# HG changeset patch
# User zhangx
# Date 1452126793 -28800
# Node ID 633b1fc8631b1e0efa22f0184cec6254cb002cc0
# Parent  b4bcd1edbb6d1b9fb20819f493b32030c996e4ba
Reorganization completed, added "scripts_structure.pdf" and "scirpts_structure.pptx".

diff -r b4bcd1edbb6d -r 633b1fc8631b Correctness.thy
--- a/Correctness.thy	Wed Jan 06 16:34:26 2016 +0000
+++ b/Correctness.thy	Thu Jan 07 08:33:13 2016 +0800
@@ -1,5 +1,5 @@
 theory Correctness
-imports PIPBasics Implementation
+imports PIPBasics
 begin
 
 text {* 
@@ -467,7 +467,7 @@
   a thread is running or not.
 *}
 
-lemma pv_blocked_pre:
+lemma pv_blocked_pre: (* ddd *)
   assumes th'_in: "th' \<in> threads (t@s)"
   and neq_th': "th' \<noteq> th"
   and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
@@ -496,7 +496,7 @@
 
 lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
 
-lemma runing_precond_pre:
+lemma runing_precond_pre: (* ddd *)
   fixes th'
   assumes th'_in: "th' \<in> threads s"
   and eq_pv: "cntP s th' = cntV s th'"
@@ -600,7 +600,7 @@
 lemmas runing_precond_pre_dtc = runing_precond_pre
          [folded vat_t.detached_eq vat_s.detached_eq]
 
-lemma runing_precond:
+lemma runing_precond: (* ddd *)
   fixes th'
   assumes th'_in: "th' \<in> threads s"
   and neq_th': "th' \<noteq> th"
@@ -660,7 +660,7 @@
           moment_plus_split neq_th' th'_in)
 qed
 
-lemma moment_blocked_eqpv:
+lemma moment_blocked_eqpv: (* ddd *)
   assumes neq_th': "th' \<noteq> th"
   and th'_in: "th' \<in> threads ((moment i t)@s)"
   and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
@@ -830,7 +830,6 @@
   apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
   by (metis neq_th runing' runing_inversion_3)
 
-
 text {* 
   Suppose @{term th} is not running, it is first shown that
   there is a path in RAG leading from node @{term th} to another thread @{text "th'"} 
diff -r b4bcd1edbb6d -r 633b1fc8631b Correctness.thy~
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Correctness.thy~	Thu Jan 07 08:33:13 2016 +0800
@@ -0,0 +1,921 @@
+theory Correctness
+imports PIPBasics Implementation
+begin
+
+text {* 
+  The following two auxiliary lemmas are used to reason about @{term Max}.
+*}
+lemma image_Max_eqI: 
+  assumes "finite B"
+  and "b \<in> B"
+  and "\<forall> x \<in> B. f x \<le> f b"
+  shows "Max (f ` B) = f b"
+  using assms
+  using Max_eqI by blast 
+
+lemma image_Max_subset:
+  assumes "finite A"
+  and "B \<subseteq> A"
+  and "a \<in> B"
+  and "Max (f ` A) = f a"
+  shows "Max (f ` B) = f a"
+proof(rule image_Max_eqI)
+  show "finite B"
+    using assms(1) assms(2) finite_subset by auto 
+next
+  show "a \<in> B" using assms by simp
+next
+  show "\<forall>x\<in>B. f x \<le> f a"
+    by (metis Max_ge assms(1) assms(2) assms(4) 
+            finite_imageI image_eqI subsetCE) 
+qed
+
+text {*
+  The following locale @{text "highest_gen"} sets the basic context for our
+  investigation: supposing thread @{text th} holds the highest @{term cp}-value
+  in state @{text s}, which means the task for @{text th} is the 
+  most urgent. We want to show that  
+  @{text th} is treated correctly by PIP, which means
+  @{text th} will not be blocked unreasonably by other less urgent
+  threads. 
+*}
+locale highest_gen =
+  fixes s th prio tm
+  assumes vt_s: "vt s"
+  and threads_s: "th \<in> threads s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+  -- {* The internal structure of @{term th}'s precedence is exposed:*}
+  and preced_th: "preced th s = Prc prio tm" 
+
+-- {* @{term s} is a valid trace, so it will inherit all results derived for
+      a valid trace: *}
+sublocale highest_gen < vat_s: valid_trace "s"
+  by (unfold_locales, insert vt_s, simp)
+
+context highest_gen
+begin
+
+text {*
+  @{term tm} is the time when the precedence of @{term th} is set, so 
+  @{term tm} must be a valid moment index into @{term s}.
+*}
+lemma lt_tm: "tm < length s"
+  by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+text {*
+  Since @{term th} holds the highest precedence and @{text "cp"}
+  is the highest precedence of all threads in the sub-tree of 
+  @{text "th"} and @{text th} is among these threads, 
+  its @{term cp} must equal to its precedence:
+*}
+lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
+proof -
+  have "?L \<le> ?R"
+  by (unfold highest, rule Max_ge, 
+        auto simp:threads_s finite_threads)
+  moreover have "?R \<le> ?L"
+    by (unfold vat_s.cp_rec, rule Max_ge, 
+        auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
+  ultimately show ?thesis by auto
+qed
+
+(* ccc *)
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq, unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[symmetric]
+  show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen + 
+  fixes t 
+  assumes vt_t: "vt (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+sublocale extend_highest_gen < vat_t: valid_trace "t@s"
+  by (unfold_locales, insert vt_t, simp)
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt (t@s)" 
+  shows "vt s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt (t @ s) \<Longrightarrow> vt s"
+      and vt_et: "vt ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+
+locale red_extend_highest_gen = extend_highest_gen +
+   fixes i::nat
+
+sublocale red_extend_highest_gen <   red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
+  apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+  by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+
+context extend_highest_gen
+begin
+
+ lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e; 
+                    extend_highest_gen s th prio tm t; 
+                    extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_gen_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt ((e # t') @ s)"
+      and et: "extend_highest_gen s th prio tm (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_gen s th prio tm (e # t')" .
+    next
+      from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+                 preced th (t@s) = preced th s" (is "?Q t") 
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show ?case
+      by auto
+  next
+    case (Cons e t)
+    interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+    interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      show ?thesis
+      proof -
+        from Cons and Create have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          case thread_create
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold Create, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:Create)
+      qed
+    next
+      case (Exit thread)
+      from h_e.exit_diff and Exit
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold Exit, auto simp:preced_def)
+    next
+      case (P thread cs)
+      with Cons
+      show ?thesis 
+        by (auto simp:P preced_def)
+    next
+      case (V thread cs)
+      with Cons
+      show ?thesis 
+        by (auto simp:V preced_def)
+    next
+      case (Set thread prio')
+      show ?thesis
+      proof -
+        from h_e.set_diff_low and Set
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold Set, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:Set)
+      qed
+    qed
+  qed
+qed
+
+text {*
+  According to @{thm th_kept}, thread @{text "th"} has its living status
+  and precedence kept along the way of @{text "t"}. The following lemma
+  shows that this preserved precedence of @{text "th"} remains as the highest
+  along the way of @{text "t"}.
+
+  The proof goes by induction over @{text "t"} using the specialized
+  induction rule @{thm ind}, followed by case analysis of each possible 
+  operations of PIP. All cases follow the same pattern rendered by the 
+  generalized introduction rule @{thm "image_Max_eqI"}. 
+
+  The very essence is to show that precedences, no matter whether they are newly introduced 
+  or modified, are always lower than the one held by @{term "th"},
+  which by @{thm th_kept} is preserved along the way.
+*}
+lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show ?case
+    by (unfold the_preced_def, simp)
+next
+  case (Cons e t)
+    interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+    interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      -- {* The following is the common pattern of each branch of the case analysis. *}
+      -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume "x \<in> ?A"
+          hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
+          thus "?f x \<le> ?f th"
+          proof
+            assume "x = thread"
+            thus ?thesis 
+              apply (simp add:Create the_preced_def preced_def, fold preced_def)
+              using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force
+          next
+            assume h: "x \<in> threads (t @ s)"
+            from Cons(2)[unfolded Create] 
+            have "x \<noteq> thread" using h by (cases, auto)
+            hence "?f x = the_preced (t@s) x" 
+              by (simp add:Create the_preced_def preced_def)
+            hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
+              by (simp add: h_t.finite_threads h)
+            also have "... = ?f th"
+              by (metis Cons.hyps(5) h_e.th_kept the_preced_def) 
+            finally show ?thesis .
+          qed
+        qed
+      qed
+     -- {* The minor part is to show that the precedence of @{text "th"} 
+           equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      -- {* Then it follows trivially that the precedence preserved
+            for @{term "th"} remains the maximum of all living threads along the way. *}
+      finally show ?thesis .
+    qed 
+  next 
+    case (Exit thread)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume "x \<in> ?A"
+          hence "x \<in> threads (t@s)" by (simp add: Exit) 
+          hence "?f x \<le> Max (?f ` threads (t@s))" 
+            by (simp add: h_t.finite_threads) 
+          also have "... \<le> ?f th" 
+            apply (simp add:Exit the_preced_def preced_def, fold preced_def)
+            using Cons.hyps(5) h_t.th_kept the_preced_def by auto
+          finally show "?f x \<le> ?f th" .
+        qed
+      qed
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      finally show ?thesis .
+    qed 
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def the_preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def the_preced_def)
+  next 
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume h: "x \<in> ?A"
+          show "?f x \<le> ?f th"
+          proof(cases "x = thread")
+            case True
+            moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
+            proof -
+              have "the_preced (t @ s) th = Prc prio tm"  
+                using h_t.th_kept preced_th by (simp add:the_preced_def)
+              moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
+              ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
+            qed
+            ultimately show ?thesis
+              by (unfold Set, simp add:the_preced_def preced_def)
+          next
+            case False
+            then have "?f x  = the_preced (t@s) x"
+              by (simp add:the_preced_def preced_def Set)
+            also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
+              using Set h h_t.finite_threads by auto 
+            also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) 
+            finally show ?thesis .
+          qed
+        qed
+      qed
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      finally show ?thesis .
+    qed 
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+text {*
+  The reason behind the following lemma is that:
+  Since @{term "cp"} is defined as the maximum precedence 
+  of those threads contained in the sub-tree of node @{term "Th th"} 
+  in @{term "RAG (t@s)"}, and all these threads are living threads, and 
+  @{term "th"} is also among them, the maximum precedence of 
+  them all must be the one for @{text "th"}.
+*}
+lemma th_cp_max_preced: 
+  "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") 
+proof -
+  let ?f = "the_preced (t@s)"
+  have "?L = ?f th"
+  proof(unfold cp_alt_def, rule image_Max_eqI)
+    show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+    proof -
+      have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} = 
+            the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
+                            (\<exists> th'. n = Th th')}"
+      by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
+      moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) 
+      ultimately show ?thesis by simp
+    qed
+  next
+    show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+      by (auto simp:subtree_def)
+  next
+    show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
+               the_preced (t @ s) x \<le> the_preced (t @ s) th"
+    proof
+      fix th'
+      assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+      hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
+      moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
+        by (meson subtree_Field)
+      ultimately have "Th th' \<in> ..." by auto
+      hence "th' \<in> threads (t@s)" 
+      proof
+        assume "Th th' \<in> {Th th}"
+        thus ?thesis using th_kept by auto 
+      next
+        assume "Th th' \<in> Field (RAG (t @ s))"
+        thus ?thesis using vat_t.not_in_thread_isolated by blast 
+      qed
+      thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
+        by (metis Max_ge finite_imageI finite_threads image_eqI 
+               max_kept th_kept the_preced_def)
+    qed
+  qed
+  also have "... = ?R" by (simp add: max_preced the_preced_def) 
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less:
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+  using assms
+by (metis Max.coboundedI finite_imageI highest not_le order.trans 
+    preced_linorder rev_image_eqI threads_s vat_s.finite_threads 
+    vat_s.le_cp)
+
+text {*
+  Counting of the number of @{term "P"} and @{term "V"} operations 
+  is the cornerstone of a large number of the following proofs. 
+  The reason is that this counting is quite easy to calculate and 
+  convenient to use in the reasoning. 
+
+  The following lemma shows that the counting controls whether 
+  a thread is running or not.
+*}
+
+lemma pv_blocked_pre: (* ddd *)
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume otherwise: "th' \<in> runing (t@s)"
+  show False
+  proof -
+    have "th' = th"
+    proof(rule preced_unique)
+      show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+      proof -
+        have "?L = cp (t@s) th'"
+          by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
+        also have "... = cp (t @ s) th" using otherwise 
+          by (metis (mono_tags, lifting) mem_Collect_eq 
+                    runing_def th_cp_max vat_t.max_cp_readys_threads)
+        also have "... = ?R" by (metis th_cp_preced th_kept) 
+        finally show ?thesis .
+      qed
+    qed (auto simp: th'_in th_kept)
+    moreover have "th' \<noteq> th" using neq_th' .
+    ultimately show ?thesis by simp
+ qed
+qed
+
+lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
+
+lemma runing_precond_pre: (* ddd *)
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof(induct rule:ind)
+  case (Cons e t)
+    interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
+    interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
+    show ?case
+    proof(cases e)
+      case (P thread cs)
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (P thread cs)" using Cons P by auto
+            thus ?thesis
+            proof(cases)
+              assume "thread \<in> runing (t@s)"
+              moreover have "th' \<notin> runing (t@s)" using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+              ultimately show ?thesis by auto
+            qed
+          qed with Cons show ?thesis
+            by (unfold P, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (V thread cs)
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (V thread cs)" using Cons V by auto
+            thus ?thesis
+            proof(cases)
+              assume "thread \<in> runing (t@s)"
+              moreover have "th' \<notin> runing (t@s)" using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+              ultimately show ?thesis by auto
+            qed
+          qed with Cons show ?thesis
+            by (unfold V, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Create thread prio')
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (Create thread prio')" using Cons Create by auto
+            thus ?thesis using Cons(5) by (cases, auto)
+          qed with Cons show ?thesis
+            by (unfold Create, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Exit thread)
+      show ?thesis
+      proof -
+        have neq_thread: "thread \<noteq> th'"
+        proof -
+          have "step (t@s) (Exit thread)" using Cons Exit by auto
+          thus ?thesis apply (cases) using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+        qed 
+        hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons
+            by (unfold Exit, simp add:cntP_def cntV_def count_def)
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread 
+          by (unfold Exit, simp) 
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+next
+  case Nil
+  with assms
+  show ?case by auto
+qed
+
+text {* Changing counting balance to detachedness *}
+lemmas runing_precond_pre_dtc = runing_precond_pre
+         [folded vat_t.detached_eq vat_s.detached_eq]
+
+lemma runing_precond: (* ddd *)
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+  using assms
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+    by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)
+  moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof -
+  interpret h_i: red_extend_highest_gen _ _ _ _ _ i
+      by (unfold_locales)
+  interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"
+      by (unfold_locales)
+  interpret h:  extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"
+  proof(unfold_locales)
+    show "vt (moment i t @ s)" by (metis h_i.vt_t) 
+  next
+    show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)
+  next
+    show "preced th (moment i t @ s) = 
+            Max (cp (moment i t @ s) ` threads (moment i t @ s))"
+              by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)
+  next
+    show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th) 
+  next
+    show "vt (moment j (restm i t) @ moment i t @ s)"
+      using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)
+  next
+    fix th' prio'
+    assume "Create th' prio' \<in> set (moment j (restm i t))"
+    thus "prio' \<le> prio" using assms
+       by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)
+  next
+    fix th' prio'
+    assume "Set th' prio' \<in> set (moment j (restm i t))"
+    thus "th' \<noteq> th \<and> prio' \<le> prio"
+    by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)
+  next
+    fix th'
+    assume "Exit th' \<in> set (moment j (restm i t))"
+    thus "th' \<noteq> th"
+      by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)
+  qed
+  show ?thesis 
+    by (metis add.commute append_assoc eq_pv h.runing_precond_pre
+          moment_plus_split neq_th' th'_in)
+qed
+
+lemma moment_blocked_eqpv: (* ddd *)
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+   and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  moreover have "th' \<notin> runing ((moment j t)@s)"
+  proof -
+    interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+    show ?thesis
+      using h.pv_blocked_pre h1 h2 neq_th' by auto 
+  qed
+  ultimately show ?thesis by auto
+qed
+
+(* The foregoing two lemmas are preparation for this one, but
+   in long run can be combined. Maybe I am wrong.
+*)
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and dtc: "detached (moment i t @ s) th'"
+  and le_ij: "i \<le> j"
+  shows "detached (moment j t @ s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+  interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales) 
+  have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"
+                by (metis dtc h_i.detached_elim)
+  from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]
+  show ?thesis by (metis h_j.detached_intro) 
+qed
+
+lemma runing_preced_inversion:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "cp (t@s) th' = preced th s" (is "?L = ?R")
+proof -
+  have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
+      by (unfold runing_def, auto)
+  also have "\<dots> = ?R"
+      by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) 
+  finally show ?thesis .
+qed
+
+text {*
+  The situation when @{term "th"} is blocked is analyzed by the following lemmas.
+*}
+
+text {*
+  The following lemmas shows the running thread @{text "th'"}, if it is different from
+  @{term th}, must be live at the very beginning. By the term {\em the very beginning},
+  we mean the moment where the formal investigation starts, i.e. the moment (or state)
+  @{term s}. 
+*}
+
+lemma runing_inversion_0:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s"
+proof -
+    -- {* The proof is by contradiction: *}
+    { assume otherwise: "\<not> ?thesis"
+      have "th' \<notin> runing (t @ s)"
+      proof -
+        -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}
+        have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)
+        -- {* However, @{text "th'"} does not exist at very beginning. *}
+        have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise
+          by (metis append.simps(1) moment_zero)
+        -- {* Therefore, there must be a moment during @{text "t"}, when 
+              @{text "th'"} came into being. *}
+        -- {* Let us suppose the moment being @{text "i"}: *}
+        from p_split_gen[OF th'_in th'_notin]
+        obtain i where lt_its: "i < length t"
+                 and le_i: "0 \<le> i"
+                 and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+                 and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by (auto)
+        interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+        interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)
+        from lt_its have "Suc i \<le> length t" by auto
+        -- {* Let us also suppose the event which makes this change is @{text e}: *}
+        from moment_head[OF this] obtain e where 
+          eq_me: "moment (Suc i) t = e # moment i t" by blast
+        hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t) 
+        hence "PIP (moment i t @ s) e" by (cases, simp)
+        -- {* It can be derived that this event @{text "e"}, which 
+              gives birth to @{term "th'"} must be a @{term "Create"}: *}
+        from create_pre[OF this, of th']
+        obtain prio where eq_e: "e = Create th' prio"
+            by (metis append_Cons eq_me lessI post pre) 
+        have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto 
+        have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+        proof -
+          have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+            by (metis h_i.cnp_cnv_eq pre)
+          thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)
+        qed
+        show ?thesis 
+          using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge
+            by auto
+      qed
+      with `th' \<in> runing (t@s)`
+      have False by simp
+    } thus ?thesis by auto
+qed
+
+text {* 
+  The second lemma says, if the running thread @{text th'} is different from 
+  @{term th}, then this @{text th'} must in the possession of some resources
+  at the very beginning. 
+
+  To ease the reasoning of resource possession of one particular thread, 
+  we used two auxiliary functions @{term cntV} and @{term cntP}, 
+  which are the counters of @{term P}-operations and 
+  @{term V}-operations respectively. 
+  If the number of @{term V}-operation is less than the number of 
+  @{term "P"}-operations, the thread must have some unreleased resource. 
+*}
+
+lemma runing_inversion_1: (* ddd *)
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  -- {* thread @{term "th'"} is a live on in state @{term "s"} and 
+        it has some unreleased resource. *}
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof -
+  -- {* The proof is a simple composition of @{thm runing_inversion_0} and 
+        @{thm runing_precond}: *}
+  -- {* By applying @{thm runing_inversion_0} to assumptions,
+        it can be shown that @{term th'} is live in state @{term s}: *}
+  have "th' \<in> threads s"  using runing_inversion_0[OF assms(1,2)] .
+  -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+qed
+
+text {* 
+  The following lemma is just a rephrasing of @{thm runing_inversion_1}:
+*}
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_3:
+  assumes runing': "th' \<in> runing (t@s)"
+  and neq_th: "th' \<noteq> th"
+  shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
+  by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
+
+lemma runing_inversion_4:
+  assumes runing': "th' \<in> runing (t@s)"
+  and neq_th: "th' \<noteq> th"
+  shows "th' \<in> threads s"
+  and    "\<not>detached s th'"
+  and    "cp (t@s) th' = preced th s"
+  apply (metis neq_th runing' runing_inversion_2)
+  apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
+  by (metis neq_th runing' runing_inversion_3)
+
+text {* 
+  Suppose @{term th} is not running, it is first shown that
+  there is a path in RAG leading from node @{term th} to another thread @{text "th'"} 
+  in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
+
+  Now, since @{term readys}-set is non-empty, there must be
+  one in it which holds the highest @{term cp}-value, which, by definition, 
+  is the @{term runing}-thread. However, we are going to show more: this running thread
+  is exactly @{term "th'"}.
+     *}
+lemma th_blockedE: (* ddd *)
+  assumes "th \<notin> runing (t@s)"
+  obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+                    "th' \<in> runing (t@s)"
+proof -
+  -- {* According to @{thm vat_t.th_chain_to_ready}, either 
+        @{term "th"} is in @{term "readys"} or there is path leading from it to 
+        one thread in @{term "readys"}. *}
+  have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)" 
+    using th_kept vat_t.th_chain_to_ready by auto
+  -- {* However, @{term th} can not be in @{term readys}, because otherwise, since 
+       @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
+  moreover have "th \<notin> readys (t@s)" 
+    using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto 
+  -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in 
+        term @{term readys}: *}
+  ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
+                          and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
+  -- {* We are going to show that this @{term th'} is running. *}
+  have "th' \<in> runing (t@s)"
+  proof -
+    -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
+    have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
+    proof -
+      have "?L =  Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
+        by (unfold cp_alt_def1, simp)
+      also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
+      proof(rule image_Max_subset)
+        show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
+      next
+        show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
+          by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) 
+      next
+        show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
+                    by (unfold tRAG_subtree_eq, auto simp:subtree_def)
+      next
+        show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
+                      (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
+        proof -
+          have "?L = the_preced (t @ s) `  threads (t @ s)" 
+                     by (unfold image_comp, rule image_cong, auto)
+          thus ?thesis using max_preced the_preced_def by auto
+        qed
+      qed
+      also have "... = ?R"
+        using th_cp_max th_cp_preced th_kept 
+              the_preced_def vat_t.max_cp_readys_threads by auto
+      finally show ?thesis .
+    qed 
+    -- {* Now, since @{term th'} holds the highest @{term cp} 
+          and we have already show it is in @{term readys},
+          it is @{term runing} by definition. *}
+    with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def) 
+  qed
+  -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
+  moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" 
+    using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
+  ultimately show ?thesis using that by metis
+qed
+
+text {*
+  Now it is easy to see there is always a thread to run by case analysis
+  on whether thread @{term th} is running: if the answer is Yes, the 
+  the running thread is obviously @{term th} itself; otherwise, the running
+  thread is the @{text th'} given by lemma @{thm th_blockedE}.
+*}
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)") 
+  case True thus ?thesis by auto
+next
+  case False
+  thus ?thesis using th_blockedE by auto
+qed
+
+end
+end
+
+
+
diff -r b4bcd1edbb6d -r 633b1fc8631b CpsG.thy~
--- a/CpsG.thy~	Wed Jan 06 16:34:26 2016 +0000
+++ b/CpsG.thy~	Thu Jan 07 08:33:13 2016 +0800
@@ -401,6 +401,29 @@
   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"
diff -r b4bcd1edbb6d -r 633b1fc8631b ExtGG.thy~
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ExtGG.thy~	Thu Jan 07 08:33:13 2016 +0800
@@ -0,0 +1,922 @@
+theory ExtGG
+imports PrioG CpsG
+begin
+
+text {* 
+  The following two auxiliary lemmas are used to reason about @{term Max}.
+*}
+lemma image_Max_eqI: 
+  assumes "finite B"
+  and "b \<in> B"
+  and "\<forall> x \<in> B. f x \<le> f b"
+  shows "Max (f ` B) = f b"
+  using assms
+  using Max_eqI by blast 
+
+lemma image_Max_subset:
+  assumes "finite A"
+  and "B \<subseteq> A"
+  and "a \<in> B"
+  and "Max (f ` A) = f a"
+  shows "Max (f ` B) = f a"
+proof(rule image_Max_eqI)
+  show "finite B"
+    using assms(1) assms(2) finite_subset by auto 
+next
+  show "a \<in> B" using assms by simp
+next
+  show "\<forall>x\<in>B. f x \<le> f a"
+    by (metis Max_ge assms(1) assms(2) assms(4) 
+            finite_imageI image_eqI subsetCE) 
+qed
+
+text {*
+  The following locale @{text "highest_gen"} sets the basic context for our
+  investigation: supposing thread @{text th} holds the highest @{term cp}-value
+  in state @{text s}, which means the task for @{text th} is the 
+  most urgent. We want to show that  
+  @{text th} is treated correctly by PIP, which means
+  @{text th} will not be blocked unreasonably by other less urgent
+  threads. 
+*}
+locale highest_gen =
+  fixes s th prio tm
+  assumes vt_s: "vt s"
+  and threads_s: "th \<in> threads s"
+  and highest: "preced th s = Max ((cp s)`threads s)"
+  -- {* The internal structure of @{term th}'s precedence is exposed:*}
+  and preced_th: "preced th s = Prc prio tm" 
+
+-- {* @{term s} is a valid trace, so it will inherit all results derived for
+      a valid trace: *}
+sublocale highest_gen < vat_s: valid_trace "s"
+  by (unfold_locales, insert vt_s, simp)
+
+context highest_gen
+begin
+
+text {*
+  @{term tm} is the time when the precedence of @{term th} is set, so 
+  @{term tm} must be a valid moment index into @{term s}.
+*}
+lemma lt_tm: "tm < length s"
+  by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+text {*
+  Since @{term th} holds the highest precedence and @{text "cp"}
+  is the highest precedence of all threads in the sub-tree of 
+  @{text "th"} and @{text th} is among these threads, 
+  its @{term cp} must equal to its precedence:
+*}
+lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
+proof -
+  have "?L \<le> ?R"
+  by (unfold highest, rule Max_ge, 
+        auto simp:threads_s finite_threads)
+  moreover have "?R \<le> ?L"
+    by (unfold vat_s.cp_rec, rule Max_ge, 
+        auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
+  ultimately show ?thesis by auto
+qed
+
+(* ccc *)
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold max_cp_eq, unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+  by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+  from highest_cp_preced max_cp_eq[symmetric]
+  show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen + 
+  fixes t 
+  assumes vt_t: "vt (t@s)"
+  and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+  and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+  and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+sublocale extend_highest_gen < vat_t: valid_trace "t@s"
+  by (unfold_locales, insert vt_t, simp)
+
+lemma step_back_vt_app: 
+  assumes vt_ts: "vt (t@s)" 
+  shows "vt s"
+proof -
+  from vt_ts show ?thesis
+  proof(induct t)
+    case Nil
+    from Nil show ?case by auto
+  next
+    case (Cons e t)
+    assume ih: " vt (t @ s) \<Longrightarrow> vt s"
+      and vt_et: "vt ((e # t) @ s)"
+    show ?case
+    proof(rule ih)
+      show "vt (t @ s)"
+      proof(rule step_back_vt)
+        from vt_et show "vt (e # t @ s)" by simp
+      qed
+    qed
+  qed
+qed
+
+
+locale red_extend_highest_gen = extend_highest_gen +
+   fixes i::nat
+
+sublocale red_extend_highest_gen <   red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
+  apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+  apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+  by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+
+context extend_highest_gen
+begin
+
+ lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes 
+    h0: "R []"
+  and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e; 
+                    extend_highest_gen s th prio tm t; 
+                    extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+  shows "R t"
+proof -
+  from vt_t extend_highest_gen_axioms show ?thesis
+  proof(induct t)
+    from h0 show "R []" .
+  next
+    case (Cons e t')
+    assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+      and vt_e: "vt ((e # t') @ s)"
+      and et: "extend_highest_gen s th prio tm (e # t')"
+    from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+    from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
+    show ?case
+    proof(rule h2 [OF vt_ts stp _ _ _ ])
+      show "R t'"
+      proof(rule ih)
+        from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+      next
+        from vt_ts show "vt (t' @ s)" .
+      qed
+    next
+      from et show "extend_highest_gen s th prio tm (e # t')" .
+    next
+      from et show ext': "extend_highest_gen s th prio tm t'"
+          by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+    qed
+  qed
+qed
+
+
+lemma th_kept: "th \<in> threads (t @ s) \<and> 
+                 preced th (t@s) = preced th s" (is "?Q t") 
+proof -
+  show ?thesis
+  proof(induct rule:ind)
+    case Nil
+    from threads_s
+    show ?case
+      by auto
+  next
+    case (Cons e t)
+    interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+    interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+    show ?case
+    proof(cases e)
+      case (Create thread prio)
+      show ?thesis
+      proof -
+        from Cons and Create have "step (t@s) (Create thread prio)" by auto
+        hence "th \<noteq> thread"
+        proof(cases)
+          case thread_create
+          with Cons show ?thesis by auto
+        qed
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold Create, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:Create)
+      qed
+    next
+      case (Exit thread)
+      from h_e.exit_diff and Exit
+      have neq_th: "thread \<noteq> th" by auto
+      with Cons
+      show ?thesis
+        by (unfold Exit, auto simp:preced_def)
+    next
+      case (P thread cs)
+      with Cons
+      show ?thesis 
+        by (auto simp:P preced_def)
+    next
+      case (V thread cs)
+      with Cons
+      show ?thesis 
+        by (auto simp:V preced_def)
+    next
+      case (Set thread prio')
+      show ?thesis
+      proof -
+        from h_e.set_diff_low and Set
+        have "th \<noteq> thread" by auto
+        hence "preced th ((e # t) @ s)  = preced th (t @ s)"
+          by (unfold Set, auto simp:preced_def)
+        moreover note Cons
+        ultimately show ?thesis
+          by (auto simp:Set)
+      qed
+    qed
+  qed
+qed
+
+text {*
+  According to @{thm th_kept}, thread @{text "th"} has its living status
+  and precedence kept along the way of @{text "t"}. The following lemma
+  shows that this preserved precedence of @{text "th"} remains as the highest
+  along the way of @{text "t"}.
+
+  The proof goes by induction over @{text "t"} using the specialized
+  induction rule @{thm ind}, followed by case analysis of each possible 
+  operations of PIP. All cases follow the same pattern rendered by the 
+  generalized introduction rule @{thm "image_Max_eqI"}. 
+
+  The very essence is to show that precedences, no matter whether they are newly introduced 
+  or modified, are always lower than the one held by @{term "th"},
+  which by @{thm th_kept} is preserved along the way.
+*}
+lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+  case Nil
+  from highest_preced_thread
+  show ?case
+    by (unfold the_preced_def, simp)
+next
+  case (Cons e t)
+    interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+    interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+  show ?case
+  proof(cases e)
+    case (Create thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      -- {* The following is the common pattern of each branch of the case analysis. *}
+      -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume "x \<in> ?A"
+          hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
+          thus "?f x \<le> ?f th"
+          proof
+            assume "x = thread"
+            thus ?thesis 
+              apply (simp add:Create the_preced_def preced_def, fold preced_def)
+              using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force
+          next
+            assume h: "x \<in> threads (t @ s)"
+            from Cons(2)[unfolded Create] 
+            have "x \<noteq> thread" using h by (cases, auto)
+            hence "?f x = the_preced (t@s) x" 
+              by (simp add:Create the_preced_def preced_def)
+            hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
+              by (simp add: h_t.finite_threads h)
+            also have "... = ?f th"
+              by (metis Cons.hyps(5) h_e.th_kept the_preced_def) 
+            finally show ?thesis .
+          qed
+        qed
+      qed
+     -- {* The minor part is to show that the precedence of @{text "th"} 
+           equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      -- {* Then it follows trivially that the precedence preserved
+            for @{term "th"} remains the maximum of all living threads along the way. *}
+      finally show ?thesis .
+    qed 
+  next 
+    case (Exit thread)
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume "x \<in> ?A"
+          hence "x \<in> threads (t@s)" by (simp add: Exit) 
+          hence "?f x \<le> Max (?f ` threads (t@s))" 
+            by (simp add: h_t.finite_threads) 
+          also have "... \<le> ?f th" 
+            apply (simp add:Exit the_preced_def preced_def, fold preced_def)
+            using Cons.hyps(5) h_t.th_kept the_preced_def by auto
+          finally show "?f x \<le> ?f th" .
+        qed
+      qed
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      finally show ?thesis .
+    qed 
+  next
+    case (P thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def the_preced_def)
+  next
+    case (V thread cs)
+    with Cons
+    show ?thesis by (auto simp:preced_def the_preced_def)
+  next 
+    case (Set thread prio')
+    show ?thesis (is "Max (?f ` ?A) = ?t")
+    proof -
+      have "Max (?f ` ?A) = ?f th"
+      proof(rule image_Max_eqI)
+        show "finite ?A" using h_e.finite_threads by auto 
+      next
+        show "th \<in> ?A" using h_e.th_kept by auto 
+      next
+        show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+        proof 
+          fix x
+          assume h: "x \<in> ?A"
+          show "?f x \<le> ?f th"
+          proof(cases "x = thread")
+            case True
+            moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
+            proof -
+              have "the_preced (t @ s) th = Prc prio tm"  
+                using h_t.th_kept preced_th by (simp add:the_preced_def)
+              moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
+              ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
+            qed
+            ultimately show ?thesis
+              by (unfold Set, simp add:the_preced_def preced_def)
+          next
+            case False
+            then have "?f x  = the_preced (t@s) x"
+              by (simp add:the_preced_def preced_def Set)
+            also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
+              using Set h h_t.finite_threads by auto 
+            also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) 
+            finally show ?thesis .
+          qed
+        qed
+      qed
+      also have "... = ?t" using h_e.th_kept the_preced_def by auto
+      finally show ?thesis .
+    qed 
+  qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
+  by (insert th_kept max_kept, auto)
+
+text {*
+  The reason behind the following lemma is that:
+  Since @{term "cp"} is defined as the maximum precedence 
+  of those threads contained in the sub-tree of node @{term "Th th"} 
+  in @{term "RAG (t@s)"}, and all these threads are living threads, and 
+  @{term "th"} is also among them, the maximum precedence of 
+  them all must be the one for @{text "th"}.
+*}
+lemma th_cp_max_preced: 
+  "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") 
+proof -
+  let ?f = "the_preced (t@s)"
+  have "?L = ?f th"
+  proof(unfold cp_alt_def, rule image_Max_eqI)
+    show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+    proof -
+      have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} = 
+            the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
+                            (\<exists> th'. n = Th th')}"
+      by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
+      moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) 
+      ultimately show ?thesis by simp
+    qed
+  next
+    show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+      by (auto simp:subtree_def)
+  next
+    show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
+               the_preced (t @ s) x \<le> the_preced (t @ s) th"
+    proof
+      fix th'
+      assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+      hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
+      moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
+        by (meson subtree_Field)
+      ultimately have "Th th' \<in> ..." by auto
+      hence "th' \<in> threads (t@s)" 
+      proof
+        assume "Th th' \<in> {Th th}"
+        thus ?thesis using th_kept by auto 
+      next
+        assume "Th th' \<in> Field (RAG (t @ s))"
+        thus ?thesis using vat_t.not_in_thread_isolated by blast 
+      qed
+      thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
+        by (metis Max_ge finite_imageI finite_threads image_eqI 
+               max_kept th_kept the_preced_def)
+    qed
+  qed
+  also have "... = ?R" by (simp add: max_preced the_preced_def) 
+  finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+  using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+  by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less:
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  shows "preced th' s < preced th s"
+  using assms
+by (metis Max.coboundedI finite_imageI highest not_le order.trans 
+    preced_linorder rev_image_eqI threads_s vat_s.finite_threads 
+    vat_s.le_cp)
+
+text {*
+  Counting of the number of @{term "P"} and @{term "V"} operations 
+  is the cornerstone of a large number of the following proofs. 
+  The reason is that this counting is quite easy to calculate and 
+  convenient to use in the reasoning. 
+
+  The following lemma shows that the counting controls whether 
+  a thread is running or not.
+*}
+
+lemma pv_blocked_pre:
+  assumes th'_in: "th' \<in> threads (t@s)"
+  and neq_th': "th' \<noteq> th"
+  and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+  shows "th' \<notin> runing (t@s)"
+proof
+  assume otherwise: "th' \<in> runing (t@s)"
+  show False
+  proof -
+    have "th' = th"
+    proof(rule preced_unique)
+      show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+      proof -
+        have "?L = cp (t@s) th'"
+          by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
+        also have "... = cp (t @ s) th" using otherwise 
+          by (metis (mono_tags, lifting) mem_Collect_eq 
+                    runing_def th_cp_max vat_t.max_cp_readys_threads)
+        also have "... = ?R" by (metis th_cp_preced th_kept) 
+        finally show ?thesis .
+      qed
+    qed (auto simp: th'_in th_kept)
+    moreover have "th' \<noteq> th" using neq_th' .
+    ultimately show ?thesis by simp
+ qed
+qed
+
+lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
+
+lemma runing_precond_pre:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and eq_pv: "cntP s th' = cntV s th'"
+  and neq_th': "th' \<noteq> th"
+  shows "th' \<in> threads (t@s) \<and>
+         cntP (t@s) th' = cntV (t@s) th'"
+proof(induct rule:ind)
+  case (Cons e t)
+    interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
+    interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
+    show ?case
+    proof(cases e)
+      case (P thread cs)
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (P thread cs)" using Cons P by auto
+            thus ?thesis
+            proof(cases)
+              assume "thread \<in> runing (t@s)"
+              moreover have "th' \<notin> runing (t@s)" using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+              ultimately show ?thesis by auto
+            qed
+          qed with Cons show ?thesis
+            by (unfold P, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (V thread cs)
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (V thread cs)" using Cons V by auto
+            thus ?thesis
+            proof(cases)
+              assume "thread \<in> runing (t@s)"
+              moreover have "th' \<notin> runing (t@s)" using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+              ultimately show ?thesis by auto
+            qed
+          qed with Cons show ?thesis
+            by (unfold V, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Create thread prio')
+      show ?thesis
+      proof -
+        have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+        proof -
+          have "thread \<noteq> th'"
+          proof -
+            have "step (t@s) (Create thread prio')" using Cons Create by auto
+            thus ?thesis using Cons(5) by (cases, auto)
+          qed with Cons show ?thesis
+            by (unfold Create, simp add:cntP_def cntV_def count_def)
+        qed
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Exit thread)
+      show ?thesis
+      proof -
+        have neq_thread: "thread \<noteq> th'"
+        proof -
+          have "step (t@s) (Exit thread)" using Cons Exit by auto
+          thus ?thesis apply (cases) using Cons(5)
+                by (metis neq_th' vat_t.pv_blocked_pre) 
+        qed 
+        hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons
+            by (unfold Exit, simp add:cntP_def cntV_def count_def)
+        moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread 
+          by (unfold Exit, simp) 
+        ultimately show ?thesis by auto
+      qed
+    next
+      case (Set thread prio')
+      with Cons
+      show ?thesis 
+        by (auto simp:cntP_def cntV_def count_def)
+    qed
+next
+  case Nil
+  with assms
+  show ?case by auto
+qed
+
+text {* Changing counting balance to detachedness *}
+lemmas runing_precond_pre_dtc = runing_precond_pre
+         [folded vat_t.detached_eq vat_s.detached_eq]
+
+lemma runing_precond:
+  fixes th'
+  assumes th'_in: "th' \<in> threads s"
+  and neq_th': "th' \<noteq> th"
+  and is_runing: "th' \<in> runing (t@s)"
+  shows "cntP s th' > cntV s th'"
+  using assms
+proof -
+  have "cntP s th' \<noteq> cntV s th'"
+    by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)
+  moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
+  ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+         th' \<in> threads ((moment (i+j) t)@s)"
+proof -
+  interpret h_i: red_extend_highest_gen _ _ _ _ _ i
+      by (unfold_locales)
+  interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"
+      by (unfold_locales)
+  interpret h:  extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"
+  proof(unfold_locales)
+    show "vt (moment i t @ s)" by (metis h_i.vt_t) 
+  next
+    show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)
+  next
+    show "preced th (moment i t @ s) = 
+            Max (cp (moment i t @ s) ` threads (moment i t @ s))"
+              by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)
+  next
+    show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th) 
+  next
+    show "vt (moment j (restm i t) @ moment i t @ s)"
+      using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)
+  next
+    fix th' prio'
+    assume "Create th' prio' \<in> set (moment j (restm i t))"
+    thus "prio' \<le> prio" using assms
+       by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)
+  next
+    fix th' prio'
+    assume "Set th' prio' \<in> set (moment j (restm i t))"
+    thus "th' \<noteq> th \<and> prio' \<le> prio"
+    by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)
+  next
+    fix th'
+    assume "Exit th' \<in> set (moment j (restm i t))"
+    thus "th' \<noteq> th"
+      by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)
+  qed
+  show ?thesis 
+    by (metis add.commute append_assoc eq_pv h.runing_precond_pre
+          moment_plus_split neq_th' th'_in)
+qed
+
+lemma moment_blocked_eqpv:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+  and le_ij: "i \<le> j"
+  shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+  have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+   and h2: "th' \<in> threads ((moment j t)@s)" by auto
+  moreover have "th' \<notin> runing ((moment j t)@s)"
+  proof -
+    interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+    show ?thesis
+      using h.pv_blocked_pre h1 h2 neq_th' by auto 
+  qed
+  ultimately show ?thesis by auto
+qed
+
+(* The foregoing two lemmas are preparation for this one, but
+   in long run can be combined. Maybe I am wrong.
+*)
+lemma moment_blocked:
+  assumes neq_th': "th' \<noteq> th"
+  and th'_in: "th' \<in> threads ((moment i t)@s)"
+  and dtc: "detached (moment i t @ s) th'"
+  and le_ij: "i \<le> j"
+  shows "detached (moment j t @ s) th' \<and>
+         th' \<in> threads ((moment j t)@s) \<and>
+         th' \<notin> runing ((moment j t)@s)"
+proof -
+  interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+  interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales) 
+  have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"
+                by (metis dtc h_i.detached_elim)
+  from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]
+  show ?thesis by (metis h_j.detached_intro) 
+qed
+
+lemma runing_preced_inversion:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "cp (t@s) th' = preced th s" (is "?L = ?R")
+proof -
+  have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
+      by (unfold runing_def, auto)
+  also have "\<dots> = ?R"
+      by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) 
+  finally show ?thesis .
+qed
+
+text {*
+  The situation when @{term "th"} is blocked is analyzed by the following lemmas.
+*}
+
+text {*
+  The following lemmas shows the running thread @{text "th'"}, if it is different from
+  @{term th}, must be live at the very beginning. By the term {\em the very beginning},
+  we mean the moment where the formal investigation starts, i.e. the moment (or state)
+  @{term s}. 
+*}
+
+lemma runing_inversion_0:
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  shows "th' \<in> threads s"
+proof -
+    -- {* The proof is by contradiction: *}
+    { assume otherwise: "\<not> ?thesis"
+      have "th' \<notin> runing (t @ s)"
+      proof -
+        -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}
+        have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)
+        -- {* However, @{text "th'"} does not exist at very beginning. *}
+        have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise
+          by (metis append.simps(1) moment_zero)
+        -- {* Therefore, there must be a moment during @{text "t"}, when 
+              @{text "th'"} came into being. *}
+        -- {* Let us suppose the moment being @{text "i"}: *}
+        from p_split_gen[OF th'_in th'_notin]
+        obtain i where lt_its: "i < length t"
+                 and le_i: "0 \<le> i"
+                 and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+                 and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by (auto)
+        interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+        interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)
+        from lt_its have "Suc i \<le> length t" by auto
+        -- {* Let us also suppose the event which makes this change is @{text e}: *}
+        from moment_head[OF this] obtain e where 
+          eq_me: "moment (Suc i) t = e # moment i t" by blast
+        hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t) 
+        hence "PIP (moment i t @ s) e" by (cases, simp)
+        -- {* It can be derived that this event @{text "e"}, which 
+              gives birth to @{term "th'"} must be a @{term "Create"}: *}
+        from create_pre[OF this, of th']
+        obtain prio where eq_e: "e = Create th' prio"
+            by (metis append_Cons eq_me lessI post pre) 
+        have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto 
+        have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+        proof -
+          have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+            by (metis h_i.cnp_cnv_eq pre)
+          thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)
+        qed
+        show ?thesis 
+          using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge
+            by auto
+      qed
+      with `th' \<in> runing (t@s)`
+      have False by simp
+    } thus ?thesis by auto
+qed
+
+text {* 
+  The second lemma says, if the running thread @{text th'} is different from 
+  @{term th}, then this @{text th'} must in the possession of some resources
+  at the very beginning. 
+
+  To ease the reasoning of resource possession of one particular thread, 
+  we used two auxiliary functions @{term cntV} and @{term cntP}, 
+  which are the counters of @{term P}-operations and 
+  @{term V}-operations respectively. 
+  If the number of @{term V}-operation is less than the number of 
+  @{term "P"}-operations, the thread must have some unreleased resource. 
+*}
+
+lemma runing_inversion_1: (* ddd *)
+  assumes neq_th': "th' \<noteq> th"
+  and runing': "th' \<in> runing (t@s)"
+  -- {* thread @{term "th'"} is a live on in state @{term "s"} and 
+        it has some unreleased resource. *}
+  shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof -
+  -- {* The proof is a simple composition of @{thm runing_inversion_0} and 
+        @{thm runing_precond}: *}
+  -- {* By applying @{thm runing_inversion_0} to assumptions,
+        it can be shown that @{term th'} is live in state @{term s}: *}
+  have "th' \<in> threads s"  using runing_inversion_0[OF assms(1,2)] .
+  -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}
+  with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+qed
+
+text {* 
+  The following lemma is just a rephrasing of @{thm runing_inversion_1}:
+*}
+lemma runing_inversion_2:
+  assumes runing': "th' \<in> runing (t@s)"
+  shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+  from runing_inversion_1[OF _ runing']
+  show ?thesis by auto
+qed
+
+lemma runing_inversion_3:
+  assumes runing': "th' \<in> runing (t@s)"
+  and neq_th: "th' \<noteq> th"
+  shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
+  by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
+
+lemma runing_inversion_4:
+  assumes runing': "th' \<in> runing (t@s)"
+  and neq_th: "th' \<noteq> th"
+  shows "th' \<in> threads s"
+  and    "\<not>detached s th'"
+  and    "cp (t@s) th' = preced th s"
+  apply (metis neq_th runing' runing_inversion_2)
+  apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
+  by (metis neq_th runing' runing_inversion_3)
+
+
+text {* 
+  Suppose @{term th} is not running, it is first shown that
+  there is a path in RAG leading from node @{term th} to another thread @{text "th'"} 
+  in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
+
+  Now, since @{term readys}-set is non-empty, there must be
+  one in it which holds the highest @{term cp}-value, which, by definition, 
+  is the @{term runing}-thread. However, we are going to show more: this running thread
+  is exactly @{term "th'"}.
+     *}
+lemma th_blockedE: (* ddd *)
+  assumes "th \<notin> runing (t@s)"
+  obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+                    "th' \<in> runing (t@s)"
+proof -
+  -- {* According to @{thm vat_t.th_chain_to_ready}, either 
+        @{term "th"} is in @{term "readys"} or there is path leading from it to 
+        one thread in @{term "readys"}. *}
+  have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)" 
+    using th_kept vat_t.th_chain_to_ready by auto
+  -- {* However, @{term th} can not be in @{term readys}, because otherwise, since 
+       @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
+  moreover have "th \<notin> readys (t@s)" 
+    using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto 
+  -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in 
+        term @{term readys}: *}
+  ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
+                          and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
+  -- {* We are going to show that this @{term th'} is running. *}
+  have "th' \<in> runing (t@s)"
+  proof -
+    -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
+    have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
+    proof -
+      have "?L =  Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
+        by (unfold cp_alt_def1, simp)
+      also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
+      proof(rule image_Max_subset)
+        show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
+      next
+        show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
+          by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) 
+      next
+        show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
+                    by (unfold tRAG_subtree_eq, auto simp:subtree_def)
+      next
+        show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
+                      (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
+        proof -
+          have "?L = the_preced (t @ s) `  threads (t @ s)" 
+                     by (unfold image_comp, rule image_cong, auto)
+          thus ?thesis using max_preced the_preced_def by auto
+        qed
+      qed
+      also have "... = ?R"
+        using th_cp_max th_cp_preced th_kept 
+              the_preced_def vat_t.max_cp_readys_threads by auto
+      finally show ?thesis .
+    qed 
+    -- {* Now, since @{term th'} holds the highest @{term cp} 
+          and we have already show it is in @{term readys},
+          it is @{term runing} by definition. *}
+    with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def) 
+  qed
+  -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
+  moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" 
+    using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
+  ultimately show ?thesis using that by metis
+qed
+
+text {*
+  Now it is easy to see there is always a thread to run by case analysis
+  on whether thread @{term th} is running: if the answer is Yes, the 
+  the running thread is obviously @{term th} itself; otherwise, the running
+  thread is the @{text th'} given by lemma @{thm th_blockedE}.
+*}
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)") 
+  case True thus ?thesis by auto
+next
+  case False
+  thus ?thesis using th_blockedE by auto
+qed
+
+end
+end
+
+
+
diff -r b4bcd1edbb6d -r 633b1fc8631b Implementation.thy
--- a/Implementation.thy	Wed Jan 06 16:34:26 2016 +0000
+++ b/Implementation.thy	Thu Jan 07 08:33:13 2016 +0800
@@ -3,733 +3,9 @@
   after every system call (or system operation)
 *}
 theory Implementation
-imports PIPBasics Max RTree
-begin
-
-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
+imports PIPBasics
 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
-
-
-sublocale valid_trace < rtree_RAG: rtree "RAG s"
-proof
-  show "single_valued (RAG s)"
-  apply (intro_locales)
-    by (unfold single_valued_def, 
-        auto intro:unique_RAG)
-
-  show "acyclic (RAG s)"
-     by (rule acyclic_RAG)
-qed
-
-sublocale valid_trace < rtree_s: rtree "tRAG s"
-proof(unfold_locales)
-  from sgv_tRAG show "single_valued (tRAG s)" .
-next
-  from acyclic_tRAG show "acyclic (tRAG s)" .
-qed
-
-sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
-proof -
-  show "fsubtree (RAG s)"
-  proof(intro_locales)
-    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
-  next
-    show "fsubtree_axioms (RAG s)"
-    proof(unfold fsubtree_axioms_def)
-    find_theorems wf RAG
-      from wf_RAG show "wf (RAG s)" .
-    qed
-  qed
-qed
-
-sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
-proof -
-  have "fsubtree (tRAG s)"
-  proof -
-    have "fbranch (tRAG s)"
-    proof(unfold tRAG_def, rule fbranch_compose)
-        show "fbranch (wRAG s)"
-        proof(rule finite_fbranchI)
-           from finite_RAG show "finite (wRAG s)"
-           by (unfold RAG_split, auto)
-        qed
-    next
-        show "fbranch (hRAG s)"
-        proof(rule finite_fbranchI)
-           from finite_RAG 
-           show "finite (hRAG s)" by (unfold RAG_split, auto)
-        qed
-    qed
-    moreover have "wf (tRAG s)"
-    proof(rule wf_subset)
-      show "wf (RAG s O RAG s)" using wf_RAG
-        by (fold wf_comp_self, simp)
-    next
-      show "tRAG s \<subseteq> (RAG s O RAG s)"
-        by (unfold tRAG_alt_def, auto)
-    qed
-    ultimately show ?thesis
-      by (unfold fsubtree_def fsubtree_axioms_def,auto)
-  qed
-  from this[folded tRAG_def] show "fsubtree (tRAG s)" .
-qed
-
-lemma Max_UNION: 
-  assumes "finite A"
-  and "A \<noteq> {}"
-  and "\<forall> M \<in> f ` A. finite M"
-  and "\<forall> M \<in> f ` A. M \<noteq> {}"
-  shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
-  using assms[simp]
-proof -
-  have "?L = Max (\<Union>(f ` A))"
-    by (fold Union_image_eq, simp)
-  also have "... = ?R"
-    by (subst Max_Union, simp+)
-  finally show ?thesis .
-qed
-
-lemma max_Max_eq:
-  assumes "finite A"
-    and "A \<noteq> {}"
-    and "x = y"
-  shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
-proof -
-  have "?R = Max (insert y A)" by simp
-  also from assms have "... = ?L"
-      by (subst Max.insert, simp+)
-  finally show ?thesis by simp
-qed
-
-context valid_trace
-begin
-
-(* ddd *)
-lemma cp_gen_rec:
-  assumes "x = Th th"
-  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
-proof(cases "children (tRAG s) x = {}")
-  case True
-  show ?thesis
-    by (unfold True cp_gen_def subtree_children, simp add:assms)
-next
-  case False
-  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
-  note fsbttRAGs.finite_subtree[simp]
-  have [simp]: "finite (children (tRAG s) x)"
-     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 
-            rule children_subtree)
-  { fix r x
-    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
-  } note this[simp]
-  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
-  proof -
-    from False obtain q where "q \<in> children (tRAG s) x" by blast
-    moreover have "subtree (tRAG s) q \<noteq> {}" by simp
-    ultimately show ?thesis by blast
-  qed
-  have h: "Max ((the_preced s \<circ> the_thread) `
-                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
-        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
-                     (is "?L = ?R")
-  proof -
-    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
-    let "Max (_ \<union> (?h ` ?B))" = ?R
-    let ?L1 = "?f ` \<Union>(?g ` ?B)"
-    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
-    proof -
-      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
-      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
-      finally have "Max ?L1 = Max ..." by simp
-      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
-        by (subst Max_UNION, simp+)
-      also have "... = Max (cp_gen s ` children (tRAG s) x)"
-          by (unfold image_comp cp_gen_alt_def, simp)
-      finally show ?thesis .
-    qed
-    show ?thesis
-    proof -
-      have "?L = Max (?f ` ?A \<union> ?L1)" by simp
-      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
-            by (subst Max_Un, simp+)
-      also have "... = max (?f x) (Max (?h ` ?B))"
-        by (unfold eq_Max_L1, simp)
-      also have "... =?R"
-        by (rule max_Max_eq, (simp)+, unfold assms, simp)
-      finally show ?thesis .
-    qed
-  qed  thus ?thesis 
-          by (fold h subtree_children, unfold cp_gen_def, simp) 
-qed
-
-lemma cp_rec:
-  "cp s th = Max ({the_preced s th} \<union> 
-                     (cp s o the_thread) ` children (tRAG s) (Th th))"
-proof -
-  have "Th th = Th th" by simp
-  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
-  show ?thesis 
-  proof -
-    have "cp_gen s ` children (tRAG s) (Th th) = 
-                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
-    proof(rule cp_gen_over_set)
-      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
-        by (unfold tRAG_alt_def, auto simp:children_def)
-    qed
-    thus ?thesis by (subst (1) h(1), unfold h(2), simp)
-  qed
-qed
-
-end
-
-(* keep *)
-lemma next_th_holding:
-  assumes vt: "vt s"
-  and nxt: "next_th s th cs th'"
-  shows "holding (wq s) th cs"
-proof -
-  from nxt[unfolded next_th_def]
-  obtain rest where h: "wq s cs = th # rest"
-                       "rest \<noteq> []" 
-                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
-  thus ?thesis
-    by (unfold cs_holding_def, auto)
-qed
-
-context valid_trace
-begin
-
-lemma next_th_waiting:
-  assumes nxt: "next_th s th cs th'"
-  shows "waiting (wq s) th' cs"
-proof -
-  from nxt[unfolded next_th_def]
-  obtain rest where h: "wq s cs = th # rest"
-                       "rest \<noteq> []" 
-                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
-  from wq_distinct[of cs, unfolded h]
-  have dst: "distinct (th # rest)" .
-  have in_rest: "th' \<in> set rest"
-  proof(unfold h, rule someI2)
-    show "distinct rest \<and> set rest = set rest" using dst by auto
-  next
-    fix x assume "distinct x \<and> set x = set rest"
-    with h(2)
-    show "hd x \<in> set (rest)" by (cases x, auto)
-  qed
-  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
-  moreover have "th' \<noteq> hd (wq s cs)"
-    by (unfold h(1), insert in_rest dst, auto)
-  ultimately show ?thesis by (auto simp:cs_waiting_def)
-qed
-
-lemma next_th_RAG:
-  assumes nxt: "next_th (s::event list) th cs th'"
-  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
-  using vt assms next_th_holding next_th_waiting
-  by (unfold s_RAG_def, simp)
-
-end
-
--- {* A useless definition *}
-definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
-where "cps s = {(th, cp s th) | th . th \<in> threads s}"
-
-
 text {* (* ddd *)
   One beauty of our modelling is that we follow the definitional extension tradition of HOL.
   The benefit of such a concise and miniature model is that  large number of intuitively 
@@ -861,7 +137,6 @@
     hence "th \<in> runing s'" by (cases, simp)
     thus ?thesis by (simp add:readys_def runing_def)
   qed
-  find_theorems readys subtree
   from vat_s'.readys_in_no_subtree[OF this assms(1)]
   show ?thesis by blast
 qed
@@ -1143,7 +418,6 @@
 
 lemma subtree_th: 
   "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
-find_theorems "subtree" "_ - _" RAG
 proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
   from edge_of_th
   show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"
@@ -1620,7 +894,6 @@
     qed auto
     have neq_th_a: "th_a \<noteq> th"
     proof -
-    find_theorems readys subtree s'
       from vat_s'.readys_in_no_subtree[OF th_ready assms]
       have "(Th th) \<notin> subtree (RAG s') (Th th')" .
       with tRAG_subtree_RAG[of s' "Th th'"]
diff -r b4bcd1edbb6d -r 633b1fc8631b Implementation.thy~
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Implementation.thy~	Thu Jan 07 08:33:13 2016 +0800
@@ -0,0 +1,1636 @@
+section {*
+  This file contains lemmas used to guide the recalculation of current precedence 
+  after every system call (or system operation)
+*}
+theory Implementation
+imports PIPBasics Max RTree
+begin
+
+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
+
+
+sublocale valid_trace < rtree_RAG: rtree "RAG s"
+proof
+  show "single_valued (RAG s)"
+  apply (intro_locales)
+    by (unfold single_valued_def, 
+        auto intro:unique_RAG)
+
+  show "acyclic (RAG s)"
+     by (rule acyclic_RAG)
+qed
+
+sublocale valid_trace < rtree_s: rtree "tRAG s"
+proof(unfold_locales)
+  from sgv_tRAG show "single_valued (tRAG s)" .
+next
+  from acyclic_tRAG show "acyclic (tRAG s)" .
+qed
+
+sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
+proof -
+  show "fsubtree (RAG s)"
+  proof(intro_locales)
+    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
+  next
+    show "fsubtree_axioms (RAG s)"
+    proof(unfold fsubtree_axioms_def)
+      from wf_RAG show "wf (RAG s)" .
+    qed
+  qed
+qed
+
+sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
+proof -
+  have "fsubtree (tRAG s)"
+  proof -
+    have "fbranch (tRAG s)"
+    proof(unfold tRAG_def, rule fbranch_compose)
+        show "fbranch (wRAG s)"
+        proof(rule finite_fbranchI)
+           from finite_RAG show "finite (wRAG s)"
+           by (unfold RAG_split, auto)
+        qed
+    next
+        show "fbranch (hRAG s)"
+        proof(rule finite_fbranchI)
+           from finite_RAG 
+           show "finite (hRAG s)" by (unfold RAG_split, auto)
+        qed
+    qed
+    moreover have "wf (tRAG s)"
+    proof(rule wf_subset)
+      show "wf (RAG s O RAG s)" using wf_RAG
+        by (fold wf_comp_self, simp)
+    next
+      show "tRAG s \<subseteq> (RAG s O RAG s)"
+        by (unfold tRAG_alt_def, auto)
+    qed
+    ultimately show ?thesis
+      by (unfold fsubtree_def fsubtree_axioms_def,auto)
+  qed
+  from this[folded tRAG_def] show "fsubtree (tRAG s)" .
+qed
+
+lemma Max_UNION: 
+  assumes "finite A"
+  and "A \<noteq> {}"
+  and "\<forall> M \<in> f ` A. finite M"
+  and "\<forall> M \<in> f ` A. M \<noteq> {}"
+  shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
+  using assms[simp]
+proof -
+  have "?L = Max (\<Union>(f ` A))"
+    by (fold Union_image_eq, simp)
+  also have "... = ?R"
+    by (subst Max_Union, simp+)
+  finally show ?thesis .
+qed
+
+lemma max_Max_eq:
+  assumes "finite A"
+    and "A \<noteq> {}"
+    and "x = y"
+  shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
+proof -
+  have "?R = Max (insert y A)" by simp
+  also from assms have "... = ?L"
+      by (subst Max.insert, simp+)
+  finally show ?thesis by simp
+qed
+
+context valid_trace
+begin
+
+(* ddd *)
+lemma cp_gen_rec:
+  assumes "x = Th th"
+  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
+proof(cases "children (tRAG s) x = {}")
+  case True
+  show ?thesis
+    by (unfold True cp_gen_def subtree_children, simp add:assms)
+next
+  case False
+  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
+  note fsbttRAGs.finite_subtree[simp]
+  have [simp]: "finite (children (tRAG s) x)"
+     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 
+            rule children_subtree)
+  { fix r x
+    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
+  } note this[simp]
+  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
+  proof -
+    from False obtain q where "q \<in> children (tRAG s) x" by blast
+    moreover have "subtree (tRAG s) q \<noteq> {}" by simp
+    ultimately show ?thesis by blast
+  qed
+  have h: "Max ((the_preced s \<circ> the_thread) `
+                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
+        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
+                     (is "?L = ?R")
+  proof -
+    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
+    let "Max (_ \<union> (?h ` ?B))" = ?R
+    let ?L1 = "?f ` \<Union>(?g ` ?B)"
+    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
+    proof -
+      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
+      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
+      finally have "Max ?L1 = Max ..." by simp
+      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
+        by (subst Max_UNION, simp+)
+      also have "... = Max (cp_gen s ` children (tRAG s) x)"
+          by (unfold image_comp cp_gen_alt_def, simp)
+      finally show ?thesis .
+    qed
+    show ?thesis
+    proof -
+      have "?L = Max (?f ` ?A \<union> ?L1)" by simp
+      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
+            by (subst Max_Un, simp+)
+      also have "... = max (?f x) (Max (?h ` ?B))"
+        by (unfold eq_Max_L1, simp)
+      also have "... =?R"
+        by (rule max_Max_eq, (simp)+, unfold assms, simp)
+      finally show ?thesis .
+    qed
+  qed  thus ?thesis 
+          by (fold h subtree_children, unfold cp_gen_def, simp) 
+qed
+
+lemma cp_rec:
+  "cp s th = Max ({the_preced s th} \<union> 
+                     (cp s o the_thread) ` children (tRAG s) (Th th))"
+proof -
+  have "Th th = Th th" by simp
+  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
+  show ?thesis 
+  proof -
+    have "cp_gen s ` children (tRAG s) (Th th) = 
+                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
+    proof(rule cp_gen_over_set)
+      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
+        by (unfold tRAG_alt_def, auto simp:children_def)
+    qed
+    thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+  qed
+qed
+
+end
+
+(* keep *)
+lemma next_th_holding:
+  assumes vt: "vt s"
+  and nxt: "next_th s th cs th'"
+  shows "holding (wq s) th cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  thus ?thesis
+    by (unfold cs_holding_def, auto)
+qed
+
+context valid_trace
+begin
+
+lemma next_th_waiting:
+  assumes nxt: "next_th s th cs th'"
+  shows "waiting (wq s) th' cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  from wq_distinct[of cs, unfolded h]
+  have dst: "distinct (th # rest)" .
+  have in_rest: "th' \<in> set rest"
+  proof(unfold h, rule someI2)
+    show "distinct rest \<and> set rest = set rest" using dst by auto
+  next
+    fix x assume "distinct x \<and> set x = set rest"
+    with h(2)
+    show "hd x \<in> set (rest)" by (cases x, auto)
+  qed
+  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
+  moreover have "th' \<noteq> hd (wq s cs)"
+    by (unfold h(1), insert in_rest dst, auto)
+  ultimately show ?thesis by (auto simp:cs_waiting_def)
+qed
+
+lemma next_th_RAG:
+  assumes nxt: "next_th (s::event list) th cs th'"
+  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
+  using vt assms next_th_holding next_th_waiting
+  by (unfold s_RAG_def, simp)
+
+end
+
+-- {* A useless definition *}
+definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+
+
+text {* (* ddd *)
+  One beauty of our modelling is that we follow the definitional extension tradition of HOL.
+  The benefit of such a concise and miniature model is that  large number of intuitively 
+  obvious facts are derived as lemmas, rather than asserted as axioms.
+*}
+
+text {*
+  However, the lemmas in the forthcoming several locales are no longer 
+  obvious. These lemmas show how the current precedences should be recalculated 
+  after every execution step (in our model, every step is represented by an event, 
+  which in turn, represents a system call, or operation). Each operation is 
+  treated in a separate locale.
+
+  The complication of current precedence recalculation comes 
+  because the changing of RAG needs to be taken into account, 
+  in addition to the changing of precedence. 
+  The reason RAG changing affects current precedence is that,
+  according to the definition, current precedence 
+  of a thread is the maximum of the precedences of its dependants, 
+  where the dependants are defined in terms of RAG.
+
+  Therefore, each operation, lemmas concerning the change of the precedences 
+  and RAG are derived first, so that the lemmas about
+  current precedence recalculation can be based on.
+*}
+
+text {* (* ddd *)
+  The following locale @{text "step_set_cps"} investigates the recalculation 
+  after the @{text "Set"} operation.
+*}
+locale step_set_cps =
+  fixes s' th prio s 
+  -- {* @{text "s'"} is the system state before the operation *}
+  -- {* @{text "s"} is the system state after the operation *}
+  defines s_def : "s \<equiv> (Set th prio#s')" 
+  -- {* @{text "s"} is assumed to be a legitimate state, from which
+         the legitimacy of @{text "s"} can be derived. *}
+  assumes vt_s: "vt s"
+
+sublocale step_set_cps < vat_s : valid_trace "s"
+proof
+  from vt_s show "vt s" .
+qed
+
+sublocale step_set_cps < vat_s' : valid_trace "s'"
+proof
+  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_set_cps 
+begin
+
+text {* (* ddd *)
+  The following two lemmas confirm that @{text "Set"}-operating only changes the precedence 
+  of the initiating thread.
+*}
+
+lemma eq_preced:
+  assumes "th' \<noteq> th"
+  shows "preced th' s = preced th' s'"
+proof -
+  from assms show ?thesis 
+    by (unfold s_def, auto simp:preced_def)
+qed
+
+lemma eq_the_preced: 
+  fixes th'
+  assumes "th' \<noteq> th"
+  shows "the_preced s th' = the_preced s' th'"
+  using assms
+  by (unfold the_preced_def, intro eq_preced, simp)
+
+text {*
+  The following lemma assures that the resetting of priority does not change the RAG. 
+*}
+
+lemma eq_dep: "RAG s = RAG s'"
+  by (unfold s_def RAG_set_unchanged, auto)
+
+text {* (* ddd *)
+  Th following lemma @{text "eq_cp_pre"} says the priority change of @{text "th"}
+  only affects those threads, which as @{text "Th th"} in their sub-trees.
+  
+  The proof of this lemma is simplified by using the alternative definition of @{text "cp"}. 
+*}
+
+lemma eq_cp_pre:
+  fixes th' 
+  assumes nd: "Th th \<notin> subtree (RAG s') (Th th')"
+  shows "cp s th' = cp s' th'"
+proof -
+  -- {* After unfolding using the alternative definition, elements 
+        affecting the @{term "cp"}-value of threads become explicit. 
+        We only need to prove the following: *}
+  have "Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
+        Max (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+        (is "Max (?f ` ?S1) = Max (?g ` ?S2)")
+  proof -
+    -- {* The base sets are equal. *}
+    have "?S1 = ?S2" using eq_dep by simp
+    -- {* The function values on the base set are equal as well. *}
+    moreover have "\<forall> e \<in> ?S2. ?f e = ?g e"
+    proof
+      fix th1
+      assume "th1 \<in> ?S2"
+      with nd have "th1 \<noteq> th" by (auto)
+      from eq_the_preced[OF this]
+      show "the_preced s th1 = the_preced s' th1" .
+    qed
+    -- {* Therefore, the image of the functions are equal. *}
+    ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq)
+    thus ?thesis by simp
+  qed
+  thus ?thesis by (simp add:cp_alt_def)
+qed
+
+text {*
+  The following lemma shows that @{term "th"} is not in the 
+  sub-tree of any other thread. 
+*}
+lemma th_in_no_subtree:
+  assumes "th' \<noteq> th"
+  shows "Th th \<notin> subtree (RAG s') (Th th')"
+proof -
+  have "th \<in> readys s'"
+  proof -
+    from step_back_step [OF vt_s[unfolded s_def]]
+    have "step s' (Set th prio)" .
+    hence "th \<in> runing s'" by (cases, simp)
+    thus ?thesis by (simp add:readys_def runing_def)
+  qed
+  from vat_s'.readys_in_no_subtree[OF this assms(1)]
+  show ?thesis by blast
+qed
+
+text {* 
+  By combining @{thm "eq_cp_pre"} and @{thm "th_in_no_subtree"}, 
+  it is obvious that the change of priority only affects the @{text "cp"}-value 
+  of the initiating thread @{text "th"}.
+*}
+lemma eq_cp:
+  fixes th' 
+  assumes "th' \<noteq> th"
+  shows "cp s th' = cp s' th'"
+  by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])
+
+end
+
+text {*
+  The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.
+*}
+
+locale step_v_cps =
+  -- {* @{text "th"} is the initiating thread *}
+  -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *}
+  fixes s' th cs s    -- {* @{text "s'"} is the state before operation*}
+  defines s_def : "s \<equiv> (V th cs#s')" -- {* @{text "s"} is the state after operation*}
+  -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *}
+  assumes vt_s: "vt s"
+
+sublocale step_v_cps < vat_s : valid_trace "s"
+proof
+  from vt_s show "vt s" .
+qed
+
+sublocale step_v_cps < vat_s' : valid_trace "s'"
+proof
+  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_v_cps
+begin
+
+lemma ready_th_s': "th \<in> readys s'"
+  using step_back_step[OF vt_s[unfolded s_def]]
+  by (cases, simp add:runing_def)
+
+lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"
+proof -
+  from vat_s'.readys_root[OF ready_th_s']
+  show ?thesis
+  by (unfold root_def, simp)
+qed
+
+lemma holding_th: "holding s' th cs"
+proof -
+  from vt_s[unfolded s_def]
+  have " PIP s' (V th cs)" by (cases, simp)
+  thus ?thesis by (cases, auto)
+qed
+
+lemma edge_of_th:
+    "(Cs cs, Th th) \<in> RAG s'" 
+proof -
+ from holding_th
+ show ?thesis 
+    by (unfold s_RAG_def holding_eq, auto)
+qed
+
+lemma ancestors_cs: 
+  "ancestors (RAG s') (Cs cs) = {Th th}"
+proof -
+  have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th)  \<union>  {Th th}"
+  proof(rule vat_s'.rtree_RAG.ancestors_accum)
+    from vt_s[unfolded s_def]
+    have " PIP s' (V th cs)" by (cases, simp)
+    thus "(Cs cs, Th th) \<in> RAG s'" 
+    proof(cases)
+      assume "holding s' th cs"
+      from this[unfolded holding_eq]
+      show ?thesis by (unfold s_RAG_def, auto)
+    qed
+  qed
+  from this[unfolded ancestors_th] show ?thesis by simp
+qed
+
+lemma preced_kept: "the_preced s = the_preced s'"
+  by (auto simp: s_def the_preced_def preced_def)
+
+end
+
+text {*
+  The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation, 
+  which represents the case when there is another thread @{text "th'"}
+  to take over the critical resource released by the initiating thread @{text "th"}.
+*}
+locale step_v_cps_nt = step_v_cps +
+  fixes th'
+  -- {* @{text "th'"} is assumed to take over @{text "cs"} *}
+  assumes nt: "next_th s' th cs th'" 
+
+context step_v_cps_nt
+begin
+
+text {*
+  Lemma @{text "RAG_s"} confirms the change of RAG:
+  two edges removed and one added, as shown by the following diagram.
+*}
+
+(*
+  RAG before the V-operation
+    th1 ----|
+            |
+    th' ----|
+            |----> cs -----|
+    th2 ----|              |
+            |              |
+    th3 ----|              |
+                           |------> th
+    th4 ----|              |
+            |              |
+    th5 ----|              |
+            |----> cs'-----|
+    th6 ----|
+            |
+    th7 ----|
+
+ RAG after the V-operation
+    th1 ----|
+            |
+            |----> cs ----> th'
+    th2 ----|              
+            |              
+    th3 ----|              
+                           
+    th4 ----|              
+            |              
+    th5 ----|              
+            |----> cs'----> th
+    th6 ----|
+            |
+    th7 ----|
+*)
+
+lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"
+                using next_th_RAG[OF nt]  .
+
+lemma ancestors_th': 
+  "ancestors (RAG s') (Th th') = {Th th, Cs cs}" 
+proof -
+  have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"
+  proof(rule  vat_s'.rtree_RAG.ancestors_accum)
+    from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto
+  qed
+  thus ?thesis using ancestors_th ancestors_cs by auto
+qed
+
+lemma RAG_s:
+  "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
+                                         {(Cs cs, Th th')}"
+proof -
+  from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
+    and nt show ?thesis  by (auto intro:next_th_unique)
+qed
+
+lemma subtree_kept:
+  assumes "th1 \<notin> {th, th'}"
+  shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R")
+proof -
+  let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})"
+  let ?RAG'' = "?RAG' \<union> {(Cs cs, Th th')}"
+  have "subtree ?RAG' (Th th1) = ?R" 
+  proof(rule subset_del_subtree_outside)
+    show "Range {(Cs cs, Th th), (Th th', Cs cs)} \<inter> subtree (RAG s') (Th th1) = {}"
+    proof -
+      have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+      proof(rule subtree_refute)
+        show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+          by (unfold ancestors_th, simp)
+      next
+        from assms show "Th th1 \<noteq> Th th" by simp
+      qed
+      moreover have "(Cs cs) \<notin>  subtree (RAG s') (Th th1)"
+      proof(rule subtree_refute)
+        show "Th th1 \<notin> ancestors (RAG s') (Cs cs)"
+          by (unfold ancestors_cs, insert assms, auto)
+      qed simp
+      ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s') (Th th1) = {}" by auto
+      thus ?thesis by simp
+     qed
+  qed
+  moreover have "subtree ?RAG'' (Th th1) =  subtree ?RAG' (Th th1)"
+  proof(rule subtree_insert_next)
+    show "Th th' \<notin> subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)"
+    proof(rule subtree_refute)
+      show "Th th1 \<notin> ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')"
+            (is "_ \<notin> ?R")
+      proof -
+          have "?R \<subseteq> ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto)
+          moreover have "Th th1 \<notin> ..." using ancestors_th' assms by simp
+          ultimately show ?thesis by auto
+      qed
+    next
+      from assms show "Th th1 \<noteq> Th th'" by simp
+    qed
+  qed
+  ultimately show ?thesis by (unfold RAG_s, simp)
+qed
+
+lemma cp_kept:
+  assumes "th1 \<notin> {th, th'}"
+  shows "cp s th1 = cp s' th1"
+    by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+
+end
+
+locale step_v_cps_nnt = step_v_cps +
+  assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
+
+context step_v_cps_nnt
+begin
+
+lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"
+proof -
+  from nnt and  step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
+  show ?thesis by auto
+qed
+
+lemma subtree_kept:
+  assumes "th1 \<noteq> th"
+  shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)"
+proof(unfold RAG_s, rule subset_del_subtree_outside)
+  show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s') (Th th1) = {}"
+  proof -
+    have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+    proof(rule subtree_refute)
+      show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+          by (unfold ancestors_th, simp)
+    next
+      from assms show "Th th1 \<noteq> Th th" by simp
+    qed
+    thus ?thesis by auto
+  qed
+qed
+
+lemma cp_kept_1:
+  assumes "th1 \<noteq> th"
+  shows "cp s th1 = cp s' th1"
+    by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+
+lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}"
+proof -
+  { fix n
+    have "(Cs cs) \<notin> ancestors (RAG s') n"
+    proof
+      assume "Cs cs \<in> ancestors (RAG s') n"
+      hence "(n, Cs cs) \<in> (RAG s')^+" by (auto simp:ancestors_def)
+      from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s'" by auto
+      then obtain th' where "nn = Th th'"
+        by (unfold s_RAG_def, auto)
+      from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s'" .
+      from this[unfolded s_RAG_def]
+      have "waiting (wq s') th' cs" by auto
+      from this[unfolded cs_waiting_def]
+      have "1 < length (wq s' cs)"
+          by (cases "wq s' cs", auto)
+      from holding_next_thI[OF holding_th this]
+      obtain th' where "next_th s' th cs th'" by auto
+      with nnt show False by auto
+    qed
+  } note h = this
+  {  fix n
+     assume "n \<in> subtree (RAG s') (Cs cs)"
+     hence "n = (Cs cs)"
+     by (elim subtreeE, insert h, auto)
+  } moreover have "(Cs cs) \<in> subtree (RAG s') (Cs cs)"
+      by (auto simp:subtree_def)
+  ultimately show ?thesis by auto 
+qed
+
+lemma subtree_th: 
+  "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
+proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
+  from edge_of_th
+  show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"
+    by (unfold edges_in_def, auto simp:subtree_def)
+qed
+
+lemma cp_kept_2: 
+  shows "cp s th = cp s' th" 
+ by (unfold cp_alt_def subtree_th preced_kept, auto)
+
+lemma eq_cp:
+  fixes th' 
+  shows "cp s th' = cp s' th'"
+  using cp_kept_1 cp_kept_2
+  by (cases "th' = th", auto)
+end
+
+
+locale step_P_cps =
+  fixes s' th cs s 
+  defines s_def : "s \<equiv> (P th cs#s')"
+  assumes vt_s: "vt s"
+
+sublocale step_P_cps < vat_s : valid_trace "s"
+proof
+  from vt_s show "vt s" .
+qed
+
+sublocale step_P_cps < vat_s' : valid_trace "s'"
+proof
+  from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
+context step_P_cps
+begin
+
+lemma readys_th: "th \<in> readys s'"
+proof -
+    from step_back_step [OF vt_s[unfolded s_def]]
+    have "PIP s' (P th cs)" .
+    hence "th \<in> runing s'" by (cases, simp)
+    thus ?thesis by (simp add:readys_def runing_def)
+qed
+
+lemma root_th: "root (RAG s') (Th th)"
+  using readys_root[OF readys_th] .
+
+lemma in_no_others_subtree:
+  assumes "th' \<noteq> th"
+  shows "Th th \<notin> subtree (RAG s') (Th th')"
+proof
+  assume "Th th \<in> subtree (RAG s') (Th th')"
+  thus False
+  proof(cases rule:subtreeE)
+    case 1
+    with assms show ?thesis by auto
+  next
+    case 2
+    with root_th show ?thesis by (auto simp:root_def)
+  qed
+qed
+
+lemma preced_kept: "the_preced s = the_preced s'"
+  by (auto simp: s_def the_preced_def preced_def)
+
+end
+
+locale step_P_cps_ne =step_P_cps +
+  fixes th'
+  assumes ne: "wq s' cs \<noteq> []"
+  defines th'_def: "th' \<equiv> hd (wq s' cs)"
+
+locale step_P_cps_e =step_P_cps +
+  assumes ee: "wq s' cs = []"
+
+context step_P_cps_e
+begin
+
+lemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"
+proof -
+  from ee and  step_RAG_p[OF vt_s[unfolded s_def], folded s_def]
+  show ?thesis by auto
+qed
+
+lemma subtree_kept:
+  assumes "th' \<noteq> th"
+  shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')"
+proof(unfold RAG_s, rule subtree_insert_next)
+  from in_no_others_subtree[OF assms] 
+  show "Th th \<notin> subtree (RAG s') (Th th')" .
+qed
+
+lemma cp_kept: 
+  assumes "th' \<noteq> th"
+  shows "cp s th' = cp s' th'"
+proof -
+  have "(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
+        (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+        by (unfold preced_kept subtree_kept[OF assms], simp)
+  thus ?thesis by (unfold cp_alt_def, simp)
+qed
+
+end
+
+context step_P_cps_ne 
+begin
+
+lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+proof -
+  from step_RAG_p[OF vt_s[unfolded s_def]] and ne
+  show ?thesis by (simp add:s_def)
+qed
+
+lemma cs_held: "(Cs cs, Th th') \<in> RAG s'"
+proof -
+  have "(Cs cs, Th th') \<in> hRAG s'"
+  proof -
+    from ne
+    have " holding s' th' cs"
+      by (unfold th'_def holding_eq cs_holding_def, auto)
+    thus ?thesis 
+      by (unfold hRAG_def, auto)
+  qed
+  thus ?thesis by (unfold RAG_split, auto)
+qed
+
+lemma tRAG_s: 
+  "tRAG s = tRAG s' \<union> {(Th th, Th th')}"
+  using RAG_tRAG_transfer[OF RAG_s cs_held] .
+
+lemma cp_kept:
+  assumes "Th th'' \<notin> ancestors (tRAG s) (Th th)"
+  shows "cp s th'' = cp s' th''"
+proof -
+  have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')"
+  proof -
+    have "Th th' \<notin> subtree (tRAG s') (Th th'')"
+    proof
+      assume "Th th' \<in> subtree (tRAG s') (Th th'')"
+      thus False
+      proof(rule subtreeE)
+         assume "Th th' = Th th''"
+         from assms[unfolded tRAG_s ancestors_def, folded this]
+         show ?thesis by auto
+      next
+         assume "Th th'' \<in> ancestors (tRAG s') (Th th')"
+         moreover have "... \<subseteq> ancestors (tRAG s) (Th th')"
+         proof(rule ancestors_mono)
+            show "tRAG s' \<subseteq> tRAG s" by (unfold tRAG_s, auto)
+         qed 
+         ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th')" by auto
+         moreover have "Th th' \<in> ancestors (tRAG s) (Th th)"
+           by (unfold tRAG_s, auto simp:ancestors_def)
+         ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th)"
+                       by (auto simp:ancestors_def)
+         with assms show ?thesis by auto
+      qed
+    qed
+    from subtree_insert_next[OF this]
+    have "subtree (tRAG s' \<union> {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" .
+    from this[folded tRAG_s] show ?thesis .
+  qed
+  show ?thesis by (unfold cp_alt_def1 h preced_kept, simp)
+qed
+
+lemma cp_gen_update_stop: (* ddd *)
+  assumes "u \<in> ancestors (tRAG s) (Th th)"
+  and "cp_gen s u = cp_gen s' u"
+  and "y \<in> ancestors (tRAG s) u"
+  shows "cp_gen s y = cp_gen s' y"
+  using assms(3)
+proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf])
+  case (1 x)
+  show ?case (is "?L = ?R")
+  proof -
+    from tRAG_ancestorsE[OF 1(2)]
+    obtain th2 where eq_x: "x = Th th2" by blast
+    from vat_s.cp_gen_rec[OF this]
+    have "?L = 
+          Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)" .
+    also have "... = 
+          Max ({the_preced s' th2} \<union> cp_gen s' ` RTree.children (tRAG s') x)"
+  
+    proof -
+      from preced_kept have "the_preced s th2 = the_preced s' th2" by simp
+      moreover have "cp_gen s ` RTree.children (tRAG s) x =
+                     cp_gen s' ` RTree.children (tRAG s') x"
+      proof -
+        have "RTree.children (tRAG s) x =  RTree.children (tRAG s') x"
+        proof(unfold tRAG_s, rule children_union_kept)
+          have start: "(Th th, Th th') \<in> tRAG s"
+            by (unfold tRAG_s, auto)
+          note x_u = 1(2)
+          show "x \<notin> Range {(Th th, Th th')}"
+          proof
+            assume "x \<in> Range {(Th th, Th th')}"
+            hence eq_x: "x = Th th'" using RangeE by auto
+            show False
+            proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start])
+              case 1
+              from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG
+              show ?thesis by (auto simp:ancestors_def acyclic_def)
+            next
+              case 2
+              with x_u[unfolded eq_x]
+              have "(Th th', Th th') \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+              with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
+            qed
+          qed
+        qed
+        moreover have "cp_gen s ` RTree.children (tRAG s) x =
+                       cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A")
+        proof(rule f_image_eq)
+          fix a
+          assume a_in: "a \<in> ?A"
+          from 1(2)
+          show "?f a = ?g a"
+          proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
+             case in_ch
+             show ?thesis
+             proof(cases "a = u")
+                case True
+                from assms(2)[folded this] show ?thesis .
+             next
+                case False
+                have a_not_in: "a \<notin> ancestors (tRAG s) (Th th)"
+                proof
+                  assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+                  have "a = u"
+                  proof(rule vat_s.rtree_s.ancestors_children_unique)
+                    from a_in' a_in show "a \<in> ancestors (tRAG s) (Th th) \<inter> 
+                                          RTree.children (tRAG s) x" by auto
+                  next 
+                    from assms(1) in_ch show "u \<in> ancestors (tRAG s) (Th th) \<inter> 
+                                      RTree.children (tRAG s) x" by auto
+                  qed
+                  with False show False by simp
+                qed
+                from a_in obtain th_a where eq_a: "a = Th th_a" 
+                    by (unfold RTree.children_def tRAG_alt_def, auto)
+                from cp_kept[OF a_not_in[unfolded eq_a]]
+                have "cp s th_a = cp s' th_a" .
+                from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
+                show ?thesis .
+             qed
+          next
+            case (out_ch z)
+            hence h: "z \<in> ancestors (tRAG s) u" "z \<in> RTree.children (tRAG s) x" by auto
+            show ?thesis
+            proof(cases "a = z")
+              case True
+              from h(2) have zx_in: "(z, x) \<in> (tRAG s)" by (auto simp:RTree.children_def)
+              from 1(1)[rule_format, OF this h(1)]
+              have eq_cp_gen: "cp_gen s z = cp_gen s' z" .
+              with True show ?thesis by metis
+            next
+              case False
+              from a_in obtain th_a where eq_a: "a = Th th_a"
+                by (auto simp:RTree.children_def tRAG_alt_def)
+              have "a \<notin> ancestors (tRAG s) (Th th)"
+              proof
+                assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+                have "a = z"
+                proof(rule vat_s.rtree_s.ancestors_children_unique)
+                  from assms(1) h(1) have "z \<in> ancestors (tRAG s) (Th th)"
+                      by (auto simp:ancestors_def)
+                  with h(2) show " z \<in> ancestors (tRAG s) (Th th) \<inter> 
+                                       RTree.children (tRAG s) x" by auto
+                next
+                  from a_in a_in'
+                  show "a \<in> ancestors (tRAG s) (Th th) \<inter> RTree.children (tRAG s) x"
+                    by auto
+                qed
+                with False show False by auto
+              qed
+              from cp_kept[OF this[unfolded eq_a]]
+              have "cp s th_a = cp s' th_a" .
+              from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
+              show ?thesis .
+            qed
+          qed
+        qed
+        ultimately show ?thesis by metis
+      qed
+      ultimately show ?thesis by simp
+    qed
+    also have "... = ?R"
+      by (fold vat_s'.cp_gen_rec[OF eq_x], simp)
+    finally show ?thesis .
+  qed
+qed
+
+lemma cp_up:
+  assumes "(Th th') \<in> ancestors (tRAG s) (Th th)"
+  and "cp s th' = cp s' th'"
+  and "(Th th'') \<in> ancestors (tRAG s) (Th th')"
+  shows "cp s th'' = cp s' th''"
+proof -
+  have "cp_gen s (Th th'') = cp_gen s' (Th th'')"
+  proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)])
+    from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]]
+    show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis
+  qed
+  with cp_gen_def_cond[OF refl[of "Th th''"]]
+  show ?thesis by metis
+qed
+
+end
+
+locale step_create_cps =
+  fixes s' th prio s 
+  defines s_def : "s \<equiv> (Create th prio#s')"
+  assumes vt_s: "vt s"
+
+sublocale step_create_cps < vat_s: valid_trace "s"
+  by (unfold_locales, insert vt_s, simp)
+
+sublocale step_create_cps < vat_s': valid_trace "s'"
+  by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
+
+context step_create_cps
+begin
+
+lemma RAG_kept: "RAG s = RAG s'"
+  by (unfold s_def RAG_create_unchanged, auto)
+
+lemma tRAG_kept: "tRAG s = tRAG s'"
+  by (unfold tRAG_alt_def RAG_kept, auto)
+
+lemma preced_kept:
+  assumes "th' \<noteq> th"
+  shows "the_preced s th' = the_preced s' th'"
+  by (unfold s_def the_preced_def preced_def, insert assms, auto)
+
+lemma th_not_in: "Th th \<notin> Field (tRAG s')"
+proof -
+  from vt_s[unfolded s_def]
+  have "PIP s' (Create th prio)" by (cases, simp)
+  hence "th \<notin> threads s'" by(cases, simp)
+  from vat_s'.not_in_thread_isolated[OF this]
+  have "Th th \<notin> Field (RAG s')" .
+  with tRAG_Field show ?thesis by auto
+qed
+
+lemma eq_cp:
+  assumes neq_th: "th' \<noteq> th"
+  shows "cp s th' = cp s' th'"
+proof -
+  have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
+        (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+  proof(unfold tRAG_kept, rule f_image_eq)
+    fix a
+    assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+    then obtain th_a where eq_a: "a = Th th_a" 
+    proof(cases rule:subtreeE)
+      case 2
+      from ancestors_Field[OF 2(2)]
+      and that show ?thesis by (unfold tRAG_alt_def, auto)
+    qed auto
+    have neq_th_a: "th_a \<noteq> th"
+    proof -
+      have "(Th th) \<notin> subtree (tRAG s') (Th th')"
+      proof
+        assume "Th th \<in> subtree (tRAG s') (Th th')"
+        thus False
+        proof(cases rule:subtreeE)
+          case 2
+          from ancestors_Field[OF this(2)]
+          and th_not_in[unfolded Field_def]
+          show ?thesis by auto
+        qed (insert assms, auto)
+      qed
+      with a_in[unfolded eq_a] show ?thesis by auto
+    qed
+    from preced_kept[OF this]
+    show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+      by (unfold eq_a, simp)
+  qed
+  thus ?thesis by (unfold cp_alt_def1, simp)
+qed
+
+lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}"
+proof -
+  { fix a
+    assume "a \<in> RTree.children (tRAG s) (Th th)"
+    hence "(a, Th th) \<in> tRAG s" by (auto simp:RTree.children_def)
+    with th_not_in have False 
+     by (unfold Field_def tRAG_kept, auto)
+  } thus ?thesis by auto
+qed
+
+lemma eq_cp_th: "cp s th = preced th s"
+ by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def)
+
+end
+
+locale step_exit_cps =
+  fixes s' th prio s 
+  defines s_def : "s \<equiv> Exit th # s'"
+  assumes vt_s: "vt s"
+
+sublocale step_exit_cps < vat_s: valid_trace "s"
+  by (unfold_locales, insert vt_s, simp)
+
+sublocale step_exit_cps < vat_s': valid_trace "s'"
+  by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
+
+context step_exit_cps
+begin
+
+lemma preced_kept:
+  assumes "th' \<noteq> th"
+  shows "the_preced s th' = the_preced s' th'"
+  by (unfold s_def the_preced_def preced_def, insert assms, auto)
+
+lemma RAG_kept: "RAG s = RAG s'"
+  by (unfold s_def RAG_exit_unchanged, auto)
+
+lemma tRAG_kept: "tRAG s = tRAG s'"
+  by (unfold tRAG_alt_def RAG_kept, auto)
+
+lemma th_ready: "th \<in> readys s'"
+proof -
+  from vt_s[unfolded s_def]
+  have "PIP s' (Exit th)" by (cases, simp)
+  hence h: "th \<in> runing s' \<and> holdents s' th = {}" by (cases, metis)
+  thus ?thesis by (unfold runing_def, auto)
+qed
+
+lemma th_holdents: "holdents s' th = {}"
+proof -
+ from vt_s[unfolded s_def]
+  have "PIP s' (Exit th)" by (cases, simp)
+  thus ?thesis by (cases, metis)
+qed
+
+lemma th_RAG: "Th th \<notin> Field (RAG s')"
+proof -
+  have "Th th \<notin> Range (RAG s')"
+  proof
+    assume "Th th \<in> Range (RAG s')"
+    then obtain cs where "holding (wq s') th cs"
+      by (unfold Range_iff s_RAG_def, auto)
+    with th_holdents[unfolded holdents_def]
+    show False by (unfold eq_holding, auto)
+  qed
+  moreover have "Th th \<notin> Domain (RAG s')"
+  proof
+    assume "Th th \<in> Domain (RAG s')"
+    then obtain cs where "waiting (wq s') th cs"
+      by (unfold Domain_iff s_RAG_def, auto)
+    with th_ready show False by (unfold readys_def eq_waiting, auto)
+  qed
+  ultimately show ?thesis by (auto simp:Field_def)
+qed
+
+lemma th_tRAG: "(Th th) \<notin> Field (tRAG s')"
+  using th_RAG tRAG_Field[of s'] by auto
+
+lemma eq_cp:
+  assumes neq_th: "th' \<noteq> th"
+  shows "cp s th' = cp s' th'"
+proof -
+  have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
+        (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+  proof(unfold tRAG_kept, rule f_image_eq)
+    fix a
+    assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+    then obtain th_a where eq_a: "a = Th th_a" 
+    proof(cases rule:subtreeE)
+      case 2
+      from ancestors_Field[OF 2(2)]
+      and that show ?thesis by (unfold tRAG_alt_def, auto)
+    qed auto
+    have neq_th_a: "th_a \<noteq> th"
+    proof -
+      from vat_s'.readys_in_no_subtree[OF th_ready assms]
+      have "(Th th) \<notin> subtree (RAG s') (Th th')" .
+      with tRAG_subtree_RAG[of s' "Th th'"]
+      have "(Th th) \<notin> subtree (tRAG s') (Th th')" by auto
+      with a_in[unfolded eq_a] show ?thesis by auto
+    qed
+    from preced_kept[OF this]
+    show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+      by (unfold eq_a, simp)
+  qed
+  thus ?thesis by (unfold cp_alt_def1, simp)
+qed
+
+end
+
+end
+
diff -r b4bcd1edbb6d -r 633b1fc8631b PIPBasics.thy
--- a/PIPBasics.thy	Wed Jan 06 16:34:26 2016 +0000
+++ b/PIPBasics.thy	Thu Jan 07 08:33:13 2016 +0800
@@ -3048,4 +3048,693 @@
   apply (drule_tac th_in_ne)
   by (unfold preced_def, auto intro: birth_time_lt)
 
+lemma inj_the_preced: 
+  "inj_on (the_preced s) (threads s)"
+  by (metis inj_onI preced_unique the_preced_def)
+
+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
+
+sublocale valid_trace < rtree_RAG: rtree "RAG s"
+proof
+  show "single_valued (RAG s)"
+  apply (intro_locales)
+    by (unfold single_valued_def, 
+        auto intro:unique_RAG)
+
+  show "acyclic (RAG s)"
+     by (rule acyclic_RAG)
+qed
+
+sublocale valid_trace < rtree_s: rtree "tRAG s"
+proof(unfold_locales)
+  from sgv_tRAG show "single_valued (tRAG s)" .
+next
+  from acyclic_tRAG show "acyclic (tRAG s)" .
+qed
+
+sublocale valid_trace < fsbtRAGs : fsubtree "RAG s"
+proof -
+  show "fsubtree (RAG s)"
+  proof(intro_locales)
+    show "fbranch (RAG s)" using finite_fbranchI[OF finite_RAG] .
+  next
+    show "fsubtree_axioms (RAG s)"
+    proof(unfold fsubtree_axioms_def)
+      from wf_RAG show "wf (RAG s)" .
+    qed
+  qed
+qed
+
+sublocale valid_trace < fsbttRAGs: fsubtree "tRAG s"
+proof -
+  have "fsubtree (tRAG s)"
+  proof -
+    have "fbranch (tRAG s)"
+    proof(unfold tRAG_def, rule fbranch_compose)
+        show "fbranch (wRAG s)"
+        proof(rule finite_fbranchI)
+           from finite_RAG show "finite (wRAG s)"
+           by (unfold RAG_split, auto)
+        qed
+    next
+        show "fbranch (hRAG s)"
+        proof(rule finite_fbranchI)
+           from finite_RAG 
+           show "finite (hRAG s)" by (unfold RAG_split, auto)
+        qed
+    qed
+    moreover have "wf (tRAG s)"
+    proof(rule wf_subset)
+      show "wf (RAG s O RAG s)" using wf_RAG
+        by (fold wf_comp_self, simp)
+    next
+      show "tRAG s \<subseteq> (RAG s O RAG s)"
+        by (unfold tRAG_alt_def, auto)
+    qed
+    ultimately show ?thesis
+      by (unfold fsubtree_def fsubtree_axioms_def,auto)
+  qed
+  from this[folded tRAG_def] show "fsubtree (tRAG s)" .
+qed
+
+lemma Max_UNION: 
+  assumes "finite A"
+  and "A \<noteq> {}"
+  and "\<forall> M \<in> f ` A. finite M"
+  and "\<forall> M \<in> f ` A. M \<noteq> {}"
+  shows "Max (\<Union>x\<in> A. f x) = Max (Max ` f ` A)" (is "?L = ?R")
+  using assms[simp]
+proof -
+  have "?L = Max (\<Union>(f ` A))"
+    by (fold Union_image_eq, simp)
+  also have "... = ?R"
+    by (subst Max_Union, simp+)
+  finally show ?thesis .
+qed
+
+lemma max_Max_eq:
+  assumes "finite A"
+    and "A \<noteq> {}"
+    and "x = y"
+  shows "max x (Max A) = Max ({y} \<union> A)" (is "?L = ?R")
+proof -
+  have "?R = Max (insert y A)" by simp
+  also from assms have "... = ?L"
+      by (subst Max.insert, simp+)
+  finally show ?thesis by simp
+qed
+
+context valid_trace
+begin
+
+(* ddd *)
+lemma cp_gen_rec:
+  assumes "x = Th th"
+  shows "cp_gen s x = Max ({the_preced s th} \<union> (cp_gen s) ` children (tRAG s) x)"
+proof(cases "children (tRAG s) x = {}")
+  case True
+  show ?thesis
+    by (unfold True cp_gen_def subtree_children, simp add:assms)
+next
+  case False
+  hence [simp]: "children (tRAG s) x \<noteq> {}" by auto
+  note fsbttRAGs.finite_subtree[simp]
+  have [simp]: "finite (children (tRAG s) x)"
+     by (intro rev_finite_subset[OF fsbttRAGs.finite_subtree], 
+            rule children_subtree)
+  { fix r x
+    have "subtree r x \<noteq> {}" by (auto simp:subtree_def)
+  } note this[simp]
+  have [simp]: "\<exists>x\<in>children (tRAG s) x. subtree (tRAG s) x \<noteq> {}"
+  proof -
+    from False obtain q where "q \<in> children (tRAG s) x" by blast
+    moreover have "subtree (tRAG s) q \<noteq> {}" by simp
+    ultimately show ?thesis by blast
+  qed
+  have h: "Max ((the_preced s \<circ> the_thread) `
+                ({x} \<union> \<Union>(subtree (tRAG s) ` children (tRAG s) x))) =
+        Max ({the_preced s th} \<union> cp_gen s ` children (tRAG s) x)"
+                     (is "?L = ?R")
+  proof -
+    let "Max (?f ` (?A \<union> \<Union> (?g ` ?B)))" = ?L
+    let "Max (_ \<union> (?h ` ?B))" = ?R
+    let ?L1 = "?f ` \<Union>(?g ` ?B)"
+    have eq_Max_L1: "Max ?L1 = Max (?h ` ?B)"
+    proof -
+      have "?L1 = ?f ` (\<Union> x \<in> ?B.(?g x))" by simp
+      also have "... =  (\<Union> x \<in> ?B. ?f ` (?g x))" by auto
+      finally have "Max ?L1 = Max ..." by simp
+      also have "... = Max (Max ` (\<lambda>x. ?f ` subtree (tRAG s) x) ` ?B)"
+        by (subst Max_UNION, simp+)
+      also have "... = Max (cp_gen s ` children (tRAG s) x)"
+          by (unfold image_comp cp_gen_alt_def, simp)
+      finally show ?thesis .
+    qed
+    show ?thesis
+    proof -
+      have "?L = Max (?f ` ?A \<union> ?L1)" by simp
+      also have "... = max (the_preced s (the_thread x)) (Max ?L1)"
+            by (subst Max_Un, simp+)
+      also have "... = max (?f x) (Max (?h ` ?B))"
+        by (unfold eq_Max_L1, simp)
+      also have "... =?R"
+        by (rule max_Max_eq, (simp)+, unfold assms, simp)
+      finally show ?thesis .
+    qed
+  qed  thus ?thesis 
+          by (fold h subtree_children, unfold cp_gen_def, simp) 
+qed
+
+lemma cp_rec:
+  "cp s th = Max ({the_preced s th} \<union> 
+                     (cp s o the_thread) ` children (tRAG s) (Th th))"
+proof -
+  have "Th th = Th th" by simp
+  note h =  cp_gen_def_cond[OF this] cp_gen_rec[OF this]
+  show ?thesis 
+  proof -
+    have "cp_gen s ` children (tRAG s) (Th th) = 
+                (cp s \<circ> the_thread) ` children (tRAG s) (Th th)"
+    proof(rule cp_gen_over_set)
+      show " \<forall>x\<in>children (tRAG s) (Th th). \<exists>th. x = Th th"
+        by (unfold tRAG_alt_def, auto simp:children_def)
+    qed
+    thus ?thesis by (subst (1) h(1), unfold h(2), simp)
+  qed
+qed
+
+end
+
+(* keep *)
+lemma next_th_holding:
+  assumes vt: "vt s"
+  and nxt: "next_th s th cs th'"
+  shows "holding (wq s) th cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  thus ?thesis
+    by (unfold cs_holding_def, auto)
+qed
+
+context valid_trace
+begin
+
+lemma next_th_waiting:
+  assumes nxt: "next_th s th cs th'"
+  shows "waiting (wq s) th' cs"
+proof -
+  from nxt[unfolded next_th_def]
+  obtain rest where h: "wq s cs = th # rest"
+                       "rest \<noteq> []" 
+                       "th' = hd (SOME q. distinct q \<and> set q = set rest)" by auto
+  from wq_distinct[of cs, unfolded h]
+  have dst: "distinct (th # rest)" .
+  have in_rest: "th' \<in> set rest"
+  proof(unfold h, rule someI2)
+    show "distinct rest \<and> set rest = set rest" using dst by auto
+  next
+    fix x assume "distinct x \<and> set x = set rest"
+    with h(2)
+    show "hd x \<in> set (rest)" by (cases x, auto)
+  qed
+  hence "th' \<in> set (wq s cs)" by (unfold h(1), auto)
+  moreover have "th' \<noteq> hd (wq s cs)"
+    by (unfold h(1), insert in_rest dst, auto)
+  ultimately show ?thesis by (auto simp:cs_waiting_def)
+qed
+
+lemma next_th_RAG:
+  assumes nxt: "next_th (s::event list) th cs th'"
+  shows "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s"
+  using vt assms next_th_holding next_th_waiting
+  by (unfold s_RAG_def, simp)
+
+end
+
+-- {* A useless definition *}
+definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"
+where "cps s = {(th, cp s th) | th . th \<in> threads s}"
+
+end
diff -r b4bcd1edbb6d -r 633b1fc8631b PIPBasics.thy~
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/PIPBasics.thy~	Thu Jan 07 08:33:13 2016 +0800
@@ -0,0 +1,3051 @@
+theory PIPBasics
+imports PIPDefs 
+begin
+
+locale valid_trace = 
+  fixes s
+  assumes vt : "vt s"
+
+locale valid_trace_e = valid_trace +
+  fixes e
+  assumes vt_e: "vt (e#s)"
+begin
+
+lemma pip_e: "PIP s e"
+  using vt_e by (cases, simp)  
+
+end
+
+lemma runing_ready: 
+  shows "runing s \<subseteq> readys s"
+  unfolding runing_def readys_def
+  by auto 
+
+lemma readys_threads:
+  shows "readys s \<subseteq> threads s"
+  unfolding readys_def
+  by auto
+
+lemma wq_v_neq:
+   "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
+  by (auto simp:wq_def Let_def cp_def split:list.splits)
+
+context valid_trace
+begin
+
+lemma ind [consumes 0, case_names Nil Cons, induct type]:
+  assumes "PP []"
+     and "(\<And>s e. valid_trace s \<Longrightarrow> valid_trace (e#s) \<Longrightarrow>
+                   PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
+     shows "PP s"
+proof(rule vt.induct[OF vt])
+  from assms(1) show "PP []" .
+next
+  fix s e
+  assume h: "vt s" "PP s" "PIP s e"
+  show "PP (e # s)"
+  proof(cases rule:assms(2))
+    from h(1) show v1: "valid_trace s" by (unfold_locales, simp)
+  next
+    from h(1,3) have "vt (e#s)" by auto
+    thus "valid_trace (e # s)" by (unfold_locales, simp)
+  qed (insert h, auto)
+qed
+
+lemma wq_distinct: "distinct (wq s cs)"
+proof(rule ind, simp add:wq_def)
+  fix s e
+  assume h1: "step s e"
+  and h2: "distinct (wq s cs)"
+  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> (RAG s)\<^sup>+"
+      and h2: "thread \<in> set (wq_fun (schs s) cs)"
+      and h3: "thread \<in> runing s"
+    show "False" 
+    proof -
+      from h3 have "\<And> cs. thread \<in>  set (wq_fun (schs s) cs) \<Longrightarrow> 
+                             thread = hd ((wq_fun (schs s) cs))" 
+        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> (RAG s)"
+        by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
+      with h1 show False by auto
+    qed
+  next
+    fix thread s a list
+    assume dst: "distinct list"
+    show "distinct (SOME q. distinct q \<and> set q = set list)"
+    proof(rule someI2)
+      from dst show  "distinct list \<and> set list = set list" by auto
+    next
+      fix q assume "distinct q \<and> set q = set list"
+      thus "distinct q" by auto
+    qed
+  qed
+qed
+
+end
+
+
+context valid_trace_e
+begin
+
+text {*
+  The following lemma shows that only the @{text "P"}
+  operation can add new thread into waiting queues. 
+  Such kind of lemmas are very obvious, but need to be checked formally.
+  This is a kind of confirmation that our modelling is correct.
+*}
+
+lemma block_pre: 
+  assumes s_ni: "thread \<notin>  set (wq s cs)"
+  and s_i: "thread \<in> set (wq (e#s) cs)"
+  shows "e = P thread cs"
+proof -
+  show ?thesis
+  proof(cases e)
+    case (P th cs)
+    with assms
+    show ?thesis
+      by (auto simp:wq_def Let_def split:if_splits)
+  next
+    case (Create th prio)
+    with assms show ?thesis
+      by (auto simp:wq_def Let_def split:if_splits)
+  next
+    case (Exit th)
+    with assms show ?thesis
+      by (auto simp:wq_def Let_def split:if_splits)
+  next
+    case (Set th prio)
+    with assms show ?thesis
+      by (auto simp:wq_def Let_def split:if_splits)
+  next
+    case (V th cs)
+    with vt_e assms show ?thesis
+      apply (auto simp:wq_def Let_def split:if_splits)
+    proof -
+      fix q qs
+      assume h1: "thread \<notin> set (wq_fun (schs s) cs)"
+        and h2: "q # qs = wq_fun (schs s) cs"
+        and h3: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+        and vt: "vt (V th cs # s)"
+      from h1 and h2[symmetric] have "thread \<notin> set (q # qs)" by simp
+      moreover have "thread \<in> set qs"
+      proof -
+        have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
+        proof(rule someI2)
+          from wq_distinct [of cs]
+          and h2[symmetric, folded wq_def]
+          show "distinct qs \<and> set qs = set qs" by auto
+        next
+          fix x assume "distinct x \<and> set x = set qs"
+          thus "set x = set qs" by auto
+        qed
+        with h3 show ?thesis by simp
+      qed
+      ultimately show "False" by auto
+      qed
+  qed
+qed
+
+end
+
+text {*
+  The following lemmas is also obvious and shallow. It says
+  that only running thread can request for a critical resource 
+  and that the requested resource must be one which is
+  not current held by the thread.
+*}
+
+lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow> 
+  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
+
+lemma abs1:
+  assumes ein: "e \<in> set es"
+  and neq: "hd es \<noteq> hd (es @ [x])"
+  shows "False"
+proof -
+  from ein have "es \<noteq> []" by auto
+  then obtain e ess where "es = e # ess" by (cases es, auto)
+  with neq show ?thesis by auto
+qed
+
+lemma q_head: "Q (hd es) \<Longrightarrow> hd es = hd [th\<leftarrow>es . Q th]"
+  by (cases es, auto)
+
+inductive_cases evt_cons: "vt (a#s)"
+
+context valid_trace_e
+begin
+
+lemma abs2:
+  assumes inq: "thread \<in> set (wq s cs)"
+  and nh: "thread = hd (wq s cs)"
+  and qt: "thread \<noteq> hd (wq (e#s) cs)"
+  and inq': "thread \<in> set (wq (e#s) cs)"
+  shows "False"
+proof -
+  from vt_e assms show "False"
+    apply (cases e)
+    apply ((simp split:if_splits add:Let_def wq_def)[1])+
+    apply (insert abs1, fast)[1]
+    apply (auto simp:wq_def simp:Let_def split:if_splits list.splits)
+  proof -
+    fix th qs
+    assume vt: "vt (V th cs # s)"
+      and th_in: "thread \<in> set (SOME q. distinct q \<and> set q = set qs)"
+      and eq_wq: "wq_fun (schs s) cs = thread # qs"
+    show "False"
+    proof -
+      from wq_distinct[of cs]
+        and eq_wq[folded wq_def] have "distinct (thread#qs)" by simp
+      moreover have "thread \<in> set qs"
+      proof -
+        have "set (SOME q. distinct q \<and> set q = set qs) = set qs"
+        proof(rule someI2)
+          from wq_distinct [of cs]
+          and eq_wq [folded wq_def]
+          show "distinct qs \<and> set qs = set qs" by auto
+        next
+          fix x assume "distinct x \<and> set x = set qs"
+          thus "set x = set qs" by auto
+        qed
+        with th_in show ?thesis by auto
+      qed
+      ultimately show ?thesis by auto
+    qed
+  qed
+qed
+
+end
+
+context valid_trace
+begin
+
+lemma vt_moment: "\<And> t. vt (moment t s)"
+proof(induct rule:ind)
+  case Nil
+  thus ?case by (simp add:vt_nil)
+next
+  case (Cons s e t)
+  show ?case
+  proof(cases "t \<ge> length (e#s)")
+    case True
+    from True have "moment t (e#s) = e#s" by simp
+    thus ?thesis using Cons
+      by (simp add:valid_trace_def)
+  next
+    case False
+    from Cons have "vt (moment t s)" by simp
+    moreover have "moment t (e#s) = moment t s"
+    proof -
+      from False have "t \<le> length s" by simp
+      from moment_app [OF this, of "[e]"] 
+      show ?thesis by simp
+    qed
+    ultimately show ?thesis by simp
+  qed
+qed
+
+(* Wrong:
+    lemma \<lbrakk>thread \<in> set (wq_fun cs1 s); thread \<in> set (wq_fun cs2 s)\<rbrakk> \<Longrightarrow> cs1 = cs2"
+*)
+
+text {* (* ddd *)
+  The nature of the work is like this: since it starts from a very simple and basic 
+  model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
+  For instance, the fact 
+  that one thread can not be blocked by two critical resources at the same time
+  is obvious, because only running threads can make new requests, if one is waiting for 
+  a critical resource and get blocked, it can not make another resource request and get 
+  blocked the second time (because it is not running). 
+
+  To derive this fact, one needs to prove by contraction and 
+  reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
+  named @{text "p_split"}, which is about status changing along the time axis. It says if 
+  a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
+  but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"} 
+  in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history 
+  of events leading to it), such that @{text "Q"} switched 
+  from being @{text "False"} to @{text "True"} and kept being @{text "True"}
+  till the last moment of @{text "s"}.
+
+  Suppose a thread @{text "th"} is blocked
+  on @{text "cs1"} and @{text "cs2"} in some state @{text "s"}, 
+  since no thread is blocked at the very beginning, by applying 
+  @{text "p_split"} to these two blocking facts, there exist 
+  two moments @{text "t1"} and @{text "t2"}  in @{text "s"}, such that 
+  @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"} 
+  and kept on blocked on them respectively ever since.
+ 
+  Without lose of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
+  However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
+  in blocked state at moment @{text "t2"} and could not
+  make any request and get blocked the second time: Contradiction.
+*}
+
+lemma waiting_unique_pre:
+  assumes h11: "thread \<in> set (wq s cs1)"
+  and h12: "thread \<noteq> hd (wq s cs1)"
+  assumes h21: "thread \<in> set (wq s cs2)"
+  and h22: "thread \<noteq> hd (wq s cs2)"
+  and neq12: "cs1 \<noteq> cs2"
+  shows "False"
+proof -
+  let "?Q cs s" = "thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
+  from h11 and h12 have q1: "?Q cs1 s" by simp
+  from h21 and h22 have q2: "?Q cs2 s" by simp
+  have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
+  have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
+  from p_split [of "?Q cs1", OF q1 nq1]
+  obtain t1 where lt1: "t1 < length s"
+    and np1: "\<not>(thread \<in> set (wq (moment t1 s) cs1) \<and>
+        thread \<noteq> hd (wq (moment t1 s) cs1))"
+    and nn1: "(\<forall>i'>t1. thread \<in> set (wq (moment i' s) cs1) \<and>
+             thread \<noteq> hd (wq (moment i' s) cs1))" by auto
+  from p_split [of "?Q cs2", OF q2 nq2]
+  obtain t2 where lt2: "t2 < length s"
+    and np2: "\<not>(thread \<in> set (wq (moment t2 s) cs2) \<and>
+        thread \<noteq> hd (wq (moment t2 s) cs2))"
+    and nn2: "(\<forall>i'>t2. thread \<in> set (wq (moment i' s) cs2) \<and>
+             thread \<noteq> hd (wq (moment i' s) cs2))" by auto
+  show ?thesis
+  proof -
+    { 
+      assume lt12: "t1 < t2"
+      let ?t3 = "Suc t2"
+      from lt2 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
+      have "t2 < ?t3" by simp
+      from nn2 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+        h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+      have "vt (e#moment t2 s)"
+      proof -
+        from vt_moment 
+        have "vt (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      then interpret vt_e: valid_trace_e "moment t2 s" "e"
+        by (unfold_locales, auto, cases, simp)
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+        case True
+        from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+          by auto 
+        from vt_e.abs2 [OF True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from vt_e.block_pre[OF False h1]
+        have "e = P thread cs2" .
+        with vt_e.vt_e have "vt ((P thread cs2)# moment t2 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t2 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t2 s)" by auto
+        with nn1 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def wq_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume lt12: "t2 < t1"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have "vt  (e#moment t1 s)"
+      proof -
+        from vt_moment
+        have "vt (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      then interpret vt_e: valid_trace_e "moment t1 s" e
+        by (unfold_locales, auto, cases, auto)
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+          by auto
+        from vt_e.abs2 True eq_th h2 h1
+        show ?thesis by auto
+      next
+        case False
+        from vt_e.block_pre [OF False h1]
+        have "e = P thread cs1" .
+        with vt_e.vt_e have "vt ((P thread cs1)# moment t1 s)" by simp
+        from p_pre [OF this] have "thread \<in> runing (moment t1 s)" by simp
+        with runing_ready have "thread \<in> readys (moment t1 s)" by auto
+        with nn2 [rule_format, OF lt12]
+        show ?thesis  by (simp add:readys_def wq_def s_waiting_def, auto)
+      qed
+    } moreover {
+      assume eqt12: "t1 = t2"
+      let ?t3 = "Suc t1"
+      from lt1 have le_t3: "?t3 \<le> length s" by auto
+      from moment_plus [OF this] 
+      obtain e where eq_m: "moment ?t3 s = e#moment t1 s" by auto
+      have lt_t3: "t1 < ?t3" by simp
+      from nn1 [rule_format, OF this] and eq_m
+      have h1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
+        h2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
+      have vt_e: "vt (e#moment t1 s)"
+      proof -
+        from vt_moment
+        have "vt (moment ?t3 s)" .
+        with eq_m show ?thesis by simp
+      qed
+      then interpret vt_e: valid_trace_e "moment t1 s" e
+        by (unfold_locales, auto, cases, auto)
+      have ?thesis
+      proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
+        case True
+        from True and np1 have eq_th: "thread = hd (wq (moment t1 s) cs1)"
+          by auto
+        from vt_e.abs2 [OF True eq_th h2 h1]
+        show ?thesis by auto
+      next
+        case False
+        from vt_e.block_pre [OF False h1]
+        have eq_e1: "e = P thread cs1" .
+        have lt_t3: "t1 < ?t3" by simp
+        with eqt12 have "t2 < ?t3" by simp
+        from nn2 [rule_format, OF this] and eq_m and eqt12
+        have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
+          h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
+        show ?thesis
+        proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
+          case True
+          from True and np2 have eq_th: "thread = hd (wq (moment t2 s) cs2)"
+            by auto
+          from vt_e and eqt12 have "vt (e#moment t2 s)" by simp 
+          then interpret vt_e2: valid_trace_e "moment t2 s" e
+            by (unfold_locales, auto, cases, auto)
+          from vt_e2.abs2 [OF True eq_th h2 h1]
+          show ?thesis .
+        next
+          case False
+          have "vt (e#moment t2 s)"
+          proof -
+            from vt_moment eqt12
+            have "vt (moment (Suc t2) s)" by auto
+            with eq_m eqt12 show ?thesis by simp
+          qed
+          then interpret vt_e2: valid_trace_e "moment t2 s" e
+            by (unfold_locales, auto, cases, auto)
+          from vt_e2.block_pre [OF False h1]
+          have "e = P thread cs2" .
+          with eq_e1 neq12 show ?thesis by auto
+        qed
+      qed
+    } ultimately show ?thesis by arith
+  qed
+qed
+
+text {*
+  This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
+*}
+
+lemma waiting_unique:
+  assumes "waiting s th cs1"
+  and "waiting s th cs2"
+  shows "cs1 = cs2"
+using waiting_unique_pre assms
+unfolding wq_def s_waiting_def
+by auto
+
+end
+
+(* not used *)
+text {*
+  Every thread can only be blocked on one critical resource, 
+  symmetrically, every critical resource can only be held by one thread. 
+  This fact is much more easier according to our definition. 
+*}
+lemma held_unique:
+  assumes "holding (s::event list) th1 cs"
+  and "holding s th2 cs"
+  shows "th1 = th2"
+ by (insert assms, unfold s_holding_def, auto)
+
+
+lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits)
+
+lemma last_set_unique: 
+  "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
+          \<Longrightarrow> th1 = th2"
+  apply (induct s, auto)
+  by (case_tac a, auto split:if_splits dest:last_set_lt)
+
+lemma preced_unique : 
+  assumes pcd_eq: "preced th1 s = preced th2 s"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "th1 = th2"
+proof -
+  from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
+  from last_set_unique [OF this th_in1 th_in2]
+  show ?thesis .
+qed
+
+lemma preced_linorder: 
+  assumes neq_12: "th1 \<noteq> th2"
+  and th_in1: "th1 \<in> threads s"
+  and th_in2: " th2 \<in> threads s"
+  shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
+proof -
+  from preced_unique [OF _ th_in1 th_in2] and neq_12 
+  have "preced th1 s \<noteq> preced th2 s" by auto
+  thus ?thesis by auto
+qed
+
+(* An aux lemma used later *)
+lemma unique_minus:
+  fixes x y z r
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r"
+  and xz: "(x, z) \<in> r^+"
+  and neq: "y \<noteq> z"
+  shows "(y, z) \<in> r^+"
+proof -
+ from xz and neq show ?thesis
+ proof(induct)
+   case (base ya)
+   have "(x, ya) \<in> r" by fact
+   from unique [OF xy this] have "y = ya" .
+   with base show ?case by auto
+ next
+   case (step ya z)
+   show ?case
+   proof(cases "y = ya")
+     case True
+     from step True show ?thesis by simp
+   next
+     case False
+     from step False
+     show ?thesis by auto
+   qed
+ qed
+qed
+
+lemma unique_base:
+  fixes r x y z
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r"
+  and xz: "(x, z) \<in> r^+"
+  and neq_yz: "y \<noteq> z"
+  shows "(y, z) \<in> r^+"
+proof -
+  from xz neq_yz show ?thesis
+  proof(induct)
+    case (base ya)
+    from xy unique base show ?case by auto
+  next
+    case (step ya z)
+    show ?case
+    proof(cases "y = ya")
+      case True
+      from True step show ?thesis by auto
+    next
+      case False
+      from False step 
+      have "(y, ya) \<in> r\<^sup>+" by auto
+      with step show ?thesis by auto
+    qed
+  qed
+qed
+
+lemma unique_chain:
+  fixes r x y z
+  assumes unique: "\<And> a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"
+  and xy: "(x, y) \<in> r^+"
+  and xz: "(x, z) \<in> r^+"
+  and neq_yz: "y \<noteq> z"
+  shows "(y, z) \<in> r^+ \<or> (z, y) \<in> r^+"
+proof -
+  from xy xz neq_yz show ?thesis
+  proof(induct)
+    case (base y)
+    have h1: "(x, y) \<in> r" and h2: "(x, z) \<in> r\<^sup>+" and h3: "y \<noteq> z" using base by auto
+    from unique_base [OF _ h1 h2 h3] and unique show ?case by auto
+  next
+    case (step y za)
+    show ?case
+    proof(cases "y = z")
+      case True
+      from True step show ?thesis by auto
+    next
+      case False
+      from False step have "(y, z) \<in> r\<^sup>+ \<or> (z, y) \<in> r\<^sup>+" by auto
+      thus ?thesis
+      proof
+        assume "(z, y) \<in> r\<^sup>+"
+        with step have "(z, za) \<in> r\<^sup>+" by auto
+        thus ?thesis by auto
+      next
+        assume h: "(y, z) \<in> r\<^sup>+"
+        from step have yza: "(y, za) \<in> r" by simp
+        from step have "za \<noteq> z" by simp
+        from unique_minus [OF _ yza h this] and unique
+        have "(za, z) \<in> r\<^sup>+" by auto
+        thus ?thesis by auto
+      qed
+    qed
+  qed
+qed
+
+text {*
+  The following three lemmas show that @{text "RAG"} does not change
+  by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
+  events, respectively.
+*}
+
+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 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 RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
+by (simp add:Let_def)
+
+
+text {* 
+  The following lemmas are used in the proof of 
+  lemma @{text "step_RAG_v"}, which characterizes how the @{text "RAG"} is changed
+  by @{text "V"}-events. 
+  However, since our model is very concise, such  seemingly obvious lemmas need to be derived from scratch,
+  starting from the model definitions.
+*}
+lemma step_v_hold_inv[elim_format]:
+  "\<And>c t. \<lbrakk>vt (V th cs # s); 
+          \<not> holding (wq s) t c; holding (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> 
+            next_th s th cs t \<and> c = cs"
+proof -
+  fix c t
+  assume vt: "vt (V th cs # s)"
+    and nhd: "\<not> holding (wq s) t c"
+    and hd: "holding (wq (V th cs # s)) t c"
+  show "next_th s th cs t \<and> c = cs"
+  proof(cases "c = cs")
+    case False
+    with nhd hd show ?thesis
+      by (unfold cs_holding_def wq_def, auto simp:Let_def)
+  next
+    case True
+    with step_back_step [OF vt] 
+    have "step s (V th c)" by simp
+    hence "next_th s th cs t"
+    proof(cases)
+      assume "holding s th c"
+      with nhd hd show ?thesis
+        apply (unfold s_holding_def cs_holding_def wq_def next_th_def,
+               auto simp:Let_def split:list.splits if_splits)
+        proof -
+          assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
+          moreover have "\<dots> = set []"
+          proof(rule someI2)
+            show "distinct [] \<and> [] = []" by auto
+          next
+            fix x assume "distinct x \<and> x = []"
+            thus "set x = set []" by auto
+          qed
+          ultimately show False by auto
+        next
+          assume " hd (SOME q. distinct q \<and> q = []) \<in> set (SOME q. distinct q \<and> q = [])"
+          moreover have "\<dots> = set []"
+          proof(rule someI2)
+            show "distinct [] \<and> [] = []" by auto
+          next
+            fix x assume "distinct x \<and> x = []"
+            thus "set x = set []" by auto
+          qed
+          ultimately show False by auto
+        qed
+    qed
+    with True show ?thesis by auto
+  qed
+qed
+
+text {* 
+  The following @{text "step_v_wait_inv"} is also an obvious lemma, which, however, needs to be
+  derived from scratch, which confirms the correctness of the definition of @{text "next_th"}.
+*}
+lemma step_v_wait_inv[elim_format]:
+    "\<And>t c. \<lbrakk>vt (V th cs # s); \<not> waiting (wq (V th cs # s)) t c; waiting (wq s) t c
+           \<rbrakk>
+          \<Longrightarrow> (next_th s th cs t \<and> cs = c)"
+proof -
+  fix t c 
+  assume vt: "vt (V th cs # s)"
+    and nw: "\<not> waiting (wq (V th cs # s)) t c"
+    and wt: "waiting (wq s) t c"
+  from vt interpret vt_v: valid_trace_e s "V th cs" 
+    by  (cases, unfold_locales, simp)
+  show "next_th s th cs t \<and> cs = c"
+  proof(cases "cs = c")
+    case False
+    with nw wt show ?thesis
+      by (auto simp:cs_waiting_def wq_def Let_def)
+  next
+    case True
+    from nw[folded True] wt[folded True]
+    have "next_th s th cs t"
+      apply (unfold next_th_def, auto simp:cs_waiting_def wq_def Let_def split:list.splits)
+    proof -
+      fix a list
+      assume t_in: "t \<in> set list"
+        and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
+        and eq_wq: "wq_fun (schs s) cs = a # list"
+      have " set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
+        show "distinct list \<and> set list = set list" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+          by auto
+      qed
+      with t_ni and t_in show "a = th" by auto
+    next
+      fix a list
+      assume t_in: "t \<in> set list"
+        and t_ni: "t \<notin> set (SOME q. distinct q \<and> set q = set list)"
+        and eq_wq: "wq_fun (schs s) cs = a # list"
+      have " set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from vt_v.wq_distinct[of cs] and eq_wq[folded wq_def]
+        show "distinct list \<and> set list = set list" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+          by auto
+      qed
+      with t_ni and t_in show "t = hd (SOME q. distinct q \<and> set q = set list)" by auto
+    next
+      fix a list
+      assume eq_wq: "wq_fun (schs s) cs = a # list"
+      from step_back_step[OF vt]
+      show "a = th"
+      proof(cases)
+        assume "holding s th cs"
+        with eq_wq show ?thesis
+          by (unfold s_holding_def wq_def, auto)
+      qed
+    qed
+    with True show ?thesis by simp
+  qed
+qed
+
+lemma step_v_not_wait[consumes 3]:
+  "\<lbrakk>vt (V th cs # s); next_th s th cs t; waiting (wq (V th cs # s)) t cs\<rbrakk> \<Longrightarrow> False"
+  by (unfold next_th_def cs_waiting_def wq_def, auto simp:Let_def)
+
+lemma step_v_release:
+  "\<lbrakk>vt (V th cs # s); holding (wq (V th cs # s)) th cs\<rbrakk> \<Longrightarrow> False"
+proof -
+  assume vt: "vt (V th cs # s)"
+    and hd: "holding (wq (V th cs # s)) th cs"
+  from vt interpret vt_v: valid_trace_e s "V th cs"
+    by (cases, unfold_locales, simp+)
+  from step_back_step [OF vt] and hd
+  show "False"
+  proof(cases)
+    assume "holding (wq (V th cs # s)) th cs" and "holding s th cs"
+    thus ?thesis
+      apply (unfold s_holding_def wq_def cs_holding_def)
+      apply (auto simp:Let_def split:list.splits)
+    proof -
+      fix list
+      assume eq_wq[folded wq_def]: 
+        "wq_fun (schs s) cs = hd (SOME q. distinct q \<and> set q = set list) # list"
+      and hd_in: "hd (SOME q. distinct q \<and> set q = set list)
+            \<in> set (SOME q. distinct q \<and> set q = set list)"
+      have "set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from vt_v.wq_distinct[of cs] and eq_wq
+        show "distinct list \<and> set list = set list" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set list \<Longrightarrow> set x = set list"
+          by auto
+      qed
+      moreover have "distinct  (hd (SOME q. distinct q \<and> set q = set list) # list)"
+      proof -
+        from vt_v.wq_distinct[of cs] and eq_wq
+        show ?thesis by auto
+      qed
+      moreover note eq_wq and hd_in
+      ultimately show "False" by auto
+    qed
+  qed
+qed
+
+lemma step_v_get_hold:
+  "\<And>th'. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) th' cs; next_th s th cs th'\<rbrakk> \<Longrightarrow> False"
+  apply (unfold cs_holding_def next_th_def wq_def,
+         auto simp:Let_def)
+proof -
+  fix rest
+  assume vt: "vt (V th cs # s)"
+    and eq_wq[folded wq_def]: " wq_fun (schs s) cs = th # rest"
+    and nrest: "rest \<noteq> []"
+    and ni: "hd (SOME q. distinct q \<and> set q = set rest)
+            \<notin> set (SOME q. distinct q \<and> set q = set rest)"
+  from vt interpret vt_v: valid_trace_e s "V th cs"
+    by (cases, unfold_locales, simp+)
+  have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+  proof(rule someI2)
+    from vt_v.wq_distinct[of cs] and eq_wq
+    show "distinct rest \<and> set rest = set rest" by auto
+  next
+    fix x assume "distinct x \<and> set x = set rest"
+    hence "set x = set rest" by auto
+    with nrest
+    show "x \<noteq> []" by (case_tac x, auto)
+  qed
+  with ni show "False" by auto
+qed
+
+lemma step_v_release_inv[elim_format]:
+"\<And>c t. \<lbrakk>vt (V th cs # s); \<not> holding (wq (V th cs # s)) t c; holding (wq s) t c\<rbrakk> \<Longrightarrow> 
+  c = cs \<and> t = th"
+  apply (unfold cs_holding_def wq_def, auto simp:Let_def split:if_splits list.splits)
+  proof -
+    fix a list
+    assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
+    from step_back_step [OF vt] show "a = th"
+    proof(cases)
+      assume "holding s th cs" with eq_wq
+      show ?thesis
+        by (unfold s_holding_def wq_def, auto)
+    qed
+  next
+    fix a list
+    assume vt: "vt (V th cs # s)" and eq_wq: "wq_fun (schs s) cs = a # list"
+    from step_back_step [OF vt] show "a = th"
+    proof(cases)
+      assume "holding s th cs" with eq_wq
+      show ?thesis
+        by (unfold s_holding_def wq_def, auto)
+    qed
+  qed
+
+lemma step_v_waiting_mono:
+  "\<And>t c. \<lbrakk>vt (V th cs # s); waiting (wq (V th cs # s)) t c\<rbrakk> \<Longrightarrow> waiting (wq s) t c"
+proof -
+  fix t c
+  let ?s' = "(V th cs # s)"
+  assume vt: "vt ?s'" 
+    and wt: "waiting (wq ?s') t c"
+  from vt interpret vt_v: valid_trace_e s "V th cs"
+    by (cases, unfold_locales, simp+)
+  show "waiting (wq s) t c"
+  proof(cases "c = cs")
+    case False
+    assume neq_cs: "c \<noteq> cs"
+    hence "waiting (wq ?s') t c = waiting (wq s) t c"
+      by (unfold cs_waiting_def wq_def, auto simp:Let_def)
+    with wt show ?thesis by simp
+  next
+    case True
+    with wt show ?thesis
+      apply (unfold cs_waiting_def wq_def, auto simp:Let_def split:list.splits)
+    proof -
+      fix a list
+      assume not_in: "t \<notin> set list"
+        and is_in: "t \<in> set (SOME q. distinct q \<and> set q = set list)"
+        and eq_wq: "wq_fun (schs s) cs = a # list"
+      have "set (SOME q. distinct q \<and> set q = set list) = set list"
+      proof(rule someI2)
+        from vt_v.wq_distinct [of cs]
+        and eq_wq[folded wq_def]
+        show "distinct list \<and> set list = set list" by auto
+      next
+        fix x assume "distinct x \<and> set x = set list"
+        thus "set x = set list" by auto
+      qed
+      with not_in is_in show "t = a" by auto
+    next
+      fix list
+      assume is_waiting: "waiting (wq (V th cs # s)) t cs"
+      and eq_wq: "wq_fun (schs s) cs = t # list"
+      hence "t \<in> set list"
+        apply (unfold wq_def, auto simp:Let_def cs_waiting_def)
+      proof -
+        assume " t \<in> set (SOME q. distinct q \<and> set q = set list)"
+        moreover have "\<dots> = set list" 
+        proof(rule someI2)
+          from vt_v.wq_distinct [of cs]
+            and eq_wq[folded wq_def]
+          show "distinct list \<and> set list = set list" by auto
+        next
+          fix x assume "distinct x \<and> set x = set list" 
+          thus "set x = set list" by auto
+        qed
+        ultimately show "t \<in> set list" by simp
+      qed
+      with eq_wq and vt_v.wq_distinct [of cs, unfolded wq_def]
+      show False by auto
+    qed
+  qed
+qed
+
+text {* (* ddd *) 
+  The following @{text "step_RAG_v"} lemma charaterizes how @{text "RAG"} is changed
+  with the happening of @{text "V"}-events:
+*}
+lemma step_RAG_v:
+fixes th::thread
+assumes vt:
+  "vt (V th cs#s)"
+shows "
+  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_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
+              step_v_get_hold step_v_release_inv)
+  apply (erule_tac step_v_not_wait, auto)
+  done
+
+text {* 
+  The following @{text "step_RAG_p"} lemma charaterizes how @{text "RAG"} is changed
+  with the happening of @{text "P"}-events:
+*}
+lemma step_RAG_p:
+  "vt (P th cs#s) \<Longrightarrow>
+  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(simp add:s_RAG_def wq_def cs_holding_def)
+  apply(auto)
+  done
+
+
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+  by (unfold s_RAG_def, auto)
+
+context valid_trace
+begin
+
+text {*
+  The following lemma shows that @{text "RAG"} is acyclic.
+  The overall structure is by induction on the formation of @{text "vt s"}
+  and then case analysis on event @{text "e"}, where the non-trivial cases 
+  for those for @{text "V"} and @{text "P"} events.
+*}
+lemma acyclic_RAG:
+  shows "acyclic (RAG s)"
+using vt
+proof(induct)
+  case (vt_cons s e)
+  interpret vt_s: valid_trace s using vt_cons(1)
+    by (unfold_locales, simp)
+  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:RAG_create_unchanged)
+  next
+    case (Exit th)
+    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_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)
+    from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
+    from step_back_step [OF vtt]
+    have "step s (V th cs)" .
+    thus ?thesis
+    proof(cases)
+      assume "holding s th cs"
+      hence th_in: "th \<in> set (wq s cs)" and
+        eq_hd: "th = hd (wq s cs)" unfolding s_holding_def wq_def by auto
+      then obtain rest where
+        eq_wq: "wq s cs = th#rest"
+        by (cases "wq s cs", auto)
+      show ?thesis
+      proof(cases "rest = []")
+        case False
+        let ?th' = "hd (SOME q. distinct q \<and> set q = set rest)"
+        from eq_wq False have eq_D: "?D = {(Cs cs, Th ?th')}"
+          by (unfold next_th_def, auto)
+        let ?E = "(?A - ?B - ?C)"
+        have "(Th ?th', Cs cs) \<notin> ?E\<^sup>*"
+        proof
+          assume "(Th ?th', Cs cs) \<in> ?E\<^sup>*"
+          hence " (Th ?th', Cs cs) \<in> ?E\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+          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 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_RAG_def s_waiting_def cs_waiting_def wq_def by simp
+          hence "cs' = cs"
+          proof(rule vt_s.waiting_unique)
+            from eq_wq vt_s.wq_distinct[of cs]
+            show "waiting s ?th' cs" 
+              apply (unfold s_waiting_def wq_def, auto)
+            proof -
+              assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq_fun (schs s) cs = th # rest"
+              have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+              next
+                fix x assume "distinct x \<and> set x = set rest"
+                with False show "x \<noteq> []" by auto
+              qed
+              hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
+                set (SOME q. distinct q \<and> set q = set rest)" by auto
+              moreover have "\<dots> = set rest" 
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" unfolding wq_def by auto
+              next
+                show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+              qed
+              moreover note hd_in
+              ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto
+            next
+              assume hd_in: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                and eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
+              have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" by auto
+              next
+                fix x assume "distinct x \<and> set x = set rest"
+                with False show "x \<noteq> []" by auto
+              qed
+              hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> 
+                set (SOME q. distinct q \<and> set q = set rest)" by auto
+              moreover have "\<dots> = set rest" 
+              proof(rule someI2)
+                from vt_s.wq_distinct[of cs] and eq_wq
+                show "distinct rest \<and> set rest = set rest" by auto
+              next
+                show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+              qed
+              moreover note hd_in
+              ultimately show False by auto
+            qed
+          qed
+          with th'_e eq_x have "(Th ?th', Cs cs) \<in> ?E" by simp
+          with False
+          show "False" by (auto simp: next_th_def eq_wq)
+        qed
+        with acyclic_insert[symmetric] and ac
+          and eq_de eq_D show ?thesis by auto
+      next
+        case True
+        with eq_wq
+        have eq_D: "?D = {}"
+          by (unfold next_th_def, auto)
+        with eq_de ac
+        show ?thesis by auto
+      qed 
+    qed
+  next
+    case (P th cs)
+    from P vt stp have vtt: "vt (P th cs#s)" by auto
+    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 =  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> (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> 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 =  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> (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> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+            by (ind_cases "step s (P th cs)", simp)
+        qed
+      qed
+      with acyclic_insert ih eq_r show ?thesis by auto
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (Set thread prio)
+      with ih
+      thm RAG_set_unchanged
+      show ?thesis by (simp add:RAG_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "acyclic (RAG ([]::state))"
+      by (auto simp: s_RAG_def cs_waiting_def 
+        cs_holding_def wq_def acyclic_def)
+qed
+
+
+lemma finite_RAG:
+  shows "finite (RAG s)"
+proof -
+  from vt show ?thesis
+  proof(induct)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s using vt_cons(1)
+      by (unfold_locales, simp)
+    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:RAG_create_unchanged)
+    next
+      case (Exit th)
+      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_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)
+      moreover from ih have ac: "finite (?A - ?B - ?C)" by simp
+      moreover have "finite ?D"
+      proof -
+        have "?D = {} \<or> (\<exists> a. ?D = {a})" 
+          by (unfold next_th_def, auto)
+        thus ?thesis
+        proof
+          assume h: "?D = {}"
+          show ?thesis by (unfold h, simp)
+        next
+          assume "\<exists> a. ?D = {a}"
+          thus ?thesis
+            by (metis finite.simps)
+        qed
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (P th cs)
+      from P vt stp have vtt: "vt (P th cs#s)" by auto
+      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 =  RAG s \<union> {(Cs cs, Th th)}" by simp
+        with True and ih show ?thesis by auto
+      next
+        case False
+        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:RAG_set_unchanged)
+    qed
+  next
+    case vt_nil
+    show "finite (RAG ([]::state))"
+      by (auto simp: s_RAG_def cs_waiting_def 
+                   cs_holding_def wq_def acyclic_def)
+  qed
+qed
+
+text {* Several useful lemmas *}
+
+lemma wf_dep_converse: 
+  shows "wf ((RAG s)^-1)"
+proof(rule finite_acyclic_wf_converse)
+  from finite_RAG 
+  show "finite (RAG s)" .
+next
+  from acyclic_RAG
+  show "acyclic (RAG s)" .
+qed
+
+end
+
+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> 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)
+
+context valid_trace
+begin
+
+lemma wq_threads: 
+  assumes h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+proof -
+ from vt and h show ?thesis
+  proof(induct arbitrary: th cs)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s
+      using vt_cons(1) by (unfold_locales, auto)
+    assume ih: "\<And>th cs. th \<in> set (wq s cs) \<Longrightarrow> th \<in> threads s"
+      and stp: "step s e"
+      and vt: "vt s"
+      and h: "th \<in> set (wq (e # s) cs)"
+    show ?case
+    proof(cases e)
+      case (Create th' prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    next
+      case (Exit th')
+      with stp ih h show ?thesis
+        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_RAG_def s_holding_def cs_holding_def)
+        done
+    next
+      case (V th' cs')
+      show ?thesis
+      proof(cases "cs' = cs")
+        case False
+        with h
+        show ?thesis
+          apply(unfold wq_def V, auto simp:Let_def V split:prod.splits, fold wq_def)
+          by (drule_tac ih, simp)
+      next
+        case True
+        from h
+        show ?thesis
+        proof(unfold V wq_def)
+          assume th_in: "th \<in> set (wq_fun (schs (V th' cs' # s)) cs)" (is "th \<in> set ?l")
+          show "th \<in> threads (V th' cs' # s)"
+          proof(cases "cs = cs'")
+            case False
+            hence "?l = wq_fun (schs s) cs" by (simp add:Let_def)
+            with th_in have " th \<in> set (wq s cs)" 
+              by (fold wq_def, simp)
+            from ih [OF this] show ?thesis by simp
+          next
+            case True
+            show ?thesis
+            proof(cases "wq_fun (schs s) cs'")
+              case Nil
+              with h V show ?thesis
+                apply (auto simp:wq_def Let_def split:if_splits)
+                by (fold wq_def, drule_tac ih, simp)
+            next
+              case (Cons a rest)
+              assume eq_wq: "wq_fun (schs s) cs' = a # rest"
+              with h V show ?thesis
+                apply (auto simp:Let_def wq_def split:if_splits)
+              proof -
+                assume th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                have "set (SOME q. distinct q \<and> set q = set rest) = set rest" 
+                proof(rule someI2)
+                  from vt_s.wq_distinct[of cs'] and eq_wq[folded wq_def]
+                  show "distinct rest \<and> set rest = set rest" by auto
+                next
+                  show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+                    by auto
+                qed
+                with eq_wq th_in have "th \<in> set (wq_fun (schs s) cs')" by auto
+                from ih[OF this[folded wq_def]] show "th \<in> threads s" .
+              next
+                assume th_in: "th \<in> set (wq_fun (schs s) cs)"
+                from ih[OF this[folded wq_def]]
+                show "th \<in> threads s" .
+              qed
+            qed
+          qed
+        qed
+      qed
+    next
+      case (P th' cs')
+      from h stp
+      show ?thesis
+        apply (unfold P wq_def)
+        apply (auto simp:Let_def split:if_splits, fold wq_def)
+        apply (auto intro:ih)
+        apply(ind_cases "step s (P th' cs')")
+        by (unfold runing_def readys_def, auto)
+    next
+      case (Set thread prio)
+      with ih h show ?thesis
+        by (auto simp:wq_def Let_def)
+    qed
+  next
+    case vt_nil
+    thus ?case by (auto simp:wq_def)
+  qed
+qed
+
+lemma range_in: "\<lbrakk>(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:
+  fixes th thread cs rest
+  assumes neq_th: "th \<noteq> thread"
+  and eq_wq: "wq s cs = thread#rest"
+  and not_in: "th \<notin>  set rest"
+  shows "(th \<in> readys (V thread cs#s)) = (th \<in> readys s)"
+proof -
+  from assms show ?thesis
+    apply (auto simp:readys_def)
+    apply(simp add:s_waiting_def[folded wq_def])
+    apply (erule_tac x = csa in allE)
+    apply (simp add:s_waiting_def wq_def Let_def split:if_splits)
+    apply (case_tac "csa = cs", simp)
+    apply (erule_tac x = cs in allE)
+    apply(auto simp add: s_waiting_def[folded wq_def] Let_def split: list.splits)
+    apply(auto simp add: wq_def)
+    apply (auto simp:s_waiting_def wq_def Let_def split:list.splits)
+    proof -
+       assume th_nin: "th \<notin> set rest"
+        and th_in: "th \<in> set (SOME q. distinct q \<and> set q = set rest)"
+        and eq_wq: "wq_fun (schs s) cs = thread # rest"
+      have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+      proof(rule someI2)
+        from wq_distinct[of cs, unfolded wq_def] and eq_wq[unfolded wq_def]
+        show "distinct rest \<and> set rest = set rest" by auto
+      next
+        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+      qed
+      with th_nin th_in show False by auto
+    qed
+qed
+
+text {* \noindent
+  The following lemmas shows that: starting from any node in @{text "RAG"}, 
+  by chasing out-going edges, it is always possible to reach a node representing a ready
+  thread. In this lemma, it is the @{text "th'"}.
+*}
+
+lemma chain_building:
+  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
+  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> (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 (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> 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> (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> (RAG s)^-1" by (auto simp:s_RAG_def)
+        show ?thesis
+        proof(cases "th' \<in> readys s")
+          case True
+          from True and th'_d show ?thesis by auto
+        next
+          case False
+          from th'_d and range_in  have "th' \<in> threads s" by auto
+          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> (RAG s)\<^sup>+" by auto
+          from th'_d and th''_in 
+          have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
+          with th''_r show ?thesis by auto
+        qed
+      qed
+    qed
+  qed
+qed
+
+text {* \noindent
+  The following is just an instance of @{text "chain_building"}.
+*}
+lemma th_chain_to_ready:
+  assumes th_in: "th \<in> threads 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 (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 this]
+  show ?thesis by auto
+qed
+
+end
+
+lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
+  by  (unfold s_waiting_def cs_waiting_def wq_def, auto)
+
+lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
+  by (unfold s_holding_def wq_def cs_holding_def, simp)
+
+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)
+
+context valid_trace
+begin
+
+lemma unique_RAG: "\<lbrakk>(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)
+
+end
+
+
+lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
+by (induct rule:trancl_induct, auto)
+
+context valid_trace
+begin
+
+lemma dchain_unique:
+  assumes th1_d: "(n, Th th1) \<in> (RAG s)^+"
+  and th1_r: "th1 \<in> readys 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_RAG 
+    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> (RAG s)\<^sup>+"
+      from trancl_split [OF this]
+      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_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_RAG_def wq_def s_waiting_def cs_waiting_def)
+      with th1_r show ?thesis by auto
+    next
+      assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
+      from trancl_split [OF this]
+      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_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_RAG_def s_waiting_def cs_waiting_def)
+      with th2_r show ?thesis by auto
+    qed
+  } thus ?thesis by auto
+qed
+
+end
+             
+
+lemma step_holdents_p_add:
+  fixes th cs s
+  assumes vt: "vt (P th cs#s)"
+  and "wq s cs = []"
+  shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
+proof -
+  from assms show ?thesis
+  unfolding  holdents_test step_RAG_p[OF vt] by (auto)
+qed
+
+lemma step_holdents_p_eq:
+  fixes th cs s
+  assumes vt: "vt (P th cs#s)"
+  and "wq s cs \<noteq> []"
+  shows "holdents (P th cs#s) th = holdents s th"
+proof -
+  from assms show ?thesis
+  unfolding  holdents_test step_RAG_p[OF vt] by auto
+qed
+
+
+lemma (in valid_trace) finite_holding :
+  shows "finite (holdents s th)"
+proof -
+  let ?F = "\<lambda> (x, y). the_cs x"
+  from finite_RAG 
+  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> 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) ` RAG s" by simp 
+    } thus ?thesis by auto
+  qed
+  ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
+qed
+
+lemma cntCS_v_dec: 
+  fixes s thread cs
+  assumes vtv: "vt (V thread cs#s)"
+  shows "(cntCS (V thread cs#s) thread + 1) = cntCS s thread"
+proof -
+  from vtv interpret vt_s: valid_trace s
+    by (cases, unfold_locales, simp)
+  from vtv interpret vt_v: valid_trace "V thread cs#s"
+     by (unfold_locales, simp)
+  from step_back_step[OF vtv]
+  have cs_in: "cs \<in> holdents s thread" 
+    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 vt_s.wq_distinct[of cs])
+    apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
+            auto simp:next_th_def)
+  proof -
+    fix rest
+    assume dst: "distinct (rest::thread list)"
+      and ne: "rest \<noteq> []"
+    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+    qed
+    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                     set (SOME q. distinct q \<and> set q = set rest)" by simp
+    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      fix x assume " distinct x \<and> set x = set rest" with ne
+      show "x \<noteq> []" by auto
+    qed
+    ultimately 
+    show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
+      by auto
+  next
+    fix rest
+    assume dst: "distinct (rest::thread list)"
+      and ne: "rest \<noteq> []"
+    and hd_ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+    moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto
+    qed
+    ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                     set (SOME q. distinct q \<and> set q = set rest)" by simp
+    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+    proof(rule someI2)
+      from dst show "distinct rest \<and> set rest = set rest" by auto
+    next
+      fix x assume " distinct x \<and> set x = set rest" with ne
+      show "x \<noteq> []" by auto
+    qed
+    ultimately show "False" by auto 
+  qed
+  ultimately 
+  have "holdents s thread = insert cs (holdents (V thread cs#s) thread)"
+    by auto
+  moreover have "card \<dots> = 
+                    Suc (card ((holdents (V thread cs#s) thread) - {cs}))"
+  proof(rule card_insert)
+    from vt_v.finite_holding
+    show " finite (holdents (V thread cs # s) thread)" .
+  qed
+  moreover from cs_not_in 
+  have "cs \<notin> (holdents (V thread cs#s) thread)" by auto
+  ultimately show ?thesis by (simp add:cntCS_def)
+qed 
+
+context valid_trace
+begin
+
+text {* (* ddd *) \noindent
+  The relationship between @{text "cntP"}, @{text "cntV"} and @{text "cntCS"} 
+  of one particular thread. 
+*} 
+
+lemma cnp_cnv_cncs:
+  shows "cntP s th = cntV s th + (if (th \<in> readys s \<or> th \<notin> threads s) 
+                                       then cntCS s th else cntCS s th + 1)"
+proof -
+  from vt show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e)
+    interpret vt_s: valid_trace s using vt_cons(1) by (unfold_locales, simp)
+    assume vt: "vt s"
+    and ih: "\<And>th. cntP s th  = cntV s th +
+               (if (th \<in> readys s \<or> th \<notin> threads s) then cntCS s th else cntCS s th + 1)"
+    and stp: "step s e"
+    from stp show ?case
+    proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+        and not_in: "thread \<notin> threads s"
+      show ?thesis
+      proof -
+        { fix cs 
+          assume "thread \<in> set (wq s cs)"
+          from vt_s.wq_threads [OF this] have "thread \<in> threads s" .
+          with not_in have "False" by simp
+        } with eq_e have eq_readys: "readys (e#s) = readys s \<union> {thread}"
+          by (auto simp:readys_def threads.simps s_waiting_def 
+            wq_def cs_waiting_def Let_def)
+        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+        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: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)) = 
+                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+            by (simp add:threads.simps)
+          with eq_cnp eq_cnv eq_cncs ih not_in
+          have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          with not_in ih have " cntP s th  = cntV s th + cntCS s th" by simp
+          moreover from eq_th and eq_readys have "th \<in> readys (e#s)" by simp
+          moreover note eq_cnp eq_cnv eq_cncs
+          ultimately have ?thesis by auto
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread" 
+      and is_runing: "thread \<in> runing s"
+      and no_hold: "holdents s thread = {}"
+      from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+      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:RAG_exit_unchanged eq_e)
+      { assume "th \<noteq> thread"
+        with eq_e
+        have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+          (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+          apply (simp add:threads.simps readys_def)
+          apply (subst s_waiting_def)
+          apply (simp add:Let_def)
+          apply (subst s_waiting_def, simp)
+          done
+        with eq_cnp eq_cnv eq_cncs ih
+        have ?thesis by simp
+      } moreover {
+        assume eq_th: "th = thread"
+        with ih is_runing have " cntP s th = cntV s th + cntCS s th" 
+          by (simp add:runing_def)
+        moreover from eq_th eq_e have "th \<notin> threads (e#s)"
+          by simp
+        moreover note eq_cnp eq_cnv eq_cncs
+        ultimately have ?thesis by auto
+      } ultimately show ?thesis by blast
+    next
+      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> (RAG s)\<^sup>+"
+      from thread_P vt stp ih  have vtp: "vt (P thread cs#s)" by auto
+      then interpret vt_p: valid_trace "(P thread cs#s)"
+        by (unfold_locales, simp)
+      show ?thesis 
+      proof -
+        { have hh: "\<And> A B C. (B = C) \<Longrightarrow> (A \<and> B) = (A \<and> C)" by blast
+          assume neq_th: "th \<noteq> thread"
+          with eq_e
+          have eq_readys: "(th \<in> readys (e#s)) = (th \<in> readys (s))"
+            apply (simp add:readys_def s_waiting_def wq_def Let_def)
+            apply (rule_tac hh)
+             apply (intro iffI allI, clarify)
+            apply (erule_tac x = csa in allE, auto)
+            apply (subgoal_tac "wq_fun (schs s) cs \<noteq> []", auto)
+            apply (erule_tac x = cs in allE, auto)
+            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_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"
+            by (simp add:cntV_def count_def)
+          moreover from eq_e neq_th have "threads (e#s) = threads s" by simp
+          moreover note ih [of th] 
+          ultimately have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          have ?thesis
+          proof -
+            from eq_e eq_th have eq_cnp: "cntP (e # s) th  = 1 + (cntP s th)" 
+              by (simp add:cntP_def count_def)
+            from eq_e eq_th have eq_cnv: "cntV (e#s) th = cntV s th"
+              by (simp add:cntV_def count_def)
+            show ?thesis
+            proof (cases "wq s cs = []")
+              case True
+              with is_runing
+              have "th \<in> readys (e#s)"
+                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> 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 vt_s.finite_holding [of thread]
+                    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> RAG s}"
+                      with no_dep show False by auto
+                    qed
+                    thus ?thesis by auto
+                  qed
+                  ultimately show ?thesis by auto
+                qed
+                thus ?thesis
+                  apply (unfold eq_e eq_th cntCS_def)
+                  apply (simp add: holdents_test)
+                  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)
+              moreover note eq_cnp eq_cnv ih [of th]
+              ultimately show ?thesis by auto
+            next
+              case False
+              have eq_wq: "wq (e#s) cs = wq s cs @ [th]"
+                    by (unfold eq_th eq_e wq_def, auto simp:Let_def)
+              have "th \<notin> readys (e#s)"
+              proof
+                assume "th \<in> readys (e#s)"
+                hence "\<forall>cs. \<not> waiting (e # s) th cs" by (simp add:readys_def)
+                from this[rule_format, of cs] have " \<not> waiting (e # s) th cs" .
+                hence "th \<in> set (wq (e#s) cs) \<Longrightarrow> th = hd (wq (e#s) cs)" 
+                  by (simp add:s_waiting_def wq_def)
+                moreover from eq_wq have "th \<in> set (wq (e#s) cs)" by auto
+                ultimately have "th = hd (wq (e#s) cs)" by blast
+                with eq_wq have "th = hd (wq s cs @ [th])" by simp
+                hence "th = hd (wq s cs)" using False by auto
+                with False eq_wq vt_p.wq_distinct [of cs]
+                show False by (fold eq_e, auto)
+              qed
+              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_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"
+                by (simp add:runing_def eq_th)
+              ultimately show ?thesis by auto
+            qed
+          qed
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_V thread cs)
+      from assms vt stp ih thread_V have vtv: "vt (V thread cs # s)" by auto
+      then interpret vt_v: valid_trace "(V thread cs # s)" by (unfold_locales, simp)
+      assume eq_e: "e = V thread cs"
+        and is_runing: "thread \<in> runing s"
+        and hold: "holding s thread cs"
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+      have eq_threads: "threads (e#s) = threads s" by (simp add: eq_e)
+      have eq_set: "set (SOME q. distinct q \<and> set q = set rest) = set rest"
+      proof(rule someI2)
+        from vt_v.wq_distinct[of cs] and eq_wq
+        show "distinct rest \<and> set rest = set rest"
+          by (metis distinct.simps(2) vt_s.wq_distinct)
+      next
+        show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest"
+          by auto
+      qed
+      show ?thesis
+      proof -
+        { assume eq_th: "th = thread"
+          from eq_th have eq_cnp: "cntP (e # s) th = cntP s th"
+            by (unfold eq_e, simp add:cntP_def count_def)
+          moreover from eq_th have eq_cnv: "cntV (e#s) th = 1 + cntV s th"
+            by (unfold eq_e, simp add:cntV_def count_def)
+          moreover from cntCS_v_dec [OF vtv] 
+          have "cntCS (e # s) thread + 1 = cntCS s thread"
+            by (simp add:eq_e)
+          moreover from is_runing have rd_before: "thread \<in> readys s"
+            by (unfold runing_def, simp)
+          moreover have "thread \<in> readys (e # s)"
+          proof -
+            from is_runing
+            have "thread \<in> threads (e#s)" 
+              by (unfold eq_e, auto simp:runing_def readys_def)
+            moreover have "\<forall> cs1. \<not> waiting (e#s) thread cs1"
+            proof
+              fix cs1
+              { assume eq_cs: "cs1 = cs" 
+                have "\<not> waiting (e # s) thread cs1"
+                proof -
+                  from eq_wq
+                  have "thread \<notin> set (wq (e#s) cs1)"
+                    apply(unfold eq_e wq_def eq_cs s_holding_def)
+                    apply (auto simp:Let_def)
+                  proof -
+                    assume "thread \<in> set (SOME q. distinct q \<and> set q = set rest)"
+                    with eq_set have "thread \<in> set rest" by simp
+                    with vt_v.wq_distinct[of cs]
+                    and eq_wq show False
+                        by (metis distinct.simps(2) vt_s.wq_distinct)
+                  qed
+                  thus ?thesis by (simp add:wq_def s_waiting_def)
+                qed
+              } moreover {
+                assume neq_cs: "cs1 \<noteq> cs"
+                  have "\<not> waiting (e # s) thread cs1" 
+                  proof -
+                    from wq_v_neq [OF neq_cs[symmetric]]
+                    have "wq (V thread cs # s) cs1 = wq s cs1" .
+                    moreover have "\<not> waiting s thread cs1" 
+                    proof -
+                      from runing_ready and is_runing
+                      have "thread \<in> readys s" by auto
+                      thus ?thesis by (simp add:readys_def)
+                    qed
+                    ultimately show ?thesis 
+                      by (auto simp:wq_def s_waiting_def eq_e)
+                  qed
+              } ultimately show "\<not> waiting (e # s) thread cs1" by blast
+            qed
+            ultimately show ?thesis by (simp add:readys_def)
+          qed
+          moreover note eq_th ih
+          ultimately have ?thesis by auto
+        } moreover {
+          assume neq_th: "th \<noteq> thread"
+          from neq_th eq_e have eq_cnp: "cntP (e # s) th = cntP s th" 
+            by (simp add:cntP_def count_def)
+          from neq_th eq_e have eq_cnv: "cntV (e # s) th = cntV s th" 
+            by (simp add:cntV_def count_def)
+          have ?thesis
+          proof(cases "th \<in> set rest")
+            case False
+            have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+              apply (insert step_back_vt[OF vtv])
+              by (simp add: False eq_e eq_wq neq_th vt_s.readys_v_eq)
+            moreover have "cntCS (e#s) th = cntCS s th"
+              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> 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> RAG s"
+                    apply (unfold next_th_def, auto)
+                  proof -
+                    assume ne: "rest \<noteq> []"
+                      and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest"
+                      and eq_wq: "wq s cs = thread # rest"
+                    from eq_set ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> 
+                                  set (SOME q. distinct q \<and> set q = set rest)
+                                  " by simp
+                    moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []"
+                    proof(rule someI2)
+                      from vt_s.wq_distinct[ of cs] and eq_wq
+                      show "distinct rest \<and> set rest = set rest" by auto
+                    next
+                      fix x assume "distinct x \<and> set x = set rest"
+                      with ne show "x \<noteq> []" by auto
+                    qed
+                    ultimately show 
+                      "(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> 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
+          next
+            case True
+            assume th_in: "th \<in> set rest"
+            show ?thesis
+            proof(cases "next_th s thread cs th")
+              case False
+              with eq_wq and th_in have 
+                neq_hd: "th \<noteq> hd (SOME q. distinct q \<and> set q = set rest)" (is "th \<noteq> hd ?rest")
+                by (auto simp:next_th_def)
+              have "(th \<in> readys (e # s)) = (th \<in> readys s)"
+              proof -
+                from eq_wq and th_in
+                have "\<not> th \<in> readys s"
+                  apply (auto simp:readys_def s_waiting_def)
+                  apply (rule_tac x = cs in exI, auto)
+                  by (insert vt_s.wq_distinct[of cs], auto simp add: wq_def)
+                moreover 
+                from eq_wq and th_in and neq_hd
+                have "\<not> (th \<in> readys (e # s))"
+                  apply (auto simp:readys_def s_waiting_def eq_e wq_def Let_def split:list.splits)
+                  by (rule_tac x = cs in exI, auto simp:eq_set)
+                ultimately show ?thesis by auto
+              qed
+              moreover have "cntCS (e#s) th = cntCS s th" 
+              proof -
+                from eq_wq and  th_in and neq_hd
+                have "(holdents (e # s) th) = (holdents s th)"
+                  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 vt_s.wq_distinct[of cs], auto simp:wq_def)
+                thus ?thesis by (simp add:cntCS_def)
+              qed
+              moreover note ih eq_cnp eq_cnv eq_threads
+              ultimately show ?thesis by auto
+            next
+              case True
+              let ?rest = " (SOME q. distinct q \<and> set q = set rest)"
+              let ?t = "hd ?rest"
+              from True eq_wq th_in neq_th
+              have "th \<in> readys (e # s)"
+                apply (auto simp:eq_e readys_def s_waiting_def wq_def
+                        Let_def next_th_def)
+              proof -
+                assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                show "?t \<in> threads s"
+                proof(rule vt_s.wq_threads)
+                  from eq_wq and t_in
+                  show "?t \<in> set (wq s cs)" by (auto simp:wq_def)
+                qed
+              next
+                fix csa
+                assume eq_wq: "wq_fun (schs s) cs = thread # rest"
+                  and t_in: "?t \<in> set rest"
+                  and neq_cs: "csa \<noteq> cs"
+                  and t_in': "?t \<in>  set (wq_fun (schs s) csa)"
+                show "?t = hd (wq_fun (schs s) csa)"
+                proof -
+                  { assume neq_hd': "?t \<noteq> hd (wq_fun (schs s) csa)"
+                    from vt_s.wq_distinct[of cs] and 
+                    eq_wq[folded wq_def] and t_in eq_wq
+                    have "?t \<noteq> thread" by auto
+                    with eq_wq and t_in
+                    have w1: "waiting s ?t cs"
+                      by (auto simp:s_waiting_def wq_def)
+                    from t_in' neq_hd'
+                    have w2: "waiting s ?t csa"
+                      by (auto simp:s_waiting_def wq_def)
+                    from vt_s.waiting_unique[OF w1 w2]
+                    and neq_cs have "False" by auto
+                  } thus ?thesis by auto
+                qed
+              qed
+              moreover have "cntP s th = cntV s th + cntCS s th + 1"
+              proof -
+                have "th \<notin> readys s" 
+                proof -
+                  from True eq_wq neq_th th_in
+                  show ?thesis
+                    apply (unfold readys_def s_waiting_def, auto)
+                    by (rule_tac x = cs in exI, auto simp add: wq_def)
+                qed
+                moreover have "th \<in> threads s"
+                proof -
+                  from th_in eq_wq
+                  have "th \<in> set (wq s cs)" by simp
+                  from vt_s.wq_threads [OF this] 
+                  show ?thesis .
+                qed
+                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_RAG_v[OF vtv], auto)
+              proof -
+                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) ` RAG s)" 
+                      apply (auto simp:image_def)
+                      by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
+                    with vt_s.finite_RAG
+                    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> RAG s}"
+                    proof
+                      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_RAG_def cs_holding_def)
+                    qed
+                  qed
+                  finally show ?thesis .
+                qed
+              qed
+              moreover note eq_cnp eq_cnv
+              ultimately show ?thesis by simp
+            qed
+          qed
+        } ultimately show ?thesis by blast
+      qed
+    next
+      case (thread_set thread prio)
+      assume eq_e: "e = Set thread prio"
+        and is_runing: "thread \<in> runing s"
+      show ?thesis
+      proof -
+        from eq_e have eq_cnp: "cntP (e#s) th = cntP s th" by (simp add:cntP_def count_def)
+        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: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)
+        { assume "th \<noteq> thread"
+          with eq_readys eq_e
+          have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) = 
+                      (th \<in> readys (s) \<or> th \<notin> threads (s))" 
+            by (simp add:threads.simps)
+          with eq_cnp eq_cnv eq_cncs ih is_runing
+          have ?thesis by simp
+        } moreover {
+          assume eq_th: "th = thread"
+          with is_runing ih have " cntP s th  = cntV s th + cntCS s th" 
+            by (unfold runing_def, auto)
+          moreover from eq_th and eq_readys is_runing have "th \<in> readys (e#s)"
+            by (simp add:runing_def)
+          moreover note eq_cnp eq_cnv eq_cncs
+          ultimately have ?thesis by auto
+        } ultimately show ?thesis by blast
+      qed   
+    qed
+  next
+    case vt_nil
+    show ?case 
+      by (unfold cntP_def cntV_def cntCS_def, 
+        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+  qed
+qed
+
+lemma not_thread_cncs:
+  assumes not_in: "th \<notin> threads s" 
+  shows "cntCS s th = 0"
+proof -
+  from vt not_in show ?thesis
+  proof(induct arbitrary:th)
+    case (vt_cons s e th)
+    interpret vt_s: valid_trace s using vt_cons(1)
+       by (unfold_locales, simp)
+    assume vt: "vt s"
+      and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> cntCS s th = 0"
+      and stp: "step s e"
+      and not_in: "th \<notin> threads (e # s)"
+    from stp show ?case
+    proof(cases)
+      case (thread_create thread prio)
+      assume eq_e: "e = Create thread prio"
+        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:RAG_create_unchanged)
+      moreover have "th \<notin> threads s" 
+      proof -
+        from not_in eq_e show ?thesis by simp
+      qed
+      moreover note ih ultimately show ?thesis by auto
+    next
+      case (thread_exit thread)
+      assume eq_e: "e = Exit thread"
+      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:RAG_exit_unchanged)
+      show ?thesis
+      proof(cases "th = thread")
+        case True
+        have "cntCS s th = 0" by (unfold cntCS_def, auto simp:nh True)
+        with eq_cns show ?thesis by simp
+      next
+        case False
+        with not_in and eq_e
+        have "th \<notin> threads s" by simp
+        from ih[OF this] and eq_cns show ?thesis by simp
+      qed
+    next
+      case (thread_P thread cs)
+      assume eq_e: "e = P thread cs"
+      and is_runing: "thread \<in> runing s"
+      from assms thread_P ih vt stp thread_P have vtp: "vt (P thread cs#s)" by auto
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      hence "cntCS (e # s) th  = cntCS s th "
+        apply (unfold cntCS_def holdents_test eq_e)
+        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
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (thread_V thread cs)
+      assume eq_e: "e = V thread cs"
+        and is_runing: "thread \<in> runing s"
+        and hold: "holding s thread cs"
+      have neq_th: "th \<noteq> thread" 
+      proof -
+        from not_in eq_e have "th \<notin> threads s" by simp
+        moreover from is_runing have "thread \<in> threads s"
+          by (simp add:runing_def readys_def)
+        ultimately show ?thesis by auto
+      qed
+      from assms thread_V vt stp ih 
+      have vtv: "vt (V thread cs#s)" by auto
+      then interpret vt_v: valid_trace "(V thread cs#s)"
+        by (unfold_locales, simp)
+      from hold obtain rest 
+        where eq_wq: "wq s cs = thread # rest"
+        by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
+      from not_in eq_e eq_wq
+      have "\<not> next_th s thread cs th"
+        apply (auto simp:next_th_def)
+      proof -
+        assume ne: "rest \<noteq> []"
+          and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
+        have "?t \<in> set rest"
+        proof(rule someI2)
+          from vt_v.wq_distinct[of cs] and eq_wq
+          show "distinct rest \<and> set rest = set rest"
+            by (metis distinct.simps(2) vt_s.wq_distinct) 
+        next
+          fix x assume "distinct x \<and> set x = set rest" with ne
+          show "hd x \<in> set rest" by (cases x, auto)
+        qed
+        with eq_wq have "?t \<in> set (wq s cs)" by simp
+        from vt_s.wq_threads[OF this] and ni
+        show False
+          using `hd (SOME q. distinct q \<and> set q = set rest) \<in> set (wq s cs)` 
+            ni vt_s.wq_threads by blast 
+      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_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
+      qed
+      ultimately show ?thesis by simp
+    next
+      case (thread_set thread prio)
+      print_facts
+      assume eq_e: "e = Set thread prio"
+        and is_runing: "thread \<in> runing s"
+      from not_in and eq_e have "th \<notin> threads s" by auto
+      from ih [OF this] and eq_e
+      show ?thesis 
+        apply (unfold eq_e cntCS_def holdents_test)
+        by (simp add:RAG_set_unchanged)
+    qed
+    next
+      case vt_nil
+      show ?case
+      by (unfold cntCS_def, 
+        auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
+  qed
+qed
+
+end
+
+lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
+  by (auto simp:s_waiting_def cs_waiting_def wq_def)
+
+context valid_trace
+begin
+
+lemma dm_RAG_threads:
+  assumes in_dom: "(Th th) \<in> Domain (RAG s)"
+  shows "th \<in> threads s"
+proof -
+  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_RAG_def, auto simp:cs_waiting_def)
+  from wq_threads [OF this] show ?thesis .
+qed
+
+end
+
+lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
+unfolding cp_def wq_def
+apply(induct s rule: schs.induct)
+thm cpreced_initial
+apply(simp add: Let_def cpreced_initial)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+
+context valid_trace
+begin
+
+lemma runing_unique:
+  assumes runing_1: "th1 \<in> runing s"
+  and runing_2: "th2 \<in> runing s"
+  shows "th1 = th2"
+proof -
+  from runing_1 and runing_2 have "cp s th1 = cp s th2"
+    unfolding runing_def
+    apply(simp)
+    done
+  hence eq_max: "Max ((\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1)) =
+                 Max ((\<lambda>th. preced th s) ` ({th2} \<union> dependants (wq s) th2))"
+    (is "Max (?f ` ?A) = Max (?f ` ?B)")
+    unfolding cp_eq_cpreced 
+    unfolding cpreced_def .
+  obtain th1' where th1_in: "th1' \<in> ?A" and eq_f_th1: "?f th1' = Max (?f ` ?A)"
+  proof -
+    have h1: "finite (?f ` ?A)"
+    proof -
+      have "finite ?A" 
+      proof -
+        have "finite (dependants (wq s) th1)"
+        proof-
+          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> (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_RAG have "finite (RAG s)" .
+              hence "finite ((RAG (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_RAG_def cs_RAG_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependants_def)
+        qed
+        thus ?thesis by simp
+      qed
+      thus ?thesis by auto
+    qed
+    moreover have h2: "(?f ` ?A) \<noteq> {}"
+    proof -
+      have "?A \<noteq> {}" by simp
+      thus ?thesis by simp
+    qed
+    from Max_in [OF h1 h2]
+    have "Max (?f ` ?A) \<in> (?f ` ?A)" .
+    thus ?thesis 
+      thm cpreced_def
+      unfolding cpreced_def[symmetric] 
+      unfolding cp_eq_cpreced[symmetric] 
+      unfolding cpreced_def 
+      using that[intro] by (auto)
+  qed
+  obtain th2' where th2_in: "th2' \<in> ?B" and eq_f_th2: "?f th2' = Max (?f ` ?B)"
+  proof -
+    have h1: "finite (?f ` ?B)"
+    proof -
+      have "finite ?B" 
+      proof -
+        have "finite (dependants (wq s) th2)"
+        proof-
+          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> (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_RAG have "finite (RAG s)" .
+              hence "finite ((RAG (wq s))\<^sup>+)"
+                apply (unfold finite_trancl)
+                by (auto simp: s_RAG_def cs_RAG_def wq_def)
+              thus ?thesis by auto
+            qed
+            ultimately show ?thesis by (auto intro:finite_subset)
+          qed
+          thus ?thesis by (simp add:cs_dependants_def)
+        qed
+        thus ?thesis by simp
+      qed
+      thus ?thesis by auto
+    qed
+    moreover have h2: "(?f ` ?B) \<noteq> {}"
+    proof -
+      have "?B \<noteq> {}" by simp
+      thus ?thesis by simp
+    qed
+    from Max_in [OF h1 h2]
+    have "Max (?f ` ?B) \<in> (?f ` ?B)" .
+    thus ?thesis by (auto intro:that)
+  qed
+  from eq_f_th1 eq_f_th2 eq_max 
+  have eq_preced: "preced th1' s = preced th2' s" by auto
+  hence eq_th12: "th1' = th2'"
+  proof (rule preced_unique)
+    from th1_in have "th1' = th1 \<or> (th1' \<in> dependants (wq s) th1)" by simp
+    thus "th1' \<in> threads s"
+    proof
+      assume "th1' \<in> dependants (wq s) th1"
+      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 (RAG s)" by (simp add:trancl_domain)
+      from dm_RAG_threads[OF this] show ?thesis .
+    next
+      assume "th1' = th1"
+      with runing_1 show ?thesis
+        by (unfold runing_def readys_def, auto)
+    qed
+  next
+    from th2_in have "th2' = th2 \<or> (th2' \<in> dependants (wq s) th2)" by simp
+    thus "th2' \<in> threads s"
+    proof
+      assume "th2' \<in> dependants (wq s) th2"
+      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 (RAG s)" by (simp add:trancl_domain)
+      from dm_RAG_threads[OF this] show ?thesis .
+    next
+      assume "th2' = th2"
+      with runing_2 show ?thesis
+        by (unfold runing_def readys_def, auto)
+    qed
+  qed
+  from th1_in have "th1' = th1 \<or> th1' \<in> dependants (wq s) th1" by simp
+  thus ?thesis
+  proof
+    assume eq_th': "th1' = th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+    thus ?thesis
+    proof
+      assume "th2' = th2" thus ?thesis using eq_th' eq_th12 by simp
+    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> (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 (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> RAG s" by simp
+      with runing_1 have "False"
+        apply (unfold runing_def readys_def s_RAG_def)
+        by (auto simp:eq_waiting)
+      thus ?thesis by simp
+    qed
+  next
+    assume th1'_in: "th1' \<in> dependants (wq s) th1"
+    from th2_in have "th2' = th2 \<or> th2' \<in> dependants (wq s) th2" by simp
+    thus ?thesis 
+    proof
+      assume "th2' = th2"
+      with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
+      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 (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> RAG s" by simp
+      with runing_2 have "False"
+        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> (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 h1 _ h2, symmetric])
+        from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
+        from runing_2 show "th2 \<in> readys s" by (simp add:runing_def) 
+      qed
+    qed
+  qed
+qed
+
+
+lemma "card (runing s) \<le> 1"
+apply(subgoal_tac "finite (runing s)")
+prefer 2
+apply (metis finite_nat_set_iff_bounded lessI runing_unique)
+apply(rule ccontr)
+apply(simp)
+apply(case_tac "Suc (Suc 0) \<le> card (runing s)")
+apply(subst (asm) card_le_Suc_iff)
+apply(simp)
+apply(auto)[1]
+apply (metis insertCI runing_unique)
+apply(auto) 
+done
+
+end
+
+
+lemma create_pre:
+  assumes stp: "step s e"
+  and not_in: "th \<notin> threads s"
+  and is_in: "th \<in> threads (e#s)"
+  obtains prio where "e = Create th prio"
+proof -
+  from assms  
+  show ?thesis
+  proof(cases)
+    case (thread_create thread prio)
+    with is_in not_in have "e = Create th prio" by simp
+    from that[OF this] show ?thesis .
+  next
+    case (thread_exit thread)
+    with assms show ?thesis by (auto intro!:that)
+  next
+    case (thread_P thread)
+    with assms show ?thesis by (auto intro!:that)
+  next
+    case (thread_V thread)
+    with assms show ?thesis by (auto intro!:that)
+  next 
+    case (thread_set thread)
+    with assms show ?thesis by (auto intro!:that)
+  qed
+qed
+
+lemma length_down_to_in: 
+  assumes le_ij: "i \<le> j"
+    and le_js: "j \<le> length s"
+  shows "length (down_to j i s) = j - i"
+proof -
+  have "length (down_to j i s) = length (from_to i j (rev s))"
+    by (unfold down_to_def, auto)
+  also have "\<dots> = j - i"
+  proof(rule length_from_to_in[OF le_ij])
+    from le_js show "j \<le> length (rev s)" by simp
+  qed
+  finally show ?thesis .
+qed
+
+
+lemma moment_head: 
+  assumes le_it: "Suc i \<le> length t"
+  obtains e where "moment (Suc i) t = e#moment i t"
+proof -
+  have "i \<le> Suc i" by simp
+  from length_down_to_in [OF this le_it]
+  have "length (down_to (Suc i) i t) = 1" by auto
+  then obtain e where "down_to (Suc i) i t = [e]"
+    apply (cases "(down_to (Suc i) i t)") by auto
+  moreover have "down_to (Suc i) 0 t = down_to (Suc i) i t @ down_to i 0 t"
+    by (rule down_to_conc[symmetric], auto)
+  ultimately have eq_me: "moment (Suc i) t = e#(moment i t)"
+    by (auto simp:down_to_moment)
+  from that [OF this] show ?thesis .
+qed
+
+context valid_trace
+begin
+
+lemma cnp_cnv_eq:
+  assumes "th \<notin> threads s"
+  shows "cntP s th = cntV s th"
+  using assms
+  using cnp_cnv_cncs not_thread_cncs by auto
+
+end
+
+
+lemma eq_RAG: 
+  "RAG (wq s) = RAG s"
+by (unfold cs_RAG_def s_RAG_def, auto)
+
+context valid_trace
+begin
+
+lemma count_eq_dependants:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "dependants (wq s) th = {}"
+proof -
+  from cnp_cnv_cncs and eq_pv
+  have "cntCS s th = 0" 
+    by (auto split:if_splits)
+  moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
+  proof -
+    from finite_holding[of th] show ?thesis
+      by (simp add:holdents_test)
+  qed
+  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> (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> RAG (wq s)"
+        thus "False" by (auto simp:cs_RAG_def)
+      next
+        fix c
+        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> (RAG (wq s))\<^sup>+} = {}" by auto
+  qed
+qed
+
+lemma dependants_threads:
+  shows "dependants (wq s) th \<subseteq> threads s"
+proof
+  { fix th th'
+    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> (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_RAG_threads[OF 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> (RAG (wq s))\<^sup>+}"
+    by (unfold cs_dependants_def, simp)
+  from hh [OF this] show "th1 \<in> threads s" .
+qed
+
+lemma finite_threads:
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+end
+
+lemma Max_f_mono:
+  assumes seq: "A \<subseteq> B"
+  and np: "A \<noteq> {}"
+  and fnt: "finite B"
+  shows "Max (f ` A) \<le> Max (f ` B)"
+proof(rule Max_mono)
+  from seq show "f ` A \<subseteq> f ` B" by auto
+next
+  from np show "f ` A \<noteq> {}" by auto
+next
+  from fnt and seq show "finite (f ` B)" by auto
+qed
+
+context valid_trace
+begin
+
+lemma cp_le:
+  assumes 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> (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> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
+  next
+    from finite_threads
+    show "finite (threads s)" .
+  next
+    from th_in
+    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_RAG_threads)
+      apply (unfold trancl_domain [of "RAG s", symmetric])
+      by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
+  qed
+qed
+
+lemma le_cp:
+  shows "preced th s \<le> cp s th"
+proof(unfold cp_eq_cpreced preced_def cpreced_def, simp)
+  show "Prc (priority th s) (last_set th s)
+    \<le> Max (insert (Prc (priority th s) (last_set th s))
+            ((\<lambda>th. Prc (priority th s) (last_set th s)) ` dependants (wq s) th))"
+    (is "?l \<le> Max (insert ?l ?A)")
+  proof(cases "?A = {}")
+    case False
+    have "finite ?A" (is "finite (?f ` ?B)")
+    proof -
+      have "finite ?B" 
+      proof-
+        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> (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_RAG have "finite (RAG s)" .
+            hence "finite ((RAG (wq s))\<^sup>+)"
+              apply (unfold finite_trancl)
+              by (auto simp: s_RAG_def cs_RAG_def wq_def)
+            thus ?thesis by auto
+          qed
+          ultimately show ?thesis by (auto intro:finite_subset)
+        qed
+        thus ?thesis by (simp add:cs_dependants_def)
+      qed
+      thus ?thesis by simp
+    qed
+    from Max_insert [OF this False, of ?l] show ?thesis by auto
+  next
+    case True
+    thus ?thesis by auto
+  qed
+qed
+
+lemma max_cp_eq: 
+  shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
+  (is "?l = ?r")
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  have "?l \<in> ((cp s) ` threads s)"
+  proof(rule Max_in)
+    from finite_threads
+    show "finite (cp s ` threads s)" by auto
+  next
+    from False show "cp s ` threads s \<noteq> {}" by auto
+  qed
+  then obtain th 
+    where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
+  have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
+  moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
+  proof -
+    have "?r \<in> (?f ` ?A)"
+    proof(rule Max_in)
+      from finite_threads
+      show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
+    next
+      from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
+    qed
+    then obtain th' where 
+      th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
+    from le_cp [of th']  eq_r
+    have "?r \<le> cp s th'" by auto
+    moreover have "\<dots> \<le> cp s th"
+    proof(fold eq_l)
+      show " cp s th' \<le> Max (cp s ` threads s)"
+      proof(rule Max_ge)
+        from th_in' show "cp s th' \<in> cp s ` threads s"
+          by auto
+      next
+        from finite_threads
+        show "finite (cp s ` threads s)" by auto
+      qed
+    qed
+    ultimately show ?thesis by auto
+  qed
+  ultimately show ?thesis using eq_l by auto
+qed
+
+lemma max_cp_readys_threads_pre:
+  assumes np: "threads s \<noteq> {}"
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(unfold max_cp_eq)
+  show "Max (cp s ` readys s) = Max ((\<lambda>th. preced th s) ` threads s)"
+  proof -
+    let ?p = "Max ((\<lambda>th. preced th s) ` threads s)" 
+    let ?f = "(\<lambda>th. preced th s)"
+    have "?p \<in> ((\<lambda>th. preced th s) ` threads s)"
+    proof(rule Max_in)
+      from finite_threads show "finite (?f ` threads s)" by simp
+    next
+      from np show "?f ` threads s \<noteq> {}" by simp
+    qed
+    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 tm_in]
+    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> (RAG s)\<^sup>+ "
+      then obtain th' where th'_in: "th' \<in> readys s" 
+        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 finite_threads
+        show "finite ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th'))" 
+          by (auto intro:finite_subset)
+      next
+        fix p assume p_in: "p \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+        from tm_max have " preced tm s = Max ((\<lambda>th. preced th s) ` threads s)" .
+        moreover have "p \<le> \<dots>"
+        proof(rule Max_ge)
+          from finite_threads
+          show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        next
+          from p_in and th'_in and dependants_threads[of th']
+          show "p \<in> (\<lambda>th. preced th s) ` threads s"
+            by (auto simp:readys_def)
+        qed
+        ultimately show "p \<le> preced tm s" by auto
+      next
+        show "preced tm s \<in> (\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')"
+        proof -
+          from tm_chain
+          have "tm \<in> dependants (wq s) th'"
+            by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
+          thus ?thesis by auto
+        qed
+      qed
+      with tm_max
+      have h: "cp s th' = Max ((\<lambda>th. preced th s) ` threads s)" by simp
+      show ?thesis
+      proof (fold h, rule Max_eqI)
+        fix q 
+        assume "q \<in> cp s ` readys s"
+        then obtain th1 where th1_in: "th1 \<in> readys s"
+          and eq_q: "q = cp s th1" by auto
+        show "q \<le> cp s th'"
+          apply (unfold h eq_q)
+          apply (unfold cp_eq_cpreced cpreced_def)
+          apply (rule Max_mono)
+        proof -
+          from dependants_threads [of th1] th1_in
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<subseteq> 
+                 (\<lambda>th. preced th s) ` threads s"
+            by (auto simp:readys_def)
+        next
+          show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}" by simp
+        next
+          from finite_threads 
+          show " finite ((\<lambda>th. preced th s) ` threads s)" by simp
+        qed
+      next
+        from finite_threads
+        show "finite (cp s ` readys s)" by (auto simp:readys_def)
+      next
+        from th'_in
+        show "cp s th' \<in> cp s ` readys s" by simp
+      qed
+    next
+      assume tm_ready: "tm \<in> readys s"
+      show ?thesis
+      proof(fold tm_max)
+        have cp_eq_p: "cp s tm = preced tm s"
+        proof(unfold cp_eq_cpreced cpreced_def, rule Max_eqI)
+          fix y 
+          assume hy: "y \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+          show "y \<le> preced tm s"
+          proof -
+            { fix y'
+              assume hy' : "y' \<in> ((\<lambda>th. preced th s) ` dependants (wq s) tm)"
+              have "y' \<le> preced tm s"
+              proof(unfold tm_max, rule Max_ge)
+                from hy' dependants_threads[of tm]
+                show "y' \<in> (\<lambda>th. preced th s) ` threads s" by auto
+              next
+                from finite_threads
+                show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+              qed
+            } with hy show ?thesis by auto
+          qed
+        next
+          from dependants_threads[of tm] finite_threads
+          show "finite ((\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm))"
+            by (auto intro:finite_subset)
+        next
+          show "preced tm s \<in> (\<lambda>th. preced th s) ` ({tm} \<union> dependants (wq s) tm)"
+            by simp
+        qed 
+        moreover have "Max (cp s ` readys s) = cp s tm"
+        proof(rule Max_eqI)
+          from tm_ready show "cp s tm \<in> cp s ` readys s" by simp
+        next
+          from finite_threads
+          show "finite (cp s ` readys s)" by (auto simp:readys_def)
+        next
+          fix y assume "y \<in> cp s ` readys s"
+          then obtain th1 where th1_readys: "th1 \<in> readys s"
+            and h: "y = cp s th1" by auto
+          show "y \<le> cp s tm"
+            apply(unfold cp_eq_p h)
+            apply(unfold cp_eq_cpreced cpreced_def tm_max, rule Max_mono)
+          proof -
+            from finite_threads
+            show "finite ((\<lambda>th. preced th s) ` threads s)" by simp
+          next
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) \<noteq> {}"
+              by simp
+          next
+            from dependants_threads[of th1] th1_readys
+            show "(\<lambda>th. preced th s) ` ({th1} \<union> dependants (wq s) th1) 
+                    \<subseteq> (\<lambda>th. preced th s) ` threads s"
+              by (auto simp:readys_def)
+          qed
+        qed
+        ultimately show " Max (cp s ` readys s) = preced tm s" by simp
+      qed 
+    qed
+  qed
+qed
+
+text {* (* ccc *) \noindent
+  Since the current precedence of the threads in ready queue will always be boosted,
+  there must be one inside it has the maximum precedence of the whole system. 
+*}
+lemma max_cp_readys_threads:
+  shows "Max (cp s ` readys s) = Max (cp s ` threads s)"
+proof(cases "threads s = {}")
+  case True
+  thus ?thesis 
+    by (auto simp:readys_def)
+next
+  case False
+  show ?thesis by (rule max_cp_readys_threads_pre[OF False])
+qed
+
+end
+
+lemma eq_holding: "holding (wq s) th cs = holding s th cs"
+  apply (unfold s_holding_def cs_holding_def wq_def, simp)
+  done
+
+lemma f_image_eq:
+  assumes h: "\<And> a. a \<in> A \<Longrightarrow> f a = g a"
+  shows "f ` A = g ` A"
+proof
+  show "f ` A \<subseteq> g ` A"
+    by(rule image_subsetI, auto intro:h)
+next
+  show "g ` A \<subseteq> f ` A"
+   by (rule image_subsetI, auto intro:h[symmetric])
+qed
+
+
+definition detached :: "state \<Rightarrow> thread \<Rightarrow> bool"
+  where "detached s th \<equiv> (\<not>(\<exists> cs. holding s th cs)) \<and> (\<not>(\<exists>cs. waiting s th cs))"
+
+
+lemma detached_test:
+  shows "detached s th = (Th th \<notin> Field (RAG s))"
+apply(simp add: detached_def Field_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)
+apply(auto)
+done
+
+context valid_trace
+begin
+
+lemma detached_intro:
+  assumes eq_pv: "cntP s th = cntV s th"
+  shows "detached s th"
+proof -
+ from cnp_cnv_cncs
+  have eq_cnt: "cntP s th =
+    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+  hence cncs_zero: "cntCS s th = 0"
+    by (auto simp:eq_pv split:if_splits)
+  with eq_cnt
+  have "th \<in> readys s \<or> th \<notin> threads s" by (auto simp:eq_pv)
+  thus ?thesis
+  proof
+    assume "th \<notin> threads s"
+    with range_in dm_RAG_threads
+    show ?thesis
+      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 (RAG s)"
+    proof -
+      from card_0_eq [OF finite_holding] and cncs_zero
+      have "holdents s th = {}"
+        by (simp add:cntCS_def)
+      thus ?thesis
+        apply(auto simp:holdents_test)
+        apply(case_tac a)
+        apply(auto simp:holdents_test s_RAG_def)
+        done
+    qed
+    ultimately show ?thesis
+      by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
+  qed
+qed
+
+lemma detached_elim:
+  assumes dtc: "detached s th"
+  shows "cntP s th = cntV s th"
+proof -
+  from cnp_cnv_cncs
+  have eq_pv: " cntP s th =
+    cntV s th + (if th \<in> readys s \<or> th \<notin> threads s then cntCS s th else cntCS s th + 1)" .
+  have cncs_z: "cntCS s th = 0"
+  proof -
+    from dtc have "holdents s th = {}"
+      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
+  show ?thesis
+  proof(cases "th \<in> threads s")
+    case True
+    with dtc 
+    have "th \<in> readys s"
+      by (unfold readys_def detached_def Field_def Domain_def Range_def, 
+           auto simp:eq_waiting s_RAG_def)
+    with cncs_z and eq_pv show ?thesis by simp
+  next
+    case False
+    with cncs_z and eq_pv show ?thesis by simp
+  qed
+qed
+
+lemma detached_eq:
+  shows "(detached s th) = (cntP s th = cntV s th)"
+  by (insert vt, auto intro:detached_intro detached_elim)
+
+end
+
+text {* 
+  The lemmas in this .thy file are all obvious lemmas, however, they still needs to be derived
+  from the concise and miniature model of PIP given in PrioGDef.thy.
+*}
+
+lemma eq_dependants: "dependants (wq s) = dependants s"
+  by (simp add: s_dependants_abv wq_def)
+
+lemma next_th_unique: 
+  assumes nt1: "next_th s th cs th1"
+  and nt2: "next_th s th cs th2"
+  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)
+
+end
diff -r b4bcd1edbb6d -r 633b1fc8631b PIPDefs.thy
--- a/PIPDefs.thy	Wed Jan 06 16:34:26 2016 +0000
+++ b/PIPDefs.thy	Thu Jan 07 08:33:13 2016 +0800
@@ -1,7 +1,7 @@
 chapter {* Definitions *}
 (*<*)
 theory PIPDefs
-imports Precedence_ord Moment
+imports Precedence_ord Moment RTree Max
 begin
 (*>*)
 
@@ -607,6 +607,37 @@
   *}
 definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
   where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+
+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"
+
+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 @{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"
+
+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)
+
+definition "cp_gen s x =
+                  Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"
+
 (*<*)
 
 end
diff -r b4bcd1edbb6d -r 633b1fc8631b PIPDefs.thy~
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/PIPDefs.thy~	Thu Jan 07 08:33:13 2016 +0800
@@ -0,0 +1,614 @@
+chapter {* Definitions *}
+(*<*)
+theory PIPDefs
+imports Precedence_ord Moment
+begin
+(*>*)
+
+text {*
+  In this section, the formal model of  Priority Inheritance Protocol (PIP) is presented. 
+  The model is based on Paulson's inductive protocol verification method, where 
+  the state of the system is modelled as a list of events happened so far with the latest 
+  event put at the head. 
+*}
+
+text {*
+  To define events, the identifiers of {\em threads},
+  {\em priority} and {\em critical resources } (abbreviated as @{text "cs"}) 
+  need to be represented. All three are represetned using standard 
+  Isabelle/HOL type @{typ "nat"}:
+*}
+
+type_synonym thread = nat -- {* Type for thread identifiers. *}
+type_synonym priority = nat  -- {* Type for priorities. *}
+type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
+
+text {*
+  \noindent
+  The abstraction of Priority Inheritance Protocol (PIP) is set at the system call level.
+  Every system call is represented as an event. The format of events is defined 
+  defined as follows:
+  *}
+
+datatype event = 
+  Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *}
+  Exit thread | -- {* Thread @{text "thread"} finishing its execution. *}
+  P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *}
+  V thread cs | -- {* Thread @{text "thread"}  releasing critical resource @{text "cs"}. *}
+  Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
+
+
+text {* 
+  As mentioned earlier, in Paulson's inductive method, the states of system are represented as lists of events,
+  which is defined by the following type @{text "state"}:
+  *}
+type_synonym state = "event list"
+
+
+text {* 
+\noindent
+  Resource Allocation Graph (RAG for short) is used extensively in our formal analysis. 
+  The following type @{text "node"} is used to represent nodes in RAG.
+  *}
+datatype node = 
+   Th "thread" | -- {* Node for thread. *}
+   Cs "cs" -- {* Node for critical resource. *}
+
+text {*
+  \noindent
+  The following function
+  @{text "threads"} is used to calculate the set of live threads (@{text "threads s"})
+  in state @{text "s"}.
+  *}
+fun threads :: "state \<Rightarrow> thread set"
+  where 
+  -- {* At the start of the system, the set of threads is empty: *}
+  "threads [] = {}" | 
+  -- {* New thread is added to the @{text "threads"}: *}
+  "threads (Create thread prio#s) = {thread} \<union> threads s" | 
+  -- {* Finished thread is removed: *}
+  "threads (Exit thread # s) = (threads s) - {thread}" | 
+  -- {* Other kind of events does not affect the value of @{text "threads"}: *}
+  "threads (e#s) = threads s" 
+
+text {* 
+  \noindent
+  The function @{text "threads"} defined above is one of 
+  the so called {\em observation function}s which forms 
+  the very basis of Paulson's inductive protocol verification method.
+  Each observation function {\em observes} one particular aspect (or attribute)
+  of the system. For example, the attribute observed by  @{text "threads s"}
+  is the set of threads living in state @{text "s"}. 
+  The protocol being modelled 
+  The decision made the protocol being modelled is based on the {\em observation}s
+  returned by {\em observation function}s. Since {\observation function}s forms 
+  the very basis on which Paulson's inductive method is based, there will be 
+  a lot of such observation functions introduced in the following. In fact, any function 
+  which takes event list as argument is a {\em observation function}.
+  *}
+
+text {* \noindent
+  Observation @{text "priority th s"} is
+  the {\em original priority} of thread @{text "th"} in state @{text "s"}. 
+  The {\em original priority} is the priority 
+  assigned to a thread when it is created or when it is reset by system call 
+  (represented by event @{text "Set thread priority"}).
+*}
+
+fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
+  where
+  -- {* @{text "0"} is assigned to threads which have never been created: *}
+  "priority thread [] = 0" |
+  "priority thread (Create thread' prio#s) = 
+     (if thread' = thread then prio else priority thread s)" |
+  "priority thread (Set thread' prio#s) = 
+     (if thread' = thread then prio else priority thread s)" |
+  "priority thread (e#s) = priority thread s"
+
+text {*
+  \noindent
+  Observation @{text "last_set th s"} is the last time when the priority of thread @{text "th"} is set, 
+  observed from state @{text "s"}.
+  The time in the system is measured by the number of events happened so far since the very beginning.
+*}
+fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
+  where
+  "last_set thread [] = 0" |
+  "last_set thread ((Create thread' prio)#s) = 
+       (if (thread = thread') then length s else last_set thread s)" |
+  "last_set thread ((Set thread' prio)#s) = 
+       (if (thread = thread') then length s else last_set thread s)" |
+  "last_set thread (_#s) = last_set thread s"
+
+text {*
+  \noindent 
+  The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of 
+  a thread is the combination of its {\em original priority} and {\em time} the priority is set. 
+  The intention is to discriminate threads with the same priority by giving threads whose priority
+  is assigned earlier higher precedences, becasue such threads are more urgent to finish. 
+  This explains the following definition:
+  *}
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+  where "preced thread s \<equiv> Prc (priority thread s) (last_set thread s)"
+
+
+text {*
+  \noindent
+  A number of important notions in PIP are represented as the following functions, 
+  defined in terms of the waiting queues of the system, where the waiting queues 
+  , as a whole, is represented by the @{text "wq"} argument of every notion function.
+  The @{text "wq"} argument is itself a functions which maps every critical resource 
+  @{text "cs"} to the list of threads which are holding or waiting for it. 
+  The thread at the head of this list is designated as the thread which is current 
+  holding the resrouce, which is slightly different from tradition where
+  all threads in the waiting queue are considered as waiting for the resource.
+  *}
+
+consts 
+  holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool" 
+  waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+  RAG :: "'b \<Rightarrow> (node \<times> node) set"
+  dependants :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+defs (overloaded) 
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  This meaning of @{text "wq"} is reflected in the following definition of @{text "holding wq th cs"},
+  where @{text "holding wq th cs"} means thread @{text "th"} is holding the critical 
+  resource @{text "cs"}. This decision is based on @{text "wq"}.
+  \end{minipage}
+  *}
+
+  cs_holding_def: 
+  "holding wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  In accordance with the definition of @{text "holding wq th cs"}, 
+  a thread @{text "th"} is considered waiting for @{text "cs"} if 
+  it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head.
+  This is reflected in the definition of @{text "waiting wq th cs"} as follows:
+  \end{minipage}
+  *}
+  cs_waiting_def: 
+  "waiting wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  @{text "RAG wq"} generates RAG (a binary relations on @{text "node"})
+  out of waiting queues of the system (represented by the @{text "wq"} argument):
+  \end{minipage}
+  *}
+  cs_RAG_def: 
+  "RAG (wq::cs \<Rightarrow> thread list) \<equiv>
+      {(Th th, Cs cs) | th cs. waiting wq th cs} \<union> {(Cs cs, Th th) | cs th. holding wq th cs}"
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  The following @{text "dependants wq th"} represents the set of threads which are RAGing on
+  thread @{text "th"} in Resource Allocation Graph @{text "RAG wq"}. 
+  Here, "RAGing" means waiting directly or indirectly on the critical resource. 
+  \end{minipage}
+  *}
+  cs_dependants_def: 
+  "dependants (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
+
+
+text {* \noindent 
+  The following
+  @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under
+  state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of 
+  Priority Inheritance that the {\em current precedence} of a thread is the precedence 
+  inherited from the maximum of all its dependants, i.e. the threads which are waiting 
+  directly or indirectly waiting for some resources from it. If no such thread exits, 
+  @{text "th"}'s {\em current precedence} equals its original precedence, i.e. 
+  @{text "preced th s"}.
+  *}
+
+definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
+  where "cpreced wq s = (\<lambda>th. Max ((\<lambda>th'. preced th' s) ` ({th} \<union> dependants wq th)))"
+
+text {*
+  Notice that the current precedence (@{text "cpreced"}) of one thread @{text "th"} can be boosted 
+  (becoming larger than its own precedence) by those threads in 
+  the @{text "dependants wq th"}-set. If one thread get boosted, we say 
+  it inherits the priority (or, more precisely, the precedence) of 
+  its dependants. This is how the word "Inheritance" in 
+  Priority Inheritance Protocol comes.
+*}
+
+(*<*)
+lemma 
+  cpreced_def2:
+  "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
+  unfolding cpreced_def image_def
+  apply(rule eq_reflection)
+  apply(rule_tac f="Max" in arg_cong)
+  by (auto)
+(*>*)
+
+
+text {* \noindent
+  Assuming @{text "qs"} be the waiting queue of a critical resource, 
+  the following abbreviation "release qs" is the waiting queue after the thread 
+  holding the resource (which is thread at the head of @{text "qs"}) released
+  the resource:
+*}
+abbreviation
+  "release qs \<equiv> case qs of
+             [] => [] 
+          |  (_#qs') => (SOME q. distinct q \<and> set q = set qs')"
+text {* \noindent
+  It can be seen from the definition that the thread at the head of @{text "qs"} is removed
+  from the return value, and the value @{term "q"} is an reordering of @{text "qs'"}, the 
+  tail of @{text "qs"}. Through this reordering, one of the waiting threads (those in @{text "qs'"} }
+  is chosen nondeterministically to be the head of the new queue @{text "q"}. 
+  Therefore, this thread is the one who takes over the resource. This is a little better different 
+  from common sense that the thread who comes the earliest should take over.  
+  The intention of this definition is to show that the choice of which thread to take over the 
+  release resource does not affect the correctness of the PIP protocol. 
+*}
+
+text {*
+  The data structure used by the operating system for scheduling is referred to as 
+  {\em schedule state}. It is represented as a record consisting of 
+  a function assigning waiting queue to resources 
+  (to be used as the @{text "wq"} argument in @{text "holding"}, @{text "waiting"} 
+  and  @{text "RAG"}, etc) and a function assigning precedence to threads:
+  *}
+
+record schedule_state = 
+    wq_fun :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
+    cprec_fun :: "thread \<Rightarrow> precedence" -- {* The function assigning precedence. *}
+
+text {* \noindent
+  The following two abbreviations (@{text "all_unlocked"} and @{text "initial_cprec"}) 
+  are used to set the initial values of the @{text "wq_fun"} @{text "cprec_fun"} fields 
+  respectively of the @{text "schedule_state"} record by the following function @{text "sch"},
+  which is used to calculate the system's {\em schedule state}.
+
+  Since there is no thread at the very beginning to make request, all critical resources 
+  are free (or unlocked). This status is represented by the abbreviation
+  @{text "all_unlocked"}. 
+  *}
+abbreviation
+  "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
+
+
+text {* \noindent
+  The initial current precedence for a thread can be anything, because there is no thread then. 
+  We simply assume every thread has precedence @{text "Prc 0 0"}.
+  *}
+
+abbreviation 
+  "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
+
+
+text {* \noindent
+  The following function @{text "schs"} is used to calculate the system's schedule state @{text "schs s"}
+  out of the current system state @{text "s"}. It is the central function to model Priority Inheritance:
+  *}
+fun schs :: "state \<Rightarrow> schedule_state"
+  where 
+  -- {*
+  \begin{minipage}{0.9\textwidth}
+    Setting the initial value of the @{text "schedule_state"} record (see the explanations above).
+  \end{minipage}
+  *}
+  "schs [] = (| wq_fun = all_unlocked,  cprec_fun = initial_cprec |)" |
+
+  -- {*
+  \begin{minipage}{0.9\textwidth}
+  \begin{enumerate}
+  \item @{text "ps"} is the schedule state of last moment.
+  \item @{text "pwq"} is the waiting queue function of last moment.
+  \item @{text "pcp"} is the precedence function of last moment (NOT USED). 
+  \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement:
+  \begin{enumerate}
+      \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to 
+            the end of @{text "cs"}'s waiting queue.
+      \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state,
+            @{text "th'"} must equal to @{text "thread"}, 
+            because @{text "thread"} is the one currently holding @{text "cs"}. 
+            The case @{text "[] \<Longrightarrow> []"} may never be executed in a legal state.
+            the @{text "(SOME q. distinct q \<and> set q = set qs)"} is used to choose arbitrarily one 
+            thread in waiting to take over the released resource @{text "cs"}. In our representation,
+            this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
+      \item For other happening event, the schedule state just does not change.
+  \end{enumerate}
+  \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue 
+        function. The RAGency of precedence function on waiting queue function is the reason to 
+        put them in the same record so that they can evolve together.
+  \end{enumerate}
+  
+
+  The calculation of @{text "cprec_fun"} depends on the value of @{text "wq_fun"}. 
+  Therefore, in the following cases, @{text "wq_fun"} is always calculated first, in 
+  the name of @{text "wq"} (if  @{text "wq_fun"} is not changed
+  by the happening event) or @{text "new_wq"} (if the value of @{text "wq_fun"} is changed).
+  \end{minipage}
+     *}
+   "schs (Create th prio # s) = 
+       (let wq = wq_fun (schs s) in
+          (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"
+|  "schs (Exit th # s) = 
+       (let wq = wq_fun (schs s) in
+          (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"
+|  "schs (Set th prio # s) = 
+       (let wq = wq_fun (schs s) in
+          (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"
+   -- {*
+   \begin{minipage}{0.9\textwidth}
+      Different from the forth coming cases, the @{text "wq_fun"} field of the schedule state 
+      is changed. So, the new value is calculated first, in the name of @{text "new_wq"}.
+   \end{minipage}   
+   *}
+|  "schs (P th cs # s) = 
+       (let wq = wq_fun (schs s) in
+        let new_wq = wq(cs := (wq cs @ [th])) in
+          (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"
+|  "schs (V th cs # s) = 
+       (let wq = wq_fun (schs s) in
+        let new_wq = wq(cs := release (wq cs)) in
+          (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"
+
+lemma cpreced_initial: 
+  "cpreced (\<lambda> cs. []) [] = (\<lambda>_. (Prc 0 0))"
+apply(simp add: cpreced_def)
+apply(simp add: cs_dependants_def cs_RAG_def cs_waiting_def cs_holding_def)
+apply(simp add: preced_def)
+done
+
+lemma sch_old_def:
+  "schs (e#s) = (let ps = schs s in 
+                  let pwq = wq_fun ps in 
+                  let nwq = case e of
+                             P th cs \<Rightarrow>  pwq(cs:=(pwq cs @ [th])) |
+                             V th cs \<Rightarrow> let nq = case (pwq cs) of
+                                                      [] \<Rightarrow> [] | 
+                                                      (_#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
+                                            in pwq(cs:=nq)                 |
+                              _ \<Rightarrow> pwq
+                  in let ncp = cpreced nwq (e#s) in 
+                     \<lparr>wq_fun = nwq, cprec_fun = ncp\<rparr>
+                 )"
+apply(cases e)
+apply(simp_all)
+done
+
+
+text {* 
+  \noindent
+  The following @{text "wq"} is a shorthand for @{text "wq_fun"}. 
+  *}
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
+  where "wq s = wq_fun (schs s)"
+
+text {* \noindent 
+  The following @{text "cp"} is a shorthand for @{text "cprec_fun"}. 
+  *}
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+  where "cp s \<equiv> cprec_fun (schs s)"
+
+text {* \noindent
+  Functions @{text "holding"}, @{text "waiting"}, @{text "RAG"} and 
+  @{text "dependants"} still have the 
+  same meaning, but redefined so that they no longer RAG on the 
+  fictitious {\em waiting queue function}
+  @{text "wq"}, but on system state @{text "s"}.
+  *}
+defs (overloaded) 
+  s_holding_abv: 
+  "holding (s::state) \<equiv> holding (wq_fun (schs s))"
+  s_waiting_abv: 
+  "waiting (s::state) \<equiv> waiting (wq_fun (schs s))"
+  s_RAG_abv: 
+  "RAG (s::state) \<equiv> RAG (wq_fun (schs s))"
+  s_dependants_abv: 
+  "dependants (s::state) \<equiv> dependants (wq_fun (schs s))"
+
+
+text {* 
+  The following lemma can be proved easily, and the meaning is obvious.
+  *}
+lemma
+  s_holding_def: 
+  "holding (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th = hd (wq_fun (schs s) cs))"
+  by (auto simp:s_holding_abv wq_def cs_holding_def)
+
+lemma s_waiting_def: 
+  "waiting (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th \<noteq> hd (wq_fun (schs s) cs))"
+  by (auto simp:s_waiting_abv wq_def cs_waiting_def)
+
+lemma s_RAG_def: 
+  "RAG (s::state) =
+    {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \<union> {(Cs cs, Th th) | cs th. holding (wq s) th cs}"
+  by (auto simp:s_RAG_abv wq_def cs_RAG_def)
+
+lemma
+  s_dependants_def: 
+  "dependants (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (RAG (wq s))^+}"
+  by (auto simp:s_dependants_abv wq_def cs_dependants_def)
+
+text {*
+  The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready} 
+  for running if it is a live thread and it is not waiting for any critical resource.
+  *}
+definition readys :: "state \<Rightarrow> thread set"
+  where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waiting s th cs)}"
+
+text {* \noindent
+  The following function @{text "runing"} calculates the set of running thread, which is the ready 
+  thread with the highest precedence.  
+  *}
+definition runing :: "state \<Rightarrow> thread set"
+  where "runing s \<equiv> {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
+
+text {* \noindent
+  Notice that the definition of @{text "running"} reflects the preemptive scheduling strategy,  
+  because, if the @{text "running"}-thread (the one in @{text "runing"} set) 
+  lowered its precedence by resetting its own priority to a lower
+  one, it will lose its status of being the max in @{text "ready"}-set and be superseded.
+*}
+
+text {* \noindent
+  The following function @{text "holdents s th"} returns the set of resources held by thread 
+  @{text "th"} in state @{text "s"}.
+  *}
+definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+  where "holdents s th \<equiv> {cs . holding s th cs}"
+
+lemma holdents_test: 
+  "holdents s th = {cs . (Cs cs, Th th) \<in> RAG s}"
+unfolding holdents_def
+unfolding s_RAG_def
+unfolding s_holding_abv
+unfolding wq_def
+by (simp)
+
+text {* \noindent
+  Observation @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in
+  state @{text "s"}:
+  *}
+definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
+  where "cntCS s th = card (holdents s th)"
+
+text {* \noindent
+  According to the convention of Paulson's inductive method,
+  the decision made by a protocol that event @{text "e"} is eligible to happen next under state @{text "s"} 
+  is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as 
+  follows (notice how the decision is based on the {\em observation function}s 
+  defined above, and also notice how a complicated protocol is modeled by a few simple 
+  observations, and how such a kind of simplicity gives rise to improved trust on
+  faithfulness):
+  *}
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+  where
+  -- {* 
+  A thread can be created if it is not a live thread:
+  *}
+  thread_create: "\<lbrakk>thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
+  -- {*
+  A thread can exit if it no longer hold any resource:
+  *}
+  thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
+  -- {*
+  \begin{minipage}{0.9\textwidth}
+  A thread can request for an critical resource @{text "cs"}, if it is running and 
+  the request does not form a loop in the current RAG. The latter condition 
+  is set up to avoid deadlock. The condition also reflects our assumption all threads are 
+  carefully programmed so that deadlock can not happen:
+  \end{minipage}
+  *}
+  thread_P: "\<lbrakk>thread \<in> runing s;  (Cs cs, Th thread)  \<notin> (RAG s)^+\<rbrakk> \<Longrightarrow> 
+                                                                step s (P thread cs)" |
+  -- {*
+  \begin{minipage}{0.9\textwidth}
+  A thread can release a critical resource @{text "cs"} 
+  if it is running and holding that resource:
+  \end{minipage}
+  *}
+  thread_V: "\<lbrakk>thread \<in> runing s;  holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
+  -- {*
+  \begin{minipage}{0.9\textwidth}
+  A thread can adjust its own priority as long as it is current running. 
+  With the resetting of one thread's priority, its precedence may change. 
+  If this change lowered the precedence, according to the definition of @{text "running"}
+  function, 
+  \end{minipage}
+  *}  
+  thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+text {*
+  In Paulson's inductive method, every protocol is defined by such a @{text "step"}
+  predicate. For instance, the predicate @{text "step"} given above 
+  defines the PIP protocol. So, it can also be called "PIP".
+*}
+
+abbreviation
+  "PIP \<equiv> step"
+
+
+text {* \noindent
+  For any protocol defined by a @{text "step"} predicate, 
+  the fact that @{text "s"} is a legal state in 
+  the protocol is expressed as: @{text "vt step s"}, where
+  the predicate @{text "vt"} can be defined as the following:
+  *}
+inductive vt :: "state \<Rightarrow> bool"
+  where
+  -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
+  vt_nil[intro]: "vt []" |
+  -- {* 
+  \begin{minipage}{0.9\textwidth}
+  If @{text "s"} a legal state of the protocol defined by predicate @{text "step"}, 
+  and event @{text "e"} is allowed to happen under state @{text "s"} by the protocol 
+  predicate @{text "step"}, then @{text "e#s"} is a new legal state rendered by the 
+  happening of @{text "e"}:
+  \end{minipage}
+  *}
+  vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
+
+text {*  \noindent
+  It is easy to see that the definition of @{text "vt"} is generic. It can be applied to 
+  any specific protocol specified by a @{text "step"}-predicate to get the set of
+  legal states of that particular protocol.
+  *}
+
+text {* 
+  The following are two very basic properties of @{text "vt"}.
+*}
+
+lemma step_back_vt: "vt (e#s) \<Longrightarrow> vt s"
+  by(ind_cases "vt (e#s)", simp)
+
+lemma step_back_step: "vt (e#s) \<Longrightarrow> step s e"
+  by(ind_cases "vt (e#s)", simp)
+
+text {* \noindent
+  The following two auxiliary functions @{text "the_cs"} and @{text "the_th"} are used to extract
+  critical resource and thread respectively out of RAG nodes.
+  *}
+fun the_cs :: "node \<Rightarrow> cs"
+  where "the_cs (Cs cs) = cs"
+
+fun the_th :: "node \<Rightarrow> thread"
+  where "the_th (Th th) = th"
+
+text {* \noindent
+  The following predicate @{text "next_th"} describe the next thread to 
+  take over when a critical resource is released. In @{text "next_th s th cs t"}, 
+  @{text "th"} is the thread to release, @{text "t"} is the one to take over.
+  Notice how this definition is backed up by the @{text "release"} function and its use 
+  in the @{text "V"}-branch of @{text "schs"} function. This @{text "next_th"} function
+  is not needed for the execution of PIP. It is introduced as an auxiliary function 
+  to state lemmas. The correctness of this definition will be confirmed by 
+  lemmas @{text "step_v_hold_inv"}, @{text " step_v_wait_inv"}, 
+  @{text "step_v_get_hold"} and @{text "step_v_not_wait"}.
+  *}
+definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
+  where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and> 
+                                     t = hd (SOME q. distinct q \<and> set q = set rest))"
+
+text {* \noindent
+  The aux function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
+  in list @{text "l"}:
+  *}
+definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
+  where "count Q l = length (filter Q l)"
+
+text {* \noindent
+  The following observation @{text "cntP s"} returns the number of operation @{text "P"} happened 
+  before reaching state @{text "s"}.
+  *}
+definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
+  where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
+
+text {* \noindent
+  The following observation @{text "cntV s"} returns the number of operation @{text "V"} happened 
+  before reaching state @{text "s"}.
+  *}
+definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
+  where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+(*<*)
+
+end
+(*>*)
+
diff -r b4bcd1edbb6d -r 633b1fc8631b Precedence_ord.thy~
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Precedence_ord.thy~	Thu Jan 07 08:33:13 2016 +0800
@@ -0,0 +1,45 @@
+header {* Order on product types *}
+
+theory Precedence_ord
+imports Main
+begin
+
+datatype precedence = Prc nat nat
+
+instantiation precedence :: order
+begin
+
+definition
+  precedence_le_def: "x \<le> y \<longleftrightarrow> (case (x, y) of
+                                   (Prc fx sx, Prc fy sy) \<Rightarrow> 
+                                 fx < fy \<or> (fx \<le> fy \<and> sy \<le> sx))"
+
+definition
+  precedence_less_def: "x < y \<longleftrightarrow> (case (x, y) of
+                               (Prc fx sx, Prc fy sy) \<Rightarrow> 
+                                     fx < fy \<or> (fx \<le> fy \<and> sy < sx))"
+
+instance
+proof
+qed (auto simp: precedence_le_def precedence_less_def 
+              intro: order_trans split:precedence.splits)
+end
+
+instance precedence :: preorder ..
+
+instance precedence :: linorder 
+proof
+qed (auto simp: precedence_le_def precedence_less_def 
+              intro: order_trans split:precedence.splits)
+
+instantiation precedence :: zero
+begin
+
+definition Zero_precedence_def:
+  "0 = Prc 0 0"
+
+instance ..
+
+end
+
+end
diff -r b4bcd1edbb6d -r 633b1fc8631b RTree.thy~
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/RTree.thy~	Thu Jan 07 08:33:13 2016 +0800
@@ -0,0 +1,1748 @@
+theory RTree
+imports "~~/src/HOL/Library/Transitive_Closure_Table" Max
+begin
+
+section {* A theory of relational trees *}
+
+inductive_cases path_nilE [elim!]: "rtrancl_path r x [] y"
+inductive_cases path_consE [elim!]: "rtrancl_path r x (z#zs) y"
+
+subsection {* Definitions *}
+
+text {*
+  In this theory, we are going to give a notion of of `Relational Graph` and
+  its derived notion `Relational Tree`. Given a binary relation @{text "r"},
+  the `Relational Graph of @{text "r"}` is the graph, the edges of which
+  are those in @{text "r"}. In this way, any binary relation can be viewed
+  as a `Relational Graph`. Note, this notion of graph includes infinite graphs. 
+
+  A `Relation Graph` @{text "r"} is said to be a `Relational Tree` if it is both
+  {\em single valued} and {\em acyclic}. 
+*}
+
+text {*
+  The following @{text "sgv"} specifies that relation @{text "r"} is {\em single valued}.
+*}
+locale sgv = 
+  fixes r
+  assumes sgv: "single_valued r"
+
+text {*
+  The following @{text "rtree"} specifies that @{text "r"} is a 
+  {\em Relational Tree}.
+*}
+locale rtree = sgv + 
+  assumes acl: "acyclic r"
+
+text {* 
+  The following two auxiliary functions @{text "rel_of"} and @{text "pred_of"} 
+  transfer between the predicate and set representation of binary relations.
+*}
+
+definition "rel_of r = {(x, y) | x y. r x y}"
+
+definition "pred_of r = (\<lambda> x y. (x, y) \<in> r)"
+
+text {*
+  To reason about {\em Relational Graph}, a notion of path is 
+  needed, which is given by the following @{text "rpath"} (short 
+  for `relational path`). 
+  The path @{text "xs"} in proposition @{text "rpath r x xs y"} is 
+  a path leading from @{text "x"} to @{text "y"}, which serves as a 
+  witness of the fact @{text "(x, y) \<in> r^*"}. 
+
+  @{text "rpath"}
+  is simply a wrapper of the @{text "rtrancl_path"} defined in the imported 
+  theory @{text "Transitive_Closure_Table"}, which defines 
+  a notion of path for the predicate form of binary relations. 
+*}
+definition "rpath r x xs y = rtrancl_path (pred_of r) x xs y"
+
+text {*
+  Given a path @{text "ps"}, @{text "edges_on ps"} is the 
+  set of edges along the path, which is defined as follows:
+*}
+
+definition "edges_on ps = {(a,b) | a b. \<exists> xs ys. ps = xs@[a,b]@ys}"
+
+text {*
+   The following @{text "indep"} defines a notion of independence. 
+   Two nodes @{text "x"} and @{text "y"} are said to be independent
+   (expressed as @{text "indep x y"}),  if neither one is reachable 
+   from the other in relational graph @{text "r"}.
+*}
+definition "indep r x y = (((x, y) \<notin> r^*) \<and> ((y, x) \<notin> r^*))"
+
+text {*
+  In relational tree @{text "r"}, the sub tree of node @{text "x"} is written
+  @{text "subtree r x"}, which is defined to be the set of nodes (including itself) 
+  which can reach @{text "x"} by following some path in @{text "r"}:
+*}
+
+definition "subtree r x = {y . (y, x) \<in> r^*}"
+
+definition "ancestors r x = {y. (x, y) \<in> r^+}"
+
+definition "root r x = (ancestors r x = {})"
+
+text {*
+  The following @{text "edge_in r x"} is the set of edges
+  contained in the sub-tree of @{text "x"}, with @{text "r"} as the underlying graph.
+*}
+
+definition "edges_in r x = {(a, b) | a b. (a, b) \<in> r \<and> b \<in> subtree r x}"
+
+text {*
+  The following lemma @{text "edges_in_meaning"} shows the intuitive meaning 
+  of `an edge @{text "(a, b)"} is in the sub-tree of @{text "x"}`, 
+  i.e., both @{text "a"} and @{text "b"} are in the sub-tree.
+*}
+lemma edges_in_meaning: 
+  "edges_in r x = {(a, b) | a b. (a, b) \<in> r \<and> a \<in> subtree r x \<and> b \<in> subtree r x}"
+proof -
+  { fix a b
+    assume h: "(a, b) \<in> r" "b \<in> subtree r x"
+    moreover have "a \<in> subtree r x"
+    proof -
+      from h(2)[unfolded subtree_def] have "(b, x) \<in> r^*" by simp
+      with h(1) have "(a, x) \<in> r^*" by auto
+      thus ?thesis by (auto simp:subtree_def)
+    qed
+    ultimately have "((a, b) \<in> r \<and> a \<in> subtree r x \<and> b \<in> subtree r x)" 
+      by (auto)
+  } thus ?thesis by (auto simp:edges_in_def)
+qed
+
+text {*
+  The following lemma shows the meaning of @{term "edges_in"} from the other side, 
+  which says: for the edge @{text "(a,b)"} to be outside of the sub-tree of @{text "x"}, 
+  it is sufficient to show that @{text "b"} is.
+*}
+lemma edges_in_refutation:
+  assumes "b \<notin> subtree r x"
+  shows "(a, b) \<notin> edges_in r x"
+  using assms by (unfold edges_in_def subtree_def, auto)
+
+definition "children r x = {y. (y, x) \<in> r}"
+
+locale fbranch =
+  fixes r
+  assumes fb: "\<forall> x \<in> Range r . finite (children r x)"
+begin
+
+lemma finite_children: "finite (children r x)"
+proof(cases "children r x = {}")
+  case True
+  thus ?thesis by auto
+next
+  case False
+  then obtain y where "(y, x) \<in> r" by (auto simp:children_def)
+  hence "x \<in> Range r" by auto
+  from fb[rule_format, OF this]
+  show ?thesis .
+qed
+
+end
+
+locale fsubtree = fbranch + 
+   assumes wf: "wf r"
+
+(* ccc *)
+
+subsection {* Auxiliary lemmas *}
+
+lemma index_minimize:
+  assumes "P (i::nat)"
+  obtains j where "P j" and "\<forall> k < j. \<not> P k" 
+proof -
+  have "\<exists> j. P j \<and> (\<forall> k < j. \<not> P k)"
+  using assms
+  proof(induct i rule:less_induct)
+    case (less t)
+    show ?case
+    proof(cases "\<forall> j < t. \<not> P j")
+      case True
+      with less (2) show ?thesis by blast
+    next
+      case False
+      then obtain j where "j < t" "P j" by auto
+      from less(1)[OF this]
+      show ?thesis .
+    qed
+  qed 
+  with that show ?thesis by metis
+qed
+
+subsection {* Properties of Relational Graphs and Relational Trees *}
+
+subsubsection {* Properties of @{text "rel_of"} and @{text "pred_of"} *}
+
+text {* The following lemmas establish bijectivity of the two functions *}
+
+lemma pred_rel_eq: "pred_of (rel_of r) = r" by (auto simp:rel_of_def pred_of_def)
+
+lemma rel_pred_eq: "rel_of (pred_of r) = r" by (auto simp:rel_of_def pred_of_def)
+
+lemma rel_of_star: "rel_of (r^**) = (rel_of r)^*"
+  by (unfold rel_of_def rtranclp_rtrancl_eq, auto)
+
+lemma pred_of_star: "pred_of (r^*) = (pred_of r)^**"
+proof -
+  { fix x y
+    have "pred_of (r^*) x y = (pred_of r)^** x y"
+    by (unfold pred_of_def rtranclp_rtrancl_eq, auto)
+  } thus ?thesis by auto
+qed
+
+lemma star_2_pstar: "(x, y) \<in> r^* = (pred_of (r^*)) x y"
+  by (simp add: pred_of_def)
+
+subsubsection {* Properties of @{text "rpath"} *}
+
+text {* Induction rule for @{text "rpath"}: *}
+
+lemma rpath_induct [consumes 1, case_names rbase rstep, induct pred: rpath]:
+  assumes "rpath r x1 x2 x3"
+    and "\<And>x. P x [] x"
+    and "\<And>x y ys z. (x, y) \<in> r \<Longrightarrow> rpath r y ys z \<Longrightarrow> P y ys z \<Longrightarrow> P x (y # ys) z"
+  shows "P x1 x2 x3"
+  using assms[unfolded rpath_def]
+  by (induct, auto simp:pred_of_def rpath_def)
+
+lemma rpathE: 
+  assumes "rpath r x xs y"
+  obtains (base) "y = x" "xs = []"
+     | (step) z zs where "(x, z) \<in> r" "rpath r z zs y" "xs = z#zs"
+  using assms
+  by (induct, auto)
+
+text {* Introduction rule for empty path *}
+lemma rbaseI [intro!]: 
+  assumes "x = y"
+  shows "rpath r x [] y"
+  by  (unfold rpath_def assms, 
+         rule Transitive_Closure_Table.rtrancl_path.base)
+
+text {* Introduction rule for non-empty path *}
+lemma rstepI [intro!]: 
+  assumes "(x, y) \<in> r"
+    and "rpath r y ys z"
+  shows "rpath r x (y#ys) z" 
+proof(unfold rpath_def, rule Transitive_Closure_Table.rtrancl_path.step)
+  from assms(1) show "pred_of r x y" by (auto simp:pred_of_def)
+next
+  from assms(2) show "rtrancl_path (pred_of r) y ys z"  
+  by (auto simp:pred_of_def rpath_def)
+qed
+
+text {* Introduction rule for @{text "@"}-path *}
+lemma rpath_appendI [intro]: 
+  assumes "rpath r x xs a" and "rpath r a ys y"
+  shows "rpath r x (xs @ ys) y"
+  using assms 
+  by (unfold rpath_def, auto intro:rtrancl_path_trans)
+
+text {* Elimination rule for empty path *}
+
+lemma rpath_cases [cases pred:rpath]:
+  assumes "rpath r a1 a2 a3"
+  obtains (rbase)  "a1 = a3" and "a2 = []"
+    | (rstep)  y :: "'a" and ys :: "'a list"  
+         where "(a1, y) \<in> r" and "a2 = y # ys" and "rpath r y ys a3"
+  using assms [unfolded rpath_def]
+  by (cases, auto simp:rpath_def pred_of_def)
+
+lemma rpath_nilE [elim!, cases pred:rpath]: 
+  assumes "rpath r x [] y"
+  obtains "y = x"
+  using assms[unfolded rpath_def] by auto
+
+-- {* This is a auxiliary lemmas used only in the proof of @{text "rpath_nnl_lastE"} *}
+lemma rpath_nnl_last:
+  assumes "rtrancl_path r x xs y"
+  and "xs \<noteq> []"
+  obtains xs' where "xs = xs'@[y]"
+proof -
+  from append_butlast_last_id[OF `xs \<noteq> []`, symmetric] 
+  obtain xs' y' where eq_xs: "xs = (xs' @ y' # [])" by simp
+  with assms(1)
+  have "rtrancl_path r x ... y" by simp
+  hence "y = y'" by (rule rtrancl_path_appendE, auto)
+  with eq_xs have "xs = xs'@[y]" by simp
+  from that[OF this] show ?thesis .
+qed
+
+text {*
+  Elimination rule for non-empty paths constructed with @{text "#"}.
+*}
+
+lemma rpath_ConsE [elim!, cases pred:rpath]:
+  assumes "rpath r x (y # ys) x2"
+  obtains (rstep) "(x, y) \<in> r" and "rpath r y ys x2"
+  using assms[unfolded rpath_def]
+  by (cases, auto simp:rpath_def pred_of_def)
+
+text {*
+  Elimination rule for non-empty path, where the destination node 
+  @{text "y"} is shown to be at the end of the path.
+*}
+lemma rpath_nnl_lastE: 
+  assumes "rpath r x xs y"
+  and "xs \<noteq> []"
+  obtains xs' where "xs = xs'@[y]"
+  using assms[unfolded rpath_def]
+  by (rule rpath_nnl_last, auto)
+
+text {* Other elimination rules of @{text "rpath"} *}
+
+lemma rpath_appendE:
+  assumes "rpath r x (xs @ [a] @ ys) y"
+  obtains "rpath r x (xs @ [a]) a" and "rpath r a ys y"
+  using rtrancl_path_appendE[OF assms[unfolded rpath_def, simplified], folded rpath_def]
+  by auto
+
+lemma rpath_subE: 
+  assumes "rpath r x (xs @ [a] @ ys @ [b] @ zs) y"
+  obtains "rpath r x (xs @ [a]) a" and "rpath r a (ys @ [b]) b" and "rpath r b zs y" 
+  using assms
+ by (elim rpath_appendE, auto)
+
+text {* Every path has a unique end point. *}
+lemma rpath_dest_eq:
+  assumes "rpath r x xs x1"
+  and "rpath r x xs x2"
+  shows "x1 = x2"
+  using assms
+  by (induct, auto)
+
+subsubsection {* Properites of @{text "edges_on"} *}
+
+lemma edges_on_unfold:
+  "edges_on (a # b # xs) = {(a, b)} \<union> edges_on (b # xs)" (is "?L = ?R")
+proof -
+  { fix c d
+    assume "(c, d) \<in> ?L"
+    then obtain l1 l2 where h: "(a # b # xs) = l1 @ [c, d] @ l2" 
+        by (auto simp:edges_on_def)
+    have "(c, d) \<in> ?R"
+    proof(cases "l1")
+      case Nil
+      with h have "(c, d) = (a, b)" by auto
+      thus ?thesis by auto
+    next
+      case (Cons e es)
+      from h[unfolded this] have "b#xs = es@[c, d]@l2" by auto
+      thus ?thesis by (auto simp:edges_on_def)
+    qed
+  } moreover
+  { fix c d
+    assume "(c, d) \<in> ?R"
+    moreover have "(a, b) \<in> ?L" 
+    proof -
+      have "(a # b # xs) = []@[a,b]@xs" by simp
+      hence "\<exists> l1 l2. (a # b # xs) = l1@[a,b]@l2" by auto
+      thus ?thesis by (unfold edges_on_def, simp)
+    qed
+    moreover {
+        assume "(c, d) \<in> edges_on (b#xs)"
+        then obtain l1 l2 where "b#xs = l1@[c, d]@l2" by (unfold edges_on_def, auto)
+        hence "a#b#xs = (a#l1)@[c,d]@l2" by simp
+        hence "\<exists> l1 l2. (a # b # xs) = l1@[c,d]@l2" by metis
+        hence "(c,d) \<in> ?L" by (unfold edges_on_def, simp)
+    }
+    ultimately have "(c, d) \<in> ?L" by auto
+  } ultimately show ?thesis by auto
+qed
+
+lemma edges_on_len:
+  assumes "(a,b) \<in> edges_on l"
+  shows "length l \<ge> 2"
+  using assms
+  by (unfold edges_on_def, auto)
+
+text {* Elimination of @{text "edges_on"} for non-empty path *}
+
+lemma edges_on_consE [elim, cases set:edges_on]:
+  assumes "(a,b) \<in> edges_on (x#xs)"
+  obtains (head)  xs' where "x = a" and "xs = b#xs'"
+      |  (tail)  "(a,b) \<in> edges_on xs"
+proof -
+  from assms obtain l1 l2 
+  where h: "(x#xs) = l1 @ [a,b] @ l2" by (unfold edges_on_def, blast)
+  have "(\<exists> xs'. x = a \<and> xs = b#xs') \<or> ((a,b) \<in> edges_on xs)"
+  proof(cases "l1")
+    case Nil with h 
+    show ?thesis by auto
+  next
+    case (Cons e el)
+    from h[unfolded this] 
+    have "xs = el @ [a,b] @ l2" by auto
+    thus ?thesis 
+      by (unfold edges_on_def, auto)
+  qed
+  thus ?thesis 
+  proof
+    assume "(\<exists>xs'. x = a \<and> xs = b # xs')"
+    then obtain xs' where "x = a" "xs = b#xs'" by blast
+    from that(1)[OF this] show ?thesis .
+  next
+    assume "(a, b) \<in> edges_on xs"
+    from that(2)[OF this] show ?thesis .
+  qed
+qed
+
+text {*
+  Every edges on the path is a graph edges:
+*}
+lemma rpath_edges_on:
+  assumes "rpath r x xs y"
+  shows "(edges_on (x#xs)) \<subseteq> r"
+  using assms
+proof(induct arbitrary:y)
+  case (rbase x)
+  thus ?case by (unfold edges_on_def, auto)
+next
+  case (rstep x y ys z)
+  show ?case
+  proof -
+    { fix a b
+      assume "(a, b) \<in> edges_on (x # y # ys)"
+      hence "(a, b) \<in> r" by (cases, insert rstep, auto)
+    } thus ?thesis by auto
+  qed
+qed
+
+text {* @{text "edges_on"} is mono with respect to @{text "#"}-operation: *}
+lemma edges_on_Cons_mono:
+   shows "edges_on xs \<subseteq> edges_on (x#xs)"
+proof -
+  { fix a b
+    assume "(a, b) \<in> edges_on xs"
+    then obtain l1 l2 where "xs = l1 @ [a,b] @ l2" 
+      by (auto simp:edges_on_def)
+    hence "x # xs = (x#l1) @ [a, b] @ l2" by auto
+    hence "(a, b) \<in> edges_on (x#xs)" 
+      by (unfold edges_on_def, blast)
+  } thus ?thesis by auto
+qed
+
+text {*
+  The following rule @{text "rpath_transfer"} is used to show 
+  that one path is intact as long as all the edges on it are intact
+  with the change of graph.
+
+  If @{text "x#xs"} is path in graph @{text "r1"} and 
+  every edges along the path is also in @{text "r2"}, 
+  then @{text "x#xs"} is also a edge in graph @{text "r2"}:
+*}
+
+lemma rpath_transfer:
+  assumes "rpath r1 x xs y"
+  and "edges_on (x#xs) \<subseteq> r2"
+  shows "rpath r2 x xs y"
+  using assms
+proof(induct)
+  case (rstep x y ys z)
+  show ?case 
+  proof(rule rstepI)
+    show "(x, y) \<in> r2"
+    proof -
+      have "(x, y) \<in> edges_on  (x # y # ys)"
+          by (unfold edges_on_def, auto)
+     with rstep(4) show ?thesis by auto
+    qed
+  next
+    show "rpath r2 y ys z" 
+     using rstep edges_on_Cons_mono[of "y#ys" "x"] by (auto)
+  qed
+qed (unfold rpath_def, auto intro!:Transitive_Closure_Table.rtrancl_path.base)
+
+lemma edges_on_rpathI:
+  assumes "edges_on (a#xs@[b]) \<subseteq> r"
+  shows "rpath r a (xs@[b]) b"
+  using assms
+proof(induct xs arbitrary: a b)
+  case Nil
+  moreover have "(a, b) \<in> edges_on (a # [] @ [b])"
+      by (unfold edges_on_def, auto)
+  ultimately have "(a, b) \<in> r" by auto
+  thus ?case by auto
+next
+  case (Cons x xs a b)
+  from this(2) have "edges_on (x # xs @ [b]) \<subseteq> r" by (simp add:edges_on_unfold)
+  from Cons(1)[OF this] have " rpath r x (xs @ [b]) b" .
+  moreover from Cons(2) have "(a, x) \<in> r" by (auto simp:edges_on_unfold)
+  ultimately show ?case by (auto)
+qed
+
+text {*
+  The following lemma extracts the path from @{text "x"} to @{text "y"}
+  from proposition @{text "(x, y) \<in> r^*"}
+*}
+lemma star_rpath:
+  assumes "(x, y) \<in> r^*"
+  obtains xs where "rpath r x xs y"
+proof -
+  have "\<exists> xs. rpath r x xs y"
+  proof(unfold rpath_def, rule iffD1[OF rtranclp_eq_rtrancl_path])
+    from assms
+    show "(pred_of r)\<^sup>*\<^sup>* x y"
+      apply (fold pred_of_star)
+      by (auto simp:pred_of_def)
+  qed
+  from that and this show ?thesis by blast
+qed
+
+text {*
+  The following lemma uses the path @{text "xs"} from @{text "x"} to @{text "y"}
+  as a witness to show @{text "(x, y) \<in> r^*"}.
+*}
+lemma rpath_star: 
+  assumes "rpath r x xs y"
+  shows "(x, y) \<in> r^*"
+proof -
+  from iffD2[OF rtranclp_eq_rtrancl_path] and assms[unfolded rpath_def]
+  have "(pred_of r)\<^sup>*\<^sup>* x y" by metis
+  thus ?thesis by (simp add: pred_of_star star_2_pstar)
+qed  
+
+lemma subtree_transfer:
+  assumes "a \<in> subtree r1 a'"
+  and "r1 \<subseteq> r2"
+  shows "a \<in> subtree r2 a'"
+proof -
+  from assms(1)[unfolded subtree_def] 
+  have "(a, a') \<in> r1^*" by auto
+  from star_rpath[OF this]
+  obtain xs where rp: "rpath r1 a xs a'" by blast
+  hence "rpath r2 a xs a'"
+  proof(rule rpath_transfer)
+    from rpath_edges_on[OF rp] and assms(2)
+    show "edges_on (a # xs) \<subseteq> r2" by simp
+  qed
+  from rpath_star[OF this]
+  show ?thesis by (auto simp:subtree_def)
+qed
+
+lemma subtree_rev_transfer:
+  assumes "a \<notin> subtree r2 a'"
+  and "r1 \<subseteq> r2"
+  shows "a \<notin> subtree r1 a'"
+  using assms and subtree_transfer by metis
+
+text {*
+  The following lemmas establishes a relation from paths in @{text "r"}
+  to @{text "r^+"} relation.
+*}
+lemma rpath_plus: 
+  assumes "rpath r x xs y"
+  and "xs \<noteq> []"
+  shows "(x, y) \<in> r^+"
+proof -
+  from assms(2) obtain e es where "xs = e#es" by (cases xs, auto)
+  from assms(1)[unfolded this]
+  show ?thesis
+  proof(cases)
+    case rstep
+    show ?thesis
+    proof -
+      from rpath_star[OF rstep(2)] have "(e, y) \<in> r\<^sup>*" .
+      with rstep(1) show "(x, y) \<in> r^+" by auto
+    qed
+  qed
+qed
+
+lemma plus_rpath: 
+  assumes "(x, y) \<in> r^+"
+  obtains xs where "rpath r x xs y" and "xs \<noteq> []"
+proof -
+  from assms
+  show ?thesis
+  proof(cases rule:converse_tranclE[consumes 1])
+    case 1
+    hence "rpath r x [y] y" by auto
+    from that[OF this] show ?thesis by auto
+  next
+    case (2 z)
+    from 2(2) have "(z, y) \<in> r^*" by auto
+    from star_rpath[OF this] obtain xs where "rpath r z xs y" by auto
+    from rstepI[OF 2(1) this]
+    have "rpath r x (z # xs) y" .
+    from that[OF this] show ?thesis by auto
+  qed
+qed
+
+subsubsection {* Properties of @{text "subtree"} and @{term "ancestors"}*}
+
+lemma ancestors_subtreeI:
+  assumes "b \<in> ancestors r a"
+  shows "a \<in> subtree r b"
+  using assms by (auto simp:subtree_def ancestors_def)
+
+lemma ancestors_Field:
+  assumes "b \<in> ancestors r a"
+  obtains "a \<in> Domain r" "b \<in> Range r"
+  using assms 
+  apply (unfold ancestors_def, simp)
+  by (metis Domain.DomainI Range.intros trancl_domain trancl_range)
+
+lemma subtreeE:
+  assumes "a \<in> subtree r b"
+  obtains "a = b"
+      | "a \<noteq> b" and "b \<in> ancestors r a"
+proof -
+  from assms have "(a, b) \<in> r^*" by (auto simp:subtree_def)
+  from rtranclD[OF this]
+  have " a = b \<or> a \<noteq> b \<and> (a, b) \<in> r\<^sup>+" .
+  with that[unfolded ancestors_def] show ?thesis by auto
+qed
+
+lemma subtree_Field:
+  assumes "a \<in> Field r"
+  shows "subtree r a \<subseteq> Field r"
+by (metis Field_def UnI1 ancestors_Field assms subsetI subtreeE)
+
+lemma subtree_Field:
+  "subtree r x \<subseteq> Field r \<union> {x}"
+proof
+  fix y
+  assume "y \<in> subtree r x"
+  thus "y \<in> Field r \<union> {x}"
+  proof(cases rule:subtreeE)
+    case 1
+    thus ?thesis by auto
+  next
+    case 2
+    thus ?thesis apply (auto simp:ancestors_def)
+    using Field_def tranclD by fastforce 
+  qed
+qed
+
+lemma subtree_ancestorsI:
+  assumes "a \<in> subtree r b"
+  and "a \<noteq> b"
+  shows "b \<in> ancestors r a"
+  using assms
+  by (auto elim!:subtreeE)
+
+text {*
+  @{text "subtree"} is mono with respect to the underlying graph.
+*}
+lemma subtree_mono:
+  assumes "r1 \<subseteq> r2"
+  shows "subtree r1 x \<subseteq> subtree r2 x"
+proof
+  fix c
+  assume "c \<in> subtree r1 x"
+  hence "(c, x) \<in> r1^*" by (auto simp:subtree_def)
+  from star_rpath[OF this] obtain xs 
+  where rp:"rpath r1 c xs x" by metis
+  hence "rpath r2 c xs x"
+  proof(rule rpath_transfer)
+    from rpath_edges_on[OF rp] have "edges_on (c # xs) \<subseteq> r1" .
+    with assms show "edges_on (c # xs) \<subseteq> r2" by auto
+  qed
+  thus "c \<in> subtree r2 x"
+    by (rule rpath_star[elim_format], auto simp:subtree_def)
+qed
+
+text {*
+  The following lemma characterizes the change of sub-tree of @{text "x"}
+  with the removal of an outside edge @{text "(a,b)"}. 
+
+  Note that, according to lemma @{thm edges_in_refutation}, the assumption
+  @{term "b \<notin> subtree r x"} amounts to saying @{text "(a, b)"} 
+  is outside the sub-tree of @{text "x"}.
+*}
+lemma subtree_del_outside: (* ddd *)
+    assumes "b \<notin> subtree r x" 
+    shows "subtree (r - {(a, b)}) x = (subtree r x)" 
+proof -
+  { fix c
+    assume "c \<in> (subtree r x)"
+    hence "(c, x) \<in> r^*" by (auto simp:subtree_def)
+    hence "c \<in> subtree (r - {(a, b)}) x"
+    proof(rule star_rpath)
+      fix xs
+      assume rp: "rpath r c xs x"
+      show ?thesis
+      proof -
+        from rp
+        have "rpath  (r - {(a, b)}) c xs x"
+        proof(rule rpath_transfer)
+          from rpath_edges_on[OF rp] have "edges_on (c # xs) \<subseteq> r" .
+          moreover have "(a, b) \<notin> edges_on (c#xs)"
+          proof
+            assume "(a, b) \<in> edges_on (c # xs)"
+            then obtain l1 l2 where h: "c#xs = l1@[a,b]@l2" by (auto simp:edges_on_def)
+            hence "tl (c#xs) = tl (l1@[a,b]@l2)" by simp
+            then obtain l1' where eq_xs_b: "xs = l1'@[b]@l2" by (cases l1, auto)
+            from rp[unfolded this]
+            show False
+            proof(rule rpath_appendE)
+              assume "rpath r b l2 x"
+              thus ?thesis
+              by(rule rpath_star[elim_format], insert assms(1), auto simp:subtree_def)
+            qed
+          qed
+          ultimately show "edges_on (c # xs) \<subseteq> r - {(a,b)}" by auto
+        qed
+        thus ?thesis by (rule rpath_star[elim_format], auto simp:subtree_def)
+      qed
+    qed
+  } moreover {
+    fix c
+    assume "c \<in> subtree (r - {(a, b)}) x"
+    moreover have "... \<subseteq> (subtree r x)" by (rule subtree_mono, auto)
+    ultimately have "c \<in> (subtree r x)" by auto
+  } ultimately show ?thesis by auto
+qed
+
+(* ddd *)
+lemma subset_del_subtree_outside: (* ddd *)
+    assumes "Range r' \<inter> subtree r x = {}" 
+    shows "subtree (r - r') x = (subtree r x)" 
+proof -
+  { fix c
+    assume "c \<in> (subtree r x)"
+    hence "(c, x) \<in> r^*" by (auto simp:subtree_def)
+    hence "c \<in> subtree (r - r') x"
+    proof(rule star_rpath)
+      fix xs
+      assume rp: "rpath r c xs x"
+      show ?thesis
+      proof -
+        from rp
+        have "rpath  (r - r') c xs x"
+        proof(rule rpath_transfer)
+          from rpath_edges_on[OF rp] have "edges_on (c # xs) \<subseteq> r" .
+          moreover {
+              fix a b
+              assume h: "(a, b) \<in> r'"
+              have "(a, b) \<notin> edges_on (c#xs)"
+              proof
+                assume "(a, b) \<in> edges_on (c # xs)"
+                then obtain l1 l2 where "c#xs = (l1@[a])@[b]@l2" by (auto simp:edges_on_def)
+                hence "tl (c#xs) = tl (l1@[a,b]@l2)" by simp
+                then obtain l1' where eq_xs_b: "xs = l1'@[b]@l2" by (cases l1, auto)
+                from rp[unfolded this]
+                show False
+                proof(rule rpath_appendE)
+                  assume "rpath r b l2 x"
+                  from rpath_star[OF this]
+                  have "b \<in> subtree r x" by (auto simp:subtree_def)
+                  with assms (1) and h show ?thesis by (auto)
+                qed
+             qed
+         } ultimately show "edges_on (c # xs) \<subseteq> r - r'" by auto
+        qed
+        thus ?thesis by (rule rpath_star[elim_format], auto simp:subtree_def)
+      qed
+    qed
+  } moreover {
+    fix c
+    assume "c \<in> subtree (r - r') x"
+    moreover have "... \<subseteq> (subtree r x)" by (rule subtree_mono, auto)
+    ultimately have "c \<in> (subtree r x)" by auto
+  } ultimately show ?thesis by auto
+qed
+
+lemma subtree_insert_ext:
+    assumes "b \<in> subtree r x"
+    shows "subtree (r \<union> {(a, b)}) x = (subtree r x) \<union> (subtree r a)" 
+    using assms by (auto simp:subtree_def rtrancl_insert)
+
+lemma subtree_insert_next:
+    assumes "b \<notin> subtree r x"
+    shows "subtree (r \<union> {(a, b)}) x = (subtree r x)" 
+    using assms
+    by (auto simp:subtree_def rtrancl_insert)
+
+lemma set_add_rootI:
+  assumes "root r a"
+  and "a \<notin> Domain r1"
+  shows "root (r \<union> r1) a"
+proof -
+  let ?r = "r \<union> r1"
+  { fix a'
+    assume "a' \<in> ancestors ?r a"
+    hence "(a, a') \<in> ?r^+" by (auto simp:ancestors_def)
+    from tranclD[OF this] obtain z where "(a, z) \<in> ?r" by auto
+    moreover have "(a, z) \<notin> r"
+    proof
+      assume "(a, z) \<in> r"
+      with assms(1) show False 
+        by (auto simp:root_def ancestors_def)
+    qed
+    ultimately have "(a, z) \<in> r1" by auto
+    with assms(2) 
+    have False by (auto)
+  } thus ?thesis by (auto simp:root_def)
+qed
+
+lemma ancestors_mono:
+  assumes "r1 \<subseteq> r2"
+  shows "ancestors r1 x \<subseteq> ancestors r2 x"
+proof
+ fix a
+ assume "a \<in> ancestors r1 x"
+ hence "(x, a) \<in> r1^+" by (auto simp:ancestors_def)
+ from plus_rpath[OF this] obtain xs where 
+    h: "rpath r1 x xs a" "xs \<noteq> []" .
+ have "rpath r2 x xs a"
+ proof(rule rpath_transfer[OF h(1)])
+  from rpath_edges_on[OF h(1)] and assms
+  show "edges_on (x # xs) \<subseteq> r2" by auto
+ qed
+ from rpath_plus[OF this h(2)]
+ show "a \<in> ancestors r2 x" by (auto simp:ancestors_def)
+qed
+
+lemma subtree_refute:
+  assumes "x \<notin> ancestors r y"
+  and "x \<noteq> y"
+  shows "y \<notin> subtree r x"
+proof
+   assume "y \<in> subtree r x"
+   thus False
+     by(elim subtreeE, insert assms, auto)
+qed
+
+subsubsection {* Properties about relational trees *}
+
+context rtree 
+begin
+
+lemma ancestors_headE:
+  assumes "c \<in> ancestors r a"
+  assumes "(a, b) \<in> r"
+  obtains "b = c"
+     |   "c \<in> ancestors r b"
+proof -
+  from assms(1) 
+  have "(a, c) \<in> r^+" by (auto simp:ancestors_def)
+  hence "b = c \<or> c \<in> ancestors r b"
+  proof(cases rule:converse_tranclE[consumes 1])
+    case 1
+    with assms(2) and sgv have "b = c" by (auto simp:single_valued_def)
+    thus ?thesis by auto
+  next
+    case (2 y)
+    from 2(1) and assms(2) and sgv have "y = b" by (auto simp:single_valued_def)
+    from 2(2)[unfolded this] have "c \<in> ancestors r b" by (auto simp:ancestors_def)
+    thus ?thesis by auto
+  qed
+  with that show ?thesis by metis
+qed
+
+lemma ancestors_accum:
+  assumes "(a, b) \<in> r"
+  shows "ancestors r a = ancestors r b \<union> {b}"
+proof -
+  { fix c
+    assume "c \<in> ancestors r a"
+    hence "(a, c) \<in> r^+" by (auto simp:ancestors_def)
+    hence "c \<in> ancestors r b \<union> {b}"
+    proof(cases rule:converse_tranclE[consumes 1])
+      case 1
+      with sgv assms have "c = b" by (unfold single_valued_def, auto)
+      thus ?thesis by auto
+    next
+      case (2 c')
+      with sgv assms have "c' = b" by (unfold single_valued_def, auto)
+      from 2(2)[unfolded this]
+      show ?thesis by (auto simp:ancestors_def)
+    qed
+  } moreover {
+    fix c
+    assume "c \<in> ancestors r b \<union> {b}"
+    hence "c = b \<or> c \<in> ancestors r b" by auto
+    hence "c \<in> ancestors r a"
+    proof
+      assume "c = b"
+      from assms[folded this] 
+      show ?thesis by (auto simp:ancestors_def)
+    next
+      assume "c \<in> ancestors r b"
+      with assms show ?thesis by (auto simp:ancestors_def)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+lemma rootI:
+  assumes h: "\<And> x'. x' \<noteq> x \<Longrightarrow> x \<notin> subtree r' x'"
+  and "r' \<subseteq> r"
+  shows "root r' x"
+proof -
+  from acyclic_subset[OF acl assms(2)]
+  have acl': "acyclic r'" .
+  { fix x'
+    assume "x' \<in> ancestors r' x"
+    hence h1: "(x, x') \<in> r'^+" by (auto simp:ancestors_def)
+    have "x' \<noteq> x"
+    proof
+      assume eq_x: "x' = x"
+      from h1[unfolded this] and acl'
+      show False by (auto simp:acyclic_def)
+    qed
+    moreover from h1 have "x \<in> subtree r' x'" by (auto simp:subtree_def)
+    ultimately have False using h by auto
+  } thus ?thesis by (auto simp:root_def)
+qed
+
+lemma rpath_overlap_oneside: (* ddd *)
+  assumes "rpath r x xs1 x1"
+  and "rpath r x xs2 x2"
+  and "length xs1 \<le> length xs2"
+  obtains xs3 where "xs2 = xs1 @ xs3"
+proof(cases "xs1 = []")
+  case True
+  with that show ?thesis by auto
+next
+  case False
+  have "\<forall> i \<le> length xs1. take i xs1 = take i xs2"
+  proof -
+     { assume "\<not> (\<forall> i \<le> length xs1. take i xs1 = take i xs2)"
+       then obtain i where "i \<le> length xs1 \<and> take i xs1 \<noteq> take i xs2" by auto
+       from this(1) have "False"
+       proof(rule index_minimize)
+          fix j
+          assume h1: "j \<le> length xs1 \<and> take j xs1 \<noteq> take j xs2"
+          and h2: " \<forall>k<j. \<not> (k \<le> length xs1 \<and> take k xs1 \<noteq> take k xs2)"
+          -- {* @{text "j - 1"} is the branch point between @{text "xs1"} and @{text "xs2"} *}
+          let ?idx = "j - 1"
+          -- {* A number of inequalities concerning @{text "j - 1"} are derived first *}
+          have lt_i: "?idx < length xs1" using False h1 
+            by (metis Suc_diff_1 le_neq_implies_less length_greater_0_conv lessI less_imp_diff_less)
+          have lt_i': "?idx < length xs2" using lt_i and assms(3) by auto
+          have lt_j: "?idx < j" using h1 by (cases j, auto)
+          -- {* From thesis inequalities, a number of equations concerning @{text "xs1"}
+                 and @{text "xs2"} are derived *}
+          have eq_take: "take ?idx xs1 = take ?idx xs2"
+            using h2[rule_format, OF lt_j] and h1 by auto
+          have eq_xs1: " xs1 = take ?idx xs1 @ xs1 ! (?idx) # drop (Suc (?idx)) xs1" 
+            using id_take_nth_drop[OF lt_i] .
+          have eq_xs2: "xs2 = take ?idx xs2 @ xs2 ! (?idx) # drop (Suc (?idx)) xs2" 
+              using id_take_nth_drop[OF lt_i'] .
+          -- {* The branch point along the path is finally pinpointed *}
+          have neq_idx: "xs1!?idx \<noteq> xs2!?idx" 
+          proof -
+            have "take j xs1 = take ?idx xs1 @ [xs1 ! ?idx]"
+                using eq_xs1 Suc_diff_1 lt_i lt_j take_Suc_conv_app_nth by fastforce 
+            moreover have eq_tk2: "take j xs2 = take ?idx xs2 @ [xs2 ! ?idx]"
+                using Suc_diff_1 lt_i' lt_j take_Suc_conv_app_nth by fastforce 
+            ultimately show ?thesis using eq_take h1 by auto
+          qed
+          show ?thesis
+          proof(cases " take (j - 1) xs1 = []")
+            case True
+            have "(x, xs1!?idx) \<in> r"
+            proof -
+                from eq_xs1[unfolded True, simplified, symmetric] assms(1) 
+                have "rpath r x ( xs1 ! ?idx # drop (Suc ?idx) xs1) x1" by simp
+                from this[unfolded rpath_def]
+                show ?thesis by (auto simp:pred_of_def)
+            qed
+            moreover have "(x, xs2!?idx) \<in> r"
+            proof -
+              from eq_xs2[folded eq_take, unfolded True, simplified, symmetric] assms(2)
+              have "rpath r x ( xs2 ! ?idx # drop (Suc ?idx) xs2) x2" by simp
+              from this[unfolded rpath_def]
+              show ?thesis by (auto simp:pred_of_def)
+            qed
+            ultimately show ?thesis using neq_idx sgv[unfolded single_valued_def] by metis
+        next
+           case False
+           then obtain e es where eq_es: "take ?idx xs1 = es@[e]" 
+            using rev_exhaust by blast 
+           have "(e, xs1!?idx) \<in> r"
+           proof -
+            from eq_xs1[unfolded eq_es] 
+            have "xs1 = es@[e, xs1!?idx]@drop (Suc ?idx) xs1" by simp
+            hence "(e, xs1!?idx) \<in> edges_on xs1" by (simp add:edges_on_def, metis)
+            with rpath_edges_on[OF assms(1)] edges_on_Cons_mono[of xs1 x]
+            show ?thesis by auto
+           qed moreover have "(e, xs2!?idx) \<in> r"
+           proof -
+            from eq_xs2[folded eq_take, unfolded eq_es]
+            have "xs2 = es@[e, xs2!?idx]@drop (Suc ?idx) xs2" by simp
+            hence "(e, xs2!?idx) \<in> edges_on xs2" by (simp add:edges_on_def, metis)
+            with rpath_edges_on[OF assms(2)] edges_on_Cons_mono[of xs2 x]
+            show ?thesis by auto
+           qed
+           ultimately show ?thesis 
+              using sgv[unfolded single_valued_def] neq_idx by metis
+        qed
+       qed
+     } thus ?thesis by auto
+  qed
+  from this[rule_format, of "length xs1"]
+  have "take (length xs1) xs1 = take (length xs1) xs2" by simp
+  moreover have "xs2 = take (length xs1) xs2 @ drop (length xs1) xs2" by simp
+  ultimately have "xs2 = xs1 @ drop (length xs1) xs2" by auto
+  from that[OF this] show ?thesis .
+qed
+
+lemma rpath_overlap [consumes 2, cases pred:rpath]:
+  assumes "rpath r x xs1 x1"
+  and "rpath r x xs2 x2"
+  obtains (less_1) xs3 where "xs2 = xs1 @ xs3"
+     |    (less_2) xs3 where "xs1 = xs2 @ xs3"
+proof -
+  have "length xs1 \<le> length xs2 \<or> length xs2 \<le> length xs1" by auto
+  with assms rpath_overlap_oneside that show ?thesis by metis
+qed
+
+text {*
+  As a corollary of @{thm "rpath_overlap_oneside"}, 
+  the following two lemmas gives one important property of relation tree, 
+  i.e. there is at most one path between any two nodes.
+  Similar to the proof of @{thm rpath_overlap}, we starts with
+  the one side version first.
+*}
+
+lemma rpath_unique_oneside:
+  assumes "rpath r x xs1 y"
+    and "rpath r x xs2 y"
+    and "length xs1 \<le> length xs2"
+  shows "xs1 = xs2"
+proof -
+  from rpath_overlap_oneside[OF assms] 
+  obtain xs3 where less_1: "xs2 = xs1 @ xs3" by blast
+  show ?thesis
+  proof(cases "xs3 = []") 
+    case True
+    from less_1[unfolded this] show ?thesis by simp
+  next
+    case False
+    note FalseH = this
+    show ?thesis
+    proof(cases "xs1 = []")
+      case True
+      have "(x, x) \<in> r^+"
+      proof(rule rpath_plus)
+        from assms(1)[unfolded True] 
+        have "y = x" by (cases rule:rpath_nilE, simp)
+        from assms(2)[unfolded this] show "rpath r x xs2 x" .
+      next
+        from less_1 and False show "xs2 \<noteq> []" by simp
+      qed
+      with acl show ?thesis by (unfold acyclic_def, auto)
+    next 
+      case False
+      then obtain e es where eq_xs1: "xs1 = es@[e]" using rev_exhaust by auto
+      from assms(2)[unfolded less_1 this]
+      have "rpath r x (es @ [e] @ xs3) y" by simp
+      thus ?thesis
+      proof(cases rule:rpath_appendE)
+        case 1
+        from rpath_dest_eq [OF 1(1)[folded eq_xs1] assms(1)]
+        have "e = y" .
+        from rpath_plus [OF 1(2)[unfolded this] FalseH]
+        have "(y, y) \<in> r^+" .
+        with acl show ?thesis by (unfold acyclic_def, auto)
+      qed
+    qed
+  qed
+qed
+
+text {*
+  The following is the full version of path uniqueness.
+*}
+lemma rpath_unique:
+  assumes "rpath r x xs1 y"
+    and "rpath r x xs2 y"
+  shows "xs1 = xs2"
+proof(cases "length xs1 \<le> length xs2")
+   case True
+   from rpath_unique_oneside[OF assms this] show ?thesis .
+next
+  case False
+  hence "length xs2 \<le> length xs1" by simp
+  from rpath_unique_oneside[OF assms(2,1) this]
+  show ?thesis by simp
+qed
+
+text {*
+  The following lemma shows that the `independence` relation is symmetric.
+  It is an obvious auxiliary lemma which will be used later. 
+*}
+lemma sym_indep: "indep r x y \<Longrightarrow> indep r y x"
+  by (unfold indep_def, auto)
+
+text {*
+  This is another `obvious` lemma about trees, which says trees rooted at 
+  independent nodes are disjoint.
+*}
+lemma subtree_disjoint:
+  assumes "indep r x y"
+  shows "subtree r x \<inter> subtree r y = {}"
+proof -
+  { fix z x y xs1 xs2 xs3
+      assume ind: "indep r x y"
+      and rp1: "rpath r z xs1 x"
+      and rp2: "rpath r z xs2 y"
+      and h: "xs2 = xs1 @ xs3"
+      have False
+      proof(cases "xs1 = []")
+        case True
+        from rp1[unfolded this] have "x = z" by auto
+        from rp2[folded this] rpath_star ind[unfolded indep_def]
+        show ?thesis by metis
+      next
+        case False
+        then obtain e es where eq_xs1: "xs1 = es@[e]" using rev_exhaust by blast
+        from rp2[unfolded h this]
+        have "rpath r z (es @ [e] @ xs3) y" by simp
+        thus ?thesis
+        proof(cases rule:rpath_appendE)
+          case 1
+          have "e = x" using 1(1)[folded eq_xs1] rp1 rpath_dest_eq by metis
+          from rpath_star[OF 1(2)[unfolded this]] ind[unfolded indep_def]
+          show ?thesis by auto
+        qed
+      qed
+  } note my_rule = this
+  { fix z
+    assume h: "z \<in> subtree r x" "z \<in> subtree r y"
+    from h(1) have "(z, x) \<in> r^*" by (unfold subtree_def, auto)
+    then obtain xs1 where rp1: "rpath r z xs1 x" using star_rpath by metis
+    from h(2) have "(z, y) \<in> r^*" by (unfold subtree_def, auto)
+    then obtain xs2 where rp2: "rpath r z xs2 y" using star_rpath by metis
+    from rp1 rp2
+    have False
+    by (cases, insert my_rule[OF sym_indep[OF assms(1)] rp2 rp1] 
+                  my_rule[OF assms(1) rp1 rp2], auto)
+  } thus ?thesis by auto
+qed
+
+text {*
+  The following lemma @{text "subtree_del"} characterizes the change of sub-tree of 
+  @{text "x"} with the removal of an inside edge @{text "(a, b)"}. 
+  Note that, the case for the removal of an outside edge has already been dealt with
+  in lemma @{text "subtree_del_outside"}). 
+
+  This lemma is underpinned by the following two `obvious` facts:
+  \begin{enumearte}
+  \item
+  In graph @{text "r"}, for an inside edge @{text "(a,b) \<in> edges_in r x"},  
+  every node @{text "c"} in the sub-tree of @{text "a"} has a path
+  which goes first from @{text "c"} to @{text "a"}, then through edge @{text "(a, b)"}, and 
+  finally reaches @{text "x"}. By the uniqueness of path in a tree,
+  all paths from sub-tree of @{text "a"} to @{text "x"} are such constructed, therefore 
+  must go through @{text "(a, b)"}. The consequence is: with the removal of @{text "(a,b)"},
+  all such paths will be broken. 
+
+  \item
+  On the other hand, all paths not originate from within the sub-tree of @{text "a"}
+  will not be affected by the removal of edge @{text "(a, b)"}. 
+  The reason is simple: if the path is affected by the removal, it must 
+  contain @{text "(a, b)"}, then it must originate from within the sub-tree of @{text "a"}.
+  \end{enumearte}
+*}
+
+lemma subtree_del_inside: (* ddd *)
+    assumes "(a,b) \<in> edges_in r x"
+    shows "subtree (r - {(a, b)}) x = (subtree r x) - subtree r a"
+proof -
+  from assms have asm: "b \<in> subtree r x" "(a, b) \<in> r" by (auto simp:edges_in_def)
+  -- {* The proof follows a common pattern to prove the equality of sets. *}
+  { -- {* The `left to right` direction.
+       *}
+    fix c
+    -- {* Assuming @{text "c"} is inside the sub-tree of @{text "x"} in the reduced graph *}
+    assume h: "c \<in> subtree (r - {(a, b)}) x" 
+    -- {* We are going to show that @{text "c"} can not be in the sub-tree of @{text "a"} in 
+          the original graph. *}
+    -- {* In other words, all nodes inside the sub-tree of @{text "a"} in the original 
+          graph will be removed from the sub-tree of @{text "x"} in the reduced graph. *}
+    -- {* The reason, as analyzed before, is that all paths from within the 
+          sub-tree of @{text "a"} are broken with the removal of edge @{text "(a,b)"}.
+       *}
+    have "c \<in> (subtree r x) - subtree r a" 
+    proof -
+      let ?r' = "r - {(a, b)}" -- {* The reduced graph is abbreviated as @{text "?r'"} *}
+      from h have "(c, x) \<in> ?r'^*" by (auto simp:subtree_def)
+      -- {* Extract from the reduced graph the path @{text "xs"} from @{text "c"} to @{text "x"}. *}
+      then obtain xs where rp0: "rpath ?r' c xs x" by (rule star_rpath, auto)
+      -- {* It is easy to show @{text "xs"} is also a path in the original graph *}
+      hence rp1: "rpath r c xs x"
+      proof(rule rpath_transfer)
+          from rpath_edges_on[OF rp0] 
+          show "edges_on (c # xs) \<subseteq> r" by auto
+      qed
+      -- {* @{text "xs"} is used as the witness to show that @{text "c"} 
+                   in the sub-tree of @{text "x"} in the original graph. *}
+      hence "c \<in> subtree r x"
+         by (rule rpath_star[elim_format], auto simp:subtree_def)
+      -- {* The next step is to show that @{text "c"} can not be in the sub-tree of @{text "a"}
+            in the original graph. *}
+      -- {* We need to use the fact that all paths originate from within sub-tree of @{text "a"}
+             are broken. *}
+      moreover have "c \<notin> subtree r a"
+      proof
+        -- {* Proof by contradiction, suppose otherwise *}
+        assume otherwise: "c \<in> subtree r a"
+        -- {* Then there is a path in original graph leading from @{text "c"} to @{text "a"} *}
+        obtain xs1 where rp_c: "rpath r c xs1 a" 
+        proof -
+          from otherwise have "(c, a) \<in> r^*" by (auto simp:subtree_def)
+          thus ?thesis by (rule star_rpath, auto intro!:that)
+        qed
+        -- {* Starting from this path, we are going to construct a fictional 
+                  path from @{text "c"} to @{text "x"}, which, as explained before,
+              is broken, so that contradiction can be derived. *}
+        -- {* First, there is a path from @{text "b"} to @{text "x"} *}
+        obtain ys where rp_b: "rpath r b ys x" 
+        proof -
+          from asm have "(b, x) \<in> r^*" by (auto simp:subtree_def)
+          thus ?thesis by (rule star_rpath, auto intro!:that)
+        qed
+        -- {* The paths @{text "xs1"} and @{text "ys"} can be 
+                 tied together using @{text "(a,b)"} to form a path 
+               from @{text "c"} to @{text "x"}: *}
+        have "rpath r c (xs1 @ b # ys) x"
+        proof -
+          from rstepI[OF asm(2) rp_b] have "rpath r a (b # ys) x" .
+          from rpath_appendI[OF rp_c this]
+          show ?thesis .
+        qed
+        -- {* By the uniqueness of path between two nodes of a tree, we have: *}
+        from rpath_unique[OF rp1 this] have eq_xs: "xs = xs1 @ b # ys" .
+        -- {* Contradiction can be derived from from this fictional path . *}
+        show False
+        proof -
+          -- {* It can be shown that @{term "(a,b)"} is on this fictional path. *}
+          have "(a, b) \<in> edges_on (c#xs)"
+          proof(cases "xs1 = []")
+            case True
+            from rp_c[unfolded this] have "rpath r c [] a" .
+            hence eq_c: "c = a" by (rule rpath_nilE, simp)
+            hence "c#xs = a#xs" by simp
+            from this and eq_xs have "c#xs = a # xs1 @ b # ys" by simp
+            from this[unfolded True] have "c#xs = []@[a,b]@ys" by simp
+            thus ?thesis by (auto simp:edges_on_def)
+          next
+            case False
+            from rpath_nnl_lastE[OF rp_c this]
+            obtain xs' where "xs1 = xs'@[a]" by auto
+            from eq_xs[unfolded this] have "c#xs = (c#xs')@[a,b]@ys" by simp
+            thus ?thesis by (unfold edges_on_def, blast)
+          qed
+          -- {* It can also be shown that @{term "(a,b)"} is not on this fictional path. *}
+          moreover have "(a, b) \<notin> edges_on (c#xs)"
+              using rpath_edges_on[OF rp0] by auto
+          -- {* Contradiction is thus derived. *}
+          ultimately show False by auto
+        qed
+      qed
+      ultimately show ?thesis by auto
+    qed
+  } moreover {
+    -- {* The `right to left` direction.
+       *} 
+     fix c
+   -- {* Assuming that @{text "c"} is in the sub-tree of @{text "x"}, but
+         outside of the sub-tree of @{text "a"} in the original graph, *}
+   assume h: "c \<in> (subtree r x) - subtree r a"
+   -- {* we need to show that in the reduced graph, @{text "c"} is still in 
+         the sub-tree of @{text "x"}. *}
+   have "c \<in> subtree (r - {(a, b)}) x"
+   proof -
+      -- {* The proof goes by showing that the path from @{text "c"} to @{text "x"}
+            in the original graph is not affected by the removal of @{text "(a,b)"}.
+         *}
+      from h have "(c, x) \<in> r^*" by (unfold subtree_def, auto)
+      -- {* Extract the path @{text "xs"} from @{text "c"} to @{text "x"} in the original graph. *}
+      from star_rpath[OF this] obtain xs where rp: "rpath r c xs x" by auto
+      -- {* Show that it is also a path in the reduced graph. *}
+      hence "rpath (r - {(a, b)}) c xs x"
+      -- {* The proof goes by using rule @{thm rpath_transfer} *} 
+      proof(rule rpath_transfer)
+        -- {* We need to show all edges on the path are still in the reduced graph. *}
+        show "edges_on (c # xs) \<subseteq> r - {(a, b)}"
+        proof -
+          -- {* It is easy to show that all the edges are in the original graph. *}
+          from rpath_edges_on [OF rp] have " edges_on (c # xs) \<subseteq> r" .
+          -- {* The essential part is to show that @{text "(a, b)"} is not on the path. *}
+          moreover have "(a,b) \<notin> edges_on (c#xs)"
+          proof
+            -- {* Proof by contradiction, suppose otherwise: *}
+            assume otherwise: "(a, b) \<in> edges_on (c#xs)"
+            -- {* Then @{text "(a, b)"} is in the middle of the path. 
+                  with @{text "l1"} and @{text "l2"} be the nodes in 
+                  the front and rear respectively. *}
+              then obtain l1 l2 where eq_xs: 
+                "c#xs = l1 @ [a, b] @ l2" by (unfold edges_on_def, blast)
+            -- {* From this, it can be shown that @{text "c"} is 
+                      in the sub-tree of @{text "a"} *}
+            have "c \<in> subtree r a" 
+            proof(cases "l1 = []")
+              case True
+              -- {* If @{text "l1"} is null, it can be derived that @{text "c = a"}. *}
+              with eq_xs have "c = a" by auto
+              -- {* So, @{text "c"} is obviously in the sub-tree of @{text "a"}. *}
+              thus ?thesis by (unfold subtree_def, auto)
+            next
+              case False
+              -- {* When @{text "l1"} is not null, it must have a tail @{text "es"}: *}
+              then obtain e es where "l1 = e#es" by (cases l1, auto)
+              -- {* The relation of this tail with @{text "xs"} is derived: *}
+              with eq_xs have "xs = es@[a,b]@l2" by auto
+              -- {* From this, a path from @{text "c"} to @{text "a"} is made visible: *}
+              from rp[unfolded this] have "rpath r c (es @ [a] @ (b#l2)) x" by simp
+              thus ?thesis
+              proof(cases rule:rpath_appendE)
+                -- {* The path from @{text "c"} to @{text "a"} is extraced 
+                             using @{thm "rpath_appendE"}: *}
+                case 1
+                from rpath_star[OF this(1)] 
+                -- {* The extracted path servers as a witness that @{text "c"} is 
+                          in the sub-tree of @{text "a"}: *}
+                show ?thesis by (simp add:subtree_def)
+            qed
+          qed with h show False by auto         
+         qed ultimately show ?thesis by auto
+       qed
+     qed
+     -- {* From , it is shown that @{text "c"} is in the sub-tree of @{text "x"}
+           inthe reduced graph. *}
+     from rpath_star[OF this] show ?thesis by (auto simp:subtree_def)
+    qed
+  } 
+  -- {* The equality of sets is derived from the two directions just proved. *}
+  ultimately show ?thesis by auto
+qed 
+
+lemma  set_del_rootI:
+  assumes "r1 \<subseteq> r"
+  and "a \<in> Domain r1"
+  shows "root (r - r1) a"
+proof -
+   let ?r = "r - r1"
+  { fix a' 
+    assume neq: "a' \<noteq> a"
+    have "a \<notin> subtree ?r a'"
+    proof
+      assume "a \<in> subtree ?r a'"
+      hence "(a, a') \<in> ?r^*" by (auto simp:subtree_def)
+      from star_rpath[OF this] obtain xs
+      where rp: "rpath ?r a xs a'" by auto
+      from rpathE[OF this] and neq
+      obtain z zs where h: "(a, z) \<in> ?r" "rpath ?r z zs a'" "xs = z#zs" by auto
+      from assms(2) obtain z' where z'_in: "(a, z') \<in> r1" by (auto simp:DomainE)
+      with assms(1) have "(a, z') \<in> r" by auto
+      moreover from h(1) have "(a, z) \<in> r" by simp 
+      ultimately have "z' = z" using sgv by (auto simp:single_valued_def)
+      from z'_in[unfolded this] and h(1) show False by auto
+   qed
+  } thus ?thesis by (intro rootI, auto)
+qed
+
+lemma edge_del_no_rootI:
+  assumes "(a, b) \<in> r"
+  shows "root (r - {(a, b)}) a"
+  by (rule set_del_rootI, insert assms, auto)
+
+lemma ancestors_children_unique:
+  assumes "z1 \<in> ancestors r x \<inter> children r y"
+  and "z2 \<in> ancestors r x \<inter> children r y"
+  shows "z1 = z2"
+proof -
+  from assms have h:
+     "(x, z1) \<in> r^+" "(z1, y) \<in> r" 
+     "(x, z2) \<in> r^+" "(z2, y) \<in> r" 
+  by (auto simp:ancestors_def children_def)
+
+  -- {* From this, a path containing @{text "z1"} is obtained. *}
+  from plus_rpath[OF h(1)] obtain xs1 
+     where h1: "rpath r x xs1 z1" "xs1 \<noteq> []" by auto
+  from rpath_nnl_lastE[OF this] obtain xs1' where eq_xs1: "xs1 = xs1' @ [z1]"
+    by auto
+  from h(2) have h2: "rpath r z1 [y] y" by auto
+  from rpath_appendI[OF h1(1) h2, unfolded eq_xs1]
+  have rp1: "rpath r x (xs1' @ [z1, y]) y" by simp
+
+  -- {* Then, another path containing @{text "z2"} is obtained. *}
+  from plus_rpath[OF h(3)] obtain xs2
+     where h3: "rpath r x xs2 z2" "xs2 \<noteq> []" by auto
+  from rpath_nnl_lastE[OF this] obtain xs2' where eq_xs2: "xs2 = xs2' @ [z2]"
+    by auto
+  from h(4) have h4: "rpath r z2 [y] y" by auto
+  from rpath_appendI[OF h3(1) h4, unfolded eq_xs2]
+     have "rpath r x (xs2' @ [z2, y]) y" by simp
+
+  -- {* Finally @{text "z1 = z2"} is proved by uniqueness of path. *}
+  from rpath_unique[OF rp1 this]
+  have "xs1' @ [z1, y] = xs2' @ [z2, y]" .
+  thus ?thesis by auto
+qed
+
+lemma ancestors_childrenE:
+  assumes "y \<in> ancestors r x"
+  obtains "x \<in> children r y"
+      | z where "z \<in> ancestors r x \<inter> children r y"
+proof -
+  from assms(1) have "(x, y) \<in> r^+" by (auto simp:ancestors_def)
+  from tranclD2[OF this] obtain z where 
+     h: "(x, z) \<in> r\<^sup>*" "(z, y) \<in> r" by auto
+  from h(1)
+  show ?thesis
+  proof(cases rule:rtranclE)
+    case base
+    from h(2)[folded this] have "x \<in> children r y" 
+              by (auto simp:children_def)
+    thus ?thesis by (intro that, auto)
+  next
+    case (step u)
+    hence "z \<in> ancestors r x" by (auto simp:ancestors_def)
+    moreover from h(2) have "z \<in> children r y" 
+              by (auto simp:children_def)
+    ultimately show ?thesis by (intro that, auto)
+  qed
+qed
+
+
+end (* of rtree *)
+
+lemma subtree_children:
+  "subtree r x = {x} \<union> (\<Union> (subtree r ` (children r x)))" (is "?L = ?R")
+proof -
+  { fix z
+    assume "z \<in> ?L"
+    hence "z \<in> ?R"
+    proof(cases rule:subtreeE[consumes 1])
+      case 2
+      hence "(z, x) \<in> r^+" by (auto simp:ancestors_def)
+      thus ?thesis
+      proof(rule tranclE)
+        assume "(z, x) \<in> r"
+        hence "z \<in> children r x" by (unfold children_def, auto)
+        moreover have "z \<in> subtree r z" by (auto simp:subtree_def)
+        ultimately show ?thesis by auto
+      next
+        fix c
+        assume h: "(z, c) \<in> r\<^sup>+" "(c, x) \<in> r"
+        hence "c \<in> children r x" by (auto simp:children_def)
+        moreover from h have "z \<in> subtree r c" by (auto simp:subtree_def)
+        ultimately show ?thesis by auto
+      qed
+    qed auto
+  } moreover {
+    fix z
+    assume h: "z \<in> ?R"
+    have "x \<in> subtree r x" by (auto simp:subtree_def)
+    moreover {
+       assume "z \<in> \<Union>(subtree r ` children r x)"
+       then obtain y where "(y, x) \<in> r" "(z, y) \<in> r^*" 
+        by (auto simp:subtree_def children_def)
+       hence "(z, x) \<in> r^*" by auto
+       hence "z \<in> ?L" by (auto simp:subtree_def)
+    } ultimately have "z \<in> ?L" using h by auto
+  } ultimately show ?thesis by auto
+qed
+
+context fsubtree 
+begin
+  
+lemma finite_subtree:
+  shows "finite (subtree r x)"
+proof(induct rule:wf_induct[OF wf])
+  case (1 x)
+  have "finite (\<Union>(subtree r ` children r x))"
+  proof(rule finite_Union)
+    show "finite (subtree r ` children r x)"
+    proof(cases "children r x = {}")
+      case True
+      thus ?thesis by auto
+    next
+      case False
+      hence "x \<in> Range r" by (auto simp:children_def)
+      from fb[rule_format, OF this] 
+      have "finite (children r x)" .
+      thus ?thesis by (rule finite_imageI)
+    qed
+  next
+    fix M 
+    assume "M \<in> subtree r ` children r x"
+    then obtain y where h: "y \<in> children r x" "M = subtree r y" by auto
+    hence "(y, x) \<in> r" by (auto simp:children_def)
+    from 1[rule_format, OF this, folded h(2)]
+    show "finite M" .
+  qed
+  thus ?case
+    by (unfold subtree_children finite_Un, auto)
+qed
+
+end
+
+definition "pairself f = (\<lambda>(a, b). (f a, f b))"
+
+definition "rel_map f r = (pairself f ` r)"
+
+lemma rel_mapE: 
+  assumes "(a, b) \<in> rel_map f r"
+  obtains c d 
+  where "(c, d) \<in> r" "(a, b) = (f c, f d)"
+  using assms
+  by (unfold rel_map_def pairself_def, auto)
+
+lemma rel_mapI: 
+  assumes "(a, b) \<in> r"
+    and "c = f a"
+    and "d = f b"
+  shows "(c, d) \<in> rel_map f r"
+  using assms
+  by (unfold rel_map_def pairself_def, auto)
+
+lemma map_appendE:
+  assumes "map f zs = xs @ ys"
+  obtains xs' ys' 
+  where "zs = xs' @ ys'" "xs = map f xs'" "ys = map f ys'"
+proof -
+  have "\<exists> xs' ys'. zs = xs' @ ys' \<and> xs = map f xs' \<and> ys = map f ys'"
+  using assms
+  proof(induct xs arbitrary:zs ys)
+    case (Nil zs ys)
+    thus ?case by auto
+  next
+    case (Cons x xs zs ys)
+    note h = this
+    show ?case
+    proof(cases zs)
+      case (Cons e es)
+      with h have eq_x: "map f es = xs @ ys" "x = f e" by auto
+      from h(1)[OF this(1)]
+      obtain xs' ys' where "es = xs' @ ys'" "xs = map f xs'" "ys = map f ys'"
+        by blast
+      with Cons eq_x
+      have "zs = (e#xs') @ ys' \<and> x # xs = map f (e#xs') \<and> ys = map f ys'" by auto
+      thus ?thesis by metis
+    qed (insert h, auto)
+  qed
+  thus ?thesis by (auto intro!:that)
+qed
+
+lemma rel_map_mono:
+  assumes "r1 \<subseteq> r2"
+  shows "rel_map f r1 \<subseteq> rel_map f r2"
+  using assms
+  by (auto simp:rel_map_def pairself_def)
+
+lemma rel_map_compose [simp]:
+    shows "rel_map f1 (rel_map f2 r) = rel_map (f1 o f2) r"
+    by (auto simp:rel_map_def pairself_def)
+
+lemma edges_on_map: "edges_on (map f xs) = rel_map f (edges_on xs)"
+proof -
+  { fix a b
+    assume "(a, b) \<in> edges_on (map f xs)"
+    then obtain l1 l2 where eq_map: "map f xs = l1 @ [a, b] @ l2" 
+      by (unfold edges_on_def, auto)
+    hence "(a, b) \<in> rel_map f (edges_on xs)"
+      by (auto elim!:map_appendE intro!:rel_mapI simp:edges_on_def)
+  } moreover { 
+    fix a b
+    assume "(a, b) \<in> rel_map f (edges_on xs)"
+    then obtain c d where 
+        h: "(c, d) \<in> edges_on xs" "(a, b) = (f c, f d)" 
+             by (elim rel_mapE, auto)
+    then obtain l1 l2 where
+        eq_xs: "xs = l1 @ [c, d] @ l2" 
+             by (auto simp:edges_on_def)
+    hence eq_map: "map f xs = map f l1 @ [f c, f d] @ map f l2" by auto
+    have "(a, b) \<in> edges_on (map f xs)"
+    proof -
+      from h(2) have "[f c, f d] = [a, b]" by simp
+      from eq_map[unfolded this] show ?thesis by (auto simp:edges_on_def)
+    qed
+  } ultimately show ?thesis by auto
+qed
+
+lemma image_id:
+  assumes "\<And> x. x \<in> A \<Longrightarrow> f x = x"
+  shows "f ` A = A"
+  using assms by (auto simp:image_def)
+
+lemma rel_map_inv_id:
+  assumes "inj_on f ((Domain r) \<union> (Range r))"
+  shows "(rel_map (inv_into ((Domain r) \<union> (Range r)) f \<circ> f) r) = r"
+proof -
+ let ?f = "(inv_into (Domain r \<union> Range r) f \<circ> f)"
+ {
+  fix a b
+  assume h0: "(a, b) \<in> r"
+  have "pairself ?f (a, b) = (a, b)"
+  proof -
+    from assms h0 have "?f a = a" by (auto intro:inv_into_f_f)
+    moreover have "?f b = b"
+      by (insert h0, simp, intro inv_into_f_f[OF assms], auto intro!:RangeI)
+    ultimately show ?thesis by (auto simp:pairself_def)
+  qed
+ } thus ?thesis by (unfold rel_map_def, intro image_id, case_tac x, auto)
+qed 
+
+lemma rel_map_acyclic:
+  assumes "acyclic r"
+  and "inj_on f ((Domain r) \<union> (Range r))"
+  shows "acyclic (rel_map f r)"
+proof -
+  let ?D = "Domain r \<union> Range r"
+  { fix a 
+    assume "(a, a) \<in> (rel_map f r)^+" 
+    from plus_rpath[OF this]
+    obtain xs where rp: "rpath (rel_map f r) a xs a" "xs \<noteq> []" by auto
+    from rpath_nnl_lastE[OF this] obtain xs' where eq_xs: "xs = xs'@[a]" by auto
+    from rpath_edges_on[OF rp(1)]
+    have h: "edges_on (a # xs) \<subseteq> rel_map f r" .
+    from edges_on_map[of "inv_into ?D f" "a#xs"]
+    have "edges_on (map (inv_into ?D f) (a # xs)) = rel_map (inv_into ?D f) (edges_on (a # xs))" .
+    with rel_map_mono[OF h, of "inv_into ?D f"]
+    have "edges_on (map (inv_into ?D f) (a # xs)) \<subseteq> rel_map ((inv_into ?D f) o f) r" by simp
+    from this[unfolded eq_xs]
+    have subr: "edges_on (map (inv_into ?D f) (a # xs' @ [a])) \<subseteq> rel_map (inv_into ?D f \<circ> f) r" .
+    have "(map (inv_into ?D f) (a # xs' @ [a])) = (inv_into ?D f a) # map (inv_into ?D f) xs' @ [inv_into ?D f a]"
+      by simp
+    from edges_on_rpathI[OF subr[unfolded this]]
+    have "rpath (rel_map (inv_into ?D f \<circ> f) r) 
+                      (inv_into ?D f a) (map (inv_into ?D f) xs' @ [inv_into ?D f a]) (inv_into ?D f a)" .
+    hence "(inv_into ?D f a, inv_into ?D f a) \<in> (rel_map (inv_into ?D f \<circ> f) r)^+"
+        by (rule rpath_plus, simp)
+    moreover have "(rel_map (inv_into ?D f \<circ> f) r) = r" by (rule rel_map_inv_id[OF assms(2)])
+    moreover note assms(1) 
+    ultimately have False by (unfold acyclic_def, auto)
+  } thus ?thesis by (auto simp:acyclic_def)
+qed
+
+lemma relpow_mult: 
+  "((r::'a rel) ^^ m) ^^ n = r ^^ (m*n)"
+proof(induct n arbitrary:m)
+  case (Suc k m)
+  thus ?case
+  proof -
+    have h: "(m * k + m) = (m + m * k)" by auto
+    show ?thesis 
+      apply (simp add:Suc relpow_add[symmetric])
+      by (unfold h, simp)
+  qed
+qed simp
+
+lemma compose_relpow_2:
+  assumes "r1 \<subseteq> r"
+  and "r2 \<subseteq> r"
+  shows "r1 O r2 \<subseteq> r ^^ (2::nat)"
+proof -
+  { fix a b
+    assume "(a, b) \<in> r1 O r2"
+    then obtain e where "(a, e) \<in> r1" "(e, b) \<in> r2"
+      by auto
+    with assms have "(a, e) \<in> r" "(e, b) \<in> r" by auto
+    hence "(a, b) \<in> r ^^ (Suc (Suc 0))" by auto
+  } thus ?thesis by (auto simp:numeral_2_eq_2)
+qed
+
+lemma acyclic_compose:
+  assumes "acyclic r"
+  and "r1 \<subseteq> r"
+  and "r2 \<subseteq> r"
+  shows "acyclic (r1 O r2)"
+proof -
+  { fix a
+    assume "(a, a) \<in> (r1 O r2)^+"
+    from trancl_mono[OF this compose_relpow_2[OF assms(2, 3)]]
+    have "(a, a) \<in> (r ^^ 2) ^+" .
+    from trancl_power[THEN iffD1, OF this]
+    obtain n where h: "(a, a) \<in> (r ^^ 2) ^^ n" "n > 0" by blast
+    from this(1)[unfolded relpow_mult] have h2: "(a, a) \<in> r ^^ (2 * n)" .
+    have "(a, a) \<in> r^+" 
+    proof(cases rule:trancl_power[THEN iffD2])
+      from h(2) h2 show "\<exists>n>0. (a, a) \<in> r ^^ n" 
+        by (rule_tac x = "2*n" in exI, auto)
+    qed
+    with assms have "False" by (auto simp:acyclic_def)
+  } thus ?thesis by (auto simp:acyclic_def)
+qed
+
+lemma children_compose_unfold: 
+  "children (r1 O r2) x = \<Union> (children r1 ` (children r2 x))"
+  by (auto simp:children_def)
+
+lemma fbranch_compose:
+  assumes "fbranch r1"
+  and "fbranch r2"
+  shows "fbranch (r1 O r2)"
+proof -
+  {  fix x
+     assume "x\<in>Range (r1 O r2)"
+     then obtain y z where h: "(y, z) \<in> r1" "(z, x) \<in> r2" by auto
+     have "finite (children (r1 O r2) x)"
+     proof(unfold children_compose_unfold, rule finite_Union)
+      show "finite (children r1 ` children r2 x)"
+      proof(rule finite_imageI)
+        from h(2) have "x \<in> Range r2" by auto
+        from assms(2)[unfolded fbranch_def, rule_format, OF this]
+        show "finite (children r2 x)" .
+      qed
+     next
+       fix M
+       assume "M \<in> children r1 ` children r2 x"
+       then obtain y where h1: "y \<in> children r2 x" "M = children r1 y" by auto
+       show "finite M"
+       proof(cases "children r1 y = {}")
+          case True
+          with h1(2) show ?thesis by auto
+       next
+          case False
+          hence "y \<in> Range r1" by (unfold children_def, auto)
+          from assms(1)[unfolded fbranch_def, rule_format, OF this, folded h1(2)]
+          show ?thesis .
+       qed
+     qed
+  } thus ?thesis by (unfold fbranch_def, auto)
+qed
+
+lemma finite_fbranchI:
+  assumes "finite r"
+  shows "fbranch r"
+proof -
+  { fix x 
+    assume "x \<in>Range r"
+    have "finite (children r x)"
+    proof -
+      have "{y. (y, x) \<in> r} \<subseteq> Domain r" by (auto)
+      from rev_finite_subset[OF finite_Domain[OF assms] this]
+      have "finite {y. (y, x) \<in> r}" .
+      thus ?thesis by (unfold children_def, simp)
+    qed
+  } thus ?thesis by (auto simp:fbranch_def)
+qed
+
+lemma subset_fbranchI:
+  assumes "fbranch r1"
+  and "r2 \<subseteq> r1"
+  shows "fbranch r2"
+proof -
+  { fix x
+    assume "x \<in>Range r2"
+    with assms(2) have "x \<in> Range r1" by auto
+    from assms(1)[unfolded fbranch_def, rule_format, OF this]
+    have "finite (children r1 x)" .
+    hence "finite (children r2 x)"
+    proof(rule rev_finite_subset)
+      from assms(2)
+      show "children r2 x \<subseteq> children r1 x" by (auto simp:children_def)
+    qed
+  } thus ?thesis by (auto simp:fbranch_def)
+qed
+
+lemma children_subtree: 
+  shows "children r x \<subseteq> subtree r x"
+  by (auto simp:children_def subtree_def)
+
+lemma children_union_kept:
+  assumes "x \<notin> Range r'"
+  shows "children (r \<union> r') x = children r x"
+  using assms
+  by (auto simp:children_def)
+
+end
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