--- a/PrioG.thy Wed Jan 06 20:46:14 2016 +0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,3051 +0,0 @@
-theory PrioG
-imports PrioGDef
-begin
-
-locale valid_trace =
- fixes s
- assumes vt : "vt s"
-
-locale valid_trace_e = valid_trace +
- fixes e
- assumes vt_e: "vt (e#s)"
-begin
-
-lemma pip_e: "PIP s e"
- using vt_e by (cases, simp)
-
-end
-
-lemma runing_ready:
- shows "runing s \<subseteq> readys s"
- unfolding runing_def readys_def
- by auto
-
-lemma readys_threads:
- shows "readys s \<subseteq> threads s"
- unfolding readys_def
- by auto
-
-lemma wq_v_neq:
- "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
- by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-context valid_trace
-begin
-
-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(rule ind, simp add:wq_def)
- fix s e
- assume h1: "step s e"
- and h2: "distinct (wq s cs)"
- thus "distinct (wq (e # s) cs)"
- proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
- fix thread s
- assume h1: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
- and h2: "thread \<in> set (wq_fun (schs s) cs)"
- and h3: "thread \<in> runing s"
- show "False"
- proof -
- from h3 have "\<And> cs. thread \<in> set (wq_fun (schs s) cs) \<Longrightarrow>
- thread = hd ((wq_fun (schs s) cs))"
- by (simp add:runing_def readys_def s_waiting_def wq_def)
- from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" .
- with h2
- have "(Cs cs, Th thread) \<in> (RAG s)"
- by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
- with h1 show False by auto
- qed
- next
- fix thread s a list
- assume dst: "distinct list"
- show "distinct (SOME q. distinct q \<and> set q = set list)"
- proof(rule someI2)
- from dst show "distinct list \<and> set list = set list" by auto
- next
- fix q assume "distinct q \<and> set q = set list"
- thus "distinct q" by auto
- qed
- qed
-qed
-
-end
-
-
-context valid_trace_e
-begin
-
-text {*
- The following lemma shows that only the @{text "P"}
- operation can add new thread into waiting queues.
- Such kind of lemmas are very obvious, but need to be checked formally.
- This is a kind of confirmation that our modelling is correct.
-*}
-
-lemma block_pre:
- assumes s_ni: "thread \<notin> set (wq s cs)"
- and s_i: "thread \<in> set (wq (e#s) cs)"
- shows "e = P thread cs"
-proof -
- show ?thesis
- proof(cases e)
- case (P th cs)
- with assms
- show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Create th prio)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Exit th)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (Set th prio)
- with assms show ?thesis
- by (auto simp:wq_def Let_def split:if_splits)
- next
- case (V th cs)
- with vt_e assms show ?thesis
- apply (auto simp:wq_def Let_def split:if_splits)
- proof -
- fix q qs
- assume h1: "thread \<notin> set (wq_fun (schs s) cs)"
- and h2: "q # qs = wq_fun (schs s) cs"
- and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
- and vt: "vt (V th cs # s)"
- from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp
- moreover have "thread \<in> set qs"
- proof -
- have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
- proof(rule someI2)
- from wq_distinct [of cs]
- and h2[symmetric, folded wq_def]
- show "distinct qs \<and> set qs = set qs" by auto
- next
- fix x assume "distinct x \<and> set x = set qs"
- thus "set x = set qs" by auto
- qed
- with h3 show ?thesis by simp
- qed
- ultimately show "False" by auto
- qed
- qed
-qed
-
-end
-
-text {*
- The following lemmas is also obvious and shallow. It says
- that only running thread can request for a critical resource
- and that the requested resource must be one which is
- not current held by the thread.
-*}
-
-lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
- thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (RAG s)^+"
-apply (ind_cases "vt ((P thread cs)#s)")
-apply (ind_cases "step s (P thread cs)")
-by auto
-
-lemma abs1:
- assumes ein: "e \<in> set es"
- and neq: "hd es \<noteq> hd (es @ [x])"
- shows "False"
-proof -
- from ein have "es \<noteq> []" by auto
- then obtain e ess where "es = e # ess" by (cases es, auto)
- with neq show ?thesis by auto
-qed
-
-lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
- by (cases es, auto)
-
-inductive_cases evt_cons: "vt (a#s)"
-
-context valid_trace_e
-begin
-
-lemma abs2:
- assumes inq: "thread \<in> set (wq s cs)"
- and nh: "thread = hd (wq s cs)"
- and qt: "thread \<noteq> hd (wq (e#s) cs)"
- and inq': "thread \<in> set (wq (e#s) cs)"
- shows "False"
-proof -
- from vt_e assms show "False"
- apply (cases e)
- apply ((simp split:if_splits add:Let_def wq_def)[1])+
- apply (insert abs1, fast)[1]
- apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)
- proof -
- fix th qs
- assume vt: "vt (V th cs # s)"
- and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
- and eq_wq: "wq_fun (schs s) cs = thread # qs"
- show "False"
- proof -
- from wq_distinct[of cs]
- and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
- moreover have "thread \<in> set qs"
- proof -
- have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
- proof(rule someI2)
- from wq_distinct [of cs]
- and eq_wq [folded wq_def]
- show "distinct qs \<and> set qs = set qs" by auto
- next
- fix x assume "distinct x \<and> set x = set qs"
- thus "set x = set qs" by auto
- qed
- with th_in show ?thesis by auto
- qed
- ultimately show ?thesis by auto
- qed
- qed
-qed
-
-end
-
-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
-
-(* Wrong:
- lemma \<lbrakk>thread \<in> set (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
-*)
-
-text {* (* ddd *)
- The nature of the work is like this: since it starts from a very simple and basic
- model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
- For instance, the fact
- that one thread can not be blocked by two critical resources at the same time
- is obvious, because only running threads can make new requests, if one is waiting for
- a critical resource and get blocked, it can not make another resource request and get
- blocked the second time (because it is not running).
-
- To derive this fact, one needs to prove by contraction and
- reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
- named @{text "p_split"}, which is about status changing along the time axis. It says if
- a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
- but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"}
- in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history
- of events leading to it), such that @{text "Q"} switched
- from being @{text "False"} to @{text "True"} and kept being @{text "True"}
- till the last moment of @{text "s"}.
-
- Suppose a thread @{text "th"} is blocked
- on @{text "cs1"} and @{text "cs2"} in some state @{text "s"},
- since no thread is blocked at the very beginning, by applying
- @{text "p_split"} to these two blocking facts, there exist
- two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that
- @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"}
- and kept on blocked on them respectively ever since.
-
- Without lose of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
- However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
- in blocked state at moment @{text "t2"} and could not
- make any request and get blocked the second time: Contradiction.
-*}
-
-lemma waiting_unique_pre:
- assumes h11: "thread \<in> set (wq s cs1)"
- and h12: "thread \<noteq> hd (wq s cs1)"
- assumes h21: "thread \<in> set (wq s cs2)"
- and h22: "thread \<noteq> hd (wq s cs2)"
- and neq12: "cs1 \<noteq> cs2"
- shows "False"
-proof -
- let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
- from h11 and h12 have q1: "?Q cs1 s" by simp
- from h21 and h22 have q2: "?Q cs2 s" by simp
- have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
- have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
- from p_split [of "?Q cs1", OF q1 nq1]
- obtain t1 where lt1: "t1 < length s"
- and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
- thread \<noteq> hd (wq (moment t1 s) cs1))"
- and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
- thread \<noteq> hd (wq (moment i' s) cs1))" by auto
- from p_split [of "?Q cs2", OF q2 nq2]
- obtain t2 where lt2: "t2 < length s"
- and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
- thread \<noteq> hd (wq (moment t2 s) cs2))"
- and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
- thread \<noteq> hd (wq (moment i' s) cs2))" by auto
- show ?thesis
- proof -
- {
- assume lt12: "t1 < t2"
- let ?t3 = "Suc t2"
- from lt2 have le_t3: "?t3 \<le> length s" by auto
- from moment_plus [OF this]
- obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
- have "t2 < ?t3" by simp
- from nn2 [rule_format, OF this] and eq_m
- have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- have "vt (e#moment t2 s)"
- proof -
- from vt_moment
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- then interpret vt_e: valid_trace_e "moment t2 s" "e"
- by (unfold_locales, auto, cases, simp)
- have ?thesis
- proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
- case True
- from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
- by auto
- from vt_e.abs2 [OF True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from vt_e.block_pre[OF False h1]
- have "e = P thread cs2" .
- with vt_e.vt_e have "vt ((P thread cs2)# moment t2 s)" by simp
- from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
- with runing_ready have "thread \<in> readys (moment t2 s)" by auto
- with nn1 [rule_format, OF lt12]
- show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto)
- qed
- } moreover {
- assume lt12: "t2 < t1"
- let ?t3 = "Suc t1"
- from lt1 have le_t3: "?t3 \<le> length s" by auto
- from moment_plus [OF this]
- obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
- have lt_t3: "t1 < ?t3" by simp
- from nn1 [rule_format, OF this] and eq_m
- have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
- h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have "vt (e#moment t1 s)"
- proof -
- from vt_moment
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- then interpret vt_e: valid_trace_e "moment t1 s" e
- by (unfold_locales, auto, cases, auto)
- have ?thesis
- proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
- case True
- from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
- by auto
- from vt_e.abs2 True eq_th h2 h1
- show ?thesis by auto
- next
- case False
- from vt_e.block_pre [OF False h1]
- have "e = P thread cs1" .
- with vt_e.vt_e have "vt ((P thread cs1)# moment t1 s)" by simp
- from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
- with runing_ready have "thread \<in> readys (moment t1 s)" by auto
- with nn2 [rule_format, OF lt12]
- show ?thesis by (simp add:readys_def wq_def s_waiting_def, auto)
- qed
- } moreover {
- assume eqt12: "t1 = t2"
- let ?t3 = "Suc t1"
- from lt1 have le_t3: "?t3 \<le> length s" by auto
- from moment_plus [OF this]
- obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
- have lt_t3: "t1 < ?t3" by simp
- from nn1 [rule_format, OF this] and eq_m
- have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
- h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have vt_e: "vt (e#moment t1 s)"
- proof -
- from vt_moment
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- then interpret vt_e: valid_trace_e "moment t1 s" e
- by (unfold_locales, auto, cases, auto)
- have ?thesis
- proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
- case True
- from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
- by auto
- from vt_e.abs2 [OF True eq_th h2 h1]
- show ?thesis by auto
- next
- case False
- from vt_e.block_pre [OF False h1]
- have eq_e1: "e = P thread cs1" .
- have lt_t3: "t1 < ?t3" by simp
- with eqt12 have "t2 < ?t3" by simp
- from nn2 [rule_format, OF this] and eq_m and eqt12
- have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- show ?thesis
- proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
- case True
- from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
- by auto
- from vt_e and eqt12 have "vt (e#moment t2 s)" by simp
- then interpret vt_e2: valid_trace_e "moment t2 s" e
- by (unfold_locales, auto, cases, auto)
- from vt_e2.abs2 [OF True eq_th h2 h1]
- show ?thesis .
- next
- case False
- have "vt (e#moment t2 s)"
- proof -
- from vt_moment eqt12
- have "vt (moment (Suc t2) s)" by auto
- with eq_m eqt12 show ?thesis by simp
- qed
- then interpret vt_e2: valid_trace_e "moment t2 s" e
- by (unfold_locales, auto, cases, auto)
- from vt_e2.block_pre [OF False h1]
- have "e = P thread cs2" .
- with eq_e1 neq12 show ?thesis by auto
- qed
- qed
- } ultimately show ?thesis by arith
- qed
-qed
-
-text {*
- This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
-*}
-
-lemma waiting_unique:
- assumes "waiting s th cs1"
- and "waiting s th cs2"
- shows "cs1 = cs2"
-using waiting_unique_pre assms
-unfolding wq_def s_waiting_def
-by auto
-
-end
-
-(* not used *)
-text {*
- Every thread can only be blocked on one critical resource,
- symmetrically, every critical resource can only be held by one thread.
- This fact is much more easier according to our definition.
-*}
-lemma held_unique:
- assumes "holding (s::event list) th1 cs"
- and "holding s th2 cs"
- shows "th1 = th2"
- by (insert assms, unfold s_holding_def, auto)
-
-
-lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits)
-
-lemma last_set_unique:
- "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
- \<Longrightarrow> th1 = th2"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits dest:last_set_lt)
-
-lemma preced_unique :
- assumes pcd_eq: "preced th1 s = preced th2 s"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "th1 = th2"
-proof -
- from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
- from last_set_unique [OF this th_in1 th_in2]
- show ?thesis .
-qed
-
-lemma preced_linorder:
- assumes neq_12: "th1 \<noteq> th2"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
- from preced_unique [OF _ th_in1 th_in2] and neq_12
- have "preced th1 s \<noteq> preced th2 s" by auto
- thus ?thesis by auto
-qed
-
-(* An aux lemma used later *)
-lemma unique_minus:
- fixes x y z r
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq: "y \<noteq> z"
- shows "(y, z) \<in> r^+"
-proof -
- from xz and neq show ?thesis
- proof(induct)
- case (base ya)
- have "(x, ya) \<in> r" by fact
- from unique [OF xy this] have "y = ya" .
- with base show ?case by auto
- next
- case (step ya z)
- show ?case
- proof(cases "y = ya")
- case True
- from step True show ?thesis by simp
- next
- case False
- from step False
- show ?thesis by auto
- qed
- qed
-qed
-
-lemma unique_base:
- fixes r x y z
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> r^+"
-proof -
- from xz neq_yz show ?thesis
- proof(induct)
- case (base ya)
- from xy unique base show ?case by auto
- next
- case (step ya z)
- show ?case
- proof(cases "y = ya")
- case True
- from True step show ?thesis by auto
- next
- case False
- from False step
- have "(y, ya) \<in> r\<^sup>+" by auto
- with step show ?thesis by auto
- qed
- qed
-qed
-
-lemma unique_chain:
- fixes r x y z
- assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
- and xy: "(x, y) \<in> r^+"
- and xz: "(x, z) \<in> r^+"
- and neq_yz: "y \<noteq> z"
- shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
-proof -
- from xy xz neq_yz show ?thesis
- proof(induct)
- case (base y)
- have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
- from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
- next
- case (step y za)
- show ?case
- proof(cases "y = z")
- case True
- from True step show ?thesis by auto
- next
- case False
- from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
- thus ?thesis
- proof
- assume "(z, y) \<in> r\<^sup>+"
- with step have "(z, za) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- next
- assume h: "(y, z) \<in> r\<^sup>+"
- from step have yza: "(y, za) \<in> r" by simp
- from step have "za \<noteq> z" by simp
- from unique_minus [OF _ yza h this] and unique
- have "(za, z) \<in> r\<^sup>+" by auto
- thus ?thesis by auto
- qed
- qed
- qed
-qed
-
-text {*
- The following three lemmas show that @{text "RAG"} does not change
- by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
- events, respectively.
-*}
-
-lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-
-text {*
- The following lemmas are used in the proof of
- lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed
- by @{text "V"}-events.
- However, since our model is very concise, such seemingly obvious lemmas need to be derived from scratch,
- starting from the model definitions.
-*}
-lemma step_v_hold_inv[elim_format]:
- "\<And>c t. \<lbrakk>vt (V th cs # s);
- \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow>
- next_th s th cs t \<and> c = cs"
-proof -
- fix c t
- assume vt: "vt (V th cs # s)"
- and nhd: "\<not> holding (wq s) t c"
- and hd: "holding (wq (V th cs # s)) t c"
- show "next_th s th cs t \<and> c = cs"
- proof(cases "c = cs")
- case False
- with nhd hd show ?thesis
- by (unfold cs_holding_def wq_def, auto simp:Let_def)
- next
- case True
- with step_back_step [OF vt]
- have "step s (V th c)" by simp
- hence "next_th s th cs t"
- proof(cases)
- assume "holding s th c"
- with nhd hd show ?thesis
- apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
- auto simp:Let_def split:list.splits if_splits)
- proof -
- assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
- moreover have "\<dots> = set []"
- proof(rule someI2)
- show "distinct [] \<and> [] = []" by auto
- next
- fix x assume "distinct x \<and> x = []"
- thus "set x = set []" by auto
- qed
- ultimately show False by auto
- next
- assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
- moreover have "\<dots> = set []"
- proof(rule someI2)
- show "distinct [] \<and> [] = []" by auto
- next
- fix x assume "distinct x \<and> x = []"
- thus "set x = set []" by auto
- qed
- ultimately show False by auto
- qed
- qed
- with True show ?thesis by auto
- qed
-qed
-
-text {*
- The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be
- derived from scratch, which confirms the correctness of the definition of @{text "next_th"}.
-*}
-lemma step_v_wait_inv[elim_format]:
- "\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
- \<rbrakk>
- \<Longrightarrow> (next_th s th cs t \<and> cs = c)"
-proof -
- fix t c
- assume vt: "vt (V th cs # s)"
- and nw: "\<not> waiting (wq (V th cs # s)) t c"
- and wt: "waiting (wq s) t c"
- from vt interpret vt_v: valid_trace_e s "V th cs"
- by (cases, unfold_locales, simp)
- show "next_th s th cs t \<and> cs = c"
- proof(cases "cs = c")
- case False
- with nw wt show ?thesis
- by (auto simp:cs_waiting_def wq_def Let_def)
- next
- case True
- from nw[folded True] wt[folded True]
- have "next_th s th cs t"
- apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
- proof -
- fix a list
- assume t_in: "t \<in> set list"
- and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have " set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
- by auto
- qed
- with t_ni and t_in show "a = th" by auto
- next
- fix a list
- assume t_in: "t \<in> set list"
- and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have " set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
- by auto
- qed
- with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto
- next
- fix a list
- assume eq_wq: "wq_fun (schs s) cs = a # list"
- from step_back_step[OF vt]
- show "a = th"
- proof(cases)
- assume "holding s th cs"
- with eq_wq show ?thesis
- by (unfold s_holding_def wq_def, auto)
- qed
- qed
- with True show ?thesis by simp
- qed
-qed
-
-lemma step_v_not_wait[consumes 3]:
- "\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
- by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
-
-lemma step_v_release:
- "\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
-proof -
- assume vt: "vt (V th cs # s)"
- and hd: "holding (wq (V th cs # s)) th cs"
- from vt interpret vt_v: valid_trace_e s "V th cs"
- by (cases, unfold_locales, simp+)
- from step_back_step [OF vt] and hd
- show "False"
- proof(cases)
- assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"
- thus ?thesis
- apply (unfold s_holding_def wq_def cs_holding_def)
- apply (auto simp:Let_def split:list.splits)
- proof -
- fix list
- assume eq_wq[folded wq_def]:
- "wq_fun (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
- and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
- \<in> set (SOME q. distinct q \<and> set q = set list)"
- have "set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from vt_v.wq_distinct[of cs] and eq_wq
- show "distinct list \<and> set list = set list" by auto
- next
- show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
- by auto
- qed
- moreover have "distinct (hd (SOME q. distinct q \<and> set q = set list) # list)"
- proof -
- from vt_v.wq_distinct[of cs] and eq_wq
- show ?thesis by auto
- qed
- moreover note eq_wq and hd_in
- ultimately show "False" by auto
- qed
- qed
-qed
-
-lemma step_v_get_hold:
- "\<And>th'. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
- apply (unfold cs_holding_def next_th_def wq_def,
- auto simp:Let_def)
-proof -
- fix rest
- assume vt: "vt (V th cs # s)"
- and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest"
- and nrest: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest)
- \<notin> set (SOME q. distinct q \<and> set q = set rest)"
- from vt interpret vt_v: valid_trace_e s "V th cs"
- by (cases, unfold_locales, simp+)
- have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
- proof(rule someI2)
- from vt_v.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"
- hence "set x = set rest" by auto
- with nrest
- show "x \<noteq> []" by (case_tac x, auto)
- qed
- with ni show "False" by auto
-qed
-
-lemma step_v_release_inv[elim_format]:
-"\<And>c t. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow>
- c = cs \<and> t = th"
- apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
- proof -
- fix a list
- assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
- from step_back_step [OF vt] show "a = th"
- proof(cases)
- assume "holding s th cs" with eq_wq
- show ?thesis
- by (unfold s_holding_def wq_def, auto)
- qed
- next
- fix a list
- assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
- from step_back_step [OF vt] show "a = th"
- proof(cases)
- assume "holding s th cs" with eq_wq
- show ?thesis
- by (unfold s_holding_def wq_def, auto)
- qed
- qed
-
-lemma step_v_waiting_mono:
- "\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
-proof -
- fix t c
- let ?s' = "(V th cs # s)"
- assume vt: "vt ?s'"
- and wt: "waiting (wq ?s') t c"
- from vt interpret vt_v: valid_trace_e s "V th cs"
- by (cases, unfold_locales, simp+)
- show "waiting (wq s) t c"
- proof(cases "c = cs")
- case False
- assume neq_cs: "c \<noteq> cs"
- hence "waiting (wq ?s') t c = waiting (wq s) t c"
- by (unfold cs_waiting_def wq_def, auto simp:Let_def)
- with wt show ?thesis by simp
- next
- case True
- with wt show ?thesis
- apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
- proof -
- fix a list
- assume not_in: "t \<notin> set list"
- and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
- and eq_wq: "wq_fun (schs s) cs = a # list"
- have "set (SOME q. distinct q \<and> set q = set list) = set list"
- proof(rule someI2)
- from vt_v.wq_distinct [of cs]
- and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- fix x assume "distinct x \<and> set x = set list"
- thus "set x = set list" by auto
- qed
- with not_in is_in show "t = a" by auto
- next
- fix list
- assume is_waiting: "waiting (wq (V th cs # s)) t cs"
- and eq_wq: "wq_fun (schs s) cs = t # list"
- hence "t \<in> set list"
- apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
- proof -
- assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
- moreover have "\<dots> = set list"
- proof(rule someI2)
- from vt_v.wq_distinct [of cs]
- and eq_wq[folded wq_def]
- show "distinct list \<and> set list = set list" by auto
- next
- fix x assume "distinct x \<and> set x = set list"
- thus "set x = set list" by auto
- qed
- ultimately show "t \<in> set list" by simp
- qed
- with eq_wq and vt_v.wq_distinct [of cs, unfolded wq_def]
- show False by auto
- qed
- qed
-qed
-
-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:
-fixes th::thread
-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
-
-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 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
-
-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)
-qed
-
-
-lemma finite_RAG:
- shows "finite (RAG s)"
-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)" .
-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)"
- 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 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:
- fixes th thread cs rest
- assumes neq_th: "th \<noteq> thread"
- and eq_wq: "wq s cs = thread#rest"
- and not_in: "th \<notin> set rest"
- shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
-proof -
- from 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 waiting_eq: "waiting s th cs = waiting (wq s) th cs"
- by (unfold s_waiting_def cs_waiting_def wq_def, auto)
-
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
- by (unfold s_holding_def wq_def cs_holding_def, simp)
-
-lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
- by (unfold s_holding_def cs_holding_def, auto)
-
-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)
-
-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:
- fixes th cs s
- assumes vt: "vt (P th cs#s)"
- and "wq s cs = []"
- shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
-proof -
- from assms show ?thesis
- unfolding holdents_test step_RAG_p[OF vt] by (auto)
-qed
-
-lemma step_holdents_p_eq:
- fixes th cs s
- assumes vt: "vt (P th cs#s)"
- and "wq s cs \<noteq> []"
- shows "holdents (P th cs#s) th = holdents s th"
-proof -
- from assms show ?thesis
- unfolding holdents_test step_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:
- fixes s thread cs
- 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
-
-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
-
-lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
- by (auto simp:s_waiting_def cs_waiting_def wq_def)
-
-context valid_trace
-begin
-
-lemma dm_RAG_threads:
- assumes in_dom: "(Th th) \<in> Domain (RAG s)"
- shows "th \<in> threads s"
-proof -
- from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
- moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
- ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
- hence "th \<in> set (wq s cs)"
- by (unfold s_RAG_def, auto simp:cs_waiting_def)
- from wq_threads [OF this] show ?thesis .
-qed
-
-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:eq_waiting)
- 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:eq_waiting)
- 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
-
-lemma length_down_to_in:
- assumes le_ij: "i \<le> j"
- and le_js: "j \<le> length s"
- shows "length (down_to j i s) = j - i"
-proof -
- have "length (down_to j i s) = length (from_to i j (rev s))"
- by (unfold down_to_def, auto)
- also have "\<dots> = j - i"
- proof(rule length_from_to_in[OF le_ij])
- from le_js show "j \<le> length (rev s)" by simp
- qed
- finally show ?thesis .
-qed
-
-
-lemma moment_head:
- assumes le_it: "Suc i \<le> length t"
- obtains e where "moment (Suc i) t = e#moment i t"
-proof -
- have "i \<le> Suc i" by simp
- from length_down_to_in [OF this le_it]
- have "length (down_to (Suc i) i t) = 1" by auto
- then obtain e where "down_to (Suc i) i t = [e]"
- apply (cases "(down_to (Suc i) i t)") by auto
- moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
- by (rule down_to_conc[symmetric], auto)
- ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
- by (auto simp:down_to_moment)
- from that [OF this] show ?thesis .
-qed
-
-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:
- 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_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:eq_waiting 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)
-
-end