section {*+
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  This file contains lemmas used to guide the recalculation of current precedence +
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  after every system call (or system operation)+
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*}+
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theory CpsG+
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imports PrioG Max RTree+
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begin+
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+
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text {* @{text "the_preced"} is also the same as @{text "preced"}, the only+
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       difference is the order of arguemts. *}+
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definition "the_preced s th = preced th s"+
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+
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lemma inj_the_preced: +
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  "inj_on (the_preced s) (threads s)"+
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  by (metis inj_onI preced_unique the_preced_def)+
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+
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text {* @{term "the_thread"} extracts thread out of RAG node. *}+
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fun the_thread :: "node \<Rightarrow> thread" where+
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   "the_thread (Th th) = th"+
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+
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text {* The following @{text "wRAG"} is the waiting sub-graph of @{text "RAG"}. *}+
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definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"+
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+
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text {* The following @{text "hRAG"} is the holding sub-graph of @{text "RAG"}. *}+
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definition "hRAG (s::state) =  {(Cs cs, Th th) | th cs. holding s th cs}"+
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+
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text {* The following lemma splits @{term "RAG"} graph into the above two sub-graphs. *}+
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lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"+
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  by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv +
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             s_holding_abv cs_RAG_def, auto)+
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+
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text {* +
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  The following @{text "tRAG"} is the thread-graph derived from @{term "RAG"}.+
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  It characterizes the dependency between threads when calculating current+
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  precedences. It is defined as the composition of the above two sub-graphs, +
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  names @{term "wRAG"} and @{term "hRAG"}.+
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 *}+
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definition "tRAG s = wRAG s O hRAG s"+
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+
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(* ccc *)+
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+
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definition "cp_gen s x =+
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                  Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"+
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+
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lemma tRAG_alt_def: +
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  "tRAG s = {(Th th1, Th th2) | th1 th2. +
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                  \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"+
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 by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)+
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+
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lemma tRAG_Field:+
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  "Field (tRAG s) \<subseteq> Field (RAG s)"+
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  by (unfold tRAG_alt_def Field_def, auto)+
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+
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lemma tRAG_ancestorsE:+
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  assumes "x \<in> ancestors (tRAG s) u"+
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  obtains th where "x = Th th"+
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proof -+
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  from assms have "(u, x) \<in> (tRAG s)^+" +
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      by (unfold ancestors_def, auto)+
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  from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto+
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  then obtain th where "x = Th th"+
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    by (unfold tRAG_alt_def, auto)+
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  from that[OF this] show ?thesis .+
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qed+
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+
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lemma tRAG_mono:+
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  assumes "RAG s' \<subseteq> RAG s"+
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  shows "tRAG s' \<subseteq> tRAG s"+
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  using assms +
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  by (unfold tRAG_alt_def, auto)+
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+
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lemma holding_next_thI:+
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  assumes "holding s th cs"+
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  and "length (wq s cs) > 1"+
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  obtains th' where "next_th s th cs th'"+
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proof -+
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  from assms(1)[folded eq_holding, unfolded cs_holding_def]+
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  have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .+
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  then obtain rest where h1: "wq s cs = th#rest" +
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    by (cases "wq s cs", auto)+
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  with assms(2) have h2: "rest \<noteq> []" by auto+
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  let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"+
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  have "next_th s th cs ?th'" using  h1(1) h2 +
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    by (unfold next_th_def, auto)+
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  from that[OF this] show ?thesis .+
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qed+
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+
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lemma RAG_tRAG_transfer:+
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  assumes "vt s'"+
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  assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"+
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  and "(Cs cs, Th th'') \<in> RAG s'"+
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  shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")+
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proof -+
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  interpret vt_s': valid_trace "s'" using assms(1)+
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    by (unfold_locales, simp)+
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  interpret rtree: rtree "RAG s'"+
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  proof+
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  show "single_valued (RAG s')"+
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  apply (intro_locales)+
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    by (unfold single_valued_def, +
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        auto intro:vt_s'.unique_RAG)+
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+
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  show "acyclic (RAG s')"+
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     by (rule vt_s'.acyclic_RAG)+
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  qed+
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  { fix n1 n2+
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    assume "(n1, n2) \<in> ?L"+
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    from this[unfolded tRAG_alt_def]+
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    obtain th1 th2 cs' where +
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      h: "n1 = Th th1" "n2 = Th th2" +
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         "(Th th1, Cs cs') \<in> RAG s"+
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         "(Cs cs', Th th2) \<in> RAG s" by auto+
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    from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto+
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    from h(3) and assms(2) +
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    have "(Th th1, Cs cs') = (Th th, Cs cs) \<or> +
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          (Th th1, Cs cs') \<in> RAG s'" by auto+
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    hence "(n1, n2) \<in> ?R"+
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    proof+
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      assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"+
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      hence eq_th1: "th1 = th" by simp+
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      moreover have "th2 = th''"+
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      proof -+
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        from h1 have "cs' = cs" by simp+
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        from assms(3) cs_in[unfolded this] rtree.sgv+
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        show ?thesis+
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          by (unfold single_valued_def, auto)+
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      qed+
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      ultimately show ?thesis using h(1,2) by auto+
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    next+
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      assume "(Th th1, Cs cs') \<in> RAG s'"+
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      with cs_in have "(Th th1, Th th2) \<in> tRAG s'"+
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        by (unfold tRAG_alt_def, auto)+
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      from this[folded h(1, 2)] show ?thesis by auto+
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    qed+
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  } moreover {+
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    fix n1 n2+
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    assume "(n1, n2) \<in> ?R"+
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    hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto+
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    hence "(n1, n2) \<in> ?L" +
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    proof+
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      assume "(n1, n2) \<in> tRAG s'"+
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      moreover have "... \<subseteq> ?L"+
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      proof(rule tRAG_mono)+
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        show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)+
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      qed+
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      ultimately show ?thesis by auto+
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    next+
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      assume eq_n: "(n1, n2) = (Th th, Th th'')"+
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      from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto+
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      moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto+
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      ultimately show ?thesis +
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        by (unfold eq_n tRAG_alt_def, auto)+
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    qed+
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  } ultimately show ?thesis by auto+
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qed+
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+
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context valid_trace+
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begin+
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+
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lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]+
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+
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end+
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+
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lemma cp_alt_def:+
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  "cp s th =  +
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           Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"+
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proof -+
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  have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =+
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        Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})" +
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          (is "Max (_ ` ?L) = Max (_ ` ?R)")+
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  proof -+
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    have "?L = ?R" +
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    by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)+
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    thus ?thesis by simp+
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  qed+
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  thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)+
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qed+
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+
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lemma cp_gen_alt_def:+
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  "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"+
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    by (auto simp:cp_gen_def)+
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+
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lemma tRAG_nodeE:+
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  assumes "(n1, n2) \<in> tRAG s"+
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  obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"+
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  using assms+
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  by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)+
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+
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lemma subtree_nodeE:+
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  assumes "n \<in> subtree (tRAG s) (Th th)"+
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  obtains th1 where "n = Th th1"+
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proof -+
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  show ?thesis+
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  proof(rule subtreeE[OF assms])+
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    assume "n = Th th"+
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    from that[OF this] show ?thesis .+
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  next+
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    assume "Th th \<in> ancestors (tRAG s) n"+
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    hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)+
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    hence "\<exists> th1. n = Th th1"+
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    proof(induct)+
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      case (base y)+
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      from tRAG_nodeE[OF this] show ?case by metis+
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    next+
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      case (step y z)+
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      thus ?case by auto+
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    qed+
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    with that show ?thesis by auto+
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  qed+
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qed+
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+
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lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"+
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proof -+
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  have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*" +
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    by (rule rtrancl_mono, auto simp:RAG_split)+
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  also have "... \<subseteq> ((RAG s)^*)^*"+
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    by (rule rtrancl_mono, auto)+
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  also have "... = (RAG s)^*" by simp+
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  finally show ?thesis by (unfold tRAG_def, simp)+
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qed+
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+
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lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"+
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proof -+
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  { fix a+
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    assume "a \<in> subtree (tRAG s) x"+
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    hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)+
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    with tRAG_star_RAG[of s]+
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    have "(a, x) \<in> (RAG s)^*" by auto+
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    hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)+
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  } thus ?thesis by auto+
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qed+
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+
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lemma tRAG_trancl_eq:+
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   "{th'. (Th th', Th th)  \<in> (tRAG s)^+} = +
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    {th'. (Th th', Th th)  \<in> (RAG s)^+}"+
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   (is "?L = ?R")+
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proof -+
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  { fix th'+
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    assume "th' \<in> ?L"+
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    hence "(Th th', Th th) \<in> (tRAG s)^+" by auto+
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    from tranclD[OF this]+
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    obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto+
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    from tRAG_subtree_RAG[of s] and this(2)+
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    have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG) +
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    moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto +
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    ultimately have "th' \<in> ?R"  by auto +
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  } moreover +
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  { fix th'+
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    assume "th' \<in> ?R"+
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    hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)+
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    from plus_rpath[OF this]+
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    obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto+
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    hence "(Th th', Th th) \<in> (tRAG s)^+"+
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    proof(induct xs arbitrary:th' th rule:length_induct)+
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      case (1 xs th' th)+
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      then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)+
−
      show ?case+
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      proof(cases "xs1")+
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        case Nil+
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        from 1(2)[unfolded Cons1 Nil]+
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        have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .+
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        hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)+
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        then obtain cs where "x1 = Cs cs" +
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              by (unfold s_RAG_def, auto)+
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        from rpath_nnl_lastE[OF rp[unfolded this]]+
−
        show ?thesis by auto+
−
      next+
−
        case (Cons x2 xs2)+
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        from 1(2)[unfolded Cons1[unfolded this]]+
−
        have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .+
−
        from rpath_edges_on[OF this]+
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        have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .+
−
        have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"+
−
            by (simp add: edges_on_unfold)+
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        with eds have rg1: "(Th th', x1) \<in> RAG s" by auto+
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        then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)+
−
        have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"+
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            by (simp add: edges_on_unfold)+
−
        from this eds+
−
        have rg2: "(x1, x2) \<in> RAG s" by auto+
−
        from this[unfolded eq_x1] +
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        obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)+
−
        from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]+
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        have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)+
−
        from rp have "rpath (RAG s) x2 xs2 (Th th)"+
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           by  (elim rpath_ConsE, simp)+
−
        from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .+
−
        show ?thesis+
−
        proof(cases "xs2 = []")+
−
          case True+
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          from rpath_nilE[OF rp'[unfolded this]]+
−
          have "th1 = th" by auto+
−
          from rt1[unfolded this] show ?thesis by auto+
−
        next+
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          case False+
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          from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]+
−
          have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp+
−
          with rt1 show ?thesis by auto+
−
        qed+
−
      qed+
−
    qed+
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    hence "th' \<in> ?L" by auto+
−
  } ultimately show ?thesis by blast+
−
qed+
−
+
−
lemma tRAG_trancl_eq_Th:+
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   "{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} = +
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    {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}"+
−
    using tRAG_trancl_eq by auto+
−
+
−
lemma dependants_alt_def:+
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  "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"+
−
  by (metis eq_RAG s_dependants_def tRAG_trancl_eq)+
−
  +
−
context valid_trace+
−
begin+
−
+
−
lemma count_eq_tRAG_plus:+
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  assumes "cntP s th = cntV s th"+
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  shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"+
−
  using assms count_eq_dependants dependants_alt_def eq_dependants by auto +
−
+
−
lemma count_eq_RAG_plus:+
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  assumes "cntP s th = cntV s th"+
−
  shows "{th'. (Th th', Th th) \<in> (RAG s)^+} = {}"+
−
  using assms count_eq_dependants cs_dependants_def eq_RAG by auto+
−
+
−
lemma count_eq_RAG_plus_Th:+
−
  assumes "cntP s th = cntV s th"+
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  shows "{Th th' | th'. (Th th', Th th) \<in> (RAG s)^+} = {}"+
−
  using count_eq_RAG_plus[OF assms] by auto+
−
+
−
lemma count_eq_tRAG_plus_Th:+
−
  assumes "cntP s th = cntV s th"+
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  shows "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"+
−
   using count_eq_tRAG_plus[OF assms] by auto+
−
+
−
end+
−
+
−
lemma tRAG_subtree_eq: +
−
   "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th'  \<in> (subtree (RAG s) (Th th))}"+
−
   (is "?L = ?R")+
−
proof -+
−
  { fix n +
−
    assume h: "n \<in> ?L"+
−
    hence "n \<in> ?R"+
−
    by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG) +
−
  } moreover {+
−
    fix n+
−
    assume "n \<in> ?R"+
−
    then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"+
−
      by (auto simp:subtree_def)+
−
    from rtranclD[OF this(2)]+
−
    have "n \<in> ?L"+
−
    proof+
−
      assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"+
−
      with h have "n \<in> {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}" by auto+
−
      thus ?thesis using subtree_def tRAG_trancl_eq by fastforce+
−
    qed (insert h, auto simp:subtree_def)+
−
  } ultimately show ?thesis by auto+
−
qed+
−
+
−
lemma threads_set_eq: +
−
   "the_thread ` (subtree (tRAG s) (Th th)) = +
−
                  {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")+
−
   by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)+
−
+
−
lemma cp_alt_def1: +
−
  "cp s th = Max ((the_preced s o the_thread) ` (subtree (tRAG s) (Th th)))"+
−
proof -+
−
  have "(the_preced s ` the_thread ` subtree (tRAG s) (Th th)) =+
−
       ((the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th))"+
−
       by auto+
−
  thus ?thesis by (unfold cp_alt_def, fold threads_set_eq, auto)+
−
qed+
−
+
−
lemma cp_gen_def_cond: +
−
  assumes "x = Th th"+
−
  shows "cp s th = cp_gen s (Th th)"+
−
by (unfold cp_alt_def1 cp_gen_def, simp)+
−
+
−
lemma cp_gen_over_set:+
−
  assumes "\<forall> x \<in> A. \<exists> th. x = Th th"+
−
  shows "cp_gen s ` A = (cp s \<circ> the_thread) ` A"+
−
proof(rule f_image_eq)+
−
  fix a+
−
  assume "a \<in> A"+
−
  from assms[rule_format, OF this]+
−
  obtain th where eq_a: "a = Th th" by auto+
−
  show "cp_gen s a = (cp s \<circ> the_thread) a"+
−
    by  (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)+
−
qed+
−
+
−
+
−
context valid_trace+
−
begin+
−
+
−
lemma RAG_threads:+
−
  assumes "(Th th) \<in> Field (RAG s)"+
−
  shows "th \<in> threads s"+
−
  using assms+
−
  by (metis Field_def UnE dm_RAG_threads range_in vt)+
−
+
−
lemma subtree_tRAG_thread:+
−
  assumes "th \<in> threads s"+
−
  shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")+
−
proof -+
−
  have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"+
−
    by (unfold tRAG_subtree_eq, simp)+
−
  also have "... \<subseteq> ?R"+
−
  proof+
−
    fix x+
−
    assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"+
−
    then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto+
−
    from this(2)+
−
    show "x \<in> ?R"+
−
    proof(cases rule:subtreeE)+
−
      case 1+
−
      thus ?thesis by (simp add: assms h(1)) +
−
    next+
−
      case 2+
−
      thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI) +
−
    qed+
−
  qed+
−
  finally show ?thesis .+
−
qed+
−
+
−
lemma readys_root:+
−
  assumes "th \<in> readys s"+
−
  shows "root (RAG s) (Th th)"+
−
proof -+
−
  { fix x+
−
    assume "x \<in> ancestors (RAG s) (Th th)"+
−
    hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)+
−
    from tranclD[OF this]+
−
    obtain z where "(Th th, z) \<in> RAG s" by auto+
−
    with assms(1) have False+
−
         apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)+
−
         by (fold wq_def, blast)+
−
  } thus ?thesis by (unfold root_def, auto)+
−
qed+
−
+
−
lemma readys_in_no_subtree:+
−
  assumes "th \<in> readys s"+
−
  and "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 readys_root[OF assms(1)]+
−
      show ?thesis by (auto simp:root_def)+
−
   qed+
−
qed+
−
+
−
lemma not_in_thread_isolated:+
−
  assumes "th \<notin> threads s"+
−
  shows "(Th th) \<notin> Field (RAG s)"+
−
proof+
−
  assume "(Th th) \<in> Field (RAG s)"+
−
  with dm_RAG_threads and range_in assms+
−
  show False by (unfold Field_def, blast)+
−
qed+
−
+
−
lemma wf_RAG: "wf (RAG s)"+
−
proof(rule finite_acyclic_wf)+
−
  from finite_RAG show "finite (RAG s)" .+
−
next+
−
  from acyclic_RAG show "acyclic (RAG s)" .+
−
qed+
−
+
−
lemma sgv_wRAG: "single_valued (wRAG s)"+
−
  using waiting_unique+
−
  by (unfold single_valued_def wRAG_def, auto)+
−
+
−
lemma sgv_hRAG: "single_valued (hRAG s)"+
−
  using holding_unique +
−
  by (unfold single_valued_def hRAG_def, auto)+
−
+
−
lemma sgv_tRAG: "single_valued (tRAG s)"+
−
  by (unfold tRAG_def, rule single_valued_relcomp, +
−
              insert sgv_wRAG sgv_hRAG, auto)+
−
+
−
lemma acyclic_tRAG: "acyclic (tRAG s)"+
−
proof(unfold tRAG_def, rule acyclic_compose)+
−
  show "acyclic (RAG s)" using acyclic_RAG .+
−
next+
−
  show "wRAG s \<subseteq> RAG s" unfolding RAG_split by auto+
−
next+
−
  show "hRAG s \<subseteq> RAG s" unfolding RAG_split by auto+
−
qed+
−
+
−
lemma sgv_RAG: "single_valued (RAG s)"+
−
  using unique_RAG by (auto simp:single_valued_def)+
−
+
−
lemma rtree_RAG: "rtree (RAG s)"+
−
  using sgv_RAG acyclic_RAG+
−
  by (unfold rtree_def rtree_axioms_def sgv_def, auto)+
−
+
−
end+
−
+
−
+
−
sublocale valid_trace < rtree_RAG: rtree "RAG s"+
−
proof+
−
  show "single_valued (RAG s)"+
−
  apply (intro_locales)+
−
    by (unfold single_valued_def, +
−
        auto intro:unique_RAG)+
−
+
−
  show "acyclic (RAG s)"+
−
     by (rule acyclic_RAG)+
−
qed+
−
+
−
sublocale valid_trace < rtree_s: rtree "tRAG s"+
−
proof(unfold_locales)+
−
  from sgv_tRAG show "single_valued (tRAG s)" .+
−
next+
−
  from acyclic_tRAG show "acyclic (tRAG s)" .+
−
qed+
−
+
−
sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"+
−
proof -+
−
  show "fsubtree (RAG s)"+
−
  proof(intro_locales)+
−
    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .+
−
  next+
−
    show "fsubtree_axioms (RAG s)"+
−
    proof(unfold fsubtree_axioms_def)+
−
    find_theorems wf RAG+
−
      from wf_RAG show "wf (RAG s)" .+
−
    qed+
−
  qed+
−
qed+
−
+
−
sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"+
−
proof -+
−
  have "fsubtree (tRAG s)"+
−
  proof -+
−
    have "fbranch (tRAG s)"+
−
    proof(unfold tRAG_def, rule fbranch_compose)+
−
        show "fbranch (wRAG s)"+
−
        proof(rule finite_fbranchI)+
−
           from finite_RAG show "finite (wRAG s)"+
−
           by (unfold RAG_split, auto)+
−
        qed+
−
    next+
−
        show "fbranch (hRAG s)"+
−
        proof(rule finite_fbranchI)+
−
           from finite_RAG +
−
           show "finite (hRAG s)" by (unfold RAG_split, auto)+
−
        qed+
−
    qed+
−
    moreover have "wf (tRAG s)"+
−
    proof(rule wf_subset)+
−
      show "wf (RAG s O RAG s)" using wf_RAG+
−
        by (fold wf_comp_self, simp)+
−
    next+
−
      show "tRAG s \<subseteq> (RAG s O RAG s)"+
−
        by (unfold tRAG_alt_def, auto)+
−
    qed+
−
    ultimately show ?thesis+
−
      by (unfold fsubtree_def fsubtree_axioms_def,auto)+
−
  qed+
−
  from this[folded tRAG_def] show "fsubtree (tRAG s)" .+
−
qed+
−
+
−
lemma Max_UNION: +
−
  assumes "finite A"+
−
  and "A \<noteq> {}"+
−
  and "\<forall> M \<in> f ` A. finite M"+
−
  and "\<forall> M \<in> f ` A. M \<noteq> {}"+
−
  shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")+
−
  using assms[simp]+
−
proof -+
−
  have "?L = Max (\<Union>(f ` A))"+
−
    by (fold Union_image_eq, simp)+
−
  also have "... = ?R"+
−
    by (subst Max_Union, simp+)+
−
  finally show ?thesis .+
−
qed+
−
+
−
lemma max_Max_eq:+
−
  assumes "finite A"+
−
    and "A \<noteq> {}"+
−
    and "x = y"+
−
  shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")+
−
proof -+
−
  have "?R = Max (insert y A)" by simp+
−
  also from assms have "... = ?L"+
−
      by (subst Max.insert, simp+)+
−
  finally show ?thesis by simp+
−
qed+
−
+
−
context valid_trace+
−
begin+
−
+
−
(* ddd *)+
−
lemma cp_gen_rec:+
−
  assumes "x = Th th"+
−
  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"+
−
proof(cases "children (tRAG s) x = {}")+
−
  case True+
−
  show ?thesis+
−
    by (unfold True cp_gen_def subtree_children, simp add:assms)+
−
next+
−
  case False+
−
  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto+
−
  note fsbttRAGs.finite_subtree[simp]+
−
  have [simp]: "finite (children (tRAG s) x)"+
−
     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], +
−
            rule children_subtree)+
−
  { fix r x+
−
    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)+
−
  } note this[simp]+
−
  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"+
−
  proof -+
−
    from False obtain q where "q \<in> children (tRAG s) x" by blast+
−
    moreover have "subtree (tRAG s) q \<noteq> {}" by simp+
−
    ultimately show ?thesis by blast+
−
  qed+
−
  have h: "Max ((the_preced s \<circ> the_thread) `+
−
                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =+
−
        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"+
−
                     (is "?L = ?R")+
−
  proof -+
−
    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L+
−
    let "Max (_ \<union> (?h ` ?B))" = ?R+
−
    let ?L1 = "?f ` \<Union>(?g ` ?B)"+
−
    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"+
−
    proof -+
−
      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp+
−
      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto+
−
      finally have "Max ?L1 = Max ..." by simp+
−
      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"+
−
        by (subst Max_UNION, simp+)+
−
      also have "... = Max (cp_gen s ` children (tRAG s) x)"+
−
          by (unfold image_comp cp_gen_alt_def, simp)+
−
      finally show ?thesis .+
−
    qed+
−
    show ?thesis+
−
    proof -+
−
      have "?L = Max (?f ` ?A \<union> ?L1)" by simp+
−
      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"+
−
            by (subst Max_Un, simp+)+
−
      also have "... = max (?f x) (Max (?h ` ?B))"+
−
        by (unfold eq_Max_L1, simp)+
−
      also have "... =?R"+
−
        by (rule max_Max_eq, (simp)+, unfold assms, simp)+
−
      finally show ?thesis .+
−
    qed+
−
  qed  thus ?thesis +
−
          by (fold h subtree_children, unfold cp_gen_def, simp) +
−
qed+
−
+
−
lemma cp_rec:+
−
  "cp s th = Max ({the_preced s th} \<union> +
−
                     (cp s o the_thread) ` children (tRAG s) (Th th))"+
−
proof -+
−
  have "Th th = Th th" by simp+
−
  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]+
−
  show ?thesis +
−
  proof -+
−
    have "cp_gen s ` children (tRAG s) (Th th) = +
−
                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"+
−
    proof(rule cp_gen_over_set)+
−
      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"+
−
        by (unfold tRAG_alt_def, auto simp:children_def)+
−
    qed+
−
    thus ?thesis by (subst (1) h(1), unfold h(2), simp)+
−
  qed+
−
qed+
−
+
−
end+
−
+
−
(* keep *)+
−
lemma next_th_holding:+
−
  assumes vt: "vt s"+
−
  and nxt: "next_th s th cs th'"+
−
  shows "holding (wq s) th cs"+
−
proof -+
−
  from nxt[unfolded next_th_def]+
−
  obtain rest where h: "wq s cs = th # rest"+
−
                       "rest \<noteq> []" +
−
                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto+
−
  thus ?thesis+
−
    by (unfold cs_holding_def, auto)+
−
qed+
−
+
−
context valid_trace+
−
begin+
−
+
−
lemma next_th_waiting:+
−
  assumes nxt: "next_th s th cs th'"+
−
  shows "waiting (wq s) th' cs"+
−
proof -+
−
  from nxt[unfolded next_th_def]+
−
  obtain rest where h: "wq s cs = th # rest"+
−
                       "rest \<noteq> []" +
−
                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto+
−
  from wq_distinct[of cs, unfolded h]+
−
  have dst: "distinct (th # rest)" .+
−
  have in_rest: "th' \<in> set rest"+
−
  proof(unfold h, rule someI2)+
−
    show "distinct rest \<and> set rest = set rest" using dst by auto+
−
  next+
−
    fix x assume "distinct x \<and> set x = set rest"+
−
    with h(2)+
−
    show "hd x \<in> set (rest)" by (cases x, auto)+
−
  qed+
−
  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)+
−
  moreover have "th' \<noteq> hd (wq s cs)"+
−
    by (unfold h(1), insert in_rest dst, auto)+
−
  ultimately show ?thesis by (auto simp:cs_waiting_def)+
−
qed+
−
+
−
lemma next_th_RAG:+
−
  assumes nxt: "next_th (s::event list) th cs th'"+
−
  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"+
−
  using vt assms next_th_holding next_th_waiting+
−
  by (unfold s_RAG_def, simp)+
−
+
−
end+
−
+
−
-- {* A useless definition *}+
−
definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"+
−
where "cps s = {(th, cp s th) | th . th \<in> threads s}"+
−
+
−
+
−
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 its dependants, +
−
  where the dependants are defined in terms of RAG.+
−
+
−
  Therefore, each operation, lemmas concerning the change of the precedences +
−
  and RAG are derived first, so that the lemmas about+
−
  current precedence recalculation can be based on.+
−
*}+
−
+
−
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"}-operating only changes the precedence +
−
  of the initiating thread.+
−
*}+
−
+
−
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: +
−
  fixes th'+
−
  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 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:+
−
  fixes th' +
−
  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+
−
  find_theorems readys subtree+
−
  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:+
−
  fixes th' +
−
  assumes "th' \<noteq> th"+
−
  shows "cp s th' = cp s' th'"+
−
  by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])+
−
+
−
end+
−
+
−
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:+
−
  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}"+
−
find_theorems "subtree" "_ - _" RAG+
−
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:+
−
  fixes th' +
−
  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+
−
+
−
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+
−
+
−
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 -+
−
    find_theorems readys subtree s'+
−
      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+
−
+
−