Theory AxSem

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theory AxSem
imports State

(*  Title:      HOL/NanoJava/AxSem.thy
ID: $Id$
Author: David von Oheimb
Copyright 2001 Technische Universitaet Muenchen
*)


header "Axiomatic Semantics"

theory AxSem imports State begin

types assn = "state => bool"
vassn = "val => assn"
triple = "assn × stmt × assn"
etriple = "assn × expr × vassn"

translations
(type) "assn" \<leftharpoondown> (type) "state => bool"
(type) "vassn" \<leftharpoondown> (type) "val => assn"
(type) "triple" \<leftharpoondown> (type) "assn × stmt × assn"
(type) "etriple" \<leftharpoondown> (type) "assn × expr × vassn"



subsection "Hoare Logic Rules"

inductive
hoare :: "[triple set, triple set] => bool" ("_ |\<turnstile>/ _" [61, 61] 60)
and ehoare :: "[triple set, etriple] => bool" ("_ |\<turnstile>e/ _" [61, 61] 60)
and hoare1 :: "[triple set, assn,stmt,assn] => bool"
("_ \<turnstile>/ ({(1_)}/ (_)/ {(1_)})" [61, 3, 90, 3] 60)
and ehoare1 :: "[triple set, assn,expr,vassn]=> bool"
("_ \<turnstile>e/ ({(1_)}/ (_)/ {(1_)})" [61, 3, 90, 3] 60)
where

"A \<turnstile> {P}c{Q} ≡ A |\<turnstile> {(P,c,Q)}"
| "A \<turnstile>e {P}e{Q} ≡ A |\<turnstile>e (P,e,Q)"

| Skip: "A \<turnstile> {P} Skip {P}"

| Comp: "[| A \<turnstile> {P} c1 {Q}; A \<turnstile> {Q} c2 {R} |] ==> A \<turnstile> {P} c1;;c2 {R}"

| Cond: "[| A \<turnstile>e {P} e {Q};
∀v. A \<turnstile> {Q v} (if v ≠ Null then c1 else c2) {R} |] ==>
A \<turnstile> {P} If(e) c1 Else c2 {R}"


| Loop: "A \<turnstile> {λs. P s ∧ s<x> ≠ Null} c {P} ==>
A \<turnstile> {P} While(x) c {λs. P s ∧ s<x> = Null}"


| LAcc: "A \<turnstile>e {λs. P (s<x>) s} LAcc x {P}"

| LAss: "A \<turnstile>e {P} e {λv s. Q (lupd(x\<mapsto>v) s)} ==>
A \<turnstile> {P} x:==e {Q}"


| FAcc: "A \<turnstile>e {P} e {λv s. ∀a. v=Addr a --> Q (get_field s a f) s} ==>
A \<turnstile>e {P} e..f {Q}"


| FAss: "[| A \<turnstile>e {P} e1 {λv s. ∀a. v=Addr a --> Q a s};
∀a. A \<turnstile>e {Q a} e2 {λv s. R (upd_obj a f v s)} |] ==>
A \<turnstile> {P} e1..f:==e2 {R}"


| NewC: "A \<turnstile>e {λs. ∀a. new_Addr s = Addr a --> P (Addr a) (new_obj a C s)}
new C {P}"


| Cast: "A \<turnstile>e {P} e {λv s. (case v of Null => True
| Addr a => obj_class s a <=C C) --> Q v s} ==>
A \<turnstile>e {P} Cast C e {Q}"


| Call: "[| A \<turnstile>e {P} e1 {Q}; ∀a. A \<turnstile>e {Q a} e2 {R a};
∀a p ls. A \<turnstile> {λs'. ∃s. R a p s ∧ ls = s ∧
s' = lupd(This\<mapsto>a)(lupd(Par\<mapsto>p)(del_locs s))}
Meth (C,m) {λs. S (s<Res>) (set_locs ls s)} |] ==>
A \<turnstile>e {P} {C}e1..m(e2) {S}"


| Meth: "∀D. A \<turnstile> {λs'. ∃s a. s<This> = Addr a ∧ D = obj_class s a ∧ D <=C C ∧
P s ∧ s' = init_locs D m s}
Impl (D,m) {Q} ==>
A \<turnstile> {P} Meth (C,m) {Q}"


--{* @{text "\<Union>Z"} instead of @{text "∀Z"} in the conclusion and\\
Z restricted to type state due to limitations of the inductive package *}

| Impl: "∀Z::state. A∪ (\<Union>Z. (λCm. (P Z Cm, Impl Cm, Q Z Cm))`Ms) |\<turnstile>
(λCm. (P Z Cm, body Cm, Q Z Cm))`Ms ==>
A |\<turnstile> (λCm. (P Z Cm, Impl Cm, Q Z Cm))`Ms"


--{* structural rules *}

| Asm: " a ∈ A ==> A |\<turnstile> {a}"

| ConjI: " ∀c ∈ C. A |\<turnstile> {c} ==> A |\<turnstile> C"

| ConjE: "[|A |\<turnstile> C; c ∈ C |] ==> A |\<turnstile> {c}"

--{* Z restricted to type state due to limitations of the inductive package *}
| Conseq:"[| ∀Z::state. A \<turnstile> {P' Z} c {Q' Z};
∀s t. (∀Z. P' Z s --> Q' Z t) --> (P s --> Q t) |] ==>
A \<turnstile> {P} c {Q }"


--{* Z restricted to type state due to limitations of the inductive package *}
| eConseq:"[| ∀Z::state. A \<turnstile>e {P' Z} e {Q' Z};
∀s v t. (∀Z. P' Z s --> Q' Z v t) --> (P s --> Q v t) |] ==>
A \<turnstile>e {P} e {Q }"



subsection "Fully polymorphic variants, required for Example only"

axioms

Conseq:"[| ∀Z. A \<turnstile> {P' Z} c {Q' Z};
∀s t. (∀Z. P' Z s --> Q' Z t) --> (P s --> Q t) |] ==>
A \<turnstile> {P} c {Q }"


eConseq:"[| ∀Z. A \<turnstile>e {P' Z} e {Q' Z};
∀s v t. (∀Z. P' Z s --> Q' Z v t) --> (P s --> Q v t) |] ==>
A \<turnstile>e {P} e {Q }"


Impl: "∀Z. A∪ (\<Union>Z. (λCm. (P Z Cm, Impl Cm, Q Z Cm))`Ms) |\<turnstile>
(λCm. (P Z Cm, body Cm, Q Z Cm))`Ms ==>
A |\<turnstile> (λCm. (P Z Cm, Impl Cm, Q Z Cm))`Ms"


subsection "Derived Rules"

lemma Conseq1: "[|A \<turnstile> {P'} c {Q}; ∀s. P s --> P' s|] ==> A \<turnstile> {P} c {Q}"
apply (rule hoare_ehoare.Conseq)
apply (rule allI, assumption)
apply fast
done

lemma Conseq2: "[|A \<turnstile> {P} c {Q'}; ∀t. Q' t --> Q t|] ==> A \<turnstile> {P} c {Q}"
apply (rule hoare_ehoare.Conseq)
apply (rule allI, assumption)
apply fast
done

lemma eConseq1: "[|A \<turnstile>e {P'} e {Q}; ∀s. P s --> P' s|] ==> A \<turnstile>e {P} e {Q}"
apply (rule hoare_ehoare.eConseq)
apply (rule allI, assumption)
apply fast
done

lemma eConseq2: "[|A \<turnstile>e {P} e {Q'}; ∀v t. Q' v t --> Q v t|] ==> A \<turnstile>e {P} e {Q}"
apply (rule hoare_ehoare.eConseq)
apply (rule allI, assumption)
apply fast
done

lemma Weaken: "[|A |\<turnstile> C'; C ⊆ C'|] ==> A |\<turnstile> C"
apply (rule hoare_ehoare.ConjI)
apply clarify
apply (drule hoare_ehoare.ConjE)
apply fast
apply assumption
done

lemma Thin_lemma:
"(A' |\<turnstile> C --> (∀A. A' ⊆ A --> A |\<turnstile> C )) ∧
(A' \<turnstile>e {P} e {Q} --> (∀A. A' ⊆ A --> A \<turnstile>e {P} e {Q}))"

apply (rule hoare_ehoare.induct)
apply (tactic "ALLGOALS(EVERY'[clarify_tac @{claset}, REPEAT o smp_tac 1])")
apply (blast intro: hoare_ehoare.Skip)
apply (blast intro: hoare_ehoare.Comp)
apply (blast intro: hoare_ehoare.Cond)
apply (blast intro: hoare_ehoare.Loop)
apply (blast intro: hoare_ehoare.LAcc)
apply (blast intro: hoare_ehoare.LAss)
apply (blast intro: hoare_ehoare.FAcc)
apply (blast intro: hoare_ehoare.FAss)
apply (blast intro: hoare_ehoare.NewC)
apply (blast intro: hoare_ehoare.Cast)
apply (erule hoare_ehoare.Call)
apply (rule, drule spec, erule conjE, tactic "smp_tac 1 1", assumption)
apply blast
apply (blast intro!: hoare_ehoare.Meth)
apply (blast intro!: hoare_ehoare.Impl)
apply (blast intro!: hoare_ehoare.Asm)
apply (blast intro: hoare_ehoare.ConjI)
apply (blast intro: hoare_ehoare.ConjE)
apply (rule hoare_ehoare.Conseq)
apply (rule, drule spec, erule conjE, tactic "smp_tac 1 1", assumption+)
apply (rule hoare_ehoare.eConseq)
apply (rule, drule spec, erule conjE, tactic "smp_tac 1 1", assumption+)
done

lemma cThin: "[|A' |\<turnstile> C; A' ⊆ A|] ==> A |\<turnstile> C"
by (erule (1) conjunct1 [OF Thin_lemma, rule_format])

lemma eThin: "[|A' \<turnstile>e {P} e {Q}; A' ⊆ A|] ==> A \<turnstile>e {P} e {Q}"
by (erule (1) conjunct2 [OF Thin_lemma, rule_format])


lemma Union: "A |\<turnstile> (\<Union>Z. C Z) = (∀Z. A |\<turnstile> C Z)"
by (auto intro: hoare_ehoare.ConjI hoare_ehoare.ConjE)

lemma Impl1':
"[|∀Z::state. A∪ (\<Union>Z. (λCm. (P Z Cm, Impl Cm, Q Z Cm))`Ms) |\<turnstile>
(λCm. (P Z Cm, body Cm, Q Z Cm))`Ms;
Cm ∈ Ms|] ==>
A \<turnstile> {P Z Cm} Impl Cm {Q Z Cm}"

apply (drule AxSem.Impl)
apply (erule Weaken)
apply (auto del: image_eqI intro: rev_image_eqI)
done

lemmas Impl1 = AxSem.Impl [of _ _ _ "{Cm}", simplified, standard]

end