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262
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theory ExtGG
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imports PrioG
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begin
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lemma birth_time_lt: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
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apply (induct s, simp)
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proof -
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fix a s
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assume ih: "s \<noteq> [] \<Longrightarrow> birthtime th s < length s"
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and eq_as: "a # s \<noteq> []"
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show "birthtime th (a # s) < length (a # s)"
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proof(cases "s \<noteq> []")
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case False
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from False show ?thesis
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by (cases a, auto simp:birthtime.simps)
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next
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case True
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from ih [OF True] show ?thesis
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by (cases a, auto simp:birthtime.simps)
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qed
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qed
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lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
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by (induct s, auto simp:threads.simps)
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lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
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apply (drule_tac th_in_ne)
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by (unfold preced_def, auto intro: birth_time_lt)
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locale highest_gen =
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fixes s th prio tm
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assumes vt_s: "vt step 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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and preced_th: "preced th s = Prc prio tm"
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context highest_gen
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begin
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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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lemma eq_cp_s_th: "cp s th = preced th s"
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proof -
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from highest and max_cp_eq[OF vt_s]
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have is_max: "preced th s = Max ((\<lambda>th. preced th s) ` threads s)" by simp
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have sbs: "({th} \<union> dependents (wq s) th) \<subseteq> threads s"
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proof -
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from threads_s and dependents_threads[OF vt_s, of th]
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show ?thesis by auto
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qed
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show ?thesis
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proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
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show "preced th s \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)" by simp
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next
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fix y
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assume "y \<in> (\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th)"
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then obtain th1 where th1_in: "th1 \<in> ({th} \<union> dependents (wq s) th)"
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and eq_y: "y = preced th1 s" by auto
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show "y \<le> preced th s"
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proof(unfold is_max, rule Max_ge)
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from finite_threads[OF vt_s]
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show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
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next
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from sbs th1_in and eq_y
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show "y \<in> (\<lambda>th. preced th s) ` threads s" by auto
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qed
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next
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from sbs and finite_threads[OF vt_s]
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show "finite ((\<lambda>th. preced th s) ` ({th} \<union> dependents (wq s) th))"
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by (auto intro:finite_subset)
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qed
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qed
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lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
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by (fold max_cp_eq[OF vt_s], unfold eq_cp_s_th, insert highest, simp)
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lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' 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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proof -
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from highest_cp_preced max_cp_eq[OF vt_s, symmetric]
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show ?thesis by simp
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qed
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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 step (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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lemma step_back_vt_app:
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assumes vt_ts: "vt cs (t@s)"
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shows "vt cs 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 cs (t @ s) \<Longrightarrow> vt cs s"
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and vt_et: "vt cs ((e # t) @ s)"
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show ?case
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proof(rule ih)
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show "vt cs (t @ s)"
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proof(rule step_back_vt)
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from vt_et show "vt cs (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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context extend_highest_gen
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begin
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lemma red_moment:
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"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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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 step (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 step (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
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and vt_e: "vt step ((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 step (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 step (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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264
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preced th (t@s) = preced th s" (is "?Q t")
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262
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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 "th \<in> threads ([] @ s) \<and> preced th ([] @ s) = preced th s"
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by auto
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next
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case (Cons e t)
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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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assume eq_e: " e = Create thread prio"
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show ?thesis
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proof -
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from Cons and eq_e 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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assume "thread \<notin> threads (t @ s)"
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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 eq_e, 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:eq_e)
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qed
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next
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case (Exit thread)
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assume eq_e: "e = Exit thread"
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from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
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from extend_highest_gen.exit_diff [OF this] and eq_e
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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 eq_e, auto simp:preced_def)
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next
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case (P thread cs)
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assume eq_e: "e = P thread cs"
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with Cons
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show ?thesis
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by (auto simp:eq_e preced_def)
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next
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case (V thread cs)
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assume eq_e: "e = V thread cs"
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with Cons
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show ?thesis
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by (auto simp:eq_e preced_def)
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next
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case (Set thread prio')
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assume eq_e: " e = Set thread prio'"
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show ?thesis
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proof -
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from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
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from extend_highest_gen.set_diff_low[OF this] and eq_e
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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 eq_e, 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:eq_e)
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qed
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qed
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qed
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qed
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lemma max_kept: "Max ((\<lambda> th'. preced th' (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 "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"
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by simp
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next
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case (Cons e t)
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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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assume eq_e: " e = Create thread prio'"
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from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto
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hence neq_thread: "thread \<noteq> th"
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proof(cases)
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assume "thread \<notin> threads (t @ s)"
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moreover have "th \<in> threads (t@s)"
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proof -
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from Cons have "extend_highest_gen s th prio tm t" by auto
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from extend_highest_gen.th_kept[OF this] show ?thesis by (simp)
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qed
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ultimately show ?thesis by auto
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qed
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from Cons have "extend_highest_gen s th prio tm t" by auto
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from extend_highest_gen.th_kept[OF this]
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have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s"
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by (auto)
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from stp
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have thread_ts: "thread \<notin> threads (t @ s)"
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by (cases, auto)
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show ?thesis (is "Max (?f ` ?A) = ?t")
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proof -
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have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"
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by (unfold eq_e, simp)
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moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"
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proof(rule Max_insert)
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from Cons have "vt step (t @ s)" by auto
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from finite_threads[OF this]
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show "finite (?f ` (threads (t@s)))" by simp
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next
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from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto
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qed
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moreover have "(Max (?f ` (threads (t@s)))) = ?t"
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proof -
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have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) =
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(\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")
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proof -
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{ fix th'
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assume "th' \<in> ?B"
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with thread_ts eq_e
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have "?f1 th' = ?f2 th'" by (auto simp:preced_def)
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} thus ?thesis
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apply (auto simp:Image_def)
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proof -
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fix th'
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assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow>
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preced th' (e # t @ s) = preced th' (t @ s)"
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and h1: "th' \<in> threads (t @ s)"
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show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"
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proof -
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from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto
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moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp
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ultimately show ?thesis by simp
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qed
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qed
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296 |
qed
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with Cons show ?thesis by auto
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298 |
qed
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moreover have "?f thread < ?t"
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proof -
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from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
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from extend_highest_gen.create_low[OF this] and eq_e
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have "prio' \<le> prio" by auto
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thus ?thesis
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by (unfold preced_th, unfold eq_e, insert lt_tm,
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auto simp:preced_def precedence_less_def preced_th)
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qed
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ultimately show ?thesis by (auto simp:max_def)
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qed
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next
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311 |
case (Exit thread)
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assume eq_e: "e = Exit thread"
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from Cons have vt_e: "vt step (e#(t @ s))" by auto
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from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto
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from stp have thread_ts: "thread \<in> threads (t @ s)"
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by(cases, unfold runing_def readys_def, auto)
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from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
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|
318 |
from extend_highest_gen.exit_diff[OF this] and eq_e
|
|
|
319 |
have neq_thread: "thread \<noteq> th" by auto
|
|
|
320 |
from Cons have "extend_highest_gen s th prio tm t" by auto
|
|
|
321 |
from extend_highest_gen.th_kept[OF this]
|
|
|
322 |
have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
|
|
|
323 |
show ?thesis (is "Max (?f ` ?A) = ?t")
|
|
|
324 |
proof -
|
|
|
325 |
have "threads (t@s) = insert thread ?A"
|
|
|
326 |
by (insert stp thread_ts, unfold eq_e, auto)
|
|
|
327 |
hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp
|
|
|
328 |
also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp
|
|
|
329 |
also have "\<dots> = max (?f thread) (Max (?f ` ?A))"
|
|
|
330 |
proof(rule Max_insert)
|
|
|
331 |
from finite_threads [OF vt_e]
|
|
|
332 |
show "finite (?f ` ?A)" by simp
|
|
|
333 |
next
|
|
|
334 |
from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
|
|
|
335 |
from extend_highest_gen.th_kept[OF this]
|
|
|
336 |
show "?f ` ?A \<noteq> {}" by auto
|
|
|
337 |
qed
|
|
|
338 |
finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .
|
|
|
339 |
moreover have "Max (?f ` (threads (t@s))) = ?t"
|
|
|
340 |
proof -
|
|
|
341 |
from Cons show ?thesis
|
|
|
342 |
by (unfold eq_e, auto simp:preced_def)
|
|
|
343 |
qed
|
|
|
344 |
ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp
|
|
|
345 |
moreover have "?f thread < ?t"
|
|
|
346 |
proof(unfold eq_e, simp add:preced_def, fold preced_def)
|
|
|
347 |
show "preced thread (t @ s) < ?t"
|
|
|
348 |
proof -
|
|
|
349 |
have "preced thread (t @ s) \<le> ?t"
|
|
|
350 |
proof -
|
|
|
351 |
from Cons
|
|
|
352 |
have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
|
|
|
353 |
(is "?t = Max (?g ` ?B)") by simp
|
|
|
354 |
moreover have "?g thread \<le> \<dots>"
|
|
|
355 |
proof(rule Max_ge)
|
|
|
356 |
have "vt step (t@s)" by fact
|
|
|
357 |
from finite_threads [OF this]
|
|
|
358 |
show "finite (?g ` ?B)" by simp
|
|
|
359 |
next
|
|
|
360 |
from thread_ts
|
|
|
361 |
show "?g thread \<in> (?g ` ?B)" by auto
|
|
|
362 |
qed
|
|
|
363 |
ultimately show ?thesis by auto
|
|
|
364 |
qed
|
|
|
365 |
moreover have "preced thread (t @ s) \<noteq> ?t"
|
|
|
366 |
proof
|
|
|
367 |
assume "preced thread (t @ s) = preced th s"
|
|
|
368 |
with h' have "preced thread (t @ s) = preced th (t@s)" by simp
|
|
|
369 |
from preced_unique [OF this] have "thread = th"
|
|
|
370 |
proof
|
|
|
371 |
from h' show "th \<in> threads (t @ s)" by simp
|
|
|
372 |
next
|
|
|
373 |
from thread_ts show "thread \<in> threads (t @ s)" .
|
|
|
374 |
qed(simp)
|
|
|
375 |
with neq_thread show "False" by simp
|
|
|
376 |
qed
|
|
|
377 |
ultimately show ?thesis by auto
|
|
|
378 |
qed
|
|
|
379 |
qed
|
|
|
380 |
ultimately show ?thesis
|
|
|
381 |
by (auto simp:max_def split:if_splits)
|
|
|
382 |
qed
|
|
|
383 |
next
|
|
|
384 |
case (P thread cs)
|
|
|
385 |
with Cons
|
|
|
386 |
show ?thesis by (auto simp:preced_def)
|
|
|
387 |
next
|
|
|
388 |
case (V thread cs)
|
|
|
389 |
with Cons
|
|
|
390 |
show ?thesis by (auto simp:preced_def)
|
|
|
391 |
next
|
|
|
392 |
case (Set thread prio')
|
|
|
393 |
show ?thesis (is "Max (?f ` ?A) = ?t")
|
|
|
394 |
proof -
|
|
|
395 |
let ?B = "threads (t@s)"
|
|
|
396 |
from Cons have "extend_highest_gen s th prio tm (e # t)" by auto
|
|
|
397 |
from extend_highest_gen.set_diff_low[OF this] and Set
|
|
|
398 |
have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto
|
|
|
399 |
from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp
|
|
|
400 |
also have "\<dots> = ?t"
|
|
|
401 |
proof(rule Max_eqI)
|
|
|
402 |
fix y
|
|
|
403 |
assume y_in: "y \<in> ?f ` ?B"
|
|
|
404 |
then obtain th1 where
|
|
|
405 |
th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto
|
|
|
406 |
show "y \<le> ?t"
|
|
|
407 |
proof(cases "th1 = thread")
|
|
|
408 |
case True
|
|
|
409 |
with neq_thread le_p eq_y Set
|
|
|
410 |
show ?thesis
|
|
|
411 |
apply (subst preced_th, insert lt_tm)
|
|
|
412 |
by (auto simp:preced_def precedence_le_def)
|
|
|
413 |
next
|
|
|
414 |
case False
|
|
|
415 |
with Set eq_y
|
|
|
416 |
have "y = preced th1 (t@s)"
|
|
|
417 |
by (simp add:preced_def)
|
|
|
418 |
moreover have "\<dots> \<le> ?t"
|
|
|
419 |
proof -
|
|
|
420 |
from Cons
|
|
|
421 |
have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"
|
|
|
422 |
by auto
|
|
|
423 |
moreover have "preced th1 (t@s) \<le> \<dots>"
|
|
|
424 |
proof(rule Max_ge)
|
|
|
425 |
from th1_in
|
|
|
426 |
show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"
|
|
|
427 |
by simp
|
|
|
428 |
next
|
|
|
429 |
show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
|
|
|
430 |
proof -
|
|
|
431 |
from Cons have "vt step (t @ s)" by auto
|
|
|
432 |
from finite_threads[OF this] show ?thesis by auto
|
|
|
433 |
qed
|
|
|
434 |
qed
|
|
|
435 |
ultimately show ?thesis by auto
|
|
|
436 |
qed
|
|
|
437 |
ultimately show ?thesis by auto
|
|
|
438 |
qed
|
|
|
439 |
next
|
|
|
440 |
from Cons and finite_threads
|
|
|
441 |
show "finite (?f ` ?B)" by auto
|
|
|
442 |
next
|
|
|
443 |
from Cons have "extend_highest_gen s th prio tm t" by auto
|
|
|
444 |
from extend_highest_gen.th_kept [OF this]
|
|
|
445 |
have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .
|
|
|
446 |
show "?t \<in> (?f ` ?B)"
|
|
|
447 |
proof -
|
|
|
448 |
from neq_thread Set h
|
|
|
449 |
have "?t = ?f th" by (auto simp:preced_def)
|
|
|
450 |
with h show ?thesis by auto
|
|
|
451 |
qed
|
|
|
452 |
qed
|
|
|
453 |
finally show ?thesis .
|
|
|
454 |
qed
|
|
|
455 |
qed
|
|
|
456 |
qed
|
|
|
457 |
|
|
|
458 |
lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
|
|
|
459 |
by (insert th_kept max_kept, auto)
|
|
|
460 |
|
|
|
461 |
lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"
|
|
|
462 |
(is "?L = ?R")
|
|
|
463 |
proof -
|
|
290
|
464 |
have "?L = cpreced (wq (t@s)) (t@s) th"
|
|
262
|
465 |
by (unfold cp_eq_cpreced, simp)
|
|
|
466 |
also have "\<dots> = ?R"
|
|
|
467 |
proof(unfold cpreced_def)
|
|
|
468 |
show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =
|
|
|
469 |
Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"
|
|
|
470 |
(is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")
|
|
|
471 |
proof(cases "?A = {}")
|
|
|
472 |
case False
|
|
|
473 |
have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp
|
|
|
474 |
moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"
|
|
|
475 |
proof(rule Max_insert)
|
|
|
476 |
show "finite (?f ` ?A)"
|
|
|
477 |
proof -
|
|
|
478 |
from dependents_threads[OF vt_t]
|
|
|
479 |
have "?A \<subseteq> threads (t@s)" .
|
|
|
480 |
moreover from finite_threads[OF vt_t] have "finite \<dots>" .
|
|
|
481 |
ultimately show ?thesis
|
|
|
482 |
by (auto simp:finite_subset)
|
|
|
483 |
qed
|
|
|
484 |
next
|
|
|
485 |
from False show "(?f ` ?A) \<noteq> {}" by simp
|
|
|
486 |
qed
|
|
|
487 |
moreover have "\<dots> = Max (?f ` ?B)"
|
|
|
488 |
proof -
|
|
|
489 |
from max_preced have "?f th = Max (?f ` ?B)" .
|
|
|
490 |
moreover have "Max (?f ` ?A) \<le> \<dots>"
|
|
|
491 |
proof(rule Max_mono)
|
|
|
492 |
from False show "(?f ` ?A) \<noteq> {}" by simp
|
|
|
493 |
next
|
|
|
494 |
show "?f ` ?A \<subseteq> ?f ` ?B"
|
|
|
495 |
proof -
|
|
|
496 |
have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])
|
|
|
497 |
thus ?thesis by auto
|
|
|
498 |
qed
|
|
|
499 |
next
|
|
|
500 |
from finite_threads[OF vt_t]
|
|
|
501 |
show "finite (?f ` ?B)" by simp
|
|
|
502 |
qed
|
|
|
503 |
ultimately show ?thesis
|
|
|
504 |
by (auto simp:max_def)
|
|
|
505 |
qed
|
|
|
506 |
ultimately show ?thesis by auto
|
|
|
507 |
next
|
|
|
508 |
case True
|
|
|
509 |
with max_preced show ?thesis by auto
|
|
|
510 |
qed
|
|
|
511 |
qed
|
|
|
512 |
finally show ?thesis .
|
|
|
513 |
qed
|
|
|
514 |
|
|
|
515 |
lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
|
|
|
516 |
by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)
|
|
|
517 |
|
|
|
518 |
lemma th_cp_preced: "cp (t@s) th = preced th s"
|
|
|
519 |
by (fold max_kept, unfold th_cp_max_preced, simp)
|
|
|
520 |
|
|
264
|
521 |
lemma preced_less:
|
|
262
|
522 |
fixes th'
|
|
|
523 |
assumes th'_in: "th' \<in> threads s"
|
|
|
524 |
and neq_th': "th' \<noteq> th"
|
|
|
525 |
shows "preced th' s < preced th s"
|
|
|
526 |
proof -
|
|
|
527 |
have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"
|
|
|
528 |
proof(rule Max_ge)
|
|
|
529 |
from finite_threads [OF vt_s]
|
|
|
530 |
show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp
|
|
|
531 |
next
|
|
|
532 |
from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"
|
|
|
533 |
by simp
|
|
|
534 |
qed
|
|
|
535 |
moreover have "preced th' s \<noteq> preced th s"
|
|
|
536 |
proof
|
|
|
537 |
assume "preced th' s = preced th s"
|
|
|
538 |
from preced_unique[OF this th'_in] neq_th' threads_s
|
|
|
539 |
show "False" by (auto simp:readys_def)
|
|
|
540 |
qed
|
|
|
541 |
ultimately show ?thesis using highest_preced_thread
|
|
|
542 |
by auto
|
|
|
543 |
qed
|
|
|
544 |
|
|
|
545 |
lemma pv_blocked:
|
|
|
546 |
fixes th'
|
|
|
547 |
assumes th'_in: "th' \<in> threads (t@s)"
|
|
|
548 |
and neq_th': "th' \<noteq> th"
|
|
|
549 |
and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
|
|
|
550 |
shows "th' \<notin> runing (t@s)"
|
|
|
551 |
proof
|
|
|
552 |
assume "th' \<in> runing (t@s)"
|
|
|
553 |
hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))"
|
|
|
554 |
by (auto simp:runing_def)
|
|
|
555 |
with max_cp_readys_threads [OF vt_t]
|
|
|
556 |
have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"
|
|
|
557 |
by auto
|
|
|
558 |
moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp
|
|
|
559 |
ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp
|
|
|
560 |
moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"
|
|
|
561 |
by simp
|
|
|
562 |
finally have h: "cp (t @ s) th' = preced th (t @ s)" .
|
|
|
563 |
show False
|
|
|
564 |
proof -
|
|
|
565 |
have "dependents (wq (t @ s)) th' = {}"
|
|
|
566 |
by (rule count_eq_dependents [OF vt_t eq_pv])
|
|
|
567 |
moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"
|
|
|
568 |
proof
|
|
|
569 |
assume "preced th' (t @ s) = preced th (t @ s)"
|
|
|
570 |
hence "th' = th"
|
|
|
571 |
proof(rule preced_unique)
|
|
|
572 |
from th_kept show "th \<in> threads (t @ s)" by simp
|
|
|
573 |
next
|
|
|
574 |
from th'_in show "th' \<in> threads (t @ s)" by simp
|
|
|
575 |
qed
|
|
|
576 |
with assms show False by simp
|
|
|
577 |
qed
|
|
|
578 |
ultimately show ?thesis
|
|
|
579 |
by (insert h, unfold cp_eq_cpreced cpreced_def, simp)
|
|
|
580 |
qed
|
|
|
581 |
qed
|
|
|
582 |
|
|
|
583 |
lemma runing_precond_pre:
|
|
|
584 |
fixes th'
|
|
|
585 |
assumes th'_in: "th' \<in> threads s"
|
|
|
586 |
and eq_pv: "cntP s th' = cntV s th'"
|
|
|
587 |
and neq_th': "th' \<noteq> th"
|
|
|
588 |
shows "th' \<in> threads (t@s) \<and>
|
|
|
589 |
cntP (t@s) th' = cntV (t@s) th'"
|
|
|
590 |
proof -
|
|
|
591 |
show ?thesis
|
|
|
592 |
proof(induct rule:ind)
|
|
|
593 |
case (Cons e t)
|
|
|
594 |
from Cons
|
|
|
595 |
have in_thread: "th' \<in> threads (t @ s)"
|
|
|
596 |
and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
|
|
|
597 |
from Cons have "extend_highest_gen s th prio tm t" by auto
|
|
|
598 |
from extend_highest_gen.pv_blocked
|
|
|
599 |
[OF this, OF in_thread neq_th' not_holding]
|
|
|
600 |
have not_runing: "th' \<notin> runing (t @ s)" .
|
|
|
601 |
show ?case
|
|
|
602 |
proof(cases e)
|
|
|
603 |
case (V thread cs)
|
|
|
604 |
from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto
|
|
|
605 |
|
|
|
606 |
show ?thesis
|
|
|
607 |
proof -
|
|
|
608 |
from Cons and V have "step (t@s) (V thread cs)" by auto
|
|
|
609 |
hence neq_th': "thread \<noteq> th'"
|
|
|
610 |
proof(cases)
|
|
|
611 |
assume "thread \<in> runing (t@s)"
|
|
|
612 |
moreover have "th' \<notin> runing (t@s)" by fact
|
|
|
613 |
ultimately show ?thesis by auto
|
|
|
614 |
qed
|
|
|
615 |
with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
|
|
|
616 |
by (unfold V, simp add:cntP_def cntV_def count_def)
|
|
|
617 |
moreover from in_thread
|
|
|
618 |
have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)
|
|
|
619 |
ultimately show ?thesis by auto
|
|
|
620 |
qed
|
|
|
621 |
next
|
|
|
622 |
case (P thread cs)
|
|
|
623 |
from Cons and P have "step (t@s) (P thread cs)" by auto
|
|
|
624 |
hence neq_th': "thread \<noteq> th'"
|
|
|
625 |
proof(cases)
|
|
|
626 |
assume "thread \<in> runing (t@s)"
|
|
|
627 |
moreover note not_runing
|
|
|
628 |
ultimately show ?thesis by auto
|
|
|
629 |
qed
|
|
|
630 |
with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
|
|
|
631 |
by (auto simp:cntP_def cntV_def count_def)
|
|
|
632 |
moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"
|
|
|
633 |
by auto
|
|
|
634 |
ultimately show ?thesis by auto
|
|
|
635 |
next
|
|
|
636 |
case (Create thread prio')
|
|
|
637 |
from Cons and Create have "step (t@s) (Create thread prio')" by auto
|
|
|
638 |
hence neq_th': "thread \<noteq> th'"
|
|
|
639 |
proof(cases)
|
|
|
640 |
assume "thread \<notin> threads (t @ s)"
|
|
|
641 |
moreover have "th' \<in> threads (t@s)" by fact
|
|
|
642 |
ultimately show ?thesis by auto
|
|
|
643 |
qed
|
|
|
644 |
with Cons and Create
|
|
|
645 |
have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
|
|
|
646 |
by (auto simp:cntP_def cntV_def count_def)
|
|
|
647 |
moreover from Cons and Create
|
|
|
648 |
have in_thread': "th' \<in> threads ((e # t) @ s)" by auto
|
|
|
649 |
ultimately show ?thesis by auto
|
|
|
650 |
next
|
|
|
651 |
case (Exit thread)
|
|
|
652 |
from Cons and Exit have "step (t@s) (Exit thread)" by auto
|
|
|
653 |
hence neq_th': "thread \<noteq> th'"
|
|
|
654 |
proof(cases)
|
|
|
655 |
assume "thread \<in> runing (t @ s)"
|
|
|
656 |
moreover note not_runing
|
|
|
657 |
ultimately show ?thesis by auto
|
|
|
658 |
qed
|
|
|
659 |
with Cons and Exit
|
|
|
660 |
have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
|
|
|
661 |
by (auto simp:cntP_def cntV_def count_def)
|
|
|
662 |
moreover from Cons and Exit and neq_th'
|
|
|
663 |
have in_thread': "th' \<in> threads ((e # t) @ s)"
|
|
|
664 |
by auto
|
|
|
665 |
ultimately show ?thesis by auto
|
|
|
666 |
next
|
|
|
667 |
case (Set thread prio')
|
|
|
668 |
with Cons
|
|
|
669 |
show ?thesis
|
|
|
670 |
by (auto simp:cntP_def cntV_def count_def)
|
|
|
671 |
qed
|
|
|
672 |
next
|
|
|
673 |
case Nil
|
|
|
674 |
with assms
|
|
|
675 |
show ?case by auto
|
|
|
676 |
qed
|
|
|
677 |
qed
|
|
|
678 |
|
|
|
679 |
(*
|
|
|
680 |
lemma runing_precond:
|
|
|
681 |
fixes th'
|
|
|
682 |
assumes th'_in: "th' \<in> threads s"
|
|
|
683 |
and eq_pv: "cntP s th' = cntV s th'"
|
|
|
684 |
and neq_th': "th' \<noteq> th"
|
|
|
685 |
shows "th' \<notin> runing (t@s)"
|
|
|
686 |
proof -
|
|
|
687 |
from runing_precond_pre[OF th'_in eq_pv neq_th']
|
|
|
688 |
have h1: "th' \<in> threads (t @ s)" and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
|
|
|
689 |
from pv_blocked[OF h1 neq_th' h2]
|
|
|
690 |
show ?thesis .
|
|
|
691 |
qed
|
|
|
692 |
*)
|
|
|
693 |
|
|
|
694 |
lemma runing_precond:
|
|
|
695 |
fixes th'
|
|
|
696 |
assumes th'_in: "th' \<in> threads s"
|
|
|
697 |
and neq_th': "th' \<noteq> th"
|
|
|
698 |
and is_runing: "th' \<in> runing (t@s)"
|
|
|
699 |
shows "cntP s th' > cntV s th'"
|
|
|
700 |
proof -
|
|
|
701 |
have "cntP s th' \<noteq> cntV s th'"
|
|
|
702 |
proof
|
|
|
703 |
assume eq_pv: "cntP s th' = cntV s th'"
|
|
|
704 |
from runing_precond_pre[OF th'_in eq_pv neq_th']
|
|
|
705 |
have h1: "th' \<in> threads (t @ s)"
|
|
|
706 |
and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto
|
|
|
707 |
from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .
|
|
|
708 |
with is_runing show "False" by simp
|
|
|
709 |
qed
|
|
|
710 |
moreover from cnp_cnv_cncs[OF vt_s, of th']
|
|
|
711 |
have "cntV s th' \<le> cntP s th'" by auto
|
|
|
712 |
ultimately show ?thesis by auto
|
|
|
713 |
qed
|
|
|
714 |
|
|
|
715 |
lemma moment_blocked_pre:
|
|
|
716 |
assumes neq_th': "th' \<noteq> th"
|
|
|
717 |
and th'_in: "th' \<in> threads ((moment i t)@s)"
|
|
|
718 |
and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
|
|
|
719 |
shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
|
|
|
720 |
th' \<in> threads ((moment (i+j) t)@s)"
|
|
|
721 |
proof(induct j)
|
|
|
722 |
case (Suc k)
|
|
|
723 |
show ?case
|
|
|
724 |
proof -
|
|
|
725 |
{ assume True: "Suc (i+k) \<le> length t"
|
|
|
726 |
from moment_head [OF this]
|
|
|
727 |
obtain e where
|
|
|
728 |
eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"
|
|
|
729 |
by blast
|
|
|
730 |
from red_moment[of "Suc(i+k)"]
|
|
|
731 |
and eq_me have "extend_highest_gen s th prio tm (e # moment (i + k) t)" by simp
|
|
|
732 |
hence vt_e: "vt step (e#(moment (i + k) t)@s)"
|
|
|
733 |
by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
|
|
|
734 |
highest_gen_def, auto)
|
|
|
735 |
have not_runing': "th' \<notin> runing (moment (i + k) t @ s)"
|
|
|
736 |
proof -
|
|
|
737 |
show "th' \<notin> runing (moment (i + k) t @ s)"
|
|
|
738 |
proof(rule extend_highest_gen.pv_blocked)
|
|
|
739 |
from Suc show "th' \<in> threads (moment (i + k) t @ s)"
|
|
|
740 |
by simp
|
|
|
741 |
next
|
|
|
742 |
from neq_th' show "th' \<noteq> th" .
|
|
|
743 |
next
|
|
|
744 |
from red_moment show "extend_highest_gen s th prio tm (moment (i + k) t)" .
|
|
|
745 |
next
|
|
|
746 |
from Suc show "cntP (moment (i + k) t @ s) th' = cntV (moment (i + k) t @ s) th'"
|
|
|
747 |
by (auto)
|
|
|
748 |
qed
|
|
|
749 |
qed
|
|
|
750 |
from step_back_step[OF vt_e]
|
|
|
751 |
have "step ((moment (i + k) t)@s) e" .
|
|
|
752 |
hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>
|
|
|
753 |
th' \<in> threads (e#(moment (i + k) t)@s)
|
|
|
754 |
"
|
|
|
755 |
proof(cases)
|
|
|
756 |
case (thread_create thread prio)
|
|
|
757 |
with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)
|
|
|
758 |
next
|
|
|
759 |
case (thread_exit thread)
|
|
|
760 |
moreover have "thread \<noteq> th'"
|
|
|
761 |
proof -
|
|
|
762 |
have "thread \<in> runing (moment (i + k) t @ s)" by fact
|
|
|
763 |
moreover note not_runing'
|
|
|
764 |
ultimately show ?thesis by auto
|
|
|
765 |
qed
|
|
|
766 |
moreover note Suc
|
|
|
767 |
ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
|
|
|
768 |
next
|
|
|
769 |
case (thread_P thread cs)
|
|
|
770 |
moreover have "thread \<noteq> th'"
|
|
|
771 |
proof -
|
|
|
772 |
have "thread \<in> runing (moment (i + k) t @ s)" by fact
|
|
|
773 |
moreover note not_runing'
|
|
|
774 |
ultimately show ?thesis by auto
|
|
|
775 |
qed
|
|
|
776 |
moreover note Suc
|
|
|
777 |
ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
|
|
|
778 |
next
|
|
|
779 |
case (thread_V thread cs)
|
|
|
780 |
moreover have "thread \<noteq> th'"
|
|
|
781 |
proof -
|
|
|
782 |
have "thread \<in> runing (moment (i + k) t @ s)" by fact
|
|
|
783 |
moreover note not_runing'
|
|
|
784 |
ultimately show ?thesis by auto
|
|
|
785 |
qed
|
|
|
786 |
moreover note Suc
|
|
|
787 |
ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)
|
|
|
788 |
next
|
|
|
789 |
case (thread_set thread prio')
|
|
|
790 |
with Suc show ?thesis
|
|
|
791 |
by (auto simp:cntP_def cntV_def count_def)
|
|
|
792 |
qed
|
|
|
793 |
with eq_me have ?thesis using eq_me by auto
|
|
|
794 |
} note h = this
|
|
|
795 |
show ?thesis
|
|
|
796 |
proof(cases "Suc (i+k) \<le> length t")
|
|
|
797 |
case True
|
|
|
798 |
from h [OF this] show ?thesis .
|
|
|
799 |
next
|
|
|
800 |
case False
|
|
|
801 |
with moment_ge
|
|
|
802 |
have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto
|
|
|
803 |
with Suc show ?thesis by auto
|
|
|
804 |
qed
|
|
|
805 |
qed
|
|
|
806 |
next
|
|
|
807 |
case 0
|
|
|
808 |
from assms show ?case by auto
|
|
|
809 |
qed
|
|
|
810 |
|
|
|
811 |
lemma moment_blocked:
|
|
|
812 |
assumes neq_th': "th' \<noteq> th"
|
|
|
813 |
and th'_in: "th' \<in> threads ((moment i t)@s)"
|
|
|
814 |
and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
|
|
|
815 |
and le_ij: "i \<le> j"
|
|
|
816 |
shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
|
|
|
817 |
th' \<in> threads ((moment j t)@s) \<and>
|
|
|
818 |
th' \<notin> runing ((moment j t)@s)"
|
|
|
819 |
proof -
|
|
|
820 |
from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
|
|
|
821 |
have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
|
|
|
822 |
and h2: "th' \<in> threads ((moment j t)@s)" by auto
|
|
|
823 |
with extend_highest_gen.pv_blocked [OF red_moment [of j], OF h2 neq_th' h1]
|
|
|
824 |
show ?thesis by auto
|
|
|
825 |
qed
|
|
|
826 |
|
|
|
827 |
lemma runing_inversion_1:
|
|
|
828 |
assumes neq_th': "th' \<noteq> th"
|
|
|
829 |
and runing': "th' \<in> runing (t@s)"
|
|
|
830 |
shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
|
|
|
831 |
proof(cases "th' \<in> threads s")
|
|
|
832 |
case True
|
|
|
833 |
with runing_precond [OF this neq_th' runing'] show ?thesis by simp
|
|
|
834 |
next
|
|
|
835 |
case False
|
|
|
836 |
let ?Q = "\<lambda> t. th' \<in> threads (t@s)"
|
|
|
837 |
let ?q = "moment 0 t"
|
|
|
838 |
from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp
|
|
|
839 |
from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)
|
|
|
840 |
from p_split_gen [of ?Q, OF this not_thread]
|
|
|
841 |
obtain i where lt_its: "i < length t"
|
|
|
842 |
and le_i: "0 \<le> i"
|
|
|
843 |
and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
|
|
|
844 |
and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by auto
|
|
|
845 |
from lt_its have "Suc i \<le> length t" by auto
|
|
|
846 |
from moment_head[OF this] obtain e where
|
|
|
847 |
eq_me: "moment (Suc i) t = e # moment i t" by blast
|
|
|
848 |
from red_moment[of "Suc i"] and eq_me
|
|
|
849 |
have "extend_highest_gen s th prio tm (e # moment i t)" by simp
|
|
|
850 |
hence vt_e: "vt step (e#(moment i t)@s)"
|
|
|
851 |
by (unfold extend_highest_gen_def extend_highest_gen_axioms_def
|
|
|
852 |
highest_gen_def, auto)
|
|
|
853 |
from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .
|
|
|
854 |
from post[rule_format, of "Suc i"] and eq_me
|
|
|
855 |
have not_in': "th' \<in> threads (e # moment i t@s)" by auto
|
|
|
856 |
from create_pre[OF stp_i pre this]
|
|
|
857 |
obtain prio where eq_e: "e = Create th' prio" .
|
|
|
858 |
have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
|
|
|
859 |
proof(rule cnp_cnv_eq)
|
|
|
860 |
from step_back_vt [OF vt_e]
|
|
|
861 |
show "vt step (moment i t @ s)" .
|
|
|
862 |
next
|
|
|
863 |
from eq_e and stp_i
|
|
|
864 |
have "step (moment i t @ s) (Create th' prio)" by simp
|
|
|
865 |
thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)
|
|
|
866 |
qed
|
|
|
867 |
with eq_e
|
|
|
868 |
have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'"
|
|
|
869 |
by (simp add:cntP_def cntV_def count_def)
|
|
|
870 |
with eq_me[symmetric]
|
|
|
871 |
have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
|
|
|
872 |
by simp
|
|
|
873 |
from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp
|
|
|
874 |
with eq_me [symmetric]
|
|
|
875 |
have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp
|
|
|
876 |
from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its
|
|
|
877 |
and moment_ge
|
|
|
878 |
have "th' \<notin> runing (t @ s)" by auto
|
|
|
879 |
with runing'
|
|
|
880 |
show ?thesis by auto
|
|
|
881 |
qed
|
|
|
882 |
|
|
|
883 |
lemma runing_inversion_2:
|
|
|
884 |
assumes runing': "th' \<in> runing (t@s)"
|
|
|
885 |
shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
|
|
|
886 |
proof -
|
|
|
887 |
from runing_inversion_1[OF _ runing']
|
|
|
888 |
show ?thesis by auto
|
|
|
889 |
qed
|
|
|
890 |
|
|
|
891 |
lemma live: "runing (t@s) \<noteq> {}"
|
|
|
892 |
proof(cases "th \<in> runing (t@s)")
|
|
|
893 |
case True thus ?thesis by auto
|
|
|
894 |
next
|
|
|
895 |
case False
|
|
|
896 |
then have not_ready: "th \<notin> readys (t@s)"
|
|
|
897 |
apply (unfold runing_def,
|
|
|
898 |
insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])
|
|
|
899 |
by auto
|
|
|
900 |
from th_kept have "th \<in> threads (t@s)" by auto
|
|
|
901 |
from th_chain_to_ready[OF vt_t this] and not_ready
|
|
|
902 |
obtain th' where th'_in: "th' \<in> readys (t@s)"
|
|
|
903 |
and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto
|
|
|
904 |
have "th' \<in> runing (t@s)"
|
|
|
905 |
proof -
|
|
|
906 |
have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"
|
|
|
907 |
proof -
|
|
|
908 |
have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) =
|
|
|
909 |
preced th (t@s)"
|
|
|
910 |
proof(rule Max_eqI)
|
|
|
911 |
fix y
|
|
|
912 |
assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
|
|
|
913 |
then obtain th1 where
|
|
|
914 |
h1: "th1 = th' \<or> th1 \<in> dependents (wq (t @ s)) th'"
|
|
|
915 |
and eq_y: "y = preced th1 (t@s)" by auto
|
|
|
916 |
show "y \<le> preced th (t @ s)"
|
|
|
917 |
proof -
|
|
|
918 |
from max_preced
|
|
|
919 |
have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .
|
|
|
920 |
moreover have "y \<le> \<dots>"
|
|
|
921 |
proof(rule Max_ge)
|
|
|
922 |
from h1
|
|
|
923 |
have "th1 \<in> threads (t@s)"
|
|
|
924 |
proof
|
|
|
925 |
assume "th1 = th'"
|
|
|
926 |
with th'_in show ?thesis by (simp add:readys_def)
|
|
|
927 |
next
|
|
|
928 |
assume "th1 \<in> dependents (wq (t @ s)) th'"
|
|
|
929 |
with dependents_threads [OF vt_t]
|
|
|
930 |
show "th1 \<in> threads (t @ s)" by auto
|
|
|
931 |
qed
|
|
|
932 |
with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp
|
|
|
933 |
next
|
|
|
934 |
from finite_threads[OF vt_t]
|
|
|
935 |
show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp
|
|
|
936 |
qed
|
|
|
937 |
ultimately show ?thesis by auto
|
|
|
938 |
qed
|
|
|
939 |
next
|
|
|
940 |
from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']
|
|
|
941 |
show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"
|
|
|
942 |
by (auto intro:finite_subset)
|
|
|
943 |
next
|
|
|
944 |
from dp
|
|
|
945 |
have "th \<in> dependents (wq (t @ s)) th'"
|
|
|
946 |
by (unfold cs_dependents_def, auto simp:eq_depend)
|
|
|
947 |
thus "preced th (t @ s) \<in>
|
|
|
948 |
(\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"
|
|
|
949 |
by auto
|
|
|
950 |
qed
|
|
|
951 |
moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"
|
|
|
952 |
proof -
|
|
|
953 |
from max_preced and max_cp_eq[OF vt_t, symmetric]
|
|
|
954 |
have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp
|
|
|
955 |
with max_cp_readys_threads[OF vt_t] show ?thesis by simp
|
|
|
956 |
qed
|
|
|
957 |
ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)
|
|
|
958 |
qed
|
|
|
959 |
with th'_in show ?thesis by (auto simp:runing_def)
|
|
|
960 |
qed
|
|
|
961 |
thus ?thesis by auto
|
|
|
962 |
qed
|
|
|
963 |
|
|
|
964 |
end
|
|
|
965 |
end
|
|
|
966 |
|
|
|
967 |
|