theory ExtSG+
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imports PrioG+
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begin+
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+
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locale highest_set =+
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  fixes s' th prio fixes s +
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  defines s_def : "s \<equiv> (Set th prio#s')"+
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  assumes vt_s: "vt step s"+
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  and highest: "preced th s = Max ((cp s)`threads s)"+
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+
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context highest_set+
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begin+
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+
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lemma vt_s': "vt step s'"+
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  by (insert vt_s, unfold s_def, drule_tac step_back_vt, simp)+
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+
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lemma step_set: "step s' (Set th prio)"+
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  by (insert vt_s, unfold s_def, drule_tac step_back_step, simp)+
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+
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lemma step_set_elim: +
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  "\<lbrakk>\<lbrakk>th \<in> runing s'\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"+
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  by (insert step_set, ind_cases "step s' (Set th prio)", auto)+
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+
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+
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lemma threads_s: "th \<in> threads s"+
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  by (rule step_set_elim, unfold runing_def readys_def, auto simp:s_def)+
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+
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lemma same_depend: "depend s = depend s'"+
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  by (insert depend_set_unchanged, unfold s_def, simp)+
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+
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lemma same_dependents:+
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  "dependents (wq s) th = dependents (wq s') th"+
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  apply (unfold cs_dependents_def)+
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  by (unfold eq_depend same_depend, simp)+
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+
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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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+
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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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+
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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 is_ready: "th \<in> readys s"+
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proof -+
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  have "\<forall>cs. \<not> waiting s th cs"+
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    apply (rule step_set_elim, unfold s_def, insert depend_set_unchanged[of th prio s'])+
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    apply (unfold s_depend_def, unfold runing_def readys_def)+
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    apply (auto, fold s_def)+
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    apply (erule_tac x = cs in allE, auto simp:waiting_eq)+
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  proof -+
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    fix cs+
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    assume h: +
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      "{(Th t, Cs c) |t c. waiting (wq s) t c} \<union> {(Cs c, Th t) |c t. holding (wq s) t c} =+
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          {(Th t, Cs c) |t c. waiting (wq s') t c} \<union> {(Cs c, Th t) |c t. holding (wq s') t c}"+
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            (is "?L = ?R")+
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    and wt: "waiting (wq s) th cs" and nwt: "\<not> waiting (wq s') th cs"+
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    from wt have "(Th th, Cs cs) \<in> ?L" by auto+
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    with h have "(Th th, Cs cs) \<in> ?R" by simp+
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    hence "waiting (wq s') th cs" by auto with nwt+
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    show False by auto+
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  qed    +
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  with threads_s show ?thesis +
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    by (unfold readys_def, auto)+
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qed+
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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[OF vt_s, symmetric]+
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  show ?thesis by simp+
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qed+
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+
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lemma is_runing: "th \<in> runing s"+
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proof -+
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  have "Max (cp s ` threads s) = Max (cp s ` readys s)"+
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  proof -+
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    have " Max (cp s ` readys s) = cp s th"+
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    proof(rule Max_eqI)+
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      from finite_threads[OF vt_s] readys_threads finite_subset+
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      have "finite (readys s)" by blast+
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      thus "finite (cp s ` readys s)" by auto+
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    next+
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      from is_ready show "cp s th \<in> cp s ` readys s" by auto+
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    next+
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      fix y+
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      assume "y \<in> cp s ` readys s"+
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      then obtain th1 where +
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        eq_y: "y = cp s th1" and th1_in: "th1 \<in> readys s" by auto+
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      show  "y \<le> cp s th" +
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      proof -+
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        have "y \<le> Max (cp s ` threads s)"+
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        proof(rule Max_ge)+
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          from eq_y and th1_in+
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          show "y \<in> cp s ` threads s"+
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            by (auto simp:readys_def)+
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        next+
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          from finite_threads[OF vt_s]+
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          show "finite (cp s ` threads s)" by auto+
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        qed+
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        with highest' show ?thesis by auto+
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      qed+
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    qed+
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    with highest' show ?thesis by auto+
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  qed+
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  thus ?thesis+
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    by (unfold runing_def, insert highest' is_ready, auto)+
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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_set = highest_set + +
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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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+
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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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+
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context extend_highest_set+
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begin+
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+
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lemma red_moment:+
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  "extend_highest_set s' th prio (moment i t)"+
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  apply (insert extend_highest_set_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])+
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  apply (unfold extend_highest_set_def extend_highest_set_axioms_def, clarsimp)+
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  by (unfold highest_set_def, auto dest:step_back_vt_app)+
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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 step (t@s); step (t@s) e; +
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                    extend_highest_set s' th prio t; +
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                    extend_highest_set s' th prio (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_set_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_set s' th prio 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_set s' th prio (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_set s' th prio t'"+
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          by (unfold extend_highest_set_def extend_highest_set_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_set s' th prio (e # t')" .+
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    next+
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      from et show ext': "extend_highest_set s' th prio t'"+
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          by (unfold extend_highest_set_def extend_highest_set_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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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 "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+
−
    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_set s' th prio (e # t)" by auto+
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      from extend_highest_set.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+
−
      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'"+
−
      show ?thesis+
−
      proof -+
−
        from Cons have "extend_highest_set s' th prio (e # t)" by auto+
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        from extend_highest_set.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+
−
          by (auto simp:eq_e)+
−
      qed+
−
    qed+
−
  qed+
−
qed+
−
+
−
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+
−
  from highest_preced_thread+
−
  show "Max ((\<lambda>th'. preced th' ([] @ s)) ` threads ([] @ s)) = preced th s"+
−
    by simp+
−
next+
−
  case (Cons e t)+
−
  show ?case+
−
  proof(cases e)+
−
    case (Create thread prio')+
−
    assume eq_e: " e = Create thread prio'"+
−
    from Cons and eq_e have stp: "step (t@s) (Create thread prio')" by auto+
−
    hence neq_thread: "thread \<noteq> th"+
−
    proof(cases)+
−
      assume "thread \<notin> threads (t @ s)"+
−
      moreover have "th \<in> threads (t@s)"+
−
      proof -+
−
        from Cons have "extend_highest_set s' th prio t" by auto+
−
        from extend_highest_set.th_kept[OF this] show ?thesis by (simp add:s_def)+
−
      qed+
−
      ultimately show ?thesis by auto+
−
    qed+
−
    from Cons have "extend_highest_set s' th prio t" by auto+
−
    from extend_highest_set.th_kept[OF this]+
−
    have h': " th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" +
−
      by (auto simp:s_def)+
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    from stp+
−
    have thread_ts: "thread \<notin> threads (t @ s)"+
−
      by (cases, auto)+
−
    show ?thesis (is "Max (?f ` ?A) = ?t")+
−
    proof -+
−
      have "Max (?f ` ?A) = Max(insert (?f thread) (?f ` (threads (t@s))))"+
−
        by (unfold eq_e, simp)+
−
      moreover have "\<dots> = max (?f thread) (Max (?f ` (threads (t@s))))"+
−
      proof(rule Max_insert)+
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        from Cons have "vt step (t @ s)" by auto+
−
        from finite_threads[OF this]+
−
        show "finite (?f ` (threads (t@s)))" by simp+
−
      next+
−
        from h' show "(?f ` (threads (t@s))) \<noteq> {}" by auto+
−
      qed+
−
      moreover have "(Max (?f ` (threads (t@s)))) = ?t"+
−
      proof -+
−
        have "(\<lambda>th'. preced th' ((e # t) @ s)) ` threads (t @ s) = +
−
          (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" (is "?f1 ` ?B = ?f2 ` ?B")+
−
        proof -+
−
          { fix th' +
−
            assume "th' \<in> ?B"+
−
            with thread_ts eq_e+
−
            have "?f1 th' = ?f2 th'" by (auto simp:preced_def)+
−
          } thus ?thesis +
−
            apply (auto simp:Image_def)+
−
          proof -+
−
            fix th'+
−
            assume h: "\<And>th'. th' \<in> threads (t @ s) \<Longrightarrow> +
−
              preced th' (e # t @ s) = preced th' (t @ s)"+
−
              and h1: "th' \<in> threads (t @ s)"+
−
            show "preced th' (t @ s) \<in> (\<lambda>th'. preced th' (e # t @ s)) ` threads (t @ s)"+
−
            proof -+
−
              from h1 have "?f1 th' \<in> ?f1 ` ?B" by auto+
−
              moreover from h[OF h1] have "?f1 th' = ?f2 th'" by simp+
−
              ultimately show ?thesis by simp+
−
            qed+
−
          qed+
−
        qed+
−
        with Cons show ?thesis by auto+
−
      qed+
−
      moreover have "?f thread < ?t"+
−
      proof -+
−
        from Cons have " extend_highest_set s' th prio (e # t)" by auto+
−
        from extend_highest_set.create_low[OF this] and eq_e+
−
        have "prio' \<le> prio" by auto+
−
        thus ?thesis+
−
        by (unfold eq_e, auto simp:preced_def s_def precedence_less_def)+
−
    qed+
−
    ultimately show ?thesis by (auto simp:max_def)+
−
  qed+
−
next+
−
    case (Exit thread)+
−
    assume eq_e: "e = Exit thread"+
−
    from Cons have vt_e: "vt step (e#(t @ s))" by auto+
−
    from Cons and eq_e have stp: "step (t@s) (Exit thread)" by auto+
−
    from stp have thread_ts: "thread \<in> threads (t @ s)"+
−
      by(cases, unfold runing_def readys_def, auto)+
−
    from Cons have "extend_highest_set s' th prio (e # t)" by auto+
−
    from extend_highest_set.exit_diff[OF this] and eq_e+
−
    have neq_thread: "thread \<noteq> th" by auto+
−
    from Cons have "extend_highest_set s' th prio t" by auto+
−
    from extend_highest_set.th_kept[OF this, folded s_def]+
−
    have h': "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .+
−
    show ?thesis (is "Max (?f ` ?A) = ?t")+
−
    proof -+
−
      have "threads (t@s) = insert thread ?A"+
−
        by (insert stp thread_ts, unfold eq_e, auto)+
−
      hence "Max (?f ` (threads (t@s))) = Max (?f ` \<dots>)" by simp+
−
      also from this have "\<dots> = Max (insert (?f thread) (?f ` ?A))" by simp+
−
      also have "\<dots> = max (?f thread) (Max (?f ` ?A))"+
−
      proof(rule Max_insert)+
−
        from finite_threads [OF vt_e]+
−
        show "finite (?f ` ?A)" by simp+
−
      next+
−
        from Cons have "extend_highest_set s' th prio (e # t)" by auto+
−
        from extend_highest_set.th_kept[OF this]+
−
        show "?f ` ?A \<noteq> {}" by  (auto simp:s_def)+
−
      qed+
−
      finally have "Max (?f ` (threads (t@s))) = max (?f thread) (Max (?f ` ?A))" .+
−
      moreover have "Max (?f ` (threads (t@s))) = ?t"+
−
      proof -+
−
        from Cons show ?thesis+
−
          by (unfold eq_e, auto simp:preced_def)+
−
      qed+
−
      ultimately have "max (?f thread) (Max (?f ` ?A)) = ?t" by simp+
−
      moreover have "?f thread < ?t"+
−
      proof(unfold eq_e, simp add:preced_def, fold preced_def)+
−
        show "preced thread (t @ s) < ?t"+
−
        proof -+
−
          have "preced thread (t @ s) \<le> ?t" +
−
          proof -+
−
            from Cons+
−
            have "?t = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" +
−
              (is "?t = Max (?g ` ?B)") by simp+
−
            moreover have "?g thread \<le> \<dots>"+
−
            proof(rule Max_ge)+
−
              have "vt step (t@s)" by fact+
−
              from finite_threads [OF this]+
−
              show "finite (?g ` ?B)" by simp+
−
            next+
−
              from thread_ts+
−
              show "?g thread \<in> (?g ` ?B)" by auto+
−
            qed+
−
            ultimately show ?thesis by auto+
−
          qed+
−
          moreover have "preced thread (t @ s) \<noteq> ?t"+
−
          proof+
−
            assume "preced thread (t @ s) = preced th s"+
−
            with h' have "preced thread (t @ s) = preced th (t@s)" by simp+
−
            from preced_unique [OF this] have "thread = th"+
−
            proof+
−
              from h' show "th \<in> threads (t @ s)" by simp+
−
            next+
−
              from thread_ts show "thread \<in> threads (t @ s)" .+
−
            qed(simp)+
−
            with neq_thread show "False" by simp+
−
          qed+
−
          ultimately show ?thesis by auto+
−
        qed+
−
      qed+
−
      ultimately show ?thesis +
−
        by (auto simp:max_def split:if_splits)+
−
    qed+
−
  next+
−
    case (P thread cs)+
−
    with Cons+
−
    show ?thesis by (auto simp:preced_def)+
−
  next+
−
    case (V thread cs)+
−
    with Cons+
−
    show ?thesis by (auto simp:preced_def)+
−
  next+
−
    case (Set thread prio')+
−
    show ?thesis (is "Max (?f ` ?A) = ?t")+
−
    proof -+
−
      let ?B = "threads (t@s)"+
−
      from Cons have "extend_highest_set s' th prio (e # t)" by auto+
−
      from extend_highest_set.set_diff_low[OF this] and Set+
−
      have neq_thread: "thread \<noteq> th" and le_p: "prio' \<le> prio" by auto+
−
      from Set have "Max (?f ` ?A) = Max (?f ` ?B)" by simp+
−
      also have "\<dots> = ?t"+
−
      proof(rule Max_eqI)+
−
        fix y+
−
        assume y_in: "y \<in> ?f ` ?B"+
−
        then obtain th1 where +
−
          th1_in: "th1 \<in> ?B" and eq_y: "y = ?f th1" by auto+
−
        show "y \<le> ?t"+
−
        proof(cases "th1 = thread")+
−
          case True+
−
          with neq_thread le_p eq_y s_def Set+
−
          show ?thesis+
−
            by (auto simp:preced_def precedence_le_def)+
−
        next+
−
          case False+
−
          with Set eq_y+
−
          have "y  = preced th1 (t@s)"+
−
            by (simp add:preced_def)+
−
          moreover have "\<dots> \<le> ?t"+
−
          proof -+
−
            from Cons+
−
            have "?t = Max ((\<lambda> th'. preced th' (t@s)) ` (threads (t@s)))"+
−
              by auto+
−
            moreover have "preced th1 (t@s) \<le> \<dots>"+
−
            proof(rule Max_ge)+
−
              from th1_in +
−
              show "preced th1 (t @ s) \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)"+
−
                by simp+
−
            next+
−
              show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"+
−
              proof -+
−
                from Cons have "vt step (t @ s)" by auto+
−
                from finite_threads[OF this] show ?thesis by auto+
−
              qed+
−
            qed+
−
            ultimately show ?thesis by auto+
−
          qed+
−
          ultimately show ?thesis by auto+
−
        qed+
−
      next+
−
        from Cons and finite_threads+
−
        show "finite (?f ` ?B)" by auto+
−
      next+
−
        from Cons have "extend_highest_set s' th prio t" by auto+
−
        from extend_highest_set.th_kept [OF this, folded s_def]+
−
        have h: "th \<in> threads (t @ s) \<and> preced th (t @ s) = preced th s" .+
−
        show "?t \<in> (?f ` ?B)" +
−
        proof -+
−
          from neq_thread Set h+
−
          have "?t = ?f th" by (auto simp:preced_def)+
−
          with h show ?thesis by auto+
−
        qed+
−
      qed+
−
      finally show ?thesis .+
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma max_preced: "preced th (t@s) = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))"+
−
  by (insert th_kept max_kept, auto)+
−
+
−
lemma th_cp_max_preced: "cp (t@s) th = Max ((\<lambda> th'. preced th' (t @ s)) ` (threads (t@s)))" +
−
  (is "?L = ?R")+
−
proof -+
−
  have "?L = cpreced (t@s) (wq (t@s)) th" +
−
    by (unfold cp_eq_cpreced, simp)+
−
  also have "\<dots> = ?R"+
−
  proof(unfold cpreced_def)+
−
    show "Max ((\<lambda>th. preced th (t @ s)) ` ({th} \<union> dependents (wq (t @ s)) th)) =+
−
          Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))"+
−
      (is "Max (?f ` ({th} \<union> ?A)) = Max (?f ` ?B)")+
−
    proof(cases "?A = {}")+
−
      case False+
−
      have "Max (?f ` ({th} \<union> ?A)) = Max (insert (?f th) (?f ` ?A))" by simp+
−
      moreover have "\<dots> = max (?f th) (Max (?f ` ?A))"+
−
      proof(rule Max_insert)+
−
        show "finite (?f ` ?A)"+
−
        proof -+
−
          from dependents_threads[OF vt_t]+
−
          have "?A \<subseteq> threads (t@s)" .+
−
          moreover from finite_threads[OF vt_t] have "finite \<dots>" .+
−
          ultimately show ?thesis +
−
            by (auto simp:finite_subset)+
−
        qed+
−
      next+
−
        from False show "(?f ` ?A) \<noteq> {}" by simp+
−
      qed+
−
      moreover have "\<dots> = Max (?f ` ?B)"+
−
      proof -+
−
        from max_preced have "?f th = Max (?f ` ?B)" .+
−
        moreover have "Max (?f ` ?A) \<le> \<dots>" +
−
        proof(rule Max_mono)+
−
          from False show "(?f ` ?A) \<noteq> {}" by simp+
−
        next+
−
          show "?f ` ?A \<subseteq> ?f ` ?B" +
−
          proof -+
−
            have "?A \<subseteq> ?B" by (rule dependents_threads[OF vt_t])+
−
            thus ?thesis by auto+
−
          qed+
−
        next+
−
          from finite_threads[OF vt_t] +
−
          show "finite (?f ` ?B)" by simp+
−
        qed+
−
        ultimately show ?thesis+
−
          by (auto simp:max_def)+
−
      qed+
−
      ultimately show ?thesis by auto+
−
    next+
−
      case True+
−
      with max_preced show ?thesis by auto+
−
    qed+
−
  qed+
−
  finally show ?thesis .+
−
qed+
−
+
−
lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"+
−
  by (unfold max_cp_eq[OF vt_t] th_cp_max_preced, simp)+
−
+
−
lemma th_cp_preced: "cp (t@s) th = preced th s"+
−
  by (fold max_kept, unfold th_cp_max_preced, simp)+
−
+
−
lemma preced_less':+
−
  fixes th'+
−
  assumes th'_in: "th' \<in> threads s"+
−
  and neq_th': "th' \<noteq> th"+
−
  shows "preced th' s < preced th s"+
−
proof -+
−
  have "preced th' s \<le> Max ((\<lambda>th'. preced th' s) ` threads s)"+
−
  proof(rule Max_ge)+
−
    from finite_threads [OF vt_s]+
−
    show "finite ((\<lambda>th'. preced th' s) ` threads s)" by simp+
−
  next+
−
    from th'_in show "preced th' s \<in> (\<lambda>th'. preced th' s) ` threads s"+
−
      by simp+
−
  qed+
−
  moreover have "preced th' s \<noteq> preced th s"+
−
  proof+
−
    assume "preced th' s = preced th s"+
−
    from preced_unique[OF this th'_in] neq_th' is_ready+
−
    show "False" by  (auto simp:readys_def)+
−
  qed+
−
  ultimately show ?thesis using highest_preced_thread+
−
    by auto+
−
qed+
−
+
−
lemma pv_blocked:+
−
  fixes th'+
−
  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 "th' \<in> runing (t@s)"+
−
  hence "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" +
−
    by (auto simp:runing_def)+
−
  with max_cp_readys_threads [OF vt_t]+
−
  have "cp (t @ s) th' = Max (cp (t@s) ` threads (t@s))"+
−
    by auto+
−
  moreover from th_cp_max have "cp (t @ s) th = \<dots>" by simp+
−
  ultimately have "cp (t @ s) th' = cp (t @ s) th" by simp+
−
  moreover from th_cp_preced and th_kept have "\<dots> = preced th (t @ s)"+
−
    by simp+
−
  finally have h: "cp (t @ s) th' = preced th (t @ s)" .+
−
  show False+
−
  proof -+
−
    have "dependents (wq (t @ s)) th' = {}" +
−
      by (rule count_eq_dependents [OF vt_t eq_pv])+
−
    moreover have "preced th' (t @ s) \<noteq> preced th (t @ s)"+
−
    proof+
−
      assume "preced th' (t @ s) = preced th (t @ s)"+
−
      hence "th' = th"+
−
      proof(rule preced_unique)+
−
        from th_kept show "th \<in> threads (t @ s)" by simp+
−
      next+
−
        from th'_in show "th' \<in> threads (t @ s)" by simp+
−
      qed+
−
      with assms show False by simp+
−
    qed+
−
    ultimately show ?thesis+
−
      by (insert h, unfold cp_eq_cpreced cpreced_def, simp)+
−
  qed+
−
qed+
−
+
−
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 -+
−
  show ?thesis+
−
  proof(induct rule:ind)+
−
    case (Cons e t)+
−
    from Cons+
−
    have in_thread: "th' \<in> threads (t @ s)"+
−
      and not_holding: "cntP (t @ s) th' = cntV (t @ s) th'" by auto+
−
    have "extend_highest_set s' th prio t" by fact+
−
    from extend_highest_set.pv_blocked +
−
    [OF this, folded s_def, OF in_thread neq_th' not_holding]+
−
    have not_runing: "th' \<notin> runing (t @ s)" .+
−
    show ?case+
−
    proof(cases e)+
−
      case (V thread cs)+
−
      from Cons and V have vt_v: "vt step (V thread cs#(t@s))" by auto+
−
+
−
      show ?thesis+
−
      proof -+
−
        from Cons and V have "step (t@s) (V thread cs)" by auto+
−
        hence neq_th': "thread \<noteq> th'"+
−
        proof(cases)+
−
          assume "thread \<in> runing (t@s)"+
−
          moreover have "th' \<notin> runing (t@s)" by fact+
−
          ultimately show ?thesis by auto+
−
        qed+
−
        with not_holding have cnt_eq: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" +
−
          by (unfold V, simp add:cntP_def cntV_def count_def)+
−
        moreover from in_thread+
−
        have in_thread': "th' \<in> threads ((e # t) @ s)" by (unfold V, simp)+
−
        ultimately show ?thesis by auto+
−
      qed+
−
    next+
−
      case (P thread cs)+
−
      from Cons and P have "step (t@s) (P thread cs)" by auto+
−
      hence neq_th': "thread \<noteq> th'"+
−
      proof(cases)+
−
        assume "thread \<in> runing (t@s)"+
−
        moreover note not_runing+
−
        ultimately show ?thesis by auto+
−
      qed+
−
      with Cons and P have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"+
−
        by (auto simp:cntP_def cntV_def count_def)+
−
      moreover from Cons and P have in_thread': "th' \<in> threads ((e # t) @ s)"+
−
        by auto+
−
      ultimately show ?thesis by auto+
−
    next+
−
      case (Create thread prio')+
−
      from Cons and Create have "step (t@s) (Create thread prio')" by auto+
−
      hence neq_th': "thread \<noteq> th'"+
−
      proof(cases)+
−
        assume "thread \<notin> threads (t @ s)"+
−
        moreover have "th' \<in> threads (t@s)" by fact+
−
        ultimately show ?thesis by auto+
−
      qed+
−
      with Cons and Create +
−
      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"+
−
        by (auto simp:cntP_def cntV_def count_def)+
−
      moreover from Cons and Create +
−
      have in_thread': "th' \<in> threads ((e # t) @ s)" by auto+
−
      ultimately show ?thesis by auto+
−
    next+
−
      case (Exit thread)+
−
      from Cons and Exit have "step (t@s) (Exit thread)" by auto+
−
      hence neq_th': "thread \<noteq> th'"+
−
      proof(cases)+
−
        assume "thread \<in> runing (t @ s)"+
−
        moreover note not_runing+
−
        ultimately show ?thesis by auto+
−
      qed+
−
      with Cons and Exit +
−
      have eq_cnt: "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"+
−
        by (auto simp:cntP_def cntV_def count_def)+
−
      moreover from Cons and Exit and neq_th' +
−
      have in_thread': "th' \<in> threads ((e # t) @ s)"+
−
        by auto+
−
      ultimately show ?thesis by auto+
−
    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+
−
qed+
−
+
−
(*+
−
lemma runing_precond:+
−
  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' \<notin> runing (t@s)"+
−
proof -+
−
  from runing_precond_pre[OF th'_in eq_pv neq_th']+
−
  have h1: "th' \<in> threads (t @ s)"  and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto+
−
  from pv_blocked[OF h1 neq_th' h2] +
−
  show ?thesis .+
−
qed+
−
*)+
−
+
−
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'"+
−
proof -+
−
  have "cntP s th' \<noteq> cntV s th'"+
−
  proof+
−
    assume eq_pv: "cntP s th' = cntV s th'"+
−
    from runing_precond_pre[OF th'_in eq_pv neq_th']+
−
    have h1: "th' \<in> threads (t @ s)"  +
−
      and h2: "cntP (t @ s) th' = cntV (t @ s) th'" by auto+
−
    from pv_blocked[OF h1 neq_th' h2] have " th' \<notin> runing (t @ s)" .+
−
    with is_runing show "False" by simp+
−
  qed+
−
  moreover from cnp_cnv_cncs[OF vt_s, of th'] +
−
  have "cntV s th' \<le> cntP s th'" 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(induct j)+
−
  case (Suc k)+
−
  show ?case+
−
  proof -+
−
    { assume True: "Suc (i+k) \<le> length t"+
−
      from moment_head [OF this] +
−
      obtain e where+
−
        eq_me: "moment (Suc(i+k)) t = e#(moment (i+k) t)"+
−
        by blast+
−
      from red_moment[of "Suc(i+k)"]+
−
      and eq_me have "extend_highest_set s' th prio (e # moment (i + k) t)" by simp+
−
      hence vt_e: "vt step (e#(moment (i + k) t)@s)"+
−
        by (unfold extend_highest_set_def extend_highest_set_axioms_def +
−
                          highest_set_def s_def, auto)+
−
      have not_runing': "th' \<notin>  runing (moment (i + k) t @ s)"+
−
      proof(unfold s_def)+
−
        show "th' \<notin> runing (moment (i + k) t @ Set th prio # s')"+
−
        proof(rule extend_highest_set.pv_blocked)+
−
          from Suc show "th' \<in> threads (moment (i + k) t @ Set th prio # s')"+
−
            by (simp add:s_def)+
−
        next+
−
          from neq_th' show "th' \<noteq> th" .+
−
        next+
−
          from red_moment show "extend_highest_set s' th prio (moment (i + k) t)" .+
−
        next+
−
          from Suc show "cntP (moment (i + k) t @ Set th prio # s') th' =+
−
            cntV (moment (i + k) t @ Set th prio # s') th'"+
−
            by (auto simp:s_def)+
−
        qed+
−
      qed+
−
      from step_back_step[OF vt_e]+
−
      have "step ((moment (i + k) t)@s) e" .+
−
      hence "cntP (e#(moment (i + k) t)@s) th' = cntV (e#(moment (i + k) t)@s) th' \<and>+
−
        th' \<in> threads (e#(moment (i + k) t)@s)+
−
        "+
−
      proof(cases)+
−
        case (thread_create thread prio)+
−
        with Suc show ?thesis by (auto simp:cntP_def cntV_def count_def)+
−
      next+
−
        case (thread_exit thread)+
−
        moreover have "thread \<noteq> th'"+
−
        proof -+
−
          have "thread \<in> runing (moment (i + k) t @ s)" by fact+
−
          moreover note not_runing'+
−
          ultimately show ?thesis by auto+
−
        qed+
−
        moreover note Suc +
−
        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)+
−
      next+
−
        case (thread_P thread cs)+
−
        moreover have "thread \<noteq> th'"+
−
        proof -+
−
          have "thread \<in> runing (moment (i + k) t @ s)" by fact+
−
          moreover note not_runing'+
−
          ultimately show ?thesis by auto+
−
        qed+
−
        moreover note Suc +
−
        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)+
−
      next+
−
        case (thread_V thread cs)+
−
        moreover have "thread \<noteq> th'"+
−
        proof -+
−
          have "thread \<in> runing (moment (i + k) t @ s)" by fact+
−
          moreover note not_runing'+
−
          ultimately show ?thesis by auto+
−
        qed+
−
        moreover note Suc +
−
        ultimately show ?thesis by (auto simp:cntP_def cntV_def count_def)+
−
      next+
−
        case (thread_set thread prio')+
−
        with Suc show ?thesis+
−
          by (auto simp:cntP_def cntV_def count_def)+
−
      qed+
−
      with eq_me have ?thesis using eq_me by auto +
−
    } note h = this+
−
    show ?thesis+
−
    proof(cases "Suc (i+k) \<le> length t")+
−
      case True+
−
      from h [OF this] show ?thesis .+
−
    next+
−
      case False+
−
      with moment_ge+
−
      have eq_m: "moment (i + Suc k) t = moment (i+k) t" by auto+
−
      with Suc show ?thesis by auto+
−
    qed+
−
  qed+
−
next+
−
  case 0+
−
  from assms show ?case by auto+
−
qed+
−
+
−
lemma moment_blocked:+
−
  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+
−
  with extend_highest_set.pv_blocked [OF  red_moment [of j], folded s_def, OF h2 neq_th' h1]+
−
  show ?thesis by auto+
−
qed+
−
+
−
lemma runing_inversion_1:+
−
  assumes neq_th': "th' \<noteq> th"+
−
  and runing': "th' \<in> runing (t@s)"+
−
  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"+
−
proof(cases "th' \<in> threads s")+
−
  case True+
−
  with runing_precond [OF this neq_th' runing'] show ?thesis by simp+
−
next+
−
  case False+
−
  let ?Q = "\<lambda> t. th' \<in> threads (t@s)"+
−
  let ?q = "moment 0 t"+
−
  from moment_eq and False have not_thread: "\<not> ?Q ?q" by simp+
−
  from runing' have "th' \<in> threads (t@s)" by (simp add:runing_def readys_def)+
−
  from p_split_gen [of ?Q, OF this not_thread]+
−
  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+
−
  from lt_its have "Suc i \<le> length t" by auto+
−
  from moment_head[OF this] obtain e where +
−
   eq_me: "moment (Suc i) t = e # moment i t" by blast+
−
  from red_moment[of "Suc i"] and eq_me+
−
  have "extend_highest_set s' th prio (e # moment i t)" by simp+
−
  hence vt_e: "vt step (e#(moment i t)@s)"+
−
    by (unfold extend_highest_set_def extend_highest_set_axioms_def +
−
      highest_set_def s_def, auto)+
−
  from step_back_step[OF this] have stp_i: "step (moment i t @ s) e" .+
−
  from post[rule_format, of "Suc i"] and eq_me +
−
  have not_in': "th' \<in> threads (e # moment i t@s)" by auto+
−
  from create_pre[OF stp_i pre this] +
−
  obtain prio where eq_e: "e = Create th' prio" .+
−
  have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"+
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  proof(rule cnp_cnv_eq)+
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    from step_back_vt [OF vt_e] +
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    show "vt step (moment i t @ s)" .+
−
  next+
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    from eq_e and stp_i +
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    have "step (moment i t @ s) (Create th' prio)" by simp+
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    thus "th' \<notin> threads (moment i t @ s)" by (cases, simp)+
−
  qed+
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  with eq_e+
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  have "cntP ((e#moment i t)@s) th' = cntV ((e#moment i t)@s) th'" +
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    by (simp add:cntP_def cntV_def count_def)+
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  with eq_me[symmetric]+
−
  have h1: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"+
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    by simp+
−
  from eq_e have "th' \<in> threads ((e#moment i t)@s)" by simp+
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  with eq_me [symmetric]+
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  have h2: "th' \<in> threads (moment (Suc i) t @ s)" by simp+
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  from moment_blocked [OF neq_th' h2 h1, of "length t"] and lt_its+
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  and moment_ge+
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  have "th' \<notin> runing (t @ s)" by auto+
−
  with runing'+
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  show ?thesis by auto+
−
qed+
−
+
−
lemma runing_inversion_2:+
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  assumes runing': "th' \<in> runing (t@s)"+
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  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+
−
+
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lemma live: "runing (t@s) \<noteq> {}"+
−
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+
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  then have not_ready: "th \<notin> readys (t@s)"+
−
    apply (unfold runing_def, +
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            insert th_cp_max max_cp_readys_threads[OF vt_t, symmetric])+
−
    by auto+
−
  from th_kept have "th \<in> threads (t@s)" by auto+
−
  from th_chain_to_ready[OF vt_t this] and not_ready+
−
  obtain th' where th'_in: "th' \<in> readys (t@s)"+
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    and dp: "(Th th, Th th') \<in> (depend (t @ s))\<^sup>+" by auto+
−
  have "th' \<in> runing (t@s)"+
−
  proof -+
−
    have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))"+
−
    proof -+
−
      have " Max ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')) = +
−
               preced th (t@s)"+
−
      proof(rule Max_eqI)+
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        fix y+
−
        assume "y \<in> (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"+
−
        then obtain th1 where+
−
          h1: "th1 = th' \<or> th1 \<in>  dependents (wq (t @ s)) th'"+
−
          and eq_y: "y = preced th1 (t@s)" by auto+
−
        show "y \<le> preced th (t @ s)"+
−
        proof -+
−
          from max_preced+
−
          have "preced th (t @ s) = Max ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" .+
−
          moreover have "y \<le> \<dots>"+
−
          proof(rule Max_ge)+
−
            from h1+
−
            have "th1 \<in> threads (t@s)"+
−
            proof+
−
              assume "th1 = th'"+
−
              with th'_in show ?thesis by (simp add:readys_def)+
−
            next+
−
              assume "th1 \<in> dependents (wq (t @ s)) th'"+
−
              with dependents_threads [OF vt_t]+
−
              show "th1 \<in> threads (t @ s)" by auto+
−
            qed+
−
            with eq_y show " y \<in> (\<lambda>th'. preced th' (t @ s)) ` threads (t @ s)" by simp+
−
          next+
−
            from finite_threads[OF vt_t]+
−
            show "finite ((\<lambda>th'. preced th' (t @ s)) ` threads (t @ s))" by simp+
−
          qed+
−
          ultimately show ?thesis by auto+
−
        qed+
−
      next+
−
        from finite_threads[OF vt_t] dependents_threads [OF vt_t, of th']+
−
        show "finite ((\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th'))"+
−
          by (auto intro:finite_subset)+
−
      next+
−
        from dp+
−
        have "th \<in> dependents (wq (t @ s)) th'" +
−
          by (unfold cs_dependents_def, auto simp:eq_depend)+
−
        thus "preced th (t @ s) \<in> +
−
                (\<lambda>th. preced th (t @ s)) ` ({th'} \<union> dependents (wq (t @ s)) th')"+
−
          by auto+
−
      qed+
−
      moreover have "\<dots> = Max (cp (t @ s) ` readys (t @ s))"+
−
      proof -+
−
        from max_preced and max_cp_eq[OF vt_t, symmetric]+
−
        have "preced th (t @ s) = Max (cp (t @ s) ` threads (t @ s))" by simp+
−
        with max_cp_readys_threads[OF vt_t] show ?thesis by simp+
−
      qed+
−
      ultimately show ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)+
−
    qed+
−
    with th'_in show ?thesis by (auto simp:runing_def)+
−
  qed+
−
  thus ?thesis by auto+
−
qed+
−
+
−
end+
−
+
−
end+
−
+
−
+
−