(*  Title:      Tools/nbe.ML

    Authors:    Klaus Aehlig, LMU Muenchen; Tobias Nipkow, Florian Haftmann, TU Muenchen

Normalization by evaluation, based on generic code generator.

*)

signature NBE =

sig

  val norm_conv: cterm -> thm

  val norm: theory -> term -> term

  datatype Univ =

      Const of int * Univ list               (*named (uninterpreted) constants*)

    | DFree of string * int                  (*free (uninterpreted) dictionary parameters*)

    | BVar of int * Univ list

    | Abs of (int * (Univ list -> Univ)) * Univ list

  val apps: Univ -> Univ list -> Univ        (*explicit applications*)

  val abss: int -> (Univ list -> Univ) -> Univ

                                             (*abstractions as closures*)

  val same: Univ -> Univ -> bool

  val univs_ref: (unit -> Univ list -> Univ list) option ref

  val trace: bool ref

  val setup: theory -> theory

end;

structure Nbe: NBE =

struct

(* generic non-sense *)

val trace = ref false;

fun tracing f x = if !trace then (Output.tracing (f x); x) else x;

(** the semantical universe **)

(*

   Functions are given by their semantical function value. To avoid

   trouble with the ML-type system, these functions have the most

   generic type, that is "Univ list -> Univ". The calling convention is

   that the arguments come as a list, the last argument first. In

   other words, a function call that usually would look like

   f x_1 x_2 ... x_n   or   f(x_1,x_2, ..., x_n)

   would be in our convention called as

              f [x_n,..,x_2,x_1]

   Moreover, to handle functions that are still waiting for some

   arguments we have additionally a list of arguments collected to far

   and the number of arguments we're still waiting for.

*)

datatype Univ =

    Const of int * Univ list           (*named (uninterpreted) constants*)

  | DFree of string * int              (*free (uninterpreted) dictionary parameters*)

  | BVar of int * Univ list            (*bound variables, named*)

  | Abs of (int * (Univ list -> Univ)) * Univ list

                                       (*abstractions as closures*);

fun same (Const (k, xs)) (Const (l, ys)) = k = l andalso sames xs ys

  | same (DFree (s, k)) (DFree (t, l)) = s = t andalso k = l

  | same (BVar (k, xs)) (BVar (l, ys)) = k = l andalso sames xs ys

  | same _ _ = false

and sames xs ys = length xs = length ys andalso forall (uncurry same) (xs ~~ ys);

(* constructor functions *)

fun abss n f = Abs ((n, f), []);

fun apps (Abs ((n, f), xs)) ys = let val k = n - length ys in

      case int_ord (k, 0)

       of EQUAL => f (ys @ xs)

        | LESS => let val (zs, ws) = chop (~ k) ys in apps (f (ws @ xs)) zs end

        | GREATER => Abs ((k, f), ys @ xs) (*note: reverse convention also for apps!*)

      end

  | apps (Const (name, xs)) ys = Const (name, ys @ xs)

  | apps (BVar (n, xs)) ys = BVar (n, ys @ xs);

(** assembling and compiling ML code from terms **)

(* abstract ML syntax *)

infix 9 `$` `$$`;

fun e1 `$` e2 = "(" ^ e1 ^ " " ^ e2 ^ ")";

fun e `$$` [] = e

  | e `$$` es = "(" ^ e ^ " " ^ space_implode " " es ^ ")";

fun ml_abs v e = "(fn " ^ v ^ " => " ^ e ^ ")";

fun ml_cases t cs =

  "(case " ^ t ^ " of " ^ space_implode " | " (map (fn (p, t) => p ^ " => " ^ t) cs) ^ ")";

fun ml_Let d e = "let\n" ^ d ^ " in " ^ e ^ " end";

fun ml_as v t = "(" ^ v ^ " as " ^ t ^ ")";

fun ml_and [] = "true"

  | ml_and [x] = x

  | ml_and xs = "(" ^ space_implode " andalso " xs ^ ")";

fun ml_if b x y = "(if " ^ b ^ " then " ^ x ^ " else " ^ y ^ ")";

fun ml_list es = "[" ^ commas es ^ "]";

fun ml_fundefs ([(name, [([], e)])]) =

      "val " ^ name ^ " = " ^ e ^ "\n"

  | ml_fundefs (eqs :: eqss) =

      let

        fun fundef (name, eqs) =

          let

            fun eqn (es, e) = name ^ " " ^ space_implode " " es ^ " = " ^ e

          in space_implode "\n  | " (map eqn eqs) end;

      in

        (prefix "fun " o fundef) eqs :: map (prefix "and " o fundef) eqss

        |> cat_lines

        |> suffix "\n"

      end;

(* nbe specific syntax and sandbox communication *)

val univs_ref = ref (NONE : (unit -> Univ list -> Univ list) option);

local

  val prefix =      "Nbe.";

  val name_ref =    prefix ^ "univs_ref";

  val name_const =  prefix ^ "Const";

  val name_abss =   prefix ^ "abss";

  val name_apps =   prefix ^ "apps";

  val name_same =   prefix ^ "same";

in

val univs_cookie = (name_ref, univs_ref);

fun nbe_fun 0 "" = "nbe_value"

  | nbe_fun i c = "c_" ^ translate_string (fn "." => "_" | c => c) c ^ "_" ^ string_of_int i;

fun nbe_dict v n = "d_" ^ v ^ "_" ^ string_of_int n;

fun nbe_bound v = "v_" ^ v;

fun nbe_default v = "w_" ^ v;

(*note: these three are the "turning spots" where proper argument order is established!*)

fun nbe_apps t [] = t

  | nbe_apps t ts = name_apps `$$` [t, ml_list (rev ts)];

fun nbe_apps_local i c ts = nbe_fun i c `$` ml_list (rev ts);

fun nbe_apps_constr idx_of c ts =

  let

    val c' = if !trace then string_of_int (idx_of c) ^ " (*" ^ c ^ "*)"

      else string_of_int (idx_of c);

  in name_const `$` ("(" ^ c' ^ ", " ^ ml_list (rev ts) ^ ")") end;

fun nbe_abss 0 f = f `$` ml_list []

  | nbe_abss n f = name_abss `$$` [string_of_int n, f];

fun nbe_same v1 v2 = "(" ^ name_same ^ " " ^ nbe_bound v1 ^ " " ^ nbe_bound v2 ^ ")";

end;

open Basic_Code_Thingol;

(* code generation *)

fun assemble_eqnss idx_of deps eqnss =

  let

    fun prep_eqns (c, (vs, eqns)) =

      let

        val dicts = maps (fn (v, sort) => map_index (nbe_dict v o fst) sort) vs;

        val num_args = length dicts + ((length o fst o hd) eqns);

      in (c, (num_args, (dicts, eqns))) end;

    val eqnss' = map prep_eqns eqnss;

    fun assemble_constapp c dss ts = 

      let

        val ts' = (maps o map) assemble_idict dss @ ts;

      in case AList.lookup (op =) eqnss' c

       of SOME (num_args, _) => if num_args <= length ts'

            then let val (ts1, ts2) = chop num_args ts'

            in nbe_apps (nbe_apps_local 0 c ts1) ts2

            end else nbe_apps (nbe_abss num_args (nbe_fun 0 c)) ts'

        | NONE => if member (op =) deps c

            then nbe_apps (nbe_fun 0 c) ts'

            else nbe_apps_constr idx_of c ts'

      end

    and assemble_idict (DictConst (inst, dss)) =

          assemble_constapp inst dss []

      | assemble_idict (DictVar (supers, (v, (n, _)))) =

          fold_rev (fn super => assemble_constapp super [] o single) supers (nbe_dict v n);

    fun assemble_iterm constapp =

      let

        fun of_iterm match_cont t =

          let

            val (t', ts) = Code_Thingol.unfold_app t

          in of_iapp match_cont t' (fold_rev (cons o of_iterm NONE) ts []) end

        and of_iapp match_cont (IConst (c, ((_, dss), _))) ts = constapp c dss ts

          | of_iapp match_cont (IVar v) ts = nbe_apps (nbe_bound v) ts

          | of_iapp match_cont ((v, _) `|-> t) ts =

              nbe_apps (nbe_abss 1 (ml_abs (ml_list [nbe_bound v]) (of_iterm NONE t))) ts

          | of_iapp match_cont (ICase (((t, _), cs), t0)) ts =

              nbe_apps (ml_cases (of_iterm NONE t)

                (map (fn (p, t) => (of_iterm NONE p, of_iterm match_cont t)) cs

                  @ [("_", case match_cont of SOME s => s | NONE => of_iterm NONE t0)])) ts

      in of_iterm end;

    fun subst_nonlin_vars args =

      let

        val vs = (fold o Code_Thingol.fold_varnames)

          (fn v => AList.map_default (op =) (v, 0) (curry (op +) 1)) args [];

        val names = Name.make_context (map fst vs);

        fun declare v k ctxt = let val vs = Name.invents ctxt v k

          in (vs, fold Name.declare vs ctxt) end;

        val (vs_renames, _) = fold_map (fn (v, k) => if k > 1

          then declare v (k - 1) #>> (fn vs => (v, vs))

          else pair (v, [])) vs names;

        val samepairs = maps (fn (v, vs) => map (pair v) vs) vs_renames;

        fun subst_vars (t as IConst _) samepairs = (t, samepairs)

          | subst_vars (t as IVar v) samepairs = (case AList.lookup (op =) samepairs v

             of SOME v' => (IVar v', AList.delete (op =) v samepairs)

              | NONE => (t, samepairs))

          | subst_vars (t1 `$ t2) samepairs = samepairs

              |> subst_vars t1

              ||>> subst_vars t2

              |>> (op `$)

          | subst_vars (ICase (_, t)) samepairs = subst_vars t samepairs;

        val (args', _) = fold_map subst_vars args samepairs;

      in (samepairs, args') end;

    fun assemble_eqn c dicts default_args (i, (args, rhs)) =

      let

        val is_eval = (c = "");

        val default_rhs = nbe_apps_local (i+1) c (dicts @ default_args);

        val match_cont = if is_eval then NONE else SOME default_rhs;

        val assemble_arg = assemble_iterm

          (fn c => fn _ => fn ts => nbe_apps_constr idx_of c ts) NONE;

        val assemble_rhs = assemble_iterm assemble_constapp match_cont ;

        val (samepairs, args') = subst_nonlin_vars args;

        val s_args = map assemble_arg args';

        val s_rhs = if null samepairs then assemble_rhs rhs

          else ml_if (ml_and (map (uncurry nbe_same) samepairs))

            (assemble_rhs rhs) default_rhs;

        val eqns = if is_eval then

            [([ml_list (rev (dicts @ s_args))], s_rhs)]

          else

            [([ml_list (rev (dicts @ map2 ml_as default_args s_args))], s_rhs),

              ([ml_list (rev (dicts @ default_args))], default_rhs)]

      in (nbe_fun i c, eqns) end;

    fun assemble_eqns (c, (num_args, (dicts, eqns))) =

      let

        val default_args = map nbe_default

          (Name.invent_list [] "a" (num_args - length dicts));

        val eqns' = map_index (assemble_eqn c dicts default_args) eqns

          @ (if c = "" then [] else [(nbe_fun (length eqns) c,

            [([ml_list (rev (dicts @ default_args))],

              nbe_apps_constr idx_of c (dicts @ default_args))])]);

      in (eqns', nbe_abss num_args (nbe_fun 0 c)) end;

    val (fun_vars, fun_vals) = map_split assemble_eqns eqnss';

    val deps_vars = ml_list (map (nbe_fun 0) deps);

  in ml_abs deps_vars (ml_Let (ml_fundefs (flat fun_vars)) (ml_list fun_vals)) end;

(* code compilation *)

fun compile_eqnss _ gr raw_deps [] = []

  | compile_eqnss ctxt gr raw_deps eqnss =

      let

        val (deps, deps_vals) = split_list (map_filter

          (fn dep => Option.map (fn univ => (dep, univ)) (fst ((Graph.get_node gr dep)))) raw_deps);

        val idx_of = raw_deps

          |> map (fn dep => (dep, snd (Graph.get_node gr dep)))

          |> AList.lookup (op =)

          |> (fn f => the o f);

        val s = assemble_eqnss idx_of deps eqnss;

        val cs = map fst eqnss;

      in

        s

        |> tracing (fn s => "\n--- code to be evaluated:\n" ^ s)

        |> ML_Context.evaluate ctxt (!trace) univs_cookie

        |> (fn f => f deps_vals)

        |> (fn univs => cs ~~ univs)

      end;

(* preparing function equations *)

fun eqns_of_stmt (_, Code_Thingol.Fun (_, (_, []))) =

      []

  | eqns_of_stmt (const, Code_Thingol.Fun (_, ((vs, _), eqns))) =

      [(const, (vs, map fst eqns))]

  | eqns_of_stmt (_, Code_Thingol.Datatypecons _) =

      []

  | eqns_of_stmt (_, Code_Thingol.Datatype _) =

      []

  | eqns_of_stmt (class, Code_Thingol.Class (_, (v, (superclasses, classops)))) =

      let

        val names = map snd superclasses @ map fst classops;

        val params = Name.invent_list [] "d" (length names);

        fun mk (k, name) =

          (name, ([(v, [])],

            [([IConst (class, (([], []), [])) `$$ map IVar params], IVar (nth params k))]));

      in map_index mk names end

  | eqns_of_stmt (_, Code_Thingol.Classrel _) =

      []

  | eqns_of_stmt (_, Code_Thingol.Classparam _) =

      []

  | eqns_of_stmt (inst, Code_Thingol.Classinst ((class, (_, arities)), (superinsts, instops))) =

      [(inst, (arities, [([], IConst (class, (([], []), [])) `$$

        map (fn (_, (_, (inst, dicts))) => IConst (inst, (([], dicts), []))) superinsts

        @ map (IConst o snd o fst) instops)]))];

fun compile_stmts ctxt stmts_deps =

  let

    val names = map (fst o fst) stmts_deps;

    val names_deps = map (fn ((name, _), deps) => (name, deps)) stmts_deps;

    val eqnss = maps (eqns_of_stmt o fst) stmts_deps;

    val refl_deps = names_deps

      |> maps snd

      |> distinct (op =)

      |> fold (insert (op =)) names;

    fun new_node name (gr, (maxidx, idx_tab)) = if can (Graph.get_node gr) name

      then (gr, (maxidx, idx_tab))

      else (Graph.new_node (name, (NONE, maxidx)) gr,

        (maxidx + 1, Inttab.update_new (maxidx, name) idx_tab));

    fun compile gr = eqnss

      |> compile_eqnss ctxt gr refl_deps

      |> rpair gr;

  in

    fold new_node refl_deps

    #> apfst (fold (fn (name, deps) => fold (curry Graph.add_edge name) deps) names_deps

      #> compile

      #-> fold (fn (name, univ) => (Graph.map_node name o apfst) (K (SOME univ))))

  end;

fun ensure_stmts ctxt naming program =

  let

    fun add_stmts names (gr, (maxidx, idx_tab)) = if exists ((can o Graph.get_node) gr) names

      then (gr, (maxidx, idx_tab))

      else (gr, (maxidx, idx_tab))

        |> compile_stmts ctxt (map (fn name => ((name, Graph.get_node program name),

          Graph.imm_succs program name)) names);

  in

    fold_rev add_stmts (Graph.strong_conn program)

    #> pair naming

  end;

(** evaluation **)

(* term evaluation *)

fun eval_term ctxt gr deps (vs : (string * sort) list, t) =

  let 

    val dict_frees = maps (fn (v, sort) => map_index (curry DFree v o fst) sort) vs;

  in

    ("", (vs, [([], t)]))

    |> singleton (compile_eqnss ctxt gr deps)

    |> snd

    |> (fn t => apps t (rev dict_frees))

  end;

(* reification *)

fun typ_of_itype program vs (ityco `%% itys) =

      let

        val Code_Thingol.Datatype (tyco, _) = Graph.get_node program ityco;

      in Type (tyco, map (typ_of_itype program vs) itys) end

  | typ_of_itype program vs (ITyVar v) =

      let

        val sort = (the o AList.lookup (op =) vs) v;

      in TFree ("'" ^ v, sort) end;

fun term_of_univ thy program idx_tab t =

  let

    fun take_until f [] = []

      | take_until f (x::xs) = if f x then [] else x :: take_until f xs;

    fun is_dict (Const (idx, _)) = (case (Graph.get_node program o the o Inttab.lookup idx_tab) idx

         of Code_Thingol.Class _ => true

          | Code_Thingol.Classrel _ => true

          | Code_Thingol.Classinst _ => true

          | _ => false)

      | is_dict (DFree _) = true

      | is_dict _ = false;

    fun const_of_idx idx = (case (Graph.get_node program o the o Inttab.lookup idx_tab) idx

         of Code_Thingol.Fun (c, _) => c

          | Code_Thingol.Datatypecons (c, _) => c

          | Code_Thingol.Classparam (c, _) => c);

    fun of_apps bounds (t, ts) =

      fold_map (of_univ bounds) ts

      #>> (fn ts' => list_comb (t, rev ts'))

    and of_univ bounds (Const (idx, ts)) typidx =

          let

            val ts' = take_until is_dict ts;

            val c = const_of_idx idx;

            val (_, T) = Code.default_typscheme thy c;

            val T' = map_type_tfree (fn (v, _) => TypeInfer.param typidx (v, [])) T;

            val typidx' = typidx + 1;

          in of_apps bounds (Term.Const (c, T'), ts') typidx' end

      | of_univ bounds (BVar (n, ts)) typidx =

          of_apps bounds (Bound (bounds - n - 1), ts) typidx

      | of_univ bounds (t as Abs _) typidx =

          typidx

          |> of_univ (bounds + 1) (apps t [BVar (bounds, [])])

          |-> (fn t' => pair (Term.Abs ("u", dummyT, t')))

  in of_univ 0 t 0 |> fst end;

(* function store *)

structure Nbe_Functions = CodeDataFun

(

  type T = Code_Thingol.naming * ((Univ option * int) Graph.T * (int * string Inttab.table));

  val empty = (Code_Thingol.empty_naming, (Graph.empty, (0, Inttab.empty)));

  fun purge thy cs (naming, (gr, (maxidx, idx_tab))) =

    let

      val names_delete = cs

        |> map_filter (Code_Thingol.lookup_const naming)

        |> filter (can (Graph.get_node gr))

        |> Graph.all_preds gr;

      val gr' = Graph.del_nodes names_delete gr;

    in (naming, (gr', (maxidx, idx_tab))) end;

);

(* compilation, evaluation and reification *)

fun compile_eval thy naming program vs_t deps =

  let

    val ctxt = ProofContext.init thy;

    val (_, (gr, (_, idx_tab))) =

      Nbe_Functions.change thy (ensure_stmts ctxt naming program o snd);

  in

    vs_t

    |> eval_term ctxt gr deps

    |> term_of_univ thy program idx_tab

  end;

(* evaluation with type reconstruction *)

fun normalize thy naming program ((vs0, (vs, ty)), t) deps =

  let

    fun subst_const f = map_aterms (fn t as Term.Const (c, ty) => Term.Const (f c, ty)

      | t => t);

    val resubst_triv_consts = subst_const (Code.resubst_alias thy);

    val ty' = typ_of_itype program vs0 ty;

    fun type_infer t =

      singleton (TypeInfer.infer_types (Syntax.pp_global thy) (Sign.tsig_of thy) I

        (try (Type.strip_sorts o Sign.the_const_type thy)) (K NONE) Name.context 0)

      (TypeInfer.constrain ty' t);

    fun check_tvars t = if null (Term.add_tvars t []) then t else

      error ("Illegal schematic type variables in normalized term: "

        ^ setmp show_types true (Syntax.string_of_term_global thy) t);

    val string_of_term = setmp show_types true (Syntax.string_of_term_global thy);

  in

    compile_eval thy naming program (vs, t) deps

    |> tracing (fn t => "Normalized:\n" ^ string_of_term t)

    |> resubst_triv_consts

    |> type_infer

    |> tracing (fn t => "Types inferred:\n" ^ string_of_term t)

    |> check_tvars

    |> tracing (fn t => "---\n")

  end;

(* evaluation oracle *)

fun add_triv_classes thy = curry (Sorts.inter_sort (Sign.classes_of thy))

  (Code.triv_classes thy);

fun mk_equals thy lhs raw_rhs =

  let

    val ty = Thm.typ_of (Thm.ctyp_of_term lhs);

    val eq = Thm.cterm_of thy (Term.Const ("==", ty --> ty --> propT));

    val rhs = Thm.cterm_of thy raw_rhs;

  in Thm.mk_binop eq lhs rhs end;

val (_, raw_norm_oracle) = Context.>>> (Context.map_theory_result

  (Thm.add_oracle (Binding.name "norm", fn (thy, naming, program, vsp_ty_t, deps, ct) =>

    mk_equals thy ct (normalize thy naming program vsp_ty_t deps))));

fun norm_oracle thy naming program vsp_ty_t deps ct =

  raw_norm_oracle (thy, naming, program, vsp_ty_t, deps, ct);

fun no_frees_conv conv ct =

  let

    val frees = Thm.add_cterm_frees ct [];

    fun apply_beta free thm = Thm.combination thm (Thm.reflexive free)

      |> Conv.fconv_rule (Conv.arg_conv (Conv.try_conv (Thm.beta_conversion false)))

      |> Conv.fconv_rule (Conv.arg1_conv (Thm.beta_conversion false));

  in

    ct

    |> fold_rev Thm.cabs frees

    |> conv

    |> fold apply_beta frees

  end;

fun no_frees_rew rew t =

  let

    val frees = map Free (Term.add_frees t []);

  in

    t

    |> fold_rev lambda frees

    |> rew

    |> (fn t' => Term.betapplys (t', frees))

  end;

val norm_conv = no_frees_conv (fn ct =>

  let

    val thy = Thm.theory_of_cterm ct;

  in Code_Thingol.eval_conv thy (add_triv_classes thy) (norm_oracle thy) ct end);

fun norm thy = no_frees_rew (Code_Thingol.eval thy (add_triv_classes thy) I (normalize thy));

(* evaluation command *)

fun norm_print_term ctxt modes t =

  let

    val thy = ProofContext.theory_of ctxt;

    val t' = norm thy t;

    val ty' = Term.type_of t';

    val ctxt' = Variable.auto_fixes t ctxt;

    val p = PrintMode.with_modes modes (fn () =>

      Pretty.block [Pretty.quote (Syntax.pretty_term ctxt' t'), Pretty.fbrk,

        Pretty.str "::", Pretty.brk 1, Pretty.quote (Syntax.pretty_typ ctxt' ty')]) ();

  in Pretty.writeln p end;

(** Isar setup **)

fun norm_print_term_cmd (modes, s) state =

  let val ctxt = Toplevel.context_of state

  in norm_print_term ctxt modes (Syntax.read_term ctxt s) end;

val setup = Value.add_evaluator ("nbe", norm o ProofContext.theory_of);

local structure P = OuterParse and K = OuterKeyword in

val opt_modes = Scan.optional (P.$$$ "(" |-- P.!!! (Scan.repeat1 P.xname --| P.$$$ ")")) [];

val _ =

  OuterSyntax.improper_command "normal_form" "normalize term by evaluation" K.diag

    (opt_modes -- P.term >> (Toplevel.keep o norm_print_term_cmd));

end;

end;