# Theory TranslCompTp

theory TranslCompTp
imports Index JVMType
```(*  Title:      HOL/MicroJava/Comp/TranslCompTp.thy
Author:     Martin Strecker
*)

theory TranslCompTp
imports Index "../BV/JVMType"
begin

(**********************************************************************)

definition comb :: "['a ⇒ 'b list × 'c, 'c ⇒ 'b list × 'd, 'a] ⇒ 'b list × 'd"
(infixr "□" 55)
where
"comb == (λ f1 f2 x0. let (xs1, x1) = f1 x0;
(xs2, x2) = f2 x1
in  (xs1 @ xs2, x2))"

definition comb_nil :: "'a ⇒ 'b list × 'a" where
"comb_nil a == ([], a)"

lemma comb_nil_left [simp]: "comb_nil □ f = f"
by (simp add: comb_def comb_nil_def split_beta)

lemma comb_nil_right [simp]: "f □ comb_nil = f"
by (simp add: comb_def comb_nil_def split_beta)

lemma comb_assoc [simp]: "(fa □ fb) □ fc = fa □ (fb □ fc)"

lemma comb_inv:
"(xs', x') = (f1 □ f2) x0 ⟹
∃xs1 x1 xs2 x2. (xs1, x1) = (f1 x0) ∧ (xs2, x2) = f2 x1 ∧ xs'= xs1 @ xs2 ∧ x'=x2"
by (case_tac "f1 x0") (simp add: comb_def split_beta)

(**********************************************************************)

abbreviation (input)
mt_of :: "method_type × state_type ⇒ method_type"
where "mt_of == fst"

abbreviation (input)
sttp_of :: "method_type × state_type ⇒ state_type"
where "sttp_of == snd"

definition nochangeST :: "state_type ⇒ method_type × state_type" where
"nochangeST sttp == ([Some sttp], sttp)"

definition pushST :: "[ty list, state_type] ⇒ method_type × state_type" where
"pushST tps == (λ (ST, LT). ([Some (ST, LT)], (tps @ ST, LT)))"

definition dupST :: "state_type ⇒ method_type × state_type" where
"dupST == (λ (ST, LT). ([Some (ST, LT)], (hd ST # ST, LT)))"

definition dup_x1ST :: "state_type ⇒ method_type × state_type" where
"dup_x1ST == (λ (ST, LT). ([Some (ST, LT)],
(hd ST # hd (tl ST) # hd ST # (tl (tl ST)), LT)))"

definition popST :: "[nat, state_type] ⇒ method_type × state_type" where
"popST n == (λ (ST, LT). ([Some (ST, LT)], (drop n ST, LT)))"

definition replST :: "[nat, ty, state_type] ⇒ method_type × state_type" where
"replST n tp == (λ (ST, LT). ([Some (ST, LT)], (tp # (drop n ST), LT)))"

definition storeST :: "[nat, ty, state_type] ⇒ method_type × state_type" where
"storeST i tp == (λ (ST, LT). ([Some (ST, LT)], (tl ST, LT [i:= OK tp])))"

(* Expressions *)

primrec compTpExpr :: "java_mb ⇒ java_mb prog ⇒ expr ⇒
state_type ⇒ method_type × state_type"
and compTpExprs :: "java_mb ⇒ java_mb prog ⇒ expr list ⇒
state_type ⇒ method_type × state_type"
where
"compTpExpr jmb G (NewC c) = pushST [Class c]"
| "compTpExpr jmb G (Cast c e) = (compTpExpr jmb G e) □ (replST 1 (Class c))"
| "compTpExpr jmb G (Lit val) = pushST [the (typeof (λv. None) val)]"
| "compTpExpr jmb G (BinOp bo e1 e2) =
(compTpExpr jmb G e1) □ (compTpExpr jmb G e2) □
(case bo of
Eq => popST 2 □ pushST [PrimT Boolean] □ popST 1 □ pushST [PrimT Boolean]
| Add => replST 2 (PrimT Integer))"
| "compTpExpr jmb G (LAcc vn) = (λ (ST, LT).
pushST [ok_val (LT ! (index jmb vn))] (ST, LT))"
| "compTpExpr jmb G (vn::=e) =
(compTpExpr jmb G e) □ dupST □ (popST 1)"
| "compTpExpr jmb G ( {cn}e..fn ) =
(compTpExpr jmb G e) □ replST 1 (snd (the (field (G,cn) fn)))"
| "compTpExpr jmb G (FAss cn e1 fn e2 ) =
(compTpExpr jmb G e1) □ (compTpExpr jmb G e2) □ dup_x1ST □ (popST 2)"
| "compTpExpr jmb G ({C}a..mn({fpTs}ps)) =
(compTpExpr jmb G a) □ (compTpExprs jmb G ps) □
(replST ((length ps) + 1) (method_rT (the (method (G,C) (mn,fpTs)))))"
| "compTpExprs jmb G [] = comb_nil"
| "compTpExprs jmb G (e#es) = (compTpExpr jmb G e) □ (compTpExprs jmb G es)"

(* Statements *)
primrec compTpStmt :: "java_mb ⇒ java_mb prog ⇒ stmt ⇒
state_type ⇒ method_type × state_type"
where
"compTpStmt jmb G Skip = comb_nil"
| "compTpStmt jmb G (Expr e) =  (compTpExpr jmb G e) □ popST 1"
| "compTpStmt jmb G (c1;; c2) = (compTpStmt jmb G c1) □ (compTpStmt jmb G c2)"
| "compTpStmt jmb G (If(e) c1 Else c2) =
(pushST [PrimT Boolean]) □ (compTpExpr jmb G e) □ popST 2 □
(compTpStmt jmb G c1) □ nochangeST □ (compTpStmt jmb G c2)"
| "compTpStmt jmb G (While(e) c) =
(pushST [PrimT Boolean]) □ (compTpExpr jmb G e) □ popST 2 □
(compTpStmt jmb G c) □ nochangeST"

definition compTpInit :: "java_mb ⇒ (vname * ty)
⇒ state_type ⇒ method_type × state_type" where
"compTpInit jmb == (λ (vn,ty). (pushST [ty]) □  (storeST (index jmb vn) ty))"

primrec compTpInitLvars :: "[java_mb, (vname × ty) list] ⇒
state_type ⇒ method_type × state_type"
where
"compTpInitLvars jmb [] = comb_nil"
| "compTpInitLvars jmb (lv#lvars) = (compTpInit jmb lv) □ (compTpInitLvars jmb lvars)"

definition start_ST :: "opstack_type" where
"start_ST == []"

definition start_LT :: "cname ⇒ ty list ⇒ nat ⇒ locvars_type" where
"start_LT C pTs n ==  (OK (Class C))#((map OK pTs))@(replicate n Err)"

definition compTpMethod  :: "[java_mb prog, cname, java_mb mdecl] ⇒ method_type" where
"compTpMethod G C == λ ((mn,pTs),rT, jmb).
let (pns,lvars,blk,res) = jmb
in (mt_of
((compTpInitLvars jmb lvars □
compTpStmt jmb G blk □
compTpExpr jmb G res □
nochangeST)
(start_ST, start_LT C pTs (length lvars))))"

definition compTp :: "java_mb prog => prog_type" where
"compTp G C sig == let (D, rT, jmb) = (the (method (G, C) sig))
in compTpMethod G C (sig, rT, jmb)"

(**********************************************************************)
(* Computing the maximum stack size from the method_type *)

definition ssize_sto :: "(state_type option) ⇒ nat" where
"ssize_sto sto ==  case sto of None ⇒ 0 | (Some (ST, LT)) ⇒ length ST"

definition max_of_list :: "nat list ⇒ nat" where
"max_of_list xs == foldr max xs 0"

definition max_ssize :: "method_type ⇒ nat" where
"max_ssize mt == max_of_list (map ssize_sto mt)"

end
```