theory PrioG
imports PrioGDef
begin
lemma runing_ready:
shows "runing s \<subseteq> readys s"
unfolding runing_def readys_def
by auto
lemma readys_threads:
shows "readys s \<subseteq> threads s"
unfolding readys_def
by auto
lemma wq_v_neq:
"cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
by (auto simp:wq_def Let_def cp_def split:list.splits)
lemma wq_distinct: "vt s \<Longrightarrow> distinct (wq s cs)"
proof(erule_tac vt.induct, simp add:wq_def)
fix s e
assume h1: "step s e"
and h2: "distinct (wq s cs)"
thus "distinct (wq (e # s) cs)"
proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
fix thread s
assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
and h2: "thread \<in> set (wq_fun (schs s) cs)"
and h3: "thread \<in> runing s"
show "False"
proof -
from h3 have "\<And> cs. thread \<in> set (wq_fun (schs s) cs) \<Longrightarrow>
thread = hd ((wq_fun (schs s) cs))"
by (simp add:runing_def readys_def s_waiting_def wq_def)
from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" .
with h2
have "(Cs cs, Th thread) \<in> (depend s)"
by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
with h1 show False by auto
qed
next
fix thread s a list
assume dst: "distinct list"
show "distinct (SOME q. distinct q \<and> set q = set list)"
proof(rule someI2)
from dst show "distinct list \<and> set list = set list" by auto
next
fix q assume "distinct q \<and> set q = set list"
thus "distinct q" by auto
qed
qed
qed
lemma step_back_vt: "vt (e#s) \<Longrightarrow> vt s"
by(ind_cases "vt (e#s)", simp)
lemma step_back_step: "vt (e#s) \<Longrightarrow> step s e"
by(ind_cases "vt (e#s)", simp)
lemma block_pre:
fixes thread cs s
assumes vt_e: "vt (e#s)"
and s_ni: "thread \<notin> set (wq s cs)"
and s_i: "thread \<in> set (wq (e#s) cs)"
shows "e = P thread cs"
proof -
show ?thesis
proof(cases e)
case (P th cs)
with assms
show ?thesis
by (auto simp:wq_def Let_def split:if_splits)
next
case (Create th prio)
with assms show ?thesis
by (auto simp:wq_def Let_def split:if_splits)
next
case (Exit th)
with assms show ?thesis
by (auto simp:wq_def Let_def split:if_splits)
next
case (Set th prio)
with assms show ?thesis
by (auto simp:wq_def Let_def split:if_splits)
next
case (V th cs)
with assms show ?thesis
apply (auto simp:wq_def Let_def split:if_splits)
proof -
fix q qs
assume h1: "thread \<notin> set (wq_fun (schs s) cs)"
and h2: "q # qs = wq_fun (schs s) cs"
and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
and vt: "vt (V th cs # s)"
from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp
moreover have "thread \<in> set qs"
proof -
have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
proof(rule someI2)
from wq_distinct [OF step_back_vt[OF vt], of cs]
and h2[symmetric, folded wq_def]
show "distinct qs \<and> set qs = set qs" by auto
next
fix x assume "distinct x \<and> set x = set qs"
thus "set x = set qs" by auto
qed
with h3 show ?thesis by simp
qed
ultimately show "False" by auto
qed
qed
qed
lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (depend s)^+"
apply (ind_cases "vt ((P thread cs)#s)")
apply (ind_cases "step s (P thread cs)")
by auto
lemma abs1:
fixes e es
assumes ein: "e \<in> set es"
and neq: "hd es \<noteq> hd (es @ [x])"
shows "False"
proof -
from ein have "es \<noteq> []" by auto
then obtain e ess where "es = e # ess" by (cases es, auto)
with neq show ?thesis by auto
qed
lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
by (cases es, auto)
inductive_cases evt_cons: "vt (a#s)"
lemma abs2:
assumes vt: "vt (e#s)"
and inq: "thread \<in> set (wq s cs)"
and nh: "thread = hd (wq s cs)"
and qt: "thread \<noteq> hd (wq (e#s) cs)"
and inq': "thread \<in> set (wq (e#s) cs)"
shows "False"
proof -
from assms show "False"
apply (cases e)
apply ((simp split:if_splits add:Let_def wq_def)[1])+
apply (insert abs1, fast)[1]
apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)
proof -
fix th qs
assume vt: "vt (V th cs # s)"
and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
and eq_wq: "wq_fun (schs s) cs = thread # qs"
show "False"
proof -
from wq_distinct[OF step_back_vt[OF vt], of cs]
and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
moreover have "thread \<in> set qs"
proof -
have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
proof(rule someI2)
from wq_distinct [OF step_back_vt[OF vt], of cs]
and eq_wq [folded wq_def]
show "distinct qs \<and> set qs = set qs" by auto
next
fix x assume "distinct x \<and> set x = set qs"
thus "set x = set qs" by auto
qed
with th_in show ?thesis by auto
qed
ultimately show ?thesis by auto
qed
qed
qed
lemma vt_moment: "\<And> t. \<lbrakk>vt s\<rbrakk> \<Longrightarrow> vt (moment t s)"
proof(induct s, simp)
fix a s t
assume h: "\<And>t.\<lbrakk>vt s\<rbrakk> \<Longrightarrow> vt (moment t s)"
and vt_a: "vt (a # s)"
show "vt (moment t (a # s))"
proof(cases "t \<ge> length (a#s)")
case True
from True have "moment t (a#s) = a#s" by simp
with vt_a show ?thesis by simp
next
case False
hence le_t1: "t \<le> length s" by simp
from vt_a have "vt s"
by (erule_tac evt_cons, simp)
from h [OF this] have "vt (moment t s)" .
moreover have "moment t (a#s) = moment t s"
proof -
from moment_app [OF le_t1, of "[a]"]
show ?thesis by simp
qed
ultimately show ?thesis by auto
qed
qed
(* Wrong:
lemma \<lbrakk>thread \<in> set (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
*)
lemma waiting_unique_pre:
fixes cs1 cs2 s thread
assumes vt: "vt s"
and h11: "thread \<in> set (wq s cs1)"
and h12: "thread \<noteq> hd (wq s cs1)"
assumes h21: "thread \<in> set (wq s cs2)"
and h22: "thread \<noteq> hd (wq s cs2)"
and neq12: "cs1 \<noteq> cs2"
shows "False"
proof -
let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
from h11 and h12 have q1: "?Q cs1 s" by simp
from h21 and h22 have q2: "?Q cs2 s" by simp
have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
from p_split [of "?Q cs1", OF q1 nq1]
obtain t1 where lt1: "t1 < length s"
and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
thread \<noteq> hd (wq (moment t1 s) cs1))"
and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
thread \<noteq> hd (wq (moment i' s) cs1))" by auto
from p_split [of "?Q cs2", OF q2 nq2]
obtain t2 where lt2: "t2 < length s"
and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
thread \<noteq> hd (wq (moment t2 s) cs2))"
and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
thread \<noteq> hd (wq (moment i' s) cs2))" by auto
show ?thesis
proof -
{
assume lt12: "t1 < t2"
let ?t3 = "Suc t2"
from lt2 have le_t3: "?t3 \<le> length s" by auto
from moment_plus [OF this]
obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
have "t2 < ?t3" by simp
from nn2 [rule_format, OF this] and eq_m
have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
have vt_e: "vt (e#moment t2 s)"
proof -
from vt_moment [OF vt]
have "vt (moment ?t3 s)" .
with eq_m show ?thesis by simp
qed
have ?thesis
proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
case True
from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
by auto
from abs2 [OF vt_e True eq_th h2 h1]
show ?thesis by auto
next
case False
from block_pre [OF vt_e False h1]
have "e = P thread cs2" .
with vt_e have "vt ((P thread cs2)# moment t2 s)" by simp
from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
with runing_ready have "thread \<in> readys (moment t2 s)" by auto
with nn1 [rule_format, OF lt12]
show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto)
qed
} moreover {
assume lt12: "t2 < t1"
let ?t3 = "Suc t1"
from lt1 have le_t3: "?t3 \<le> length s" by auto
from moment_plus [OF this]
obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
have lt_t3: "t1 < ?t3" by simp
from nn1 [rule_format, OF this] and eq_m
have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
have vt_e: "vt (e#moment t1 s)"
proof -
from vt_moment [OF vt]
have "vt (moment ?t3 s)" .
with eq_m show ?thesis by simp
qed
have ?thesis
proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
case True
from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
by auto
from abs2 [OF vt_e True eq_th h2 h1]
show ?thesis by auto
next
case False
from block_pre [OF vt_e False h1]
have "e = P thread cs1" .
with vt_e have "vt ((P thread cs1)# moment t1 s)" by simp
from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
with runing_ready have "thread \<in> readys (moment t1 s)" by auto
with nn2 [rule_format, OF lt12]
show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto)
qed
} moreover {
assume eqt12: "t1 = t2"
let ?t3 = "Suc t1"
from lt1 have le_t3: "?t3 \<le> length s" by auto
from moment_plus [OF this]
obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
have lt_t3: "t1 < ?t3" by simp
from nn1 [rule_format, OF this] and eq_m
have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
have vt_e: "vt (e#moment t1 s)"
proof -
from vt_moment [OF vt]
have "vt (moment ?t3 s)" .
with eq_m show ?thesis by simp
qed
have ?thesis
proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
case True
from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
by auto
from abs2 [OF vt_e True eq_th h2 h1]
show ?thesis by auto
next
case False
from block_pre [OF vt_e False h1]
have eq_e1: "e = P thread cs1" .
have lt_t3: "t1 < ?t3" by simp
with eqt12 have "t2 < ?t3" by simp
from nn2 [rule_format, OF this] and eq_m and eqt12
have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
show ?thesis
proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
case True
from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
by auto
from vt_e and eqt12 have "vt (e#moment t2 s)" by simp
from abs2 [OF this True eq_th h2 h1]
show ?thesis .
next
case False
have vt_e: "vt (e#moment t2 s)"
proof -
from vt_moment [OF vt] eqt12
have "vt (moment (Suc t2) s)" by auto
with eq_m eqt12 show ?thesis by simp
qed
from block_pre [OF vt_e False h1]
have "e = P thread cs2" .
with eq_e1 neq12 show ?thesis by auto
qed
qed
} ultimately show ?thesis by arith
qed
qed
lemma waiting_unique:
fixes s cs1 cs2
assumes "vt s"
and "waiting s th cs1"
and "waiting s th cs2"
shows "cs1 = cs2"
using waiting_unique_pre assms
unfolding wq_def s_waiting_def
by auto
(* not used *)
lemma held_unique:
fixes s::"state"
assumes "holding s th1 cs"
and "holding s th2 cs"
shows "th1 = th2"
using assms
unfolding s_holding_def
by auto
lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
apply (induct s, auto)
by (case_tac a, auto split:if_splits)
lemma birthtime_unique:
"\<lbrakk>birthtime th1 s = birthtime th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
\<Longrightarrow> th1 = th2"
apply (induct s, auto)
by (case_tac a, auto split:if_splits dest:birthtime_lt)
lemma preced_unique :
assumes pcd_eq: "preced th1 s = preced th2 s"
and th_in1: "th1 \<in> threads s"
and th_in2: " th2 \<in> threads s"
shows "th1 = th2"
proof -
from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
from birthtime_unique [OF this th_in1 th_in2]
show ?thesis .
qed
lemma preced_linorder:
assumes neq_12: "th1 \<noteq> th2"
and th_in1: "th1 \<in> threads s"
and th_in2: " th2 \<in> threads s"
shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
proof -
from preced_unique [OF _ th_in1 th_in2] and neq_12
have "preced th1 s \<noteq> preced th2 s" by auto
thus ?thesis by auto
qed
lemma unique_minus:
fixes x y z r
assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
and xy: "(x, y) \<in> r"
and xz: "(x, z) \<in> r^+"
and neq: "y \<noteq> z"
shows "(y, z) \<in> r^+"
proof -
from xz and neq show ?thesis
proof(induct)
case (base ya)
have "(x, ya) \<in> r" by fact
from unique [OF xy this] have "y = ya" .
with base show ?case by auto
next
case (step ya z)
show ?case
proof(cases "y = ya")
case True
from step True show ?thesis by simp
next
case False
from step False
show ?thesis by auto
qed
qed
qed
lemma unique_base:
fixes r x y z
assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
and xy: "(x, y) \<in> r"
and xz: "(x, z) \<in> r^+"
and neq_yz: "y \<noteq> z"
shows "(y, z) \<in> r^+"
proof -
from xz neq_yz show ?thesis
proof(induct)
case (base ya)
from xy unique base show ?case by auto
next
case (step ya z)
show ?case
proof(cases "y = ya")
case True
from True step show ?thesis by auto
next
case False
from False step
have "(y, ya) \<in> r\<^sup>+" by auto
with step show ?thesis by auto
qed
qed
qed
lemma unique_chain:
fixes r x y z
assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
and xy: "(x, y) \<in> r^+"
and xz: "(x, z) \<in> r^+"
and neq_yz: "y \<noteq> z"
shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
proof -
from xy xz neq_yz show ?thesis
proof(induct)
case (base y)
have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
next
case (step y za)
show ?case
proof(cases "y = z")
case True
from True step show ?thesis by auto
next
case False
from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
thus ?thesis
proof
assume "(z, y) \<in> r\<^sup>+"
with step have "(z, za) \<in> r\<^sup>+" by auto
thus ?thesis by auto
next
assume h: "(y, z) \<in> r\<^sup>+"
from step have yza: "(y, za) \<in> r" by simp
from step have "za \<noteq> z" by simp
from unique_minus [OF _ yza h this] and unique
have "(za, z) \<in> r\<^sup>+" by auto
thus ?thesis by auto
qed
qed
qed
qed
lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
apply (unfold s_depend_def s_waiting_def wq_def)
by (simp add:Let_def)
lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
apply (unfold s_depend_def s_waiting_def wq_def)
by (simp add:Let_def)
lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
apply (unfold s_depend_def s_waiting_def wq_def)
by (simp add:Let_def)
lemma step_v_hold_inv[elim_format]:
"\<And>c t. \<lbrakk>vt (V th cs # s);
\<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> next_th s th cs t \<and> c = cs"
proof -
fix c t
assume vt: "vt (V th cs # s)"
and nhd: "\<not> holding (wq s) t c"
and hd: "holding (wq (V th cs # s)) t c"
show "next_th s th cs t \<and> c = cs"
proof(cases "c = cs")
case False
with nhd hd show ?thesis
by (unfold cs_holding_def wq_def, auto simp:Let_def)
next
case True
with step_back_step [OF vt]
have "step s (V th c)" by simp
hence "next_th s th cs t"
proof(cases)
assume "holding s th c"
with nhd hd show ?thesis
apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
auto simp:Let_def split:list.splits if_splits)
proof -
assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
moreover have "\<dots> = set []"
proof(rule someI2)
show "distinct [] \<and> [] = []" by auto
next
fix x assume "distinct x \<and> x = []"
thus "set x = set []" by auto
qed
ultimately show False by auto
next
assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
moreover have "\<dots> = set []"
proof(rule someI2)
show "distinct [] \<and> [] = []" by auto
next
fix x assume "distinct x \<and> x = []"
thus "set x = set []" by auto
qed
ultimately show False by auto
qed
qed
with True show ?thesis by auto
qed
qed
lemma step_v_wait_inv[elim_format]:
"\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
\<rbrakk>
\<Longrightarrow> (next_th s th cs t \<and> cs = c)"
proof -
fix t c
assume vt: "vt (V th cs # s)"
and nw: "\<not> waiting (wq (V th cs # s)) t c"
and wt: "waiting (wq s) t c"
show "next_th s th cs t \<and> cs = c"
proof(cases "cs = c")
case False
with nw wt show ?thesis
by (auto simp:cs_waiting_def wq_def Let_def)
next
case True
from nw[folded True] wt[folded True]
have "next_th s th cs t"
apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
proof -
fix a list
assume t_in: "t \<in> set list"
and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
and eq_wq: "wq_fun (schs s) cs = a # list"
have " set (SOME q. distinct q \<and> set q = set list) = set list"
proof(rule someI2)
from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def]
show "distinct list \<and> set list = set list" by auto
next
show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
by auto
qed
with t_ni and t_in show "a = th" by auto
next
fix a list
assume t_in: "t \<in> set list"
and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
and eq_wq: "wq_fun (schs s) cs = a # list"
have " set (SOME q. distinct q \<and> set q = set list) = set list"
proof(rule someI2)
from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def]
show "distinct list \<and> set list = set list" by auto
next
show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
by auto
qed
with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto
next
fix a list
assume eq_wq: "wq_fun (schs s) cs = a # list"
from step_back_step[OF vt]
show "a = th"
proof(cases)
assume "holding s th cs"
with eq_wq show ?thesis
by (unfold s_holding_def wq_def, auto)
qed
qed
with True show ?thesis by simp
qed
qed
lemma step_v_not_wait[consumes 3]:
"\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
lemma step_v_release:
"\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
proof -
assume vt: "vt (V th cs # s)"
and hd: "holding (wq (V th cs # s)) th cs"
from step_back_step [OF vt] and hd
show "False"
proof(cases)
assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"
thus ?thesis
apply (unfold s_holding_def wq_def cs_holding_def)
apply (auto simp:Let_def split:list.splits)
proof -
fix list
assume eq_wq[folded wq_def]:
"wq_fun (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
\<in> set (SOME q. distinct q \<and> set q = set list)"
have "set (SOME q. distinct q \<and> set q = set list) = set list"
proof(rule someI2)
from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
show "distinct list \<and> set list = set list" by auto
next
show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
by auto
qed
moreover have "distinct (hd (SOME q. distinct q \<and> set q = set list) # list)"
proof -
from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
show ?thesis by auto
qed
moreover note eq_wq and hd_in
ultimately show "False" by auto
qed
qed
qed
lemma step_v_get_hold:
"\<And>th'. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
apply (unfold cs_holding_def next_th_def wq_def,
auto simp:Let_def)
proof -
fix rest
assume vt: "vt (V th cs # s)"
and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest"
and nrest: "rest \<noteq> []"
and ni: "hd (SOME q. distinct q \<and> set q = set rest)
\<notin> set (SOME q. distinct q \<and> set q = set rest)"
have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
proof(rule someI2)
from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq
show "distinct rest \<and> set rest = set rest" by auto
next
fix x assume "distinct x \<and> set x = set rest"
hence "set x = set rest" by auto
with nrest
show "x \<noteq> []" by (case_tac x, auto)
qed
with ni show "False" by auto
qed
lemma step_v_release_inv[elim_format]:
"\<And>c t. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow>
c = cs \<and> t = th"
apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
proof -
fix a list
assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
from step_back_step [OF vt] show "a = th"
proof(cases)
assume "holding s th cs" with eq_wq
show ?thesis
by (unfold s_holding_def wq_def, auto)
qed
next
fix a list
assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
from step_back_step [OF vt] show "a = th"
proof(cases)
assume "holding s th cs" with eq_wq
show ?thesis
by (unfold s_holding_def wq_def, auto)
qed
qed
lemma step_v_waiting_mono:
"\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
proof -
fix t c
let ?s' = "(V th cs # s)"
assume vt: "vt ?s'"
and wt: "waiting (wq ?s') t c"
show "waiting (wq s) t c"
proof(cases "c = cs")
case False
assume neq_cs: "c \<noteq> cs"
hence "waiting (wq ?s') t c = waiting (wq s) t c"
by (unfold cs_waiting_def wq_def, auto simp:Let_def)
with wt show ?thesis by simp
next
case True
with wt show ?thesis
apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
proof -
fix a list
assume not_in: "t \<notin> set list"
and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
and eq_wq: "wq_fun (schs s) cs = a # list"
have "set (SOME q. distinct q \<and> set q = set list) = set list"
proof(rule someI2)
from wq_distinct [OF step_back_vt[OF vt], of cs]
and eq_wq[folded wq_def]
show "distinct list \<and> set list = set list" by auto
next
fix x assume "distinct x \<and> set x = set list"
thus "set x = set list" by auto
qed
with not_in is_in show "t = a" by auto
next
fix list
assume is_waiting: "waiting (wq (V th cs # s)) t cs"
and eq_wq: "wq_fun (schs s) cs = t # list"
hence "t \<in> set list"
apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
proof -
assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
moreover have "\<dots> = set list"
proof(rule someI2)
from wq_distinct [OF step_back_vt[OF vt], of cs]
and eq_wq[folded wq_def]
show "distinct list \<and> set list = set list" by auto
next
fix x assume "distinct x \<and> set x = set list"
thus "set x = set list" by auto
qed
ultimately show "t \<in> set list" by simp
qed
with eq_wq and wq_distinct [OF step_back_vt[OF vt], of cs, unfolded wq_def]
show False by auto
qed
qed
qed
lemma step_depend_v:
fixes th::thread
assumes vt:
"vt (V th cs#s)"
shows "
depend (V th cs # s) =
depend s - {(Cs cs, Th th)} -
{(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
{(Cs cs, Th th') |th'. next_th s th cs th'}"
apply (insert vt, unfold s_depend_def)
apply (auto split:if_splits list.splits simp:Let_def)
apply (auto elim: step_v_waiting_mono step_v_hold_inv
step_v_release step_v_wait_inv
step_v_get_hold step_v_release_inv)
apply (erule_tac step_v_not_wait, auto)
done
lemma step_depend_p:
"vt (P th cs#s) \<Longrightarrow>
depend (P th cs # s) = (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
else depend s \<union> {(Th th, Cs cs)})"
apply(simp only: s_depend_def wq_def)
apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
apply(case_tac "csa = cs", auto)
apply(fold wq_def)
apply(drule_tac step_back_step)
apply(ind_cases " step s (P (hd (wq s cs)) cs)")
apply(auto simp:s_depend_def wq_def cs_holding_def)
done
lemma simple_A:
fixes A
assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
shows "A = {} \<or> (\<exists> a. A = {a})"
proof(cases "A = {}")
case True thus ?thesis by simp
next
case False then obtain a where "a \<in> A" by auto
with h have "A = {a}" by auto
thus ?thesis by simp
qed
lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
by (unfold s_depend_def, auto)
lemma acyclic_depend:
fixes s
assumes vt: "vt s"
shows "acyclic (depend s)"
proof -
from vt show ?thesis
proof(induct)
case (vt_cons s e)
assume ih: "acyclic (depend s)"
and stp: "step s e"
and vt: "vt s"
show ?case
proof(cases e)
case (Create th prio)
with ih
show ?thesis by (simp add:depend_create_unchanged)
next
case (Exit th)
with ih show ?thesis by (simp add:depend_exit_unchanged)
next
case (V th cs)
from V vt stp have vtt: "vt (V th cs#s)" by auto
from step_depend_v [OF this]
have eq_de:
"depend (e # s) =
depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
{(Cs cs, Th th') |th'. next_th s th cs th'}"
(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
from step_back_step [OF vtt]
have "step s (V th cs)" .
thus ?thesis
proof(cases)
assume "holding s th cs"
hence th_in: "th \<in> set (wq s cs)" and
eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
then obtain rest where
eq_wq: "wq s cs = th#rest"
by (cases "wq s cs", auto)
show ?thesis
proof(cases "rest = []")
case False
let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
by (unfold next_th_def, auto)
let ?E = "(?A - ?B - ?C)"
have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
proof
assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
from tranclD [OF this]
obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
hence th_d: "(Th ?th', x) \<in> ?A" by simp
from depend_target_th [OF this]
obtain cs' where eq_x: "x = Cs cs'" by auto
with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
hence wt_th': "waiting s ?th' cs'"
unfolding s_depend_def s_waiting_def cs_waiting_def wq_def by simp
hence "cs' = cs"
proof(rule waiting_unique [OF vt])
from eq_wq wq_distinct[OF vt, of cs]
show "waiting s ?th' cs"
apply (unfold s_waiting_def wq_def, auto)
proof -
assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
and eq_wq: "wq_fun (schs s) cs = th # rest"
have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
proof(rule someI2)
from wq_distinct[OF vt, of cs] and eq_wq
show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
next
fix x assume "distinct x \<and> set x = set rest"
with False show "x \<noteq> []" by auto
qed
hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
set (SOME q. distinct q \<and> set q = set rest)" by auto
moreover have "\<dots> = set rest"
proof(rule someI2)
from wq_distinct[OF vt, of cs] and eq_wq
show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
next
show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
qed
moreover note hd_in
ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
next
assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
proof(rule someI2)
from wq_distinct[OF vt, of cs] and eq_wq
show "distinct rest \<and> set rest = set rest" by auto
next
fix x assume "distinct x \<and> set x = set rest"
with False show "x \<noteq> []" by auto
qed
hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
set (SOME q. distinct q \<and> set q = set rest)" by auto
moreover have "\<dots> = set rest"
proof(rule someI2)
from wq_distinct[OF vt, of cs] and eq_wq
show "distinct rest \<and> set rest = set rest" by auto
next
show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
qed
moreover note hd_in
ultimately show False by auto
qed
qed
with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
with False
show "False" by (auto simp: next_th_def eq_wq)
qed
with acyclic_insert[symmetric] and ac
and eq_de eq_D show ?thesis by auto
next
case True
with eq_wq
have eq_D: "?D = {}"
by (unfold next_th_def, auto)
with eq_de ac
show ?thesis by auto
qed
qed
next
case (P th cs)
from P vt stp have vtt: "vt (P th cs#s)" by auto
from step_depend_p [OF this] P
have "depend (e # s) =
(if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
by simp
moreover have "acyclic ?R"
proof(cases "wq s cs = []")
case True
hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
proof
assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
from tranclD2 [OF this]
obtain x where "(x, Cs cs) \<in> depend s" by auto
with True show False by (auto simp:s_depend_def cs_waiting_def)
qed
with acyclic_insert ih eq_r show ?thesis by auto
next
case False
hence eq_r: "?R = depend s \<union> {(Th th, Cs cs)}" by simp
have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
proof
assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
hence "(Cs cs, Th th) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
moreover from step_back_step [OF vtt] have "step s (P th cs)" .
ultimately show False
proof -
show " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
by (ind_cases "step s (P th cs)", simp)
qed
qed
with acyclic_insert ih eq_r show ?thesis by auto
qed
ultimately show ?thesis by simp
next
case (Set thread prio)
with ih
thm depend_set_unchanged
show ?thesis by (simp add:depend_set_unchanged)
qed
next
case vt_nil
show "acyclic (depend ([]::state))"
by (auto simp: s_depend_def cs_waiting_def
cs_holding_def wq_def acyclic_def)
qed
qed
lemma finite_depend:
fixes s
assumes vt: "vt s"
shows "finite (depend s)"
proof -
from vt show ?thesis
proof(induct)
case (vt_cons s e)
assume ih: "finite (depend s)"
and stp: "step s e"
and vt: "vt s"
show ?case
proof(cases e)
case (Create th prio)
with ih
show ?thesis by (simp add:depend_create_unchanged)
next
case (Exit th)
with ih show ?thesis by (simp add:depend_exit_unchanged)
next
case (V th cs)
from V vt stp have vtt: "vt (V th cs#s)" by auto
from step_depend_v [OF this]
have eq_de: "depend (e # s) =
depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
{(Cs cs, Th th') |th'. next_th s th cs th'}
"
(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
moreover have "finite ?D"
proof -
have "?D = {} \<or> (\<exists> a. ?D = {a})"
by (unfold next_th_def, auto)
thus ?thesis
proof
assume h: "?D = {}"
show ?thesis by (unfold h, simp)
next
assume "\<exists> a. ?D = {a}"
thus ?thesis
by (metis finite.simps)
qed
qed
ultimately show ?thesis by simp
next
case (P th cs)
from P vt stp have vtt: "vt (P th cs#s)" by auto
from step_depend_p [OF this] P
have "depend (e # s) =
(if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
by simp
moreover have "finite ?R"
proof(cases "wq s cs = []")
case True
hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
with True and ih show ?thesis by auto
next
case False
hence "?R = depend s \<union> {(Th th, Cs cs)}" by simp
with False and ih show ?thesis by auto
qed
ultimately show ?thesis by auto
next
case (Set thread prio)
with ih
show ?thesis by (simp add:depend_set_unchanged)
qed
next
case vt_nil
show "finite (depend ([]::state))"
by (auto simp: s_depend_def cs_waiting_def
cs_holding_def wq_def acyclic_def)
qed
qed
text {* Several useful lemmas *}
lemma wf_dep_converse:
fixes s
assumes vt: "vt s"
shows "wf ((depend s)^-1)"
proof(rule finite_acyclic_wf_converse)
from finite_depend [OF vt]
show "finite (depend s)" .
next
from acyclic_depend[OF vt]
show "acyclic (depend s)" .
qed
lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
by (induct l, auto)
lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
lemma wq_threads:
fixes s cs
assumes vt: "vt s"
and h: "th \<in> set (wq s cs)"
shows "th \<in> threads s"
proof -
from vt and h show ?thesis
proof(induct arbitrary: th cs)
case (vt_cons s e)
assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
and stp: "step s e"
and vt: "vt s"
and h: "th \<in> set (wq (e # s) cs)"
show ?case
proof(cases e)
case (Create th' prio)
with ih h show ?thesis
by (auto simp:wq_def Let_def)
next
case (Exit th')
with stp ih h show ?thesis
apply (auto simp:wq_def Let_def)
apply (ind_cases "step s (Exit th')")
apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
s_depend_def s_holding_def cs_holding_def)
done
next
case (V th' cs')
show ?thesis
proof(cases "cs' = cs")
case False
with h
show ?thesis
apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
by (drule_tac ih, simp)
next
case True
from h
show ?thesis
proof(unfold V wq_def)
assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
show "th \<in> threads (V th' cs' # s)"
proof(cases "cs = cs'")
case False
hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
with th_in have " th \<in> set (wq s cs)"
by (fold wq_def, simp)
from ih [OF this] show ?thesis by simp
next
case True
show ?thesis
proof(cases "wq_fun (schs s) cs'")
case Nil
with h V show ?thesis
apply (auto simp:wq_def Let_def split:if_splits)
by (fold wq_def, drule_tac ih, simp)
next
case (Cons a rest)
assume eq_wq: "wq_fun (schs s) cs' = a # rest"
with h V show ?thesis
apply (auto simp:Let_def wq_def split:if_splits)
proof -
assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
proof(rule someI2)
from wq_distinct[OF vt, of cs'] and eq_wq[folded wq_def]
show "distinct rest \<and> set rest = set rest" by auto
next
show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
by auto
qed
with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
from ih[OF this[folded wq_def]] show "th \<in> threads s" .
next
assume th_in: "th \<in> set (wq_fun (schs s) cs)"
from ih[OF this[folded wq_def]]
show "th \<in> threads s" .
qed
qed
qed
qed
qed
next
case (P th' cs')
from h stp
show ?thesis
apply (unfold P wq_def)
apply (auto simp:Let_def split:if_splits, fold wq_def)
apply (auto intro:ih)
apply(ind_cases "step s (P th' cs')")
by (unfold runing_def readys_def, auto)
next
case (Set thread prio)
with ih h show ?thesis
by (auto simp:wq_def Let_def)
qed
next
case vt_nil
thus ?case by (auto simp:wq_def)
qed
qed
lemma range_in: "\<lbrakk>vt s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
apply(unfold s_depend_def cs_waiting_def cs_holding_def)
by (auto intro:wq_threads)
lemma readys_v_eq:
fixes th thread cs rest
assumes vt: "vt s"
and neq_th: "th \<noteq> thread"
and eq_wq: "wq s cs = thread#rest"
and not_in: "th \<notin> set rest"
shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
proof -
from assms show ?thesis
apply (auto simp:readys_def)
apply(simp add:s_waiting_def[folded wq_def])
apply (erule_tac x = csa in allE)
apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
apply (case_tac "csa = cs", simp)
apply (erule_tac x = cs in allE)
apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
apply(auto simp add: wq_def)
apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
proof -
assume th_nin: "th \<notin> set rest"
and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
and eq_wq: "wq_fun (schs s) cs = thread # rest"
have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
proof(rule someI2)
from wq_distinct[OF vt, of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
show "distinct rest \<and> set rest = set rest" by auto
next
show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
qed
with th_nin th_in show False by auto
qed
qed
lemma chain_building:
assumes vt: "vt s"
shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
proof -
from wf_dep_converse [OF vt]
have h: "wf ((depend s)\<inverse>)" .
show ?thesis
proof(induct rule:wf_induct [OF h])
fix x
assume ih [rule_format]:
"\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow>
y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
proof
assume x_d: "x \<in> Domain (depend s)"
show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
proof(cases x)
case (Th th)
from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
hence "Cs cs \<in> Domain (depend s)" by auto
from ih [OF x_in_r this] obtain th'
where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto
have "(x, Th th') \<in> (depend s)\<^sup>+" using Th x_in cs_in by auto
with th'_ready show ?thesis by auto
next
case (Cs cs)
from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (depend s)^-1" by (auto simp:s_depend_def)
show ?thesis
proof(cases "th' \<in> readys s")
case True
from True and th'_d show ?thesis by auto
next
case False
from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto
with False have "Th th' \<in> Domain (depend s)"
by (auto simp:readys_def wq_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
from ih [OF th'_d this]
obtain th'' where
th''_r: "th'' \<in> readys s" and
th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto
from th'_d and th''_in
have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
with th''_r show ?thesis by auto
qed
qed
qed
qed
qed
lemma th_chain_to_ready:
fixes s th
assumes vt: "vt s"
and th_in: "th \<in> threads s"
shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"
proof(cases "th \<in> readys s")
case True
thus ?thesis by auto
next
case False
from False and th_in have "Th th \<in> Domain (depend s)"
by (auto simp:readys_def s_waiting_def s_depend_def wq_def cs_waiting_def Domain_def)
from chain_building [rule_format, OF vt this]
show ?thesis by auto
qed
lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
by (unfold s_waiting_def cs_waiting_def wq_def, auto)
lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
by (unfold s_holding_def wq_def cs_holding_def, simp)
lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
by (unfold s_holding_def cs_holding_def, auto)
lemma unique_depend: "\<lbrakk>vt s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
by(auto elim:waiting_unique holding_unique)
lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
by (induct rule:trancl_induct, auto)
lemma dchain_unique:
assumes vt: "vt s"
and th1_d: "(n, Th th1) \<in> (depend s)^+"
and th1_r: "th1 \<in> readys s"
and th2_d: "(n, Th th2) \<in> (depend s)^+"
and th2_r: "th2 \<in> readys s"
shows "th1 = th2"
proof -
{ assume neq: "th1 \<noteq> th2"
hence "Th th1 \<noteq> Th th2" by simp
from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]
have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
hence "False"
proof
assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
from trancl_split [OF this]
obtain n where dd: "(Th th1, n) \<in> depend s" by auto
then obtain cs where eq_n: "n = Cs cs"
by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
from dd eq_n have "th1 \<notin> readys s"
by (auto simp:readys_def s_depend_def wq_def s_waiting_def cs_waiting_def)
with th1_r show ?thesis by auto
next
assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
from trancl_split [OF this]
obtain n where dd: "(Th th2, n) \<in> depend s" by auto
then obtain cs where eq_n: "n = Cs cs"
by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
from dd eq_n have "th2 \<notin> readys s"
by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def)
with th2_r show ?thesis by auto
qed
} thus ?thesis by auto
qed
lemma step_holdents_p_add:
fixes th cs s
assumes vt: "vt (P th cs#s)"
and "wq s cs = []"
shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
proof -
from assms show ?thesis
unfolding holdents_test step_depend_p[OF vt] by (auto)
qed
lemma step_holdents_p_eq:
fixes th cs s
assumes vt: "vt (P th cs#s)"
and "wq s cs \<noteq> []"
shows "holdents (P th cs#s) th = holdents s th"
proof -
from assms show ?thesis
unfolding holdents_test step_depend_p[OF vt] by auto
qed
lemma finite_holding:
fixes s th cs
assumes vt: "vt s"
shows "finite (holdents s th)"
proof -
let ?F = "\<lambda> (x, y). the_cs x"
from finite_depend [OF vt]
have "finite (depend s)" .
hence "finite (?F `(depend s))" by simp
moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>"
proof -
{ have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
fix x assume "(Cs x, Th th) \<in> depend s"
hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
moreover have "?F (Cs x, Th th) = x" by simp
ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp
} thus ?thesis by auto
qed
ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
qed
lemma cntCS_v_dec:
fixes s thread cs
assumes vtv: "vt (V thread cs#s)"
shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
proof -
from step_back_step[OF vtv]
have cs_in: "cs \<in> holdents s thread"
apply (cases, unfold holdents_test s_depend_def, simp)
by (unfold cs_holding_def s_holding_def wq_def, auto)
moreover have cs_not_in:
"(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
apply (unfold holdents_test, unfold step_depend_v[OF vtv],
auto simp:next_th_def)
proof -
fix rest
assume dst: "distinct (rest::thread list)"
and ne: "rest \<noteq> []"
and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
proof(rule someI2)
from dst show "distinct rest \<and> set rest = set rest" by auto
next
show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
qed
ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
set (SOME q. distinct q \<and> set q = set rest)" by simp
moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
proof(rule someI2)
from dst show "distinct rest \<and> set rest = set rest" by auto
next
fix x assume " distinct x \<and> set x = set rest" with ne
show "x \<noteq> []" by auto
qed
ultimately
show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"
by auto
next
fix rest
assume dst: "distinct (rest::thread list)"
and ne: "rest \<noteq> []"
and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
proof(rule someI2)
from dst show "distinct rest \<and> set rest = set rest" by auto
next
show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
qed
ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
set (SOME q. distinct q \<and> set q = set rest)" by simp
moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
proof(rule someI2)
from dst show "distinct rest \<and> set rest = set rest" by auto
next
fix x assume " distinct x \<and> set x = set rest" with ne
show "x \<noteq> []" by auto
qed
ultimately show "False" by auto
qed
ultimately
have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
by auto
moreover have "card \<dots> =
Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
proof(rule card_insert)
from finite_holding [OF vtv]
show " finite (holdents (V thread cs # s) thread)" .
qed
moreover from cs_not_in
have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
ultimately show ?thesis by (simp add:cntCS_def)
qed
lemma cnp_cnv_cncs:
fixes s th
assumes vt: "vt s"
shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
then cntCS s th else cntCS s th + 1)"
proof -
from vt show ?thesis
proof(induct arbitrary:th)
case (vt_cons s e)
assume vt: "vt s"
and ih: "\<And>th. cntP s th = cntV s th +
(if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
and stp: "step s e"
from stp show ?case
proof(cases)
case (thread_create thread prio)
assume eq_e: "e = Create thread prio"
and not_in: "thread \<notin> threads s"
show ?thesis
proof -
{ fix cs
assume "thread \<in> set (wq s cs)"
from wq_threads [OF vt this] have "thread \<in> threads s" .
with not_in have "False" by simp
} with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
by (auto simp:readys_def threads.simps s_waiting_def
wq_def cs_waiting_def Let_def)
from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_test
by (simp add:depend_create_unchanged eq_e)
{ assume "th \<noteq> thread"
with eq_readys eq_e
have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
(th \<in> readys (s) \<or> th \<notin> threads (s))"
by (simp add:threads.simps)
with eq_cnp eq_cnv eq_cncs ih not_in
have ?thesis by simp
} moreover {
assume eq_th: "th = thread"
with not_in ih have " cntP s th = cntV s th + cntCS s th" by simp
moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
moreover note eq_cnp eq_cnv eq_cncs
ultimately have ?thesis by auto
} ultimately show ?thesis by blast
qed
next
case (thread_exit thread)
assume eq_e: "e = Exit thread"
and is_runing: "thread \<in> runing s"
and no_hold: "holdents s thread = {}"
from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_test
by (simp add:depend_exit_unchanged eq_e)
{ assume "th \<noteq> thread"
with eq_e
have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
(th \<in> readys (s) \<or> th \<notin> threads (s))"
apply (simp add:threads.simps readys_def)
apply (subst s_waiting_def)
apply (simp add:Let_def)
apply (subst s_waiting_def, simp)
done
with eq_cnp eq_cnv eq_cncs ih
have ?thesis by simp
} moreover {
assume eq_th: "th = thread"
with ih is_runing have " cntP s th = cntV s th + cntCS s th"
by (simp add:runing_def)
moreover from eq_th eq_e have "th \<notin> threads (e#s)"
by simp
moreover note eq_cnp eq_cnv eq_cncs
ultimately have ?thesis by auto
} ultimately show ?thesis by blast
next
case (thread_P thread cs)
assume eq_e: "e = P thread cs"
and is_runing: "thread \<in> runing s"
and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto
show ?thesis
proof -
{ have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
assume neq_th: "th \<noteq> thread"
with eq_e
have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
apply (simp add:readys_def s_waiting_def wq_def Let_def)
apply (rule_tac hh, clarify)
apply (intro iffI allI, clarify)
apply (erule_tac x = csa in allE, auto)
apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
apply (erule_tac x = cs in allE, auto)
by (case_tac "(wq_fun (schs s) cs)", auto)
moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
apply (simp add:cntCS_def holdents_test)
by (unfold step_depend_p [OF vtp], auto)
moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
by (simp add:cntP_def count_def)
moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
by (simp add:cntV_def count_def)
moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
moreover note ih [of th]
ultimately have ?thesis by simp
} moreover {
assume eq_th: "th = thread"
have ?thesis
proof -
from eq_e eq_th have eq_cnp: "cntP (e # s) th = 1 + (cntP s th)"
by (simp add:cntP_def count_def)
from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
by (simp add:cntV_def count_def)
show ?thesis
proof (cases "wq s cs = []")
case True
with is_runing
have "th \<in> readys (e#s)"
apply (unfold eq_e wq_def, unfold readys_def s_depend_def)
apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
moreover have "cntCS (e # s) th = 1 + cntCS s th"
proof -
have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> depend s} =
Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")
proof -
have "?L = insert cs ?R" by auto
moreover have "card \<dots> = Suc (card (?R - {cs}))"
proof(rule card_insert)
from finite_holding [OF vt, of thread]
show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
by (unfold holdents_test, simp)
qed
moreover have "?R - {cs} = ?R"
proof -
have "cs \<notin> ?R"
proof
assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend s}"
with no_dep show False by auto
qed
thus ?thesis by auto
qed
ultimately show ?thesis by auto
qed
thus ?thesis
apply (unfold eq_e eq_th cntCS_def)
apply (simp add: holdents_test)
by (unfold step_depend_p [OF vtp], auto simp:True)
qed
moreover from is_runing have "th \<in> readys s"
by (simp add:runing_def eq_th)
moreover note eq_cnp eq_cnv ih [of th]
ultimately show ?thesis by auto
next
case False
have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
by (unfold eq_th eq_e wq_def, auto simp:Let_def)
have "th \<notin> readys (e#s)"
proof
assume "th \<in> readys (e#s)"
hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
by (simp add:s_waiting_def wq_def)
moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
ultimately have "th = hd (wq (e#s) cs)" by blast
with eq_wq have "th = hd (wq s cs @ [th])" by simp
hence "th = hd (wq s cs)" using False by auto
with False eq_wq wq_distinct [OF vtp, of cs]
show False by (fold eq_e, auto)
qed
moreover from is_runing have "th \<in> threads (e#s)"
by (unfold eq_e, auto simp:runing_def readys_def eq_th)
moreover have "cntCS (e # s) th = cntCS s th"
apply (unfold cntCS_def holdents_test eq_e step_depend_p[OF vtp])
by (auto simp:False)
moreover note eq_cnp eq_cnv ih[of th]
moreover from is_runing have "th \<in> readys s"
by (simp add:runing_def eq_th)
ultimately show ?thesis by auto
qed
qed
} ultimately show ?thesis by blast
qed
next
case (thread_V thread cs)
from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
assume eq_e: "e = V thread cs"
and is_runing: "thread \<in> runing s"
and hold: "holding s thread cs"
from hold obtain rest
where eq_wq: "wq s cs = thread # rest"
by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
proof(rule someI2)
from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq
show "distinct rest \<and> set rest = set rest" by auto
next
show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
by auto
qed
show ?thesis
proof -
{ assume eq_th: "th = thread"
from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
by (unfold eq_e, simp add:cntP_def count_def)
moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
by (unfold eq_e, simp add:cntV_def count_def)
moreover from cntCS_v_dec [OF vtv]
have "cntCS (e # s) thread + 1 = cntCS s thread"
by (simp add:eq_e)
moreover from is_runing have rd_before: "thread \<in> readys s"
by (unfold runing_def, simp)
moreover have "thread \<in> readys (e # s)"
proof -
from is_runing
have "thread \<in> threads (e#s)"
by (unfold eq_e, auto simp:runing_def readys_def)
moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
proof
fix cs1
{ assume eq_cs: "cs1 = cs"
have "\<not> waiting (e # s) thread cs1"
proof -
from eq_wq
have "thread \<notin> set (wq (e#s) cs1)"
apply(unfold eq_e wq_def eq_cs s_holding_def)
apply (auto simp:Let_def)
proof -
assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
with eq_set have "thread \<in> set rest" by simp
with wq_distinct[OF step_back_vt[OF vtv], of cs]
and eq_wq show False by auto
qed
thus ?thesis by (simp add:wq_def s_waiting_def)
qed
} moreover {
assume neq_cs: "cs1 \<noteq> cs"
have "\<not> waiting (e # s) thread cs1"
proof -
from wq_v_neq [OF neq_cs[symmetric]]
have "wq (V thread cs # s) cs1 = wq s cs1" .
moreover have "\<not> waiting s thread cs1"
proof -
from runing_ready and is_runing
have "thread \<in> readys s" by auto
thus ?thesis by (simp add:readys_def)
qed
ultimately show ?thesis
by (auto simp:wq_def s_waiting_def eq_e)
qed
} ultimately show "\<not> waiting (e # s) thread cs1" by blast
qed
ultimately show ?thesis by (simp add:readys_def)
qed
moreover note eq_th ih
ultimately have ?thesis by auto
} moreover {
assume neq_th: "th \<noteq> thread"
from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th"
by (simp add:cntP_def count_def)
from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th"
by (simp add:cntV_def count_def)
have ?thesis
proof(cases "th \<in> set rest")
case False
have "(th \<in> readys (e # s)) = (th \<in> readys s)"
apply (insert step_back_vt[OF vtv])
by (unfold eq_e, rule readys_v_eq [OF _ neq_th eq_wq False], auto)
moreover have "cntCS (e#s) th = cntCS s th"
apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
proof -
have "{csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
{cs. (Cs cs, Th th) \<in> depend s}"
proof -
from False eq_wq
have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> depend s"
apply (unfold next_th_def, auto)
proof -
assume ne: "rest \<noteq> []"
and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
and eq_wq: "wq s cs = thread # rest"
from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
set (SOME q. distinct q \<and> set q = set rest)
" by simp
moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
proof(rule someI2)
from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq
show "distinct rest \<and> set rest = set rest" by auto
next
fix x assume "distinct x \<and> set x = set rest"
with ne show "x \<noteq> []" by auto
qed
ultimately show
"(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"
by auto
qed
thus ?thesis by auto
qed
thus "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
card {cs. (Cs cs, Th th) \<in> depend s}" by simp
qed
moreover note ih eq_cnp eq_cnv eq_threads
ultimately show ?thesis by auto
next
case True
assume th_in: "th \<in> set rest"
show ?thesis
proof(cases "next_th s thread cs th")
case False
with eq_wq and th_in have
neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
by (auto simp:next_th_def)
have "(th \<in> readys (e # s)) = (th \<in> readys s)"
proof -
from eq_wq and th_in
have "\<not> th \<in> readys s"
apply (auto simp:readys_def s_waiting_def)
apply (rule_tac x = cs in exI, auto)
by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto simp add: wq_def)
moreover
from eq_wq and th_in and neq_hd
have "\<not> (th \<in> readys (e # s))"
apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
by (rule_tac x = cs in exI, auto simp:eq_set)
ultimately show ?thesis by auto
qed
moreover have "cntCS (e#s) th = cntCS s th"
proof -
from eq_wq and th_in and neq_hd
have "(holdents (e # s) th) = (holdents s th)"
apply (unfold eq_e step_depend_v[OF vtv],
auto simp:next_th_def eq_set s_depend_def holdents_test wq_def
Let_def cs_holding_def)
by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto simp:wq_def)
thus ?thesis by (simp add:cntCS_def)
qed
moreover note ih eq_cnp eq_cnv eq_threads
ultimately show ?thesis by auto
next
case True
let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
let ?t = "hd ?rest"
from True eq_wq th_in neq_th
have "th \<in> readys (e # s)"
apply (auto simp:eq_e readys_def s_waiting_def wq_def
Let_def next_th_def)
proof -
assume eq_wq: "wq_fun (schs s) cs = thread # rest"
and t_in: "?t \<in> set rest"
show "?t \<in> threads s"
proof(rule wq_threads[OF step_back_vt[OF vtv]])
from eq_wq and t_in
show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
qed
next
fix csa
assume eq_wq: "wq_fun (schs s) cs = thread # rest"
and t_in: "?t \<in> set rest"
and neq_cs: "csa \<noteq> cs"
and t_in': "?t \<in> set (wq_fun (schs s) csa)"
show "?t = hd (wq_fun (schs s) csa)"
proof -
{ assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
from wq_distinct[OF step_back_vt[OF vtv], of cs] and
eq_wq[folded wq_def] and t_in eq_wq
have "?t \<noteq> thread" by auto
with eq_wq and t_in
have w1: "waiting s ?t cs"
by (auto simp:s_waiting_def wq_def)
from t_in' neq_hd'
have w2: "waiting s ?t csa"
by (auto simp:s_waiting_def wq_def)
from waiting_unique[OF step_back_vt[OF vtv] w1 w2]
and neq_cs have "False" by auto
} thus ?thesis by auto
qed
qed
moreover have "cntP s th = cntV s th + cntCS s th + 1"
proof -
have "th \<notin> readys s"
proof -
from True eq_wq neq_th th_in
show ?thesis
apply (unfold readys_def s_waiting_def, auto)
by (rule_tac x = cs in exI, auto simp add: wq_def)
qed
moreover have "th \<in> threads s"
proof -
from th_in eq_wq
have "th \<in> set (wq s cs)" by simp
from wq_threads [OF step_back_vt[OF vtv] this]
show ?thesis .
qed
ultimately show ?thesis using ih by auto
qed
moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
apply (unfold cntCS_def holdents_test eq_e step_depend_v[OF vtv], auto)
proof -
show "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs} =
Suc (card {cs. (Cs cs, Th th) \<in> depend s})"
(is "card ?A = Suc (card ?B)")
proof -
have "?A = insert cs ?B" by auto
hence "card ?A = card (insert cs ?B)" by simp
also have "\<dots> = Suc (card ?B)"
proof(rule card_insert_disjoint)
have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` depend s)"
apply (auto simp:image_def)
by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
with finite_depend[OF step_back_vt[OF vtv]]
show "finite {cs. (Cs cs, Th th) \<in> depend s}" by (auto intro:finite_subset)
next
show "cs \<notin> {cs. (Cs cs, Th th) \<in> depend s}"
proof
assume "cs \<in> {cs. (Cs cs, Th th) \<in> depend s}"
hence "(Cs cs, Th th) \<in> depend s" by simp
with True neq_th eq_wq show False
by (auto simp:next_th_def s_depend_def cs_holding_def)
qed
qed
finally show ?thesis .
qed
qed
moreover note eq_cnp eq_cnv
ultimately show ?thesis by simp
qed
qed
} ultimately show ?thesis by blast
qed
next
case (thread_set thread prio)
assume eq_e: "e = Set thread prio"
and is_runing: "thread \<in> runing s"
show ?thesis
proof -
from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_test
by (simp add:depend_set_unchanged eq_e)
from eq_e have eq_readys: "readys (e#s) = readys s"
by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
auto simp:Let_def)
{ assume "th \<noteq> thread"
with eq_readys eq_e
have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
(th \<in> readys (s) \<or> th \<notin> threads (s))"
by (simp add:threads.simps)
with eq_cnp eq_cnv eq_cncs ih is_runing
have ?thesis by simp
} moreover {
assume eq_th: "th = thread"
with is_runing ih have " cntP s th = cntV s th + cntCS s th"
by (unfold runing_def, auto)
moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
by (simp add:runing_def)
moreover note eq_cnp eq_cnv eq_cncs
ultimately have ?thesis by auto
} ultimately show ?thesis by blast
qed
qed
next
case vt_nil
show ?case
by (unfold cntP_def cntV_def cntCS_def,
auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
qed
qed
lemma not_thread_cncs:
fixes th s
assumes vt: "vt s"
and not_in: "th \<notin> threads s"
shows "cntCS s th = 0"
proof -
from vt not_in show ?thesis
proof(induct arbitrary:th)
case (vt_cons s e th)
assume vt: "vt s"
and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
and stp: "step s e"
and not_in: "th \<notin> threads (e # s)"
from stp show ?case
proof(cases)
case (thread_create thread prio)
assume eq_e: "e = Create thread prio"
and not_in': "thread \<notin> threads s"
have "cntCS (e # s) th = cntCS s th"
apply (unfold eq_e cntCS_def holdents_test)
by (simp add:depend_create_unchanged)
moreover have "th \<notin> threads s"
proof -
from not_in eq_e show ?thesis by simp
qed
moreover note ih ultimately show ?thesis by auto
next
case (thread_exit thread)
assume eq_e: "e = Exit thread"
and nh: "holdents s thread = {}"
have eq_cns: "cntCS (e # s) th = cntCS s th"
apply (unfold eq_e cntCS_def holdents_test)
by (simp add:depend_exit_unchanged)
show ?thesis
proof(cases "th = thread")
case True
have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
with eq_cns show ?thesis by simp
next
case False
with not_in and eq_e
have "th \<notin> threads s" by simp
from ih[OF this] and eq_cns show ?thesis by simp
qed
next
case (thread_P thread cs)
assume eq_e: "e = P thread cs"
and is_runing: "thread \<in> runing s"
from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
have neq_th: "th \<noteq> thread"
proof -
from not_in eq_e have "th \<notin> threads s" by simp
moreover from is_runing have "thread \<in> threads s"
by (simp add:runing_def readys_def)
ultimately show ?thesis by auto
qed
hence "cntCS (e # s) th = cntCS s th "
apply (unfold cntCS_def holdents_test eq_e)
by (unfold step_depend_p[OF vtp], auto)
moreover have "cntCS s th = 0"
proof(rule ih)
from not_in eq_e show "th \<notin> threads s" by simp
qed
ultimately show ?thesis by simp
next
case (thread_V thread cs)
assume eq_e: "e = V thread cs"
and is_runing: "thread \<in> runing s"
and hold: "holding s thread cs"
have neq_th: "th \<noteq> thread"
proof -
from not_in eq_e have "th \<notin> threads s" by simp
moreover from is_runing have "thread \<in> threads s"
by (simp add:runing_def readys_def)
ultimately show ?thesis by auto
qed
from assms thread_V vt stp ih have vtv: "vt (V thread cs#s)" by auto
from hold obtain rest
where eq_wq: "wq s cs = thread # rest"
by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
from not_in eq_e eq_wq
have "\<not> next_th s thread cs th"
apply (auto simp:next_th_def)
proof -
assume ne: "rest \<noteq> []"
and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
have "?t \<in> set rest"
proof(rule someI2)
from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq
show "distinct rest \<and> set rest = set rest" by auto
next
fix x assume "distinct x \<and> set x = set rest" with ne
show "hd x \<in> set rest" by (cases x, auto)
qed
with eq_wq have "?t \<in> set (wq s cs)" by simp
from wq_threads[OF step_back_vt[OF vtv], OF this] and ni
show False by auto
qed
moreover note neq_th eq_wq
ultimately have "cntCS (e # s) th = cntCS s th"
by (unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
moreover have "cntCS s th = 0"
proof(rule ih)
from not_in eq_e show "th \<notin> threads s" by simp
qed
ultimately show ?thesis by simp
next
case (thread_set thread prio)
print_facts
assume eq_e: "e = Set thread prio"
and is_runing: "thread \<in> runing s"
from not_in and eq_e have "th \<notin> threads s" by auto
from ih [OF this] and eq_e
show ?thesis
apply (unfold eq_e cntCS_def holdents_test)
by (simp add:depend_set_unchanged)
qed
next
case vt_nil
show ?case
by (unfold cntCS_def,
auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
qed
qed
lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
by (auto simp:s_waiting_def cs_waiting_def wq_def)
lemma dm_depend_threads:
fixes th s
assumes vt: "vt s"
and in_dom: "(Th th) \<in> Domain (depend s)"
shows "th \<in> threads s"
proof -
from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
ultimately have "(Th th, Cs cs) \<in> depend s" by simp
hence "th \<in> set (wq s cs)"
by (unfold s_depend_def, auto simp:cs_waiting_def)
from wq_threads [OF vt this] show ?thesis .
qed
lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
unfolding cp_def wq_def
apply(induct s rule: schs.induct)
apply(simp add: Let_def cpreced_initial)
apply(simp add: Let_def)
apply(simp add: Let_def)
apply(simp add: Let_def)
apply(subst (2) schs.simps)
apply(simp add: Let_def)
apply(subst (2) schs.simps)
apply(simp add: Let_def)
done
lemma runing_unique:
fixes th1 th2 s
assumes vt: "vt s"
and runing_1: "th1 \<in> runing s"
and runing_2: "th2 \<in> runing s"
shows "th1 = th2"
proof -
from runing_1 and runing_2 have "cp s th1 = cp s th2"
by (unfold runing_def, simp)
hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
(is "Max (?f ` ?A) = Max (?f ` ?B)")
by (unfold cp_eq_cpreced cpreced_def)
obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
proof -
have h1: "finite (?f ` ?A)"
proof -
have "finite ?A"
proof -
have "finite (dependents (wq s) th1)"
proof-
have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
moreover have "finite \<dots>"
proof -
from finite_depend[OF vt] have "finite (depend s)" .
hence "finite ((depend (wq s))\<^sup>+)"
apply (unfold finite_trancl)
by (auto simp: s_depend_def cs_depend_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
qed
thus ?thesis by (simp add:cs_dependents_def)
qed
thus ?thesis by simp
qed
thus ?thesis by auto
qed
moreover have h2: "(?f ` ?A) \<noteq> {}"
proof -
have "?A \<noteq> {}" by simp
thus ?thesis by simp
qed
from Max_in [OF h1 h2]
have "Max (?f ` ?A) \<in> (?f ` ?A)" .
thus ?thesis by (auto intro:that)
qed
obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
proof -
have h1: "finite (?f ` ?B)"
proof -
have "finite ?B"
proof -
have "finite (dependents (wq s) th2)"
proof-
have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
moreover have "finite \<dots>"
proof -
from finite_depend[OF vt] have "finite (depend s)" .
hence "finite ((depend (wq s))\<^sup>+)"
apply (unfold finite_trancl)
by (auto simp: s_depend_def cs_depend_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
qed
thus ?thesis by (simp add:cs_dependents_def)
qed
thus ?thesis by simp
qed
thus ?thesis by auto
qed
moreover have h2: "(?f ` ?B) \<noteq> {}"
proof -
have "?B \<noteq> {}" by simp
thus ?thesis by simp
qed
from Max_in [OF h1 h2]
have "Max (?f ` ?B) \<in> (?f ` ?B)" .
thus ?thesis by (auto intro:that)
qed
from eq_f_th1 eq_f_th2 eq_max
have eq_preced: "preced th1' s = preced th2' s" by auto
hence eq_th12: "th1' = th2'"
proof (rule preced_unique)
from th1_in have "th1' = th1 \<or> (th1' \<in> dependents (wq s) th1)" by simp
thus "th1' \<in> threads s"
proof
assume "th1' \<in> dependents (wq s) th1"
hence "(Th th1') \<in> Domain ((depend s)^+)"
apply (unfold cs_dependents_def cs_depend_def s_depend_def)
by (auto simp:Domain_def)
hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
from dm_depend_threads[OF vt this] show ?thesis .
next
assume "th1' = th1"
with runing_1 show ?thesis
by (unfold runing_def readys_def, auto)
qed
next
from th2_in have "th2' = th2 \<or> (th2' \<in> dependents (wq s) th2)" by simp
thus "th2' \<in> threads s"
proof
assume "th2' \<in> dependents (wq s) th2"
hence "(Th th2') \<in> Domain ((depend s)^+)"
apply (unfold cs_dependents_def cs_depend_def s_depend_def)
by (auto simp:Domain_def)
hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
from dm_depend_threads[OF vt this] show ?thesis .
next
assume "th2' = th2"
with runing_2 show ?thesis
by (unfold runing_def readys_def, auto)
qed
qed
from th1_in have "th1' = th1 \<or> th1' \<in> dependents (wq s) th1" by simp
thus ?thesis
proof
assume eq_th': "th1' = th1"
from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
thus ?thesis
proof
assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
next
assume "th2' \<in> dependents (wq s) th2"
with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
hence "(Th th1, Th th2) \<in> (depend s)^+"
by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
hence "Th th1 \<in> Domain ((depend s)^+)"
apply (unfold cs_dependents_def cs_depend_def s_depend_def)
by (auto simp:Domain_def)
hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
from depend_target_th [OF this]
obtain cs' where "n = Cs cs'" by auto
with d have "(Th th1, Cs cs') \<in> depend s" by simp
with runing_1 have "False"
apply (unfold runing_def readys_def s_depend_def)
by (auto simp:eq_waiting)
thus ?thesis by simp
qed
next
assume th1'_in: "th1' \<in> dependents (wq s) th1"
from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
thus ?thesis
proof
assume "th2' = th2"
with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
hence "(Th th2, Th th1) \<in> (depend s)^+"
by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
hence "Th th2 \<in> Domain ((depend s)^+)"
apply (unfold cs_dependents_def cs_depend_def s_depend_def)
by (auto simp:Domain_def)
hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
from depend_target_th [OF this]
obtain cs' where "n = Cs cs'" by auto
with d have "(Th th2, Cs cs') \<in> depend s" by simp
with runing_2 have "False"
apply (unfold runing_def readys_def s_depend_def)
by (auto simp:eq_waiting)
thus ?thesis by simp
next
assume "th2' \<in> dependents (wq s) th2"
with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
show ?thesis
proof(rule dchain_unique[OF vt h1 _ h2, symmetric])
from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
qed
qed
qed
qed
lemma create_pre:
assumes stp: "step s e"
and not_in: "th \<notin> threads s"
and is_in: "th \<in> threads (e#s)"
obtains prio where "e = Create th prio"
proof -
from assms
show ?thesis
proof(cases)
case (thread_create thread prio)
with is_in not_in have "e = Create th prio" by simp
from that[OF this] show ?thesis .
next
case (thread_exit thread)
with assms show ?thesis by (auto intro!:that)
next
case (thread_P thread)
with assms show ?thesis by (auto intro!:that)
next
case (thread_V thread)
with assms show ?thesis by (auto intro!:that)
next
case (thread_set thread)
with assms show ?thesis by (auto intro!:that)
qed
qed
lemma length_down_to_in:
assumes le_ij: "i \<le> j"
and le_js: "j \<le> length s"
shows "length (down_to j i s) = j - i"
proof -
have "length (down_to j i s) = length (from_to i j (rev s))"
by (unfold down_to_def, auto)
also have "\<dots> = j - i"
proof(rule length_from_to_in[OF le_ij])
from le_js show "j \<le> length (rev s)" by simp
qed
finally show ?thesis .
qed
lemma moment_head:
assumes le_it: "Suc i \<le> length t"
obtains e where "moment (Suc i) t = e#moment i t"
proof -
have "i \<le> Suc i" by simp
from length_down_to_in [OF this le_it]
have "length (down_to (Suc i) i t) = 1" by auto
then obtain e where "down_to (Suc i) i t = [e]"
apply (cases "(down_to (Suc i) i t)") by auto
moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
by (rule down_to_conc[symmetric], auto)
ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
by (auto simp:down_to_moment)
from that [OF this] show ?thesis .
qed
lemma cnp_cnv_eq:
fixes th s
assumes "vt s"
and "th \<notin> threads s"
shows "cntP s th = cntV s th"
proof -
from assms show ?thesis
proof(induct)
case (vt_cons s e)
have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
have not_in: "th \<notin> threads (e # s)" by fact
have "step s e" by fact
thus ?case proof(cases)
case (thread_create thread prio)
assume eq_e: "e = Create thread prio"
hence "thread \<in> threads (e#s)" by simp
with not_in and eq_e have "th \<notin> threads s" by auto
from ih [OF this] show ?thesis using eq_e
by (auto simp:cntP_def cntV_def count_def)
next
case (thread_exit thread)
assume eq_e: "e = Exit thread"
and not_holding: "holdents s thread = {}"
have vt_s: "vt s" by fact
from finite_holding[OF vt_s] have "finite (holdents s thread)" .
with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
moreover note cnp_cnv_cncs[OF vt_s, of thread]
ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
show ?thesis
proof(cases "th = thread")
case True
with eq_thread eq_e show ?thesis
by (auto simp:cntP_def cntV_def count_def)
next
case False
with not_in and eq_e have "th \<notin> threads s" by simp
from ih[OF this] and eq_e show ?thesis
by (auto simp:cntP_def cntV_def count_def)
qed
next
case (thread_P thread cs)
assume eq_e: "e = P thread cs"
have "thread \<in> runing s" by fact
with not_in eq_e have neq_th: "thread \<noteq> th"
by (auto simp:runing_def readys_def)
from not_in eq_e have "th \<notin> threads s" by simp
from ih[OF this] and neq_th and eq_e show ?thesis
by (auto simp:cntP_def cntV_def count_def)
next
case (thread_V thread cs)
assume eq_e: "e = V thread cs"
have "thread \<in> runing s" by fact
with not_in eq_e have neq_th: "thread \<noteq> th"
by (auto simp:runing_def readys_def)
from not_in eq_e have "th \<notin> threads s" by simp
from ih[OF this] and neq_th and eq_e show ?thesis
by (auto simp:cntP_def cntV_def count_def)
next
case (thread_set thread prio)
assume eq_e: "e = Set thread prio"
and "thread \<in> runing s"
hence "thread \<in> threads (e#s)"
by (simp add:runing_def readys_def)
with not_in and eq_e have "th \<notin> threads s" by auto
from ih [OF this] show ?thesis using eq_e
by (auto simp:cntP_def cntV_def count_def)
qed
next
case vt_nil
show ?case by (auto simp:cntP_def cntV_def count_def)
qed
qed
lemma eq_depend:
"depend (wq s) = depend s"
by (unfold cs_depend_def s_depend_def, auto)
lemma count_eq_dependents:
assumes vt: "vt s"
and eq_pv: "cntP s th = cntV s th"
shows "dependents (wq s) th = {}"
proof -
from cnp_cnv_cncs[OF vt] and eq_pv
have "cntCS s th = 0"
by (auto split:if_splits)
moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
proof -
from finite_holding[OF vt, of th] show ?thesis
by (simp add:holdents_test)
qed
ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
by (unfold cntCS_def holdents_test cs_dependents_def, auto)
show ?thesis
proof(unfold cs_dependents_def)
{ assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
hence "False"
proof(cases)
assume "(Th th', Th th) \<in> depend (wq s)"
thus "False" by (auto simp:cs_depend_def)
next
fix c
assume "(c, Th th) \<in> depend (wq s)"
with h and eq_depend show "False"
by (cases c, auto simp:cs_depend_def)
qed
} thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
qed
qed
lemma dependents_threads:
fixes s th
assumes vt: "vt s"
shows "dependents (wq s) th \<subseteq> threads s"
proof
{ fix th th'
assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
have "Th th \<in> Domain (depend s)"
proof -
from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
thus ?thesis using eq_depend by simp
qed
from dm_depend_threads[OF vt this]
have "th \<in> threads s" .
} note hh = this
fix th1
assume "th1 \<in> dependents (wq s) th"
hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
by (unfold cs_dependents_def, simp)
from hh [OF this] show "th1 \<in> threads s" .
qed
lemma finite_threads:
assumes vt: "vt s"
shows "finite (threads s)"
using vt
by (induct) (auto elim: step.cases)
lemma Max_f_mono:
assumes seq: "A \<subseteq> B"
and np: "A \<noteq> {}"
and fnt: "finite B"
shows "Max (f ` A) \<le> Max (f ` B)"
proof(rule Max_mono)
from seq show "f ` A \<subseteq> f ` B" by auto
next
from np show "f ` A \<noteq> {}" by auto
next
from fnt and seq show "finite (f ` B)" by auto
qed
lemma cp_le:
assumes vt: "vt s"
and th_in: "th \<in> threads s"
shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def)
show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))
\<le> Max ((\<lambda>th. preced th s) ` threads s)"
(is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
proof(rule Max_f_mono)
show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
next
from finite_threads [OF vt]
show "finite (threads s)" .
next
from th_in
show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
apply (auto simp:Domain_def)
apply (rule_tac dm_depend_threads[OF vt])
apply (unfold trancl_domain [of "depend s", symmetric])
by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
qed
qed
lemma le_cp:
assumes vt: "vt s"
shows "preced th s \<le> cp s th"
proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
show "Prc (original_priority th s) (birthtime th s)
\<le> Max (insert (Prc (original_priority th s) (birthtime th s))
((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))"
(is "?l \<le> Max (insert ?l ?A)")
proof(cases "?A = {}")
case False
have "finite ?A" (is "finite (?f ` ?B)")
proof -
have "finite ?B"
proof-
have "finite {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th)" in bexI, auto)
moreover have "finite \<dots>"
proof -
from finite_depend[OF vt] have "finite (depend s)" .
hence "finite ((depend (wq s))\<^sup>+)"
apply (unfold finite_trancl)
by (auto simp: s_depend_def cs_depend_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
qed
thus ?thesis by (simp add:cs_dependents_def)
qed
thus ?thesis by simp
qed
from Max_insert [OF this False, of ?l] show ?thesis by auto
next
case True
thus ?thesis by auto
qed
qed
lemma max_cp_eq:
assumes vt: "vt s"
shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
(is "?l = ?r")
proof(cases "threads s = {}")
case True
thus ?thesis by auto
next
case False
have "?l \<in> ((cp s) ` threads s)"
proof(rule Max_in)
from finite_threads[OF vt]
show "finite (cp s ` threads s)" by auto
next
from False show "cp s ` threads s \<noteq> {}" by auto
qed
then obtain th
where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
have "\<dots> \<le> ?r" by (rule cp_le[OF vt th_in])
moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
proof -
have "?r \<in> (?f ` ?A)"
proof(rule Max_in)
from finite_threads[OF vt]
show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
next
from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
qed
then obtain th' where
th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
from le_cp [OF vt, of th'] eq_r
have "?r \<le> cp s th'" by auto
moreover have "\<dots> \<le> cp s th"
proof(fold eq_l)
show " cp s th' \<le> Max (cp s ` threads s)"
proof(rule Max_ge)
from th_in' show "cp s th' \<in> cp s ` threads s"
by auto
next
from finite_threads[OF vt]
show "finite (cp s ` threads s)" by auto
qed
qed
ultimately show ?thesis by auto
qed
ultimately show ?thesis using eq_l by auto
qed
lemma max_cp_readys_threads_pre:
assumes vt: "vt s"
and np: "threads s \<noteq> {}"
shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
proof(unfold max_cp_eq[OF vt])
show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
proof -
let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
let ?f = "(\<lambda>th. preced th s)"
have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
proof(rule Max_in)
from finite_threads[OF vt] show "finite (?f ` threads s)" by simp
next
from np show "?f ` threads s \<noteq> {}" by simp
qed
then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
by (auto simp:Image_def)
from th_chain_to_ready [OF vt tm_in]
have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
thus ?thesis
proof
assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
then obtain th' where th'_in: "th' \<in> readys s"
and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
have "cp s th' = ?f tm"
proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
from dependents_threads[OF vt] finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))"
by (auto intro:finite_subset)
next
fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
moreover have "p \<le> \<dots>"
proof(rule Max_ge)
from finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
next
from p_in and th'_in and dependents_threads[OF vt, of th']
show "p \<in> (\<lambda>th. preced th s) ` threads s"
by (auto simp:readys_def)
qed
ultimately show "p \<le> preced tm s" by auto
next
show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
proof -
from tm_chain
have "tm \<in> dependents (wq s) th'"
by (unfold cs_dependents_def s_depend_def cs_depend_def, auto)
thus ?thesis by auto
qed
qed
with tm_max
have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
show ?thesis
proof (fold h, rule Max_eqI)
fix q
assume "q \<in> cp s ` readys s"
then obtain th1 where th1_in: "th1 \<in> readys s"
and eq_q: "q = cp s th1" by auto
show "q \<le> cp s th'"
apply (unfold h eq_q)
apply (unfold cp_eq_cpreced cpreced_def)
apply (rule Max_mono)
proof -
from dependents_threads [OF vt, of th1] th1_in
show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq>
(\<lambda>th. preced th s) ` threads s"
by (auto simp:readys_def)
next
show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp
next
from finite_threads[OF vt]
show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
qed
next
from finite_threads[OF vt]
show "finite (cp s ` readys s)" by (auto simp:readys_def)
next
from th'_in
show "cp s th' \<in> cp s ` readys s" by simp
qed
next
assume tm_ready: "tm \<in> readys s"
show ?thesis
proof(fold tm_max)
have cp_eq_p: "cp s tm = preced tm s"
proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
fix y
assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
show "y \<le> preced tm s"
proof -
{ fix y'
assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
have "y' \<le> preced tm s"
proof(unfold tm_max, rule Max_ge)
from hy' dependents_threads[OF vt, of tm]
show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
next
from finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
qed
} with hy show ?thesis by auto
qed
next
from dependents_threads[OF vt, of tm] finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
by (auto intro:finite_subset)
next
show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
by simp
qed
moreover have "Max (cp s ` readys s) = cp s tm"
proof(rule Max_eqI)
from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
next
from finite_threads[OF vt]
show "finite (cp s ` readys s)" by (auto simp:readys_def)
next
fix y assume "y \<in> cp s ` readys s"
then obtain th1 where th1_readys: "th1 \<in> readys s"
and h: "y = cp s th1" by auto
show "y \<le> cp s tm"
apply(unfold cp_eq_p h)
apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
proof -
from finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
next
show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
by simp
next
from dependents_threads[OF vt, of th1] th1_readys
show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)
\<subseteq> (\<lambda>th. preced th s) ` threads s"
by (auto simp:readys_def)
qed
qed
ultimately show " Max (cp s ` readys s) = preced tm s" by simp
qed
qed
qed
qed
lemma max_cp_readys_threads:
assumes vt: "vt s"
shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
proof(cases "threads s = {}")
case True
thus ?thesis
by (auto simp:readys_def)
next
case False
show ?thesis by (rule max_cp_readys_threads_pre[OF vt False])
qed
lemma eq_holding: "holding (wq s) th cs = holding s th cs"
apply (unfold s_holding_def cs_holding_def wq_def, simp)
done
lemma f_image_eq:
assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
shows "f ` A = g ` A"
proof
show "f ` A \<subseteq> g ` A"
by(rule image_subsetI, auto intro:h)
next
show "g ` A \<subseteq> f ` A"
by (rule image_subsetI, auto intro:h[symmetric])
qed
definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
lemma detached_test:
shows "detached s th = (Th th \<notin> Field (depend s))"
apply(simp add: detached_def Field_def)
apply(simp add: s_depend_def)
apply(simp add: s_holding_abv s_waiting_abv)
apply(simp add: Domain_iff Range_iff)
apply(simp add: wq_def)
apply(auto)
done
lemma detached_intro:
fixes s th
assumes vt: "vt s"
and eq_pv: "cntP s th = cntV s th"
shows "detached s th"
proof -
from cnp_cnv_cncs[OF vt]
have eq_cnt: "cntP s th =
cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
hence cncs_zero: "cntCS s th = 0"
by (auto simp:eq_pv split:if_splits)
with eq_cnt
have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
thus ?thesis
proof
assume "th \<notin> threads s"
with range_in[OF vt] dm_depend_threads[OF vt]
show ?thesis
by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
next
assume "th \<in> readys s"
moreover have "Th th \<notin> Range (depend s)"
proof -
from card_0_eq [OF finite_holding [OF vt]] and cncs_zero
have "holdents s th = {}"
by (simp add:cntCS_def)
thus ?thesis
apply(auto simp:holdents_test)
apply(case_tac a)
apply(auto simp:holdents_test s_depend_def)
done
qed
ultimately show ?thesis
by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def readys_def)
qed
qed
lemma detached_elim:
fixes s th
assumes vt: "vt s"
and dtc: "detached s th"
shows "cntP s th = cntV s th"
proof -
from cnp_cnv_cncs[OF vt]
have eq_pv: " cntP s th =
cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
have cncs_z: "cntCS s th = 0"
proof -
from dtc have "holdents s th = {}"
unfolding detached_def holdents_test s_depend_def
by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
thus ?thesis by (auto simp:cntCS_def)
qed
show ?thesis
proof(cases "th \<in> threads s")
case True
with dtc
have "th \<in> readys s"
by (unfold readys_def detached_def Field_def Domain_def Range_def,
auto simp:eq_waiting s_depend_def)
with cncs_z and eq_pv show ?thesis by simp
next
case False
with cncs_z and eq_pv show ?thesis by simp
qed
qed
lemma detached_eq:
fixes s th
assumes vt: "vt s"
shows "(detached s th) = (cntP s th = cntV s th)"
by (insert vt, auto intro:detached_intro detached_elim)
end