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theory Eval = State + WellType:(* Title: HOL/MicroJava/J/Eval.thy
ID: $Id: Eval.thy,v 1.15.2.3 2002/02/21 13:08:43 kleing Exp $
Author: David von Oheimb
Copyright 1999 Technische Universitaet Muenchen
*)
header {* \isaheader{Operational Evaluation (big step) Semantics} *}
theory Eval = State + WellType:
consts
eval :: "java_mb prog => (xstate × expr × val × xstate) set"
evals :: "java_mb prog => (xstate × expr list × val list × xstate) set"
exec :: "java_mb prog => (xstate × stmt × xstate) set"
syntax (xsymbols)
eval :: "[java_mb prog,xstate,expr,val,xstate] => bool "
("_ \<turnstile> _ -_\<succ>_-> _" [51,82,60,82,82] 81)
evals:: "[java_mb prog,xstate,expr list,
val list,xstate] => bool "
("_ \<turnstile> _ -_[\<succ>]_-> _" [51,82,60,51,82] 81)
exec :: "[java_mb prog,xstate,stmt, xstate] => bool "
("_ \<turnstile> _ -_-> _" [51,82,60,82] 81)
syntax
eval :: "[java_mb prog,xstate,expr,val,xstate] => bool "
("_ |- _ -_>_-> _" [51,82,60,82,82] 81)
evals:: "[java_mb prog,xstate,expr list,
val list,xstate] => bool "
("_ |- _ -_[>]_-> _" [51,82,60,51,82] 81)
exec :: "[java_mb prog,xstate,stmt, xstate] => bool "
("_ |- _ -_-> _" [51,82,60,82] 81)
translations
"G\<turnstile>s -e \<succ> v-> (x,s')" <= "(s, e, v, x, s') \<in> eval G"
"G\<turnstile>s -e \<succ> v-> s' " == "(s, e, v, s') \<in> eval G"
"G\<turnstile>s -e[\<succ>]v-> (x,s')" <= "(s, e, v, x, s') \<in> evals G"
"G\<turnstile>s -e[\<succ>]v-> s' " == "(s, e, v, s') \<in> evals G"
"G\<turnstile>s -c -> (x,s')" <= "(s, c, x, s') \<in> exec G"
"G\<turnstile>s -c -> s' " == "(s, c, s') \<in> exec G"
inductive "eval G" "evals G" "exec G" intros
(* evaluation of expressions *)
XcptE:"G\<turnstile>(Some xc,s) -e\<succ>arbitrary-> (Some xc,s)" -- "cf. 15.5"
-- "cf. 15.8.1"
NewC: "[| h = heap s; (a,x) = new_Addr h;
h'= h(a\<mapsto>(C,init_vars (fields (G,C)))) |] ==>
G\<turnstile>Norm s -NewC C\<succ>Addr a-> c_hupd h' (x,s)"
-- "cf. 15.15"
Cast: "[| G\<turnstile>Norm s0 -e\<succ>v-> (x1,s1);
x2 = raise_if (¬ cast_ok G C (heap s1) v) ClassCast x1 |] ==>
G\<turnstile>Norm s0 -Cast C e\<succ>v-> (x2,s1)"
-- "cf. 15.7.1"
Lit: "G\<turnstile>Norm s -Lit v\<succ>v-> Norm s"
BinOp:"[| G\<turnstile>Norm s -e1\<succ>v1-> s1;
G\<turnstile>s1 -e2\<succ>v2-> s2;
v = (case bop of Eq => Bool (v1 = v2)
| Add => Intg (the_Intg v1 + the_Intg v2)) |] ==>
G\<turnstile>Norm s -BinOp bop e1 e2\<succ>v-> s2"
-- "cf. 15.13.1, 15.2"
LAcc: "G\<turnstile>Norm s -LAcc v\<succ>the (locals s v)-> Norm s"
-- "cf. 15.25.1"
LAss: "[| G\<turnstile>Norm s -e\<succ>v-> (x,(h,l));
l' = (if x = None then l(va\<mapsto>v) else l) |] ==>
G\<turnstile>Norm s -va::=e\<succ>v-> (x,(h,l'))"
-- "cf. 15.10.1, 15.2"
FAcc: "[| G\<turnstile>Norm s0 -e\<succ>a'-> (x1,s1);
v = the (snd (the (heap s1 (the_Addr a'))) (fn,T)) |] ==>
G\<turnstile>Norm s0 -{T}e..fn\<succ>v-> (np a' x1,s1)"
-- "cf. 15.25.1"
FAss: "[| G\<turnstile> Norm s0 -e1\<succ>a'-> (x1,s1); a = the_Addr a';
G\<turnstile>(np a' x1,s1) -e2\<succ>v -> (x2,s2);
h = heap s2; (c,fs) = the (h a);
h' = h(a\<mapsto>(c,(fs((fn,T)\<mapsto>v)))) |] ==>
G\<turnstile>Norm s0 -{T}e1..fn:=e2\<succ>v-> c_hupd h' (x2,s2)"
-- "cf. 15.11.4.1, 15.11.4.2, 15.11.4.4, 15.11.4.5, 14.15"
Call: "[| G\<turnstile>Norm s0 -e\<succ>a'-> s1; a = the_Addr a';
G\<turnstile>s1 -ps[\<succ>]pvs-> (x,(h,l)); dynT = fst (the (h a));
(md,rT,pns,lvars,blk,res) = the (method (G,dynT) (mn,pTs));
G\<turnstile>(np a' x,(h,(init_vars lvars)(pns[\<mapsto>]pvs)(This\<mapsto>a'))) -blk-> s3;
G\<turnstile> s3 -res\<succ>v -> (x4,s4) |] ==>
G\<turnstile>Norm s0 -{C}e..mn({pTs}ps)\<succ>v-> (x4,(heap s4,l))"
-- "evaluation of expression lists"
-- "cf. 15.5"
XcptEs:"G\<turnstile>(Some xc,s) -e[\<succ>]arbitrary-> (Some xc,s)"
-- "cf. 15.11.???"
Nil: "G\<turnstile>Norm s0 -[][\<succ>][]-> Norm s0"
-- "cf. 15.6.4"
Cons: "[| G\<turnstile>Norm s0 -e \<succ> v -> s1;
G\<turnstile> s1 -es[\<succ>]vs-> s2 |] ==>
G\<turnstile>Norm s0 -e#es[\<succ>]v#vs-> s2"
-- "execution of statements"
-- "cf. 14.1"
XcptS:"G\<turnstile>(Some xc,s) -c-> (Some xc,s)"
-- "cf. 14.5"
Skip: "G\<turnstile>Norm s -Skip-> Norm s"
-- "cf. 14.7"
Expr: "[| G\<turnstile>Norm s0 -e\<succ>v-> s1 |] ==>
G\<turnstile>Norm s0 -Expr e-> s1"
-- "cf. 14.2"
Comp: "[| G\<turnstile>Norm s0 -c1-> s1;
G\<turnstile> s1 -c2-> s2|] ==>
G\<turnstile>Norm s0 -c1;; c2-> s2"
-- "cf. 14.8.2"
Cond: "[| G\<turnstile>Norm s0 -e\<succ>v-> s1;
G\<turnstile> s1 -(if the_Bool v then c1 else c2)-> s2|] ==>
G\<turnstile>Norm s0 -If(e) c1 Else c2-> s2"
-- "cf. 14.10, 14.10.1"
LoopF:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1; ¬the_Bool v |] ==>
G\<turnstile>Norm s0 -While(e) c-> s1"
LoopT:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1; the_Bool v;
G\<turnstile>s1 -c-> s2; G\<turnstile>s2 -While(e) c-> s3 |] ==>
G\<turnstile>Norm s0 -While(e) c-> s3"
lemmas eval_evals_exec_induct = eval_evals_exec.induct [split_format (complete)]
lemma NewCI: "[|new_Addr (heap s) = (a,x);
s' = c_hupd (heap s(a\<mapsto>(C,init_vars (fields (G,C))))) (x,s)|] ==>
G\<turnstile>Norm s -NewC C\<succ>Addr a-> s'"
apply (simp (no_asm_simp))
apply (rule eval_evals_exec.NewC)
apply auto
done
lemma eval_evals_exec_no_xcpt:
"!!s s'. (G\<turnstile>(x,s) -e \<succ> v -> (x',s') --> x'=None --> x=None) \<and>
(G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x'=None --> x=None) \<and>
(G\<turnstile>(x,s) -c -> (x',s') --> x'=None --> x=None)"
apply(simp (no_asm_simp) only: split_tupled_all)
apply(rule eval_evals_exec_induct)
apply(unfold c_hupd_def)
apply(simp_all)
done
lemma eval_no_xcpt: "G\<turnstile>(x,s) -e\<succ>v-> (None,s') ==> x=None"
apply (drule eval_evals_exec_no_xcpt [THEN conjunct1, THEN mp])
apply (fast)
done
lemma evals_no_xcpt: "G\<turnstile>(x,s) -e[\<succ>]v-> (None,s') ==> x=None"
apply (drule eval_evals_exec_no_xcpt [THEN conjunct2, THEN conjunct1, THEN mp])
apply (fast)
done
lemma eval_evals_exec_xcpt:
"!!s s'. (G\<turnstile>(x,s) -e \<succ> v -> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s) \<and>
(G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s) \<and>
(G\<turnstile>(x,s) -c -> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s)"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (rule eval_evals_exec_induct)
apply (unfold c_hupd_def)
apply (simp_all)
done
lemma eval_xcpt: "G\<turnstile>(Some xc,s) -e\<succ>v-> (x',s') ==> x'=Some xc \<and> s'=s"
apply (drule eval_evals_exec_xcpt [THEN conjunct1, THEN mp])
apply (fast)
done
lemma exec_xcpt: "G\<turnstile>(Some xc,s) -s0-> (x',s') ==> x'=Some xc \<and> s'=s"
apply (drule eval_evals_exec_xcpt [THEN conjunct2, THEN conjunct2, THEN mp])
apply (fast)
done
end
lemmas eval_evals_exec_induct:
[| !!e a b xc. P1 (Some xc) a b e arbitrary (Some xc) a b;
!!C a h h' aa b x.
[| h = fst (aa, b); (a, x) = new_Addr h;
h' = h(a|->(C, init_vars (fields (G, C)))) |]
==> split (%v. split (P1 None aa b (NewC C) (Addr a) v))
(c_hupd h' (x, aa, b));
!!C e a b aa ba v x1 x2.
[| (Norm (a, b), e, v, x1, aa, ba) : eval G; P1 None a b e v x1 aa ba;
x2 = raise_if (¬ cast_ok G C (fst (aa, ba)) v) ClassCast x1 |]
==> P1 None a b (Cast C e) v x2 aa ba;
!!a b v. P1 None a b (Lit v) v None a b;
!!bop e1 e2 a b aa ab ba ac ad bb v v1 v2.
[| (Norm (a, b), e1, v1, aa, ab, ba) : eval G; P1 None a b e1 v1 aa ab ba;
((aa, ab, ba), e2, v2, ac, ad, bb) : eval G;
P1 aa ab ba e2 v2 ac ad bb;
v = (case bop of Eq => Bool (v1 = v2)
| Add => Intg (the_Intg v1 + the_Intg v2)) |]
==> P1 None a b (BinOp bop e1 e2) v ac ad bb;
!!a b v. P1 None a b (LAcc v) (the (snd (a, b) v)) None a b;
!!e h l l' a b v va x.
[| (Norm (a, b), e, v, x, h, l) : eval G; P1 None a b e v x h l;
l' = (if x = None then l(va|->v) else l) |]
==> P1 None a b (va::=e) v x h l';
!!T a' e fn a b aa ba v x1.
[| (Norm (a, b), e, a', x1, aa, ba) : eval G; P1 None a b e a' x1 aa ba;
v = the (snd (the (fst (aa, ba) (the_Addr a'))) (fn, T)) |]
==> P1 None a b ({T}e..fn) v (np a' x1) aa ba;
!!T a a' c e1 e2 fn fs h h' aa b ab ba ac bb v x1 x2.
[| (Norm (aa, b), e1, a', x1, ab, ba) : eval G;
P1 None aa b e1 a' x1 ab ba; a = the_Addr a';
((np a' x1, ab, ba), e2, v, x2, ac, bb) : eval G;
P1 (np a' x1) ab ba e2 v x2 ac bb; h = fst (ac, bb);
(c, fs) = the (h a); h' = h(a|->(c, fs((fn, T)|->v))) |]
==> split (%va. split (P1 None aa b ({T}e1..fn:=e2) v va))
(c_hupd h' (x2, ac, bb));
!!C a a' blk dynT e h l lvars md mn pTs pns ps pvs rT res aa b ab ac ba ad ae
bb af bc v x x4.
[| (Norm (aa, b), e, a', ab, ac, ba) : eval G; P1 None aa b e a' ab ac ba;
a = the_Addr a'; ((ab, ac, ba), ps, pvs, x, h, l) : evals G;
P2 ab ac ba ps pvs x h l; dynT = fst (the (h a));
(md, rT, pns, lvars, blk, res) = the (method (G, dynT) (mn, pTs));
((np a' x, h, init_vars lvars(pns[|->]pvs)(This|->a')), blk, ad, ae,
bb)
: exec G;
P3 (np a' x) h (init_vars lvars(pns[|->]pvs)(This|->a')) blk ad ae bb;
((ad, ae, bb), res, v, x4, af, bc) : eval G;
P1 ad ae bb res v x4 af bc |]
==> P1 None aa b ({C}e..mn( {pTs}ps)) v x4 (fst (af, bc)) l;
!!e a b xc. P2 (Some xc) a b e arbitrary (Some xc) a b;
!!a b. P2 None a b [] [] None a b;
!!e es a b aa ab ba ac ad bb v vs.
[| (Norm (a, b), e, v, aa, ab, ba) : eval G; P1 None a b e v aa ab ba;
((aa, ab, ba), es, vs, ac, ad, bb) : evals G;
P2 aa ab ba es vs ac ad bb |]
==> P2 None a b (e # es) (v # vs) ac ad bb;
!!c a b xc. P3 (Some xc) a b c (Some xc) a b;
!!a b. P3 None a b Skip None a b;
!!e a b aa ab ba v.
[| (Norm (a, b), e, v, aa, ab, ba) : eval G; P1 None a b e v aa ab ba |]
==> P3 None a b (Expr e) aa ab ba;
!!c1 c2 a b aa ab ba ac ad bb.
[| (Norm (a, b), c1, aa, ab, ba) : exec G; P3 None a b c1 aa ab ba;
((aa, ab, ba), c2, ac, ad, bb) : exec G; P3 aa ab ba c2 ac ad bb |]
==> P3 None a b (c1;; c2) ac ad bb;
!!c1 c2 e a b aa ab ba ac ad bb v.
[| (Norm (a, b), e, v, aa, ab, ba) : eval G; P1 None a b e v aa ab ba;
((aa, ab, ba), if the_Bool v then c1 else c2, ac, ad, bb) : exec G;
P3 aa ab ba (if the_Bool v then c1 else c2) ac ad bb |]
==> P3 None a b (If (e) c1 Else c2) ac ad bb;
!!c e a b aa ab ba v.
[| (Norm (a, b), e, v, aa, ab, ba) : eval G; P1 None a b e v aa ab ba;
¬ the_Bool v |]
==> P3 None a b (While (e) c) aa ab ba;
!!c e a b aa ab ba ac ad bb ae af bc v.
[| (Norm (a, b), e, v, aa, ab, ba) : eval G; P1 None a b e v aa ab ba;
the_Bool v; ((aa, ab, ba), c, ac, ad, bb) : exec G;
P3 aa ab ba c ac ad bb;
((ac, ad, bb), While (e) c, ae, af, bc) : exec G;
P3 ac ad bb (While (e) c) ae af bc |]
==> P3 None a b (While (e) c) ae af bc |]
==> (((xka, xkb, xkc), xja, xia, xha, xhb, xhc) : eval G -->
P1 xka xkb xkc xja xia xha xhb xhc) &
(((xga, xgb, xgc), xfa, xea, xda, xdb, xdc) : evals G -->
P2 xga xgb xgc xfa xea xda xdb xdc) &
(((xca, xcb, xcc), xba, xaa, xab, xac) : exec G -->
P3 xca xcb xcc xba xaa xab xac)
lemma NewCI:
[| new_Addr (fst s) = (a, x);
s' = c_hupd (fst s(a|->(C, init_vars (fields (G, C))))) (x, s) |]
==> G |- Norm s -NewC C>Addr a-> s'
lemma eval_evals_exec_no_xcpt:
(G |- (x, s) -e>v-> (x', s') --> x' = None --> x = None) & (G |- (x, s) -es[>]vs-> (x', s') --> x' = None --> x = None) & (G |- (x, s) -c-> (x', s') --> x' = None --> x = None)
lemma eval_no_xcpt:
G |- (x, s) -e>v-> Norm s' ==> x = None
lemma evals_no_xcpt:
G |- (x, s) -e[>]v-> Norm s' ==> x = None
lemma eval_evals_exec_xcpt:
(G |- (x, s) -e>v-> (x', s') --> x = Some xc --> x' = Some xc & s' = s) & (G |- (x, s) -es[>]vs-> (x', s') --> x = Some xc --> x' = Some xc & s' = s) & (G |- (x, s) -c-> (x', s') --> x = Some xc --> x' = Some xc & s' = s)
lemma eval_xcpt:
G |- (Some xc, s) -e>v-> (x', s') ==> x' = Some xc & s' = s
lemma exec_xcpt:
G |- (Some xc, s) -s0-> (x', s') ==> x' = Some xc & s' = s