \renewcommand\newblock{\par}%
+ \else
+ \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+ \fi
+ \sloppy\clubpenalty4000\widowpenalty4000%
+ \sfcode`\.=\@m}
+ {\def\@noitemerr
+ {\@latex@warning{Empty `thebibliography' environment}}%
+ \endlist}
+\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
+ {\let\protect\noexpand\immediate
+ \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+\newcount\@tempcntc
+\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
+ \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
+ {\@ifundefined
+ {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
+ ?}\@warning
+ {Citation `\@citeb' on page \thepage \space undefined}}%
+ {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
+ \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
+ \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
+ \else
+ \advance\@tempcntb\@ne
+ \ifnum\@tempcntb=\@tempcntc
+ \else\advance\@tempcntb\m@ne\@citeo
+ \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
+\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
+ \@citea\def\@citea{,\,\hskip\z@skip}%
+ \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
+ {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
+ \def\@citea{--}\fi
+ \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
+\else
+\renewenvironment{thebibliography}[1]
+ {\section*{\refname}
+ \small
+ \list{}%
+ {\settowidth\labelwidth{}%
+ \leftmargin\parindent
+ \itemindent=-\parindent
+ \labelsep=\z@
+ \if@openbib
+ \advance\leftmargin\bibindent
+ \itemindent -\bibindent
+ \listparindent \itemindent
+ \parsep \z@
+ \fi
+ \usecounter{enumiv}%
+ \let\p@enumiv\@empty
+ \renewcommand\theenumiv{}}%
+ \if@openbib
+ \renewcommand\newblock{\par}%
+ \else
+ \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+ \fi
+ \sloppy\clubpenalty4000\widowpenalty4000%
+ \sfcode`\.=\@m}
+ {\def\@noitemerr
+ {\@latex@warning{Empty `thebibliography' environment}}%
+ \endlist}
+ \def\@cite#1{#1}%
+ \def\@lbibitem[#1]#2{\item[]\if@filesw
+ {\def\protect##1{\string ##1\space}\immediate
+ \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+ \fi
+\else
+\@cons\@openbib@code{\noexpand\small}
+\fi
+
+\def\idxquad{\hskip 10\p@}% space that divides entry from number
+
+\def\@idxitem{\par\hangindent 10\p@}
+
+\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
+ \noindent\hangindent\wd0\box0}% index entry
+
+\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
+ \noindent\hangindent\wd0\box0}% order index entry
+
+\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
+
+\renewenvironment{theindex}
+ {\@mkboth{\indexname}{\indexname}%
+ \thispagestyle{empty}\parindent\z@
+ \parskip\z@ \@plus .3\p@\relax
+ \let\item\par
+ \def\,{\relax\ifmmode\mskip\thinmuskip
+ \else\hskip0.2em\ignorespaces\fi}%
+ \normalfont\small
+ \begin{multicols}{2}[\@makeschapterhead{\indexname}]%
+ }
+ {\end{multicols}}
+
+\renewcommand\footnoterule{%
+ \kern-3\p@
+ \hrule\@width 2truecm
+ \kern2.6\p@}
+ \newdimen\fnindent
+ \fnindent1em
+\long\def\@makefntext#1{%
+ \parindent \fnindent%
+ \leftskip \fnindent%
+ \noindent
+ \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
+
+\long\def\@makecaption#1#2{%
+ \vskip\abovecaptionskip
+ \sbox\@tempboxa{{\bfseries #1.} #2}%
+ \ifdim \wd\@tempboxa >\hsize
+ {\bfseries #1.} #2\par
+ \else
+ \global \@minipagefalse
+ \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
+ \fi
+ \vskip\belowcaptionskip}
+
+\def\fps@figure{htbp}
+\def\fnum@figure{\figurename\thinspace\thefigure}
+\def \@floatboxreset {%
+ \reset@font
+ \small
+ \@setnobreak
+ \@setminipage
+}
+\def\fps@table{htbp}
+\def\fnum@table{\tablename~\thetable}
+\renewenvironment{table}
+ {\setlength\abovecaptionskip{0\p@}%
+ \setlength\belowcaptionskip{10\p@}%
+ \@float{table}}
+ {\end@float}
+\renewenvironment{table*}
+ {\setlength\abovecaptionskip{0\p@}%
+ \setlength\belowcaptionskip{10\p@}%
+ \@dblfloat{table}}
+ {\end@dblfloat}
+
+\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
+ ext@#1\endcsname}{#1}{\protect\numberline{\csname
+ the#1\endcsname}{\ignorespaces #2}}\begingroup
+ \@parboxrestore
+ \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+ \endgroup}
+
+% LaTeX does not provide a command to enter the authors institute
+% addresses. The \institute command is defined here.
+
+\newcounter{@inst}
+\newcounter{@auth}
+\newcounter{auco}
+\newdimen\instindent
+\newbox\authrun
+\newtoks\authorrunning
+\newtoks\tocauthor
+\newbox\titrun
+\newtoks\titlerunning
+\newtoks\toctitle
+
+\def\clearheadinfo{\gdef\@author{No Author Given}%
+ \gdef\@title{No Title Given}%
+ \gdef\@subtitle{}%
+ \gdef\@institute{No Institute Given}%
+ \gdef\@thanks{}%
+ \global\titlerunning={}\global\authorrunning={}%
+ \global\toctitle={}\global\tocauthor={}}
+
+\def\institute#1{\gdef\@institute{#1}}
+
+\def\institutename{\par
+ \begingroup
+ \parskip=\z@
+ \parindent=\z@
+ \setcounter{@inst}{1}%
+ \def\and{\par\stepcounter{@inst}%
+ \noindent$^{\the@inst}$\enspace\ignorespaces}%
+ \setbox0=\vbox{\def\thanks##1{}\@institute}%
+ \ifnum\c@@inst=1\relax
+ \gdef\fnnstart{0}%
+ \else
+ \xdef\fnnstart{\c@@inst}%
+ \setcounter{@inst}{1}%
+ \noindent$^{\the@inst}$\enspace
+ \fi
+ \ignorespaces
+ \@institute\par
+ \endgroup}
+
+\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
+ {\star\star\star}\or \dagger\or \ddagger\or
+ \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
+ \or \ddagger\ddagger \else\@ctrerr\fi}}
+
+\def\inst#1{\unskip$^{#1}$}
+\def\fnmsep{\unskip$^,$}
+\def\email#1{{\tt#1}}
+\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
+\@ifpackageloaded{babel}{%
+\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
+\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
+\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
+}{\switcht@@therlang}%
+}
+\def\homedir{\~{ }}
+
+\def\subtitle#1{\gdef\@subtitle{#1}}
+\clearheadinfo
+
+\renewcommand\maketitle{\newpage
+ \refstepcounter{chapter}%
+ \stepcounter{section}%
+ \setcounter{section}{0}%
+ \setcounter{subsection}{0}%
+ \setcounter{figure}{0}
+ \setcounter{table}{0}
+ \setcounter{equation}{0}
+ \setcounter{footnote}{0}%
+ \begingroup
+ \parindent=\z@
+ \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
+ \if@twocolumn
+ \ifnum \col@number=\@ne
+ \@maketitle
+ \else
+ \twocolumn[\@maketitle]%
+ \fi
+ \else
+ \newpage
+ \global\@topnum\z@ % Prevents figures from going at top of page.
+ \@maketitle
+ \fi
+ \thispagestyle{empty}\@thanks
+%
+ \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
+ \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
+ \instindent=\hsize
+ \advance\instindent by-\headlineindent
+% \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
+% \addcontentsline{toc}{title}{\the\toctitle}\fi
+ \if@runhead
+ \if!\the\titlerunning!\else
+ \edef\@title{\the\titlerunning}%
+ \fi
+ \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
+ \ifdim\wd\titrun>\instindent
+ \typeout{Title too long for running head. Please supply}%
+ \typeout{a shorter form with \string\titlerunning\space prior to
+ \string\maketitle}%
+ \global\setbox\titrun=\hbox{\small\rm
+ Title Suppressed Due to Excessive Length}%
+ \fi
+ \xdef\@title{\copy\titrun}%
+ \fi
+%
+ \if!\the\tocauthor!\relax
+ {\def\and{\noexpand\protect\noexpand\and}%
+ \protected@xdef\toc@uthor{\@author}}%
+ \else
+ \def\\{\noexpand\protect\noexpand\newline}%
+ \protected@xdef\scratch{\the\tocauthor}%
+ \protected@xdef\toc@uthor{\scratch}%
+ \fi
+% \addcontentsline{toc}{author}{\toc@uthor}%
+ \if@runhead
+ \if!\the\authorrunning!
+ \value{@inst}=\value{@auth}%
+ \setcounter{@auth}{1}%
+ \else
+ \edef\@author{\the\authorrunning}%
+ \fi
+ \global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}%
+ \ifdim\wd\authrun>\instindent
+ \typeout{Names of authors too long for running head. Please supply}%
+ \typeout{a shorter form with \string\authorrunning\space prior to
+ \string\maketitle}%
+ \global\setbox\authrun=\hbox{\small\rm
+ Authors Suppressed Due to Excessive Length}%
+ \fi
+ \xdef\@author{\copy\authrun}%
+ \markboth{\@author}{\@title}%
+ \fi
+ \endgroup
+ \setcounter{footnote}{\fnnstart}%
+ \clearheadinfo}
+%
+\def\@maketitle{\newpage
+ \markboth{}{}%
+ \def\lastand{\ifnum\value{@inst}=2\relax
+ \unskip{} \andname\
+ \else
+ \unskip \lastandname\
+ \fi}%
+ \def\and{\stepcounter{@auth}\relax
+ \ifnum\value{@auth}=\value{@inst}%
+ \lastand
+ \else
+ \unskip,
+ \fi}%
+ \begin{center}%
+ \let\newline\\
+ {\Large \bfseries\boldmath
+ \pretolerance=10000
+ \@title \par}\vskip .8cm
+\if!\@subtitle!\else {\large \bfseries\boldmath
+ \vskip -.65cm
+ \pretolerance=10000
+ \@subtitle \par}\vskip .8cm\fi
+ \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
+ \def\thanks##1{}\@author}%
+ \global\value{@inst}=\value{@auth}%
+ \global\value{auco}=\value{@auth}%
+ \setcounter{@auth}{1}%
+{\lineskip .5em
+\noindent\ignorespaces
+\@author\vskip.35cm}
+ {\small\institutename}
+ \end{center}%
+ }
+
+% definition of the "\spnewtheorem" command.
+%
+% Usage:
+%
+% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
+% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
+% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
+%
+% New is "cap_font" and "body_font". It stands for
+% fontdefinition of the caption and the text itself.
+%
+% "\spnewtheorem*" gives a theorem without number.
+%
+% A defined spnewthoerem environment is used as described
+% by Lamport.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\@thmcountersep{}
+\def\@thmcounterend{.}
+
+\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
+
+% definition of \spnewtheorem with number
+
+\def\@spnthm#1#2{%
+ \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
+\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
+
+\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}\@addtoreset{#1}{#3}%
+ \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
+ \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}%
+ \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spothm#1[#2]#3#4#5{%
+ \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
+ {\expandafter\@ifdefinable\csname #1\endcsname
+ {\global\@namedef{the#1}{\@nameuse{the#2}}%
+ \expandafter\xdef\csname #1name\endcsname{#3}%
+ \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
+ \global\@namedef{end#1}{\@endtheorem}}}}
+
+\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\refstepcounter{#1}%
+\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
+
+\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
+ \ignorespaces}
+
+\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
+ the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
+
+\def\@spbegintheorem#1#2#3#4{\trivlist
+ \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
+
+\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
+ \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
+
+% definition of \spnewtheorem* without number
+
+\def\@sthm#1#2{\@Ynthm{#1}{#2}}
+
+\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+ {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
+
+\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
+
+\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
+ {#4}{#2}{#3}\ignorespaces}
+
+\def\@Begintheorem#1#2#3{#3\trivlist
+ \item[\hskip\labelsep{#2#1\@thmcounterend}]}
+
+\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
+ \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
+
+\if@envcntsect
+ \def\@thmcountersep{.}
+ \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
+\else
+ \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
+ \if@envcntreset
+ \@addtoreset{theorem}{section}
+ \else
+ \@addtoreset{theorem}{chapter}
+ \fi
+\fi
+
+%definition of divers theorem environments
+\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
+\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
+\if@envcntsame % alle Umgebungen wie Theorem.
+ \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
+\else % alle Umgebungen mit eigenem Zaehler
+ \if@envcntsect % mit section numeriert
+ \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
+ \else % nicht mit section numeriert
+ \if@envcntreset
+ \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+ \@addtoreset{#1}{section}}
+ \else
+ \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+ \@addtoreset{#1}{chapter}}%
+ \fi
+ \fi
+\fi
+\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
+\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
+\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
+\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
+\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
+\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
+\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
+\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
+\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
+\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
+\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
+\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
+\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
+\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
+
+\def\@takefromreset#1#2{%
+ \def\@tempa{#1}%
+ \let\@tempd\@elt
+ \def\@elt##1{%
+ \def\@tempb{##1}%
+ \ifx\@tempa\@tempb\else
+ \@addtoreset{##1}{#2}%
+ \fi}%
+ \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
+ \expandafter\def\csname cl@#2\endcsname{}%
+ \@tempc
+ \let\@elt\@tempd}
+
+\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+ \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
+ \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
+ \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
+ }
+
+\renewenvironment{abstract}{%
+ \list{}{\advance\topsep by0.35cm\relax\small
+ \leftmargin=1cm
+ \labelwidth=\z@
+ \listparindent=\z@
+ \itemindent\listparindent
+ \rightmargin\leftmargin}\item[\hskip\labelsep
+ \bfseries\abstractname]}
+ {\endlist}
+
+\newdimen\headlineindent % dimension for space between
+\headlineindent=1.166cm % number and text of headings.
+
+\def\ps@headings{\let\@mkboth\@gobbletwo
+ \let\@oddfoot\@empty\let\@evenfoot\@empty
+ \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+ \leftmark\hfil}
+ \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
+ \llap{\thepage}}
+ \def\chaptermark##1{}%
+ \def\sectionmark##1{}%
+ \def\subsectionmark##1{}}
+
+\def\ps@titlepage{\let\@mkboth\@gobbletwo
+ \let\@oddfoot\@empty\let\@evenfoot\@empty
+ \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+ \hfil}
+ \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
+ \llap{\thepage}}
+ \def\chaptermark##1{}%
+ \def\sectionmark##1{}%
+ \def\subsectionmark##1{}}
+
+\if@runhead\ps@headings\else
+\ps@empty\fi
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\endinput
+%end of file llncs.cls
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/document/root.bib Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,152 @@
+@Article{Lampson:Redell:cacm:1980,
+ author = "B. Lampson and D. Redell",
+ title = "{Experience with processes and monitors in Mesa}",
+ journal = "Communications of the ACM",
+ volume = "23",
+ number = "2",
+ pages = "105--117",
+ month = feb,
+ year = "1980",
+ keywords = "Mesa, processes, monitors",
+}
+
+@Article{journals/tc/ShaRL90,
+ title = "Priority Inheritance Protocols: An Approach to
+ Real-Time Synchronization",
+ author = "S. Liu and R. Rajkumar and J. P. Lehoczky",
+ journal = "IEEE Trans. Computers",
+ year = "1990",
+ number = "9",
+ volume = "39",
+ bibdate = "2011-10-27",
+ bibsource = "DBLP,
+ http://dblp.uni-trier.de/db/journals/tc/tc39.html#ShaRL90",
+ pages = "1175--1185",
+ URL = "http://doi.ieeecomputersociety.org/10.1109/12.57058",
+}
+
+@MISC{yodaiken-july02,
+author = {V. Yodaiken},
+title = {Against Priority Inheritance},
+month = July,
+year = {2002},
+howpublished={\url{http://www.linuxfordevices.com/files/misc/yodaiken-july02.pdf}},
+}
+
+@MISC{locke-july02,
+author = {D. Locke},
+title = {Priority Inheritance: The Real Story},
+month = July,
+year = {2002},
+howpublished={\url{http://www.math.unipd.it/~tullio/SCD/2007/Materiale/Locke.pdf}},
+}
+
+@MISC{Faria08,
+author = {J. M. S. Faria},
+title = {Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC},
+year = {2008},
+howpublished={\url{http://repositorio-aberto.up.pt/bitstream/10216/11466/2/Texto%20integral.pdf}},
+}
+
+
+http://repositorio-aberto.up.pt/bitstream/10216/11466/2/Texto%20integral.pdf
+
+@Article{Reeves-Glenn-1998,
+ title = "Re: What Really Happened on Mars?",
+ author = "G. Reeves",
+ journal = "Risks-Forum Digest",
+ year = "1998",
+ month = "January",
+ number = "58",
+ volume = "19",
+}
+
+@TechReport{dutertre99b,
+ title = "The {Priority Ceiling Protocol}: Formalization and
+ Analysis Using {PVS}",
+ author = "B. Dutertre",
+ month = Oct,
+ year = "1999",
+ institution = "System Design Laboratory, SRI International",
+ address = "Menlo Park, CA",
+ note = "Available at
+ \url{http://www.sdl.sri.com/dsa/publis/prio-ceiling.html}",
+}
+
+@InProceedings{conf/fase/JahierHR09,
+ title = "Synchronous Modeling and Validation of Priority
+ Inheritance Schedulers",
+ author = "E. Jahier and B. Halbwachs and P.
+ Raymond",
+ bibdate = "2009-04-01",
+ bibsource = "DBLP,
+ http://dblp.uni-trier.de/db/conf/fase/fase2009.html#JahierHR09",
+ booktitle = "FASE",
+ booktitle = "Fundamental Approaches to Software Engineering, 12th
+ International Conference, {FASE} 2009, Held as Part of
+ the Joint European Conferences on Theory and Practice
+ of Software, {ETAPS} 2009, York, {UK}, March 22-29,
+ 2009. Proceedings",
+ publisher = "Springer",
+ year = "2009",
+ volume = "5503",
+ editor = "Marsha Chechik and Martin Wirsing",
+ ISBN = "978-3-642-00592-3",
+ pages = "140--154",
+ series = "Lecture Notes in Computer Science",
+ URL = "http://dx.doi.org/10.1007/978-3-642-00593-0",
+}
+
+@InProceedings{WellingsBSB07,
+ title = "Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java",
+ author = "A. J. Wellings and A. Burns and O. M. Santos and B. M. Brosgol",
+ publisher = "IEEE Computer Society",
+ year = "2007",
+ booktitle = "Proceedings of the 10th IEEE International Symposium on Object
+ and Component-Oriented Real-Time Distributed Computing",
+ pages = "115--123",
+}
+
+@Article{Wang:2002:SGP,
+ author = "Y. Wang and E. Anceaume and F. Brasileiro and F.
+ Greve and M. Hurfin",
+ title = "Solving the group priority inversion problem in a
+ timed asynchronous system",
+ journal = "IEEE Transactions on Computers",
+ volume = "51",
+ number = "8",
+ pages = "900--915",
+ month = aug,
+ year = "2002",
+ CODEN = "ITCOB4",
+ doi = "http://dx.doi.org/10.1109/TC.2002.1024738",
+ ISSN = "0018-9340 (print), 1557-9956 (electronic)",
+ issn-l = "0018-9340",
+ bibdate = "Tue Jul 5 09:41:56 MDT 2011",
+ bibsource = "http://www.computer.org/tc/;
+ http://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib",
+ URL = "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1024738",
+ acknowledgement = "Nelson H. F. Beebe, University of Utah, Department
+ of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake
+ City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1
+ 801 581 4148, e-mail: \path|beebe@math.utah.edu|,
+ \path|beebe@acm.org|, \path|beebe@computer.org|
+ (Internet), URL:
+ \path|http://www.math.utah.edu/~beebe/|",
+ fjournal = "IEEE Transactions on Computers",
+ doi-url = "http://dx.doi.org/10.1109/TC.2002.1024738",
+}
+
+@Article{journals/rts/BabaogluMS93,
+ title = "A Formalization of Priority Inversion",
+ author = "{\"O} Babaoglu and K. Marzullo and F. B. Schneider",
+ journal = "Real-Time Systems",
+ year = "1993",
+ number = "4",
+ volume = "5",
+ bibdate = "2011-06-03",
+ bibsource = "DBLP,
+ http://dblp.uni-trier.de/db/journals/rts/rts5.html#BabaogluMS93",
+ pages = "285--303",
+ URL = "http://dx.doi.org/10.1007/BF01088832",
+}
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/document/root.tex Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,74 @@
+\documentclass[runningheads]{llncs}
+\usepackage{isabelle}
+\usepackage{isabellesym}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{tikz}
+\usepackage{pgf}
+%\usetikzlibrary{arrows,automata,decorations,fit,calc}
+%\usetikzlibrary{shapes,shapes.arrows,snakes,positioning}
+%\usepgflibrary{shapes.misc} % LATEX and plain TEX and pure pgf
+%\usetikzlibrary{matrix}
+\usepackage{pdfsetup}
+\usepackage{ot1patch}
+\usepackage{times}
+%%\usepackage{proof}
+%%\usepackage{mathabx}
+\usepackage{stmaryrd}
+\usepackage{url}
+
+\titlerunning{Myhill-Nerode using Regular Expressions}
+
+
+\urlstyle{rm}
+\isabellestyle{it}
+\renewcommand{\isastyleminor}{\it}%
+\renewcommand{\isastyle}{\normalsize\it}%
+
+
+\def\dn{\,\stackrel{\mbox{\scriptsize def}}{=}\,}
+\renewcommand{\isasymequiv}{$\dn$}
+\renewcommand{\isasymemptyset}{$\varnothing$}
+\renewcommand{\isacharunderscore}{\mbox{$\_\!\_$}}
+
+\newcommand{\isasymcalL}{\ensuremath{\cal{L}}}
+\newcommand{\isasymbigplus}{\ensuremath{\bigplus}}
+
+\newcommand{\bigplus}{\mbox{\Large\bf$+$}}
+\begin{document}
+
+\title{A Formalisation of Priority Inheritance Protocol \\
+ for Correct and Efficient Implementation}
+\author{Xingyuan Zhang\inst{1} \and Christian Urban\inst{2} \and Chunhan Wu\inst{1}}
+\institute{PLA University of Science and Technology, China \and
+ King's College, University of London, U.K.}
+\maketitle
+
+%\mbox{}\\[-10mm]
+\begin{abstract}
+Despite the wide use of Priority Inheritance Protocol in real time operating
+system, it's correctness has never been formally proved and mechanically checked.
+All existing verification are based on model checking technology. Full automatic
+verification gives little help to understand why the protocol is correct.
+And results such obtained only apply to models of limited size.
+This paper presents a formal verification based on theorem proving.
+Machine checked formal proof does help to get deeper understanding. We found
+the fact which is not mentioned in the literature, that the choice of next
+thread to take over when an critical resource is release does not affect the correctness
+of the protocol. The paper also shows how formal proof can help to construct
+correct and efficient implementation.
+\end{abstract}
+
+
+\input{session}
+
+%%\mbox{}\\[-10mm]
+\bibliographystyle{plain}
+\bibliography{root}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Paper/tt.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,94 @@
+
+There are several works on inversion avoidance:
+\begin{enumerate}
+\item {\em Solving the group priority inversion problem in a timed asynchronous system}.
+The notion of \<exclamdown>\<degree>Group Priority Inversion\<exclamdown>\<plusminus> is introduced. The main strategy is still inversion avoidance.
+The method is by reordering requests in the setting of Client-Server.
+\item {\em Examples of inaccurate specification of the protocol}.
+\end{enumerate}
+
+
+
+
+
+
+section{* Related works *}
+
+text {*
+1. <<Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java>> models and
+verifies the combination of Priority Inheritance (PI) and Priority Ceiling Emulation (PCE) protocols in
+the setting of Java virtual machine using extended Timed Automata(TA) formalism of the UPPAAL tool.
+Although a detailed formal model of combined PI and PCE is given, the number of properties is quite
+small and the focus is put on the harmonious working of PI and PCE. Most key features of PI
+(as well as PCE) are not shown. Because of the limitation of the model checking technique
+ used there, properties are shown only for a small number of scenarios. Therefore, the verification
+does not show the correctness of the formal model itself in a convincing way.
+2. << Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC>>. A formal model
+of PI is given in TLA+. Only 3 properties are shown for PI using model checking. The limitation of
+model checking is intrinsic to the work.
+3. <<Synchronous modeling and validation of priority inheritance schedulers>>. Gives a formal model
+of PI and PCE in AADL (Architecture Analysis & Design Language) and checked several properties
+using model checking. The number of properties shown there is less than here and the scale
+is also limited by the model checking technique.
+
+
+There are several works on inversion avoidance:
+1. <<Solving the group priority inversion problem in a timed asynchronous system>>.
+The notion of \<exclamdown>\<degree>Group Priority Inversion\<exclamdown>\<plusminus> is introduced. The main strategy is still inversion avoidance.
+The method is by reordering requests in the setting of Client-Server.
+
+
+<<Examples of inaccurate specification of the protocol>>.
+
+*}
+
+text {*
+
+\section{An overview of priority inversion and priority inheritance}
+
+Priority inversion refers to the phenomenon when a thread with high priority is blocked
+by a thread with low priority. Priority happens when the high priority thread requests
+for some critical resource already taken by the low priority thread. Since the high
+priority thread has to wait for the low priority thread to complete, it is said to be
+blocked by the low priority thread. Priority inversion might prevent high priority
+thread from fulfill its task in time if the duration of priority inversion is indefinite
+and unpredictable. Indefinite priority inversion happens when indefinite number
+of threads with medium priorities is activated during the period when the high
+priority thread is blocked by the low priority thread. Although these medium
+priority threads can not preempt the high priority thread directly, they are able
+to preempt the low priority threads and cause it to stay in critical section for
+an indefinite long duration. In this way, the high priority thread may be blocked indefinitely.
+
+Priority inheritance is one protocol proposed to avoid indefinite priority inversion.
+The basic idea is to let the high priority thread donate its priority to the low priority
+thread holding the critical resource, so that it will not be preempted by medium priority
+threads. The thread with highest priority will not be blocked unless it is requesting
+some critical resource already taken by other threads. Viewed from a different angle,
+any thread which is able to block the highest priority threads must already hold some
+critical resource. Further more, it must have hold some critical resource at the
+moment the highest priority is created, otherwise, it may never get change to run and
+get hold. Since the number of such resource holding lower priority threads is finite,
+if every one of them finishes with its own critical section in a definite duration,
+the duration the highest priority thread is blocked is definite as well. The key to
+guarantee lower priority threads to finish in definite is to donate them the highest
+priority. In such cases, the lower priority threads is said to have inherited the
+highest priority. And this explains the name of the protocol:
+{\em Priority Inheritance} and how Priority Inheritance prevents indefinite delay.
+
+The objectives of this paper are:
+\begin{enumerate}
+\item Build the above mentioned idea into formal model and prove a series of properties
+until we are convinced that the formal model does fulfill the original idea.
+\item Show how formally derived properties can be used as guidelines for correct
+and efficient implementation.
+\end{enumerate}.
+The proof is totally formal in the sense that every detail is reduced to the
+very first principles of Higher Order Logic. The nature of interactive theorem
+proving is for the human user to persuade computer program to accept its arguments.
+A clear and simple understanding of the problem at hand is both a prerequisite and a
+byproduct of such an effort, because everything has finally be reduced to the very
+first principle to be checked mechanically. The former intuitive explanation of
+Priority Inheritance is just such a byproduct.
+*}
+
+*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Precedence_ord.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,38 @@
+(* Title: HOL/Library/Product_ord.thy
+ Author: Norbert Voelker
+*)
+
+header {* Order on product types *}
+
+theory Precedence_ord
+imports Main
+begin
+
+datatype precedence = Prc nat nat
+
+instantiation precedence :: order
+begin
+
+definition
+ precedence_le_def: "x \<le> y \<longleftrightarrow> (case (x, y) of
+ (Prc fx sx, Prc fy sy) \<Rightarrow>
+ fx < fy \<or> (fx \<le> fy \<and> sy \<le> sx))"
+
+definition
+ precedence_less_def: "x < y \<longleftrightarrow> (case (x, y) of
+ (Prc fx sx, Prc fy sy) \<Rightarrow>
+ fx < fy \<or> (fx \<le> fy \<and> sy < sx))"
+
+instance
+proof
+qed (auto simp: precedence_le_def precedence_less_def
+ intro: order_trans split:precedence.splits)
+end
+
+instance precedence :: preorder ..
+
+instance precedence :: linorder proof
+qed (auto simp: precedence_le_def precedence_less_def
+ intro: order_trans split:precedence.splits)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Prio.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,2813 @@
+theory Prio
+imports Precedence_ord Moment Lsp Happen_within
+begin
+
+type_synonym thread = nat
+type_synonym priority = nat
+type_synonym cs = nat
+
+datatype event =
+ Create thread priority |
+ Exit thread |
+ P thread cs |
+ V thread cs |
+ Set thread priority
+
+datatype node =
+ Th "thread" |
+ Cs "cs"
+
+type_synonym state = "event list"
+
+fun threads :: "state \<Rightarrow> thread set"
+where
+ "threads [] = {}" |
+ "threads (Create thread prio#s) = {thread} \<union> threads s" |
+ "threads (Exit thread # s) = (threads s) - {thread}" |
+ "threads (e#s) = threads s"
+
+fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> nat"
+where
+ "original_priority thread [] = 0" |
+ "original_priority thread (Create thread' prio#s) =
+ (if thread' = thread then prio else original_priority thread s)" |
+ "original_priority thread (Set thread' prio#s) =
+ (if thread' = thread then prio else original_priority thread s)" |
+ "original_priority thread (e#s) = original_priority thread s"
+
+fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
+where
+ "birthtime thread [] = 0" |
+ "birthtime thread ((Create thread' prio)#s) = (if (thread = thread') then length s
+ else birthtime thread s)" |
+ "birthtime thread ((Set thread' prio)#s) = (if (thread = thread') then length s
+ else birthtime thread s)" |
+ "birthtime thread (e#s) = birthtime thread s"
+
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+ where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
+
+consts holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ depend :: "'b \<Rightarrow> (node \<times> node) set"
+ dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+defs (overloaded) cs_holding_def: "holding wq thread cs == (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
+ cs_waiting_def: "waiting wq thread cs == (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
+ cs_depend_def: "depend (wq::cs \<Rightarrow> thread list) == {(Th t, Cs c) | t c. waiting wq t c} \<union>
+ {(Cs c, Th t) | c t. holding wq t c}"
+ cs_dependents_def: "dependents (wq::cs \<Rightarrow> thread list) th == {th' . (Th th', Th th) \<in> (depend wq)^+}"
+
+record schedule_state =
+ waiting_queue :: "cs \<Rightarrow> thread list"
+ cur_preced :: "thread \<Rightarrow> precedence"
+
+
+definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
+where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
+
+fun schs :: "state \<Rightarrow> schedule_state"
+where
+ "schs [] = \<lparr>waiting_queue = \<lambda> cs. [],
+ cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
+ "schs (e#s) = (let ps = schs s in
+ let pwq = waiting_queue ps in
+ let pcp = cur_preced ps in
+ let nwq = case e of
+ P thread cs \<Rightarrow> pwq(cs:=(pwq cs @ [thread])) |
+ V thread cs \<Rightarrow> let nq = case (pwq cs) of
+ [] \<Rightarrow> [] |
+ (th#pq) \<Rightarrow> case (lsp pcp pq) of
+ (l, [], r) \<Rightarrow> []
+ | (l, m#ms, r) \<Rightarrow> m#(l@ms@r)
+ in pwq(cs:=nq) |
+ _ \<Rightarrow> pwq
+ in let ncp = cpreced (e#s) nwq in
+ \<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
+ )"
+
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
+where "wq s == waiting_queue (schs s)"
+
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+where "cp s = cur_preced (schs s)"
+
+defs (overloaded) s_holding_def: "holding (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
+ s_waiting_def: "waiting (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
+ s_depend_def: "depend (s::state) == {(Th t, Cs c) | t c. waiting (wq s) t c} \<union>
+ {(Cs c, Th t) | c t. holding (wq s) t c}"
+ s_dependents_def: "dependents (s::state) th == {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
+
+definition readys :: "state \<Rightarrow> thread set"
+where
+ "readys s =
+ {thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
+
+definition runing :: "state \<Rightarrow> thread set"
+where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
+
+definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+ where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
+
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+where
+ thread_create: "\<lbrakk>prio \<le> max_prio; thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
+ thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
+ thread_P: "\<lbrakk>thread \<in> runing s; (Cs cs, Th thread) \<notin> (depend s)^+\<rbrakk> \<Longrightarrow> step s (P thread cs)" |
+ thread_V: "\<lbrakk>thread \<in> runing s; holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
+ thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
+ for cs
+where
+ vt_nil[intro]: "vt cs []" |
+ vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
+
+lemma runing_ready: "runing s \<subseteq> readys s"
+ by (auto simp only:runing_def readys_def)
+
+lemma wq_v_eq_nil:
+ fixes s cs thread rest
+ assumes eq_wq: "wq s cs = thread # rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [], r)"
+ shows "wq (V thread cs#s) cs = []"
+proof -
+ from prems show ?thesis
+ by (auto simp:wq_def Let_def cp_def split:list.splits)
+qed
+
+lemma wq_v_eq:
+ fixes s cs thread rest
+ assumes eq_wq: "wq s cs = thread # rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+ shows "wq (V thread cs#s) cs = th'#l@r"
+proof -
+ from prems show ?thesis
+ by (auto simp:wq_def Let_def cp_def split:list.splits)
+qed
+
+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 h1: "thread \<in> runing s"
+ and h2: "holding s thread cs"
+ and h3: "waiting_queue (schs s) cs = a # list"
+ and h4: "a \<notin> set list"
+ and h5: "distinct list"
+ thus "distinct
+ ((\<lambda>(l, a, r). case a of [] \<Rightarrow> [] | m # ms \<Rightarrow> m # l @ ms @ r)
+ (lsp (cur_preced (schs s)) list))"
+ apply (cases "(lsp (cur_preced (schs s)) list)", simp)
+ apply (case_tac b, simp)
+ by (drule_tac lsp_set_eq, simp)
+ qed
+qed
+
+lemma block_pre:
+ fixes thread cs s
+ assumes s_ni: "thread \<notin> set (wq s cs)"
+ and s_i: "thread \<in> set (wq (e#s) cs)"
+ shows "e = P thread cs"
+proof -
+ have ee: "\<And> x y. \<lbrakk>x = y\<rbrakk> \<Longrightarrow> set x = set y"
+ by auto
+ from s_ni s_i show ?thesis
+ proof (cases e, auto split:if_splits simp add:Let_def wq_def)
+ fix uu uub uuc uud uue
+ assume h: "(uuc, thread # uu, uub) = lsp (cur_preced (schs s)) uud"
+ and h1 [symmetric]: "uue # uud = waiting_queue (schs s) cs"
+ and h2: "thread \<notin> set (waiting_queue (schs s) cs)"
+ from lsp_set [OF h] have "set (uuc @ (thread # uu) @ uub) = set uud" .
+ hence "thread \<in> set uud" by auto
+ with h1 have "thread \<in> set (waiting_queue (schs s) cs)" by auto
+ with h2 show False by auto
+ next
+ fix uu uua uub uuc uud uue
+ assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+ and h2: "uue # uud = waiting_queue (schs s) cs"
+ and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+ and h4: "thread \<in> set uuc"
+ from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+ with h4 have "thread \<in> set uud" by auto
+ with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
+ apply (drule_tac ee) by auto
+ with h1 show "False" by fast
+ next
+ fix uu uua uub uuc uud uue
+ assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+ and h2: "uue # uud = waiting_queue (schs s) cs"
+ and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+ and h4: "thread \<in> set uu"
+ from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+ with h4 have "thread \<in> set uud" by auto
+ with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
+ apply (drule_tac ee) by auto
+ with h1 show "False" by fast
+ next
+ fix uu uua uub uuc uud uue
+ assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
+ and h2: "uue # uud = waiting_queue (schs s) cs"
+ and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
+ and h4: "thread \<in> set uub"
+ from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
+ with h4 have "thread \<in> set uud" by auto
+ with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
+ apply (drule_tac ee) by auto
+ with h1 show "False" by fast
+ 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 -
+ have ee: "\<And> uuc thread uu uub s list. (uuc, thread # uu, uub) = lsp (cur_preced (schs s)) list \<Longrightarrow>
+ lsp (cur_preced (schs s)) list = (uuc, thread # uu, uub)
+ " by simp
+ from prems show "False"
+ apply (cases e)
+ apply ((simp split:if_splits add:Let_def wq_def)[1])+
+ apply (insert abs1, fast)[1]
+ apply ((simp split:if_splits add:Let_def)[1])+
+ apply (simp split:if_splits list.splits add:Let_def wq_def)
+ apply (auto dest!:ee)
+ apply (drule_tac lsp_set_eq, simp)
+ apply (subgoal_tac "distinct (waiting_queue (schs s) cs)", simp, fold wq_def)
+ apply (rule_tac wq_distinct, auto)
+ apply (erule_tac evt_cons, auto)
+ apply (drule_tac lsp_set_eq, simp)
+ apply (subgoal_tac "distinct (wq s cs)", simp)
+ apply (rule_tac wq_distinct, auto)
+ apply (erule_tac evt_cons, auto)
+ apply (drule_tac lsp_set_eq, simp)
+ apply (subgoal_tac "distinct (wq s cs)", simp)
+ apply (rule_tac wq_distinct, auto)
+ apply (erule_tac evt_cons, auto)
+ apply (auto simp:wq_def Let_def split:if_splits prod.splits)
+ done
+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 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 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 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
+ from block_pre [OF False h1]
+ have "e = P thread cs2" .
+ with eq_e1 neq12 show ?thesis by auto
+ qed
+ qed
+ } ultimately show ?thesis by arith
+ qed
+qed
+
+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)
+
+definition head_of :: "('a \<Rightarrow> 'b::linorder) \<Rightarrow> 'a set \<Rightarrow> 'a set"
+ where "head_of f A = {a . a \<in> A \<and> f a = Max (f ` A)}"
+
+definition wq_head :: "state \<Rightarrow> cs \<Rightarrow> thread set"
+ where "wq_head s cs = head_of (cp s) (set (wq s cs))"
+
+lemma f_nil_simp: "\<lbrakk>f cs = []\<rbrakk> \<Longrightarrow> f(cs:=[]) = f"
+proof
+ fix x
+ assume h:"f cs = []"
+ show "(f(cs := [])) x = f x"
+ proof(cases "cs = x")
+ case True
+ with h show ?thesis by simp
+ next
+ case False
+ with h show ?thesis by simp
+ 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 holding_nil:
+ "\<lbrakk>wq s cs = []; holding (wq s) th cs\<rbrakk> \<Longrightarrow> False"
+ by (unfold cs_holding_def, auto)
+
+lemma waiting_kept_1: "
+ \<lbrakk>vt step (V th cs#s); wq s cs = a # list; waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c;
+ lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
+ \<Longrightarrow> waiting (wq s) t c"
+ apply (drule_tac step_back_vt, drule_tac wq_distinct[of _ cs])
+ apply(drule_tac lsp_set_eq)
+ by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_kept_2:
+ "\<And>a list t c aa ca.
+ \<lbrakk>wq s cs = a # list; waiting ((wq s)(cs := [])) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
+ \<Longrightarrow> waiting (wq s) t c"
+ apply(drule_tac lsp_set_eq)
+ by (unfold cs_waiting_def, auto split:if_splits)
+
+
+lemma holding_nil_simp: "\<lbrakk>holding ((wq s)(cs := [])) t c\<rbrakk> \<Longrightarrow> holding (wq s) t c"
+ by(unfold cs_holding_def, auto)
+
+lemma step_wq_elim: "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; a \<noteq> th\<rbrakk> \<Longrightarrow> False"
+ apply(drule_tac step_back_step)
+ apply(ind_cases "step s (V th cs)")
+ by(unfold s_holding_def, auto)
+
+lemma holding_cs_neq_simp: "c \<noteq> cs \<Longrightarrow> holding ((wq s)(cs := u)) t c = holding (wq s) t c"
+ by (unfold cs_holding_def, auto)
+
+lemma holding_th_neq_elim:
+ "\<And>a list c t aa ca ab lista.
+ \<lbrakk>\<not> holding (wq s) t c; holding ((wq s)(cs := ab # aa @ lista @ ca)) t c;
+ ab \<noteq> t\<rbrakk>
+ \<Longrightarrow> False"
+ by (unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_nil_abs:
+ "\<not> holding ((wq s)(cs := [])) th cs"
+ by (unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_abs: "\<lbrakk>holding ((wq s)(cs := ab # aa @ lista @ c)) th cs; ab \<noteq> th\<rbrakk>
+ \<Longrightarrow> False"
+ by (unfold cs_holding_def, auto split:if_splits)
+
+lemma waiting_abs: "\<not> waiting ((wq s)(cs := t # l @ r)) t cs"
+ by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_1:
+ "\<lbrakk>\<not> waiting ((wq s)(cs := [])) t c; waiting (wq s) t c; c \<noteq> cs\<rbrakk>
+ \<Longrightarrow> False"
+ by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_2: "
+ \<lbrakk>\<not> waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c; waiting (wq s) t c;
+ c \<noteq> cs\<rbrakk>
+ \<Longrightarrow> False"
+ by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_abs_3:
+ "\<lbrakk>wq s cs = a # list; \<not> waiting ((wq s)(cs := [])) t c;
+ waiting (wq s) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
+ \<Longrightarrow> False"
+ apply(drule_tac lsp_mid_nil, simp)
+ by(unfold cs_waiting_def, auto split:if_splits)
+
+lemma waiting_simp: "c \<noteq> cs \<Longrightarrow> waiting ((wq s)(cs:=z)) t c = waiting (wq s) t c"
+ by(unfold cs_waiting_def, auto split:if_splits)
+
+lemma holding_cs_eq:
+ "\<lbrakk>\<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> c = cs"
+ by(unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_cs_eq_1:
+ "\<lbrakk>\<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c\<rbrakk>
+ \<Longrightarrow> c = cs"
+ by(unfold cs_holding_def, auto split:if_splits)
+
+lemma holding_th_eq:
+ "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; \<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c;
+ lsp (cp s) list = (aa, [], ca)\<rbrakk>
+ \<Longrightarrow> t = th"
+ apply(drule_tac lsp_mid_nil, simp)
+ apply(unfold cs_holding_def, auto split:if_splits)
+ apply(drule_tac step_back_step)
+ apply(ind_cases "step s (V th cs)")
+ by (unfold s_holding_def, auto split:if_splits)
+
+lemma holding_th_eq_1:
+ "\<lbrakk>vt step (V th cs#s);
+ wq s cs = a # list; \<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c;
+ lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
+ \<Longrightarrow> t = th"
+ apply(drule_tac step_back_step)
+ apply(ind_cases "step s (V th cs)")
+ apply(unfold s_holding_def cs_holding_def)
+ by (auto split:if_splits)
+
+lemma holding_th_eq_2: "\<lbrakk>holding ((wq s)(cs := ac # x)) th cs\<rbrakk>
+ \<Longrightarrow> ac = th"
+ by (unfold cs_holding_def, auto)
+
+lemma holding_th_eq_3: "
+ \<lbrakk>\<not> holding (wq s) t c;
+ holding ((wq s)(cs := ac # x)) t c\<rbrakk>
+ \<Longrightarrow> ac = t"
+ by (unfold cs_holding_def, auto)
+
+lemma holding_wq_eq: "holding ((wq s)(cs := th' # l @ r)) th' cs"
+ by (unfold cs_holding_def, auto)
+
+lemma waiting_th_eq: "
+ \<lbrakk>waiting (wq s) t c; wq s cs = a # list;
+ lsp (cp s) list = (aa, ac # lista, ba); \<not> waiting ((wq s)(cs := ac # aa @ lista @ ba)) t c\<rbrakk>
+ \<Longrightarrow> ac = t"
+ apply(drule_tac lsp_set_eq, simp)
+ by (unfold cs_waiting_def, auto split:if_splits)
+
+lemma step_depend_v:
+ "vt step (V th cs#s) \<Longrightarrow>
+ depend (V th cs # s) =
+ depend s - {(Cs cs, Th th)} -
+ {(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+ {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+ apply (unfold s_depend_def wq_def,
+ auto split:list.splits simp:Let_def f_nil_simp holding_wq_eq, fold wq_def cp_def)
+ apply (auto split:list.splits prod.splits
+ simp:Let_def f_nil_simp holding_nil_simp holding_cs_neq_simp holding_nil_abs
+ waiting_abs waiting_simp holding_wq_eq
+ elim:holding_nil waiting_kept_1 waiting_kept_2 step_wq_elim holding_th_neq_elim
+ holding_abs waiting_abs_1 waiting_abs_3 holding_cs_eq holding_cs_eq_1
+ holding_th_eq holding_th_eq_1 holding_th_eq_2 holding_th_eq_3 waiting_th_eq
+ dest:lsp_mid_length)
+ 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'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+ {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+ (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
+ from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+ have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
+ thus ?thesis
+ proof(cases "wq s cs")
+ case Nil
+ hence "?D = {}" by simp
+ with ac and eq_de show ?thesis by simp
+ next
+ case (Cons tth rest)
+ from stp and V have "step s (V th cs)" by simp
+ hence eq_wq: "wq s cs = th # rest"
+ proof -
+ show "step s (V th cs) \<Longrightarrow> wq s cs = th # rest"
+ apply(ind_cases "step s (V th cs)")
+ by(insert Cons, unfold s_holding_def, simp)
+ qed
+ show ?thesis
+ proof(cases "lsp (cp s) rest")
+ fix l b r
+ assume eq_lsp: "lsp (cp s) rest = (l, b, r) "
+ show ?thesis
+ proof(cases "b")
+ case Nil
+ with eq_lsp and eq_wq have "?D = {}" by simp
+ with ac and eq_de show ?thesis by simp
+ next
+ case (Cons th' m)
+ with eq_lsp
+ have eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+ apply simp
+ by (drule_tac lsp_mid_length, simp)
+ from eq_wq and eq_lsp
+ have eq_D: "?D = {(Cs cs, Th th')}" by auto
+ from eq_wq and eq_lsp
+ have eq_C: "?C = {(Th th', Cs cs)}" by 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 eq_lsp wq_distinct[OF vt, of cs]
+ show "waiting s th' cs" by(unfold s_waiting_def, auto dest:lsp_set_eq)
+ qed
+ with th'_e eq_x have "(Th th', Cs cs) \<in> ?E" by simp
+ with eq_C show "False" by simp
+ qed
+ with acyclic_insert[symmetric] and ac and eq_D
+ and eq_de show ?thesis by simp
+ qed
+ 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'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
+ {(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
+ (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 (rule simple_A, 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 *}
+
+thm wf_trancl
+thm finite_acyclic_wf
+thm finite_acyclic_wf_converse
+thm wf_induct
+
+
+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
+ proof(cases "(lsp (cur_preced (schs s)) rest)", unfold V)
+ fix l m r
+ assume eq_lsp: "lsp (cur_preced (schs s)) rest = (l, m, r)"
+ and eq_wq: "waiting_queue (schs s) cs' = a # rest"
+ and th_in_set: "th \<in> set (wq (V th' cs' # s) cs)"
+ show ?thesis
+ proof(cases "m")
+ case Nil
+ with eq_lsp have "rest = []" using lsp_mid_nil by auto
+ with eq_wq have "waiting_queue (schs s) cs' = [a]" by simp
+ with h[unfolded V wq_def] True
+ show ?thesis
+ by (simp add:Let_def)
+ next
+ case (Cons b rb)
+ with lsp_mid_length[OF eq_lsp] have eq_m: "m = [b]" by auto
+ with eq_lsp have "lsp (cur_preced (schs s)) rest = (l, [b], r)" by simp
+ with h[unfolded V wq_def] True lsp_set_eq [OF this] eq_wq
+ show ?thesis
+ apply (auto simp:Let_def, fold wq_def)
+ by (rule_tac ih [of _ cs'], auto)+
+ qed
+ 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 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)
+ by (auto simp:s_waiting_def wq_def Let_def split:list.splits prod.splits
+ dest:lsp_set_eq)
+qed
+
+lemma readys_v_eq_1:
+ fixes th thread cs rest
+ assumes neq_th: "th \<noteq> thread"
+ and eq_wq: "wq s cs = thread#rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+ and neq_th': "th \<noteq> th'"
+ 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 = cs in allE)
+ apply (simp add:s_waiting_def wq_def Let_def split:prod.splits list.splits)
+ apply (drule_tac lsp_mid_nil,simp, clarify, fold cp_def, clarsimp)
+ apply (frule_tac lsp_set_eq, simp)
+ apply (erule_tac x = csa in allE)
+ apply (subst (asm) (2) s_waiting_def, unfold wq_def)
+ apply (auto simp:Let_def split:list.splits prod.splits if_splits
+ dest:lsp_set_eq)
+ apply (unfold s_waiting_def)
+ apply (fold wq_def, clarsimp)
+ apply (clarsimp)+
+ apply (case_tac "csa = cs", simp)
+ apply (erule_tac x = cs in allE, simp)
+ apply (unfold wq_def)
+ by (auto simp:Let_def split:list.splits prod.splits if_splits
+ dest:lsp_set_eq)
+qed
+
+lemma readys_v_eq_2:
+ fixes th thread cs rest
+ assumes neq_th: "th \<noteq> thread"
+ and eq_wq: "wq s cs = thread#rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+ and neq_th': "th = th'"
+ and vt: "vt step s"
+ shows "(th \<in> readys (V thread cs#s))"
+proof -
+ from prems show ?thesis
+ apply (auto simp:readys_def)
+ apply (rule_tac wq_threads [of s _ cs], auto dest:lsp_set_eq)
+ apply (unfold s_waiting_def wq_def)
+ apply (auto simp:Let_def split:list.splits prod.splits if_splits
+ dest:lsp_set_eq lsp_mid_nil lsp_mid_length)
+ apply (fold cp_def, simp+, clarsimp)
+ apply (frule_tac lsp_set_eq, simp)
+ apply (fold wq_def)
+ apply (subgoal_tac "csa = cs", simp)
+ apply (rule_tac waiting_unique [of s th'], simp)
+ by (auto simp:s_waiting_def)
+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
+
+definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
+where "count Q l = length (filter Q l)"
+
+definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
+
+definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+
+
+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 step_holdents_v_minus:
+ fixes th cs s
+ assumes vt: "vt step (V th cs#s)"
+ shows "holdents (V th cs#s) th = holdents s th - {cs}"
+proof -
+ { fix rest l r
+ assume eq_wq: "wq s cs = th # rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
+ have "False"
+ proof -
+ from lsp_set_eq [OF eq_lsp] have " rest = l @ [th] @ r" .
+ with eq_wq have "wq s cs = th#\<dots>" by simp
+ with wq_distinct [OF step_back_vt[OF vt], of cs]
+ show ?thesis by auto
+ qed
+ } thus ?thesis unfolding holdents_def step_depend_v[OF vt] by auto
+qed
+
+lemma step_holdents_v_add:
+ fixes th th' cs s rest l r
+ assumes vt: "vt step (V th' cs#s)"
+ and neq_th: "th \<noteq> th'"
+ and eq_wq: "wq s cs = th' # rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
+ shows "holdents (V th' cs#s) th = holdents s th \<union> {cs}"
+proof -
+ from prems show ?thesis
+ unfolding holdents_def step_depend_v[OF vt] by auto
+qed
+
+lemma step_holdents_v_eq:
+ fixes th th' cs s rest l r th''
+ assumes vt: "vt step (V th' cs#s)"
+ and neq_th: "th \<noteq> th'"
+ and eq_wq: "wq s cs = th' # rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [th''], r)"
+ and neq_th': "th \<noteq> th''"
+ shows "holdents (V th' cs#s) th = holdents s th"
+proof -
+ from prems show ?thesis
+ unfolding holdents_def step_depend_v[OF vt] by auto
+qed
+
+definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
+where "cntCS s th = card (holdents s th)"
+
+lemma cntCS_v_eq:
+ fixes th thread cs rest
+ assumes neq_th: "th \<noteq> thread"
+ and eq_wq: "wq s cs = thread#rest"
+ and not_in: "th \<notin> set rest"
+ and vtv: "vt step (V thread cs#s)"
+ shows "cntCS (V thread cs#s) th = cntCS s th"
+proof -
+ from prems show ?thesis
+ apply (unfold cntCS_def holdents_def step_depend_v)
+ apply auto
+ apply (subgoal_tac "\<not> (\<exists>l r. lsp (cp s) rest = (l, [th], r))", auto)
+ by (drule_tac lsp_set_eq, auto)
+qed
+
+lemma cntCS_v_eq_1:
+ fixes th thread cs rest
+ assumes neq_th: "th \<noteq> thread"
+ and eq_wq: "wq s cs = thread#rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+ and neq_th': "th \<noteq> th'"
+ and vtv: "vt step (V thread cs#s)"
+ shows "cntCS (V thread cs#s) th = cntCS s th"
+proof -
+ from prems show ?thesis
+ apply (unfold cntCS_def holdents_def step_depend_v)
+ by auto
+qed
+
+fun the_cs :: "node \<Rightarrow> cs"
+where "the_cs (Cs cs) = cs"
+
+lemma cntCS_v_eq_2:
+ fixes th thread cs rest
+ assumes neq_th: "th \<noteq> thread"
+ and eq_wq: "wq s cs = thread#rest"
+ and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+ and neq_th': "th = th'"
+ and vtv: "vt step (V thread cs#s)"
+ shows "cntCS (V thread cs#s) th = 1 + cntCS s th"
+proof -
+ have "card {csa. csa = cs \<or> (Cs csa, Th th') \<in> depend s} =
+ Suc (card {cs. (Cs cs, Th th') \<in> depend s})"
+ (is "card ?A = Suc (card ?B)")
+ proof -
+ have h: "?A = insert cs ?B" by auto
+ moreover have h1: "?B = ?B - {cs}"
+ proof -
+ { assume "(Cs cs, Th th') \<in> depend s"
+ moreover have "(Th th', Cs cs) \<in> depend s"
+ proof -
+ from wq_distinct [OF step_back_vt[OF vtv], of cs]
+ eq_wq lsp_set_eq [OF eq_lsp] show ?thesis
+ apply (auto simp:s_depend_def)
+ by (unfold cs_waiting_def, auto)
+ qed
+ moreover note acyclic_depend [OF step_back_vt[OF vtv]]
+ ultimately have "False"
+ apply (auto simp:acyclic_def)
+ apply (erule_tac x="Cs cs" in allE)
+ apply (subgoal_tac "(Cs cs, Cs cs) \<in> (depend s)\<^sup>+", simp)
+ by (rule_tac trancl_into_trancl [where b = "Th th'"], auto)
+ } thus ?thesis by auto
+ qed
+ moreover have "card (insert cs ?B) = Suc (card (?B - {cs}))"
+ proof(rule card_insert)
+ from finite_depend [OF step_back_vt [OF vtv]]
+ have fnt: "finite (depend s)" .
+ show " finite {cs. (Cs cs, Th th') \<in> depend s}" (is "finite ?B")
+ proof -
+ have "?B \<subseteq> (\<lambda> (a, b). the_cs a) ` (depend s)"
+ apply (auto simp:image_def)
+ by (rule_tac x = "(Cs x, Th th')" in bexI, auto)
+ with fnt show ?thesis by (auto intro:finite_subset)
+ qed
+ qed
+ ultimately show ?thesis by simp
+ qed
+ with prems show ?thesis
+ apply (unfold cntCS_def holdents_def step_depend_v[OF vtv])
+ 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
+
+inductive_cases case_step_v: "step s (V thread cs)"
+
+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 -
+ have cs_in: "cs \<in> holdents s thread" using step_back_step[OF vtv]
+ apply (erule_tac case_step_v)
+ apply (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])
+ by (unfold holdents_def, unfold step_depend_v[OF vtv],
+ auto dest:lsp_set_eq)
+ 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 prio max_prio thread)
+ 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)
+ 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 -
+ have "thread \<notin> set (wq (e#s) cs1)"
+ proof(cases "lsp (cp s) rest")
+ fix l m r
+ assume h: "lsp (cp s) rest = (l, m, r)"
+ show ?thesis
+ proof(cases "m")
+ case Nil
+ from wq_v_eq_nil [OF eq_wq] h Nil eq_e
+ have " wq (e # s) cs = []" by auto
+ thus ?thesis using eq_cs by auto
+ next
+ case (Cons th' l')
+ from lsp_mid_length [OF h] and Cons h
+ have eqh: "lsp (cp s) rest = (l, [th'], r)" by auto
+ from wq_v_eq [OF eq_wq this]
+ have "wq (V thread cs # s) cs = th' # l @ r" .
+ moreover from lsp_set_eq [OF eqh]
+ have "set rest = set \<dots>" by auto
+ moreover have "thread \<notin> set rest"
+ proof -
+ from wq_distinct [OF step_back_vt[OF vtv], of cs]
+ and eq_wq show ?thesis by auto
+ qed
+ moreover note eq_e eq_cs
+ ultimately show ?thesis by simp
+ qed
+ 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)"
+ by(unfold eq_e, rule readys_v_eq [OF neq_th eq_wq False])
+ moreover have "cntCS (e#s) th = cntCS s th"
+ by(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq False vtv])
+ moreover note ih eq_cnp eq_cnv eq_threads
+ ultimately show ?thesis by auto
+ next
+ case True
+ obtain l m r where eq_lsp: "lsp (cp s) rest = (l, m, r)"
+ by (cases "lsp (cp s) rest", auto)
+ with True have "m \<noteq> []" by (auto dest:lsp_mid_nil)
+ with eq_lsp obtain th' where eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
+ by (case_tac m, auto dest:lsp_mid_length)
+ show ?thesis
+ proof(cases "th = th'")
+ case False
+ have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+ by (unfold eq_e, rule readys_v_eq_1 [OF neq_th eq_wq eq_lsp False])
+ moreover have "cntCS (e#s) th = cntCS s th"
+ by (unfold eq_e, rule cntCS_v_eq_1[OF neq_th eq_wq eq_lsp False vtv])
+ moreover note ih eq_cnp eq_cnv eq_threads
+ ultimately show ?thesis by auto
+ next
+ case True
+ have "th \<in> readys (e # s)"
+ by (unfold eq_e, rule readys_v_eq_2 [OF neq_th eq_wq eq_lsp True vt])
+ moreover have "cntP s th = cntV s th + cntCS s th + 1"
+ proof -
+ have "th \<notin> readys s"
+ proof -
+ from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
+ 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 True eq_wq lsp_set_eq [OF eq_lsp] neq_th
+ 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 have "cntCS (e # s) th = 1 + cntCS s th"
+ by (unfold eq_e, rule cntCS_v_eq_2 [OF neq_th eq_wq eq_lsp True vtv])
+ 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 prio max_prio thread)
+ 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)
+ have "cntCS (e # s) th = cntCS s th"
+ proof(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq _ vtv])
+ show "th \<notin> set rest"
+ proof
+ assume "th \<in> set rest"
+ with eq_wq have "th \<in> set (wq s cs)" by simp
+ from wq_threads [OF vt this] eq_e not_in
+ show False by simp
+ qed
+ qed
+ 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
+
+fun the_th :: "node \<Rightarrow> thread"
+ where "the_th (Th th) = th"
+
+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 prio max_prio thread)
+ 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 prio max_prio thread)
+ 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 prio max_prio thread)
+ 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
--- /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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/PrioGDef.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,401 @@
+(*<*)
+theory PrioGDef
+imports Precedence_ord Moment
+begin
+(*>*)
+
+text {*
+ In this section, the formal model of Priority Inheritance is presented.
+ The model is based on Paulson's inductive protocol verification method, where
+ the state of the system is modelled as a list of events happened so far with the latest
+ event put at the head.
+
+ To define events, the identifiers of {\em threads},
+ {\em priority} and {\em critical resources } (abbreviated as @{text "cs"})
+ need to be represented. All three are represetned using standard
+ Isabelle/HOL type @{typ "nat"}:
+*}
+
+type_synonym thread = nat -- {* Type for thread identifiers. *}
+type_synonym priority = nat -- {* Type for priorities. *}
+type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
+
+text {*
+ \noindent
+ Every event in the system corresponds to a system call, the formats of which are
+ defined as follows:
+ *}
+
+datatype event =
+ Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *}
+ Exit thread | -- {* Thread @{text "thread"} finishing its execution. *}
+ P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *}
+ V thread cs | -- {* Thread @{text "thread"} releasing critical resource @{text "cs"}. *}
+ Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
+
+text {*
+\noindent
+ Resource Allocation Graph (RAG for short) is used extensively in our formal analysis.
+ The following type @{text "node"} is used to represent nodes in RAG.
+ *}
+datatype node =
+ Th "thread" | -- {* Node for thread. *}
+ Cs "cs" -- {* Node for critical resource. *}
+
+text {*
+ In Paulson's inductive method, the states of system are represented as lists of events,
+ which is defined by the following type @{text "state"}:
+ *}
+type_synonym state = "event list"
+
+text {*
+ \noindent
+ The following function
+ @{text "threads"} is used to calculate the set of live threads (@{text "threads s"})
+ in state @{text "s"}.
+ *}
+fun threads :: "state \<Rightarrow> thread set"
+ where
+ -- {* At the start of the system, the set of threads is empty: *}
+ "threads [] = {}" |
+ -- {* New thread is added to the @{text "threads"}: *}
+ "threads (Create thread prio#s) = {thread} \<union> threads s" |
+ -- {* Finished thread is removed: *}
+ "threads (Exit thread # s) = (threads s) - {thread}" |
+ -- {* Other kind of events does not affect the value of @{text "threads"}: *}
+ "threads (e#s) = threads s"
+text {* \noindent
+ Functions such as @{text "threads"}, which extract information out of system states, are called
+ {\em observing functions}. A series of observing functions will be defined in the sequel in order to
+ model the protocol.
+ Observing function @{text "original_priority"} calculates
+ the {\em original priority} of thread @{text "th"} in state @{text "s"}, expressed as
+ : @{text "original_priority th s" }. The {\em original priority} is the priority
+ assigned to a thread when it is created or when it is reset by system call
+ @{text "Set thread priority"}.
+*}
+
+fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
+ where
+ -- {* @{text "0"} is assigned to threads which have never been created: *}
+ "original_priority thread [] = 0" |
+ "original_priority thread (Create thread' prio#s) =
+ (if thread' = thread then prio else original_priority thread s)" |
+ "original_priority thread (Set thread' prio#s) =
+ (if thread' = thread then prio else original_priority thread s)" |
+ "original_priority thread (e#s) = original_priority thread s"
+
+text {*
+ \noindent
+ In the following,
+ @{text "birthtime th s"} is the time when thread @{text "th"} is created,
+ observed from state @{text "s"}.
+ The time in the system is measured by the number of events happened so far since the very beginning.
+*}
+fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
+ where
+ "birthtime thread [] = 0" |
+ "birthtime thread ((Create thread' prio)#s) =
+ (if (thread = thread') then length s else birthtime thread s)" |
+ "birthtime thread ((Set thread' prio)#s) =
+ (if (thread = thread') then length s else birthtime thread s)" |
+ "birthtime thread (e#s) = birthtime thread s"
+
+text {*
+ \noindent
+ The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of
+ a thread is the combination of its {\em original priority} and {\em birth time}. The intention is
+ to discriminate threads with the same priority by giving threads whose priority
+ is assigned earlier higher precedences, becasue such threads are more urgent to finish.
+ This explains the following definition:
+ *}
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+ where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
+
+
+text {*
+ \noindent
+ A number of important notions are defined here:
+ *}
+
+consts
+ holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ depend :: "'b \<Rightarrow> (node \<times> node) set"
+ dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+text {*
+ \noindent
+ In the definition of the following several functions, it is supposed that
+ the waiting queue of every critical resource is given by a waiting queue
+ function @{text "wq"}, which servers as arguments of these functions.
+ *}
+defs (overloaded)
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ We define that the thread which is at the head of waiting queue of resource @{text "cs"}
+ is holding the resource. This definition is slightly different from tradition where
+ all threads in the waiting queue are considered as waiting for the resource.
+ This notion is reflected in the definition of @{text "holding wq th cs"} as follows:
+ \end{minipage}
+ *}
+ cs_holding_def:
+ "holding wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ In accordance with the definition of @{text "holding wq th cs"},
+ a thread @{text "th"} is considered waiting for @{text "cs"} if
+ it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head.
+ This is reflected in the definition of @{text "waiting wq th cs"} as follows:
+ \end{minipage}
+ *}
+ cs_waiting_def:
+ "waiting wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ @{text "depend wq"} represents the Resource Allocation Graph of the system under the waiting
+ queue function @{text "wq"}.
+ \end{minipage}
+ *}
+ cs_depend_def:
+ "depend (wq::cs \<Rightarrow> thread list) \<equiv>
+ {(Th t, Cs c) | t c. waiting wq t c} \<union> {(Cs c, Th t) | c t. holding wq t c}"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ The following @{text "dependents wq th"} represents the set of threads which are depending on
+ thread @{text "th"} in Resource Allocation Graph @{text "depend wq"}:
+ \end{minipage}
+ *}
+ cs_dependents_def:
+ "dependents (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (depend wq)^+}"
+
+text {*
+ The data structure used by the operating system for scheduling is referred to as
+ {\em schedule state}. It is represented as a record consisting of
+ a function assigning waiting queue to resources and a function assigning precedence to
+ threads:
+ *}
+record schedule_state =
+ waiting_queue :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
+ cur_preced :: "thread \<Rightarrow> precedence" -- {* The function assigning precedence. *}
+
+text {* \noindent
+ The following
+ @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under
+ state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of
+ Priority Inheritance that the {\em current precedence} of a thread is the precedence
+ inherited from the maximum of all its dependents, i.e. the threads which are waiting
+ directly or indirectly waiting for some resources from it. If no such thread exits,
+ @{text "th"}'s {\em current precedence} equals its original precedence, i.e.
+ @{text "preced th s"}.
+ *}
+definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
+ where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
+
+text {* \noindent
+ The following function @{text "schs"} is used to calculate the schedule state @{text "schs s"}.
+ It is the key function to model Priority Inheritance:
+ *}
+fun schs :: "state \<Rightarrow> schedule_state"
+ where "schs [] = \<lparr>waiting_queue = \<lambda> cs. [], cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ \begin{enumerate}
+ \item @{text "ps"} is the schedule state of last moment.
+ \item @{text "pwq"} is the waiting queue function of last moment.
+ \item @{text "pcp"} is the precedence function of last moment.
+ \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement:
+ \begin{enumerate}
+ \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to
+ the end of @{text "cs"}'s waiting queue.
+ \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state,
+ @{text "th'"} must equal to @{text "thread"},
+ because @{text "thread"} is the one currently holding @{text "cs"}.
+ The case @{text "[] \<Longrightarrow> []"} may never be executed in a legal state.
+ the @{text "(SOME q. distinct q \<and> set q = set qs)"} is used to choose arbitrarily one
+ thread in waiting to take over the released resource @{text "cs"}. In our representation,
+ this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
+ \item For other happening event, the schedule state just does not change.
+ \end{enumerate}
+ \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue
+ function. The dependency of precedence function on waiting queue function is the reason to
+ put them in the same record so that they can evolve together.
+ \end{enumerate}
+ \end{minipage}
+ *}
+ "schs (e#s) = (let ps = schs s in
+ let pwq = waiting_queue ps in
+ let pcp = cur_preced ps in
+ let nwq = case e of
+ P thread cs \<Rightarrow> pwq(cs:=(pwq cs @ [thread])) |
+ V thread cs \<Rightarrow> let nq = case (pwq cs) of
+ [] \<Rightarrow> [] |
+ (th'#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
+ in pwq(cs:=nq) |
+ _ \<Rightarrow> pwq
+ in let ncp = cpreced (e#s) nwq in
+ \<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
+ )"
+
+text {*
+ \noindent
+ The following @{text "wq"} is a shorthand for @{text "waiting_queue"}.
+ *}
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
+ where "wq s = waiting_queue (schs s)"
+
+text {* \noindent
+ The following @{text "cp"} is a shorthand for @{text "cur_preced"}.
+ *}
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+ where "cp s = cur_preced (schs s)"
+
+text {* \noindent
+ Functions @{text "holding"}, @{text "waiting"}, @{text "depend"} and
+ @{text "dependents"} still have the
+ same meaning, but redefined so that they no longer depend on the
+ fictitious {\em waiting queue function}
+ @{text "wq"}, but on system state @{text "s"}.
+ *}
+defs (overloaded)
+ s_holding_def:
+ "holding (s::state) thread cs \<equiv> (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
+ s_waiting_def:
+ "waiting (s::state) thread cs \<equiv> (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
+ s_depend_def:
+ "depend (s::state) \<equiv>
+ {(Th t, Cs c) | t c. waiting (wq s) t c} \<union> {(Cs c, Th t) | c t. holding (wq s) t c}"
+ s_dependents_def:
+ "dependents (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
+
+text {*
+ The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready}
+ for running if it is a live thread and it is not waiting for any critical resource.
+ *}
+definition readys :: "state \<Rightarrow> thread set"
+ where "readys s = {thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
+
+text {* \noindent
+ The following function @{text "runing"} calculates the set of running thread, which is the ready
+ thread with the highest precedence.
+ *}
+definition runing :: "state \<Rightarrow> thread set"
+ where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
+
+text {* \noindent
+ The following function @{text "holdents s th"} returns the set of resources held by thread
+ @{text "th"} in state @{text "s"}.
+ *}
+definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+ where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
+
+text {* \noindent
+ @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in
+ state @{text "s"}:
+ *}
+definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntCS s th = card (holdents s th)"
+
+text {* \noindent
+ The fact that event @{text "e"} is eligible to happen next in state @{text "s"}
+ is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as
+ follows:
+ *}
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+ where
+ -- {*
+ A thread can be created if it is not a live thread:
+ *}
+ thread_create: "\<lbrakk>thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
+ -- {*
+ A thread can exit if it no longer hold any resource:
+ *}
+ thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ A thread can request for an critical resource @{text "cs"}, if it is running and
+ the request does not form a loop in the current RAG. The latter condition
+ is set up to avoid deadlock. The condition also reflects our assumption all threads are
+ carefully programmed so that deadlock can not happen:
+ \end{minipage}
+ *}
+ thread_P: "\<lbrakk>thread \<in> runing s; (Cs cs, Th thread) \<notin> (depend s)^+\<rbrakk> \<Longrightarrow>
+ step s (P thread cs)" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ A thread can release a critical resource @{text "cs"}
+ if it is running and holding that resource:
+ \end{minipage}
+ *}
+ thread_V: "\<lbrakk>thread \<in> runing s; holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
+ -- {*
+ A thread can adjust its own priority as long as it is current running:
+ *}
+ thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+text {* \noindent
+ With predicate @{text "step"}, the fact that @{text "s"} is a legal state in
+ Priority Inheritance protocol can be expressed as: @{text "vt step s"}, where
+ the predicate @{text "vt"} can be defined as the following:
+ *}
+inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
+ for cs -- {* @{text "cs"} is an argument representing any step predicate. *}
+ where
+ -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
+ vt_nil[intro]: "vt cs []" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ If @{text "s"} a legal state, and event @{text "e"} is eligible to happen
+ in state @{text "s"}, then @{text "e#s"} is a legal state as well:
+ \end{minipage}
+ *}
+ vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
+
+text {* \noindent
+ It is easy to see that the definition of @{text "vt"} is generic. It can be applied to
+ any step predicate to get the set of legal states.
+ *}
+
+text {* \noindent
+ The following two functions @{text "the_cs"} and @{text "the_th"} are used to extract
+ critical resource and thread respectively out of RAG nodes.
+ *}
+fun the_cs :: "node \<Rightarrow> cs"
+ where "the_cs (Cs cs) = cs"
+
+fun the_th :: "node \<Rightarrow> thread"
+ where "the_th (Th th) = th"
+
+text {* \noindent
+ The following predicate @{text "next_th"} describe the next thread to
+ take over when a critical resource is released. In @{text "next_th s th cs t"},
+ @{text "th"} is the thread to release, @{text "t"} is the one to take over.
+ *}
+definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
+ where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and>
+ t = hd (SOME q. distinct q \<and> set q = set rest))"
+
+text {* \noindent
+ The function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
+ in list @{text "l"}:
+ *}
+definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
+ where "count Q l = length (filter Q l)"
+
+text {* \noindent
+ The following @{text "cntP s"} returns the number of operation @{text "P"} happened
+ before reaching state @{text "s"}.
+ *}
+definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
+
+text {* \noindent
+ The following @{text "cntV s"} returns the number of operation @{text "V"} happened
+ before reaching state @{text "s"}.
+ *}
+definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+(*<*)
+end
+(*>*)
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/ROOT.ML Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,2 @@
+use_thy "CpsG";
+use_thy "ExtGG";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/document/llncs.cls Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,1189 @@
+% LLNCS DOCUMENT CLASS -- version 2.13 (28-Jan-2002)
+% Springer Verlag LaTeX2e support for Lecture Notes in Computer Science
+%
+%%
+%% \CharacterTable
+%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%% Digits \0\1\2\3\4\5\6\7\8\9
+%% Exclamation \! Double quote \" Hash (number) \#
+%% Dollar \$ Percent \% Ampersand \&
+%% Acute accent \' Left paren \( Right paren \)
+%% Asterisk \* Plus \+ Comma \,
+%% Minus \- Point \. Solidus \/
+%% Colon \: Semicolon \; Less than \<
+%% Equals \= Greater than \> Question mark \?
+%% Commercial at \@ Left bracket \[ Backslash \\
+%% Right bracket \] Circumflex \^ Underscore \_
+%% Grave accent \` Left brace \{ Vertical bar \|
+%% Right brace \} Tilde \~}
+%%
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesClass{llncs}[2002/01/28 v2.13
+^^J LaTeX document class for Lecture Notes in Computer Science]
+% Options
+\let\if@envcntreset\iffalse
+\DeclareOption{envcountreset}{\let\if@envcntreset\iftrue}
+\DeclareOption{citeauthoryear}{\let\citeauthoryear=Y}
+\DeclareOption{oribibl}{\let\oribibl=Y}
+\let\if@custvec\iftrue
+\DeclareOption{orivec}{\let\if@custvec\iffalse}
+\let\if@envcntsame\iffalse
+\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue}
+\let\if@envcntsect\iffalse
+\DeclareOption{envcountsect}{\let\if@envcntsect\iftrue}
+\let\if@runhead\iffalse
+\DeclareOption{runningheads}{\let\if@runhead\iftrue}
+
+\let\if@openbib\iffalse
+\DeclareOption{openbib}{\let\if@openbib\iftrue}
+
+% languages
+\let\switcht@@therlang\relax
+\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}}
+\def\ds@francais{\def\switcht@@therlang{\switcht@francais}}
+
+\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
+
+\ProcessOptions
+
+\LoadClass[twoside]{article}
+\RequirePackage{multicol} % needed for the list of participants, index
+
+\setlength{\textwidth}{12.2cm}
+\setlength{\textheight}{19.3cm}
+\renewcommand\@pnumwidth{2em}
+\renewcommand\@tocrmarg{3.5em}
+%
+\def\@dottedtocline#1#2#3#4#5{%
+ \ifnum #1>\c@tocdepth \else
+ \vskip \z@ \@plus.2\p@
+ {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \parindent #2\relax\@afterindenttrue
+ \interlinepenalty\@M
+ \leavevmode
+ \@tempdima #3\relax
+ \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
+ {#4}\nobreak
+ \leaders\hbox{$\m@th
+ \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
+ mu$}\hfill
+ \nobreak
+ \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}%
+ \par}%
+ \fi}
+%
+\def\switcht@albion{%
+\def\abstractname{Abstract.}
+\def\ackname{Acknowledgement.}
+\def\andname{and}
+\def\lastandname{\unskip, and}
+\def\appendixname{Appendix}
+\def\chaptername{Chapter}
+\def\claimname{Claim}
+\def\conjecturename{Conjecture}
+\def\contentsname{Table of Contents}
+\def\corollaryname{Corollary}
+\def\definitionname{Definition}
+\def\examplename{Example}
+\def\exercisename{Exercise}
+\def\figurename{Fig.}
+\def\keywordname{{\bf Key words:}}
+\def\indexname{Index}
+\def\lemmaname{Lemma}
+\def\contriblistname{List of Contributors}
+\def\listfigurename{List of Figures}
+\def\listtablename{List of Tables}
+\def\mailname{{\it Correspondence to\/}:}
+\def\noteaddname{Note added in proof}
+\def\notename{Note}
+\def\partname{Part}
+\def\problemname{Problem}
+\def\proofname{Proof}
+\def\propertyname{Property}
+\def\propositionname{Proposition}
+\def\questionname{Question}
+\def\remarkname{Remark}
+\def\seename{see}
+\def\solutionname{Solution}
+\def\subclassname{{\it Subject Classifications\/}:}
+\def\tablename{Table}
+\def\theoremname{Theorem}}
+\switcht@albion
+% Names of theorem like environments are already defined
+% but must be translated if another language is chosen
+%
+% French section
+\def\switcht@francais{%\typeout{On parle francais.}%
+ \def\abstractname{R\'esum\'e.}%
+ \def\ackname{Remerciements.}%
+ \def\andname{et}%
+ \def\lastandname{ et}%
+ \def\appendixname{Appendice}
+ \def\chaptername{Chapitre}%
+ \def\claimname{Pr\'etention}%
+ \def\conjecturename{Hypoth\`ese}%
+ \def\contentsname{Table des mati\`eres}%
+ \def\corollaryname{Corollaire}%
+ \def\definitionname{D\'efinition}%
+ \def\examplename{Exemple}%
+ \def\exercisename{Exercice}%
+ \def\figurename{Fig.}%
+ \def\keywordname{{\bf Mots-cl\'e:}}
+ \def\indexname{Index}
+ \def\lemmaname{Lemme}%
+ \def\contriblistname{Liste des contributeurs}
+ \def\listfigurename{Liste des figures}%
+ \def\listtablename{Liste des tables}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}%
+ \def\notename{Remarque}%
+ \def\partname{Partie}%
+ \def\problemname{Probl\`eme}%
+ \def\proofname{Preuve}%
+ \def\propertyname{Caract\'eristique}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Question}%
+ \def\remarkname{Remarque}%
+ \def\seename{voir}
+ \def\solutionname{Solution}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tableau}%
+ \def\theoremname{Th\'eor\`eme}%
+}
+%
+% German section
+\def\switcht@deutsch{%\typeout{Man spricht deutsch.}%
+ \def\abstractname{Zusammenfassung.}%
+ \def\ackname{Danksagung.}%
+ \def\andname{und}%
+ \def\lastandname{ und}%
+ \def\appendixname{Anhang}%
+ \def\chaptername{Kapitel}%
+ \def\claimname{Behauptung}%
+ \def\conjecturename{Hypothese}%
+ \def\contentsname{Inhaltsverzeichnis}%
+ \def\corollaryname{Korollar}%
+%\def\definitionname{Definition}%
+ \def\examplename{Beispiel}%
+ \def\exercisename{\"Ubung}%
+ \def\figurename{Abb.}%
+ \def\keywordname{{\bf Schl\"usselw\"orter:}}
+ \def\indexname{Index}
+%\def\lemmaname{Lemma}%
+ \def\contriblistname{Mitarbeiter}
+ \def\listfigurename{Abbildungsverzeichnis}%
+ \def\listtablename{Tabellenverzeichnis}%
+ \def\mailname{{\it Correspondence to\/}:}
+ \def\noteaddname{Nachtrag}%
+ \def\notename{Anmerkung}%
+ \def\partname{Teil}%
+%\def\problemname{Problem}%
+ \def\proofname{Beweis}%
+ \def\propertyname{Eigenschaft}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Frage}%
+ \def\remarkname{Anmerkung}%
+ \def\seename{siehe}
+ \def\solutionname{L\"osung}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tabelle}%
+%\def\theoremname{Theorem}%
+}
+
+% Ragged bottom for the actual page
+\def\thisbottomragged{\def\@textbottom{\vskip\z@ plus.0001fil
+\global\let\@textbottom\relax}}
+
+\renewcommand\small{%
+ \@setfontsize\small\@ixpt{11}%
+ \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@
+ \abovedisplayshortskip \z@ \@plus2\p@
+ \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@
+ \def\@listi{\leftmargin\leftmargini
+ \parsep 0\p@ \@plus1\p@ \@minus\p@
+ \topsep 8\p@ \@plus2\p@ \@minus4\p@
+ \itemsep0\p@}%
+ \belowdisplayskip \abovedisplayskip
+}
+
+\frenchspacing
+\widowpenalty=10000
+\clubpenalty=10000
+
+\setlength\oddsidemargin {63\p@}
+\setlength\evensidemargin {63\p@}
+\setlength\marginparwidth {90\p@}
+
+\setlength\headsep {16\p@}
+
+\setlength\footnotesep{7.7\p@}
+\setlength\textfloatsep{8mm\@plus 2\p@ \@minus 4\p@}
+\setlength\intextsep {8mm\@plus 2\p@ \@minus 2\p@}
+
+\setcounter{secnumdepth}{2}
+
+\newcounter {chapter}
+\renewcommand\thechapter {\@arabic\c@chapter}
+
+\newif\if@mainmatter \@mainmattertrue
+\newcommand\frontmatter{\cleardoublepage
+ \@mainmatterfalse\pagenumbering{Roman}}
+\newcommand\mainmatter{\cleardoublepage
+ \@mainmattertrue\pagenumbering{arabic}}
+\newcommand\backmatter{\if@openright\cleardoublepage\else\clearpage\fi
+ \@mainmatterfalse}
+
+\renewcommand\part{\cleardoublepage
+ \thispagestyle{empty}%
+ \if@twocolumn
+ \onecolumn
+ \@tempswatrue
+ \else
+ \@tempswafalse
+ \fi
+ \null\vfil
+ \secdef\@part\@spart}
+
+\def\@part[#1]#2{%
+ \ifnum \c@secnumdepth >-2\relax
+ \refstepcounter{part}%
+ \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
+ \else
+ \addcontentsline{toc}{part}{#1}%
+ \fi
+ \markboth{}{}%
+ {\centering
+ \interlinepenalty \@M
+ \normalfont
+ \ifnum \c@secnumdepth >-2\relax
+ \huge\bfseries \partname~\thepart
+ \par
+ \vskip 20\p@
+ \fi
+ \Huge \bfseries #2\par}%
+ \@endpart}
+\def\@spart#1{%
+ {\centering
+ \interlinepenalty \@M
+ \normalfont
+ \Huge \bfseries #1\par}%
+ \@endpart}
+\def\@endpart{\vfil\newpage
+ \if@twoside
+ \null
+ \thispagestyle{empty}%
+ \newpage
+ \fi
+ \if@tempswa
+ \twocolumn
+ \fi}
+
+\newcommand\chapter{\clearpage
+ \thispagestyle{empty}%
+ \global\@topnum\z@
+ \@afterindentfalse
+ \secdef\@chapter\@schapter}
+\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
+ \if@mainmatter
+ \refstepcounter{chapter}%
+ \typeout{\@chapapp\space\thechapter.}%
+ \addcontentsline{toc}{chapter}%
+ {\protect\numberline{\thechapter}#1}%
+ \else
+ \addcontentsline{toc}{chapter}{#1}%
+ \fi
+ \else
+ \addcontentsline{toc}{chapter}{#1}%
+ \fi
+ \chaptermark{#1}%
+ \addtocontents{lof}{\protect\addvspace{10\p@}}%
+ \addtocontents{lot}{\protect\addvspace{10\p@}}%
+ \if@twocolumn
+ \@topnewpage[\@makechapterhead{#2}]%
+ \else
+ \@makechapterhead{#2}%
+ \@afterheading
+ \fi}
+\def\@makechapterhead#1{%
+% \vspace*{50\p@}%
+ {\centering
+ \ifnum \c@secnumdepth >\m@ne
+ \if@mainmatter
+ \large\bfseries \@chapapp{} \thechapter
+ \par\nobreak
+ \vskip 20\p@
+ \fi
+ \fi
+ \interlinepenalty\@M
+ \Large \bfseries #1\par\nobreak
+ \vskip 40\p@
+ }}
+\def\@schapter#1{\if@twocolumn
+ \@topnewpage[\@makeschapterhead{#1}]%
+ \else
+ \@makeschapterhead{#1}%
+ \@afterheading
+ \fi}
+\def\@makeschapterhead#1{%
+% \vspace*{50\p@}%
+ {\centering
+ \normalfont
+ \interlinepenalty\@M
+ \Large \bfseries #1\par\nobreak
+ \vskip 40\p@
+ }}
+
+\renewcommand\section{\@startsection{section}{1}{\z@}%
+ {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+ {12\p@ \@plus 4\p@ \@minus 4\p@}%
+ {\normalfont\large\bfseries\boldmath
+ \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+ {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+ {8\p@ \@plus 4\p@ \@minus 4\p@}%
+ {\normalfont\normalsize\bfseries\boldmath
+ \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+ {-18\p@ \@plus -4\p@ \@minus -4\p@}%
+ {-0.5em \@plus -0.22em \@minus -0.1em}%
+ {\normalfont\normalsize\bfseries\boldmath}}
+\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
+ {-12\p@ \@plus -4\p@ \@minus -4\p@}%
+ {-0.5em \@plus -0.22em \@minus -0.1em}%
+ {\normalfont\normalsize\itshape}}
+\renewcommand\subparagraph[1]{\typeout{LLNCS warning: You should not use
+ \string\subparagraph\space with this class}\vskip0.5cm
+You should not use \verb|\subparagraph| with this class.\vskip0.5cm}
+
+\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
+\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
+\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
+\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
+\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
+\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
+\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
+\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
+\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
+\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
+\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}
+
+\let\footnotesize\small
+
+\if@custvec
+\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}}
+{\mbox{\boldmath$\textstyle#1$}}
+{\mbox{\boldmath$\scriptstyle#1$}}
+{\mbox{\boldmath$\scriptscriptstyle#1$}}}
+\fi
+
+\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}}
+\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil
+\penalty50\hskip1em\null\nobreak\hfil\squareforqed
+\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
+
+\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+\gets\cr\to\cr}}}}}
+\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+<\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
+>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.8pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.3pt}<\cr}}}}}
+\def\bbbr{{\rm I\!R}} %reelle Zahlen
+\def\bbbm{{\rm I\!M}}
+\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
+\def\bbbf{{\rm I\!F}}
+\def\bbbh{{\rm I\!H}}
+\def\bbbk{{\rm I\!K}}
+\def\bbbp{{\rm I\!P}}
+\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
+{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
+\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
+\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbs{{\mathchoice
+{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
+\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
+{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
+
+\let\ts\,
+
+\setlength\leftmargini {17\p@}
+\setlength\leftmargin {\leftmargini}
+\setlength\leftmarginii {\leftmargini}
+\setlength\leftmarginiii {\leftmargini}
+\setlength\leftmarginiv {\leftmargini}
+\setlength \labelsep {.5em}
+\setlength \labelwidth{\leftmargini}
+\addtolength\labelwidth{-\labelsep}
+
+\def\@listI{\leftmargin\leftmargini
+ \parsep 0\p@ \@plus1\p@ \@minus\p@
+ \topsep 8\p@ \@plus2\p@ \@minus4\p@
+ \itemsep0\p@}
+\let\@listi\@listI
+\@listi
+\def\@listii {\leftmargin\leftmarginii
+ \labelwidth\leftmarginii
+ \advance\labelwidth-\labelsep
+ \topsep 0\p@ \@plus2\p@ \@minus\p@}
+\def\@listiii{\leftmargin\leftmarginiii
+ \labelwidth\leftmarginiii
+ \advance\labelwidth-\labelsep
+ \topsep 0\p@ \@plus\p@\@minus\p@
+ \parsep \z@
+ \partopsep \p@ \@plus\z@ \@minus\p@}
+
+\renewcommand\labelitemi{\normalfont\bfseries --}
+\renewcommand\labelitemii{$\m@th\bullet$}
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
+ {{\contentsname}}}
+ \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
+ \def\lastand{\ifnum\value{auco}=2\relax
+ \unskip{} \andname\
+ \else
+ \unskip \lastandname\
+ \fi}%
+ \def\and{\stepcounter{@auth}\relax
+ \ifnum\value{@auth}=\value{auco}%
+ \lastand
+ \else
+ \unskip,
+ \fi}%
+ \@starttoc{toc}\if@restonecol\twocolumn\fi}
+
+\def\l@part#1#2{\addpenalty{\@secpenalty}%
+ \addvspace{2em plus\p@}% % space above part line
+ \begingroup
+ \parindent \z@
+ \rightskip \z@ plus 5em
+ \hrule\vskip5pt
+ \large % same size as for a contribution heading
+ \bfseries\boldmath % set line in boldface
+ \leavevmode % TeX command to enter horizontal mode.
+ #1\par
+ \vskip5pt
+ \hrule
+ \vskip1pt
+ \nobreak % Never break after part entry
+ \endgroup}
+
+\def\@dotsep{2}
+
+\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
+{chapter.\thechapter}\fi}
+
+\def\addnumcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
+ {\thechapter}#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}
+\def\addcontentsmarkwop#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}
+
+\def\@adcmk[#1]{\ifcase #1 \or
+\def\@gtempa{\addnumcontentsmark}%
+ \or \def\@gtempa{\addcontentsmark}%
+ \or \def\@gtempa{\addcontentsmarkwop}%
+ \fi\@gtempa{toc}{chapter}}
+\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
+
+\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
+ \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ {\large\bfseries\boldmath#1}\ifx0#2\hfil\null
+ \else
+ \nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hss #2}%
+ \fi\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@title#1#2{\addpenalty{-\@highpenalty}
+ \addvspace{8pt plus 1pt}
+ \@tempdima \z@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ #1\nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hss #2}\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@author#1#2{\addpenalty{\@highpenalty}
+ \@tempdima=\z@ %15\p@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
+ \textit{#1}\par
+ \penalty\@highpenalty \endgroup}
+
+%\setcounter{tocdepth}{0}
+\newdimen\tocchpnum
+\newdimen\tocsecnum
+\newdimen\tocsectotal
+\newdimen\tocsubsecnum
+\newdimen\tocsubsectotal
+\newdimen\tocsubsubsecnum
+\newdimen\tocsubsubsectotal
+\newdimen\tocparanum
+\newdimen\tocparatotal
+\newdimen\tocsubparanum
+\tocchpnum=\z@ % no chapter numbers
+\tocsecnum=15\p@ % section 88. plus 2.222pt
+\tocsubsecnum=23\p@ % subsection 88.8 plus 2.222pt
+\tocsubsubsecnum=27\p@ % subsubsection 88.8.8 plus 1.444pt
+\tocparanum=35\p@ % paragraph 88.8.8.8 plus 1.666pt
+\tocsubparanum=43\p@ % subparagraph 88.8.8.8.8 plus 1.888pt
+\def\calctocindent{%
+\tocsectotal=\tocchpnum
+\advance\tocsectotal by\tocsecnum
+\tocsubsectotal=\tocsectotal
+\advance\tocsubsectotal by\tocsubsecnum
+\tocsubsubsectotal=\tocsubsectotal
+\advance\tocsubsubsectotal by\tocsubsubsecnum
+\tocparatotal=\tocsubsubsectotal
+\advance\tocparatotal by\tocparanum}
+\calctocindent
+
+\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
+\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
+\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
+\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
+\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
+
+\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
+ \@starttoc{lof}\if@restonecol\twocolumn\fi}
+\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
+
+\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
+ \@starttoc{lot}\if@restonecol\twocolumn\fi}
+\let\l@table\l@figure
+
+\renewcommand\listoffigures{%
+ \section*{\listfigurename
+ \@mkboth{\listfigurename}{\listfigurename}}%
+ \@starttoc{lof}%
+ }
+
+\renewcommand\listoftables{%
+ \section*{\listtablename
+ \@mkboth{\listtablename}{\listtablename}}%
+ \@starttoc{lot}%
+ }
+
+\ifx\oribibl\undefined
+\ifx\citeauthoryear\undefined
+\renewenvironment{thebibliography}[1]
+ {\section*{\refname}
+ \def\@biblabel##1{##1.}
+ \small
+ \list{\@biblabel{\@arabic\c@enumiv}}%
+ {\settowidth\labelwidth{\@biblabel{#1}}%
+ \leftmargin\labelwidth
+ \advance\leftmargin\labelsep
+ \if@openbib
+ \advance\leftmargin\bibindent
+ \itemindent -\bibindent
+ \listparindent \itemindent
+ \parsep \z@
+ \fi
+ \usecounter{enumiv}%
+ \let\p@enumiv\@empty
+ \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+ \if@openbib
+ \renewcommand\newblock{\par}%
+ \else
+ \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+ \fi
+ \sloppy\clubpenalty4000\widowpenalty4000%
+ \sfcode`\.=\@m}
+ {\def\@noitemerr
+ {\@latex@warning{Empty `thebibliography' environment}}%
+ \endlist}
+\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
+ {\let\protect\noexpand\immediate
+ \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+\newcount\@tempcntc
+\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
+ \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
+ {\@ifundefined
+ {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
+ ?}\@warning
+ {Citation `\@citeb' on page \thepage \space undefined}}%
+ {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
+ \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
+ \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
+ \else
+ \advance\@tempcntb\@ne
+ \ifnum\@tempcntb=\@tempcntc
+ \else\advance\@tempcntb\m@ne\@citeo
+ \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
+\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
+ \@citea\def\@citea{,\,\hskip\z@skip}%
+ \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
+ {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
+ \def\@citea{--}\fi
+ \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
+\else
+\renewenvironment{thebibliography}[1]
+ {\section*{\refname}
+ \small
+ \list{}%
+ {\settowidth\labelwidth{}%
+ \leftmargin\parindent
+ \itemindent=-\parindent
+ \labelsep=\z@
+ \if@openbib
+ \advance\leftmargin\bibindent
+ \itemindent -\bibindent
+ \listparindent \itemindent
+ \parsep \z@
+ \fi
+ \usecounter{enumiv}%
+ \let\p@enumiv\@empty
+ \renewcommand\theenumiv{}}%
+ \if@openbib
+ \renewcommand\newblock{\par}%
+ \else
+ \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+ \fi
+ \sloppy\clubpenalty4000\widowpenalty4000%
+ \sfcode`\.=\@m}
+ {\def\@noitemerr
+ {\@latex@warning{Empty `thebibliography' environment}}%
+ \endlist}
+ \def\@cite#1{#1}%
+ \def\@lbibitem[#1]#2{\item[]\if@filesw
+ {\def\protect##1{\string ##1\space}\immediate
+ \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+ \fi
+\else
+\@cons\@openbib@code{\noexpand\small}
+\fi
+
+\def\idxquad{\hskip 10\p@}% space that divides entry from number
+
+\def\@idxitem{\par\hangindent 10\p@}
+
+\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
+ \noindent\hangindent\wd0\box0}% index entry
+
+\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
+ \noindent\hangindent\wd0\box0}% order index entry
+
+\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
+
+\renewenvironment{theindex}
+ {\@mkboth{\indexname}{\indexname}%
+ \thispagestyle{empty}\parindent\z@
+ \parskip\z@ \@plus .3\p@\relax
+ \let\item\par
+ \def\,{\relax\ifmmode\mskip\thinmuskip
+ \else\hskip0.2em\ignorespaces\fi}%
+ \normalfont\small
+ \begin{multicols}{2}[\@makeschapterhead{\indexname}]%
+ }
+ {\end{multicols}}
+
+\renewcommand\footnoterule{%
+ \kern-3\p@
+ \hrule\@width 2truecm
+ \kern2.6\p@}
+ \newdimen\fnindent
+ \fnindent1em
+\long\def\@makefntext#1{%
+ \parindent \fnindent%
+ \leftskip \fnindent%
+ \noindent
+ \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
+
+\long\def\@makecaption#1#2{%
+ \vskip\abovecaptionskip
+ \sbox\@tempboxa{{\bfseries #1.} #2}%
+ \ifdim \wd\@tempboxa >\hsize
+ {\bfseries #1.} #2\par
+ \else
+ \global \@minipagefalse
+ \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
+ \fi
+ \vskip\belowcaptionskip}
+
+\def\fps@figure{htbp}
+\def\fnum@figure{\figurename\thinspace\thefigure}
+\def \@floatboxreset {%
+ \reset@font
+ \small
+ \@setnobreak
+ \@setminipage
+}
+\def\fps@table{htbp}
+\def\fnum@table{\tablename~\thetable}
+\renewenvironment{table}
+ {\setlength\abovecaptionskip{0\p@}%
+ \setlength\belowcaptionskip{10\p@}%
+ \@float{table}}
+ {\end@float}
+\renewenvironment{table*}
+ {\setlength\abovecaptionskip{0\p@}%
+ \setlength\belowcaptionskip{10\p@}%
+ \@dblfloat{table}}
+ {\end@dblfloat}
+
+\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
+ ext@#1\endcsname}{#1}{\protect\numberline{\csname
+ the#1\endcsname}{\ignorespaces #2}}\begingroup
+ \@parboxrestore
+ \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+ \endgroup}
+
+% LaTeX does not provide a command to enter the authors institute
+% addresses. The \institute command is defined here.
+
+\newcounter{@inst}
+\newcounter{@auth}
+\newcounter{auco}
+\newdimen\instindent
+\newbox\authrun
+\newtoks\authorrunning
+\newtoks\tocauthor
+\newbox\titrun
+\newtoks\titlerunning
+\newtoks\toctitle
+
+\def\clearheadinfo{\gdef\@author{No Author Given}%
+ \gdef\@title{No Title Given}%
+ \gdef\@subtitle{}%
+ \gdef\@institute{No Institute Given}%
+ \gdef\@thanks{}%
+ \global\titlerunning={}\global\authorrunning={}%
+ \global\toctitle={}\global\tocauthor={}}
+
+\def\institute#1{\gdef\@institute{#1}}
+
+\def\institutename{\par
+ \begingroup
+ \parskip=\z@
+ \parindent=\z@
+ \setcounter{@inst}{1}%
+ \def\and{\par\stepcounter{@inst}%
+ \noindent$^{\the@inst}$\enspace\ignorespaces}%
+ \setbox0=\vbox{\def\thanks##1{}\@institute}%
+ \ifnum\c@@inst=1\relax
+ \gdef\fnnstart{0}%
+ \else
+ \xdef\fnnstart{\c@@inst}%
+ \setcounter{@inst}{1}%
+ \noindent$^{\the@inst}$\enspace
+ \fi
+ \ignorespaces
+ \@institute\par
+ \endgroup}
+
+\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
+ {\star\star\star}\or \dagger\or \ddagger\or
+ \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
+ \or \ddagger\ddagger \else\@ctrerr\fi}}
+
+\def\inst#1{\unskip$^{#1}$}
+\def\fnmsep{\unskip$^,$}
+\def\email#1{{\tt#1}}
+\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
+\@ifpackageloaded{babel}{%
+\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
+\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
+\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
+}{\switcht@@therlang}%
+}
+\def\homedir{\~{ }}
+
+\def\subtitle#1{\gdef\@subtitle{#1}}
+\clearheadinfo
+
+\renewcommand\maketitle{\newpage
+ \refstepcounter{chapter}%
+ \stepcounter{section}%
+ \setcounter{section}{0}%
+ \setcounter{subsection}{0}%
+ \setcounter{figure}{0}
+ \setcounter{table}{0}
+ \setcounter{equation}{0}
+ \setcounter{footnote}{0}%
+ \begingroup
+ \parindent=\z@
+ \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
+ \if@twocolumn
+ \ifnum \col@number=\@ne
+ \@maketitle
+ \else
+ \twocolumn[\@maketitle]%
+ \fi
+ \else
+ \newpage
+ \global\@topnum\z@ % Prevents figures from going at top of page.
+ \@maketitle
+ \fi
+ \thispagestyle{empty}\@thanks
+%
+ \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
+ \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
+ \instindent=\hsize
+ \advance\instindent by-\headlineindent
+% \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
+% \addcontentsline{toc}{title}{\the\toctitle}\fi
+ \if@runhead
+ \if!\the\titlerunning!\else
+ \edef\@title{\the\titlerunning}%
+ \fi
+ \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
+ \ifdim\wd\titrun>\instindent
+ \typeout{Title too long for running head. Please supply}%
+ \typeout{a shorter form with \string\titlerunning\space prior to
+ \string\maketitle}%
+ \global\setbox\titrun=\hbox{\small\rm
+ Title Suppressed Due to Excessive Length}%
+ \fi
+ \xdef\@title{\copy\titrun}%
+ \fi
+%
+ \if!\the\tocauthor!\relax
+ {\def\and{\noexpand\protect\noexpand\and}%
+ \protected@xdef\toc@uthor{\@author}}%
+ \else
+ \def\\{\noexpand\protect\noexpand\newline}%
+ \protected@xdef\scratch{\the\tocauthor}%
+ \protected@xdef\toc@uthor{\scratch}%
+ \fi
+% \addcontentsline{toc}{author}{\toc@uthor}%
+ \if@runhead
+ \if!\the\authorrunning!
+ \value{@inst}=\value{@auth}%
+ \setcounter{@auth}{1}%
+ \else
+ \edef\@author{\the\authorrunning}%
+ \fi
+ \global\setbox\authrun=\hbox{\small\unboldmath\@author\unskip}%
+ \ifdim\wd\authrun>\instindent
+ \typeout{Names of authors too long for running head. Please supply}%
+ \typeout{a shorter form with \string\authorrunning\space prior to
+ \string\maketitle}%
+ \global\setbox\authrun=\hbox{\small\rm
+ Authors Suppressed Due to Excessive Length}%
+ \fi
+ \xdef\@author{\copy\authrun}%
+ \markboth{\@author}{\@title}%
+ \fi
+ \endgroup
+ \setcounter{footnote}{\fnnstart}%
+ \clearheadinfo}
+%
+\def\@maketitle{\newpage
+ \markboth{}{}%
+ \def\lastand{\ifnum\value{@inst}=2\relax
+ \unskip{} \andname\
+ \else
+ \unskip \lastandname\
+ \fi}%
+ \def\and{\stepcounter{@auth}\relax
+ \ifnum\value{@auth}=\value{@inst}%
+ \lastand
+ \else
+ \unskip,
+ \fi}%
+ \begin{center}%
+ \let\newline\\
+ {\Large \bfseries\boldmath
+ \pretolerance=10000
+ \@title \par}\vskip .8cm
+\if!\@subtitle!\else {\large \bfseries\boldmath
+ \vskip -.65cm
+ \pretolerance=10000
+ \@subtitle \par}\vskip .8cm\fi
+ \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
+ \def\thanks##1{}\@author}%
+ \global\value{@inst}=\value{@auth}%
+ \global\value{auco}=\value{@auth}%
+ \setcounter{@auth}{1}%
+{\lineskip .5em
+\noindent\ignorespaces
+\@author\vskip.35cm}
+ {\small\institutename}
+ \end{center}%
+ }
+
+% definition of the "\spnewtheorem" command.
+%
+% Usage:
+%
+% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
+% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
+% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
+%
+% New is "cap_font" and "body_font". It stands for
+% fontdefinition of the caption and the text itself.
+%
+% "\spnewtheorem*" gives a theorem without number.
+%
+% A defined spnewthoerem environment is used as described
+% by Lamport.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\@thmcountersep{}
+\def\@thmcounterend{.}
+
+\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
+
+% definition of \spnewtheorem with number
+
+\def\@spnthm#1#2{%
+ \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
+\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
+
+\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}\@addtoreset{#1}{#3}%
+ \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
+ \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}%
+ \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spothm#1[#2]#3#4#5{%
+ \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
+ {\expandafter\@ifdefinable\csname #1\endcsname
+ {\global\@namedef{the#1}{\@nameuse{the#2}}%
+ \expandafter\xdef\csname #1name\endcsname{#3}%
+ \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
+ \global\@namedef{end#1}{\@endtheorem}}}}
+
+\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\refstepcounter{#1}%
+\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
+
+\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
+ \ignorespaces}
+
+\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
+ the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
+
+\def\@spbegintheorem#1#2#3#4{\trivlist
+ \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
+
+\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
+ \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
+
+% definition of \spnewtheorem* without number
+
+\def\@sthm#1#2{\@Ynthm{#1}{#2}}
+
+\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+ {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
+
+\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
+
+\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
+ {#4}{#2}{#3}\ignorespaces}
+
+\def\@Begintheorem#1#2#3{#3\trivlist
+ \item[\hskip\labelsep{#2#1\@thmcounterend}]}
+
+\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
+ \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
+
+\if@envcntsect
+ \def\@thmcountersep{.}
+ \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
+\else
+ \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
+ \if@envcntreset
+ \@addtoreset{theorem}{section}
+ \else
+ \@addtoreset{theorem}{chapter}
+ \fi
+\fi
+
+%definition of divers theorem environments
+\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
+\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
+\if@envcntsame % alle Umgebungen wie Theorem.
+ \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
+\else % alle Umgebungen mit eigenem Zaehler
+ \if@envcntsect % mit section numeriert
+ \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
+ \else % nicht mit section numeriert
+ \if@envcntreset
+ \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+ \@addtoreset{#1}{section}}
+ \else
+ \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+ \@addtoreset{#1}{chapter}}%
+ \fi
+ \fi
+\fi
+\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
+\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
+\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
+\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
+\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
+\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
+\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
+\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
+\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
+\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
+\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
+\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
+\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
+\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
+
+\def\@takefromreset#1#2{%
+ \def\@tempa{#1}%
+ \let\@tempd\@elt
+ \def\@elt##1{%
+ \def\@tempb{##1}%
+ \ifx\@tempa\@tempb\else
+ \@addtoreset{##1}{#2}%
+ \fi}%
+ \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
+ \expandafter\def\csname cl@#2\endcsname{}%
+ \@tempc
+ \let\@elt\@tempd}
+
+\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+ \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
+ \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
+ \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
+ }
+
+\renewenvironment{abstract}{%
+ \list{}{\advance\topsep by0.35cm\relax\small
+ \leftmargin=1cm
+ \labelwidth=\z@
+ \listparindent=\z@
+ \itemindent\listparindent
+ \rightmargin\leftmargin}\item[\hskip\labelsep
+ \bfseries\abstractname]}
+ {\endlist}
+
+\newdimen\headlineindent % dimension for space between
+\headlineindent=1.166cm % number and text of headings.
+
+\def\ps@headings{\let\@mkboth\@gobbletwo
+ \let\@oddfoot\@empty\let\@evenfoot\@empty
+ \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+ \leftmark\hfil}
+ \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
+ \llap{\thepage}}
+ \def\chaptermark##1{}%
+ \def\sectionmark##1{}%
+ \def\subsectionmark##1{}}
+
+\def\ps@titlepage{\let\@mkboth\@gobbletwo
+ \let\@oddfoot\@empty\let\@evenfoot\@empty
+ \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+ \hfil}
+ \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
+ \llap{\thepage}}
+ \def\chaptermark##1{}%
+ \def\sectionmark##1{}%
+ \def\subsectionmark##1{}}
+
+\if@runhead\ps@headings\else
+\ps@empty\fi
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\endinput
+%end of file llncs.cls
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/document/root.bib Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,111 @@
+@article{OwensReppyTuron09,
+ author = {S.~Owens and J.~Reppy and A.~Turon},
+ title = {{R}egular-{E}xpression {D}erivatives {R}e-{E}xamined},
+ journal = {Journal of Functional Programming},
+ volume = 19,
+ number = {2},
+ year = 2009,
+ pages = {173--190}
+}
+
+
+
+@Unpublished{KraussNipkow11,
+ author = {A.~Kraus and T.~Nipkow},
+ title = {{P}roof {P}earl: {R}egular {E}xpression {E}quivalence and {R}elation {A}lgebra},
+ note = {To appear in Journal of Automated Reasoning},
+ year = {2011}
+}
+
+@Book{Kozen97,
+ author = {D.~Kozen},
+ title = {{A}utomata and {C}omputability},
+ publisher = {Springer Verlag},
+ year = {1997}
+}
+
+
+@incollection{Constable00,
+ author = {R.~L.~Constable and
+ P.~B.~Jackson and
+ P.~Naumov and
+ J.~C.~Uribe},
+ title = {{C}onstructively {F}ormalizing {A}utomata {T}heory},
+ booktitle = {Proof, Language, and Interaction},
+ year = {2000},
+ publisher = {MIT Press},
+ pages = {213-238}
+}
+
+
+@techreport{Filliatre97,
+ author = {J.-C. Filli\^atre},
+ institution = {LIP - ENS Lyon},
+ number = {97--04},
+ title = {{F}inite {A}utomata {T}heory in {C}oq:
+ {A} {C}onstructive {P}roof of {K}leene's {T}heorem},
+ type = {Research Report},
+ year = {1997}
+}
+
+@article{OwensSlind08,
+ author = {S.~Owens and K.~Slind},
+ title = {{A}dapting {F}unctional {P}rograms to {H}igher {O}rder {L}ogic},
+ journal = {Higher-Order and Symbolic Computation},
+ volume = {21},
+ number = {4},
+ year = {2008},
+ pages = {377--409}
+}
+
+@article{Brzozowski64,
+ author = {J.~A.~Brzozowski},
+ title = {{D}erivatives of {R}egular {E}xpressions},
+ journal = {J.~ACM},
+ volume = {11},
+ issue = {4},
+ year = {1964},
+ pages = {481--494},
+ publisher = {ACM}
+}
+
+@inproceedings{Nipkow98,
+ author={T.~Nipkow},
+ title={{V}erified {L}exical {A}nalysis},
+ booktitle={Proc.~of the 11th International Conference on Theorem Proving in Higher Order Logics},
+ series={LNCS},
+ volume=1479,
+ pages={1--15},
+ year=1998
+}
+
+@inproceedings{BerghoferNipkow00,
+ author={S.~Berghofer and T.~Nipkow},
+ title={{E}xecuting {H}igher {O}rder {L}ogic},
+ booktitle={Proc.~of the International Workshop on Types for Proofs and Programs},
+ year=2002,
+ series={LNCS},
+ volume=2277,
+ pages="24--40"
+}
+
+@book{HopcroftUllman69,
+ author = {J.~E.~Hopcroft and
+ J.~D.~Ullman},
+ title = {{F}ormal {L}anguages and {T}heir {R}elation to {A}utomata},
+ publisher = {Addison-Wesley},
+ year = {1969}
+}
+
+
+@inproceedings{BerghoferReiter09,
+ author = {S.~Berghofer and
+ M.~Reiter},
+ title = {{F}ormalizing the {L}ogic-{A}utomaton {C}onnection},
+ booktitle = {Proc.~of the 22nd International
+ Conference on Theorem Proving in Higher Order Logics},
+ year = {2009},
+ pages = {147-163},
+ series = {LNCS},
+ volume = {5674}
+}
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/document/root.tex Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,73 @@
+\documentclass[runningheads]{llncs}
+\usepackage{isabelle}
+\usepackage{isabellesym}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{tikz}
+\usepackage{pgf}
+\usetikzlibrary{arrows,automata,decorations,fit,calc}
+\usetikzlibrary{shapes,shapes.arrows,snakes,positioning}
+\usepgflibrary{shapes.misc} % LATEX and plain TEX and pure pgf
+\usetikzlibrary{matrix}
+\usepackage{pdfsetup}
+\usepackage{ot1patch}
+\usepackage{times}
+%%\usepackage{proof}
+%%\usepackage{mathabx}
+\usepackage{stmaryrd}
+
+\titlerunning{Myhill-Nerode using Regular Expressions}
+
+
+\urlstyle{rm}
+\isabellestyle{it}
+\renewcommand{\isastyleminor}{\it}%
+\renewcommand{\isastyle}{\normalsize\it}%
+
+
+\def\dn{\,\stackrel{\mbox{\scriptsize def}}{=}\,}
+\renewcommand{\isasymequiv}{$\dn$}
+\renewcommand{\isasymemptyset}{$\varnothing$}
+\renewcommand{\isacharunderscore}{\mbox{$\_\!\_$}}
+
+\newcommand{\isasymcalL}{\ensuremath{\cal{L}}}
+\newcommand{\isasymbigplus}{\ensuremath{\bigplus}}
+
+\newcommand{\bigplus}{\mbox{\Large\bf$+$}}
+\begin{document}
+
+\title{A Formalisation of the Myhill-Nerode Theorem\\ based on Regular
+ Expressions (Proof Pearl)}
+\author{Chunhan Wu\inst{1} \and Xingyuan Zhang\inst{1} \and Christian Urban\inst{2}}
+\institute{PLA University of Science and Technology, China \and TU Munich, Germany}
+\maketitle
+
+%\mbox{}\\[-10mm]
+\begin{abstract}
+There are numerous textbooks on regular languages. Nearly all of them
+introduce the subject by describing finite automata and only mentioning on the
+side a connection with regular expressions. Unfortunately, automata are difficult
+to formalise in HOL-based theorem provers. The reason is that
+they need to be represented as graphs, matrices or functions, none of which
+are inductive datatypes. Also convenient operations for disjoint unions of
+graphs and functions are not easily formalisiable in HOL. In contrast, regular
+expressions can be defined conveniently as a datatype and a corresponding
+reasoning infrastructure comes for free. We show in this paper that a central
+result from formal language theory---the Myhill-Nerode theorem---can be
+recreated using only regular expressions.
+
+\end{abstract}
+
+
+\input{session}
+
+%%\mbox{}\\[-10mm]
+\bibliographystyle{plain}
+\bibliography{root}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
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