(*  Title:      Pure/type_infer.ML

    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen

Simple type inference.

*)

signature TYPE_INFER =

sig

  val anyT: sort -> typ

  val polymorphicT: typ -> typ

  val constrain: typ -> term -> term

  val is_param: indexname -> bool

  val param: int -> string * sort -> typ

  val paramify_vars: typ -> typ

  val paramify_dummies: typ -> int -> typ * int

  val fixate_params: Name.context -> term list -> term list

  val appl_error: Pretty.pp -> string -> term -> typ -> term -> typ -> string list

  val infer_types: Pretty.pp -> Type.tsig -> (typ list -> typ list) ->

    (string -> typ option) -> (indexname -> typ option) -> Name.context -> int ->

    term list -> term list

end;

structure TypeInfer: TYPE_INFER =

struct

(** type parameters and constraints **)

fun anyT S = TFree ("'_dummy_", S);

(*indicate polymorphic Vars*)

fun polymorphicT T = Type ("_polymorphic_", [T]);

val constrain = Syntax.type_constraint;

(* user parameters *)

fun is_param (x, _: int) = String.isPrefix "?" x;

fun param i (x, S) = TVar (("?" ^ x, i), S);

val paramify_vars =

  Same.commit

    (Term_Subst.map_atypsT_same

      (fn TVar ((x, i), S) => (param i (x, S)) | _ => raise Same.SAME));

val paramify_dummies =

  let

    fun dummy S maxidx = (param (maxidx + 1) ("'dummy", S), maxidx + 1);

    fun paramify (TFree ("'_dummy_", S)) maxidx = dummy S maxidx

      | paramify (Type ("dummy", _)) maxidx = dummy [] maxidx

      | paramify (Type (a, Ts)) maxidx =

          let val (Ts', maxidx') = fold_map paramify Ts maxidx

          in (Type (a, Ts'), maxidx') end

      | paramify T maxidx = (T, maxidx);

  in paramify end;

fun fixate_params name_context ts =

  let

    fun subst_param (xi, S) (inst, used) =

      if is_param xi then

        let

          val [a] = Name.invents used Name.aT 1;

          val used' = Name.declare a used;

        in (((xi, S), TFree (a, S)) :: inst, used') end

      else (inst, used);

    val name_context' = (fold o fold_types) Term.declare_typ_names ts name_context;

    val (inst, _) = fold_rev subst_param (fold Term.add_tvars ts []) ([], name_context');

  in (map o map_types) (Term_Subst.instantiateT inst) ts end;

(** pretyps and preterms **)

(*parameters may get instantiated, anything else is rigid*)

datatype pretyp =

  PType of string * pretyp list |

  PTFree of string * sort |

  PTVar of indexname * sort |

  Param of int * sort;

datatype preterm =

  PConst of string * pretyp |

  PFree of string * pretyp |

  PVar of indexname * pretyp |

  PBound of int |

  PAbs of string * pretyp * preterm |

  PAppl of preterm * preterm |

  Constraint of preterm * pretyp;

(* utils *)

fun deref tye (T as Param (i, S)) =

      (case Inttab.lookup tye i of

        NONE => T

      | SOME U => deref tye U)

  | deref tye T = T;

fun fold_pretyps f (PConst (_, T)) x = f T x

  | fold_pretyps f (PFree (_, T)) x = f T x

  | fold_pretyps f (PVar (_, T)) x = f T x

  | fold_pretyps _ (PBound _) x = x

  | fold_pretyps f (PAbs (_, T, t)) x = fold_pretyps f t (f T x)

  | fold_pretyps f (PAppl (t, u)) x = fold_pretyps f u (fold_pretyps f t x)

  | fold_pretyps f (Constraint (t, T)) x = f T (fold_pretyps f t x);

(** raw typs/terms to pretyps/preterms **)

(* pretyp_of *)

fun pretyp_of is_para typ params_idx =

  let

    val (params', idx) = fold_atyps

      (fn TVar (xi as (x, _), S) =>

          (fn ps_idx as (ps, idx) =>

            if is_para xi andalso not (Vartab.defined ps xi)

            then (Vartab.update (xi, Param (idx, S)) ps, idx + 1) else ps_idx)

        | _ => I) typ params_idx;

    fun pre_of (TVar (v as (xi, _))) idx =

          (case Vartab.lookup params' xi of

            NONE => PTVar v

          | SOME p => p, idx)

      | pre_of (TFree ("'_dummy_", S)) idx = (Param (idx, S), idx + 1)

      | pre_of (TFree v) idx = (PTFree v, idx)

      | pre_of (T as Type (a, Ts)) idx =

          if T = dummyT then (Param (idx, []), idx + 1)

          else

            let val (Ts', idx') = fold_map pre_of Ts idx

            in (PType (a, Ts'), idx') end;

    val (ptyp, idx') = pre_of typ idx;

  in (ptyp, (params', idx')) end;

(* preterm_of *)

fun preterm_of const_type is_para tm (vparams, params, idx) =

  let

    fun add_vparm xi (ps_idx as (ps, idx)) =

      if not (Vartab.defined ps xi) then

        (Vartab.update (xi, Param (idx, [])) ps, idx + 1)

      else ps_idx;

    val (vparams', idx') = fold_aterms

      (fn Var (_, Type ("_polymorphic_", _)) => I

        | Var (xi, _) => add_vparm xi

        | Free (x, _) => add_vparm (x, ~1)

        | _ => I)

      tm (vparams, idx);

    fun var_param xi = the (Vartab.lookup vparams' xi);

    val preT_of = pretyp_of is_para;

    fun polyT_of T idx = apsnd snd (pretyp_of (K true) T (Vartab.empty, idx));

    fun constraint T t ps =

      if T = dummyT then (t, ps)

      else

        let val (T', ps') = preT_of T ps

        in (Constraint (t, T'), ps') end;

    fun pre_of (Const (c, T)) (ps, idx) =

          (case const_type c of

            SOME U =>

              let val (pU, idx') = polyT_of U idx

              in constraint T (PConst (c, pU)) (ps, idx') end

          | NONE => raise TYPE ("No such constant: " ^ quote c, [], []))

      | pre_of (Var (xi, Type ("_polymorphic_", [T]))) (ps, idx) =

          let val (pT, idx') = polyT_of T idx

          in (PVar (xi, pT), (ps, idx')) end

      | pre_of (Var (xi, T)) ps_idx = constraint T (PVar (xi, var_param xi)) ps_idx

      | pre_of (Free (x, T)) ps_idx = constraint T (PFree (x, var_param (x, ~1))) ps_idx

      | pre_of (Const ("_type_constraint_", Type ("fun", [T, _])) $ t) ps_idx =

          pre_of t ps_idx |-> constraint T

      | pre_of (Bound i) ps_idx = (PBound i, ps_idx)

      | pre_of (Abs (x, T, t)) ps_idx =

          let

            val (T', ps_idx') = preT_of T ps_idx;

            val (t', ps_idx'') = pre_of t ps_idx';

          in (PAbs (x, T', t'), ps_idx'') end

      | pre_of (t $ u) ps_idx =

          let

            val (t', ps_idx') = pre_of t ps_idx;

            val (u', ps_idx'') = pre_of u ps_idx';

          in (PAppl (t', u'), ps_idx'') end;

    val (tm', (params', idx'')) = pre_of tm (params, idx');

  in (tm', (vparams', params', idx'')) end;

(** pretyps/terms to typs/terms **)

(* add_parms *)

fun add_parmsT tye T =

  (case deref tye T of

    PType (_, Ts) => fold (add_parmsT tye) Ts

  | Param (i, _) => insert (op =) i

  | _ => I);

fun add_parms tye = fold_pretyps (add_parmsT tye);

(* add_names *)

fun add_namesT tye T =

  (case deref tye T of

    PType (_, Ts) => fold (add_namesT tye) Ts

  | PTFree (x, _) => Name.declare x

  | PTVar ((x, _), _) => Name.declare x

  | Param _ => I);

fun add_names tye = fold_pretyps (add_namesT tye);

(* simple_typ/term_of *)

fun simple_typ_of tye f T =

  (case deref tye T of

    PType (a, Ts) => Type (a, map (simple_typ_of tye f) Ts)

  | PTFree v => TFree v

  | PTVar v => TVar v

  | Param (i, S) => TVar (f i, S));

(*convert types, drop constraints*)

fun simple_term_of tye f (PConst (c, T)) = Const (c, simple_typ_of tye f T)

  | simple_term_of tye f (PFree (x, T)) = Free (x, simple_typ_of tye f T)

  | simple_term_of tye f (PVar (xi, T)) = Var (xi, simple_typ_of tye f T)

  | simple_term_of tye f (PBound i) = Bound i

  | simple_term_of tye f (PAbs (x, T, t)) =

      Abs (x, simple_typ_of tye f T, simple_term_of tye f t)

  | simple_term_of tye f (PAppl (t, u)) =

      simple_term_of tye f t $ simple_term_of tye f u

  | simple_term_of tye f (Constraint (t, _)) = simple_term_of tye f t;

(* typs_terms_of *)

fun typs_terms_of tye used maxidx (Ts, ts) =

  let

    val used' = fold (add_names tye) ts (fold (add_namesT tye) Ts used);

    val parms = rev (fold (add_parms tye) ts (fold (add_parmsT tye) Ts []));

    val names = Name.invents used' ("?" ^ Name.aT) (length parms);

    val tab = Inttab.make (parms ~~ names);

    fun f i = (the (Inttab.lookup tab i), maxidx + 1);

  in (map (simple_typ_of tye f) Ts, map (simple_term_of tye f) ts) end;

(** order-sorted unification of types **)

exception NO_UNIFIER of string * pretyp Inttab.table;

fun unify pp tsig =

  let

    (* adjust sorts of parameters *)

    fun not_of_sort x S' S =

      "Variable " ^ x ^ "::" ^ Pretty.string_of_sort pp S' ^ " not of sort " ^

        Pretty.string_of_sort pp S;

    fun meet (_, []) tye_idx = tye_idx

      | meet (Param (i, S'), S) (tye_idx as (tye, idx)) =

          if Type.subsort tsig (S', S) then tye_idx

          else (Inttab.update_new (i,

            Param (idx, Type.inter_sort tsig (S', S))) tye, idx + 1)

      | meet (PType (a, Ts), S) (tye_idx as (tye, _)) =

          meets (Ts, Type.arity_sorts pp tsig a S

            handle ERROR msg => raise NO_UNIFIER (msg, tye)) tye_idx

      | meet (PTFree (x, S'), S) (tye_idx as (tye, _)) =

          if Type.subsort tsig (S', S) then tye_idx

          else raise NO_UNIFIER (not_of_sort x S' S, tye)

      | meet (PTVar (xi, S'), S) (tye_idx as (tye, _)) =

          if Type.subsort tsig (S', S) then tye_idx

          else raise NO_UNIFIER (not_of_sort (Term.string_of_vname xi) S' S, tye)

    and meets (T :: Ts, S :: Ss) (tye_idx as (tye, _)) =

          meets (Ts, Ss) (meet (deref tye T, S) tye_idx)

      | meets _ tye_idx = tye_idx;

    (* occurs check and assignment *)

    fun occurs_check tye i (Param (i', S)) =

          if i = i' then raise NO_UNIFIER ("Occurs check!", tye)

          else

            (case Inttab.lookup tye i' of

              NONE => ()

            | SOME T => occurs_check tye i T)

      | occurs_check tye i (PType (_, Ts)) = List.app (occurs_check tye i) Ts

      | occurs_check _ _ _ = ();

    fun assign i (T as Param (i', _)) S tye_idx =

          if i = i' then tye_idx

          else tye_idx |> meet (T, S) |>> Inttab.update_new (i, T)

      | assign i T S (tye_idx as (tye, _)) =

          (occurs_check tye i T; tye_idx |> meet (T, S) |>> Inttab.update_new (i, T));

    (* unification *)

    fun unif (T1, T2) (tye_idx as (tye, idx)) =

      (case (deref tye T1, deref tye T2) of

        (Param (i, S), T) => assign i T S tye_idx

      | (T, Param (i, S)) => assign i T S tye_idx

      | (PType (a, Ts), PType (b, Us)) =>

          if a <> b then

            raise NO_UNIFIER ("Clash of types " ^ quote a ^ " and " ^ quote b, tye)

          else fold unif (Ts ~~ Us) tye_idx

      | (T, U) => if T = U then tye_idx else raise NO_UNIFIER ("", tye));

  in unif end;

(** type inference **)

(* appl_error *)

fun appl_error pp why t T u U =

 ["Type error in application: " ^ why,

  "",

  Pretty.string_of (Pretty.block

    [Pretty.str "Operator:", Pretty.brk 2, Pretty.term pp t,

      Pretty.str " ::", Pretty.brk 1, Pretty.typ pp T]),

  Pretty.string_of (Pretty.block

    [Pretty.str "Operand:", Pretty.brk 3, Pretty.term pp u,

      Pretty.str " ::", Pretty.brk 1, Pretty.typ pp U]),

  ""];

(* infer *)

fun infer pp tsig =

  let

    (* errors *)

    fun unif_failed msg =

      "Type unification failed" ^ (if msg = "" then "" else ": " ^ msg) ^ "\n";

    fun prep_output tye bs ts Ts =

      let

        val (Ts_bTs', ts') = typs_terms_of tye Name.context ~1 (Ts @ map snd bs, ts);

        val (Ts', Ts'') = chop (length Ts) Ts_bTs';

        fun prep t =

          let val xs = rev (Term.variant_frees t (rev (map fst bs ~~ Ts'')))

          in Term.subst_bounds (map Syntax.mark_boundT xs, t) end;

      in (map prep ts', Ts') end;

    fun err_loose i =

      raise TYPE ("Loose bound variable: B." ^ string_of_int i, [], []);

    fun err_appl msg tye bs t T u U =

      let

        val ([t', u'], [T', U']) = prep_output tye bs [t, u] [T, U];

        val why =

          (case T' of

            Type ("fun", _) => "Incompatible operand type"

          | _ => "Operator not of function type");

        val text = unif_failed msg ^ cat_lines (appl_error pp why t' T' u' U');

      in raise TYPE (text, [T', U'], [t', u']) end;

    fun err_constraint msg tye bs t T U =

      let

        val ([t'], [T', U']) = prep_output tye bs [t] [T, U];

        val text = cat_lines

         [unif_failed msg,

          "Cannot meet type constraint:", "",

          Pretty.string_of (Pretty.block

           [Pretty.str "Term:", Pretty.brk 2, Pretty.term pp t',

            Pretty.str " ::", Pretty.brk 1, Pretty.typ pp T']),

          Pretty.string_of (Pretty.block

           [Pretty.str "Type:", Pretty.brk 2, Pretty.typ pp U']), ""];

      in raise TYPE (text, [T', U'], [t']) end;

    (* main *)

    val unif = unify pp tsig;

    fun inf _ (PConst (_, T)) tye_idx = (T, tye_idx)

      | inf _ (PFree (_, T)) tye_idx = (T, tye_idx)

      | inf _ (PVar (_, T)) tye_idx = (T, tye_idx)

      | inf bs (PBound i) tye_idx =

          (snd (nth bs i handle Subscript => err_loose i), tye_idx)

      | inf bs (PAbs (x, T, t)) tye_idx =

          let val (U, tye_idx') = inf ((x, T) :: bs) t tye_idx

          in (PType ("fun", [T, U]), tye_idx') end

      | inf bs (PAppl (t, u)) tye_idx =

          let

            val (T, tye_idx') = inf bs t tye_idx;

            val (U, (tye, idx)) = inf bs u tye_idx';

            val V = Param (idx, []);

            val U_to_V = PType ("fun", [U, V]);

            val tye_idx'' = unif (U_to_V, T) (tye, idx + 1)

              handle NO_UNIFIER (msg, tye') => err_appl msg tye' bs t T u U;

          in (V, tye_idx'') end

      | inf bs (Constraint (t, U)) tye_idx =

          let val (T, tye_idx') = inf bs t tye_idx in

            (T,

             unif (T, U) tye_idx'

               handle NO_UNIFIER (msg, tye) => err_constraint msg tye bs t T U)

          end;

  in inf [] end;

(* infer_types *)

fun infer_types pp tsig check_typs const_type var_type used maxidx raw_ts =

  let

    (*constrain vars*)

    val get_type = the_default dummyT o var_type;

    val constrain_vars = Term.map_aterms

      (fn Free (x, T) => constrain T (Free (x, get_type (x, ~1)))

        | Var (xi, T) => constrain T (Var (xi, get_type xi))

        | t => t);

    (*convert to preterms*)

    val ts = burrow_types check_typs raw_ts;

    val (ts', (_, _, idx)) =

      fold_map (preterm_of const_type is_param o constrain_vars) ts

      (Vartab.empty, Vartab.empty, 0);

    (*do type inference*)

    val (tye, _) = fold (snd oo infer pp tsig) ts' (Inttab.empty, idx);

  in #2 (typs_terms_of tye used maxidx ([], ts')) end;

end;