--- a/ExtGG.thy~ Wed Jan 27 19:28:42 2016 +0800
+++ b/ExtGG.thy~ Wed Jan 27 23:34:23 2016 +0800
@@ -1,922 +1,920 @@
-theory ExtGG
-imports PrioG CpsG
+section {*
+ This file contains lemmas used to guide the recalculation of current precedence
+ after every system call (or system operation)
+*}
+theory Implementation
+imports PIPBasics
+begin
+
+text {* (* ddd *)
+ One beauty of our modelling is that we follow the definitional extension tradition of HOL.
+ The benefit of such a concise and miniature model is that large number of intuitively
+ obvious facts are derived as lemmas, rather than asserted as axioms.
+*}
+
+text {*
+ However, the lemmas in the forthcoming several locales are no longer
+ obvious. These lemmas show how the current precedences should be recalculated
+ after every execution step (in our model, every step is represented by an event,
+ which in turn, represents a system call, or operation). Each operation is
+ treated in a separate locale.
+
+ The complication of current precedence recalculation comes
+ because the changing of RAG needs to be taken into account,
+ in addition to the changing of precedence.
+
+ The reason RAG changing affects current precedence is that,
+ according to the definition, current precedence
+ of a thread is the maximum of the precedences of every threads in its subtree,
+ where the notion of sub-tree in RAG is defined in RTree.thy.
+
+ Therefore, for each operation, lemmas about the change of precedences
+ and RAG are derived first, on which lemmas about current precedence
+ recalculation are based on.
+*}
+
+section {* The @{term Set} operation *}
+
+text {* (* ddd *)
+ The following locale @{text "step_set_cps"} investigates the recalculation
+ after the @{text "Set"} operation.
+*}
+locale step_set_cps =
+ fixes s' th prio s
+ -- {* @{text "s'"} is the system state before the operation *}
+ -- {* @{text "s"} is the system state after the operation *}
+ defines s_def : "s \<equiv> (Set th prio#s')"
+ -- {* @{text "s"} is assumed to be a legitimate state, from which
+ the legitimacy of @{text "s"} can be derived. *}
+ assumes vt_s: "vt s"
+
+sublocale step_set_cps < vat_s : valid_trace "s"
+proof
+ from vt_s show "vt s" .
+qed
+
+sublocale step_set_cps < vat_s' : valid_trace "s'"
+proof
+ from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_set_cps
begin
-text {*
- The following two auxiliary lemmas are used to reason about @{term Max}.
+text {* (* ddd *)
+ The following two lemmas confirm that @{text "Set"}-operation
+ only changes the precedence of the initiating thread (or actor)
+ of the operation (or event).
*}
-lemma image_Max_eqI:
- assumes "finite B"
- and "b \<in> B"
- and "\<forall> x \<in> B. f x \<le> f b"
- shows "Max (f ` B) = f b"
+
+lemma eq_preced:
+ assumes "th' \<noteq> th"
+ shows "preced th' s = preced th' s'"
+proof -
+ from assms show ?thesis
+ by (unfold s_def, auto simp:preced_def)
+qed
+
+lemma eq_the_preced:
+ assumes "th' \<noteq> th"
+ shows "the_preced s th' = the_preced s' th'"
using assms
- using Max_eqI by blast
+ by (unfold the_preced_def, intro eq_preced, simp)
+
+text {*
+ The following lemma assures that the resetting of priority does not change the RAG.
+*}
+
+lemma eq_dep: "RAG s = RAG s'"
+ by (unfold s_def RAG_set_unchanged, auto)
-lemma image_Max_subset:
- assumes "finite A"
- and "B \<subseteq> A"
- and "a \<in> B"
- and "Max (f ` A) = f a"
- shows "Max (f ` B) = f a"
-proof(rule image_Max_eqI)
- show "finite B"
- using assms(1) assms(2) finite_subset by auto
-next
- show "a \<in> B" using assms by simp
-next
- show "\<forall>x\<in>B. f x \<le> f a"
- by (metis Max_ge assms(1) assms(2) assms(4)
- finite_imageI image_eqI subsetCE)
+text {* (* ddd *)
+ Th following lemma @{text "eq_cp_pre"} says that the priority change of @{text "th"}
+ only affects those threads, which as @{text "Th th"} in their sub-trees.
+
+ The proof of this lemma is simplified by using the alternative definition
+ of @{text "cp"}.
+*}
+
+lemma eq_cp_pre:
+ assumes nd: "Th th \<notin> subtree (RAG s') (Th th')"
+ shows "cp s th' = cp s' th'"
+proof -
+ -- {* After unfolding using the alternative definition, elements
+ affecting the @{term "cp"}-value of threads become explicit.
+ We only need to prove the following: *}
+ have "Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
+ Max (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+ (is "Max (?f ` ?S1) = Max (?g ` ?S2)")
+ proof -
+ -- {* The base sets are equal. *}
+ have "?S1 = ?S2" using eq_dep by simp
+ -- {* The function values on the base set are equal as well. *}
+ moreover have "\<forall> e \<in> ?S2. ?f e = ?g e"
+ proof
+ fix th1
+ assume "th1 \<in> ?S2"
+ with nd have "th1 \<noteq> th" by (auto)
+ from eq_the_preced[OF this]
+ show "the_preced s th1 = the_preced s' th1" .
+ qed
+ -- {* Therefore, the image of the functions are equal. *}
+ ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq)
+ thus ?thesis by simp
+ qed
+ thus ?thesis by (simp add:cp_alt_def)
qed
text {*
- The following locale @{text "highest_gen"} sets the basic context for our
- investigation: supposing thread @{text th} holds the highest @{term cp}-value
- in state @{text s}, which means the task for @{text th} is the
- most urgent. We want to show that
- @{text th} is treated correctly by PIP, which means
- @{text th} will not be blocked unreasonably by other less urgent
- threads.
+ The following lemma shows that @{term "th"} is not in the
+ sub-tree of any other thread.
*}
-locale highest_gen =
- fixes s th prio tm
+lemma th_in_no_subtree:
+ assumes "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s') (Th th')"
+proof -
+ have "th \<in> readys s'"
+ proof -
+ from step_back_step [OF vt_s[unfolded s_def]]
+ have "step s' (Set th prio)" .
+ hence "th \<in> runing s'" by (cases, simp)
+ thus ?thesis by (simp add:readys_def runing_def)
+ qed
+ from vat_s'.readys_in_no_subtree[OF this assms(1)]
+ show ?thesis by blast
+qed
+
+text {*
+ By combining @{thm "eq_cp_pre"} and @{thm "th_in_no_subtree"},
+ it is obvious that the change of priority only affects the @{text "cp"}-value
+ of the initiating thread @{text "th"}.
+*}
+lemma eq_cp:
+ assumes "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+ by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])
+
+end
+
+section {* The @{term V} operation *}
+
+text {*
+ The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.
+*}
+
+locale step_v_cps =
+ -- {* @{text "th"} is the initiating thread *}
+ -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *}
+ fixes s' th cs s -- {* @{text "s'"} is the state before operation*}
+ defines s_def : "s \<equiv> (V th cs#s')" -- {* @{text "s"} is the state after operation*}
+ -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *}
assumes vt_s: "vt s"
- and threads_s: "th \<in> threads s"
- and highest: "preced th s = Max ((cp s)`threads s)"
- -- {* The internal structure of @{term th}'s precedence is exposed:*}
- and preced_th: "preced th s = Prc prio tm"
+
+sublocale step_v_cps < vat_s : valid_trace "s"
+proof
+ from vt_s show "vt s" .
+qed
+
+sublocale step_v_cps < vat_s' : valid_trace "s'"
+proof
+ from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_v_cps
+begin
+
+lemma ready_th_s': "th \<in> readys s'"
+ using step_back_step[OF vt_s[unfolded s_def]]
+ by (cases, simp add:runing_def)
+
+lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"
+proof -
+ from vat_s'.readys_root[OF ready_th_s']
+ show ?thesis
+ by (unfold root_def, simp)
+qed
+
+lemma holding_th: "holding s' th cs"
+proof -
+ from vt_s[unfolded s_def]
+ have " PIP s' (V th cs)" by (cases, simp)
+ thus ?thesis by (cases, auto)
+qed
--- {* @{term s} is a valid trace, so it will inherit all results derived for
- a valid trace: *}
-sublocale highest_gen < vat_s: valid_trace "s"
- by (unfold_locales, insert vt_s, simp)
+lemma edge_of_th:
+ "(Cs cs, Th th) \<in> RAG s'"
+proof -
+ from holding_th
+ show ?thesis
+ by (unfold s_RAG_def holding_eq, auto)
+qed
-context highest_gen
+lemma ancestors_cs:
+ "ancestors (RAG s') (Cs cs) = {Th th}"
+proof -
+ have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th) \<union> {Th th}"
+ proof(rule vat_s'.rtree_RAG.ancestors_accum)
+ from vt_s[unfolded s_def]
+ have " PIP s' (V th cs)" by (cases, simp)
+ thus "(Cs cs, Th th) \<in> RAG s'"
+ proof(cases)
+ assume "holding s' th cs"
+ from this[unfolded holding_eq]
+ show ?thesis by (unfold s_RAG_def, auto)
+ qed
+ qed
+ from this[unfolded ancestors_th] show ?thesis by simp
+qed
+
+lemma preced_kept: "the_preced s = the_preced s'"
+ by (auto simp: s_def the_preced_def preced_def)
+
+end
+
+text {*
+ The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation,
+ which represents the case when there is another thread @{text "th'"}
+ to take over the critical resource released by the initiating thread @{text "th"}.
+*}
+locale step_v_cps_nt = step_v_cps +
+ fixes th'
+ -- {* @{text "th'"} is assumed to take over @{text "cs"} *}
+ assumes nt: "next_th s' th cs th'"
+
+context step_v_cps_nt
begin
text {*
- @{term tm} is the time when the precedence of @{term th} is set, so
- @{term tm} must be a valid moment index into @{term s}.
+ Lemma @{text "RAG_s"} confirms the change of RAG:
+ two edges removed and one added, as shown by the following diagram.
*}
-lemma lt_tm: "tm < length s"
- by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+(*
+ RAG before the V-operation
+ th1 ----|
+ |
+ th' ----|
+ |----> cs -----|
+ th2 ----| |
+ | |
+ th3 ----| |
+ |------> th
+ th4 ----| |
+ | |
+ th5 ----| |
+ |----> cs'-----|
+ th6 ----|
+ |
+ th7 ----|
+
+ RAG after the V-operation
+ th1 ----|
+ |
+ |----> cs ----> th'
+ th2 ----|
+ |
+ th3 ----|
+
+ th4 ----|
+ |
+ th5 ----|
+ |----> cs'----> th
+ th6 ----|
+ |
+ th7 ----|
+*)
+
+lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"
+ using next_th_RAG[OF nt] .
+
+lemma ancestors_th':
+ "ancestors (RAG s') (Th th') = {Th th, Cs cs}"
+proof -
+ have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"
+ proof(rule vat_s'.rtree_RAG.ancestors_accum)
+ from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto
+ qed
+ thus ?thesis using ancestors_th ancestors_cs by auto
+qed
+
+lemma RAG_s:
+ "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
+ {(Cs cs, Th th')}"
+proof -
+ from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
+ and nt show ?thesis by (auto intro:next_th_unique)
+qed
-text {*
- Since @{term th} holds the highest precedence and @{text "cp"}
- is the highest precedence of all threads in the sub-tree of
- @{text "th"} and @{text th} is among these threads,
- its @{term cp} must equal to its precedence:
-*}
-lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
+lemma subtree_kept: (* ddd *)
+ assumes "th1 \<notin> {th, th'}"
+ shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R")
proof -
- have "?L \<le> ?R"
- by (unfold highest, rule Max_ge,
- auto simp:threads_s finite_threads)
- moreover have "?R \<le> ?L"
- by (unfold vat_s.cp_rec, rule Max_ge,
- auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
- ultimately show ?thesis by auto
+ let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})"
+ let ?RAG'' = "?RAG' \<union> {(Cs cs, Th th')}"
+ have "subtree ?RAG' (Th th1) = ?R"
+ proof(rule subset_del_subtree_outside)
+ show "Range {(Cs cs, Th th), (Th th', Cs cs)} \<inter> subtree (RAG s') (Th th1) = {}"
+ proof -
+ have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+ proof(rule subtree_refute)
+ show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+ by (unfold ancestors_th, simp)
+ next
+ from assms show "Th th1 \<noteq> Th th" by simp
+ qed
+ moreover have "(Cs cs) \<notin> subtree (RAG s') (Th th1)"
+ proof(rule subtree_refute)
+ show "Th th1 \<notin> ancestors (RAG s') (Cs cs)"
+ by (unfold ancestors_cs, insert assms, auto)
+ qed simp
+ ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s') (Th th1) = {}" by auto
+ thus ?thesis by simp
+ qed
+ qed
+ moreover have "subtree ?RAG'' (Th th1) = subtree ?RAG' (Th th1)"
+ proof(rule subtree_insert_next)
+ show "Th th' \<notin> subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)"
+ proof(rule subtree_refute)
+ show "Th th1 \<notin> ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')"
+ (is "_ \<notin> ?R")
+ proof -
+ have "?R \<subseteq> ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto)
+ moreover have "Th th1 \<notin> ..." using ancestors_th' assms by simp
+ ultimately show ?thesis by auto
+ qed
+ next
+ from assms show "Th th1 \<noteq> Th th'" by simp
+ qed
+ qed
+ ultimately show ?thesis by (unfold RAG_s, simp)
+qed
+
+lemma cp_kept:
+ assumes "th1 \<notin> {th, th'}"
+ shows "cp s th1 = cp s' th1"
+ by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+
+end
+
+locale step_v_cps_nnt = step_v_cps +
+ assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
+
+context step_v_cps_nnt
+begin
+
+lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"
+proof -
+ from nnt and step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
+ show ?thesis by auto
qed
-(* ccc *)
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold max_cp_eq, unfold eq_cp_s_th, insert highest, simp)
+lemma subtree_kept:
+ assumes "th1 \<noteq> th"
+ shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)"
+proof(unfold RAG_s, rule subset_del_subtree_outside)
+ show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s') (Th th1) = {}"
+ proof -
+ have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+ proof(rule subtree_refute)
+ show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+ by (unfold ancestors_th, simp)
+ next
+ from assms show "Th th1 \<noteq> Th th" by simp
+ qed
+ thus ?thesis by auto
+ qed
+qed
+
+lemma cp_kept_1:
+ assumes "th1 \<noteq> th"
+ shows "cp s th1 = cp s' th1"
+ by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+
+lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}"
+proof -
+ { fix n
+ have "(Cs cs) \<notin> ancestors (RAG s') n"
+ proof
+ assume "Cs cs \<in> ancestors (RAG s') n"
+ hence "(n, Cs cs) \<in> (RAG s')^+" by (auto simp:ancestors_def)
+ from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s'" by auto
+ then obtain th' where "nn = Th th'"
+ by (unfold s_RAG_def, auto)
+ from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s'" .
+ from this[unfolded s_RAG_def]
+ have "waiting (wq s') th' cs" by auto
+ from this[unfolded cs_waiting_def]
+ have "1 < length (wq s' cs)"
+ by (cases "wq s' cs", auto)
+ from holding_next_thI[OF holding_th this]
+ obtain th' where "next_th s' th cs th'" by auto
+ with nnt show False by auto
+ qed
+ } note h = this
+ { fix n
+ assume "n \<in> subtree (RAG s') (Cs cs)"
+ hence "n = (Cs cs)"
+ by (elim subtreeE, insert h, auto)
+ } moreover have "(Cs cs) \<in> subtree (RAG s') (Cs cs)"
+ by (auto simp:subtree_def)
+ ultimately show ?thesis by auto
+qed
+
+lemma subtree_th:
+ "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
+proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
+ from edge_of_th
+ show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"
+ by (unfold edges_in_def, auto simp:subtree_def)
+qed
+
+lemma cp_kept_2:
+ shows "cp s th = cp s' th"
+ by (unfold cp_alt_def subtree_th preced_kept, auto)
+
+lemma eq_cp:
+ shows "cp s th' = cp s' th'"
+ using cp_kept_1 cp_kept_2
+ by (cases "th' = th", auto)
+end
+
+
+locale step_P_cps =
+ fixes s' th cs s
+ defines s_def : "s \<equiv> (P th cs#s')"
+ assumes vt_s: "vt s"
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+sublocale step_P_cps < vat_s : valid_trace "s"
+proof
+ from vt_s show "vt s" .
+qed
+
+section {* The @{term P} operation *}
+
+sublocale step_P_cps < vat_s' : valid_trace "s'"
+proof
+ from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_P_cps
+begin
+
+lemma readys_th: "th \<in> readys s'"
+proof -
+ from step_back_step [OF vt_s[unfolded s_def]]
+ have "PIP s' (P th cs)" .
+ hence "th \<in> runing s'" by (cases, simp)
+ thus ?thesis by (simp add:readys_def runing_def)
+qed
+
+lemma root_th: "root (RAG s') (Th th)"
+ using readys_root[OF readys_th] .
-lemma highest': "cp s th = Max (cp s ` threads s)"
+lemma in_no_others_subtree:
+ assumes "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s') (Th th')"
+proof
+ assume "Th th \<in> subtree (RAG s') (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 1
+ with assms show ?thesis by auto
+ next
+ case 2
+ with root_th show ?thesis by (auto simp:root_def)
+ qed
+qed
+
+lemma preced_kept: "the_preced s = the_preced s'"
+ by (auto simp: s_def the_preced_def preced_def)
+
+end
+
+locale step_P_cps_ne =step_P_cps +
+ fixes th'
+ assumes ne: "wq s' cs \<noteq> []"
+ defines th'_def: "th' \<equiv> hd (wq s' cs)"
+
+locale step_P_cps_e =step_P_cps +
+ assumes ee: "wq s' cs = []"
+
+context step_P_cps_e
+begin
+
+lemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"
proof -
- from highest_cp_preced max_cp_eq[symmetric]
- show ?thesis by simp
+ from ee and step_RAG_p[OF vt_s[unfolded s_def], folded s_def]
+ show ?thesis by auto
+qed
+
+lemma subtree_kept:
+ assumes "th' \<noteq> th"
+ shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')"
+proof(unfold RAG_s, rule subtree_insert_next)
+ from in_no_others_subtree[OF assms]
+ show "Th th \<notin> subtree (RAG s') (Th th')" .
+qed
+
+lemma cp_kept:
+ assumes "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+proof -
+ have "(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
+ (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+ by (unfold preced_kept subtree_kept[OF assms], simp)
+ thus ?thesis by (unfold cp_alt_def, simp)
qed
end
-locale extend_highest_gen = highest_gen +
- fixes t
- assumes vt_t: "vt (t@s)"
- and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
- and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
- and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-sublocale extend_highest_gen < vat_t: valid_trace "t@s"
- by (unfold_locales, insert vt_t, simp)
-
-lemma step_back_vt_app:
- assumes vt_ts: "vt (t@s)"
- shows "vt s"
-proof -
- from vt_ts show ?thesis
- proof(induct t)
- case Nil
- from Nil show ?case by auto
- next
- case (Cons e t)
- assume ih: " vt (t @ s) \<Longrightarrow> vt s"
- and vt_et: "vt ((e # t) @ s)"
- show ?case
- proof(rule ih)
- show "vt (t @ s)"
- proof(rule step_back_vt)
- from vt_et show "vt (e # t @ s)" by simp
- qed
- qed
- qed
-qed
-
-
-locale red_extend_highest_gen = extend_highest_gen +
- fixes i::nat
-
-sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
- apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
- apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
- by (unfold highest_gen_def, auto dest:step_back_vt_app)
-
-
-context extend_highest_gen
+context step_P_cps_ne
begin
- lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes
- h0: "R []"
- and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e;
- extend_highest_gen s th prio tm t;
- extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
- shows "R t"
+lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+proof -
+ from step_RAG_p[OF vt_s[unfolded s_def]] and ne
+ show ?thesis by (simp add:s_def)
+qed
+
+lemma cs_held: "(Cs cs, Th th') \<in> RAG s'"
proof -
- from vt_t extend_highest_gen_axioms show ?thesis
- proof(induct t)
- from h0 show "R []" .
- next
- case (Cons e t')
- assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
- and vt_e: "vt ((e # t') @ s)"
- and et: "extend_highest_gen s th prio tm (e # t')"
- from vt_e and step_back_step have stp: "step (t'@s) e" by auto
- from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
- show ?case
- proof(rule h2 [OF vt_ts stp _ _ _ ])
- show "R t'"
- proof(rule ih)
- from et show ext': "extend_highest_gen s th prio tm t'"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
- next
- from vt_ts show "vt (t' @ s)" .
- qed
- next
- from et show "extend_highest_gen s th prio tm (e # t')" .
- next
- from et show ext': "extend_highest_gen s th prio tm t'"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
- qed
+ have "(Cs cs, Th th') \<in> hRAG s'"
+ proof -
+ from ne
+ have " holding s' th' cs"
+ by (unfold th'_def holding_eq cs_holding_def, auto)
+ thus ?thesis
+ by (unfold hRAG_def, auto)
qed
+ thus ?thesis by (unfold RAG_split, auto)
qed
+lemma tRAG_s:
+ "tRAG s = tRAG s' \<union> {(Th th, Th th')}"
+ using RAG_tRAG_transfer[OF RAG_s cs_held] .
-lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
+lemma cp_kept:
+ assumes "Th th'' \<notin> ancestors (tRAG s) (Th th)"
+ shows "cp s th'' = cp s' th''"
proof -
- show ?thesis
- proof(induct rule:ind)
- case Nil
- from threads_s
- show ?case
- by auto
- next
- case (Cons e t)
- interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
- interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
- show ?case
- proof(cases e)
- case (Create thread prio)
- show ?thesis
- proof -
- from Cons and Create have "step (t@s) (Create thread prio)" by auto
- hence "th \<noteq> thread"
- proof(cases)
- case thread_create
- with Cons show ?thesis by auto
- qed
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold Create, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:Create)
- qed
- next
- case (Exit thread)
- from h_e.exit_diff and Exit
- have neq_th: "thread \<noteq> th" by auto
- with Cons
- show ?thesis
- by (unfold Exit, auto simp:preced_def)
- next
- case (P thread cs)
- with Cons
- show ?thesis
- by (auto simp:P preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis
- by (auto simp:V preced_def)
- next
- case (Set thread prio')
- show ?thesis
- proof -
- from h_e.set_diff_low and Set
- have "th \<noteq> thread" by auto
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold Set, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:Set)
+ have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')"
+ proof -
+ have "Th th' \<notin> subtree (tRAG s') (Th th'')"
+ proof
+ assume "Th th' \<in> subtree (tRAG s') (Th th'')"
+ thus False
+ proof(rule subtreeE)
+ assume "Th th' = Th th''"
+ from assms[unfolded tRAG_s ancestors_def, folded this]
+ show ?thesis by auto
+ next
+ assume "Th th'' \<in> ancestors (tRAG s') (Th th')"
+ moreover have "... \<subseteq> ancestors (tRAG s) (Th th')"
+ proof(rule ancestors_mono)
+ show "tRAG s' \<subseteq> tRAG s" by (unfold tRAG_s, auto)
+ qed
+ ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th')" by auto
+ moreover have "Th th' \<in> ancestors (tRAG s) (Th th)"
+ by (unfold tRAG_s, auto simp:ancestors_def)
+ ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th)"
+ by (auto simp:ancestors_def)
+ with assms show ?thesis by auto
qed
qed
+ from subtree_insert_next[OF this]
+ have "subtree (tRAG s' \<union> {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" .
+ from this[folded tRAG_s] show ?thesis .
qed
+ show ?thesis by (unfold cp_alt_def1 h preced_kept, simp)
qed
-text {*
- According to @{thm th_kept}, thread @{text "th"} has its living status
- and precedence kept along the way of @{text "t"}. The following lemma
- shows that this preserved precedence of @{text "th"} remains as the highest
- along the way of @{text "t"}.
-
- The proof goes by induction over @{text "t"} using the specialized
- induction rule @{thm ind}, followed by case analysis of each possible
- operations of PIP. All cases follow the same pattern rendered by the
- generalized introduction rule @{thm "image_Max_eqI"}.
-
- The very essence is to show that precedences, no matter whether they are newly introduced
- or modified, are always lower than the one held by @{term "th"},
- which by @{thm th_kept} is preserved along the way.
-*}
-lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
- case Nil
- from highest_preced_thread
- show ?case
- by (unfold the_preced_def, simp)
-next
- case (Cons e t)
- interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
- interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
- show ?case
- proof(cases e)
- case (Create thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
+lemma cp_gen_update_stop: (* ddd *)
+ assumes "u \<in> ancestors (tRAG s) (Th th)"
+ and "cp_gen s u = cp_gen s' u"
+ and "y \<in> ancestors (tRAG s) u"
+ shows "cp_gen s y = cp_gen s' y"
+ using assms(3)
+proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf])
+ case (1 x)
+ show ?case (is "?L = ?R")
+ proof -
+ from tRAG_ancestorsE[OF 1(2)]
+ obtain th2 where eq_x: "x = Th th2" by blast
+ from vat_s.cp_gen_rec[OF this]
+ have "?L =
+ Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)" .
+ also have "... =
+ Max ({the_preced s' th2} \<union> cp_gen s' ` RTree.children (tRAG s') x)"
+
proof -
- -- {* The following is the common pattern of each branch of the case analysis. *}
- -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
- have "Max (?f ` ?A) = ?f th"
- proof(rule image_Max_eqI)
- show "finite ?A" using h_e.finite_threads by auto
- next
- show "th \<in> ?A" using h_e.th_kept by auto
- next
- show "\<forall>x\<in>?A. ?f x \<le> ?f th"
- proof
- fix x
- assume "x \<in> ?A"
- hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
- thus "?f x \<le> ?f th"
+ from preced_kept have "the_preced s th2 = the_preced s' th2" by simp
+ moreover have "cp_gen s ` RTree.children (tRAG s) x =
+ cp_gen s' ` RTree.children (tRAG s') x"
+ proof -
+ have "RTree.children (tRAG s) x = RTree.children (tRAG s') x"
+ proof(unfold tRAG_s, rule children_union_kept)
+ have start: "(Th th, Th th') \<in> tRAG s"
+ by (unfold tRAG_s, auto)
+ note x_u = 1(2)
+ show "x \<notin> Range {(Th th, Th th')}"
proof
- assume "x = thread"
- thus ?thesis
- apply (simp add:Create the_preced_def preced_def, fold preced_def)
- using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force
- next
- assume h: "x \<in> threads (t @ s)"
- from Cons(2)[unfolded Create]
- have "x \<noteq> thread" using h by (cases, auto)
- hence "?f x = the_preced (t@s) x"
- by (simp add:Create the_preced_def preced_def)
- hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
- by (simp add: h_t.finite_threads h)
- also have "... = ?f th"
- by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
- finally show ?thesis .
+ assume "x \<in> Range {(Th th, Th th')}"
+ hence eq_x: "x = Th th'" using RangeE by auto
+ show False
+ proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start])
+ case 1
+ from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG
+ show ?thesis by (auto simp:ancestors_def acyclic_def)
+ next
+ case 2
+ with x_u[unfolded eq_x]
+ have "(Th th', Th th') \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+ with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
+ qed
qed
qed
- qed
- -- {* The minor part is to show that the precedence of @{text "th"}
- equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
- also have "... = ?t" using h_e.th_kept the_preced_def by auto
- -- {* Then it follows trivially that the precedence preserved
- for @{term "th"} remains the maximum of all living threads along the way. *}
- finally show ?thesis .
- qed
- next
- case (Exit thread)
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = ?f th"
- proof(rule image_Max_eqI)
- show "finite ?A" using h_e.finite_threads by auto
- next
- show "th \<in> ?A" using h_e.th_kept by auto
- next
- show "\<forall>x\<in>?A. ?f x \<le> ?f th"
- proof
- fix x
- assume "x \<in> ?A"
- hence "x \<in> threads (t@s)" by (simp add: Exit)
- hence "?f x \<le> Max (?f ` threads (t@s))"
- by (simp add: h_t.finite_threads)
- also have "... \<le> ?f th"
- apply (simp add:Exit the_preced_def preced_def, fold preced_def)
- using Cons.hyps(5) h_t.th_kept the_preced_def by auto
- finally show "?f x \<le> ?f th" .
- qed
- qed
- also have "... = ?t" using h_e.th_kept the_preced_def by auto
- finally show ?thesis .
- qed
- next
- case (P thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def the_preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def the_preced_def)
- next
- case (Set thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = ?f th"
- proof(rule image_Max_eqI)
- show "finite ?A" using h_e.finite_threads by auto
- next
- show "th \<in> ?A" using h_e.th_kept by auto
- next
- show "\<forall>x\<in>?A. ?f x \<le> ?f th"
- proof
- fix x
- assume h: "x \<in> ?A"
- show "?f x \<le> ?f th"
- proof(cases "x = thread")
- case True
- moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
- proof -
- have "the_preced (t @ s) th = Prc prio tm"
- using h_t.th_kept preced_th by (simp add:the_preced_def)
- moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
- ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
+ moreover have "cp_gen s ` RTree.children (tRAG s) x =
+ cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A")
+ proof(rule f_image_eq)
+ fix a
+ assume a_in: "a \<in> ?A"
+ from 1(2)
+ show "?f a = ?g a"
+ proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
+ case in_ch
+ show ?thesis
+ proof(cases "a = u")
+ case True
+ from assms(2)[folded this] show ?thesis .
+ next
+ case False
+ have a_not_in: "a \<notin> ancestors (tRAG s) (Th th)"
+ proof
+ assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+ have "a = u"
+ proof(rule vat_s.rtree_s.ancestors_children_unique)
+ from a_in' a_in show "a \<in> ancestors (tRAG s) (Th th) \<inter>
+ RTree.children (tRAG s) x" by auto
+ next
+ from assms(1) in_ch show "u \<in> ancestors (tRAG s) (Th th) \<inter>
+ RTree.children (tRAG s) x" by auto
+ qed
+ with False show False by simp
+ qed
+ from a_in obtain th_a where eq_a: "a = Th th_a"
+ by (unfold RTree.children_def tRAG_alt_def, auto)
+ from cp_kept[OF a_not_in[unfolded eq_a]]
+ have "cp s th_a = cp s' th_a" .
+ from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
+ show ?thesis .
+ qed
+ next
+ case (out_ch z)
+ hence h: "z \<in> ancestors (tRAG s) u" "z \<in> RTree.children (tRAG s) x" by auto
+ show ?thesis
+ proof(cases "a = z")
+ case True
+ from h(2) have zx_in: "(z, x) \<in> (tRAG s)" by (auto simp:RTree.children_def)
+ from 1(1)[rule_format, OF this h(1)]
+ have eq_cp_gen: "cp_gen s z = cp_gen s' z" .
+ with True show ?thesis by metis
+ next
+ case False
+ from a_in obtain th_a where eq_a: "a = Th th_a"
+ by (auto simp:RTree.children_def tRAG_alt_def)
+ have "a \<notin> ancestors (tRAG s) (Th th)"
+ proof
+ assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+ have "a = z"
+ proof(rule vat_s.rtree_s.ancestors_children_unique)
+ from assms(1) h(1) have "z \<in> ancestors (tRAG s) (Th th)"
+ by (auto simp:ancestors_def)
+ with h(2) show " z \<in> ancestors (tRAG s) (Th th) \<inter>
+ RTree.children (tRAG s) x" by auto
+ next
+ from a_in a_in'
+ show "a \<in> ancestors (tRAG s) (Th th) \<inter> RTree.children (tRAG s) x"
+ by auto
+ qed
+ with False show False by auto
+ qed
+ from cp_kept[OF this[unfolded eq_a]]
+ have "cp s th_a = cp s' th_a" .
+ from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
+ show ?thesis .
qed
- ultimately show ?thesis
- by (unfold Set, simp add:the_preced_def preced_def)
- next
- case False
- then have "?f x = the_preced (t@s) x"
- by (simp add:the_preced_def preced_def Set)
- also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
- using Set h h_t.finite_threads by auto
- also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
- finally show ?thesis .
qed
qed
+ ultimately show ?thesis by metis
qed
- also have "... = ?t" using h_e.th_kept the_preced_def by auto
- finally show ?thesis .
- qed
+ ultimately show ?thesis by simp
+ qed
+ also have "... = ?R"
+ by (fold vat_s'.cp_gen_rec[OF eq_x], simp)
+ finally show ?thesis .
qed
qed
-lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
- by (insert th_kept max_kept, auto)
-
-text {*
- The reason behind the following lemma is that:
- Since @{term "cp"} is defined as the maximum precedence
- of those threads contained in the sub-tree of node @{term "Th th"}
- in @{term "RAG (t@s)"}, and all these threads are living threads, and
- @{term "th"} is also among them, the maximum precedence of
- them all must be the one for @{text "th"}.
-*}
-lemma th_cp_max_preced:
- "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R")
-proof -
- let ?f = "the_preced (t@s)"
- have "?L = ?f th"
- proof(unfold cp_alt_def, rule image_Max_eqI)
- show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
- proof -
- have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} =
- the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
- (\<exists> th'. n = Th th')}"
- by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
- moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree)
- ultimately show ?thesis by simp
- qed
- next
- show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
- by (auto simp:subtree_def)
- next
- show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
- the_preced (t @ s) x \<le> the_preced (t @ s) th"
- proof
- fix th'
- assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
- hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
- moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
- by (meson subtree_Field)
- ultimately have "Th th' \<in> ..." by auto
- hence "th' \<in> threads (t@s)"
- proof
- assume "Th th' \<in> {Th th}"
- thus ?thesis using th_kept by auto
- next
- assume "Th th' \<in> Field (RAG (t @ s))"
- thus ?thesis using vat_t.not_in_thread_isolated by blast
- qed
- thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
- by (metis Max_ge finite_imageI finite_threads image_eqI
- max_kept th_kept the_preced_def)
- qed
- qed
- also have "... = ?R" by (simp add: max_preced the_preced_def)
- finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
- using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
- by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less:
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- shows "preced th' s < preced th s"
- using assms
-by (metis Max.coboundedI finite_imageI highest not_le order.trans
- preced_linorder rev_image_eqI threads_s vat_s.finite_threads
- vat_s.le_cp)
-
-text {*
- Counting of the number of @{term "P"} and @{term "V"} operations
- is the cornerstone of a large number of the following proofs.
- The reason is that this counting is quite easy to calculate and
- convenient to use in the reasoning.
-
- The following lemma shows that the counting controls whether
- a thread is running or not.
-*}
-
-lemma pv_blocked_pre:
- assumes th'_in: "th' \<in> threads (t@s)"
- and neq_th': "th' \<noteq> th"
- and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
- shows "th' \<notin> runing (t@s)"
-proof
- assume otherwise: "th' \<in> runing (t@s)"
- show False
- proof -
- have "th' = th"
- proof(rule preced_unique)
- show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
- proof -
- have "?L = cp (t@s) th'"
- by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
- also have "... = cp (t @ s) th" using otherwise
- by (metis (mono_tags, lifting) mem_Collect_eq
- runing_def th_cp_max vat_t.max_cp_readys_threads)
- also have "... = ?R" by (metis th_cp_preced th_kept)
- finally show ?thesis .
- qed
- qed (auto simp: th'_in th_kept)
- moreover have "th' \<noteq> th" using neq_th' .
- ultimately show ?thesis by simp
- qed
-qed
-
-lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
-
-lemma runing_precond_pre:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads (t@s) \<and>
- cntP (t@s) th' = cntV (t@s) th'"
-proof(induct rule:ind)
- case (Cons e t)
- interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
- interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
- show ?case
- proof(cases e)
- case (P thread cs)
- show ?thesis
- proof -
- have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- proof -
- have "thread \<noteq> th'"
- proof -
- have "step (t@s) (P thread cs)" using Cons P by auto
- thus ?thesis
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" using Cons(5)
- by (metis neq_th' vat_t.pv_blocked_pre)
- ultimately show ?thesis by auto
- qed
- qed with Cons show ?thesis
- by (unfold P, simp add:cntP_def cntV_def count_def)
- qed
- moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (V thread cs)
- show ?thesis
- proof -
- have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- proof -
- have "thread \<noteq> th'"
- proof -
- have "step (t@s) (V thread cs)" using Cons V by auto
- thus ?thesis
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" using Cons(5)
- by (metis neq_th' vat_t.pv_blocked_pre)
- ultimately show ?thesis by auto
- qed
- qed with Cons show ?thesis
- by (unfold V, simp add:cntP_def cntV_def count_def)
- qed
- moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (Create thread prio')
- show ?thesis
- proof -
- have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- proof -
- have "thread \<noteq> th'"
- proof -
- have "step (t@s) (Create thread prio')" using Cons Create by auto
- thus ?thesis using Cons(5) by (cases, auto)
- qed with Cons show ?thesis
- by (unfold Create, simp add:cntP_def cntV_def count_def)
- qed
- moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (Exit thread)
- show ?thesis
- proof -
- have neq_thread: "thread \<noteq> th'"
- proof -
- have "step (t@s) (Exit thread)" using Cons Exit by auto
- thus ?thesis apply (cases) using Cons(5)
- by (metis neq_th' vat_t.pv_blocked_pre)
- qed
- hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons
- by (unfold Exit, simp add:cntP_def cntV_def count_def)
- moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread
- by (unfold Exit, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (Set thread prio')
- with Cons
- show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
-next
- case Nil
- with assms
- show ?case by auto
-qed
-
-text {* Changing counting balance to detachedness *}
-lemmas runing_precond_pre_dtc = runing_precond_pre
- [folded vat_t.detached_eq vat_s.detached_eq]
-
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- and is_runing: "th' \<in> runing (t@s)"
- shows "cntP s th' > cntV s th'"
- using assms
-proof -
- have "cntP s th' \<noteq> cntV s th'"
- by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)
- moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
- ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
- th' \<in> threads ((moment (i+j) t)@s)"
+lemma cp_up:
+ assumes "(Th th') \<in> ancestors (tRAG s) (Th th)"
+ and "cp s th' = cp s' th'"
+ and "(Th th'') \<in> ancestors (tRAG s) (Th th')"
+ shows "cp s th'' = cp s' th''"
proof -
- interpret h_i: red_extend_highest_gen _ _ _ _ _ i
- by (unfold_locales)
- interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"
- by (unfold_locales)
- interpret h: extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"
- proof(unfold_locales)
- show "vt (moment i t @ s)" by (metis h_i.vt_t)
- next
- show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)
- next
- show "preced th (moment i t @ s) =
- Max (cp (moment i t @ s) ` threads (moment i t @ s))"
- by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)
- next
- show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th)
- next
- show "vt (moment j (restm i t) @ moment i t @ s)"
- using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)
- next
- fix th' prio'
- assume "Create th' prio' \<in> set (moment j (restm i t))"
- thus "prio' \<le> prio" using assms
- by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)
- next
- fix th' prio'
- assume "Set th' prio' \<in> set (moment j (restm i t))"
- thus "th' \<noteq> th \<and> prio' \<le> prio"
- by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)
- next
- fix th'
- assume "Exit th' \<in> set (moment j (restm i t))"
- thus "th' \<noteq> th"
- by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)
- qed
- show ?thesis
- by (metis add.commute append_assoc eq_pv h.runing_precond_pre
- moment_plus_split neq_th' th'_in)
-qed
-
-lemma moment_blocked_eqpv:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- and le_ij: "i \<le> j"
- shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
- have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
- and h2: "th' \<in> threads ((moment j t)@s)" by auto
- moreover have "th' \<notin> runing ((moment j t)@s)"
- proof -
- interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
- show ?thesis
- using h.pv_blocked_pre h1 h2 neq_th' by auto
+ have "cp_gen s (Th th'') = cp_gen s' (Th th'')"
+ proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)])
+ from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]]
+ show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis
qed
- ultimately show ?thesis by auto
-qed
-
-(* The foregoing two lemmas are preparation for this one, but
- in long run can be combined. Maybe I am wrong.
-*)
-lemma moment_blocked:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and dtc: "detached (moment i t @ s) th'"
- and le_ij: "i \<le> j"
- shows "detached (moment j t @ s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
- interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
- have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"
- by (metis dtc h_i.detached_elim)
- from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]
- show ?thesis by (metis h_j.detached_intro)
-qed
-
-lemma runing_preced_inversion:
- assumes runing': "th' \<in> runing (t@s)"
- shows "cp (t@s) th' = preced th s" (is "?L = ?R")
-proof -
- have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
- by (unfold runing_def, auto)
- also have "\<dots> = ?R"
- by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads)
- finally show ?thesis .
-qed
-
-text {*
- The situation when @{term "th"} is blocked is analyzed by the following lemmas.
-*}
-
-text {*
- The following lemmas shows the running thread @{text "th'"}, if it is different from
- @{term th}, must be live at the very beginning. By the term {\em the very beginning},
- we mean the moment where the formal investigation starts, i.e. the moment (or state)
- @{term s}.
-*}
-
-lemma runing_inversion_0:
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- shows "th' \<in> threads s"
-proof -
- -- {* The proof is by contradiction: *}
- { assume otherwise: "\<not> ?thesis"
- have "th' \<notin> runing (t @ s)"
- proof -
- -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}
- have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)
- -- {* However, @{text "th'"} does not exist at very beginning. *}
- have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise
- by (metis append.simps(1) moment_zero)
- -- {* Therefore, there must be a moment during @{text "t"}, when
- @{text "th'"} came into being. *}
- -- {* Let us suppose the moment being @{text "i"}: *}
- from p_split_gen[OF th'_in th'_notin]
- obtain i where lt_its: "i < length t"
- and le_i: "0 \<le> i"
- and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
- and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by (auto)
- interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
- interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)
- from lt_its have "Suc i \<le> length t" by auto
- -- {* Let us also suppose the event which makes this change is @{text e}: *}
- from moment_head[OF this] obtain e where
- eq_me: "moment (Suc i) t = e # moment i t" by blast
- hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t)
- hence "PIP (moment i t @ s) e" by (cases, simp)
- -- {* It can be derived that this event @{text "e"}, which
- gives birth to @{term "th'"} must be a @{term "Create"}: *}
- from create_pre[OF this, of th']
- obtain prio where eq_e: "e = Create th' prio"
- by (metis append_Cons eq_me lessI post pre)
- have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto
- have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
- proof -
- have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
- by (metis h_i.cnp_cnv_eq pre)
- thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)
- qed
- show ?thesis
- using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge
- by auto
- qed
- with `th' \<in> runing (t@s)`
- have False by simp
- } thus ?thesis by auto
-qed
-
-text {*
- The second lemma says, if the running thread @{text th'} is different from
- @{term th}, then this @{text th'} must in the possession of some resources
- at the very beginning.
-
- To ease the reasoning of resource possession of one particular thread,
- we used two auxiliary functions @{term cntV} and @{term cntP},
- which are the counters of @{term P}-operations and
- @{term V}-operations respectively.
- If the number of @{term V}-operation is less than the number of
- @{term "P"}-operations, the thread must have some unreleased resource.
-*}
-
-lemma runing_inversion_1: (* ddd *)
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- -- {* thread @{term "th'"} is a live on in state @{term "s"} and
- it has some unreleased resource. *}
- shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof -
- -- {* The proof is a simple composition of @{thm runing_inversion_0} and
- @{thm runing_precond}: *}
- -- {* By applying @{thm runing_inversion_0} to assumptions,
- it can be shown that @{term th'} is live in state @{term s}: *}
- have "th' \<in> threads s" using runing_inversion_0[OF assms(1,2)] .
- -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}
- with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-qed
-
-text {*
- The following lemma is just a rephrasing of @{thm runing_inversion_1}:
-*}
-lemma runing_inversion_2:
- assumes runing': "th' \<in> runing (t@s)"
- shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
- from runing_inversion_1[OF _ runing']
- show ?thesis by auto
-qed
-
-lemma runing_inversion_3:
- assumes runing': "th' \<in> runing (t@s)"
- and neq_th: "th' \<noteq> th"
- shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
- by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
-
-lemma runing_inversion_4:
- assumes runing': "th' \<in> runing (t@s)"
- and neq_th: "th' \<noteq> th"
- shows "th' \<in> threads s"
- and "\<not>detached s th'"
- and "cp (t@s) th' = preced th s"
- apply (metis neq_th runing' runing_inversion_2)
- apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
- by (metis neq_th runing' runing_inversion_3)
-
-
-text {*
- Suppose @{term th} is not running, it is first shown that
- there is a path in RAG leading from node @{term th} to another thread @{text "th'"}
- in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
-
- Now, since @{term readys}-set is non-empty, there must be
- one in it which holds the highest @{term cp}-value, which, by definition,
- is the @{term runing}-thread. However, we are going to show more: this running thread
- is exactly @{term "th'"}.
- *}
-lemma th_blockedE: (* ddd *)
- assumes "th \<notin> runing (t@s)"
- obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
- "th' \<in> runing (t@s)"
-proof -
- -- {* According to @{thm vat_t.th_chain_to_ready}, either
- @{term "th"} is in @{term "readys"} or there is path leading from it to
- one thread in @{term "readys"}. *}
- have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)"
- using th_kept vat_t.th_chain_to_ready by auto
- -- {* However, @{term th} can not be in @{term readys}, because otherwise, since
- @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
- moreover have "th \<notin> readys (t@s)"
- using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto
- -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in
- term @{term readys}: *}
- ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
- and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
- -- {* We are going to show that this @{term th'} is running. *}
- have "th' \<in> runing (t@s)"
- proof -
- -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
- have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
- proof -
- have "?L = Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
- by (unfold cp_alt_def1, simp)
- also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
- proof(rule image_Max_subset)
- show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
- next
- show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
- by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread)
- next
- show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
- by (unfold tRAG_subtree_eq, auto simp:subtree_def)
- next
- show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
- (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
- proof -
- have "?L = the_preced (t @ s) ` threads (t @ s)"
- by (unfold image_comp, rule image_cong, auto)
- thus ?thesis using max_preced the_preced_def by auto
- qed
- qed
- also have "... = ?R"
- using th_cp_max th_cp_preced th_kept
- the_preced_def vat_t.max_cp_readys_threads by auto
- finally show ?thesis .
- qed
- -- {* Now, since @{term th'} holds the highest @{term cp}
- and we have already show it is in @{term readys},
- it is @{term runing} by definition. *}
- with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def)
- qed
- -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
- moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
- using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
- ultimately show ?thesis using that by metis
-qed
-
-text {*
- Now it is easy to see there is always a thread to run by case analysis
- on whether thread @{term th} is running: if the answer is Yes, the
- the running thread is obviously @{term th} itself; otherwise, the running
- thread is the @{text th'} given by lemma @{thm th_blockedE}.
-*}
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
- case True thus ?thesis by auto
-next
- case False
- thus ?thesis using th_blockedE by auto
+ with cp_gen_def_cond[OF refl[of "Th th''"]]
+ show ?thesis by metis
qed
end
+
+section {* The @{term Create} operation *}
+
+locale step_create_cps =
+ fixes s' th prio s
+ defines s_def : "s \<equiv> (Create th prio#s')"
+ assumes vt_s: "vt s"
+
+sublocale step_create_cps < vat_s: valid_trace "s"
+ by (unfold_locales, insert vt_s, simp)
+
+sublocale step_create_cps < vat_s': valid_trace "s'"
+ by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
+
+context step_create_cps
+begin
+
+lemma RAG_kept: "RAG s = RAG s'"
+ by (unfold s_def RAG_create_unchanged, auto)
+
+lemma tRAG_kept: "tRAG s = tRAG s'"
+ by (unfold tRAG_alt_def RAG_kept, auto)
+
+lemma preced_kept:
+ assumes "th' \<noteq> th"
+ shows "the_preced s th' = the_preced s' th'"
+ by (unfold s_def the_preced_def preced_def, insert assms, auto)
+
+lemma th_not_in: "Th th \<notin> Field (tRAG s')"
+proof -
+ from vt_s[unfolded s_def]
+ have "PIP s' (Create th prio)" by (cases, simp)
+ hence "th \<notin> threads s'" by(cases, simp)
+ from vat_s'.not_in_thread_isolated[OF this]
+ have "Th th \<notin> Field (RAG s')" .
+ with tRAG_Field show ?thesis by auto
+qed
+
+lemma eq_cp:
+ assumes neq_th: "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+proof -
+ have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
+ (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+ proof(unfold tRAG_kept, rule f_image_eq)
+ fix a
+ assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+ then obtain th_a where eq_a: "a = Th th_a"
+ proof(cases rule:subtreeE)
+ case 2
+ from ancestors_Field[OF 2(2)]
+ and that show ?thesis by (unfold tRAG_alt_def, auto)
+ qed auto
+ have neq_th_a: "th_a \<noteq> th"
+ proof -
+ have "(Th th) \<notin> subtree (tRAG s') (Th th')"
+ proof
+ assume "Th th \<in> subtree (tRAG s') (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 2
+ from ancestors_Field[OF this(2)]
+ and th_not_in[unfolded Field_def]
+ show ?thesis by auto
+ qed (insert assms, auto)
+ qed
+ with a_in[unfolded eq_a] show ?thesis by auto
+ qed
+ from preced_kept[OF this]
+ show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+ by (unfold eq_a, simp)
+ qed
+ thus ?thesis by (unfold cp_alt_def1, simp)
+qed
+
+lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}"
+proof -
+ { fix a
+ assume "a \<in> RTree.children (tRAG s) (Th th)"
+ hence "(a, Th th) \<in> tRAG s" by (auto simp:RTree.children_def)
+ with th_not_in have False
+ by (unfold Field_def tRAG_kept, auto)
+ } thus ?thesis by auto
+qed
+
+lemma eq_cp_th: "cp s th = preced th s"
+ by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def)
+
end
+locale step_exit_cps =
+ fixes s' th prio s
+ defines s_def : "s \<equiv> Exit th # s'"
+ assumes vt_s: "vt s"
+sublocale step_exit_cps < vat_s: valid_trace "s"
+ by (unfold_locales, insert vt_s, simp)
+sublocale step_exit_cps < vat_s': valid_trace "s'"
+ by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
+
+context step_exit_cps
+begin
+
+lemma preced_kept:
+ assumes "th' \<noteq> th"
+ shows "the_preced s th' = the_preced s' th'"
+ by (unfold s_def the_preced_def preced_def, insert assms, auto)
+
+lemma RAG_kept: "RAG s = RAG s'"
+ by (unfold s_def RAG_exit_unchanged, auto)
+
+lemma tRAG_kept: "tRAG s = tRAG s'"
+ by (unfold tRAG_alt_def RAG_kept, auto)
+
+lemma th_ready: "th \<in> readys s'"
+proof -
+ from vt_s[unfolded s_def]
+ have "PIP s' (Exit th)" by (cases, simp)
+ hence h: "th \<in> runing s' \<and> holdents s' th = {}" by (cases, metis)
+ thus ?thesis by (unfold runing_def, auto)
+qed
+
+lemma th_holdents: "holdents s' th = {}"
+proof -
+ from vt_s[unfolded s_def]
+ have "PIP s' (Exit th)" by (cases, simp)
+ thus ?thesis by (cases, metis)
+qed
+
+lemma th_RAG: "Th th \<notin> Field (RAG s')"
+proof -
+ have "Th th \<notin> Range (RAG s')"
+ proof
+ assume "Th th \<in> Range (RAG s')"
+ then obtain cs where "holding (wq s') th cs"
+ by (unfold Range_iff s_RAG_def, auto)
+ with th_holdents[unfolded holdents_def]
+ show False by (unfold eq_holding, auto)
+ qed
+ moreover have "Th th \<notin> Domain (RAG s')"
+ proof
+ assume "Th th \<in> Domain (RAG s')"
+ then obtain cs where "waiting (wq s') th cs"
+ by (unfold Domain_iff s_RAG_def, auto)
+ with th_ready show False by (unfold readys_def eq_waiting, auto)
+ qed
+ ultimately show ?thesis by (auto simp:Field_def)
+qed
+
+lemma th_tRAG: "(Th th) \<notin> Field (tRAG s')"
+ using th_RAG tRAG_Field[of s'] by auto
+
+lemma eq_cp:
+ assumes neq_th: "th' \<noteq> th"
+ shows "cp s th' = cp s' th'"
+proof -
+ have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
+ (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+ proof(unfold tRAG_kept, rule f_image_eq)
+ fix a
+ assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+ then obtain th_a where eq_a: "a = Th th_a"
+ proof(cases rule:subtreeE)
+ case 2
+ from ancestors_Field[OF 2(2)]
+ and that show ?thesis by (unfold tRAG_alt_def, auto)
+ qed auto
+ have neq_th_a: "th_a \<noteq> th"
+ proof -
+ from vat_s'.readys_in_no_subtree[OF th_ready assms]
+ have "(Th th) \<notin> subtree (RAG s') (Th th')" .
+ with tRAG_subtree_RAG[of s' "Th th'"]
+ have "(Th th) \<notin> subtree (tRAG s') (Th th')" by auto
+ with a_in[unfolded eq_a] show ?thesis by auto
+ qed
+ from preced_kept[OF this]
+ show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+ by (unfold eq_a, simp)
+ qed
+ thus ?thesis by (unfold cp_alt_def1, simp)
+qed
+
+end
+
+end
+