Added unary and binary operations like (+,-,<, ...); Added smallstep semantics (no proofs about it yet).
(* Title: HOL/Bali/Evaln.thy
ID: $Id$
Author: David von Oheimb and Norbert Schirmer
License: GPL (GNU GENERAL PUBLIC LICENSE)
*)
header {* Operational evaluation (big-step) semantics of Java expressions and
statements
*}
theory Evaln = Eval + TypeSafe:
text {*
Variant of eval relation with counter for bounded recursive depth.
Evaln omits the technical accessibility tests @{term check_field_access}
and @{term check_method_access}, since we proved the absence of errors for
wellformed programs.
*}
consts
evaln :: "prog \<Rightarrow> (state \<times> term \<times> nat \<times> vals \<times> state) set"
syntax
evaln :: "[prog, state, term, nat, vals * state] => bool"
("_|-_ -_>-_-> _" [61,61,80, 61,61] 60)
evarn :: "[prog, state, var , vvar , nat, state] => bool"
("_|-_ -_=>_-_-> _" [61,61,90,61,61,61] 60)
eval_n:: "[prog, state, expr , val , nat, state] => bool"
("_|-_ -_->_-_-> _" [61,61,80,61,61,61] 60)
evalsn:: "[prog, state, expr list, val list, nat, state] => bool"
("_|-_ -_#>_-_-> _" [61,61,61,61,61,61] 60)
execn :: "[prog, state, stmt , nat, state] => bool"
("_|-_ -_-_-> _" [61,61,65, 61,61] 60)
syntax (xsymbols)
evaln :: "[prog, state, term, nat, vals \<times> state] \<Rightarrow> bool"
("_\<turnstile>_ \<midarrow>_\<succ>\<midarrow>_\<rightarrow> _" [61,61,80, 61,61] 60)
evarn :: "[prog, state, var , vvar , nat, state] \<Rightarrow> bool"
("_\<turnstile>_ \<midarrow>_=\<succ>_\<midarrow>_\<rightarrow> _" [61,61,90,61,61,61] 60)
eval_n:: "[prog, state, expr , val , nat, state] \<Rightarrow> bool"
("_\<turnstile>_ \<midarrow>_-\<succ>_\<midarrow>_\<rightarrow> _" [61,61,80,61,61,61] 60)
evalsn:: "[prog, state, expr list, val list, nat, state] \<Rightarrow> bool"
("_\<turnstile>_ \<midarrow>_\<doteq>\<succ>_\<midarrow>_\<rightarrow> _" [61,61,61,61,61,61] 60)
execn :: "[prog, state, stmt , nat, state] \<Rightarrow> bool"
("_\<turnstile>_ \<midarrow>_\<midarrow>_\<rightarrow> _" [61,61,65, 61,61] 60)
translations
"G\<turnstile>s \<midarrow>t \<succ>\<midarrow>n\<rightarrow> w___s' " == "(s,t,n,w___s') \<in> evaln G"
"G\<turnstile>s \<midarrow>t \<succ>\<midarrow>n\<rightarrow> (w, s')" <= "(s,t,n,w, s') \<in> evaln G"
"G\<turnstile>s \<midarrow>t \<succ>\<midarrow>n\<rightarrow> (w,x,s')" <= "(s,t,n,w,x,s') \<in> evaln G"
"G\<turnstile>s \<midarrow>c \<midarrow>n\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In1r c\<succ>\<midarrow>n\<rightarrow> (\<diamondsuit> ,x,s')"
"G\<turnstile>s \<midarrow>c \<midarrow>n\<rightarrow> s' " == "G\<turnstile>s \<midarrow>In1r c\<succ>\<midarrow>n\<rightarrow> (\<diamondsuit> , s')"
"G\<turnstile>s \<midarrow>e-\<succ>v \<midarrow>n\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In1l e\<succ>\<midarrow>n\<rightarrow> (In1 v ,x,s')"
"G\<turnstile>s \<midarrow>e-\<succ>v \<midarrow>n\<rightarrow> s' " == "G\<turnstile>s \<midarrow>In1l e\<succ>\<midarrow>n\<rightarrow> (In1 v , s')"
"G\<turnstile>s \<midarrow>e=\<succ>vf \<midarrow>n\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In2 e\<succ>\<midarrow>n\<rightarrow> (In2 vf,x,s')"
"G\<turnstile>s \<midarrow>e=\<succ>vf \<midarrow>n\<rightarrow> s' " == "G\<turnstile>s \<midarrow>In2 e\<succ>\<midarrow>n\<rightarrow> (In2 vf, s')"
"G\<turnstile>s \<midarrow>e\<doteq>\<succ>v \<midarrow>n\<rightarrow> (x,s')" <= "G\<turnstile>s \<midarrow>In3 e\<succ>\<midarrow>n\<rightarrow> (In3 v ,x,s')"
"G\<turnstile>s \<midarrow>e\<doteq>\<succ>v \<midarrow>n\<rightarrow> s' " == "G\<turnstile>s \<midarrow>In3 e\<succ>\<midarrow>n\<rightarrow> (In3 v , s')"
inductive "evaln G" intros
(* propagation of abrupt completion *)
Abrupt: "G\<turnstile>(Some xc,s) \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (arbitrary3 t,(Some xc,s))"
(* evaluation of variables *)
LVar: "G\<turnstile>Norm s \<midarrow>LVar vn=\<succ>lvar vn s\<midarrow>n\<rightarrow> Norm s"
FVar: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s2;
(v,s2') = fvar statDeclC stat fn a' s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>{accC,statDeclC,stat}e..fn=\<succ>v\<midarrow>n\<rightarrow> s2'"
AVar: "\<lbrakk>G\<turnstile> Norm s0 \<midarrow>e1-\<succ>a\<midarrow>n\<rightarrow> s1 ; G\<turnstile>s1 \<midarrow>e2-\<succ>i\<midarrow>n\<rightarrow> s2;
(v,s2') = avar G i a s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>e1.[e2]=\<succ>v\<midarrow>n\<rightarrow> s2'"
(* evaluation of expressions *)
NewC: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s1;
G\<turnstile> s1 \<midarrow>halloc (CInst C)\<succ>a\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>NewC C-\<succ>Addr a\<midarrow>n\<rightarrow> s2"
NewA: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>e-\<succ>i'\<midarrow>n\<rightarrow> s2;
G\<turnstile>abupd (check_neg i') s2 \<midarrow>halloc (Arr T (the_Intg i'))\<succ>a\<rightarrow> s3\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>New T[e]-\<succ>Addr a\<midarrow>n\<rightarrow> s3"
Cast: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1;
s2 = abupd (raise_if (\<not>G,snd s1\<turnstile>v fits T) ClassCast) s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Cast T e-\<succ>v\<midarrow>n\<rightarrow> s2"
Inst: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1;
b = (v\<noteq>Null \<and> G,store s1\<turnstile>v fits RefT T)\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>e InstOf T-\<succ>Bool b\<midarrow>n\<rightarrow> s1"
Lit: "G\<turnstile>Norm s \<midarrow>Lit v-\<succ>v\<midarrow>n\<rightarrow> Norm s"
UnOp: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1\<rbrakk>
\<Longrightarrow> G\<turnstile>Norm s0 \<midarrow>UnOp unop e-\<succ>(eval_unop unop v)\<midarrow>n\<rightarrow> s1"
BinOp: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>e2-\<succ>v2\<midarrow>n\<rightarrow> s2\<rbrakk>
\<Longrightarrow> G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>(eval_binop binop v1 v2)\<midarrow>n\<rightarrow> s2"
Super: "G\<turnstile>Norm s \<midarrow>Super-\<succ>val_this s\<midarrow>n\<rightarrow> Norm s"
Acc: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>va=\<succ>(v,f)\<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Acc va-\<succ>v\<midarrow>n\<rightarrow> s1"
Ass: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>va=\<succ>(w,f)\<midarrow>n\<rightarrow> s1;
G\<turnstile> s1 \<midarrow>e-\<succ>v \<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>va:=e-\<succ>v\<midarrow>n\<rightarrow> assign f v s2"
Cond: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<midarrow>n\<rightarrow> s1;
G\<turnstile> s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<midarrow>n\<rightarrow> s2"
Call:
"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2;
D = invocation_declclass G mode (store s2) a' statT \<lparr>name=mn,parTs=pTs\<rparr>;
G\<turnstile>init_lvars G D \<lparr>name=mn,parTs=pTs\<rparr> mode a' vs s2
\<midarrow>Methd D \<lparr>name=mn,parTs=pTs\<rparr>-\<succ>v\<midarrow>n\<rightarrow> s3\<rbrakk>
\<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>{accC,statT,mode}e\<cdot>mn({pTs}args)-\<succ>v\<midarrow>n\<rightarrow> (restore_lvars s2 s3)"
Methd:"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>body G D sig-\<succ>v\<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Methd D sig-\<succ>v\<midarrow>Suc n\<rightarrow> s1"
Body: "\<lbrakk>G\<turnstile>Norm s0\<midarrow>Init D\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>c\<midarrow>n\<rightarrow> s2\<rbrakk>\<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Body D c
-\<succ>the (locals (store s2) Result)\<midarrow>n\<rightarrow>abupd (absorb Ret) s2"
(* evaluation of expression lists *)
Nil:
"G\<turnstile>Norm s0 \<midarrow>[]\<doteq>\<succ>[]\<midarrow>n\<rightarrow> Norm s0"
Cons: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e -\<succ> v \<midarrow>n\<rightarrow> s1;
G\<turnstile> s1 \<midarrow>es\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>e#es\<doteq>\<succ>v#vs\<midarrow>n\<rightarrow> s2"
(* execution of statements *)
Skip: "G\<turnstile>Norm s \<midarrow>Skip\<midarrow>n\<rightarrow> Norm s"
Expr: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Expr e\<midarrow>n\<rightarrow> s1"
Lab: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c \<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>l\<bullet> c\<midarrow>n\<rightarrow> abupd (absorb l) s1"
Comp: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1 \<midarrow>n\<rightarrow> s1;
G\<turnstile> s1 \<midarrow>c2 \<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>c1;; c2\<midarrow>n\<rightarrow> s2"
If: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1;
G\<turnstile> s1\<midarrow>(if the_Bool b then c1 else c2)\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>If(e) c1 Else c2 \<midarrow>n\<rightarrow> s2"
Loop: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1;
if normal s1 \<and> the_Bool b
then (G\<turnstile>s1 \<midarrow>c\<midarrow>n\<rightarrow> s2 \<and>
G\<turnstile>(abupd (absorb (Cont l)) s2) \<midarrow>l\<bullet> While(e) c\<midarrow>n\<rightarrow> s3)
else s3 = s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>l\<bullet> While(e) c\<midarrow>n\<rightarrow> s3"
Do: "G\<turnstile>Norm s \<midarrow>Do j\<midarrow>n\<rightarrow> (Some (Jump j), s)"
Throw:"\<lbrakk>G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Throw e\<midarrow>n\<rightarrow> abupd (throw a') s1"
Try: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2;
if G,s2\<turnstile>catch tn then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n\<rightarrow> s3 else s3 = s2\<rbrakk>
\<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Try c1 Catch(tn vn) c2\<midarrow>n\<rightarrow> s3"
Fin: "\<lbrakk>G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n\<rightarrow> (x1,s1);
G\<turnstile>Norm s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<midarrow>n\<rightarrow> abupd (abrupt_if (x1\<noteq>None) x1) s2"
Init: "\<lbrakk>the (class G C) = c;
if inited C (globs s0) then s3 = Norm s0
else (G\<turnstile>Norm (init_class_obj G C s0)
\<midarrow>(if C = Object then Skip else Init (super c))\<midarrow>n\<rightarrow> s1 \<and>
G\<turnstile>set_lvars empty s1 \<midarrow>init c\<midarrow>n\<rightarrow> s2 \<and>
s3 = restore_lvars s1 s2)\<rbrakk>
\<Longrightarrow>
G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s3"
monos
if_def2
declare split_if [split del] split_if_asm [split del]
option.split [split del] option.split_asm [split del]
inductive_cases evaln_cases: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> vs'"
inductive_cases evaln_elim_cases:
"G\<turnstile>(Some xc, s) \<midarrow>t \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r Skip \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (Do j) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (l\<bullet> c) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In3 ([]) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In3 (e#es) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Lit w) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (UnOp unop e) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (BinOp binop e1 e2) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In2 (LVar vn) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Cast T e) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (e InstOf T) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Super) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Acc va) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r (Expr e) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (c1;; c2) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1l (Methd C sig) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1l (Body D c) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1l (e0 ? e1 : e2) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r (If(e) c1 Else c2) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (l\<bullet> While(e) c) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (c1 Finally c2) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1r (Throw e) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In1l (NewC C) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (New T[e]) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l (Ass va e) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r (Try c1 Catch(tn vn) c2) \<succ>\<midarrow>n\<rightarrow> xs'"
"G\<turnstile>Norm s \<midarrow>In2 ({accC,statDeclC,stat}e..fn) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In2 (e1.[e2]) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1l ({accC,statT,mode}e\<cdot>mn({pT}p)) \<succ>\<midarrow>n\<rightarrow> vs'"
"G\<turnstile>Norm s \<midarrow>In1r (Init C) \<succ>\<midarrow>n\<rightarrow> xs'"
declare split_if [split] split_if_asm [split]
option.split [split] option.split_asm [split]
lemma evaln_Inj_elim: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (w,s') \<Longrightarrow> case t of In1 ec \<Rightarrow>
(case ec of Inl e \<Rightarrow> (\<exists>v. w = In1 v) | Inr c \<Rightarrow> w = \<diamondsuit>)
| In2 e \<Rightarrow> (\<exists>v. w = In2 v) | In3 e \<Rightarrow> (\<exists>v. w = In3 v)"
apply (erule evaln_cases , auto)
apply (induct_tac "t")
apply (induct_tac "a")
apply auto
done
ML_setup {*
fun enf nam inj rhs =
let
val name = "evaln_" ^ nam ^ "_eq"
val lhs = "G\<turnstile>s \<midarrow>" ^ inj ^ " t\<succ>\<midarrow>n\<rightarrow> (w, s')"
val () = qed_goal name (the_context()) (lhs ^ " = (" ^ rhs ^ ")")
(K [Auto_tac, ALLGOALS (ftac (thm "evaln_Inj_elim")) THEN Auto_tac])
fun is_Inj (Const (inj,_) $ _) = true
| is_Inj _ = false
fun pred (_ $ (Const ("Pair",_) $ _ $ (Const ("Pair", _) $ _ $
(Const ("Pair", _) $ _ $ (Const ("Pair", _) $ x $ _ )))) $ _ ) = is_Inj x
in
make_simproc name lhs pred (thm name)
end;
val evaln_expr_proc = enf "expr" "In1l" "\<exists>v. w=In1 v \<and> G\<turnstile>s \<midarrow>t-\<succ>v \<midarrow>n\<rightarrow> s'";
val evaln_var_proc = enf "var" "In2" "\<exists>vf. w=In2 vf \<and> G\<turnstile>s \<midarrow>t=\<succ>vf\<midarrow>n\<rightarrow> s'";
val evaln_exprs_proc= enf "exprs""In3" "\<exists>vs. w=In3 vs \<and> G\<turnstile>s \<midarrow>t\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s'";
val evaln_stmt_proc = enf "stmt" "In1r" " w=\<diamondsuit> \<and> G\<turnstile>s \<midarrow>t \<midarrow>n\<rightarrow> s'";
Addsimprocs [evaln_expr_proc,evaln_var_proc,evaln_exprs_proc,evaln_stmt_proc];
bind_thms ("evaln_AbruptIs", sum3_instantiate (thm "evaln.Abrupt"))
*}
declare evaln_AbruptIs [intro!]
lemma evaln_Callee: "G\<turnstile>Norm s\<midarrow>In1l (Callee l e)\<succ>\<midarrow>n\<rightarrow> (v,s') = False"
proof -
{ fix s t v s'
assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s')" and
normal: "normal s" and
callee: "t=In1l (Callee l e)"
then have "False"
proof (induct)
qed (auto)
}
then show ?thesis
by (cases s') fastsimp
qed
lemma evaln_InsInitE: "G\<turnstile>Norm s\<midarrow>In1l (InsInitE c e)\<succ>\<midarrow>n\<rightarrow> (v,s') = False"
proof -
{ fix s t v s'
assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s')" and
normal: "normal s" and
callee: "t=In1l (InsInitE c e)"
then have "False"
proof (induct)
qed (auto)
}
then show ?thesis
by (cases s') fastsimp
qed
lemma evaln_InsInitV: "G\<turnstile>Norm s\<midarrow>In2 (InsInitV c w)\<succ>\<midarrow>n\<rightarrow> (v,s') = False"
proof -
{ fix s t v s'
assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s')" and
normal: "normal s" and
callee: "t=In2 (InsInitV c w)"
then have "False"
proof (induct)
qed (auto)
}
then show ?thesis
by (cases s') fastsimp
qed
lemma evaln_FinA: "G\<turnstile>Norm s\<midarrow>In1r (FinA a c)\<succ>\<midarrow>n\<rightarrow> (v,s') = False"
proof -
{ fix s t v s'
assume eval: "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s')" and
normal: "normal s" and
callee: "t=In1r (FinA a c)"
then have "False"
proof (induct)
qed (auto)
}
then show ?thesis
by (cases s') fastsimp
qed
lemma evaln_abrupt_lemma: "G\<turnstile>s \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (v,s') \<Longrightarrow>
fst s = Some xc \<longrightarrow> s' = s \<and> v = arbitrary3 e"
apply (erule evaln_cases , auto)
done
lemma evaln_abrupt:
"\<And>s'. G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (w,s') = (s' = (Some xc,s) \<and>
w=arbitrary3 e \<and> G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (arbitrary3 e,(Some xc,s)))"
apply auto
apply (frule evaln_abrupt_lemma, auto)+
done
ML {*
local
fun is_Some (Const ("Pair",_) $ (Const ("Datatype.option.Some",_) $ _)$ _) =true
| is_Some _ = false
fun pred (_ $ (Const ("Pair",_) $
_ $ (Const ("Pair", _) $ _ $ (Const ("Pair", _) $ _ $
(Const ("Pair", _) $ _ $ x)))) $ _ ) = is_Some x
in
val evaln_abrupt_proc =
make_simproc "evaln_abrupt" "G\<turnstile>(Some xc,s) \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (w,s')" pred (thm "evaln_abrupt")
end;
Addsimprocs [evaln_abrupt_proc]
*}
lemma evaln_LitI: "G\<turnstile>s \<midarrow>Lit v-\<succ>(if normal s then v else arbitrary)\<midarrow>n\<rightarrow> s"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Lit)
lemma CondI:
"\<And>s1. \<lbrakk>G\<turnstile>s \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>s \<midarrow>e ? e1 : e2-\<succ>(if normal s1 then v else arbitrary)\<midarrow>n\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Cond)
lemma evaln_SkipI [intro!]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Skip)
lemma evaln_ExprI: "G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s' \<Longrightarrow> G\<turnstile>s \<midarrow>Expr e\<midarrow>n\<rightarrow> s'"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Expr)
lemma evaln_CompI: "\<lbrakk>G\<turnstile>s \<midarrow>c1\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow> G\<turnstile>s \<midarrow>c1;; c2\<midarrow>n\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.Comp)
lemma evaln_IfI:
"\<lbrakk>G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1; G\<turnstile>s1 \<midarrow>(if the_Bool v then c1 else c2)\<midarrow>n\<rightarrow> s2\<rbrakk> \<Longrightarrow>
G\<turnstile>s \<midarrow>If(e) c1 Else c2\<midarrow>n\<rightarrow> s2"
apply (case_tac "s", case_tac "a = None")
by (auto intro!: evaln.If)
lemma evaln_SkipD [dest!]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s' \<Longrightarrow> s' = s"
by (erule evaln_cases, auto)
lemma evaln_Skip_eq [simp]: "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s' = (s = s')"
apply auto
done
lemma evaln_eval:
assumes evaln: "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1)" and
wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T" and
conf_s0: "s0\<Colon>\<preceq>(G, L)" and
wf: "wf_prog G"
shows "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)"
proof -
from evaln
show "\<And> L accC T. \<lbrakk>s0\<Colon>\<preceq>(G, L);\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T\<rbrakk>
\<Longrightarrow> G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)"
(is "PROP ?EqEval s0 s1 t v")
proof (induct)
case Abrupt
show ?case by (rule eval.Abrupt)
next
case LVar
show ?case by (rule eval.LVar)
next
case (FVar a accC' e fn n s0 s1 s2 s2' stat statDeclC v L accC T)
have eval_initn: "G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<midarrow>n\<rightarrow> s1" .
have eval_en: "G\<turnstile>s1 \<midarrow>e-\<succ>a\<midarrow>n\<rightarrow> s2" .
have hyp_init: "PROP ?EqEval (Norm s0) s1 (In1r (Init statDeclC)) \<diamondsuit>" .
have hyp_e: "PROP ?EqEval s1 s2 (In1l e) (In1 a)" .
have fvar: "(v, s2') = fvar statDeclC stat fn a s2" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>In2 ({accC',statDeclC,stat}e..fn)\<Colon>T" .
then obtain statC f where
wt_e: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>e\<Colon>-Class statC" and
accfield: "accfield G accC statC fn = Some (statDeclC,f)" and
stat: "stat=is_static f" and
accC': "accC'=accC" and
T: "T=(Inl (type f))"
by (rule wt_elim_cases) (auto simp add: member_is_static_simp)
from wf wt_e
have iscls_statC: "is_class G statC"
by (auto dest: ty_expr_is_type type_is_class)
with wf accfield
have iscls_statDeclC: "is_class G statDeclC"
by (auto dest!: accfield_fields dest: fields_declC)
then
have wt_init: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>(Init statDeclC)\<Colon>\<surd>"
by simp
from conf_s0 wt_init
have eval_init: "G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<rightarrow> s1"
by (rule hyp_init)
with wt_init conf_s0 wf
have conf_s1: "s1\<Colon>\<preceq>(G, L)"
by (blast dest: exec_ts)
with hyp_e wt_e
have eval_e: "G\<turnstile>s1 \<midarrow>e-\<succ>a\<rightarrow> s2"
by blast
with wf conf_s1 wt_e
obtain conf_s2: "s2\<Colon>\<preceq>(G, L)" and
conf_a: "normal s2 \<longrightarrow> G,store s2\<turnstile>a\<Colon>\<preceq>Class statC"
by (auto dest!: eval_type_sound)
obtain s3 where
check: "s3 = check_field_access G accC statDeclC fn stat a s2'"
by simp
from accfield wt_e eval_init eval_e conf_s2 conf_a fvar stat check wf
have eq_s3_s2': "s3=s2'"
by (auto dest!: error_free_field_access)
with eval_init eval_e fvar check accC'
show "G\<turnstile>Norm s0 \<midarrow>{accC',statDeclC,stat}e..fn=\<succ>v\<rightarrow> s2'"
by (auto intro: eval.FVar)
next
case AVar
with wf show ?case
apply -
apply (erule wt_elim_cases)
apply (blast intro!: eval.AVar dest: eval_type_sound)
done
next
case NewC
with wf show ?case
apply -
apply (erule wt_elim_cases)
apply (blast intro!: eval.NewC dest: eval_type_sound is_acc_classD)
done
next
case (NewA T a e i n s0 s1 s2 s3 L accC Ta)
have hyp_init: "PROP ?EqEval (Norm s0) s1 (In1r (init_comp_ty T)) \<diamondsuit>" .
have hyp_size: "PROP ?EqEval s1 s2 (In1l e) (In1 i)" .
have "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1l (New T[e])\<Colon>Ta" .
then obtain
wt_init: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>init_comp_ty T\<Colon>\<surd>" and
wt_size: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-PrimT Integer"
by (rule wt_elim_cases) (auto intro: wt_init_comp_ty dest: is_acc_typeD)
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
from this wt_init
have eval_init: "G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<rightarrow> s1"
by (rule hyp_init)
moreover
from eval_init wt_init wf conf_s0
have "s1\<Colon>\<preceq>(G, L)"
by (auto dest: eval_type_sound)
from this wt_size
have "G\<turnstile>s1 \<midarrow>e-\<succ>i\<rightarrow> s2"
by (rule hyp_size)
moreover note NewA
ultimately show ?case
by (blast intro!: eval.NewA)
next
case Cast
with wf show ?case
by - (erule wt_elim_cases, rule eval.Cast,auto dest: eval_type_sound)
next
case Inst
with wf show ?case
by - (erule wt_elim_cases, rule eval.Inst,auto dest: eval_type_sound)
next
case Lit
show ?case by (rule eval.Lit)
next
case UnOp
with wf show ?case
by - (erule wt_elim_cases, rule eval.UnOp,auto dest: eval_type_sound)
next
case BinOp
with wf show ?case
by - (erule wt_elim_cases, blast intro!: eval.BinOp dest: eval_type_sound)
next
case Super
show ?case by (rule eval.Super)
next
case Acc
then show ?case
by - (erule wt_elim_cases, rule eval.Acc,auto dest: eval_type_sound)
next
case Ass
with wf show ?case
by - (erule wt_elim_cases, blast intro!: eval.Ass dest: eval_type_sound)
next
case (Cond b e0 e1 e2 n s0 s1 s2 v L accC T)
have hyp_e0: "PROP ?EqEval (Norm s0) s1 (In1l e0) (In1 b)" .
have hyp_if: "PROP ?EqEval s1 s2
(In1l (if the_Bool b then e1 else e2)) (In1 v)" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1l (e0 ? e1 : e2)\<Colon>T" .
then obtain T1 T2 statT where
wt_e0: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e0\<Colon>-PrimT Boolean" and
wt_e1: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e1\<Colon>-T1" and
wt_e2: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e2\<Colon>-T2" and
statT: "G\<turnstile>T1\<preceq>T2 \<and> statT = T2 \<or> G\<turnstile>T2\<preceq>T1 \<and> statT = T1" and
T : "T=Inl statT"
by (rule wt_elim_cases) auto
from conf_s0 wt_e0
have eval_e0: "G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<rightarrow> s1"
by (rule hyp_e0)
moreover
from eval_e0 conf_s0 wf wt_e0
have "s1\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
with wt_e1 wt_e2 statT hyp_if
have "G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<rightarrow> s2"
by (cases "the_Bool b") auto
ultimately
show ?case
by (rule eval.Cond)
next
case (Call invDeclC a' accC' args e mn mode n pTs' s0 s1 s2 s4 statT
v vs L accC T)
txt {* Repeats large parts of the type soundness proof. One should factor
out some lemmata about the relations and conformance of @{text
s2}, @{text s3} and @{text s3'} *}
have evaln_e: "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1" .
have evaln_args: "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2" .
have invDeclC: "invDeclC
= invocation_declclass G mode (store s2) a' statT
\<lparr>name = mn, parTs = pTs'\<rparr>" .
let ?InitLvars
= "init_lvars G invDeclC \<lparr>name = mn, parTs = pTs'\<rparr> mode a' vs s2"
obtain s3 s3' where
init_lvars: "s3 =
init_lvars G invDeclC \<lparr>name = mn, parTs = pTs'\<rparr> mode a' vs s2" and
check: "s3' =
check_method_access G accC' statT mode \<lparr>name = mn, parTs = pTs'\<rparr> a' s3"
by simp
have evaln_methd:
"G\<turnstile>?InitLvars \<midarrow>Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>-\<succ>v\<midarrow>n\<rightarrow> s4" .
have hyp_e: "PROP ?EqEval (Norm s0) s1 (In1l e) (In1 a')" .
have hyp_args: "PROP ?EqEval s1 s2 (In3 args) (In3 vs)" .
have hyp_methd: "PROP ?EqEval ?InitLvars s4
(In1l (Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>)) (In1 v)".
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>
\<turnstile>In1l ({accC',statT,mode}e\<cdot>mn( {pTs'}args))\<Colon>T" .
from wt obtain pTs statDeclT statM where
wt_e: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT" and
wt_args: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>args\<Colon>\<doteq>pTs" and
statM: "max_spec G accC statT \<lparr>name=mn,parTs=pTs\<rparr>
= {((statDeclT,statM),pTs')}" and
mode: "mode = invmode statM e" and
T: "T =Inl (resTy statM)" and
eq_accC_accC': "accC=accC'"
by (rule wt_elim_cases) auto
from conf_s0 wt_e hyp_e
have eval_e: "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<rightarrow> s1"
by blast
with wf conf_s0 wt_e
obtain conf_s1: "s1\<Colon>\<preceq>(G, L)" and
conf_a': "normal s1 \<Longrightarrow> G, store s1\<turnstile>a'\<Colon>\<preceq>RefT statT"
by (auto dest!: eval_type_sound)
from conf_s1 wt_args hyp_args
have eval_args: "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<rightarrow> s2"
by blast
with wt_args conf_s1 wf
obtain conf_s2: "s2\<Colon>\<preceq>(G, L)" and
conf_args: "normal s2
\<Longrightarrow> list_all2 (conf G (store s2)) vs pTs"
by (auto dest!: eval_type_sound)
from statM
obtain
statM': "(statDeclT,statM)\<in>mheads G accC statT \<lparr>name=mn,parTs=pTs'\<rparr>" and
pTs_widen: "G\<turnstile>pTs[\<preceq>]pTs'"
by (blast dest: max_spec2mheads)
from check
have eq_store_s3'_s3: "store s3'=store s3"
by (cases s3) (simp add: check_method_access_def Let_def)
obtain invC
where invC: "invC = invocation_class mode (store s2) a' statT"
by simp
with init_lvars
have invC': "invC = (invocation_class mode (store s3) a' statT)"
by (cases s2,cases mode) (auto simp add: init_lvars_def2 )
show "G\<turnstile>Norm s0 \<midarrow>{accC',statT,mode}e\<cdot>mn( {pTs'}args)
-\<succ>v\<rightarrow> (set_lvars (locals (store s2))) s4"
proof (cases "normal s2")
case False
with init_lvars
obtain keep_abrupt: "abrupt s3 = abrupt s2" and
"store s3 = store (init_lvars G invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>
mode a' vs s2)"
by (auto simp add: init_lvars_def2)
moreover
from keep_abrupt False check
have eq_s3'_s3: "s3'=s3"
by (auto simp add: check_method_access_def Let_def)
moreover
from eq_s3'_s3 False keep_abrupt evaln_methd init_lvars
obtain "s4=s3'"
"In1 v=arbitrary3 (In1l (Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>))"
by auto
moreover note False
ultimately have
"G\<turnstile>s3' \<midarrow>Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>-\<succ>v\<rightarrow> s4"
by (auto)
from eval_e eval_args invDeclC init_lvars check this
show ?thesis
by (rule eval.Call)
next
case True
note normal_s2 = True
with eval_args
have normal_s1: "normal s1"
by (cases "normal s1") auto
with conf_a' eval_args
have conf_a'_s2: "G, store s2\<turnstile>a'\<Colon>\<preceq>RefT statT"
by (auto dest: eval_gext intro: conf_gext)
show ?thesis
proof (cases "a'=Null \<longrightarrow> is_static statM")
case False
then obtain not_static: "\<not> is_static statM" and Null: "a'=Null"
by blast
with normal_s2 init_lvars mode
obtain np: "abrupt s3 = Some (Xcpt (Std NullPointer))" and
"store s3 = store (init_lvars G invDeclC
\<lparr>name = mn, parTs = pTs'\<rparr> mode a' vs s2)"
by (auto simp add: init_lvars_def2)
moreover
from np check
have eq_s3'_s3: "s3'=s3"
by (auto simp add: check_method_access_def Let_def)
moreover
from eq_s3'_s3 np evaln_methd init_lvars
obtain "s4=s3'"
"In1 v=arbitrary3 (In1l (Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>))"
by auto
moreover note np
ultimately have
"G\<turnstile>s3' \<midarrow>Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>-\<succ>v\<rightarrow> s4"
by (auto)
from eval_e eval_args invDeclC init_lvars check this
show ?thesis
by (rule eval.Call)
next
case True
with mode have notNull: "mode = IntVir \<longrightarrow> a' \<noteq> Null"
by (auto dest!: Null_staticD)
with conf_s2 conf_a'_s2 wf invC
have dynT_prop: "G\<turnstile>mode\<rightarrow>invC\<preceq>statT"
by (cases s2) (auto intro: DynT_propI)
with wt_e statM' invC mode wf
obtain dynM where
dynM: "dynlookup G statT invC \<lparr>name=mn,parTs=pTs'\<rparr> = Some dynM" and
acc_dynM: "G \<turnstile>Methd \<lparr>name=mn,parTs=pTs'\<rparr> dynM
in invC dyn_accessible_from accC"
by (force dest!: call_access_ok)
with invC' check eq_accC_accC'
have eq_s3'_s3: "s3'=s3"
by (auto simp add: check_method_access_def Let_def)
from dynT_prop wf wt_e statM' mode invC invDeclC dynM
obtain
wf_dynM: "wf_mdecl G invDeclC (\<lparr>name=mn,parTs=pTs'\<rparr>,mthd dynM)" and
dynM': "methd G invDeclC \<lparr>name=mn,parTs=pTs'\<rparr> = Some dynM" and
iscls_invDeclC: "is_class G invDeclC" and
invDeclC': "invDeclC = declclass dynM" and
invC_widen: "G\<turnstile>invC\<preceq>\<^sub>C invDeclC" and
is_static_eq: "is_static dynM = is_static statM" and
involved_classes_prop:
"(if invmode statM e = IntVir
then \<forall>statC. statT = ClassT statC \<longrightarrow> G\<turnstile>invC\<preceq>\<^sub>C statC
else ((\<exists>statC. statT = ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C invDeclC) \<or>
(\<forall>statC. statT \<noteq> ClassT statC \<and> invDeclC = Object)) \<and>
statDeclT = ClassT invDeclC)"
by (auto dest: DynT_mheadsD)
obtain L' where
L':"L'=(\<lambda> k.
(case k of
EName e
\<Rightarrow> (case e of
VNam v
\<Rightarrow>(table_of (lcls (mbody (mthd dynM)))
(pars (mthd dynM)[\<mapsto>]pTs')) v
| Res \<Rightarrow> Some (resTy dynM))
| This \<Rightarrow> if is_static statM
then None else Some (Class invDeclC)))"
by simp
from wf_dynM [THEN wf_mdeclD1, THEN conjunct1] normal_s2 conf_s2 wt_e
wf eval_args conf_a' mode notNull wf_dynM involved_classes_prop
have conf_s3: "s3\<Colon>\<preceq>(G,L')"
apply -
(*FIXME confomrs_init_lvars should be
adjusted to be more directy applicable *)
apply (drule conforms_init_lvars [of G invDeclC
"\<lparr>name=mn,parTs=pTs'\<rparr>" dynM "store s2" vs pTs "abrupt s2"
L statT invC a' "(statDeclT,statM)" e])
apply (rule wf)
apply (rule conf_args,assumption)
apply (simp add: pTs_widen)
apply (cases s2,simp)
apply (rule dynM')
apply (force dest: ty_expr_is_type)
apply (rule invC_widen)
apply (force intro: conf_gext dest: eval_gext)
apply simp
apply simp
apply (simp add: invC)
apply (simp add: invDeclC)
apply (force dest: wf_mdeclD1 is_acc_typeD)
apply (cases s2, simp add: L' init_lvars
cong add: lname.case_cong ename.case_cong)
done
from is_static_eq wf_dynM L'
obtain mthdT where
"\<lparr>prg=G,cls=invDeclC,lcl=L'\<rparr>
\<turnstile>Body invDeclC (stmt (mbody (mthd dynM)))\<Colon>-mthdT" and
mthdT_widen: "G\<turnstile>mthdT\<preceq>resTy dynM"
by - (drule wf_mdecl_bodyD,
auto simp: cong add: lname.case_cong ename.case_cong)
with dynM' iscls_invDeclC invDeclC'
have
"\<lparr>prg=G,cls=invDeclC,lcl=L'\<rparr>
\<turnstile>(Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>)\<Colon>-mthdT"
by (auto intro: wt.Methd)
with conf_s3 hyp_methd init_lvars eq_s3'_s3
have "G\<turnstile>s3' \<midarrow>Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>-\<succ>v\<rightarrow> s4"
by auto
from eval_e eval_args invDeclC init_lvars check this
show ?thesis
by (rule eval.Call)
qed
qed
next
case Methd
with wf show ?case
by - (erule wt_elim_cases, rule eval.Methd,
auto dest: eval_type_sound simp add: body_def2)
next
case Body
with wf show ?case
by - (erule wt_elim_cases, blast intro!: eval.Body dest: eval_type_sound)
next
case Nil
show ?case by (rule eval.Nil)
next
case Cons
with wf show ?case
by - (erule wt_elim_cases, blast intro!: eval.Cons dest: eval_type_sound)
next
case Skip
show ?case by (rule eval.Skip)
next
case Expr
with wf show ?case
by - (erule wt_elim_cases, rule eval.Expr,auto dest: eval_type_sound)
next
case Lab
with wf show ?case
by - (erule wt_elim_cases, rule eval.Lab,auto dest: eval_type_sound)
next
case Comp
with wf show ?case
by - (erule wt_elim_cases, blast intro!: eval.Comp dest: eval_type_sound)
next
case (If b c1 c2 e n s0 s1 s2 L accC T)
have hyp_e: "PROP ?EqEval (Norm s0) s1 (In1l e) (In1 b)" .
have hyp_then_else:
"PROP ?EqEval s1 s2 (In1r (if the_Bool b then c1 else c2)) \<diamondsuit>" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1r (If(e) c1 Else c2)\<Colon>T" .
then obtain
wt_e: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>e\<Colon>-PrimT Boolean" and
wt_then_else: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>(if the_Bool b then c1 else c2)\<Colon>\<surd>"
by (rule wt_elim_cases) (auto split add: split_if)
from conf_s0 wt_e
have eval_e: "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<rightarrow> s1"
by (rule hyp_e)
moreover
from eval_e wt_e conf_s0 wf
have "s1\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
from this wt_then_else
have "G\<turnstile>s1 \<midarrow>(if the_Bool b then c1 else c2)\<rightarrow> s2"
by (rule hyp_then_else)
ultimately
show ?case
by (rule eval.If)
next
case (Loop b c e l n s0 s1 s2 s3 L accC T)
have hyp_e: "PROP ?EqEval (Norm s0) s1 (In1l e) (In1 b)" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1r (l\<bullet> While(e) c)\<Colon>T" .
then obtain wt_e: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-PrimT Boolean" and
wt_c: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>"
by (rule wt_elim_cases) (blast)
from conf_s0 wt_e
have eval_e: "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<rightarrow> s1"
by (rule hyp_e)
moreover
from eval_e wt_e conf_s0 wf
have conf_s1: "s1\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
have "if normal s1 \<and> the_Bool b
then (G\<turnstile>s1 \<midarrow>c\<rightarrow> s2 \<and>
G\<turnstile>(abupd (absorb (Cont l)) s2) \<midarrow>l\<bullet> While(e) c\<rightarrow> s3)
else s3 = s1"
proof (cases "normal s1 \<and> the_Bool b")
case True
from Loop True have hyp_c: "PROP ?EqEval s1 s2 (In1r c) \<diamondsuit>"
by (auto)
from Loop True have hyp_w: "PROP ?EqEval (abupd (absorb (Cont l)) s2)
s3 (In1r (l\<bullet> While(e) c)) \<diamondsuit>"
by (auto)
from conf_s1 wt_c
have eval_c: "G\<turnstile>s1 \<midarrow>c\<rightarrow> s2"
by (rule hyp_c)
moreover
from eval_c conf_s1 wt_c wf
have "s2\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
then
have "abupd (absorb (Cont l)) s2 \<Colon>\<preceq>(G, L)"
by (cases s2) (auto intro: conforms_absorb)
from this and wt
have "G\<turnstile>abupd (absorb (Cont l)) s2 \<midarrow>l\<bullet> While(e) c\<rightarrow> s3"
by (rule hyp_w)
moreover note True
ultimately
show ?thesis
by simp
next
case False
with Loop have "s3 = s1" by simp
with False
show ?thesis
by auto
qed
ultimately
show ?case
by (rule eval.Loop)
next
case Do
show ?case by (rule eval.Do)
next
case Throw
with wf show ?case
by - (erule wt_elim_cases, rule eval.Throw,auto dest: eval_type_sound)
next
case (Try c1 c2 n s0 s1 s2 s3 catchC vn L accC T)
have hyp_c1: "PROP ?EqEval (Norm s0) s1 (In1r c1) \<diamondsuit>" .
have conf_s0:"Norm s0\<Colon>\<preceq>(G, L)" .
have wt:"\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>In1r (Try c1 Catch(catchC vn) c2)\<Colon>T" .
then obtain
wt_c1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c1\<Colon>\<surd>" and
wt_c2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<lparr>lcl := L(VName vn\<mapsto>Class catchC)\<rparr>\<turnstile>c2\<Colon>\<surd>"
by (rule wt_elim_cases) (auto)
from conf_s0 wt_c1
have eval_c1: "G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> s1"
by (rule hyp_c1)
moreover
have sxalloc: "G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2" .
moreover
from eval_c1 wt_c1 conf_s0 wf
have "s1\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
with sxalloc wf
have conf_s2: "s2\<Colon>\<preceq>(G, L)"
by (auto dest: sxalloc_type_sound split: option.splits)
have "if G,s2\<turnstile>catch catchC then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<rightarrow> s3 else s3 = s2"
proof (cases "G,s2\<turnstile>catch catchC")
case True
note Catch = this
with Try have hyp_c2: "PROP ?EqEval (new_xcpt_var vn s2) s3 (In1r c2) \<diamondsuit>"
by auto
show ?thesis
proof (cases "normal s1")
case True
with sxalloc wf
have eq_s2_s1: "s2=s1"
by (auto dest: sxalloc_type_sound split: option.splits)
with True
have "\<not> G,s2\<turnstile>catch catchC"
by (simp add: catch_def)
with Catch show ?thesis
by (contradiction)
next
case False
with sxalloc wf
obtain a
where xcpt_s2: "abrupt s2 = Some (Xcpt (Loc a))"
by (auto dest!: sxalloc_type_sound split: option.splits)
with Catch
have "G\<turnstile>obj_ty (the (globs (store s2) (Heap a)))\<preceq>Class catchC"
by (cases s2) simp
with xcpt_s2 conf_s2 wf
have "new_xcpt_var vn s2\<Colon>\<preceq>(G, L(VName vn\<mapsto>Class catchC))"
by (auto dest: Try_lemma)
from this wt_c2
have "G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<rightarrow> s3"
by (auto intro: hyp_c2)
with Catch
show ?thesis
by simp
qed
next
case False
with Try
have "s3=s2"
by simp
with False
show ?thesis
by simp
qed
ultimately
show ?case
by (rule eval.Try)
next
case (Fin c1 c2 n s0 s1 s2 x1 L accC T)
have hyp_c1: "PROP ?EqEval (Norm s0) (x1,s1) (In1r c1) \<diamondsuit>" .
have hyp_c2: "PROP ?EqEval (Norm s1) (s2) (In1r c2) \<diamondsuit>" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1r (c1 Finally c2)\<Colon>T" .
then obtain
wt_c1: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c1\<Colon>\<surd>" and
wt_c2: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c2\<Colon>\<surd>"
by (rule wt_elim_cases) blast
from conf_s0 wt_c1
have eval_c1: "G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> (x1, s1)"
by (rule hyp_c1)
with wf wt_c1 conf_s0
obtain conf_s1: "Norm s1\<Colon>\<preceq>(G, L)" and
error_free_s1: "error_free (x1,s1)"
by (auto dest!: eval_type_sound intro: conforms_NormI)
from conf_s1 wt_c2
have eval_c2: "G\<turnstile>Norm s1 \<midarrow>c2\<rightarrow> s2"
by (rule hyp_c2)
with eval_c1 error_free_s1
show "G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<rightarrow> abupd (abrupt_if (x1 \<noteq> None) x1) s2"
by (auto intro: eval.Fin simp add: error_free_def)
next
case (Init C c n s0 s1 s2 s3 L accC T)
have cls: "the (class G C) = c" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1r (Init C)\<Colon>T" .
with cls
have cls_C: "class G C = Some c"
by - (erule wt_elim_cases,auto)
have "if inited C (globs s0) then s3 = Norm s0
else (G\<turnstile>Norm (init_class_obj G C s0)
\<midarrow>(if C = Object then Skip else Init (super c))\<rightarrow> s1 \<and>
G\<turnstile>set_lvars empty s1 \<midarrow>init c\<rightarrow> s2 \<and> s3 = restore_lvars s1 s2)"
proof (cases "inited C (globs s0)")
case True
with Init have "s3 = Norm s0"
by simp
with True show ?thesis
by simp
next
case False
with Init
obtain
hyp_init_super:
"PROP ?EqEval (Norm ((init_class_obj G C) s0)) s1
(In1r (if C = Object then Skip else Init (super c))) \<diamondsuit>"
and
hyp_init_c:
"PROP ?EqEval ((set_lvars empty) s1) s2 (In1r (init c)) \<diamondsuit>" and
s3: "s3 = (set_lvars (locals (store s1))) s2"
by (simp only: if_False)
from conf_s0 wf cls_C False
have conf_s0': "(Norm ((init_class_obj G C) s0))\<Colon>\<preceq>(G, L)"
by (auto dest: conforms_init_class_obj)
moreover
from wf cls_C
have wt_init_super:
"\<lparr>prg = G, cls = accC, lcl = L\<rparr>
\<turnstile>(if C = Object then Skip else Init (super c))\<Colon>\<surd>"
by (cases "C=Object")
(auto dest: wf_prog_cdecl wf_cdecl_supD is_acc_classD)
ultimately
have eval_init_super:
"G\<turnstile>Norm ((init_class_obj G C) s0)
\<midarrow>(if C = Object then Skip else Init (super c))\<rightarrow> s1"
by (rule hyp_init_super)
with conf_s0' wt_init_super wf
have "s1\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
then
have "(set_lvars empty) s1\<Colon>\<preceq>(G, empty)"
by (cases s1) (auto dest: conforms_set_locals )
with wf cls_C
have eval_init_c: "G\<turnstile>(set_lvars empty) s1 \<midarrow>init c\<rightarrow> s2"
by (auto intro!: hyp_init_c dest: wf_prog_cdecl wf_cdecl_wt_init)
from False eval_init_super eval_init_c s3
show ?thesis
by simp
qed
with cls show ?case
by (rule eval.Init)
qed
qed
lemma Suc_le_D_lemma: "\<lbrakk>Suc n <= m'; (\<And>m. n <= m \<Longrightarrow> P (Suc m)) \<rbrakk> \<Longrightarrow> P m'"
apply (frule Suc_le_D)
apply fast
done
lemma evaln_nonstrict [rule_format (no_asm), elim]:
"\<And>ws. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> ws \<Longrightarrow> \<forall>m. n\<le>m \<longrightarrow> G\<turnstile>s \<midarrow>t\<succ>\<midarrow>m\<rightarrow> ws"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (erule evaln.induct)
apply (tactic {* ALLGOALS (EVERY'[strip_tac, TRY o etac (thm "Suc_le_D_lemma"),
REPEAT o smp_tac 1,
resolve_tac (thms "evaln.intros") THEN_ALL_NEW TRY o atac]) *})
(* 3 subgoals *)
apply (auto split del: split_if)
done
lemmas evaln_nonstrict_Suc = evaln_nonstrict [OF _ le_refl [THEN le_SucI]]
lemma evaln_max2: "\<lbrakk>G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1\<rightarrow> ws1; G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2\<rightarrow> ws2\<rbrakk> \<Longrightarrow>
G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max n1 n2\<rightarrow> ws1 \<and> G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max n1 n2\<rightarrow> ws2"
apply (fast intro: le_maxI1 le_maxI2)
done
lemma evaln_max3:
"\<lbrakk>G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1\<rightarrow> ws1; G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2\<rightarrow> ws2; G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>n3\<rightarrow> ws3\<rbrakk> \<Longrightarrow>
G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> ws1 \<and>
G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> ws2 \<and>
G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> ws3"
apply (drule (1) evaln_max2, erule thin_rl)
apply (fast intro!: le_maxI1 le_maxI2)
done
lemma le_max3I1: "(n2::nat) \<le> max n1 (max n2 n3)"
proof -
have "n2 \<le> max n2 n3"
by (rule le_maxI1)
also
have "max n2 n3 \<le> max n1 (max n2 n3)"
by (rule le_maxI2)
finally
show ?thesis .
qed
lemma le_max3I2: "(n3::nat) \<le> max n1 (max n2 n3)"
proof -
have "n3 \<le> max n2 n3"
by (rule le_maxI2)
also
have "max n2 n3 \<le> max n1 (max n2 n3)"
by (rule le_maxI2)
finally
show ?thesis .
qed
lemma eval_evaln:
assumes eval: "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)" and
wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T" and
conf_s0: "s0\<Colon>\<preceq>(G, L)" and
wf: "wf_prog G"
shows "\<exists>n. G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1)"
proof -
from eval
show "\<And> L accC T. \<lbrakk>s0\<Colon>\<preceq>(G, L);\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T\<rbrakk>
\<Longrightarrow> \<exists> n. G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1)"
(is "PROP ?EqEval s0 s1 t v")
proof (induct)
case (Abrupt s t xc L accC T)
obtain n where
"G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (arbitrary3 t, Some xc, s)"
by (rules intro: evaln.Abrupt)
then show ?case ..
next
case Skip
show ?case by (blast intro: evaln.Skip)
next
case (Expr e s0 s1 v L accC T)
then obtain n where
"G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"
by (rules elim!: wt_elim_cases)
then have "G\<turnstile>Norm s0 \<midarrow>Expr e\<midarrow>n\<rightarrow> s1"
by (rule evaln.Expr)
then show ?case ..
next
case (Lab c l s0 s1 L accC T)
then obtain n where
"G\<turnstile>Norm s0 \<midarrow>c\<midarrow>n\<rightarrow> s1"
by (rules elim!: wt_elim_cases)
then have "G\<turnstile>Norm s0 \<midarrow>l\<bullet> c\<midarrow>n\<rightarrow> abupd (absorb l) s1"
by (rule evaln.Lab)
then show ?case ..
next
case (Comp c1 c2 s0 s1 s2 L accC T)
with wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n1\<rightarrow> s1"
"G\<turnstile>s1 \<midarrow>c2\<midarrow>n2\<rightarrow> s2"
by (blast elim!: wt_elim_cases dest: eval_type_sound)
then have "G\<turnstile>Norm s0 \<midarrow>c1;; c2\<midarrow>max n1 n2\<rightarrow> s2"
by (blast intro: evaln.Comp dest: evaln_max2 )
then show ?case ..
next
case (If b c1 c2 e s0 s1 s2 L accC T)
with wf obtain
"\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-PrimT Boolean"
"\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>(if the_Bool b then c1 else c2)\<Colon>\<surd>"
by (cases "the_Bool b") (auto elim!: wt_elim_cases)
with If wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n1\<rightarrow> s1"
"G\<turnstile>s1 \<midarrow>(if the_Bool b then c1 else c2)\<midarrow>n2\<rightarrow> s2"
by (blast dest: eval_type_sound)
then have "G\<turnstile>Norm s0 \<midarrow>If(e) c1 Else c2\<midarrow>max n1 n2\<rightarrow> s2"
by (blast intro: evaln.If dest: evaln_max2)
then show ?case ..
next
case (Loop b c e l s0 s1 s2 s3 L accC T)
have eval_e: "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<rightarrow> s1" .
have hyp_e: "PROP ?EqEval (Norm s0) s1 (In1l e) (In1 b)" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1r (l\<bullet> While(e) c)\<Colon>T" .
then obtain wt_e: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-PrimT Boolean" and
wt_c: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>"
by (rule wt_elim_cases) (blast)
from conf_s0 wt_e
obtain n1 where
"G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n1\<rightarrow> s1"
by (rules dest: hyp_e)
moreover
from eval_e wt_e conf_s0 wf
have conf_s1: "s1\<Colon>\<preceq>(G, L)"
by (rules dest: eval_type_sound)
obtain n2 where
"if normal s1 \<and> the_Bool b
then (G\<turnstile>s1 \<midarrow>c\<midarrow>n2\<rightarrow> s2 \<and>
G\<turnstile>(abupd (absorb (Cont l)) s2)\<midarrow>l\<bullet> While(e) c\<midarrow>n2\<rightarrow> s3)
else s3 = s1"
proof (cases "normal s1 \<and> the_Bool b")
case True
from Loop True have hyp_c: "PROP ?EqEval s1 s2 (In1r c) \<diamondsuit>"
by (auto)
from Loop True have hyp_w: "PROP ?EqEval (abupd (absorb (Cont l)) s2)
s3 (In1r (l\<bullet> While(e) c)) \<diamondsuit>"
by (auto)
from Loop True have eval_c: "G\<turnstile>s1 \<midarrow>c\<rightarrow> s2"
by simp
from conf_s1 wt_c
obtain m1 where
evaln_c: "G\<turnstile>s1 \<midarrow>c\<midarrow>m1\<rightarrow> s2"
by (rules dest: hyp_c)
moreover
from eval_c conf_s1 wt_c wf
have "s2\<Colon>\<preceq>(G, L)"
by (rules dest: eval_type_sound)
then
have "abupd (absorb (Cont l)) s2 \<Colon>\<preceq>(G, L)"
by (cases s2) (auto intro: conforms_absorb)
from this and wt
obtain m2 where
"G\<turnstile>abupd (absorb (Cont l)) s2 \<midarrow>l\<bullet> While(e) c\<midarrow>m2\<rightarrow> s3"
by (blast dest: hyp_w)
moreover note True and that
ultimately show ?thesis
by simp (rules intro: evaln_nonstrict le_maxI1 le_maxI2)
next
case False
with Loop have "s3 = s1"
by simp
with False that
show ?thesis
by auto
qed
ultimately
have "G\<turnstile>Norm s0 \<midarrow>l\<bullet> While(e) c\<midarrow>max n1 n2\<rightarrow> s3"
apply -
apply (rule evaln.Loop)
apply (rules intro: evaln_nonstrict intro: le_maxI1)
apply (auto intro: evaln_nonstrict intro: le_maxI2)
done
then show ?case ..
next
case (Do j s L accC T)
have "G\<turnstile>Norm s \<midarrow>Do j\<midarrow>n\<rightarrow> (Some (Jump j), s)"
by (rule evaln.Do)
then show ?case ..
next
case (Throw a e s0 s1 L accC T)
then obtain n where
"G\<turnstile>Norm s0 \<midarrow>e-\<succ>a\<midarrow>n\<rightarrow> s1"
by (rules elim!: wt_elim_cases)
then have "G\<turnstile>Norm s0 \<midarrow>Throw e\<midarrow>n\<rightarrow> abupd (throw a) s1"
by (rule evaln.Throw)
then show ?case ..
next
case (Try catchC c1 c2 s0 s1 s2 s3 vn L accC T)
have hyp_c1: "PROP ?EqEval (Norm s0) s1 (In1r c1) \<diamondsuit>" .
have eval_c1: "G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> s1" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>In1r (Try c1 Catch(catchC vn) c2)\<Colon>T" .
then obtain
wt_c1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c1\<Colon>\<surd>" and
wt_c2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<lparr>lcl := L(VName vn\<mapsto>Class catchC)\<rparr>\<turnstile>c2\<Colon>\<surd>"
by (rule wt_elim_cases) (auto)
from conf_s0 wt_c1
obtain n1 where
"G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n1\<rightarrow> s1"
by (blast dest: hyp_c1)
moreover
have sxalloc: "G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2" .
moreover
from eval_c1 wt_c1 conf_s0 wf
have "s1\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
with sxalloc wf
have conf_s2: "s2\<Colon>\<preceq>(G, L)"
by (auto dest: sxalloc_type_sound split: option.splits)
obtain n2 where
"if G,s2\<turnstile>catch catchC then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n2\<rightarrow> s3 else s3 = s2"
proof (cases "G,s2\<turnstile>catch catchC")
case True
note Catch = this
with Try have hyp_c2: "PROP ?EqEval (new_xcpt_var vn s2) s3 (In1r c2) \<diamondsuit>"
by auto
show ?thesis
proof (cases "normal s1")
case True
with sxalloc wf
have eq_s2_s1: "s2=s1"
by (auto dest: sxalloc_type_sound split: option.splits)
with True
have "\<not> G,s2\<turnstile>catch catchC"
by (simp add: catch_def)
with Catch show ?thesis
by (contradiction)
next
case False
with sxalloc wf
obtain a
where xcpt_s2: "abrupt s2 = Some (Xcpt (Loc a))"
by (auto dest!: sxalloc_type_sound split: option.splits)
with Catch
have "G\<turnstile>obj_ty (the (globs (store s2) (Heap a)))\<preceq>Class catchC"
by (cases s2) simp
with xcpt_s2 conf_s2 wf
have "new_xcpt_var vn s2\<Colon>\<preceq>(G, L(VName vn\<mapsto>Class catchC))"
by (auto dest: Try_lemma)
(* FIXME extract lemma for this conformance, also useful for
eval_type_sound and evaln_eval *)
from this wt_c2
obtain m where "G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>m\<rightarrow> s3"
by (auto dest: hyp_c2)
with True that
show ?thesis
by simp
qed
next
case False
with Try
have "s3=s2"
by simp
with False and that
show ?thesis
by simp
qed
ultimately
have "G\<turnstile>Norm s0 \<midarrow>Try c1 Catch(catchC vn) c2\<midarrow>max n1 n2\<rightarrow> s3"
by (auto intro!: evaln.Try le_maxI1 le_maxI2)
then show ?case ..
next
case (Fin c1 c2 s0 s1 s2 s3 x1 L accC T)
have s3: "s3 = (if \<exists>err. x1 = Some (Error err)
then (x1, s1)
else abupd (abrupt_if (x1 \<noteq> None) x1) s2)" .
from Fin wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n1\<rightarrow> (x1, s1)"
"G\<turnstile>Norm s1 \<midarrow>c2\<midarrow>n2\<rightarrow> s2" and
error_free_s1: "error_free (x1,s1)"
by (blast elim!: wt_elim_cases
dest: eval_type_sound intro: conforms_NormI)
then have
"G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<midarrow>max n1 n2\<rightarrow> abupd (abrupt_if (x1 \<noteq> None) x1) s2"
by (blast intro: evaln.Fin dest: evaln_max2)
with error_free_s1 s3
show "\<exists>n. G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<midarrow>n\<rightarrow> s3"
by (auto simp add: error_free_def)
next
case (Init C c s0 s1 s2 s3 L accC T)
have cls: "the (class G C) = c" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1r (Init C)\<Colon>T" .
with cls
have cls_C: "class G C = Some c"
by - (erule wt_elim_cases,auto)
obtain n where
"if inited C (globs s0) then s3 = Norm s0
else (G\<turnstile>Norm (init_class_obj G C s0)
\<midarrow>(if C = Object then Skip else Init (super c))\<midarrow>n\<rightarrow> s1 \<and>
G\<turnstile>set_lvars empty s1 \<midarrow>init c\<midarrow>n\<rightarrow> s2 \<and>
s3 = restore_lvars s1 s2)"
proof (cases "inited C (globs s0)")
case True
with Init have "s3 = Norm s0"
by simp
with True that show ?thesis
by simp
next
case False
with Init
obtain
hyp_init_super:
"PROP ?EqEval (Norm ((init_class_obj G C) s0)) s1
(In1r (if C = Object then Skip else Init (super c))) \<diamondsuit>"
and
hyp_init_c:
"PROP ?EqEval ((set_lvars empty) s1) s2 (In1r (init c)) \<diamondsuit>" and
s3: "s3 = (set_lvars (locals (store s1))) s2" and
eval_init_super:
"G\<turnstile>Norm ((init_class_obj G C) s0)
\<midarrow>(if C = Object then Skip else Init (super c))\<rightarrow> s1"
by (simp only: if_False)
from conf_s0 wf cls_C False
have conf_s0': "(Norm ((init_class_obj G C) s0))\<Colon>\<preceq>(G, L)"
by (auto dest: conforms_init_class_obj)
moreover
from wf cls_C
have wt_init_super:
"\<lparr>prg = G, cls = accC, lcl = L\<rparr>
\<turnstile>(if C = Object then Skip else Init (super c))\<Colon>\<surd>"
by (cases "C=Object")
(auto dest: wf_prog_cdecl wf_cdecl_supD is_acc_classD)
ultimately
obtain m1 where
"G\<turnstile>Norm ((init_class_obj G C) s0)
\<midarrow>(if C = Object then Skip else Init (super c))\<midarrow>m1\<rightarrow> s1"
by (rules dest: hyp_init_super)
moreover
from eval_init_super conf_s0' wt_init_super wf
have "s1\<Colon>\<preceq>(G, L)"
by (rules dest: eval_type_sound)
then
have "(set_lvars empty) s1\<Colon>\<preceq>(G, empty)"
by (cases s1) (auto dest: conforms_set_locals )
with wf cls_C
obtain m2 where
"G\<turnstile>(set_lvars empty) s1 \<midarrow>init c\<midarrow>m2\<rightarrow> s2"
by (blast dest!: hyp_init_c
dest: wf_prog_cdecl intro!: wf_cdecl_wt_init)
moreover note s3 and False and that
ultimately show ?thesis
by simp (rules intro: evaln_nonstrict le_maxI1 le_maxI2)
qed
from cls this have "G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s3"
by (rule evaln.Init)
then show ?case ..
next
case (NewC C a s0 s1 s2 L accC T)
with wf obtain n where
"G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s1"
by (blast elim!: wt_elim_cases dest: is_acc_classD)
with NewC
have "G\<turnstile>Norm s0 \<midarrow>NewC C-\<succ>Addr a\<midarrow>n\<rightarrow> s2"
by (rules intro: evaln.NewC)
then show ?case ..
next
case (NewA T a e i s0 s1 s2 s3 L accC Ta)
hence "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>init_comp_ty T\<Colon>\<surd>"
by (auto elim!: wt_elim_cases
intro!: wt_init_comp_ty dest: is_acc_typeD)
with NewA wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<midarrow>n1\<rightarrow> s1"
"G\<turnstile>s1 \<midarrow>e-\<succ>i\<midarrow>n2\<rightarrow> s2"
by (blast elim!: wt_elim_cases dest: eval_type_sound)
moreover
have "G\<turnstile>abupd (check_neg i) s2 \<midarrow>halloc Arr T (the_Intg i)\<succ>a\<rightarrow> s3" .
ultimately
have "G\<turnstile>Norm s0 \<midarrow>New T[e]-\<succ>Addr a\<midarrow>max n1 n2\<rightarrow> s3"
by (blast intro: evaln.NewA dest: evaln_max2)
then show ?case ..
next
case (Cast castT e s0 s1 s2 v L accC T)
with wf obtain n where
"G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"
by (rules elim!: wt_elim_cases)
moreover
have "s2 = abupd (raise_if (\<not> G,snd s1\<turnstile>v fits castT) ClassCast) s1" .
ultimately
have "G\<turnstile>Norm s0 \<midarrow>Cast castT e-\<succ>v\<midarrow>n\<rightarrow> s2"
by (rule evaln.Cast)
then show ?case ..
next
case (Inst T b e s0 s1 v L accC T')
with wf obtain n where
"G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"
by (rules elim!: wt_elim_cases)
moreover
have "b = (v \<noteq> Null \<and> G,snd s1\<turnstile>v fits RefT T)" .
ultimately
have "G\<turnstile>Norm s0 \<midarrow>e InstOf T-\<succ>Bool b\<midarrow>n\<rightarrow> s1"
by (rule evaln.Inst)
then show ?case ..
next
case (Lit s v L accC T)
have "G\<turnstile>Norm s \<midarrow>Lit v-\<succ>v\<midarrow>n\<rightarrow> Norm s"
by (rule evaln.Lit)
then show ?case ..
next
case (UnOp e s0 s1 unop v L accC T)
with wf obtain n where
"G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"
by (rules elim!: wt_elim_cases)
hence "G\<turnstile>Norm s0 \<midarrow>UnOp unop e-\<succ>eval_unop unop v\<midarrow>n\<rightarrow> s1"
by (rule evaln.UnOp)
then show ?case ..
next
case (BinOp binop e1 e2 s0 s1 s2 v1 v2 L accC T)
with wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<midarrow>n1\<rightarrow> s1"
"G\<turnstile>s1 \<midarrow>e2-\<succ>v2\<midarrow>n2\<rightarrow> s2"
by (blast elim!: wt_elim_cases dest: eval_type_sound)
hence "G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>(eval_binop binop v1 v2)\<midarrow>max n1 n2
\<rightarrow> s2"
by (blast intro!: evaln.BinOp dest: evaln_max2)
then show ?case ..
next
case (Super s L accC T)
have "G\<turnstile>Norm s \<midarrow>Super-\<succ>val_this s\<midarrow>n\<rightarrow> Norm s"
by (rule evaln.Super)
then show ?case ..
next
case (Acc f s0 s1 v va L accC T)
with wf obtain n where
"G\<turnstile>Norm s0 \<midarrow>va=\<succ>(v, f)\<midarrow>n\<rightarrow> s1"
by (rules elim!: wt_elim_cases)
then
have "G\<turnstile>Norm s0 \<midarrow>Acc va-\<succ>v\<midarrow>n\<rightarrow> s1"
by (rule evaln.Acc)
then show ?case ..
next
case (Ass e f s0 s1 s2 v var w L accC T)
with wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>var=\<succ>(w, f)\<midarrow>n1\<rightarrow> s1"
"G\<turnstile>s1 \<midarrow>e-\<succ>v\<midarrow>n2\<rightarrow> s2"
by (blast elim!: wt_elim_cases dest: eval_type_sound)
then
have "G\<turnstile>Norm s0 \<midarrow>var:=e-\<succ>v\<midarrow>max n1 n2\<rightarrow> assign f v s2"
by (blast intro: evaln.Ass dest: evaln_max2)
then show ?case ..
next
case (Cond b e0 e1 e2 s0 s1 s2 v L accC T)
have hyp_e0: "PROP ?EqEval (Norm s0) s1 (In1l e0) (In1 b)" .
have hyp_if: "PROP ?EqEval s1 s2
(In1l (if the_Bool b then e1 else e2)) (In1 v)" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>In1l (e0 ? e1 : e2)\<Colon>T" .
then obtain T1 T2 statT where
wt_e0: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e0\<Colon>-PrimT Boolean" and
wt_e1: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e1\<Colon>-T1" and
wt_e2: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e2\<Colon>-T2" and
statT: "G\<turnstile>T1\<preceq>T2 \<and> statT = T2 \<or> G\<turnstile>T2\<preceq>T1 \<and> statT = T1" and
T : "T=Inl statT"
by (rule wt_elim_cases) auto
have eval_e0: "G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<rightarrow> s1" .
from conf_s0 wt_e0
obtain n1 where
"G\<turnstile>Norm s0 \<midarrow>e0-\<succ>b\<midarrow>n1\<rightarrow> s1"
by (rules dest: hyp_e0)
moreover
from eval_e0 conf_s0 wf wt_e0
have "s1\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
with wt_e1 wt_e2 statT hyp_if obtain n2 where
"G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n2\<rightarrow> s2"
by (cases "the_Bool b") force+
ultimately
have "G\<turnstile>Norm s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<midarrow>max n1 n2\<rightarrow> s2"
by (blast intro: evaln.Cond dest: evaln_max2)
then show ?case ..
next
case (Call invDeclC a' accC' args e mn mode pTs' s0 s1 s2 s3 s3' s4 statT
v vs L accC T)
(* Repeats large parts of the type soundness proof. One should factor
out some lemmata about the relations and conformance of s2, s3 and s3'*)
have eval_e: "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<rightarrow> s1" .
have eval_args: "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<rightarrow> s2" .
have invDeclC: "invDeclC
= invocation_declclass G mode (store s2) a' statT
\<lparr>name = mn, parTs = pTs'\<rparr>" .
have
init_lvars: "s3 =
init_lvars G invDeclC \<lparr>name = mn, parTs = pTs'\<rparr> mode a' vs s2" .
have
check: "s3' =
check_method_access G accC' statT mode \<lparr>name = mn, parTs = pTs'\<rparr> a' s3" .
have eval_methd:
"G\<turnstile>s3' \<midarrow>Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>-\<succ>v\<rightarrow> s4" .
have hyp_e: "PROP ?EqEval (Norm s0) s1 (In1l e) (In1 a')" .
have hyp_args: "PROP ?EqEval s1 s2 (In3 args) (In3 vs)" .
have hyp_methd: "PROP ?EqEval s3' s4
(In1l (Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>)) (In1 v)".
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>
\<turnstile>In1l ({accC',statT,mode}e\<cdot>mn( {pTs'}args))\<Colon>T" .
from wt obtain pTs statDeclT statM where
wt_e: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT" and
wt_args: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>args\<Colon>\<doteq>pTs" and
statM: "max_spec G accC statT \<lparr>name=mn,parTs=pTs\<rparr>
= {((statDeclT,statM),pTs')}" and
mode: "mode = invmode statM e" and
T: "T =Inl (resTy statM)" and
eq_accC_accC': "accC=accC'"
by (rule wt_elim_cases) auto
from conf_s0 wt_e
obtain n1 where
evaln_e: "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<midarrow>n1\<rightarrow> s1"
by (rules dest: hyp_e)
from wf eval_e conf_s0 wt_e
obtain conf_s1: "s1\<Colon>\<preceq>(G, L)" and
conf_a': "normal s1 \<Longrightarrow> G, store s1\<turnstile>a'\<Colon>\<preceq>RefT statT"
by (auto dest!: eval_type_sound)
from conf_s1 wt_args
obtain n2 where
evaln_args: "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<midarrow>n2\<rightarrow> s2"
by (blast dest: hyp_args)
from wt_args conf_s1 eval_args wf
obtain conf_s2: "s2\<Colon>\<preceq>(G, L)" and
conf_args: "normal s2
\<Longrightarrow> list_all2 (conf G (store s2)) vs pTs"
by (auto dest!: eval_type_sound)
from statM
obtain
statM': "(statDeclT,statM)\<in>mheads G accC statT \<lparr>name=mn,parTs=pTs'\<rparr>" and
pTs_widen: "G\<turnstile>pTs[\<preceq>]pTs'"
by (blast dest: max_spec2mheads)
from check
have eq_store_s3'_s3: "store s3'=store s3"
by (cases s3) (simp add: check_method_access_def Let_def)
obtain invC
where invC: "invC = invocation_class mode (store s2) a' statT"
by simp
with init_lvars
have invC': "invC = (invocation_class mode (store s3) a' statT)"
by (cases s2,cases mode) (auto simp add: init_lvars_def2 )
obtain n3 where
"G\<turnstile>Norm s0 \<midarrow>{accC',statT,mode}e\<cdot>mn( {pTs'}args)-\<succ>v\<midarrow>n3\<rightarrow>
(set_lvars (locals (store s2))) s4"
proof (cases "normal s2")
case False
with init_lvars
obtain keep_abrupt: "abrupt s3 = abrupt s2" and
"store s3 = store (init_lvars G invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>
mode a' vs s2)"
by (auto simp add: init_lvars_def2)
moreover
from keep_abrupt False check
have eq_s3'_s3: "s3'=s3"
by (auto simp add: check_method_access_def Let_def)
moreover
from eq_s3'_s3 False keep_abrupt eval_methd init_lvars
obtain "s4=s3'"
"In1 v=arbitrary3 (In1l (Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>))"
by auto
moreover note False evaln.Abrupt
ultimately obtain m where
"G\<turnstile>s3' \<midarrow>Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>-\<succ>v\<midarrow>m\<rightarrow> s4"
by force
from evaln_e evaln_args invDeclC init_lvars eq_s3'_s3 this
have
"G\<turnstile>Norm s0 \<midarrow>{accC',statT,mode}e\<cdot>mn( {pTs'}args)-\<succ>v\<midarrow>max n1 (max n2 m)\<rightarrow>
(set_lvars (locals (store s2))) s4"
by (auto intro!: evaln.Call le_maxI1 le_max3I1 le_max3I2)
with that show ?thesis
by rules
next
case True
note normal_s2 = True
with eval_args
have normal_s1: "normal s1"
by (cases "normal s1") auto
with conf_a' eval_args
have conf_a'_s2: "G, store s2\<turnstile>a'\<Colon>\<preceq>RefT statT"
by (auto dest: eval_gext intro: conf_gext)
show ?thesis
proof (cases "a'=Null \<longrightarrow> is_static statM")
case False
then obtain not_static: "\<not> is_static statM" and Null: "a'=Null"
by blast
with normal_s2 init_lvars mode
obtain np: "abrupt s3 = Some (Xcpt (Std NullPointer))" and
"store s3 = store (init_lvars G invDeclC
\<lparr>name = mn, parTs = pTs'\<rparr> mode a' vs s2)"
by (auto simp add: init_lvars_def2)
moreover
from np check
have eq_s3'_s3: "s3'=s3"
by (auto simp add: check_method_access_def Let_def)
moreover
from eq_s3'_s3 np eval_methd init_lvars
obtain "s4=s3'"
"In1 v=arbitrary3 (In1l (Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>))"
by auto
moreover note np
ultimately obtain m where
"G\<turnstile>s3' \<midarrow>Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>-\<succ>v\<midarrow>m\<rightarrow> s4"
by force
from evaln_e evaln_args invDeclC init_lvars eq_s3'_s3 this
have
"G\<turnstile>Norm s0 \<midarrow>{accC',statT,mode}e\<cdot>mn( {pTs'}args)-\<succ>v\<midarrow>max n1 (max n2 m)\<rightarrow>
(set_lvars (locals (store s2))) s4"
by (auto intro!: evaln.Call le_maxI1 le_max3I1 le_max3I2)
with that show ?thesis
by rules
next
case True
with mode have notNull: "mode = IntVir \<longrightarrow> a' \<noteq> Null"
by (auto dest!: Null_staticD)
with conf_s2 conf_a'_s2 wf invC
have dynT_prop: "G\<turnstile>mode\<rightarrow>invC\<preceq>statT"
by (cases s2) (auto intro: DynT_propI)
with wt_e statM' invC mode wf
obtain dynM where
dynM: "dynlookup G statT invC \<lparr>name=mn,parTs=pTs'\<rparr> = Some dynM" and
acc_dynM: "G \<turnstile>Methd \<lparr>name=mn,parTs=pTs'\<rparr> dynM
in invC dyn_accessible_from accC"
by (force dest!: call_access_ok)
with invC' check eq_accC_accC'
have eq_s3'_s3: "s3'=s3"
by (auto simp add: check_method_access_def Let_def)
from dynT_prop wf wt_e statM' mode invC invDeclC dynM
obtain
wf_dynM: "wf_mdecl G invDeclC (\<lparr>name=mn,parTs=pTs'\<rparr>,mthd dynM)" and
dynM': "methd G invDeclC \<lparr>name=mn,parTs=pTs'\<rparr> = Some dynM" and
iscls_invDeclC: "is_class G invDeclC" and
invDeclC': "invDeclC = declclass dynM" and
invC_widen: "G\<turnstile>invC\<preceq>\<^sub>C invDeclC" and
is_static_eq: "is_static dynM = is_static statM" and
involved_classes_prop:
"(if invmode statM e = IntVir
then \<forall>statC. statT = ClassT statC \<longrightarrow> G\<turnstile>invC\<preceq>\<^sub>C statC
else ((\<exists>statC. statT = ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C invDeclC) \<or>
(\<forall>statC. statT \<noteq> ClassT statC \<and> invDeclC = Object)) \<and>
statDeclT = ClassT invDeclC)"
by (auto dest: DynT_mheadsD)
obtain L' where
L':"L'=(\<lambda> k.
(case k of
EName e
\<Rightarrow> (case e of
VNam v
\<Rightarrow>(table_of (lcls (mbody (mthd dynM)))
(pars (mthd dynM)[\<mapsto>]pTs')) v
| Res \<Rightarrow> Some (resTy dynM))
| This \<Rightarrow> if is_static statM
then None else Some (Class invDeclC)))"
by simp
from wf_dynM [THEN wf_mdeclD1, THEN conjunct1] normal_s2 conf_s2 wt_e
wf eval_args conf_a' mode notNull wf_dynM involved_classes_prop
have conf_s3: "s3\<Colon>\<preceq>(G,L')"
apply -
(*FIXME confomrs_init_lvars should be
adjusted to be more directy applicable *)
apply (drule conforms_init_lvars [of G invDeclC
"\<lparr>name=mn,parTs=pTs'\<rparr>" dynM "store s2" vs pTs "abrupt s2"
L statT invC a' "(statDeclT,statM)" e])
apply (rule wf)
apply (rule conf_args,assumption)
apply (simp add: pTs_widen)
apply (cases s2,simp)
apply (rule dynM')
apply (force dest: ty_expr_is_type)
apply (rule invC_widen)
apply (force intro: conf_gext dest: eval_gext)
apply simp
apply simp
apply (simp add: invC)
apply (simp add: invDeclC)
apply (force dest: wf_mdeclD1 is_acc_typeD)
apply (cases s2, simp add: L' init_lvars
cong add: lname.case_cong ename.case_cong)
done
with is_static_eq wf_dynM L'
obtain mthdT where
"\<lparr>prg=G,cls=invDeclC,lcl=L'\<rparr>
\<turnstile>Body invDeclC (stmt (mbody (mthd dynM)))\<Colon>-mthdT"
by - (drule wf_mdecl_bodyD,
auto simp: cong add: lname.case_cong ename.case_cong)
with dynM' iscls_invDeclC invDeclC'
have
"\<lparr>prg=G,cls=invDeclC,lcl=L'\<rparr>
\<turnstile>(Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>)\<Colon>-mthdT"
by (auto intro: wt.Methd)
with conf_s3 eq_s3'_s3 hyp_methd
obtain m where
"G\<turnstile>s3' \<midarrow>Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>-\<succ>v\<midarrow>m\<rightarrow> s4"
by (blast)
from evaln_e evaln_args invDeclC init_lvars eq_s3'_s3 this
have
"G\<turnstile>Norm s0 \<midarrow>{accC',statT,mode}e\<cdot>mn( {pTs'}args)-\<succ>v\<midarrow>max n1 (max n2 m)\<rightarrow>
(set_lvars (locals (store s2))) s4"
by (auto intro!: evaln.Call le_maxI1 le_max3I1 le_max3I2)
with that show ?thesis
by rules
qed
qed
then show ?case ..
next
case (Methd D s0 s1 sig v L accC T)
then obtain n where
"G\<turnstile>Norm s0 \<midarrow>body G D sig-\<succ>v\<midarrow>n\<rightarrow> s1"
by - (erule wt_elim_cases, force simp add: body_def2)
then have "G\<turnstile>Norm s0 \<midarrow>Methd D sig-\<succ>v\<midarrow>Suc n\<rightarrow> s1"
by (rule evaln.Methd)
then show ?case ..
next
case (Body D c s0 s1 s2 L accC T)
with wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>Init D\<midarrow>n1\<rightarrow> s1"
"G\<turnstile>s1 \<midarrow>c\<midarrow>n2\<rightarrow> s2"
by (blast elim!: wt_elim_cases dest: eval_type_sound)
then have
"G\<turnstile>Norm s0 \<midarrow>Body D c-\<succ>the (locals (store s2) Result)\<midarrow>max n1 n2
\<rightarrow> abupd (absorb Ret) s2"
by (blast intro: evaln.Body dest: evaln_max2)
then show ?case ..
next
case (LVar s vn L accC T)
obtain n where
"G\<turnstile>Norm s \<midarrow>LVar vn=\<succ>lvar vn s\<midarrow>n\<rightarrow> Norm s"
by (rules intro: evaln.LVar)
then show ?case ..
next
case (FVar a accC e fn s0 s1 s2 s2' s3 stat statDeclC v L accC' T)
have eval_init: "G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<rightarrow> s1" .
have eval_e: "G\<turnstile>s1 \<midarrow>e-\<succ>a\<rightarrow> s2" .
have check: "s3 = check_field_access G accC statDeclC fn stat a s2'" .
have hyp_init: "PROP ?EqEval (Norm s0) s1 (In1r (Init statDeclC)) \<diamondsuit>" .
have hyp_e: "PROP ?EqEval s1 s2 (In1l e) (In1 a)" .
have fvar: "(v, s2') = fvar statDeclC stat fn a s2" .
have conf_s0: "Norm s0\<Colon>\<preceq>(G, L)" .
have wt: "\<lparr>prg=G, cls=accC', lcl=L\<rparr>\<turnstile>In2 ({accC,statDeclC,stat}e..fn)\<Colon>T" .
then obtain statC f where
wt_e: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>e\<Colon>-Class statC" and
accfield: "accfield G accC statC fn = Some (statDeclC,f)" and
stat: "stat=is_static f" and
accC': "accC'=accC" and
T: "T=(Inl (type f))"
by (rule wt_elim_cases) (auto simp add: member_is_static_simp)
from wf wt_e
have iscls_statC: "is_class G statC"
by (auto dest: ty_expr_is_type type_is_class)
with wf accfield
have iscls_statDeclC: "is_class G statDeclC"
by (auto dest!: accfield_fields dest: fields_declC)
then
have wt_init: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>(Init statDeclC)\<Colon>\<surd>"
by simp
from conf_s0 wt_init
obtain n1 where
evaln_init: "G\<turnstile>Norm s0 \<midarrow>Init statDeclC\<midarrow>n1\<rightarrow> s1"
by (rules dest: hyp_init)
from eval_init wt_init conf_s0 wf
have conf_s1: "s1\<Colon>\<preceq>(G, L)"
by (blast dest: eval_type_sound)
with wt_e
obtain n2 where
evaln_e: "G\<turnstile>s1 \<midarrow>e-\<succ>a\<midarrow>n2\<rightarrow> s2"
by (blast dest: hyp_e)
from eval_e wf conf_s1 wt_e
obtain conf_s2: "s2\<Colon>\<preceq>(G, L)" and
conf_a: "normal s2 \<longrightarrow> G,store s2\<turnstile>a\<Colon>\<preceq>Class statC"
by (auto dest!: eval_type_sound)
from accfield wt_e eval_init eval_e conf_s2 conf_a fvar stat check wf
have eq_s3_s2': "s3=s2'"
by (auto dest!: error_free_field_access)
with evaln_init evaln_e fvar accC'
have "G\<turnstile>Norm s0 \<midarrow>{accC,statDeclC,stat}e..fn=\<succ>v\<midarrow>max n1 n2\<rightarrow> s3"
by (auto intro: evaln.FVar dest: evaln_max2)
then show ?case ..
next
case (AVar a e1 e2 i s0 s1 s2 s2' v L accC T)
with wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>e1-\<succ>a\<midarrow>n1\<rightarrow> s1"
"G\<turnstile>s1 \<midarrow>e2-\<succ>i\<midarrow>n2\<rightarrow> s2"
by (blast elim!: wt_elim_cases dest: eval_type_sound)
moreover
have "(v, s2') = avar G i a s2" .
ultimately
have "G\<turnstile>Norm s0 \<midarrow>e1.[e2]=\<succ>v\<midarrow>max n1 n2\<rightarrow> s2'"
by (blast intro!: evaln.AVar dest: evaln_max2)
then show ?case ..
next
case (Nil s0 L accC T)
show ?case by (rules intro: evaln.Nil)
next
case (Cons e es s0 s1 s2 v vs L accC T)
with wf obtain n1 n2 where
"G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n1\<rightarrow> s1"
"G\<turnstile>s1 \<midarrow>es\<doteq>\<succ>vs\<midarrow>n2\<rightarrow> s2"
by (blast elim!: wt_elim_cases dest: eval_type_sound)
then
have "G\<turnstile>Norm s0 \<midarrow>e # es\<doteq>\<succ>v # vs\<midarrow>max n1 n2\<rightarrow> s2"
by (blast intro!: evaln.Cons dest: evaln_max2)
then show ?case ..
qed
qed
end