--- a/PIPBasics.thy Fri Jan 29 17:08:54 2016 +0000
+++ b/PIPBasics.thy Wed Feb 03 21:51:57 2016 +0800
@@ -2,20 +2,664 @@
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.
+*}
+=======
+>>>>>>> other
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
+<<<<<<< local
+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"
+=======
lemma runing_ready:
shows "runing s \<subseteq> readys s"
unfolding runing_def readys_def
@@ -38,7 +682,170 @@
by (simp add:runing_def readys_def s_waiting_def wq_def)
context valid_trace
+>>>>>>> other
begin
+<<<<<<< local
+ 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 actor_inv:
assumes "PIP s e"
@@ -65,6 +872,7 @@
thus "valid_trace (e # s)" by (unfold_locales, simp)
qed (insert h, auto)
qed
+>>>>>>> other
lemma wq_distinct: "distinct (wq s cs)"
proof(induct rule:ind)
@@ -134,10 +942,35 @@
end
-
+<<<<<<< local
+section {* Waiting queues and threads *}
+
+=======
+
+>>>>>>> other
context valid_trace_e
begin
+<<<<<<< local
+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
+=======
text {*
The following lemma shows that only the @{text "P"}
operation can add new thread into waiting queues.
@@ -243,12 +1076,146 @@
with th_in show ?thesis by auto
qed
ultimately show ?thesis by auto
+>>>>>>> other
qed
qed
qed
+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
@@ -287,6 +1254,11 @@
context valid_trace
begin
+<<<<<<< local
+lemma dm_RAG_threads:
+ assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+ shows "th \<in> threads s"
+=======
text {* (* ddd *)
The nature of the work is like this: since it starts from a very simple and basic
@@ -328,7 +1300,16 @@
and h22: "thread \<noteq> hd (wq s cs2)"
and neq12: "cs1 \<noteq> cs2"
shows "False"
-proof -
+>>>>>>> other
+proof -
+<<<<<<< local
+ 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 .
+=======
let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
from h11 and h12 have q1: "?Q cs1 s" by simp
from h21 and h22 have q2: "?Q cs2 s" by simp
@@ -479,8 +1460,23 @@
qed
} ultimately show ?thesis by arith
qed
-qed
-
+>>>>>>> other
+qed
+
+<<<<<<< local
+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)
+=======
text {*
This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
*}
@@ -492,9 +1488,13 @@
using waiting_unique_pre assms
unfolding wq_def s_waiting_def
by auto
+>>>>>>> other
end
+<<<<<<< local
+section {* The change of @{term RAG} *}
+=======
(* not used *)
text {*
Every thread can only be blocked on one critical resource,
@@ -539,6 +1539,7 @@
have "preced th1 s \<noteq> preced th2 s" by auto
thus ?thesis by auto
qed
+>>>>>>> other
(* An aux lemma used later *)
lemma unique_minus:
@@ -638,6 +1639,16 @@
events, respectively.
*}
+<<<<<<< local
+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)
+=======
lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
apply (unfold s_RAG_def s_waiting_def wq_def)
by (simp add:Let_def)
@@ -650,7 +1661,16 @@
apply (unfold s_RAG_def s_waiting_def wq_def)
by (simp add:Let_def)
-
+>>>>>>> other
+
+<<<<<<< local
+context valid_trace_v
+begin
+
+lemma holding_cs_eq_th:
+ assumes "holding s t cs"
+ shows "t = th"
+=======
text {*
The following lemmas are used in the proof of
lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed
@@ -662,6 +1682,7 @@
"\<And>c t. \<lbrakk>vt (V th cs # s);
\<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow>
next_th s th cs t \<and> c = cs"
+>>>>>>> other
proof -
fix c t
assume vt: "vt (V th cs # s)"
@@ -708,6 +1729,33 @@
qed
qed
+<<<<<<< local
+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")
+=======
text {*
The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be
derived from scratch, which confirms the correctness of the definition of @{text "next_th"}.
@@ -729,7 +1777,25 @@
with nw wt show ?thesis
by (auto simp:cs_waiting_def wq_def Let_def)
next
+>>>>>>> other
case True
+<<<<<<< local
+ 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
+=======
from nw[folded True] wt[folded True]
have "next_th s th cs t"
apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
@@ -773,6 +1839,7 @@
qed
qed
with True show ?thesis by simp
+>>>>>>> other
qed
qed
@@ -929,6 +1996,1547 @@
qed
qed
+<<<<<<< local
+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
+=======
text {* (* ddd *)
The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
with the happening of @{text "V"}-events:
@@ -1242,6 +3850,7 @@
from acyclic_RAG
show "acyclic (RAG s)" .
qed
+>>>>>>> other
end
@@ -1449,7 +4058,7 @@
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)^+)"
@@ -1464,6 +4073,10 @@
show ?thesis by auto
qed
+<<<<<<< local
+lemma finite_subtree_threads:
+ "finite {th'. Th th' \<in> subtree (RAG s) (Th th)}" (is "finite ?A")
+=======
end
lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
@@ -1699,7 +4312,22 @@
lemma cnp_cnv_cncs:
shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s)
then cntCS s th else cntCS s th + 1)"
-proof -
+>>>>>>> other
+proof -
+<<<<<<< local
+ 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
+=======
from vt show ?thesis
proof(induct arbitrary:th)
case (vt_cons s e)
@@ -2170,8 +4798,11 @@
by (unfold cntP_def cntV_def cntCS_def,
auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
qed
-qed
-
+>>>>>>> other
+qed
+
+<<<<<<< local
+=======
lemma not_thread_cncs:
assumes not_in: "th \<notin> threads s"
shows "cntCS s th = 0"
@@ -2342,12 +4973,26 @@
context valid_trace
begin
+>>>>>>> other
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"
+<<<<<<< local
+ 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
+=======
unfolding runing_def
apply(simp)
done
@@ -2399,6 +5044,7 @@
unfolding cp_eq_cpreced[symmetric]
unfolding cpreced_def
using that[intro] by (auto)
+>>>>>>> other
qed
obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
proof -
@@ -2554,33 +5200,1415 @@
end
-lemma create_pre:
- assumes stp: "step s e"
- and not_in: "th \<notin> threads s"
- and is_in: "th \<in> threads (e#s)"
- obtains prio where "e = Create th prio"
-proof -
- from assms
+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
- proof(cases)
- case (thread_create thread prio)
- with is_in not_in have "e = Create th prio" by simp
- from that[OF this] show ?thesis .
+ 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 (thread_exit thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_P thread)
- with assms show ?thesis by (auto intro!:that)
+ 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
+
+<<<<<<< local
+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 (thread_V thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_set thread)
- with assms show ?thesis by (auto intro!:that)
+ 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:
+=======
context valid_trace
begin
@@ -2601,6 +6629,7 @@
begin
lemma count_eq_dependants:
+>>>>>>> other
assumes eq_pv: "cntP s th = cntV s th"
shows "dependants (wq s) th = {}"
proof -
@@ -2658,6 +6687,29 @@
shows "finite (threads s)"
using vt by (induct) (auto elim: step.cases)
+<<<<<<< local
+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
+
+=======
end
lemma Max_f_mono:
@@ -3362,6 +7414,7 @@
context valid_trace
begin
+>>>>>>> other
lemma count_eq_tRAG_plus:
assumes "cntP s th = cntV s th"
shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
@@ -3381,6 +7434,8 @@
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
+<<<<<<< local
+=======
end
@@ -3512,6 +7567,7 @@
with dm_RAG_threads and range_in assms
show False by (unfold Field_def, blast)
qed
+>>>>>>> other
lemma wf_RAG: "wf (RAG s)"
proof(rule finite_acyclic_wf)
@@ -3557,6 +7613,17 @@
by (unfold single_valued_def,
auto intro:unique_RAG)
+<<<<<<< local
+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
+=======
show "acyclic (RAG s)"
by (rule acyclic_RAG)
qed
@@ -3639,10 +7706,104 @@
by (subst Max.insert, simp+)
finally show ?thesis by simp
qed
+>>>>>>> other
context valid_trace
begin
+<<<<<<< local
+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
+=======
+>>>>>>> other
(* ddd *)
lemma cp_gen_rec:
assumes "x = Th th"
@@ -3718,6 +7879,63 @@
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
end
@@ -3770,8 +7988,11 @@
end
+<<<<<<< local
+end=======
-- {* A useless definition *}
definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
where "cps s = {(th, cp s th) | th . th \<in> threads s}"
end
+>>>>>>> other