Newer version.
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