--- a/Correctness.thy Thu Jan 14 03:29:22 2016 +0000
+++ b/Correctness.thy Fri Jan 15 02:05:29 2016 +0000
@@ -503,29 +503,29 @@
qed
text {*
- The following lemma shows how the counting of
- @{term "P"} and @{term "V"} operations affects
- the running state of threads in @{term t}.
- The lemma shows that if a thread's @{term "P"}-count equals
- its @{term "V"}-count (which means it no longer has any
- resource in its possession), it can not be in running thread.
+ The following lemma shows how the counters for @{term "P"} and
+ @{term "V"} operations relate to the running threads in the states
+ @{term s} and @{term "t @ s"}. The lemma shows that if a thread's
+ @{term "P"}-count equals its @{term "V"}-count (which means it no
+ longer has any resource in its possession), it cannot be a running
+ thread.
- The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
- The key is the use of @{thm count_eq_dependants}
- to derive the emptiness of @{text th'}s @{term dependants}-set
- from the balance of its @{term P} @{term V} counts.
- From this, it can be shown @{text th'}s @{term cp}-value
- equals to its own precedence.
+ The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
+ The key is the use of @{thm count_eq_dependants} to derive the
+ emptiness of @{text th'}s @{term dependants}-set from the balance of
+ its @{term P} and @{term V} counts. From this, it can be shown
+ @{text th'}s @{term cp}-value equals to its own precedence.
- On the other hand, since @{text th'} is running, by
- @{thm runing_preced_inversion}, its @{term cp}-value
- equals to the precedence of @{term th}.
+ On the other hand, since @{text th'} is running, by @{thm
+ runing_preced_inversion}, its @{term cp}-value equals to the
+ precedence of @{term th}.
- Combining the above two we have that @{text th'} and
- @{term th} have the same precedence. By uniqueness of precedence, we
- have @{text "th' = th"}, which is in contradiction with the
- assumption @{text "th' \<noteq> th"}.
+ Combining the above two resukts we have that @{text th'} and @{term
+ th} have the same precedence. By uniqueness of precedences, we have
+ @{text "th' = th"}, which is in contradiction with the assumption
+ @{text "th' \<noteq> th"}.
+
*}
lemma eq_pv_blocked: (* ddd *)
--- a/Correctness.thy~ Thu Jan 14 03:29:22 2016 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,795 +0,0 @@
-theory Correctness
-imports PIPBasics
-begin
-
-
-text {*
- The following two auxiliary lemmas are used to reason about @{term Max}.
-*}
-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"
- using assms
- using Max_eqI by blast
-
-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)
-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.
-*}
-locale highest_gen =
- fixes s th prio tm
- 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"
-
--- {* @{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)
-
-context highest_gen
-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 lt_tm: "tm < length s"
- by (insert preced_tm_lt[OF threads_s preced_th], simp)
-
-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")
-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
-qed
-
-lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)"
- using eq_cp_s_th highest max_cp_eq the_preced_def by presburger
-
-
-lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
- by (simp add: eq_cp_s_th highest)
-
-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
-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"
-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
- qed
-qed
-
-
-lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
-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)
- qed
- qed
- qed
-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 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")
- 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"
- 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 .
- 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)
- 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
- qed
- also have "... = ?t" using h_e.th_kept the_preced_def by auto
- finally show ?thesis .
- qed
- 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[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th"
- using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
-
-lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))"
- by (simp add: th_cp_max_preced)
-
-lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th"
- using max_kept th_kept the_preced_def by auto
-
-lemma [simp]: "the_preced (t@s) th = preced th (t@s)"
- using the_preced_def by auto
-
-lemma [simp]: "preced th (t@s) = preced th s"
- by (simp add: th_kept)
-
-lemma [simp]: "cp s th = preced th s"
- by (simp add: eq_cp_s_th)
-
-lemma th_cp_preced [simp]: "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)
-
-section {* The `blocking thread` *}
-
-text {*
- The purpose of PIP is to ensure that the most
- urgent thread @{term th} is not blocked unreasonably.
- Therefore, a clear picture of the blocking thread is essential
- to assure people that the purpose is fulfilled.
-
- In this section, we are going to derive a series of lemmas
- with finally give rise to a picture of the blocking thread.
-
- By `blocking thread`, we mean a thread in running state but
- different from thread @{term th}.
-*}
-
-text {*
- The following lemmas shows that the @{term cp}-value
- of the blocking thread @{text th'} equals to the highest
- precedence in the whole system.
-*}
-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 following lemma shows how the counting of
- @{term "P"} and @{term "V"} operations affects
- the running state of threads in @{term t}.
-
- The lemma shows that if a thread's @{term "P"}-count equals
- its @{term "V"}-count (which means it no longer has any
- resource in its possession), it can not be in running thread.
-
- The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
- The key is the use of @{thm count_eq_dependants}
- to derive the emptiness of @{text th'}s @{term dependants}-set
- from the balance of its @{term P} @{term V} counts.
- From this, it can be shown @{text th'}s @{term cp}-value
- equals to its own precedence.
-
- On the other hand, since @{text th'} is running, by
- @{thm runing_preced_inversion}, its @{term cp}-value
- equals to the precedence of @{term th}.
-
- Combining the above two we have that @{text th'} and
- @{term th} have the same precedence. By uniqueness of precedence, we
- have @{text "th' = th"}, which is in contradiction with the
- assumption @{text "th' \<noteq> th"}.
-*}
-
-lemma eq_pv_blocked: (* ddd *)
- assumes 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'_in: "th' \<in> threads (t@s)"
- using otherwise readys_threads runing_def by auto
- have "th' = th"
- proof(rule preced_unique)
- -- {* The proof goes like this:
- it is first shown that the @{term preced}-value of @{term th'}
- equals to that of @{term th}, then by uniqueness
- of @{term preced}-values (given by lemma @{thm preced_unique}),
- @{term th'} equals to @{term th}: *}
- show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
- proof -
- -- {* Since the counts of @{term th'} are balanced, the subtree
- of it contains only itself, so, its @{term cp}-value
- equals its @{term preced}-value: *}
- have "?L = cp (t@s) th'"
- by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
- -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion},
- its @{term cp}-value equals @{term "preced th s"},
- which equals to @{term "?R"} by simplification: *}
- also have "... = ?R"
- thm runing_preced_inversion
- using runing_preced_inversion[OF otherwise] by simp
- finally show ?thesis .
- qed
- qed (auto simp: th'_in th_kept)
- with `th' \<noteq> th` show ?thesis by simp
- qed
-qed
-
-text {*
- The following lemma is the extrapolation of @{thm eq_pv_blocked}.
- It says if a thread, different from @{term th},
- does not hold any resource at the very beginning,
- it will keep hand-emptied in the future @{term "t@s"}.
-*}
-lemma eq_pv_persist: (* ddd *)
- assumes neq_th': "th' \<noteq> th"
- and eq_pv: "cntP s th' = cntV s th'"
- shows "cntP (t@s) th' = cntV (t@s) th'"
-proof(induction rule:ind) -- {* The proof goes by induction. *}
- -- {* The nontrivial case is for the @{term Cons}: *}
- case (Cons e t)
- -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *}
- 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 -
- -- {* It can be proved that @{term cntP}-value of @{term th'} does not change
- by the happening of event @{term e}: *}
- have "cntP ((e#t)@s) th' = cntP (t@s) th'"
- proof(rule ccontr) -- {* Proof by contradiction. *}
- -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *}
- assume otherwise: "cntP ((e # t) @ s) th' \<noteq> cntP (t @ s) th'"
- -- {* Then the actor of @{term e} must be @{term th'} and @{term e}
- must be a @{term P}-event: *}
- hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv)
- with vat_t.actor_inv[OF Cons(2)]
- -- {* According to @{thm actor_inv}, @{term th'} must be running at
- the moment @{term "t@s"}: *}
- have "th' \<in> runing (t@s)" by (cases e, auto)
- -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis
- shows @{term th'} can not be running at moment @{term "t@s"}: *}
- moreover have "th' \<notin> runing (t@s)"
- using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
- -- {* Contradiction is finally derived: *}
- ultimately show False by simp
- qed
- -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change
- by the happening of event @{term e}: *}
- -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *}
- moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'"
- proof(rule ccontr) -- {* Proof by contradiction. *}
- assume otherwise: "cntV ((e # t) @ s) th' \<noteq> cntV (t @ s) th'"
- hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv)
- with vat_t.actor_inv[OF Cons(2)]
- have "th' \<in> runing (t@s)" by (cases e, auto)
- moreover have "th' \<notin> runing (t@s)"
- using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
- ultimately show False by simp
- qed
- -- {* Finally, it can be shown that the @{term cntP} and @{term cntV}
- value for @{term th'} are still in balance, so @{term th'}
- is still hand-emptied after the execution of event @{term e}: *}
- ultimately show ?thesis using Cons(5) by metis
- qed
-qed (auto simp:eq_pv)
-
-text {*
- By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist},
- it can be derived easily that @{term th'} can not be running in the future:
-*}
-lemma eq_pv_blocked_persist:
- assumes neq_th': "th' \<noteq> th"
- and eq_pv: "cntP s th' = cntV s th'"
- shows "th' \<notin> runing (t@s)"
- using assms
- by (simp add: eq_pv_blocked eq_pv_persist)
-
-text {*
- The following lemma shows the blocking thread @{term th'}
- must hold some resource in the very beginning.
-*}
-lemma runing_cntP_cntV_inv: (* ddd *)
- assumes is_runing: "th' \<in> runing (t@s)"
- and neq_th': "th' \<noteq> th"
- shows "cntP s th' > cntV s th'"
- using assms
-proof -
- -- {* First, it can be shown that the number of @{term P} and
- @{term V} operations can not be equal for thred @{term th'} *}
- have "cntP s th' \<noteq> cntV s th'"
- proof
- -- {* The proof goes by contradiction, suppose otherwise: *}
- assume otherwise: "cntP s th' = cntV s th'"
- -- {* By applying @{thm eq_pv_blocked_persist} to this: *}
- from eq_pv_blocked_persist[OF neq_th' otherwise]
- -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *}
- have "th' \<notin> runing (t@s)" .
- -- {* This is obvious in contradiction with assumption @{thm is_runing} *}
- thus False using is_runing by simp
- qed
- -- {* However, the number of @{term V} is always less or equal to @{term P}: *}
- moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
- -- {* Thesis is finally derived by combining the these two results: *}
- ultimately show ?thesis by auto
-qed
-
-
-text {*
- The following lemmas shows the blocking thread @{text th'} must be live
- at the very beginning, i.e. the moment (or state) @{term s}.
-
- The proof is a simple combination of the results above:
-*}
-lemma runing_threads_inv:
- assumes runing': "th' \<in> runing (t@s)"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads s"
-proof(rule ccontr) -- {* Proof by contradiction: *}
- assume otherwise: "th' \<notin> threads s"
- have "th' \<notin> runing (t @ s)"
- proof -
- from vat_s.cnp_cnv_eq[OF otherwise]
- have "cntP s th' = cntV s th'" .
- from eq_pv_blocked_persist[OF neq_th' this]
- show ?thesis .
- qed
- with runing' show False by simp
-qed
-
-text {*
- The following lemma summarizes several foregoing
- lemmas to give an overall picture of the blocking thread @{text "th'"}:
-*}
-lemma runing_inversion: (* ddd, one of the main lemmas to present *)
- 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"
-proof -
- from runing_threads_inv[OF assms]
- show "th' \<in> threads s" .
-next
- from runing_cntP_cntV_inv[OF runing' neq_th]
- show "\<not>detached s th'" using vat_s.detached_eq by simp
-next
- from runing_preced_inversion[OF runing']
- show "cp (t@s) th' = preced th s" .
-qed
-
-section {* The existence of `blocking thread` *}
-
-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, the other main lemma to be presented: *)
- 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
-qed
-
-
-end
-end
--- a/Journal/Paper.thy Thu Jan 14 03:29:22 2016 +0000
+++ b/Journal/Paper.thy Fri Jan 15 02:05:29 2016 +0000
@@ -23,7 +23,9 @@
cpreced ("cprec") and
cp ("cprec") and
holdents ("resources") and
- DUMMY ("\<^raw:\mbox{$\_\!\_$}>")
+ DUMMY ("\<^raw:\mbox{$\_\!\_$}>") and
+ cntP ("c\<^bsub>P\<^esub>") and
+ cntV ("c\<^bsub>V\<^esub>")
(*>*)
@@ -178,17 +180,18 @@
priority.}'' The same error is also repeated later in this textbook.
- While \cite{Laplante11,Liu00,book,Sha90,Silberschatz13} are the only formal publications we have
- found that specify the incorrect behaviour, it seems also many
- informal descriptions of PIP overlook the possibility that another
- high-priority might wait for a low-priority process to finish.
- A notable exception is the texbook \cite{buttazzo}, which gives the correct
- behaviour of resetting the priority of a thread to the highest remaining priority of the
- threads it blocks. This textbook also gives an informal proof for
- the correctness of PIP in the style of Sha et al. Unfortunately, this informal
- proof is too vague to be useful for formalising the correctness of PIP
- and the specification leaves out nearly all details in order
- to implement PIP efficiently.\medskip\smallskip
+ While \cite{Laplante11,Liu00,book,Sha90,Silberschatz13} are the only
+ formal publications we have found that specify the incorrect
+ behaviour, it seems also many informal descriptions of PIP overlook
+ the possibility that another high-priority might wait for a
+ low-priority process to finish. A notable exception is the texbook
+ \cite{buttazzo}, which gives the correct behaviour of resetting the
+ priority of a thread to the highest remaining priority of the
+ threads it blocks. This textbook also gives an informal proof for
+ the correctness of PIP in the style of Sha et al. Unfortunately,
+ this informal proof is too vague to be useful for formalising the
+ correctness of PIP and the specification leaves out nearly all
+ details in order to implement PIP efficiently.\medskip\smallskip
%
%The advantage of formalising the
%correctness of a high-level specification of PIP in a theorem prover
@@ -196,27 +199,29 @@
%informal reasoning (since we have to analyse all possible behaviours
%of threads, i.e.~\emph{traces}, that could possibly happen).
- \noindent
- {\bf Contributions:} There have been earlier formal investigations
- into PIP \cite{Faria08,Jahier09,Wellings07}, but they employ model
- checking techniques. This paper presents a formalised and
- mechanically checked proof for the correctness of PIP. For this we
- needed to design a new correctness criterion for PIP. In contrast to model checking, our
- formalisation provides insight into why PIP is correct and allows us
- to prove stronger properties that, as we will show, can help with an
- efficient implementation of PIP in the educational PINTOS operating
- system \cite{PINTOS}. For example, we found by ``playing'' with the
- formalisation that the choice of the next thread to take over a lock
- when a resource is released is irrelevant for PIP being correct---a
- fact that has not been mentioned in the literature and not been used
- in the reference implementation of PIP in PINTOS. This fact, however, is important
- for an efficient implementation of PIP, because we can give the lock
- to the thread with the highest priority so that it terminates more
- quickly. We were also being able to generalise the scheduler of Sha
- et al.~\cite{Sha90} to the practically relevant case where critical
- sections can overlap; see Figure~\ref{overlap} \emph{a)} below for an example of
- this restriction. %In the existing literature there is no
- %proof and also no proving method that cover this generalised case.
+ \noindent {\bf Contributions:} There have been earlier formal
+ investigations into PIP \cite{Faria08,Jahier09,Wellings07}, but they
+ employ model checking techniques. This paper presents a formalised
+ and mechanically checked proof for the correctness of PIP. For this
+ we needed to design a new correctness criterion for PIP. In contrast
+ to model checking, our formalisation provides insight into why PIP
+ is correct and allows us to prove stronger properties that, as we
+ will show, can help with an efficient implementation of PIP. We
+ illustrate this with an implementation of PIP in the educational
+ PINTOS operating system \cite{PINTOS}. For example, we found by
+ ``playing'' with the formalisation that the choice of the next
+ thread to take over a lock when a resource is released is irrelevant
+ for PIP being correct---a fact that has not been mentioned in the
+ literature and not been used in the reference implementation of PIP
+ in PINTOS. This fact, however, is important for an efficient
+ implementation of PIP, because we can give the lock to the thread
+ with the highest priority so that it terminates more quickly. We
+ were also being able to generalise the scheduler of Sha et
+ al.~\cite{Sha90} to the practically relevant case where critical
+ sections can overlap; see Figure~\ref{overlap} \emph{a)} below for
+ an example of this restriction. In the existing literature there is
+ no proof and also no proving method that cover this generalised
+ case.
\begin{figure}
\begin{center}
@@ -382,6 +387,18 @@
tasks involved in the inheritance process and thus minimises the number
of potentially expensive thread-switches.
+ We will also need counters for @{term P} and @{term V} events of a thread @{term th}
+ in a state @{term s}. This can be straightforwardly defined in Isabelle as
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \mbox{\begin{tabular}{@ {}l}
+ @{thm cntP_def}\\
+ @{thm cntV_def}
+ \end{tabular}}
+ \end{isabelle}
+
+ \noindent using the predefined function @{const count} for lists.
+
Next, we introduce the concept of \emph{waiting queues}. They are
lists of threads associated with every resource. The first thread in
this list (i.e.~the head, or short @{term hd}) is chosen to be the one
@@ -943,8 +960,39 @@
From these two lemmas we can see the correctness of PIP, which is
that: the blockage of th is reasonable and under control.
+ Lemmas we want to describe:
+
+ \begin{lemma}
+ @{thm eq_pv_persist}
+ \end{lemma}
+
+ \begin{lemma}
+ @{thm eq_pv_blocked}
+ \end{lemma}
+
+ \begin{lemma}
+ @{thm runing_cntP_cntV_inv}
+ \end{lemma}
+
+ \noindent
+ Remember we do not have the well-nestedness restriction in our
+ proof, which means the difference between the counters @{const cntV}
+ and @{const cntP} can be larger than @{term 1}.
+
+ \begin{lemma}
+ @{thm runing_inversion}
+ \end{lemma}
+
+
+ \begin{lemma}
+ @{thm th_blockedE}
+ \end{lemma}
+
\subsection*{END OUTLINE}
+
+
+
In what follows we will describe properties of PIP that allow us to prove
Theorem~\ref{mainthm} and, when instructive, briefly describe our argument.
It is relatively easy to see that:
@@ -957,7 +1005,7 @@
\end{isabelle}
\noindent
- The second property is by induction of @{term vt}. The next three
+ The second property is by induction on @{term vt}. The next three
properties are:
\begin{isabelle}\ \ \ \ \ %%%
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