Theory TranslCompTp (Isabelle2005: October 2005)

# Theory TranslCompTp

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theory TranslCompTp
imports Index JVMType
begin

    (*  Title:      HOL/MicroJava/Comp/TranslCompTp.thy
        ID:         $Id: TranslCompTp.thy,v 1.3 2005/02/01 17:02:05 paulson Exp $
        Author:     Martin Strecker
    *)

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

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

    constdefs
      comb :: "['a => 'b list × 'c, 'c => 'b list × 'd, 'a] => 'b list × 'd"
      "comb == (λ f1 f2 x0. let (xs1, x1) = f1 x0;
                                (xs2, x2) = f2 x1
                            in  (xs1 @ xs2, x2))"
      comb_nil :: "'a => 'b list × 'a"
      "comb_nil a == ([], a)"

    syntax (xsymbols)
      "comb" :: "['a => 'b list × 'c, 'c => 'b list × 'd, 'a] => 'b list × 'd"
                (infixr "\<box>" 55)

    lemma comb_nil_left [simp]: "comb_nil \<box> f = f"
    by (simp add: comb_def comb_nil_def split_beta)

    lemma comb_nil_right [simp]: "f \<box> comb_nil = f"
    by (simp add: comb_def comb_nil_def split_beta)

    lemma comb_assoc [simp]: "(fa \<box> fb) \<box> fc = fa \<box> (fb \<box> fc)"
    by (simp add: comb_def split_beta)

    lemma comb_inv: "(xs', x') = (f1 \<box> f2) x0 ==>
      ∃ xs1 x1 xs2 x2. (xs1, x1) = (f1 x0) ∧ (xs2, x2) = f2 x1 ∧ xs'= xs1 @ xs2 ∧ x'=x2"
    apply (case_tac "f1 x0")
    apply (case_tac "f2 x1")
    apply (simp add: comb_def split_beta)
    done

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

    syntax
      mt_of         :: "method_type × state_type => method_type"
      sttp_of       :: "method_type × state_type => state_type"

    translations
      "mt_of"     => "fst"
      "sttp_of"   => "snd"

    consts
      compTpExpr  :: "java_mb => java_mb prog => expr
                       => state_type => method_type × state_type"

      compTpExprs :: "java_mb => java_mb prog => expr list
                       => state_type => method_type × state_type"

      compTpStmt  :: "java_mb => java_mb prog => stmt
                       => state_type => method_type × state_type"

    constdefs
      nochangeST :: "state_type => method_type × state_type"
      "nochangeST sttp == ([Some sttp], sttp)"
      pushST :: "[ty list, state_type] => method_type × state_type"
      "pushST tps == (λ (ST, LT). ([Some (ST, LT)], (tps @ ST, LT)))"
      dupST :: "state_type => method_type × state_type"
      "dupST == (λ (ST, LT). ([Some (ST, LT)], (hd ST # ST, LT)))"
      dup_x1ST :: "state_type => method_type × state_type"
      "dup_x1ST == (λ (ST, LT). ([Some (ST, LT)],
                                 (hd ST # hd (tl ST) # hd ST # (tl (tl ST)), LT)))"
      popST :: "[nat, state_type] => method_type × state_type"
      "popST n == (λ (ST, LT). ([Some (ST, LT)], (drop n ST, LT)))"
      replST :: "[nat, ty, state_type] => method_type × state_type"
      "replST n tp == (λ (ST, LT). ([Some (ST, LT)], (tp # (drop n ST), LT)))"

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

    (* Expressions *)

    primrec

      "compTpExpr jmb G (NewC c) = pushST [Class c]"

      "compTpExpr jmb G (Cast c e) =
      (compTpExpr jmb G e) \<box> (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) \<box> (compTpExpr jmb G e2) \<box>
         (case bo of
           Eq => popST 2 \<box> pushST [PrimT Boolean] \<box> popST 1 \<box> 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) \<box> dupST \<box> (popST 1)"

      "compTpExpr jmb G ( {cn}e..fn ) =
          (compTpExpr jmb G e) \<box> replST 1 (snd (the (field (G,cn) fn)))"

      "compTpExpr jmb G (FAss cn e1 fn e2 ) =
          (compTpExpr jmb G e1) \<box> (compTpExpr jmb G e2) \<box> dup_x1ST \<box> (popST 2)"

      "compTpExpr jmb G ({C}a..mn({fpTs}ps)) =
           (compTpExpr jmb G a) \<box> (compTpExprs jmb G ps) \<box>
           (replST ((length ps) + 1) (method_rT (the (method (G,C) (mn,fpTs)))))"

    (* Expression lists *)

      "compTpExprs jmb G [] = comb_nil"

      "compTpExprs jmb G (e#es) = (compTpExpr jmb G e) \<box> (compTpExprs jmb G es)"

    (* Statements *)
    primrec
       "compTpStmt jmb G Skip = comb_nil"

       "compTpStmt jmb G (Expr e) =  (compTpExpr jmb G e) \<box> popST 1"

       "compTpStmt jmb G (c1;; c2) = (compTpStmt jmb G c1) \<box> (compTpStmt jmb G c2)"

       "compTpStmt jmb G (If(e) c1 Else c2) =
          (pushST [PrimT Boolean]) \<box> (compTpExpr jmb G e) \<box> popST 2 \<box>
          (compTpStmt jmb G c1) \<box> nochangeST \<box> (compTpStmt jmb G c2)"

       "compTpStmt jmb G (While(e) c) =
          (pushST [PrimT Boolean]) \<box> (compTpExpr jmb G e) \<box> popST 2 \<box>
          (compTpStmt jmb G c) \<box> nochangeST"

    constdefs
      compTpInit  :: "java_mb => (vname * ty)
                       => state_type => method_type × state_type"
      "compTpInit jmb == (λ (vn,ty). (pushST [ty]) \<box>  (storeST (index jmb vn) ty))"

    consts
      compTpInitLvars :: "[java_mb, (vname × ty) list]
                       => state_type => method_type × state_type"

    primrec
      "compTpInitLvars jmb [] = comb_nil"
      "compTpInitLvars jmb (lv#lvars) = (compTpInit jmb lv) \<box> (compTpInitLvars jmb lvars)"

    constdefs
       start_ST :: "opstack_type"
      "start_ST == []"

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

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

      compTp :: "java_mb prog => prog_type"
      "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 *)

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

      max_of_list :: "nat list => nat"
      "max_of_list xs == foldr max xs 0"

      max_ssize :: "method_type => nat"
      "max_ssize mt == max_of_list (map ssize_sto mt)"

    end

lemma comb_nil_left:

      comb comb_nil f = f

lemma comb_nil_right:

      comb f comb_nil = f

lemma comb_assoc:

      comb (comb fa fb) fc = comb fa (comb fb fc)

lemma comb_inv:

      (xs', x') = comb f1.0 f2.0 x0.0
      ==> ∃xs1 x1 xs2 x2.
             (xs1, x1) = f1.0 x0.0 ∧ (xs2, x2) = f2.0 x1 ∧ xs' = xs1 @ xs2 ∧ x' = x2