
diff -r b3795b1f030b -r ce85c3c4e5bf PIPBasics.thy
--- a/PIPBasics.thy	Sun Feb 07 21:21:53 2016 +0800
+++ b/PIPBasics.thy	Mon Feb 08 10:57:01 2016 +0800
@@ -401,6 +401,274 @@
         insert assms V, auto simp:cntV_def)
 qed (insert assms, auto simp:cntV_def)
 
+lemma eq_RAG: 
+  "RAG (wq s) = RAG s"
+  by (unfold cs_RAG_def s_RAG_def, auto)
+
+text {* 
+  The following three lemmas shows the shape
+  of nodes in @{term tRAG}.
+*}
+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)
+
+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 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
+
+text {*
+  The following lemmas relate @{term tRAG} with 
+  @{term RAG} from different perspectives. 
+*}
+
+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
+    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 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, auto)
+        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 tRAG_Field:
+  "Field (tRAG s) \<subseteq> Field (RAG s)"
+  by (unfold tRAG_alt_def Field_def, auto)
+
+lemma tRAG_mono:
+  assumes "RAG s' \<subseteq> RAG s"
+  shows "tRAG s' \<subseteq> tRAG s"
+  using assms 
+  by (unfold tRAG_alt_def, auto)
+
+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 RAG_tRAG_transfer:
+  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 -
+  { 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(1) have cs_in: "(Cs cs', Th th2) \<in> RAG s" by auto
+    from h(3) and assms(1) 
+    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)"
+      with assms(1) have "(Th th1, Cs cs) \<in> RAG s'" by auto
+      moreover have "th2 = th''"
+      proof -
+        from h1 have "cs' = cs" by simp
+        from assms(2) cs_in[unfolded this]
+        have "holding s th'' cs" "holding s th2 cs"
+          by (unfold s_RAG_def, fold holding_eq, auto)
+        from held_unique[OF this]
+        show ?thesis by simp 
+      qed
+      ultimately show ?thesis using h(1,2) h1 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(1), auto)
+      qed
+      ultimately show ?thesis by auto
+    next
+      assume eq_n: "(n1, n2) = (Th th, Th th'')"
+      from assms(1, 2) have "(Cs cs, Th th'') \<in> RAG s'" by auto
+      moreover have "(Th th, Cs cs) \<in> RAG s'" using assms(1) by auto
+      ultimately show ?thesis 
+        by (unfold eq_n tRAG_alt_def, auto)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+text {* 
+  The following lemmas gives an alternative definition @{term dependants}
+  in terms of @{term tRAG}.
+*}
+
+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)
+
+text {* 
+  The following lemmas gives another alternative definition @{term dependants}
+  in terms of @{term RAG}.
+*}
+
+lemma dependants_alt_def1:
+  "dependants (s::state) th = {th'. (Th th', Th th) \<in> (RAG s)^+}"
+  using dependants_alt_def tRAG_trancl_eq by auto
+
 section {* Locales used to investigate the execution of PIP *}
 
 text {* 
@@ -1180,6 +1448,34 @@
   using assms
   by (metis Field_def UnE dm_RAG_threads rg_RAG_threads)
 
+text {*
+  As a corollary, this lemma shows that @{term tRAG}
+  is also covered by the @{term threads}-set.
+*}
+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
+
 end
 
 section {* The change of @{term RAG} *}
@@ -2157,7 +2453,7 @@
 proof -
   from assms(2)
   show ?thesis
-  by (cases rule:waiting_esE, insert assms, auto) (* ccc *)
+  by (cases rule:waiting_esE, insert assms, auto)
 qed
 
 end
@@ -2519,6 +2815,13 @@
 
 section {* RAG is well-founded *}
 
+text {*
+  In this section, it is proved that both @{term RAG} and 
+  its converse @{term "(RAG s)^-1"} are well-founded.
+  The proof is very simple with the help of
+  already proved fact that @{term RAG} is finite.
+*}
+
 context valid_trace
 begin
 
@@ -2541,7 +2844,12 @@
 
 end
 
-section {* RAG forms a forest (or tree) *}
+section {* RAG forms a finite-branching forest (or tree) *}
+
+text {*
+  With all the well-formedness proofs about @{term RAG} in place, 
+  it is easy to show.
+*}
 
 context valid_trace
 begin
@@ -2569,7 +2877,7 @@
 qed
 
 
-section {* Derived properties for parts of RAG *}
+section {* Derived properties for sub-graphs of RAG *}
 
 context valid_trace
 begin
@@ -2597,6 +2905,11 @@
 
 end
 
+text {*
+  It can be shown that @{term tRAG} is also a 
+  finite-branch relational tree (or forest):  
+*}
+
 sublocale valid_trace < rtree_s: rtree "tRAG s"
 proof(unfold_locales)
   from sgv_tRAG show "single_valued (tRAG s)" .
@@ -2636,282 +2949,6 @@
   from this[folded tRAG_def] show "fsubtree (tRAG s)" .
 qed
 
-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)
-
-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 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
-    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 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, auto)
-        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 tRAG_Field:
-  "Field (tRAG s) \<subseteq> Field (RAG s)"
-  by (unfold tRAG_alt_def Field_def, auto)
-
-lemma tRAG_mono:
-  assumes "RAG s' \<subseteq> RAG s"
-  shows "tRAG s' \<subseteq> tRAG s"
-  using assms 
-  by (unfold tRAG_alt_def, auto)
-
-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)
-
-context valid_trace
-begin
-
-lemma RAG_tRAG_transfer:
-  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 -
-  { 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(1) have cs_in: "(Cs cs', Th th2) \<in> RAG s" by auto
-    from h(3) and assms(1) 
-    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(2) cs_in[unfolded this]
-        show ?thesis using unique_RAG by 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(1), auto)
-      qed
-      ultimately show ?thesis by auto
-    next
-      assume eq_n: "(n1, n2) = (Th th, Th th'')"
-      from assms(1, 2) have "(Cs cs, Th th'') \<in> RAG s'" by auto
-      moreover have "(Th th, Cs cs) \<in> RAG s'" using assms(1) by auto
-      ultimately show ?thesis 
-        by (unfold eq_n tRAG_alt_def, auto)
-    qed
-  } ultimately show ?thesis by auto
-qed
-
-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 eq_RAG: 
-  "RAG (wq s) = RAG s"
-  by (unfold cs_RAG_def s_RAG_def, 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)
-
-lemma dependants_alt_def1:
-  "dependants (s::state) th = {th'. (Th th', Th th) \<in> (RAG s)^+}"
-  using dependants_alt_def tRAG_trancl_eq by auto
-
-end
-
 section {* Chain to readys *}
 
 context valid_trace
@@ -2961,7 +2998,7 @@
 qed
 
 text {* \noindent
-  The following is just an instance of @{text "chain_building"}.
+  The following lemma is proved easily by instantiating @{thm "chain_building"}.
 *}                    
 lemma th_chain_to_ready:
   assumes th_in: "th \<in> threads s"




