(* Title: HOL/NanoJava/AxSem.thy
ID: $Id$
Author: David von Oheimb
Copyright 2001 Technische Universitaet Muenchen
*)
header "Axiomatic Semantics (Hoare Logic)"
theory AxSem = State:
types assn = "state => bool"
triple = "assn \<times> stmt \<times> assn"
translations
"assn" \<leftharpoondown> (type)"state => bool"
"triple" \<leftharpoondown> (type)"assn \<times> stmt \<times> assn"
consts hoare :: "(triple set \<times> triple set) set"
syntax (xsymbols)
"@hoare" :: "[triple set, triple set ] => bool" ("_ |\<turnstile>/ _" [61,61] 60)
"@hoare1" :: "[triple set, assn,stmt,assn] => bool"
("_ \<turnstile>/ ({(1_)}/ (_)/ {(1_)})" [61,3,90,3]60)
syntax
"@hoare" :: "[triple set, triple set ] => bool" ("_ ||-/ _" [61,61] 60)
"@hoare1" :: "[triple set, assn,stmt,assn] => bool"
("_ |-/ ({(1_)}/ (_)/ {(1_)})" [61,3,90,3] 60)
translations "A |\<turnstile> C" \<rightleftharpoons> "(A,C) \<in> hoare"
"A \<turnstile> {P}c{Q}" \<rightleftharpoons> "A |\<turnstile> {(P,c,Q)}"
inductive hoare
intros
Skip: "A |- {P} Skip {P}"
Comp: "[| A |- {P} c1 {Q}; A |- {Q} c2 {R} |] ==> A |- {P} c1;;c2 {R}"
Cond: "[| A |- {\<lambda>s. P s \<and> s<e> \<noteq> Null} c1 {Q};
A |- {\<lambda>s. P s \<and> s<e> = Null} c2 {Q} |] ==>
A |- {P} If(e) c1 Else c2 {Q}"
Loop: "A |- {\<lambda>s. P s \<and> s<e> \<noteq> Null} c {P} ==>
A |- {P} While(e) c {\<lambda>s. P s \<and> s<e> = Null}"
NewC: "A |- {\<lambda>s.\<forall>a. new_Addr s=Addr a--> P (lupd(x|->Addr a)(new_obj a C s))}
x:=new C {P}"
Cast: "A |- {\<lambda>s.(case s<y> of Null=> True | Addr a=> obj_class s a <=C C) -->
P (lupd(x|->s<y>) s)} x:=(C)y {P}"
FAcc: "A |- {\<lambda>s.\<forall>a. s<y>=Addr a-->P(lupd(x|->get_field s a f) s)} x:=y..f{P}"
FAss: "A |- {\<lambda>s. \<forall>a. s<y>=Addr a --> P (upd_obj a f (s<x>) s)} y..f:=x {P}"
Call: "\<forall>l. A |- {\<lambda>s'. \<exists>s. P s \<and> l = s \<and>
s' = lupd(This|->s<y>)(lupd(Param|->s<p>)(init_locs C m s))}
Meth C m {\<lambda>s. Q (lupd(x|->s<Res>)(set_locs l s))} ==>
A |- {P} x:={C}y..m(p) {Q}"
Meth: "\<forall>D. A |- {\<lambda>s. \<exists>a. s<This> = Addr a \<and> D=obj_class s a \<and> D <=C C \<and> P s}
Impl D m {Q} ==>
A |- {P} Meth C m {Q}"
(*\<Union>z instead of \<forall>z in the conclusion and
z restricted to type state due to limitations of the inductive paackage *)
Impl: "A\<union> (\<Union>z::state. (\<lambda>(C,m). (P z C m, Impl C m, Q z C m))`ms) ||-
(\<Union>z::state. (\<lambda>(C,m). (P z C m, body C m, Q z C m))`ms) ==>
A ||- (\<Union>z::state. (\<lambda>(C,m). (P z C m, Impl C m, Q z C m))`ms)"
(* structural rules *)
(* z restricted to type state due to limitations of the inductive paackage *)
Conseq:"[| \<forall>z::state. A |- {P' z} c {Q' z};
\<forall>s t. (\<forall>z::state. P' z s --> Q' z t) --> (P s --> Q t) |] ==>
A |- {P} c {Q }"
Asm: " a \<in> A ==> A ||- {a}"
ConjI: " \<forall>c \<in> C. A ||- {c} ==> A ||- C"
ConjE: "[|A ||- C; c \<in> C |] ==> A ||- {c}";
subsection "Derived Rules"
lemma Conseq1: "\<lbrakk>A \<turnstile> {P'} c {Q}; \<forall>s. P s \<longrightarrow> P' s\<rbrakk> \<Longrightarrow> A \<turnstile> {P} c {Q}"
apply (rule hoare.Conseq)
apply (rule allI, assumption)
apply fast
done
lemma Weaken: "\<lbrakk>A |\<turnstile> C'; C \<subseteq> C'\<rbrakk> \<Longrightarrow> A |\<turnstile> C"
apply (rule hoare.ConjI)
apply clarify
apply (drule hoare.ConjE)
apply fast
apply assumption
done
lemma Union: "A |\<turnstile> (\<Union>z. C z) = (\<forall>z. A |\<turnstile> C z)"
by (auto intro: hoare.ConjI hoare.ConjE)
lemma Impl':
"\<forall>z. A\<union> (\<Union>z. (\<lambda>(C,m). (P z C m, Impl C m, Q (z::state) C m))`ms) ||-
(\<lambda>(C,m). (P z C m, body C m, Q (z::state) C m))`ms ==>
A ||- (\<lambda>(C,m). (P z C m, Impl C m, Q (z::state) C m))`ms"
apply (drule Union[THEN iffD2])
apply (drule hoare.Impl)
apply (drule Union[THEN iffD1])
apply (erule spec)
done
lemma Impl1:
"\<lbrakk>\<forall>z. A\<union> (\<Union>z. (\<lambda>(C,m). (P z C m, Impl C m, Q (z::state) C m))`ms) ||-
(\<lambda>(C,m). (P z C m, body C m, Q (z::state) C m))`ms;
(C,m)\<in> ms\<rbrakk> \<Longrightarrow>
A |- {P z C m} Impl C m {Q (z::state) C m}"
apply (drule Impl')
apply (erule Weaken)
apply (auto del: image_eqI intro: rev_image_eqI)
done
end