# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1505999755 -3600
# Node ID b32b3bd99150e73b15985a6118b413a1918431d5
# Parent  c3a42076b1648e44a2328031f02ec92912ad6d6c
cleaned up

diff -r c3a42076b164 -r b32b3bd99150 Attic/Test.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Test.thy	Thu Sep 21 14:15:55 2017 +0100
@@ -0,0 +1,789 @@
+theory Test 
+imports Precedence_ord Graphs
+begin
+
+type_synonym thread = nat -- {* Type for thread identifiers. *}
+type_synonym priority = nat  -- {* Type for priorities. *}
+type_synonym cs = nat -- {* Type for critical sections (or resources). *}
+
+-- {* Schedulling Events *}
+
+datatype event = 
+  Create thread priority 
+| Exit thread 
+| P thread cs 
+| V thread cs 
+| Set thread priority 
+
+type_synonym state = "event list"
+
+fun threads :: "state \<Rightarrow> thread set"
+  where 
+  "threads [] = {}" 
+| "threads (Create th prio#s) = {th} \<union> threads s" 
+| "threads (Exit th # s) = (threads s) - {th}" 
+| "threads (_#s) = threads s" 
+
+fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
+  where
+  "priority th [] = 0" 
+| "priority th (Create th' prio#s) = (if th' = th then prio else priority th s)" 
+| "priority th (Set th' prio#s) = (if th' = th then prio else priority th s)" 
+| "priority th (_#s) = priority th s"
+
+fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
+  where
+  "last_set th [] = 0" 
+| "last_set th ((Create th' prio)#s) = (if (th = th') then length s else last_set th s)" 
+| "last_set th ((Set th' prio)#s) = (if (th = th') then length s else last_set th s)" 
+| "last_set th (_#s) = last_set th s"
+
+
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+  where "preced th s \<equiv> Prc (priority th s) (last_set th s)"
+
+abbreviation 
+  "preceds s ths \<equiv> {preced th s | th. th \<in> ths}"
+ 
+definition
+  "holds wq th cs \<equiv> th \<in> set (wq cs) \<and> th = hd (wq cs)"
+
+definition
+  "waits wq th cs \<equiv> th \<in> set (wq cs) \<and> th \<noteq> hd (wq cs)"
+
+--{* Nodes in Resource Graph *}
+datatype node = 
+  Th "thread" 
+| Cs "cs" 
+
+definition
+  "RAG wq \<equiv> {(Th th, Cs cs) | th cs. waits wq th cs} \<union> {(Cs cs, Th th) | cs th. holds wq th cs}"
+
+definition
+  "dependants wq th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
+
+record schedule_state = 
+  wq_fun :: "cs \<Rightarrow> thread list" 
+  cprec_fun :: "thread \<Rightarrow> precedence" 
+
+definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
+  where 
+  "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
+
+abbreviation
+  "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
+
+abbreviation 
+  "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
+ 
+abbreviation
+  "release qs \<equiv> case qs of
+             [] => [] 
+          |  (_ # qs) => SOME q. distinct q \<and> set q = set qs"
+
+lemma [simp]: 
+  "(SOME q. distinct q \<and> q = []) = []"
+by auto
+
+lemma [simp]: 
+  "(x \<in> set (SOME q. distinct q \<and> set q = set p)) = (x \<in> set p)"
+apply(rule iffI)
+apply (metis (mono_tags, lifting) List.finite_set finite_distinct_list some_eq_ex)+
+done
+
+abbreviation
+  "next_to_run ths \<equiv> hd (SOME q::thread list. distinct q \<and> set q = set ths)"
+
+
+fun schs :: "state \<Rightarrow> schedule_state"
+  where 
+  "schs [] = (| wq_fun = \<lambda> cs. [],  cprec_fun = (\<lambda>_. Prc 0 0) |)" 
+| "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)|))"
+|  "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)|))"
+
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
+  where "wq s = wq_fun (schs s)"
+
+definition cpreced2 :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+  where "cpreced2 s \<equiv> cprec_fun (schs s)"
+
+abbreviation
+  "cpreceds2 s ths \<equiv> {cpreced2 s th | th. th \<in> ths}"
+
+definition
+  "holds2 s \<equiv> holds (wq_fun (schs s))"
+
+definition
+  "waits2 s \<equiv> waits (wq_fun (schs s))"
+
+definition
+  "RAG2 s \<equiv> RAG (wq_fun (schs s))"
+  
+definition
+  "dependants2 s \<equiv> dependants (wq_fun (schs s))"
+
+(* ready -> is a thread that is not waiting for any resource *) 
+definition readys :: "state \<Rightarrow> thread set"
+  where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waits2 s th cs)}"
+
+definition runing :: "state \<Rightarrow> thread set"
+  where "runing s \<equiv> {th . th \<in> readys s \<and> cpreced2 s th = Max (cpreceds2 s (readys s))}"
+
+(* all resources a thread hols in a state *)
+definition holding :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+  where "holding s th \<equiv> {cs . holds2 s th cs}"
+
+
+lemma exists_distinct:
+  obtains ys where "distinct ys" "set ys = set xs"
+by (metis List.finite_set finite_distinct_list)
+
+lemma next_to_run_set [simp]:
+  "wts \<noteq> [] \<Longrightarrow> next_to_run wts \<in> set wts"
+apply(rule exists_distinct[of wts])
+by (metis (mono_tags, lifting) hd_in_set set_empty some_eq_ex)
+
+lemma holding_RAG: 
+  "holding s th = {cs . (Cs cs, Th th) \<in> RAG2 s}"
+unfolding holding_def RAG2_def RAG_def
+unfolding holds2_def holds_def waits_def
+by auto
+
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+  where
+  step_Create: "\<lbrakk>th \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create th prio)" 
+| step_Exit: "\<lbrakk>th \<in> runing s; holding s th = {}\<rbrakk> \<Longrightarrow> step s (Exit th)" 
+| step_P: "\<lbrakk>th \<in> runing s;  (Cs cs, Th th)  \<notin> (RAG2 s)^+\<rbrakk> \<Longrightarrow> step s (P th cs)" 
+| step_V: "\<lbrakk>th \<in> runing s;  holds2 s th cs\<rbrakk> \<Longrightarrow> step s (V th cs)" 
+| step_Set: "\<lbrakk>th \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set th prio)"
+
+(* valid states *)
+inductive vt :: "state \<Rightarrow> bool"
+  where
+  vt_nil[intro]: "vt []" 
+| vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
+
+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_threads: 
+  assumes vt: "vt s"
+  and h: "th \<in> set (wq s cs)"
+  shows "th \<in> threads s"
+using assms
+apply(induct)
+apply(simp add: wq_def)
+apply(erule step.cases)
+apply(auto simp add: wq_def Let_def holding_def holds2_def holds_def waits2_def runing_def readys_def)
+apply(simp add: waits_def)
+apply(auto simp add: waits_def split: if_splits)[1]
+apply(auto split: if_splits)
+apply(simp only: waits_def)
+by (metis insert_iff set_simps(2))
+
+
+
+lemma Domain_RAG_threads:
+  assumes vt: "vt s"
+  and in_dom: "(Th th) \<in> Domain (RAG2 s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where "(Th th, n) \<in> RAG2 s" by auto
+  then obtain cs where "n = Cs cs" "(Th th, Cs cs) \<in> RAG2 s"  
+    unfolding RAG2_def RAG_def by auto
+  then have "th \<in> set (wq s cs)"
+    unfolding wq_def RAG_def RAG2_def waits_def by auto
+  with wq_threads [OF vt] show ?thesis .
+qed
+
+lemma dependants_threads:
+  assumes vt: "vt s"
+  shows "dependants2 s th \<subseteq> threads s"
+proof
+  fix th1 
+  assume "th1 \<in> dependants2 s th"
+  then have h: "(Th th1, Th th) \<in> (RAG2 s)\<^sup>+"
+    unfolding dependants2_def dependants_def RAG2_def by simp
+  then have "Th th1 \<in> Domain ((RAG2 s)\<^sup>+)" unfolding Domain_def by auto
+  then have "Th th1 \<in> Domain (RAG2 s)" using trancl_domain by simp
+  then show "th1 \<in> threads s" using vt by (rule_tac Domain_RAG_threads)
+qed
+
+lemma finite_threads:
+  assumes vt: "vt s"
+  shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+
+section {* Distinctness of @{const wq} *}
+
+lemma wq_distinct_step: 
+  assumes "step s e" "distinct (wq s cs)" 
+  shows "distinct (wq (e # s) cs)"
+using assms
+unfolding wq_def
+apply(erule_tac step.cases)
+apply(auto simp add: RAG2_def RAG_def Let_def)[1]
+apply(auto simp add: wq_def Let_def RAG2_def RAG_def holds_def runing_def waits2_def waits_def readys_def)
+apply(auto split: list.split)
+apply(rule someI2)
+apply(auto)
+done
+
+lemma wq_distinct: 
+  assumes "vt s" 
+  shows "distinct (wq s cs)"
+using assms
+apply(induct)
+apply(simp add: wq_def)
+apply(simp add: wq_distinct_step)
+done
+
+
+section {* Single_Valuedness of @{const waits2}, @{const holds2}, @{const RAG2} *}
+
+lemma waits2_unique:
+  assumes "vt s"
+  and "waits2 s th cs1"
+  and "waits2 s th cs2"
+  shows "cs1 = cs2"
+using assms
+apply(induct)
+apply(simp add: waits2_def waits_def)
+apply(erule step.cases)
+apply(auto simp add: Let_def waits2_def waits_def holds_def RAG2_def RAG_def 
+ readys_def runing_def split: if_splits)
+apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
+apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
+apply (metis distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
+by (metis (full_types, hide_lams) distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
+
+lemma single_valued_waits2:
+  assumes "vt s"
+  shows "single_valuedP (waits2 s)"
+using assms
+unfolding single_valued_def
+by (simp add: Product_Type.Collect_case_prodD waits2_unique)
+
+lemma single_valued_holds2:
+  assumes "vt s"
+  shows "single_valuedP (\<lambda>cs th. holds2 s th cs)"
+unfolding single_valued_def holds2_def holds_def by simp
+
+lemma single_valued_RAG2:
+  assumes "vt s"
+  shows "single_valued (RAG2 s)"
+using single_valued_waits2[OF assms] single_valued_holds2[OF assms] 
+unfolding RAG2_def RAG_def
+apply(rule_tac single_valued_union)
+unfolding holds2_def[symmetric] waits2_def[symmetric]
+apply(rule single_valued_Collect)
+apply(simp)
+apply(simp add: inj_on_def)
+apply(rule single_valued_Collect)
+apply(simp)
+apply(simp add: inj_on_def)
+apply(auto)
+done
+
+
+section {* Properties of @{const RAG2} under events *}
+
+lemma RAG_Set [simp]: 
+  shows "RAG2 (Set th prio # s) = RAG2 s"
+unfolding RAG2_def
+by (simp add: Let_def)
+
+lemma RAG_Create [simp]: 
+  "RAG2 (Create th prio # s) = RAG2 s"
+unfolding RAG2_def
+by (simp add: Let_def)
+
+lemma RAG_Exit [simp]: 
+  shows "RAG2 (Exit th # s) = RAG2 s"
+unfolding RAG2_def
+by (simp add: Let_def)
+
+lemma RAG_P1:
+  assumes "wq s cs = []"
+  shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Cs cs, Th th)}"
+using assms
+unfolding  RAG2_def RAG_def wq_def Let_def waits_def holds_def
+by (auto simp add: Let_def)
+
+lemma RAG_P2:
+  assumes "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" "wq s cs \<noteq> []"
+  shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Th th, Cs cs)}"
+using assms
+unfolding RAG2_def RAG_def wq_def Let_def waits_def holds_def
+by (auto simp add: Let_def)
+
+
+lemma RAG_V1:
+assumes vt: "wq s cs = [th]"
+shows "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}"
+using assms
+unfolding RAG2_def RAG_def waits_def holds_def wq_def
+by (auto simp add: Let_def)
+
+lemma RAG_V2:
+assumes vt:"vt s" "wq s cs = th # wts \<and> wts \<noteq> []"
+shows "RAG2 (V th cs # s) \<subseteq>
+  RAG2 s - {(Cs cs, Th th), (Th (next_to_run wts), Cs cs)} \<union> {(Cs cs, Th (next_to_run wts))}"
+unfolding RAG2_def RAG_def waits_def holds_def
+using assms wq_distinct[OF vt(1), of"cs"]
+by (auto simp add: Let_def wq_def)
+
+
+
+section {* Acyclicity of @{const RAG2} *}
+
+lemma acyclic_RAG_step: 
+  assumes vt: "vt s"
+  and     stp: "step s e"
+  and     ac: "acyclic (RAG2 s)"
+  shows "acyclic (RAG2 (e # s))"
+using stp vt ac
+proof (induct)
+  case (step_P th s cs)
+  have ac: "acyclic (RAG2 s)" by fact
+  have ds: "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" by fact
+  { assume wq_empty: "wq s cs = []" -- "case waiting queue is empty"
+    then have "(Th th, Cs cs) \<notin> (RAG2 s)\<^sup>+"
+    proof (rule_tac notI)
+      assume "(Th th, Cs cs) \<in> (RAG2 s)\<^sup>+"
+      then obtain x where "(x, Cs cs) \<in> RAG2 s" using tranclD2 by metis 
+      with wq_empty show False by (auto simp: RAG2_def RAG_def wq_def waits_def)
+    qed
+    with ac have "acyclic (RAG2 s \<union> {(Cs cs, Th th)})" by simp
+    then have "acyclic (RAG2 (P th cs # s))" using RAG_P1[OF wq_empty] 
+      by (rule acyclic_subset)
+  }
+  moreover
+  { assume wq_not_empty: "wq s cs \<noteq> []" -- "case waiting queue is not empty"
+    from ac ds
+    have "acyclic (RAG2 s \<union> {(Th th, Cs cs)})" by simp
+    then have "acyclic (RAG2 (P th cs # s))" using RAG_P2[OF ds wq_not_empty] 
+      by (rule acyclic_subset)
+  }
+  ultimately show "acyclic (RAG2 (P th cs # s))" by metis
+next    
+  case (step_V th s cs) -- "case for release of a lock" 
+  have vt: "vt s" by fact
+  have ac: "acyclic (RAG2 s)" by fact
+  have hd: "holds2 s th cs" by fact
+  from vt have wq_distinct:"distinct (wq s cs)" by (rule wq_distinct)
+  from hd have "th \<in> set (wq s cs)" "th = hd (wq s cs)" unfolding holds2_def holds_def wq_def by auto
+  then obtain wts where eq_wq: "wq s cs = th # wts"  by (cases "wq s cs") (auto)
+  -- "case no thread present in the waiting queue to take over"
+  { assume "wts = []" 
+    with eq_wq have "wq s cs = [th]" by simp
+    then have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}" by (rule RAG_V1)
+    moreover have "acyclic (RAG2 s - {(Cs cs, Th th)})" using ac by (auto intro: acyclic_subset)
+    ultimately 
+    have "acyclic (RAG2 (V th cs # s))" by (auto intro: acyclic_subset)
+  }
+  moreover
+  -- "at least one thread present to take over"
+  { def nth \<equiv> "next_to_run wts"
+    assume wq_not_empty: "wts \<noteq> []" 
+    have "waits2 s nth cs" 
+      using eq_wq wq_not_empty wq_distinct
+      unfolding nth_def waits2_def waits_def wq_def[symmetric] by auto
+    then have cs_in_RAG: "(Th nth, Cs cs) \<in> RAG2 s" 
+      unfolding RAG2_def RAG_def waits2_def by auto
+    have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)}" 
+      unfolding nth_def using  vt wq_not_empty eq_wq by (rule_tac RAG_V2) (auto)
+    moreover 
+    have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)})" 
+    proof -
+      have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})" using ac by (auto intro: acyclic_subset)
+      moreover 
+      have "(Th nth, Cs cs) \<notin> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+" 
+      proof (rule notI)
+        assume "(Th nth, Cs cs) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+"
+        then obtain z where a: "(Th nth, z) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})"
+          by (metis converse_tranclE)
+        then have "(Th nth, z) \<in> RAG2 s" by simp
+        then have "z = Cs cs" using cs_in_RAG single_valued_RAG2[OF vt]
+          by (simp add: single_valued_def)
+        then show "False" using a by simp
+      qed
+      ultimately 
+      show "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth) })" by simp
+    qed
+    ultimately have "acyclic (RAG2 (V th cs # s))" 
+      by (rule_tac acyclic_subset)
+  }
+  ultimately show "acyclic (RAG2 (V th cs # s))" by metis
+qed (simp_all)
+
+
+lemma finite_RAG:
+  assumes "vt s"
+  shows "finite (RAG2 s)"
+using assms
+apply(induct)
+apply(simp add: RAG2_def RAG_def waits_def holds_def)
+apply(erule step.cases)
+apply(auto)
+apply(case_tac "wq sa cs = []")
+apply(rule finite_subset)
+apply(rule RAG_P1)
+apply(simp)
+apply(simp)
+apply(rule finite_subset)
+apply(rule RAG_P2)
+apply(simp)
+apply(simp)
+apply(simp)
+apply(subgoal_tac "\<exists>wts. wq sa cs = th # wts")
+apply(erule exE)
+apply(case_tac "wts = []")
+apply(rule finite_subset)
+apply(rule RAG_V1)
+apply(simp)
+apply(simp)
+apply(rule finite_subset)
+apply(rule RAG_V2)
+apply(simp)
+apply(simp)
+apply(simp)
+apply(subgoal_tac "th \<in> set (wq sa cs) \<and> th = hd (wq sa cs)") 
+apply(case_tac "wq sa cs") 
+apply(auto)[2]
+apply(auto simp add: holds2_def holds_def wq_def)
+done
+
+
+
+lemma dchain_unique:
+  assumes vt: "vt s"
+  and th1_d: "(n, Th th1) \<in> (RAG2 s)^+"
+  and th1_r: "th1 \<in> readys s"
+  and th2_d: "(n, Th th2) \<in> (RAG2 s)^+"
+  and th2_r: "th2 \<in> readys s"
+  shows "th1 = th2"
+proof(rule ccontr)
+  assume neq: "th1 \<noteq> th2"
+   with single_valued_confluent2 [OF single_valued_RAG2 [OF vt]] th1_d th2_d
+  have "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG2 s)\<^sup>+" by auto
+  moreover
+  { assume "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+"      
+    then obtain n where dd: "(Th th1, n) \<in> RAG2 s" by (metis converse_tranclE)
+    then obtain cs where eq_n: "n = Cs cs"
+      unfolding RAG2_def RAG_def by (case_tac n) (auto)
+    from dd eq_n have "th1 \<notin> readys s"
+      unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
+    with th1_r have "False" by auto
+  }
+  moreover
+  { assume "(Th th2, Th th1) \<in> (RAG2 s)\<^sup>+"
+    then obtain n where dd: "(Th th2, n) \<in> RAG2 s" by (metis converse_tranclE)
+    then obtain cs where eq_n: "n = Cs cs"
+      unfolding RAG2_def RAG_def by (case_tac n) (auto)
+    from dd eq_n have "th2 \<notin> readys s"
+      unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
+    with th2_r have "False" by auto
+  }
+  ultimately show "False" by metis
+qed
+
+lemma cpreced2_cpreced: "cpreced2 s th = cpreced (wq s) s th"
+unfolding cpreced2_def wq_def
+apply(induct s rule: schs.induct)
+apply(simp add: Let_def cpreced_def dependants_def RAG_def waits_def holds_def preced_def)
+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
+
+lemma cpreced_Exit:
+  shows "cpreced2 (Exit th # s) th' = cpreced2 s th'"
+by (simp add: cpreced2_cpreced cpreced_def preced_def wq_def Let_def)
+
+lemma readys_Exit:
+  shows "readys (Exit th # s) = readys s - {th}"
+by (auto simp add: readys_def waits2_def Let_def)
+
+lemma readys_Create:
+  shows "readys (Create th prio # s) \<subseteq> {th} \<union> readys s"
+apply (auto simp add: readys_def waits2_def Let_def waits_def)
+done
+
+lemma readys_Set:
+  shows "readys (Set th prio # s) = readys s"
+by (auto simp add: readys_def waits2_def Let_def)
+
+
+lemma readys_P:
+  shows "readys (P th cs # s) \<subseteq> readys s"
+apply(auto simp add: readys_def waits2_def Let_def)
+apply(simp add: waits_def)
+apply(case_tac "csa = cs")
+apply(simp)
+apply(drule_tac x="cs" in spec)
+apply(simp)
+apply (metis hd_append2 in_set_insert insert_Nil list.sel(1))
+apply(drule_tac x="csa" in spec)
+apply(simp)
+done
+
+lemma readys_V:
+  shows "readys (V th cs # s) \<subseteq> readys s \<union> set (wq s cs)"
+apply(auto simp add: readys_def waits2_def waits_def Let_def wq_def)
+done
+
+
+fun the_th :: "node \<Rightarrow> thread"
+  where "the_th (Th th) = th"
+
+lemma image_Collect2:
+  "f ` A = {f x | x. x \<in> A}"
+apply(auto)
+done
+
+lemma Collect_disj_eq2:
+  "{f x | x. x = y \<or> x \<in> A} = {f y} \<union> {f x | x. x \<in> A}"
+by (auto)
+
+lemma last_set_lt: 
+  "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+  apply(induct rule: threads.induct)
+  apply(auto)
+  done
+
+lemma last_set_eq_iff: 
+  assumes "th1 \<in> threads s" "th2 \<in> threads s"
+  shows "last_set th1 s = last_set th2 s \<longleftrightarrow> th1 = th2"
+  using assms
+  apply(induct s rule: threads.induct) 
+  apply(auto split:if_splits dest:last_set_lt)
+  done
+
+lemma preced_eq_iff: 
+  assumes th_in1: "th1 \<in> threads s"
+  and th_in2: "th2 \<in> threads s"
+  shows "preced th1 s = preced th2 s \<longleftrightarrow> th1 = th2"
+using assms
+by (auto simp add: preced_def last_set_eq_iff)
+
+lemma dm_RAG_threads:
+  assumes vt: "vt s"
+  and in_dom: "(Th th) \<in> Domain (RAG2 s)"
+  shows "th \<in> threads s"
+proof -
+  from in_dom obtain n where a: "(Th th, n) \<in> RAG2 s" by auto
+  then obtain cs where "n = Cs cs" 
+    unfolding RAG2_def RAG_def
+    by auto
+  then have "(Th th, Cs cs) \<in> RAG2 s" using a by simp
+  hence "th \<in> set (wq s cs)"
+    unfolding RAG2_def wq_def RAG_def waits_def
+    by (auto)
+  then show ?thesis
+    apply(rule_tac wq_threads)
+    apply(rule assms)
+    apply(simp)
+    done
+qed
+
+lemma cpreced_eq_iff:
+  assumes "th1 \<in> readys s" "th2 \<in> readys s" "vt s"
+  shows "cpreced2 s th1 = cpreced2 s th2 \<longleftrightarrow> th1 = th2" 
+proof 
+  def S1\<equiv>"({th1} \<union> dependants (wq s) th1)"
+  def S2\<equiv>"({th2} \<union> dependants (wq s) th2)"
+  have fin: "finite ((the_th o fst) ` ((RAG (wq s))\<^sup>+))" 
+      apply(rule)
+      apply(simp add: finite_trancl)
+      apply(simp add: wq_def)
+      apply(rule finite_RAG[simplified RAG2_def])
+      apply(rule assms)
+      done
+
+  from fin have h: "finite (preceds s S1)" "finite (preceds s S2)"
+      apply(simp_all add: S2_def S1_def Collect_disj_eq2 image_Collect[symmetric])
+      apply(rule)
+      apply(simp add: dependants_def)  
+      apply(rule rev_finite_subset)
+      apply(assumption)
+      apply(auto simp add: image_def)[1]
+      apply(metis fst_conv the_th.simps)
+      apply(rule)
+      apply(simp add: dependants_def)  
+      apply(rule rev_finite_subset)
+      apply(assumption)
+      apply(auto simp add: image_def)[1]
+      apply(metis fst_conv the_th.simps)
+      done
+    moreover have "S1 \<noteq> {}" "S2 \<noteq> {}" by (simp_all add: S1_def S2_def)
+    then have "(preceds s S1) \<noteq> {}" "(preceds s S2) \<noteq> {}" by simp_all
+    ultimately have m: "Max (preceds s S1) \<in> (preceds s S1)" "Max (preceds s S2) \<in> (preceds s S2)"
+      apply(rule_tac [!] Max_in)
+      apply(simp_all)
+      done
+
+  assume q: "cpreced2 s th1 = cpreced2 s th2"
+  then have eq_max: "Max (preceds s S1) = Max (preceds s S2)"
+    unfolding cpreced2_cpreced cpreced_def
+    apply(simp only: S1_def S2_def)
+    apply(simp add: Collect_disj_eq2)
+    done
+ 
+  obtain th0 where th0_in: "th0 \<in> S1" "th0 \<in> S2" and 
+      eq_f_th1: "preced th0 s = Max (preceds s S1)"
+                "preced th0 s = Max (preceds s S2)"
+    using m 
+      apply(clarify)
+      apply(simp add: eq_max)
+      apply(subst (asm) (2)  preced_eq_iff)
+      apply(insert assms(2))[1]
+      apply(simp add: S2_def)
+      apply(auto)[1]
+      apply (metis contra_subsetD readys_threads)
+      apply(simp add: dependants_def)
+      apply(subgoal_tac "Th tha \<in> Domain ((RAG2 s)^+)")
+      apply(simp add: trancl_domain)
+      apply (metis Domain_RAG_threads assms(3))
+      apply(simp only: RAG2_def wq_def)
+      apply (metis Domain_iff)
+      apply(insert assms(1))[1]
+      apply(simp add: S1_def)
+      apply(auto)[1]
+      apply (metis contra_subsetD readys_threads)
+      apply(simp add: dependants_def)
+      apply(subgoal_tac "Th th \<in> Domain ((RAG2 s)^+)")
+      apply(simp add: trancl_domain)
+      apply (metis Domain_RAG_threads assms(3))
+      apply(simp only: RAG2_def wq_def)
+      apply (metis Domain_iff)
+      apply(simp)
+    done
+  then show "th1 = th2"
+    apply -
+    apply(insert th0_in assms(1, 2))[1]
+    apply(simp add: S1_def S2_def)
+    apply(auto)
+    --"first case"
+    prefer 2
+    apply(subgoal_tac "Th th2 \<in> Domain (RAG2 s)")
+    apply(subgoal_tac "\<exists>cs. (Th th2, Cs cs) \<in> RAG2 s")
+    apply(erule exE)
+    apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
+    apply(auto simp add: RAG2_def RAG_def)[1]
+    apply(subgoal_tac "Th th2 \<in> Domain ((RAG2 s)^+)")
+    apply (metis trancl_domain)
+    apply(subgoal_tac "(Th th2, Th th1) \<in> (RAG2 s)^+")
+    apply (metis Domain_iff)
+    apply(simp add: dependants_def RAG2_def wq_def)
+    --"second case"
+    apply(subgoal_tac "Th th1 \<in> Domain (RAG2 s)")
+    apply(subgoal_tac "\<exists>cs. (Th th1, Cs cs) \<in> RAG2 s")
+    apply(erule exE)
+    apply(insert assms(1))[1]
+    apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
+    apply(auto simp add: RAG2_def RAG_def)[1]
+    apply(subgoal_tac "Th th1 \<in> Domain ((RAG2 s)^+)")
+    apply (metis trancl_domain)
+    apply(subgoal_tac "(Th th1, Th th2) \<in> (RAG2 s)^+")
+    apply (metis Domain_iff)
+    apply(simp add: dependants_def RAG2_def wq_def)
+    --"third case"
+    apply(rule dchain_unique)
+    apply(rule assms(3))
+    apply(simp add: dependants_def RAG2_def wq_def)
+    apply(simp)
+    apply(simp add: dependants_def RAG2_def wq_def)
+    apply(simp)
+    done
+next
+  assume "th1 = th2"
+  then show "cpreced2 s th1 = cpreced2 s th2" by simp
+qed
+
+lemma at_most_one_running:
+  assumes "vt s"
+  shows "card (runing s) \<le> 1"
+proof (rule ccontr)
+  assume "\<not> card (runing s) \<le> 1"
+  then have "2 \<le> card (runing s)" by auto
+  moreover 
+  have "finite (runing s)"
+    by (metis `\<not> card (runing s) \<le> 1` card_infinite le0)
+  ultimately obtain th1 th2 where a: 
+    "th1 \<noteq> th2" "th1 \<in> runing s" "th2 \<in> runing s" 
+    "cpreced2 s th1 = cpreced2 s th2" 
+    apply(auto simp add: numerals card_le_Suc_iff runing_def) 
+    apply(blast)
+    done
+  then have "th1 = th2"
+    apply(subst (asm) cpreced_eq_iff)
+    apply(auto intro: assms a)
+    apply (metis contra_subsetD runing_ready)+
+    done
+  then show "False" using a(1) by auto
+qed
+
+
+
+  (*
+  obtain th0 where th0_in: "th0 \<in> S1 \<and> th0 \<in> S2" 
+    and eq_f_th0: "preced th0 s = Max ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
+  proof -
+    from fin have h1: "finite ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
+      apply(simp only: S1_def S2_def)
+      apply(rule)
+      apply(rule)
+      apply(rule)
+      apply(simp add: dependants_def) 
+      apply(rule rev_finite_subset)
+      apply(assumption)
+      apply(auto simp add: image_def)
+      apply (metis fst_conv the_th.simps)
+      done
+    moreover 
+    have "S1 \<inter> S2 \<noteq> {}" apply (simp add: S1_def S2_def) 
+      apply(auto) 
+      
+      done
+    then have h2: "((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<noteq> {}" by simp
+    ultimately have "Max ((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<in> ((\<lambda>th. preced th s) ` (S1 \<union> S2))"
+      apply(rule Max_in)
+      done
+    then show ?thesis using that[intro] apply(auto) 
+      
+      apply(erule_tac preced_unique)
+      done
+  qed
+  *)
+
+thm waits_def waits2_def
+
+end
diff -r c3a42076b164 -r b32b3bd99150 ExtGG.ty
--- a/ExtGG.ty	Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,922 +0,0 @@
-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 c3a42076b164 -r b32b3bd99150 Paper/ExtGG.ty
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Paper/ExtGG.ty	Thu Sep 21 14:15:55 2017 +0100
@@ -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 c3a42076b164 -r b32b3bd99150 Test.thy
--- a/Test.thy	Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,789 +0,0 @@
-theory Test 
-imports Precedence_ord Graphs
-begin
-
-type_synonym thread = nat -- {* Type for thread identifiers. *}
-type_synonym priority = nat  -- {* Type for priorities. *}
-type_synonym cs = nat -- {* Type for critical sections (or resources). *}
-
--- {* Schedulling Events *}
-
-datatype event = 
-  Create thread priority 
-| Exit thread 
-| P thread cs 
-| V thread cs 
-| Set thread priority 
-
-type_synonym state = "event list"
-
-fun threads :: "state \<Rightarrow> thread set"
-  where 
-  "threads [] = {}" 
-| "threads (Create th prio#s) = {th} \<union> threads s" 
-| "threads (Exit th # s) = (threads s) - {th}" 
-| "threads (_#s) = threads s" 
-
-fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
-  where
-  "priority th [] = 0" 
-| "priority th (Create th' prio#s) = (if th' = th then prio else priority th s)" 
-| "priority th (Set th' prio#s) = (if th' = th then prio else priority th s)" 
-| "priority th (_#s) = priority th s"
-
-fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
-  where
-  "last_set th [] = 0" 
-| "last_set th ((Create th' prio)#s) = (if (th = th') then length s else last_set th s)" 
-| "last_set th ((Set th' prio)#s) = (if (th = th') then length s else last_set th s)" 
-| "last_set th (_#s) = last_set th s"
-
-
-definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
-  where "preced th s \<equiv> Prc (priority th s) (last_set th s)"
-
-abbreviation 
-  "preceds s ths \<equiv> {preced th s | th. th \<in> ths}"
- 
-definition
-  "holds wq th cs \<equiv> th \<in> set (wq cs) \<and> th = hd (wq cs)"
-
-definition
-  "waits wq th cs \<equiv> th \<in> set (wq cs) \<and> th \<noteq> hd (wq cs)"
-
---{* Nodes in Resource Graph *}
-datatype node = 
-  Th "thread" 
-| Cs "cs" 
-
-definition
-  "RAG wq \<equiv> {(Th th, Cs cs) | th cs. waits wq th cs} \<union> {(Cs cs, Th th) | cs th. holds wq th cs}"
-
-definition
-  "dependants wq th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
-
-record schedule_state = 
-  wq_fun :: "cs \<Rightarrow> thread list" 
-  cprec_fun :: "thread \<Rightarrow> precedence" 
-
-definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
-  where 
-  "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
-
-abbreviation
-  "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
-
-abbreviation 
-  "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
- 
-abbreviation
-  "release qs \<equiv> case qs of
-             [] => [] 
-          |  (_ # qs) => SOME q. distinct q \<and> set q = set qs"
-
-lemma [simp]: 
-  "(SOME q. distinct q \<and> q = []) = []"
-by auto
-
-lemma [simp]: 
-  "(x \<in> set (SOME q. distinct q \<and> set q = set p)) = (x \<in> set p)"
-apply(rule iffI)
-apply (metis (mono_tags, lifting) List.finite_set finite_distinct_list some_eq_ex)+
-done
-
-abbreviation
-  "next_to_run ths \<equiv> hd (SOME q::thread list. distinct q \<and> set q = set ths)"
-
-
-fun schs :: "state \<Rightarrow> schedule_state"
-  where 
-  "schs [] = (| wq_fun = \<lambda> cs. [],  cprec_fun = (\<lambda>_. Prc 0 0) |)" 
-| "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)|))"
-|  "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)|))"
-
-definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list" 
-  where "wq s = wq_fun (schs s)"
-
-definition cpreced2 :: "state \<Rightarrow> thread \<Rightarrow> precedence"
-  where "cpreced2 s \<equiv> cprec_fun (schs s)"
-
-abbreviation
-  "cpreceds2 s ths \<equiv> {cpreced2 s th | th. th \<in> ths}"
-
-definition
-  "holds2 s \<equiv> holds (wq_fun (schs s))"
-
-definition
-  "waits2 s \<equiv> waits (wq_fun (schs s))"
-
-definition
-  "RAG2 s \<equiv> RAG (wq_fun (schs s))"
-  
-definition
-  "dependants2 s \<equiv> dependants (wq_fun (schs s))"
-
-(* ready -> is a thread that is not waiting for any resource *) 
-definition readys :: "state \<Rightarrow> thread set"
-  where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waits2 s th cs)}"
-
-definition runing :: "state \<Rightarrow> thread set"
-  where "runing s \<equiv> {th . th \<in> readys s \<and> cpreced2 s th = Max (cpreceds2 s (readys s))}"
-
-(* all resources a thread hols in a state *)
-definition holding :: "state \<Rightarrow> thread \<Rightarrow> cs set"
-  where "holding s th \<equiv> {cs . holds2 s th cs}"
-
-
-lemma exists_distinct:
-  obtains ys where "distinct ys" "set ys = set xs"
-by (metis List.finite_set finite_distinct_list)
-
-lemma next_to_run_set [simp]:
-  "wts \<noteq> [] \<Longrightarrow> next_to_run wts \<in> set wts"
-apply(rule exists_distinct[of wts])
-by (metis (mono_tags, lifting) hd_in_set set_empty some_eq_ex)
-
-lemma holding_RAG: 
-  "holding s th = {cs . (Cs cs, Th th) \<in> RAG2 s}"
-unfolding holding_def RAG2_def RAG_def
-unfolding holds2_def holds_def waits_def
-by auto
-
-inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
-  where
-  step_Create: "\<lbrakk>th \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create th prio)" 
-| step_Exit: "\<lbrakk>th \<in> runing s; holding s th = {}\<rbrakk> \<Longrightarrow> step s (Exit th)" 
-| step_P: "\<lbrakk>th \<in> runing s;  (Cs cs, Th th)  \<notin> (RAG2 s)^+\<rbrakk> \<Longrightarrow> step s (P th cs)" 
-| step_V: "\<lbrakk>th \<in> runing s;  holds2 s th cs\<rbrakk> \<Longrightarrow> step s (V th cs)" 
-| step_Set: "\<lbrakk>th \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set th prio)"
-
-(* valid states *)
-inductive vt :: "state \<Rightarrow> bool"
-  where
-  vt_nil[intro]: "vt []" 
-| vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
-
-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_threads: 
-  assumes vt: "vt s"
-  and h: "th \<in> set (wq s cs)"
-  shows "th \<in> threads s"
-using assms
-apply(induct)
-apply(simp add: wq_def)
-apply(erule step.cases)
-apply(auto simp add: wq_def Let_def holding_def holds2_def holds_def waits2_def runing_def readys_def)
-apply(simp add: waits_def)
-apply(auto simp add: waits_def split: if_splits)[1]
-apply(auto split: if_splits)
-apply(simp only: waits_def)
-by (metis insert_iff set_simps(2))
-
-
-
-lemma Domain_RAG_threads:
-  assumes vt: "vt s"
-  and in_dom: "(Th th) \<in> Domain (RAG2 s)"
-  shows "th \<in> threads s"
-proof -
-  from in_dom obtain n where "(Th th, n) \<in> RAG2 s" by auto
-  then obtain cs where "n = Cs cs" "(Th th, Cs cs) \<in> RAG2 s"  
-    unfolding RAG2_def RAG_def by auto
-  then have "th \<in> set (wq s cs)"
-    unfolding wq_def RAG_def RAG2_def waits_def by auto
-  with wq_threads [OF vt] show ?thesis .
-qed
-
-lemma dependants_threads:
-  assumes vt: "vt s"
-  shows "dependants2 s th \<subseteq> threads s"
-proof
-  fix th1 
-  assume "th1 \<in> dependants2 s th"
-  then have h: "(Th th1, Th th) \<in> (RAG2 s)\<^sup>+"
-    unfolding dependants2_def dependants_def RAG2_def by simp
-  then have "Th th1 \<in> Domain ((RAG2 s)\<^sup>+)" unfolding Domain_def by auto
-  then have "Th th1 \<in> Domain (RAG2 s)" using trancl_domain by simp
-  then show "th1 \<in> threads s" using vt by (rule_tac Domain_RAG_threads)
-qed
-
-lemma finite_threads:
-  assumes vt: "vt s"
-  shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-
-section {* Distinctness of @{const wq} *}
-
-lemma wq_distinct_step: 
-  assumes "step s e" "distinct (wq s cs)" 
-  shows "distinct (wq (e # s) cs)"
-using assms
-unfolding wq_def
-apply(erule_tac step.cases)
-apply(auto simp add: RAG2_def RAG_def Let_def)[1]
-apply(auto simp add: wq_def Let_def RAG2_def RAG_def holds_def runing_def waits2_def waits_def readys_def)
-apply(auto split: list.split)
-apply(rule someI2)
-apply(auto)
-done
-
-lemma wq_distinct: 
-  assumes "vt s" 
-  shows "distinct (wq s cs)"
-using assms
-apply(induct)
-apply(simp add: wq_def)
-apply(simp add: wq_distinct_step)
-done
-
-
-section {* Single_Valuedness of @{const waits2}, @{const holds2}, @{const RAG2} *}
-
-lemma waits2_unique:
-  assumes "vt s"
-  and "waits2 s th cs1"
-  and "waits2 s th cs2"
-  shows "cs1 = cs2"
-using assms
-apply(induct)
-apply(simp add: waits2_def waits_def)
-apply(erule step.cases)
-apply(auto simp add: Let_def waits2_def waits_def holds_def RAG2_def RAG_def 
- readys_def runing_def split: if_splits)
-apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
-apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
-apply (metis distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
-by (metis (full_types, hide_lams) distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
-
-lemma single_valued_waits2:
-  assumes "vt s"
-  shows "single_valuedP (waits2 s)"
-using assms
-unfolding single_valued_def
-by (simp add: Product_Type.Collect_case_prodD waits2_unique)
-
-lemma single_valued_holds2:
-  assumes "vt s"
-  shows "single_valuedP (\<lambda>cs th. holds2 s th cs)"
-unfolding single_valued_def holds2_def holds_def by simp
-
-lemma single_valued_RAG2:
-  assumes "vt s"
-  shows "single_valued (RAG2 s)"
-using single_valued_waits2[OF assms] single_valued_holds2[OF assms] 
-unfolding RAG2_def RAG_def
-apply(rule_tac single_valued_union)
-unfolding holds2_def[symmetric] waits2_def[symmetric]
-apply(rule single_valued_Collect)
-apply(simp)
-apply(simp add: inj_on_def)
-apply(rule single_valued_Collect)
-apply(simp)
-apply(simp add: inj_on_def)
-apply(auto)
-done
-
-
-section {* Properties of @{const RAG2} under events *}
-
-lemma RAG_Set [simp]: 
-  shows "RAG2 (Set th prio # s) = RAG2 s"
-unfolding RAG2_def
-by (simp add: Let_def)
-
-lemma RAG_Create [simp]: 
-  "RAG2 (Create th prio # s) = RAG2 s"
-unfolding RAG2_def
-by (simp add: Let_def)
-
-lemma RAG_Exit [simp]: 
-  shows "RAG2 (Exit th # s) = RAG2 s"
-unfolding RAG2_def
-by (simp add: Let_def)
-
-lemma RAG_P1:
-  assumes "wq s cs = []"
-  shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Cs cs, Th th)}"
-using assms
-unfolding  RAG2_def RAG_def wq_def Let_def waits_def holds_def
-by (auto simp add: Let_def)
-
-lemma RAG_P2:
-  assumes "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" "wq s cs \<noteq> []"
-  shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Th th, Cs cs)}"
-using assms
-unfolding RAG2_def RAG_def wq_def Let_def waits_def holds_def
-by (auto simp add: Let_def)
-
-
-lemma RAG_V1:
-assumes vt: "wq s cs = [th]"
-shows "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}"
-using assms
-unfolding RAG2_def RAG_def waits_def holds_def wq_def
-by (auto simp add: Let_def)
-
-lemma RAG_V2:
-assumes vt:"vt s" "wq s cs = th # wts \<and> wts \<noteq> []"
-shows "RAG2 (V th cs # s) \<subseteq>
-  RAG2 s - {(Cs cs, Th th), (Th (next_to_run wts), Cs cs)} \<union> {(Cs cs, Th (next_to_run wts))}"
-unfolding RAG2_def RAG_def waits_def holds_def
-using assms wq_distinct[OF vt(1), of"cs"]
-by (auto simp add: Let_def wq_def)
-
-
-
-section {* Acyclicity of @{const RAG2} *}
-
-lemma acyclic_RAG_step: 
-  assumes vt: "vt s"
-  and     stp: "step s e"
-  and     ac: "acyclic (RAG2 s)"
-  shows "acyclic (RAG2 (e # s))"
-using stp vt ac
-proof (induct)
-  case (step_P th s cs)
-  have ac: "acyclic (RAG2 s)" by fact
-  have ds: "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" by fact
-  { assume wq_empty: "wq s cs = []" -- "case waiting queue is empty"
-    then have "(Th th, Cs cs) \<notin> (RAG2 s)\<^sup>+"
-    proof (rule_tac notI)
-      assume "(Th th, Cs cs) \<in> (RAG2 s)\<^sup>+"
-      then obtain x where "(x, Cs cs) \<in> RAG2 s" using tranclD2 by metis 
-      with wq_empty show False by (auto simp: RAG2_def RAG_def wq_def waits_def)
-    qed
-    with ac have "acyclic (RAG2 s \<union> {(Cs cs, Th th)})" by simp
-    then have "acyclic (RAG2 (P th cs # s))" using RAG_P1[OF wq_empty] 
-      by (rule acyclic_subset)
-  }
-  moreover
-  { assume wq_not_empty: "wq s cs \<noteq> []" -- "case waiting queue is not empty"
-    from ac ds
-    have "acyclic (RAG2 s \<union> {(Th th, Cs cs)})" by simp
-    then have "acyclic (RAG2 (P th cs # s))" using RAG_P2[OF ds wq_not_empty] 
-      by (rule acyclic_subset)
-  }
-  ultimately show "acyclic (RAG2 (P th cs # s))" by metis
-next    
-  case (step_V th s cs) -- "case for release of a lock" 
-  have vt: "vt s" by fact
-  have ac: "acyclic (RAG2 s)" by fact
-  have hd: "holds2 s th cs" by fact
-  from vt have wq_distinct:"distinct (wq s cs)" by (rule wq_distinct)
-  from hd have "th \<in> set (wq s cs)" "th = hd (wq s cs)" unfolding holds2_def holds_def wq_def by auto
-  then obtain wts where eq_wq: "wq s cs = th # wts"  by (cases "wq s cs") (auto)
-  -- "case no thread present in the waiting queue to take over"
-  { assume "wts = []" 
-    with eq_wq have "wq s cs = [th]" by simp
-    then have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}" by (rule RAG_V1)
-    moreover have "acyclic (RAG2 s - {(Cs cs, Th th)})" using ac by (auto intro: acyclic_subset)
-    ultimately 
-    have "acyclic (RAG2 (V th cs # s))" by (auto intro: acyclic_subset)
-  }
-  moreover
-  -- "at least one thread present to take over"
-  { def nth \<equiv> "next_to_run wts"
-    assume wq_not_empty: "wts \<noteq> []" 
-    have "waits2 s nth cs" 
-      using eq_wq wq_not_empty wq_distinct
-      unfolding nth_def waits2_def waits_def wq_def[symmetric] by auto
-    then have cs_in_RAG: "(Th nth, Cs cs) \<in> RAG2 s" 
-      unfolding RAG2_def RAG_def waits2_def by auto
-    have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)}" 
-      unfolding nth_def using  vt wq_not_empty eq_wq by (rule_tac RAG_V2) (auto)
-    moreover 
-    have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)})" 
-    proof -
-      have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})" using ac by (auto intro: acyclic_subset)
-      moreover 
-      have "(Th nth, Cs cs) \<notin> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+" 
-      proof (rule notI)
-        assume "(Th nth, Cs cs) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+"
-        then obtain z where a: "(Th nth, z) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})"
-          by (metis converse_tranclE)
-        then have "(Th nth, z) \<in> RAG2 s" by simp
-        then have "z = Cs cs" using cs_in_RAG single_valued_RAG2[OF vt]
-          by (simp add: single_valued_def)
-        then show "False" using a by simp
-      qed
-      ultimately 
-      show "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth) })" by simp
-    qed
-    ultimately have "acyclic (RAG2 (V th cs # s))" 
-      by (rule_tac acyclic_subset)
-  }
-  ultimately show "acyclic (RAG2 (V th cs # s))" by metis
-qed (simp_all)
-
-
-lemma finite_RAG:
-  assumes "vt s"
-  shows "finite (RAG2 s)"
-using assms
-apply(induct)
-apply(simp add: RAG2_def RAG_def waits_def holds_def)
-apply(erule step.cases)
-apply(auto)
-apply(case_tac "wq sa cs = []")
-apply(rule finite_subset)
-apply(rule RAG_P1)
-apply(simp)
-apply(simp)
-apply(rule finite_subset)
-apply(rule RAG_P2)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(subgoal_tac "\<exists>wts. wq sa cs = th # wts")
-apply(erule exE)
-apply(case_tac "wts = []")
-apply(rule finite_subset)
-apply(rule RAG_V1)
-apply(simp)
-apply(simp)
-apply(rule finite_subset)
-apply(rule RAG_V2)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(subgoal_tac "th \<in> set (wq sa cs) \<and> th = hd (wq sa cs)") 
-apply(case_tac "wq sa cs") 
-apply(auto)[2]
-apply(auto simp add: holds2_def holds_def wq_def)
-done
-
-
-
-lemma dchain_unique:
-  assumes vt: "vt s"
-  and th1_d: "(n, Th th1) \<in> (RAG2 s)^+"
-  and th1_r: "th1 \<in> readys s"
-  and th2_d: "(n, Th th2) \<in> (RAG2 s)^+"
-  and th2_r: "th2 \<in> readys s"
-  shows "th1 = th2"
-proof(rule ccontr)
-  assume neq: "th1 \<noteq> th2"
-   with single_valued_confluent2 [OF single_valued_RAG2 [OF vt]] th1_d th2_d
-  have "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG2 s)\<^sup>+" by auto
-  moreover
-  { assume "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+"      
-    then obtain n where dd: "(Th th1, n) \<in> RAG2 s" by (metis converse_tranclE)
-    then obtain cs where eq_n: "n = Cs cs"
-      unfolding RAG2_def RAG_def by (case_tac n) (auto)
-    from dd eq_n have "th1 \<notin> readys s"
-      unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
-    with th1_r have "False" by auto
-  }
-  moreover
-  { assume "(Th th2, Th th1) \<in> (RAG2 s)\<^sup>+"
-    then obtain n where dd: "(Th th2, n) \<in> RAG2 s" by (metis converse_tranclE)
-    then obtain cs where eq_n: "n = Cs cs"
-      unfolding RAG2_def RAG_def by (case_tac n) (auto)
-    from dd eq_n have "th2 \<notin> readys s"
-      unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
-    with th2_r have "False" by auto
-  }
-  ultimately show "False" by metis
-qed
-
-lemma cpreced2_cpreced: "cpreced2 s th = cpreced (wq s) s th"
-unfolding cpreced2_def wq_def
-apply(induct s rule: schs.induct)
-apply(simp add: Let_def cpreced_def dependants_def RAG_def waits_def holds_def preced_def)
-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
-
-lemma cpreced_Exit:
-  shows "cpreced2 (Exit th # s) th' = cpreced2 s th'"
-by (simp add: cpreced2_cpreced cpreced_def preced_def wq_def Let_def)
-
-lemma readys_Exit:
-  shows "readys (Exit th # s) = readys s - {th}"
-by (auto simp add: readys_def waits2_def Let_def)
-
-lemma readys_Create:
-  shows "readys (Create th prio # s) \<subseteq> {th} \<union> readys s"
-apply (auto simp add: readys_def waits2_def Let_def waits_def)
-done
-
-lemma readys_Set:
-  shows "readys (Set th prio # s) = readys s"
-by (auto simp add: readys_def waits2_def Let_def)
-
-
-lemma readys_P:
-  shows "readys (P th cs # s) \<subseteq> readys s"
-apply(auto simp add: readys_def waits2_def Let_def)
-apply(simp add: waits_def)
-apply(case_tac "csa = cs")
-apply(simp)
-apply(drule_tac x="cs" in spec)
-apply(simp)
-apply (metis hd_append2 in_set_insert insert_Nil list.sel(1))
-apply(drule_tac x="csa" in spec)
-apply(simp)
-done
-
-lemma readys_V:
-  shows "readys (V th cs # s) \<subseteq> readys s \<union> set (wq s cs)"
-apply(auto simp add: readys_def waits2_def waits_def Let_def wq_def)
-done
-
-
-fun the_th :: "node \<Rightarrow> thread"
-  where "the_th (Th th) = th"
-
-lemma image_Collect2:
-  "f ` A = {f x | x. x \<in> A}"
-apply(auto)
-done
-
-lemma Collect_disj_eq2:
-  "{f x | x. x = y \<or> x \<in> A} = {f y} \<union> {f x | x. x \<in> A}"
-by (auto)
-
-lemma last_set_lt: 
-  "th \<in> threads s \<Longrightarrow> last_set th s < length s"
-  apply(induct rule: threads.induct)
-  apply(auto)
-  done
-
-lemma last_set_eq_iff: 
-  assumes "th1 \<in> threads s" "th2 \<in> threads s"
-  shows "last_set th1 s = last_set th2 s \<longleftrightarrow> th1 = th2"
-  using assms
-  apply(induct s rule: threads.induct) 
-  apply(auto split:if_splits dest:last_set_lt)
-  done
-
-lemma preced_eq_iff: 
-  assumes th_in1: "th1 \<in> threads s"
-  and th_in2: "th2 \<in> threads s"
-  shows "preced th1 s = preced th2 s \<longleftrightarrow> th1 = th2"
-using assms
-by (auto simp add: preced_def last_set_eq_iff)
-
-lemma dm_RAG_threads:
-  assumes vt: "vt s"
-  and in_dom: "(Th th) \<in> Domain (RAG2 s)"
-  shows "th \<in> threads s"
-proof -
-  from in_dom obtain n where a: "(Th th, n) \<in> RAG2 s" by auto
-  then obtain cs where "n = Cs cs" 
-    unfolding RAG2_def RAG_def
-    by auto
-  then have "(Th th, Cs cs) \<in> RAG2 s" using a by simp
-  hence "th \<in> set (wq s cs)"
-    unfolding RAG2_def wq_def RAG_def waits_def
-    by (auto)
-  then show ?thesis
-    apply(rule_tac wq_threads)
-    apply(rule assms)
-    apply(simp)
-    done
-qed
-
-lemma cpreced_eq_iff:
-  assumes "th1 \<in> readys s" "th2 \<in> readys s" "vt s"
-  shows "cpreced2 s th1 = cpreced2 s th2 \<longleftrightarrow> th1 = th2" 
-proof 
-  def S1\<equiv>"({th1} \<union> dependants (wq s) th1)"
-  def S2\<equiv>"({th2} \<union> dependants (wq s) th2)"
-  have fin: "finite ((the_th o fst) ` ((RAG (wq s))\<^sup>+))" 
-      apply(rule)
-      apply(simp add: finite_trancl)
-      apply(simp add: wq_def)
-      apply(rule finite_RAG[simplified RAG2_def])
-      apply(rule assms)
-      done
-
-  from fin have h: "finite (preceds s S1)" "finite (preceds s S2)"
-      apply(simp_all add: S2_def S1_def Collect_disj_eq2 image_Collect[symmetric])
-      apply(rule)
-      apply(simp add: dependants_def)  
-      apply(rule rev_finite_subset)
-      apply(assumption)
-      apply(auto simp add: image_def)[1]
-      apply(metis fst_conv the_th.simps)
-      apply(rule)
-      apply(simp add: dependants_def)  
-      apply(rule rev_finite_subset)
-      apply(assumption)
-      apply(auto simp add: image_def)[1]
-      apply(metis fst_conv the_th.simps)
-      done
-    moreover have "S1 \<noteq> {}" "S2 \<noteq> {}" by (simp_all add: S1_def S2_def)
-    then have "(preceds s S1) \<noteq> {}" "(preceds s S2) \<noteq> {}" by simp_all
-    ultimately have m: "Max (preceds s S1) \<in> (preceds s S1)" "Max (preceds s S2) \<in> (preceds s S2)"
-      apply(rule_tac [!] Max_in)
-      apply(simp_all)
-      done
-
-  assume q: "cpreced2 s th1 = cpreced2 s th2"
-  then have eq_max: "Max (preceds s S1) = Max (preceds s S2)"
-    unfolding cpreced2_cpreced cpreced_def
-    apply(simp only: S1_def S2_def)
-    apply(simp add: Collect_disj_eq2)
-    done
- 
-  obtain th0 where th0_in: "th0 \<in> S1" "th0 \<in> S2" and 
-      eq_f_th1: "preced th0 s = Max (preceds s S1)"
-                "preced th0 s = Max (preceds s S2)"
-    using m 
-      apply(clarify)
-      apply(simp add: eq_max)
-      apply(subst (asm) (2)  preced_eq_iff)
-      apply(insert assms(2))[1]
-      apply(simp add: S2_def)
-      apply(auto)[1]
-      apply (metis contra_subsetD readys_threads)
-      apply(simp add: dependants_def)
-      apply(subgoal_tac "Th tha \<in> Domain ((RAG2 s)^+)")
-      apply(simp add: trancl_domain)
-      apply (metis Domain_RAG_threads assms(3))
-      apply(simp only: RAG2_def wq_def)
-      apply (metis Domain_iff)
-      apply(insert assms(1))[1]
-      apply(simp add: S1_def)
-      apply(auto)[1]
-      apply (metis contra_subsetD readys_threads)
-      apply(simp add: dependants_def)
-      apply(subgoal_tac "Th th \<in> Domain ((RAG2 s)^+)")
-      apply(simp add: trancl_domain)
-      apply (metis Domain_RAG_threads assms(3))
-      apply(simp only: RAG2_def wq_def)
-      apply (metis Domain_iff)
-      apply(simp)
-    done
-  then show "th1 = th2"
-    apply -
-    apply(insert th0_in assms(1, 2))[1]
-    apply(simp add: S1_def S2_def)
-    apply(auto)
-    --"first case"
-    prefer 2
-    apply(subgoal_tac "Th th2 \<in> Domain (RAG2 s)")
-    apply(subgoal_tac "\<exists>cs. (Th th2, Cs cs) \<in> RAG2 s")
-    apply(erule exE)
-    apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
-    apply(auto simp add: RAG2_def RAG_def)[1]
-    apply(subgoal_tac "Th th2 \<in> Domain ((RAG2 s)^+)")
-    apply (metis trancl_domain)
-    apply(subgoal_tac "(Th th2, Th th1) \<in> (RAG2 s)^+")
-    apply (metis Domain_iff)
-    apply(simp add: dependants_def RAG2_def wq_def)
-    --"second case"
-    apply(subgoal_tac "Th th1 \<in> Domain (RAG2 s)")
-    apply(subgoal_tac "\<exists>cs. (Th th1, Cs cs) \<in> RAG2 s")
-    apply(erule exE)
-    apply(insert assms(1))[1]
-    apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
-    apply(auto simp add: RAG2_def RAG_def)[1]
-    apply(subgoal_tac "Th th1 \<in> Domain ((RAG2 s)^+)")
-    apply (metis trancl_domain)
-    apply(subgoal_tac "(Th th1, Th th2) \<in> (RAG2 s)^+")
-    apply (metis Domain_iff)
-    apply(simp add: dependants_def RAG2_def wq_def)
-    --"third case"
-    apply(rule dchain_unique)
-    apply(rule assms(3))
-    apply(simp add: dependants_def RAG2_def wq_def)
-    apply(simp)
-    apply(simp add: dependants_def RAG2_def wq_def)
-    apply(simp)
-    done
-next
-  assume "th1 = th2"
-  then show "cpreced2 s th1 = cpreced2 s th2" by simp
-qed
-
-lemma at_most_one_running:
-  assumes "vt s"
-  shows "card (runing s) \<le> 1"
-proof (rule ccontr)
-  assume "\<not> card (runing s) \<le> 1"
-  then have "2 \<le> card (runing s)" by auto
-  moreover 
-  have "finite (runing s)"
-    by (metis `\<not> card (runing s) \<le> 1` card_infinite le0)
-  ultimately obtain th1 th2 where a: 
-    "th1 \<noteq> th2" "th1 \<in> runing s" "th2 \<in> runing s" 
-    "cpreced2 s th1 = cpreced2 s th2" 
-    apply(auto simp add: numerals card_le_Suc_iff runing_def) 
-    apply(blast)
-    done
-  then have "th1 = th2"
-    apply(subst (asm) cpreced_eq_iff)
-    apply(auto intro: assms a)
-    apply (metis contra_subsetD runing_ready)+
-    done
-  then show "False" using a(1) by auto
-qed
-
-
-
-  (*
-  obtain th0 where th0_in: "th0 \<in> S1 \<and> th0 \<in> S2" 
-    and eq_f_th0: "preced th0 s = Max ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
-  proof -
-    from fin have h1: "finite ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
-      apply(simp only: S1_def S2_def)
-      apply(rule)
-      apply(rule)
-      apply(rule)
-      apply(simp add: dependants_def) 
-      apply(rule rev_finite_subset)
-      apply(assumption)
-      apply(auto simp add: image_def)
-      apply (metis fst_conv the_th.simps)
-      done
-    moreover 
-    have "S1 \<inter> S2 \<noteq> {}" apply (simp add: S1_def S2_def) 
-      apply(auto) 
-      
-      done
-    then have h2: "((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<noteq> {}" by simp
-    ultimately have "Max ((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<in> ((\<lambda>th. preced th s) ` (S1 \<union> S2))"
-      apply(rule Max_in)
-      done
-    then show ?thesis using that[intro] apply(auto) 
-      
-      apply(erule_tac preced_unique)
-      done
-  qed
-  *)
-
-thm waits_def waits2_def
-
-end
diff -r c3a42076b164 -r b32b3bd99150 draf.txt
--- a/draf.txt	Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,11 +0,0 @@
-There are low priority threads, 
-which do not hold any resources, 
-such thread will not block th. 
-Theorem 3 does not exclude such threads.
-
-There are resources, which are not held by any low prioirty threads,
-such resources can not cause blockage of th neither. And similiary, 
-theorem 6 does not exlude them.
-
-Our one bound excudle them by using a different formaulation. 
-
diff -r c3a42076b164 -r b32b3bd99150 red_1.thy
--- a/red_1.thy	Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,359 +0,0 @@
-section {*
-  This file contains lemmas used to guide the recalculation of current precedence 
-  after every system call (or system operation)
-*}
-theory CpsG
-imports PrioG Max RTree
-begin
-
-
-definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"
-
-definition "hRAG (s::state) =  {(Cs cs, Th th) | th cs. holding s th cs}"
-
-definition "tRAG s = wRAG s O hRAG s"
-
-definition "pairself f = (\<lambda>(a, b). (f a, f b))"
-
-definition "rel_map f r = (pairself f ` r)"
-
-fun the_thread :: "node \<Rightarrow> thread" where
-   "the_thread (Th th) = th"
-
-definition "tG s = rel_map the_thread (tRAG s)"
-
-locale pip = 
-  fixes s
-  assumes vt: "vt s"
-
-
-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)
-
-lemma relpow_mult: 
-  "((r::'a rel) ^^ m) ^^ n = r ^^ (m*n)"
-proof(induct n arbitrary:m)
-  case (Suc k m)
-  thus ?case (is "?L = ?R")
-  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 range_tRAG: "Range (tRAG s) \<subseteq> {Th th | th. True}"
-proof -
-  have "Range (wRAG s O hRAG s) \<subseteq> {Th th |th. True}" (is "?L \<subseteq> ?R")
-  proof -
-    have "?L \<subseteq> Range (hRAG s)" by auto
-    also have "... \<subseteq> ?R" 
-      by (unfold hRAG_def, auto)
-    finally show ?thesis by auto
-  qed
-  thus ?thesis by (simp add:tRAG_def)
-qed
-
-lemma domain_tRAG: "Domain (tRAG s) \<subseteq> {Th th | th. True}"
-proof -
-  have "Domain (wRAG s O hRAG s) \<subseteq> {Th th |th. True}" (is "?L \<subseteq> ?R")
-  proof -
-    have "?L \<subseteq> Domain (wRAG s)" by auto
-    also have "... \<subseteq> ?R" 
-      by (unfold wRAG_def, auto)
-    finally show ?thesis by auto
-  qed
-  thus ?thesis by (simp add:tRAG_def)
-qed
-
-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 plus_rpath: 
-  assumes "(a, b) \<in> r^+"
-  obtains xs where "rpath r a xs b" "xs \<noteq> []"
-proof -
-  from assms obtain m where h: "(a, m) \<in> r" "(m, b) \<in> r^*"
-      by (auto dest!:tranclD)
-  from star_rpath[OF this(2)] obtain xs where "rpath r m xs b" by auto
-  from rstepI[OF h(1) this] have "rpath r a (m # xs) b" .
-  from that[OF this] show ?thesis by auto
-qed
-
-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_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 intro!:rstepI)
-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
-
-context pip
-begin
-
-interpretation rtree_RAG: rtree "RAG s"
-proof
-  show "single_valued (RAG s)"
-    by (unfold single_valued_def, auto intro: unique_RAG[OF vt])
-
-  show "acyclic (RAG s)"
-     by (rule acyclic_RAG[OF vt])
-qed
-
-lemma sgv_wRAG: 
-  shows "single_valued (wRAG s)"
-  using waiting_unique[OF vt]
-  by (unfold single_valued_def wRAG_def, auto)
-
-lemma sgv_hRAG: 
-  shows "single_valued (hRAG s)"
-  using held_unique
-  by (unfold single_valued_def hRAG_def, auto)
-
-lemma sgv_tRAG: shows "single_valued (tRAG s)"
-  by (unfold tRAG_def, rule Relation.single_valued_relcomp, 
-        insert sgv_hRAG sgv_wRAG, auto)
-
-lemma acyclic_hRAG: 
-  shows "acyclic (hRAG s)"
-  by (rule acyclic_subset[OF acyclic_RAG[OF vt]], insert RAG_split, auto)
-
-lemma acyclic_wRAG: 
-  shows "acyclic (wRAG s)"
-  by (rule acyclic_subset[OF acyclic_RAG[OF vt]], insert RAG_split, auto)
-
-lemma acyclic_tRAG: 
-  shows "acyclic (tRAG s)"
-  by (unfold tRAG_def, rule acyclic_compose[OF acyclic_RAG[OF vt]],
-         unfold RAG_split, auto)
-
-lemma acyclic_tG:
-  shows "acyclic (tG s)"
-proof(unfold tG_def, rule rel_map_acyclic[OF acyclic_tRAG])
-  show "inj_on the_thread (Domain (tRAG s) \<union> Range (tRAG s))"
-  proof(rule subset_inj_on)
-    show " inj_on the_thread {Th th |th. True}" by (unfold inj_on_def, auto)
-  next
-    from domain_tRAG range_tRAG 
-    show " Domain (tRAG s) \<union> Range (tRAG s) \<subseteq> {Th th |th. True}" by auto
-  qed
-qed
-
-end