theory ExtGG+
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imports PrioG CpsG+
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begin+
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+
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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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+
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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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+
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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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+
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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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+
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context highest_gen+
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begin+
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+
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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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+
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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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+
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(* ccc *)+
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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, unfold eq_cp_s_th, insert highest, simp)+
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+
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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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+
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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[symmetric]+
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  show ?thesis by simp+
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qed+
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+
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end+
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+
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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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+
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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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+
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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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+
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+
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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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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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+
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context extend_highest_gen+
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begin+
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+
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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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+
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+
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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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+
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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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+
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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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+
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  The very essence is to show that precedences, no matter whether they are newly introduced +
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  or modified, are always lower than the one held by @{term "th"},+
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  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+
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    by (unfold the_preced_def, simp)+
−
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 +
−
      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"+
−
        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"+
−
          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 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] +
−
            have "x \<noteq> thread" using h by (cases, auto)+
−
            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) +
−
            finally show ?thesis .+
−
          qed+
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        qed+
−
      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} *}+
−
      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+
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            for @{term "th"} remains the maximum of all living threads along the way. *}+
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      finally show ?thesis .+
−
    qed +
−
  next +
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    case (Exit thread)+
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    show ?thesis (is "Max (?f ` ?A) = ?t")+
−
    proof -+
−
      have "Max (?f ` ?A) = ?f th"+
−
      proof(rule image_Max_eqI)+
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        show "finite ?A" using h_e.finite_threads by auto +
−
      next+
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        show "th \<in> ?A" using h_e.th_kept by auto +
−
      next+
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        show "\<forall>x\<in>?A. ?f x \<le> ?f th"+
−
        proof +
−
          fix x+
−
          assume "x \<in> ?A"+
−
          hence "x \<in> threads (t@s)" by (simp add: Exit) +
−
          hence "?f x \<le> Max (?f ` threads (t@s))" +
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            by (simp add: h_t.finite_threads) +
−
          also have "... \<le> ?f th" +
−
            apply (simp add:Exit the_preced_def preced_def, fold preced_def)+
−
            using Cons.hyps(5) h_t.th_kept the_preced_def by auto+
−
          finally show "?f x \<le> ?f th" .+
−
        qed+
−
      qed+
−
      also have "... = ?t" using h_e.th_kept the_preced_def by auto+
−
      finally show ?thesis .+
−
    qed +
−
  next+
−
    case (P thread cs)+
−
    with Cons+
−
    show ?thesis by (auto simp:preced_def the_preced_def)+
−
  next+
−
    case (V thread cs)+
−
    with Cons+
−
    show ?thesis by (auto simp:preced_def the_preced_def)+
−
  next +
−
    case (Set thread prio')+
−
    show ?thesis (is "Max (?f ` ?A) = ?t")+
−
    proof -+
−
      have "Max (?f ` ?A) = ?f th"+
−
      proof(rule image_Max_eqI)+
−
        show "finite ?A" using h_e.finite_threads by auto +
−
      next+
−
        show "th \<in> ?A" using h_e.th_kept by auto +
−
      next+
−
        show "\<forall>x\<in>?A. ?f x \<le> ?f th"+
−
        proof +
−
          fix x+
−
          assume h: "x \<in> ?A"+
−
          show "?f x \<le> ?f th"+
−
          proof(cases "x = thread")+
−
            case True+
−
            moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"+
−
            proof -+
−
              have "the_preced (t @ s) th = Prc prio tm"  +
−
                using h_t.th_kept preced_th by (simp add:the_preced_def)+
−
              moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto+
−
              ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)+
−
            qed+
−
            ultimately show ?thesis+
−
              by (unfold Set, simp add:the_preced_def preced_def)+
−
          next+
−
            case False+
−
            then have "?f x  = the_preced (t@s) x"+
−
              by (simp add:the_preced_def preced_def Set)+
−
            also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"+
−
              using Set h h_t.finite_threads by auto +
−
            also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) +
−
            finally show ?thesis .+
−
          qed+
−
        qed+
−
      qed+
−
      also have "... = ?t" using h_e.th_kept the_preced_def by auto+
−
      finally show ?thesis .+
−
    qed +
−
  qed+
−
qed+
−
+
−
lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"+
−
  by (insert th_kept max_kept, auto)+
−
+
−
text {*+
−
  The reason behind the following lemma is that:+
−
  Since @{term "cp"} is defined as the maximum precedence +
−
  of those threads contained in the sub-tree of node @{term "Th th"} +
−
  in @{term "RAG (t@s)"}, and all these threads are living threads, and +
−
  @{term "th"} is also among them, the maximum precedence of +
−
  them all must be the one for @{text "th"}.+
−
*}+
−
lemma th_cp_max_preced: +
−
  "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") +
−
proof -+
−
  let ?f = "the_preced (t@s)"+
−
  have "?L = ?f th"+
−
  proof(unfold cp_alt_def, rule image_Max_eqI)+
−
    show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"+
−
    proof -+
−
      have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} = +
−
            the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>+
−
                            (\<exists> th'. n = Th th')}"+
−
      by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)+
−
      moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) +
−
      ultimately show ?thesis by simp+
−
    qed+
−
  next+
−
    show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"+
−
      by (auto simp:subtree_def)+
−
  next+
−
    show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.+
−
               the_preced (t @ s) x \<le> the_preced (t @ s) th"+
−
    proof+
−
      fix th'+
−
      assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"+
−
      hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto+
−
      moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"+
−
        by (meson subtree_Field)+
−
      ultimately have "Th th' \<in> ..." by auto+
−
      hence "th' \<in> threads (t@s)" +
−
      proof+
−
        assume "Th th' \<in> {Th th}"+
−
        thus ?thesis using th_kept by auto +
−
      next+
−
        assume "Th th' \<in> Field (RAG (t @ s))"+
−
        thus ?thesis using vat_t.not_in_thread_isolated by blast +
−
      qed+
−
      thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"+
−
        by (metis Max_ge finite_imageI finite_threads image_eqI +
−
               max_kept th_kept the_preced_def)+
−
    qed+
−
  qed+
−
  also have "... = ?R" by (simp add: max_preced the_preced_def) +
−
  finally show ?thesis .+
−
qed+
−
+
−
lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"+
−
  using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger+
−
+
−
lemma th_cp_preced: "cp (t@s) th = preced th s"+
−
  by (fold max_kept, unfold th_cp_max_preced, simp)+
−
+
−
lemma preced_less:+
−
  assumes th'_in: "th' \<in> threads s"+
−
  and neq_th': "th' \<noteq> th"+
−
  shows "preced th' s < preced th s"+
−
  using assms+
−
by (metis Max.coboundedI finite_imageI highest not_le order.trans +
−
    preced_linorder rev_image_eqI threads_s vat_s.finite_threads +
−
    vat_s.le_cp)+
−
+
−
text {*+
−
  Counting of the number of @{term "P"} and @{term "V"} operations +
−
  is the cornerstone of a large number of the following proofs. +
−
  The reason is that this counting is quite easy to calculate and +
−
  convenient to use in the reasoning. +
−
+
−
  The following lemma shows that the counting controls whether +
−
  a thread is running or not.+
−
*}+
−
+
−
lemma pv_blocked_pre:+
−
  assumes th'_in: "th' \<in> threads (t@s)"+
−
  and neq_th': "th' \<noteq> th"+
−
  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"+
−
  shows "th' \<notin> runing (t@s)"+
−
proof+
−
  assume otherwise: "th' \<in> runing (t@s)"+
−
  show False+
−
  proof -+
−
    have "th' = th"+
−
    proof(rule preced_unique)+
−
      show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")+
−
      proof -+
−
        have "?L = cp (t@s) th'"+
−
          by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)+
−
        also have "... = cp (t @ s) th" using otherwise +
−
          by (metis (mono_tags, lifting) mem_Collect_eq +
−
                    runing_def th_cp_max vat_t.max_cp_readys_threads)+
−
        also have "... = ?R" by (metis th_cp_preced th_kept) +
−
        finally show ?thesis .+
−
      qed+
−
    qed (auto simp: th'_in th_kept)+
−
    moreover have "th' \<noteq> th" using neq_th' .+
−
    ultimately show ?thesis by simp+
−
 qed+
−
qed+
−
+
−
lemmas pv_blocked = pv_blocked_pre[folded detached_eq]+
−
+
−
lemma runing_precond_pre:+
−
  fixes th'+
−
  assumes th'_in: "th' \<in> threads s"+
−
  and eq_pv: "cntP s th' = cntV s th'"+
−
  and neq_th': "th' \<noteq> th"+
−
  shows "th' \<in> threads (t@s) \<and>+
−
         cntP (t@s) th' = cntV (t@s) th'"+
−
proof(induct rule:ind)+
−
  case (Cons e t)+
−
    interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp+
−
    interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp+
−
    show ?case+
−
    proof(cases e)+
−
      case (P thread cs)+
−
      show ?thesis+
−
      proof -+
−
        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"+
−
        proof -+
−
          have "thread \<noteq> th'"+
−
          proof -+
−
            have "step (t@s) (P thread cs)" using Cons P by auto+
−
            thus ?thesis+
−
            proof(cases)+
−
              assume "thread \<in> runing (t@s)"+
−
              moreover have "th' \<notin> runing (t@s)" using Cons(5)+
−
                by (metis neq_th' vat_t.pv_blocked_pre) +
−
              ultimately show ?thesis by auto+
−
            qed+
−
          qed with Cons show ?thesis+
−
            by (unfold P, simp add:cntP_def cntV_def count_def)+
−
        qed+
−
        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)+
−
        ultimately show ?thesis by auto+
−
      qed+
−
    next+
−
      case (V thread cs)+
−
      show ?thesis+
−
      proof -+
−
        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"+
−
        proof -+
−
          have "thread \<noteq> th'"+
−
          proof -+
−
            have "step (t@s) (V thread cs)" using Cons V by auto+
−
            thus ?thesis+
−
            proof(cases)+
−
              assume "thread \<in> runing (t@s)"+
−
              moreover have "th' \<notin> runing (t@s)" using Cons(5)+
−
                by (metis neq_th' vat_t.pv_blocked_pre) +
−
              ultimately show ?thesis by auto+
−
            qed+
−
          qed with Cons show ?thesis+
−
            by (unfold V, simp add:cntP_def cntV_def count_def)+
−
        qed+
−
        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)+
−
        ultimately show ?thesis by auto+
−
      qed+
−
    next+
−
      case (Create thread prio')+
−
      show ?thesis+
−
      proof -+
−
        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"+
−
        proof -+
−
          have "thread \<noteq> th'"+
−
          proof -+
−
            have "step (t@s) (Create thread prio')" using Cons Create by auto+
−
            thus ?thesis using Cons(5) by (cases, auto)+
−
          qed with Cons show ?thesis+
−
            by (unfold Create, simp add:cntP_def cntV_def count_def)+
−
        qed+
−
        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)+
−
        ultimately show ?thesis by auto+
−
      qed+
−
    next+
−
      case (Exit thread)+
−
      show ?thesis+
−
      proof -+
−
        have neq_thread: "thread \<noteq> th'"+
−
        proof -+
−
          have "step (t@s) (Exit thread)" using Cons Exit by auto+
−
          thus ?thesis apply (cases) using Cons(5)+
−
                by (metis neq_th' vat_t.pv_blocked_pre) +
−
        qed +
−
        hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons+
−
            by (unfold Exit, simp add:cntP_def cntV_def count_def)+
−
        moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread +
−
          by (unfold Exit, simp) +
−
        ultimately show ?thesis by auto+
−
      qed+
−
    next+
−
      case (Set thread prio')+
−
      with Cons+
−
      show ?thesis +
−
        by (auto simp:cntP_def cntV_def count_def)+
−
    qed+
−
next+
−
  case Nil+
−
  with assms+
−
  show ?case by auto+
−
qed+
−
+
−
text {* Changing counting balance to detachedness *}+
−
lemmas runing_precond_pre_dtc = runing_precond_pre+
−
         [folded vat_t.detached_eq vat_s.detached_eq]+
−
+
−
lemma runing_precond:+
−
  fixes th'+
−
  assumes th'_in: "th' \<in> threads s"+
−
  and neq_th': "th' \<noteq> th"+
−
  and is_runing: "th' \<in> runing (t@s)"+
−
  shows "cntP s th' > cntV s th'"+
−
  using assms+
−
proof -+
−
  have "cntP s th' \<noteq> cntV s th'"+
−
    by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)+
−
  moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto+
−
  ultimately show ?thesis by auto+
−
qed+
−
+
−
lemma moment_blocked_pre:+
−
  assumes neq_th': "th' \<noteq> th"+
−
  and th'_in: "th' \<in> threads ((moment i t)@s)"+
−
  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"+
−
  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>+
−
         th' \<in> threads ((moment (i+j) t)@s)"+
−
proof -+
−
  interpret h_i: red_extend_highest_gen _ _ _ _ _ i+
−
      by (unfold_locales)+
−
  interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"+
−
      by (unfold_locales)+
−
  interpret h:  extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"+
−
  proof(unfold_locales)+
−
    show "vt (moment i t @ s)" by (metis h_i.vt_t) +
−
  next+
−
    show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)+
−
  next+
−
    show "preced th (moment i t @ s) = +
−
            Max (cp (moment i t @ s) ` threads (moment i t @ s))"+
−
              by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)+
−
  next+
−
    show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th) +
−
  next+
−
    show "vt (moment j (restm i t) @ moment i t @ s)"+
−
      using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)+
−
  next+
−
    fix th' prio'+
−
    assume "Create th' prio' \<in> set (moment j (restm i t))"+
−
    thus "prio' \<le> prio" using assms+
−
       by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)+
−
  next+
−
    fix th' prio'+
−
    assume "Set th' prio' \<in> set (moment j (restm i t))"+
−
    thus "th' \<noteq> th \<and> prio' \<le> prio"+
−
    by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)+
−
  next+
−
    fix th'+
−
    assume "Exit th' \<in> set (moment j (restm i t))"+
−
    thus "th' \<noteq> th"+
−
      by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)+
−
  qed+
−
  show ?thesis +
−
    by (metis add.commute append_assoc eq_pv h.runing_precond_pre+
−
          moment_plus_split neq_th' th'_in)+
−
qed+
−
+
−
lemma moment_blocked_eqpv:+
−
  assumes neq_th': "th' \<noteq> th"+
−
  and th'_in: "th' \<in> threads ((moment i t)@s)"+
−
  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"+
−
  and le_ij: "i \<le> j"+
−
  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>+
−
         th' \<in> threads ((moment j t)@s) \<and>+
−
         th' \<notin> runing ((moment j t)@s)"+
−
proof -+
−
  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij+
−
  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"+
−
   and h2: "th' \<in> threads ((moment j t)@s)" by auto+
−
  moreover have "th' \<notin> runing ((moment j t)@s)"+
−
  proof -+
−
    interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)+
−
    show ?thesis+
−
      using h.pv_blocked_pre h1 h2 neq_th' by auto +
−
  qed+
−
  ultimately show ?thesis by auto+
−
qed+
−
+
−
(* The foregoing two lemmas are preparation for this one, but+
−
   in long run can be combined. Maybe I am wrong.+
−
*)+
−
lemma moment_blocked:+
−
  assumes neq_th': "th' \<noteq> th"+
−
  and th'_in: "th' \<in> threads ((moment i t)@s)"+
−
  and dtc: "detached (moment i t @ s) th'"+
−
  and le_ij: "i \<le> j"+
−
  shows "detached (moment j t @ s) th' \<and>+
−
         th' \<in> threads ((moment j t)@s) \<and>+
−
         th' \<notin> runing ((moment j t)@s)"+
−
proof -+
−
  interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)+
−
  interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales) +
−
  have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"+
−
                by (metis dtc h_i.detached_elim)+
−
  from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]+
−
  show ?thesis by (metis h_j.detached_intro) +
−
qed+
−
+
−
lemma runing_preced_inversion:+
−
  assumes runing': "th' \<in> runing (t@s)"+
−
  shows "cp (t@s) th' = preced th s" (is "?L = ?R")+
−
proof -+
−
  have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms+
−
      by (unfold runing_def, auto)+
−
  also have "\<dots> = ?R"+
−
      by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) +
−
  finally show ?thesis .+
−
qed+
−
+
−
text {*+
−
  The situation when @{term "th"} is blocked is analyzed by the following lemmas.+
−
*}+
−
+
−
text {*+
−
  The following lemmas shows the running thread @{text "th'"}, if it is different from+
−
  @{term th}, must be live at the very beginning. By the term {\em the very beginning},+
−
  we mean the moment where the formal investigation starts, i.e. the moment (or state)+
−
  @{term s}. +
−
*}+
−
+
−
lemma runing_inversion_0:+
−
  assumes neq_th': "th' \<noteq> th"+
−
  and runing': "th' \<in> runing (t@s)"+
−
  shows "th' \<in> threads s"+
−
proof -+
−
    -- {* The proof is by contradiction: *}+
−
    { assume otherwise: "\<not> ?thesis"+
−
      have "th' \<notin> runing (t @ s)"+
−
      proof -+
−
        -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}+
−
        have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)+
−
        -- {* However, @{text "th'"} does not exist at very beginning. *}+
−
        have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise+
−
          by (metis append.simps(1) moment_zero)+
−
        -- {* Therefore, there must be a moment during @{text "t"}, when +
−
              @{text "th'"} came into being. *}+
−
        -- {* Let us suppose the moment being @{text "i"}: *}+
−
        from p_split_gen[OF th'_in th'_notin]+
−
        obtain i where lt_its: "i < length t"+
−
                 and le_i: "0 \<le> i"+
−
                 and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")+
−
                 and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by (auto)+
−
        interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)+
−
        interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)+
−
        from lt_its have "Suc i \<le> length t" by auto+
−
        -- {* Let us also suppose the event which makes this change is @{text e}: *}+
−
        from moment_head[OF this] obtain e where +
−
          eq_me: "moment (Suc i) t = e # moment i t" by blast+
−
        hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t) +
−
        hence "PIP (moment i t @ s) e" by (cases, simp)+
−
        -- {* It can be derived that this event @{text "e"}, which +
−
              gives birth to @{term "th'"} must be a @{term "Create"}: *}+
−
        from create_pre[OF this, of th']+
−
        obtain prio where eq_e: "e = Create th' prio"+
−
            by (metis append_Cons eq_me lessI post pre) +
−
        have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto +
−
        have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"+
−
        proof -+
−
          have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"+
−
            by (metis h_i.cnp_cnv_eq pre)+
−
          thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)+
−
        qed+
−
        show ?thesis +
−
          using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge+
−
            by auto+
−
      qed+
−
      with `th' \<in> runing (t@s)`+
−
      have False by simp+
−
    } thus ?thesis by auto+
−
qed+
−
+
−
text {* +
−
  The second lemma says, if the running thread @{text th'} is different from +
−
  @{term th}, then this @{text th'} must in the possession of some resources+
−
  at the very beginning. +
−
+
−
  To ease the reasoning of resource possession of one particular thread, +
−
  we used two auxiliary functions @{term cntV} and @{term cntP}, +
−
  which are the counters of @{term P}-operations and +
−
  @{term V}-operations respectively. +
−
  If the number of @{term V}-operation is less than the number of +
−
  @{term "P"}-operations, the thread must have some unreleased resource. +
−
*}+
−
+
−
lemma runing_inversion_1: (* ddd *)+
−
  assumes neq_th': "th' \<noteq> th"+
−
  and runing': "th' \<in> runing (t@s)"+
−
  -- {* thread @{term "th'"} is a live on in state @{term "s"} and +
−
        it has some unreleased resource. *}+
−
  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"+
−
proof -+
−
  -- {* The proof is a simple composition of @{thm runing_inversion_0} and +
−
        @{thm runing_precond}: *}+
−
  -- {* By applying @{thm runing_inversion_0} to assumptions,+
−
        it can be shown that @{term th'} is live in state @{term s}: *}+
−
  have "th' \<in> threads s"  using runing_inversion_0[OF assms(1,2)] .+
−
  -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}+
−
  with runing_precond [OF this neq_th' runing'] show ?thesis by simp+
−
qed+
−
+
−
text {* +
−
  The following lemma is just a rephrasing of @{thm runing_inversion_1}:+
−
*}+
−
lemma runing_inversion_2:+
−
  assumes runing': "th' \<in> runing (t@s)"+
−
  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"+
−
proof -+
−
  from runing_inversion_1[OF _ runing']+
−
  show ?thesis by auto+
−
qed+
−
+
−
lemma runing_inversion_3:+
−
  assumes runing': "th' \<in> runing (t@s)"+
−
  and neq_th: "th' \<noteq> th"+
−
  shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"+
−
  by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)+
−
+
−
lemma runing_inversion_4:+
−
  assumes runing': "th' \<in> runing (t@s)"+
−
  and neq_th: "th' \<noteq> th"+
−
  shows "th' \<in> threads s"+
−
  and    "\<not>detached s th'"+
−
  and    "cp (t@s) th' = preced th s"+
−
  apply (metis neq_th runing' runing_inversion_2)+
−
  apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)+
−
  by (metis neq_th runing' runing_inversion_3)+
−
+
−
+
−
text {* +
−
  Suppose @{term th} is not running, it is first shown that+
−
  there is a path in RAG leading from node @{term th} to another thread @{text "th'"} +
−
  in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).+
−
+
−
  Now, since @{term readys}-set is non-empty, there must be+
−
  one in it which holds the highest @{term cp}-value, which, by definition, +
−
  is the @{term runing}-thread. However, we are going to show more: this running thread+
−
  is exactly @{term "th'"}.+
−
     *}+
−
lemma th_blockedE: (* ddd *)+
−
  assumes "th \<notin> runing (t@s)"+
−
  obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"+
−
                    "th' \<in> runing (t@s)"+
−
proof -+
−
  -- {* According to @{thm vat_t.th_chain_to_ready}, either +
−
        @{term "th"} is in @{term "readys"} or there is path leading from it to +
−
        one thread in @{term "readys"}. *}+
−
  have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)" +
−
    using th_kept vat_t.th_chain_to_ready by auto+
−
  -- {* However, @{term th} can not be in @{term readys}, because otherwise, since +
−
       @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}+
−
  moreover have "th \<notin> readys (t@s)" +
−
    using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto +
−
  -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in +
−
        term @{term readys}: *}+
−
  ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"+
−
                          and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto+
−
  -- {* We are going to show that this @{term th'} is running. *}+
−
  have "th' \<in> runing (t@s)"+
−
  proof -+
−
    -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}+
−
    have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")+
−
    proof -+
−
      have "?L =  Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"+
−
        by (unfold cp_alt_def1, simp)+
−
      also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"+
−
      proof(rule image_Max_subset)+
−
        show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)+
−
      next+
−
        show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"+
−
          by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) +
−
      next+
−
        show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp+
−
                    by (unfold tRAG_subtree_eq, auto simp:subtree_def)+
−
      next+
−
        show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =+
−
                      (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")+
−
        proof -+
−
          have "?L = the_preced (t @ s) `  threads (t @ s)" +
−
                     by (unfold image_comp, rule image_cong, auto)+
−
          thus ?thesis using max_preced the_preced_def by auto+
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        qed+
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      qed+
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      also have "... = ?R"+
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        using th_cp_max th_cp_preced th_kept +
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              the_preced_def vat_t.max_cp_readys_threads by auto+
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      finally show ?thesis .+
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    qed +
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    -- {* Now, since @{term th'} holds the highest @{term cp} +
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          and we have already show it is in @{term readys},+
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          it is @{term runing} by definition. *}+
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    with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def) +
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  qed+
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  -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}+
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  moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" +
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    using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)+
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  ultimately show ?thesis using that by metis+
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qed+
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+
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text {*+
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  Now it is easy to see there is always a thread to run by case analysis+
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  on whether thread @{term th} is running: if the answer is Yes, the +
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  the running thread is obviously @{term th} itself; otherwise, the running+
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  thread is the @{text th'} given by lemma @{thm th_blockedE}.+
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*}+
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lemma live: "runing (t@s) \<noteq> {}"+
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proof(cases "th \<in> runing (t@s)") +
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  case True thus ?thesis by auto+
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next+
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  case False+
−
  thus ?thesis using th_blockedE by auto+
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qed+
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+
−
end+
−
end+
−
+
−
+
−
+
−