section {*+
−
  This file contains lemmas used to guide the recalculation of current precedence +
−
  after every system call (or system operation)+
−
*}+
−
theory Implementation+
−
imports PIPBasics+
−
begin+
−
+
−
text {* (* ddd *)+
−
  One beauty of our modelling is that we follow the definitional extension tradition of HOL.+
−
  The benefit of such a concise and miniature model is that  large number of intuitively +
−
  obvious facts are derived as lemmas, rather than asserted as axioms.+
−
*}+
−
+
−
text {*+
−
  However, the lemmas in the forthcoming several locales are no longer +
−
  obvious. These lemmas show how the current precedences should be recalculated +
−
  after every execution step (in our model, every step is represented by an event, +
−
  which in turn, represents a system call, or operation). Each operation is +
−
  treated in a separate locale.+
−
+
−
  The complication of current precedence recalculation comes +
−
  because the changing of RAG needs to be taken into account, +
−
  in addition to the changing of precedence. +
−
+
−
  The reason RAG changing affects current precedence is that,+
−
  according to the definition, current precedence +
−
  of a thread is the maximum of the precedences of every threads in its subtree, +
−
  where the notion of sub-tree in RAG is defined in RTree.thy.+
−
+
−
  Therefore, for each operation, lemmas about the change of precedences +
−
  and RAG are derived first, on which lemmas about current precedence +
−
  recalculation are based on.+
−
*}+
−
+
−
section {* The @{term Set} operation *}+
−
+
−
text {* (* ddd *)+
−
  The following locale @{text "step_set_cps"} investigates the recalculation +
−
  after the @{text "Set"} operation.+
−
*}+
−
locale step_set_cps =+
−
  fixes s' th prio s +
−
  -- {* @{text "s'"} is the system state before the operation *}+
−
  -- {* @{text "s"} is the system state after the operation *}+
−
  defines s_def : "s \<equiv> (Set th prio#s')" +
−
  -- {* @{text "s"} is assumed to be a legitimate state, from which+
−
         the legitimacy of @{text "s"} can be derived. *}+
−
  assumes vt_s: "vt s"+
−
+
−
sublocale step_set_cps < vat_s : valid_trace "s"+
−
proof+
−
  from vt_s show "vt s" .+
−
qed+
−
+
−
sublocale step_set_cps < vat_s' : valid_trace "s'"+
−
proof+
−
  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .+
−
qed+
−
+
−
context step_set_cps +
−
begin+
−
+
−
text {* (* ddd *)+
−
  The following two lemmas confirm that @{text "Set"}-operation+
−
  only changes the precedence of the initiating thread (or actor)+
−
  of the operation (or event).+
−
*}+
−
+
−
lemma eq_preced:+
−
  assumes "th' \<noteq> th"+
−
  shows "preced th' s = preced th' s'"+
−
proof -+
−
  from assms show ?thesis +
−
    by (unfold s_def, auto simp:preced_def)+
−
qed+
−
+
−
lemma eq_the_preced: +
−
  assumes "th' \<noteq> th"+
−
  shows "the_preced s th' = the_preced s' th'"+
−
  using assms+
−
  by (unfold the_preced_def, intro eq_preced, simp)+
−
+
−
text {*+
−
  The following lemma assures that the resetting of priority does not change the RAG. +
−
*}+
−
+
−
lemma eq_dep: "RAG s = RAG s'"+
−
  by (unfold s_def RAG_set_unchanged, auto)+
−
+
−
text {* (* ddd *)+
−
  Th following lemma @{text "eq_cp_pre"} says that the priority change of @{text "th"}+
−
  only affects those threads, which as @{text "Th th"} in their sub-trees.+
−
  +
−
  The proof of this lemma is simplified by using the alternative definition +
−
  of @{text "cp"}. +
−
*}+
−
+
−
lemma eq_cp_pre:+
−
  assumes nd: "Th th \<notin> subtree (RAG s') (Th th')"+
−
  shows "cp s th' = cp s' th'"+
−
proof -+
−
  -- {* After unfolding using the alternative definition, elements +
−
        affecting the @{term "cp"}-value of threads become explicit. +
−
        We only need to prove the following: *}+
−
  have "Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =+
−
        Max (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"+
−
        (is "Max (?f ` ?S1) = Max (?g ` ?S2)")+
−
  proof -+
−
    -- {* The base sets are equal. *}+
−
    have "?S1 = ?S2" using eq_dep by simp+
−
    -- {* The function values on the base set are equal as well. *}+
−
    moreover have "\<forall> e \<in> ?S2. ?f e = ?g e"+
−
    proof+
−
      fix th1+
−
      assume "th1 \<in> ?S2"+
−
      with nd have "th1 \<noteq> th" by (auto)+
−
      from eq_the_preced[OF this]+
−
      show "the_preced s th1 = the_preced s' th1" .+
−
    qed+
−
    -- {* Therefore, the image of the functions are equal. *}+
−
    ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq)+
−
    thus ?thesis by simp+
−
  qed+
−
  thus ?thesis by (simp add:cp_alt_def)+
−
qed+
−
+
−
text {*+
−
  The following lemma shows that @{term "th"} is not in the +
−
  sub-tree of any other thread. +
−
*}+
−
lemma th_in_no_subtree:+
−
  assumes "th' \<noteq> th"+
−
  shows "Th th \<notin> subtree (RAG s') (Th th')"+
−
proof -+
−
  have "th \<in> readys s'"+
−
  proof -+
−
    from step_back_step [OF vt_s[unfolded s_def]]+
−
    have "step s' (Set th prio)" .+
−
    hence "th \<in> runing s'" by (cases, simp)+
−
    thus ?thesis by (simp add:readys_def runing_def)+
−
  qed+
−
  from vat_s'.readys_in_no_subtree[OF this assms(1)]+
−
  show ?thesis by blast+
−
qed+
−
+
−
text {* +
−
  By combining @{thm "eq_cp_pre"} and @{thm "th_in_no_subtree"}, +
−
  it is obvious that the change of priority only affects the @{text "cp"}-value +
−
  of the initiating thread @{text "th"}.+
−
*}+
−
lemma eq_cp:+
−
  assumes "th' \<noteq> th"+
−
  shows "cp s th' = cp s' th'"+
−
  by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])+
−
+
−
end+
−
+
−
section {* The @{term V} operation *}+
−
+
−
text {*+
−
  The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.+
−
*}+
−
+
−
locale step_v_cps =+
−
  -- {* @{text "th"} is the initiating thread *}+
−
  -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *}+
−
  fixes s' th cs s    -- {* @{text "s'"} is the state before operation*}+
−
  defines s_def : "s \<equiv> (V th cs#s')" -- {* @{text "s"} is the state after operation*}+
−
  -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *}+
−
  assumes vt_s: "vt s"+
−
+
−
sublocale step_v_cps < vat_s : valid_trace "s"+
−
proof+
−
  from vt_s show "vt s" .+
−
qed+
−
+
−
sublocale step_v_cps < vat_s' : valid_trace "s'"+
−
proof+
−
  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .+
−
qed+
−
+
−
context step_v_cps+
−
begin+
−
+
−
lemma ready_th_s': "th \<in> readys s'"+
−
  using step_back_step[OF vt_s[unfolded s_def]]+
−
  by (cases, simp add:runing_def)+
−
+
−
lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"+
−
proof -+
−
  from vat_s'.readys_root[OF ready_th_s']+
−
  show ?thesis+
−
  by (unfold root_def, simp)+
−
qed+
−
+
−
lemma holding_th: "holding s' th cs"+
−
proof -+
−
  from vt_s[unfolded s_def]+
−
  have " PIP s' (V th cs)" by (cases, simp)+
−
  thus ?thesis by (cases, auto)+
−
qed+
−
+
−
lemma edge_of_th:+
−
    "(Cs cs, Th th) \<in> RAG s'" +
−
proof -+
−
 from holding_th+
−
 show ?thesis +
−
    by (unfold s_RAG_def holding_eq, auto)+
−
qed+
−
+
−
lemma ancestors_cs: +
−
  "ancestors (RAG s') (Cs cs) = {Th th}"+
−
proof -+
−
  have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th)  \<union>  {Th th}"+
−
  proof(rule vat_s'.rtree_RAG.ancestors_accum)+
−
    from vt_s[unfolded s_def]+
−
    have " PIP s' (V th cs)" by (cases, simp)+
−
    thus "(Cs cs, Th th) \<in> RAG s'" +
−
    proof(cases)+
−
      assume "holding s' th cs"+
−
      from this[unfolded holding_eq]+
−
      show ?thesis by (unfold s_RAG_def, auto)+
−
    qed+
−
  qed+
−
  from this[unfolded ancestors_th] show ?thesis by simp+
−
qed+
−
+
−
lemma preced_kept: "the_preced s = the_preced s'"+
−
  by (auto simp: s_def the_preced_def preced_def)+
−
+
−
end+
−
+
−
text {*+
−
  The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation, +
−
  which represents the case when there is another thread @{text "th'"}+
−
  to take over the critical resource released by the initiating thread @{text "th"}.+
−
*}+
−
locale step_v_cps_nt = step_v_cps ++
−
  fixes th'+
−
  -- {* @{text "th'"} is assumed to take over @{text "cs"} *}+
−
  assumes nt: "next_th s' th cs th'" +
−
+
−
context step_v_cps_nt+
−
begin+
−
+
−
text {*+
−
  Lemma @{text "RAG_s"} confirms the change of RAG:+
−
  two edges removed and one added, as shown by the following diagram.+
−
*}+
−
+
−
(*+
−
  RAG before the V-operation+
−
    th1 ----|+
−
            |+
−
    th' ----|+
−
            |----> cs -----|+
−
    th2 ----|              |+
−
            |              |+
−
    th3 ----|              |+
−
                           |------> th+
−
    th4 ----|              |+
−
            |              |+
−
    th5 ----|              |+
−
            |----> cs'-----|+
−
    th6 ----|+
−
            |+
−
    th7 ----|+
−
+
−
 RAG after the V-operation+
−
    th1 ----|+
−
            |+
−
            |----> cs ----> th'+
−
    th2 ----|              +
−
            |              +
−
    th3 ----|              +
−
                           +
−
    th4 ----|              +
−
            |              +
−
    th5 ----|              +
−
            |----> cs'----> th+
−
    th6 ----|+
−
            |+
−
    th7 ----|+
−
*)+
−
+
−
lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"+
−
                using next_th_RAG[OF nt]  .+
−
+
−
lemma ancestors_th': +
−
  "ancestors (RAG s') (Th th') = {Th th, Cs cs}" +
−
proof -+
−
  have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"+
−
  proof(rule  vat_s'.rtree_RAG.ancestors_accum)+
−
    from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto+
−
  qed+
−
  thus ?thesis using ancestors_th ancestors_cs by auto+
−
qed+
−
+
−
lemma RAG_s:+
−
  "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>+
−
                                         {(Cs cs, Th th')}"+
−
proof -+
−
  from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]+
−
    and nt show ?thesis  by (auto intro:next_th_unique)+
−
qed+
−
+
−
lemma subtree_kept: (* ddd *)+
−
  assumes "th1 \<notin> {th, th'}"+
−
  shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R")+
−
proof -+
−
  let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})"+
−
  let ?RAG'' = "?RAG' \<union> {(Cs cs, Th th')}"+
−
  have "subtree ?RAG' (Th th1) = ?R" +
−
  proof(rule subset_del_subtree_outside)+
−
    show "Range {(Cs cs, Th th), (Th th', Cs cs)} \<inter> subtree (RAG s') (Th th1) = {}"+
−
    proof -+
−
      have "(Th th) \<notin> subtree (RAG s') (Th th1)"+
−
      proof(rule subtree_refute)+
−
        show "Th th1 \<notin> ancestors (RAG s') (Th th)"+
−
          by (unfold ancestors_th, simp)+
−
      next+
−
        from assms show "Th th1 \<noteq> Th th" by simp+
−
      qed+
−
      moreover have "(Cs cs) \<notin>  subtree (RAG s') (Th th1)"+
−
      proof(rule subtree_refute)+
−
        show "Th th1 \<notin> ancestors (RAG s') (Cs cs)"+
−
          by (unfold ancestors_cs, insert assms, auto)+
−
      qed simp+
−
      ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s') (Th th1) = {}" by auto+
−
      thus ?thesis by simp+
−
     qed+
−
  qed+
−
  moreover have "subtree ?RAG'' (Th th1) =  subtree ?RAG' (Th th1)"+
−
  proof(rule subtree_insert_next)+
−
    show "Th th' \<notin> subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)"+
−
    proof(rule subtree_refute)+
−
      show "Th th1 \<notin> ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')"+
−
            (is "_ \<notin> ?R")+
−
      proof -+
−
          have "?R \<subseteq> ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto)+
−
          moreover have "Th th1 \<notin> ..." using ancestors_th' assms by simp+
−
          ultimately show ?thesis by auto+
−
      qed+
−
    next+
−
      from assms show "Th th1 \<noteq> Th th'" by simp+
−
    qed+
−
  qed+
−
  ultimately show ?thesis by (unfold RAG_s, simp)+
−
qed+
−
+
−
lemma cp_kept:+
−
  assumes "th1 \<notin> {th, th'}"+
−
  shows "cp s th1 = cp s' th1"+
−
    by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)+
−
+
−
end+
−
+
−
locale step_v_cps_nnt = step_v_cps ++
−
  assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"+
−
+
−
context step_v_cps_nnt+
−
begin+
−
+
−
lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"+
−
proof -+
−
  from nnt and  step_RAG_v[OF vt_s[unfolded s_def], folded s_def]+
−
  show ?thesis by auto+
−
qed+
−
+
−
lemma subtree_kept:+
−
  assumes "th1 \<noteq> th"+
−
  shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)"+
−
proof(unfold RAG_s, rule subset_del_subtree_outside)+
−
  show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s') (Th th1) = {}"+
−
  proof -+
−
    have "(Th th) \<notin> subtree (RAG s') (Th th1)"+
−
    proof(rule subtree_refute)+
−
      show "Th th1 \<notin> ancestors (RAG s') (Th th)"+
−
          by (unfold ancestors_th, simp)+
−
    next+
−
      from assms show "Th th1 \<noteq> Th th" by simp+
−
    qed+
−
    thus ?thesis by auto+
−
  qed+
−
qed+
−
+
−
lemma cp_kept_1:+
−
  assumes "th1 \<noteq> th"+
−
  shows "cp s th1 = cp s' th1"+
−
    by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)+
−
+
−
lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}"+
−
proof -+
−
  { fix n+
−
    have "(Cs cs) \<notin> ancestors (RAG s') n"+
−
    proof+
−
      assume "Cs cs \<in> ancestors (RAG s') n"+
−
      hence "(n, Cs cs) \<in> (RAG s')^+" by (auto simp:ancestors_def)+
−
      from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s'" by auto+
−
      then obtain th' where "nn = Th th'"+
−
        by (unfold s_RAG_def, auto)+
−
      from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s'" .+
−
      from this[unfolded s_RAG_def]+
−
      have "waiting (wq s') th' cs" by auto+
−
      from this[unfolded cs_waiting_def]+
−
      have "1 < length (wq s' cs)"+
−
          by (cases "wq s' cs", auto)+
−
      from holding_next_thI[OF holding_th this]+
−
      obtain th' where "next_th s' th cs th'" by auto+
−
      with nnt show False by auto+
−
    qed+
−
  } note h = this+
−
  {  fix n+
−
     assume "n \<in> subtree (RAG s') (Cs cs)"+
−
     hence "n = (Cs cs)"+
−
     by (elim subtreeE, insert h, auto)+
−
  } moreover have "(Cs cs) \<in> subtree (RAG s') (Cs cs)"+
−
      by (auto simp:subtree_def)+
−
  ultimately show ?thesis by auto +
−
qed+
−
+
−
lemma subtree_th: +
−
  "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"+
−
proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)+
−
  from edge_of_th+
−
  show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"+
−
    by (unfold edges_in_def, auto simp:subtree_def)+
−
qed+
−
+
−
lemma cp_kept_2: +
−
  shows "cp s th = cp s' th" +
−
 by (unfold cp_alt_def subtree_th preced_kept, auto)+
−
+
−
lemma eq_cp:+
−
  shows "cp s th' = cp s' th'"+
−
  using cp_kept_1 cp_kept_2+
−
  by (cases "th' = th", auto)+
−
end+
−
+
−
+
−
locale step_P_cps =+
−
  fixes s' th cs s +
−
  defines s_def : "s \<equiv> (P th cs#s')"+
−
  assumes vt_s: "vt s"+
−
+
−
sublocale step_P_cps < vat_s : valid_trace "s"+
−
proof+
−
  from vt_s show "vt s" .+
−
qed+
−
+
−
section {* The @{term P} operation *}+
−
+
−
sublocale step_P_cps < vat_s' : valid_trace "s'"+
−
proof+
−
  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .+
−
qed+
−
+
−
context step_P_cps+
−
begin+
−
+
−
lemma readys_th: "th \<in> readys s'"+
−
proof -+
−
    from step_back_step [OF vt_s[unfolded s_def]]+
−
    have "PIP s' (P th cs)" .+
−
    hence "th \<in> runing s'" by (cases, simp)+
−
    thus ?thesis by (simp add:readys_def runing_def)+
−
qed+
−
+
−
lemma root_th: "root (RAG s') (Th th)"+
−
  using readys_root[OF readys_th] .+
−
+
−
lemma in_no_others_subtree:+
−
  assumes "th' \<noteq> th"+
−
  shows "Th th \<notin> subtree (RAG s') (Th th')"+
−
proof+
−
  assume "Th th \<in> subtree (RAG s') (Th th')"+
−
  thus False+
−
  proof(cases rule:subtreeE)+
−
    case 1+
−
    with assms show ?thesis by auto+
−
  next+
−
    case 2+
−
    with root_th show ?thesis by (auto simp:root_def)+
−
  qed+
−
qed+
−
+
−
lemma preced_kept: "the_preced s = the_preced s'"+
−
  by (auto simp: s_def the_preced_def preced_def)+
−
+
−
end+
−
+
−
locale step_P_cps_ne =step_P_cps ++
−
  fixes th'+
−
  assumes ne: "wq s' cs \<noteq> []"+
−
  defines th'_def: "th' \<equiv> hd (wq s' cs)"+
−
+
−
locale step_P_cps_e =step_P_cps ++
−
  assumes ee: "wq s' cs = []"+
−
+
−
context step_P_cps_e+
−
begin+
−
+
−
lemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"+
−
proof -+
−
  from ee and  step_RAG_p[OF vt_s[unfolded s_def], folded s_def]+
−
  show ?thesis by auto+
−
qed+
−
+
−
lemma subtree_kept:+
−
  assumes "th' \<noteq> th"+
−
  shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')"+
−
proof(unfold RAG_s, rule subtree_insert_next)+
−
  from in_no_others_subtree[OF assms] +
−
  show "Th th \<notin> subtree (RAG s') (Th th')" .+
−
qed+
−
+
−
lemma cp_kept: +
−
  assumes "th' \<noteq> th"+
−
  shows "cp s th' = cp s' th'"+
−
proof -+
−
  have "(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =+
−
        (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"+
−
        by (unfold preced_kept subtree_kept[OF assms], simp)+
−
  thus ?thesis by (unfold cp_alt_def, simp)+
−
qed+
−
+
−
end+
−
+
−
context step_P_cps_ne +
−
begin+
−
+
−
lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"+
−
proof -+
−
  from step_RAG_p[OF vt_s[unfolded s_def]] and ne+
−
  show ?thesis by (simp add:s_def)+
−
qed+
−
+
−
lemma cs_held: "(Cs cs, Th th') \<in> RAG s'"+
−
proof -+
−
  have "(Cs cs, Th th') \<in> hRAG s'"+
−
  proof -+
−
    from ne+
−
    have " holding s' th' cs"+
−
      by (unfold th'_def holding_eq cs_holding_def, auto)+
−
    thus ?thesis +
−
      by (unfold hRAG_def, auto)+
−
  qed+
−
  thus ?thesis by (unfold RAG_split, auto)+
−
qed+
−
+
−
lemma tRAG_s: +
−
  "tRAG s = tRAG s' \<union> {(Th th, Th th')}"+
−
  using RAG_tRAG_transfer[OF RAG_s cs_held] .+
−
+
−
lemma cp_kept:+
−
  assumes "Th th'' \<notin> ancestors (tRAG s) (Th th)"+
−
  shows "cp s th'' = cp s' th''"+
−
proof -+
−
  have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')"+
−
  proof -+
−
    have "Th th' \<notin> subtree (tRAG s') (Th th'')"+
−
    proof+
−
      assume "Th th' \<in> subtree (tRAG s') (Th th'')"+
−
      thus False+
−
      proof(rule subtreeE)+
−
         assume "Th th' = Th th''"+
−
         from assms[unfolded tRAG_s ancestors_def, folded this]+
−
         show ?thesis by auto+
−
      next+
−
         assume "Th th'' \<in> ancestors (tRAG s') (Th th')"+
−
         moreover have "... \<subseteq> ancestors (tRAG s) (Th th')"+
−
         proof(rule ancestors_mono)+
−
            show "tRAG s' \<subseteq> tRAG s" by (unfold tRAG_s, auto)+
−
         qed +
−
         ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th')" by auto+
−
         moreover have "Th th' \<in> ancestors (tRAG s) (Th th)"+
−
           by (unfold tRAG_s, auto simp:ancestors_def)+
−
         ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th)"+
−
                       by (auto simp:ancestors_def)+
−
         with assms show ?thesis by auto+
−
      qed+
−
    qed+
−
    from subtree_insert_next[OF this]+
−
    have "subtree (tRAG s' \<union> {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" .+
−
    from this[folded tRAG_s] show ?thesis .+
−
  qed+
−
  show ?thesis by (unfold cp_alt_def1 h preced_kept, simp)+
−
qed+
−
+
−
lemma cp_gen_update_stop: (* ddd *)+
−
  assumes "u \<in> ancestors (tRAG s) (Th th)"+
−
  and "cp_gen s u = cp_gen s' u"+
−
  and "y \<in> ancestors (tRAG s) u"+
−
  shows "cp_gen s y = cp_gen s' y"+
−
  using assms(3)+
−
proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf])+
−
  case (1 x)+
−
  show ?case (is "?L = ?R")+
−
  proof -+
−
    from tRAG_ancestorsE[OF 1(2)]+
−
    obtain th2 where eq_x: "x = Th th2" by blast+
−
    from vat_s.cp_gen_rec[OF this]+
−
    have "?L = +
−
          Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)" .+
−
    also have "... = +
−
          Max ({the_preced s' th2} \<union> cp_gen s' ` RTree.children (tRAG s') x)"+
−
  +
−
    proof -+
−
      from preced_kept have "the_preced s th2 = the_preced s' th2" by simp+
−
      moreover have "cp_gen s ` RTree.children (tRAG s) x =+
−
                     cp_gen s' ` RTree.children (tRAG s') x"+
−
      proof -+
−
        have "RTree.children (tRAG s) x =  RTree.children (tRAG s') x"+
−
        proof(unfold tRAG_s, rule children_union_kept)+
−
          have start: "(Th th, Th th') \<in> tRAG s"+
−
            by (unfold tRAG_s, auto)+
−
          note x_u = 1(2)+
−
          show "x \<notin> Range {(Th th, Th th')}"+
−
          proof+
−
            assume "x \<in> Range {(Th th, Th th')}"+
−
            hence eq_x: "x = Th th'" using RangeE by auto+
−
            show False+
−
            proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start])+
−
              case 1+
−
              from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG+
−
              show ?thesis by (auto simp:ancestors_def acyclic_def)+
−
            next+
−
              case 2+
−
              with x_u[unfolded eq_x]+
−
              have "(Th th', Th th') \<in> (tRAG s)^+" by (auto simp:ancestors_def)+
−
              with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)+
−
            qed+
−
          qed+
−
        qed+
−
        moreover have "cp_gen s ` RTree.children (tRAG s) x =+
−
                       cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A")+
−
        proof(rule f_image_eq)+
−
          fix a+
−
          assume a_in: "a \<in> ?A"+
−
          from 1(2)+
−
          show "?f a = ?g a"+
−
          proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch])+
−
             case in_ch+
−
             show ?thesis+
−
             proof(cases "a = u")+
−
                case True+
−
                from assms(2)[folded this] show ?thesis .+
−
             next+
−
                case False+
−
                have a_not_in: "a \<notin> ancestors (tRAG s) (Th th)"+
−
                proof+
−
                  assume a_in': "a \<in> ancestors (tRAG s) (Th th)"+
−
                  have "a = u"+
−
                  proof(rule vat_s.rtree_s.ancestors_children_unique)+
−
                    from a_in' a_in show "a \<in> ancestors (tRAG s) (Th th) \<inter> +
−
                                          RTree.children (tRAG s) x" by auto+
−
                  next +
−
                    from assms(1) in_ch show "u \<in> ancestors (tRAG s) (Th th) \<inter> +
−
                                      RTree.children (tRAG s) x" by auto+
−
                  qed+
−
                  with False show False by simp+
−
                qed+
−
                from a_in obtain th_a where eq_a: "a = Th th_a" +
−
                    by (unfold RTree.children_def tRAG_alt_def, auto)+
−
                from cp_kept[OF a_not_in[unfolded eq_a]]+
−
                have "cp s th_a = cp s' th_a" .+
−
                from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a]+
−
                show ?thesis .+
−
             qed+
−
          next+
−
            case (out_ch z)+
−
            hence h: "z \<in> ancestors (tRAG s) u" "z \<in> RTree.children (tRAG s) x" by auto+
−
            show ?thesis+
−
            proof(cases "a = z")+
−
              case True+
−
              from h(2) have zx_in: "(z, x) \<in> (tRAG s)" by (auto simp:RTree.children_def)+
−
              from 1(1)[rule_format, OF this h(1)]+
−
              have eq_cp_gen: "cp_gen s z = cp_gen s' z" .+
−
              with True show ?thesis by metis+
−
            next+
−
              case False+
−
              from a_in obtain th_a where eq_a: "a = Th th_a"+
−
                by (auto simp:RTree.children_def tRAG_alt_def)+
−
              have "a \<notin> ancestors (tRAG s) (Th th)"+
−
              proof+
−
                assume a_in': "a \<in> ancestors (tRAG s) (Th th)"+
−
                have "a = z"+
−
                proof(rule vat_s.rtree_s.ancestors_children_unique)+
−
                  from assms(1) h(1) have "z \<in> ancestors (tRAG s) (Th th)"+
−
                      by (auto simp:ancestors_def)+
−
                  with h(2) show " z \<in> ancestors (tRAG s) (Th th) \<inter> +
−
                                       RTree.children (tRAG s) x" by auto+
−
                next+
−
                  from a_in a_in'+
−
                  show "a \<in> ancestors (tRAG s) (Th th) \<inter> RTree.children (tRAG s) x"+
−
                    by auto+
−
                qed+
−
                with False show False by auto+
−
              qed+
−
              from cp_kept[OF this[unfolded eq_a]]+
−
              have "cp s th_a = cp s' th_a" .+
−
              from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a]+
−
              show ?thesis .+
−
            qed+
−
          qed+
−
        qed+
−
        ultimately show ?thesis by metis+
−
      qed+
−
      ultimately show ?thesis by simp+
−
    qed+
−
    also have "... = ?R"+
−
      by (fold vat_s'.cp_gen_rec[OF eq_x], simp)+
−
    finally show ?thesis .+
−
  qed+
−
qed+
−
+
−
lemma cp_up:+
−
  assumes "(Th th') \<in> ancestors (tRAG s) (Th th)"+
−
  and "cp s th' = cp s' th'"+
−
  and "(Th th'') \<in> ancestors (tRAG s) (Th th')"+
−
  shows "cp s th'' = cp s' th''"+
−
proof -+
−
  have "cp_gen s (Th th'') = cp_gen s' (Th th'')"+
−
  proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)])+
−
    from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]]+
−
    show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis+
−
  qed+
−
  with cp_gen_def_cond[OF refl[of "Th th''"]]+
−
  show ?thesis by metis+
−
qed+
−
+
−
end+
−
+
−
section {* The @{term Create} operation *}+
−
+
−
locale step_create_cps =+
−
  fixes s' th prio s +
−
  defines s_def : "s \<equiv> (Create th prio#s')"+
−
  assumes vt_s: "vt s"+
−
+
−
sublocale step_create_cps < vat_s: valid_trace "s"+
−
  by (unfold_locales, insert vt_s, simp)+
−
+
−
sublocale step_create_cps < vat_s': valid_trace "s'"+
−
  by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)+
−
+
−
context step_create_cps+
−
begin+
−
+
−
lemma RAG_kept: "RAG s = RAG s'"+
−
  by (unfold s_def RAG_create_unchanged, auto)+
−
+
−
lemma tRAG_kept: "tRAG s = tRAG s'"+
−
  by (unfold tRAG_alt_def RAG_kept, auto)+
−
+
−
lemma preced_kept:+
−
  assumes "th' \<noteq> th"+
−
  shows "the_preced s th' = the_preced s' th'"+
−
  by (unfold s_def the_preced_def preced_def, insert assms, auto)+
−
+
−
lemma th_not_in: "Th th \<notin> Field (tRAG s')"+
−
proof -+
−
  from vt_s[unfolded s_def]+
−
  have "PIP s' (Create th prio)" by (cases, simp)+
−
  hence "th \<notin> threads s'" by(cases, simp)+
−
  from vat_s'.not_in_thread_isolated[OF this]+
−
  have "Th th \<notin> Field (RAG s')" .+
−
  with tRAG_Field show ?thesis by auto+
−
qed+
−
+
−
lemma eq_cp:+
−
  assumes neq_th: "th' \<noteq> th"+
−
  shows "cp s th' = cp s' th'"+
−
proof -+
−
  have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =+
−
        (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"+
−
  proof(unfold tRAG_kept, rule f_image_eq)+
−
    fix a+
−
    assume a_in: "a \<in> subtree (tRAG s') (Th th')"+
−
    then obtain th_a where eq_a: "a = Th th_a" +
−
    proof(cases rule:subtreeE)+
−
      case 2+
−
      from ancestors_Field[OF 2(2)]+
−
      and that show ?thesis by (unfold tRAG_alt_def, auto)+
−
    qed auto+
−
    have neq_th_a: "th_a \<noteq> th"+
−
    proof -+
−
      have "(Th th) \<notin> subtree (tRAG s') (Th th')"+
−
      proof+
−
        assume "Th th \<in> subtree (tRAG s') (Th th')"+
−
        thus False+
−
        proof(cases rule:subtreeE)+
−
          case 2+
−
          from ancestors_Field[OF this(2)]+
−
          and th_not_in[unfolded Field_def]+
−
          show ?thesis by auto+
−
        qed (insert assms, auto)+
−
      qed+
−
      with a_in[unfolded eq_a] show ?thesis by auto+
−
    qed+
−
    from preced_kept[OF this]+
−
    show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"+
−
      by (unfold eq_a, simp)+
−
  qed+
−
  thus ?thesis by (unfold cp_alt_def1, simp)+
−
qed+
−
+
−
lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}"+
−
proof -+
−
  { fix a+
−
    assume "a \<in> RTree.children (tRAG s) (Th th)"+
−
    hence "(a, Th th) \<in> tRAG s" by (auto simp:RTree.children_def)+
−
    with th_not_in have False +
−
     by (unfold Field_def tRAG_kept, auto)+
−
  } thus ?thesis by auto+
−
qed+
−
+
−
lemma eq_cp_th: "cp s th = preced th s"+
−
 by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def)+
−
+
−
end+
−
+
−
locale step_exit_cps =+
−
  fixes s' th prio s +
−
  defines s_def : "s \<equiv> Exit th # s'"+
−
  assumes vt_s: "vt s"+
−
+
−
sublocale step_exit_cps < vat_s: valid_trace "s"+
−
  by (unfold_locales, insert vt_s, simp)+
−
+
−
sublocale step_exit_cps < vat_s': valid_trace "s'"+
−
  by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)+
−
+
−
context step_exit_cps+
−
begin+
−
+
−
lemma preced_kept:+
−
  assumes "th' \<noteq> th"+
−
  shows "the_preced s th' = the_preced s' th'"+
−
  by (unfold s_def the_preced_def preced_def, insert assms, auto)+
−
+
−
lemma RAG_kept: "RAG s = RAG s'"+
−
  by (unfold s_def RAG_exit_unchanged, auto)+
−
+
−
lemma tRAG_kept: "tRAG s = tRAG s'"+
−
  by (unfold tRAG_alt_def RAG_kept, auto)+
−
+
−
lemma th_ready: "th \<in> readys s'"+
−
proof -+
−
  from vt_s[unfolded s_def]+
−
  have "PIP s' (Exit th)" by (cases, simp)+
−
  hence h: "th \<in> runing s' \<and> holdents s' th = {}" by (cases, metis)+
−
  thus ?thesis by (unfold runing_def, auto)+
−
qed+
−
+
−
lemma th_holdents: "holdents s' th = {}"+
−
proof -+
−
 from vt_s[unfolded s_def]+
−
  have "PIP s' (Exit th)" by (cases, simp)+
−
  thus ?thesis by (cases, metis)+
−
qed+
−
+
−
lemma th_RAG: "Th th \<notin> Field (RAG s')"+
−
proof -+
−
  have "Th th \<notin> Range (RAG s')"+
−
  proof+
−
    assume "Th th \<in> Range (RAG s')"+
−
    then obtain cs where "holding (wq s') th cs"+
−
      by (unfold Range_iff s_RAG_def, auto)+
−
    with th_holdents[unfolded holdents_def]+
−
    show False by (unfold eq_holding, auto)+
−
  qed+
−
  moreover have "Th th \<notin> Domain (RAG s')"+
−
  proof+
−
    assume "Th th \<in> Domain (RAG s')"+
−
    then obtain cs where "waiting (wq s') th cs"+
−
      by (unfold Domain_iff s_RAG_def, auto)+
−
    with th_ready show False by (unfold readys_def eq_waiting, auto)+
−
  qed+
−
  ultimately show ?thesis by (auto simp:Field_def)+
−
qed+
−
+
−
lemma th_tRAG: "(Th th) \<notin> Field (tRAG s')"+
−
  using th_RAG tRAG_Field[of s'] by auto+
−
+
−
lemma eq_cp:+
−
  assumes neq_th: "th' \<noteq> th"+
−
  shows "cp s th' = cp s' th'"+
−
proof -+
−
  have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =+
−
        (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"+
−
  proof(unfold tRAG_kept, rule f_image_eq)+
−
    fix a+
−
    assume a_in: "a \<in> subtree (tRAG s') (Th th')"+
−
    then obtain th_a where eq_a: "a = Th th_a" +
−
    proof(cases rule:subtreeE)+
−
      case 2+
−
      from ancestors_Field[OF 2(2)]+
−
      and that show ?thesis by (unfold tRAG_alt_def, auto)+
−
    qed auto+
−
    have neq_th_a: "th_a \<noteq> th"+
−
    proof -+
−
      from vat_s'.readys_in_no_subtree[OF th_ready assms]+
−
      have "(Th th) \<notin> subtree (RAG s') (Th th')" .+
−
      with tRAG_subtree_RAG[of s' "Th th'"]+
−
      have "(Th th) \<notin> subtree (tRAG s') (Th th')" by auto+
−
      with a_in[unfolded eq_a] show ?thesis by auto+
−
    qed+
−
    from preced_kept[OF this]+
−
    show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"+
−
      by (unfold eq_a, simp)+
−
  qed+
−
  thus ?thesis by (unfold cp_alt_def1, simp)+
−
qed+
−
+
−
end+
−
+
−
end+
−
+
−