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 => ('a × 'a) set => bool" "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]" constdefs Pre :: "'a rgformula => 'a set" "Pre x ≡ fst(snd x)" Post :: "'a rgformula => 'a set" "Post x ≡ snd(snd(snd(snd x)))" Rely :: "'a rgformula => ('a × 'a) set" "Rely x ≡ fst(snd(snd x))" Guar :: "'a rgformula => ('a × 'a) set" "Guar x ≡ fst(snd(snd(snd x)))" Com :: "'a rgformula => 'a com" "Com x ≡ fst x" subsection {* Proof System for Parallel Programs *} types '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 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="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 declare map_eq_Cons_conv [simp del] Cons_eq_map_conv [simp del] 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 declare map_eq_Cons_conv [simp del] Cons_eq_map_conv [simp del] 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 length_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 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]: "[| 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 *} constdefs ParallelCom :: "('a rgformula) list => 'a par_com" "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 del: Un_subset_iff) 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 del: Un_subset_iff) 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 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="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 del: Un_subset_iff) 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 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="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 del: Un_subset_iff) 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 del: Un_subset_iff) 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
lemma tl_of_assum_in_assum:
[| (P, s) # (P, t) # xs ∈ assum (pre, rely); stable pre rely |]
==> (P, t) # xs ∈ assum (pre, rely)
lemma etran_in_comm:
(P, t) # xs ∈ comm (guar, post) ==> (P, s) # (P, t) # xs ∈ comm (guar, post)
lemma ctran_in_comm:
[| (s, s) ∈ guar; (Q, s) # xs ∈ comm (guar, post) |]
==> (P, s) # (Q, s) # xs ∈ comm (guar, post)
lemma takecptn_is_cptn:
c ∈ cptn ==> take (Suc j) c ∈ cptn
lemma dropcptn_is_cptn:
[| j < length c; c ∈ cptn |] ==> drop j c ∈ cptn
lemma takepar_cptn_is_par_cptn:
c ∈ par_cptn ==> take (Suc j) c ∈ par_cptn
lemma droppar_cptn_is_par_cptn:
[| j < length c; c ∈ par_cptn |] ==> drop j c ∈ par_cptn
lemma tl_of_cptn_is_cptn:
[| x # xs ∈ cptn; xs ≠ [] |] ==> xs ∈ cptn
lemma not_ctran_None:
[| (None, s) # xs ∈ cptn; i < length xs |] ==> ((None, s) # xs) ! i -e-> xs ! i
lemma cptn_not_empty:
[] ∉ cptn
lemma etran_or_ctran:
[| x ∈ cptn; m ≤ length x; !!i. Suc i < m ==> ¬ x ! i -c-> x ! Suc i;
Suc i < m |]
==> x ! i -e-> x ! Suc i
lemma etran_or_ctran2:
[| 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)
lemma etran_or_ctran2_disjI1:
[| x ∈ cptn; Suc i < length x; x ! i -c-> x ! Suc i |]
==> ¬ x ! i -e-> x ! Suc i
lemma etran_or_ctran2_disjI2:
[| x ∈ cptn; Suc i < length x; x ! i -e-> x ! Suc i |]
==> ¬ x ! i -c-> x ! Suc i
lemma not_ctran_None2:
[| (None, s) # xs ∈ cptn; i < length xs |]
==> ¬ ((None, s) # xs) ! i -c-> xs ! i
lemma Ex_first_occurrence:
P n ==> ∃m. P m ∧ (∀i<m. ¬ P i)
lemma stability:
[| 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)
lemma unique_ctran_Basic:
[| x ∈ cptn; x ! 0 = (Some (Basic f), s); Suc i < length x;
x ! i -c-> x ! Suc i; Suc j < length x; i ≠ j |]
==> x ! j -e-> x ! Suc j
lemma exists_ctran_Basic_None:
[| x ∈ cptn; x ! 0 = (Some (Basic f), s); i < length x; fst (x ! i) = None |]
==> ∃j<i. x ! j -c-> x ! Suc j
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]
lemma unique_ctran_Await:
[| x ∈ cptn; x ! 0 = (Some (Await b c), s); Suc i < length x;
x ! i -c-> x ! Suc i; Suc j < length x; i ≠ j |]
==> x ! j -e-> x ! Suc j
lemma exists_ctran_Await_None:
[| x ∈ cptn; x ! 0 = (Some (Await b c), s); i < length x; fst (x ! i) = None |]
==> ∃j<i. x ! j -c-> x ! Suc j
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)
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]
lemma Cond_sound:
[| stable pre rely; \<Turnstile> P1.0 sat [pre ∩ b, rely, guar, post];
\<Turnstile> P2.0 sat [pre ∩ - b, rely, guar, post]; ∀s. (s, s) ∈ guar |]
==> \<Turnstile> Cond b P1.0 P2.0 sat [pre, rely, guar, post]
lemma last_lift_not_None:
fst (lift Q ((x # xs) ! length xs)) ≠ None
lemma Seq_sound1:
[| x ∈ cptn_mod; x ! 0 = (Some (Seq P Q), s);
!!i. i < length x ==> fst (x ! i) ≠ Some Q |]
==> ∃xs∈cp (Some P) s. x = map (lift Q) xs
lemma Seq_sound2:
[| x ∈ cptn; x ! 0 = (Some (Seq P Q), s); i < length x; fst (x ! i) = Some Q;
!!j. 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
lemma last_lift_not_None2:
fst (lift Q (last (x # xs))) ≠ None
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]
lemma last_append:
ys ≠ [] ==> (xs @ ys) ! (length (xs @ ys) - Suc 0) = ys ! (length ys - Suc 0)
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)
lemma While_sound_aux:
[| pre ∩ - b ⊆ post; \<Turnstile> P sat [pre ∩ b, rely, guar, pre];
!!s. (s, s) ∈ guar; stable pre rely; stable post rely; x ∈ cptn_mod;
x = (Some (While b P), s) # xs; x ∈ assum (pre, rely) |]
==> x ∈ comm (guar, post)
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]
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]
theorem rgsound:
\<turnstile> P sat [pre, rely, guar, post]
==> \<Turnstile> P sat [pre, rely, guar, post]
lemma two:
[| ∀i<length xs.
rely ∪ (UN j:{j. j < length xs ∧ j ≠ i}. Guar (xs ! j)) ⊆ Rely (xs ! i);
pre ⊆ (INT 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)
lemma three:
[| xs ≠ [];
!!i. i < length xs
==> rely ∪ (UN j:{j. j < length xs ∧ j ≠ i}. Guar (xs ! j))
⊆ Rely (xs ! i);
pre ⊆ (INT i:{i. i < length xs}. Pre (xs ! i));
!!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. i < length clist ==> clist ! i ∈ cp (Some (Com (xs ! i))) s; x ∝ clist;
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 ∪ (UN j:{j. j < length xs ∧ j ≠ i}. Guar (xs ! j))
lemma four:
[| xs ≠ [];
∀i<length xs.
rely ∪ (UN j:{j. j < length xs ∧ j ≠ i}. Guar (xs ! j)) ⊆ Rely (xs ! i);
(UN j:{j. j < length xs}. Guar (xs ! j)) ⊆ guar;
pre ⊆ (INT 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
lemma parcptn_not_empty:
[] ∉ par_cptn
lemma five:
[| xs ≠ [];
∀i<length xs.
rely ∪ (UN j:{j. j < length xs ∧ j ≠ i}. Guar (xs ! j)) ⊆ Rely (xs ! i);
pre ⊆ (INT i:{i. i < length xs}. Pre (xs ! i));
(INT 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
lemma ParallelEmpty:
[| x ∈ par_cp (ParallelCom []) s; Suc i < length x |]
==> ¬ x ! i -pc-> x ! Suc i
theorem par_rgsound:
\<turnstile> c SAT [pre, rely, guar, post]
==> \<Turnstile> ParallelCom c SAT [pre, rely, guar, post]