--- a/CpsG.thy~ Sat Jan 16 11:02:17 2016 +0800
+++ b/CpsG.thy~ Sun Jan 17 22:18:35 2016 +0800
@@ -15,21 +15,11 @@
from fnt and seq show "finite (f ` B)" by auto
qed
-(* I am going to use this file as a start point to retrofiting
- PIPBasics.thy, which is originally called CpsG.ghy *)
locale valid_trace =
fixes s
assumes vt : "vt s"
-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)
-
-thm s_waiting_def cs_waiting_def wq_def
-
locale valid_trace_e = valid_trace +
fixes e
assumes vt_e: "vt (e#s)"
@@ -40,6 +30,118 @@
end
+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"
+
+locale valid_trace_v = valid_trace_e +
+ fixes th cs
+ assumes is_v: "e = V th cs"
+begin
+ definition "rest = tl (wq s cs)"
+ definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
+end
+
+locale valid_trace_v_n = valid_trace_v +
+ assumes rest_nnl: "rest \<noteq> []"
+
+locale valid_trace_v_e = valid_trace_v +
+ assumes rest_nil: "rest = []"
+
+locale valid_trace_set= valid_trace_e +
+ fixes th prio
+ assumes is_set: "e = Set th prio"
+
+context valid_trace
+begin
+
+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
+
+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
+
+lemma finite_threads:
+ shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+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 RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+ by (unfold s_RAG_def, auto)
+
+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)
+
+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 runing_ready:
shows "runing s \<subseteq> readys s"
unfolding runing_def readys_def
@@ -64,6 +166,199 @@
context valid_trace
begin
+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
+
+end
+
+context valid_trace_create
+begin
+
+lemma wq_neq_simp [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_neq_simp [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_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
+
+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 runing_th_s:
+ shows "th \<in> runing s"
+proof -
+ from pip_e[unfolded is_v]
+ show ?thesis by (cases, simp)
+qed
+
+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
+
+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_neq_simp [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 actor_inv:
assumes "PIP s e"
and "\<not> isCreate e"
@@ -71,7 +366,6 @@
using assms
by (induct, auto)
-
lemma isP_E:
assumes "isP e"
obtains cs where "e = P (actor e) cs"
@@ -82,90 +376,35 @@
obtains cs where "e = V (actor e) cs"
using assms by (cases e, auto)
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes "PP []"
- and "(\<And>s e. valid_trace s \<Longrightarrow> valid_trace (e#s) \<Longrightarrow>
- PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
- shows "PP s"
-proof(rule vt.induct[OF vt])
- from assms(1) show "PP []" .
-next
- fix s e
- assume h: "vt s" "PP s" "PIP s e"
- show "PP (e # s)"
- proof(cases rule:assms(2))
- from h(1) show v1: "valid_trace s" by (unfold_locales, simp)
- next
- from h(1,3) have "vt (e#s)" by auto
- thus "valid_trace (e # s)" by (unfold_locales, simp)
- qed (insert h, auto)
-qed
-
lemma wq_distinct: "distinct (wq s cs)"
proof(induct rule:ind)
case (Cons s e)
- from Cons(4,3)
+ interpret vt_e: valid_trace_e s e using Cons by simp
show ?case
- proof(induct)
- case (thread_P th s cs1)
- show ?case
- proof(cases "cs = cs1")
- case True
- thus ?thesis (is "distinct ?L")
- proof -
- have "?L = wq_fun (schs s) cs1 @ [th]" using True
- by (simp add:wq_def wf_def Let_def split:list.splits)
- moreover have "distinct ..."
- proof -
- have "th \<notin> set (wq_fun (schs s) cs1)"
- proof
- assume otherwise: "th \<in> set (wq_fun (schs s) cs1)"
- from runing_head[OF thread_P(1) this]
- have "th = hd (wq_fun (schs s) cs1)" .
- hence "(Cs cs1, Th th) \<in> (RAG s)" using otherwise
- by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
- with thread_P(2) show False by auto
- qed
- moreover have "distinct (wq_fun (schs s) cs1)"
- using True thread_P wq_def by auto
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis by simp
- qed
- next
- case False
- with thread_P(3)
- show ?thesis
- by (auto simp:wq_def wf_def Let_def split:list.splits)
- qed
+ proof(cases e)
+ case (Create th prio)
+ interpret vt_create: valid_trace_create s e th prio
+ using Create by (unfold_locales, simp)
+ show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept)
+ next
+ case (Exit th)
+ interpret vt_exit: valid_trace_exit s e th
+ using Exit by (unfold_locales, simp)
+ show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept)
next
- case (thread_V th s cs1)
- thus ?case
- proof(cases "cs = cs1")
- case True
- show ?thesis (is "distinct ?L")
- proof(cases "(wq s cs)")
- case Nil
- thus ?thesis
- by (auto simp:wq_def wf_def Let_def split:list.splits)
- next
- case (Cons w_hd w_tl)
- moreover have "distinct (SOME q. distinct q \<and> set q = set w_tl)"
- proof(rule someI2)
- from thread_V(3)[unfolded Cons]
- show "distinct w_tl \<and> set w_tl = set w_tl" by auto
- qed auto
- ultimately show ?thesis
- by (auto simp:wq_def wf_def Let_def True split:list.splits)
- qed
- next
- case False
- with thread_V(3)
- show ?thesis
- by (auto simp:wq_def wf_def Let_def split:list.splits)
- qed
- qed (insert Cons, auto simp: wq_def Let_def split:list.splits)
+ case (P th cs)
+ interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
+ show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept)
+ next
+ case (V th cs)
+ interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
+ show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept)
+ next
+ case (Set th prio)
+ interpret vt_set: valid_trace_set s e th prio
+ using Set by (unfold_locales, simp)
+ show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept)
+ qed
qed (unfold wq_def Let_def, simp)
end
@@ -248,41 +487,6 @@
context valid_trace
begin
-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)
- 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
-
-
-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)
-
-context valid_trace
-begin
text {* (* ddd *)
@@ -489,7 +693,6 @@
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)
@@ -522,98 +725,6 @@
thus ?thesis by auto
qed
-(* An aux lemma used later *)
-lemma unique_minus:
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq: "y \<noteq> z"
- shows "(y, z) \<in> r^+"
-proof -
- from xz and neq show ?thesis
- proof(induct)
- case (base ya)
- have "(x, ya) \<in> r" by fact
- from unique [OF xy this] have "y = ya" .
- with base show ?case by auto
- next
- case (step ya z)
- show ?case
- proof(cases "y = ya")
- case True
- from step True show ?thesis by simp
- next
- case False
- from step False
- show ?thesis by auto
- qed
- qed
-qed
-
-lemma unique_base:
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> r^+"
-proof -
- from xz neq_yz show ?thesis
- proof(induct)
- case (base ya)
- from xy unique base show ?case by auto
- next
- case (step ya z)
- show ?case
- proof(cases "y = ya")
- case True
- from True step show ?thesis by auto
- next
- case False
- from False step
- have "(y, ya) \<in> r\<^sup>+" by auto
- with step show ?thesis by auto
- qed
- qed
-qed
-
-lemma unique_chain:
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r^+"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
-proof -
- from xy xz neq_yz show ?thesis
- proof(induct)
- case (base y)
- have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
- from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
- next
- case (step y za)
- show ?case
- proof(cases "y = z")
- case True
- from True step show ?thesis by auto
- next
- case False
- from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
- thus ?thesis
- proof
- assume "(z, y) \<in> r\<^sup>+"
- with step have "(z, za) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- next
- assume h: "(y, z) \<in> r\<^sup>+"
- from step have yza: "(y, za) \<in> r" by simp
- from step have "za \<noteq> z" by simp
- from unique_minus [OF _ yza h this] and unique
- have "(za, z) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- qed
- qed
- qed
-qed
-
text {*
The following three lemmas show that @{text "RAG"} does not change
by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
@@ -632,291 +743,13 @@
apply (unfold s_RAG_def s_waiting_def wq_def)
by (simp add:Let_def)
-context valid_trace
-begin
-
-lemma finite_threads:
- shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
-unfolding cp_def wq_def
-apply(induct s rule: schs.induct)
-thm cpreced_initial
-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 RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
- by (unfold s_RAG_def, auto)
-
-lemma wq_threads:
- assumes h: "th \<in> set (wq s cs)"
- shows "th \<in> threads s"
-proof -
- from vt and h show ?thesis
- proof(induct arbitrary: th cs)
- case (vt_cons s e)
- interpret vt_s: valid_trace s
- using vt_cons(1) by (unfold_locales, auto)
- assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
- and stp: "step s e"
- and vt: "vt s"
- and h: "th \<in> set (wq (e # s) cs)"
- show ?case
- proof(cases e)
- case (Create th' prio)
- with ih h show ?thesis
- by (auto simp:wq_def Let_def)
- next
- case (Exit th')
- with stp ih h show ?thesis
- apply (auto simp:wq_def Let_def)
- apply (ind_cases "step s (Exit th')")
- apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
- s_RAG_def s_holding_def cs_holding_def)
- done
- next
- case (V th' cs')
- show ?thesis
- proof(cases "cs' = cs")
- case False
- with h
- show ?thesis
- apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
- by (drule_tac ih, simp)
- next
- case True
- from h
- show ?thesis
- proof(unfold V wq_def)
- assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
- show "th \<in> threads (V th' cs' # s)"
- proof(cases "cs = cs'")
- case False
- hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
- with th_in have " th \<in> set (wq s cs)"
- by (fold wq_def, simp)
- from ih [OF this] show ?thesis by simp
- next
- case True
- show ?thesis
- proof(cases "wq_fun (schs s) cs'")
- case Nil
- with h V show ?thesis
- apply (auto simp:wq_def Let_def split:if_splits)
- by (fold wq_def, drule_tac ih, simp)
- next
- case (Cons a rest)
- assume eq_wq: "wq_fun (schs s) cs' = a # rest"
- with h V show ?thesis
- apply (auto simp:Let_def wq_def split:if_splits)
- proof -
- assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
- have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
- by auto
- qed
- with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
- from ih[OF this[folded wq_def]] show "th \<in> threads s" .
- next
- assume th_in: "th \<in> set (wq_fun (schs s) cs)"
- from ih[OF this[folded wq_def]]
- show "th \<in> threads s" .
- qed
- qed
- qed
- qed
- qed
- next
- case (P th' cs')
- from h stp
- show ?thesis
- apply (unfold P wq_def)
- apply (auto simp:Let_def split:if_splits, fold wq_def)
- apply (auto intro:ih)
- apply(ind_cases "step s (P th' cs')")
- by (unfold runing_def readys_def, auto)
- next
- case (Set thread prio)
- with ih h show ?thesis
- by (auto simp:wq_def Let_def)
- qed
- next
- case vt_nil
- thus ?case by (auto simp:wq_def)
- qed
-qed
-
-lemma dm_RAG_threads:
- assumes in_dom: "(Th th) \<in> Domain (RAG s)"
- shows "th \<in> threads s"
-proof -
- from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
- moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
- ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
- hence "th \<in> set (wq s cs)"
- by (unfold s_RAG_def, auto simp:cs_waiting_def)
- from wq_threads [OF this] show ?thesis .
-qed
-
-
-lemma cp_le:
- assumes th_in: "th \<in> threads s"
- shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
- show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
- \<le> Max ((\<lambda>th. preced th s) ` threads s)"
- (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
- proof(rule Max_f_mono)
- show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
- next
- from finite_threads
- show "finite (threads s)" .
- next
- from th_in
- show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
- apply (auto simp:Domain_def)
- apply (rule_tac dm_RAG_threads)
- apply (unfold trancl_domain [of "RAG s", symmetric])
- by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
- qed
-qed
-
-lemma le_cp:
- shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
- show "Prc (priority th s) (last_set th s)
- \<le> Max (insert (Prc (priority th s) (last_set th s))
- ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
- (is "?l \<le> Max (insert ?l ?A)")
- proof(cases "?A = {}")
- case False
- have "finite ?A" (is "finite (?f ` ?B)")
- proof -
- have "finite ?B"
- proof-
- have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_RAG have "finite (RAG s)" .
- hence "finite ((RAG (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_RAG_def cs_RAG_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependants_def)
- qed
- thus ?thesis by simp
- qed
- from Max_insert [OF this False, of ?l] show ?thesis by auto
- next
- case True
- thus ?thesis by auto
- qed
-qed
-
-lemma max_cp_eq:
- shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
- (is "?l = ?r")
-proof(cases "threads s = {}")
- case True
- thus ?thesis by auto
-next
- case False
- have "?l \<in> ((cp s) ` threads s)"
- proof(rule Max_in)
- from finite_threads
- show "finite (cp s ` threads s)" by auto
- next
- from False show "cp s ` threads s \<noteq> {}" by auto
- qed
- then obtain th
- where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
- have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
- moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
- proof -
- have "?r \<in> (?f ` ?A)"
- proof(rule Max_in)
- from finite_threads
- show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
- next
- from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
- qed
- then obtain th' where
- th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
- from le_cp [of th'] eq_r
- have "?r \<le> cp s th'" by auto
- moreover have "\<dots> \<le> cp s th"
- proof(fold eq_l)
- show " cp s th' \<le> Max (cp s ` threads s)"
- proof(rule Max_ge)
- from th_in' show "cp s th' \<in> cp s ` threads s"
- by auto
- next
- from finite_threads
- show "finite (cp s ` threads s)" by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis using eq_l by auto
-qed
-
-lemma max_cp_eq_the_preced:
- shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
- using max_cp_eq using the_preced_def by presburger
-
-end
-
-lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
- by (unfold 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
-
-locale valid_trace_v = valid_trace_e +
- fixes th cs
- assumes is_v: "e = V th cs"
context valid_trace_v
begin
-definition "rest = tl (wq s cs)"
-
-definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
-
lemma distinct_rest: "distinct rest"
by (simp add: distinct_tl rest_def wq_distinct)
-lemma runing_th_s:
- shows "th \<in> runing s"
-proof -
- from pip_e[unfolded is_v]
- show ?thesis by (cases, simp)
-qed
-
lemma holding_cs_eq_th:
assumes "holding s t cs"
shows "t = th"
@@ -930,38 +763,6 @@
qed
qed
-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
-
-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 distinct_wq': "distinct wq'"
by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def)
@@ -1020,10 +821,19 @@
show ?thesis .
qed
+lemma holding_esI1:
+ assumes "holding s t c"
+ and "t \<noteq> th"
+ shows "holding (e#s) t c"
+proof -
+ have "c \<noteq> cs" using assms using holding_cs_eq_th by blast
+ from holding_esI2[OF this assms(1)]
+ show ?thesis .
+qed
+
end
-locale valid_trace_v_n = valid_trace_v +
- assumes rest_nnl: "rest \<noteq> []"
+context valid_trace_v_n
begin
lemma neq_wq': "wq' \<noteq> []"
@@ -1148,8 +958,8 @@
end
-locale valid_trace_v_e = valid_trace_v +
- assumes rest_nil: "rest = []"
+
+context valid_trace_v_e
begin
lemma nil_wq': "wq' = []"
@@ -1200,6 +1010,16 @@
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"
@@ -1230,7 +1050,7 @@
from that[OF False this] show ?thesis .
qed
-end (* ccc *)
+end
lemma rel_eqI:
assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
@@ -1248,7 +1068,7 @@
context valid_trace_v
begin
-lemma
+lemma RAG_es:
"RAG (e # s) =
RAG s - {(Cs cs, Th th)} -
{(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
@@ -1356,16 +1176,83 @@
thus ?thesis
proof(cases rule:in_RAG_E)
case (waiting th' cs')
- thus ?thesis
+ 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
- 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
end
-
-lemma step_RAG_v: (* ccc *)
+lemma step_RAG_v:
assumes vt:
"vt (V th cs#s)"
shows "
@@ -1373,1385 +1260,115 @@
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"
- show "(n1, n2) \<in> ?R" sorry
-next
- fix n1 n2
- assume "(n1, n2) \<in> ?R"
- show "(n1, n2) \<in> ?L" sorry
+proof -
+ interpret vt_v: valid_trace_v s "V th cs"
+ using assms step_back_vt by (unfold_locales, auto)
+ show ?thesis using vt_v.RAG_es .
+qed
+
+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)
-text {* (* ddd *)
- The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
- with the happening of @{text "V"}-events:
-*}
-lemma step_RAG_v:
-assumes vt:
- "vt (V th cs#s)"
-shows "
- RAG (V th cs # 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'}"
- apply (insert vt, unfold s_RAG_def)
- apply (auto split:if_splits list.splits simp:Let_def)
- apply (auto elim: step_v_waiting_mono step_v_hold_inv
- step_v_release step_v_wait_inv
- step_v_get_hold step_v_release_inv)
- apply (erule_tac step_v_not_wait, auto)
- done
+lemma (in valid_trace_exit)
+ threads_es [simp]: "threads (e#s) = threads s - {th}"
+ by (unfold is_exit, simp)
-text {*
- The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
- with the happening of @{text "P"}-events:
-*}
-lemma step_RAG_p:
- "vt (P th cs#s) \<Longrightarrow>
- RAG (P th cs # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
- else RAG s \<union> {(Th th, Cs cs)})"
- apply(simp only: s_RAG_def wq_def)
- apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
- apply(case_tac "csa = cs", auto)
- apply(fold wq_def)
- apply(drule_tac step_back_step)
- apply(ind_cases " step s (P (hd (wq s cs)) cs)")
- apply(simp add:s_RAG_def wq_def cs_holding_def)
- apply(auto)
- done
+lemma (in valid_trace_p)
+ threads_es [simp]: "threads (e#s) = threads s"
+ by (unfold is_p, simp)
-
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
- by (unfold s_RAG_def, auto)
-
-context valid_trace
-begin
+lemma (in valid_trace_v)
+ threads_es [simp]: "threads (e#s) = threads s"
+ by (unfold is_v, simp)
-text {*
- The following lemma shows that @{text "RAG"} is acyclic.
- The overall structure is by induction on the formation of @{text "vt s"}
- and then case analysis on event @{text "e"}, where the non-trivial cases
- for those for @{text "V"} and @{text "P"} events.
-*}
-lemma acyclic_RAG:
- shows "acyclic (RAG s)"
-using vt
-proof(induct)
- case (vt_cons s e)
- interpret vt_s: valid_trace s using vt_cons(1)
- by (unfold_locales, simp)
- assume ih: "acyclic (RAG s)"
- and stp: "step s e"
- and vt: "vt s"
- show ?case
- proof(cases e)
- case (Create th prio)
- with ih
- show ?thesis by (simp add:RAG_create_unchanged)
- next
- case (Exit th)
- with ih show ?thesis by (simp add:RAG_exit_unchanged)
- next
- case (V th cs)
- from V vt stp have vtt: "vt (V th cs#s)" by auto
- from step_RAG_v [OF this]
- have eq_de:
- "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 = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
- from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
- from step_back_step [OF vtt]
- have "step s (V th cs)" .
- thus ?thesis
- proof(cases)
- assume "holding s th cs"
- hence th_in: "th \<in> set (wq s cs)" and
- eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
- then obtain rest where
- eq_wq: "wq s cs = th#rest"
- by (cases "wq s cs", auto)
- show ?thesis
- proof(cases "rest = []")
- case False
- let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
- from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
- by (unfold next_th_def, auto)
- let ?E = "(?A - ?B - ?C)"
- have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
- proof
- assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
- hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- from tranclD [OF this]
- obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
- hence th_d: "(Th ?th', x) \<in> ?A" by simp
- from RAG_target_th [OF this]
- obtain cs' where eq_x: "x = Cs cs'" by auto
- with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
- hence wt_th': "waiting s ?th' cs'"
- unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
- hence "cs' = cs"
- proof(rule vt_s.waiting_unique)
- from eq_wq vt_s.wq_distinct[of cs]
- show "waiting s ?th' cs"
- apply (unfold s_waiting_def wq_def, auto)
- proof -
- assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq_fun (schs s) cs = th # rest"
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with False show "x \<noteq> []" by auto
- qed
- hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
- set (SOME q. distinct q \<and> set q = set rest)" by auto
- moreover have "\<dots> = set rest"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- moreover note hd_in
- ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
- next
- assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with False show "x \<noteq> []" by auto
- qed
- hence "hd (SOME q. distinct q \<and> set q = set rest) \<in>
- set (SOME q. distinct q \<and> set q = set rest)" by auto
- moreover have "\<dots> = set rest"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- moreover note hd_in
- ultimately show False by auto
- qed
- qed
- with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
- with False
- show "False" by (auto simp: next_th_def eq_wq)
- qed
- with acyclic_insert[symmetric] and ac
- and eq_de eq_D show ?thesis by auto
- next
- case True
- with eq_wq
- have eq_D: "?D = {}"
- by (unfold next_th_def, auto)
- with eq_de ac
- show ?thesis by auto
- qed
- qed
- next
- case (P th cs)
- from P vt stp have vtt: "vt (P th cs#s)" by auto
- from step_RAG_p [OF this] P
- have "RAG (e # s) =
- (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
- RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
- by simp
- moreover have "acyclic ?R"
- proof(cases "wq s cs = []")
- case True
- hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
- have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
- proof
- assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
- hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- from tranclD2 [OF this]
- obtain x where "(x, Cs cs) \<in> RAG s" by auto
- with True show False by (auto simp:s_RAG_def cs_waiting_def)
- qed
- with acyclic_insert ih eq_r show ?thesis by auto
- next
- case False
- hence eq_r: "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
- have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
- proof
- assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
- hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
- moreover from step_back_step [OF vtt] have "step s (P th cs)" .
- ultimately show False
- proof -
- show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
- by (ind_cases "step s (P th cs)", simp)
- qed
- qed
- with acyclic_insert ih eq_r show ?thesis by auto
- qed
- ultimately show ?thesis by simp
- next
- case (Set thread prio)
- with ih
- thm RAG_set_unchanged
- show ?thesis by (simp add:RAG_set_unchanged)
- qed
- next
- case vt_nil
- show "acyclic (RAG ([]::state))"
- by (auto simp: s_RAG_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
+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)
+ 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 finite_RAG:
- shows "finite (RAG s)"
+lemma (in valid_trace_exit)
+ th_not_in_wq: "th \<notin> set (wq s cs)"
proof -
- from vt show ?thesis
- proof(induct)
- case (vt_cons s e)
- interpret vt_s: valid_trace s using vt_cons(1)
- by (unfold_locales, simp)
- assume ih: "finite (RAG s)"
- and stp: "step s e"
- and vt: "vt s"
- show ?case
- proof(cases e)
- case (Create th prio)
- with ih
- show ?thesis by (simp add:RAG_create_unchanged)
- next
- case (Exit th)
- with ih show ?thesis by (simp add:RAG_exit_unchanged)
- next
- case (V th cs)
- from V vt stp have vtt: "vt (V th cs#s)" by auto
- from step_RAG_v [OF this]
- have eq_de: "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 = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
- moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
- moreover have "finite ?D"
- proof -
- have "?D = {} \<or> (\<exists> a. ?D = {a})"
- by (unfold next_th_def, auto)
- thus ?thesis
- proof
- assume h: "?D = {}"
- show ?thesis by (unfold h, simp)
- next
- assume "\<exists> a. ?D = {a}"
- thus ?thesis
- by (metis finite.simps)
- qed
- qed
- ultimately show ?thesis by simp
- next
- case (P th cs)
- from P vt stp have vtt: "vt (P th cs#s)" by auto
- from step_RAG_p [OF this] P
- have "RAG (e # s) =
- (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
- RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
- by simp
- moreover have "finite ?R"
- proof(cases "wq s cs = []")
- case True
- hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
- with True and ih show ?thesis by auto
- next
- case False
- hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
- with False and ih show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- next
- case (Set thread prio)
- with ih
- show ?thesis by (simp add:RAG_set_unchanged)
- qed
- next
- case vt_nil
- show "finite (RAG ([]::state))"
- by (auto simp: s_RAG_def cs_waiting_def
- cs_holding_def wq_def acyclic_def)
- qed
-qed
-
-text {* Several useful lemmas *}
-
-lemma wf_dep_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)" .
+ from pip_e[unfolded is_exit]
+ show ?thesis
+ by (cases, unfold holdents_def s_holding_def, fold wq_def,
+ auto elim!:runing_wqE)
qed
-end
-
-lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
- by (induct l, auto)
-
-lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
- by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
-
-context valid_trace
-begin
-
-lemma wq_threads:
- assumes h: "th \<in> set (wq s cs)"
+lemma (in valid_trace) wq_threads:
+ assumes "th \<in> set (wq s cs)"
shows "th \<in> threads s"
-proof -
- from vt and h show ?thesis
- proof(induct arbitrary: th cs)
- case (vt_cons s e)
- interpret vt_s: valid_trace s
- using vt_cons(1) by (unfold_locales, auto)
- assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
- and stp: "step s e"
- and vt: "vt s"
- and h: "th \<in> set (wq (e # s) cs)"
- show ?case
- proof(cases e)
- case (Create th' prio)
- with ih h show ?thesis
+ 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)
- next
- case (Exit th')
- with stp ih h show ?thesis
- apply (auto simp:wq_def Let_def)
- apply (ind_cases "step s (Exit th')")
- apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
- s_RAG_def s_holding_def cs_holding_def)
- done
- next
- case (V th' cs')
- show ?thesis
- proof(cases "cs' = cs")
- case False
- with h
- show ?thesis
- apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
- by (drule_tac ih, simp)
- next
- case True
- from h
- show ?thesis
- proof(unfold V wq_def)
- assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
- show "th \<in> threads (V th' cs' # s)"
- proof(cases "cs = cs'")
- case False
- hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
- with th_in have " th \<in> set (wq s cs)"
- by (fold wq_def, simp)
- from ih [OF this] show ?thesis by simp
- next
- case True
- show ?thesis
- proof(cases "wq_fun (schs s) cs'")
- case Nil
- with h V show ?thesis
- apply (auto simp:wq_def Let_def split:if_splits)
- by (fold wq_def, drule_tac ih, simp)
- next
- case (Cons a rest)
- assume eq_wq: "wq_fun (schs s) cs' = a # rest"
- with h V show ?thesis
- apply (auto simp:Let_def wq_def split:if_splits)
- proof -
- assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
- have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
- by auto
- qed
- with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
- from ih[OF this[folded wq_def]] show "th \<in> threads s" .
- next
- assume th_in: "th \<in> set (wq_fun (schs s) cs)"
- from ih[OF this[folded wq_def]]
- show "th \<in> threads s" .
- qed
- qed
- qed
- qed
- qed
- next
- case (P th' cs')
- from h stp
- show ?thesis
- apply (unfold P wq_def)
- apply (auto simp:Let_def split:if_splits, fold wq_def)
- apply (auto intro:ih)
- apply(ind_cases "step s (P th' cs')")
- by (unfold runing_def readys_def, auto)
- next
- case (Set thread prio)
- with ih h show ?thesis
- by (auto simp:wq_def Let_def)
- qed
- next
- case vt_nil
- thus ?case by (auto simp:wq_def)
qed
-qed
-
-lemma range_in: "\<lbrakk>(Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
- apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
- by (auto intro:wq_threads)
-
-lemma readys_v_eq:
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and not_in: "th \<notin> set rest"
- shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
- from assms show ?thesis
- apply (auto simp:readys_def)
- apply(simp add:s_waiting_def[folded wq_def])
- apply (erule_tac x = csa in allE)
- apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
- apply (case_tac "csa = cs", simp)
- apply (erule_tac x = cs in allE)
- apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
- apply(auto simp add: wq_def)
- apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
- proof -
- assume th_nin: "th \<notin> set rest"
- and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
- and eq_wq: "wq_fun (schs s) cs = thread # rest"
- have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
- show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- with th_nin th_in show False by auto
- qed
-qed
-
-text {* \noindent
- The following lemmas shows that: starting from any node in @{text "RAG"},
- by chasing out-going edges, it is always possible to reach a node representing a ready
- thread. In this lemma, it is the @{text "th'"}.
-*}
-
-lemma chain_building:
- shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
-proof -
- from wf_dep_converse
- have h: "wf ((RAG s)\<inverse>)" .
- show ?thesis
- proof(induct rule:wf_induct [OF h])
- fix x
- assume ih [rule_format]:
- "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow>
- y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
- show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
- proof
- assume x_d: "x \<in> Domain (RAG s)"
- show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
- proof(cases x)
- case (Th th)
- from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
- with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
- from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
- hence "Cs cs \<in> Domain (RAG s)" by auto
- from ih [OF x_in_r this] obtain th'
- where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
- have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
- with th'_ready show ?thesis by auto
- next
- case (Cs cs)
- from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
- show ?thesis
- proof(cases "th' \<in> readys s")
- case True
- from True and th'_d show ?thesis by auto
- next
- case False
- from th'_d and range_in have "th' \<in> threads s" by auto
- with False have "Th th' \<in> Domain (RAG s)"
- by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
- from ih [OF th'_d this]
- obtain th'' where
- th''_r: "th'' \<in> readys s" and
- th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
- from th'_d and th''_in
- have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
- with th''_r show ?thesis by auto
- qed
- qed
- qed
- qed
-qed
-
-text {* \noindent
- The following is just an instance of @{text "chain_building"}.
-*}
-lemma th_chain_to_ready:
- assumes th_in: "th \<in> threads s"
- shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
-proof(cases "th \<in> readys s")
- case True
- thus ?thesis by auto
-next
- case False
- from False and th_in have "Th th \<in> Domain (RAG s)"
- by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
- from chain_building [rule_format, OF this]
- show ?thesis by auto
-qed
-
-end
-
-
-
-lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
- by (unfold s_holding_def cs_holding_def, auto)
-
-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 holding_unique)
-
-end
-
-
-lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
-by (induct rule:trancl_induct, auto)
+qed
context valid_trace
begin
-lemma dchain_unique:
- assumes th1_d: "(n, Th th1) \<in> (RAG s)^+"
- and th1_r: "th1 \<in> readys s"
- and th2_d: "(n, Th th2) \<in> (RAG s)^+"
- and th2_r: "th2 \<in> readys s"
- shows "th1 = th2"
-proof -
- { assume neq: "th1 \<noteq> th2"
- hence "Th th1 \<noteq> Th th2" by simp
- from unique_chain [OF _ th1_d th2_d this] and unique_RAG
- have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
- hence "False"
- proof
- assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
- from trancl_split [OF this]
- obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
- then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
- from dd eq_n have "th1 \<notin> readys s"
- by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
- with th1_r show ?thesis by auto
- next
- assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
- from trancl_split [OF this]
- obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
- then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
- from dd eq_n have "th2 \<notin> readys s"
- by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
- with th2_r show ?thesis by auto
- qed
- } thus ?thesis by auto
-qed
-
-end
-
-
-lemma step_holdents_p_add:
- assumes vt: "vt (P th cs#s)"
- and "wq s cs = []"
- shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
-proof -
- from assms show ?thesis
- unfolding holdents_test step_RAG_p[OF vt] by (auto)
-qed
-
-lemma step_holdents_p_eq:
- assumes vt: "vt (P th cs#s)"
- and "wq s cs \<noteq> []"
- shows "holdents (P th cs#s) th = holdents s th"
-proof -
- from assms show ?thesis
- unfolding holdents_test step_RAG_p[OF vt] by auto
-qed
-
-
-lemma (in valid_trace) finite_holding :
- shows "finite (holdents s th)"
-proof -
- let ?F = "\<lambda> (x, y). the_cs x"
- from finite_RAG
- have "finite (RAG s)" .
- hence "finite (?F `(RAG s))" by simp
- moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>"
- proof -
- { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
- fix x assume "(Cs x, Th th) \<in> RAG s"
- hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
- moreover have "?F (Cs x, Th th) = x" by simp
- ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp
- } thus ?thesis by auto
- qed
- ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
-qed
-
-lemma cntCS_v_dec:
- assumes vtv: "vt (V thread cs#s)"
- shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
-proof -
- from vtv interpret vt_s: valid_trace s
- by (cases, unfold_locales, simp)
- from vtv interpret vt_v: valid_trace "V thread cs#s"
- by (unfold_locales, simp)
- from step_back_step[OF vtv]
- have cs_in: "cs \<in> holdents s thread"
- apply (cases, unfold holdents_test s_RAG_def, simp)
- by (unfold cs_holding_def s_holding_def wq_def, auto)
- moreover have cs_not_in:
- "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
- apply (insert vt_s.wq_distinct[of cs])
- apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
- auto simp:next_th_def)
- proof -
- fix rest
- assume dst: "distinct (rest::thread list)"
- and ne: "rest \<noteq> []"
- and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)" by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume " distinct x \<and> set x = set rest" with ne
- show "x \<noteq> []" by auto
- qed
- ultimately
- show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
- by auto
- next
- fix rest
- assume dst: "distinct (rest::thread list)"
- and ne: "rest \<noteq> []"
- and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
- qed
- ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)" by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from dst show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume " distinct x \<and> set x = set rest" with ne
- show "x \<noteq> []" by auto
- qed
- ultimately show "False" by auto
- qed
- ultimately
- have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
- by auto
- moreover have "card \<dots> =
- Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
- proof(rule card_insert)
- from vt_v.finite_holding
- show " finite (holdents (V thread cs # s) thread)" .
- qed
- moreover from cs_not_in
- have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
- ultimately show ?thesis by (simp add:cntCS_def)
-qed
-
-lemma 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_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)
-
-context valid_trace
-begin
-
-text {* (* ddd *) \noindent
- The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"}
- of one particular thread.
-*}
-
-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 -
- from vt show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e)
- interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
- assume vt: "vt s"
- and ih: "\<And>th. cntP s th = cntV s th +
- (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
- and stp: "step s e"
- from stp show ?case
- proof(cases)
- case (thread_create thread prio)
- assume eq_e: "e = Create thread prio"
- and not_in: "thread \<notin> threads s"
- show ?thesis
- proof -
- { fix cs
- assume "thread \<in> set (wq s cs)"
- from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
- with not_in have "False" by simp
- } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
- by (auto simp:readys_def threads.simps s_waiting_def
- wq_def cs_waiting_def Let_def)
- from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
- from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
- have eq_cncs: "cntCS (e#s) th = cntCS s th"
- unfolding cntCS_def holdents_test
- by (simp add:RAG_create_unchanged eq_e)
- { assume "th \<noteq> thread"
- with eq_readys eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> threads (s))"
- by (simp add:threads.simps)
- with eq_cnp eq_cnv eq_cncs ih not_in
- have ?thesis by simp
- } moreover {
- assume eq_th: "th = thread"
- with not_in ih have " cntP s th = cntV s th + cntCS s th" by simp
- moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
- moreover note eq_cnp eq_cnv eq_cncs
- ultimately have ?thesis by auto
- } ultimately show ?thesis by blast
- qed
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- and is_runing: "thread \<in> runing s"
- and no_hold: "holdents s thread = {}"
- from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
- from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
- have eq_cncs: "cntCS (e#s) th = cntCS s th"
- unfolding cntCS_def holdents_test
- by (simp add:RAG_exit_unchanged eq_e)
- { assume "th \<noteq> thread"
- with eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> threads (s))"
- apply (simp add:threads.simps readys_def)
- apply (subst s_waiting_def)
- apply (simp add:Let_def)
- apply (subst s_waiting_def, simp)
- done
- with eq_cnp eq_cnv eq_cncs ih
- have ?thesis by simp
- } moreover {
- assume eq_th: "th = thread"
- with ih is_runing have " cntP s th = cntV s th + cntCS s th"
- by (simp add:runing_def)
- moreover from eq_th eq_e have "th \<notin> threads (e#s)"
- by simp
- moreover note eq_cnp eq_cnv eq_cncs
- ultimately have ?thesis by auto
- } ultimately show ?thesis by blast
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- and is_runing: "thread \<in> runing s"
- and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
- from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto
- then interpret vt_p: valid_trace "(P thread cs#s)"
- by (unfold_locales, simp)
- show ?thesis
- proof -
- { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
- assume neq_th: "th \<noteq> thread"
- with eq_e
- have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
- apply (simp add:readys_def s_waiting_def wq_def Let_def)
- apply (rule_tac hh)
- apply (intro iffI allI, clarify)
- apply (erule_tac x = csa in allE, auto)
- apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
- apply (erule_tac x = cs in allE, auto)
- by (case_tac "(wq_fun (schs s) cs)", auto)
- moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
- apply (simp add:cntCS_def holdents_test)
- by (unfold step_RAG_p [OF vtp], auto)
- moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
- by (simp add:cntP_def count_def)
- moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
- by (simp add:cntV_def count_def)
- moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
- moreover note ih [of th]
- ultimately have ?thesis by simp
- } moreover {
- assume eq_th: "th = thread"
- have ?thesis
- proof -
- from eq_e eq_th have eq_cnp: "cntP (e # s) th = 1 + (cntP s th)"
- by (simp add:cntP_def count_def)
- from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
- by (simp add:cntV_def count_def)
- show ?thesis
- proof (cases "wq s cs = []")
- case True
- with is_runing
- have "th \<in> readys (e#s)"
- apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
- apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
- by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
- moreover have "cntCS (e # s) th = 1 + cntCS s th"
- proof -
- have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
- Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
- proof -
- have "?L = insert cs ?R" by auto
- moreover have "card \<dots> = Suc (card (?R - {cs}))"
- proof(rule card_insert)
- from vt_s.finite_holding [of thread]
- show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
- by (unfold holdents_test, simp)
- qed
- moreover have "?R - {cs} = ?R"
- proof -
- have "cs \<notin> ?R"
- proof
- assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
- with no_dep show False by auto
- qed
- thus ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- thus ?thesis
- apply (unfold eq_e eq_th cntCS_def)
- apply (simp add: holdents_test)
- by (unfold step_RAG_p [OF vtp], auto simp:True)
- qed
- moreover from is_runing have "th \<in> readys s"
- by (simp add:runing_def eq_th)
- moreover note eq_cnp eq_cnv ih [of th]
- ultimately show ?thesis by auto
- next
- case False
- have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
- by (unfold eq_th eq_e wq_def, auto simp:Let_def)
- have "th \<notin> readys (e#s)"
- proof
- assume "th \<in> readys (e#s)"
- hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
- from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
- hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)"
- by (simp add:s_waiting_def wq_def)
- moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
- ultimately have "th = hd (wq (e#s) cs)" by blast
- with eq_wq have "th = hd (wq s cs @ [th])" by simp
- hence "th = hd (wq s cs)" using False by auto
- with False eq_wq vt_p.wq_distinct [of cs]
- show False by (fold eq_e, auto)
- qed
- moreover from is_runing have "th \<in> threads (e#s)"
- by (unfold eq_e, auto simp:runing_def readys_def eq_th)
- moreover have "cntCS (e # s) th = cntCS s th"
- apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
- by (auto simp:False)
- moreover note eq_cnp eq_cnv ih[of th]
- moreover from is_runing have "th \<in> readys s"
- by (simp add:runing_def eq_th)
- ultimately show ?thesis by auto
- qed
- qed
- } ultimately show ?thesis by blast
- qed
- next
- case (thread_V thread cs)
- from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
- then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
- assume eq_e: "e = V thread cs"
- and is_runing: "thread \<in> runing s"
- and hold: "holding s thread cs"
- from hold obtain rest
- where eq_wq: "wq s cs = thread # rest"
- by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
- have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
- have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest"
- by (metis distinct.simps(2) vt_s.wq_distinct)
- next
- show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
- by auto
- qed
- show ?thesis
- proof -
- { assume eq_th: "th = thread"
- from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
- by (unfold eq_e, simp add:cntP_def count_def)
- moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
- by (unfold eq_e, simp add:cntV_def count_def)
- moreover from cntCS_v_dec [OF vtv]
- have "cntCS (e # s) thread + 1 = cntCS s thread"
- by (simp add:eq_e)
- moreover from is_runing have rd_before: "thread \<in> readys s"
- by (unfold runing_def, simp)
- moreover have "thread \<in> readys (e # s)"
- proof -
- from is_runing
- have "thread \<in> threads (e#s)"
- by (unfold eq_e, auto simp:runing_def readys_def)
- moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
- proof
- fix cs1
- { assume eq_cs: "cs1 = cs"
- have "\<not> waiting (e # s) thread cs1"
- proof -
- from eq_wq
- have "thread \<notin> set (wq (e#s) cs1)"
- apply(unfold eq_e wq_def eq_cs s_holding_def)
- apply (auto simp:Let_def)
- proof -
- assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
- with eq_set have "thread \<in> set rest" by simp
- with vt_v.wq_distinct[of cs]
- and eq_wq show False
- by (metis distinct.simps(2) vt_s.wq_distinct)
- qed
- thus ?thesis by (simp add:wq_def s_waiting_def)
- qed
- } moreover {
- assume neq_cs: "cs1 \<noteq> cs"
- have "\<not> waiting (e # s) thread cs1"
- proof -
- from wq_v_neq [OF neq_cs[symmetric]]
- have "wq (V thread cs # s) cs1 = wq s cs1" .
- moreover have "\<not> waiting s thread cs1"
- proof -
- from runing_ready and is_runing
- have "thread \<in> readys s" by auto
- thus ?thesis by (simp add:readys_def)
- qed
- ultimately show ?thesis
- by (auto simp:wq_def s_waiting_def eq_e)
- qed
- } ultimately show "\<not> waiting (e # s) thread cs1" by blast
- qed
- ultimately show ?thesis by (simp add:readys_def)
- qed
- moreover note eq_th ih
- ultimately have ?thesis by auto
- } moreover {
- assume neq_th: "th \<noteq> thread"
- from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th"
- by (simp add:cntP_def count_def)
- from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th"
- by (simp add:cntV_def count_def)
- have ?thesis
- proof(cases "th \<in> set rest")
- case False
- have "(th \<in> readys (e # s)) = (th \<in> readys s)"
- apply (insert step_back_vt[OF vtv])
- by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
- moreover have "cntCS (e#s) th = cntCS s th"
- apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
- proof -
- have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
- {cs. (Cs cs, Th th) \<in> RAG s}"
- proof -
- from False eq_wq
- have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
- apply (unfold next_th_def, auto)
- proof -
- assume ne: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
- and eq_wq: "wq s cs = thread # rest"
- from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin>
- set (SOME q. distinct q \<and> set q = set rest)
- " by simp
- moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from vt_s.wq_distinct[ of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with ne show "x \<noteq> []" by auto
- qed
- ultimately show
- "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
- by auto
- qed
- thus ?thesis by auto
- qed
- thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
- card {cs. (Cs cs, Th th) \<in> RAG s}" by simp
- qed
- moreover note ih eq_cnp eq_cnv eq_threads
- ultimately show ?thesis by auto
- next
- case True
- assume th_in: "th \<in> set rest"
- show ?thesis
- proof(cases "next_th s thread cs th")
- case False
- with eq_wq and th_in have
- neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
- by (auto simp:next_th_def)
- have "(th \<in> readys (e # s)) = (th \<in> readys s)"
- proof -
- from eq_wq and th_in
- have "\<not> th \<in> readys s"
- apply (auto simp:readys_def s_waiting_def)
- apply (rule_tac x = cs in exI, auto)
- by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
- moreover
- from eq_wq and th_in and neq_hd
- have "\<not> (th \<in> readys (e # s))"
- apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
- by (rule_tac x = cs in exI, auto simp:eq_set)
- ultimately show ?thesis by auto
- qed
- moreover have "cntCS (e#s) th = cntCS s th"
- proof -
- from eq_wq and th_in and neq_hd
- have "(holdents (e # s) th) = (holdents s th)"
- apply (unfold eq_e step_RAG_v[OF vtv],
- auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
- Let_def cs_holding_def)
- by (insert vt_s.wq_distinct[of cs], auto simp:wq_def)
- thus ?thesis by (simp add:cntCS_def)
- qed
- moreover note ih eq_cnp eq_cnv eq_threads
- ultimately show ?thesis by auto
- next
- case True
- let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
- let ?t = "hd ?rest"
- from True eq_wq th_in neq_th
- have "th \<in> readys (e # s)"
- apply (auto simp:eq_e readys_def s_waiting_def wq_def
- Let_def next_th_def)
- proof -
- assume eq_wq: "wq_fun (schs s) cs = thread # rest"
- and t_in: "?t \<in> set rest"
- show "?t \<in> threads s"
- proof(rule vt_s.wq_threads)
- from eq_wq and t_in
- show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
- qed
- next
- fix csa
- assume eq_wq: "wq_fun (schs s) cs = thread # rest"
- and t_in: "?t \<in> set rest"
- and neq_cs: "csa \<noteq> cs"
- and t_in': "?t \<in> set (wq_fun (schs s) csa)"
- show "?t = hd (wq_fun (schs s) csa)"
- proof -
- { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
- from vt_s.wq_distinct[of cs] and
- eq_wq[folded wq_def] and t_in eq_wq
- have "?t \<noteq> thread" by auto
- with eq_wq and t_in
- have w1: "waiting s ?t cs"
- by (auto simp:s_waiting_def wq_def)
- from t_in' neq_hd'
- have w2: "waiting s ?t csa"
- by (auto simp:s_waiting_def wq_def)
- from vt_s.waiting_unique[OF w1 w2]
- and neq_cs have "False" by auto
- } thus ?thesis by auto
- qed
- qed
- moreover have "cntP s th = cntV s th + cntCS s th + 1"
- proof -
- have "th \<notin> readys s"
- proof -
- from True eq_wq neq_th th_in
- show ?thesis
- apply (unfold readys_def s_waiting_def, auto)
- by (rule_tac x = cs in exI, auto simp add: wq_def)
- qed
- moreover have "th \<in> threads s"
- proof -
- from th_in eq_wq
- have "th \<in> set (wq s cs)" by simp
- from vt_s.wq_threads [OF this]
- show ?thesis .
- qed
- ultimately show ?thesis using ih by auto
- qed
- moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
- apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
- proof -
- show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
- Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
- (is "card ?A = Suc (card ?B)")
- proof -
- have "?A = insert cs ?B" by auto
- hence "card ?A = card (insert cs ?B)" by simp
- also have "\<dots> = Suc (card ?B)"
- proof(rule card_insert_disjoint)
- have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
- with vt_s.finite_RAG
- show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
- next
- show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
- proof
- assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
- hence "(Cs cs, Th th) \<in> RAG s" by simp
- with True neq_th eq_wq show False
- by (auto simp:next_th_def s_RAG_def cs_holding_def)
- qed
- qed
- finally show ?thesis .
- qed
- qed
- moreover note eq_cnp eq_cnv
- ultimately show ?thesis by simp
- qed
- qed
- } ultimately show ?thesis by blast
- qed
- next
- case (thread_set thread prio)
- assume eq_e: "e = Set thread prio"
- and is_runing: "thread \<in> runing s"
- show ?thesis
- proof -
- from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
- from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
- have eq_cncs: "cntCS (e#s) th = cntCS s th"
- unfolding cntCS_def holdents_test
- by (simp add:RAG_set_unchanged eq_e)
- from eq_e have eq_readys: "readys (e#s) = readys s"
- by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
- auto simp:Let_def)
- { assume "th \<noteq> thread"
- with eq_readys eq_e
- have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
- (th \<in> readys (s) \<or> th \<notin> threads (s))"
- by (simp add:threads.simps)
- with eq_cnp eq_cnv eq_cncs ih is_runing
- have ?thesis by simp
- } moreover {
- assume eq_th: "th = thread"
- with is_runing ih have " cntP s th = cntV s th + cntCS s th"
- by (unfold runing_def, auto)
- moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
- by (simp add:runing_def)
- moreover note eq_cnp eq_cnv eq_cncs
- ultimately have ?thesis by auto
- } ultimately show ?thesis by blast
- qed
- qed
- next
- case vt_nil
- show ?case
- by (unfold cntP_def cntV_def cntCS_def,
- auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
- qed
-qed
-
-lemma not_thread_cncs:
- assumes not_in: "th \<notin> threads s"
- shows "cntCS s th = 0"
-proof -
- from vt not_in show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e th)
- interpret vt_s: valid_trace s using vt_cons(1)
- by (unfold_locales, simp)
- assume vt: "vt s"
- and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
- and stp: "step s e"
- and not_in: "th \<notin> threads (e # s)"
- from stp show ?case
- proof(cases)
- case (thread_create thread prio)
- assume eq_e: "e = Create thread prio"
- and not_in': "thread \<notin> threads s"
- have "cntCS (e # s) th = cntCS s th"
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:RAG_create_unchanged)
- moreover have "th \<notin> threads s"
- proof -
- from not_in eq_e show ?thesis by simp
- qed
- moreover note ih ultimately show ?thesis by auto
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- and nh: "holdents s thread = {}"
- have eq_cns: "cntCS (e # s) th = cntCS s th"
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:RAG_exit_unchanged)
- show ?thesis
- proof(cases "th = thread")
- case True
- have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
- with eq_cns show ?thesis by simp
- next
- case False
- with not_in and eq_e
- have "th \<notin> threads s" by simp
- from ih[OF this] and eq_cns show ?thesis by simp
- qed
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- and is_runing: "thread \<in> runing s"
- from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- hence "cntCS (e # s) th = cntCS s th "
- apply (unfold cntCS_def holdents_test eq_e)
- by (unfold step_RAG_p[OF vtp], auto)
- moreover have "cntCS s th = 0"
- proof(rule ih)
- from not_in eq_e show "th \<notin> threads s" by simp
- qed
- ultimately show ?thesis by simp
- next
- case (thread_V thread cs)
- assume eq_e: "e = V thread cs"
- and is_runing: "thread \<in> runing s"
- and hold: "holding s thread cs"
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- from assms thread_V vt stp ih
- have vtv: "vt (V thread cs#s)" by auto
- then interpret vt_v: valid_trace "(V thread cs#s)"
- by (unfold_locales, simp)
- from hold obtain rest
- where eq_wq: "wq s cs = thread # rest"
- by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
- from not_in eq_e eq_wq
- have "\<not> next_th s thread cs th"
- apply (auto simp:next_th_def)
- proof -
- assume ne: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
- have "?t \<in> set rest"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest"
- by (metis distinct.simps(2) vt_s.wq_distinct)
- next
- fix x assume "distinct x \<and> set x = set rest" with ne
- show "hd x \<in> set rest" by (cases x, auto)
- qed
- with eq_wq have "?t \<in> set (wq s cs)" by simp
- from vt_s.wq_threads[OF this] and ni
- show False
- using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)`
- ni vt_s.wq_threads by blast
- qed
- moreover note neq_th eq_wq
- ultimately have "cntCS (e # s) th = cntCS s th"
- by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
- moreover have "cntCS s th = 0"
- proof(rule ih)
- from not_in eq_e show "th \<notin> threads s" by simp
- qed
- ultimately show ?thesis by simp
- next
- case (thread_set thread prio)
- print_facts
- assume eq_e: "e = Set thread prio"
- and is_runing: "thread \<in> runing s"
- from not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] and eq_e
- show ?thesis
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:RAG_set_unchanged)
- qed
- next
- case vt_nil
- show ?case
- by (unfold cntCS_def,
- auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
- qed
-qed
-
-end
-
-
-context valid_trace
-begin
-
-lemma dm_RAG_threads:
+lemma dm_RAG_threads:
assumes in_dom: "(Th th) \<in> Domain (RAG s)"
shows "th \<in> threads s"
proof -
@@ -2763,360 +1380,7 @@
from wq_threads [OF this] show ?thesis .
qed
-end
-
-lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
-unfolding cp_def wq_def
-apply(induct s rule: schs.induct)
-thm cpreced_initial
-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
-
-context valid_trace
-begin
-
-lemma runing_unique:
- assumes runing_1: "th1 \<in> runing s"
- and runing_2: "th2 \<in> runing s"
- shows "th1 = th2"
-proof -
- from runing_1 and runing_2 have "cp s th1 = cp s th2"
- unfolding runing_def
- apply(simp)
- done
- hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
- Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
- (is "Max (?f ` ?A) = Max (?f ` ?B)")
- unfolding cp_eq_cpreced
- unfolding cpreced_def .
- obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
- proof -
- have h1: "finite (?f ` ?A)"
- proof -
- have "finite ?A"
- proof -
- have "finite (dependants (wq s) th1)"
- proof-
- have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_RAG have "finite (RAG s)" .
- hence "finite ((RAG (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_RAG_def cs_RAG_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependants_def)
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by auto
- qed
- moreover have h2: "(?f ` ?A) \<noteq> {}"
- proof -
- have "?A \<noteq> {}" by simp
- thus ?thesis by simp
- qed
- from Max_in [OF h1 h2]
- have "Max (?f ` ?A) \<in> (?f ` ?A)" .
- thus ?thesis
- thm cpreced_def
- unfolding cpreced_def[symmetric]
- unfolding cp_eq_cpreced[symmetric]
- unfolding cpreced_def
- using that[intro] by (auto)
- qed
- obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
- proof -
- have h1: "finite (?f ` ?B)"
- proof -
- have "finite ?B"
- proof -
- have "finite (dependants (wq s) th2)"
- proof-
- have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_RAG have "finite (RAG s)" .
- hence "finite ((RAG (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_RAG_def cs_RAG_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependants_def)
- qed
- thus ?thesis by simp
- qed
- thus ?thesis by auto
- qed
- moreover have h2: "(?f ` ?B) \<noteq> {}"
- proof -
- have "?B \<noteq> {}" by simp
- thus ?thesis by simp
- qed
- from Max_in [OF h1 h2]
- have "Max (?f ` ?B) \<in> (?f ` ?B)" .
- thus ?thesis by (auto intro:that)
- qed
- from eq_f_th1 eq_f_th2 eq_max
- have eq_preced: "preced th1' s = preced th2' s" by auto
- hence eq_th12: "th1' = th2'"
- proof (rule preced_unique)
- from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
- thus "th1' \<in> threads s"
- proof
- assume "th1' \<in> dependants (wq s) th1"
- hence "(Th th1') \<in> Domain ((RAG s)^+)"
- apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
- by (auto simp:Domain_def)
- hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
- from dm_RAG_threads[OF this] show ?thesis .
- next
- assume "th1' = th1"
- with runing_1 show ?thesis
- by (unfold runing_def readys_def, auto)
- qed
- next
- from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
- thus "th2' \<in> threads s"
- proof
- assume "th2' \<in> dependants (wq s) th2"
- hence "(Th th2') \<in> Domain ((RAG s)^+)"
- apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
- by (auto simp:Domain_def)
- hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
- from dm_RAG_threads[OF this] show ?thesis .
- next
- assume "th2' = th2"
- with runing_2 show ?thesis
- by (unfold runing_def readys_def, auto)
- qed
- qed
- from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
- thus ?thesis
- proof
- assume eq_th': "th1' = th1"
- from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
- thus ?thesis
- proof
- assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
- next
- assume "th2' \<in> dependants (wq s) th2"
- with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
- hence "(Th th1, Th th2) \<in> (RAG s)^+"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
- hence "Th th1 \<in> Domain ((RAG s)^+)"
- apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
- by (auto simp:Domain_def)
- hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
- from RAG_target_th [OF this]
- obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th1, Cs cs') \<in> RAG s" by simp
- with runing_1 have "False"
- apply (unfold runing_def readys_def s_RAG_def)
- by (auto simp:waiting_eq)
- thus ?thesis by simp
- qed
- next
- assume th1'_in: "th1' \<in> dependants (wq s) th1"
- from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
- thus ?thesis
- proof
- assume "th2' = th2"
- with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
- hence "(Th th2, Th th1) \<in> (RAG s)^+"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
- hence "Th th2 \<in> Domain ((RAG s)^+)"
- apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
- by (auto simp:Domain_def)
- hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
- from RAG_target_th [OF this]
- obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th2, Cs cs') \<in> RAG s" by simp
- with runing_2 have "False"
- apply (unfold runing_def readys_def s_RAG_def)
- by (auto simp:waiting_eq)
- thus ?thesis by simp
- next
- assume "th2' \<in> dependants (wq s) th2"
- with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
- hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
- from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
- show ?thesis
- proof(rule dchain_unique[OF h1 _ h2, symmetric])
- from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
- from runing_2 show "th2 \<in> readys s" by (simp add:runing_def)
- qed
- qed
- qed
-qed
-
-
-lemma "card (runing s) \<le> 1"
-apply(subgoal_tac "finite (runing s)")
-prefer 2
-apply (metis finite_nat_set_iff_bounded lessI runing_unique)
-apply(rule ccontr)
-apply(simp)
-apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
-apply(subst (asm) card_le_Suc_iff)
-apply(simp)
-apply(auto)[1]
-apply (metis insertCI runing_unique)
-apply(auto)
-done
-
-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
- show ?thesis
- proof(cases)
- case (thread_create thread prio)
- with is_in not_in have "e = Create th prio" by simp
- from that[OF this] show ?thesis .
- next
- case (thread_exit thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_P thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_V thread)
- with assms show ?thesis by (auto intro!:that)
- next
- case (thread_set thread)
- with assms show ?thesis by (auto intro!:that)
- qed
-qed
-
-context valid_trace
-begin
-
-lemma cnp_cnv_eq:
- assumes "th \<notin> threads s"
- shows "cntP s th = cntV s th"
- using assms
- using cnp_cnv_cncs not_thread_cncs by auto
-
-end
-
-
-lemma eq_RAG:
- "RAG (wq s) = RAG s"
-by (unfold cs_RAG_def s_RAG_def, auto)
-
-context valid_trace
-begin
-
-lemma count_eq_dependants:
- assumes eq_pv: "cntP s th = cntV s th"
- shows "dependants (wq s) th = {}"
-proof -
- from cnp_cnv_cncs and eq_pv
- have "cntCS s th = 0"
- by (auto split:if_splits)
- moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
- proof -
- from finite_holding[of th] show ?thesis
- by (simp add:holdents_test)
- qed
- ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
- by (unfold cntCS_def holdents_test cs_dependants_def, auto)
- show ?thesis
- proof(unfold cs_dependants_def)
- { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
- then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
- hence "False"
- proof(cases)
- assume "(Th th', Th th) \<in> RAG (wq s)"
- thus "False" by (auto simp:cs_RAG_def)
- next
- fix c
- assume "(c, Th th) \<in> RAG (wq s)"
- with h and eq_RAG show "False"
- by (cases c, auto simp:cs_RAG_def)
- qed
- } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
- qed
-qed
-
-lemma dependants_threads:
- shows "dependants (wq s) th \<subseteq> threads s"
-proof
- { fix th th'
- assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
- have "Th th \<in> Domain (RAG s)"
- proof -
- from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
- hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
- with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
- thus ?thesis using eq_RAG by simp
- qed
- from dm_RAG_threads[OF this]
- have "th \<in> threads s" .
- } note hh = this
- fix th1
- assume "th1 \<in> dependants (wq s) th"
- hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
- by (unfold cs_dependants_def, simp)
- from hh [OF this] show "th1 \<in> threads s" .
-qed
-
-lemma finite_threads:
- shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-end
-
-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
-
-context valid_trace
-begin
-
-lemma cp_le:
+lemma cp_le:
assumes th_in: "th \<in> threads s"
shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
@@ -3138,46 +1402,6 @@
qed
qed
-lemma le_cp:
- shows "preced th s \<le> cp s th"
-proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
- show "Prc (priority th s) (last_set th s)
- \<le> Max (insert (Prc (priority th s) (last_set th s))
- ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
- (is "?l \<le> Max (insert ?l ?A)")
- proof(cases "?A = {}")
- case False
- have "finite ?A" (is "finite (?f ` ?B)")
- proof -
- have "finite ?B"
- proof-
- have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
- proof -
- let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
- apply (auto simp:image_def)
- by (rule_tac x = "(Th x, Th th)" in bexI, auto)
- moreover have "finite \<dots>"
- proof -
- from finite_RAG have "finite (RAG s)" .
- hence "finite ((RAG (wq s))\<^sup>+)"
- apply (unfold finite_trancl)
- by (auto simp: s_RAG_def cs_RAG_def wq_def)
- thus ?thesis by auto
- qed
- ultimately show ?thesis by (auto intro:finite_subset)
- qed
- thus ?thesis by (simp add:cs_dependants_def)
- qed
- thus ?thesis by simp
- qed
- from Max_insert [OF this False, of ?l] show ?thesis by auto
- next
- case True
- thus ?thesis by auto
- qed
-qed
-
lemma max_cp_eq:
shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
(is "?l = ?r")
@@ -3225,998 +1449,198 @@
ultimately show ?thesis using eq_l by auto
qed
-lemma max_cp_readys_threads_pre:
- assumes np: "threads s \<noteq> {}"
- shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
-proof(unfold max_cp_eq)
- show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
- proof -
- let ?p = "Max ((\<lambda>th. preced th s) ` threads s)"
- let ?f = "(\<lambda>th. preced th s)"
- have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
- proof(rule Max_in)
- from finite_threads show "finite (?f ` threads s)" by simp
- next
- from np show "?f ` threads s \<noteq> {}" by simp
- qed
- then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
- by (auto simp:Image_def)
- from th_chain_to_ready [OF tm_in]
- have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
- thus ?thesis
- proof
- assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
- then obtain th' where th'_in: "th' \<in> readys s"
- and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
- have "cp s th' = ?f tm"
- proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
- from dependants_threads finite_threads
- show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))"
- by (auto intro:finite_subset)
- next
- fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
- from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
- moreover have "p \<le> \<dots>"
- proof(rule Max_ge)
- from finite_threads
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- from p_in and th'_in and dependants_threads[of th']
- show "p \<in> (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- qed
- ultimately show "p \<le> preced tm s" by auto
- next
- show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
- proof -
- from tm_chain
- have "tm \<in> dependants (wq s) th'"
- by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
- thus ?thesis by auto
- qed
- qed
- with tm_max
- have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
- show ?thesis
- proof (fold h, rule Max_eqI)
- fix q
- assume "q \<in> cp s ` readys s"
- then obtain th1 where th1_in: "th1 \<in> readys s"
- and eq_q: "q = cp s th1" by auto
- show "q \<le> cp s th'"
- apply (unfold h eq_q)
- apply (unfold cp_eq_cpreced cpreced_def)
- apply (rule Max_mono)
- proof -
- from dependants_threads [of th1] th1_in
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq>
- (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- next
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
- next
- from finite_threads
- show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
- qed
- next
- from finite_threads
- show "finite (cp s ` readys s)" by (auto simp:readys_def)
- next
- from th'_in
- show "cp s th' \<in> cp s ` readys s" by simp
- qed
- next
- assume tm_ready: "tm \<in> readys s"
- show ?thesis
- proof(fold tm_max)
- have cp_eq_p: "cp s tm = preced tm s"
- proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
- fix y
- assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
- show "y \<le> preced tm s"
- proof -
- { fix y'
- assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
- have "y' \<le> preced tm s"
- proof(unfold tm_max, rule Max_ge)
- from hy' dependants_threads[of tm]
- show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
- next
- from finite_threads
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- qed
- } with hy show ?thesis by auto
- qed
- next
- from dependants_threads[of tm] finite_threads
- show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
- by (auto intro:finite_subset)
- next
- show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
- by simp
- qed
- moreover have "Max (cp s ` readys s) = cp s tm"
- proof(rule Max_eqI)
- from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
- next
- from finite_threads
- show "finite (cp s ` readys s)" by (auto simp:readys_def)
- next
- fix y assume "y \<in> cp s ` readys s"
- then obtain th1 where th1_readys: "th1 \<in> readys s"
- and h: "y = cp s th1" by auto
- show "y \<le> cp s tm"
- apply(unfold cp_eq_p h)
- apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
- proof -
- from finite_threads
- show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
- next
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
- by simp
- next
- from dependants_threads[of th1] th1_readys
- show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)
- \<subseteq> (\<lambda>th. preced th s) ` threads s"
- by (auto simp:readys_def)
- qed
- qed
- ultimately show " Max (cp s ` readys s) = preced tm s" by simp
- qed
- qed
- qed
-qed
-
-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)"
-proof(cases "threads s = {}")
- case True
- thus ?thesis
- by (auto simp:readys_def)
-next
- case False
- show ?thesis by (rule max_cp_readys_threads_pre[OF False])
-qed
-
-end
-
-lemma eq_holding: "holding (wq s) th cs = holding s th cs"
- apply (unfold s_holding_def cs_holding_def wq_def, simp)
- done
-
-lemma f_image_eq:
- assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
- shows "f ` A = g ` A"
-proof
- show "f ` A \<subseteq> g ` A"
- by(rule image_subsetI, auto intro:h)
-next
- show "g ` A \<subseteq> f ` A"
- by (rule image_subsetI, auto intro:h[symmetric])
-qed
-
-
-definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
- where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
-
-lemma detached_test:
- shows "detached s th = (Th th \<notin> Field (RAG s))"
-apply(simp add: detached_def Field_def)
-apply(simp add: s_RAG_def)
-apply(simp add: s_holding_abv s_waiting_abv)
-apply(simp add: Domain_iff Range_iff)
-apply(simp add: wq_def)
-apply(auto)
-done
-
-context valid_trace
-begin
-
-lemma detached_intro:
- assumes eq_pv: "cntP s th = cntV s th"
- shows "detached s th"
-proof -
- from cnp_cnv_cncs
- have eq_cnt: "cntP s th =
- cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
- hence cncs_zero: "cntCS s th = 0"
- by (auto simp:eq_pv split:if_splits)
- with eq_cnt
- have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
- thus ?thesis
- proof
- assume "th \<notin> threads s"
- with range_in 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 card_0_eq [OF finite_holding] and cncs_zero
- have "holdents s th = {}"
- by (simp add:cntCS_def)
- thus ?thesis
- apply(auto simp:holdents_test)
- apply(case_tac a)
- apply(auto simp:holdents_test s_RAG_def)
- done
- 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 -
- from cnp_cnv_cncs
- have eq_pv: " cntP s th =
- cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
- have cncs_z: "cntCS s th = 0"
- proof -
- from dtc have "holdents s th = {}"
- unfolding detached_def holdents_test s_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 and eq_pv show ?thesis by simp
- next
- case False
- with cncs_z and eq_pv show ?thesis by simp
- qed
-qed
-
-lemma detached_eq:
- shows "(detached s th) = (cntP s th = cntV s th)"
- by (insert vt, auto intro:detached_intro detached_elim)
-
-end
-
-text {*
- The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived
- from the concise and miniature model of PIP given in PrioGDef.thy.
-*}
-
-lemma eq_dependants: "dependants (wq s) = dependants s"
- by (simp add: s_dependants_abv wq_def)
-
-lemma next_th_unique:
- assumes nt1: "next_th s th cs th1"
- and nt2: "next_th s th cs th2"
- shows "th1 = th2"
-using assms by (unfold next_th_def, auto)
-
-lemma birth_time_lt: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
- apply (induct s, simp)
-proof -
- fix a s
- assume ih: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
- and eq_as: "a # s \<noteq> []"
- show "last_set th (a # s) < length (a # s)"
- proof(cases "s \<noteq> []")
- case False
- from False show ?thesis
- by (cases a, auto simp:last_set.simps)
- next
- case True
- from ih [OF True] show ?thesis
- by (cases a, auto simp:last_set.simps)
- qed
-qed
-
-lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
- by (induct s, auto simp:threads.simps)
-
-lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
- apply (drule_tac th_in_ne)
- by (unfold preced_def, auto intro: birth_time_lt)
-
-lemma inj_the_preced:
- "inj_on (the_preced s) (threads s)"
- by (metis inj_onI preced_unique the_preced_def)
-
-lemma tRAG_alt_def:
- "tRAG s = {(Th th1, Th th2) | th1 th2.
- \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
- by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
-
-lemma tRAG_Field:
- "Field (tRAG s) \<subseteq> Field (RAG s)"
- by (unfold tRAG_alt_def Field_def, auto)
-
-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 tRAG_mono:
- assumes "RAG s' \<subseteq> RAG s"
- shows "tRAG s' \<subseteq> tRAG s"
- using assms
- by (unfold tRAG_alt_def, auto)
-
-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 eq_holding, unfolded cs_holding_def]
- have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
- 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
-
-lemma RAG_tRAG_transfer:
- assumes "vt s'"
- 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 -
- interpret vt_s': valid_trace "s'" using assms(1)
- by (unfold_locales, simp)
- interpret rtree: rtree "RAG s'"
- proof
- show "single_valued (RAG s')"
- apply (intro_locales)
- by (unfold single_valued_def,
- auto intro:vt_s'.unique_RAG)
-
- show "acyclic (RAG s')"
- by (rule vt_s'.acyclic_RAG)
- qed
- { 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(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
- from h(3) and assms(2)
- 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(3) cs_in[unfolded this] rtree.sgv
- show ?thesis
- by (unfold single_valued_def, 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(2), auto)
- qed
- ultimately show ?thesis by auto
- next
- assume eq_n: "(n1, n2) = (Th th, Th th'')"
- from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
- moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
- ultimately show ?thesis
- by (unfold eq_n tRAG_alt_def, auto)
- qed
- } ultimately show ?thesis by auto
-qed
-
-context valid_trace
-begin
-
-lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
+lemma max_cp_eq_the_preced:
+ shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
+ using max_cp_eq using the_preced_def by presburger
end
-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 cp_gen_alt_def:
- "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
- by (auto simp:cp_gen_def)
-
-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 tRAG_def)
+lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
+ by (unfold preced_def, simp)
-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 (in valid_trace_v)
+ preced_es: "preced th (e#s) = preced th s"
+ by (unfold is_v preced_def, simp)
-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[of s]
- have "(a, x) \<in> (RAG s)^*" by auto
- hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
- } thus ?thesis by auto
+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 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[of s] 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, simp)
- 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 (in valid_trace_v)
+ the_preced_es: "the_preced (e#s) = the_preced s"
+ by (unfold is_v preced_def, simp)
-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)
-
-context valid_trace
-begin
-
-lemma count_eq_tRAG_plus:
- assumes "cntP s th = cntV s th"
- shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
- using assms count_eq_dependants dependants_alt_def eq_dependants by auto
-
-lemma count_eq_RAG_plus:
- assumes "cntP s th = cntV s th"
- shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
- using assms count_eq_dependants cs_dependants_def eq_RAG by auto
-
-lemma count_eq_RAG_plus_Th:
- assumes "cntP s th = cntV s th"
- shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"
- using count_eq_RAG_plus[OF assms] by auto
-
-lemma count_eq_tRAG_plus_Th:
- assumes "cntP s th = cntV s th"
- shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
- using count_eq_tRAG_plus[OF assms] by auto
-
-end
-
-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)
-
-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
+context valid_trace_p
begin
-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 range_in vt)
-
-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 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 range_in assms
- show False by (unfold Field_def, blast)
-qed
-
-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 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 holding_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)
-
-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_RAG: "single_valued (RAG s)"
- using unique_RAG by (auto simp:single_valued_def)
-
-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"
+lemma not_holding_es_th_cs: "\<not> holding s th cs"
proof
- show "single_valued (RAG s)"
- apply (intro_locales)
- by (unfold single_valued_def,
- auto intro:unique_RAG)
-
- show "acyclic (RAG s)"
- by (rule acyclic_RAG)
-qed
-
-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 < 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
-
-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 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
-
-context valid_trace
-begin
-
-(* ddd *)
-lemma cp_gen_rec:
- assumes "x = Th th"
- shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
-proof(cases "children (tRAG s) x = {}")
- case True
- show ?thesis
- by (unfold True cp_gen_def subtree_children, simp add:assms)
-next
- case False
- hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
- note fsbttRAGs.finite_subtree[simp]
- have [simp]: "finite (children (tRAG s) x)"
- by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree],
- rule children_subtree)
- { fix r x
- have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
- } note this[simp]
- have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
- proof -
- from False obtain q where "q \<in> children (tRAG s) x" by blast
- moreover have "subtree (tRAG s) q \<noteq> {}" by simp
- ultimately show ?thesis by blast
- qed
- have h: "Max ((the_preced s \<circ> the_thread) `
- ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
- Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
- (is "?L = ?R")
- proof -
- let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
- let "Max (_ \<union> (?h ` ?B))" = ?R
- let ?L1 = "?f ` \<Union>(?g ` ?B)"
- have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
- proof -
- have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
- also have "... = (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
- finally have "Max ?L1 = Max ..." by simp
- also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
- by (subst Max_UNION, simp+)
- also have "... = Max (cp_gen s ` children (tRAG s) x)"
- by (unfold image_comp cp_gen_alt_def, simp)
- finally show ?thesis .
- qed
- show ?thesis
- proof -
- have "?L = Max (?f ` ?A \<union> ?L1)" by simp
- also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
- by (subst Max_Un, simp+)
- also have "... = max (?f x) (Max (?h ` ?B))"
- by (unfold eq_Max_L1, simp)
- also have "... =?R"
- by (rule max_Max_eq, (simp)+, unfold assms, simp)
- finally show ?thesis .
- qed
- qed thus ?thesis
- by (fold h subtree_children, unfold cp_gen_def, simp)
-qed
-
-lemma cp_rec:
- "cp s th = Max ({the_preced s th} \<union>
- (cp s o the_thread) ` children (tRAG s) (Th th))"
-proof -
- have "Th th = Th th" by simp
- note h = cp_gen_def_cond[OF this] cp_gen_rec[OF this]
- show ?thesis
- proof -
- have "cp_gen s ` children (tRAG s) (Th th) =
- (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
- proof(rule cp_gen_over_set)
- show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
- by (unfold tRAG_alt_def, auto simp:children_def)
- qed
- thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+ 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
-(* keep *)
-lemma next_th_holding:
- assumes vt: "vt s"
- and nxt: "next_th s th cs th'"
- shows "holding (wq s) th cs"
-proof -
- from nxt[unfolded next_th_def]
- obtain rest where h: "wq s cs = th # rest"
- "rest \<noteq> []"
- "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
- thus ?thesis
- by (unfold cs_holding_def, auto)
-qed
+locale valid_trace_p_h = valid_trace_p +
+ assumes we: "wq s cs = []"
-context valid_trace
+locale valid_trace_p_w = valid_trace_p +
+ assumes we: "wq s cs \<noteq> []"
begin
-lemma next_th_waiting:
- assumes nxt: "next_th s th cs th'"
- shows "waiting (wq s) th' cs"
-proof -
- from nxt[unfolded next_th_def]
- obtain rest where h: "wq s cs = th # rest"
- "rest \<noteq> []"
- "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
- from wq_distinct[of cs, unfolded h]
- have dst: "distinct (th # rest)" .
- have in_rest: "th' \<in> set rest"
- proof(unfold h, rule someI2)
- show "distinct rest \<and> set rest = set rest" using dst by auto
- next
- fix x assume "distinct x \<and> set x = set rest"
- with h(2)
- show "hd x \<in> set (rest)" by (cases x, auto)
- qed
- hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
- moreover have "th' \<noteq> hd (wq s cs)"
- by (unfold h(1), insert in_rest dst, auto)
- ultimately show ?thesis by (auto simp:cs_waiting_def)
-qed
+definition "holder = hd (wq s cs)"
+definition "waiters = tl (wq s cs)"
+definition "waiters' = waiters @ [th]"
-lemma next_th_RAG:
- assumes nxt: "next_th (s::event list) th cs th'"
- shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
- using vt assms next_th_holding next_th_waiting
- by (unfold s_RAG_def, simp)
+lemma wq_s_cs: "wq s cs = holder#waiters"
+ by (simp add: holder_def waiters_def we)
+
+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)
end
--- {* A useless definition *}
-definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
-where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+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
-find_theorems release
+lemma waiting_kept:
+ assumes "waiting s th' cs'"
+ shows "waiting (e#s) th' cs'"
+ using assms
+ by (metis cs_waiting_def list.sel(1) list.set_intros(2)
+ th_not_in_wq waiting_eq we wq_es_cs' wq_neq_simp)
+
-lemma "wq (V th cs # s) cs1 = ttt"
- apply (unfold wq_def, auto simp:Let_def)
+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')
+ find_theorems waiting e s
+ qed
+ qed
+qed
end
+
+
+lemma "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
+ from wq_es_cs[unfolded this]
+ have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto
+ hence "holding (e#s) th cs"
+ using cs_holding_def holding_eq by blast
+ thus
+ qed
+end
+
+text {*
+ The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
+ with the happening of @{text "P"}-events:
+*}
+lemma step_RAG_p:
+ "vt (P th cs#s) \<Longrightarrow>
+ RAG (P th cs # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+ else RAG s \<union> {(Th th, Cs cs)})"
+ apply(simp only: s_RAG_def wq_def)
+ apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
+ apply(case_tac "csa = cs", auto)
+ apply(fold wq_def)
+ apply(drule_tac step_back_step)
+ apply(ind_cases " step s (P (hd (wq s cs)) cs)")
+ apply(simp add:s_RAG_def wq_def cs_holding_def)
+ apply(auto)
+ done
+
+
+
+end
\ No newline at end of file