theory PrioG+
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imports PrioGDef +
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begin+
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+
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lemma runing_ready: "runing s \<subseteq> readys s"+
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  by (auto simp only:runing_def readys_def)+
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+
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lemma wq_v_neq:+
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   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"+
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  by (auto simp:wq_def Let_def cp_def split:list.splits)+
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+
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lemma wq_distinct: "vt step s \<Longrightarrow> distinct (wq s cs)"+
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proof(erule_tac vt.induct, simp add:wq_def)+
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  fix s e+
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  assume h1: "step s e"+
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  and h2: "distinct (wq s cs)"+
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  thus "distinct (wq (e # s) cs)"+
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  proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)+
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    fix thread s+
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    assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"+
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      and h2: "thread \<in> set (waiting_queue (schs s) cs)"+
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      and h3: "thread \<in> runing s"+
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    show "False" +
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    proof -+
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      from h3 have "\<And> cs. thread \<in>  set (waiting_queue (schs s) cs) \<Longrightarrow> +
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                             thread = hd ((waiting_queue (schs s) cs))" +
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        by (simp add:runing_def readys_def s_waiting_def wq_def)+
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      from this [OF h2] have "thread = hd (waiting_queue (schs s) cs)" .+
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      with h2+
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      have "(Cs cs, Th thread) \<in> (depend s)"+
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        by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)+
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      with h1 show False by auto+
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    qed+
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  next+
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    fix thread s a list+
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    assume dst: "distinct list"+
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    show "distinct (SOME q. distinct q \<and> set q = set list)"+
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    proof(rule someI2)+
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      from dst show  "distinct list \<and> set list = set list" by auto+
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    next+
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      fix q assume "distinct q \<and> set q = set list"+
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      thus "distinct q" by auto+
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    qed+
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  qed+
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qed+
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+
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lemma step_back_vt: "vt ccs (e#s) \<Longrightarrow> vt ccs s"+
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  by(ind_cases "vt ccs (e#s)", simp)+
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+
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lemma step_back_step: "vt ccs (e#s) \<Longrightarrow> ccs s e"+
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  by(ind_cases "vt ccs (e#s)", simp)+
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+
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lemma block_pre: +
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  fixes thread cs s+
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  assumes vt_e: "vt step (e#s)"+
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  and s_ni: "thread \<notin>  set (wq s cs)"+
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  and s_i: "thread \<in> set (wq (e#s) cs)"+
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  shows "e = P thread cs"+
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proof -+
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  show ?thesis+
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  proof(cases e)+
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    case (P th cs)+
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    with assms+
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    show ?thesis+
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      by (auto simp:wq_def Let_def split:if_splits)+
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  next+
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    case (Create th prio)+
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    with assms show ?thesis+
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      by (auto simp:wq_def Let_def split:if_splits)+
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  next+
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    case (Exit th)+
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    with assms show ?thesis+
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      by (auto simp:wq_def Let_def split:if_splits)+
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  next+
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    case (Set th prio)+
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    with assms show ?thesis+
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      by (auto simp:wq_def Let_def split:if_splits)+
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  next+
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    case (V th cs)+
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    with assms show ?thesis+
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      apply (auto simp:wq_def Let_def split:if_splits)+
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    proof -+
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      fix q qs+
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      assume h1: "thread \<notin> set (waiting_queue (schs s) cs)"+
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        and h2: "q # qs = waiting_queue (schs s) cs"+
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        and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"+
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        and vt: "vt step (V th cs # s)"+
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      from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp+
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      moreover have "thread \<in> set qs"+
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      proof -+
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        have "set (SOME q. distinct q \<and> set q = set qs) = set qs"+
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        proof(rule someI2)+
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          from wq_distinct [OF step_back_vt[OF vt], of cs]+
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          and h2[symmetric, folded wq_def]+
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          show "distinct qs \<and> set qs = set qs" by auto+
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        next+
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          fix x assume "distinct x \<and> set x = set qs"+
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          thus "set x = set qs" by auto+
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        qed+
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        with h3 show ?thesis by simp+
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      qed+
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      ultimately show "False" by auto+
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      qed+
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  qed+
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qed+
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+
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lemma p_pre: "\<lbrakk>vt step ((P thread cs)#s)\<rbrakk> \<Longrightarrow> +
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  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (depend s)^+"+
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apply (ind_cases "vt step ((P thread cs)#s)")+
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apply (ind_cases "step s (P thread cs)")+
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by auto+
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+
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lemma abs1:+
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  fixes e es+
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  assumes ein: "e \<in> set es"+
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  and neq: "hd es \<noteq> hd (es @ [x])"+
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  shows "False"+
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proof -+
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  from ein have "es \<noteq> []" by auto+
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  then obtain e ess where "es = e # ess" by (cases es, auto)+
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  with neq show ?thesis by auto+
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qed+
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+
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lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"+
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  by (cases es, auto)+
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+
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inductive_cases evt_cons: "vt cs (a#s)"+
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+
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lemma abs2:+
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  assumes vt: "vt step (e#s)"+
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  and inq: "thread \<in> set (wq s cs)"+
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  and nh: "thread = hd (wq s cs)"+
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  and qt: "thread \<noteq> hd (wq (e#s) cs)"+
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  and inq': "thread \<in> set (wq (e#s) cs)"+
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  shows "False"+
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proof -+
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  from assms show "False"+
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    apply (cases e)+
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    apply ((simp split:if_splits add:Let_def wq_def)[1])++
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    apply (insert abs1, fast)[1]+
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    apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)+
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  proof -+
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    fix th qs+
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    assume vt: "vt step (V th cs # s)"+
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      and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"+
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      and eq_wq: "waiting_queue (schs s) cs = thread # qs"+
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    show "False"+
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    proof -+
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      from wq_distinct[OF step_back_vt[OF vt], of cs]+
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        and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp+
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      moreover have "thread \<in> set qs"+
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      proof -+
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        have "set (SOME q. distinct q \<and> set q = set qs) = set qs"+
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        proof(rule someI2)+
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          from wq_distinct [OF step_back_vt[OF vt], of cs]+
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          and eq_wq [folded wq_def]+
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          show "distinct qs \<and> set qs = set qs" by auto+
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        next+
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          fix x assume "distinct x \<and> set x = set qs"+
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          thus "set x = set qs" by auto+
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        qed+
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        with th_in show ?thesis by auto+
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      qed+
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      ultimately show ?thesis by auto+
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    qed+
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  qed+
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qed+
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+
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lemma vt_moment: "\<And> t. \<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"+
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proof(induct s, simp)+
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  fix a s t+
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  assume h: "\<And>t.\<lbrakk>vt cs s; t \<le> length s\<rbrakk> \<Longrightarrow> vt cs (moment t s)"+
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    and vt_a: "vt cs (a # s)"+
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    and le_t: "t \<le> length (a # s)"+
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  show "vt cs (moment t (a # s))"+
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  proof(cases "t = length (a#s)")+
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    case True+
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    from True have "moment t (a#s) = a#s" by simp+
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    with vt_a show ?thesis by simp+
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  next+
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    case False+
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    with le_t have le_t1: "t \<le> length s" by simp+
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    from vt_a have "vt cs s"+
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      by (erule_tac evt_cons, simp)+
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    from h [OF this le_t1] have "vt cs (moment t s)" .+
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    moreover have "moment t (a#s) = moment t s"+
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    proof -+
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      from moment_app [OF le_t1, of "[a]"] +
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      show ?thesis by simp+
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    qed+
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    ultimately show ?thesis by auto+
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  qed+
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qed+
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+
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(* Wrong:+
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    lemma \<lbrakk>thread \<in> set (waiting_queue cs1 s); thread \<in> set (waiting_queue cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"+
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*)+
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+
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lemma waiting_unique_pre:+
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  fixes cs1 cs2 s thread+
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  assumes vt: "vt step s"+
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  and h11: "thread \<in> set (wq s cs1)"+
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  and h12: "thread \<noteq> hd (wq s cs1)"+
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  assumes h21: "thread \<in> set (wq s cs2)"+
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  and h22: "thread \<noteq> hd (wq s cs2)"+
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  and neq12: "cs1 \<noteq> cs2"+
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  shows "False"+
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proof -+
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  let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"+
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  from h11 and h12 have q1: "?Q cs1 s" by simp+
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  from h21 and h22 have q2: "?Q cs2 s" by simp+
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  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)+
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  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)+
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  from p_split [of "?Q cs1", OF q1 nq1]+
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  obtain t1 where lt1: "t1 < length s"+
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    and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>+
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        thread \<noteq> hd (wq (moment t1 s) cs1))"+
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    and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>+
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             thread \<noteq> hd (wq (moment i' s) cs1))" by auto+
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  from p_split [of "?Q cs2", OF q2 nq2]+
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  obtain t2 where lt2: "t2 < length s"+
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    and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>+
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        thread \<noteq> hd (wq (moment t2 s) cs2))"+
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    and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>+
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             thread \<noteq> hd (wq (moment i' s) cs2))" by auto+
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  show ?thesis+
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  proof -+
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    { +
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      assume lt12: "t1 < t2"+
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      let ?t3 = "Suc t2"+
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      from lt2 have le_t3: "?t3 \<le> length s" by auto+
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      from moment_plus [OF this] +
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      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto+
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      have "t2 < ?t3" by simp+
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      from nn2 [rule_format, OF this] and eq_m+
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      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and+
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        h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto+
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      have vt_e: "vt step (e#moment t2 s)"+
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      proof -+
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        from vt_moment [OF vt le_t3]+
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        have "vt step (moment ?t3 s)" .+
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        with eq_m show ?thesis by simp+
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      qed+
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      have ?thesis+
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      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")+
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        case True+
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        from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"+
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          by auto+
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        from abs2 [OF vt_e True eq_th h2 h1]+
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        show ?thesis by auto+
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      next+
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        case False+
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        from block_pre [OF vt_e False h1]+
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        have "e = P thread cs2" .+
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        with vt_e have "vt step ((P thread cs2)# moment t2 s)" by simp+
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        from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp+
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        with runing_ready have "thread \<in> readys (moment t2 s)" by auto+
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        with nn1 [rule_format, OF lt12]+
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        show ?thesis  by (simp add:readys_def s_waiting_def, auto)+
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      qed+
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    } moreover {+
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      assume lt12: "t2 < t1"+
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      let ?t3 = "Suc t1"+
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      from lt1 have le_t3: "?t3 \<le> length s" by auto+
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      from moment_plus [OF this] +
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      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto+
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      have lt_t3: "t1 < ?t3" by simp+
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      from nn1 [rule_format, OF this] and eq_m+
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      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and+
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        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto+
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      have vt_e: "vt step (e#moment t1 s)"+
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      proof -+
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        from vt_moment [OF vt le_t3]+
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        have "vt step (moment ?t3 s)" .+
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        with eq_m show ?thesis by simp+
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      qed+
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      have ?thesis+
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      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")+
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        case True+
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        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"+
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          by auto+
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        from abs2 [OF vt_e True eq_th h2 h1]+
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        show ?thesis by auto+
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      next+
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        case False+
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        from block_pre [OF vt_e False h1]+
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        have "e = P thread cs1" .+
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        with vt_e have "vt step ((P thread cs1)# moment t1 s)" by simp+
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        from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp+
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        with runing_ready have "thread \<in> readys (moment t1 s)" by auto+
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        with nn2 [rule_format, OF lt12]+
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        show ?thesis  by (simp add:readys_def s_waiting_def, auto)+
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      qed+
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    } moreover {+
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      assume eqt12: "t1 = t2"+
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      let ?t3 = "Suc t1"+
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      from lt1 have le_t3: "?t3 \<le> length s" by auto+
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      from moment_plus [OF this] +
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      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto+
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      have lt_t3: "t1 < ?t3" by simp+
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      from nn1 [rule_format, OF this] and eq_m+
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      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and+
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        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto+
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      have vt_e: "vt step (e#moment t1 s)"+
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      proof -+
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        from vt_moment [OF vt le_t3]+
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        have "vt step (moment ?t3 s)" .+
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        with eq_m show ?thesis by simp+
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      qed+
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      have ?thesis+
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      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")+
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        case True+
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        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"+
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          by auto+
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        from abs2 [OF vt_e True eq_th h2 h1]+
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        show ?thesis by auto+
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      next+
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        case False+
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        from block_pre [OF vt_e False h1]+
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        have eq_e1: "e = P thread cs1" .+
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        have lt_t3: "t1 < ?t3" by simp+
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        with eqt12 have "t2 < ?t3" by simp+
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        from nn2 [rule_format, OF this] and eq_m and eqt12+
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        have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and+
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          h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto+
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        show ?thesis+
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        proof(cases "thread \<in> set (wq (moment t2 s) cs2)")+
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          case True+
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          from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"+
−
            by auto+
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          from vt_e and eqt12 have "vt step (e#moment t2 s)" by simp +
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          from abs2 [OF this True eq_th h2 h1]+
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          show ?thesis .+
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        next+
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          case False+
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          have vt_e: "vt step (e#moment t2 s)"+
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          proof -+
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            from vt_moment [OF vt le_t3] eqt12+
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            have "vt step (moment (Suc t2) s)" by auto+
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            with eq_m eqt12 show ?thesis by simp+
−
          qed+
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          from block_pre [OF vt_e False h1]+
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          have "e = P thread cs2" .+
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          with eq_e1 neq12 show ?thesis by auto+
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        qed+
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      qed+
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    } ultimately show ?thesis by arith+
−
  qed+
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qed+
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+
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lemma waiting_unique:+
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  fixes s cs1 cs2+
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  assumes "vt step s"+
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  and "waiting s th cs1"+
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  and "waiting s th cs2"+
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  shows "cs1 = cs2"+
−
proof -+
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  from waiting_unique_pre and prems+
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  show ?thesis+
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    by (auto simp add:s_waiting_def)+
−
qed+
−
+
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lemma held_unique:+
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  assumes "vt step s"+
−
  and "holding s th1 cs"+
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  and "holding s th2 cs"+
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  shows "th1 = th2"+
−
proof -+
−
  from prems show ?thesis+
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    unfolding s_holding_def+
−
    by auto+
−
qed+
−
+
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lemma birthtime_lt: "th \<in> threads s \<Longrightarrow> birthtime th s < length s"+
−
  apply (induct s, auto)+
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  by (case_tac a, auto split:if_splits)+
−
+
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lemma birthtime_unique: +
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  "\<lbrakk>birthtime th1 s = birthtime th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>+
−
          \<Longrightarrow> th1 = th2"+
−
  apply (induct s, auto)+
−
  by (case_tac a, auto split:if_splits dest:birthtime_lt)+
−
+
−
lemma preced_unique : +
−
  assumes pcd_eq: "preced th1 s = preced th2 s"+
−
  and th_in1: "th1 \<in> threads s"+
−
  and th_in2: " th2 \<in> threads s"+
−
  shows "th1 = th2"+
−
proof -+
−
  from pcd_eq have "birthtime th1 s = birthtime th2 s" by (simp add:preced_def)+
−
  from birthtime_unique [OF this th_in1 th_in2]+
−
  show ?thesis .+
−
qed+
−
+
−
lemma preced_linorder: +
−
  assumes neq_12: "th1 \<noteq> th2"+
−
  and th_in1: "th1 \<in> threads s"+
−
  and th_in2: " th2 \<in> threads s"+
−
  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"+
−
proof -+
−
  from preced_unique [OF _ th_in1 th_in2] and neq_12 +
−
  have "preced th1 s \<noteq> preced th2 s" by auto+
−
  thus ?thesis by auto+
−
qed+
−
+
−
lemma unique_minus:+
−
  fixes x y z r+
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  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"+
−
  and xy: "(x, y) \<in> r"+
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  and xz: "(x, z) \<in> r^+"+
−
  and neq: "y \<noteq> z"+
−
  shows "(y, z) \<in> r^+"+
−
proof -+
−
 from xz and neq show ?thesis+
−
 proof(induct)+
−
   case (base ya)+
−
   have "(x, ya) \<in> r" by fact+
−
   from unique [OF xy this] have "y = ya" .+
−
   with base show ?case by auto+
−
 next+
−
   case (step ya z)+
−
   show ?case+
−
   proof(cases "y = ya")+
−
     case True+
−
     from step True show ?thesis by simp+
−
   next+
−
     case False+
−
     from step False+
−
     show ?thesis by auto+
−
   qed+
−
 qed+
−
qed+
−
+
−
lemma unique_base:+
−
  fixes r x y z+
−
  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"+
−
  and xy: "(x, y) \<in> r"+
−
  and xz: "(x, z) \<in> r^+"+
−
  and neq_yz: "y \<noteq> z"+
−
  shows "(y, z) \<in> r^+"+
−
proof -+
−
  from xz neq_yz show ?thesis+
−
  proof(induct)+
−
    case (base ya)+
−
    from xy unique base show ?case by auto+
−
  next+
−
    case (step ya z)+
−
    show ?case+
−
    proof(cases "y = ya")+
−
      case True+
−
      from True step show ?thesis by auto+
−
    next+
−
      case False+
−
      from False step +
−
      have "(y, ya) \<in> r\<^sup>+" by auto+
−
      with step show ?thesis by auto+
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma unique_chain:+
−
  fixes r x y z+
−
  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"+
−
  and xy: "(x, y) \<in> r^+"+
−
  and xz: "(x, z) \<in> r^+"+
−
  and neq_yz: "y \<noteq> z"+
−
  shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"+
−
proof -+
−
  from xy xz neq_yz show ?thesis+
−
  proof(induct)+
−
    case (base y)+
−
    have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto+
−
    from unique_base [OF _ h1 h2 h3] and unique show ?case by auto+
−
  next+
−
    case (step y za)+
−
    show ?case+
−
    proof(cases "y = z")+
−
      case True+
−
      from True step show ?thesis by auto+
−
    next+
−
      case False+
−
      from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto+
−
      thus ?thesis+
−
      proof+
−
        assume "(z, y) \<in> r\<^sup>+"+
−
        with step have "(z, za) \<in> r\<^sup>+" by auto+
−
        thus ?thesis by auto+
−
      next+
−
        assume h: "(y, z) \<in> r\<^sup>+"+
−
        from step have yza: "(y, za) \<in> r" by simp+
−
        from step have "za \<noteq> z" by simp+
−
        from unique_minus [OF _ yza h this] and unique+
−
        have "(za, z) \<in> r\<^sup>+" by auto+
−
        thus ?thesis by auto+
−
      qed+
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"+
−
apply (unfold s_depend_def s_waiting_def wq_def)+
−
by (simp add:Let_def)+
−
+
−
lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"+
−
apply (unfold s_depend_def s_waiting_def wq_def)+
−
by (simp add:Let_def)+
−
+
−
lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"+
−
apply (unfold s_depend_def s_waiting_def wq_def)+
−
by (simp add:Let_def)+
−
+
−
+
−
+
−
lemma step_v_hold_inv[elim_format]:+
−
  "\<And>c t. \<lbrakk>vt step (V th cs # s); +
−
  \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> next_th s th cs t \<and> c = cs"+
−
proof -+
−
  fix c t+
−
  assume vt: "vt step (V th cs # s)"+
−
    and nhd: "\<not> holding (wq s) t c"+
−
    and hd: "holding (wq (V th cs # s)) t c"+
−
  show "next_th s th cs t \<and> c = cs"+
−
  proof(cases "c = cs")+
−
    case False+
−
    with nhd hd show ?thesis+
−
      by (unfold cs_holding_def wq_def, auto simp:Let_def)+
−
  next+
−
    case True+
−
    with step_back_step [OF vt] +
−
    have "step s (V th c)" by simp+
−
    hence "next_th s th cs t"+
−
    proof(cases)+
−
      assume "holding s th c"+
−
      with nhd hd show ?thesis+
−
        apply (unfold s_holding_def cs_holding_def wq_def next_th_def,+
−
               auto simp:Let_def split:list.splits if_splits)+
−
        proof -+
−
          assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"+
−
          moreover have "\<dots> = set []"+
−
          proof(rule someI2)+
−
            show "distinct [] \<and> [] = []" by auto+
−
          next+
−
            fix x assume "distinct x \<and> x = []"+
−
            thus "set x = set []" by auto+
−
          qed+
−
          ultimately show False by auto+
−
        next+
−
          assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"+
−
          moreover have "\<dots> = set []"+
−
          proof(rule someI2)+
−
            show "distinct [] \<and> [] = []" by auto+
−
          next+
−
            fix x assume "distinct x \<and> x = []"+
−
            thus "set x = set []" by auto+
−
          qed+
−
          ultimately show False by auto+
−
        qed+
−
    qed+
−
    with True show ?thesis by auto+
−
  qed+
−
qed+
−
+
−
lemma step_v_wait_inv[elim_format]:+
−
    "\<And>t c. \<lbrakk>vt step (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c+
−
           \<rbrakk>+
−
          \<Longrightarrow> (next_th s th cs t \<and> cs = c)"+
−
proof -+
−
  fix t c +
−
  assume vt: "vt step (V th cs # s)"+
−
    and nw: "\<not> waiting (wq (V th cs # s)) t c"+
−
    and wt: "waiting (wq s) t c"+
−
  show "next_th s th cs t \<and> cs = c"+
−
  proof(cases "cs = c")+
−
    case False+
−
    with nw wt show ?thesis+
−
      by (auto simp:cs_waiting_def wq_def Let_def)+
−
  next+
−
    case True+
−
    from nw[folded True] wt[folded True]+
−
    have "next_th s th cs t"+
−
      apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)+
−
    proof -+
−
      fix a list+
−
      assume t_in: "t \<in> set list"+
−
        and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"+
−
        and eq_wq: "waiting_queue (schs s) cs = a # list"+
−
      have " set (SOME q. distinct q \<and> set q = set list) = set list"+
−
      proof(rule someI2)+
−
        from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def]+
−
        show "distinct list \<and> set list = set list" by auto+
−
      next+
−
        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"+
−
          by auto+
−
      qed+
−
      with t_ni and t_in show "a = th" by auto+
−
    next+
−
      fix a list+
−
      assume t_in: "t \<in> set list"+
−
        and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"+
−
        and eq_wq: "waiting_queue (schs s) cs = a # list"+
−
      have " set (SOME q. distinct q \<and> set q = set list) = set list"+
−
      proof(rule someI2)+
−
        from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq[folded wq_def]+
−
        show "distinct list \<and> set list = set list" by auto+
−
      next+
−
        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"+
−
          by auto+
−
      qed+
−
      with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto+
−
    next+
−
      fix a list+
−
      assume eq_wq: "waiting_queue (schs s) cs = a # list"+
−
      from step_back_step[OF vt]+
−
      show "a = th"+
−
      proof(cases)+
−
        assume "holding s th cs"+
−
        with eq_wq show ?thesis+
−
          by (unfold s_holding_def wq_def, auto)+
−
      qed+
−
    qed+
−
    with True show ?thesis by simp+
−
  qed+
−
qed+
−
+
−
lemma step_v_not_wait[consumes 3]:+
−
  "\<lbrakk>vt step (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"+
−
  by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)+
−
+
−
lemma step_v_release:+
−
  "\<lbrakk>vt step (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"+
−
proof -+
−
  assume vt: "vt step (V th cs # s)"+
−
    and hd: "holding (wq (V th cs # s)) th cs"+
−
  from step_back_step [OF vt] and hd+
−
  show "False"+
−
  proof(cases)+
−
    assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"+
−
    thus ?thesis+
−
      apply (unfold s_holding_def wq_def cs_holding_def)+
−
      apply (auto simp:Let_def split:list.splits)+
−
    proof -+
−
      fix list+
−
      assume eq_wq[folded wq_def]: +
−
        "waiting_queue (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"+
−
      and hd_in: "hd (SOME q. distinct q \<and> set q = set list)+
−
            \<in> set (SOME q. distinct q \<and> set q = set list)"+
−
      have "set (SOME q. distinct q \<and> set q = set list) = set list"+
−
      proof(rule someI2)+
−
        from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq+
−
        show "distinct list \<and> set list = set list" by auto+
−
      next+
−
        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"+
−
          by auto+
−
      qed+
−
      moreover have "distinct  (hd (SOME q. distinct q \<and> set q = set list) # list)"+
−
      proof -+
−
        from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq+
−
        show ?thesis by auto+
−
      qed+
−
      moreover note eq_wq and hd_in+
−
      ultimately show "False" by auto+
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma step_v_get_hold:+
−
  "\<And>th'. \<lbrakk>vt step (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"+
−
  apply (unfold cs_holding_def next_th_def wq_def,+
−
         auto simp:Let_def)+
−
proof -+
−
  fix rest+
−
  assume vt: "vt step (V th cs # s)"+
−
    and eq_wq[folded wq_def]: " waiting_queue (schs s) cs = th # rest"+
−
    and nrest: "rest \<noteq> []"+
−
    and ni: "hd (SOME q. distinct q \<and> set q = set rest)+
−
            \<notin> set (SOME q. distinct q \<and> set q = set rest)"+
−
  have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"+
−
  proof(rule someI2)+
−
    from wq_distinct[OF step_back_vt[OF vt], of cs] and eq_wq+
−
    show "distinct rest \<and> set rest = set rest" by auto+
−
  next+
−
    fix x assume "distinct x \<and> set x = set rest"+
−
    hence "set x = set rest" by auto+
−
    with nrest+
−
    show "x \<noteq> []" by (case_tac x, auto)+
−
  qed+
−
  with ni show "False" by auto+
−
qed+
−
+
−
lemma step_v_release_inv[elim_format]:+
−
"\<And>c t. \<lbrakk>vt step (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> +
−
  c = cs \<and> t = th"+
−
  apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)+
−
  proof -+
−
    fix a list+
−
    assume vt: "vt step (V th cs # s)" and eq_wq: "waiting_queue (schs s) cs = a # list"+
−
    from step_back_step [OF vt] show "a = th"+
−
    proof(cases)+
−
      assume "holding s th cs" with eq_wq+
−
      show ?thesis+
−
        by (unfold s_holding_def wq_def, auto)+
−
    qed+
−
  next+
−
    fix a list+
−
    assume vt: "vt step (V th cs # s)" and eq_wq: "waiting_queue (schs s) cs = a # list"+
−
    from step_back_step [OF vt] show "a = th"+
−
    proof(cases)+
−
      assume "holding s th cs" with eq_wq+
−
      show ?thesis+
−
        by (unfold s_holding_def wq_def, auto)+
−
    qed+
−
  qed+
−
+
−
lemma step_v_waiting_mono:+
−
  "\<And>t c. \<lbrakk>vt step (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"+
−
proof -+
−
  fix t c+
−
  let ?s' = "(V th cs # s)"+
−
  assume vt: "vt step ?s'" +
−
    and wt: "waiting (wq ?s') t c"+
−
  show "waiting (wq s) t c"+
−
  proof(cases "c = cs")+
−
    case False+
−
    assume neq_cs: "c \<noteq> cs"+
−
    hence "waiting (wq ?s') t c = waiting (wq s) t c"+
−
      by (unfold cs_waiting_def wq_def, auto simp:Let_def)+
−
    with wt show ?thesis by simp+
−
  next+
−
    case True+
−
    with wt show ?thesis+
−
      apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)+
−
    proof -+
−
      fix a list+
−
      assume not_in: "t \<notin> set list"+
−
        and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"+
−
        and eq_wq: "waiting_queue (schs s) cs = a # list"+
−
      have "set (SOME q. distinct q \<and> set q = set list) = set list"+
−
      proof(rule someI2)+
−
        from wq_distinct [OF step_back_vt[OF vt], of cs]+
−
        and eq_wq[folded wq_def]+
−
        show "distinct list \<and> set list = set list" by auto+
−
      next+
−
        fix x assume "distinct x \<and> set x = set list"+
−
        thus "set x = set list" by auto+
−
      qed+
−
      with not_in is_in show "t = a" by auto+
−
    next+
−
      fix list+
−
      assume is_waiting: "waiting (wq (V th cs # s)) t cs"+
−
      and eq_wq: "waiting_queue (schs s) cs = t # list"+
−
      hence "t \<in> set list"+
−
        apply (unfold wq_def, auto simp:Let_def cs_waiting_def)+
−
      proof -+
−
        assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"+
−
        moreover have "\<dots> = set list" +
−
        proof(rule someI2)+
−
          from wq_distinct [OF step_back_vt[OF vt], of cs]+
−
            and eq_wq[folded wq_def]+
−
          show "distinct list \<and> set list = set list" by auto+
−
        next+
−
          fix x assume "distinct x \<and> set x = set list" +
−
          thus "set x = set list" by auto+
−
        qed+
−
        ultimately show "t \<in> set list" by simp+
−
      qed+
−
      with eq_wq and wq_distinct [OF step_back_vt[OF vt], of cs, unfolded wq_def]+
−
      show False by auto+
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma step_depend_v:+
−
fixes th::thread+
−
assumes vt:+
−
  "vt step (V th cs#s)"+
−
shows "+
−
  depend (V th cs # s) =+
−
  depend s - {(Cs cs, Th th)} -+
−
  {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>+
−
  {(Cs cs, Th th') |th'.  next_th s th cs th'}"+
−
  apply (insert vt, unfold s_depend_def) +
−
  apply (auto split:if_splits list.splits simp:Let_def)+
−
  apply (auto elim: step_v_waiting_mono step_v_hold_inv +
−
              step_v_release step_v_wait_inv+
−
              step_v_get_hold step_v_release_inv)+
−
  apply (erule_tac step_v_not_wait, auto)+
−
  done+
−
+
−
lemma step_depend_p:+
−
  "vt step (P th cs#s) \<Longrightarrow>+
−
  depend (P th cs # s) =  (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}+
−
                                             else depend s \<union> {(Th th, Cs cs)})"+
−
  apply(simp only: s_depend_def wq_def)+
−
  apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)+
−
  apply(case_tac "csa = cs", auto)+
−
  apply(fold wq_def)+
−
  apply(drule_tac step_back_step)+
−
  apply(ind_cases " step s (P (hd (wq s cs)) cs)")+
−
  apply(auto simp:s_depend_def wq_def cs_holding_def)+
−
  done+
−
+
−
lemma simple_A:+
−
  fixes A+
−
  assumes h: "\<And> x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y"+
−
  shows "A = {} \<or> (\<exists> a. A = {a})"+
−
proof(cases "A = {}")+
−
  case True thus ?thesis by simp+
−
next+
−
  case False then obtain a where "a \<in> A" by auto+
−
  with h have "A = {a}" by auto+
−
  thus ?thesis by simp+
−
qed+
−
+
−
lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"+
−
  by (unfold s_depend_def, auto)+
−
+
−
lemma acyclic_depend: +
−
  fixes s+
−
  assumes vt: "vt step s"+
−
  shows "acyclic (depend s)"+
−
proof -+
−
  from vt show ?thesis+
−
  proof(induct)+
−
    case (vt_cons s e)+
−
    assume ih: "acyclic (depend s)"+
−
      and stp: "step s e"+
−
      and vt: "vt step s"+
−
    show ?case+
−
    proof(cases e)+
−
      case (Create th prio)+
−
      with ih+
−
      show ?thesis by (simp add:depend_create_unchanged)+
−
    next+
−
      case (Exit th)+
−
      with ih show ?thesis by (simp add:depend_exit_unchanged)+
−
    next+
−
      case (V th cs)+
−
      from V vt stp have vtt: "vt step (V th cs#s)" by auto+
−
      from step_depend_v [OF this]+
−
      have eq_de: +
−
        "depend (e # s) = +
−
            depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>+
−
            {(Cs cs, Th th') |th'. next_th s th cs th'}"+
−
        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)+
−
      from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)+
−
      from step_back_step [OF vtt]+
−
      have "step s (V th cs)" .+
−
      thus ?thesis+
−
      proof(cases)+
−
        assume "holding s th cs"+
−
        hence th_in: "th \<in> set (wq s cs)" and+
−
          eq_hd: "th = hd (wq s cs)" by (unfold s_holding_def, auto)+
−
        then obtain rest where+
−
          eq_wq: "wq s cs = th#rest"+
−
          by (cases "wq s cs", auto)+
−
        show ?thesis+
−
        proof(cases "rest = []")+
−
          case False+
−
          let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"+
−
          from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"+
−
            by (unfold next_th_def, auto)+
−
          let ?E = "(?A - ?B - ?C)"+
−
          have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"+
−
          proof+
−
            assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"+
−
            hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)+
−
            from tranclD [OF this]+
−
            obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast+
−
            hence th_d: "(Th ?th', x) \<in> ?A" by simp+
−
            from depend_target_th [OF this]+
−
            obtain cs' where eq_x: "x = Cs cs'" by auto+
−
            with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp+
−
            hence wt_th': "waiting s ?th' cs'"+
−
              unfolding s_depend_def s_waiting_def cs_waiting_def by simp+
−
            hence "cs' = cs"+
−
            proof(rule waiting_unique [OF vt])+
−
              from eq_wq wq_distinct[OF vt, of cs]+
−
              show "waiting s ?th' cs" +
−
                apply (unfold s_waiting_def, auto)+
−
              proof -+
−
                assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"+
−
                and eq_wq: "wq s cs = th # rest"+
−
                have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"+
−
                proof(rule someI2)+
−
                  from wq_distinct[OF vt, of cs] and eq_wq+
−
                  show "distinct rest \<and> set rest = set rest" by auto+
−
                next+
−
                  fix x assume "distinct x \<and> set x = set rest"+
−
                  with False show "x \<noteq> []" by auto+
−
                qed+
−
                hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> +
−
                                  set (SOME q. distinct q \<and> set q = set rest)" by auto+
−
                moreover have "\<dots> = set rest" +
−
                proof(rule someI2)+
−
                  from wq_distinct[OF vt, of cs] and eq_wq+
−
                  show "distinct rest \<and> set rest = set rest" by auto+
−
                next+
−
                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto+
−
                qed+
−
                moreover note hd_in+
−
                ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto+
−
              next+
−
                assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"+
−
                and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"+
−
                have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"+
−
                proof(rule someI2)+
−
                  from wq_distinct[OF vt, of cs] and eq_wq+
−
                  show "distinct rest \<and> set rest = set rest" by auto+
−
                next+
−
                  fix x assume "distinct x \<and> set x = set rest"+
−
                  with False show "x \<noteq> []" by auto+
−
                qed+
−
                hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> +
−
                                  set (SOME q. distinct q \<and> set q = set rest)" by auto+
−
                moreover have "\<dots> = set rest" +
−
                proof(rule someI2)+
−
                  from wq_distinct[OF vt, of cs] and eq_wq+
−
                  show "distinct rest \<and> set rest = set rest" by auto+
−
                next+
−
                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto+
−
                qed+
−
                moreover note hd_in+
−
                ultimately show False by auto+
−
              qed+
−
            qed+
−
            with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp+
−
            with False+
−
            show "False" by (auto simp: next_th_def eq_wq)+
−
          qed+
−
          with acyclic_insert[symmetric] and ac+
−
            and eq_de eq_D show ?thesis by auto+
−
        next+
−
          case True+
−
          with eq_wq+
−
          have eq_D: "?D = {}"+
−
            by (unfold next_th_def, auto)+
−
          with eq_de ac+
−
          show ?thesis by auto+
−
        qed +
−
      qed+
−
  next+
−
    case (P th cs)+
−
    from P vt stp have vtt: "vt step (P th cs#s)" by auto+
−
    from step_depend_p [OF this] P+
−
    have "depend (e # s) = +
−
      (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else +
−
      depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")+
−
      by simp+
−
    moreover have "acyclic ?R"+
−
    proof(cases "wq s cs = []")+
−
      case True+
−
      hence eq_r: "?R =  depend s \<union> {(Cs cs, Th th)}" by simp+
−
      have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"+
−
      proof+
−
        assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"+
−
        hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)+
−
        from tranclD2 [OF this]+
−
        obtain x where "(x, Cs cs) \<in> depend s" by auto+
−
        with True show False by (auto simp:s_depend_def cs_waiting_def)+
−
      qed+
−
      with acyclic_insert ih eq_r show ?thesis by auto+
−
    next+
−
      case False+
−
      hence eq_r: "?R =  depend s \<union> {(Th th, Cs cs)}" by simp+
−
      have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"+
−
      proof+
−
        assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"+
−
        hence "(Cs cs, Th th) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)+
−
          moreover from step_back_step [OF vtt] have "step s (P th cs)" .+
−
          ultimately show False+
−
          proof -+
−
            show " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"+
−
              by (ind_cases "step s (P th cs)", simp)+
−
          qed+
−
        qed+
−
        with acyclic_insert ih eq_r show ?thesis by auto+
−
      qed+
−
      ultimately show ?thesis by simp+
−
    next+
−
      case (Set thread prio)+
−
      with ih+
−
      thm depend_set_unchanged+
−
      show ?thesis by (simp add:depend_set_unchanged)+
−
    qed+
−
  next+
−
    case vt_nil+
−
    show "acyclic (depend ([]::state))"+
−
      by (auto simp: s_depend_def cs_waiting_def +
−
                      cs_holding_def wq_def acyclic_def)+
−
  qed+
−
qed+
−
+
−
lemma finite_depend: +
−
  fixes s+
−
  assumes vt: "vt step s"+
−
  shows "finite (depend s)"+
−
proof -+
−
  from vt show ?thesis+
−
  proof(induct)+
−
    case (vt_cons s e)+
−
    assume ih: "finite (depend s)"+
−
      and stp: "step s e"+
−
      and vt: "vt step s"+
−
    show ?case+
−
    proof(cases e)+
−
      case (Create th prio)+
−
      with ih+
−
      show ?thesis by (simp add:depend_create_unchanged)+
−
    next+
−
      case (Exit th)+
−
      with ih show ?thesis by (simp add:depend_exit_unchanged)+
−
    next+
−
      case (V th cs)+
−
      from V vt stp have vtt: "vt step (V th cs#s)" by auto+
−
      from step_depend_v [OF this]+
−
      have eq_de: "depend (e # s) = +
−
                   depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>+
−
                      {(Cs cs, Th th') |th'. next_th s th cs th'}+
−
"+
−
        (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)+
−
      moreover from ih have ac: "finite (?A - ?B - ?C)" by simp+
−
      moreover have "finite ?D"+
−
      proof -+
−
        have "?D = {} \<or> (\<exists> a. ?D = {a})" +
−
          by (unfold next_th_def, auto)+
−
        thus ?thesis+
−
        proof+
−
          assume h: "?D = {}"+
−
          show ?thesis by (unfold h, simp)+
−
        next+
−
          assume "\<exists> a. ?D = {a}"+
−
          thus ?thesis by auto+
−
        qed+
−
      qed+
−
      ultimately show ?thesis by simp+
−
    next+
−
      case (P th cs)+
−
      from P vt stp have vtt: "vt step (P th cs#s)" by auto+
−
      from step_depend_p [OF this] P+
−
      have "depend (e # s) = +
−
              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else +
−
                                    depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")+
−
        by simp+
−
      moreover have "finite ?R"+
−
      proof(cases "wq s cs = []")+
−
        case True+
−
        hence eq_r: "?R =  depend s \<union> {(Cs cs, Th th)}" by simp+
−
        with True and ih show ?thesis by auto+
−
      next+
−
        case False+
−
        hence "?R = depend s \<union> {(Th th, Cs cs)}" by simp+
−
        with False and ih show ?thesis by auto+
−
      qed+
−
      ultimately show ?thesis by auto+
−
    next+
−
      case (Set thread prio)+
−
      with ih+
−
      show ?thesis by (simp add:depend_set_unchanged)+
−
    qed+
−
  next+
−
    case vt_nil+
−
    show "finite (depend ([]::state))"+
−
      by (auto simp: s_depend_def cs_waiting_def +
−
                   cs_holding_def wq_def acyclic_def)+
−
  qed+
−
qed+
−
+
−
text {* Several useful lemmas *}+
−
+
−
lemma wf_dep_converse: +
−
  fixes s+
−
  assumes vt: "vt step s"+
−
  shows "wf ((depend s)^-1)"+
−
proof(rule finite_acyclic_wf_converse)+
−
  from finite_depend [OF vt]+
−
  show "finite (depend s)" .+
−
next+
−
  from acyclic_depend[OF vt]+
−
  show "acyclic (depend s)" .+
−
qed+
−
+
−
lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"+
−
by (induct l, auto)+
−
+
−
lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"+
−
  by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)+
−
+
−
lemma wq_threads: +
−
  fixes s cs+
−
  assumes vt: "vt step s"+
−
  and h: "th \<in> set (wq s cs)"+
−
  shows "th \<in> threads s"+
−
proof -+
−
 from vt and h show ?thesis+
−
  proof(induct arbitrary: th cs)+
−
    case (vt_cons s e)+
−
    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"+
−
      and stp: "step s e"+
−
      and vt: "vt step s"+
−
      and h: "th \<in> set (wq (e # s) cs)"+
−
    show ?case+
−
    proof(cases e)+
−
      case (Create th' prio)+
−
      with ih h show ?thesis+
−
        by (auto simp:wq_def Let_def)+
−
    next+
−
      case (Exit th')+
−
      with stp ih h show ?thesis+
−
        apply (auto simp:wq_def Let_def)+
−
        apply (ind_cases "step s (Exit th')")+
−
        apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def+
−
               s_depend_def s_holding_def cs_holding_def)+
−
        by (fold wq_def, auto)+
−
    next+
−
      case (V th' cs')+
−
      show ?thesis+
−
      proof(cases "cs' = cs")+
−
        case False+
−
        with h+
−
        show ?thesis+
−
          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)+
−
          by (drule_tac ih, simp)+
−
      next+
−
        case True+
−
        from h+
−
        show ?thesis+
−
        proof(unfold V wq_def)+
−
          assume th_in: "th \<in> set (waiting_queue (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")+
−
          show "th \<in> threads (V th' cs' # s)"+
−
          proof(cases "cs = cs'")+
−
            case False+
−
            hence "?l = waiting_queue (schs s) cs" by (simp add:Let_def)+
−
            with th_in have " th \<in> set (wq s cs)" +
−
              by (fold wq_def, simp)+
−
            from ih [OF this] show ?thesis by simp+
−
          next+
−
            case True+
−
            show ?thesis+
−
            proof(cases "waiting_queue (schs s) cs'")+
−
              case Nil+
−
              with h V show ?thesis+
−
                apply (auto simp:wq_def Let_def split:if_splits)+
−
                by (fold wq_def, drule_tac ih, simp)+
−
            next+
−
              case (Cons a rest)+
−
              assume eq_wq: "waiting_queue (schs s) cs' = a # rest"+
−
              with h V show ?thesis+
−
                apply (auto simp:Let_def wq_def split:if_splits)+
−
              proof -+
−
                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"+
−
                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" +
−
                proof(rule someI2)+
−
                  from wq_distinct[OF vt, of cs'] and eq_wq[folded wq_def]+
−
                  show "distinct rest \<and> set rest = set rest" by auto+
−
                next+
−
                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"+
−
                    by auto+
−
                qed+
−
                with eq_wq th_in have "th \<in> set (waiting_queue (schs s) cs')" by auto+
−
                from ih[OF this[folded wq_def]] show "th \<in> threads s" .+
−
              next+
−
                assume th_in: "th \<in> set (waiting_queue (schs s) cs)"+
−
                from ih[OF this[folded wq_def]]+
−
                show "th \<in> threads s" .+
−
              qed+
−
            qed+
−
          qed+
−
        qed+
−
      qed+
−
    next+
−
      case (P th' cs')+
−
      from h stp+
−
      show ?thesis+
−
        apply (unfold P wq_def)+
−
        apply (auto simp:Let_def split:if_splits, fold wq_def)+
−
        apply (auto intro:ih)+
−
        apply(ind_cases "step s (P th' cs')")+
−
        by (unfold runing_def readys_def, auto)+
−
    next+
−
      case (Set thread prio)+
−
      with ih h show ?thesis+
−
        by (auto simp:wq_def Let_def)+
−
    qed+
−
  next+
−
    case vt_nil+
−
    thus ?case by (auto simp:wq_def)+
−
  qed+
−
qed+
−
+
−
lemma range_in: "\<lbrakk>vt step s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"+
−
  apply(unfold s_depend_def cs_waiting_def cs_holding_def)+
−
  by (auto intro:wq_threads)+
−
+
−
lemma readys_v_eq:+
−
  fixes th thread cs rest+
−
  assumes vt: "vt step s"+
−
  and neq_th: "th \<noteq> thread"+
−
  and eq_wq: "wq s cs = thread#rest"+
−
  and not_in: "th \<notin>  set rest"+
−
  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"+
−
proof -+
−
  from prems show ?thesis+
−
    apply (auto simp:readys_def)+
−
    apply (case_tac "cs = csa", simp add:s_waiting_def)+
−
    apply (erule_tac x = csa in allE)+
−
    apply (simp add:s_waiting_def wq_def Let_def split:if_splits)+
−
    apply (case_tac "csa = cs", simp)+
−
    apply (erule_tac x = cs in allE)+
−
    apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)+
−
    proof -+
−
      assume th_nin: "th \<notin> set rest"+
−
        and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"+
−
        and eq_wq: "waiting_queue (schs s) cs = thread # rest"+
−
      have "set (SOME q. distinct q \<and> set q = set rest) = set rest"+
−
      proof(rule someI2)+
−
        from wq_distinct[OF vt, of cs] and eq_wq[folded wq_def]+
−
        show "distinct rest \<and> set rest = set rest" by auto+
−
      next+
−
        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto+
−
      qed+
−
      with th_nin th_in show False by auto+
−
    qed+
−
qed+
−
+
−
lemma chain_building:+
−
  assumes vt: "vt step s"+
−
  shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"+
−
proof -+
−
  from wf_dep_converse [OF vt]+
−
  have h: "wf ((depend s)\<inverse>)" .+
−
  show ?thesis+
−
  proof(induct rule:wf_induct [OF h])+
−
    fix x+
−
    assume ih [rule_format]: +
−
      "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow> +
−
           y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"+
−
    show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"+
−
    proof+
−
      assume x_d: "x \<in> Domain (depend s)"+
−
      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"+
−
      proof(cases x)+
−
        case (Th th)+
−
        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)+
−
        with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp+
−
        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast+
−
        hence "Cs cs \<in> Domain (depend s)" by auto+
−
        from ih [OF x_in_r this] obtain th'+
−
          where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto+
−
        have "(x, Th th') \<in> (depend s)\<^sup>+" using Th x_in cs_in by auto+
−
        with th'_ready show ?thesis by auto+
−
      next+
−
        case (Cs cs)+
−
        from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (depend s)^-1" by (auto simp:s_depend_def)+
−
        show ?thesis+
−
        proof(cases "th' \<in> readys s")+
−
          case True+
−
          from True and th'_d show ?thesis by auto+
−
        next+
−
          case False+
−
          from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto+
−
          with False have "Th th' \<in> Domain (depend s)" +
−
            by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)+
−
          from ih [OF th'_d this]+
−
          obtain th'' where +
−
            th''_r: "th'' \<in> readys s" and +
−
            th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto+
−
          from th'_d and th''_in +
−
          have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto+
−
          with th''_r show ?thesis by auto+
−
        qed+
−
      qed+
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma th_chain_to_ready:+
−
  fixes s th+
−
  assumes vt: "vt step s"+
−
  and th_in: "th \<in> threads s"+
−
  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"+
−
proof(cases "th \<in> readys s")+
−
  case True+
−
  thus ?thesis by auto+
−
next+
−
  case False+
−
  from False and th_in have "Th th \<in> Domain (depend s)" +
−
    by (auto simp:readys_def s_waiting_def s_depend_def cs_waiting_def Domain_def)+
−
  from chain_building [rule_format, OF vt this]+
−
  show ?thesis by auto+
−
qed+
−
+
−
lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"+
−
  by  (unfold s_waiting_def cs_waiting_def, auto)+
−
+
−
lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"+
−
  by (unfold s_holding_def cs_holding_def, simp)+
−
+
−
lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"+
−
  by (unfold s_holding_def cs_holding_def, auto)+
−
+
−
lemma unique_depend: "\<lbrakk>vt step s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"+
−
  apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)+
−
  by(auto elim:waiting_unique holding_unique)+
−
+
−
lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"+
−
by (induct rule:trancl_induct, auto)+
−
+
−
lemma dchain_unique:+
−
  assumes vt: "vt step s"+
−
  and th1_d: "(n, Th th1) \<in> (depend s)^+"+
−
  and th1_r: "th1 \<in> readys s"+
−
  and th2_d: "(n, Th th2) \<in> (depend s)^+"+
−
  and th2_r: "th2 \<in> readys s"+
−
  shows "th1 = th2"+
−
proof -+
−
  { assume neq: "th1 \<noteq> th2"+
−
    hence "Th th1 \<noteq> Th th2" by simp+
−
    from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]+
−
    have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto+
−
    hence "False"+
−
    proof+
−
      assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"+
−
      from trancl_split [OF this]+
−
      obtain n where dd: "(Th th1, n) \<in> depend s" by auto+
−
      then obtain cs where eq_n: "n = Cs cs"+
−
        by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)+
−
      from dd eq_n have "th1 \<notin> readys s"+
−
        by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)+
−
      with th1_r show ?thesis by auto+
−
    next+
−
      assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"+
−
      from trancl_split [OF this]+
−
      obtain n where dd: "(Th th2, n) \<in> depend s" by auto+
−
      then obtain cs where eq_n: "n = Cs cs"+
−
        by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)+
−
      from dd eq_n have "th2 \<notin> readys s"+
−
        by (auto simp:readys_def s_depend_def s_waiting_def cs_waiting_def)+
−
      with th2_r show ?thesis by auto+
−
    qed+
−
  } thus ?thesis by auto+
−
qed+
−
             +
−
+
−
lemma step_holdents_p_add:+
−
  fixes th cs s+
−
  assumes vt: "vt step (P th cs#s)"+
−
  and "wq s cs = []"+
−
  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"+
−
proof -+
−
  from prems show ?thesis+
−
  unfolding  holdents_def step_depend_p[OF vt] by auto+
−
qed+
−
+
−
lemma step_holdents_p_eq:+
−
  fixes th cs s+
−
  assumes vt: "vt step (P th cs#s)"+
−
  and "wq s cs \<noteq> []"+
−
  shows "holdents (P th cs#s) th = holdents s th"+
−
proof -+
−
  from prems show ?thesis+
−
  unfolding  holdents_def step_depend_p[OF vt] by auto+
−
qed+
−
+
−
+
−
lemma finite_holding:+
−
  fixes s th cs+
−
  assumes vt: "vt step s"+
−
  shows "finite (holdents s th)"+
−
proof -+
−
  let ?F = "\<lambda> (x, y). the_cs x"+
−
  from finite_depend [OF vt]+
−
  have "finite (depend s)" .+
−
  hence "finite (?F `(depend s))" by simp+
−
  moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>" +
−
  proof -+
−
    { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto+
−
      fix x assume "(Cs x, Th th) \<in> depend s"+
−
      hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)+
−
      moreover have "?F (Cs x, Th th) = x" by simp+
−
      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp +
−
    } thus ?thesis by auto+
−
  qed+
−
  ultimately show ?thesis by (unfold holdents_def, auto intro:finite_subset)+
−
qed+
−
+
−
lemma cntCS_v_dec: +
−
  fixes s thread cs+
−
  assumes vtv: "vt step (V thread cs#s)"+
−
  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"+
−
proof -+
−
  from step_back_step[OF vtv]+
−
  have cs_in: "cs \<in> holdents s thread" +
−
    apply (cases, unfold holdents_def s_depend_def, simp)+
−
    by (unfold cs_holding_def s_holding_def, auto)+
−
  moreover have cs_not_in: +
−
    "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"+
−
    apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])+
−
    apply (unfold holdents_def, unfold step_depend_v[OF vtv],+
−
            auto simp:next_th_def)+
−
  proof -+
−
    fix rest+
−
    assume dst: "distinct (rest::thread list)"+
−
      and ne: "rest \<noteq> []"+
−
    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"+
−
    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"+
−
    proof(rule someI2)+
−
      from dst show "distinct rest \<and> set rest = set rest" by auto+
−
    next+
−
      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto+
−
    qed+
−
    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> +
−
                     set (SOME q. distinct q \<and> set q = set rest)" by simp+
−
    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"+
−
    proof(rule someI2)+
−
      from dst show "distinct rest \<and> set rest = set rest" by auto+
−
    next+
−
      fix x assume " distinct x \<and> set x = set rest" with ne+
−
      show "x \<noteq> []" by auto+
−
    qed+
−
    ultimately +
−
    show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"+
−
      by auto+
−
  next+
−
    fix rest+
−
    assume dst: "distinct (rest::thread list)"+
−
      and ne: "rest \<noteq> []"+
−
    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"+
−
    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"+
−
    proof(rule someI2)+
−
      from dst show "distinct rest \<and> set rest = set rest" by auto+
−
    next+
−
      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto+
−
    qed+
−
    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> +
−
                     set (SOME q. distinct q \<and> set q = set rest)" by simp+
−
    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"+
−
    proof(rule someI2)+
−
      from dst show "distinct rest \<and> set rest = set rest" by auto+
−
    next+
−
      fix x assume " distinct x \<and> set x = set rest" with ne+
−
      show "x \<noteq> []" by auto+
−
    qed+
−
    ultimately show "False" by auto +
−
  qed+
−
  ultimately +
−
  have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"+
−
    by auto+
−
  moreover have "card \<dots> = +
−
                    Suc (card ((holdents (V thread cs#s) thread) - {cs}))"+
−
  proof(rule card_insert)+
−
    from finite_holding [OF vtv]+
−
    show " finite (holdents (V thread cs # s) thread)" .+
−
  qed+
−
  moreover from cs_not_in +
−
  have "cs \<notin> (holdents (V thread cs#s) thread)" by auto+
−
  ultimately show ?thesis by (simp add:cntCS_def)+
−
qed +
−
+
−
lemma cnp_cnv_cncs:+
−
  fixes s th+
−
  assumes vt: "vt step s"+
−
  shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) +
−
                                       then cntCS s th else cntCS s th + 1)"+
−
proof -+
−
  from vt show ?thesis+
−
  proof(induct arbitrary:th)+
−
    case (vt_cons s e)+
−
    assume vt: "vt step s"+
−
    and ih: "\<And>th. cntP s th  = cntV s th ++
−
               (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"+
−
    and stp: "step s e"+
−
    from stp show ?case+
−
    proof(cases)+
−
      case (thread_create thread prio)+
−
      assume eq_e: "e = Create thread prio"+
−
        and not_in: "thread \<notin> threads s"+
−
      show ?thesis+
−
      proof -+
−
        { fix cs +
−
          assume "thread \<in> set (wq s cs)"+
−
          from wq_threads [OF vt this] have "thread \<in> threads s" .+
−
          with not_in have "False" by simp+
−
        } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"+
−
          by (auto simp:readys_def threads.simps s_waiting_def +
−
            wq_def cs_waiting_def Let_def)+
−
        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)+
−
        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)+
−
        have eq_cncs: "cntCS (e#s) th = cntCS s th"+
−
          unfolding cntCS_def holdents_def+
−
          by (simp add:depend_create_unchanged eq_e)+
−
        { assume "th \<noteq> thread"+
−
          with eq_readys eq_e+
−
          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = +
−
                      (th \<in> readys (s) \<or> th \<notin> threads (s))" +
−
            by (simp add:threads.simps)+
−
          with eq_cnp eq_cnv eq_cncs ih not_in+
−
          have ?thesis by simp+
−
        } moreover {+
−
          assume eq_th: "th = thread"+
−
          with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp+
−
          moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp+
−
          moreover note eq_cnp eq_cnv eq_cncs+
−
          ultimately have ?thesis by auto+
−
        } ultimately show ?thesis by blast+
−
      qed+
−
    next+
−
      case (thread_exit thread)+
−
      assume eq_e: "e = Exit thread" +
−
      and is_runing: "thread \<in> runing s"+
−
      and no_hold: "holdents s thread = {}"+
−
      from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)+
−
      from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)+
−
      have eq_cncs: "cntCS (e#s) th = cntCS s th"+
−
        unfolding cntCS_def holdents_def+
−
        by (simp add:depend_exit_unchanged eq_e)+
−
      { assume "th \<noteq> thread"+
−
        with eq_e+
−
        have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = +
−
          (th \<in> readys (s) \<or> th \<notin> threads (s))" +
−
          apply (simp add:threads.simps readys_def)+
−
          apply (subst s_waiting_def)+
−
          apply (subst (1 2) wq_def)+
−
          apply (simp add:Let_def)+
−
          apply (subst s_waiting_def, simp)+
−
          by (fold wq_def, simp)+
−
        with eq_cnp eq_cnv eq_cncs ih+
−
        have ?thesis by simp+
−
      } moreover {+
−
        assume eq_th: "th = thread"+
−
        with ih is_runing have " cntP s th = cntV s th + cntCS s th" +
−
          by (simp add:runing_def)+
−
        moreover from eq_th eq_e have "th \<notin> threads (e#s)"+
−
          by simp+
−
        moreover note eq_cnp eq_cnv eq_cncs+
−
        ultimately have ?thesis by auto+
−
      } ultimately show ?thesis by blast+
−
    next+
−
      case (thread_P thread cs)+
−
      assume eq_e: "e = P thread cs"+
−
        and is_runing: "thread \<in> runing s"+
−
        and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"+
−
      from prems have vtp: "vt step (P thread cs#s)" by auto+
−
      show ?thesis +
−
      proof -+
−
        { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast+
−
          assume neq_th: "th \<noteq> thread"+
−
          with eq_e+
−
          have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"+
−
            apply (simp add:readys_def s_waiting_def wq_def Let_def)+
−
            apply (rule_tac hh, clarify)+
−
            apply (intro iffI allI, clarify)+
−
            apply (erule_tac x = csa in allE, auto)+
−
            apply (subgoal_tac "waiting_queue (schs s) cs \<noteq> []", auto)+
−
            apply (erule_tac x = cs in allE, auto)+
−
            by (case_tac "(waiting_queue (schs s) cs)", auto)+
−
          moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"+
−
            apply (simp add:cntCS_def holdents_def)+
−
            by (unfold  step_depend_p [OF vtp], auto)+
−
          moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"+
−
            by (simp add:cntP_def count_def)+
−
          moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"+
−
            by (simp add:cntV_def count_def)+
−
          moreover from eq_e neq_th have "threads (e#s) = threads s" by simp+
−
          moreover note ih [of th] +
−
          ultimately have ?thesis by simp+
−
        } moreover {+
−
          assume eq_th: "th = thread"+
−
          have ?thesis+
−
          proof -+
−
            from eq_e eq_th have eq_cnp: "cntP (e # s) th  = 1 + (cntP s th)" +
−
              by (simp add:cntP_def count_def)+
−
            from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"+
−
              by (simp add:cntV_def count_def)+
−
            show ?thesis+
−
            proof (cases "wq s cs = []")+
−
              case True+
−
              with is_runing+
−
              have "th \<in> readys (e#s)"+
−
                apply (unfold eq_e wq_def, unfold readys_def s_depend_def)+
−
                apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)+
−
                by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)+
−
              moreover have "cntCS (e # s) th = 1 + cntCS s th"+
−
              proof -+
−
                have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> depend s} =+
−
                  Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")+
−
                proof -+
−
                  have "?L = insert cs ?R" by auto+
−
                  moreover have "card \<dots> = Suc (card (?R - {cs}))" +
−
                  proof(rule card_insert)+
−
                    from finite_holding [OF vt, of thread]+
−
                    show " finite {cs. (Cs cs, Th thread) \<in> depend s}"+
−
                      by (unfold holdents_def, simp)+
−
                  qed+
−
                  moreover have "?R - {cs} = ?R"+
−
                  proof -+
−
                    have "cs \<notin> ?R"+
−
                    proof+
−
                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend s}"+
−
                      with no_dep show False by auto+
−
                    qed+
−
                    thus ?thesis by auto+
−
                  qed+
−
                  ultimately show ?thesis by auto+
−
                qed+
−
                thus ?thesis+
−
                  apply (unfold eq_e eq_th cntCS_def)+
−
                  apply (simp add: holdents_def)+
−
                  by (unfold step_depend_p [OF vtp], auto simp:True)+
−
              qed+
−
              moreover from is_runing have "th \<in> readys s"+
−
                by (simp add:runing_def eq_th)+
−
              moreover note eq_cnp eq_cnv ih [of th]+
−
              ultimately show ?thesis by auto+
−
            next+
−
              case False+
−
              have eq_wq: "wq (e#s) cs = wq s cs @ [th]"+
−
                    by (unfold eq_th eq_e wq_def, auto simp:Let_def)+
−
              have "th \<notin> readys (e#s)"+
−
              proof+
−
                assume "th \<in> readys (e#s)"+
−
                hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)+
−
                from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .+
−
                hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)" +
−
                  by (simp add:s_waiting_def)+
−
                moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto+
−
                ultimately have "th = hd (wq (e#s) cs)" by blast+
−
                with eq_wq have "th = hd (wq s cs @ [th])" by simp+
−
                hence "th = hd (wq s cs)" using False by auto+
−
                with False eq_wq wq_distinct [OF vtp, of cs]+
−
                show False by (fold eq_e, auto)+
−
              qed+
−
              moreover from is_runing have "th \<in> threads (e#s)" +
−
                by (unfold eq_e, auto simp:runing_def readys_def eq_th)+
−
              moreover have "cntCS (e # s) th = cntCS s th"+
−
                apply (unfold cntCS_def holdents_def eq_e step_depend_p[OF vtp])+
−
                by (auto simp:False)+
−
              moreover note eq_cnp eq_cnv ih[of th]+
−
              moreover from is_runing have "th \<in> readys s"+
−
                by (simp add:runing_def eq_th)+
−
              ultimately show ?thesis by auto+
−
            qed+
−
          qed+
−
        } ultimately show ?thesis by blast+
−
      qed+
−
    next+
−
      case (thread_V thread cs)+
−
      from prems have vtv: "vt step (V thread cs # s)" by auto+
−
      assume eq_e: "e = V thread cs"+
−
        and is_runing: "thread \<in> runing s"+
−
        and hold: "holding s thread cs"+
−
      from hold obtain rest +
−
        where eq_wq: "wq s cs = thread # rest"+
−
        by (case_tac "wq s cs", auto simp:s_holding_def)+
−
      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)+
−
      have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"+
−
      proof(rule someI2)+
−
        from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq+
−
        show "distinct rest \<and> set rest = set rest" by auto+
−
      next+
−
        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"+
−
          by auto+
−
      qed+
−
      show ?thesis+
−
      proof -+
−
        { assume eq_th: "th = thread"+
−
          from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"+
−
            by (unfold eq_e, simp add:cntP_def count_def)+
−
          moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"+
−
            by (unfold eq_e, simp add:cntV_def count_def)+
−
          moreover from cntCS_v_dec [OF vtv] +
−
          have "cntCS (e # s) thread + 1 = cntCS s thread"+
−
            by (simp add:eq_e)+
−
          moreover from is_runing have rd_before: "thread \<in> readys s"+
−
            by (unfold runing_def, simp)+
−
          moreover have "thread \<in> readys (e # s)"+
−
          proof -+
−
            from is_runing+
−
            have "thread \<in> threads (e#s)" +
−
              by (unfold eq_e, auto simp:runing_def readys_def)+
−
            moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"+
−
            proof+
−
              fix cs1+
−
              { assume eq_cs: "cs1 = cs" +
−
                have "\<not> waiting (e # s) thread cs1"+
−
                proof -+
−
                  from eq_wq+
−
                  have "thread \<notin> set (wq (e#s) cs1)"+
−
                    apply(unfold eq_e wq_def eq_cs s_holding_def)+
−
                    apply (auto simp:Let_def)+
−
                  proof -+
−
                    assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"+
−
                    with eq_set have "thread \<in> set rest" by simp+
−
                    with wq_distinct[OF step_back_vt[OF vtv], of cs]+
−
                    and eq_wq show False by auto+
−
                  qed+
−
                  thus ?thesis by (simp add:s_waiting_def)+
−
                qed+
−
              } moreover {+
−
                assume neq_cs: "cs1 \<noteq> cs"+
−
                  have "\<not> waiting (e # s) thread cs1" +
−
                  proof -+
−
                    from wq_v_neq [OF neq_cs[symmetric]]+
−
                    have "wq (V thread cs # s) cs1 = wq s cs1" .+
−
                    moreover have "\<not> waiting s thread cs1" +
−
                    proof -+
−
                      from runing_ready and is_runing+
−
                      have "thread \<in> readys s" by auto+
−
                      thus ?thesis by (simp add:readys_def)+
−
                    qed+
−
                    ultimately show ?thesis +
−
                      by (auto simp:s_waiting_def eq_e)+
−
                  qed+
−
              } ultimately show "\<not> waiting (e # s) thread cs1" by blast+
−
            qed+
−
            ultimately show ?thesis by (simp add:readys_def)+
−
          qed+
−
          moreover note eq_th ih+
−
          ultimately have ?thesis by auto+
−
        } moreover {+
−
          assume neq_th: "th \<noteq> thread"+
−
          from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" +
−
            by (simp add:cntP_def count_def)+
−
          from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" +
−
            by (simp add:cntV_def count_def)+
−
          have ?thesis+
−
          proof(cases "th \<in> set rest")+
−
            case False+
−
            have "(th \<in> readys (e # s)) = (th \<in> readys s)"+
−
              apply (insert step_back_vt[OF vtv])+
−
              by (unfold eq_e, rule readys_v_eq [OF _ neq_th eq_wq False], auto)+
−
            moreover have "cntCS (e#s) th = cntCS s th"+
−
              apply (insert neq_th, unfold eq_e cntCS_def holdents_def step_depend_v[OF vtv], auto)+
−
              proof -+
−
                have "{csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =+
−
                      {cs. (Cs cs, Th th) \<in> depend s}"+
−
                proof -+
−
                  from False eq_wq+
−
                  have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> depend s"+
−
                    apply (unfold next_th_def, auto)+
−
                  proof -+
−
                    assume ne: "rest \<noteq> []"+
−
                      and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"+
−
                      and eq_wq: "wq s cs = thread # rest"+
−
                    from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> +
−
                                  set (SOME q. distinct q \<and> set q = set rest)+
−
                                  " by simp+
−
                    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"+
−
                    proof(rule someI2)+
−
                      from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq+
−
                      show "distinct rest \<and> set rest = set rest" by auto+
−
                    next+
−
                      fix x assume "distinct x \<and> set x = set rest"+
−
                      with ne show "x \<noteq> []" by auto+
−
                    qed+
−
                    ultimately show +
−
                      "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"+
−
                      by auto+
−
                  qed    +
−
                  thus ?thesis by auto+
−
                qed+
−
                thus "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =+
−
                             card {cs. (Cs cs, Th th) \<in> depend s}" by simp +
−
              qed+
−
            moreover note ih eq_cnp eq_cnv eq_threads+
−
            ultimately show ?thesis by auto+
−
          next+
−
            case True+
−
            assume th_in: "th \<in> set rest"+
−
            show ?thesis+
−
            proof(cases "next_th s thread cs th")+
−
              case False+
−
              with eq_wq and th_in have +
−
                neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")+
−
                by (auto simp:next_th_def)+
−
              have "(th \<in> readys (e # s)) = (th \<in> readys s)"+
−
              proof -+
−
                from eq_wq and th_in+
−
                have "\<not> th \<in> readys s"+
−
                  apply (auto simp:readys_def s_waiting_def)+
−
                  apply (rule_tac x = cs in exI, auto)+
−
                  by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto)+
−
                moreover +
−
                from eq_wq and th_in and neq_hd+
−
                have "\<not> (th \<in> readys (e # s))"+
−
                  apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)+
−
                  by (rule_tac x = cs in exI, auto simp:eq_set)+
−
                ultimately show ?thesis by auto+
−
              qed+
−
              moreover have "cntCS (e#s) th = cntCS s th" +
−
              proof -+
−
                from eq_wq and  th_in and neq_hd+
−
                have "(holdents (e # s) th) = (holdents s th)"+
−
                  apply (unfold eq_e step_depend_v[OF vtv], +
−
                         auto simp:next_th_def eq_set s_depend_def holdents_def wq_def+
−
                                   Let_def cs_holding_def)+
−
                  by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto simp:wq_def)+
−
                thus ?thesis by (simp add:cntCS_def)+
−
              qed+
−
              moreover note ih eq_cnp eq_cnv eq_threads+
−
              ultimately show ?thesis by auto+
−
            next+
−
              case True+
−
              let ?rest = " (SOME q. distinct q \<and> set q = set rest)"+
−
              let ?t = "hd ?rest"+
−
              from True eq_wq th_in neq_th+
−
              have "th \<in> readys (e # s)"+
−
                apply (auto simp:eq_e readys_def s_waiting_def wq_def+
−
                        Let_def next_th_def)+
−
              proof -+
−
                assume eq_wq: "waiting_queue (schs s) cs = thread # rest"+
−
                  and t_in: "?t \<in> set rest"+
−
                show "?t \<in> threads s"+
−
                proof(rule wq_threads[OF step_back_vt[OF vtv]])+
−
                  from eq_wq and t_in+
−
                  show "?t \<in> set (wq s cs)" by (auto simp:wq_def)+
−
                qed+
−
              next+
−
                fix csa+
−
                assume eq_wq: "waiting_queue (schs s) cs = thread # rest"+
−
                  and t_in: "?t \<in> set rest"+
−
                  and neq_cs: "csa \<noteq> cs"+
−
                  and t_in': "?t \<in>  set (waiting_queue (schs s) csa)"+
−
                show "?t = hd (waiting_queue (schs s) csa)"+
−
                proof -+
−
                  { assume neq_hd': "?t \<noteq> hd (waiting_queue (schs s) csa)"+
−
                    from wq_distinct[OF step_back_vt[OF vtv], of cs] and +
−
                    eq_wq[folded wq_def] and t_in eq_wq+
−
                    have "?t \<noteq> thread" by auto+
−
                    with eq_wq and t_in+
−
                    have w1: "waiting s ?t cs"+
−
                      by (auto simp:s_waiting_def wq_def)+
−
                    from t_in' neq_hd'+
−
                    have w2: "waiting s ?t csa"+
−
                      by (auto simp:s_waiting_def wq_def)+
−
                    from waiting_unique[OF step_back_vt[OF vtv] w1 w2]+
−
                    and neq_cs have "False" by auto+
−
                  } thus ?thesis by auto+
−
                qed+
−
              qed+
−
              moreover have "cntP s th = cntV s th + cntCS s th + 1"+
−
              proof -+
−
                have "th \<notin> readys s" +
−
                proof -+
−
                  from True eq_wq neq_th th_in+
−
                  show ?thesis+
−
                    apply (unfold readys_def s_waiting_def, auto)+
−
                    by (rule_tac x = cs in exI, auto)+
−
                qed+
−
                moreover have "th \<in> threads s"+
−
                proof -+
−
                  from th_in eq_wq+
−
                  have "th \<in> set (wq s cs)" by simp+
−
                  from wq_threads [OF step_back_vt[OF vtv] this] +
−
                  show ?thesis .+
−
                qed+
−
                ultimately show ?thesis using ih by auto+
−
              qed+
−
              moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"+
−
                apply (unfold cntCS_def holdents_def eq_e step_depend_v[OF vtv], auto)+
−
              proof -+
−
                show "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs} =+
−
                               Suc (card {cs. (Cs cs, Th th) \<in> depend s})"+
−
                  (is "card ?A = Suc (card ?B)")+
−
                proof -+
−
                  have "?A = insert cs ?B" by auto+
−
                  hence "card ?A = card (insert cs ?B)" by simp+
−
                  also have "\<dots> = Suc (card ?B)"+
−
                  proof(rule card_insert_disjoint)+
−
                    have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` depend s)" +
−
                      apply (auto simp:image_def)+
−
                      by (rule_tac x = "(Cs x, Th th)" in bexI, auto)+
−
                    with finite_depend[OF step_back_vt[OF vtv]]+
−
                    show "finite {cs. (Cs cs, Th th) \<in> depend s}" by (auto intro:finite_subset)+
−
                  next+
−
                    show "cs \<notin> {cs. (Cs cs, Th th) \<in> depend s}"+
−
                    proof+
−
                      assume "cs \<in> {cs. (Cs cs, Th th) \<in> depend s}"+
−
                      hence "(Cs cs, Th th) \<in> depend s" by simp+
−
                      with True neq_th eq_wq show False+
−
                        by (auto simp:next_th_def s_depend_def cs_holding_def)+
−
                    qed+
−
                  qed+
−
                  finally show ?thesis .+
−
                qed+
−
              qed+
−
              moreover note eq_cnp eq_cnv+
−
              ultimately show ?thesis by simp+
−
            qed+
−
          qed+
−
        } ultimately show ?thesis by blast+
−
      qed+
−
    next+
−
      case (thread_set thread prio)+
−
      assume eq_e: "e = Set thread prio"+
−
        and is_runing: "thread \<in> runing s"+
−
      show ?thesis+
−
      proof -+
−
        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)+
−
        from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)+
−
        have eq_cncs: "cntCS (e#s) th = cntCS s th"+
−
          unfolding cntCS_def holdents_def+
−
          by (simp add:depend_set_unchanged eq_e)+
−
        from eq_e have eq_readys: "readys (e#s) = readys s" +
−
          by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,+
−
                  auto simp:Let_def)+
−
        { assume "th \<noteq> thread"+
−
          with eq_readys eq_e+
−
          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = +
−
                      (th \<in> readys (s) \<or> th \<notin> threads (s))" +
−
            by (simp add:threads.simps)+
−
          with eq_cnp eq_cnv eq_cncs ih is_runing+
−
          have ?thesis by simp+
−
        } moreover {+
−
          assume eq_th: "th = thread"+
−
          with is_runing ih have " cntP s th  = cntV s th + cntCS s th" +
−
            by (unfold runing_def, auto)+
−
          moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"+
−
            by (simp add:runing_def)+
−
          moreover note eq_cnp eq_cnv eq_cncs+
−
          ultimately have ?thesis by auto+
−
        } ultimately show ?thesis by blast+
−
      qed   +
−
    qed+
−
  next+
−
    case vt_nil+
−
    show ?case +
−
      by (unfold cntP_def cntV_def cntCS_def, +
−
        auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)+
−
  qed+
−
qed+
−
+
−
lemma not_thread_cncs:+
−
  fixes th s+
−
  assumes vt: "vt step s"+
−
  and not_in: "th \<notin> threads s" +
−
  shows "cntCS s th = 0"+
−
proof -+
−
  from vt not_in show ?thesis+
−
  proof(induct arbitrary:th)+
−
    case (vt_cons s e th)+
−
    assume vt: "vt step s"+
−
      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"+
−
      and stp: "step s e"+
−
      and not_in: "th \<notin> threads (e # s)"+
−
    from stp show ?case+
−
    proof(cases)+
−
      case (thread_create thread prio)+
−
      assume eq_e: "e = Create thread prio"+
−
        and not_in': "thread \<notin> threads s"+
−
      have "cntCS (e # s) th = cntCS s th"+
−
        apply (unfold eq_e cntCS_def holdents_def)+
−
        by (simp add:depend_create_unchanged)+
−
      moreover have "th \<notin> threads s" +
−
      proof -+
−
        from not_in eq_e show ?thesis by simp+
−
      qed+
−
      moreover note ih ultimately show ?thesis by auto+
−
    next+
−
      case (thread_exit thread)+
−
      assume eq_e: "e = Exit thread"+
−
      and nh: "holdents s thread = {}"+
−
      have eq_cns: "cntCS (e # s) th = cntCS s th"+
−
        apply (unfold eq_e cntCS_def holdents_def)+
−
        by (simp add:depend_exit_unchanged)+
−
      show ?thesis+
−
      proof(cases "th = thread")+
−
        case True+
−
        have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)+
−
        with eq_cns show ?thesis by simp+
−
      next+
−
        case False+
−
        with not_in and eq_e+
−
        have "th \<notin> threads s" by simp+
−
        from ih[OF this] and eq_cns show ?thesis by simp+
−
      qed+
−
    next+
−
      case (thread_P thread cs)+
−
      assume eq_e: "e = P thread cs"+
−
      and is_runing: "thread \<in> runing s"+
−
      from prems have vtp: "vt step (P thread cs#s)" by auto+
−
      have neq_th: "th \<noteq> thread" +
−
      proof -+
−
        from not_in eq_e have "th \<notin> threads s" by simp+
−
        moreover from is_runing have "thread \<in> threads s"+
−
          by (simp add:runing_def readys_def)+
−
        ultimately show ?thesis by auto+
−
      qed+
−
      hence "cntCS (e # s) th  = cntCS s th "+
−
        apply (unfold cntCS_def holdents_def eq_e)+
−
        by (unfold step_depend_p[OF vtp], auto)+
−
      moreover have "cntCS s th = 0"+
−
      proof(rule ih)+
−
        from not_in eq_e show "th \<notin> threads s" by simp+
−
      qed+
−
      ultimately show ?thesis by simp+
−
    next+
−
      case (thread_V thread cs)+
−
      assume eq_e: "e = V thread cs"+
−
        and is_runing: "thread \<in> runing s"+
−
        and hold: "holding s thread cs"+
−
      have neq_th: "th \<noteq> thread" +
−
      proof -+
−
        from not_in eq_e have "th \<notin> threads s" by simp+
−
        moreover from is_runing have "thread \<in> threads s"+
−
          by (simp add:runing_def readys_def)+
−
        ultimately show ?thesis by auto+
−
      qed+
−
      from prems have vtv: "vt step (V thread cs#s)" by auto+
−
      from hold obtain rest +
−
        where eq_wq: "wq s cs = thread # rest"+
−
        by (case_tac "wq s cs", auto simp:s_holding_def)+
−
      from not_in eq_e eq_wq+
−
      have "\<not> next_th s thread cs th"+
−
        apply (auto simp:next_th_def)+
−
      proof -+
−
        assume ne: "rest \<noteq> []"+
−
          and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")+
−
        have "?t \<in> set rest"+
−
        proof(rule someI2)+
−
          from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq+
−
          show "distinct rest \<and> set rest = set rest" by auto+
−
        next+
−
          fix x assume "distinct x \<and> set x = set rest" with ne+
−
          show "hd x \<in> set rest" by (cases x, auto)+
−
        qed+
−
        with eq_wq have "?t \<in> set (wq s cs)" by simp+
−
        from wq_threads[OF step_back_vt[OF vtv], OF this] and ni+
−
        show False by auto+
−
      qed+
−
      moreover note neq_th eq_wq+
−
      ultimately have "cntCS (e # s) th  = cntCS s th"+
−
        by (unfold eq_e cntCS_def holdents_def step_depend_v[OF vtv], auto)+
−
      moreover have "cntCS s th = 0"+
−
      proof(rule ih)+
−
        from not_in eq_e show "th \<notin> threads s" by simp+
−
      qed+
−
      ultimately show ?thesis by simp+
−
    next+
−
      case (thread_set thread prio)+
−
      print_facts+
−
      assume eq_e: "e = Set thread prio"+
−
        and is_runing: "thread \<in> runing s"+
−
      from not_in and eq_e have "th \<notin> threads s" by auto+
−
      from ih [OF this] and eq_e+
−
      show ?thesis +
−
        apply (unfold eq_e cntCS_def holdents_def)+
−
        by (simp add:depend_set_unchanged)+
−
    qed+
−
    next+
−
      case vt_nil+
−
      show ?case+
−
      by (unfold cntCS_def, +
−
        auto simp:count_def holdents_def s_depend_def wq_def cs_holding_def)+
−
  qed+
−
qed+
−
+
−
lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"+
−
  by (auto simp:s_waiting_def cs_waiting_def)+
−
+
−
lemma dm_depend_threads:+
−
  fixes th s+
−
  assumes vt: "vt step s"+
−
  and in_dom: "(Th th) \<in> Domain (depend s)"+
−
  shows "th \<in> threads s"+
−
proof -+
−
  from in_dom obtain n where "(Th th, n) \<in> depend s" by auto+
−
  moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto+
−
  ultimately have "(Th th, Cs cs) \<in> depend s" by simp+
−
  hence "th \<in> set (wq s cs)"+
−
    by (unfold s_depend_def, auto simp:cs_waiting_def)+
−
  from wq_threads [OF vt this] show ?thesis .+
−
qed+
−
+
−
lemma cp_eq_cpreced: "cp s th = cpreced s (wq s) th"+
−
proof(unfold cp_def wq_def, induct s)+
−
  case (Cons e s')+
−
  show ?case+
−
    by (auto simp:Let_def)+
−
next+
−
  case Nil+
−
  show ?case by (auto simp:Let_def)+
−
qed+
−
+
−
+
−
lemma runing_unique:+
−
  fixes th1 th2 s+
−
  assumes vt: "vt step s"+
−
  and runing_1: "th1 \<in> runing s"+
−
  and runing_2: "th2 \<in> runing s"+
−
  shows "th1 = th2"+
−
proof -+
−
  from runing_1 and runing_2 have "cp s th1 = cp s th2"+
−
    by (unfold runing_def, simp)+
−
  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1)) =+
−
                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependents (wq s) th2))"+
−
    (is "Max (?f ` ?A) = Max (?f ` ?B)")+
−
    by (unfold cp_eq_cpreced cpreced_def)+
−
  obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"+
−
  proof -+
−
    have h1: "finite (?f ` ?A)"+
−
    proof -+
−
      have "finite ?A" +
−
      proof -+
−
        have "finite (dependents (wq s) th1)"+
−
        proof-+
−
          have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"+
−
          proof -+
−
            let ?F = "\<lambda> (x, y). the_th x"+
−
            have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"+
−
              apply (auto simp:image_def)+
−
              by (rule_tac x = "(Th x, Th th1)" in bexI, auto)+
−
            moreover have "finite \<dots>"+
−
            proof -+
−
              from finite_depend[OF vt] have "finite (depend s)" .+
−
              hence "finite ((depend (wq s))\<^sup>+)"+
−
                apply (unfold finite_trancl)+
−
                by (auto simp: s_depend_def cs_depend_def wq_def)+
−
              thus ?thesis by auto+
−
            qed+
−
            ultimately show ?thesis by (auto intro:finite_subset)+
−
          qed+
−
          thus ?thesis by (simp add:cs_dependents_def)+
−
        qed+
−
        thus ?thesis by simp+
−
      qed+
−
      thus ?thesis by auto+
−
    qed+
−
    moreover have h2: "(?f ` ?A) \<noteq> {}"+
−
    proof -+
−
      have "?A \<noteq> {}" by simp+
−
      thus ?thesis by simp+
−
    qed+
−
    from Max_in [OF h1 h2]+
−
    have "Max (?f ` ?A) \<in> (?f ` ?A)" .+
−
    thus ?thesis by (auto intro:that)+
−
  qed+
−
  obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"+
−
  proof -+
−
    have h1: "finite (?f ` ?B)"+
−
    proof -+
−
      have "finite ?B" +
−
      proof -+
−
        have "finite (dependents (wq s) th2)"+
−
        proof-+
−
          have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"+
−
          proof -+
−
            let ?F = "\<lambda> (x, y). the_th x"+
−
            have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"+
−
              apply (auto simp:image_def)+
−
              by (rule_tac x = "(Th x, Th th2)" in bexI, auto)+
−
            moreover have "finite \<dots>"+
−
            proof -+
−
              from finite_depend[OF vt] have "finite (depend s)" .+
−
              hence "finite ((depend (wq s))\<^sup>+)"+
−
                apply (unfold finite_trancl)+
−
                by (auto simp: s_depend_def cs_depend_def wq_def)+
−
              thus ?thesis by auto+
−
            qed+
−
            ultimately show ?thesis by (auto intro:finite_subset)+
−
          qed+
−
          thus ?thesis by (simp add:cs_dependents_def)+
−
        qed+
−
        thus ?thesis by simp+
−
      qed+
−
      thus ?thesis by auto+
−
    qed+
−
    moreover have h2: "(?f ` ?B) \<noteq> {}"+
−
    proof -+
−
      have "?B \<noteq> {}" by simp+
−
      thus ?thesis by simp+
−
    qed+
−
    from Max_in [OF h1 h2]+
−
    have "Max (?f ` ?B) \<in> (?f ` ?B)" .+
−
    thus ?thesis by (auto intro:that)+
−
  qed+
−
  from eq_f_th1 eq_f_th2 eq_max +
−
  have eq_preced: "preced th1' s = preced th2' s" by auto+
−
  hence eq_th12: "th1' = th2'"+
−
  proof (rule preced_unique)+
−
    from th1_in have "th1' = th1 \<or> (th1' \<in> dependents (wq s) th1)" by simp+
−
    thus "th1' \<in> threads s"+
−
    proof+
−
      assume "th1' \<in> dependents (wq s) th1"+
−
      hence "(Th th1') \<in> Domain ((depend s)^+)"+
−
        apply (unfold cs_dependents_def cs_depend_def s_depend_def)+
−
        by (auto simp:Domain_def)+
−
      hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)+
−
      from dm_depend_threads[OF vt this] show ?thesis .+
−
    next+
−
      assume "th1' = th1"+
−
      with runing_1 show ?thesis+
−
        by (unfold runing_def readys_def, auto)+
−
    qed+
−
  next+
−
    from th2_in have "th2' = th2 \<or> (th2' \<in> dependents (wq s) th2)" by simp+
−
    thus "th2' \<in> threads s"+
−
    proof+
−
      assume "th2' \<in> dependents (wq s) th2"+
−
      hence "(Th th2') \<in> Domain ((depend s)^+)"+
−
        apply (unfold cs_dependents_def cs_depend_def s_depend_def)+
−
        by (auto simp:Domain_def)+
−
      hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)+
−
      from dm_depend_threads[OF vt this] show ?thesis .+
−
    next+
−
      assume "th2' = th2"+
−
      with runing_2 show ?thesis+
−
        by (unfold runing_def readys_def, auto)+
−
    qed+
−
  qed+
−
  from th1_in have "th1' = th1 \<or> th1' \<in> dependents (wq s) th1" by simp+
−
  thus ?thesis+
−
  proof+
−
    assume eq_th': "th1' = th1"+
−
    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp+
−
    thus ?thesis+
−
    proof+
−
      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp+
−
    next+
−
      assume "th2' \<in> dependents (wq s) th2"+
−
      with eq_th12 eq_th' have "th1 \<in> dependents (wq s) th2" by simp+
−
      hence "(Th th1, Th th2) \<in> (depend s)^+"+
−
        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)+
−
      hence "Th th1 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]+
−
        by auto+
−
      hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)+
−
      then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)+
−
      from depend_target_th [OF this]+
−
      obtain cs' where "n = Cs cs'" by auto+
−
      with d have "(Th th1, Cs cs') \<in> depend s" by simp+
−
      with runing_1 have "False"+
−
        apply (unfold runing_def readys_def s_depend_def)+
−
        by (auto simp:eq_waiting)+
−
      thus ?thesis by simp+
−
    qed+
−
  next+
−
    assume th1'_in: "th1' \<in> dependents (wq s) th1"+
−
    from th2_in have "th2' = th2 \<or> th2' \<in> dependents (wq s) th2" by simp+
−
    thus ?thesis +
−
    proof+
−
      assume "th2' = th2"+
−
      with th1'_in eq_th12 have "th2 \<in> dependents (wq s) th1" by simp+
−
      hence "(Th th2, Th th1) \<in> (depend s)^+"+
−
        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)+
−
      hence "Th th2 \<in> Domain ((depend s)^+)" using Domain_def [of "(depend s)^+"]+
−
        by auto+
−
      hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)+
−
      then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)+
−
      from depend_target_th [OF this]+
−
      obtain cs' where "n = Cs cs'" by auto+
−
      with d have "(Th th2, Cs cs') \<in> depend s" by simp+
−
      with runing_2 have "False"+
−
        apply (unfold runing_def readys_def s_depend_def)+
−
        by (auto simp:eq_waiting)+
−
      thus ?thesis by simp+
−
    next+
−
      assume "th2' \<in> dependents (wq s) th2"+
−
      with eq_th12 have "th1' \<in> dependents (wq s) th2" by simp+
−
      hence h1: "(Th th1', Th th2) \<in> (depend s)^+"+
−
        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)+
−
      from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"+
−
        by (unfold cs_dependents_def s_depend_def cs_depend_def, simp)+
−
      show ?thesis+
−
      proof(rule dchain_unique[OF vt h1 _ h2, symmetric])+
−
        from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)+
−
        from runing_2 show "th2 \<in> readys s" by (simp add:runing_def) +
−
      qed+
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma create_pre:+
−
  assumes stp: "step s e"+
−
  and not_in: "th \<notin> threads s"+
−
  and is_in: "th \<in> threads (e#s)"+
−
  obtains prio where "e = Create th prio"+
−
proof -+
−
  from assms  +
−
  show ?thesis+
−
  proof(cases)+
−
    case (thread_create thread prio)+
−
    with is_in not_in have "e = Create th prio" by simp+
−
    from that[OF this] show ?thesis .+
−
  next+
−
    case (thread_exit thread)+
−
    with assms show ?thesis by (auto intro!:that)+
−
  next+
−
    case (thread_P thread)+
−
    with assms show ?thesis by (auto intro!:that)+
−
  next+
−
    case (thread_V thread)+
−
    with assms show ?thesis by (auto intro!:that)+
−
  next +
−
    case (thread_set thread)+
−
    with assms show ?thesis by (auto intro!:that)+
−
  qed+
−
qed+
−
+
−
lemma length_down_to_in: +
−
  assumes le_ij: "i \<le> j"+
−
    and le_js: "j \<le> length s"+
−
  shows "length (down_to j i s) = j - i"+
−
proof -+
−
  have "length (down_to j i s) = length (from_to i j (rev s))"+
−
    by (unfold down_to_def, auto)+
−
  also have "\<dots> = j - i"+
−
  proof(rule length_from_to_in[OF le_ij])+
−
    from le_js show "j \<le> length (rev s)" by simp+
−
  qed+
−
  finally show ?thesis .+
−
qed+
−
+
−
+
−
lemma moment_head: +
−
  assumes le_it: "Suc i \<le> length t"+
−
  obtains e where "moment (Suc i) t = e#moment i t"+
−
proof -+
−
  have "i \<le> Suc i" by simp+
−
  from length_down_to_in [OF this le_it]+
−
  have "length (down_to (Suc i) i t) = 1" by auto+
−
  then obtain e where "down_to (Suc i) i t = [e]"+
−
    apply (cases "(down_to (Suc i) i t)") by auto+
−
  moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"+
−
    by (rule down_to_conc[symmetric], auto)+
−
  ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"+
−
    by (auto simp:down_to_moment)+
−
  from that [OF this] show ?thesis .+
−
qed+
−
+
−
lemma cnp_cnv_eq:+
−
  fixes th s+
−
  assumes "vt step s"+
−
  and "th \<notin> threads s"+
−
  shows "cntP s th = cntV s th"+
−
proof -+
−
  from assms show ?thesis+
−
  proof(induct)+
−
    case (vt_cons s e)+
−
    have ih: "th \<notin> threads s \<Longrightarrow> cntP s th = cntV s th" by fact+
−
    have not_in: "th \<notin> threads (e # s)" by fact+
−
    have "step s e" by fact+
−
    thus ?case proof(cases)+
−
      case (thread_create thread prio)+
−
      assume eq_e: "e = Create thread prio"+
−
      hence "thread \<in> threads (e#s)" by simp+
−
      with not_in and eq_e have "th \<notin> threads s" by auto+
−
      from ih [OF this] show ?thesis using eq_e+
−
        by (auto simp:cntP_def cntV_def count_def)+
−
    next+
−
      case (thread_exit thread)+
−
      assume eq_e: "e = Exit thread"+
−
        and not_holding: "holdents s thread = {}"+
−
      have vt_s: "vt step s" by fact+
−
      from finite_holding[OF vt_s] have "finite (holdents s thread)" .+
−
      with not_holding have "cntCS s thread = 0" by (unfold cntCS_def, auto)+
−
      moreover have "thread \<in> readys s" using thread_exit by (auto simp:runing_def)+
−
      moreover note cnp_cnv_cncs[OF vt_s, of thread]+
−
      ultimately have eq_thread: "cntP s thread = cntV s thread" by auto+
−
      show ?thesis+
−
      proof(cases "th = thread")+
−
        case True+
−
        with eq_thread eq_e show ?thesis +
−
          by (auto simp:cntP_def cntV_def count_def)+
−
      next+
−
        case False+
−
        with not_in and eq_e have "th \<notin> threads s" by simp+
−
        from ih[OF this] and eq_e show ?thesis +
−
           by (auto simp:cntP_def cntV_def count_def)+
−
      qed+
−
    next+
−
      case (thread_P thread cs)+
−
      assume eq_e: "e = P thread cs"+
−
      have "thread \<in> runing s" by fact+
−
      with not_in eq_e have neq_th: "thread \<noteq> th" +
−
        by (auto simp:runing_def readys_def)+
−
      from not_in eq_e have "th \<notin> threads s" by simp+
−
      from ih[OF this] and neq_th and eq_e show ?thesis+
−
        by (auto simp:cntP_def cntV_def count_def)+
−
    next+
−
      case (thread_V thread cs)+
−
      assume eq_e: "e = V thread cs"+
−
      have "thread \<in> runing s" by fact+
−
      with not_in eq_e have neq_th: "thread \<noteq> th" +
−
        by (auto simp:runing_def readys_def)+
−
      from not_in eq_e have "th \<notin> threads s" by simp+
−
      from ih[OF this] and neq_th and eq_e show ?thesis+
−
        by (auto simp:cntP_def cntV_def count_def)+
−
    next+
−
      case (thread_set thread prio)+
−
      assume eq_e: "e = Set thread prio"+
−
        and "thread \<in> runing s"+
−
      hence "thread \<in> threads (e#s)" +
−
        by (simp add:runing_def readys_def)+
−
      with not_in and eq_e have "th \<notin> threads s" by auto+
−
      from ih [OF this] show ?thesis using eq_e+
−
        by (auto simp:cntP_def cntV_def count_def)  +
−
    qed+
−
  next+
−
    case vt_nil+
−
    show ?case by (auto simp:cntP_def cntV_def count_def)+
−
  qed+
−
qed+
−
+
−
lemma eq_depend: +
−
  "depend (wq s) = depend s"+
−
by (unfold cs_depend_def s_depend_def, auto)+
−
+
−
lemma count_eq_dependents:+
−
  assumes vt: "vt step s"+
−
  and eq_pv: "cntP s th = cntV s th"+
−
  shows "dependents (wq s) th = {}"+
−
proof -+
−
  from cnp_cnv_cncs[OF vt] and eq_pv+
−
  have "cntCS s th = 0" +
−
    by (auto split:if_splits)+
−
  moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"+
−
  proof -+
−
    from finite_holding[OF vt, of th] show ?thesis+
−
      by (simp add:holdents_def)+
−
  qed+
−
  ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"+
−
    by (unfold cntCS_def holdents_def cs_dependents_def, auto)+
−
  show ?thesis+
−
  proof(unfold cs_dependents_def)+
−
    { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"+
−
      then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto+
−
      hence "False"+
−
      proof(cases)+
−
        assume "(Th th', Th th) \<in> depend (wq s)"+
−
        thus "False" by (auto simp:cs_depend_def)+
−
      next+
−
        fix c+
−
        assume "(c, Th th) \<in> depend (wq s)"+
−
        with h and eq_depend show "False"+
−
          by (cases c, auto simp:cs_depend_def)+
−
      qed+
−
    } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto+
−
  qed+
−
qed+
−
+
−
lemma dependents_threads:+
−
  fixes s th+
−
  assumes vt: "vt step s"+
−
  shows "dependents (wq s) th \<subseteq> threads s"+
−
proof+
−
  { fix th th'+
−
    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"+
−
    have "Th th \<in> Domain (depend s)"+
−
    proof -+
−
      from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto+
−
      hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)+
−
      with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp+
−
      thus ?thesis using eq_depend by simp+
−
    qed+
−
    from dm_depend_threads[OF vt this]+
−
    have "th \<in> threads s" .+
−
  } note hh = this+
−
  fix th1 +
−
  assume "th1 \<in> dependents (wq s) th"+
−
  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"+
−
    by (unfold cs_dependents_def, simp)+
−
  from hh [OF this] show "th1 \<in> threads s" .+
−
qed+
−
+
−
lemma finite_threads:+
−
  assumes vt: "vt step s"+
−
  shows "finite (threads s)"+
−
proof -+
−
  from vt show ?thesis+
−
  proof(induct)+
−
    case (vt_cons s e)+
−
    assume vt: "vt step s"+
−
    and step: "step s e"+
−
    and ih: "finite (threads s)"+
−
    from step+
−
    show ?case+
−
    proof(cases)+
−
      case (thread_create thread prio)+
−
      assume eq_e: "e = Create thread prio"+
−
      with ih+
−
      show ?thesis by (unfold eq_e, auto)+
−
    next+
−
      case (thread_exit thread)+
−
      assume eq_e: "e = Exit thread"+
−
      with ih show ?thesis +
−
        by (unfold eq_e, auto)+
−
    next+
−
      case (thread_P thread cs)+
−
      assume eq_e: "e = P thread cs"+
−
      with ih show ?thesis by (unfold eq_e, auto)+
−
    next+
−
      case (thread_V thread cs)+
−
      assume eq_e: "e = V thread cs"+
−
      with ih show ?thesis by (unfold eq_e, auto)+
−
    next +
−
      case (thread_set thread prio)+
−
      from vt_cons thread_set show ?thesis by simp+
−
    qed+
−
  next+
−
    case vt_nil+
−
    show ?case by (auto)+
−
  qed+
−
qed+
−
+
−
lemma Max_f_mono:+
−
  assumes seq: "A \<subseteq> B"+
−
  and np: "A \<noteq> {}"+
−
  and fnt: "finite B"+
−
  shows "Max (f ` A) \<le> Max (f ` B)"+
−
proof(rule Max_mono)+
−
  from seq show "f ` A \<subseteq> f ` B" by auto+
−
next+
−
  from np show "f ` A \<noteq> {}" by auto+
−
next+
−
  from fnt and seq show "finite (f ` B)" by auto+
−
qed+
−
+
−
lemma cp_le:+
−
  assumes vt: "vt step s"+
−
  and th_in: "th \<in> threads s"+
−
  shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"+
−
proof(unfold cp_eq_cpreced cpreced_def cs_dependents_def)+
−
  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))+
−
         \<le> Max ((\<lambda>th. preced th s) ` threads s)"+
−
    (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")+
−
  proof(rule Max_f_mono)+
−
    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp+
−
  next+
−
    from finite_threads [OF vt]+
−
    show "finite (threads s)" .+
−
  next+
−
    from th_in+
−
    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"+
−
      apply (auto simp:Domain_def)+
−
      apply (rule_tac dm_depend_threads[OF vt])+
−
      apply (unfold trancl_domain [of "depend s", symmetric])+
−
      by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)+
−
  qed+
−
qed+
−
+
−
lemma le_cp:+
−
  assumes vt: "vt step s"+
−
  shows "preced th s \<le> cp s th"+
−
proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)+
−
  show "Prc (original_priority th s) (birthtime th s)+
−
    \<le> Max (insert (Prc (original_priority th s) (birthtime th s))+
−
            ((\<lambda>th. Prc (original_priority th s) (birthtime th s)) ` dependents (wq s) th))"+
−
    (is "?l \<le> Max (insert ?l ?A)")+
−
  proof(cases "?A = {}")+
−
    case False+
−
    have "finite ?A" (is "finite (?f ` ?B)")+
−
    proof -+
−
      have "finite ?B" +
−
      proof-+
−
        have "finite {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}"+
−
        proof -+
−
          let ?F = "\<lambda> (x, y). the_th x"+
−
          have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"+
−
            apply (auto simp:image_def)+
−
            by (rule_tac x = "(Th x, Th th)" in bexI, auto)+
−
          moreover have "finite \<dots>"+
−
          proof -+
−
            from finite_depend[OF vt] have "finite (depend s)" .+
−
            hence "finite ((depend (wq s))\<^sup>+)"+
−
              apply (unfold finite_trancl)+
−
              by (auto simp: s_depend_def cs_depend_def wq_def)+
−
            thus ?thesis by auto+
−
          qed+
−
          ultimately show ?thesis by (auto intro:finite_subset)+
−
        qed+
−
        thus ?thesis by (simp add:cs_dependents_def)+
−
      qed+
−
      thus ?thesis by simp+
−
    qed+
−
    from Max_insert [OF this False, of ?l] show ?thesis by auto+
−
  next+
−
    case True+
−
    thus ?thesis by auto+
−
  qed+
−
qed+
−
+
−
lemma max_cp_eq: +
−
  assumes vt: "vt step s"+
−
  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"+
−
  (is "?l = ?r")+
−
proof(cases "threads s = {}")+
−
  case True+
−
  thus ?thesis by auto+
−
next+
−
  case False+
−
  have "?l \<in> ((cp s) ` threads s)"+
−
  proof(rule Max_in)+
−
    from finite_threads[OF vt] +
−
    show "finite (cp s ` threads s)" by auto+
−
  next+
−
    from False show "cp s ` threads s \<noteq> {}" by auto+
−
  qed+
−
  then obtain th +
−
    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto+
−
  have "\<dots> \<le> ?r" by (rule cp_le[OF vt th_in])+
−
  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")+
−
  proof -+
−
    have "?r \<in> (?f ` ?A)"+
−
    proof(rule Max_in)+
−
      from finite_threads[OF vt]+
−
      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto+
−
    next+
−
      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto+
−
    qed+
−
    then obtain th' where +
−
      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto+
−
    from le_cp [OF vt, of th']  eq_r+
−
    have "?r \<le> cp s th'" by auto+
−
    moreover have "\<dots> \<le> cp s th"+
−
    proof(fold eq_l)+
−
      show " cp s th' \<le> Max (cp s ` threads s)"+
−
      proof(rule Max_ge)+
−
        from th_in' show "cp s th' \<in> cp s ` threads s"+
−
          by auto+
−
      next+
−
        from finite_threads[OF vt]+
−
        show "finite (cp s ` threads s)" by auto+
−
      qed+
−
    qed+
−
    ultimately show ?thesis by auto+
−
  qed+
−
  ultimately show ?thesis using eq_l by auto+
−
qed+
−
+
−
lemma max_cp_readys_threads_pre:+
−
  assumes vt: "vt step s"+
−
  and np: "threads s \<noteq> {}"+
−
  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"+
−
proof(unfold max_cp_eq[OF vt])+
−
  show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"+
−
  proof -+
−
    let ?p = "Max ((\<lambda>th. preced th s) ` threads s)" +
−
    let ?f = "(\<lambda>th. preced th s)"+
−
    have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"+
−
    proof(rule Max_in)+
−
      from finite_threads[OF vt] show "finite (?f ` threads s)" by simp+
−
    next+
−
      from np show "?f ` threads s \<noteq> {}" by simp+
−
    qed+
−
    then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"+
−
      by (auto simp:Image_def)+
−
    from th_chain_to_ready [OF vt tm_in]+
−
    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .+
−
    thus ?thesis+
−
    proof+
−
      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "+
−
      then obtain th' where th'_in: "th' \<in> readys s" +
−
        and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto+
−
      have "cp s th' = ?f tm"+
−
      proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)+
−
        from dependents_threads[OF vt] finite_threads[OF vt]+
−
        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th'))" +
−
          by (auto intro:finite_subset)+
−
      next+
−
        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"+
−
        from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .+
−
        moreover have "p \<le> \<dots>"+
−
        proof(rule Max_ge)+
−
          from finite_threads[OF vt]+
−
          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp+
−
        next+
−
          from p_in and th'_in and dependents_threads[OF vt, of th']+
−
          show "p \<in> (\<lambda>th. preced th s) ` threads s"+
−
            by (auto simp:readys_def)+
−
        qed+
−
        ultimately show "p \<le> preced tm s" by auto+
−
      next+
−
        show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependents (wq s) th')"+
−
        proof -+
−
          from tm_chain+
−
          have "tm \<in> dependents (wq s) th'"+
−
            by (unfold cs_dependents_def s_depend_def cs_depend_def, auto)+
−
          thus ?thesis by auto+
−
        qed+
−
      qed+
−
      with tm_max+
−
      have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp+
−
      show ?thesis+
−
      proof (fold h, rule Max_eqI)+
−
        fix q +
−
        assume "q \<in> cp s ` readys s"+
−
        then obtain th1 where th1_in: "th1 \<in> readys s"+
−
          and eq_q: "q = cp s th1" by auto+
−
        show "q \<le> cp s th'"+
−
          apply (unfold h eq_q)+
−
          apply (unfold cp_eq_cpreced cpreced_def)+
−
          apply (rule Max_mono)+
−
        proof -+
−
          from dependents_threads [OF vt, of th1] th1_in+
−
          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<subseteq> +
−
                 (\<lambda>th. preced th s) ` threads s"+
−
            by (auto simp:readys_def)+
−
        next+
−
          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}" by simp+
−
        next+
−
          from finite_threads[OF vt] +
−
          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp+
−
        qed+
−
      next+
−
        from finite_threads[OF vt]+
−
        show "finite (cp s ` readys s)" by (auto simp:readys_def)+
−
      next+
−
        from th'_in+
−
        show "cp s th' \<in> cp s ` readys s" by simp+
−
      qed+
−
    next+
−
      assume tm_ready: "tm \<in> readys s"+
−
      show ?thesis+
−
      proof(fold tm_max)+
−
        have cp_eq_p: "cp s tm = preced tm s"+
−
        proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)+
−
          fix y +
−
          assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"+
−
          show "y \<le> preced tm s"+
−
          proof -+
−
            { fix y'+
−
              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependents (wq s) tm)"+
−
              have "y' \<le> preced tm s"+
−
              proof(unfold tm_max, rule Max_ge)+
−
                from hy' dependents_threads[OF vt, of tm]+
−
                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto+
−
              next+
−
                from finite_threads[OF vt] +
−
                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp+
−
              qed+
−
            } with hy show ?thesis by auto+
−
          qed+
−
        next+
−
          from dependents_threads[OF vt, of tm] finite_threads[OF vt]+
−
          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm))"+
−
            by (auto intro:finite_subset)+
−
        next+
−
          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependents (wq s) tm)"+
−
            by simp+
−
        qed +
−
        moreover have "Max (cp s ` readys s) = cp s tm"+
−
        proof(rule Max_eqI)+
−
          from tm_ready show "cp s tm \<in> cp s ` readys s" by simp+
−
        next+
−
          from finite_threads[OF vt]+
−
          show "finite (cp s ` readys s)" by (auto simp:readys_def)+
−
        next+
−
          fix y assume "y \<in> cp s ` readys s"+
−
          then obtain th1 where th1_readys: "th1 \<in> readys s"+
−
            and h: "y = cp s th1" by auto+
−
          show "y \<le> cp s tm"+
−
            apply(unfold cp_eq_p h)+
−
            apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)+
−
          proof -+
−
            from finite_threads[OF vt]+
−
            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp+
−
          next+
−
            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) \<noteq> {}"+
−
              by simp+
−
          next+
−
            from dependents_threads[OF vt, of th1] th1_readys+
−
            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependents (wq s) th1) +
−
                    \<subseteq> (\<lambda>th. preced th s) ` threads s"+
−
              by (auto simp:readys_def)+
−
          qed+
−
        qed+
−
        ultimately show " Max (cp s ` readys s) = preced tm s" by simp+
−
      qed +
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma max_cp_readys_threads:+
−
  assumes vt: "vt step s"+
−
  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"+
−
proof(cases "threads s = {}")+
−
  case True+
−
  thus ?thesis +
−
    by (auto simp:readys_def)+
−
next+
−
  case False+
−
  show ?thesis by (rule max_cp_readys_threads_pre[OF vt False])+
−
qed+
−
+
−
lemma readys_threads:+
−
  shows "readys s \<subseteq> threads s"+
−
proof+
−
  fix th+
−
  assume "th \<in> readys s"+
−
  thus "th \<in> threads s"+
−
    by (unfold readys_def, auto)+
−
qed+
−
+
−
lemma eq_holding: "holding (wq s) th cs = holding s th cs"+
−
  apply (unfold s_holding_def cs_holding_def, simp)+
−
  done+
−
+
−
lemma f_image_eq:+
−
  assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"+
−
  shows "f ` A = g ` A"+
−
proof+
−
  show "f ` A \<subseteq> g ` A"+
−
    by(rule image_subsetI, auto intro:h)+
−
next+
−
  show "g ` A \<subseteq> f ` A"+
−
   by(rule image_subsetI, auto intro:h[symmetric])+
−
qed+
−
+
−
end+
−