theory CpsG
imports PIPDefs
begin
section {* Generic aulxiliary lemmas *}
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
lemma Max_fg_mono:
assumes "finite A"
and "\<forall> a \<in> A. f a \<le> g a"
shows "Max (f ` A) \<le> Max (g ` A)"
proof(cases "A = {}")
case True
thus ?thesis by auto
next
case False
show ?thesis
proof(rule Max.boundedI)
from assms show "finite (f ` A)" by auto
next
from False show "f ` A \<noteq> {}" by auto
next
fix fa
assume "fa \<in> f ` A"
then obtain a where h_fa: "a \<in> A" "fa = f a" by auto
show "fa \<le> Max (g ` A)"
proof(rule Max_ge_iff[THEN iffD2])
from assms show "finite (g ` A)" by auto
next
from False show "g ` A \<noteq> {}" by auto
next
from h_fa have "g a \<in> g ` A" by auto
moreover have "fa \<le> g a" using h_fa assms(2) by auto
ultimately show "\<exists>a\<in>g ` A. fa \<le> a" by auto
qed
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 Max_UNION:
assumes "finite A"
and "A \<noteq> {}"
and "\<forall> M \<in> f ` A. finite M"
and "\<forall> M \<in> f ` A. M \<noteq> {}"
shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
using assms[simp]
proof -
have "?L = Max (\<Union>(f ` A))"
by (fold Union_image_eq, simp)
also have "... = ?R"
by (subst Max_Union, simp+)
finally show ?thesis .
qed
lemma max_Max_eq:
assumes "finite A"
and "A \<noteq> {}"
and "x = y"
shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
proof -
have "?R = Max (insert y A)" by simp
also from assms have "... = ?L"
by (subst Max.insert, simp+)
finally show ?thesis by simp
qed
lemma rel_eqI:
assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
shows "A = B"
using assms by auto
section {* Lemmas do not depend on trace validity *}
lemma birth_time_lt:
assumes "s \<noteq> []"
shows "last_set th s < length s"
using assms
proof(induct s)
case (Cons a s)
show ?case
proof(cases "s \<noteq> []")
case False
thus ?thesis
by (cases a, auto)
next
case True
show ?thesis using Cons(1)[OF True]
by (cases a, auto)
qed
qed simp
lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
by (induct s, auto)
lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
by (drule_tac th_in_ne, unfold preced_def, auto intro: birth_time_lt)
lemma eq_RAG:
"RAG (wq s) = RAG s"
by (unfold cs_RAG_def s_RAG_def, auto)
lemma waiting_holding:
assumes "waiting (s::state) th cs"
obtains th' where "holding s th' cs"
proof -
from assms[unfolded s_waiting_def, folded wq_def]
obtain th' where "th' \<in> set (wq s cs)" "th' = hd (wq s cs)"
by (metis empty_iff hd_in_set list.set(1))
hence "holding s th' cs"
by (unfold s_holding_def, fold wq_def, auto)
from that[OF 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 cp_alt_def:
"cp s th =
Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
proof -
have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
(is "Max (_ ` ?L) = Max (_ ` ?R)")
proof -
have "?L = ?R"
by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
thus ?thesis by simp
qed
thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
qed
lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
by (unfold s_RAG_def, auto)
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 children_RAG_alt_def:
"children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}"
by (unfold s_RAG_def, auto simp:children_def holding_eq)
lemma holdents_alt_def:
"holdents s th = the_cs ` (children (RAG (s::state)) (Th th))"
by (unfold children_RAG_alt_def holdents_def, simp add: image_image)
lemma cntCS_alt_def:
"cntCS s th = card (children (RAG s) (Th th))"
apply (unfold children_RAG_alt_def cntCS_def holdents_def)
by (rule card_image[symmetric], auto simp:inj_on_def)
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 [simp]:
"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 runing_head:
assumes "th \<in> runing s"
and "th \<in> set (wq_fun (schs s) cs)"
shows "th = hd (wq_fun (schs s) cs)"
using assms
by (simp add:runing_def readys_def s_waiting_def wq_def)
lemma runing_wqE:
assumes "th \<in> runing s"
and "th \<in> set (wq s cs)"
obtains rest where "wq s cs = th#rest"
proof -
from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
by (meson list.set_cases)
have "th' = th"
proof(rule ccontr)
assume "th' \<noteq> th"
hence "th \<noteq> hd (wq s cs)" using eq_wq by auto
with assms(2)
have "waiting s th cs"
by (unfold s_waiting_def, fold wq_def, auto)
with assms show False
by (unfold runing_def readys_def, auto)
qed
with eq_wq that show ?thesis by metis
qed
lemma isP_E:
assumes "isP e"
obtains cs where "e = P (actor e) cs"
using assms by (cases e, auto)
lemma isV_E:
assumes "isV e"
obtains cs where "e = V (actor e) cs"
using assms by (cases e, auto)
text {*
Every thread can only be blocked on one critical resource,
symmetrically, every critical resource can only be held by one thread.
This fact is much more easier according to our definition.
*}
lemma held_unique:
assumes "holding (s::event list) th1 cs"
and "holding s th2 cs"
shows "th1 = th2"
by (insert assms, unfold s_holding_def, auto)
lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
apply (induct s, auto)
by (case_tac a, auto split:if_splits)
lemma last_set_unique:
"\<lbrakk>last_set th1 s = last_set 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:last_set_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 "last_set th1 s = last_set th2 s" by (simp add:preced_def)
from last_set_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 in_RAG_E:
assumes "(n1, n2) \<in> RAG (s::state)"
obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
| (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
by auto
lemma count_rec1 [simp]:
assumes "Q e"
shows "count Q (e#es) = Suc (count Q es)"
using assms
by (unfold count_def, auto)
lemma count_rec2 [simp]:
assumes "\<not>Q e"
shows "count Q (e#es) = (count Q es)"
using assms
by (unfold count_def, auto)
lemma count_rec3 [simp]:
shows "count Q [] = 0"
by (unfold count_def, auto)
lemma cntP_simp1[simp]:
"cntP (P th cs'#s) th = cntP s th + 1"
by (unfold cntP_def, simp)
lemma cntP_simp2[simp]:
assumes "th' \<noteq> th"
shows "cntP (P th cs'#s) th' = cntP s th'"
using assms
by (unfold cntP_def, simp)
lemma cntP_simp3[simp]:
assumes "\<not> isP e"
shows "cntP (e#s) th' = cntP s th'"
using assms
by (unfold cntP_def, cases e, simp+)
lemma cntV_simp1[simp]:
"cntV (V th cs'#s) th = cntV s th + 1"
by (unfold cntV_def, simp)
lemma cntV_simp2[simp]:
assumes "th' \<noteq> th"
shows "cntV (V th cs'#s) th' = cntV s th'"
using assms
by (unfold cntV_def, simp)
lemma cntV_simp3[simp]:
assumes "\<not> isV e"
shows "cntV (e#s) th' = cntV s th'"
using assms
by (unfold cntV_def, cases e, simp+)
lemma cntP_diff_inv:
assumes "cntP (e#s) th \<noteq> cntP s th"
shows "isP e \<and> actor e = th"
proof(cases e)
case (P th' pty)
show ?thesis
by (cases "(\<lambda>e. \<exists>cs. e = P th cs) (P th' pty)",
insert assms P, auto simp:cntP_def)
qed (insert assms, auto simp:cntP_def)
lemma cntV_diff_inv:
assumes "cntV (e#s) th \<noteq> cntV s th"
shows "isV e \<and> actor e = th"
proof(cases e)
case (V th' pty)
show ?thesis
by (cases "(\<lambda>e. \<exists>cs. e = V th cs) (V th' pty)",
insert assms V, auto simp:cntV_def)
qed (insert assms, auto simp:cntV_def)
lemma eq_dependants: "dependants (wq s) = dependants s"
by (simp add: s_dependants_abv wq_def)
lemma inj_the_preced:
"inj_on (the_preced s) (threads s)"
by (metis inj_onI preced_unique the_preced_def)
lemma holding_next_thI:
assumes "holding s th cs"
and "length (wq s cs) > 1"
obtains th' where "next_th s th cs th'"
proof -
from assms(1)[folded holding_eq, unfolded cs_holding_def]
have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)"
by (unfold s_holding_def, fold wq_def, auto)
then obtain rest where h1: "wq s cs = th#rest"
by (cases "wq s cs", auto)
with assms(2) have h2: "rest \<noteq> []" by auto
let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
have "next_th s th cs ?th'" using h1(1) h2
by (unfold next_th_def, auto)
from that[OF this] show ?thesis .
qed
(* ccc *)
section {* Locales used to investigate the execution of PIP *}
text {*
The following locale @{text valid_trace} is used to constrain the
trace to be valid. All properties hold for valid traces are
derived under this locale.
*}
locale valid_trace =
fixes s
assumes vt : "vt s"
text {*
The following locale @{text valid_trace_e} describes
the valid extension of a valid trace. The event @{text "e"}
represents an event in the system, which corresponds
to a one step operation of the PIP protocol.
It is required that @{text "e"} is an event eligible to happen
under state @{text "s"}, which is already required to be valid
by the parent locale @{text "valid_trace"}.
This locale is used to investigate one step execution of PIP,
properties concerning the effects of @{text "e"}'s execution,
for example, how the values of observation functions are changed,
or how desirable properties are kept invariant, are derived
under this locale. The state before execution is @{text "s"}, while
the state after execution is @{text "e#s"}. Therefore, the lemmas
derived usually relate observations on @{text "e#s"} to those
on @{text "s"}.
*}
locale valid_trace_e = valid_trace +
fixes e
assumes vt_e: "vt (e#s)"
begin
text {*
The following lemma shows that @{text "e"} must be a
eligible event (or a valid step) to be taken under
the state represented by @{text "s"}.
*}
lemma pip_e: "PIP s e"
using vt_e by (cases, simp)
end
text {*
Because @{term "e#s"} is also a valid trace, properties
derived for valid trace @{term s} also hold on @{term "e#s"}.
*}
sublocale valid_trace_e < vat_es!: valid_trace "e#s"
using vt_e
by (unfold_locales, simp)
text {*
For each specific event (or operation), there is a sublocale
further constraining that the event @{text e} to be that
particular event.
For example, the following
locale @{text "valid_trace_create"} is the sublocale for
event @{term "Create"}:
*}
locale valid_trace_create = valid_trace_e +
fixes th prio
assumes is_create: "e = Create th prio"
locale valid_trace_exit = valid_trace_e +
fixes th
assumes is_exit: "e = Exit th"
locale valid_trace_p = valid_trace_e +
fixes th cs
assumes is_p: "e = P th cs"
text {*
locale @{text "valid_trace_p"} is divided further into two
sublocales, namely, @{text "valid_trace_p_h"}
and @{text "valid_trace_p_w"}.
*}
text {*
The following two sublocales @{text "valid_trace_p_h"}
and @{text "valid_trace_p_w"} represent two complementary
cases under @{text "valid_trace_p"}, where
@{text "valid_trace_p_h"} further constraints that
@{text "wq s cs = []"}, which means the waiting queue of
the requested resource @{text "cs"} is empty, in which
case, the requesting thread @{text "th"}
will take hold of @{text "cs"}.
Opposite to @{text "valid_trace_p_h"},
@{text "valid_trace_p_w"} constraints that
@{text "wq s cs \<noteq> []"}, which means the waiting queue of
the requested resource @{text "cs"} is nonempty, in which
case, the requesting thread @{text "th"} will be blocked
on @{text "cs"}:
Peculiar properties will be derived under respective
locales.
*}
locale valid_trace_p_h = valid_trace_p +
assumes we: "wq s cs = []"
locale valid_trace_p_w = valid_trace_p +
assumes wne: "wq s cs \<noteq> []"
begin
text {*
The following @{text "holder"} designates
the holder of @{text "cs"} before the @{text "P"}-operation.
*}
definition "holder = hd (wq s cs)"
text {*
The following @{text "waiters"} designates
the list of threads waiting for @{text "cs"}
before the @{text "P"}-operation.
*}
definition "waiters = tl (wq s cs)"
end
text {*
@{text "valid_trace_v"} is set for the @{term V}-operation.
*}
locale valid_trace_v = valid_trace_e +
fixes th cs
assumes is_v: "e = V th cs"
begin
-- {* The following @{text "rest"} is the tail of
waiting queue of the resource @{text "cs"}
to be released by this @{text "V"}-operation.
*}
definition "rest = tl (wq s cs)"
text {*
The following @{text "wq'"} is the waiting
queue of @{term "cs"}
after the @{text "V"}-operation, which
is simply a reordering of @{term "rest"}.
The effect of this reordering needs to be
understood by two cases:
\begin{enumerate}
\item When @{text "rest = []"},
the reordering gives rise to an empty list as well,
which means there is no thread holding or waiting
for resource @{term "cs"}, therefore, it is free.
\item When @{text "rest \<noteq> []"}, the effect of
this reordering is to arbitrarily
switch one thread in @{term "rest"} to the
head, which, by definition take over the hold
of @{term "cs"} and is designated by @{text "taker"}
in the following sublocale @{text "valid_trace_v_n"}.
*}
definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
text {*
The following @{text "rest'"} is the tail of the
waiting queue after the @{text "V"}-operation.
It plays only auxiliary role to ease reasoning.
*}
definition "rest' = tl wq'"
end
text {*
In the following, @{text "valid_trace_v"} is also
divided into two
sublocales: when @{text "rest"} is empty (represented
by @{text "valid_trace_v_e"}), which means, there is no thread waiting
for @{text "cs"}, therefore, after the @{text "V"}-operation,
it will become free; otherwise (represented
by @{text "valid_trace_v_n"}), one thread
will be picked from those in @{text "rest"} to take
over @{text "cs"}.
*}
locale valid_trace_v_e = valid_trace_v +
assumes rest_nil: "rest = []"
locale valid_trace_v_n = valid_trace_v +
assumes rest_nnl: "rest \<noteq> []"
begin
text {*
The following @{text "taker"} is the thread to
take over @{text "cs"}.
*}
definition "taker = hd wq'"
end
locale valid_trace_set = valid_trace_e +
fixes th prio
assumes is_set: "e = Set th prio"
context valid_trace
begin
text {*
Induction rule introduced to easy the
derivation of properties for valid trace @{term "s"}.
One more premises, namely @{term "valid_trace_e s e"}
is added, so that an interpretation of
@{text "valid_trace_e"} can be instantiated
so that all properties derived so far becomes
available in the proof of induction step.
You will see its use in the proofs that follows.
*}
lemma ind [consumes 0, case_names Nil Cons, induct type]:
assumes "PP []"
and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
shows "PP s"
proof(induct rule:vt.induct[OF vt, case_names Init Step])
case Init
from assms(1) show ?case .
next
case (Step s e)
show ?case
proof(rule assms(2))
show "valid_trace_e s e" using Step by (unfold_locales, auto)
next
show "PP s" using Step by simp
next
show "PIP s e" using Step by simp
qed
qed
text {*
The following lemma says that if @{text "s"} is a valid state, so
is its any postfix. Where @{term "monent t s"} is the postfix of
@{term "s"} with length @{term "t"}.
*}
lemma vt_moment: "\<And> t. vt (moment t s)"
proof(induct rule:ind)
case Nil
thus ?case by (simp add:vt_nil)
next
case (Cons s e t)
show ?case
proof(cases "t \<ge> length (e#s)")
case True
from True have "moment t (e#s) = e#s" by simp
thus ?thesis using Cons
by (simp add:valid_trace_def valid_trace_e_def, auto)
next
case False
from Cons have "vt (moment t s)" by simp
moreover have "moment t (e#s) = moment t s"
proof -
from False have "t \<le> length s" by simp
from moment_app [OF this, of "[e]"]
show ?thesis by simp
qed
ultimately show ?thesis by simp
qed
qed
end
text {*
The following locale @{text "valid_moment"} is to inherit the properties
derived on any valid state to the prefix of it, with length @{text "i"}.
*}
locale valid_moment = valid_trace +
fixes i :: nat
sublocale valid_moment < vat_moment!: valid_trace "(moment i s)"
by (unfold_locales, insert vt_moment, auto)
locale valid_moment_e = valid_moment +
assumes less_i: "i < length s"
begin
definition "next_e = hd (moment (Suc i) s)"
lemma trace_e:
"moment (Suc i) s = next_e#moment i s"
proof -
from less_i have "Suc i \<le> length s" by auto
from moment_plus[OF this, folded next_e_def]
show ?thesis .
qed
end
sublocale valid_moment_e < vat_moment_e!: valid_trace_e "moment i s" "next_e"
using vt_moment[of "Suc i", unfolded trace_e]
by (unfold_locales, simp)
section {* Distinctiveness of waiting queues *}
context valid_trace_create
begin
lemma wq_kept [simp]:
shows "wq (e#s) cs' = wq s cs'"
using assms unfolding is_create wq_def
by (auto simp:Let_def)
lemma wq_distinct_kept:
assumes "distinct (wq s cs')"
shows "distinct (wq (e#s) cs')"
using assms by simp
end
context valid_trace_exit
begin
lemma wq_kept [simp]:
shows "wq (e#s) cs' = wq s cs'"
using assms unfolding is_exit wq_def
by (auto simp:Let_def)
lemma wq_distinct_kept:
assumes "distinct (wq s cs')"
shows "distinct (wq (e#s) cs')"
using assms by simp
end
context valid_trace_p
begin
lemma wq_neq_simp [simp]:
assumes "cs' \<noteq> cs"
shows "wq (e#s) cs' = wq s cs'"
using assms unfolding is_p wq_def
by (auto simp:Let_def)
lemma runing_th_s:
shows "th \<in> runing s"
proof -
from pip_e[unfolded is_p]
show ?thesis by (cases, simp)
qed
lemma th_not_in_wq:
shows "th \<notin> set (wq s cs)"
proof
assume otherwise: "th \<in> set (wq s cs)"
from runing_wqE[OF runing_th_s this]
obtain rest where eq_wq: "wq s cs = th#rest" by blast
with otherwise
have "holding s th cs"
by (unfold s_holding_def, fold wq_def, simp)
hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
by (unfold s_RAG_def, fold holding_eq, auto)
from pip_e[unfolded is_p]
show False
proof(cases)
case (thread_P)
with cs_th_RAG show ?thesis by auto
qed
qed
lemma wq_es_cs:
"wq (e#s) cs = wq s cs @ [th]"
by (unfold is_p wq_def, auto simp:Let_def)
lemma wq_distinct_kept:
assumes "distinct (wq s cs')"
shows "distinct (wq (e#s) cs')"
proof(cases "cs' = cs")
case True
show ?thesis using True assms th_not_in_wq
by (unfold True wq_es_cs, auto)
qed (insert assms, simp)
end
context valid_trace_v
begin
lemma wq_neq_simp [simp]:
assumes "cs' \<noteq> cs"
shows "wq (e#s) cs' = wq s cs'"
using assms unfolding is_v wq_def
by (auto simp:Let_def)
lemma wq_s_cs:
"wq s cs = th#rest"
proof -
from pip_e[unfolded is_v]
show ?thesis
proof(cases)
case (thread_V)
from this(2) show ?thesis
by (unfold rest_def s_holding_def, fold wq_def,
metis empty_iff list.collapse list.set(1))
qed
qed
lemma wq_es_cs:
"wq (e#s) cs = wq'"
using wq_s_cs[unfolded wq_def]
by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp)
lemma wq_distinct_kept:
assumes "distinct (wq s cs')"
shows "distinct (wq (e#s) cs')"
proof(cases "cs' = cs")
case True
show ?thesis
proof(unfold True wq_es_cs wq'_def, rule someI2)
show "distinct rest \<and> set rest = set rest"
using assms[unfolded True wq_s_cs] by auto
qed simp
qed (insert assms, simp)
end
context valid_trace_set
begin
lemma wq_kept [simp]:
shows "wq (e#s) cs' = wq s cs'"
using assms unfolding is_set wq_def
by (auto simp:Let_def)
lemma wq_distinct_kept:
assumes "distinct (wq s cs')"
shows "distinct (wq (e#s) cs')"
using assms by simp
end
context valid_trace
begin
lemma finite_threads:
shows "finite (threads s)"
using vt by (induct) (auto elim: step.cases)
lemma finite_readys [simp]: "finite (readys s)"
using finite_threads readys_threads rev_finite_subset by blast
lemma wq_distinct: "distinct (wq s cs)"
proof(induct rule:ind)
case (Cons s e)
interpret vt_e: valid_trace_e s e using Cons by simp
show ?case
proof(cases e)
case (Create th prio)
interpret vt_create: valid_trace_create s e th prio
using Create by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept)
next
case (Exit th)
interpret vt_exit: valid_trace_exit s e th
using Exit by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept)
next
case (P th cs)
interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept)
next
case (V th cs)
interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept)
next
case (Set th prio)
interpret vt_set: valid_trace_set s e th prio
using Set by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept)
qed
qed (unfold wq_def Let_def, simp)
end
section {* Waiting queues and threads *}
context valid_trace_e
begin
lemma wq_out_inv:
assumes s_in: "thread \<in> set (wq s cs)"
and s_hd: "thread = hd (wq s cs)"
and s_i: "thread \<noteq> hd (wq (e#s) cs)"
shows "e = V thread cs"
proof(cases e)
-- {* There are only two non-trivial cases: *}
case (V th cs1)
show ?thesis
proof(cases "cs1 = cs")
case True
have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
thus ?thesis
proof(cases)
case (thread_V)
moreover have "th = thread" using thread_V(2) s_hd
by (unfold s_holding_def wq_def, simp)
ultimately show ?thesis using V True by simp
qed
qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
next
case (P th cs1)
show ?thesis
proof(cases "cs1 = cs")
case True
with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
by (auto simp:wq_def Let_def split:if_splits)
with s_i s_hd s_in have False
by (metis empty_iff hd_append2 list.set(1) wq_def)
thus ?thesis by simp
qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
qed (insert assms, auto simp:wq_def Let_def split:if_splits)
lemma wq_in_inv:
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(cases e)
-- {* This is the only non-trivial case: *}
case (V th cs1)
have False
proof(cases "cs1 = cs")
case True
show ?thesis
proof(cases "(wq s cs1)")
case (Cons w_hd w_tl)
have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
proof -
have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
using Cons V by (auto simp:wq_def Let_def True split:if_splits)
moreover have "set ... \<subseteq> set (wq s cs)"
proof(rule someI2)
show "distinct w_tl \<and> set w_tl = set w_tl"
by (metis distinct.simps(2) local.Cons wq_distinct)
qed (insert Cons True, auto)
ultimately show ?thesis by simp
qed
with assms show ?thesis by auto
qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
thus ?thesis by auto
qed (insert assms, auto simp:wq_def Let_def split:if_splits)
end
lemma (in valid_trace_create)
th_not_in_threads: "th \<notin> threads s"
proof -
from pip_e[unfolded is_create]
show ?thesis by (cases, simp)
qed
lemma (in valid_trace_create)
threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
by (unfold is_create, simp)
lemma (in valid_trace_exit)
threads_es [simp]: "threads (e#s) = threads s - {th}"
by (unfold is_exit, simp)
lemma (in valid_trace_p)
threads_es [simp]: "threads (e#s) = threads s"
by (unfold is_p, simp)
lemma (in valid_trace_v)
threads_es [simp]: "threads (e#s) = threads s"
by (unfold is_v, simp)
lemma (in valid_trace_v)
th_not_in_rest[simp]: "th \<notin> set rest"
proof
assume otherwise: "th \<in> set rest"
have "distinct (wq s cs)" by (simp add: wq_distinct)
from this[unfolded wq_s_cs] and otherwise
show False by auto
qed
lemma (in valid_trace_v) distinct_rest: "distinct rest"
by (simp add: distinct_tl rest_def wq_distinct)
lemma (in valid_trace_v)
set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
proof(unfold wq_es_cs wq'_def, rule someI2)
show "distinct rest \<and> set rest = set rest"
by (simp add: distinct_rest)
next
fix x
assume "distinct x \<and> set x = set rest"
thus "set x = set (wq s cs) - {th}"
by (unfold wq_s_cs, simp)
qed
lemma (in valid_trace_exit)
th_not_in_wq: "th \<notin> set (wq s cs)"
proof -
from pip_e[unfolded is_exit]
show ?thesis
by (cases, unfold holdents_def s_holding_def, fold wq_def,
auto elim!:runing_wqE)
qed
lemma (in valid_trace) wq_threads:
assumes "th \<in> set (wq s cs)"
shows "th \<in> threads s"
using assms
proof(induct rule:ind)
case (Nil)
thus ?case by (auto simp:wq_def)
next
case (Cons s e)
interpret vt_e: valid_trace_e s e using Cons by simp
show ?case
proof(cases e)
case (Create th' prio')
interpret vt: valid_trace_create s e th' prio'
using Create by (unfold_locales, simp)
show ?thesis
using Cons.hyps(2) Cons.prems by auto
next
case (Exit th')
interpret vt: valid_trace_exit s e th'
using Exit by (unfold_locales, simp)
show ?thesis
using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto
next
case (P th' cs')
interpret vt: valid_trace_p s e th' cs'
using P by (unfold_locales, simp)
show ?thesis
using Cons.hyps(2) Cons.prems readys_threads
runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv
by fastforce
next
case (V th' cs')
interpret vt: valid_trace_v s e th' cs'
using V by (unfold_locales, simp)
show ?thesis using Cons
using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
next
case (Set th' prio)
interpret vt: valid_trace_set s e th' prio
using Set by (unfold_locales, simp)
show ?thesis using Cons.hyps(2) Cons.prems vt.is_set
by (auto simp:wq_def Let_def)
qed
qed
section {* RAG and threads *}
context valid_trace
begin
lemma dm_RAG_threads:
assumes in_dom: "(Th th) \<in> Domain (RAG s)"
shows "th \<in> threads s"
proof -
from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
hence "th \<in> set (wq s cs)"
by (unfold s_RAG_def, auto simp:cs_waiting_def)
from wq_threads [OF this] show ?thesis .
qed
lemma rg_RAG_threads:
assumes "(Th th) \<in> Range (RAG s)"
shows "th \<in> threads s"
using assms
by (unfold s_RAG_def cs_waiting_def cs_holding_def,
auto intro:wq_threads)
lemma RAG_threads:
assumes "(Th th) \<in> Field (RAG s)"
shows "th \<in> threads s"
using assms
by (metis Field_def UnE dm_RAG_threads rg_RAG_threads)
end
section {* The change of @{term RAG} *}
text {*
The following three lemmas show that @{text "RAG"} does not change
by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
events, respectively.
*}
lemma (in valid_trace_set) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s"
by (unfold is_set s_RAG_def s_waiting_def wq_def, simp add:Let_def)
lemma (in valid_trace_create) RAG_unchanged [simp]: "(RAG (e # s)) = RAG s"
by (unfold is_create s_RAG_def s_waiting_def wq_def, simp add:Let_def)
lemma (in valid_trace_exit) RAG_unchanged[simp]: "(RAG (e # s)) = RAG s"
by (unfold is_exit s_RAG_def s_waiting_def wq_def, simp add:Let_def)
context valid_trace_v
begin
lemma holding_cs_eq_th:
assumes "holding s t cs"
shows "t = th"
proof -
from pip_e[unfolded is_v]
show ?thesis
proof(cases)
case (thread_V)
from held_unique[OF this(2) assms]
show ?thesis by simp
qed
qed
lemma distinct_wq': "distinct wq'"
by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def)
lemma set_wq': "set wq' = set rest"
by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def)
lemma th'_in_inv:
assumes "th' \<in> set wq'"
shows "th' \<in> set rest"
using assms set_wq' by simp
lemma runing_th_s:
shows "th \<in> runing s"
proof -
from pip_e[unfolded is_v]
show ?thesis by (cases, simp)
qed
lemma neq_t_th:
assumes "waiting (e#s) t c"
shows "t \<noteq> th"
proof
assume otherwise: "t = th"
show False
proof(cases "c = cs")
case True
have "t \<in> set wq'"
using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
by simp
from th'_in_inv[OF this] have "t \<in> set rest" .
with wq_s_cs[folded otherwise] wq_distinct[of cs]
show ?thesis by simp
next
case False
have "wq (e#s) c = wq s c" using False
by (unfold is_v, simp)
hence "waiting s t c" using assms
by (simp add: cs_waiting_def waiting_eq)
hence "t \<notin> readys s" by (unfold readys_def, auto)
hence "t \<notin> runing s" using runing_ready by auto
with runing_th_s[folded otherwise] show ?thesis by auto
qed
qed
lemma waiting_esI1:
assumes "waiting s t c"
and "c \<noteq> cs"
shows "waiting (e#s) t c"
proof -
have "wq (e#s) c = wq s c"
using assms(2) is_v by auto
with assms(1) show ?thesis
using cs_waiting_def waiting_eq by auto
qed
lemma holding_esI2:
assumes "c \<noteq> cs"
and "holding s t c"
shows "holding (e#s) t c"
proof -
from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
from assms(2)[unfolded s_holding_def, folded wq_def,
folded this, unfolded wq_def, folded s_holding_def]
show ?thesis .
qed
lemma holding_esI1:
assumes "holding s t c"
and "t \<noteq> th"
shows "holding (e#s) t c"
proof -
have "c \<noteq> cs" using assms using holding_cs_eq_th by blast
from holding_esI2[OF this assms(1)]
show ?thesis .
qed
end
context valid_trace_v_n
begin
lemma neq_wq': "wq' \<noteq> []"
proof (unfold wq'_def, rule someI2)
show "distinct rest \<and> set rest = set rest"
by (simp add: distinct_rest)
next
fix x
assume " distinct x \<and> set x = set rest"
thus "x \<noteq> []" using rest_nnl by auto
qed
lemma eq_wq': "wq' = taker # rest'"
by (simp add: neq_wq' rest'_def taker_def)
lemma next_th_taker:
shows "next_th s th cs taker"
using rest_nnl taker_def wq'_def wq_s_cs
by (auto simp:next_th_def)
lemma taker_unique:
assumes "next_th s th cs taker'"
shows "taker' = taker"
proof -
from assms
obtain rest' where
h: "wq s cs = th # rest'"
"taker' = hd (SOME q. distinct q \<and> set q = set rest')"
by (unfold next_th_def, auto)
with wq_s_cs have "rest' = rest" by auto
thus ?thesis using h(2) taker_def wq'_def by auto
qed
lemma waiting_set_eq:
"{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
by (smt all_not_in_conv bot.extremum insertI1 insert_subset
mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
lemma holding_set_eq:
"{(Cs cs, Th th') |th'. next_th s th cs th'} = {(Cs cs, Th taker)}"
using next_th_taker taker_def waiting_set_eq
by fastforce
lemma holding_taker:
shows "holding (e#s) taker cs"
by (unfold s_holding_def, fold wq_def, unfold wq_es_cs,
auto simp:neq_wq' taker_def)
lemma waiting_esI2:
assumes "waiting s t cs"
and "t \<noteq> taker"
shows "waiting (e#s) t cs"
proof -
have "t \<in> set wq'"
proof(unfold wq'_def, rule someI2)
show "distinct rest \<and> set rest = set rest"
by (simp add: distinct_rest)
next
fix x
assume "distinct x \<and> set x = set rest"
moreover have "t \<in> set rest"
using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto
ultimately show "t \<in> set x" by simp
qed
moreover have "t \<noteq> hd wq'"
using assms(2) taker_def by auto
ultimately show ?thesis
by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
qed
lemma waiting_esE:
assumes "waiting (e#s) t c"
obtains "c \<noteq> cs" "waiting s t c"
| "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
proof(cases "c = cs")
case False
hence "wq (e#s) c = wq s c" using is_v by auto
with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto
from that(1)[OF False this] show ?thesis .
next
case True
from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
hence "t \<noteq> taker" by (simp add: taker_def)
moreover hence "t \<noteq> th" using assms neq_t_th by blast
moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv)
ultimately have "waiting s t cs"
by (metis cs_waiting_def list.distinct(2) list.sel(1)
list.set_sel(2) rest_def waiting_eq wq_s_cs)
show ?thesis using that(2)
using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto
qed
lemma holding_esI1:
assumes "c = cs"
and "t = taker"
shows "holding (e#s) t c"
by (unfold assms, simp add: holding_taker)
lemma holding_esE:
assumes "holding (e#s) t c"
obtains "c = cs" "t = taker"
| "c \<noteq> cs" "holding s t c"
proof(cases "c = cs")
case True
from assms[unfolded True, unfolded s_holding_def,
folded wq_def, unfolded wq_es_cs]
have "t = taker" by (simp add: taker_def)
from that(1)[OF True this] show ?thesis .
next
case False
hence "wq (e#s) c = wq s c" using is_v by auto
from assms[unfolded s_holding_def, folded wq_def,
unfolded this, unfolded wq_def, folded s_holding_def]
have "holding s t c" .
from that(2)[OF False this] show ?thesis .
qed
end
context valid_trace_v_e
begin
lemma nil_wq': "wq' = []"
proof (unfold wq'_def, rule someI2)
show "distinct rest \<and> set rest = set rest"
by (simp add: distinct_rest)
next
fix x
assume " distinct x \<and> set x = set rest"
thus "x = []" using rest_nil by auto
qed
lemma no_taker:
assumes "next_th s th cs taker"
shows "False"
proof -
from assms[unfolded next_th_def]
obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
by auto
thus ?thesis using rest_def rest_nil by auto
qed
lemma waiting_set_eq:
"{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
using no_taker by auto
lemma holding_set_eq:
"{(Cs cs, Th th') |th'. next_th s th cs th'} = {}"
using no_taker by auto
lemma no_holding:
assumes "holding (e#s) taker cs"
shows False
proof -
from wq_es_cs[unfolded nil_wq']
have " wq (e # s) cs = []" .
from assms[unfolded s_holding_def, folded wq_def, unfolded this]
show ?thesis by auto
qed
lemma no_waiting:
assumes "waiting (e#s) t cs"
shows False
proof -
from wq_es_cs[unfolded nil_wq']
have " wq (e # s) cs = []" .
from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
show ?thesis by auto
qed
lemma waiting_esI2:
assumes "waiting s t c"
shows "waiting (e#s) t c"
proof -
have "c \<noteq> cs" using assms
using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto
from waiting_esI1[OF assms this]
show ?thesis .
qed
lemma waiting_esE:
assumes "waiting (e#s) t c"
obtains "c \<noteq> cs" "waiting s t c"
proof(cases "c = cs")
case False
hence "wq (e#s) c = wq s c" using is_v by auto
with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto
from that(1)[OF False this] show ?thesis .
next
case True
from no_waiting[OF assms[unfolded True]]
show ?thesis by auto
qed
lemma holding_esE:
assumes "holding (e#s) t c"
obtains "c \<noteq> cs" "holding s t c"
proof(cases "c = cs")
case True
from no_holding[OF assms[unfolded True]]
show ?thesis by auto
next
case False
hence "wq (e#s) c = wq s c" using is_v by auto
from assms[unfolded s_holding_def, folded wq_def,
unfolded this, unfolded wq_def, folded s_holding_def]
have "holding s t c" .
from that[OF False this] show ?thesis .
qed
end
context valid_trace_v
begin
lemma RAG_es:
"RAG (e # s) =
RAG 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 = ?R")
proof(rule rel_eqI)
fix n1 n2
assume "(n1, n2) \<in> ?L"
thus "(n1, n2) \<in> ?R"
proof(cases rule:in_RAG_E)
case (waiting th' cs')
show ?thesis
proof(cases "rest = []")
case False
interpret h_n: valid_trace_v_n s e th cs
by (unfold_locales, insert False, simp)
from waiting(3)
show ?thesis
proof(cases rule:h_n.waiting_esE)
case 1
with waiting(1,2)
show ?thesis
by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
fold waiting_eq, auto)
next
case 2
with waiting(1,2)
show ?thesis
by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
fold waiting_eq, auto)
qed
next
case True
interpret h_e: valid_trace_v_e s e th cs
by (unfold_locales, insert True, simp)
from waiting(3)
show ?thesis
proof(cases rule:h_e.waiting_esE)
case 1
with waiting(1,2)
show ?thesis
by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
fold waiting_eq, auto)
qed
qed
next
case (holding th' cs')
show ?thesis
proof(cases "rest = []")
case False
interpret h_n: valid_trace_v_n s e th cs
by (unfold_locales, insert False, simp)
from holding(3)
show ?thesis
proof(cases rule:h_n.holding_esE)
case 1
with holding(1,2)
show ?thesis
by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
fold waiting_eq, auto)
next
case 2
with holding(1,2)
show ?thesis
by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
fold holding_eq, auto)
qed
next
case True
interpret h_e: valid_trace_v_e s e th cs
by (unfold_locales, insert True, simp)
from holding(3)
show ?thesis
proof(cases rule:h_e.holding_esE)
case 1
with holding(1,2)
show ?thesis
by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
fold holding_eq, auto)
qed
qed
qed
next
fix n1 n2
assume h: "(n1, n2) \<in> ?R"
show "(n1, n2) \<in> ?L"
proof(cases "rest = []")
case False
interpret h_n: valid_trace_v_n s e th cs
by (unfold_locales, insert False, simp)
from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
\<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or>
(n2 = Th h_n.taker \<and> n1 = Cs cs)"
by auto
thus ?thesis
proof
assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
with h_n.holding_taker
show ?thesis
by (unfold s_RAG_def, fold holding_eq, auto)
next
assume h: "(n1, n2) \<in> RAG s \<and>
(n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
hence "(n1, n2) \<in> RAG s" by simp
thus ?thesis
proof(cases rule:in_RAG_E)
case (waiting th' cs')
from h and this(1,2)
have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
hence "waiting (e#s) th' cs'"
proof
assume "cs' \<noteq> cs"
from waiting_esI1[OF waiting(3) this]
show ?thesis .
next
assume neq_th': "th' \<noteq> h_n.taker"
show ?thesis
proof(cases "cs' = cs")
case False
from waiting_esI1[OF waiting(3) this]
show ?thesis .
next
case True
from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
show ?thesis .
qed
qed
thus ?thesis using waiting(1,2)
by (unfold s_RAG_def, fold waiting_eq, auto)
next
case (holding th' cs')
from h this(1,2)
have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
hence "holding (e#s) th' cs'"
proof
assume "cs' \<noteq> cs"
from holding_esI2[OF this holding(3)]
show ?thesis .
next
assume "th' \<noteq> th"
from holding_esI1[OF holding(3) this]
show ?thesis .
qed
thus ?thesis using holding(1,2)
by (unfold s_RAG_def, fold holding_eq, auto)
qed
qed
next
case True
interpret h_e: valid_trace_v_e s e th cs
by (unfold_locales, insert True, simp)
from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)"
by auto
from h_s(1)
show ?thesis
proof(cases rule:in_RAG_E)
case (waiting th' cs')
from h_e.waiting_esI2[OF this(3)]
show ?thesis using waiting(1,2)
by (unfold s_RAG_def, fold waiting_eq, auto)
next
case (holding th' cs')
with h_s(2)
have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
thus ?thesis
proof
assume neq_cs: "cs' \<noteq> cs"
from holding_esI2[OF this holding(3)]
show ?thesis using holding(1,2)
by (unfold s_RAG_def, fold holding_eq, auto)
next
assume "th' \<noteq> th"
from holding_esI1[OF holding(3) this]
show ?thesis using holding(1,2)
by (unfold s_RAG_def, fold holding_eq, auto)
qed
qed
qed
qed
lemma
finite_RAG_kept:
assumes "finite (RAG s)"
shows "finite (RAG (e#s))"
proof(cases "rest = []")
case True
interpret vt: valid_trace_v_e using True
by (unfold_locales, simp)
show ?thesis using assms
by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
next
case False
interpret vt: valid_trace_v_n using False
by (unfold_locales, simp)
show ?thesis using assms
by (unfold RAG_es vt.waiting_set_eq vt.holding_set_eq, simp)
qed
end
context valid_trace_p
begin
lemma waiting_kept:
assumes "waiting s th' cs'"
shows "waiting (e#s) th' cs'"
using assms
by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2)
rotate1.simps(2) self_append_conv2 set_rotate1
th_not_in_wq waiting_eq wq_es_cs wq_neq_simp)
lemma holding_kept:
assumes "holding s th' cs'"
shows "holding (e#s) th' cs'"
proof(cases "cs' = cs")
case False
hence "wq (e#s) cs' = wq s cs'" by simp
with assms show ?thesis using cs_holding_def holding_eq by auto
next
case True
from assms[unfolded s_holding_def, folded wq_def]
obtain rest where eq_wq: "wq s cs' = th'#rest"
by (metis empty_iff list.collapse list.set(1))
hence "wq (e#s) cs' = th'#(rest@[th])"
by (simp add: True wq_es_cs)
thus ?thesis
by (simp add: cs_holding_def holding_eq)
qed
end
lemma (in valid_trace_p) th_not_waiting: "\<not> waiting s th c"
proof -
have "th \<in> readys s"
using runing_ready runing_th_s by blast
thus ?thesis
by (unfold readys_def, auto)
qed
context valid_trace_p_h
begin
lemma wq_es_cs': "wq (e#s) cs = [th]"
using wq_es_cs[unfolded we] by simp
lemma holding_es_th_cs:
shows "holding (e#s) th cs"
proof -
from wq_es_cs'
have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto
thus ?thesis using cs_holding_def holding_eq by blast
qed
lemma RAG_edge: "(Cs cs, Th th) \<in> RAG (e#s)"
by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto)
lemma waiting_esE:
assumes "waiting (e#s) th' cs'"
obtains "waiting s th' cs'"
using assms
by (metis cs_waiting_def event.distinct(15) is_p list.sel(1)
set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv)
lemma holding_esE:
assumes "holding (e#s) th' cs'"
obtains "cs' \<noteq> cs" "holding s th' cs'"
| "cs' = cs" "th' = th"
proof(cases "cs' = cs")
case True
from held_unique[OF holding_es_th_cs assms[unfolded True]]
have "th' = th" by simp
from that(2)[OF True this] show ?thesis .
next
case False
have "holding s th' cs'" using assms
using False cs_holding_def holding_eq by auto
from that(1)[OF False this] show ?thesis .
qed
lemma RAG_es: "RAG (e # s) = RAG s \<union> {(Cs cs, Th th)}" (is "?L = ?R")
proof(rule rel_eqI)
fix n1 n2
assume "(n1, n2) \<in> ?L"
thus "(n1, n2) \<in> ?R"
proof(cases rule:in_RAG_E)
case (waiting th' cs')
from this(3)
show ?thesis
proof(cases rule:waiting_esE)
case 1
thus ?thesis using waiting(1,2)
by (unfold s_RAG_def, fold waiting_eq, auto)
qed
next
case (holding th' cs')
from this(3)
show ?thesis
proof(cases rule:holding_esE)
case 1
with holding(1,2)
show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
next
case 2
with holding(1,2) show ?thesis by auto
qed
qed
next
fix n1 n2
assume "(n1, n2) \<in> ?R"
hence "(n1, n2) \<in> RAG s \<or> (n1 = Cs cs \<and> n2 = Th th)" by auto
thus "(n1, n2) \<in> ?L"
proof
assume "(n1, n2) \<in> RAG s"
thus ?thesis
proof(cases rule:in_RAG_E)
case (waiting th' cs')
from waiting_kept[OF this(3)]
show ?thesis using waiting(1,2)
by (unfold s_RAG_def, fold waiting_eq, auto)
next
case (holding th' cs')
from holding_kept[OF this(3)]
show ?thesis using holding(1,2)
by (unfold s_RAG_def, fold holding_eq, auto)
qed
next
assume "n1 = Cs cs \<and> n2 = Th th"
with holding_es_th_cs
show ?thesis
by (unfold s_RAG_def, fold holding_eq, auto)
qed
qed
end
context valid_trace_p_w
begin
lemma wq_s_cs: "wq s cs = holder#waiters"
by (simp add: holder_def waiters_def wne)
lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
by (simp add: wq_es_cs wq_s_cs)
lemma waiting_es_th_cs: "waiting (e#s) th cs"
using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
lemma holding_esE:
assumes "holding (e#s) th' cs'"
obtains "holding s th' cs'"
using assms
proof(cases "cs' = cs")
case False
hence "wq (e#s) cs' = wq s cs'" by simp
with assms show ?thesis
using cs_holding_def holding_eq that by auto
next
case True
with assms show ?thesis
by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that
wq_es_cs' wq_s_cs)
qed
lemma waiting_esE:
assumes "waiting (e#s) th' cs'"
obtains "th' \<noteq> th" "waiting s th' cs'"
| "th' = th" "cs' = cs"
proof(cases "waiting s th' cs'")
case True
have "th' \<noteq> th"
proof
assume otherwise: "th' = th"
from True[unfolded this]
show False by (simp add: th_not_waiting)
qed
from that(1)[OF this True] show ?thesis .
next
case False
hence "th' = th \<and> cs' = cs"
by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2)
set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
with that(2) show ?thesis by metis
qed
lemma RAG_es: "RAG (e # s) = RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
proof(rule rel_eqI)
fix n1 n2
assume "(n1, n2) \<in> ?L"
thus "(n1, n2) \<in> ?R"
proof(cases rule:in_RAG_E)
case (waiting th' cs')
from this(3)
show ?thesis
proof(cases rule:waiting_esE)
case 1
thus ?thesis using waiting(1,2)
by (unfold s_RAG_def, fold waiting_eq, auto)
next
case 2
thus ?thesis using waiting(1,2) by auto
qed
next
case (holding th' cs')
from this(3)
show ?thesis
proof(cases rule:holding_esE)
case 1
with holding(1,2)
show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
qed
qed
next
fix n1 n2
assume "(n1, n2) \<in> ?R"
hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
thus "(n1, n2) \<in> ?L"
proof
assume "(n1, n2) \<in> RAG s"
thus ?thesis
proof(cases rule:in_RAG_E)
case (waiting th' cs')
from waiting_kept[OF this(3)]
show ?thesis using waiting(1,2)
by (unfold s_RAG_def, fold waiting_eq, auto)
next
case (holding th' cs')
from holding_kept[OF this(3)]
show ?thesis using holding(1,2)
by (unfold s_RAG_def, fold holding_eq, auto)
qed
next
assume "n1 = Th th \<and> n2 = Cs cs"
thus ?thesis using RAG_edge by auto
qed
qed
end
context valid_trace_p
begin
lemma RAG_es: "RAG (e # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
else RAG s \<union> {(Th th, Cs cs)})"
proof(cases "wq s cs = []")
case True
interpret vt_p: valid_trace_p_h using True
by (unfold_locales, simp)
show ?thesis by (simp add: vt_p.RAG_es vt_p.we)
next
case False
interpret vt_p: valid_trace_p_w using False
by (unfold_locales, simp)
show ?thesis by (simp add: vt_p.RAG_es vt_p.wne)
qed
end
section {* Finiteness of RAG *}
context valid_trace
begin
lemma finite_RAG:
shows "finite (RAG s)"
proof(induct rule:ind)
case Nil
show ?case
by (auto simp: s_RAG_def cs_waiting_def
cs_holding_def wq_def acyclic_def)
next
case (Cons s e)
interpret vt_e: valid_trace_e s e using Cons by simp
show ?case
proof(cases e)
case (Create th prio)
interpret vt: valid_trace_create s e th prio using Create
by (unfold_locales, simp)
show ?thesis using Cons by simp
next
case (Exit th)
interpret vt: valid_trace_exit s e th using Exit
by (unfold_locales, simp)
show ?thesis using Cons by simp
next
case (P th cs)
interpret vt: valid_trace_p s e th cs using P
by (unfold_locales, simp)
show ?thesis using Cons using vt.RAG_es by auto
next
case (V th cs)
interpret vt: valid_trace_v s e th cs using V
by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt.finite_RAG_kept)
next
case (Set th prio)
interpret vt: valid_trace_set s e th prio using Set
by (unfold_locales, simp)
show ?thesis using Cons by simp
qed
qed
end
section {* RAG is acyclic *}
text {* (* ddd *)
The nature of the work is like this: since it starts from a very simple and basic
model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
For instance, the fact
that one thread can not be blocked by two critical resources at the same time
is obvious, because only running threads can make new requests, if one is waiting for
a critical resource and get blocked, it can not make another resource request and get
blocked the second time (because it is not running).
To derive this fact, one needs to prove by contraction and
reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
named @{text "p_split"}, which is about status changing along the time axis. It says if
a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"}
in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history
of events leading to it), such that @{text "Q"} switched
from being @{text "False"} to @{text "True"} and kept being @{text "True"}
till the last moment of @{text "s"}.
Suppose a thread @{text "th"} is blocked
on @{text "cs1"} and @{text "cs2"} in some state @{text "s"},
since no thread is blocked at the very beginning, by applying
@{text "p_split"} to these two blocking facts, there exist
two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that
@{text "th"} got blocked on @{text "cs1"} and @{text "cs2"}
and kept on blocked on them respectively ever since.
Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
in blocked state at moment @{text "t2"} and could not
make any request and get blocked the second time: Contradiction.
*}
context valid_trace
begin
lemma waiting_unique_pre: (* ddd *)
assumes 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" = "\<lambda> 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> ?Q cs1 (moment t1 s)"
and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
from p_split [of "?Q cs2", OF q2 nq2]
obtain t2 where lt2: "t2 < length s"
and np2: "\<not> ?Q cs2 (moment t2 s)"
and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
{ fix s cs
assume q: "?Q cs s"
have "thread \<notin> runing s"
proof
assume "thread \<in> runing s"
hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and>
thread \<noteq> hd (wq_fun (schs s) cs))"
by (unfold runing_def s_waiting_def readys_def, auto)
from this[rule_format, of cs] q
show False by (simp add: wq_def)
qed
} note q_not_runing = this
{ fix t1 t2 cs1 cs2
assume lt1: "t1 < length s"
and np1: "\<not> ?Q cs1 (moment t1 s)"
and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
and lt2: "t2 < length s"
and np2: "\<not> ?Q cs2 (moment t2 s)"
and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
and lt12: "t1 < t2"
let ?t3 = "Suc t2"
interpret ve2: valid_moment_e _ t2 using lt2
by (unfold_locales, simp)
let ?e = ve2.next_e
have "t2 < ?t3" by simp
from nn2 [rule_format, OF this] and ve2.trace_e
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 ?thesis
proof -
have "thread \<in> runing (moment t2 s)"
proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
case True
have "?e = V thread cs2"
proof -
have eq_th: "thread = hd (wq (moment t2 s) cs2)"
using True and np2 by auto
thus ?thesis
using True h2 ve2.vat_moment_e.wq_out_inv by blast
qed
thus ?thesis
using step.cases ve2.vat_moment_e.pip_e by auto
next
case False
hence "?e = P thread cs2"
using h1 ve2.vat_moment_e.wq_in_inv by blast
thus ?thesis
using step.cases ve2.vat_moment_e.pip_e by auto
qed
moreover have "thread \<notin> runing (moment t2 s)"
by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
ultimately show ?thesis by simp
qed
} note lt_case = this
show ?thesis
proof -
{ assume "t1 < t2"
from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
have ?thesis .
} moreover {
assume "t2 < t1"
from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
have ?thesis .
} moreover {
assume eq_12: "t1 = t2"
let ?t3 = "Suc t2"
interpret ve2: valid_moment_e _ t2 using lt2
by (unfold_locales, simp)
let ?e = ve2.next_e
have "t2 < ?t3" by simp
from nn2 [rule_format, OF this] and ve2.trace_e
have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" by auto
have lt_2: "t2 < ?t3" by simp
from nn2 [rule_format, OF this] and ve2.trace_e
have h1: "thread \<in> set (wq (?e#moment t2 s) cs2)" and
h2: "thread \<noteq> hd (wq (?e#moment t2 s) cs2)" by auto
from nn1[rule_format, OF lt_2[folded eq_12], unfolded ve2.trace_e[folded eq_12]]
eq_12[symmetric]
have g1: "thread \<in> set (wq (?e#moment t1 s) cs1)" and
g2: "thread \<noteq> hd (wq (?e#moment t1 s) cs1)" by auto
have "?e = V thread cs2 \<or> ?e = P thread cs2"
using h1 h2 np2 ve2.vat_moment_e.wq_in_inv
ve2.vat_moment_e.wq_out_inv by blast
moreover have "?e = V thread cs1 \<or> ?e = P thread cs1"
using eq_12 g1 g2 np1 ve2.vat_moment_e.wq_in_inv
ve2.vat_moment_e.wq_out_inv by blast
ultimately have ?thesis using neq12 by auto
} ultimately show ?thesis using nat_neq_iff by blast
qed
qed
text {*
This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
*}
lemma waiting_unique:
assumes "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
end
lemma (in valid_trace_v)
preced_es [simp]: "preced th (e#s) = preced th s"
by (unfold is_v preced_def, simp)
lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
proof
fix th'
show "the_preced (V th cs # s) th' = the_preced s th'"
by (unfold the_preced_def preced_def, simp)
qed
lemma (in valid_trace_v)
the_preced_es: "the_preced (e#s) = the_preced s"
by (unfold is_v preced_def, simp)
context valid_trace_p
begin
lemma not_holding_s_th_cs: "\<not> holding s th cs"
proof
assume otherwise: "holding s th cs"
from pip_e[unfolded is_p]
show False
proof(cases)
case (thread_P)
moreover have "(Cs cs, Th th) \<in> RAG s"
using otherwise cs_holding_def
holding_eq th_not_in_wq by auto
ultimately show ?thesis by auto
qed
qed
end
lemma (in valid_trace_v_n) finite_waiting_set:
"finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
by (simp add: waiting_set_eq)
lemma (in valid_trace_v_n) finite_holding_set:
"finite {(Cs cs, Th th') |th'. next_th s th cs th'}"
by (simp add: holding_set_eq)
lemma (in valid_trace_v_e) finite_waiting_set:
"finite {(Th th', Cs cs) |th'. next_th s th cs th'}"
by (simp add: waiting_set_eq)
lemma (in valid_trace_v_e) finite_holding_set:
"finite {(Cs cs, Th th') |th'. next_th s th cs th'}"
by (simp add: holding_set_eq)
context valid_trace_v_e
begin
lemma
acylic_RAG_kept:
assumes "acyclic (RAG s)"
shows "acyclic (RAG (e#s))"
proof(rule acyclic_subset[OF assms])
show "RAG (e # s) \<subseteq> RAG s"
by (unfold RAG_es waiting_set_eq holding_set_eq, auto)
qed
end
context valid_trace_v_n
begin
lemma waiting_taker: "waiting s taker cs"
apply (unfold s_waiting_def, fold wq_def, unfold wq_s_cs taker_def)
using eq_wq' th'_in_inv wq'_def by fastforce
lemma
acylic_RAG_kept:
assumes "acyclic (RAG s)"
shows "acyclic (RAG (e#s))"
proof -
have "acyclic ((RAG s - {(Cs cs, Th th)} - {(Th taker, Cs cs)}) \<union>
{(Cs cs, Th taker)})" (is "acyclic (?A \<union> _)")
proof -
from assms
have "acyclic ?A"
by (rule acyclic_subset, auto)
moreover have "(Th taker, Cs cs) \<notin> ?A^*"
proof
assume otherwise: "(Th taker, Cs cs) \<in> ?A^*"
hence "(Th taker, Cs cs) \<in> ?A^+"
by (unfold rtrancl_eq_or_trancl, auto)
from tranclD[OF this]
obtain cs' where h: "(Th taker, Cs cs') \<in> ?A"
"(Th taker, Cs cs') \<in> RAG s"
by (unfold s_RAG_def, auto)
from this(2) have "waiting s taker cs'"
by (unfold s_RAG_def, fold waiting_eq, auto)
from waiting_unique[OF this waiting_taker]
have "cs' = cs" .
from h(1)[unfolded this] show False by auto
qed
ultimately show ?thesis by auto
qed
thus ?thesis
by (unfold RAG_es waiting_set_eq holding_set_eq, simp)
qed
end
context valid_trace_p_h
begin
lemma
acylic_RAG_kept:
assumes "acyclic (RAG s)"
shows "acyclic (RAG (e#s))"
proof -
have "acyclic (RAG s \<union> {(Cs cs, Th th)})" (is "acyclic (?A \<union> _)")
proof -
from assms
have "acyclic ?A"
by (rule acyclic_subset, auto)
moreover have "(Th th, Cs cs) \<notin> ?A^*"
proof
assume otherwise: "(Th th, Cs cs) \<in> ?A^*"
hence "(Th th, Cs cs) \<in> ?A^+"
by (unfold rtrancl_eq_or_trancl, auto)
from tranclD[OF this]
obtain cs' where h: "(Th th, Cs cs') \<in> RAG s"
by (unfold s_RAG_def, auto)
hence "waiting s th cs'"
by (unfold s_RAG_def, fold waiting_eq, auto)
with th_not_waiting show False by auto
qed
ultimately show ?thesis by auto
qed
thus ?thesis by (unfold RAG_es, simp)
qed
end
context valid_trace_p_w
begin
lemma
acylic_RAG_kept:
assumes "acyclic (RAG s)"
shows "acyclic (RAG (e#s))"
proof -
have "acyclic (RAG s \<union> {(Th th, Cs cs)})" (is "acyclic (?A \<union> _)")
proof -
from assms
have "acyclic ?A"
by (rule acyclic_subset, auto)
moreover have "(Cs cs, Th th) \<notin> ?A^*"
proof
assume otherwise: "(Cs cs, Th th) \<in> ?A^*"
from pip_e[unfolded is_p]
show False
proof(cases)
case (thread_P)
moreover from otherwise have "(Cs cs, Th th) \<in> ?A^+"
by (unfold rtrancl_eq_or_trancl, auto)
ultimately show ?thesis by auto
qed
qed
ultimately show ?thesis by auto
qed
thus ?thesis by (unfold RAG_es, simp)
qed
end
context valid_trace
begin
lemma acyclic_RAG:
shows "acyclic (RAG s)"
proof(induct rule:ind)
case Nil
show ?case
by (auto simp: s_RAG_def cs_waiting_def
cs_holding_def wq_def acyclic_def)
next
case (Cons s e)
interpret vt_e: valid_trace_e s e using Cons by simp
show ?case
proof(cases e)
case (Create th prio)
interpret vt: valid_trace_create s e th prio using Create
by (unfold_locales, simp)
show ?thesis using Cons by simp
next
case (Exit th)
interpret vt: valid_trace_exit s e th using Exit
by (unfold_locales, simp)
show ?thesis using Cons by simp
next
case (P th cs)
interpret vt: valid_trace_p s e th cs using P
by (unfold_locales, simp)
show ?thesis
proof(cases "wq s cs = []")
case True
then interpret vt_h: valid_trace_p_h s e th cs
by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_h.acylic_RAG_kept)
next
case False
then interpret vt_w: valid_trace_p_w s e th cs
by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_w.acylic_RAG_kept)
qed
next
case (V th cs)
interpret vt: valid_trace_v s e th cs using V
by (unfold_locales, simp)
show ?thesis
proof(cases "vt.rest = []")
case True
then interpret vt_e: valid_trace_v_e s e th cs
by (unfold_locales, simp)
show ?thesis by (simp add: Cons.hyps(2) vt_e.acylic_RAG_kept)
next
case False
then interpret vt_n: valid_trace_v_n s e th cs
by (unfold_locales, simp)
show ?thesis by (simp add: Cons.hyps(2) vt_n.acylic_RAG_kept)
qed
next
case (Set th prio)
interpret vt: valid_trace_set s e th prio using Set
by (unfold_locales, simp)
show ?thesis using Cons by simp
qed
qed
end
section {* RAG is single-valued *}
context valid_trace
begin
lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
by(auto elim:waiting_unique held_unique)
lemma sgv_RAG: "single_valued (RAG s)"
using unique_RAG by (auto simp:single_valued_def)
end
section {* RAG is well-founded *}
context valid_trace
begin
lemma wf_RAG: "wf (RAG s)"
proof(rule finite_acyclic_wf)
from finite_RAG show "finite (RAG s)" .
next
from acyclic_RAG show "acyclic (RAG s)" .
qed
lemma wf_RAG_converse:
shows "wf ((RAG s)^-1)"
proof(rule finite_acyclic_wf_converse)
from finite_RAG
show "finite (RAG s)" .
next
from acyclic_RAG
show "acyclic (RAG s)" .
qed
end
section {* RAG forms a forest (or tree) *}
context valid_trace
begin
lemma rtree_RAG: "rtree (RAG s)"
using sgv_RAG acyclic_RAG
by (unfold rtree_def rtree_axioms_def sgv_def, auto)
end
sublocale valid_trace < rtree_RAG: rtree "RAG s"
using rtree_RAG .
sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
proof -
show "fsubtree (RAG s)"
proof(intro_locales)
show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
next
show "fsubtree_axioms (RAG s)"
proof(unfold fsubtree_axioms_def)
from wf_RAG show "wf (RAG s)" .
qed
qed
qed
section {* Derived properties for parts of RAG *}
context valid_trace
begin
lemma acyclic_tRAG: "acyclic (tRAG s)"
proof(unfold tRAG_def, rule acyclic_compose)
show "acyclic (RAG s)" using acyclic_RAG .
next
show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto
next
show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto
qed
lemma sgv_wRAG: "single_valued (wRAG s)"
using waiting_unique
by (unfold single_valued_def wRAG_def, auto)
lemma sgv_hRAG: "single_valued (hRAG s)"
using held_unique
by (unfold single_valued_def hRAG_def, auto)
lemma sgv_tRAG: "single_valued (tRAG s)"
by (unfold tRAG_def, rule single_valued_relcomp,
insert sgv_wRAG sgv_hRAG, auto)
end
sublocale valid_trace < rtree_s: rtree "tRAG s"
proof(unfold_locales)
from sgv_tRAG show "single_valued (tRAG s)" .
next
from acyclic_tRAG show "acyclic (tRAG s)" .
qed
sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
proof -
have "fsubtree (tRAG s)"
proof -
have "fbranch (tRAG s)"
proof(unfold tRAG_def, rule fbranch_compose)
show "fbranch (wRAG s)"
proof(rule finite_fbranchI)
from finite_RAG show "finite (wRAG s)"
by (unfold RAG_split, auto)
qed
next
show "fbranch (hRAG s)"
proof(rule finite_fbranchI)
from finite_RAG
show "finite (hRAG s)" by (unfold RAG_split, auto)
qed
qed
moreover have "wf (tRAG s)"
proof(rule wf_subset)
show "wf (RAG s O RAG s)" using wf_RAG
by (fold wf_comp_self, simp)
next
show "tRAG s \<subseteq> (RAG s O RAG s)"
by (unfold tRAG_alt_def, auto)
qed
ultimately show ?thesis
by (unfold fsubtree_def fsubtree_axioms_def,auto)
qed
from this[folded tRAG_def] show "fsubtree (tRAG s)" .
qed
lemma tRAG_nodeE:
assumes "(n1, n2) \<in> tRAG s"
obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
using assms
by (auto simp: tRAG_def wRAG_def hRAG_def)
lemma tRAG_ancestorsE:
assumes "x \<in> ancestors (tRAG s) u"
obtains th where "x = Th th"
proof -
from assms have "(u, x) \<in> (tRAG s)^+"
by (unfold ancestors_def, auto)
from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
then obtain th where "x = Th th"
by (unfold tRAG_alt_def, auto)
from that[OF this] show ?thesis .
qed
lemma subtree_nodeE:
assumes "n \<in> subtree (tRAG s) (Th th)"
obtains th1 where "n = Th th1"
proof -
show ?thesis
proof(rule subtreeE[OF assms])
assume "n = Th th"
from that[OF this] show ?thesis .
next
assume "Th th \<in> ancestors (tRAG s) n"
hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
hence "\<exists> th1. n = Th th1"
proof(induct)
case (base y)
from tRAG_nodeE[OF this] show ?case by metis
next
case (step y z)
thus ?case by auto
qed
with that show ?thesis by auto
qed
qed
lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
proof -
have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*"
by (rule rtrancl_mono, auto simp:RAG_split)
also have "... \<subseteq> ((RAG s)^*)^*"
by (rule rtrancl_mono, auto)
also have "... = (RAG s)^*" by simp
finally show ?thesis by (unfold tRAG_def, simp)
qed
lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
proof -
{ fix a
assume "a \<in> subtree (tRAG s) x"
hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
with tRAG_star_RAG
have "(a, x) \<in> (RAG s)^*" by auto
hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
} thus ?thesis by auto
qed
lemma tRAG_trancl_eq:
"{th'. (Th th', Th th) \<in> (tRAG s)^+} =
{th'. (Th th', Th th) \<in> (RAG s)^+}"
(is "?L = ?R")
proof -
{ fix th'
assume "th' \<in> ?L"
hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
from tranclD[OF this]
obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
from tRAG_subtree_RAG and this(2)
have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG)
moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto
ultimately have "th' \<in> ?R" by auto
} moreover
{ fix th'
assume "th' \<in> ?R"
hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
from plus_rpath[OF this]
obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
hence "(Th th', Th th) \<in> (tRAG s)^+"
proof(induct xs arbitrary:th' th rule:length_induct)
case (1 xs th' th)
then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
show ?case
proof(cases "xs1")
case Nil
from 1(2)[unfolded Cons1 Nil]
have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
hence "(Th th', x1) \<in> (RAG s)"
by (cases, auto)
then obtain cs where "x1 = Cs cs"
by (unfold s_RAG_def, auto)
from rpath_nnl_lastE[OF rp[unfolded this]]
show ?thesis by auto
next
case (Cons x2 xs2)
from 1(2)[unfolded Cons1[unfolded this]]
have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
from rpath_edges_on[OF this]
have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
by (simp add: edges_on_unfold)
with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
by (simp add: edges_on_unfold)
from this eds
have rg2: "(x1, x2) \<in> RAG s" by auto
from this[unfolded eq_x1]
obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
from rp have "rpath (RAG s) x2 xs2 (Th th)"
by (elim rpath_ConsE, simp)
from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
show ?thesis
proof(cases "xs2 = []")
case True
from rpath_nilE[OF rp'[unfolded this]]
have "th1 = th" by auto
from rt1[unfolded this] show ?thesis by auto
next
case False
from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
with rt1 show ?thesis by auto
qed
qed
qed
hence "th' \<in> ?L" by auto
} ultimately show ?thesis by blast
qed
lemma tRAG_trancl_eq_Th:
"{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} =
{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}"
using tRAG_trancl_eq by auto
lemma tRAG_Field:
"Field (tRAG s) \<subseteq> Field (RAG s)"
by (unfold tRAG_alt_def Field_def, auto)
lemma tRAG_mono:
assumes "RAG s' \<subseteq> RAG s"
shows "tRAG s' \<subseteq> tRAG s"
using assms
by (unfold tRAG_alt_def, auto)
lemma tRAG_subtree_eq:
"(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}"
(is "?L = ?R")
proof -
{ fix n
assume h: "n \<in> ?L"
hence "n \<in> ?R"
by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG)
} moreover {
fix n
assume "n \<in> ?R"
then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
by (auto simp:subtree_def)
from rtranclD[OF this(2)]
have "n \<in> ?L"
proof
assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
with h have "n \<in> {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}" by auto
thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
qed (insert h, auto simp:subtree_def)
} ultimately show ?thesis by auto
qed
lemma threads_set_eq:
"the_thread ` (subtree (tRAG s) (Th th)) =
{th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
context valid_trace
begin
lemma RAG_tRAG_transfer:
assumes "RAG s' = RAG s \<union> {(Th th, Cs cs)}"
and "(Cs cs, Th th'') \<in> RAG s"
shows "tRAG s' = tRAG s \<union> {(Th th, Th th'')}" (is "?L = ?R")
proof -
{ fix n1 n2
assume "(n1, n2) \<in> ?L"
from this[unfolded tRAG_alt_def]
obtain th1 th2 cs' where
h: "n1 = Th th1" "n2 = Th th2"
"(Th th1, Cs cs') \<in> RAG s'"
"(Cs cs', Th th2) \<in> RAG s'" by auto
from h(4) and assms(1) have cs_in: "(Cs cs', Th th2) \<in> RAG s" by auto
from h(3) and assms(1)
have "(Th th1, Cs cs') = (Th th, Cs cs) \<or>
(Th th1, Cs cs') \<in> RAG s" by auto
hence "(n1, n2) \<in> ?R"
proof
assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
hence eq_th1: "th1 = th" by simp
moreover have "th2 = th''"
proof -
from h1 have "cs' = cs" by simp
from assms(2) cs_in[unfolded this]
show ?thesis using unique_RAG by auto
qed
ultimately show ?thesis using h(1,2) by auto
next
assume "(Th th1, Cs cs') \<in> RAG s"
with cs_in have "(Th th1, Th th2) \<in> tRAG s"
by (unfold tRAG_alt_def, auto)
from this[folded h(1, 2)] show ?thesis by auto
qed
} moreover {
fix n1 n2
assume "(n1, n2) \<in> ?R"
hence "(n1, n2) \<in>tRAG s \<or> (n1, n2) = (Th th, Th th'')" by auto
hence "(n1, n2) \<in> ?L"
proof
assume "(n1, n2) \<in> tRAG s"
moreover have "... \<subseteq> ?L"
proof(rule tRAG_mono)
show "RAG s \<subseteq> RAG s'" by (unfold assms(1), auto)
qed
ultimately show ?thesis by auto
next
assume eq_n: "(n1, n2) = (Th th, Th th'')"
from assms(1, 2) have "(Cs cs, Th th'') \<in> RAG s'" by auto
moreover have "(Th th, Cs cs) \<in> RAG s'" using assms(1) by auto
ultimately show ?thesis
by (unfold eq_n tRAG_alt_def, auto)
qed
} ultimately show ?thesis by auto
qed
lemma subtree_tRAG_thread:
assumes "th \<in> threads s"
shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
proof -
have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
by (unfold tRAG_subtree_eq, simp)
also have "... \<subseteq> ?R"
proof
fix x
assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
from this(2)
show "x \<in> ?R"
proof(cases rule:subtreeE)
case 1
thus ?thesis by (simp add: assms h(1))
next
case 2
thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI)
qed
qed
finally show ?thesis .
qed
lemma dependants_alt_def:
"dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
lemma dependants_alt_def1:
"dependants (s::state) th = {th'. (Th th', Th th) \<in> (RAG s)^+}"
using dependants_alt_def tRAG_trancl_eq by auto
end
section {* Chain to readys *}
context valid_trace
begin
lemma chain_building:
assumes "node \<in> Domain (RAG s)"
obtains th' where "th' \<in> readys s" "(node, Th th') \<in> (RAG s)^+"
proof -
from assms have "node \<in> Range ((RAG s)^-1)" by auto
from wf_base[OF wf_RAG_converse this]
obtain b where h_b: "(b, node) \<in> ((RAG s)\<inverse>)\<^sup>+" "\<forall>c. (c, b) \<notin> (RAG s)\<inverse>" by auto
obtain th' where eq_b: "b = Th th'"
proof(cases b)
case (Cs cs)
from h_b(1)[unfolded trancl_converse]
have "(node, b) \<in> ((RAG s)\<^sup>+)" by auto
from tranclE[OF this]
obtain n where "(n, b) \<in> RAG s" by auto
from this[unfolded Cs]
obtain th1 where "waiting s th1 cs"
by (unfold s_RAG_def, fold waiting_eq, auto)
from waiting_holding[OF this]
obtain th2 where "holding s th2 cs" .
hence "(Cs cs, Th th2) \<in> RAG s"
by (unfold s_RAG_def, fold holding_eq, auto)
with h_b(2)[unfolded Cs, rule_format]
have False by auto
thus ?thesis by auto
qed auto
have "th' \<in> readys s"
proof -
from h_b(2)[unfolded eq_b]
have "\<forall>cs. \<not> waiting s th' cs"
by (unfold s_RAG_def, fold waiting_eq, auto)
moreover have "th' \<in> threads s"
proof(rule rg_RAG_threads)
from tranclD[OF h_b(1), unfolded eq_b]
obtain z where "(z, Th th') \<in> (RAG s)" by auto
thus "Th th' \<in> Range (RAG s)" by auto
qed
ultimately show ?thesis by (auto simp:readys_def)
qed
moreover have "(node, Th th') \<in> (RAG s)^+"
using h_b(1)[unfolded trancl_converse] eq_b by auto
ultimately show ?thesis using that by metis
qed
text {* \noindent
The following is just an instance of @{text "chain_building"}.
*}
lemma th_chain_to_ready:
assumes th_in: "th \<in> threads s"
shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG 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 (RAG s)"
by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
from chain_building [rule_format, OF this]
show ?thesis by auto
qed
lemma finite_subtree_threads:
"finite {th'. Th th' \<in> subtree (RAG s) (Th th)}" (is "finite ?A")
proof -
have "?A = the_thread ` {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}"
by (auto, insert image_iff, fastforce)
moreover have "finite {Th th' | th' . Th th' \<in> subtree (RAG s) (Th th)}"
(is "finite ?B")
proof -
have "?B = (subtree (RAG s) (Th th)) \<inter> {Th th' | th'. True}"
by auto
moreover have "... \<subseteq> (subtree (RAG s) (Th th))" by auto
moreover have "finite ..." by (simp add: fsbtRAGs.finite_subtree)
ultimately show ?thesis by auto
qed
ultimately show ?thesis by auto
qed
lemma runing_unique:
assumes 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"
unfolding runing_def by auto
from this[unfolded cp_alt_def]
have eq_max:
"Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th1)}) =
Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th2)})"
(is "Max ?L = Max ?R") .
have "Max ?L \<in> ?L"
proof(rule Max_in)
show "finite ?L" by (simp add: finite_subtree_threads)
next
show "?L \<noteq> {}" using subtree_def by fastforce
qed
then obtain th1' where
h_1: "Th th1' \<in> subtree (RAG s) (Th th1)" "the_preced s th1' = Max ?L"
by auto
have "Max ?R \<in> ?R"
proof(rule Max_in)
show "finite ?R" by (simp add: finite_subtree_threads)
next
show "?R \<noteq> {}" using subtree_def by fastforce
qed
then obtain th2' where
h_2: "Th th2' \<in> subtree (RAG s) (Th th2)" "the_preced s th2' = Max ?R"
by auto
have "th1' = th2'"
proof(rule preced_unique)
from h_1(1)
show "th1' \<in> threads s"
proof(cases rule:subtreeE)
case 1
hence "th1' = th1" by simp
with runing_1 show ?thesis by (auto simp:runing_def readys_def)
next
case 2
from this(2)
have "(Th th1', Th th1) \<in> (RAG s)^+" by (auto simp:ancestors_def)
from tranclD[OF this]
have "(Th th1') \<in> Domain (RAG s)" by auto
from dm_RAG_threads[OF this] show ?thesis .
qed
next
from h_2(1)
show "th2' \<in> threads s"
proof(cases rule:subtreeE)
case 1
hence "th2' = th2" by simp
with runing_2 show ?thesis by (auto simp:runing_def readys_def)
next
case 2
from this(2)
have "(Th th2', Th th2) \<in> (RAG s)^+" by (auto simp:ancestors_def)
from tranclD[OF this]
have "(Th th2') \<in> Domain (RAG s)" by auto
from dm_RAG_threads[OF this] show ?thesis .
qed
next
have "the_preced s th1' = the_preced s th2'"
using eq_max h_1(2) h_2(2) by metis
thus "preced th1' s = preced th2' s" by (simp add:the_preced_def)
qed
from h_1(1)[unfolded this]
have star1: "(Th th2', Th th1) \<in> (RAG s)^*" by (auto simp:subtree_def)
from h_2(1)[unfolded this]
have star2: "(Th th2', Th th2) \<in> (RAG s)^*" by (auto simp:subtree_def)
from star_rpath[OF star1] obtain xs1
where rp1: "rpath (RAG s) (Th th2') xs1 (Th th1)"
by auto
from star_rpath[OF star2] obtain xs2
where rp2: "rpath (RAG s) (Th th2') xs2 (Th th2)"
by auto
from rp1 rp2
show ?thesis
proof(cases)
case (less_1 xs')
moreover have "xs' = []"
proof(rule ccontr)
assume otherwise: "xs' \<noteq> []"
from rpath_plus[OF less_1(3) this]
have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+" .
from tranclD[OF this]
obtain cs where "waiting s th1 cs"
by (unfold s_RAG_def, fold waiting_eq, auto)
with runing_1 show False
by (unfold runing_def readys_def, auto)
qed
ultimately have "xs2 = xs1" by simp
from rpath_dest_eq[OF rp1 rp2[unfolded this]]
show ?thesis by simp
next
case (less_2 xs')
moreover have "xs' = []"
proof(rule ccontr)
assume otherwise: "xs' \<noteq> []"
from rpath_plus[OF less_2(3) this]
have "(Th th2, Th th1) \<in> (RAG s)\<^sup>+" .
from tranclD[OF this]
obtain cs where "waiting s th2 cs"
by (unfold s_RAG_def, fold waiting_eq, auto)
with runing_2 show False
by (unfold runing_def readys_def, auto)
qed
ultimately have "xs2 = xs1" by simp
from rpath_dest_eq[OF rp1 rp2[unfolded this]]
show ?thesis by simp
qed
qed
lemma card_runing: "card (runing s) \<le> 1"
proof(cases "runing s = {}")
case True
thus ?thesis by auto
next
case False
then obtain th where [simp]: "th \<in> runing s" by auto
from runing_unique[OF this]
have "runing s = {th}" by auto
thus ?thesis by auto
qed
end
section {* Relating @{term cp} and @{term the_preced} and @{term preced} *}
context valid_trace
begin
lemma le_cp:
shows "preced th s \<le> cp s th"
proof(unfold cp_alt_def, rule Max_ge)
show "finite (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})"
by (simp add: finite_subtree_threads)
next
show "preced th s \<in> the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}"
by (simp add: subtree_def the_preced_def)
qed
lemma cp_le:
assumes th_in: "th \<in> threads s"
shows "cp s th \<le> Max (the_preced s ` threads s)"
proof(unfold cp_alt_def, rule Max_f_mono)
show "finite (threads s)" by (simp add: finite_threads)
next
show " {th'. Th th' \<in> subtree (RAG s) (Th th)} \<noteq> {}"
using subtree_def by fastforce
next
show "{th'. Th th' \<in> subtree (RAG s) (Th th)} \<subseteq> threads s"
using assms
by (smt Domain.DomainI dm_RAG_threads mem_Collect_eq
node.inject(1) rtranclD subsetI subtree_def trancl_domain)
qed
lemma max_cp_eq:
shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
(is "?L = ?R")
proof -
have "?L \<le> ?R"
proof(cases "threads s = {}")
case False
show ?thesis
by (rule Max.boundedI,
insert cp_le,
auto simp:finite_threads False)
qed auto
moreover have "?R \<le> ?L"
by (rule Max_fg_mono,
simp add: finite_threads,
simp add: le_cp the_preced_def)
ultimately show ?thesis by auto
qed
lemma threads_alt_def:
"(threads s) = (\<Union> th \<in> readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)})"
(is "?L = ?R")
proof -
{ fix th1
assume "th1 \<in> ?L"
from th_chain_to_ready[OF this]
have "th1 \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th th1, Th th') \<in> (RAG s)\<^sup>+)" .
hence "th1 \<in> ?R" by (auto simp:subtree_def)
} moreover
{ fix th'
assume "th' \<in> ?R"
then obtain th where h: "th \<in> readys s" " Th th' \<in> subtree (RAG s) (Th th)"
by auto
from this(2)
have "th' \<in> ?L"
proof(cases rule:subtreeE)
case 1
with h(1) show ?thesis by (auto simp:readys_def)
next
case 2
from tranclD[OF this(2)[unfolded ancestors_def, simplified]]
have "Th th' \<in> Domain (RAG s)" by auto
from dm_RAG_threads[OF this]
show ?thesis .
qed
} ultimately show ?thesis by auto
qed
text {* (* ccc *) \noindent
Since the current precedence of the threads in ready queue will always be boosted,
there must be one inside it has the maximum precedence of the whole system.
*}
lemma max_cp_readys_threads:
shows "Max (cp s ` readys s) = Max (cp s ` threads s)" (is "?L = ?R")
proof(cases "readys s = {}")
case False
have "?R = Max (the_preced s ` threads s)" by (unfold max_cp_eq, simp)
also have "... =
Max (the_preced s ` (\<Union>th\<in>readys s. {th'. Th th' \<in> subtree (RAG s) (Th th)}))"
by (unfold threads_alt_def, simp)
also have "... =
Max ((\<Union>th\<in>readys s. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}))"
by (unfold image_UN, simp)
also have "... =
Max (Max ` (\<lambda>th. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)}) ` readys s)"
proof(rule Max_UNION)
show "\<forall>M\<in>(\<lambda>x. the_preced s `
{th'. Th th' \<in> subtree (RAG s) (Th x)}) ` readys s. finite M"
using finite_subtree_threads by auto
qed (auto simp:False subtree_def)
also have "... =
Max ((Max \<circ> (\<lambda>th. the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})) ` readys s)"
by (unfold image_comp, simp)
also have "... = ?L" (is "Max (?f ` ?A) = Max (?g ` ?A)")
proof -
have "(?f ` ?A) = (?g ` ?A)"
proof(rule f_image_eq)
fix th1
assume "th1 \<in> ?A"
thus "?f th1 = ?g th1"
by (unfold cp_alt_def, simp)
qed
thus ?thesis by simp
qed
finally show ?thesis by simp
qed (auto simp:threads_alt_def)
end
section {* Relating @{term cntP}, @{term cntV}, @{term cntCS} and @{term pvD} *}
context valid_trace_p_w
begin
lemma holding_s_holder: "holding s holder cs"
by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
lemma holding_es_holder: "holding (e#s) holder cs"
by (unfold s_holding_def, fold wq_def, unfold wq_es_cs wq_s_cs, auto)
lemma holdents_es:
shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
proof -
{ fix cs'
assume "cs' \<in> ?L"
hence h: "holding (e#s) th' cs'" by (auto simp:holdents_def)
have "holding s th' cs'"
proof(cases "cs' = cs")
case True
from held_unique[OF h[unfolded True] holding_es_holder]
have "th' = holder" .
thus ?thesis
by (unfold True holdents_def, insert holding_s_holder, simp)
next
case False
hence "wq (e#s) cs' = wq s cs'" by simp
from h[unfolded s_holding_def, folded wq_def, unfolded this]
show ?thesis
by (unfold s_holding_def, fold wq_def, auto)
qed
hence "cs' \<in> ?R" by (auto simp:holdents_def)
} moreover {
fix cs'
assume "cs' \<in> ?R"
hence h: "holding s th' cs'" by (auto simp:holdents_def)
have "holding (e#s) th' cs'"
proof(cases "cs' = cs")
case True
from held_unique[OF h[unfolded True] holding_s_holder]
have "th' = holder" .
thus ?thesis
by (unfold True holdents_def, insert holding_es_holder, simp)
next
case False
hence "wq s cs' = wq (e#s) cs'" by simp
from h[unfolded s_holding_def, folded wq_def, unfolded this]
show ?thesis
by (unfold s_holding_def, fold wq_def, auto)
qed
hence "cs' \<in> ?L" by (auto simp:holdents_def)
} ultimately show ?thesis by auto
qed
lemma cntCS_es_th[simp]: "cntCS (e#s) th' = cntCS s th'"
by (unfold cntCS_def holdents_es, simp)
lemma th_not_ready_es:
shows "th \<notin> readys (e#s)"
using waiting_es_th_cs
by (unfold readys_def, auto)
end
lemma (in valid_trace) finite_holdents: "finite (holdents s th)"
by (unfold holdents_alt_def, insert fsbtRAGs.finite_children, auto)
context valid_trace_p
begin
lemma ready_th_s: "th \<in> readys s"
using runing_th_s
by (unfold runing_def, auto)
lemma live_th_s: "th \<in> threads s"
using readys_threads ready_th_s by auto
lemma live_th_es: "th \<in> threads (e#s)"
using live_th_s
by (unfold is_p, simp)
lemma waiting_neq_th:
assumes "waiting s t c"
shows "t \<noteq> th"
using assms using th_not_waiting by blast
end
context valid_trace_p_h
begin
lemma th_not_waiting':
"\<not> waiting (e#s) th cs'"
proof(cases "cs' = cs")
case True
show ?thesis
by (unfold True s_waiting_def, fold wq_def, unfold wq_es_cs', auto)
next
case False
from th_not_waiting[of cs', unfolded s_waiting_def, folded wq_def]
show ?thesis
by (unfold s_waiting_def, fold wq_def, insert False, simp)
qed
lemma ready_th_es:
shows "th \<in> readys (e#s)"
using th_not_waiting'
by (unfold readys_def, insert live_th_es, auto)
lemma holdents_es_th:
"holdents (e#s) th = (holdents s th) \<union> {cs}" (is "?L = ?R")
proof -
{ fix cs'
assume "cs' \<in> ?L"
hence "holding (e#s) th cs'"
by (unfold holdents_def, auto)
hence "cs' \<in> ?R"
by (cases rule:holding_esE, auto simp:holdents_def)
} moreover {
fix cs'
assume "cs' \<in> ?R"
hence "holding s th cs' \<or> cs' = cs"
by (auto simp:holdents_def)
hence "cs' \<in> ?L"
proof
assume "holding s th cs'"
from holding_kept[OF this]
show ?thesis by (auto simp:holdents_def)
next
assume "cs' = cs"
thus ?thesis using holding_es_th_cs
by (unfold holdents_def, auto)
qed
} ultimately show ?thesis by auto
qed
lemma cntCS_es_th: "cntCS (e#s) th = cntCS s th + 1"
proof -
have "card (holdents s th \<union> {cs}) = card (holdents s th) + 1"
proof(subst card_Un_disjoint)
show "holdents s th \<inter> {cs} = {}"
using not_holding_s_th_cs by (auto simp:holdents_def)
qed (auto simp:finite_holdents)
thus ?thesis
by (unfold cntCS_def holdents_es_th, simp)
qed
lemma no_holder:
"\<not> holding s th' cs"
proof
assume otherwise: "holding s th' cs"
from this[unfolded s_holding_def, folded wq_def, unfolded we]
show False by auto
qed
lemma holdents_es_th':
assumes "th' \<noteq> th"
shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
proof -
{ fix cs'
assume "cs' \<in> ?L"
hence h_e: "holding (e#s) th' cs'" by (auto simp:holdents_def)
have "cs' \<noteq> cs"
proof
assume "cs' = cs"
from held_unique[OF h_e[unfolded this] holding_es_th_cs]
have "th' = th" .
with assms show False by simp
qed
from h_e[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp[OF this]]
have "th' \<in> set (wq s cs') \<and> th' = hd (wq s cs')" .
hence "cs' \<in> ?R"
by (unfold holdents_def s_holding_def, fold wq_def, auto)
} moreover {
fix cs'
assume "cs' \<in> ?R"
hence "holding s th' cs'" by (auto simp:holdents_def)
from holding_kept[OF this]
have "holding (e # s) th' cs'" .
hence "cs' \<in> ?L"
by (unfold holdents_def, auto)
} ultimately show ?thesis by auto
qed
lemma cntCS_es_th'[simp]:
assumes "th' \<noteq> th"
shows "cntCS (e#s) th' = cntCS s th'"
by (unfold cntCS_def holdents_es_th'[OF assms], simp)
end
context valid_trace_p
begin
lemma readys_kept1:
assumes "th' \<noteq> th"
and "th' \<in> readys (e#s)"
shows "th' \<in> readys s"
proof -
{ fix cs'
assume wait: "waiting s th' cs'"
have n_wait: "\<not> waiting (e#s) th' cs'"
using assms(2)[unfolded readys_def] by auto
have False
proof(cases "cs' = cs")
case False
with n_wait wait
show ?thesis
by (unfold s_waiting_def, fold wq_def, auto)
next
case True
show ?thesis
proof(cases "wq s cs = []")
case True
then interpret vt: valid_trace_p_h
by (unfold_locales, simp)
show ?thesis using n_wait wait waiting_kept by auto
next
case False
then interpret vt: valid_trace_p_w by (unfold_locales, simp)
show ?thesis using n_wait wait waiting_kept by blast
qed
qed
} with assms(2) show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_kept2:
assumes "th' \<noteq> th"
and "th' \<in> readys s"
shows "th' \<in> readys (e#s)"
proof -
{ fix cs'
assume wait: "waiting (e#s) th' cs'"
have n_wait: "\<not> waiting s th' cs'"
using assms(2)[unfolded readys_def] by auto
have False
proof(cases "cs' = cs")
case False
with n_wait wait
show ?thesis
by (unfold s_waiting_def, fold wq_def, auto)
next
case True
show ?thesis
proof(cases "wq s cs = []")
case True
then interpret vt: valid_trace_p_h
by (unfold_locales, simp)
show ?thesis using n_wait vt.waiting_esE wait by blast
next
case False
then interpret vt: valid_trace_p_w by (unfold_locales, simp)
show ?thesis using assms(1) n_wait vt.waiting_esE wait by auto
qed
qed
} with assms(2) show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_simp [simp]:
assumes "th' \<noteq> th"
shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
using readys_kept1[OF assms] readys_kept2[OF assms]
by metis
lemma cnp_cnv_cncs_kept: (* ddd *)
assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
proof(cases "th' = th")
case True
note eq_th' = this
show ?thesis
proof(cases "wq s cs = []")
case True
then interpret vt: valid_trace_p_h by (unfold_locales, simp)
show ?thesis
using assms eq_th' is_p ready_th_s vt.cntCS_es_th vt.ready_th_es pvD_def by auto
next
case False
then interpret vt: valid_trace_p_w by (unfold_locales, simp)
show ?thesis
using add.commute add.left_commute assms eq_th' is_p live_th_s
ready_th_s vt.th_not_ready_es pvD_def
apply (auto)
by (fold is_p, simp)
qed
next
case False
note h_False = False
thus ?thesis
proof(cases "wq s cs = []")
case True
then interpret vt: valid_trace_p_h by (unfold_locales, simp)
show ?thesis using assms
by (insert True h_False pvD_def, auto split:if_splits,unfold is_p, auto)
next
case False
then interpret vt: valid_trace_p_w by (unfold_locales, simp)
show ?thesis using assms
by (insert False h_False pvD_def, auto split:if_splits,unfold is_p, auto)
qed
qed
end
context valid_trace_v
begin
lemma holding_th_cs_s:
"holding s th cs"
by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
lemma th_ready_s [simp]: "th \<in> readys s"
using runing_th_s
by (unfold runing_def readys_def, auto)
lemma th_live_s [simp]: "th \<in> threads s"
using th_ready_s by (unfold readys_def, auto)
lemma th_ready_es [simp]: "th \<in> readys (e#s)"
using runing_th_s neq_t_th
by (unfold is_v runing_def readys_def, auto)
lemma th_live_es [simp]: "th \<in> threads (e#s)"
using th_ready_es by (unfold readys_def, auto)
lemma pvD_th_s[simp]: "pvD s th = 0"
by (unfold pvD_def, simp)
lemma pvD_th_es[simp]: "pvD (e#s) th = 0"
by (unfold pvD_def, simp)
lemma cntCS_s_th [simp]: "cntCS s th > 0"
proof -
have "cs \<in> holdents s th" using holding_th_cs_s
by (unfold holdents_def, simp)
moreover have "finite (holdents s th)" using finite_holdents
by simp
ultimately show ?thesis
by (unfold cntCS_def,
auto intro!:card_gt_0_iff[symmetric, THEN iffD1])
qed
end
context valid_trace_v
begin
lemma th_not_waiting:
"\<not> waiting s th c"
proof -
have "th \<in> readys s"
using runing_ready runing_th_s by blast
thus ?thesis
by (unfold readys_def, auto)
qed
lemma waiting_neq_th:
assumes "waiting s t c"
shows "t \<noteq> th"
using assms using th_not_waiting by blast
end
context valid_trace_v_n
begin
lemma not_ready_taker_s[simp]:
"taker \<notin> readys s"
using waiting_taker
by (unfold readys_def, auto)
lemma taker_live_s [simp]: "taker \<in> threads s"
proof -
have "taker \<in> set wq'" by (simp add: eq_wq')
from th'_in_inv[OF this]
have "taker \<in> set rest" .
hence "taker \<in> set (wq s cs)" by (simp add: wq_s_cs)
thus ?thesis using wq_threads by auto
qed
lemma taker_live_es [simp]: "taker \<in> threads (e#s)"
using taker_live_s threads_es by blast
lemma taker_ready_es [simp]:
shows "taker \<in> readys (e#s)"
proof -
{ fix cs'
assume "waiting (e#s) taker cs'"
hence False
proof(cases rule:waiting_esE)
case 1
thus ?thesis using waiting_taker waiting_unique by auto
qed simp
} thus ?thesis by (unfold readys_def, auto)
qed
lemma neq_taker_th: "taker \<noteq> th"
using th_not_waiting waiting_taker by blast
lemma not_holding_taker_s_cs:
shows "\<not> holding s taker cs"
using holding_cs_eq_th neq_taker_th by auto
lemma holdents_es_taker:
"holdents (e#s) taker = holdents s taker \<union> {cs}" (is "?L = ?R")
proof -
{ fix cs'
assume "cs' \<in> ?L"
hence "holding (e#s) taker cs'" by (auto simp:holdents_def)
hence "cs' \<in> ?R"
proof(cases rule:holding_esE)
case 2
thus ?thesis by (auto simp:holdents_def)
qed auto
} moreover {
fix cs'
assume "cs' \<in> ?R"
hence "holding s taker cs' \<or> cs' = cs" by (auto simp:holdents_def)
hence "cs' \<in> ?L"
proof
assume "holding s taker cs'"
hence "holding (e#s) taker cs'"
using holding_esI2 holding_taker by fastforce
thus ?thesis by (auto simp:holdents_def)
next
assume "cs' = cs"
with holding_taker
show ?thesis by (auto simp:holdents_def)
qed
} ultimately show ?thesis by auto
qed
lemma cntCS_es_taker [simp]: "cntCS (e#s) taker = cntCS s taker + 1"
proof -
have "card (holdents s taker \<union> {cs}) = card (holdents s taker) + 1"
proof(subst card_Un_disjoint)
show "holdents s taker \<inter> {cs} = {}"
using not_holding_taker_s_cs by (auto simp:holdents_def)
qed (auto simp:finite_holdents)
thus ?thesis
by (unfold cntCS_def, insert holdents_es_taker, simp)
qed
lemma pvD_taker_s[simp]: "pvD s taker = 1"
by (unfold pvD_def, simp)
lemma pvD_taker_es[simp]: "pvD (e#s) taker = 0"
by (unfold pvD_def, simp)
lemma pvD_th_s[simp]: "pvD s th = 0"
by (unfold pvD_def, simp)
lemma pvD_th_es[simp]: "pvD (e#s) th = 0"
by (unfold pvD_def, simp)
lemma holdents_es_th:
"holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")
proof -
{ fix cs'
assume "cs' \<in> ?L"
hence "holding (e#s) th cs'" by (auto simp:holdents_def)
hence "cs' \<in> ?R"
proof(cases rule:holding_esE)
case 2
thus ?thesis by (auto simp:holdents_def)
qed (insert neq_taker_th, auto)
} moreover {
fix cs'
assume "cs' \<in> ?R"
hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def)
from holding_esI2[OF this]
have "cs' \<in> ?L" by (auto simp:holdents_def)
} ultimately show ?thesis by auto
qed
lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1"
proof -
have "card (holdents s th - {cs}) = card (holdents s th) - 1"
proof -
have "cs \<in> holdents s th" using holding_th_cs_s
by (auto simp:holdents_def)
moreover have "finite (holdents s th)"
by (simp add: finite_holdents)
ultimately show ?thesis by auto
qed
thus ?thesis by (unfold cntCS_def holdents_es_th)
qed
lemma holdents_kept:
assumes "th' \<noteq> taker"
and "th' \<noteq> th"
shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
proof -
{ fix cs'
assume h: "cs' \<in> ?L"
have "cs' \<in> ?R"
proof(cases "cs' = cs")
case False
hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
show ?thesis
by (unfold holdents_def s_holding_def, fold wq_def, auto)
next
case True
from h[unfolded this]
have "holding (e#s) th' cs" by (auto simp:holdents_def)
from held_unique[OF this holding_taker]
have "th' = taker" .
with assms show ?thesis by auto
qed
} moreover {
fix cs'
assume h: "cs' \<in> ?R"
have "cs' \<in> ?L"
proof(cases "cs' = cs")
case False
hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
from h have "holding s th' cs'" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
show ?thesis
by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
next
case True
from h[unfolded this]
have "holding s th' cs" by (auto simp:holdents_def)
from held_unique[OF this holding_th_cs_s]
have "th' = th" .
with assms show ?thesis by auto
qed
} ultimately show ?thesis by auto
qed
lemma cntCS_kept [simp]:
assumes "th' \<noteq> taker"
and "th' \<noteq> th"
shows "cntCS (e#s) th' = cntCS s th'"
by (unfold cntCS_def holdents_kept[OF assms], simp)
lemma readys_kept1:
assumes "th' \<noteq> taker"
and "th' \<in> readys (e#s)"
shows "th' \<in> readys s"
proof -
{ fix cs'
assume wait: "waiting s th' cs'"
have n_wait: "\<not> waiting (e#s) th' cs'"
using assms(2)[unfolded readys_def] by auto
have False
proof(cases "cs' = cs")
case False
with n_wait wait
show ?thesis
by (unfold s_waiting_def, fold wq_def, auto)
next
case True
have "th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)"
using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
moreover have "\<not> (th' \<in> set rest \<and> th' \<noteq> hd (taker # rest'))"
using n_wait[unfolded True s_waiting_def, folded wq_def,
unfolded wq_es_cs set_wq', unfolded eq_wq'] .
ultimately have "th' = taker" by auto
with assms(1)
show ?thesis by simp
qed
} with assms(2) show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_kept2:
assumes "th' \<noteq> taker"
and "th' \<in> readys s"
shows "th' \<in> readys (e#s)"
proof -
{ fix cs'
assume wait: "waiting (e#s) th' cs'"
have n_wait: "\<not> waiting s th' cs'"
using assms(2)[unfolded readys_def] by auto
have False
proof(cases "cs' = cs")
case False
with n_wait wait
show ?thesis
by (unfold s_waiting_def, fold wq_def, auto)
next
case True
have "th' \<in> set rest \<and> th' \<noteq> hd (taker # rest')"
using wait [unfolded True s_waiting_def, folded wq_def,
unfolded wq_es_cs set_wq', unfolded eq_wq'] .
moreover have "\<not> (th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest))"
using n_wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
ultimately have "th' = taker" by auto
with assms(1)
show ?thesis by simp
qed
} with assms(2) show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_simp [simp]:
assumes "th' \<noteq> taker"
shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
using readys_kept1[OF assms] readys_kept2[OF assms]
by metis
lemma cnp_cnv_cncs_kept:
assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
proof -
{ assume eq_th': "th' = taker"
have ?thesis
apply (unfold eq_th' pvD_taker_es cntCS_es_taker)
by (insert neq_taker_th assms[unfolded eq_th'], unfold is_v, simp)
} moreover {
assume eq_th': "th' = th"
have ?thesis
apply (unfold eq_th' pvD_th_es cntCS_es_th)
by (insert assms[unfolded eq_th'], unfold is_v, simp)
} moreover {
assume h: "th' \<noteq> taker" "th' \<noteq> th"
have ?thesis using assms
apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp)
by (fold is_v, unfold pvD_def, simp)
} ultimately show ?thesis by metis
qed
end
context valid_trace_v_e
begin
lemma holdents_es_th:
"holdents (e#s) th = holdents s th - {cs}" (is "?L = ?R")
proof -
{ fix cs'
assume "cs' \<in> ?L"
hence "holding (e#s) th cs'" by (auto simp:holdents_def)
hence "cs' \<in> ?R"
proof(cases rule:holding_esE)
case 1
thus ?thesis by (auto simp:holdents_def)
qed
} moreover {
fix cs'
assume "cs' \<in> ?R"
hence "cs' \<noteq> cs" "holding s th cs'" by (auto simp:holdents_def)
from holding_esI2[OF this]
have "cs' \<in> ?L" by (auto simp:holdents_def)
} ultimately show ?thesis by auto
qed
lemma cntCS_es_th [simp]: "cntCS (e#s) th = cntCS s th - 1"
proof -
have "card (holdents s th - {cs}) = card (holdents s th) - 1"
proof -
have "cs \<in> holdents s th" using holding_th_cs_s
by (auto simp:holdents_def)
moreover have "finite (holdents s th)"
by (simp add: finite_holdents)
ultimately show ?thesis by auto
qed
thus ?thesis by (unfold cntCS_def holdents_es_th)
qed
lemma holdents_kept:
assumes "th' \<noteq> th"
shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
proof -
{ fix cs'
assume h: "cs' \<in> ?L"
have "cs' \<in> ?R"
proof(cases "cs' = cs")
case False
hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
show ?thesis
by (unfold holdents_def s_holding_def, fold wq_def, auto)
next
case True
from h[unfolded this]
have "holding (e#s) th' cs" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def,
unfolded wq_es_cs nil_wq']
show ?thesis by auto
qed
} moreover {
fix cs'
assume h: "cs' \<in> ?R"
have "cs' \<in> ?L"
proof(cases "cs' = cs")
case False
hence eq_wq: "wq (e#s) cs' = wq s cs'" by simp
from h have "holding s th' cs'" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
show ?thesis
by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
next
case True
from h[unfolded this]
have "holding s th' cs" by (auto simp:holdents_def)
from held_unique[OF this holding_th_cs_s]
have "th' = th" .
with assms show ?thesis by auto
qed
} ultimately show ?thesis by auto
qed
lemma cntCS_kept [simp]:
assumes "th' \<noteq> th"
shows "cntCS (e#s) th' = cntCS s th'"
by (unfold cntCS_def holdents_kept[OF assms], simp)
lemma readys_kept1:
assumes "th' \<in> readys (e#s)"
shows "th' \<in> readys s"
proof -
{ fix cs'
assume wait: "waiting s th' cs'"
have n_wait: "\<not> waiting (e#s) th' cs'"
using assms(1)[unfolded readys_def] by auto
have False
proof(cases "cs' = cs")
case False
with n_wait wait
show ?thesis
by (unfold s_waiting_def, fold wq_def, auto)
next
case True
have "th' \<in> set (th # rest) \<and> th' \<noteq> hd (th # rest)"
using wait[unfolded True s_waiting_def, folded wq_def, unfolded wq_s_cs] .
hence "th' \<in> set rest" by auto
with set_wq' have "th' \<in> set wq'" by metis
with nil_wq' show ?thesis by simp
qed
} thus ?thesis using assms
by (unfold readys_def, auto)
qed
lemma readys_kept2:
assumes "th' \<in> readys s"
shows "th' \<in> readys (e#s)"
proof -
{ fix cs'
assume wait: "waiting (e#s) th' cs'"
have n_wait: "\<not> waiting s th' cs'"
using assms[unfolded readys_def] by auto
have False
proof(cases "cs' = cs")
case False
with n_wait wait
show ?thesis
by (unfold s_waiting_def, fold wq_def, auto)
next
case True
have "th' \<in> set [] \<and> th' \<noteq> hd []"
using wait[unfolded True s_waiting_def, folded wq_def,
unfolded wq_es_cs nil_wq'] .
thus ?thesis by simp
qed
} with assms show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_simp [simp]:
shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
using readys_kept1[OF assms] readys_kept2[OF assms]
by metis
lemma cnp_cnv_cncs_kept:
assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
proof -
{
assume eq_th': "th' = th"
have ?thesis
apply (unfold eq_th' pvD_th_es cntCS_es_th)
by (insert assms[unfolded eq_th'], unfold is_v, simp)
} moreover {
assume h: "th' \<noteq> th"
have ?thesis using assms
apply (unfold cntCS_kept[OF h], insert h, unfold is_v, simp)
by (fold is_v, unfold pvD_def, simp)
} ultimately show ?thesis by metis
qed
end
context valid_trace_v
begin
lemma cnp_cnv_cncs_kept:
assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
proof(cases "rest = []")
case True
then interpret vt: valid_trace_v_e by (unfold_locales, simp)
show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast
next
case False
then interpret vt: valid_trace_v_n by (unfold_locales, simp)
show ?thesis using assms using vt.cnp_cnv_cncs_kept by blast
qed
end
context valid_trace_create
begin
lemma th_not_live_s [simp]: "th \<notin> threads s"
proof -
from pip_e[unfolded is_create]
show ?thesis by (cases, simp)
qed
lemma th_not_ready_s [simp]: "th \<notin> readys s"
using th_not_live_s by (unfold readys_def, simp)
lemma th_live_es [simp]: "th \<in> threads (e#s)"
by (unfold is_create, simp)
lemma not_waiting_th_s [simp]: "\<not> waiting s th cs'"
proof
assume "waiting s th cs'"
from this[unfolded s_waiting_def, folded wq_def, unfolded wq_kept]
have "th \<in> set (wq s cs')" by auto
from wq_threads[OF this] have "th \<in> threads s" .
with th_not_live_s show False by simp
qed
lemma not_holding_th_s [simp]: "\<not> holding s th cs'"
proof
assume "holding s th cs'"
from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept]
have "th \<in> set (wq s cs')" by auto
from wq_threads[OF this] have "th \<in> threads s" .
with th_not_live_s show False by simp
qed
lemma not_waiting_th_es [simp]: "\<not> waiting (e#s) th cs'"
proof
assume "waiting (e # s) th cs'"
from this[unfolded s_waiting_def, folded wq_def, unfolded wq_kept]
have "th \<in> set (wq s cs')" by auto
from wq_threads[OF this] have "th \<in> threads s" .
with th_not_live_s show False by simp
qed
lemma not_holding_th_es [simp]: "\<not> holding (e#s) th cs'"
proof
assume "holding (e # s) th cs'"
from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept]
have "th \<in> set (wq s cs')" by auto
from wq_threads[OF this] have "th \<in> threads s" .
with th_not_live_s show False by simp
qed
lemma ready_th_es [simp]: "th \<in> readys (e#s)"
by (simp add:readys_def)
lemma holdents_th_s: "holdents s th = {}"
by (unfold holdents_def, auto)
lemma holdents_th_es: "holdents (e#s) th = {}"
by (unfold holdents_def, auto)
lemma cntCS_th_s [simp]: "cntCS s th = 0"
by (unfold cntCS_def, simp add:holdents_th_s)
lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0"
by (unfold cntCS_def, simp add:holdents_th_es)
lemma pvD_th_s [simp]: "pvD s th = 0"
by (unfold pvD_def, simp)
lemma pvD_th_es [simp]: "pvD (e#s) th = 0"
by (unfold pvD_def, simp)
lemma holdents_kept:
assumes "th' \<noteq> th"
shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
proof -
{ fix cs'
assume h: "cs' \<in> ?L"
hence "cs' \<in> ?R"
by (unfold holdents_def s_holding_def, fold wq_def,
unfold wq_kept, auto)
} moreover {
fix cs'
assume h: "cs' \<in> ?R"
hence "cs' \<in> ?L"
by (unfold holdents_def s_holding_def, fold wq_def,
unfold wq_kept, auto)
} ultimately show ?thesis by auto
qed
lemma cntCS_kept [simp]:
assumes "th' \<noteq> th"
shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
using holdents_kept[OF assms]
by (unfold cntCS_def, simp)
lemma readys_kept1:
assumes "th' \<noteq> th"
and "th' \<in> readys (e#s)"
shows "th' \<in> readys s"
proof -
{ fix cs'
assume wait: "waiting s th' cs'"
have n_wait: "\<not> waiting (e#s) th' cs'"
using assms by (auto simp:readys_def)
from wait[unfolded s_waiting_def, folded wq_def]
n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept]
have False by auto
} thus ?thesis using assms
by (unfold readys_def, auto)
qed
lemma readys_kept2:
assumes "th' \<noteq> th"
and "th' \<in> readys s"
shows "th' \<in> readys (e#s)"
proof -
{ fix cs'
assume wait: "waiting (e#s) th' cs'"
have n_wait: "\<not> waiting s th' cs'"
using assms(2) by (auto simp:readys_def)
from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept]
n_wait[unfolded s_waiting_def, folded wq_def]
have False by auto
} with assms show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_simp [simp]:
assumes "th' \<noteq> th"
shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
using readys_kept1[OF assms] readys_kept2[OF assms]
by metis
lemma pvD_kept [simp]:
assumes "th' \<noteq> th"
shows "pvD (e#s) th' = pvD s th'"
using assms
by (unfold pvD_def, simp)
lemma cnp_cnv_cncs_kept:
assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
proof -
{
assume eq_th': "th' = th"
have ?thesis using assms
by (unfold eq_th', simp, unfold is_create, simp)
} moreover {
assume h: "th' \<noteq> th"
hence ?thesis using assms
by (simp, simp add:is_create)
} ultimately show ?thesis by metis
qed
end
context valid_trace_exit
begin
lemma th_live_s [simp]: "th \<in> threads s"
proof -
from pip_e[unfolded is_exit]
show ?thesis
by (cases, unfold runing_def readys_def, simp)
qed
lemma th_ready_s [simp]: "th \<in> readys s"
proof -
from pip_e[unfolded is_exit]
show ?thesis
by (cases, unfold runing_def, simp)
qed
lemma th_not_live_es [simp]: "th \<notin> threads (e#s)"
by (unfold is_exit, simp)
lemma not_holding_th_s [simp]: "\<not> holding s th cs'"
proof -
from pip_e[unfolded is_exit]
show ?thesis
by (cases, unfold holdents_def, auto)
qed
lemma cntCS_th_s [simp]: "cntCS s th = 0"
proof -
from pip_e[unfolded is_exit]
show ?thesis
by (cases, unfold cntCS_def, simp)
qed
lemma not_holding_th_es [simp]: "\<not> holding (e#s) th cs'"
proof
assume "holding (e # s) th cs'"
from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept]
have "holding s th cs'"
by (unfold s_holding_def, fold wq_def, auto)
with not_holding_th_s
show False by simp
qed
lemma ready_th_es [simp]: "th \<notin> readys (e#s)"
by (simp add:readys_def)
lemma holdents_th_s: "holdents s th = {}"
by (unfold holdents_def, auto)
lemma holdents_th_es: "holdents (e#s) th = {}"
by (unfold holdents_def, auto)
lemma cntCS_th_es [simp]: "cntCS (e#s) th = 0"
by (unfold cntCS_def, simp add:holdents_th_es)
lemma pvD_th_s [simp]: "pvD s th = 0"
by (unfold pvD_def, simp)
lemma pvD_th_es [simp]: "pvD (e#s) th = 0"
by (unfold pvD_def, simp)
lemma holdents_kept:
assumes "th' \<noteq> th"
shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
proof -
{ fix cs'
assume h: "cs' \<in> ?L"
hence "cs' \<in> ?R"
by (unfold holdents_def s_holding_def, fold wq_def,
unfold wq_kept, auto)
} moreover {
fix cs'
assume h: "cs' \<in> ?R"
hence "cs' \<in> ?L"
by (unfold holdents_def s_holding_def, fold wq_def,
unfold wq_kept, auto)
} ultimately show ?thesis by auto
qed
lemma cntCS_kept [simp]:
assumes "th' \<noteq> th"
shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
using holdents_kept[OF assms]
by (unfold cntCS_def, simp)
lemma readys_kept1:
assumes "th' \<noteq> th"
and "th' \<in> readys (e#s)"
shows "th' \<in> readys s"
proof -
{ fix cs'
assume wait: "waiting s th' cs'"
have n_wait: "\<not> waiting (e#s) th' cs'"
using assms by (auto simp:readys_def)
from wait[unfolded s_waiting_def, folded wq_def]
n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept]
have False by auto
} thus ?thesis using assms
by (unfold readys_def, auto)
qed
lemma readys_kept2:
assumes "th' \<noteq> th"
and "th' \<in> readys s"
shows "th' \<in> readys (e#s)"
proof -
{ fix cs'
assume wait: "waiting (e#s) th' cs'"
have n_wait: "\<not> waiting s th' cs'"
using assms(2) by (auto simp:readys_def)
from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept]
n_wait[unfolded s_waiting_def, folded wq_def]
have False by auto
} with assms show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_simp [simp]:
assumes "th' \<noteq> th"
shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
using readys_kept1[OF assms] readys_kept2[OF assms]
by metis
lemma pvD_kept [simp]:
assumes "th' \<noteq> th"
shows "pvD (e#s) th' = pvD s th'"
using assms
by (unfold pvD_def, simp)
lemma cnp_cnv_cncs_kept:
assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
proof -
{
assume eq_th': "th' = th"
have ?thesis using assms
by (unfold eq_th', simp, unfold is_exit, simp)
} moreover {
assume h: "th' \<noteq> th"
hence ?thesis using assms
by (simp, simp add:is_exit)
} ultimately show ?thesis by metis
qed
end
context valid_trace_set
begin
lemma th_live_s [simp]: "th \<in> threads s"
proof -
from pip_e[unfolded is_set]
show ?thesis
by (cases, unfold runing_def readys_def, simp)
qed
lemma th_ready_s [simp]: "th \<in> readys s"
proof -
from pip_e[unfolded is_set]
show ?thesis
by (cases, unfold runing_def, simp)
qed
lemma th_not_live_es [simp]: "th \<in> threads (e#s)"
by (unfold is_set, simp)
lemma holdents_kept:
shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
proof -
{ fix cs'
assume h: "cs' \<in> ?L"
hence "cs' \<in> ?R"
by (unfold holdents_def s_holding_def, fold wq_def,
unfold wq_kept, auto)
} moreover {
fix cs'
assume h: "cs' \<in> ?R"
hence "cs' \<in> ?L"
by (unfold holdents_def s_holding_def, fold wq_def,
unfold wq_kept, auto)
} ultimately show ?thesis by auto
qed
lemma cntCS_kept [simp]:
shows "cntCS (e#s) th' = cntCS s th'" (is "?L = ?R")
using holdents_kept
by (unfold cntCS_def, simp)
lemma threads_kept[simp]:
"threads (e#s) = threads s"
by (unfold is_set, simp)
lemma readys_kept1:
assumes "th' \<in> readys (e#s)"
shows "th' \<in> readys s"
proof -
{ fix cs'
assume wait: "waiting s th' cs'"
have n_wait: "\<not> waiting (e#s) th' cs'"
using assms by (auto simp:readys_def)
from wait[unfolded s_waiting_def, folded wq_def]
n_wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept]
have False by auto
} moreover have "th' \<in> threads s"
using assms[unfolded readys_def] by auto
ultimately show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_kept2:
assumes "th' \<in> readys s"
shows "th' \<in> readys (e#s)"
proof -
{ fix cs'
assume wait: "waiting (e#s) th' cs'"
have n_wait: "\<not> waiting s th' cs'"
using assms by (auto simp:readys_def)
from wait[unfolded s_waiting_def, folded wq_def, unfolded wq_kept]
n_wait[unfolded s_waiting_def, folded wq_def]
have False by auto
} with assms show ?thesis
by (unfold readys_def, auto)
qed
lemma readys_simp [simp]:
shows "(th' \<in> readys (e#s)) = (th' \<in> readys s)"
using readys_kept1 readys_kept2
by metis
lemma pvD_kept [simp]:
shows "pvD (e#s) th' = pvD s th'"
by (unfold pvD_def, simp)
lemma cnp_cnv_cncs_kept:
assumes "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
shows "cntP (e#s) th' = cntV (e#s) th' + cntCS (e#s) th' + pvD (e#s) th'"
using assms
by (unfold is_set, simp, fold is_set, simp)
end
context valid_trace
begin
lemma cnp_cnv_cncs: "cntP s th' = cntV s th' + cntCS s th' + pvD s th'"
proof(induct rule:ind)
case Nil
thus ?case
by (unfold cntP_def cntV_def pvD_def cntCS_def holdents_def
s_holding_def, simp)
next
case (Cons s e)
interpret vt_e: valid_trace_e s e using Cons by simp
show ?case
proof(cases e)
case (Create th prio)
interpret vt_create: valid_trace_create s e th prio
using Create by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_create.cnp_cnv_cncs_kept)
next
case (Exit th)
interpret vt_exit: valid_trace_exit s e th
using Exit by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_exit.cnp_cnv_cncs_kept)
next
case (P th cs)
interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_p.cnp_cnv_cncs_kept)
next
case (V th cs)
interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_v.cnp_cnv_cncs_kept)
next
case (Set th prio)
interpret vt_set: valid_trace_set s e th prio
using Set by (unfold_locales, simp)
show ?thesis using Cons by (simp add: vt_set.cnp_cnv_cncs_kept)
qed
qed
end
section {* Corollaries of @{thm valid_trace.cnp_cnv_cncs} *}
context valid_trace
begin
lemma not_thread_holdents:
assumes not_in: "th \<notin> threads s"
shows "holdents s th = {}"
proof -
{ fix cs
assume "cs \<in> holdents s th"
hence "holding s th cs" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def]
have "th \<in> set (wq s cs)" by auto
with wq_threads have "th \<in> threads s" by auto
with assms
have False by simp
} thus ?thesis by auto
qed
lemma not_thread_cncs:
assumes not_in: "th \<notin> threads s"
shows "cntCS s th = 0"
using not_thread_holdents[OF assms]
by (simp add:cntCS_def)
lemma cnp_cnv_eq:
assumes "th \<notin> threads s"
shows "cntP s th = cntV s th"
using assms cnp_cnv_cncs not_thread_cncs pvD_def
by (auto)
lemma eq_pv_children:
assumes eq_pv: "cntP s th = cntV s th"
shows "children (RAG s) (Th th) = {}"
proof -
from cnp_cnv_cncs and eq_pv
have "cntCS s th = 0"
by (auto split:if_splits)
from this[unfolded cntCS_def holdents_alt_def]
have card_0: "card (the_cs ` children (RAG s) (Th th)) = 0" .
have "finite (the_cs ` children (RAG s) (Th th))"
by (simp add: fsbtRAGs.finite_children)
from card_0[unfolded card_0_eq[OF this]]
show ?thesis by auto
qed
lemma eq_pv_holdents:
assumes eq_pv: "cntP s th = cntV s th"
shows "holdents s th = {}"
by (unfold holdents_alt_def eq_pv_children[OF assms], simp)
lemma eq_pv_subtree:
assumes eq_pv: "cntP s th = cntV s th"
shows "subtree (RAG s) (Th th) = {Th th}"
using eq_pv_children[OF assms]
by (unfold subtree_children, simp)
lemma count_eq_RAG_plus:
assumes "cntP s th = cntV s th"
shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
proof(rule ccontr)
assume otherwise: "{th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} \<noteq> {}"
then obtain th' where "(Th th', Th th) \<in> (RAG s)^+" by auto
from tranclD2[OF this]
obtain z where "z \<in> children (RAG s) (Th th)"
by (auto simp:children_def)
with eq_pv_children[OF assms]
show False by simp
qed
lemma eq_pv_dependants:
assumes eq_pv: "cntP s th = cntV s th"
shows "dependants s th = {}"
proof -
from count_eq_RAG_plus[OF assms, folded dependants_alt_def1]
show ?thesis .
qed
lemma count_eq_tRAG_plus:
assumes "cntP s th = cntV s th"
shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
using assms eq_pv_dependants dependants_alt_def eq_dependants by auto
lemma count_eq_RAG_plus_Th:
assumes "cntP s th = cntV s th"
shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
using count_eq_RAG_plus[OF assms] by auto
lemma count_eq_tRAG_plus_Th:
assumes "cntP s th = cntV s th"
shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
using count_eq_tRAG_plus[OF assms] by auto
end
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 (RAG s))"
apply(simp add: detached_def Field_def)
apply(simp add: s_RAG_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
context valid_trace
begin
lemma detached_intro:
assumes eq_pv: "cntP s th = cntV s th"
shows "detached s th"
proof -
from eq_pv cnp_cnv_cncs
have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:pvD_def)
thus ?thesis
proof
assume "th \<notin> threads s"
with rg_RAG_threads dm_RAG_threads
show ?thesis
by (auto simp add: detached_def s_RAG_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 (RAG s)"
proof -
from eq_pv_children[OF assms]
have "children (RAG s) (Th th) = {}" .
thus ?thesis
by (unfold children_def, auto)
qed
ultimately show ?thesis
by (auto simp add: detached_def s_RAG_def s_waiting_abv
s_holding_abv wq_def readys_def)
qed
qed
lemma detached_elim:
assumes dtc: "detached s th"
shows "cntP s th = cntV s th"
proof -
have cncs_z: "cntCS s th = 0"
proof -
from dtc have "holdents s th = {}"
unfolding detached_def holdents_test s_RAG_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:waiting_eq s_RAG_def)
with cncs_z show ?thesis using cnp_cnv_cncs by (simp add:pvD_def)
next
case False
with cncs_z and cnp_cnv_cncs show ?thesis by (simp add:pvD_def)
qed
qed
lemma detached_eq:
shows "(detached s th) = (cntP s th = cntV s th)"
by (insert vt, auto intro:detached_intro detached_elim)
end
section {* Recursive definition of @{term "cp"} *}
lemma cp_alt_def1:
"cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"
proof -
have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =
((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"
by auto
thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)
qed
lemma cp_gen_def_cond:
assumes "x = Th th"
shows "cp s th = cp_gen s (Th th)"
by (unfold cp_alt_def1 cp_gen_def, simp)
lemma cp_gen_over_set:
assumes "\<forall> x \<in> A. \<exists> th. x = Th th"
shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"
proof(rule f_image_eq)
fix a
assume "a \<in> A"
from assms[rule_format, OF this]
obtain th where eq_a: "a = Th th" by auto
show "cp_gen s a = (cp s \<circ> the_thread) a"
by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
qed
context valid_trace
begin
(* ddd *)
lemma cp_gen_rec:
assumes "x = Th th"
shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
proof(cases "children (tRAG s) x = {}")
case True
show ?thesis
by (unfold True cp_gen_def subtree_children, simp add:assms)
next
case False
hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
note fsbttRAGs.finite_subtree[simp]
have [simp]: "finite (children (tRAG s) x)"
by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree],
rule children_subtree)
{ fix r x
have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
} note this[simp]
have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
proof -
from False obtain q where "q \<in> children (tRAG s) x" by blast
moreover have "subtree (tRAG s) q \<noteq> {}" by simp
ultimately show ?thesis by blast
qed
have h: "Max ((the_preced s \<circ> the_thread) `
({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
(is "?L = ?R")
proof -
let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
let "Max (_ \<union> (?h ` ?B))" = ?R
let ?L1 = "?f ` \<Union>(?g ` ?B)"
have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
proof -
have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
also have "... = (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
finally have "Max ?L1 = Max ..." by simp
also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
by (subst Max_UNION, simp+)
also have "... = Max (cp_gen s ` children (tRAG s) x)"
by (unfold image_comp cp_gen_alt_def, simp)
finally show ?thesis .
qed
show ?thesis
proof -
have "?L = Max (?f ` ?A \<union> ?L1)" by simp
also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
by (subst Max_Un, simp+)
also have "... = max (?f x) (Max (?h ` ?B))"
by (unfold eq_Max_L1, simp)
also have "... =?R"
by (rule max_Max_eq, (simp)+, unfold assms, simp)
finally show ?thesis .
qed
qed thus ?thesis
by (fold h subtree_children, unfold cp_gen_def, simp)
qed
lemma cp_rec:
"cp s th = Max ({the_preced s th} \<union>
(cp s o the_thread) ` children (tRAG s) (Th th))"
proof -
have "Th th = Th th" by simp
note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this]
show ?thesis
proof -
have "cp_gen s ` children (tRAG s) (Th th) =
(cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
proof(rule cp_gen_over_set)
show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
by (unfold tRAG_alt_def, auto simp:children_def)
qed
thus ?thesis by (subst (1) h(1), unfold h(2), simp)
qed
qed
end
section {* Other properties useful in Implementation.thy or Correctness.thy *}
context valid_trace_e
begin
lemma actor_inv:
assumes "\<not> isCreate e"
shows "actor e \<in> runing s"
using pip_e assms
by (induct, auto)
end
context valid_trace
begin
lemma readys_root:
assumes "th \<in> readys s"
shows "root (RAG s) (Th th)"
proof -
{ fix x
assume "x \<in> ancestors (RAG s) (Th th)"
hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
from tranclD[OF this]
obtain z where "(Th th, z) \<in> RAG s" by auto
with assms(1) have False
apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
by (fold wq_def, blast)
} thus ?thesis by (unfold root_def, auto)
qed
lemma readys_in_no_subtree:
assumes "th \<in> readys s"
and "th' \<noteq> th"
shows "Th th \<notin> subtree (RAG s) (Th th')"
proof
assume "Th th \<in> subtree (RAG s) (Th th')"
thus False
proof(cases rule:subtreeE)
case 1
with assms show ?thesis by auto
next
case 2
with readys_root[OF assms(1)]
show ?thesis by (auto simp:root_def)
qed
qed
lemma not_in_thread_isolated:
assumes "th \<notin> threads s"
shows "(Th th) \<notin> Field (RAG s)"
proof
assume "(Th th) \<in> Field (RAG s)"
with dm_RAG_threads and rg_RAG_threads assms
show False by (unfold Field_def, blast)
qed
lemma next_th_holding:
assumes nxt: "next_th s th cs th'"
shows "holding (wq s) th cs"
proof -
from nxt[unfolded next_th_def]
obtain rest where h: "wq s cs = th # rest"
"rest \<noteq> []"
"th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
thus ?thesis
by (unfold cs_holding_def, auto)
qed
lemma next_th_waiting:
assumes nxt: "next_th s th cs th'"
shows "waiting (wq s) th' cs"
proof -
from nxt[unfolded next_th_def]
obtain rest where h: "wq s cs = th # rest"
"rest \<noteq> []"
"th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
from wq_distinct[of cs, unfolded h]
have dst: "distinct (th # rest)" .
have in_rest: "th' \<in> set rest"
proof(unfold h, rule someI2)
show "distinct rest \<and> set rest = set rest" using dst by auto
next
fix x assume "distinct x \<and> set x = set rest"
with h(2)
show "hd x \<in> set (rest)" by (cases x, auto)
qed
hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
moreover have "th' \<noteq> hd (wq s cs)"
by (unfold h(1), insert in_rest dst, auto)
ultimately show ?thesis by (auto simp:cs_waiting_def)
qed
lemma next_th_RAG:
assumes nxt: "next_th (s::event list) th cs th'"
shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
using vt assms next_th_holding next_th_waiting
by (unfold s_RAG_def, simp)
end
context valid_trace_p
begin
find_theorems readys th
end
end