# File fixrec_package.ML

    (*  Title:      HOLCF/fixrec_package.ML
        ID:         $Id: fixrec_package.ML,v 1.10 2005/08/16 11:42:27 wenzelm Exp $
        Author:     Amber Telfer and Brian Huffman

    Recursive function definition package for HOLCF.
    *)

    signature FIXREC_PACKAGE =
    sig
      val add_fixrec: bool -> ((string * Attrib.src list) * string) list list
        -> theory -> theory
      val add_fixrec_i: bool -> ((string * theory attribute list) * term) list list
        -> theory -> theory
      val add_fixpat: (string * Attrib.src list) * string list
        -> theory -> theory
      val add_fixpat_i: (string * theory attribute list) * term list
        -> theory -> theory
    end;

    structure FixrecPackage: FIXREC_PACKAGE =
    struct

    fun fixrec_err s = error ("fixrec definition error:\n" ^ s);
    fun fixrec_eq_err thy s eq =
      fixrec_err (s ^ "\nin\n" ^ quote (Sign.string_of_term thy eq));

    (* ->> is taken from holcf_logic.ML *)
    (* TODO: fix dependencies so we can import HOLCFLogic here *)
    infixr 6 ->>;
    fun S ->> T = Type ("Cfun.->",[S,T]);

    (* extern_name is taken from domain/library.ML *)
    fun extern_name con = case Symbol.explode con of
    		   ("o"::"p"::" "::rest) => implode rest
    		   | _ => con;

    val mk_trp = HOLogic.mk_Trueprop;

    (* splits a cterm into the right and lefthand sides of equality *)
    fun dest_eqs t = HOLogic.dest_eq (HOLogic.dest_Trueprop t);
    (*
    fun dest_eqs (Const ("==", _)$lhs$rhs) = (lhs, rhs)
      | dest_eqs (Const ("Trueprop", _)$(Const ("op =", _)$lhs$rhs)) = (lhs,rhs)
      | dest_eqs t = fixrec_err (Sign.string_of_term (sign_of (the_context())) t);
    *)
    (* similar to Thm.head_of, but for continuous application *)
    fun chead_of (Const("Cfun.Rep_CFun",_)$f$t) = chead_of f
      | chead_of u = u;

    (* these are helpful functions copied from HOLCF/domain/library.ML *)
    fun %: s = Free(s,dummyT);
    fun %%: s = Const(s,dummyT);
    infix 0 ==;  fun S ==  T = %%:"==" $ S $ T;
    infix 1 ===; fun S === T = %%:"op =" $ S $ T;
    infix 9 `  ; fun f ` x = %%:"Cfun.Rep_CFun" $ f $ x;

    (* infers the type of a term *)
    (* similar to Theory.inferT_axm, but allows any type, not just propT *)
    fun infer sg t =
      fst (Sign.infer_types (Sign.pp sg) sg (K NONE) (K NONE) [] true ([t],dummyT));

    (* Similar to Term.lambda, but also allows abstraction over constants *)
    fun lambda' (v as Free (x, T)) t = Abs (x, T, abstract_over (v, t))
      | lambda' (v as Var ((x, _), T)) t = Abs (x, T, abstract_over (v, t))
      | lambda' (v as Const (x, T)) t = Abs (Sign.base_name x, T, abstract_over (v, t))
      | lambda' v t = raise TERM ("lambda'", [v, t]);

    (* builds the expression (LAM v. rhs) *)
    fun big_lambda v rhs = %%:"Cfun.Abs_CFun"$(lambda' v rhs);

    (* builds the expression (LAM v1 v2 .. vn. rhs) *)
    fun big_lambdas [] rhs = rhs
      | big_lambdas (v::vs) rhs = big_lambda v (big_lambdas vs rhs);

    (* builds the expression (LAM <v1,v2,..,vn>. rhs) *)
    fun lambda_ctuple [] rhs = big_lambda (%:"unit") rhs
      | lambda_ctuple (v::[]) rhs = big_lambda v rhs
      | lambda_ctuple (v::vs) rhs =
          %%:"Cprod.csplit"`(big_lambda v (lambda_ctuple vs rhs));

    (* builds the expression <v1,v2,..,vn> *)
    fun mk_ctuple [] = %%:"UU"
    |   mk_ctuple (t::[]) = t
    |   mk_ctuple (t::ts) = %%:"Cprod.cpair"`t`(mk_ctuple ts);

    (*************************************************************************)
    (************* fixed-point definitions and unfolding theorems ************)
    (*************************************************************************)

    fun add_fixdefs eqs thy =
      let
        val (lhss,rhss) = ListPair.unzip (map dest_eqs eqs);
        val fixpoint = %%:"Fix.fix"`lambda_ctuple lhss (mk_ctuple rhss);

        fun one_def (l as Const(n,T)) r =
              let val b = Sign.base_name n in (b, (b^"_def", l == r)) end
          | one_def _ _ = fixrec_err "fixdefs: lhs not of correct form";
        fun defs [] _ = []
          | defs (l::[]) r = [one_def l r]
          | defs (l::ls) r = one_def l (%%:"Cprod.cfst"`r) :: defs ls (%%:"Cprod.csnd"`r);
        val (names, pre_fixdefs) = ListPair.unzip (defs lhss fixpoint);

        val fixdefs = map (inferT_axm thy) pre_fixdefs;
        val (thy', fixdef_thms) =
          PureThy.add_defs_i false (map Thm.no_attributes fixdefs) thy;
        val ctuple_fixdef_thm = foldr1 (fn (x,y) => cpair_equalI OF [x,y]) fixdef_thms;

        fun mk_cterm t = cterm_of thy' (infer thy' t);
        val ctuple_unfold_ct = mk_cterm (mk_trp (mk_ctuple lhss === mk_ctuple rhss));
        val ctuple_unfold_thm = prove_goalw_cterm [] ctuple_unfold_ct
              (fn _ => [rtac (ctuple_fixdef_thm RS fix_eq2 RS trans) 1,
                        simp_tac (simpset_of thy') 1]);
        val ctuple_induct_thm =
              (space_implode "_" names ^ "_induct", ctuple_fixdef_thm RS def_fix_ind);

        fun unfolds [] thm = []
          | unfolds (n::[]) thm = [(n^"_unfold", thm)]
          | unfolds (n::ns) thm = let
              val thmL = thm RS cpair_eqD1;
              val thmR = thm RS cpair_eqD2;
            in (n^"_unfold", thmL) :: unfolds ns thmR end;
        val unfold_thms = unfolds names ctuple_unfold_thm;
        val thms = ctuple_induct_thm :: unfold_thms;
        val (thy'', _) = PureThy.add_thms (map Thm.no_attributes thms) thy';
      in
        (thy'', names, fixdef_thms, map snd unfold_thms)
      end;

    (*************************************************************************)
    (*********** monadic notation and pattern matching compilation ***********)
    (*************************************************************************)

    fun add_names (Const(a,_), bs) = Sign.base_name a ins_string bs
      | add_names (Free(a,_) , bs) = a ins_string bs
      | add_names (f $ u     , bs) = add_names (f, add_names(u, bs))
      | add_names (Abs(a,_,t), bs) = add_names (t, a ins_string bs)
      | add_names (_         , bs) = bs;

    fun add_terms ts xs = foldr add_names xs ts;

    (* builds a monadic term for matching a constructor pattern *)
    fun pre_build pat rhs vs taken =
      case pat of
        Const("Cfun.Rep_CFun",_)$f$(v as Free(n,T)) =>
          pre_build f rhs (v::vs) taken
      | Const("Cfun.Rep_CFun",_)$f$x =>
          let val (rhs', v, taken') = pre_build x rhs [] taken;
          in pre_build f rhs' (v::vs) taken' end
      | Const(c,T) =>
          let
            val n = variant taken "v";
            fun result_type (Type("Cfun.->",[_,T])) (x::xs) = result_type T xs
              | result_type T _ = T;
            val v = Free(n, result_type T vs);
            val m = "match_"^(extern_name(Sign.base_name c));
            val k = lambda_ctuple vs rhs;
          in
            (%%:"Fixrec.bind"`(%%:m`v)`k, v, n::taken)
          end
      | Free(n,_) => fixrec_err ("expected constructor, found free variable " ^ quote n)
      | _ => fixrec_err "pre_build: invalid pattern";

    (* builds a monadic term for matching a function definition pattern *)
    (* returns (name, arity, matcher) *)
    fun building pat rhs vs taken =
      case pat of
        Const("Cfun.Rep_CFun", _)$f$(v as Free(n,T)) =>
          building f rhs (v::vs) taken
      | Const("Cfun.Rep_CFun", _)$f$x =>
          let val (rhs', v, taken') = pre_build x rhs [] taken;
          in building f rhs' (v::vs) taken' end
      | Const(name,_) => (name, length vs, big_lambdas vs rhs)
      | _ => fixrec_err "function is not declared as constant in theory";

    fun match_eq eq =
      let val (lhs,rhs) = dest_eqs eq;
      in building lhs (%%:"Fixrec.return"`rhs) [] (add_terms [eq] []) end;

    (* returns the sum (using +++) of the terms in ms *)
    (* also applies "run" to the result! *)
    fun fatbar arity ms =
      let
        fun unLAM 0 t = t
          | unLAM n (_$Abs(_,_,t)) = unLAM (n-1) t
          | unLAM _ _ = fixrec_err "fatbar: internal error, not enough LAMs";
        fun reLAM 0 t = t
          | reLAM n t = reLAM (n-1) (%%:"Cfun.Abs_CFun" $ Abs("",dummyT,t));
        fun mplus (x,y) = %%:"Fixrec.mplus"`x`y;
        val msum = foldr1 mplus (map (unLAM arity) ms);
      in
        reLAM arity (%%:"Fixrec.run"`msum)
      end;

    fun unzip3 [] = ([],[],[])
      | unzip3 ((x,y,z)::ts) =
          let val (xs,ys,zs) = unzip3 ts
          in (x::xs, y::ys, z::zs) end;

    (* this is the pattern-matching compiler function *)
    fun compile_pats eqs =
      let
        val ((n::names),(a::arities),mats) = unzip3 (map match_eq eqs);
        val cname = if forall (fn x => n=x) names then n
              else fixrec_err "all equations in block must define the same function";
        val arity = if forall (fn x => a=x) arities then a
              else fixrec_err "all equations in block must have the same arity";
        val rhs = fatbar arity mats;
      in
        mk_trp (%%:cname === rhs)
      end;

    (*************************************************************************)
    (********************** Proving associated theorems **********************)
    (*************************************************************************)

    (* proves a block of pattern matching equations as theorems, using unfold *)
    fun make_simps thy (unfold_thm, eqns) =
      let
        fun tacsf prems =
          [rtac (unfold_thm RS ssubst_lhs) 1, simp_tac (simpset_of thy addsimps prems) 1];
        fun prove_term t = prove_goalw_cterm [] (cterm_of thy t) tacsf;
        fun prove_eqn ((name, eqn_t), atts) = ((name, prove_term eqn_t), atts);
      in
        map prove_eqn eqns
      end;

    (*************************************************************************)
    (************************* Main fixrec function **************************)
    (*************************************************************************)

    fun gen_add_fixrec prep_prop prep_attrib strict blocks thy =
      let
        val eqns = List.concat blocks;
        val lengths = map length blocks;

        val ((names, srcss), strings) = apfst split_list (split_list eqns);
        val atts = map (map (prep_attrib thy)) srcss;
        val eqn_ts = map (prep_prop thy) strings;
        val rec_ts = map (fn eq => chead_of (fst (dest_eqs (Logic.strip_imp_concl eq)))
          handle TERM _ => fixrec_eq_err thy "not a proper equation" eq) eqn_ts;
        val (_, eqn_ts') = InductivePackage.unify_consts thy rec_ts eqn_ts;

        fun unconcat [] _ = []
          | unconcat (n::ns) xs = List.take (xs,n) :: unconcat ns (List.drop (xs,n));
        val pattern_blocks = unconcat lengths (map Logic.strip_imp_concl eqn_ts');
        val compiled_ts = map (infer thy o compile_pats) pattern_blocks;
        val (thy', cnames, fixdef_thms, unfold_thms) = add_fixdefs compiled_ts thy;
      in
        if strict then let (* only prove simp rules if strict = true *)
          val eqn_blocks = unconcat lengths ((names ~~ eqn_ts') ~~ atts);
          val simps = List.concat (map (make_simps thy') (unfold_thms ~~ eqn_blocks));
          val (thy'', simp_thms) = PureThy.add_thms simps thy';

          val simp_names = map (fn name => name^"_simps") cnames;
          val simp_attribute = rpair [Simplifier.simp_add_global];
          val simps' = map simp_attribute (simp_names ~~ unconcat lengths simp_thms);
        in
          (#1 o PureThy.add_thmss simps') thy''
        end
        else thy'
      end;

    val add_fixrec = gen_add_fixrec Sign.read_prop Attrib.global_attribute;
    val add_fixrec_i = gen_add_fixrec Sign.cert_prop (K I);

    (*************************************************************************)
    (******************************** Fixpat *********************************)
    (*************************************************************************)

    fun fix_pat thy t =
      let
        val T = fastype_of t;
        val eq = mk_trp (HOLogic.eq_const T $ t $ Var (("x",0),T));
        val cname = case chead_of t of Const(c,_) => c | _ =>
                  fixrec_err "function is not declared as constant in theory";
        val unfold_thm = PureThy.get_thm thy (Name (cname^"_unfold"));
        val simp = prove_goalw_cterm [] (cterm_of thy eq)
              (fn _ => [stac unfold_thm 1, simp_tac (simpset_of thy) 1]);
      in simp end;

    fun gen_add_fixpat prep_term prep_attrib ((name, srcs), strings) thy =
      let
        val atts = map (prep_attrib thy) srcs;
        val ts = map (prep_term thy) strings;
        val simps = map (fix_pat thy) ts;
      in
        (#1 o PureThy.add_thmss [((name, simps), atts)]) thy
      end;

    val add_fixpat = gen_add_fixpat Sign.read_term Attrib.global_attribute;
    val add_fixpat_i = gen_add_fixpat Sign.cert_term (K I);

    (*************************************************************************)
    (******************************** Parsers ********************************)
    (*************************************************************************)

    local structure P = OuterParse and K = OuterKeyword in

    val fixrec_eqn = P.opt_thm_name ":" -- P.prop;

    val fixrec_strict = P.opt_keyword "permissive" >> not;

    val fixrec_decl = fixrec_strict -- P.and_list1 (Scan.repeat1 fixrec_eqn);

    (* this builds a parser for a new keyword, fixrec, whose functionality
    is defined by add_fixrec *)
    val fixrecP =
      OuterSyntax.command "fixrec" "define recursive functions (HOLCF)" K.thy_decl
        (fixrec_decl >> (Toplevel.theory o uncurry add_fixrec));

    (* fixpat parser *)
    val fixpat_decl = P.opt_thm_name ":" -- Scan.repeat1 P.prop;

    val fixpatP =
      OuterSyntax.command "fixpat" "define rewrites for fixrec functions" K.thy_decl
        (fixpat_decl >> (Toplevel.theory o add_fixpat));

    val _ = OuterSyntax.add_parsers [fixrecP, fixpatP];

    end; (* local structure *)

    end; (* struct *)