
diff -r 35094c8fd8bf -r a073cb249a06 src/HOL/Hoare_Parallel/RG_Hoare.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Hoare_Parallel/RG_Hoare.thy	Mon Sep 21 10:58:25 2009 +0200
@@ -0,0 +1,1375 @@
+header {* \section{The Proof System} *}
+
+theory RG_Hoare imports RG_Tran begin
+
+subsection {* Proof System for Component Programs *}
+
+declare Un_subset_iff [iff del]
+declare Cons_eq_map_conv[iff]
+
+constdefs
+  stable :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"  
+  "stable \<equiv> \<lambda>f g. (\<forall>x y. x \<in> f \<longrightarrow> (x, y) \<in> g \<longrightarrow> y \<in> f)" 
+
+inductive
+  rghoare :: "['a com, 'a set, ('a \<times> 'a) set, ('a \<times> 'a) set, 'a set] \<Rightarrow> bool"  
+    ("\<turnstile> _ sat [_, _, _, _]" [60,0,0,0,0] 45)
+where
+  Basic: "\<lbrakk> pre \<subseteq> {s. f s \<in> post}; {(s,t). s \<in> pre \<and> (t=f s \<or> t=s)} \<subseteq> guar; 
+            stable pre rely; stable post rely \<rbrakk> 
+           \<Longrightarrow> \<turnstile> Basic f sat [pre, rely, guar, post]"
+
+| Seq: "\<lbrakk> \<turnstile> P sat [pre, rely, guar, mid]; \<turnstile> Q sat [mid, rely, guar, post] \<rbrakk> 
+           \<Longrightarrow> \<turnstile> Seq P Q sat [pre, rely, guar, post]"
+
+| Cond: "\<lbrakk> stable pre rely; \<turnstile> P1 sat [pre \<inter> b, rely, guar, post];
+           \<turnstile> P2 sat [pre \<inter> -b, rely, guar, post]; \<forall>s. (s,s)\<in>guar \<rbrakk>
+          \<Longrightarrow> \<turnstile> Cond b P1 P2 sat [pre, rely, guar, post]"
+
+| While: "\<lbrakk> stable pre rely; (pre \<inter> -b) \<subseteq> post; stable post rely;
+            \<turnstile> P sat [pre \<inter> b, rely, guar, pre]; \<forall>s. (s,s)\<in>guar \<rbrakk>
+          \<Longrightarrow> \<turnstile> While b P sat [pre, rely, guar, post]"
+
+| Await: "\<lbrakk> stable pre rely; stable post rely; 
+            \<forall>V. \<turnstile> P sat [pre \<inter> b \<inter> {V}, {(s, t). s = t}, 
+                UNIV, {s. (V, s) \<in> guar} \<inter> post] \<rbrakk>
+           \<Longrightarrow> \<turnstile> Await b P sat [pre, rely, guar, post]"
+  
+| Conseq: "\<lbrakk> pre \<subseteq> pre'; rely \<subseteq> rely'; guar' \<subseteq> guar; post' \<subseteq> post;
+             \<turnstile> P sat [pre', rely', guar', post'] \<rbrakk>
+            \<Longrightarrow> \<turnstile> P sat [pre, rely, guar, post]"
+
+constdefs 
+  Pre :: "'a rgformula \<Rightarrow> 'a set"
+  "Pre x \<equiv> fst(snd x)"
+  Post :: "'a rgformula \<Rightarrow> 'a set"
+  "Post x \<equiv> snd(snd(snd(snd x)))"
+  Rely :: "'a rgformula \<Rightarrow> ('a \<times> 'a) set"
+  "Rely x \<equiv> fst(snd(snd x))"
+  Guar :: "'a rgformula \<Rightarrow> ('a \<times> 'a) set"
+  "Guar x \<equiv> fst(snd(snd(snd x)))"
+  Com :: "'a rgformula \<Rightarrow> 'a com"
+  "Com x \<equiv> fst x"
+
+subsection {* Proof System for Parallel Programs *}
+
+types 'a par_rgformula = "('a rgformula) list \<times> 'a set \<times> ('a \<times> 'a) set \<times> ('a \<times> 'a) set \<times> 'a set"
+
+inductive
+  par_rghoare :: "('a rgformula) list \<Rightarrow> 'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set \<Rightarrow> bool"
+    ("\<turnstile> _ SAT [_, _, _, _]" [60,0,0,0,0] 45)
+where
+  Parallel: 
+  "\<lbrakk> \<forall>i<length xs. rely \<union> (\<Union>j\<in>{j. j<length xs \<and> j\<noteq>i}. Guar(xs!j)) \<subseteq> Rely(xs!i);
+    (\<Union>j\<in>{j. j<length xs}. Guar(xs!j)) \<subseteq> guar;
+     pre \<subseteq> (\<Inter>i\<in>{i. i<length xs}. Pre(xs!i)); 
+    (\<Inter>i\<in>{i. i<length xs}. Post(xs!i)) \<subseteq> post;
+    \<forall>i<length xs. \<turnstile> Com(xs!i) sat [Pre(xs!i),Rely(xs!i),Guar(xs!i),Post(xs!i)] \<rbrakk>
+   \<Longrightarrow>  \<turnstile> xs SAT [pre, rely, guar, post]"
+
+section {* Soundness*}
+
+subsubsection {* Some previous lemmas *}
+
+lemma tl_of_assum_in_assum: 
+  "(P, s) # (P, t) # xs \<in> assum (pre, rely) \<Longrightarrow> stable pre rely 
+  \<Longrightarrow> (P, t) # xs \<in> assum (pre, rely)"
+apply(simp add:assum_def)
+apply clarify
+apply(rule conjI)
+ apply(erule_tac x=0 in allE)
+ apply(simp (no_asm_use)only:stable_def)
+ apply(erule allE,erule allE,erule impE,assumption,erule mp)
+ apply(simp add:Env)
+apply clarify
+apply(erule_tac x="Suc i" in allE)
+apply simp
+done
+
+lemma etran_in_comm: 
+  "(P, t) # xs \<in> comm(guar, post) \<Longrightarrow> (P, s) # (P, t) # xs \<in> comm(guar, post)"
+apply(simp add:comm_def)
+apply clarify
+apply(case_tac i,simp+)
+done
+
+lemma ctran_in_comm: 
+  "\<lbrakk>(s, s) \<in> guar; (Q, s) # xs \<in> comm(guar, post)\<rbrakk> 
+  \<Longrightarrow> (P, s) # (Q, s) # xs \<in> comm(guar, post)"
+apply(simp add:comm_def)
+apply clarify
+apply(case_tac i,simp+)
+done
+
+lemma takecptn_is_cptn [rule_format, elim!]: 
+  "\<forall>j. c \<in> cptn \<longrightarrow> take (Suc j) c \<in> cptn"
+apply(induct "c")
+ apply(force elim: cptn.cases)
+apply clarify
+apply(case_tac j) 
+ apply simp
+ apply(rule CptnOne)
+apply simp
+apply(force intro:cptn.intros elim:cptn.cases)
+done
+
+lemma dropcptn_is_cptn [rule_format,elim!]: 
+  "\<forall>j<length c. c \<in> cptn \<longrightarrow> drop j c \<in> cptn"
+apply(induct "c")
+ apply(force elim: cptn.cases)
+apply clarify
+apply(case_tac j,simp+) 
+apply(erule cptn.cases)
+  apply simp
+ apply force
+apply force
+done
+
+lemma takepar_cptn_is_par_cptn [rule_format,elim]: 
+  "\<forall>j. c \<in> par_cptn \<longrightarrow> take (Suc j) c \<in> par_cptn"
+apply(induct "c")
+ apply(force elim: cptn.cases)
+apply clarify
+apply(case_tac j,simp) 
+ apply(rule ParCptnOne)
+apply(force intro:par_cptn.intros elim:par_cptn.cases)
+done
+
+lemma droppar_cptn_is_par_cptn [rule_format]:
+  "\<forall>j<length c. c \<in> par_cptn \<longrightarrow> drop j c \<in> par_cptn"
+apply(induct "c")
+ apply(force elim: par_cptn.cases)
+apply clarify
+apply(case_tac j,simp+) 
+apply(erule par_cptn.cases)
+  apply simp
+ apply force
+apply force
+done
+
+lemma tl_of_cptn_is_cptn: "\<lbrakk>x # xs \<in> cptn; xs \<noteq> []\<rbrakk> \<Longrightarrow> xs  \<in> cptn"
+apply(subgoal_tac "1 < length (x # xs)") 
+ apply(drule dropcptn_is_cptn,simp+)
+done
+
+lemma not_ctran_None [rule_format]: 
+  "\<forall>s. (None, s)#xs \<in> cptn \<longrightarrow> (\<forall>i<length xs. ((None, s)#xs)!i -e\<rightarrow> xs!i)"
+apply(induct xs,simp+)
+apply clarify
+apply(erule cptn.cases,simp)
+ apply simp
+ apply(case_tac i,simp)
+  apply(rule Env)
+ apply simp
+apply(force elim:ctran.cases)
+done
+
+lemma cptn_not_empty [simp]:"[] \<notin> cptn"
+apply(force elim:cptn.cases)
+done
+
+lemma etran_or_ctran [rule_format]: 
+  "\<forall>m i. x\<in>cptn \<longrightarrow> m \<le> length x 
+   \<longrightarrow> (\<forall>i. Suc i < m \<longrightarrow> \<not> x!i -c\<rightarrow> x!Suc i) \<longrightarrow> Suc i < m 
+   \<longrightarrow> x!i -e\<rightarrow> x!Suc i"
+apply(induct x,simp)
+apply clarify
+apply(erule cptn.cases,simp)
+ apply(case_tac i,simp)
+  apply(rule Env)
+ apply simp
+ apply(erule_tac x="m - 1" in allE)
+ apply(case_tac m,simp,simp)
+ apply(subgoal_tac "(\<forall>i. Suc i < nata \<longrightarrow> (((P, t) # xs) ! i, xs ! i) \<notin> ctran)")
+  apply force
+ apply clarify
+ apply(erule_tac x="Suc ia" in allE,simp)
+apply(erule_tac x="0" and P="\<lambda>j. ?H j \<longrightarrow> (?J j) \<notin> ctran" in allE,simp)
+done
+
+lemma etran_or_ctran2 [rule_format]: 
+  "\<forall>i. Suc i<length x \<longrightarrow> x\<in>cptn \<longrightarrow> (x!i -c\<rightarrow> x!Suc i \<longrightarrow> \<not> x!i -e\<rightarrow> x!Suc i)
+  \<or> (x!i -e\<rightarrow> x!Suc i \<longrightarrow> \<not> x!i -c\<rightarrow> x!Suc i)"
+apply(induct x)
+ apply simp
+apply clarify
+apply(erule cptn.cases,simp)
+ apply(case_tac i,simp+)
+apply(case_tac i,simp)
+ apply(force elim:etran.cases)
+apply simp
+done
+
+lemma etran_or_ctran2_disjI1: 
+  "\<lbrakk> x\<in>cptn; Suc i<length x; x!i -c\<rightarrow> x!Suc i\<rbrakk> \<Longrightarrow> \<not> x!i -e\<rightarrow> x!Suc i"
+by(drule etran_or_ctran2,simp_all)
+
+lemma etran_or_ctran2_disjI2: 
+  "\<lbrakk> x\<in>cptn; Suc i<length x; x!i -e\<rightarrow> x!Suc i\<rbrakk> \<Longrightarrow> \<not> x!i -c\<rightarrow> x!Suc i"
+by(drule etran_or_ctran2,simp_all)
+
+lemma not_ctran_None2 [rule_format]: 
+  "\<lbrakk> (None, s) # xs \<in>cptn; i<length xs\<rbrakk> \<Longrightarrow> \<not> ((None, s) # xs) ! i -c\<rightarrow> xs ! i"
+apply(frule not_ctran_None,simp)
+apply(case_tac i,simp)
+ apply(force elim:etranE)
+apply simp
+apply(rule etran_or_ctran2_disjI2,simp_all)
+apply(force intro:tl_of_cptn_is_cptn)
+done
+
+lemma Ex_first_occurrence [rule_format]: "P (n::nat) \<longrightarrow> (\<exists>m. P m \<and> (\<forall>i<m. \<not> P i))";
+apply(rule nat_less_induct)
+apply clarify
+apply(case_tac "\<forall>m. m<n \<longrightarrow> \<not> P m")
+apply auto
+done
+ 
+lemma stability [rule_format]: 
+  "\<forall>j k. x \<in> cptn \<longrightarrow> stable p rely \<longrightarrow> j\<le>k \<longrightarrow> k<length x \<longrightarrow> snd(x!j)\<in>p  \<longrightarrow>
+  (\<forall>i. (Suc i)<length x \<longrightarrow> 
+          (x!i -e\<rightarrow> x!(Suc i)) \<longrightarrow> (snd(x!i), snd(x!(Suc i))) \<in> rely) \<longrightarrow> 
+  (\<forall>i. j\<le>i \<and> i<k \<longrightarrow> x!i -e\<rightarrow> x!Suc i) \<longrightarrow> snd(x!k)\<in>p \<and> fst(x!j)=fst(x!k)"
+apply(induct x)
+ apply clarify
+ apply(force elim:cptn.cases)
+apply clarify
+apply(erule cptn.cases,simp)
+ apply simp
+ apply(case_tac k,simp,simp)
+ apply(case_tac j,simp) 
+  apply(erule_tac x=0 in allE)
+  apply(erule_tac x="nat" and P="\<lambda>j. (0\<le>j) \<longrightarrow> (?J j)" in allE,simp)
+  apply(subgoal_tac "t\<in>p")
+   apply(subgoal_tac "(\<forall>i. i < length xs \<longrightarrow> ((P, t) # xs) ! i -e\<rightarrow> xs ! i \<longrightarrow> (snd (((P, t) # xs) ! i), snd (xs ! i)) \<in> rely)")
+    apply clarify
+    apply(erule_tac x="Suc i" and P="\<lambda>j. (?H j) \<longrightarrow> (?J j)\<in>etran" in allE,simp)
+   apply clarify
+   apply(erule_tac x="Suc i" and P="\<lambda>j. (?H j) \<longrightarrow> (?J j) \<longrightarrow> (?T j)\<in>rely" in allE,simp)
+  apply(erule_tac x=0 and P="\<lambda>j. (?H j) \<longrightarrow> (?J j)\<in>etran \<longrightarrow> ?T j" in allE,simp)
+  apply(simp(no_asm_use) only:stable_def)
+  apply(erule_tac x=s in allE)
+  apply(erule_tac x=t in allE)
+  apply simp 
+  apply(erule mp)
+  apply(erule mp)
+  apply(rule Env)
+ apply simp
+ apply(erule_tac x="nata" in allE)
+ apply(erule_tac x="nat" and P="\<lambda>j. (?s\<le>j) \<longrightarrow> (?J j)" in allE,simp)
+ apply(subgoal_tac "(\<forall>i. i < length xs \<longrightarrow> ((P, t) # xs) ! i -e\<rightarrow> xs ! i \<longrightarrow> (snd (((P, t) # xs) ! i), snd (xs ! i)) \<in> rely)")
+  apply clarify
+  apply(erule_tac x="Suc i" and P="\<lambda>j. (?H j) \<longrightarrow> (?J j)\<in>etran" in allE,simp)
+ apply clarify
+ apply(erule_tac x="Suc i" and P="\<lambda>j. (?H j) \<longrightarrow> (?J j) \<longrightarrow> (?T j)\<in>rely" in allE,simp)
+apply(case_tac k,simp,simp)
+apply(case_tac j)
+ apply(erule_tac x=0 and P="\<lambda>j. (?H j) \<longrightarrow> (?J j)\<in>etran" in allE,simp)
+ apply(erule etran.cases,simp)
+apply(erule_tac x="nata" in allE)
+apply(erule_tac x="nat" and P="\<lambda>j. (?s\<le>j) \<longrightarrow> (?J j)" in allE,simp)
+apply(subgoal_tac "(\<forall>i. i < length xs \<longrightarrow> ((Q, t) # xs) ! i -e\<rightarrow> xs ! i \<longrightarrow> (snd (((Q, t) # xs) ! i), snd (xs ! i)) \<in> rely)")
+ apply clarify
+ apply(erule_tac x="Suc i" and P="\<lambda>j. (?H j) \<longrightarrow> (?J j)\<in>etran" in allE,simp)
+apply clarify
+apply(erule_tac x="Suc i" and P="\<lambda>j. (?H j) \<longrightarrow> (?J j) \<longrightarrow> (?T j)\<in>rely" in allE,simp)
+done
+
+subsection {* Soundness of the System for Component Programs *}
+
+subsubsection {* Soundness of the Basic rule *}
+
+lemma unique_ctran_Basic [rule_format]: 
+  "\<forall>s i. x \<in> cptn \<longrightarrow> x ! 0 = (Some (Basic f), s) \<longrightarrow> 
+  Suc i<length x \<longrightarrow> x!i -c\<rightarrow> x!Suc i \<longrightarrow> 
+  (\<forall>j. Suc j<length x \<longrightarrow> i\<noteq>j \<longrightarrow> x!j -e\<rightarrow> x!Suc j)"
+apply(induct x,simp)
+apply simp
+apply clarify
+apply(erule cptn.cases,simp)
+ apply(case_tac i,simp+)
+ apply clarify
+ apply(case_tac j,simp)
+  apply(rule Env)
+ apply simp
+apply clarify
+apply simp
+apply(case_tac i)
+ apply(case_tac j,simp,simp)
+ apply(erule ctran.cases,simp_all)
+ apply(force elim: not_ctran_None)
+apply(ind_cases "((Some (Basic f), sa), Q, t) \<in> ctran" for sa Q t)
+apply simp
+apply(drule_tac i=nat in not_ctran_None,simp)
+apply(erule etranE,simp)
+done
+
+lemma exists_ctran_Basic_None [rule_format]: 
+  "\<forall>s i. x \<in> cptn \<longrightarrow> x ! 0 = (Some (Basic f), s) 
+  \<longrightarrow> i<length x \<longrightarrow> fst(x!i)=None \<longrightarrow> (\<exists>j<i. x!j -c\<rightarrow> x!Suc j)"
+apply(induct x,simp)
+apply simp
+apply clarify
+apply(erule cptn.cases,simp)
+ apply(case_tac i,simp,simp)
+ apply(erule_tac x=nat in allE,simp)
+ apply clarify
+ apply(rule_tac x="Suc j" in exI,simp,simp)
+apply clarify
+apply(case_tac i,simp,simp)
+apply(rule_tac x=0 in exI,simp)
+done
+
+lemma Basic_sound: 
+  " \<lbrakk>pre \<subseteq> {s. f s \<in> post}; {(s, t). s \<in> pre \<and> t = f s} \<subseteq> guar; 
+  stable pre rely; stable post rely\<rbrakk>
+  \<Longrightarrow> \<Turnstile> Basic f sat [pre, rely, guar, post]"
+apply(unfold com_validity_def)
+apply clarify
+apply(simp add:comm_def)
+apply(rule conjI)
+ apply clarify
+ apply(simp add:cp_def assum_def)
+ apply clarify
+ apply(frule_tac j=0 and k=i and p=pre in stability)
+       apply simp_all
+   apply(erule_tac x=ia in allE,simp)
+  apply(erule_tac i=i and f=f in unique_ctran_Basic,simp_all)
+ apply(erule subsetD,simp)
+ apply(case_tac "x!i")
+ apply clarify
+ apply(drule_tac s="Some (Basic f)" in sym,simp)
+ apply(thin_tac "\<forall>j. ?H j")
+ apply(force elim:ctran.cases)
+apply clarify
+apply(simp add:cp_def)
+apply clarify
+apply(frule_tac i="length x - 1" and f=f in exists_ctran_Basic_None,simp+)
+  apply(case_tac x,simp+)
+  apply(rule last_fst_esp,simp add:last_length)
+ apply (case_tac x,simp+)
+apply(simp add:assum_def)
+apply clarify
+apply(frule_tac j=0 and k="j" and p=pre in stability)
+      apply simp_all
+  apply(erule_tac x=i in allE,simp)
+ apply(erule_tac i=j and f=f in unique_ctran_Basic,simp_all)
+apply(case_tac "x!j")
+apply clarify
+apply simp
+apply(drule_tac s="Some (Basic f)" in sym,simp)
+apply(case_tac "x!Suc j",simp)
+apply(rule ctran.cases,simp)
+apply(simp_all)
+apply(drule_tac c=sa in subsetD,simp)
+apply clarify
+apply(frule_tac j="Suc j" and k="length x - 1" and p=post in stability,simp_all)
+ apply(case_tac x,simp+)
+ apply(erule_tac x=i in allE)
+apply(erule_tac i=j and f=f in unique_ctran_Basic,simp_all)
+  apply arith+
+apply(case_tac x)
+apply(simp add:last_length)+
+done
+
+subsubsection{* Soundness of the Await rule *}
+
+lemma unique_ctran_Await [rule_format]: 
+  "\<forall>s i. x \<in> cptn \<longrightarrow> x ! 0 = (Some (Await b c), s) \<longrightarrow> 
+  Suc i<length x \<longrightarrow> x!i -c\<rightarrow> x!Suc i \<longrightarrow> 
+  (\<forall>j. Suc j<length x \<longrightarrow> i\<noteq>j \<longrightarrow> x!j -e\<rightarrow> x!Suc j)"
+apply(induct x,simp+)
+apply clarify
+apply(erule cptn.cases,simp)
+ apply(case_tac i,simp+)
+ apply clarify
+ apply(case_tac j,simp)
+  apply(rule Env)
+ apply simp
+apply clarify
+apply simp
+apply(case_tac i)
+ apply(case_tac j,simp,simp)
+ apply(erule ctran.cases,simp_all)
+ apply(force elim: not_ctran_None)
+apply(ind_cases "((Some (Await b c), sa), Q, t) \<in> ctran" for sa Q t,simp)
+apply(drule_tac i=nat in not_ctran_None,simp)
+apply(erule etranE,simp)
+done
+
+lemma exists_ctran_Await_None [rule_format]: 
+  "\<forall>s i.  x \<in> cptn \<longrightarrow> x ! 0 = (Some (Await b c), s) 
+  \<longrightarrow> i<length x \<longrightarrow> fst(x!i)=None \<longrightarrow> (\<exists>j<i. x!j -c\<rightarrow> x!Suc j)"
+apply(induct x,simp+)
+apply clarify
+apply(erule cptn.cases,simp)
+ apply(case_tac i,simp+)
+ apply(erule_tac x=nat in allE,simp)
+ apply clarify
+ apply(rule_tac x="Suc j" in exI,simp,simp)
+apply clarify
+apply(case_tac i,simp,simp)
+apply(rule_tac x=0 in exI,simp)
+done
+
+lemma Star_imp_cptn: 
+  "(P, s) -c*\<rightarrow> (R, t) \<Longrightarrow> \<exists>l \<in> cp P s. (last l)=(R, t)
+  \<and> (\<forall>i. Suc i<length l \<longrightarrow> l!i -c\<rightarrow> l!Suc i)"
+apply (erule converse_rtrancl_induct2)
+ apply(rule_tac x="[(R,t)]" in bexI)
+  apply simp
+ apply(simp add:cp_def)
+ apply(rule CptnOne)
+apply clarify
+apply(rule_tac x="(a, b)#l" in bexI)
+ apply (rule conjI)
+  apply(case_tac l,simp add:cp_def)
+  apply(simp add:last_length)
+ apply clarify
+apply(case_tac i,simp)
+apply(simp add:cp_def)
+apply force
+apply(simp add:cp_def)
+ apply(case_tac l)
+ apply(force elim:cptn.cases)
+apply simp
+apply(erule CptnComp)
+apply clarify
+done
+ 
+lemma Await_sound: 
+  "\<lbrakk>stable pre rely; stable post rely;
+  \<forall>V. \<turnstile> P sat [pre \<inter> b \<inter> {s. s = V}, {(s, t). s = t}, 
+                 UNIV, {s. (V, s) \<in> guar} \<inter> post] \<and>
+  \<Turnstile> P sat [pre \<inter> b \<inter> {s. s = V}, {(s, t). s = t}, 
+                 UNIV, {s. (V, s) \<in> guar} \<inter> post] \<rbrakk>
+  \<Longrightarrow> \<Turnstile> Await b P sat [pre, rely, guar, post]"
+apply(unfold com_validity_def)
+apply clarify
+apply(simp add:comm_def)
+apply(rule conjI)
+ apply clarify
+ apply(simp add:cp_def assum_def)
+ apply clarify
+ apply(frule_tac j=0 and k=i and p=pre in stability,simp_all)
+   apply(erule_tac x=ia in allE,simp)
+  apply(subgoal_tac "x\<in> cp (Some(Await b P)) s")
+  apply(erule_tac i=i in unique_ctran_Await,force,simp_all)
+  apply(simp add:cp_def)
+--{* here starts the different part. *}
+ apply(erule ctran.cases,simp_all)
+ apply(drule Star_imp_cptn) 
+ apply clarify
+ apply(erule_tac x=sa in allE)
+ apply clarify
+ apply(erule_tac x=sa in allE)
+ apply(drule_tac c=l in subsetD)
+  apply (simp add:cp_def)
+  apply clarify
+  apply(erule_tac x=ia and P="\<lambda>i. ?H i \<longrightarrow> (?J i,?I i)\<in>ctran" in allE,simp)
+  apply(erule etranE,simp)
+ apply simp
+apply clarify
+apply(simp add:cp_def)
+apply clarify
+apply(frule_tac i="length x - 1" in exists_ctran_Await_None,force)
+  apply (case_tac x,simp+)
+ apply(rule last_fst_esp,simp add:last_length)
+ apply(case_tac x, (simp add:cptn_not_empty)+)
+apply clarify
+apply(simp add:assum_def)
+apply clarify
+apply(frule_tac j=0 and k="j" and p=pre in stability,simp_all)
+  apply(erule_tac x=i in allE,simp)
+ apply(erule_tac i=j in unique_ctran_Await,force,simp_all)
+apply(case_tac "x!j")
+apply clarify
+apply simp
+apply(drule_tac s="Some (Await b P)" in sym,simp)
+apply(case_tac "x!Suc j",simp)
+apply(rule ctran.cases,simp)
+apply(simp_all)
+apply(drule Star_imp_cptn) 
+apply clarify
+apply(erule_tac x=sa in allE)
+apply clarify
+apply(erule_tac x=sa in allE)
+apply(drule_tac c=l in subsetD)
+ apply (simp add:cp_def)
+ apply clarify
+ apply(erule_tac x=i and P="\<lambda>i. ?H i \<longrightarrow> (?J i,?I i)\<in>ctran" in allE,simp)
+ apply(erule etranE,simp)
+apply simp
+apply clarify
+apply(frule_tac j="Suc j" and k="length x - 1" and p=post in stability,simp_all)
+ apply(case_tac x,simp+)
+ apply(erule_tac x=i in allE)
+apply(erule_tac i=j in unique_ctran_Await,force,simp_all)
+ apply arith+
+apply(case_tac x)
+apply(simp add:last_length)+
+done
+
+subsubsection{* Soundness of the Conditional rule *}
+
+lemma Cond_sound: 
+  "\<lbrakk> stable pre rely; \<Turnstile> P1 sat [pre \<inter> b, rely, guar, post]; 
+  \<Turnstile> P2 sat [pre \<inter> - b, rely, guar, post]; \<forall>s. (s,s)\<in>guar\<rbrakk>
+  \<Longrightarrow> \<Turnstile> (Cond b P1 P2) sat [pre, rely, guar, post]"
+apply(unfold com_validity_def)
+apply clarify
+apply(simp add:cp_def comm_def)
+apply(case_tac "\<exists>i. Suc i<length x \<and> x!i -c\<rightarrow> x!Suc i")
+ prefer 2
+ apply simp
+ apply clarify
+ apply(frule_tac j="0" and k="length x - 1" and p=pre in stability,simp+)
+     apply(case_tac x,simp+)
+    apply(simp add:assum_def)
+   apply(simp add:assum_def)
+  apply(erule_tac m="length x" in etran_or_ctran,simp+)
+ apply(case_tac x, (simp add:last_length)+)
+apply(erule exE)
+apply(drule_tac n=i and P="\<lambda>i. ?H i \<and> (?J i,?I i)\<in> ctran" in Ex_first_occurrence)
+apply clarify
+apply (simp add:assum_def)
+apply(frule_tac j=0 and k="m" and p=pre in stability,simp+)
+ apply(erule_tac m="Suc m" in etran_or_ctran,simp+)
+apply(erule ctran.cases,simp_all)
+ apply(erule_tac x="sa" in allE)
+ apply(drule_tac c="drop (Suc m) x" in subsetD)
+  apply simp
+  apply clarify
+ apply simp
+ apply clarify
+ apply(case_tac "i\<le>m")
+  apply(drule le_imp_less_or_eq)
+  apply(erule disjE)
+   apply(erule_tac x=i in allE, erule impE, assumption)
+   apply simp+
+ apply(erule_tac x="i - (Suc m)" and P="\<lambda>j. ?H j \<longrightarrow> ?J j \<longrightarrow> (?I j)\<in>guar" in allE)
+ apply(subgoal_tac "(Suc m)+(i - Suc m) \<le> length x")
+  apply(subgoal_tac "(Suc m)+Suc (i - Suc m) \<le> length x")
+   apply(rotate_tac -2)
+   apply simp
+  apply arith
+ apply arith
+apply(case_tac "length (drop (Suc m) x)",simp)
+apply(erule_tac x="sa" in allE)
+back
+apply(drule_tac c="drop (Suc m) x" in subsetD,simp)
+ apply clarify
+apply simp
+apply clarify
+apply(case_tac "i\<le>m")
+ apply(drule le_imp_less_or_eq)
+ apply(erule disjE)
+  apply(erule_tac x=i in allE, erule impE, assumption)
+  apply simp
+ apply simp
+apply(erule_tac x="i - (Suc m)" and P="\<lambda>j. ?H j \<longrightarrow> ?J j \<longrightarrow> (?I j)\<in>guar" in allE)
+apply(subgoal_tac "(Suc m)+(i - Suc m) \<le> length x")
+ apply(subgoal_tac "(Suc m)+Suc (i - Suc m) \<le> length x")
+  apply(rotate_tac -2)
+  apply simp
+ apply arith
+apply arith
+done
+
+subsubsection{* Soundness of the Sequential rule *}
+
+inductive_cases Seq_cases [elim!]: "(Some (Seq P Q), s) -c\<rightarrow> t"
+
+lemma last_lift_not_None: "fst ((lift Q) ((x#xs)!(length xs))) \<noteq> None"
+apply(subgoal_tac "length xs<length (x # xs)")
+ apply(drule_tac Q=Q in  lift_nth)
+ apply(erule ssubst)
+ apply (simp add:lift_def)
+ apply(case_tac "(x # xs) ! length xs",simp)
+apply simp
+done
+
+declare map_eq_Cons_conv [simp del] Cons_eq_map_conv [simp del]
+lemma Seq_sound1 [rule_format]: 
+  "x\<in> cptn_mod \<Longrightarrow> \<forall>s P. x !0=(Some (Seq P Q), s) \<longrightarrow> 
+  (\<forall>i<length x. fst(x!i)\<noteq>Some Q) \<longrightarrow> 
+  (\<exists>xs\<in> cp (Some P) s. x=map (lift Q) xs)"
+apply(erule cptn_mod.induct)
+apply(unfold cp_def)
+apply safe
+apply simp_all
+    apply(simp add:lift_def)
+    apply(rule_tac x="[(Some Pa, sa)]" in exI,simp add:CptnOne)
+   apply(subgoal_tac "(\<forall>i < Suc (length xs). fst (((Some (Seq Pa Q), t) # xs) ! i) \<noteq> Some Q)")
+    apply clarify
+    apply(rule_tac x="(Some Pa, sa) #(Some Pa, t) # zs" in exI,simp)
+    apply(rule conjI,erule CptnEnv)
+    apply(simp (no_asm_use) add:lift_def)
+   apply clarify
+   apply(erule_tac x="Suc i" in allE, simp)
+  apply(ind_cases "((Some (Seq Pa Q), sa), None, t) \<in> ctran" for Pa sa t)
+ apply(rule_tac x="(Some P, sa) # xs" in exI, simp add:cptn_iff_cptn_mod lift_def)
+apply(erule_tac x="length xs" in allE, simp)
+apply(simp only:Cons_lift_append)
+apply(subgoal_tac "length xs < length ((Some P, sa) # xs)")
+ apply(simp only :nth_append length_map last_length nth_map)
+ apply(case_tac "last((Some P, sa) # xs)")
+ apply(simp add:lift_def)
+apply simp
+done
+declare map_eq_Cons_conv [simp del] Cons_eq_map_conv [simp del]
+
+lemma Seq_sound2 [rule_format]: 
+  "x \<in> cptn \<Longrightarrow> \<forall>s P i. x!0=(Some (Seq P Q), s) \<longrightarrow> i<length x 
+  \<longrightarrow> fst(x!i)=Some Q \<longrightarrow>
+  (\<forall>j<i. fst(x!j)\<noteq>(Some Q)) \<longrightarrow>
+  (\<exists>xs ys. xs \<in> cp (Some P) s \<and> length xs=Suc i 
+   \<and> ys \<in> cp (Some Q) (snd(xs !i)) \<and> x=(map (lift Q) xs)@tl ys)"
+apply(erule cptn.induct)
+apply(unfold cp_def)
+apply safe
+apply simp_all
+ apply(case_tac i,simp+)
+ apply(erule allE,erule impE,assumption,simp)
+ apply clarify
+ apply(subgoal_tac "(\<forall>j < nat. fst (((Some (Seq Pa Q), t) # xs) ! j) \<noteq> Some Q)",clarify)
+  prefer 2
+  apply force
+ apply(case_tac xsa,simp,simp)
+ apply(rule_tac x="(Some Pa, sa) #(Some Pa, t) # list" in exI,simp)
+ apply(rule conjI,erule CptnEnv)
+ apply(simp (no_asm_use) add:lift_def)
+ apply(rule_tac x=ys in exI,simp)
+apply(ind_cases "((Some (Seq Pa Q), sa), t) \<in> ctran" for Pa sa t)
+ apply simp
+ apply(rule_tac x="(Some Pa, sa)#[(None, ta)]" in exI,simp)
+ apply(rule conjI)
+  apply(drule_tac xs="[]" in CptnComp,force simp add:CptnOne,simp)
+ apply(case_tac i, simp+)
+ apply(case_tac nat,simp+)
+ apply(rule_tac x="(Some Q,ta)#xs" in exI,simp add:lift_def)
+ apply(case_tac nat,simp+)
+ apply(force)
+apply(case_tac i, simp+)
+apply(case_tac nat,simp+)
+apply(erule_tac x="Suc nata" in allE,simp)
+apply clarify
+apply(subgoal_tac "(\<forall>j<Suc nata. fst (((Some (Seq P2 Q), ta) # xs) ! j) \<noteq> Some Q)",clarify)
+ prefer 2
+ apply clarify
+ apply force
+apply(rule_tac x="(Some Pa, sa)#(Some P2, ta)#(tl xsa)" in exI,simp)
+apply(rule conjI,erule CptnComp)
+apply(rule nth_tl_if,force,simp+)
+apply(rule_tac x=ys in exI,simp)
+apply(rule conjI)
+apply(rule nth_tl_if,force,simp+)
+ apply(rule tl_zero,simp+)
+ apply force
+apply(rule conjI,simp add:lift_def)
+apply(subgoal_tac "lift Q (Some P2, ta) =(Some (Seq P2 Q), ta)") 
+ apply(simp add:Cons_lift del:map.simps)
+ apply(rule nth_tl_if)
+   apply force
+  apply simp+
+apply(simp add:lift_def)
+done
+(*
+lemma last_lift_not_None3: "fst (last (map (lift Q) (x#xs))) \<noteq> None"
+apply(simp only:last_length [THEN sym])
+apply(subgoal_tac "length xs<length (x # xs)")
+ apply(drule_tac Q=Q in  lift_nth)
+ apply(erule ssubst)
+ apply (simp add:lift_def)
+ apply(case_tac "(x # xs) ! length xs",simp)
+apply simp
+done
+*)
+
+lemma last_lift_not_None2: "fst ((lift Q) (last (x#xs))) \<noteq> None"
+apply(simp only:last_length [THEN sym])
+apply(subgoal_tac "length xs<length (x # xs)")
+ apply(drule_tac Q=Q in  lift_nth)
+ apply(erule ssubst)
+ apply (simp add:lift_def)
+ apply(case_tac "(x # xs) ! length xs",simp)
+apply simp
+done
+
+lemma Seq_sound: 
+  "\<lbrakk>\<Turnstile> P sat [pre, rely, guar, mid]; \<Turnstile> Q sat [mid, rely, guar, post]\<rbrakk>
+  \<Longrightarrow> \<Turnstile> Seq P Q sat [pre, rely, guar, post]"
+apply(unfold com_validity_def)
+apply clarify
+apply(case_tac "\<exists>i<length x. fst(x!i)=Some Q")
+ prefer 2
+ apply (simp add:cp_def cptn_iff_cptn_mod)
+ apply clarify
+ apply(frule_tac Seq_sound1,force)
+  apply force
+ apply clarify
+ apply(erule_tac x=s in allE,simp)
+ apply(drule_tac c=xs in subsetD,simp add:cp_def cptn_iff_cptn_mod)
+  apply(simp add:assum_def)
+  apply clarify
+  apply(erule_tac P="\<lambda>j. ?H j \<longrightarrow> ?J j \<longrightarrow> ?I j" in allE,erule impE, assumption)
+  apply(simp add:snd_lift)
+  apply(erule mp)
+  apply(force elim:etranE intro:Env simp add:lift_def)
+ apply(simp add:comm_def)
+ apply(rule conjI)
+  apply clarify
+  apply(erule_tac P="\<lambda>j. ?H j \<longrightarrow> ?J j \<longrightarrow> ?I j" in allE,erule impE, assumption)
+  apply(simp add:snd_lift)
+  apply(erule mp)
+  apply(case_tac "(xs!i)")
+  apply(case_tac "(xs! Suc i)")
+  apply(case_tac "fst(xs!i)")
+   apply(erule_tac x=i in allE, simp add:lift_def)
+  apply(case_tac "fst(xs!Suc i)")
+   apply(force simp add:lift_def)
+  apply(force simp add:lift_def)
+ apply clarify
+ apply(case_tac xs,simp add:cp_def)
+ apply clarify
+ apply (simp del:map.simps)
+ apply(subgoal_tac "(map (lift Q) ((a, b) # list))\<noteq>[]")
+  apply(drule last_conv_nth)
+  apply (simp del:map.simps)
+  apply(simp only:last_lift_not_None)
+ apply simp
+--{* @{text "\<exists>i<length x. fst (x ! i) = Some Q"} *}
+apply(erule exE)
+apply(drule_tac n=i and P="\<lambda>i. i < length x \<and> fst (x ! i) = Some Q" in Ex_first_occurrence)
+apply clarify
+apply (simp add:cp_def)
+ apply clarify
+ apply(frule_tac i=m in Seq_sound2,force)
+  apply simp+
+apply clarify
+apply(simp add:comm_def)
+apply(erule_tac x=s in allE)
+apply(drule_tac c=xs in subsetD,simp)
+ apply(case_tac "xs=[]",simp)
+ apply(simp add:cp_def assum_def nth_append)
+ apply clarify
+ apply(erule_tac x=i in allE)
+  back 
+ apply(simp add:snd_lift)
+ apply(erule mp)
+ apply(force elim:etranE intro:Env simp add:lift_def)
+apply simp
+apply clarify
+apply(erule_tac x="snd(xs!m)" in allE)
+apply(drule_tac c=ys in subsetD,simp add:cp_def assum_def)
+ apply(case_tac "xs\<noteq>[]")
+ apply(drule last_conv_nth,simp)
+ apply(rule conjI)
+  apply(erule mp)
+  apply(case_tac "xs!m")
+  apply(case_tac "fst(xs!m)",simp)
+  apply(simp add:lift_def nth_append)
+ apply clarify
+ apply(erule_tac x="m+i" in allE)
+ back
+ back
+ apply(case_tac ys,(simp add:nth_append)+)
+ apply (case_tac i, (simp add:snd_lift)+)
+  apply(erule mp)
+  apply(case_tac "xs!m")
+  apply(force elim:etran.cases intro:Env simp add:lift_def)
+ apply simp 
+apply simp
+apply clarify
+apply(rule conjI,clarify)
+ apply(case_tac "i<m",simp add:nth_append)
+  apply(simp add:snd_lift)
+  apply(erule allE, erule impE, assumption, erule mp)
+  apply(case_tac "(xs ! i)")
+  apply(case_tac "(xs ! Suc i)")   
+  apply(case_tac "fst(xs ! i)",force simp add:lift_def)   
+  apply(case_tac "fst(xs ! Suc i)")
+   apply (force simp add:lift_def)
+  apply (force simp add:lift_def)
+ apply(erule_tac x="i-m" in allE) 
+ back
+ back
+ apply(subgoal_tac "Suc (i - m) < length ys",simp)
+  prefer 2
+  apply arith
+ apply(simp add:nth_append snd_lift)
+ apply(rule conjI,clarify)
+  apply(subgoal_tac "i=m")
+   prefer 2
+   apply arith
+  apply clarify
+  apply(simp add:cp_def)
+  apply(rule tl_zero)
+    apply(erule mp)
+    apply(case_tac "lift Q (xs!m)",simp add:snd_lift)
+    apply(case_tac "xs!m",case_tac "fst(xs!m)",simp add:lift_def snd_lift)
+     apply(case_tac ys,simp+)
+    apply(simp add:lift_def)
+   apply simp 
+  apply force
+ apply clarify
+ apply(rule tl_zero)
+   apply(rule tl_zero)
+     apply (subgoal_tac "i-m=Suc(i-Suc m)")
+      apply simp
+      apply(erule mp)
+      apply(case_tac ys,simp+)
+   apply force
+  apply arith
+ apply force
+apply clarify
+apply(case_tac "(map (lift Q) xs @ tl ys)\<noteq>[]")
+ apply(drule last_conv_nth)
+ apply(simp add: snd_lift nth_append)
+ apply(rule conjI,clarify)
+  apply(case_tac ys,simp+)
+ apply clarify
+ apply(case_tac ys,simp+)
+done
+
+subsubsection{* Soundness of the While rule *}
+
+lemma last_append[rule_format]:
+  "\<forall>xs. ys\<noteq>[] \<longrightarrow> ((xs@ys)!(length (xs@ys) - (Suc 0)))=(ys!(length ys - (Suc 0)))"
+apply(induct ys)
+ apply simp
+apply clarify
+apply (simp add:nth_append length_append)
+done
+
+lemma assum_after_body: 
+  "\<lbrakk> \<Turnstile> P sat [pre \<inter> b, rely, guar, pre]; 
+  (Some P, s) # xs \<in> cptn_mod; fst (last ((Some P, s) # xs)) = None; s \<in> b;
+  (Some (While b P), s) # (Some (Seq P (While b P)), s) # 
+   map (lift (While b P)) xs @ ys \<in> assum (pre, rely)\<rbrakk>
+  \<Longrightarrow> (Some (While b P), snd (last ((Some P, s) # xs))) # ys \<in> assum (pre, rely)"
+apply(simp add:assum_def com_validity_def cp_def cptn_iff_cptn_mod)
+apply clarify
+apply(erule_tac x=s in allE)
+apply(drule_tac c="(Some P, s) # xs" in subsetD,simp)
+ apply clarify
+ apply(erule_tac x="Suc i" in allE)
+ apply simp
+ apply(simp add:Cons_lift_append nth_append snd_lift del:map.simps)
+ apply(erule mp)
+ apply(erule etranE,simp)
+ apply(case_tac "fst(((Some P, s) # xs) ! i)")
+  apply(force intro:Env simp add:lift_def)
+ apply(force intro:Env simp add:lift_def)
+apply(rule conjI)
+ apply clarify
+ apply(simp add:comm_def last_length)
+apply clarify
+apply(rule conjI)
+ apply(simp add:comm_def)
+apply clarify
+apply(erule_tac x="Suc(length xs + i)" in allE,simp)
+apply(case_tac i, simp add:nth_append Cons_lift_append snd_lift del:map.simps)
+ apply(simp add:last_length)
+ apply(erule mp)
+ apply(case_tac "last xs")
+ apply(simp add:lift_def)
+apply(simp add:Cons_lift_append nth_append snd_lift del:map.simps)
+done
+
+lemma While_sound_aux [rule_format]: 
+  "\<lbrakk> pre \<inter> - b \<subseteq> post; \<Turnstile> P sat [pre \<inter> b, rely, guar, pre]; \<forall>s. (s, s) \<in> guar;
+   stable pre rely;  stable post rely; x \<in> cptn_mod \<rbrakk> 
+  \<Longrightarrow>  \<forall>s xs. x=(Some(While b P),s)#xs \<longrightarrow> x\<in>assum(pre, rely) \<longrightarrow> x \<in> comm (guar, post)"
+apply(erule cptn_mod.induct)
+apply safe
+apply (simp_all del:last.simps)
+--{* 5 subgoals left *}
+apply(simp add:comm_def)
+--{* 4 subgoals left *}
+apply(rule etran_in_comm)
+apply(erule mp)
+apply(erule tl_of_assum_in_assum,simp)
+--{* While-None *}
+apply(ind_cases "((Some (While b P), s), None, t) \<in> ctran" for s t)
+apply(simp add:comm_def)
+apply(simp add:cptn_iff_cptn_mod [THEN sym])
+apply(rule conjI,clarify)
+ apply(force simp add:assum_def)
+apply clarify
+apply(rule conjI, clarify)
+ apply(case_tac i,simp,simp)
+ apply(force simp add:not_ctran_None2)
+apply(subgoal_tac "\<forall>i. Suc i < length ((None, t) # xs) \<longrightarrow> (((None, t) # xs) ! i, ((None, t) # xs) ! Suc i)\<in> etran")
+ prefer 2
+ apply clarify
+ apply(rule_tac m="length ((None, s) # xs)" in etran_or_ctran,simp+)
+ apply(erule not_ctran_None2,simp)
+ apply simp+
+apply(frule_tac j="0" and k="length ((None, s) # xs) - 1" and p=post in stability,simp+)
+   apply(force simp add:assum_def subsetD)
+  apply(simp add:assum_def)
+  apply clarify
+  apply(erule_tac x="i" in allE,simp) 
+  apply(erule_tac x="Suc i" in allE,simp) 
+ apply simp
+apply clarify
+apply (simp add:last_length)
+--{* WhileOne *}
+apply(thin_tac "P = While b P \<longrightarrow> ?Q")
+apply(rule ctran_in_comm,simp)
+apply(simp add:Cons_lift del:map.simps)
+apply(simp add:comm_def del:map.simps)
+apply(rule conjI)
+ apply clarify
+ apply(case_tac "fst(((Some P, sa) # xs) ! i)")
+  apply(case_tac "((Some P, sa) # xs) ! i")
+  apply (simp add:lift_def)
+  apply(ind_cases "(Some (While b P), ba) -c\<rightarrow> t" for ba t)
+   apply simp
+  apply simp
+ apply(simp add:snd_lift del:map.simps)
+ apply(simp only:com_validity_def cp_def cptn_iff_cptn_mod)
+ apply(erule_tac x=sa in allE)
+ apply(drule_tac c="(Some P, sa) # xs" in subsetD)
+  apply (simp add:assum_def del:map.simps)
+  apply clarify
+  apply(erule_tac x="Suc ia" in allE,simp add:snd_lift del:map.simps)
+  apply(erule mp)
+  apply(case_tac "fst(((Some P, sa) # xs) ! ia)")
+   apply(erule etranE,simp add:lift_def)
+   apply(rule Env)
+  apply(erule etranE,simp add:lift_def)
+  apply(rule Env)
+ apply (simp add:comm_def del:map.simps)
+ apply clarify
+ apply(erule allE,erule impE,assumption)
+ apply(erule mp)
+ apply(case_tac "((Some P, sa) # xs) ! i")
+ apply(case_tac "xs!i")
+ apply(simp add:lift_def)
+ apply(case_tac "fst(xs!i)")
+  apply force
+ apply force
+--{* last=None *}
+apply clarify
+apply(subgoal_tac "(map (lift (While b P)) ((Some P, sa) # xs))\<noteq>[]")
+ apply(drule last_conv_nth)
+ apply (simp del:map.simps)
+ apply(simp only:last_lift_not_None)
+apply simp
+--{* WhileMore *}
+apply(thin_tac "P = While b P \<longrightarrow> ?Q")
+apply(rule ctran_in_comm,simp del:last.simps)
+--{* metiendo la hipotesis antes de dividir la conclusion. *}
+apply(subgoal_tac "(Some (While b P), snd (last ((Some P, sa) # xs))) # ys \<in> assum (pre, rely)")
+ apply (simp del:last.simps)
+ prefer 2
+ apply(erule assum_after_body)
+  apply (simp del:last.simps)+
+--{* lo de antes. *}
+apply(simp add:comm_def del:map.simps last.simps)
+apply(rule conjI)
+ apply clarify
+ apply(simp only:Cons_lift_append)
+ apply(case_tac "i<length xs")
+  apply(simp add:nth_append del:map.simps last.simps)
+  apply(case_tac "fst(((Some P, sa) # xs) ! i)")
+   apply(case_tac "((Some P, sa) # xs) ! i")
+   apply (simp add:lift_def del:last.simps)
+   apply(ind_cases "(Some (While b P), ba) -c\<rightarrow> t" for ba t)
+    apply simp
+   apply simp
+  apply(simp add:snd_lift del:map.simps last.simps)
+  apply(thin_tac " \<forall>i. i < length ys \<longrightarrow> ?P i")
+  apply(simp only:com_validity_def cp_def cptn_iff_cptn_mod)
+  apply(erule_tac x=sa in allE)
+  apply(drule_tac c="(Some P, sa) # xs" in subsetD)
+   apply (simp add:assum_def del:map.simps last.simps)
+   apply clarify
+   apply(erule_tac x="Suc ia" in allE,simp add:nth_append snd_lift del:map.simps last.simps, erule mp)
+   apply(case_tac "fst(((Some P, sa) # xs) ! ia)")
+    apply(erule etranE,simp add:lift_def)
+    apply(rule Env)
+   apply(erule etranE,simp add:lift_def)
+   apply(rule Env)
+  apply (simp add:comm_def del:map.simps)
+  apply clarify
+  apply(erule allE,erule impE,assumption)
+  apply(erule mp)
+  apply(case_tac "((Some P, sa) # xs) ! i")
+  apply(case_tac "xs!i")
+  apply(simp add:lift_def)
+  apply(case_tac "fst(xs!i)")
+   apply force
+ apply force
+--{*  @{text "i \<ge> length xs"} *}
+apply(subgoal_tac "i-length xs <length ys") 
+ prefer 2
+ apply arith
+apply(erule_tac x="i-length xs" in allE,clarify)
+apply(case_tac "i=length xs")
+ apply (simp add:nth_append snd_lift del:map.simps last.simps)
+ apply(simp add:last_length del:last.simps)
+ apply(erule mp)
+ apply(case_tac "last((Some P, sa) # xs)")
+ apply(simp add:lift_def del:last.simps)
+--{* @{text "i>length xs"} *} 
+apply(case_tac "i-length xs")
+ apply arith
+apply(simp add:nth_append del:map.simps last.simps)
+apply(rotate_tac -3)
+apply(subgoal_tac "i- Suc (length xs)=nat")
+ prefer 2
+ apply arith
+apply simp
+--{* last=None *}
+apply clarify
+apply(case_tac ys)
+ apply(simp add:Cons_lift del:map.simps last.simps)
+ apply(subgoal_tac "(map (lift (While b P)) ((Some P, sa) # xs))\<noteq>[]")
+  apply(drule last_conv_nth)
+  apply (simp del:map.simps)
+  apply(simp only:last_lift_not_None)
+ apply simp
+apply(subgoal_tac "((Some (Seq P (While b P)), sa) # map (lift (While b P)) xs @ ys)\<noteq>[]")
+ apply(drule last_conv_nth)
+ apply (simp del:map.simps last.simps)
+ apply(simp add:nth_append del:last.simps)
+ apply(subgoal_tac "((Some (While b P), snd (last ((Some P, sa) # xs))) # a # list)\<noteq>[]")
+  apply(drule last_conv_nth)
+  apply (simp del:map.simps last.simps)
+ apply simp
+apply simp
+done
+
+lemma While_sound: 
+  "\<lbrakk>stable pre rely; pre \<inter> - b \<subseteq> post; stable post rely;
+    \<Turnstile> P sat [pre \<inter> b, rely, guar, pre]; \<forall>s. (s,s)\<in>guar\<rbrakk>
+  \<Longrightarrow> \<Turnstile> While b P sat [pre, rely, guar, post]"
+apply(unfold com_validity_def)
+apply clarify
+apply(erule_tac xs="tl x" in While_sound_aux)
+ apply(simp add:com_validity_def)
+ apply force
+ apply simp_all
+apply(simp add:cptn_iff_cptn_mod cp_def)
+apply(simp add:cp_def)
+apply clarify
+apply(rule nth_equalityI)
+ apply simp_all
+ apply(case_tac x,simp+)
+apply clarify
+apply(case_tac i,simp+)
+apply(case_tac x,simp+)
+done
+
+subsubsection{* Soundness of the Rule of Consequence *}
+
+lemma Conseq_sound: 
+  "\<lbrakk>pre \<subseteq> pre'; rely \<subseteq> rely'; guar' \<subseteq> guar; post' \<subseteq> post; 
+  \<Turnstile> P sat [pre', rely', guar', post']\<rbrakk>
+  \<Longrightarrow> \<Turnstile> P sat [pre, rely, guar, post]"
+apply(simp add:com_validity_def assum_def comm_def)
+apply clarify
+apply(erule_tac x=s in allE)
+apply(drule_tac c=x in subsetD)
+ apply force
+apply force
+done
+
+subsubsection {* Soundness of the system for sequential component programs *}
+
+theorem rgsound: 
+  "\<turnstile> P sat [pre, rely, guar, post] \<Longrightarrow> \<Turnstile> P sat [pre, rely, guar, post]"
+apply(erule rghoare.induct)
+ apply(force elim:Basic_sound)
+ apply(force elim:Seq_sound)
+ apply(force elim:Cond_sound)
+ apply(force elim:While_sound)
+ apply(force elim:Await_sound)
+apply(erule Conseq_sound,simp+)
+done     
+
+subsection {* Soundness of the System for Parallel Programs *}
+
+constdefs
+  ParallelCom :: "('a rgformula) list \<Rightarrow> 'a par_com"
+  "ParallelCom Ps \<equiv> map (Some \<circ> fst) Ps" 
+
+lemma two: 
+  "\<lbrakk> \<forall>i<length xs. rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j)) 
+     \<subseteq> Rely (xs ! i);
+   pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i));
+   \<forall>i<length xs. 
+   \<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)];
+   length xs=length clist; x \<in> par_cp (ParallelCom xs) s; x\<in>par_assum(pre, rely);
+  \<forall>i<length clist. clist!i\<in>cp (Some(Com(xs!i))) s; x \<propto> clist \<rbrakk>
+  \<Longrightarrow> \<forall>j i. i<length clist \<and> Suc j<length x \<longrightarrow> (clist!i!j) -c\<rightarrow> (clist!i!Suc j) 
+  \<longrightarrow> (snd(clist!i!j), snd(clist!i!Suc j)) \<in> Guar(xs!i)"
+apply(unfold par_cp_def)
+apply (rule ccontr) 
+--{* By contradiction: *}
+apply (simp del: Un_subset_iff)
+apply(erule exE)
+--{* the first c-tran that does not satisfy the guarantee-condition is from @{text "\<sigma>_i"} at step @{text "m"}. *}
+apply(drule_tac n=j and P="\<lambda>j. \<exists>i. ?H i j" in Ex_first_occurrence)
+apply(erule exE)
+apply clarify
+--{* @{text "\<sigma>_i \<in> A(pre, rely_1)"} *}
+apply(subgoal_tac "take (Suc (Suc m)) (clist!i) \<in> assum(Pre(xs!i), Rely(xs!i))")
+--{* but this contradicts @{text "\<Turnstile> \<sigma>_i sat [pre_i,rely_i,guar_i,post_i]"} *}
+ apply(erule_tac x=i and P="\<lambda>i. ?H i \<longrightarrow> \<Turnstile> (?J i) sat [?I i,?K i,?M i,?N i]" in allE,erule impE,assumption)
+ apply(simp add:com_validity_def)
+ apply(erule_tac x=s in allE)
+ apply(simp add:cp_def comm_def)
+ apply(drule_tac c="take (Suc (Suc m)) (clist ! i)" in subsetD)
+  apply simp
+  apply (blast intro: takecptn_is_cptn) 
+ apply simp
+ apply clarify
+ apply(erule_tac x=m and P="\<lambda>j. ?I j \<and> ?J j \<longrightarrow> ?H j" in allE)
+ apply (simp add:conjoin_def same_length_def)
+apply(simp add:assum_def del: Un_subset_iff)
+apply(rule conjI)
+ apply(erule_tac x=i and P="\<lambda>j. ?H j \<longrightarrow>  ?I j \<in>cp (?K j) (?J j)" in allE)
+ apply(simp add:cp_def par_assum_def)
+ apply(drule_tac c="s" in subsetD,simp)
+ apply simp
+apply clarify
+apply(erule_tac x=i and P="\<lambda>j. ?H j \<longrightarrow> ?M \<union> UNION (?S j) (?T j) \<subseteq>  (?L j)" in allE)
+apply(simp del: Un_subset_iff)
+apply(erule subsetD)
+apply simp
+apply(simp add:conjoin_def compat_label_def)
+apply clarify
+apply(erule_tac x=ia and P="\<lambda>j. ?H j \<longrightarrow> (?P j) \<or> ?Q j" in allE,simp)
+--{* each etran in @{text "\<sigma>_1[0\<dots>m]"} corresponds to  *}
+apply(erule disjE)
+--{* a c-tran in some @{text "\<sigma>_{ib}"}  *}
+ apply clarify
+ apply(case_tac "i=ib",simp)
+  apply(erule etranE,simp)
+ apply(erule_tac x="ib" and P="\<lambda>i. ?H i \<longrightarrow> (?I i) \<or> (?J i)" in allE)
+ apply (erule etranE)
+ apply(case_tac "ia=m",simp)
+ apply simp
+ apply(erule_tac x=ia and P="\<lambda>j. ?H j \<longrightarrow> (\<forall> i. ?P i j)" in allE)
+ apply(subgoal_tac "ia<m",simp)
+  prefer 2
+  apply arith
+ apply(erule_tac x=ib and P="\<lambda>j. (?I j, ?H j)\<in> ctran \<longrightarrow> (?P i j)" in allE,simp)
+ apply(simp add:same_state_def)
+ apply(erule_tac x=i and P="\<lambda>j. (?T j) \<longrightarrow> (\<forall>i. (?H j i) \<longrightarrow> (snd (?d j i))=(snd (?e j i)))" in all_dupE)
+ apply(erule_tac x=ib and P="\<lambda>j. (?T j) \<longrightarrow> (\<forall>i. (?H j i) \<longrightarrow> (snd (?d j i))=(snd (?e j i)))" in allE,simp)
+--{* or an e-tran in @{text "\<sigma>"}, 
+therefore it satisfies @{text "rely \<or> guar_{ib}"} *}
+apply (force simp add:par_assum_def same_state_def)
+done
+
+
+lemma three [rule_format]: 
+  "\<lbrakk> xs\<noteq>[]; \<forall>i<length xs. rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j)) 
+   \<subseteq> Rely (xs ! i);
+   pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i));
+   \<forall>i<length xs. 
+    \<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)];
+   length xs=length clist; x \<in> par_cp (ParallelCom xs) s; x \<in> par_assum(pre, rely);
+  \<forall>i<length clist. clist!i\<in>cp (Some(Com(xs!i))) s; x \<propto> clist \<rbrakk>
+  \<Longrightarrow> \<forall>j i. i<length clist \<and> Suc j<length x \<longrightarrow> (clist!i!j) -e\<rightarrow> (clist!i!Suc j) 
+  \<longrightarrow> (snd(clist!i!j), snd(clist!i!Suc j)) \<in> rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j))"
+apply(drule two)
+ apply simp_all
+apply clarify
+apply(simp add:conjoin_def compat_label_def)
+apply clarify
+apply(erule_tac x=j and P="\<lambda>j. ?H j \<longrightarrow> (?J j \<and> (\<exists>i. ?P i j)) \<or> ?I j" in allE,simp)
+apply(erule disjE)
+ prefer 2
+ apply(force simp add:same_state_def par_assum_def)
+apply clarify
+apply(case_tac "i=ia",simp)
+ apply(erule etranE,simp)
+apply(erule_tac x="ia" and P="\<lambda>i. ?H i \<longrightarrow> (?I i) \<or> (?J i)" in allE,simp)
+apply(erule_tac x=j and P="\<lambda>j. \<forall>i. ?S j i \<longrightarrow> (?I j i, ?H j i)\<in> ctran \<longrightarrow> (?P i j)" in allE)
+apply(erule_tac x=ia and P="\<lambda>j. ?S j \<longrightarrow> (?I j, ?H j)\<in> ctran \<longrightarrow> (?P j)" in allE)
+apply(simp add:same_state_def)
+apply(erule_tac x=i and P="\<lambda>j. (?T j) \<longrightarrow> (\<forall>i. (?H j i) \<longrightarrow> (snd (?d j i))=(snd (?e j i)))" in all_dupE)
+apply(erule_tac x=ia and P="\<lambda>j. (?T j) \<longrightarrow> (\<forall>i. (?H j i) \<longrightarrow> (snd (?d j i))=(snd (?e j i)))" in allE,simp)
+done
+
+lemma four: 
+  "\<lbrakk>xs\<noteq>[]; \<forall>i < length xs. rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j)) 
+    \<subseteq> Rely (xs ! i);
+   (\<Union>j\<in>{j. j < length xs}. Guar (xs ! j)) \<subseteq> guar; 
+   pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i));
+   \<forall>i < length xs.
+    \<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)];
+   x \<in> par_cp (ParallelCom xs) s; x \<in> par_assum (pre, rely); Suc i < length x; 
+   x ! i -pc\<rightarrow> x ! Suc i\<rbrakk>
+  \<Longrightarrow> (snd (x ! i), snd (x ! Suc i)) \<in> guar"
+apply(simp add: ParallelCom_def del: Un_subset_iff)
+apply(subgoal_tac "(map (Some \<circ> fst) xs)\<noteq>[]")
+ prefer 2
+ apply simp
+apply(frule rev_subsetD)
+ apply(erule one [THEN equalityD1])
+apply(erule subsetD)
+apply (simp del: Un_subset_iff)
+apply clarify
+apply(drule_tac pre=pre and rely=rely and  x=x and s=s and xs=xs and clist=clist in two)
+apply(assumption+)
+     apply(erule sym)
+    apply(simp add:ParallelCom_def)
+   apply assumption
+  apply(simp add:Com_def)
+ apply assumption
+apply(simp add:conjoin_def same_program_def)
+apply clarify
+apply(erule_tac x=i and P="\<lambda>j. ?H j \<longrightarrow> fst(?I j)=(?J j)" in all_dupE)
+apply(erule_tac x="Suc i" and P="\<lambda>j. ?H j \<longrightarrow> fst(?I j)=(?J j)" in allE)
+apply(erule par_ctranE,simp)
+apply(erule_tac x=i and P="\<lambda>j. \<forall>i. ?S j i \<longrightarrow> (?I j i, ?H j i)\<in> ctran \<longrightarrow> (?P i j)" in allE)
+apply(erule_tac x=ia and P="\<lambda>j. ?S j \<longrightarrow> (?I j, ?H j)\<in> ctran \<longrightarrow> (?P j)" in allE)
+apply(rule_tac x=ia in exI)
+apply(simp add:same_state_def)
+apply(erule_tac x=ia and P="\<lambda>j. (?T j) \<longrightarrow> (\<forall>i. (?H j i) \<longrightarrow> (snd (?d j i))=(snd (?e j i)))" in all_dupE,simp)
+apply(erule_tac x=ia and P="\<lambda>j. (?T j) \<longrightarrow> (\<forall>i. (?H j i) \<longrightarrow> (snd (?d j i))=(snd (?e j i)))" in allE,simp)
+apply(erule_tac x=i and P="\<lambda>j. ?H j \<longrightarrow> (snd (?d j))=(snd (?e j))" in all_dupE)
+apply(erule_tac x=i and P="\<lambda>j. ?H j \<longrightarrow> (snd (?d j))=(snd (?e j))" in all_dupE,simp)
+apply(erule_tac x="Suc i" and P="\<lambda>j. ?H j \<longrightarrow> (snd (?d j))=(snd (?e j))" in allE,simp)
+apply(erule mp)
+apply(subgoal_tac "r=fst(clist ! ia ! Suc i)",simp)
+apply(drule_tac i=ia in list_eq_if)
+back
+apply simp_all
+done
+
+lemma parcptn_not_empty [simp]:"[] \<notin> par_cptn"
+apply(force elim:par_cptn.cases)
+done
+
+lemma five: 
+  "\<lbrakk>xs\<noteq>[]; \<forall>i<length xs. rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j))
+   \<subseteq> Rely (xs ! i);
+   pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i)); 
+   (\<Inter>i\<in>{i. i < length xs}. Post (xs ! i)) \<subseteq> post;
+   \<forall>i < length xs.
+    \<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)];
+    x \<in> par_cp (ParallelCom xs) s; x \<in> par_assum (pre, rely); 
+   All_None (fst (last x)) \<rbrakk> \<Longrightarrow> snd (last x) \<in> post"
+apply(simp add: ParallelCom_def del: Un_subset_iff)
+apply(subgoal_tac "(map (Some \<circ> fst) xs)\<noteq>[]")
+ prefer 2
+ apply simp
+apply(frule rev_subsetD)
+ apply(erule one [THEN equalityD1])
+apply(erule subsetD)
+apply(simp del: Un_subset_iff)
+apply clarify
+apply(subgoal_tac "\<forall>i<length clist. clist!i\<in>assum(Pre(xs!i), Rely(xs!i))")
+ apply(erule_tac x=i and P="\<lambda>i. ?H i \<longrightarrow> \<Turnstile> (?J i) sat [?I i,?K i,?M i,?N i]" in allE,erule impE,assumption)
+ apply(simp add:com_validity_def)
+ apply(erule_tac x=s in allE)
+ apply(erule_tac x=i and P="\<lambda>j. ?H j \<longrightarrow> (?I j) \<in> cp (?J j) s" in allE,simp)
+ apply(drule_tac c="clist!i" in subsetD)
+  apply (force simp add:Com_def)
+ apply(simp add:comm_def conjoin_def same_program_def del:last.simps)
+ apply clarify
+ apply(erule_tac x="length x - 1" and P="\<lambda>j. ?H j \<longrightarrow> fst(?I j)=(?J j)" in allE)
+ apply (simp add:All_None_def same_length_def)
+ apply(erule_tac x=i and P="\<lambda>j. ?H j \<longrightarrow> length(?J j)=(?K j)" in allE)
+ apply(subgoal_tac "length x - 1 < length x",simp)
+  apply(case_tac "x\<noteq>[]")
+   apply(simp add: last_conv_nth)
+   apply(erule_tac x="clist!i" in ballE)
+    apply(simp add:same_state_def)
+    apply(subgoal_tac "clist!i\<noteq>[]")
+     apply(simp add: last_conv_nth)
+    apply(case_tac x)
+     apply (force simp add:par_cp_def)
+    apply (force simp add:par_cp_def)
+   apply force
+  apply (force simp add:par_cp_def)
+ apply(case_tac x)
+  apply (force simp add:par_cp_def)
+ apply (force simp add:par_cp_def)
+apply clarify
+apply(simp add:assum_def)
+apply(rule conjI)
+ apply(simp add:conjoin_def same_state_def par_cp_def)
+ apply clarify
+ apply(erule_tac x=ia and P="\<lambda>j. (?T j) \<longrightarrow> (\<forall>i. (?H j i) \<longrightarrow> (snd (?d j i))=(snd (?e j i)))" in allE,simp)
+ apply(erule_tac x=0 and P="\<lambda>j. ?H j \<longrightarrow> (snd (?d j))=(snd (?e j))" in allE)
+ apply(case_tac x,simp+)
+ apply (simp add:par_assum_def)
+ apply clarify
+ apply(drule_tac c="snd (clist ! ia ! 0)" in subsetD)
+ apply assumption
+ apply simp
+apply clarify
+apply(erule_tac x=ia in all_dupE)
+apply(rule subsetD, erule mp, assumption)
+apply(erule_tac pre=pre and rely=rely and x=x and s=s in  three)
+ apply(erule_tac x=ic in allE,erule mp)
+ apply simp_all
+ apply(simp add:ParallelCom_def)
+ apply(force simp add:Com_def)
+apply(simp add:conjoin_def same_length_def)
+done
+
+lemma ParallelEmpty [rule_format]: 
+  "\<forall>i s. x \<in> par_cp (ParallelCom []) s \<longrightarrow> 
+  Suc i < length x \<longrightarrow> (x ! i, x ! Suc i) \<notin> par_ctran"
+apply(induct_tac x)
+ apply(simp add:par_cp_def ParallelCom_def)
+apply clarify
+apply(case_tac list,simp,simp)
+apply(case_tac i)
+ apply(simp add:par_cp_def ParallelCom_def)
+ apply(erule par_ctranE,simp)
+apply(simp add:par_cp_def ParallelCom_def)
+apply clarify
+apply(erule par_cptn.cases,simp)
+ apply simp
+apply(erule par_ctranE)
+back
+apply simp
+done
+
+theorem par_rgsound: 
+  "\<turnstile> c SAT [pre, rely, guar, post] \<Longrightarrow> 
+   \<Turnstile> (ParallelCom c) SAT [pre, rely, guar, post]"
+apply(erule par_rghoare.induct)
+apply(case_tac xs,simp)
+ apply(simp add:par_com_validity_def par_comm_def)
+ apply clarify
+ apply(case_tac "post=UNIV",simp)
+  apply clarify
+  apply(drule ParallelEmpty)
+   apply assumption
+  apply simp
+ apply clarify
+ apply simp
+apply(subgoal_tac "xs\<noteq>[]")
+ prefer 2
+ apply simp
+apply(thin_tac "xs = a # list")
+apply(simp add:par_com_validity_def par_comm_def)
+apply clarify
+apply(rule conjI)
+ apply clarify
+ apply(erule_tac pre=pre and rely=rely and guar=guar and x=x and s=s and xs=xs in four)
+        apply(assumption+)
+     apply clarify
+     apply (erule allE, erule impE, assumption,erule rgsound)
+    apply(assumption+)
+apply clarify
+apply(erule_tac pre=pre and rely=rely and post=post and x=x and s=s and xs=xs in five)
+      apply(assumption+)
+   apply clarify
+   apply (erule allE, erule impE, assumption,erule rgsound)
+  apply(assumption+) 
+done
+
+end




