
diff -r 313acffe63b6 -r 92f61f6a0fe7 PrioG.thy
--- a/PrioG.thy	Tue May 20 12:49:21 2014 +0100
+++ b/PrioG.thy	Thu May 22 17:40:39 2014 +0100
@@ -24,7 +24,7 @@
   thus "distinct (wq (e # s) cs)"
   proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
     fix thread s
-    assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+    assume h1: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
       and h2: "thread \<in> set (wq_fun (schs s) cs)"
       and h3: "thread \<in> runing s"
     show "False" 
@@ -34,8 +34,8 @@
         by (simp add:runing_def readys_def s_waiting_def wq_def)
       from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" .
       with h2
-      have "(Cs cs, Th thread) \<in> (depend s)"
-        by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
+      have "(Cs cs, Th thread) \<in> (RAG s)"
+        by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
       with h1 show False by auto
     qed
   next
@@ -112,7 +112,7 @@
 qed
 
 lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow> 
-  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (depend s)^+"
+  thread \<in> runing s \<and> (Cs cs, Th thread)  \<notin> (RAG s)^+"
 apply (ind_cases "vt ((P thread cs)#s)")
 apply (ind_cases "step s (P thread cs)")
 by auto
@@ -501,16 +501,16 @@
   qed
 qed
 
-lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
+lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
 by (simp add:Let_def)
 
-lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
+lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
 by (simp add:Let_def)
 
-lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
+lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
 by (simp add:Let_def)
 
 
@@ -773,16 +773,16 @@
   qed
 qed
 
-lemma step_depend_v:
+lemma step_RAG_v:
 fixes th::thread
 assumes vt:
   "vt (V th cs#s)"
 shows "
-  depend (V th cs # s) =
-  depend s - {(Cs cs, Th th)} -
+  RAG (V th cs # s) =
+  RAG s - {(Cs cs, Th th)} -
   {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
   {(Cs cs, Th th') |th'.  next_th s th cs th'}"
-  apply (insert vt, unfold s_depend_def) 
+  apply (insert vt, unfold s_RAG_def) 
   apply (auto split:if_splits list.splits simp:Let_def)
   apply (auto elim: step_v_waiting_mono step_v_hold_inv 
               step_v_release step_v_wait_inv
@@ -790,17 +790,17 @@
   apply (erule_tac step_v_not_wait, auto)
   done
 
-lemma step_depend_p:
+lemma step_RAG_p:
   "vt (P th cs#s) \<Longrightarrow>
-  depend (P th cs # s) =  (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
-                                             else depend s \<union> {(Th th, Cs cs)})"
-  apply(simp only: s_depend_def wq_def)
+  RAG (P th cs # s) =  (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+                                             else RAG s \<union> {(Th th, Cs cs)})"
+  apply(simp only: s_RAG_def wq_def)
   apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
   apply(case_tac "csa = cs", auto)
   apply(fold wq_def)
   apply(drule_tac step_back_step)
   apply(ind_cases " step s (P (hd (wq s cs)) cs)")
-  apply(auto simp:s_depend_def wq_def cs_holding_def)
+  apply(auto simp:s_RAG_def wq_def cs_holding_def)
   done
 
 lemma simple_A:
@@ -815,35 +815,35 @@
   thus ?thesis by simp
 qed
 
-lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
-  by (unfold s_depend_def, auto)
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, auto)
 
-lemma acyclic_depend: 
+lemma acyclic_RAG: 
   fixes s
   assumes vt: "vt s"
-  shows "acyclic (depend s)"
+  shows "acyclic (RAG s)"
 proof -
   from vt show ?thesis
   proof(induct)
     case (vt_cons s e)
-    assume ih: "acyclic (depend s)"
+    assume ih: "acyclic (RAG s)"
       and stp: "step s e"
       and vt: "vt s"
     show ?case
     proof(cases e)
       case (Create th prio)
       with ih
-      show ?thesis by (simp add:depend_create_unchanged)
+      show ?thesis by (simp add:RAG_create_unchanged)
     next
       case (Exit th)
-      with ih show ?thesis by (simp add:depend_exit_unchanged)
+      with ih show ?thesis by (simp add:RAG_exit_unchanged)
     next
       case (V th cs)
       from V vt stp have vtt: "vt (V th cs#s)" by auto
-      from step_depend_v [OF this]
+      from step_RAG_v [OF this]
       have eq_de: 
-        "depend (e # s) = 
-            depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+        "RAG (e # s) = 
+            RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
             {(Cs cs, Th th') |th'. next_th s th cs th'}"
         (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
       from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
@@ -871,11 +871,11 @@
             from tranclD [OF this]
             obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
             hence th_d: "(Th ?th', x) \<in> ?A" by simp
-            from depend_target_th [OF this]
+            from RAG_target_th [OF this]
             obtain cs' where eq_x: "x = Cs cs'" by auto
             with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
             hence wt_th': "waiting s ?th' cs'"
-              unfolding s_depend_def s_waiting_def cs_waiting_def wq_def by simp
+              unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
             hence "cs' = cs"
             proof(rule waiting_unique [OF vt])
               from eq_wq wq_distinct[OF vt, of cs]
@@ -945,35 +945,35 @@
   next
     case (P th cs)
     from P vt stp have vtt: "vt (P th cs#s)" by auto
-    from step_depend_p [OF this] P
-    have "depend (e # s) = 
-      (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
-      depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+    from step_RAG_p [OF this] P
+    have "RAG (e # s) = 
+      (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else 
+      RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
       by simp
     moreover have "acyclic ?R"
     proof(cases "wq s cs = []")
       case True
-      hence eq_r: "?R =  depend s \<union> {(Cs cs, Th th)}" by simp
-      have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
+      hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
+      have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
       proof
-        assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
-        hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+        assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
+        hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
         from tranclD2 [OF this]
-        obtain x where "(x, Cs cs) \<in> depend s" by auto
-        with True show False by (auto simp:s_depend_def cs_waiting_def)
+        obtain x where "(x, Cs cs) \<in> RAG s" by auto
+        with True show False by (auto simp:s_RAG_def cs_waiting_def)
       qed
       with acyclic_insert ih eq_r show ?thesis by auto
     next
       case False
-      hence eq_r: "?R =  depend s \<union> {(Th th, Cs cs)}" by simp
-      have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
+      hence eq_r: "?R =  RAG s \<union> {(Th th, Cs cs)}" by simp
+      have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
       proof
-        assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
-        hence "(Cs cs, Th th) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+        assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
+        hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
           moreover from step_back_step [OF vtt] have "step s (P th cs)" .
           ultimately show False
           proof -
-            show " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+            show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
               by (ind_cases "step s (P th cs)", simp)
           qed
         qed
@@ -983,42 +983,42 @@
     next
       case (Set thread prio)
       with ih
-      thm depend_set_unchanged
-      show ?thesis by (simp add:depend_set_unchanged)
+      thm RAG_set_unchanged
+      show ?thesis by (simp add:RAG_set_unchanged)
     qed
   next
     case vt_nil
-    show "acyclic (depend ([]::state))"
-      by (auto simp: s_depend_def cs_waiting_def 
+    show "acyclic (RAG ([]::state))"
+      by (auto simp: s_RAG_def cs_waiting_def 
                       cs_holding_def wq_def acyclic_def)
   qed
 qed
 
-lemma finite_depend: 
+lemma finite_RAG: 
   fixes s
   assumes vt: "vt s"
-  shows "finite (depend s)"
+  shows "finite (RAG s)"
 proof -
   from vt show ?thesis
   proof(induct)
     case (vt_cons s e)
-    assume ih: "finite (depend s)"
+    assume ih: "finite (RAG s)"
       and stp: "step s e"
       and vt: "vt s"
     show ?case
     proof(cases e)
       case (Create th prio)
       with ih
-      show ?thesis by (simp add:depend_create_unchanged)
+      show ?thesis by (simp add:RAG_create_unchanged)
     next
       case (Exit th)
-      with ih show ?thesis by (simp add:depend_exit_unchanged)
+      with ih show ?thesis by (simp add:RAG_exit_unchanged)
     next
       case (V th cs)
       from V vt stp have vtt: "vt (V th cs#s)" by auto
-      from step_depend_v [OF this]
-      have eq_de: "depend (e # s) = 
-                   depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+      from step_RAG_v [OF this]
+      have eq_de: "RAG (e # s) = 
+                   RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
                       {(Cs cs, Th th') |th'. next_th s th cs th'}
 "
         (is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
@@ -1041,31 +1041,31 @@
     next
       case (P th cs)
       from P vt stp have vtt: "vt (P th cs#s)" by auto
-      from step_depend_p [OF this] P
-      have "depend (e # s) = 
-              (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else 
-                                    depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+      from step_RAG_p [OF this] P
+      have "RAG (e # s) = 
+              (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else 
+                                    RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
         by simp
       moreover have "finite ?R"
       proof(cases "wq s cs = []")
         case True
-        hence eq_r: "?R =  depend s \<union> {(Cs cs, Th th)}" by simp
+        hence eq_r: "?R =  RAG s \<union> {(Cs cs, Th th)}" by simp
         with True and ih show ?thesis by auto
       next
         case False
-        hence "?R = depend s \<union> {(Th th, Cs cs)}" by simp
+        hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
         with False and ih show ?thesis by auto
       qed
       ultimately show ?thesis by auto
     next
       case (Set thread prio)
       with ih
-      show ?thesis by (simp add:depend_set_unchanged)
+      show ?thesis by (simp add:RAG_set_unchanged)
     qed
   next
     case vt_nil
-    show "finite (depend ([]::state))"
-      by (auto simp: s_depend_def cs_waiting_def 
+    show "finite (RAG ([]::state))"
+      by (auto simp: s_RAG_def cs_waiting_def 
                    cs_holding_def wq_def acyclic_def)
   qed
 qed
@@ -1075,20 +1075,20 @@
 lemma wf_dep_converse: 
   fixes s
   assumes vt: "vt s"
-  shows "wf ((depend s)^-1)"
+  shows "wf ((RAG s)^-1)"
 proof(rule finite_acyclic_wf_converse)
-  from finite_depend [OF vt]
-  show "finite (depend s)" .
+  from finite_RAG [OF vt]
+  show "finite (RAG s)" .
 next
-  from acyclic_depend[OF vt]
-  show "acyclic (depend s)" .
+  from acyclic_RAG[OF vt]
+  show "acyclic (RAG s)" .
 qed
 
 lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
 by (induct l, auto)
 
-lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
-  by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
+  by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
 
 lemma wq_threads: 
   fixes s cs
@@ -1114,7 +1114,7 @@
         apply (auto simp:wq_def Let_def)
         apply (ind_cases "step s (Exit th')")
         apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
-               s_depend_def s_holding_def cs_holding_def)
+               s_RAG_def s_holding_def cs_holding_def)
         done
     next
       case (V th' cs')
@@ -1192,8 +1192,8 @@
   qed
 qed
 
-lemma range_in: "\<lbrakk>vt s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
-  apply(unfold s_depend_def cs_waiting_def cs_holding_def)
+lemma range_in: "\<lbrakk>vt s; (Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+  apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
   by (auto intro:wq_threads)
 
 lemma readys_v_eq:
@@ -1231,33 +1231,33 @@
 
 lemma chain_building:
   assumes vt: "vt s"
-  shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
+  shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
 proof -
   from wf_dep_converse [OF vt]
-  have h: "wf ((depend s)\<inverse>)" .
+  have h: "wf ((RAG s)\<inverse>)" .
   show ?thesis
   proof(induct rule:wf_induct [OF h])
     fix x
     assume ih [rule_format]: 
-      "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow> 
-           y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
-    show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
+      "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow> 
+           y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
+    show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
     proof
-      assume x_d: "x \<in> Domain (depend s)"
-      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
+      assume x_d: "x \<in> Domain (RAG s)"
+      show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
       proof(cases x)
         case (Th th)
-        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
-        with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
-        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
-        hence "Cs cs \<in> Domain (depend s)" by auto
+        from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
+        with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
+        from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
+        hence "Cs cs \<in> Domain (RAG s)" by auto
         from ih [OF x_in_r this] obtain th'
-          where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto
-        have "(x, Th th') \<in> (depend s)\<^sup>+" using Th x_in cs_in by auto
+          where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
+        have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
         with th'_ready show ?thesis by auto
       next
         case (Cs cs)
-        from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (depend s)^-1" by (auto simp:s_depend_def)
+        from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
         show ?thesis
         proof(cases "th' \<in> readys s")
           case True
@@ -1265,14 +1265,14 @@
         next
           case False
           from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto
-          with False have "Th th' \<in> Domain (depend s)" 
-            by (auto simp:readys_def wq_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
+          with False have "Th th' \<in> Domain (RAG s)" 
+            by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
           from ih [OF th'_d this]
           obtain th'' where 
             th''_r: "th'' \<in> readys s" and 
-            th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto
+            th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
           from th'_d and th''_in 
-          have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
+          have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
           with th''_r show ?thesis by auto
         qed
       qed
@@ -1284,14 +1284,14 @@
   fixes s th
   assumes vt: "vt s"
   and th_in: "th \<in> threads s"
-  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"
+  shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
 proof(cases "th \<in> readys s")
   case True
   thus ?thesis by auto
 next
   case False
-  from False and th_in have "Th th \<in> Domain (depend s)" 
-    by (auto simp:readys_def s_waiting_def s_depend_def wq_def cs_waiting_def Domain_def)
+  from False and th_in have "Th th \<in> Domain (RAG s)" 
+    by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
   from chain_building [rule_format, OF vt this]
   show ?thesis by auto
 qed
@@ -1305,8 +1305,8 @@
 lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
   by (unfold s_holding_def cs_holding_def, auto)
 
-lemma unique_depend: "\<lbrakk>vt s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
-  apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
+lemma unique_RAG: "\<lbrakk>vt s; (n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
+  apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
   by(auto elim:waiting_unique holding_unique)
 
 lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
@@ -1314,34 +1314,34 @@
 
 lemma dchain_unique:
   assumes vt: "vt s"
-  and th1_d: "(n, Th th1) \<in> (depend s)^+"
+  and th1_d: "(n, Th th1) \<in> (RAG s)^+"
   and th1_r: "th1 \<in> readys s"
-  and th2_d: "(n, Th th2) \<in> (depend s)^+"
+  and th2_d: "(n, Th th2) \<in> (RAG s)^+"
   and th2_r: "th2 \<in> readys s"
   shows "th1 = th2"
 proof -
   { assume neq: "th1 \<noteq> th2"
     hence "Th th1 \<noteq> Th th2" by simp
-    from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]
-    have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
+    from unique_chain [OF _ th1_d th2_d this] and unique_RAG [OF vt]
+    have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
     hence "False"
     proof
-      assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
+      assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
       from trancl_split [OF this]
-      obtain n where dd: "(Th th1, n) \<in> depend s" by auto
+      obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
       then obtain cs where eq_n: "n = Cs cs"
-        by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+        by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
       from dd eq_n have "th1 \<notin> readys s"
-        by (auto simp:readys_def s_depend_def wq_def s_waiting_def cs_waiting_def)
+        by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
       with th1_r show ?thesis by auto
     next
-      assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
+      assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
       from trancl_split [OF this]
-      obtain n where dd: "(Th th2, n) \<in> depend s" by auto
+      obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
       then obtain cs where eq_n: "n = Cs cs"
-        by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+        by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
       from dd eq_n have "th2 \<notin> readys s"
-        by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def)
+        by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
       with th2_r show ?thesis by auto
     qed
   } thus ?thesis by auto
@@ -1355,7 +1355,7 @@
   shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
 proof -
   from assms show ?thesis
-  unfolding  holdents_test step_depend_p[OF vt] by (auto)
+  unfolding  holdents_test step_RAG_p[OF vt] by (auto)
 qed
 
 lemma step_holdents_p_eq:
@@ -1365,7 +1365,7 @@
   shows "holdents (P th cs#s) th = holdents s th"
 proof -
   from assms show ?thesis
-  unfolding  holdents_test step_depend_p[OF vt] by auto
+  unfolding  holdents_test step_RAG_p[OF vt] by auto
 qed
 
 
@@ -1375,16 +1375,16 @@
   shows "finite (holdents s th)"
 proof -
   let ?F = "\<lambda> (x, y). the_cs x"
-  from finite_depend [OF vt]
-  have "finite (depend s)" .
-  hence "finite (?F `(depend s))" by simp
-  moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>" 
+  from finite_RAG [OF vt]
+  have "finite (RAG s)" .
+  hence "finite (?F `(RAG s))" by simp
+  moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>" 
   proof -
     { have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
-      fix x assume "(Cs x, Th th) \<in> depend s"
-      hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
+      fix x assume "(Cs x, Th th) \<in> RAG s"
+      hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
       moreover have "?F (Cs x, Th th) = x" by simp
-      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp 
+      ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp 
     } thus ?thesis by auto
   qed
   ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
@@ -1397,12 +1397,12 @@
 proof -
   from step_back_step[OF vtv]
   have cs_in: "cs \<in> holdents s thread" 
-    apply (cases, unfold holdents_test s_depend_def, simp)
+    apply (cases, unfold holdents_test s_RAG_def, simp)
     by (unfold cs_holding_def s_holding_def wq_def, auto)
   moreover have cs_not_in: 
     "(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
     apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
-    apply (unfold holdents_test, unfold step_depend_v[OF vtv],
+    apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
             auto simp:next_th_def)
   proof -
     fix rest
@@ -1425,7 +1425,7 @@
       show "x \<noteq> []" by auto
     qed
     ultimately 
-    show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"
+    show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
       by auto
   next
     fix rest
@@ -1494,7 +1494,7 @@
         from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
         have eq_cncs: "cntCS (e#s) th = cntCS s th"
           unfolding cntCS_def holdents_test
-          by (simp add:depend_create_unchanged eq_e)
+          by (simp add:RAG_create_unchanged eq_e)
         { assume "th \<noteq> thread"
           with eq_readys eq_e
           have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
@@ -1519,7 +1519,7 @@
       from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
       have eq_cncs: "cntCS (e#s) th = cntCS s th"
         unfolding cntCS_def holdents_test
-        by (simp add:depend_exit_unchanged eq_e)
+        by (simp add:RAG_exit_unchanged eq_e)
       { assume "th \<noteq> thread"
         with eq_e
         have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
@@ -1544,7 +1544,7 @@
       case (thread_P thread cs)
       assume eq_e: "e = P thread cs"
         and is_runing: "thread \<in> runing s"
-        and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+        and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
       from thread_P vt stp ih  have vtp: "vt (P thread cs#s)" by auto
       show ?thesis 
       proof -
@@ -1561,7 +1561,7 @@
             by (case_tac "(wq_fun (schs s) cs)", auto)
           moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
             apply (simp add:cntCS_def holdents_test)
-            by (unfold  step_depend_p [OF vtp], auto)
+            by (unfold  step_RAG_p [OF vtp], auto)
           moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
             by (simp add:cntP_def count_def)
           moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
@@ -1582,26 +1582,26 @@
               case True
               with is_runing
               have "th \<in> readys (e#s)"
-                apply (unfold eq_e wq_def, unfold readys_def s_depend_def)
+                apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
                 apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
                 by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
               moreover have "cntCS (e # s) th = 1 + cntCS s th"
               proof -
-                have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> depend s} =
-                  Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")
+                have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
+                  Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
                 proof -
                   have "?L = insert cs ?R" by auto
                   moreover have "card \<dots> = Suc (card (?R - {cs}))" 
                   proof(rule card_insert)
                     from finite_holding [OF vt, of thread]
-                    show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
+                    show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
                       by (unfold holdents_test, simp)
                   qed
                   moreover have "?R - {cs} = ?R"
                   proof -
                     have "cs \<notin> ?R"
                     proof
-                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend s}"
+                      assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
                       with no_dep show False by auto
                     qed
                     thus ?thesis by auto
@@ -1611,7 +1611,7 @@
                 thus ?thesis
                   apply (unfold eq_e eq_th cntCS_def)
                   apply (simp add: holdents_test)
-                  by (unfold step_depend_p [OF vtp], auto simp:True)
+                  by (unfold step_RAG_p [OF vtp], auto simp:True)
               qed
               moreover from is_runing have "th \<in> readys s"
                 by (simp add:runing_def eq_th)
@@ -1638,7 +1638,7 @@
               moreover from is_runing have "th \<in> threads (e#s)" 
                 by (unfold eq_e, auto simp:runing_def readys_def eq_th)
               moreover have "cntCS (e # s) th = cntCS s th"
-                apply (unfold cntCS_def holdents_test eq_e step_depend_p[OF vtp])
+                apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
                 by (auto simp:False)
               moreover note eq_cnp eq_cnv ih[of th]
               moreover from is_runing have "th \<in> readys s"
@@ -1735,13 +1735,13 @@
               apply (insert step_back_vt[OF vtv])
               by (unfold eq_e, rule readys_v_eq [OF _ neq_th eq_wq False], auto)
             moreover have "cntCS (e#s) th = cntCS s th"
-              apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
+              apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
               proof -
-                have "{csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
-                      {cs. (Cs cs, Th th) \<in> depend s}"
+                have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+                      {cs. (Cs cs, Th th) \<in> RAG s}"
                 proof -
                   from False eq_wq
-                  have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> depend s"
+                  have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
                     apply (unfold next_th_def, auto)
                   proof -
                     assume ne: "rest \<noteq> []"
@@ -1759,13 +1759,13 @@
                       with ne show "x \<noteq> []" by auto
                     qed
                     ultimately show 
-                      "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"
+                      "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
                       by auto
                   qed    
                   thus ?thesis by auto
                 qed
-                thus "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
-                             card {cs. (Cs cs, Th th) \<in> depend s}" by simp 
+                thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+                             card {cs. (Cs cs, Th th) \<in> RAG s}" by simp 
               qed
             moreover note ih eq_cnp eq_cnv eq_threads
             ultimately show ?thesis by auto
@@ -1796,8 +1796,8 @@
               proof -
                 from eq_wq and  th_in and neq_hd
                 have "(holdents (e # s) th) = (holdents s th)"
-                  apply (unfold eq_e step_depend_v[OF vtv], 
-                         auto simp:next_th_def eq_set s_depend_def holdents_test wq_def
+                  apply (unfold eq_e step_RAG_v[OF vtv], 
+                         auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
                                    Let_def cs_holding_def)
                   by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto simp:wq_def)
                 thus ?thesis by (simp add:cntCS_def)
@@ -1862,28 +1862,28 @@
                 ultimately show ?thesis using ih by auto
               qed
               moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
-                apply (unfold cntCS_def holdents_test eq_e step_depend_v[OF vtv], auto)
+                apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
               proof -
-                show "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs} =
-                               Suc (card {cs. (Cs cs, Th th) \<in> depend s})"
+                show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
+                               Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
                   (is "card ?A = Suc (card ?B)")
                 proof -
                   have "?A = insert cs ?B" by auto
                   hence "card ?A = card (insert cs ?B)" by simp
                   also have "\<dots> = Suc (card ?B)"
                   proof(rule card_insert_disjoint)
-                    have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` depend s)" 
+                    have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)" 
                       apply (auto simp:image_def)
                       by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
-                    with finite_depend[OF step_back_vt[OF vtv]]
-                    show "finite {cs. (Cs cs, Th th) \<in> depend s}" by (auto intro:finite_subset)
+                    with finite_RAG[OF step_back_vt[OF vtv]]
+                    show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
                   next
-                    show "cs \<notin> {cs. (Cs cs, Th th) \<in> depend s}"
+                    show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
                     proof
-                      assume "cs \<in> {cs. (Cs cs, Th th) \<in> depend s}"
-                      hence "(Cs cs, Th th) \<in> depend s" by simp
+                      assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
+                      hence "(Cs cs, Th th) \<in> RAG s" by simp
                       with True neq_th eq_wq show False
-                        by (auto simp:next_th_def s_depend_def cs_holding_def)
+                        by (auto simp:next_th_def s_RAG_def cs_holding_def)
                     qed
                   qed
                   finally show ?thesis .
@@ -1905,7 +1905,7 @@
         from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
         have eq_cncs: "cntCS (e#s) th = cntCS s th"
           unfolding cntCS_def holdents_test
-          by (simp add:depend_set_unchanged eq_e)
+          by (simp add:RAG_set_unchanged eq_e)
         from eq_e have eq_readys: "readys (e#s) = readys s" 
           by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
                   auto simp:Let_def)
@@ -1931,7 +1931,7 @@
     case vt_nil
     show ?case 
       by (unfold cntP_def cntV_def cntCS_def, 
-        auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
+        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
   qed
 qed
 
@@ -1955,7 +1955,7 @@
         and not_in': "thread \<notin> threads s"
       have "cntCS (e # s) th = cntCS s th"
         apply (unfold eq_e cntCS_def holdents_test)
-        by (simp add:depend_create_unchanged)
+        by (simp add:RAG_create_unchanged)
       moreover have "th \<notin> threads s" 
       proof -
         from not_in eq_e show ?thesis by simp
@@ -1967,7 +1967,7 @@
       and nh: "holdents s thread = {}"
       have eq_cns: "cntCS (e # s) th = cntCS s th"
         apply (unfold eq_e cntCS_def holdents_test)
-        by (simp add:depend_exit_unchanged)
+        by (simp add:RAG_exit_unchanged)
       show ?thesis
       proof(cases "th = thread")
         case True
@@ -1993,7 +1993,7 @@
       qed
       hence "cntCS (e # s) th  = cntCS s th "
         apply (unfold cntCS_def holdents_test eq_e)
-        by (unfold step_depend_p[OF vtp], auto)
+        by (unfold step_RAG_p[OF vtp], auto)
       moreover have "cntCS s th = 0"
       proof(rule ih)
         from not_in eq_e show "th \<notin> threads s" by simp
@@ -2035,7 +2035,7 @@
       qed
       moreover note neq_th eq_wq
       ultimately have "cntCS (e # s) th  = cntCS s th"
-        by (unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
+        by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
       moreover have "cntCS s th = 0"
       proof(rule ih)
         from not_in eq_e show "th \<notin> threads s" by simp
@@ -2050,30 +2050,30 @@
       from ih [OF this] and eq_e
       show ?thesis 
         apply (unfold eq_e cntCS_def holdents_test)
-        by (simp add:depend_set_unchanged)
+        by (simp add:RAG_set_unchanged)
     qed
     next
       case vt_nil
       show ?case
       by (unfold cntCS_def, 
-        auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
+        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
   qed
 qed
 
 lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
   by (auto simp:s_waiting_def cs_waiting_def wq_def)
 
-lemma dm_depend_threads:
+lemma dm_RAG_threads:
   fixes th s
   assumes vt: "vt s"
-  and in_dom: "(Th th) \<in> Domain (depend s)"
+  and in_dom: "(Th th) \<in> Domain (RAG s)"
   shows "th \<in> threads s"
 proof -
-  from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
-  moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
-  ultimately have "(Th th, Cs cs) \<in> depend s" by simp
+  from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
+  moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
+  ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
   hence "th \<in> set (wq s cs)"
-    by (unfold s_depend_def, auto simp:cs_waiting_def)
+    by (unfold s_RAG_def, auto simp:cs_waiting_def)
   from wq_threads [OF vt this] show ?thesis .
 qed
 
@@ -2112,18 +2112,18 @@
       proof -
         have "finite (dependants (wq s) th1)"
         proof-
-          have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
+          have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
           proof -
             let ?F = "\<lambda> (x, y). the_th x"
-            have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+            have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
               apply (auto simp:image_def)
               by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
             moreover have "finite \<dots>"
             proof -
-              from finite_depend[OF vt] have "finite (depend s)" .
-              hence "finite ((depend (wq s))\<^sup>+)"
+              from finite_RAG[OF vt] have "finite (RAG s)" .
+              hence "finite ((RAG (wq s))\<^sup>+)"
                 apply (unfold finite_trancl)
-                by (auto simp: s_depend_def cs_depend_def wq_def)
+                by (auto simp: s_RAG_def cs_RAG_def wq_def)
               thus ?thesis by auto
             qed
             ultimately show ?thesis by (auto intro:finite_subset)
@@ -2151,18 +2151,18 @@
       proof -
         have "finite (dependants (wq s) th2)"
         proof-
-          have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
+          have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
           proof -
             let ?F = "\<lambda> (x, y). the_th x"
-            have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+            have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
               apply (auto simp:image_def)
               by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
             moreover have "finite \<dots>"
             proof -
-              from finite_depend[OF vt] have "finite (depend s)" .
-              hence "finite ((depend (wq s))\<^sup>+)"
+              from finite_RAG[OF vt] have "finite (RAG s)" .
+              hence "finite ((RAG (wq s))\<^sup>+)"
                 apply (unfold finite_trancl)
-                by (auto simp: s_depend_def cs_depend_def wq_def)
+                by (auto simp: s_RAG_def cs_RAG_def wq_def)
               thus ?thesis by auto
             qed
             ultimately show ?thesis by (auto intro:finite_subset)
@@ -2190,11 +2190,11 @@
     thus "th1' \<in> threads s"
     proof
       assume "th1' \<in> dependants (wq s) th1"
-      hence "(Th th1') \<in> Domain ((depend s)^+)"
-        apply (unfold cs_dependants_def cs_depend_def s_depend_def)
+      hence "(Th th1') \<in> Domain ((RAG s)^+)"
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
         by (auto simp:Domain_def)
-      hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
-      from dm_depend_threads[OF vt this] show ?thesis .
+      hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      from dm_RAG_threads[OF vt this] show ?thesis .
     next
       assume "th1' = th1"
       with runing_1 show ?thesis
@@ -2205,11 +2205,11 @@
     thus "th2' \<in> threads s"
     proof
       assume "th2' \<in> dependants (wq s) th2"
-      hence "(Th th2') \<in> Domain ((depend s)^+)"
-        apply (unfold cs_dependants_def cs_depend_def s_depend_def)
+      hence "(Th th2') \<in> Domain ((RAG s)^+)"
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
         by (auto simp:Domain_def)
-      hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
-      from dm_depend_threads[OF vt this] show ?thesis .
+      hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      from dm_RAG_threads[OF vt this] show ?thesis .
     next
       assume "th2' = th2"
       with runing_2 show ?thesis
@@ -2227,18 +2227,18 @@
     next
       assume "th2' \<in> dependants (wq s) th2"
       with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
-      hence "(Th th1, Th th2) \<in> (depend s)^+"
-        by (unfold cs_dependants_def s_depend_def cs_depend_def, simp)
-      hence "Th th1 \<in> Domain ((depend s)^+)" 
-        apply (unfold cs_dependants_def cs_depend_def s_depend_def)
+      hence "(Th th1, Th th2) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      hence "Th th1 \<in> Domain ((RAG s)^+)" 
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
         by (auto simp:Domain_def)
-      hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
-      then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
-      from depend_target_th [OF this]
+      hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
+      from RAG_target_th [OF this]
       obtain cs' where "n = Cs cs'" by auto
-      with d have "(Th th1, Cs cs') \<in> depend s" by simp
+      with d have "(Th th1, Cs cs') \<in> RAG s" by simp
       with runing_1 have "False"
-        apply (unfold runing_def readys_def s_depend_def)
+        apply (unfold runing_def readys_def s_RAG_def)
         by (auto simp:eq_waiting)
       thus ?thesis by simp
     qed
@@ -2249,27 +2249,27 @@
     proof
       assume "th2' = th2"
       with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
-      hence "(Th th2, Th th1) \<in> (depend s)^+"
-        by (unfold cs_dependants_def s_depend_def cs_depend_def, simp)
-      hence "Th th2 \<in> Domain ((depend s)^+)" 
-        apply (unfold cs_dependants_def cs_depend_def s_depend_def)
+      hence "(Th th2, Th th1) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      hence "Th th2 \<in> Domain ((RAG s)^+)" 
+        apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
         by (auto simp:Domain_def)
-      hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
-      then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
-      from depend_target_th [OF this]
+      hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+      then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
+      from RAG_target_th [OF this]
       obtain cs' where "n = Cs cs'" by auto
-      with d have "(Th th2, Cs cs') \<in> depend s" by simp
+      with d have "(Th th2, Cs cs') \<in> RAG s" by simp
       with runing_2 have "False"
-        apply (unfold runing_def readys_def s_depend_def)
+        apply (unfold runing_def readys_def s_RAG_def)
         by (auto simp:eq_waiting)
       thus ?thesis by simp
     next
       assume "th2' \<in> dependants (wq s) th2"
       with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
-      hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
-        by (unfold cs_dependants_def s_depend_def cs_depend_def, simp)
-      from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
-        by (unfold cs_dependants_def s_depend_def cs_depend_def, simp)
+      hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+      from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
+        by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
       show ?thesis
       proof(rule dchain_unique[OF vt h1 _ h2, symmetric])
         from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
@@ -2411,9 +2411,9 @@
   qed
 qed
 
-lemma eq_depend: 
-  "depend (wq s) = depend s"
-by (unfold cs_depend_def s_depend_def, auto)
+lemma eq_RAG: 
+  "RAG (wq s) = RAG s"
+by (unfold cs_RAG_def s_RAG_def, auto)
 
 lemma count_eq_dependants:
   assumes vt: "vt s"
@@ -2423,28 +2423,28 @@
   from cnp_cnv_cncs[OF vt] and eq_pv
   have "cntCS s th = 0" 
     by (auto split:if_splits)
-  moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
+  moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
   proof -
     from finite_holding[OF vt, of th] show ?thesis
       by (simp add:holdents_test)
   qed
-  ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
+  ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
     by (unfold cntCS_def holdents_test cs_dependants_def, auto)
   show ?thesis
   proof(unfold cs_dependants_def)
-    { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
-      then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
+    { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
+      then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
       hence "False"
       proof(cases)
-        assume "(Th th', Th th) \<in> depend (wq s)"
-        thus "False" by (auto simp:cs_depend_def)
+        assume "(Th th', Th th) \<in> RAG (wq s)"
+        thus "False" by (auto simp:cs_RAG_def)
       next
         fix c
-        assume "(c, Th th) \<in> depend (wq s)"
-        with h and eq_depend show "False"
-          by (cases c, auto simp:cs_depend_def)
+        assume "(c, Th th) \<in> RAG (wq s)"
+        with h and eq_RAG show "False"
+          by (cases c, auto simp:cs_RAG_def)
       qed
-    } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
+    } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
   qed
 qed
 
@@ -2454,20 +2454,20 @@
   shows "dependants (wq s) th \<subseteq> threads s"
 proof
   { fix th th'
-    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
-    have "Th th \<in> Domain (depend s)"
+    assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
+    have "Th th \<in> Domain (RAG s)"
     proof -
-      from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
-      hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
-      with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
-      thus ?thesis using eq_depend by simp
+      from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
+      hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
+      with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
+      thus ?thesis using eq_RAG by simp
     qed
-    from dm_depend_threads[OF vt this]
+    from dm_RAG_threads[OF vt this]
     have "th \<in> threads s" .
   } note hh = this
   fix th1 
   assume "th1 \<in> dependants (wq s) th"
-  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
+  hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
     by (unfold cs_dependants_def, simp)
   from hh [OF this] show "th1 \<in> threads s" .
 qed
@@ -2496,21 +2496,21 @@
   and th_in: "th \<in> threads s"
   shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
 proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
-  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))
+  show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
          \<le> Max ((\<lambda>th. preced th s) ` threads s)"
     (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
   proof(rule Max_f_mono)
-    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
   next
     from finite_threads [OF vt]
     show "finite (threads s)" .
   next
     from th_in
-    show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
+    show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
       apply (auto simp:Domain_def)
-      apply (rule_tac dm_depend_threads[OF vt])
-      apply (unfold trancl_domain [of "depend s", symmetric])
-      by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
+      apply (rule_tac dm_RAG_threads[OF vt])
+      apply (unfold trancl_domain [of "RAG s", symmetric])
+      by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
   qed
 qed
 
@@ -2528,18 +2528,18 @@
     proof -
       have "finite ?B" 
       proof-
-        have "finite {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}"
+        have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
         proof -
           let ?F = "\<lambda> (x, y). the_th x"
-          have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+          have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
             apply (auto simp:image_def)
             by (rule_tac x = "(Th x, Th th)" in bexI, auto)
           moreover have "finite \<dots>"
           proof -
-            from finite_depend[OF vt] have "finite (depend s)" .
-            hence "finite ((depend (wq s))\<^sup>+)"
+            from finite_RAG[OF vt] have "finite (RAG s)" .
+            hence "finite ((RAG (wq s))\<^sup>+)"
               apply (unfold finite_trancl)
-              by (auto simp: s_depend_def cs_depend_def wq_def)
+              by (auto simp: s_RAG_def cs_RAG_def wq_def)
             thus ?thesis by auto
           qed
           ultimately show ?thesis by (auto intro:finite_subset)
@@ -2621,12 +2621,12 @@
     then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
       by (auto simp:Image_def)
     from th_chain_to_ready [OF vt tm_in]
-    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
+    have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
     thus ?thesis
     proof
-      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
+      assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
       then obtain th' where th'_in: "th' \<in> readys s" 
-        and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
+        and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
       have "cp s th' = ?f tm"
       proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
         from dependants_threads[OF vt] finite_threads[OF vt]
@@ -2650,7 +2650,7 @@
         proof -
           from tm_chain
           have "tm \<in> dependants (wq s) th'"
-            by (unfold cs_dependants_def s_depend_def cs_depend_def, auto)
+            by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
           thus ?thesis by auto
         qed
       qed
@@ -2780,9 +2780,9 @@
 
 
 lemma detached_test:
-  shows "detached s th = (Th th \<notin> Field (depend s))"
+  shows "detached s th = (Th th \<notin> Field (RAG s))"
 apply(simp add: detached_def Field_def)
-apply(simp add: s_depend_def)
+apply(simp add: s_RAG_def)
 apply(simp add: s_holding_abv s_waiting_abv)
 apply(simp add: Domain_iff Range_iff)
 apply(simp add: wq_def)
@@ -2805,12 +2805,12 @@
   thus ?thesis
   proof
     assume "th \<notin> threads s"
-    with range_in[OF vt] dm_depend_threads[OF vt]
+    with range_in[OF vt] dm_RAG_threads[OF vt]
     show ?thesis
-      by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
+      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
   next
     assume "th \<in> readys s"
-    moreover have "Th th \<notin> Range (depend s)"
+    moreover have "Th th \<notin> Range (RAG s)"
     proof -
       from card_0_eq [OF finite_holding [OF vt]] and cncs_zero
       have "holdents s th = {}"
@@ -2818,11 +2818,11 @@
       thus ?thesis
         apply(auto simp:holdents_test)
         apply(case_tac a)
-        apply(auto simp:holdents_test s_depend_def)
+        apply(auto simp:holdents_test s_RAG_def)
         done
     qed
     ultimately show ?thesis
-      by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def readys_def)
+      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
   qed
 qed
 
@@ -2838,7 +2838,7 @@
   have cncs_z: "cntCS s th = 0"
   proof -
     from dtc have "holdents s th = {}"
-      unfolding detached_def holdents_test s_depend_def
+      unfolding detached_def holdents_test s_RAG_def
       by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
     thus ?thesis by (auto simp:cntCS_def)
   qed
@@ -2848,7 +2848,7 @@
     with dtc 
     have "th \<in> readys s"
       by (unfold readys_def detached_def Field_def Domain_def Range_def, 
-           auto simp:eq_waiting s_depend_def)
+           auto simp:eq_waiting s_RAG_def)
     with cncs_z and eq_pv show ?thesis by simp
   next
     case False
@@ -2862,4 +2862,4 @@
   shows "(detached s th) = (cntP s th = cntV s th)"
   by (insert vt, auto intro:detached_intro detached_elim)
 
-end
\ No newline at end of file
+end




