89
+ − 1
theory PrioG
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imports CpsG
85
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begin
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text {*
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The following two auxiliary lemmas are used to reason about @{term Max}.
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*}
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lemma image_Max_eqI:
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assumes "finite B"
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and "b \<in> B"
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and "\<forall> x \<in> B. f x \<le> f b"
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shows "Max (f ` B) = f b"
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using assms
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using Max_eqI by blast
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lemma image_Max_subset:
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assumes "finite A"
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and "B \<subseteq> A"
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and "a \<in> B"
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and "Max (f ` A) = f a"
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shows "Max (f ` B) = f a"
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proof(rule image_Max_eqI)
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show "finite B"
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using assms(1) assms(2) finite_subset by auto
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next
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show "a \<in> B" using assms by simp
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next
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show "\<forall>x\<in>B. f x \<le> f a"
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by (metis Max_ge assms(1) assms(2) assms(4)
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finite_imageI image_eqI subsetCE)
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qed
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text {*
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The following locale @{text "highest_gen"} sets the basic context for our
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investigation: supposing thread @{text th} holds the highest @{term cp}-value
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in state @{text s}, which means the task for @{text th} is the
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most urgent. We want to show that
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@{text th} is treated correctly by PIP, which means
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@{text th} will not be blocked unreasonably by other less urgent
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threads.
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*}
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locale highest_gen =
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fixes s th prio tm
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assumes vt_s: "vt s"
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and threads_s: "th \<in> threads s"
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and highest: "preced th s = Max ((cp s)`threads s)"
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-- {* The internal structure of @{term th}'s precedence is exposed:*}
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and preced_th: "preced th s = Prc prio tm"
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-- {* @{term s} is a valid trace, so it will inherit all results derived for
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a valid trace: *}
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sublocale highest_gen < vat_s: valid_trace "s"
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by (unfold_locales, insert vt_s, simp)
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context highest_gen
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begin
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text {*
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@{term tm} is the time when the precedence of @{term th} is set, so
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@{term tm} must be a valid moment index into @{term s}.
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*}
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lemma lt_tm: "tm < length s"
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by (insert preced_tm_lt[OF threads_s preced_th], simp)
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text {*
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Since @{term th} holds the highest precedence and @{text "cp"}
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is the highest precedence of all threads in the sub-tree of
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@{text "th"} and @{text th} is among these threads,
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its @{term cp} must equal to its precedence:
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*}
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lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
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proof -
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have "?L \<le> ?R"
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by (unfold highest, rule Max_ge,
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auto simp:threads_s finite_threads)
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moreover have "?R \<le> ?L"
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by (unfold vat_s.cp_rec, rule Max_ge,
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auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
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ultimately show ?thesis by auto
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qed
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lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)"
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using eq_cp_s_th highest max_cp_eq the_preced_def by presburger
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lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)"
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by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
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lemma highest': "cp s th = Max (cp s ` threads s)"
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by (simp add: eq_cp_s_th highest)
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end
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locale extend_highest_gen = highest_gen +
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fixes t
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assumes vt_t: "vt (t@s)"
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and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
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and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
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and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
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sublocale extend_highest_gen < vat_t: valid_trace "t@s"
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by (unfold_locales, insert vt_t, simp)
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lemma step_back_vt_app:
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assumes vt_ts: "vt (t@s)"
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shows "vt s"
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proof -
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from vt_ts show ?thesis
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proof(induct t)
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case Nil
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from Nil show ?case by auto
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next
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case (Cons e t)
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assume ih: " vt (t @ s) \<Longrightarrow> vt s"
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and vt_et: "vt ((e # t) @ s)"
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show ?case
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proof(rule ih)
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show "vt (t @ s)"
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proof(rule step_back_vt)
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from vt_et show "vt (e # t @ s)" by simp
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qed
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qed
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qed
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qed
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(* locale red_extend_highest_gen = extend_highest_gen +
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fixes i::nat
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*)
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(*
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sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
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apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
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apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
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by (unfold highest_gen_def, auto dest:step_back_vt_app)
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*)
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context extend_highest_gen
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begin
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lemma ind [consumes 0, case_names Nil Cons, induct type]:
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assumes
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h0: "R []"
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and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e;
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extend_highest_gen s th prio tm t;
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extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
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shows "R t"
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proof -
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from vt_t extend_highest_gen_axioms show ?thesis
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proof(induct t)
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from h0 show "R []" .
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next
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case (Cons e t')
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assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
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and vt_e: "vt ((e # t') @ s)"
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and et: "extend_highest_gen s th prio tm (e # t')"
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from vt_e and step_back_step have stp: "step (t'@s) e" by auto
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from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
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show ?case
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proof(rule h2 [OF vt_ts stp _ _ _ ])
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show "R t'"
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proof(rule ih)
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from et show ext': "extend_highest_gen s th prio tm t'"
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by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
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next
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from vt_ts show "vt (t' @ s)" .
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qed
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next
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from et show "extend_highest_gen s th prio tm (e # t')" .
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next
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from et show ext': "extend_highest_gen s th prio tm t'"
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by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
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qed
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qed
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qed
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lemma th_kept: "th \<in> threads (t @ s) \<and>
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preced th (t@s) = preced th s" (is "?Q t")
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proof -
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show ?thesis
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proof(induct rule:ind)
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case Nil
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from threads_s
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show ?case
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by auto
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next
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case (Cons e t)
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interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
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interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
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show ?case
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proof(cases e)
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case (Create thread prio)
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show ?thesis
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proof -
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from Cons and Create have "step (t@s) (Create thread prio)" by auto
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hence "th \<noteq> thread"
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proof(cases)
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case thread_create
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with Cons show ?thesis by auto
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qed
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hence "preced th ((e # t) @ s) = preced th (t @ s)"
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by (unfold Create, auto simp:preced_def)
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moreover note Cons
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ultimately show ?thesis
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by (auto simp:Create)
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qed
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next
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case (Exit thread)
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from h_e.exit_diff and Exit
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have neq_th: "thread \<noteq> th" by auto
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with Cons
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show ?thesis
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by (unfold Exit, auto simp:preced_def)
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next
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case (P thread cs)
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with Cons
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show ?thesis
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by (auto simp:P preced_def)
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next
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case (V thread cs)
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with Cons
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show ?thesis
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by (auto simp:V preced_def)
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next
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case (Set thread prio')
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show ?thesis
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proof -
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from h_e.set_diff_low and Set
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have "th \<noteq> thread" by auto
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hence "preced th ((e # t) @ s) = preced th (t @ s)"
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by (unfold Set, auto simp:preced_def)
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moreover note Cons
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ultimately show ?thesis
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by (auto simp:Set)
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qed
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qed
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qed
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qed
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text {*
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According to @{thm th_kept}, thread @{text "th"} has its living status
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and precedence kept along the way of @{text "t"}. The following lemma
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shows that this preserved precedence of @{text "th"} remains as the highest
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along the way of @{text "t"}.
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The proof goes by induction over @{text "t"} using the specialized
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induction rule @{thm ind}, followed by case analysis of each possible
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operations of PIP. All cases follow the same pattern rendered by the
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generalized introduction rule @{thm "image_Max_eqI"}.
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The very essence is to show that precedences, no matter whether they
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are newly introduced or modified, are always lower than the one held
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by @{term "th"}, which by @{thm th_kept} is preserved along the way.
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*}
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lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
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proof(induct rule:ind)
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case Nil
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from highest_preced_thread
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show ?case by simp
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next
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case (Cons e t)
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interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
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interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
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show ?case
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proof(cases e)
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case (Create thread prio')
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show ?thesis (is "Max (?f ` ?A) = ?t")
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proof -
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-- {* The following is the common pattern of each branch of the case analysis. *}
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-- {* The major part is to show that @{text "th"} holds the highest precedence: *}
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have "Max (?f ` ?A) = ?f th"
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proof(rule image_Max_eqI)
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show "finite ?A" using h_e.finite_threads by auto
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next
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show "th \<in> ?A" using h_e.th_kept by auto
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next
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show "\<forall>x\<in>?A. ?f x \<le> ?f th"
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proof
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fix x
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assume "x \<in> ?A"
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hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
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thus "?f x \<le> ?f th"
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proof
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assume "x = thread"
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thus ?thesis
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apply (simp add:Create the_preced_def preced_def, fold preced_def)
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using Create h_e.create_low h_t.th_kept lt_tm preced_leI2
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preced_th by force
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next
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assume h: "x \<in> threads (t @ s)"
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from Cons(2)[unfolded Create]
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have "x \<noteq> thread" using h by (cases, auto)
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hence "?f x = the_preced (t@s) x"
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by (simp add:Create the_preced_def preced_def)
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hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
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by (simp add: h_t.finite_threads h)
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also have "... = ?f th"
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by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
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finally show ?thesis .
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qed
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qed
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qed
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-- {* The minor part is to show that the precedence of @{text "th"}
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equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
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also have "... = ?t" using h_e.th_kept the_preced_def by auto
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-- {* Then it follows trivially that the precedence preserved
+ − 307
for @{term "th"} remains the maximum of all living threads along the way. *}
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finally show ?thesis .
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qed
+ − 310
next
+ − 311
case (Exit thread)
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show ?thesis (is "Max (?f ` ?A) = ?t")
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proof -
+ − 314
have "Max (?f ` ?A) = ?f th"
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proof(rule image_Max_eqI)
+ − 316
show "finite ?A" using h_e.finite_threads by auto
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next
+ − 318
show "th \<in> ?A" using h_e.th_kept by auto
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next
+ − 320
show "\<forall>x\<in>?A. ?f x \<le> ?f th"
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proof
+ − 322
fix x
+ − 323
assume "x \<in> ?A"
+ − 324
hence "x \<in> threads (t@s)" by (simp add: Exit)
+ − 325
hence "?f x \<le> Max (?f ` threads (t@s))"
+ − 326
by (simp add: h_t.finite_threads)
+ − 327
also have "... \<le> ?f th"
+ − 328
apply (simp add:Exit the_preced_def preced_def, fold preced_def)
+ − 329
using Cons.hyps(5) h_t.th_kept the_preced_def by auto
+ − 330
finally show "?f x \<le> ?f th" .
+ − 331
qed
+ − 332
qed
+ − 333
also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ − 334
finally show ?thesis .
+ − 335
qed
+ − 336
next
+ − 337
case (P thread cs)
+ − 338
with Cons
+ − 339
show ?thesis by (auto simp:preced_def the_preced_def)
+ − 340
next
+ − 341
case (V thread cs)
+ − 342
with Cons
+ − 343
show ?thesis by (auto simp:preced_def the_preced_def)
+ − 344
next
+ − 345
case (Set thread prio')
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show ?thesis (is "Max (?f ` ?A) = ?t")
+ − 347
proof -
+ − 348
have "Max (?f ` ?A) = ?f th"
+ − 349
proof(rule image_Max_eqI)
+ − 350
show "finite ?A" using h_e.finite_threads by auto
+ − 351
next
+ − 352
show "th \<in> ?A" using h_e.th_kept by auto
+ − 353
next
+ − 354
show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ − 355
proof
+ − 356
fix x
+ − 357
assume h: "x \<in> ?A"
+ − 358
show "?f x \<le> ?f th"
+ − 359
proof(cases "x = thread")
+ − 360
case True
+ − 361
moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
+ − 362
proof -
+ − 363
have "the_preced (t @ s) th = Prc prio tm"
+ − 364
using h_t.th_kept preced_th by (simp add:the_preced_def)
+ − 365
moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
+ − 366
ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
+ − 367
qed
+ − 368
ultimately show ?thesis
+ − 369
by (unfold Set, simp add:the_preced_def preced_def)
+ − 370
next
+ − 371
case False
+ − 372
then have "?f x = the_preced (t@s) x"
+ − 373
by (simp add:the_preced_def preced_def Set)
+ − 374
also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
+ − 375
using Set h h_t.finite_threads by auto
+ − 376
also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
+ − 377
finally show ?thesis .
+ − 378
qed
+ − 379
qed
+ − 380
qed
+ − 381
also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ − 382
finally show ?thesis .
+ − 383
qed
+ − 384
qed
+ − 385
qed
+ − 386
+ − 387
lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
+ − 388
by (insert th_kept max_kept, auto)
+ − 389
+ − 390
text {*
+ − 391
The reason behind the following lemma is that:
+ − 392
Since @{term "cp"} is defined as the maximum precedence
+ − 393
of those threads contained in the sub-tree of node @{term "Th th"}
+ − 394
in @{term "RAG (t@s)"}, and all these threads are living threads, and
+ − 395
@{term "th"} is also among them, the maximum precedence of
+ − 396
them all must be the one for @{text "th"}.
+ − 397
*}
+ − 398
lemma th_cp_max_preced:
+ − 399
"cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R")
+ − 400
proof -
+ − 401
let ?f = "the_preced (t@s)"
+ − 402
have "?L = ?f th"
+ − 403
proof(unfold cp_alt_def, rule image_Max_eqI)
+ − 404
show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ − 405
proof -
+ − 406
have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} =
+ − 407
the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
+ − 408
(\<exists> th'. n = Th th')}"
+ − 409
by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
+ − 410
moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree)
+ − 411
ultimately show ?thesis by simp
+ − 412
qed
+ − 413
next
+ − 414
show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ − 415
by (auto simp:subtree_def)
+ − 416
next
+ − 417
show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
+ − 418
the_preced (t @ s) x \<le> the_preced (t @ s) th"
+ − 419
proof
+ − 420
fix th'
+ − 421
assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ − 422
hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
+ − 423
moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
+ − 424
by (meson subtree_Field)
+ − 425
ultimately have "Th th' \<in> ..." by auto
+ − 426
hence "th' \<in> threads (t@s)"
+ − 427
proof
+ − 428
assume "Th th' \<in> {Th th}"
+ − 429
thus ?thesis using th_kept by auto
+ − 430
next
+ − 431
assume "Th th' \<in> Field (RAG (t @ s))"
+ − 432
thus ?thesis using vat_t.not_in_thread_isolated by blast
+ − 433
qed
+ − 434
thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
+ − 435
by (metis Max_ge finite_imageI finite_threads image_eqI
+ − 436
max_kept th_kept the_preced_def)
+ − 437
qed
+ − 438
qed
+ − 439
also have "... = ?R" by (simp add: max_preced the_preced_def)
+ − 440
finally show ?thesis .
+ − 441
qed
+ − 442
+ − 443
lemma th_cp_max[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th"
+ − 444
using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
+ − 445
+ − 446
lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))"
+ − 447
by (simp add: th_cp_max_preced)
+ − 448
+ − 449
lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th"
+ − 450
using max_kept th_kept the_preced_def by auto
+ − 451
+ − 452
lemma [simp]: "the_preced (t@s) th = preced th (t@s)"
+ − 453
using the_preced_def by auto
+ − 454
+ − 455
lemma [simp]: "preced th (t@s) = preced th s"
+ − 456
by (simp add: th_kept)
+ − 457
+ − 458
lemma [simp]: "cp s th = preced th s"
+ − 459
by (simp add: eq_cp_s_th)
+ − 460
+ − 461
lemma th_cp_preced [simp]: "cp (t@s) th = preced th s"
+ − 462
by (fold max_kept, unfold th_cp_max_preced, simp)
+ − 463
+ − 464
lemma preced_less:
+ − 465
assumes th'_in: "th' \<in> threads s"
+ − 466
and neq_th': "th' \<noteq> th"
+ − 467
shows "preced th' s < preced th s"
+ − 468
using assms
+ − 469
by (metis Max.coboundedI finite_imageI highest not_le order.trans
+ − 470
preced_linorder rev_image_eqI threads_s vat_s.finite_threads
+ − 471
vat_s.le_cp)
+ − 472
+ − 473
section {* The `blocking thread` *}
+ − 474
+ − 475
text {*
+ − 476
The purpose of PIP is to ensure that the most
+ − 477
urgent thread @{term th} is not blocked unreasonably.
+ − 478
Therefore, a clear picture of the blocking thread is essential
+ − 479
to assure people that the purpose is fulfilled.
+ − 480
+ − 481
In this section, we are going to derive a series of lemmas
+ − 482
with finally give rise to a picture of the blocking thread.
+ − 483
+ − 484
By `blocking thread`, we mean a thread in running state but
+ − 485
different from thread @{term th}.
+ − 486
*}
+ − 487
+ − 488
text {*
+ − 489
The following lemmas shows that the @{term cp}-value
+ − 490
of the blocking thread @{text th'} equals to the highest
+ − 491
precedence in the whole system.
+ − 492
*}
+ − 493
lemma runing_preced_inversion:
+ − 494
assumes runing': "th' \<in> runing (t@s)"
+ − 495
shows "cp (t@s) th' = preced th s" (is "?L = ?R")
+ − 496
proof -
+ − 497
have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
+ − 498
by (unfold runing_def, auto)
+ − 499
also have "\<dots> = ?R"
+ − 500
by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads)
+ − 501
finally show ?thesis .
+ − 502
qed
+ − 503
+ − 504
text {*
+ − 505
+ − 506
The following lemma shows how the counters for @{term "P"} and
+ − 507
@{term "V"} operations relate to the running threads in the states
+ − 508
@{term s} and @{term "t @ s"}. The lemma shows that if a thread's
+ − 509
@{term "P"}-count equals its @{term "V"}-count (which means it no
+ − 510
longer has any resource in its possession), it cannot be a running
+ − 511
thread.
+ − 512
+ − 513
The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
90
+ − 514
The key is the use of @{thm eq_pv_dependants} to derive the
85
+ − 515
emptiness of @{text th'}s @{term dependants}-set from the balance of
+ − 516
its @{term P} and @{term V} counts. From this, it can be shown
+ − 517
@{text th'}s @{term cp}-value equals to its own precedence.
+ − 518
+ − 519
On the other hand, since @{text th'} is running, by @{thm
+ − 520
runing_preced_inversion}, its @{term cp}-value equals to the
+ − 521
precedence of @{term th}.
+ − 522
+ − 523
Combining the above two resukts we have that @{text th'} and @{term
+ − 524
th} have the same precedence. By uniqueness of precedences, we have
+ − 525
@{text "th' = th"}, which is in contradiction with the assumption
+ − 526
@{text "th' \<noteq> th"}.
+ − 527
+ − 528
*}
+ − 529
+ − 530
lemma eq_pv_blocked: (* ddd *)
+ − 531
assumes neq_th': "th' \<noteq> th"
+ − 532
and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+ − 533
shows "th' \<notin> runing (t@s)"
+ − 534
proof
+ − 535
assume otherwise: "th' \<in> runing (t@s)"
+ − 536
show False
+ − 537
proof -
+ − 538
have th'_in: "th' \<in> threads (t@s)"
+ − 539
using otherwise readys_threads runing_def by auto
+ − 540
have "th' = th"
+ − 541
proof(rule preced_unique)
+ − 542
-- {* The proof goes like this:
+ − 543
it is first shown that the @{term preced}-value of @{term th'}
+ − 544
equals to that of @{term th}, then by uniqueness
+ − 545
of @{term preced}-values (given by lemma @{thm preced_unique}),
+ − 546
@{term th'} equals to @{term th}: *}
+ − 547
show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+ − 548
proof -
+ − 549
-- {* Since the counts of @{term th'} are balanced, the subtree
+ − 550
of it contains only itself, so, its @{term cp}-value
+ − 551
equals its @{term preced}-value: *}
+ − 552
have "?L = cp (t@s) th'"
90
+ − 553
by (unfold cp_eq_cpreced cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp)
85
+ − 554
-- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion},
+ − 555
its @{term cp}-value equals @{term "preced th s"},
+ − 556
which equals to @{term "?R"} by simplification: *}
+ − 557
also have "... = ?R"
+ − 558
thm runing_preced_inversion
+ − 559
using runing_preced_inversion[OF otherwise] by simp
+ − 560
finally show ?thesis .
+ − 561
qed
+ − 562
qed (auto simp: th'_in th_kept)
+ − 563
with `th' \<noteq> th` show ?thesis by simp
+ − 564
qed
+ − 565
qed
+ − 566
+ − 567
text {*
+ − 568
The following lemma is the extrapolation of @{thm eq_pv_blocked}.
+ − 569
It says if a thread, different from @{term th},
+ − 570
does not hold any resource at the very beginning,
+ − 571
it will keep hand-emptied in the future @{term "t@s"}.
+ − 572
*}
+ − 573
lemma eq_pv_persist: (* ddd *)
+ − 574
assumes neq_th': "th' \<noteq> th"
+ − 575
and eq_pv: "cntP s th' = cntV s th'"
+ − 576
shows "cntP (t@s) th' = cntV (t@s) th'"
+ − 577
proof(induction rule:ind) -- {* The proof goes by induction. *}
+ − 578
-- {* The nontrivial case is for the @{term Cons}: *}
+ − 579
case (Cons e t)
+ − 580
-- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *}
+ − 581
interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
+ − 582
interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
+ − 583
show ?case
+ − 584
proof -
+ − 585
-- {* It can be proved that @{term cntP}-value of @{term th'} does not change
+ − 586
by the happening of event @{term e}: *}
+ − 587
have "cntP ((e#t)@s) th' = cntP (t@s) th'"
+ − 588
proof(rule ccontr) -- {* Proof by contradiction. *}
+ − 589
-- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *}
+ − 590
assume otherwise: "cntP ((e # t) @ s) th' \<noteq> cntP (t @ s) th'"
+ − 591
-- {* Then the actor of @{term e} must be @{term th'} and @{term e}
+ − 592
must be a @{term P}-event: *}
+ − 593
hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv)
+ − 594
with vat_t.actor_inv[OF Cons(2)]
+ − 595
-- {* According to @{thm actor_inv}, @{term th'} must be running at
+ − 596
the moment @{term "t@s"}: *}
+ − 597
have "th' \<in> runing (t@s)" by (cases e, auto)
+ − 598
-- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis
+ − 599
shows @{term th'} can not be running at moment @{term "t@s"}: *}
+ − 600
moreover have "th' \<notin> runing (t@s)"
+ − 601
using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
+ − 602
-- {* Contradiction is finally derived: *}
+ − 603
ultimately show False by simp
+ − 604
qed
+ − 605
-- {* It can also be proved that @{term cntV}-value of @{term th'} does not change
+ − 606
by the happening of event @{term e}: *}
+ − 607
-- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *}
+ − 608
moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'"
+ − 609
proof(rule ccontr) -- {* Proof by contradiction. *}
+ − 610
assume otherwise: "cntV ((e # t) @ s) th' \<noteq> cntV (t @ s) th'"
+ − 611
hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv)
+ − 612
with vat_t.actor_inv[OF Cons(2)]
+ − 613
have "th' \<in> runing (t@s)" by (cases e, auto)
+ − 614
moreover have "th' \<notin> runing (t@s)"
+ − 615
using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
+ − 616
ultimately show False by simp
+ − 617
qed
+ − 618
-- {* Finally, it can be shown that the @{term cntP} and @{term cntV}
+ − 619
value for @{term th'} are still in balance, so @{term th'}
+ − 620
is still hand-emptied after the execution of event @{term e}: *}
+ − 621
ultimately show ?thesis using Cons(5) by metis
+ − 622
qed
+ − 623
qed (auto simp:eq_pv)
+ − 624
+ − 625
text {*
+ − 626
By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist},
+ − 627
it can be derived easily that @{term th'} can not be running in the future:
+ − 628
*}
+ − 629
lemma eq_pv_blocked_persist:
+ − 630
assumes neq_th': "th' \<noteq> th"
+ − 631
and eq_pv: "cntP s th' = cntV s th'"
+ − 632
shows "th' \<notin> runing (t@s)"
+ − 633
using assms
+ − 634
by (simp add: eq_pv_blocked eq_pv_persist)
+ − 635
+ − 636
text {*
+ − 637
The following lemma shows the blocking thread @{term th'}
+ − 638
must hold some resource in the very beginning.
+ − 639
*}
+ − 640
lemma runing_cntP_cntV_inv: (* ddd *)
+ − 641
assumes is_runing: "th' \<in> runing (t@s)"
+ − 642
and neq_th': "th' \<noteq> th"
+ − 643
shows "cntP s th' > cntV s th'"
+ − 644
using assms
+ − 645
proof -
+ − 646
-- {* First, it can be shown that the number of @{term P} and
+ − 647
@{term V} operations can not be equal for thred @{term th'} *}
+ − 648
have "cntP s th' \<noteq> cntV s th'"
+ − 649
proof
+ − 650
-- {* The proof goes by contradiction, suppose otherwise: *}
+ − 651
assume otherwise: "cntP s th' = cntV s th'"
+ − 652
-- {* By applying @{thm eq_pv_blocked_persist} to this: *}
+ − 653
from eq_pv_blocked_persist[OF neq_th' otherwise]
+ − 654
-- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *}
+ − 655
have "th' \<notin> runing (t@s)" .
+ − 656
-- {* This is obvious in contradiction with assumption @{thm is_runing} *}
+ − 657
thus False using is_runing by simp
+ − 658
qed
+ − 659
-- {* However, the number of @{term V} is always less or equal to @{term P}: *}
+ − 660
moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
+ − 661
-- {* Thesis is finally derived by combining the these two results: *}
+ − 662
ultimately show ?thesis by auto
+ − 663
qed
+ − 664
+ − 665
+ − 666
text {*
+ − 667
The following lemmas shows the blocking thread @{text th'} must be live
+ − 668
at the very beginning, i.e. the moment (or state) @{term s}.
+ − 669
+ − 670
The proof is a simple combination of the results above:
+ − 671
*}
+ − 672
lemma runing_threads_inv:
+ − 673
assumes runing': "th' \<in> runing (t@s)"
+ − 674
and neq_th': "th' \<noteq> th"
+ − 675
shows "th' \<in> threads s"
+ − 676
proof(rule ccontr) -- {* Proof by contradiction: *}
+ − 677
assume otherwise: "th' \<notin> threads s"
+ − 678
have "th' \<notin> runing (t @ s)"
+ − 679
proof -
+ − 680
from vat_s.cnp_cnv_eq[OF otherwise]
+ − 681
have "cntP s th' = cntV s th'" .
+ − 682
from eq_pv_blocked_persist[OF neq_th' this]
+ − 683
show ?thesis .
+ − 684
qed
+ − 685
with runing' show False by simp
+ − 686
qed
+ − 687
+ − 688
text {*
+ − 689
The following lemma summarizes several foregoing
+ − 690
lemmas to give an overall picture of the blocking thread @{text "th'"}:
+ − 691
*}
+ − 692
lemma runing_inversion: (* ddd, one of the main lemmas to present *)
+ − 693
assumes runing': "th' \<in> runing (t@s)"
+ − 694
and neq_th: "th' \<noteq> th"
+ − 695
shows "th' \<in> threads s"
+ − 696
and "\<not>detached s th'"
+ − 697
and "cp (t@s) th' = preced th s"
+ − 698
proof -
+ − 699
from runing_threads_inv[OF assms]
+ − 700
show "th' \<in> threads s" .
+ − 701
next
+ − 702
from runing_cntP_cntV_inv[OF runing' neq_th]
+ − 703
show "\<not>detached s th'" using vat_s.detached_eq by simp
+ − 704
next
+ − 705
from runing_preced_inversion[OF runing']
+ − 706
show "cp (t@s) th' = preced th s" .
+ − 707
qed
+ − 708
+ − 709
section {* The existence of `blocking thread` *}
+ − 710
+ − 711
text {*
+ − 712
Suppose @{term th} is not running, it is first shown that
+ − 713
there is a path in RAG leading from node @{term th} to another thread @{text "th'"}
+ − 714
in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
+ − 715
+ − 716
Now, since @{term readys}-set is non-empty, there must be
+ − 717
one in it which holds the highest @{term cp}-value, which, by definition,
+ − 718
is the @{term runing}-thread. However, we are going to show more: this running thread
+ − 719
is exactly @{term "th'"}.
+ − 720
*}
+ − 721
lemma th_blockedE: (* ddd, the other main lemma to be presented: *)
+ − 722
assumes "th \<notin> runing (t@s)"
+ − 723
obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ − 724
"th' \<in> runing (t@s)"
+ − 725
proof -
+ − 726
-- {* According to @{thm vat_t.th_chain_to_ready}, either
+ − 727
@{term "th"} is in @{term "readys"} or there is path leading from it to
+ − 728
one thread in @{term "readys"}. *}
+ − 729
have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)"
+ − 730
using th_kept vat_t.th_chain_to_ready by auto
+ − 731
-- {* However, @{term th} can not be in @{term readys}, because otherwise, since
+ − 732
@{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
+ − 733
moreover have "th \<notin> readys (t@s)"
+ − 734
using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto
+ − 735
-- {* So, there must be a path from @{term th} to another thread @{text "th'"} in
+ − 736
term @{term readys}: *}
+ − 737
ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
+ − 738
and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
+ − 739
-- {* We are going to show that this @{term th'} is running. *}
+ − 740
have "th' \<in> runing (t@s)"
+ − 741
proof -
+ − 742
-- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
+ − 743
have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
+ − 744
proof -
+ − 745
have "?L = Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
+ − 746
by (unfold cp_alt_def1, simp)
+ − 747
also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
+ − 748
proof(rule image_Max_subset)
+ − 749
show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
+ − 750
next
+ − 751
show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
90
+ − 752
by (metis Range.intros dp trancl_range vat_t.rg_RAG_threads vat_t.subtree_tRAG_thread)
85
+ − 753
next
+ − 754
show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
+ − 755
by (unfold tRAG_subtree_eq, auto simp:subtree_def)
+ − 756
next
+ − 757
show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
+ − 758
(the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
+ − 759
proof -
+ − 760
have "?L = the_preced (t @ s) ` threads (t @ s)"
+ − 761
by (unfold image_comp, rule image_cong, auto)
+ − 762
thus ?thesis using max_preced the_preced_def by auto
+ − 763
qed
+ − 764
qed
+ − 765
also have "... = ?R"
+ − 766
using th_cp_max th_cp_preced th_kept
+ − 767
the_preced_def vat_t.max_cp_readys_threads by auto
+ − 768
finally show ?thesis .
+ − 769
qed
+ − 770
-- {* Now, since @{term th'} holds the highest @{term cp}
+ − 771
and we have already show it is in @{term readys},
+ − 772
it is @{term runing} by definition. *}
+ − 773
with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def)
+ − 774
qed
+ − 775
-- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
+ − 776
moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ − 777
using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
+ − 778
ultimately show ?thesis using that by metis
+ − 779
qed
+ − 780
+ − 781
text {*
+ − 782
Now it is easy to see there is always a thread to run by case analysis
+ − 783
on whether thread @{term th} is running: if the answer is Yes, the
+ − 784
the running thread is obviously @{term th} itself; otherwise, the running
+ − 785
thread is the @{text th'} given by lemma @{thm th_blockedE}.
+ − 786
*}
+ − 787
lemma live: "runing (t@s) \<noteq> {}"
+ − 788
proof(cases "th \<in> runing (t@s)")
+ − 789
case True thus ?thesis by auto
+ − 790
next
+ − 791
case False
+ − 792
thus ?thesis using th_blockedE by auto
+ − 793
qed
+ − 794
+ − 795
end
+ − 796
end