Theory RG_Hoare

theory RG_Hoare
imports RG_Tran
header {* \section{The Proof System} *}

theory RG_Hoare imports RG_Tran begin

subsection {* Proof System for Component Programs *}

declare Un_subset_iff [simp del] le_sup_iff [simp del]

definition stable :: "'a set => ('a × 'a) set => bool" where
"stable ≡ λf g. (∀x y. x ∈ f --> (x, y) ∈ g --> y ∈ f)"

inductive
rghoare :: "['a com, 'a set, ('a × 'a) set, ('a × 'a) set, 'a set] => bool"
("\<turnstile> _ sat [_, _, _, _]" [60,0,0,0,0] 45)
where
Basic: "[| pre ⊆ {s. f s ∈ post}; {(s,t). s ∈ pre ∧ (t=f s ∨ t=s)} ⊆ guar;
stable pre rely; stable post rely |]
==> \<turnstile> Basic f sat [pre, rely, guar, post]"


| Seq: "[| \<turnstile> P sat [pre, rely, guar, mid]; \<turnstile> Q sat [mid, rely, guar, post] |]
==> \<turnstile> Seq P Q sat [pre, rely, guar, post]"


| Cond: "[| stable pre rely; \<turnstile> P1 sat [pre ∩ b, rely, guar, post];
\<turnstile> P2 sat [pre ∩ -b, rely, guar, post]; ∀s. (s,s)∈guar |]
==> \<turnstile> Cond b P1 P2 sat [pre, rely, guar, post]"


| While: "[| stable pre rely; (pre ∩ -b) ⊆ post; stable post rely;
\<turnstile> P sat [pre ∩ b, rely, guar, pre]; ∀s. (s,s)∈guar |]
==> \<turnstile> While b P sat [pre, rely, guar, post]"


| Await: "[| stable pre rely; stable post rely;
∀V. \<turnstile> P sat [pre ∩ b ∩ {V}, {(s, t). s = t},
UNIV, {s. (V, s) ∈ guar} ∩ post] |]
==> \<turnstile> Await b P sat [pre, rely, guar, post]"


| Conseq: "[| pre ⊆ pre'; rely ⊆ rely'; guar' ⊆ guar; post' ⊆ post;
\<turnstile> P sat [pre', rely', guar', post'] |]
==> \<turnstile> P sat [pre, rely, guar, post]"


definition Pre :: "'a rgformula => 'a set" where
"Pre x ≡ fst(snd x)"

definition Post :: "'a rgformula => 'a set" where
"Post x ≡ snd(snd(snd(snd x)))"

definition Rely :: "'a rgformula => ('a × 'a) set" where
"Rely x ≡ fst(snd(snd x))"

definition Guar :: "'a rgformula => ('a × 'a) set" where
"Guar x ≡ fst(snd(snd(snd x)))"

definition Com :: "'a rgformula => 'a com" where
"Com x ≡ fst x"

subsection {* Proof System for Parallel Programs *}

type_synonym 'a par_rgformula =
"('a rgformula) list × 'a set × ('a × 'a) set × ('a × 'a) set × 'a set"

inductive
par_rghoare :: "('a rgformula) list => 'a set => ('a × 'a) set => ('a × 'a) set => 'a set => bool"
("\<turnstile> _ SAT [_, _, _, _]" [60,0,0,0,0] 45)
where
Parallel:
"[| ∀i<length xs. rely ∪ (\<Union>j∈{j. j<length xs ∧ j≠i}. Guar(xs!j)) ⊆ Rely(xs!i);
(\<Union>j∈{j. j<length xs}. Guar(xs!j)) ⊆ guar;
pre ⊆ (\<Inter>i∈{i. i<length xs}. Pre(xs!i));
(\<Inter>i∈{i. i<length xs}. Post(xs!i)) ⊆ post;
∀i<length xs. \<turnstile> Com(xs!i) sat [Pre(xs!i),Rely(xs!i),Guar(xs!i),Post(xs!i)] |]
==> \<turnstile> xs SAT [pre, rely, guar, post]"


section {* Soundness*}

subsubsection {* Some previous lemmas *}

lemma tl_of_assum_in_assum:
"(P, s) # (P, t) # xs ∈ assum (pre, rely) ==> stable pre rely
==> (P, t) # xs ∈ 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 ∈ comm(guar, post) ==> (P, s) # (P, t) # xs ∈ comm(guar, post)"
apply(simp add:comm_def)
apply clarify
apply(case_tac i,simp+)
done

lemma ctran_in_comm:
"[|(s, s) ∈ guar; (Q, s) # xs ∈ comm(guar, post)|]
==> (P, s) # (Q, s) # xs ∈ comm(guar, post)"

apply(simp add:comm_def)
apply clarify
apply(case_tac i,simp+)
done

lemma takecptn_is_cptn [rule_format, elim!]:
"∀j. c ∈ cptn --> take (Suc j) c ∈ 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!]:
"∀j<length c. c ∈ cptn --> drop j c ∈ 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]:
"∀j. c ∈ par_cptn --> take (Suc j) c ∈ 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]:
"∀j<length c. c ∈ par_cptn --> drop j c ∈ 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: "[|x # xs ∈ cptn; xs ≠ []|] ==> xs ∈ cptn"
apply(subgoal_tac "1 < length (x # xs)")
apply(drule dropcptn_is_cptn,simp+)
done

lemma not_ctran_None [rule_format]:
"∀s. (None, s)#xs ∈ cptn --> (∀i<length xs. ((None, s)#xs)!i -e-> 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]:"[] ∉ cptn"
apply(force elim:cptn.cases)
done

lemma etran_or_ctran [rule_format]:
"∀m i. x∈cptn --> m ≤ length x
--> (∀i. Suc i < m --> ¬ x!i -c-> x!Suc i) --> Suc i < m
--> x!i -e-> 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 "(∀i. Suc i < nata --> (((P, t) # xs) ! i, xs ! i) ∉ ctran)")
apply force
apply clarify
apply(erule_tac x="Suc ia" in allE,simp)
apply(erule_tac x="0" and P="λj. ?H j --> (?J j) ∉ ctran" in allE,simp)
done

lemma etran_or_ctran2 [rule_format]:
"∀i. Suc i<length x --> x∈cptn --> (x!i -c-> x!Suc i --> ¬ x!i -e-> x!Suc i)
∨ (x!i -e-> x!Suc i --> ¬ x!i -c-> 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:
"[| x∈cptn; Suc i<length x; x!i -c-> x!Suc i|] ==> ¬ x!i -e-> x!Suc i"
by(drule etran_or_ctran2,simp_all)

lemma etran_or_ctran2_disjI2:
"[| x∈cptn; Suc i<length x; x!i -e-> x!Suc i|] ==> ¬ x!i -c-> x!Suc i"
by(drule etran_or_ctran2,simp_all)

lemma not_ctran_None2 [rule_format]:
"[| (None, s) # xs ∈cptn; i<length xs|] ==> ¬ ((None, s) # xs) ! i -c-> 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) --> (∃m. P m ∧ (∀i<m. ¬ P i))";
apply(rule nat_less_induct)
apply clarify
apply(case_tac "∀m. m<n --> ¬ P m")
apply auto
done

lemma stability [rule_format]:
"∀j k. x ∈ cptn --> stable p rely --> j≤k --> k<length x --> snd(x!j)∈p -->
(∀i. (Suc i)<length x -->
(x!i -e-> x!(Suc i)) --> (snd(x!i), snd(x!(Suc i))) ∈ rely) -->
(∀i. j≤i ∧ i<k --> x!i -e-> x!Suc i) --> snd(x!k)∈p ∧ 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="λj. (0≤j) --> (?J j)" in allE,simp)
apply(subgoal_tac "t∈p")
apply(subgoal_tac "(∀i. i < length xs --> ((P, t) # xs) ! i -e-> xs ! i --> (snd (((P, t) # xs) ! i), snd (xs ! i)) ∈ rely)")
apply clarify
apply(erule_tac x="Suc i" and P="λj. (?H j) --> (?J j)∈etran" in allE,simp)
apply clarify
apply(erule_tac x="Suc i" and P="λj. (?H j) --> (?J j) --> (?T j)∈rely" in allE,simp)
apply(erule_tac x=0 and P="λj. (?H j) --> (?J j)∈etran --> ?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="λj. (?s≤j) --> (?J j)" in allE,simp)
apply(subgoal_tac "(∀i. i < length xs --> ((P, t) # xs) ! i -e-> xs ! i --> (snd (((P, t) # xs) ! i), snd (xs ! i)) ∈ rely)")
apply clarify
apply(erule_tac x="Suc i" and P="λj. (?H j) --> (?J j)∈etran" in allE,simp)
apply clarify
apply(erule_tac x="Suc i" and P="λj. (?H j) --> (?J j) --> (?T j)∈rely" in allE,simp)
apply(case_tac k,simp,simp)
apply(case_tac j)
apply(erule_tac x=0 and P="λj. (?H j) --> (?J j)∈etran" in allE,simp)
apply(erule etran.cases,simp)
apply(erule_tac x="nata" in allE)
apply(erule_tac x="nat" and P="λj. (?s≤j) --> (?J j)" in allE,simp)
apply(subgoal_tac "(∀i. i < length xs --> ((Q, t) # xs) ! i -e-> xs ! i --> (snd (((Q, t) # xs) ! i), snd (xs ! i)) ∈ rely)")
apply clarify
apply(erule_tac x="Suc i" and P="λj. (?H j) --> (?J j)∈etran" in allE,simp)
apply clarify
apply(erule_tac x="Suc i" and P="λj. (?H j) --> (?J j) --> (?T j)∈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]:
"∀s i. x ∈ cptn --> x ! 0 = (Some (Basic f), s) -->
Suc i<length x --> x!i -c-> x!Suc i -->
(∀j. Suc j<length x --> i≠j --> x!j -e-> 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) ∈ 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]:
"∀s i. x ∈ cptn --> x ! 0 = (Some (Basic f), s)
--> i<length x --> fst(x!i)=None --> (∃j<i. x!j -c-> 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:
" [|pre ⊆ {s. f s ∈ post}; {(s, t). s ∈ pre ∧ t = f s} ⊆ guar;
stable pre rely; stable post rely|]
==> \<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 "∀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]:
"∀s i. x ∈ cptn --> x ! 0 = (Some (Await b c), s) -->
Suc i<length x --> x!i -c-> x!Suc i -->
(∀j. Suc j<length x --> i≠j --> x!j -e-> 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) ∈ 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]:
"∀s i. x ∈ cptn --> x ! 0 = (Some (Await b c), s)
--> i<length x --> fst(x!i)=None --> (∃j<i. x!j -c-> 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*-> (R, t) ==> ∃l ∈ cp P s. (last l)=(R, t)
∧ (∀i. Suc i<length l --> l!i -c-> 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:
"[|stable pre rely; stable post rely;
∀V. \<turnstile> P sat [pre ∩ b ∩ {s. s = V}, {(s, t). s = t},
UNIV, {s. (V, s) ∈ guar} ∩ post] ∧
\<Turnstile> P sat [pre ∩ b ∩ {s. s = V}, {(s, t). s = t},
UNIV, {s. (V, s) ∈ guar} ∩ post] |]
==> \<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∈ 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="λi. ?H i --> (?J i,?I i)∈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+)
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="λi. ?H i --> (?J i,?I i)∈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:
"[| stable pre rely; \<Turnstile> P1 sat [pre ∩ b, rely, guar, post];
\<Turnstile> P2 sat [pre ∩ - b, rely, guar, post]; ∀s. (s,s)∈guar|]
==> \<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 "∃i. Suc i<length x ∧ x!i -c-> 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="λi. ?H i ∧ (?J i,?I i)∈ 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≤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="λj. ?H j --> ?J j --> (?I j)∈guar" in allE)
apply(subgoal_tac "(Suc m)+(i - Suc m) ≤ length x")
apply(subgoal_tac "(Suc m)+Suc (i - Suc m) ≤ 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≤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="λj. ?H j --> ?J j --> (?I j)∈guar" in allE)
apply(subgoal_tac "(Suc m)+(i - Suc m) ≤ length x")
apply(subgoal_tac "(Suc m)+Suc (i - Suc m) ≤ 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-> t"

lemma last_lift_not_None: "fst ((lift Q) ((x#xs)!(length xs))) ≠ 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

lemma Seq_sound1 [rule_format]:
"x∈ cptn_mod ==> ∀s P. x !0=(Some (Seq P Q), s) -->
(∀i<length x. fst(x!i)≠Some Q) -->
(∃xs∈ 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 "(∀i < Suc (length xs). fst (((Some (Seq Pa Q), t) # xs) ! i) ≠ 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) ∈ 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

lemma Seq_sound2 [rule_format]:
"x ∈ cptn ==> ∀s P i. x!0=(Some (Seq P Q), s) --> i<length x
--> fst(x!i)=Some Q -->
(∀j<i. fst(x!j)≠(Some Q)) -->
(∃xs ys. xs ∈ cp (Some P) s ∧ length xs=Suc i
∧ ys ∈ cp (Some Q) (snd(xs !i)) ∧ 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 "(∀j < nat. fst (((Some (Seq Pa Q), t) # xs) ! j) ≠ 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) ∈ 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 "(∀j<Suc nata. fst (((Some (Seq P2 Q), ta) # xs) ! j) ≠ 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))) ≠ 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))) ≠ 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:
"[|\<Turnstile> P sat [pre, rely, guar, mid]; \<Turnstile> Q sat [mid, rely, guar, post]|]
==> \<Turnstile> Seq P Q sat [pre, rely, guar, post]"

apply(unfold com_validity_def)
apply clarify
apply(case_tac "∃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="λj. ?H j --> ?J j --> ?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="λj. ?H j --> ?J j --> ?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))≠[]")
apply(drule last_conv_nth)
apply (simp del:map.simps)
apply(simp only:last_lift_not_None)
apply simp
--{* @{text "∃i<length x. fst (x ! i) = Some Q"} *}
apply(erule exE)
apply(drule_tac n=i and P="λi. i < length x ∧ 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≠[]")
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)≠[]")
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]:
"∀xs. ys≠[] --> ((xs@ys)!(length (xs@ys) - (Suc 0)))=(ys!(length ys - (Suc 0)))"
apply(induct ys)
apply simp
apply clarify
apply (simp add:nth_append)
done

lemma assum_after_body:
"[| \<Turnstile> P sat [pre ∩ b, rely, guar, pre];
(Some P, s) # xs ∈ cptn_mod; fst (last ((Some P, s) # xs)) = None; s ∈ b;
(Some (While b P), s) # (Some (Seq P (While b P)), s) #
map (lift (While b P)) xs @ ys ∈ assum (pre, rely)|]
==> (Some (While b P), snd (last ((Some P, s) # xs))) # ys ∈ 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 last_conv_nth lift_def split_def)
apply(simp add:Cons_lift_append nth_append snd_lift)
done

lemma While_sound_aux [rule_format]:
"[| pre ∩ - b ⊆ post; \<Turnstile> P sat [pre ∩ b, rely, guar, pre]; ∀s. (s, s) ∈ guar;
stable pre rely; stable post rely; x ∈ cptn_mod |]
==> ∀s xs. x=(Some(While b P),s)#xs --> x∈assum(pre, rely) --> x ∈ 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) ∈ 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 "∀i. Suc i < length ((None, t) # xs) --> (((None, t) # xs) ! i, ((None, t) # xs) ! Suc i)∈ 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 --> ?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-> 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))≠[]")
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 --> ?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 ∈ 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-> t" for ba t)
apply simp
apply simp
apply(simp add:snd_lift del:map.simps last.simps)
apply(thin_tac " ∀i. i < length ys --> ?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 ≥ 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))≠[]")
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)≠[]")
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)≠[]")
apply(drule last_conv_nth)
apply (simp del:map.simps last.simps)
apply simp
apply simp
done

lemma While_sound:
"[|stable pre rely; pre ∩ - b ⊆ post; stable post rely;
\<Turnstile> P sat [pre ∩ b, rely, guar, pre]; ∀s. (s,s)∈guar|]
==> \<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:
"[|pre ⊆ pre'; rely ⊆ rely'; guar' ⊆ guar; post' ⊆ post;
\<Turnstile> P sat [pre', rely', guar', post']|]
==> \<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] ==> \<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 *}

definition ParallelCom :: "('a rgformula) list => 'a par_com" where
"ParallelCom Ps ≡ map (Some o fst) Ps"

lemma two:
"[| ∀i<length xs. rely ∪ (\<Union>j∈{j. j < length xs ∧ j ≠ i}. Guar (xs ! j))
⊆ Rely (xs ! i);
pre ⊆ (\<Inter>i∈{i. i < length xs}. Pre (xs ! i));
∀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 ∈ par_cp (ParallelCom xs) s; x∈par_assum(pre, rely);
∀i<length clist. clist!i∈cp (Some(Com(xs!i))) s; x ∝ clist |]
==> ∀j i. i<length clist ∧ Suc j<length x --> (clist!i!j) -c-> (clist!i!Suc j)
--> (snd(clist!i!j), snd(clist!i!Suc j)) ∈ Guar(xs!i)"

apply(unfold par_cp_def)
apply (rule ccontr)
--{* By contradiction: *}
apply simp
apply(erule exE)
--{* the first c-tran that does not satisfy the guarantee-condition is from @{text "σ_i"} at step @{text "m"}. *}
apply(drule_tac n=j and P="λj. ∃i. ?H i j" in Ex_first_occurrence)
apply(erule exE)
apply clarify
--{* @{text "σ_i ∈ A(pre, rely_1)"} *}
apply(subgoal_tac "take (Suc (Suc m)) (clist!i) ∈ assum(Pre(xs!i), Rely(xs!i))")
--{* but this contradicts @{text "\<Turnstile> σ_i sat [pre_i,rely_i,guar_i,post_i]"} *}
apply(erule_tac x=i and P="λi. ?H i --> \<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="λj. ?I j ∧ ?J j --> ?H j" in allE)
apply (simp add:conjoin_def same_length_def)
apply(simp add:assum_def)
apply(rule conjI)
apply(erule_tac x=i and P="λj. ?H j --> ?I j ∈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="λj. ?H j --> ?M ∪ UNION (?S j) (?T j) ⊆ (?L j)" in allE)
apply simp
apply(erule subsetD)
apply simp
apply(simp add:conjoin_def compat_label_def)
apply clarify
apply(erule_tac x=ia and P="λj. ?H j --> (?P j) ∨ ?Q j" in allE,simp)
--{* each etran in @{text "σ_1[0…m]"} corresponds to *}
apply(erule disjE)
--{* a c-tran in some @{text "σ_{ib}"} *}
apply clarify
apply(case_tac "i=ib",simp)
apply(erule etranE,simp)
apply(erule_tac x="ib" and P="λi. ?H i --> (?I i) ∨ (?J i)" in allE)
apply (erule etranE)
apply(case_tac "ia=m",simp)
apply simp
apply(erule_tac x=ia and P="λj. ?H j --> (∀ i. ?P i j)" in allE)
apply(subgoal_tac "ia<m",simp)
prefer 2
apply arith
apply(erule_tac x=ib and P="λj. (?I j, ?H j)∈ ctran --> (?P i j)" in allE,simp)
apply(simp add:same_state_def)
apply(erule_tac x=i and P="λj. (?T j) --> (∀i. (?H j i) --> (snd (?d j i))=(snd (?e j i)))" in all_dupE)
apply(erule_tac x=ib and P="λj. (?T j) --> (∀i. (?H j i) --> (snd (?d j i))=(snd (?e j i)))" in allE,simp)
--{* or an e-tran in @{text "σ"},
therefore it satisfies @{text "rely ∨ guar_{ib}"} *}

apply (force simp add:par_assum_def same_state_def)
done


lemma three [rule_format]:
"[| xs≠[]; ∀i<length xs. rely ∪ (\<Union>j∈{j. j < length xs ∧ j ≠ i}. Guar (xs ! j))
⊆ Rely (xs ! i);
pre ⊆ (\<Inter>i∈{i. i < length xs}. Pre (xs ! i));
∀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 ∈ par_cp (ParallelCom xs) s; x ∈ par_assum(pre, rely);
∀i<length clist. clist!i∈cp (Some(Com(xs!i))) s; x ∝ clist |]
==> ∀j i. i<length clist ∧ Suc j<length x --> (clist!i!j) -e-> (clist!i!Suc j)
--> (snd(clist!i!j), snd(clist!i!Suc j)) ∈ rely ∪ (\<Union>j∈{j. j < length xs ∧ j ≠ 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="λj. ?H j --> (?J j ∧ (∃i. ?P i j)) ∨ ?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="λi. ?H i --> (?I i) ∨ (?J i)" in allE,simp)
apply(erule_tac x=j and P="λj. ∀i. ?S j i --> (?I j i, ?H j i)∈ ctran --> (?P i j)" in allE)
apply(erule_tac x=ia and P="λj. ?S j --> (?I j, ?H j)∈ ctran --> (?P j)" in allE)
apply(simp add:same_state_def)
apply(erule_tac x=i and P="λj. (?T j) --> (∀i. (?H j i) --> (snd (?d j i))=(snd (?e j i)))" in all_dupE)
apply(erule_tac x=ia and P="λj. (?T j) --> (∀i. (?H j i) --> (snd (?d j i))=(snd (?e j i)))" in allE,simp)
done

lemma four:
"[|xs≠[]; ∀i < length xs. rely ∪ (\<Union>j∈{j. j < length xs ∧ j ≠ i}. Guar (xs ! j))
⊆ Rely (xs ! i);
(\<Union>j∈{j. j < length xs}. Guar (xs ! j)) ⊆ guar;
pre ⊆ (\<Inter>i∈{i. i < length xs}. Pre (xs ! i));
∀i < length xs.
\<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)];
x ∈ par_cp (ParallelCom xs) s; x ∈ par_assum (pre, rely); Suc i < length x;
x ! i -pc-> x ! Suc i|]
==> (snd (x ! i), snd (x ! Suc i)) ∈ guar"

apply(simp add: ParallelCom_def)
apply(subgoal_tac "(map (Some o fst) xs)≠[]")
prefer 2
apply simp
apply(frule rev_subsetD)
apply(erule one [THEN equalityD1])
apply(erule subsetD)
apply simp
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="λj. ?H j --> fst(?I j)=(?J j)" in all_dupE)
apply(erule_tac x="Suc i" and P="λj. ?H j --> fst(?I j)=(?J j)" in allE)
apply(erule par_ctranE,simp)
apply(erule_tac x=i and P="λj. ∀i. ?S j i --> (?I j i, ?H j i)∈ ctran --> (?P i j)" in allE)
apply(erule_tac x=ia and P="λj. ?S j --> (?I j, ?H j)∈ ctran --> (?P j)" in allE)
apply(rule_tac x=ia in exI)
apply(simp add:same_state_def)
apply(erule_tac x=ia and P="λj. (?T j) --> (∀i. (?H j i) --> (snd (?d j i))=(snd (?e j i)))" in all_dupE,simp)
apply(erule_tac x=ia and P="λj. (?T j) --> (∀i. (?H j i) --> (snd (?d j i))=(snd (?e j i)))" in allE,simp)
apply(erule_tac x=i and P="λj. ?H j --> (snd (?d j))=(snd (?e j))" in all_dupE)
apply(erule_tac x=i and P="λj. ?H j --> (snd (?d j))=(snd (?e j))" in all_dupE,simp)
apply(erule_tac x="Suc i" and P="λj. ?H j --> (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]:"[] ∉ par_cptn"
apply(force elim:par_cptn.cases)
done

lemma five:
"[|xs≠[]; ∀i<length xs. rely ∪ (\<Union>j∈{j. j < length xs ∧ j ≠ i}. Guar (xs ! j))
⊆ Rely (xs ! i);
pre ⊆ (\<Inter>i∈{i. i < length xs}. Pre (xs ! i));
(\<Inter>i∈{i. i < length xs}. Post (xs ! i)) ⊆ post;
∀i < length xs.
\<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)];
x ∈ par_cp (ParallelCom xs) s; x ∈ par_assum (pre, rely);
All_None (fst (last x)) |] ==> snd (last x) ∈ post"

apply(simp add: ParallelCom_def)
apply(subgoal_tac "(map (Some o fst) xs)≠[]")
prefer 2
apply simp
apply(frule rev_subsetD)
apply(erule one [THEN equalityD1])
apply(erule subsetD)
apply simp
apply clarify
apply(subgoal_tac "∀i<length clist. clist!i∈assum(Pre(xs!i), Rely(xs!i))")
apply(erule_tac x=i and P="λi. ?H i --> \<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="λj. ?H j --> (?I j) ∈ 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="λj. ?H j --> fst(?I j)=(?J j)" in allE)
apply (simp add:All_None_def same_length_def)
apply(erule_tac x=i and P="λj. ?H j --> length(?J j)=(?K j)" in allE)
apply(subgoal_tac "length x - 1 < length x",simp)
apply(case_tac "x≠[]")
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≠[]")
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="λj. (?T j) --> (∀i. (?H j i) --> (snd (?d j i))=(snd (?e j i)))" in allE,simp)
apply(erule_tac x=0 and P="λj. ?H j --> (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]:
"∀i s. x ∈ par_cp (ParallelCom []) s -->
Suc i < length x --> (x ! i, x ! Suc i) ∉ 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] ==>
\<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≠[]")
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