diff -r b620a2a0806a -r b4bcd1edbb6d PrioG.thy~
--- a/PrioG.thy~	Wed Jan 06 20:46:14 2016 +0800
+++ b/PrioG.thy~	Wed Jan 06 16:34:26 2016 +0000
@@ -1,5 +1,5 @@
 theory PrioG
-imports PrioGDef 
+imports PrioGDef RTree
 begin
 
 locale valid_trace = 
@@ -3023,4 +3023,606 @@
   shows "th1 = th2"
 using assms by (unfold next_th_def, auto)
 
+lemma birth_time_lt:  "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
+  apply (induct s, simp)
+proof -
+  fix a s
+  assume ih: "s \<noteq> [] \<Longrightarrow> last_set th s < length s"
+    and eq_as: "a # s \<noteq> []"
+  show "last_set th (a # s) < length (a # s)"
+  proof(cases "s \<noteq> []")
+    case False
+    from False show ?thesis
+      by (cases a, auto simp:last_set.simps)
+  next
+    case True
+    from ih [OF True] show ?thesis
+      by (cases a, auto simp:last_set.simps)
+  qed
+qed
+
+lemma th_in_ne: "th \<in> threads s \<Longrightarrow> s \<noteq> []"
+  by (induct s, auto simp:threads.simps)
+
+lemma preced_tm_lt: "th \<in> threads s \<Longrightarrow> preced th s = Prc x y \<Longrightarrow> y < length s"
+  apply (drule_tac th_in_ne)
+  by (unfold preced_def, auto intro: birth_time_lt)
+
+text {* @{text "the_preced"} is also the same as @{text "preced"}, the only
+       difference is the order of arguemts. *}
+definition "the_preced s th = preced th s"
+
+lemma inj_the_preced: 
+  "inj_on (the_preced s) (threads s)"
+  by (metis inj_onI preced_unique the_preced_def)
+
+text {* @{term "the_thread"} extracts thread out of RAG node. *}
+fun the_thread :: "node \<Rightarrow> thread" where
+   "the_thread (Th th) = th"
+
+text {* The following @{text "wRAG"} is the waiting sub-graph of @{text "RAG"}. *}
+definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"
+
+text {* The following @{text "hRAG"} is the holding sub-graph of @{text "RAG"}. *}
+definition "hRAG (s::state) =  {(Cs cs, Th th) | th cs. holding s th cs}"
+
+text {* The following lemma splits @{term "RAG"} graph into the above two sub-graphs. *}
+lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"
+  by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv 
+             s_holding_abv cs_RAG_def, auto)
+
+text {* 
+  The following @{text "tRAG"} is the thread-graph derived from @{term "RAG"}.
+  It characterizes the dependency between threads when calculating current
+  precedences. It is defined as the composition of the above two sub-graphs, 
+  names @{term "wRAG"} and @{term "hRAG"}.
+ *}
+definition "tRAG s = wRAG s O hRAG s"
+
+(* ccc *)
+
+definition "cp_gen s x =
+                  Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"
+
+lemma tRAG_alt_def: 
+  "tRAG s = {(Th th1, Th th2) | th1 th2. 
+                  \<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
+ by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
+
+lemma tRAG_Field:
+  "Field (tRAG s) \<subseteq> Field (RAG s)"
+  by (unfold tRAG_alt_def Field_def, auto)
+
+lemma tRAG_ancestorsE:
+  assumes "x \<in> ancestors (tRAG s) u"
+  obtains th where "x = Th th"
+proof -
+  from assms have "(u, x) \<in> (tRAG s)^+" 
+      by (unfold ancestors_def, auto)
+  from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
+  then obtain th where "x = Th th"
+    by (unfold tRAG_alt_def, auto)
+  from that[OF this] show ?thesis .
+qed
+
+lemma tRAG_mono:
+  assumes "RAG s' \<subseteq> RAG s"
+  shows "tRAG s' \<subseteq> tRAG s"
+  using assms 
+  by (unfold tRAG_alt_def, auto)
+
+lemma holding_next_thI:
+  assumes "holding s th cs"
+  and "length (wq s cs) > 1"
+  obtains th' where "next_th s th cs th'"
+proof -
+  from assms(1)[folded eq_holding, unfolded cs_holding_def]
+  have " th \<in> set (wq s cs) \<and> th = hd (wq s cs)" .
+  then obtain rest where h1: "wq s cs = th#rest" 
+    by (cases "wq s cs", auto)
+  with assms(2) have h2: "rest \<noteq> []" by auto
+  let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+  have "next_th s th cs ?th'" using  h1(1) h2 
+    by (unfold next_th_def, auto)
+  from that[OF this] show ?thesis .
+qed
+
+lemma RAG_tRAG_transfer:
+  assumes "vt s'"
+  assumes "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+  and "(Cs cs, Th th'') \<in> RAG s'"
+  shows "tRAG s = tRAG s' \<union> {(Th th, Th th'')}" (is "?L = ?R")
+proof -
+  interpret vt_s': valid_trace "s'" using assms(1)
+    by (unfold_locales, simp)
+  interpret rtree: rtree "RAG s'"
+  proof
+  show "single_valued (RAG s')"
+  apply (intro_locales)
+    by (unfold single_valued_def, 
+        auto intro:vt_s'.unique_RAG)
+
+  show "acyclic (RAG s')"
+     by (rule vt_s'.acyclic_RAG)
+  qed
+  { fix n1 n2
+    assume "(n1, n2) \<in> ?L"
+    from this[unfolded tRAG_alt_def]
+    obtain th1 th2 cs' where 
+      h: "n1 = Th th1" "n2 = Th th2" 
+         "(Th th1, Cs cs') \<in> RAG s"
+         "(Cs cs', Th th2) \<in> RAG s" by auto
+    from h(4) and assms(2) have cs_in: "(Cs cs', Th th2) \<in> RAG s'" by auto
+    from h(3) and assms(2) 
+    have "(Th th1, Cs cs') = (Th th, Cs cs) \<or> 
+          (Th th1, Cs cs') \<in> RAG s'" by auto
+    hence "(n1, n2) \<in> ?R"
+    proof
+      assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
+      hence eq_th1: "th1 = th" by simp
+      moreover have "th2 = th''"
+      proof -
+        from h1 have "cs' = cs" by simp
+        from assms(3) cs_in[unfolded this] rtree.sgv
+        show ?thesis
+          by (unfold single_valued_def, auto)
+      qed
+      ultimately show ?thesis using h(1,2) by auto
+    next
+      assume "(Th th1, Cs cs') \<in> RAG s'"
+      with cs_in have "(Th th1, Th th2) \<in> tRAG s'"
+        by (unfold tRAG_alt_def, auto)
+      from this[folded h(1, 2)] show ?thesis by auto
+    qed
+  } moreover {
+    fix n1 n2
+    assume "(n1, n2) \<in> ?R"
+    hence "(n1, n2) \<in>tRAG s' \<or> (n1, n2) = (Th th, Th th'')" by auto
+    hence "(n1, n2) \<in> ?L" 
+    proof
+      assume "(n1, n2) \<in> tRAG s'"
+      moreover have "... \<subseteq> ?L"
+      proof(rule tRAG_mono)
+        show "RAG s' \<subseteq> RAG s" by (unfold assms(2), auto)
+      qed
+      ultimately show ?thesis by auto
+    next
+      assume eq_n: "(n1, n2) = (Th th, Th th'')"
+      from assms(2, 3) have "(Cs cs, Th th'') \<in> RAG s" by auto
+      moreover have "(Th th, Cs cs) \<in> RAG s" using assms(2) by auto
+      ultimately show ?thesis 
+        by (unfold eq_n tRAG_alt_def, auto)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+context valid_trace
+begin
+
+lemmas RAG_tRAG_transfer = RAG_tRAG_transfer[OF vt]
+
 end
+
+lemma cp_alt_def:
+  "cp s th =  
+           Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
+proof -
+  have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) =
+        Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})" 
+          (is "Max (_ ` ?L) = Max (_ ` ?R)")
+  proof -
+    have "?L = ?R" 
+    by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def)
+    thus ?thesis by simp
+  qed
+  thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
+qed
+
+lemma cp_gen_alt_def:
+  "cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
+    by (auto simp:cp_gen_def)
+
+lemma tRAG_nodeE:
+  assumes "(n1, n2) \<in> tRAG s"
+  obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
+  using assms
+  by (auto simp: tRAG_def wRAG_def hRAG_def tRAG_def)
+
+lemma subtree_nodeE:
+  assumes "n \<in> subtree (tRAG s) (Th th)"
+  obtains th1 where "n = Th th1"
+proof -
+  show ?thesis
+  proof(rule subtreeE[OF assms])
+    assume "n = Th th"
+    from that[OF this] show ?thesis .
+  next
+    assume "Th th \<in> ancestors (tRAG s) n"
+    hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+    hence "\<exists> th1. n = Th th1"
+    proof(induct)
+      case (base y)
+      from tRAG_nodeE[OF this] show ?case by metis
+    next
+      case (step y z)
+      thus ?case by auto
+    qed
+    with that show ?thesis by auto
+  qed
+qed
+
+lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
+proof -
+  have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*" 
+    by (rule rtrancl_mono, auto simp:RAG_split)
+  also have "... \<subseteq> ((RAG s)^*)^*"
+    by (rule rtrancl_mono, auto)
+  also have "... = (RAG s)^*" by simp
+  finally show ?thesis by (unfold tRAG_def, simp)
+qed
+
+lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
+proof -
+  { fix a
+    assume "a \<in> subtree (tRAG s) x"
+    hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
+    with tRAG_star_RAG[of s]
+    have "(a, x) \<in> (RAG s)^*" by auto
+    hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
+  } thus ?thesis by auto
+qed
+
+lemma tRAG_trancl_eq:
+   "{th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
+    {th'. (Th th', Th th)  \<in> (RAG s)^+}"
+   (is "?L = ?R")
+proof -
+  { fix th'
+    assume "th' \<in> ?L"
+    hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
+    from tranclD[OF this]
+    obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
+    from tRAG_subtree_RAG[of s] and this(2)
+    have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG) 
+    moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto 
+    ultimately have "th' \<in> ?R"  by auto 
+  } moreover 
+  { fix th'
+    assume "th' \<in> ?R"
+    hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
+    from plus_rpath[OF this]
+    obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
+    hence "(Th th', Th th) \<in> (tRAG s)^+"
+    proof(induct xs arbitrary:th' th rule:length_induct)
+      case (1 xs th' th)
+      then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
+      show ?case
+      proof(cases "xs1")
+        case Nil
+        from 1(2)[unfolded Cons1 Nil]
+        have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
+        hence "(Th th', x1) \<in> (RAG s)" by (cases, simp)
+        then obtain cs where "x1 = Cs cs" 
+              by (unfold s_RAG_def, auto)
+        from rpath_nnl_lastE[OF rp[unfolded this]]
+        show ?thesis by auto
+      next
+        case (Cons x2 xs2)
+        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]
+        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)
+        with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
+        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)"
+            by (simp add: edges_on_unfold)
+        from this eds
+        have rg2: "(x1, x2) \<in> RAG s" by auto
+        from this[unfolded eq_x1] 
+        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]
+        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)"
+           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
+          from rpath_nilE[OF rp'[unfolded this]]
+          have "th1 = th" by auto
+          from rt1[unfolded this] show ?thesis by auto
+        next
+          case False
+          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
+    hence "th' \<in> ?L" by auto
+  } ultimately show ?thesis by blast
+qed
+
+lemma tRAG_trancl_eq_Th:
+   "{Th th' | th'. (Th th', Th th)  \<in> (tRAG s)^+} = 
+    {Th th' | th'. (Th th', Th th)  \<in> (RAG s)^+}"
+    using tRAG_trancl_eq by auto
+
+lemma dependants_alt_def:
+  "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:
+  assumes "cntP s th = cntV s th"
+  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:
+  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"
+  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"
+  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
+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
+
+end