theory Prio
imports Precedence_ord Moment Lsp Happen_within
begin
type_synonym thread = nat
type_synonym priority = nat
type_synonym cs = nat
datatype event =
Create thread priority |
Exit thread |
P thread cs |
V thread cs |
Set thread priority
datatype node =
Th "thread" |
Cs "cs"
type_synonym state = "event list"
fun threads :: "state \<Rightarrow> thread set"
where
"threads [] = {}" |
"threads (Create thread prio#s) = {thread} \<union> threads s" |
"threads (Exit thread # s) = (threads s) - {thread}" |
"threads (e#s) = threads s"
fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> nat"
where
"original_priority thread [] = 0" |
"original_priority thread (Create thread' prio#s) =
(if thread' = thread then prio else original_priority thread s)" |
"original_priority thread (Set thread' prio#s) =
(if thread' = thread then prio else original_priority thread s)" |
"original_priority thread (e#s) = original_priority thread s"
fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
where
"birthtime thread [] = 0" |
"birthtime thread ((Create thread' prio)#s) = (if (thread = thread') then length s
else birthtime thread s)" |
"birthtime thread ((Set thread' prio)#s) = (if (thread = thread') then length s
else birthtime thread s)" |
"birthtime thread (e#s) = birthtime thread s"
definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
consts holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
depend :: "'b \<Rightarrow> (node \<times> node) set"
dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
defs (overloaded) cs_holding_def: "holding wq thread cs == (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
cs_waiting_def: "waiting wq thread cs == (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
cs_depend_def: "depend (wq::cs \<Rightarrow> thread list) == {(Th t, Cs c) | t c. waiting wq t c} \<union>
{(Cs c, Th t) | c t. holding wq t c}"
cs_dependents_def: "dependents (wq::cs \<Rightarrow> thread list) th == {th' . (Th th', Th th) \<in> (depend wq)^+}"
record schedule_state =
waiting_queue :: "cs \<Rightarrow> thread list"
cur_preced :: "thread \<Rightarrow> precedence"
definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
fun schs :: "state \<Rightarrow> schedule_state"
where
"schs [] = \<lparr>waiting_queue = \<lambda> cs. [],
cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
"schs (e#s) = (let ps = schs s in
let pwq = waiting_queue ps in
let pcp = cur_preced ps in
let nwq = case e of
P thread cs \<Rightarrow> pwq(cs:=(pwq cs @ [thread])) |
V thread cs \<Rightarrow> let nq = case (pwq cs) of
[] \<Rightarrow> [] |
(th#pq) \<Rightarrow> case (lsp pcp pq) of
(l, [], r) \<Rightarrow> []
| (l, m#ms, r) \<Rightarrow> m#(l@ms@r)
in pwq(cs:=nq) |
_ \<Rightarrow> pwq
in let ncp = cpreced (e#s) nwq in
\<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
)"
definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
where "wq s == waiting_queue (schs s)"
definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
where "cp s = cur_preced (schs s)"
defs (overloaded) s_holding_def: "holding (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
s_waiting_def: "waiting (s::state) thread cs == (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
s_depend_def: "depend (s::state) == {(Th t, Cs c) | t c. waiting (wq s) t c} \<union>
{(Cs c, Th t) | c t. holding (wq s) t c}"
s_dependents_def: "dependents (s::state) th == {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
definition readys :: "state \<Rightarrow> thread set"
where
"readys s =
{thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
definition runing :: "state \<Rightarrow> thread set"
where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
where
thread_create: "\<lbrakk>prio \<le> max_prio; thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
thread_P: "\<lbrakk>thread \<in> runing s; (Cs cs, Th thread) \<notin> (depend s)^+\<rbrakk> \<Longrightarrow> step s (P thread cs)" |
thread_V: "\<lbrakk>thread \<in> runing s; holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
for cs
where
vt_nil[intro]: "vt cs []" |
vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
lemma runing_ready: "runing s \<subseteq> readys s"
by (auto simp only:runing_def readys_def)
lemma wq_v_eq_nil:
fixes s cs thread rest
assumes eq_wq: "wq s cs = thread # rest"
and eq_lsp: "lsp (cp s) rest = (l, [], r)"
shows "wq (V thread cs#s) cs = []"
proof -
from prems show ?thesis
by (auto simp:wq_def Let_def cp_def split:list.splits)
qed
lemma wq_v_eq:
fixes s cs thread rest
assumes eq_wq: "wq s cs = thread # rest"
and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
shows "wq (V thread cs#s) cs = th'#l@r"
proof -
from prems show ?thesis
by (auto simp:wq_def Let_def cp_def split:list.splits)
qed
lemma wq_v_neq:
"cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
by (auto simp:wq_def Let_def cp_def split:list.splits)
lemma wq_distinct: "vt step s \<Longrightarrow> distinct (wq s cs)"
proof(erule_tac vt.induct, simp add:wq_def)
fix s e
assume h1: "step s e"
and h2: "distinct (wq s cs)"
thus "distinct (wq (e # s) cs)"
proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
fix thread s
assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
and h2: "thread \<in> set (waiting_queue (schs s) cs)"
and h3: "thread \<in> runing s"
show "False"
proof -
from h3 have "\<And> cs. thread \<in> set (waiting_queue (schs s) cs) \<Longrightarrow>
thread = hd ((waiting_queue (schs s) cs))"
by (simp add:runing_def readys_def s_waiting_def wq_def)
from this [OF h2] have "thread = hd (waiting_queue (schs s) cs)" .
with h2
have "(Cs cs, Th thread) \<in> (depend s)"
by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
with h1 show False by auto
qed
next
fix thread s a list
assume h1: "thread \<in> runing s"
and h2: "holding s thread cs"
and h3: "waiting_queue (schs s) cs = a # list"
and h4: "a \<notin> set list"
and h5: "distinct list"
thus "distinct
((\<lambda>(l, a, r). case a of [] \<Rightarrow> [] | m # ms \<Rightarrow> m # l @ ms @ r)
(lsp (cur_preced (schs s)) list))"
apply (cases "(lsp (cur_preced (schs s)) list)", simp)
apply (case_tac b, simp)
by (drule_tac lsp_set_eq, simp)
qed
qed
lemma block_pre:
fixes thread cs s
assumes s_ni: "thread \<notin> set (wq s cs)"
and s_i: "thread \<in> set (wq (e#s) cs)"
shows "e = P thread cs"
proof -
have ee: "\<And> x y. \<lbrakk>x = y\<rbrakk> \<Longrightarrow> set x = set y"
by auto
from s_ni s_i show ?thesis
proof (cases e, auto split:if_splits simp add:Let_def wq_def)
fix uu uub uuc uud uue
assume h: "(uuc, thread # uu, uub) = lsp (cur_preced (schs s)) uud"
and h1 [symmetric]: "uue # uud = waiting_queue (schs s) cs"
and h2: "thread \<notin> set (waiting_queue (schs s) cs)"
from lsp_set [OF h] have "set (uuc @ (thread # uu) @ uub) = set uud" .
hence "thread \<in> set uud" by auto
with h1 have "thread \<in> set (waiting_queue (schs s) cs)" by auto
with h2 show False by auto
next
fix uu uua uub uuc uud uue
assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
and h2: "uue # uud = waiting_queue (schs s) cs"
and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
and h4: "thread \<in> set uuc"
from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
with h4 have "thread \<in> set uud" by auto
with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
apply (drule_tac ee) by auto
with h1 show "False" by fast
next
fix uu uua uub uuc uud uue
assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
and h2: "uue # uud = waiting_queue (schs s) cs"
and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
and h4: "thread \<in> set uu"
from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
with h4 have "thread \<in> set uud" by auto
with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
apply (drule_tac ee) by auto
with h1 show "False" by fast
next
fix uu uua uub uuc uud uue
assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"
and h2: "uue # uud = waiting_queue (schs s) cs"
and h3: "(uuc, uua # uu, uub) = lsp (cur_preced (schs s)) uud"
and h4: "thread \<in> set uub"
from lsp_set [OF h3] have "set (uuc @ (uua # uu) @ uub) = set uud" .
with h4 have "thread \<in> set uud" by auto
with h2 have "thread \<in> set (waiting_queue (schs s) cs)"
apply (drule_tac ee) by auto
with h1 show "False" by fast
qed
qed
lemma p_pre: "\<lbrakk>vt step ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (depend s)^+"
apply (ind_cases "vt step ((P thread cs)#s)")
apply (ind_cases "step s (P thread cs)")
by auto
lemma abs1:
fixes e es
assumes ein: "e \<in> set es"
and neq: "hd es \<noteq> hd (es @ [x])"
shows "False"
proof -
from ein have "es \<noteq> []" by auto
then obtain e ess where "es = e # ess" by (cases es, auto)
with neq show ?thesis by auto
qed
lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
by (cases es, auto)
inductive_cases evt_cons: "vt cs (a#s)"
lemma abs2:
assumes vt: "vt step (e#s)"
and inq: "thread \<in> set (wq s cs)"
and nh: "thread = hd (wq s cs)"
and qt: "thread \<noteq> hd (wq (e#s) cs)"
and inq': "thread \<in> set (wq (e#s) cs)"
shows "False"
proof -
have ee: "\<And> uuc thread uu uub s list. (uuc, thread # uu, uub) = lsp (cur_preced (schs s)) list \<Longrightarrow>
lsp (cur_preced (schs s)) list = (uuc, thread # uu, uub)
" by simp
from prems show "False"
apply (cases e)
apply ((simp split:if_splits add:Let_def wq_def)[1])+
apply (insert abs1, fast)[1]
apply ((simp split:if_splits add:Let_def)[1])+
apply (simp split:if_splits list.splits add:Let_def wq_def)
apply (auto dest!:ee)
apply (drule_tac lsp_set_eq, simp)
apply (subgoal_tac "distinct (waiting_queue (schs s) cs)", simp, fold wq_def)
apply (rule_tac wq_distinct, auto)
apply (erule_tac evt_cons, auto)
apply (drule_tac lsp_set_eq, simp)
apply (subgoal_tac "distinct (wq s cs)", simp)
apply (rule_tac wq_distinct, auto)
apply (erule_tac evt_cons, auto)
apply (drule_tac lsp_set_eq, simp)
apply (subgoal_tac "distinct (wq s cs)", simp)
apply (rule_tac wq_distinct, auto)
apply (erule_tac evt_cons, auto)
apply (auto simp:wq_def Let_def split:if_splits prod.splits)
done
qed
lemma vt_moment: "\<And> t. \<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
proof(induct s, simp)
fix a s t
assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"
and vt_a: "vt cs (a # s)"
and le_t: "t \<le> length (a # s)"
show "vt cs (moment t (a # s))"
proof(cases "t = length (a#s)")
case True
from True have "moment t (a#s) = a#s" by simp
with vt_a show ?thesis by simp
next
case False
with le_t have le_t1: "t \<le> length s" by simp
from vt_a have "vt cs s"
by (erule_tac evt_cons, simp)
from h [OF this le_t1] have "vt cs (moment t s)" .
moreover have "moment t (a#s) = moment t s"
proof -
from moment_app [OF le_t1, of "[a]"]
show ?thesis by simp
qed
ultimately show ?thesis by auto
qed
qed
(* Wrong:
lemma \<lbrakk>thread \<in> set (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
*)
lemma waiting_unique_pre:
fixes cs1 cs2 s thread
assumes vt: "vt step s"
and h11: "thread \<in> set (wq s cs1)"
and h12: "thread \<noteq> hd (wq s cs1)"
assumes h21: "thread \<in> set (wq s cs2)"
and h22: "thread \<noteq> hd (wq s cs2)"
and neq12: "cs1 \<noteq> cs2"
shows "False"
proof -
let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
from h11 and h12 have q1: "?Q cs1 s" by simp
from h21 and h22 have q2: "?Q cs2 s" by simp
have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
from p_split [of "?Q cs1", OF q1 nq1]
obtain t1 where lt1: "t1 < length s"
and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
thread \<noteq> hd (wq (moment t1 s) cs1))"
and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
thread \<noteq> hd (wq (moment i' s) cs1))" by auto
from p_split [of "?Q cs2", OF q2 nq2]
obtain t2 where lt2: "t2 < length s"
and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
thread \<noteq> hd (wq (moment t2 s) cs2))"
and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
thread \<noteq> hd (wq (moment i' s) cs2))" by auto
show ?thesis
proof -
{
assume lt12: "t1 < t2"
let ?t3 = "Suc t2"
from lt2 have le_t3: "?t3 \<le> length s" by auto
from moment_plus [OF this]
obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
have "t2 < ?t3" by simp
from nn2 [rule_format, OF this] and eq_m
have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
have vt_e: "vt step (e#moment t2 s)"
proof -
from vt_moment [OF vt le_t3]
have "vt step (moment ?t3 s)" .
with eq_m show ?thesis by simp
qed
have ?thesis
proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
case True
from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
by auto
from abs2 [OF vt_e True eq_th h2 h1]
show ?thesis by auto
next
case False
from block_pre [OF False h1]
have "e = P thread cs2" .
with vt_e have "vt step ((P thread cs2)# moment t2 s)" by simp
from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
with runing_ready have "thread \<in> readys (moment t2 s)" by auto
with nn1 [rule_format, OF lt12]
show ?thesis by (simp add:readys_def s_waiting_def, auto)
qed
} moreover {
assume lt12: "t2 < t1"
let ?t3 = "Suc t1"
from lt1 have le_t3: "?t3 \<le> length s" by auto
from moment_plus [OF this]
obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
have lt_t3: "t1 < ?t3" by simp
from nn1 [rule_format, OF this] and eq_m
have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
have vt_e: "vt step (e#moment t1 s)"
proof -
from vt_moment [OF vt le_t3]
have "vt step (moment ?t3 s)" .
with eq_m show ?thesis by simp
qed
have ?thesis
proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
case True
from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
by auto
from abs2 [OF vt_e True eq_th h2 h1]
show ?thesis by auto
next
case False
from block_pre [OF False h1]
have "e = P thread cs1" .
with vt_e have "vt step ((P thread cs1)# moment t1 s)" by simp
from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
with runing_ready have "thread \<in> readys (moment t1 s)" by auto
with nn2 [rule_format, OF lt12]
show ?thesis by (simp add:readys_def s_waiting_def, auto)
qed
} moreover {
assume eqt12: "t1 = t2"
let ?t3 = "Suc t1"
from lt1 have le_t3: "?t3 \<le> length s" by auto
from moment_plus [OF this]
obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
have lt_t3: "t1 < ?t3" by simp
from nn1 [rule_format, OF this] and eq_m
have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
have vt_e: "vt step (e#moment t1 s)"
proof -
from vt_moment [OF vt le_t3]
have "vt step (moment ?t3 s)" .
with eq_m show ?thesis by simp
qed
have ?thesis
proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
case True
from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
by auto
from abs2 [OF vt_e True eq_th h2 h1]
show ?thesis by auto
next
case False
from block_pre [OF False h1]
have eq_e1: "e = P thread cs1" .
have lt_t3: "t1 < ?t3" by simp
with eqt12 have "t2 < ?t3" by simp
from nn2 [rule_format, OF this] and eq_m and eqt12
have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
show ?thesis
proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
case True
from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
by auto
from vt_e and eqt12 have "vt step (e#moment t2 s)" by simp
from abs2 [OF this True eq_th h2 h1]
show ?thesis .
next
case False
from block_pre [OF False h1]
have "e = P thread cs2" .
with eq_e1 neq12 show ?thesis by auto
qed
qed
} ultimately show ?thesis by arith
qed
qed
lemma waiting_unique:
assumes "vt step s"
and "waiting s th cs1"
and "waiting s th cs2"
shows "cs1 = cs2"
proof -
from waiting_unique_pre and prems
show ?thesis
by (auto simp add:s_waiting_def)
qed
lemma holded_unique:
assumes "vt step s"
and "holding s th1 cs"
and "holding s th2 cs"
shows "th1 = th2"
proof -
from prems show ?thesis
unfolding s_holding_def
by auto
qed
lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"
apply (induct s, auto)
by (case_tac a, auto split:if_splits)
lemma birthtime_unique:
"\<lbrakk>birthtime th1 s = birthtime th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
\<Longrightarrow> th1 = th2"
apply (induct s, auto)
by (case_tac a, auto split:if_splits dest:birthtime_lt)
lemma preced_unique :
assumes pcd_eq: "preced th1 s = preced th2 s"
and th_in1: "th1 \<in> threads s"
and th_in2: " th2 \<in> threads s"
shows "th1 = th2"
proof -
from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)
from birthtime_unique [OF this th_in1 th_in2]
show ?thesis .
qed
lemma preced_linorder:
assumes neq_12: "th1 \<noteq> th2"
and th_in1: "th1 \<in> threads s"
and th_in2: " th2 \<in> threads s"
shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
proof -
from preced_unique [OF _ th_in1 th_in2] and neq_12
have "preced th1 s \<noteq> preced th2 s" by auto
thus ?thesis by auto
qed
lemma unique_minus:
fixes x y z r
assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
and xy: "(x, y) \<in> r"
and xz: "(x, z) \<in> r^+"
and neq: "y \<noteq> z"
shows "(y, z) \<in> r^+"
proof -
from xz and neq show ?thesis
proof(induct)
case (base ya)
have "(x, ya) \<in> r" by fact
from unique [OF xy this] have "y = ya" .
with base show ?case by auto
next
case (step ya z)
show ?case
proof(cases "y = ya")
case True
from step True show ?thesis by simp
next
case False
from step False
show ?thesis by auto
qed
qed
qed
lemma unique_base:
fixes r x y z
assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
and xy: "(x, y) \<in> r"
and xz: "(x, z) \<in> r^+"
and neq_yz: "y \<noteq> z"
shows "(y, z) \<in> r^+"
proof -
from xz neq_yz show ?thesis
proof(induct)
case (base ya)
from xy unique base show ?case by auto
next
case (step ya z)
show ?case
proof(cases "y = ya")
case True
from True step show ?thesis by auto
next
case False
from False step
have "(y, ya) \<in> r\<^sup>+" by auto
with step show ?thesis by auto
qed
qed
qed
lemma unique_chain:
fixes r x y z
assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
and xy: "(x, y) \<in> r^+"
and xz: "(x, z) \<in> r^+"
and neq_yz: "y \<noteq> z"
shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
proof -
from xy xz neq_yz show ?thesis
proof(induct)
case (base y)
have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
next
case (step y za)
show ?case
proof(cases "y = z")
case True
from True step show ?thesis by auto
next
case False
from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
thus ?thesis
proof
assume "(z, y) \<in> r\<^sup>+"
with step have "(z, za) \<in> r\<^sup>+" by auto
thus ?thesis by auto
next
assume h: "(y, z) \<in> r\<^sup>+"
from step have yza: "(y, za) \<in> r" by simp
from step have "za \<noteq> z" by simp
from unique_minus [OF _ yza h this] and unique
have "(za, z) \<in> r\<^sup>+" by auto
thus ?thesis by auto
qed
qed
qed
qed
lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
apply (unfold s_depend_def s_waiting_def wq_def)
by (simp add:Let_def)
lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
apply (unfold s_depend_def s_waiting_def wq_def)
by (simp add:Let_def)
lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
apply (unfold s_depend_def s_waiting_def wq_def)
by (simp add:Let_def)
definition head_of :: "('a \<Rightarrow> 'b::linorder) \<Rightarrow> 'a set \<Rightarrow> 'a set"
where "head_of f A = {a . a \<in> A \<and> f a = Max (f ` A)}"
definition wq_head :: "state \<Rightarrow> cs \<Rightarrow> thread set"
where "wq_head s cs = head_of (cp s) (set (wq s cs))"
lemma f_nil_simp: "\<lbrakk>f cs = []\<rbrakk> \<Longrightarrow> f(cs:=[]) = f"
proof
fix x
assume h:"f cs = []"
show "(f(cs := [])) x = f x"
proof(cases "cs = x")
case True
with h show ?thesis by simp
next
case False
with h show ?thesis by simp
qed
qed
lemma step_back_vt: "vt ccs (e#s) \<Longrightarrow> vt ccs s"
by(ind_cases "vt ccs (e#s)", simp)
lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e"
by(ind_cases "vt ccs (e#s)", simp)
lemma holding_nil:
"\<lbrakk>wq s cs = []; holding (wq s) th cs\<rbrakk> \<Longrightarrow> False"
by (unfold cs_holding_def, auto)
lemma waiting_kept_1: "
\<lbrakk>vt step (V th cs#s); wq s cs = a # list; waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c;
lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
\<Longrightarrow> waiting (wq s) t c"
apply (drule_tac step_back_vt, drule_tac wq_distinct[of _ cs])
apply(drule_tac lsp_set_eq)
by (unfold cs_waiting_def, auto split:if_splits)
lemma waiting_kept_2:
"\<And>a list t c aa ca.
\<lbrakk>wq s cs = a # list; waiting ((wq s)(cs := [])) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
\<Longrightarrow> waiting (wq s) t c"
apply(drule_tac lsp_set_eq)
by (unfold cs_waiting_def, auto split:if_splits)
lemma holding_nil_simp: "\<lbrakk>holding ((wq s)(cs := [])) t c\<rbrakk> \<Longrightarrow> holding (wq s) t c"
by(unfold cs_holding_def, auto)
lemma step_wq_elim: "\<lbrakk>vt step (V th cs#s); wq s cs = a # list; a \<noteq> th\<rbrakk> \<Longrightarrow> False"
apply(drule_tac step_back_step)
apply(ind_cases "step s (V th cs)")
by(unfold s_holding_def, auto)
lemma holding_cs_neq_simp: "c \<noteq> cs \<Longrightarrow> holding ((wq s)(cs := u)) t c = holding (wq s) t c"
by (unfold cs_holding_def, auto)
lemma holding_th_neq_elim:
"\<And>a list c t aa ca ab lista.
\<lbrakk>\<not> holding (wq s) t c; holding ((wq s)(cs := ab # aa @ lista @ ca)) t c;
ab \<noteq> t\<rbrakk>
\<Longrightarrow> False"
by (unfold cs_holding_def, auto split:if_splits)
lemma holding_nil_abs:
"\<not> holding ((wq s)(cs := [])) th cs"
by (unfold cs_holding_def, auto split:if_splits)
lemma holding_abs: "\<lbrakk>holding ((wq s)(cs := ab # aa @ lista @ c)) th cs; ab \<noteq> th\<rbrakk>
\<Longrightarrow> False"
by (unfold cs_holding_def, auto split:if_splits)
lemma waiting_abs: "\<not> waiting ((wq s)(cs := t # l @ r)) t cs"
by (unfold cs_waiting_def, auto split:if_splits)
lemma waiting_abs_1:
"\<lbrakk>\<not> waiting ((wq s)(cs := [])) t c; waiting (wq s) t c; c \<noteq> cs\<rbrakk>
\<Longrightarrow> False"
by (unfold cs_waiting_def, auto split:if_splits)
lemma waiting_abs_2: "
\<lbrakk>\<not> waiting ((wq s)(cs := ab # aa @ lista @ ca)) t c; waiting (wq s) t c;
c \<noteq> cs\<rbrakk>
\<Longrightarrow> False"
by (unfold cs_waiting_def, auto split:if_splits)
lemma waiting_abs_3:
"\<lbrakk>wq s cs = a # list; \<not> waiting ((wq s)(cs := [])) t c;
waiting (wq s) t c; lsp (cp s) list = (aa, [], ca)\<rbrakk>
\<Longrightarrow> False"
apply(drule_tac lsp_mid_nil, simp)
by(unfold cs_waiting_def, auto split:if_splits)
lemma waiting_simp: "c \<noteq> cs \<Longrightarrow> waiting ((wq s)(cs:=z)) t c = waiting (wq s) t c"
by(unfold cs_waiting_def, auto split:if_splits)
lemma holding_cs_eq:
"\<lbrakk>\<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> c = cs"
by(unfold cs_holding_def, auto split:if_splits)
lemma holding_cs_eq_1:
"\<lbrakk>\<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c\<rbrakk>
\<Longrightarrow> c = cs"
by(unfold cs_holding_def, auto split:if_splits)
lemma holding_th_eq:
"\<lbrakk>vt step (V th cs#s); wq s cs = a # list; \<not> holding ((wq s)(cs := [])) t c; holding (wq s) t c;
lsp (cp s) list = (aa, [], ca)\<rbrakk>
\<Longrightarrow> t = th"
apply(drule_tac lsp_mid_nil, simp)
apply(unfold cs_holding_def, auto split:if_splits)
apply(drule_tac step_back_step)
apply(ind_cases "step s (V th cs)")
by (unfold s_holding_def, auto split:if_splits)
lemma holding_th_eq_1:
"\<lbrakk>vt step (V th cs#s);
wq s cs = a # list; \<not> holding ((wq s)(cs := ab # aa @ lista @ ca)) t c; holding (wq s) t c;
lsp (cp s) list = (aa, ab # lista, ca)\<rbrakk>
\<Longrightarrow> t = th"
apply(drule_tac step_back_step)
apply(ind_cases "step s (V th cs)")
apply(unfold s_holding_def cs_holding_def)
by (auto split:if_splits)
lemma holding_th_eq_2: "\<lbrakk>holding ((wq s)(cs := ac # x)) th cs\<rbrakk>
\<Longrightarrow> ac = th"
by (unfold cs_holding_def, auto)
lemma holding_th_eq_3: "
\<lbrakk>\<not> holding (wq s) t c;
holding ((wq s)(cs := ac # x)) t c\<rbrakk>
\<Longrightarrow> ac = t"
by (unfold cs_holding_def, auto)
lemma holding_wq_eq: "holding ((wq s)(cs := th' # l @ r)) th' cs"
by (unfold cs_holding_def, auto)
lemma waiting_th_eq: "
\<lbrakk>waiting (wq s) t c; wq s cs = a # list;
lsp (cp s) list = (aa, ac # lista, ba); \<not> waiting ((wq s)(cs := ac # aa @ lista @ ba)) t c\<rbrakk>
\<Longrightarrow> ac = t"
apply(drule_tac lsp_set_eq, simp)
by (unfold cs_waiting_def, auto split:if_splits)
lemma step_depend_v:
"vt step (V th cs#s) \<Longrightarrow>
depend (V th cs # s) =
depend s - {(Cs cs, Th th)} -
{(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
{(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
apply (unfold s_depend_def wq_def,
auto split:list.splits simp:Let_def f_nil_simp holding_wq_eq, fold wq_def cp_def)
apply (auto split:list.splits prod.splits
simp:Let_def f_nil_simp holding_nil_simp holding_cs_neq_simp holding_nil_abs
waiting_abs waiting_simp holding_wq_eq
elim:holding_nil waiting_kept_1 waiting_kept_2 step_wq_elim holding_th_neq_elim
holding_abs waiting_abs_1 waiting_abs_3 holding_cs_eq holding_cs_eq_1
holding_th_eq holding_th_eq_1 holding_th_eq_2 holding_th_eq_3 waiting_th_eq
dest:lsp_mid_length)
done
lemma step_depend_p:
"vt step (P th cs#s) \<Longrightarrow>
depend (P th cs # s) = (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
else depend s \<union> {(Th th, Cs cs)})"
apply(unfold s_depend_def wq_def)
apply (auto split:list.splits prod.splits simp:Let_def cs_waiting_def cs_holding_def)
apply(case_tac "c = cs", auto)
apply(fold wq_def)
apply(drule_tac step_back_step)
by (ind_cases " step s (P (hd (wq s cs)) cs)",
auto simp:s_depend_def wq_def cs_holding_def)
lemma simple_A:
fixes A
assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"
shows "A = {} \<or> (\<exists> a. A = {a})"
proof(cases "A = {}")
case True thus ?thesis by simp
next
case False then obtain a where "a \<in> A" by auto
with h have "A = {a}" by auto
thus ?thesis by simp
qed
lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
by (unfold s_depend_def, auto)
lemma acyclic_depend:
fixes s
assumes vt: "vt step s"
shows "acyclic (depend s)"
proof -
from vt show ?thesis
proof(induct)
case (vt_cons s e)
assume ih: "acyclic (depend s)"
and stp: "step s e"
and vt: "vt step s"
show ?case
proof(cases e)
case (Create th prio)
with ih
show ?thesis by (simp add:depend_create_unchanged)
next
case (Exit th)
with ih show ?thesis by (simp add:depend_exit_unchanged)
next
case (V th cs)
from V vt stp have vtt: "vt step (V th cs#s)" by auto
from step_depend_v [OF this]
have eq_de: "depend (e # s) =
depend s - {(Cs cs, Th th)} -
{(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
{(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
thus ?thesis
proof(cases "wq s cs")
case Nil
hence "?D = {}" by simp
with ac and eq_de show ?thesis by simp
next
case (Cons tth rest)
from stp and V have "step s (V th cs)" by simp
hence eq_wq: "wq s cs = th # rest"
proof -
show "step s (V th cs) \<Longrightarrow> wq s cs = th # rest"
apply(ind_cases "step s (V th cs)")
by(insert Cons, unfold s_holding_def, simp)
qed
show ?thesis
proof(cases "lsp (cp s) rest")
fix l b r
assume eq_lsp: "lsp (cp s) rest = (l, b, r) "
show ?thesis
proof(cases "b")
case Nil
with eq_lsp and eq_wq have "?D = {}" by simp
with ac and eq_de show ?thesis by simp
next
case (Cons th' m)
with eq_lsp
have eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
apply simp
by (drule_tac lsp_mid_length, simp)
from eq_wq and eq_lsp
have eq_D: "?D = {(Cs cs, Th th')}" by auto
from eq_wq and eq_lsp
have eq_C: "?C = {(Th th', Cs cs)}" by auto
let ?E = "(?A - ?B - ?C)"
have "(Th th', Cs cs) \<notin> ?E\<^sup>*"
proof
assume "(Th th', Cs cs) \<in> ?E\<^sup>*"
hence " (Th th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
from tranclD [OF this]
obtain x where th'_e: "(Th th', x) \<in> ?E" by blast
hence th_d: "(Th th', x) \<in> ?A" by simp
from depend_target_th [OF this]
obtain cs' where eq_x: "x = Cs cs'" by auto
with th_d have "(Th th', Cs cs') \<in> ?A" by simp
hence wt_th': "waiting s th' cs'"
unfolding s_depend_def s_waiting_def cs_waiting_def by simp
hence "cs' = cs"
proof(rule waiting_unique [OF vt])
from eq_wq eq_lsp wq_distinct[OF vt, of cs]
show "waiting s th' cs" by(unfold s_waiting_def, auto dest:lsp_set_eq)
qed
with th'_e eq_x have "(Th th', Cs cs) \<in> ?E" by simp
with eq_C show "False" by simp
qed
with acyclic_insert[symmetric] and ac and eq_D
and eq_de show ?thesis by simp
qed
qed
qed
next
case (P th cs)
from P vt stp have vtt: "vt step (P th cs#s)" by auto
from step_depend_p [OF this] P
have "depend (e # s) =
(if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
by simp
moreover have "acyclic ?R"
proof(cases "wq s cs = []")
case True
hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
proof
assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
from tranclD2 [OF this]
obtain x where "(x, Cs cs) \<in> depend s" by auto
with True show False by (auto simp:s_depend_def cs_waiting_def)
qed
with acyclic_insert ih eq_r show ?thesis by auto
next
case False
hence eq_r: "?R = depend s \<union> {(Th th, Cs cs)}" by simp
have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
proof
assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
hence "(Cs cs, Th th) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
moreover from step_back_step [OF vtt] have "step s (P th cs)" .
ultimately show False
proof -
show " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
by (ind_cases "step s (P th cs)", simp)
qed
qed
with acyclic_insert ih eq_r show ?thesis by auto
qed
ultimately show ?thesis by simp
next
case (Set thread prio)
with ih
thm depend_set_unchanged
show ?thesis by (simp add:depend_set_unchanged)
qed
next
case vt_nil
show "acyclic (depend ([]::state))"
by (auto simp: s_depend_def cs_waiting_def
cs_holding_def wq_def acyclic_def)
qed
qed
lemma finite_depend:
fixes s
assumes vt: "vt step s"
shows "finite (depend s)"
proof -
from vt show ?thesis
proof(induct)
case (vt_cons s e)
assume ih: "finite (depend s)"
and stp: "step s e"
and vt: "vt step s"
show ?case
proof(cases e)
case (Create th prio)
with ih
show ?thesis by (simp add:depend_create_unchanged)
next
case (Exit th)
with ih show ?thesis by (simp add:depend_exit_unchanged)
next
case (V th cs)
from V vt stp have vtt: "vt step (V th cs#s)" by auto
from step_depend_v [OF this]
have eq_de: "depend (e # s) =
depend s - {(Cs cs, Th th)} -
{(Th th', Cs cs) |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))} \<union>
{(Cs cs, Th th') |th'. \<exists>rest. wq s cs = th # rest \<and> (\<exists>l r. lsp (cp s) rest = (l, [th'], r))}"
(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
moreover have "finite ?D"
proof -
have "?D = {} \<or> (\<exists> a. ?D = {a})" by (rule simple_A, auto)
thus ?thesis
proof
assume h: "?D = {}"
show ?thesis by (unfold h, simp)
next
assume "\<exists> a. ?D = {a}"
thus ?thesis by auto
qed
qed
ultimately show ?thesis by simp
next
case (P th cs)
from P vt stp have vtt: "vt step (P th cs#s)" by auto
from step_depend_p [OF this] P
have "depend (e # s) =
(if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
by simp
moreover have "finite ?R"
proof(cases "wq s cs = []")
case True
hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
with True and ih show ?thesis by auto
next
case False
hence "?R = depend s \<union> {(Th th, Cs cs)}" by simp
with False and ih show ?thesis by auto
qed
ultimately show ?thesis by auto
next
case (Set thread prio)
with ih
show ?thesis by (simp add:depend_set_unchanged)
qed
next
case vt_nil
show "finite (depend ([]::state))"
by (auto simp: s_depend_def cs_waiting_def
cs_holding_def wq_def acyclic_def)
qed
qed
text {* Several useful lemmas *}
thm wf_trancl
thm finite_acyclic_wf
thm finite_acyclic_wf_converse
thm wf_induct
lemma wf_dep_converse:
fixes s
assumes vt: "vt step s"
shows "wf ((depend s)^-1)"
proof(rule finite_acyclic_wf_converse)
from finite_depend [OF vt]
show "finite (depend s)" .
next
from acyclic_depend[OF vt]
show "acyclic (depend s)" .
qed
lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
by (induct l, auto)
lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
lemma wq_threads:
fixes s cs
assumes vt: "vt step s"
and h: "th \<in> set (wq s cs)"
shows "th \<in> threads s"
proof -
from vt and h show ?thesis
proof(induct arbitrary: th cs)
case (vt_cons s e)
assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
and stp: "step s e"
and vt: "vt step s"
and h: "th \<in> set (wq (e # s) cs)"
show ?case
proof(cases e)
case (Create th' prio)
with ih h show ?thesis
by (auto simp:wq_def Let_def)
next
case (Exit th')
with stp ih h show ?thesis
apply (auto simp:wq_def Let_def)
apply (ind_cases "step s (Exit th')")
apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
s_depend_def s_holding_def cs_holding_def)
by (fold wq_def, auto)
next
case (V th' cs')
show ?thesis
proof(cases "cs' = cs")
case False
with h
show ?thesis
apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
by (drule_tac ih, simp)
next
case True
from h
show ?thesis
proof(unfold V wq_def)
assume th_in: "th \<in> set (waiting_queue (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
show "th \<in> threads (V th' cs' # s)"
proof(cases "cs = cs'")
case False
hence "?l = waiting_queue (schs s) cs" by (simp add:Let_def)
with th_in have " th \<in> set (wq s cs)"
by (fold wq_def, simp)
from ih [OF this] show ?thesis by simp
next
case True
show ?thesis
proof(cases "waiting_queue (schs s) cs'")
case Nil
with h V show ?thesis
apply (auto simp:wq_def Let_def split:if_splits)
by (fold wq_def, drule_tac ih, simp)
next
case (Cons a rest)
assume eq_wq: "waiting_queue (schs s) cs' = a # rest"
with h V show ?thesis
proof(cases "(lsp (cur_preced (schs s)) rest)", unfold V)
fix l m r
assume eq_lsp: "lsp (cur_preced (schs s)) rest = (l, m, r)"
and eq_wq: "waiting_queue (schs s) cs' = a # rest"
and th_in_set: "th \<in> set (wq (V th' cs' # s) cs)"
show ?thesis
proof(cases "m")
case Nil
with eq_lsp have "rest = []" using lsp_mid_nil by auto
with eq_wq have "waiting_queue (schs s) cs' = [a]" by simp
with h[unfolded V wq_def] True
show ?thesis
by (simp add:Let_def)
next
case (Cons b rb)
with lsp_mid_length[OF eq_lsp] have eq_m: "m = [b]" by auto
with eq_lsp have "lsp (cur_preced (schs s)) rest = (l, [b], r)" by simp
with h[unfolded V wq_def] True lsp_set_eq [OF this] eq_wq
show ?thesis
apply (auto simp:Let_def, fold wq_def)
by (rule_tac ih [of _ cs'], auto)+
qed
qed
qed
qed
qed
qed
next
case (P th' cs')
from h stp
show ?thesis
apply (unfold P wq_def)
apply (auto simp:Let_def split:if_splits, fold wq_def)
apply (auto intro:ih)
apply(ind_cases "step s (P th' cs')")
by (unfold runing_def readys_def, auto)
next
case (Set thread prio)
with ih h show ?thesis
by (auto simp:wq_def Let_def)
qed
next
case vt_nil
thus ?case by (auto simp:wq_def)
qed
qed
lemma range_in: "\<lbrakk>vt step s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
apply(unfold s_depend_def cs_waiting_def cs_holding_def)
by (auto intro:wq_threads)
lemma readys_v_eq:
fixes th thread cs rest
assumes neq_th: "th \<noteq> thread"
and eq_wq: "wq s cs = thread#rest"
and not_in: "th \<notin> set rest"
shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
proof -
from prems show ?thesis
apply (auto simp:readys_def)
apply (case_tac "cs = csa", simp add:s_waiting_def)
apply (erule_tac x = csa in allE)
apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
apply (case_tac "csa = cs", simp)
apply (erule_tac x = cs in allE)
by (auto simp:s_waiting_def wq_def Let_def split:list.splits prod.splits
dest:lsp_set_eq)
qed
lemma readys_v_eq_1:
fixes th thread cs rest
assumes neq_th: "th \<noteq> thread"
and eq_wq: "wq s cs = thread#rest"
and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
and neq_th': "th \<noteq> th'"
shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
proof -
from prems show ?thesis
apply (auto simp:readys_def)
apply (case_tac "cs = csa", simp add:s_waiting_def)
apply (erule_tac x = cs in allE)
apply (simp add:s_waiting_def wq_def Let_def split:prod.splits list.splits)
apply (drule_tac lsp_mid_nil,simp, clarify, fold cp_def, clarsimp)
apply (frule_tac lsp_set_eq, simp)
apply (erule_tac x = csa in allE)
apply (subst (asm) (2) s_waiting_def, unfold wq_def)
apply (auto simp:Let_def split:list.splits prod.splits if_splits
dest:lsp_set_eq)
apply (unfold s_waiting_def)
apply (fold wq_def, clarsimp)
apply (clarsimp)+
apply (case_tac "csa = cs", simp)
apply (erule_tac x = cs in allE, simp)
apply (unfold wq_def)
by (auto simp:Let_def split:list.splits prod.splits if_splits
dest:lsp_set_eq)
qed
lemma readys_v_eq_2:
fixes th thread cs rest
assumes neq_th: "th \<noteq> thread"
and eq_wq: "wq s cs = thread#rest"
and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
and neq_th': "th = th'"
and vt: "vt step s"
shows "(th \<in> readys (V thread cs#s))"
proof -
from prems show ?thesis
apply (auto simp:readys_def)
apply (rule_tac wq_threads [of s _ cs], auto dest:lsp_set_eq)
apply (unfold s_waiting_def wq_def)
apply (auto simp:Let_def split:list.splits prod.splits if_splits
dest:lsp_set_eq lsp_mid_nil lsp_mid_length)
apply (fold cp_def, simp+, clarsimp)
apply (frule_tac lsp_set_eq, simp)
apply (fold wq_def)
apply (subgoal_tac "csa = cs", simp)
apply (rule_tac waiting_unique [of s th'], simp)
by (auto simp:s_waiting_def)
qed
lemma chain_building:
assumes vt: "vt step s"
shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
proof -
from wf_dep_converse [OF vt]
have h: "wf ((depend s)\<inverse>)" .
show ?thesis
proof(induct rule:wf_induct [OF h])
fix x
assume ih [rule_format]:
"\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow>
y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
proof
assume x_d: "x \<in> Domain (depend s)"
show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
proof(cases x)
case (Th th)
from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
hence "Cs cs \<in> Domain (depend s)" by auto
from ih [OF x_in_r this] obtain th'
where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto
have "(x, Th th') \<in> (depend s)\<^sup>+" using Th x_in cs_in by auto
with th'_ready show ?thesis by auto
next
case (Cs cs)
from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (depend s)^-1" by (auto simp:s_depend_def)
show ?thesis
proof(cases "th' \<in> readys s")
case True
from True and th'_d show ?thesis by auto
next
case False
from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto
with False have "Th th' \<in> Domain (depend s)"
by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
from ih [OF th'_d this]
obtain th'' where
th''_r: "th'' \<in> readys s" and
th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto
from th'_d and th''_in
have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
with th''_r show ?thesis by auto
qed
qed
qed
qed
qed
lemma th_chain_to_ready:
fixes s th
assumes vt: "vt step s"
and th_in: "th \<in> threads s"
shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"
proof(cases "th \<in> readys s")
case True
thus ?thesis by auto
next
case False
from False and th_in have "Th th \<in> Domain (depend s)"
by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
from chain_building [rule_format, OF vt this]
show ?thesis by auto
qed
lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
by (unfold s_waiting_def cs_waiting_def, auto)
lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
by (unfold s_holding_def cs_holding_def, simp)
lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
by (unfold s_holding_def cs_holding_def, auto)
lemma unique_depend: "\<lbrakk>vt step s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
by(auto elim:waiting_unique holding_unique)
lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
by (induct rule:trancl_induct, auto)
lemma dchain_unique:
assumes vt: "vt step s"
and th1_d: "(n, Th th1) \<in> (depend s)^+"
and th1_r: "th1 \<in> readys s"
and th2_d: "(n, Th th2) \<in> (depend s)^+"
and th2_r: "th2 \<in> readys s"
shows "th1 = th2"
proof -
{ assume neq: "th1 \<noteq> th2"
hence "Th th1 \<noteq> Th th2" by simp
from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]
have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
hence "False"
proof
assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
from trancl_split [OF this]
obtain n where dd: "(Th th1, n) \<in> depend s" by auto
then obtain cs where eq_n: "n = Cs cs"
by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
from dd eq_n have "th1 \<notin> readys s"
by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
with th1_r show ?thesis by auto
next
assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
from trancl_split [OF this]
obtain n where dd: "(Th th2, n) \<in> depend s" by auto
then obtain cs where eq_n: "n = Cs cs"
by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
from dd eq_n have "th2 \<notin> readys s"
by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)
with th2_r show ?thesis by auto
qed
} thus ?thesis by auto
qed
definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
where "count Q l = length (filter Q l)"
definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
lemma step_holdents_p_add:
fixes th cs s
assumes vt: "vt step (P th cs#s)"
and "wq s cs = []"
shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
proof -
from prems show ?thesis
unfolding holdents_def step_depend_p[OF vt] by auto
qed
lemma step_holdents_p_eq:
fixes th cs s
assumes vt: "vt step (P th cs#s)"
and "wq s cs \<noteq> []"
shows "holdents (P th cs#s) th = holdents s th"
proof -
from prems show ?thesis
unfolding holdents_def step_depend_p[OF vt] by auto
qed
lemma step_holdents_v_minus:
fixes th cs s
assumes vt: "vt step (V th cs#s)"
shows "holdents (V th cs#s) th = holdents s th - {cs}"
proof -
{ fix rest l r
assume eq_wq: "wq s cs = th # rest"
and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
have "False"
proof -
from lsp_set_eq [OF eq_lsp] have " rest = l @ [th] @ r" .
with eq_wq have "wq s cs = th#\<dots>" by simp
with wq_distinct [OF step_back_vt[OF vt], of cs]
show ?thesis by auto
qed
} thus ?thesis unfolding holdents_def step_depend_v[OF vt] by auto
qed
lemma step_holdents_v_add:
fixes th th' cs s rest l r
assumes vt: "vt step (V th' cs#s)"
and neq_th: "th \<noteq> th'"
and eq_wq: "wq s cs = th' # rest"
and eq_lsp: "lsp (cp s) rest = (l, [th], r)"
shows "holdents (V th' cs#s) th = holdents s th \<union> {cs}"
proof -
from prems show ?thesis
unfolding holdents_def step_depend_v[OF vt] by auto
qed
lemma step_holdents_v_eq:
fixes th th' cs s rest l r th''
assumes vt: "vt step (V th' cs#s)"
and neq_th: "th \<noteq> th'"
and eq_wq: "wq s cs = th' # rest"
and eq_lsp: "lsp (cp s) rest = (l, [th''], r)"
and neq_th': "th \<noteq> th''"
shows "holdents (V th' cs#s) th = holdents s th"
proof -
from prems show ?thesis
unfolding holdents_def step_depend_v[OF vt] by auto
qed
definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
where "cntCS s th = card (holdents s th)"
lemma cntCS_v_eq:
fixes th thread cs rest
assumes neq_th: "th \<noteq> thread"
and eq_wq: "wq s cs = thread#rest"
and not_in: "th \<notin> set rest"
and vtv: "vt step (V thread cs#s)"
shows "cntCS (V thread cs#s) th = cntCS s th"
proof -
from prems show ?thesis
apply (unfold cntCS_def holdents_def step_depend_v)
apply auto
apply (subgoal_tac "\<not> (\<exists>l r. lsp (cp s) rest = (l, [th], r))", auto)
by (drule_tac lsp_set_eq, auto)
qed
lemma cntCS_v_eq_1:
fixes th thread cs rest
assumes neq_th: "th \<noteq> thread"
and eq_wq: "wq s cs = thread#rest"
and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
and neq_th': "th \<noteq> th'"
and vtv: "vt step (V thread cs#s)"
shows "cntCS (V thread cs#s) th = cntCS s th"
proof -
from prems show ?thesis
apply (unfold cntCS_def holdents_def step_depend_v)
by auto
qed
fun the_cs :: "node \<Rightarrow> cs"
where "the_cs (Cs cs) = cs"
lemma cntCS_v_eq_2:
fixes th thread cs rest
assumes neq_th: "th \<noteq> thread"
and eq_wq: "wq s cs = thread#rest"
and eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
and neq_th': "th = th'"
and vtv: "vt step (V thread cs#s)"
shows "cntCS (V thread cs#s) th = 1 + cntCS s th"
proof -
have "card {csa. csa = cs \<or> (Cs csa, Th th') \<in> depend s} =
Suc (card {cs. (Cs cs, Th th') \<in> depend s})"
(is "card ?A = Suc (card ?B)")
proof -
have h: "?A = insert cs ?B" by auto
moreover have h1: "?B = ?B - {cs}"
proof -
{ assume "(Cs cs, Th th') \<in> depend s"
moreover have "(Th th', Cs cs) \<in> depend s"
proof -
from wq_distinct [OF step_back_vt[OF vtv], of cs]
eq_wq lsp_set_eq [OF eq_lsp] show ?thesis
apply (auto simp:s_depend_def)
by (unfold cs_waiting_def, auto)
qed
moreover note acyclic_depend [OF step_back_vt[OF vtv]]
ultimately have "False"
apply (auto simp:acyclic_def)
apply (erule_tac x="Cs cs" in allE)
apply (subgoal_tac "(Cs cs, Cs cs) \<in> (depend s)\<^sup>+", simp)
by (rule_tac trancl_into_trancl [where b = "Th th'"], auto)
} thus ?thesis by auto
qed
moreover have "card (insert cs ?B) = Suc (card (?B - {cs}))"
proof(rule card_insert)
from finite_depend [OF step_back_vt [OF vtv]]
have fnt: "finite (depend s)" .
show " finite {cs. (Cs cs, Th th') \<in> depend s}" (is "finite ?B")
proof -
have "?B \<subseteq> (\<lambda> (a, b). the_cs a) ` (depend s)"
apply (auto simp:image_def)
by (rule_tac x = "(Cs x, Th th')" in bexI, auto)
with fnt show ?thesis by (auto intro:finite_subset)
qed
qed
ultimately show ?thesis by simp
qed
with prems show ?thesis
apply (unfold cntCS_def holdents_def step_depend_v[OF vtv])
by auto
qed
lemma finite_holding:
fixes s th cs
assumes vt: "vt step s"
shows "finite (holdents s th)"
proof -
let ?F = "\<lambda> (x, y). the_cs x"
from finite_depend [OF vt]
have "finite (depend s)" .
hence "finite (?F `(depend s))" by simp
moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>"
proof -
{ have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
fix x assume "(Cs x, Th th) \<in> depend s"
hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
moreover have "?F (Cs x, Th th) = x" by simp
ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp
} thus ?thesis by auto
qed
ultimately show ?thesis by (unfold holdents_def, auto intro:finite_subset)
qed
inductive_cases case_step_v: "step s (V thread cs)"
lemma cntCS_v_dec:
fixes s thread cs
assumes vtv: "vt step (V thread cs#s)"
shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
proof -
have cs_in: "cs \<in> holdents s thread" using step_back_step[OF vtv]
apply (erule_tac case_step_v)
apply (unfold holdents_def s_depend_def, simp)
by (unfold cs_holding_def s_holding_def, auto)
moreover have cs_not_in:
"(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
by (unfold holdents_def, unfold step_depend_v[OF vtv],
auto dest:lsp_set_eq)
ultimately
have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
by auto
moreover have "card \<dots> =
Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
proof(rule card_insert)
from finite_holding [OF vtv]
show " finite (holdents (V thread cs # s) thread)" .
qed
moreover from cs_not_in
have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
ultimately show ?thesis by (simp add:cntCS_def)
qed
lemma cnp_cnv_cncs:
fixes s th
assumes vt: "vt step s"
shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
then cntCS s th else cntCS s th + 1)"
proof -
from vt show ?thesis
proof(induct arbitrary:th)
case (vt_cons s e)
assume vt: "vt step s"
and ih: "\<And>th. cntP s th = cntV s th +
(if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
and stp: "step s e"
from stp show ?case
proof(cases)
case (thread_create prio max_prio thread)
assume eq_e: "e = Create thread prio"
and not_in: "thread \<notin> threads s"
show ?thesis
proof -
{ fix cs
assume "thread \<in> set (wq s cs)"
from wq_threads [OF vt this] have "thread \<in> threads s" .
with not_in have "False" by simp
} with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
by (auto simp:readys_def threads.simps s_waiting_def
wq_def cs_waiting_def Let_def)
from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_def
by (simp add:depend_create_unchanged eq_e)
{ assume "th \<noteq> thread"
with eq_readys eq_e
have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
(th \<in> readys (s) \<or> th \<notin> threads (s))"
by (simp add:threads.simps)
with eq_cnp eq_cnv eq_cncs ih not_in
have ?thesis by simp
} moreover {
assume eq_th: "th = thread"
with not_in ih have " cntP s th = cntV s th + cntCS s th" by simp
moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
moreover note eq_cnp eq_cnv eq_cncs
ultimately have ?thesis by auto
} ultimately show ?thesis by blast
qed
next
case (thread_exit thread)
assume eq_e: "e = Exit thread"
and is_runing: "thread \<in> runing s"
and no_hold: "holdents s thread = {}"
from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_def
by (simp add:depend_exit_unchanged eq_e)
{ assume "th \<noteq> thread"
with eq_e
have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
(th \<in> readys (s) \<or> th \<notin> threads (s))"
apply (simp add:threads.simps readys_def)
apply (subst s_waiting_def)
apply (subst (1 2) wq_def)
apply (simp add:Let_def)
apply (subst s_waiting_def, simp)
by (fold wq_def, simp)
with eq_cnp eq_cnv eq_cncs ih
have ?thesis by simp
} moreover {
assume eq_th: "th = thread"
with ih is_runing have " cntP s th = cntV s th + cntCS s th"
by (simp add:runing_def)
moreover from eq_th eq_e have "th \<notin> threads (e#s)"
by simp
moreover note eq_cnp eq_cnv eq_cncs
ultimately have ?thesis by auto
} ultimately show ?thesis by blast
next
case (thread_P thread cs)
assume eq_e: "e = P thread cs"
and is_runing: "thread \<in> runing s"
and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
from prems have vtp: "vt step (P thread cs#s)" by auto
show ?thesis
proof -
{ have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
assume neq_th: "th \<noteq> thread"
with eq_e
have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
apply (simp add:readys_def s_waiting_def wq_def Let_def)
apply (rule_tac hh, clarify)
apply (intro iffI allI, clarify)
apply (erule_tac x = csa in allE, auto)
apply (subgoal_tac "waiting_queue (schs s) cs \<noteq> []", auto)
apply (erule_tac x = cs in allE, auto)
by (case_tac "(waiting_queue (schs s) cs)", auto)
moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
apply (simp add:cntCS_def holdents_def)
by (unfold step_depend_p [OF vtp], auto)
moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
by (simp add:cntP_def count_def)
moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
by (simp add:cntV_def count_def)
moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
moreover note ih [of th]
ultimately have ?thesis by simp
} moreover {
assume eq_th: "th = thread"
have ?thesis
proof -
from eq_e eq_th have eq_cnp: "cntP (e # s) th = 1 + (cntP s th)"
by (simp add:cntP_def count_def)
from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
by (simp add:cntV_def count_def)
show ?thesis
proof (cases "wq s cs = []")
case True
with is_runing
have "th \<in> readys (e#s)"
apply (unfold eq_e wq_def, unfold readys_def s_depend_def)
apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
moreover have "cntCS (e # s) th = 1 + cntCS s th"
proof -
have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> depend s} =
Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")
proof -
have "?L = insert cs ?R" by auto
moreover have "card \<dots> = Suc (card (?R - {cs}))"
proof(rule card_insert)
from finite_holding [OF vt, of thread]
show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
by (unfold holdents_def, simp)
qed
moreover have "?R - {cs} = ?R"
proof -
have "cs \<notin> ?R"
proof
assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend s}"
with no_dep show False by auto
qed
thus ?thesis by auto
qed
ultimately show ?thesis by auto
qed
thus ?thesis
apply (unfold eq_e eq_th cntCS_def)
apply (simp add: holdents_def)
by (unfold step_depend_p [OF vtp], auto simp:True)
qed
moreover from is_runing have "th \<in> readys s"
by (simp add:runing_def eq_th)
moreover note eq_cnp eq_cnv ih [of th]
ultimately show ?thesis by auto
next
case False
have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
by (unfold eq_th eq_e wq_def, auto simp:Let_def)
have "th \<notin> readys (e#s)"
proof
assume "th \<in> readys (e#s)"
hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
by (simp add:s_waiting_def)
moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
ultimately have "th = hd (wq (e#s) cs)" by blast
with eq_wq have "th = hd (wq s cs @ [th])" by simp
hence "th = hd (wq s cs)" using False by auto
with False eq_wq wq_distinct [OF vtp, of cs]
show False by (fold eq_e, auto)
qed
moreover from is_runing have "th \<in> threads (e#s)"
by (unfold eq_e, auto simp:runing_def readys_def eq_th)
moreover have "cntCS (e # s) th = cntCS s th"
apply (unfold cntCS_def holdents_def eq_e step_depend_p[OF vtp])
by (auto simp:False)
moreover note eq_cnp eq_cnv ih[of th]
moreover from is_runing have "th \<in> readys s"
by (simp add:runing_def eq_th)
ultimately show ?thesis by auto
qed
qed
} ultimately show ?thesis by blast
qed
next
case (thread_V thread cs)
from prems have vtv: "vt step (V thread cs # s)" by auto
assume eq_e: "e = V thread cs"
and is_runing: "thread \<in> runing s"
and hold: "holding s thread cs"
from hold obtain rest
where eq_wq: "wq s cs = thread # rest"
by (case_tac "wq s cs", auto simp:s_holding_def)
have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
show ?thesis
proof -
{ assume eq_th: "th = thread"
from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
by (unfold eq_e, simp add:cntP_def count_def)
moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
by (unfold eq_e, simp add:cntV_def count_def)
moreover from cntCS_v_dec [OF vtv]
have "cntCS (e # s) thread + 1 = cntCS s thread"
by (simp add:eq_e)
moreover from is_runing have rd_before: "thread \<in> readys s"
by (unfold runing_def, simp)
moreover have "thread \<in> readys (e # s)"
proof -
from is_runing
have "thread \<in> threads (e#s)"
by (unfold eq_e, auto simp:runing_def readys_def)
moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
proof
fix cs1
{ assume eq_cs: "cs1 = cs"
have "\<not> waiting (e # s) thread cs1"
proof -
have "thread \<notin> set (wq (e#s) cs1)"
proof(cases "lsp (cp s) rest")
fix l m r
assume h: "lsp (cp s) rest = (l, m, r)"
show ?thesis
proof(cases "m")
case Nil
from wq_v_eq_nil [OF eq_wq] h Nil eq_e
have " wq (e # s) cs = []" by auto
thus ?thesis using eq_cs by auto
next
case (Cons th' l')
from lsp_mid_length [OF h] and Cons h
have eqh: "lsp (cp s) rest = (l, [th'], r)" by auto
from wq_v_eq [OF eq_wq this]
have "wq (V thread cs # s) cs = th' # l @ r" .
moreover from lsp_set_eq [OF eqh]
have "set rest = set \<dots>" by auto
moreover have "thread \<notin> set rest"
proof -
from wq_distinct [OF step_back_vt[OF vtv], of cs]
and eq_wq show ?thesis by auto
qed
moreover note eq_e eq_cs
ultimately show ?thesis by simp
qed
qed
thus ?thesis by (simp add:s_waiting_def)
qed
} moreover {
assume neq_cs: "cs1 \<noteq> cs"
have "\<not> waiting (e # s) thread cs1"
proof -
from wq_v_neq [OF neq_cs[symmetric]]
have "wq (V thread cs # s) cs1 = wq s cs1" .
moreover have "\<not> waiting s thread cs1"
proof -
from runing_ready and is_runing
have "thread \<in> readys s" by auto
thus ?thesis by (simp add:readys_def)
qed
ultimately show ?thesis
by (auto simp:s_waiting_def eq_e)
qed
} ultimately show "\<not> waiting (e # s) thread cs1" by blast
qed
ultimately show ?thesis by (simp add:readys_def)
qed
moreover note eq_th ih
ultimately have ?thesis by auto
} moreover {
assume neq_th: "th \<noteq> thread"
from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th"
by (simp add:cntP_def count_def)
from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th"
by (simp add:cntV_def count_def)
have ?thesis
proof(cases "th \<in> set rest")
case False
have "(th \<in> readys (e # s)) = (th \<in> readys s)"
by(unfold eq_e, rule readys_v_eq [OF neq_th eq_wq False])
moreover have "cntCS (e#s) th = cntCS s th"
by(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq False vtv])
moreover note ih eq_cnp eq_cnv eq_threads
ultimately show ?thesis by auto
next
case True
obtain l m r where eq_lsp: "lsp (cp s) rest = (l, m, r)"
by (cases "lsp (cp s) rest", auto)
with True have "m \<noteq> []" by (auto dest:lsp_mid_nil)
with eq_lsp obtain th' where eq_lsp: "lsp (cp s) rest = (l, [th'], r)"
by (case_tac m, auto dest:lsp_mid_length)
show ?thesis
proof(cases "th = th'")
case False
have "(th \<in> readys (e # s)) = (th \<in> readys s)"
by (unfold eq_e, rule readys_v_eq_1 [OF neq_th eq_wq eq_lsp False])
moreover have "cntCS (e#s) th = cntCS s th"
by (unfold eq_e, rule cntCS_v_eq_1[OF neq_th eq_wq eq_lsp False vtv])
moreover note ih eq_cnp eq_cnv eq_threads
ultimately show ?thesis by auto
next
case True
have "th \<in> readys (e # s)"
by (unfold eq_e, rule readys_v_eq_2 [OF neq_th eq_wq eq_lsp True vt])
moreover have "cntP s th = cntV s th + cntCS s th + 1"
proof -
have "th \<notin> readys s"
proof -
from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
show ?thesis
apply (unfold readys_def s_waiting_def, auto)
by (rule_tac x = cs in exI, auto)
qed
moreover have "th \<in> threads s"
proof -
from True eq_wq lsp_set_eq [OF eq_lsp] neq_th
have "th \<in> set (wq s cs)" by simp
from wq_threads [OF step_back_vt[OF vtv] this]
show ?thesis .
qed
ultimately show ?thesis using ih by auto
qed
moreover have "cntCS (e # s) th = 1 + cntCS s th"
by (unfold eq_e, rule cntCS_v_eq_2 [OF neq_th eq_wq eq_lsp True vtv])
moreover note eq_cnp eq_cnv
ultimately show ?thesis by simp
qed
qed
} ultimately show ?thesis by blast
qed
next
case (thread_set thread prio)
assume eq_e: "e = Set thread prio"
and is_runing: "thread \<in> runing s"
show ?thesis
proof -
from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_def
by (simp add:depend_set_unchanged eq_e)
from eq_e have eq_readys: "readys (e#s) = readys s"
by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
auto simp:Let_def)
{ assume "th \<noteq> thread"
with eq_readys eq_e
have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
(th \<in> readys (s) \<or> th \<notin> threads (s))"
by (simp add:threads.simps)
with eq_cnp eq_cnv eq_cncs ih is_runing
have ?thesis by simp
} moreover {
assume eq_th: "th = thread"
with is_runing ih have " cntP s th = cntV s th + cntCS s th"
by (unfold runing_def, auto)
moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
by (simp add:runing_def)
moreover note eq_cnp eq_cnv eq_cncs
ultimately have ?thesis by auto
} ultimately show ?thesis by blast
qed
qed
next
case vt_nil
show ?case
by (unfold cntP_def cntV_def cntCS_def,
auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
qed
qed
lemma not_thread_cncs:
fixes th s
assumes vt: "vt step s"
and not_in: "th \<notin> threads s"
shows "cntCS s th = 0"
proof -
from vt not_in show ?thesis
proof(induct arbitrary:th)
case (vt_cons s e th)
assume vt: "vt step s"
and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
and stp: "step s e"
and not_in: "th \<notin> threads (e # s)"
from stp show ?case
proof(cases)
case (thread_create prio max_prio thread)
assume eq_e: "e = Create thread prio"
and not_in': "thread \<notin> threads s"
have "cntCS (e # s) th = cntCS s th"
apply (unfold eq_e cntCS_def holdents_def)
by (simp add:depend_create_unchanged)
moreover have "th \<notin> threads s"
proof -
from not_in eq_e show ?thesis by simp
qed
moreover note ih ultimately show ?thesis by auto
next
case (thread_exit thread)
assume eq_e: "e = Exit thread"
and nh: "holdents s thread = {}"
have eq_cns: "cntCS (e # s) th = cntCS s th"
apply (unfold eq_e cntCS_def holdents_def)
by (simp add:depend_exit_unchanged)
show ?thesis
proof(cases "th = thread")
case True
have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
with eq_cns show ?thesis by simp
next
case False
with not_in and eq_e
have "th \<notin> threads s" by simp
from ih[OF this] and eq_cns show ?thesis by simp
qed
next
case (thread_P thread cs)
assume eq_e: "e = P thread cs"
and is_runing: "thread \<in> runing s"
from prems have vtp: "vt step (P thread cs#s)" by auto
have neq_th: "th \<noteq> thread"
proof -
from not_in eq_e have "th \<notin> threads s" by simp
moreover from is_runing have "thread \<in> threads s"
by (simp add:runing_def readys_def)
ultimately show ?thesis by auto
qed
hence "cntCS (e # s) th = cntCS s th "
apply (unfold cntCS_def holdents_def eq_e)
by (unfold step_depend_p[OF vtp], auto)
moreover have "cntCS s th = 0"
proof(rule ih)
from not_in eq_e show "th \<notin> threads s" by simp
qed
ultimately show ?thesis by simp
next
case (thread_V thread cs)
assume eq_e: "e = V thread cs"
and is_runing: "thread \<in> runing s"
and hold: "holding s thread cs"
have neq_th: "th \<noteq> thread"
proof -
from not_in eq_e have "th \<notin> threads s" by simp
moreover from is_runing have "thread \<in> threads s"
by (simp add:runing_def readys_def)
ultimately show ?thesis by auto
qed
from prems have vtv: "vt step (V thread cs#s)" by auto
from hold obtain rest
where eq_wq: "wq s cs = thread # rest"
by (case_tac "wq s cs", auto simp:s_holding_def)
have "cntCS (e # s) th = cntCS s th"
proof(unfold eq_e, rule cntCS_v_eq [OF neq_th eq_wq _ vtv])
show "th \<notin> set rest"
proof
assume "th \<in> set rest"
with eq_wq have "th \<in> set (wq s cs)" by simp
from wq_threads [OF vt this] eq_e not_in
show False by simp
qed
qed
moreover have "cntCS s th = 0"
proof(rule ih)
from not_in eq_e show "th \<notin> threads s" by simp
qed
ultimately show ?thesis by simp
next
case (thread_set thread prio)
print_facts
assume eq_e: "e = Set thread prio"
and is_runing: "thread \<in> runing s"
from not_in and eq_e have "th \<notin> threads s" by auto
from ih [OF this] and eq_e
show ?thesis
apply (unfold eq_e cntCS_def holdents_def)
by (simp add:depend_set_unchanged)
qed
next
case vt_nil
show ?case
by (unfold cntCS_def,
auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)
qed
qed
lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
by (auto simp:s_waiting_def cs_waiting_def)
lemma dm_depend_threads:
fixes th s
assumes vt: "vt step s"
and in_dom: "(Th th) \<in> Domain (depend s)"
shows "th \<in> threads s"
proof -
from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
ultimately have "(Th th, Cs cs) \<in> depend s" by simp
hence "th \<in> set (wq s cs)"
by (unfold s_depend_def, auto simp:cs_waiting_def)
from wq_threads [OF vt this] show ?thesis .
qed
lemma cp_eq_cpreced: "cp s th = cpreced s (wq s) th"
proof(unfold cp_def wq_def, induct s)
case (Cons e s')
show ?case
by (auto simp:Let_def)
next
case Nil
show ?case by (auto simp:Let_def)
qed
fun the_th :: "node \<Rightarrow> thread"
where "the_th (Th th) = th"
lemma runing_unique:
fixes th1 th2 s
assumes vt: "vt step s"
and runing_1: "th1 \<in> runing s"
and runing_2: "th2 \<in> runing s"
shows "th1 = th2"
proof -
from runing_1 and runing_2 have "cp s th1 = cp s th2"
by (unfold runing_def, simp)
hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =
Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"
(is "Max (?f ` ?A) = Max (?f ` ?B)")
by (unfold cp_eq_cpreced cpreced_def)
obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
proof -
have h1: "finite (?f ` ?A)"
proof -
have "finite ?A"
proof -
have "finite (dependents (wq s) th1)"
proof-
have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
moreover have "finite \<dots>"
proof -
from finite_depend[OF vt] have "finite (depend s)" .
hence "finite ((depend (wq s))\<^sup>+)"
apply (unfold finite_trancl)
by (auto simp: s_depend_def cs_depend_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
qed
thus ?thesis by (simp add:cs_dependents_def)
qed
thus ?thesis by simp
qed
thus ?thesis by auto
qed
moreover have h2: "(?f ` ?A) \<noteq> {}"
proof -
have "?A \<noteq> {}" by simp
thus ?thesis by simp
qed
from Max_in [OF h1 h2]
have "Max (?f ` ?A) \<in> (?f ` ?A)" .
thus ?thesis by (auto intro:that)
qed
obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
proof -
have h1: "finite (?f ` ?B)"
proof -
have "finite ?B"
proof -
have "finite (dependents (wq s) th2)"
proof-
have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
moreover have "finite \<dots>"
proof -
from finite_depend[OF vt] have "finite (depend s)" .
hence "finite ((depend (wq s))\<^sup>+)"
apply (unfold finite_trancl)
by (auto simp: s_depend_def cs_depend_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
qed
thus ?thesis by (simp add:cs_dependents_def)
qed
thus ?thesis by simp
qed
thus ?thesis by auto
qed
moreover have h2: "(?f ` ?B) \<noteq> {}"
proof -
have "?B \<noteq> {}" by simp
thus ?thesis by simp
qed
from Max_in [OF h1 h2]
have "Max (?f ` ?B) \<in> (?f ` ?B)" .
thus ?thesis by (auto intro:that)
qed
from eq_f_th1 eq_f_th2 eq_max
have eq_preced: "preced th1' s = preced th2' s" by auto
hence eq_th12: "th1' = th2'"
proof (rule preced_unique)
from th1_in have "th1' = th1 \<or> (th1' \<in> dependents (wq s) th1)" by simp
thus "th1' \<in> threads s"
proof
assume "th1' \<in> dependents (wq s) th1"
hence "(Th th1') \<in> Domain ((depend s)^+)"
apply (unfold cs_dependents_def cs_depend_def s_depend_def)
by (auto simp:Domain_def)
hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
from dm_depend_threads[OF vt this] show ?thesis .
next
assume "th1' = th1"
with runing_1 show ?thesis
by (unfold runing_def readys_def, auto)
qed
next
from th2_in have "th2' = th2 \<or> (th2' \<in> dependents (wq s) th2)" by simp
thus "th2' \<in> threads s"
proof
assume "th2' \<in> dependents (wq s) th2"
hence "(Th th2') \<in> Domain ((depend s)^+)"
apply (unfold cs_dependents_def cs_depend_def s_depend_def)
by (auto simp:Domain_def)
hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
from dm_depend_threads[OF vt this] show ?thesis .
next
assume "th2' = th2"
with runing_2 show ?thesis
by (unfold runing_def readys_def, auto)
qed
qed
from th1_in have "th1' = th1 \<or> th1' \<in> dependents (wq s) th1" by simp
thus ?thesis
proof
assume eq_th': "th1' = th1"
from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
thus ?thesis
proof
assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
next
assume "th2' \<in> dependents (wq s) th2"
with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp
hence "(Th th1, Th th2) \<in> (depend s)^+"
by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
hence "Th th1 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
by auto
hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
from depend_target_th [OF this]
obtain cs' where "n = Cs cs'" by auto
with d have "(Th th1, Cs cs') \<in> depend s" by simp
with runing_1 have "False"
apply (unfold runing_def readys_def s_depend_def)
by (auto simp:eq_waiting)
thus ?thesis by simp
qed
next
assume th1'_in: "th1' \<in> dependents (wq s) th1"
from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp
thus ?thesis
proof
assume "th2' = th2"
with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp
hence "(Th th2, Th th1) \<in> (depend s)^+"
by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
hence "Th th2 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]
by auto
hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
from depend_target_th [OF this]
obtain cs' where "n = Cs cs'" by auto
with d have "(Th th2, Cs cs') \<in> depend s" by simp
with runing_2 have "False"
apply (unfold runing_def readys_def s_depend_def)
by (auto simp:eq_waiting)
thus ?thesis by simp
next
assume "th2' \<in> dependents (wq s) th2"
with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp
hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)
show ?thesis
proof(rule dchain_unique[OF vt h1 _ h2, symmetric])
from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
qed
qed
qed
qed
lemma create_pre:
assumes stp: "step s e"
and not_in: "th \<notin> threads s"
and is_in: "th \<in> threads (e#s)"
obtains prio where "e = Create th prio"
proof -
from assms
show ?thesis
proof(cases)
case (thread_create prio max_prio thread)
with is_in not_in have "e = Create th prio" by simp
from that[OF this] show ?thesis .
next
case (thread_exit thread)
with assms show ?thesis by (auto intro!:that)
next
case (thread_P thread)
with assms show ?thesis by (auto intro!:that)
next
case (thread_V thread)
with assms show ?thesis by (auto intro!:that)
next
case (thread_set thread)
with assms show ?thesis by (auto intro!:that)
qed
qed
lemma length_down_to_in:
assumes le_ij: "i \<le> j"
and le_js: "j \<le> length s"
shows "length (down_to j i s) = j - i"
proof -
have "length (down_to j i s) = length (from_to i j (rev s))"
by (unfold down_to_def, auto)
also have "\<dots> = j - i"
proof(rule length_from_to_in[OF le_ij])
from le_js show "j \<le> length (rev s)" by simp
qed
finally show ?thesis .
qed
lemma moment_head:
assumes le_it: "Suc i \<le> length t"
obtains e where "moment (Suc i) t = e#moment i t"
proof -
have "i \<le> Suc i" by simp
from length_down_to_in [OF this le_it]
have "length (down_to (Suc i) i t) = 1" by auto
then obtain e where "down_to (Suc i) i t = [e]"
apply (cases "(down_to (Suc i) i t)") by auto
moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
by (rule down_to_conc[symmetric], auto)
ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
by (auto simp:down_to_moment)
from that [OF this] show ?thesis .
qed
lemma cnp_cnv_eq:
fixes th s
assumes "vt step s"
and "th \<notin> threads s"
shows "cntP s th = cntV s th"
proof -
from assms show ?thesis
proof(induct)
case (vt_cons s e)
have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact
have not_in: "th \<notin> threads (e # s)" by fact
have "step s e" by fact
thus ?case proof(cases)
case (thread_create prio max_prio thread)
assume eq_e: "e = Create thread prio"
hence "thread \<in> threads (e#s)" by simp
with not_in and eq_e have "th \<notin> threads s" by auto
from ih [OF this] show ?thesis using eq_e
by (auto simp:cntP_def cntV_def count_def)
next
case (thread_exit thread)
assume eq_e: "e = Exit thread"
and not_holding: "holdents s thread = {}"
have vt_s: "vt step s" by fact
from finite_holding[OF vt_s] have "finite (holdents s thread)" .
with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)
moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)
moreover note cnp_cnv_cncs[OF vt_s, of thread]
ultimately have eq_thread: "cntP s thread = cntV s thread" by auto
show ?thesis
proof(cases "th = thread")
case True
with eq_thread eq_e show ?thesis
by (auto simp:cntP_def cntV_def count_def)
next
case False
with not_in and eq_e have "th \<notin> threads s" by simp
from ih[OF this] and eq_e show ?thesis
by (auto simp:cntP_def cntV_def count_def)
qed
next
case (thread_P thread cs)
assume eq_e: "e = P thread cs"
have "thread \<in> runing s" by fact
with not_in eq_e have neq_th: "thread \<noteq> th"
by (auto simp:runing_def readys_def)
from not_in eq_e have "th \<notin> threads s" by simp
from ih[OF this] and neq_th and eq_e show ?thesis
by (auto simp:cntP_def cntV_def count_def)
next
case (thread_V thread cs)
assume eq_e: "e = V thread cs"
have "thread \<in> runing s" by fact
with not_in eq_e have neq_th: "thread \<noteq> th"
by (auto simp:runing_def readys_def)
from not_in eq_e have "th \<notin> threads s" by simp
from ih[OF this] and neq_th and eq_e show ?thesis
by (auto simp:cntP_def cntV_def count_def)
next
case (thread_set thread prio)
assume eq_e: "e = Set thread prio"
and "thread \<in> runing s"
hence "thread \<in> threads (e#s)"
by (simp add:runing_def readys_def)
with not_in and eq_e have "th \<notin> threads s" by auto
from ih [OF this] show ?thesis using eq_e
by (auto simp:cntP_def cntV_def count_def)
qed
next
case vt_nil
show ?case by (auto simp:cntP_def cntV_def count_def)
qed
qed
lemma eq_depend:
"depend (wq s) = depend s"
by (unfold cs_depend_def s_depend_def, auto)
lemma count_eq_dependents:
assumes vt: "vt step s"
and eq_pv: "cntP s th = cntV s th"
shows "dependents (wq s) th = {}"
proof -
from cnp_cnv_cncs[OF vt] and eq_pv
have "cntCS s th = 0"
by (auto split:if_splits)
moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
proof -
from finite_holding[OF vt, of th] show ?thesis
by (simp add:holdents_def)
qed
ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
by (unfold cntCS_def holdents_def cs_dependents_def, auto)
show ?thesis
proof(unfold cs_dependents_def)
{ assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
hence "False"
proof(cases)
assume "(Th th', Th th) \<in> depend (wq s)"
thus "False" by (auto simp:cs_depend_def)
next
fix c
assume "(c, Th th) \<in> depend (wq s)"
with h and eq_depend show "False"
by (cases c, auto simp:cs_depend_def)
qed
} thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
qed
qed
lemma dependents_threads:
fixes s th
assumes vt: "vt step s"
shows "dependents (wq s) th \<subseteq> threads s"
proof
{ fix th th'
assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
have "Th th \<in> Domain (depend s)"
proof -
from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
thus ?thesis using eq_depend by simp
qed
from dm_depend_threads[OF vt this]
have "th \<in> threads s" .
} note hh = this
fix th1
assume "th1 \<in> dependents (wq s) th"
hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
by (unfold cs_dependents_def, simp)
from hh [OF this] show "th1 \<in> threads s" .
qed
lemma finite_threads:
assumes vt: "vt step s"
shows "finite (threads s)"
proof -
from vt show ?thesis
proof(induct)
case (vt_cons s e)
assume vt: "vt step s"
and step: "step s e"
and ih: "finite (threads s)"
from step
show ?case
proof(cases)
case (thread_create prio max_prio thread)
assume eq_e: "e = Create thread prio"
with ih
show ?thesis by (unfold eq_e, auto)
next
case (thread_exit thread)
assume eq_e: "e = Exit thread"
with ih show ?thesis
by (unfold eq_e, auto)
next
case (thread_P thread cs)
assume eq_e: "e = P thread cs"
with ih show ?thesis by (unfold eq_e, auto)
next
case (thread_V thread cs)
assume eq_e: "e = V thread cs"
with ih show ?thesis by (unfold eq_e, auto)
next
case (thread_set thread prio)
from vt_cons thread_set show ?thesis by simp
qed
next
case vt_nil
show ?case by (auto)
qed
qed
lemma Max_f_mono:
assumes seq: "A \<subseteq> B"
and np: "A \<noteq> {}"
and fnt: "finite B"
shows "Max (f ` A) \<le> Max (f ` B)"
proof(rule Max_mono)
from seq show "f ` A \<subseteq> f ` B" by auto
next
from np show "f ` A \<noteq> {}" by auto
next
from fnt and seq show "finite (f ` B)" by auto
qed
lemma cp_le:
assumes vt: "vt step s"
and th_in: "th \<in> threads s"
shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def)
show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))
\<le> Max ((\<lambda>th. preced th s) ` threads s)"
(is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
proof(rule Max_f_mono)
show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
next
from finite_threads [OF vt]
show "finite (threads s)" .
next
from th_in
show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
apply (auto simp:Domain_def)
apply (rule_tac dm_depend_threads[OF vt])
apply (unfold trancl_domain [of "depend s", symmetric])
by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
qed
qed
lemma le_cp:
assumes vt: "vt step s"
shows "preced th s \<le> cp s th"
proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
show "Prc (original_priority th s) (birthtime th s)
\<le> Max (insert (Prc (original_priority th s) (birthtime th s))
((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))"
(is "?l \<le> Max (insert ?l ?A)")
proof(cases "?A = {}")
case False
have "finite ?A" (is "finite (?f ` ?B)")
proof -
have "finite ?B"
proof-
have "finite {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th)" in bexI, auto)
moreover have "finite \<dots>"
proof -
from finite_depend[OF vt] have "finite (depend s)" .
hence "finite ((depend (wq s))\<^sup>+)"
apply (unfold finite_trancl)
by (auto simp: s_depend_def cs_depend_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
qed
thus ?thesis by (simp add:cs_dependents_def)
qed
thus ?thesis by simp
qed
from Max_insert [OF this False, of ?l] show ?thesis by auto
next
case True
thus ?thesis by auto
qed
qed
lemma max_cp_eq:
assumes vt: "vt step s"
shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
(is "?l = ?r")
proof(cases "threads s = {}")
case True
thus ?thesis by auto
next
case False
have "?l \<in> ((cp s) ` threads s)"
proof(rule Max_in)
from finite_threads[OF vt]
show "finite (cp s ` threads s)" by auto
next
from False show "cp s ` threads s \<noteq> {}" by auto
qed
then obtain th
where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
have "\<dots> \<le> ?r" by (rule cp_le[OF vt th_in])
moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
proof -
have "?r \<in> (?f ` ?A)"
proof(rule Max_in)
from finite_threads[OF vt]
show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
next
from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
qed
then obtain th' where
th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
from le_cp [OF vt, of th'] eq_r
have "?r \<le> cp s th'" by auto
moreover have "\<dots> \<le> cp s th"
proof(fold eq_l)
show " cp s th' \<le> Max (cp s ` threads s)"
proof(rule Max_ge)
from th_in' show "cp s th' \<in> cp s ` threads s"
by auto
next
from finite_threads[OF vt]
show "finite (cp s ` threads s)" by auto
qed
qed
ultimately show ?thesis by auto
qed
ultimately show ?thesis using eq_l by auto
qed
lemma max_cp_readys_threads_pre:
assumes vt: "vt step s"
and np: "threads s \<noteq> {}"
shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
proof(unfold max_cp_eq[OF vt])
show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
proof -
let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
let ?f = "(\<lambda>th. preced th s)"
have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
proof(rule Max_in)
from finite_threads[OF vt] show "finite (?f ` threads s)" by simp
next
from np show "?f ` threads s \<noteq> {}" by simp
qed
then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
by (auto simp:Image_def)
from th_chain_to_ready [OF vt tm_in]
have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
thus ?thesis
proof
assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
then obtain th' where th'_in: "th' \<in> readys s"
and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
have "cp s th' = ?f tm"
proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
from dependents_threads[OF vt] finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))"
by (auto intro:finite_subset)
next
fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
moreover have "p \<le> \<dots>"
proof(rule Max_ge)
from finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
next
from p_in and th'_in and dependents_threads[OF vt, of th']
show "p \<in> (\<lambda>th. preced th s) ` threads s"
by (auto simp:readys_def)
qed
ultimately show "p \<le> preced tm s" by auto
next
show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"
proof -
from tm_chain
have "tm \<in> dependents (wq s) th'"
by (unfold cs_dependents_def s_depend_def cs_depend_def, auto)
thus ?thesis by auto
qed
qed
with tm_max
have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
show ?thesis
proof (fold h, rule Max_eqI)
fix q
assume "q \<in> cp s ` readys s"
then obtain th1 where th1_in: "th1 \<in> readys s"
and eq_q: "q = cp s th1" by auto
show "q \<le> cp s th'"
apply (unfold h eq_q)
apply (unfold cp_eq_cpreced cpreced_def)
apply (rule Max_mono)
proof -
from dependents_threads [OF vt, of th1] th1_in
show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq>
(\<lambda>th. preced th s) ` threads s"
by (auto simp:readys_def)
next
show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp
next
from finite_threads[OF vt]
show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
qed
next
from finite_threads[OF vt]
show "finite (cp s ` readys s)" by (auto simp:readys_def)
next
from th'_in
show "cp s th' \<in> cp s ` readys s" by simp
qed
next
assume tm_ready: "tm \<in> readys s"
show ?thesis
proof(fold tm_max)
have cp_eq_p: "cp s tm = preced tm s"
proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
fix y
assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
show "y \<le> preced tm s"
proof -
{ fix y'
assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"
have "y' \<le> preced tm s"
proof(unfold tm_max, rule Max_ge)
from hy' dependents_threads[OF vt, of tm]
show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
next
from finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
qed
} with hy show ?thesis by auto
qed
next
from dependents_threads[OF vt, of tm] finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"
by (auto intro:finite_subset)
next
show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"
by simp
qed
moreover have "Max (cp s ` readys s) = cp s tm"
proof(rule Max_eqI)
from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
next
from finite_threads[OF vt]
show "finite (cp s ` readys s)" by (auto simp:readys_def)
next
fix y assume "y \<in> cp s ` readys s"
then obtain th1 where th1_readys: "th1 \<in> readys s"
and h: "y = cp s th1" by auto
show "y \<le> cp s tm"
apply(unfold cp_eq_p h)
apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
proof -
from finite_threads[OF vt]
show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
next
show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"
by simp
next
from dependents_threads[OF vt, of th1] th1_readys
show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)
\<subseteq> (\<lambda>th. preced th s) ` threads s"
by (auto simp:readys_def)
qed
qed
ultimately show " Max (cp s ` readys s) = preced tm s" by simp
qed
qed
qed
qed
lemma max_cp_readys_threads:
assumes vt: "vt step s"
shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
proof(cases "threads s = {}")
case True
thus ?thesis
by (auto simp:readys_def)
next
case False
show ?thesis by (rule max_cp_readys_threads_pre[OF vt False])
qed
lemma readys_threads:
shows "readys s \<subseteq> threads s"
proof
fix th
assume "th \<in> readys s"
thus "th \<in> threads s"
by (unfold readys_def, auto)
qed
lemma eq_holding: "holding (wq s) th cs = holding s th cs"
apply (unfold s_holding_def cs_holding_def, simp)
done
lemma f_image_eq:
assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
shows "f ` A = g ` A"
proof
show "f ` A \<subseteq> g ` A"
by(rule image_subsetI, auto intro:h)
next
show "g ` A \<subseteq> f ` A"
by(rule image_subsetI, auto intro:h[symmetric])
qed
end