(*  Title:      HOL/Tools/Predicate_Compile/code_prolog.ML

     Author:     Lukas Bulwahn, TU Muenchen

 Prototype of an code generator for logic programming languages

 (a.k.a. Prolog).

 *)

 signature CODE_PROLOG =

 sig

   datatype prolog_system = SWI_PROLOG | YAP

   type code_options =

     {ensure_groundness : bool,

      limit_globally : int option,

      limited_types : (typ * int) list,

      limited_predicates : (string list * int) list,

      replacing : ((string * string) * string) list,

      manual_reorder : ((string * int) * int list) list}

   val set_ensure_groundness : code_options -> code_options

   val map_limit_predicates : ((string list * int) list -> (string list * int) list)

     -> code_options -> code_options

   val code_options_of : theory -> code_options 

   val map_code_options : (code_options -> code_options) -> theory -> theory

   datatype arith_op = Plus | Minus

   datatype prol_term = Var of string | Cons of string | AppF of string * prol_term list

     | Number of int | ArithOp of arith_op * prol_term list;

   datatype prem = Conj of prem list

     | Rel of string * prol_term list | NotRel of string * prol_term list

     | Eq of prol_term * prol_term | NotEq of prol_term * prol_term

     | ArithEq of prol_term * prol_term | NotArithEq of prol_term * prol_term

     | Ground of string * typ;

   type clause = ((string * prol_term list) * prem);

   type logic_program = clause list;

   type constant_table = (string * string) list

   val generate : Predicate_Compile_Aux.mode option * bool ->

     Proof.context -> string -> (logic_program * constant_table)

   val write_program : logic_program -> string

   val run : (Time.time * prolog_system) -> logic_program -> (string * prol_term list) ->

     string list -> int option -> prol_term list list

   val active : bool Config.T

   val test_goals :

     Proof.context -> bool -> (string * typ) list -> (term * term list) list ->

       Quickcheck.result list

   val trace : bool Unsynchronized.ref

   val replace : ((string * string) * string) -> logic_program -> logic_program

 end;

 structure Code_Prolog : CODE_PROLOG =

 struct

 (* diagnostic tracing *)

 val trace = Unsynchronized.ref false

 fun tracing s = if !trace then Output.tracing s else () 

 (* code generation options *)

 type code_options =

   {ensure_groundness : bool,

    limit_globally : int option,

    limited_types : (typ * int) list,

    limited_predicates : (string list * int) list,

    replacing : ((string * string) * string) list,

    manual_reorder : ((string * int) * int list) list}

 fun set_ensure_groundness {ensure_groundness, limit_globally, limited_types, limited_predicates,

   replacing, manual_reorder} =

   {ensure_groundness = true, limit_globally = limit_globally, limited_types = limited_types,

    limited_predicates = limited_predicates, replacing = replacing,

    manual_reorder = manual_reorder}

 fun map_limit_predicates f {ensure_groundness, limit_globally, limited_types, limited_predicates,

   replacing, manual_reorder} =

   {ensure_groundness = ensure_groundness, limit_globally = limit_globally, limited_types = limited_types,

    limited_predicates = f limited_predicates, replacing = replacing,

    manual_reorder = manual_reorder}

 fun merge_global_limit (NONE, NONE) = NONE

   | merge_global_limit (NONE, SOME n) = SOME n

   | merge_global_limit (SOME n, NONE) = SOME n

   | merge_global_limit (SOME n, SOME m) = SOME (Int.max (n, m))  (* FIXME odd merge *)

 structure Options = Theory_Data

 (

   type T = code_options

   val empty = {ensure_groundness = false, limit_globally = NONE,

     limited_types = [], limited_predicates = [], replacing = [], manual_reorder = []}

   val extend = I;

   fun merge

     ({ensure_groundness = ensure_groundness1, limit_globally = limit_globally1,

       limited_types = limited_types1, limited_predicates = limited_predicates1,

       replacing = replacing1, manual_reorder = manual_reorder1},

      {ensure_groundness = ensure_groundness2, limit_globally = limit_globally2,

       limited_types = limited_types2, limited_predicates = limited_predicates2,

       replacing = replacing2, manual_reorder = manual_reorder2}) =

     {ensure_groundness = ensure_groundness1 orelse ensure_groundness2 (* FIXME odd merge *),

      limit_globally = merge_global_limit (limit_globally1, limit_globally2),

      limited_types = AList.merge (op =) (K true) (limited_types1, limited_types2),

      limited_predicates = AList.merge (op =) (K true) (limited_predicates1, limited_predicates2),

      manual_reorder = AList.merge (op =) (K true) (manual_reorder1, manual_reorder2),

      replacing = Library.merge (op =) (replacing1, replacing2)};

 );

 val code_options_of = Options.get

 val map_code_options = Options.map

 (* system configuration *)

 datatype prolog_system = SWI_PROLOG | YAP

 fun string_of_system SWI_PROLOG = "swiprolog"

   | string_of_system YAP = "yap"

 type system_configuration = {timeout : Time.time, prolog_system : prolog_system}

 structure System_Config = Generic_Data

 (

   type T = system_configuration

   val empty = {timeout = seconds 10.0, prolog_system = SWI_PROLOG}

   val extend = I;

   fun merge (a, _) = a

 )

 (* general string functions *)

 val first_upper = implode o nth_map 0 Symbol.to_ascii_upper o raw_explode;

 val first_lower = implode o nth_map 0 Symbol.to_ascii_lower o raw_explode;

 (* internal program representation *)

 datatype arith_op = Plus | Minus

 datatype prol_term = Var of string | Cons of string | AppF of string * prol_term list

   | Number of int | ArithOp of arith_op * prol_term list;

 fun dest_Var (Var v) = v

 fun add_vars (Var v) = insert (op =) v

   | add_vars (ArithOp (_, ts)) = fold add_vars ts

   | add_vars (AppF (_, ts)) = fold add_vars ts

   | add_vars _ = I

 fun map_vars f (Var v) = Var (f v)

   | map_vars f (ArithOp (opr, ts)) = ArithOp (opr, map (map_vars f) ts)

   | map_vars f (AppF (fs, ts)) = AppF (fs, map (map_vars f) ts)

   | map_vars f t = t

 fun maybe_AppF (c, []) = Cons c

   | maybe_AppF (c, xs) = AppF (c, xs)

 fun is_Var (Var _) = true

   | is_Var _ = false

 fun is_arith_term (Var _) = true

   | is_arith_term (Number _) = true

   | is_arith_term (ArithOp (_, operands)) = forall is_arith_term operands

   | is_arith_term _ = false

 fun string_of_prol_term (Var s) = "Var " ^ s

   | string_of_prol_term (Cons s) = "Cons " ^ s

   | string_of_prol_term (AppF (f, args)) = f ^ "(" ^ commas (map string_of_prol_term args) ^ ")" 

   | string_of_prol_term (Number n) = "Number " ^ string_of_int n

 datatype prem = Conj of prem list

   | Rel of string * prol_term list | NotRel of string * prol_term list

   | Eq of prol_term * prol_term | NotEq of prol_term * prol_term

   | ArithEq of prol_term * prol_term | NotArithEq of prol_term * prol_term

   | Ground of string * typ;

 fun dest_Rel (Rel (c, ts)) = (c, ts)

 fun map_term_prem f (Conj prems) = Conj (map (map_term_prem f) prems)

   | map_term_prem f (Rel (r, ts)) = Rel (r, map f ts)

   | map_term_prem f (NotRel (r, ts)) = NotRel (r, map f ts)

   | map_term_prem f (Eq (l, r)) = Eq (f l, f r)

   | map_term_prem f (NotEq (l, r)) = NotEq (f l, f r)

   | map_term_prem f (ArithEq (l, r)) = ArithEq (f l, f r)

   | map_term_prem f (NotArithEq (l, r)) = NotArithEq (f l, f r)

   | map_term_prem f (Ground (v, T)) = Ground (dest_Var (f (Var v)), T)

 fun fold_prem_terms f (Conj prems) = fold (fold_prem_terms f) prems

   | fold_prem_terms f (Rel (_, ts)) = fold f ts

   | fold_prem_terms f (NotRel (_, ts)) = fold f ts

   | fold_prem_terms f (Eq (l, r)) = f l #> f r

   | fold_prem_terms f (NotEq (l, r)) = f l #> f r

   | fold_prem_terms f (ArithEq (l, r)) = f l #> f r

   | fold_prem_terms f (NotArithEq (l, r)) = f l #> f r

   | fold_prem_terms f (Ground (v, T)) = f (Var v)

 type clause = ((string * prol_term list) * prem);

 type logic_program = clause list;

 (* translation from introduction rules to internal representation *)

 fun mk_conform f empty avoid name =

   let

     fun dest_Char (Symbol.Char c) = c

     val name' = space_implode "" (map (dest_Char o Symbol.decode)

       (filter (fn s => Symbol.is_ascii_letter s orelse Symbol.is_ascii_digit s)

         (Symbol.explode name)))

     val name'' = f (if name' = "" then empty else name')

   in if member (op =) avoid name'' then singleton (Name.variant_list avoid) name'' else name'' end

 (** constant table **)

 type constant_table = (string * string) list

 fun declare_consts consts constant_table =

   let

     fun update' c table =

       if AList.defined (op =) table c then table else

         let

           val c' = mk_conform first_lower "pred" (map snd table) (Long_Name.base_name c)

         in

           AList.update (op =) (c, c') table

         end

   in

     fold update' consts constant_table

   end

 fun translate_const constant_table c =

   case AList.lookup (op =) constant_table c of

     SOME c' => c'

   | NONE => error ("No such constant: " ^ c)

 fun inv_lookup _ [] _ = NONE

   | inv_lookup eq ((key, value)::xs) value' =

       if eq (value', value) then SOME key

       else inv_lookup eq xs value';

 fun restore_const constant_table c =

   case inv_lookup (op =) constant_table c of

     SOME c' => c'

   | NONE => error ("No constant corresponding to "  ^ c)

 (** translation of terms, literals, premises, and clauses **)

 fun translate_arith_const @{const_name "Groups.plus_class.plus"} = SOME Plus

   | translate_arith_const @{const_name "Groups.minus_class.minus"} = SOME Minus

   | translate_arith_const _ = NONE

 fun mk_nat_term constant_table n =

   let

     val zero = translate_const constant_table @{const_name "Groups.zero_class.zero"}

     val Suc = translate_const constant_table @{const_name "Suc"}

   in funpow n (fn t => AppF (Suc, [t])) (Cons zero) end

 fun translate_term ctxt constant_table t =

   case try HOLogic.dest_number t of

     SOME (@{typ "int"}, n) => Number n

   | SOME (@{typ "nat"}, n) => mk_nat_term constant_table n

   | NONE =>

       (case strip_comb t of

         (Free (v, T), []) => Var v 

       | (Const (c, _), []) => Cons (translate_const constant_table c)

       | (Const (c, _), args) =>

         (case translate_arith_const c of

           SOME aop => ArithOp (aop, map (translate_term ctxt constant_table) args)

         | NONE =>                                                             

             AppF (translate_const constant_table c, map (translate_term ctxt constant_table) args))

       | _ => error ("illegal term for translation: " ^ Syntax.string_of_term ctxt t))

 fun translate_literal ctxt constant_table t =

   case strip_comb t of

     (Const (@{const_name HOL.eq}, _), [l, r]) =>

       let

         val l' = translate_term ctxt constant_table l

         val r' = translate_term ctxt constant_table r

       in

         (if is_Var l' andalso is_arith_term r' andalso not (is_Var r') then ArithEq else Eq) (l', r')

       end

   | (Const (c, _), args) =>

       Rel (translate_const constant_table c, map (translate_term ctxt constant_table) args)

   | _ => error ("illegal literal for translation: " ^ Syntax.string_of_term ctxt t)

 fun NegRel_of (Rel lit) = NotRel lit

   | NegRel_of (Eq eq) = NotEq eq

   | NegRel_of (ArithEq eq) = NotArithEq eq

 fun mk_groundness_prems t = map Ground (Term.add_frees t [])

 fun translate_prem ensure_groundness ctxt constant_table t =  

     case try HOLogic.dest_not t of

       SOME t =>

         if ensure_groundness then

           Conj (mk_groundness_prems t @ [NegRel_of (translate_literal ctxt constant_table t)])

         else

           NegRel_of (translate_literal ctxt constant_table t)

     | NONE => translate_literal ctxt constant_table t

 fun imp_prems_conv cv ct =

   case Thm.term_of ct of

     Const ("==>", _) $ _ $ _ => Conv.combination_conv (Conv.arg_conv cv) (imp_prems_conv cv) ct

   | _ => Conv.all_conv ct

 fun Trueprop_conv cv ct =

   case Thm.term_of ct of

     Const (@{const_name Trueprop}, _) $ _ => Conv.arg_conv cv ct  

   | _ => raise Fail "Trueprop_conv"

 fun preprocess_intro thy rule =

   Conv.fconv_rule

     (imp_prems_conv

       (Trueprop_conv (Conv.try_conv (Conv.rewr_conv @{thm Predicate.eq_is_eq}))))

     (Thm.transfer thy rule)

 fun translate_intros ensure_groundness ctxt gr const constant_table =

   let

     val intros = map (preprocess_intro (Proof_Context.theory_of ctxt)) (Graph.get_node gr const)

     val (intros', ctxt') = Variable.import_terms true (map prop_of intros) ctxt

     val constant_table' = declare_consts (fold Term.add_const_names intros' []) constant_table

     fun translate_intro intro =

       let

         val head = HOLogic.dest_Trueprop (Logic.strip_imp_concl intro)

         val prems = map HOLogic.dest_Trueprop (Logic.strip_imp_prems intro)

         val prems' = Conj (map (translate_prem ensure_groundness ctxt' constant_table') prems)

         val clause = (dest_Rel (translate_literal ctxt' constant_table' head), prems')

       in clause end

   in

     (map translate_intro intros', constant_table')

   end

 fun depending_preds_of (key, intros) =

   fold Term.add_const_names (map Thm.prop_of intros) []

 fun add_edges edges_of key G =

   let

     fun extend' key (G, visited) = 

       case try (Graph.get_node G) key of

           SOME v =>

             let

               val new_edges = filter (fn k => is_some (try (Graph.get_node G) k)) (edges_of (key, v))

               val (G', visited') = fold extend'

                 (subtract (op =) (key :: visited) new_edges) (G, key :: visited)

             in

               (fold (Graph.add_edge o (pair key)) new_edges G', visited')

             end

         | NONE => (G, visited)

   in

     fst (extend' key (G, []))

   end

 fun print_intros ctxt gr consts =

   tracing (cat_lines (map (fn const =>

     "Constant " ^ const ^ "has intros:\n" ^

     cat_lines (map (Display.string_of_thm ctxt) (Graph.get_node gr const))) consts))

 (* translation of moded predicates *)

 (** generating graph of moded predicates **)

 (* could be moved to Predicate_Compile_Core *)

 fun requires_modes polarity cls =

   let

     fun req_mode_of pol (t, derivation) =

       (case fst (strip_comb t) of

         Const (c, _) => SOME (c, (pol, Predicate_Compile_Core.head_mode_of derivation))

       | _ => NONE)

     fun req (Predicate_Compile_Aux.Prem t, derivation) = req_mode_of polarity (t, derivation)

       | req (Predicate_Compile_Aux.Negprem t, derivation) = req_mode_of (not polarity) (t, derivation)

       | req _ = NONE

   in      

     maps (fn (_, prems) => map_filter req prems) cls

   end

 structure Mode_Graph = Graph(type key = string * (bool * Predicate_Compile_Aux.mode)

   val ord = prod_ord fast_string_ord (prod_ord bool_ord Predicate_Compile_Aux.mode_ord));

 fun mk_moded_clauses_graph ctxt scc gr =

   let

     val options = Predicate_Compile_Aux.default_options

     val mode_analysis_options =

       {use_generators = true, reorder_premises = true, infer_pos_and_neg_modes = true}

     fun infer prednames (gr, (pos_modes, neg_modes, random)) =

       let

         val (lookup_modes, lookup_neg_modes, needs_random) =

           ((fn s => the (AList.lookup (op =) pos_modes s)),

            (fn s => the (AList.lookup (op =) neg_modes s)),

            (fn s => member (op =) (the (AList.lookup (op =) random s))))

         val (preds, all_vs, param_vs, all_modes, clauses) =

           Predicate_Compile_Core.prepare_intrs options ctxt prednames

             (maps (Core_Data.intros_of ctxt) prednames)

         val ((moded_clauses, random'), _) =

           Mode_Inference.infer_modes mode_analysis_options options 

             (lookup_modes, lookup_neg_modes, needs_random) ctxt preds all_modes param_vs clauses

         val modes = map (fn (p, mps) => (p, map fst mps)) moded_clauses

         val pos_modes' = map (apsnd (map_filter (fn (true, m) => SOME m | _ => NONE))) modes

         val neg_modes' = map (apsnd (map_filter (fn (false, m) => SOME m | _ => NONE))) modes

         val _ = tracing ("Inferred modes:\n" ^

           cat_lines (map (fn (s, ms) => s ^ ": " ^ commas (map

             (fn (p, m) => Predicate_Compile_Aux.string_of_mode m ^ (if p then "pos" else "neg")) ms)) modes))

         val gr' = gr

           |> fold (fn (p, mps) => fold (fn (mode, cls) =>

                 Mode_Graph.new_node ((p, mode), cls)) mps)

             moded_clauses

           |> fold (fn (p, mps) => fold (fn (mode, cls) => fold (fn req =>

               Mode_Graph.add_edge ((p, mode), req)) (requires_modes (fst mode) cls)) mps)

             moded_clauses

       in

         (gr', (AList.merge (op =) (op =) (pos_modes, pos_modes'),

           AList.merge (op =) (op =) (neg_modes, neg_modes'),

           AList.merge (op =) (op =) (random, random')))

       end

   in  

     fst (fold infer (rev scc) (Mode_Graph.empty, ([], [], []))) 

   end

 fun declare_moded_predicate moded_preds table =

   let

     fun update' (p as (pred, (pol, mode))) table =

       if AList.defined (op =) table p then table else

         let

           val name = Long_Name.base_name pred ^ (if pol then "p" else "n")

             ^ Predicate_Compile_Aux.ascii_string_of_mode mode

           val p' = mk_conform first_lower "pred" (map snd table) name

         in

           AList.update (op =) (p, p') table

         end

   in

     fold update' moded_preds table

   end

 fun mk_program ctxt moded_gr moded_preds (prog, (moded_pred_table, constant_table)) =

   let

     val moded_pred_table' = declare_moded_predicate moded_preds moded_pred_table

     fun mk_literal pol derivation constant_table' t =

       let

         val (p, args) = strip_comb t

         val mode = Predicate_Compile_Core.head_mode_of derivation 

         val name = fst (dest_Const p)

         val p' = the (AList.lookup (op =) moded_pred_table' (name, (pol, mode)))

         val args' = map (translate_term ctxt constant_table') args

       in

         Rel (p', args')

       end

     fun mk_prem pol (indprem, derivation) constant_table =

       case indprem of

         Predicate_Compile_Aux.Generator (s, T) => (Ground (s, T), constant_table)

       | _ =>

         declare_consts (Term.add_const_names (Predicate_Compile_Aux.dest_indprem indprem) []) constant_table

         |> (fn constant_table' =>

           (case indprem of Predicate_Compile_Aux.Negprem t =>

             NegRel_of (mk_literal (not pol) derivation constant_table' t)

           | _ =>

             mk_literal pol derivation constant_table' (Predicate_Compile_Aux.dest_indprem indprem), constant_table'))

     fun mk_clause pred_name pol (ts, prems) (prog, constant_table) =

     let

       val constant_table' = declare_consts (fold Term.add_const_names ts []) constant_table

       val args = map (translate_term ctxt constant_table') ts

       val (prems', constant_table'') = fold_map (mk_prem pol) prems constant_table'

     in

       (((pred_name, args), Conj prems') :: prog, constant_table'')

     end

     fun mk_clauses (pred, mode as (pol, _)) =

       let

         val clauses = Mode_Graph.get_node moded_gr (pred, mode)

         val pred_name = the (AList.lookup (op =) moded_pred_table' (pred, mode))

       in

         fold (mk_clause pred_name pol) clauses

       end

   in

     apsnd (pair moded_pred_table') (fold mk_clauses moded_preds (prog, constant_table))

   end

 fun generate (use_modes, ensure_groundness) ctxt const =

   let 

     fun strong_conn_of gr keys =

       Graph.strong_conn (Graph.restrict (member (op =) (Graph.all_succs gr keys)) gr)

     val gr = Core_Data.intros_graph_of ctxt

     val gr' = add_edges depending_preds_of const gr

     val scc = strong_conn_of gr' [const]

     val initial_constant_table = 

       declare_consts [@{const_name "Groups.zero_class.zero"}, @{const_name "Suc"}] []

   in

     case use_modes of

       SOME mode =>

         let

           val moded_gr = mk_moded_clauses_graph ctxt scc gr

           val moded_gr' = Mode_Graph.restrict

             (member (op =) (Mode_Graph.all_succs moded_gr [(const, (true, mode))])) moded_gr

           val scc = Mode_Graph.strong_conn moded_gr' 

         in

           apfst rev (apsnd snd

             (fold (mk_program ctxt moded_gr') (rev scc) ([], ([], initial_constant_table))))

         end

       | NONE =>

         let 

           val _ = print_intros ctxt gr (flat scc)

           val constant_table = declare_consts (flat scc) initial_constant_table

         in

           apfst flat (fold_map (translate_intros ensure_groundness ctxt gr) (flat scc) constant_table)

         end

   end

 (* implementation for fully enumerating predicates and

   for size-limited predicates for enumerating the values of a datatype upto a specific size *)

 fun add_ground_typ (Conj prems) = fold add_ground_typ prems

   | add_ground_typ (Ground (_, T)) = insert (op =) T

   | add_ground_typ _ = I

 fun mk_relname (Type (Tcon, Targs)) =

   first_lower (Long_Name.base_name Tcon) ^ space_implode "_" (map mk_relname Targs)

   | mk_relname _ = raise Fail "unexpected type"

 fun mk_lim_relname T = "lim_" ^  mk_relname T

 (* This is copied from "pat_completeness.ML" *)

 fun inst_constrs_of thy (T as Type (name, _)) =

   map (fn (Cn,CT) =>

     Envir.subst_term_types (Sign.typ_match thy (body_type CT, T) Vartab.empty) (Const (Cn, CT)))

     (the (Datatype.get_constrs thy name))

   | inst_constrs_of thy T = raise TYPE ("inst_constrs_of", [T], [])

 fun is_recursive_constr T (Const (constr_name, T')) = member (op =) (binder_types T') T

 fun mk_ground_impl ctxt limited_types (T as Type (Tcon, Targs)) (seen, constant_table) =

   if member (op =) seen T then ([], (seen, constant_table))

   else

     let

       val (limited, size) = case AList.lookup (op =) limited_types T of

         SOME s => (true, s)

       | NONE => (false, 0)      

       val rel_name = (if limited then mk_lim_relname else mk_relname) T

       fun mk_impl (Const (constr_name, cT), recursive) (seen, constant_table) =

         let

           val constant_table' = declare_consts [constr_name] constant_table

           val Ts = binder_types cT

           val (rec_clauses, (seen', constant_table'')) =

             fold_map (mk_ground_impl ctxt limited_types) Ts (seen, constant_table')

           val vars = map (fn i => Var ("x" ^ string_of_int i)) (1 upto (length Ts))

           val lim_var =

             if limited then

               if recursive then [AppF ("suc", [Var "Lim"])]              

               else [Var "Lim"]

             else [] 

           fun mk_prem v T' =

             if limited andalso T' = T then Rel (mk_lim_relname T', [Var "Lim", v])

             else Rel (mk_relname T', [v])

           val clause =

             ((rel_name, lim_var @ [maybe_AppF (translate_const constant_table'' constr_name, vars)]),

              Conj (map2 mk_prem vars Ts))

         in

           (clause :: flat rec_clauses, (seen', constant_table''))

         end

       val constrs = inst_constrs_of (Proof_Context.theory_of ctxt) T

       val constrs' = (constrs ~~ map (is_recursive_constr T) constrs)

         |> (fn cs => filter_out snd cs @ filter snd cs)

       val (clauses, constant_table') =

         apfst flat (fold_map mk_impl constrs' (T :: seen, constant_table))

       val size_term = funpow size (fn t => AppF ("suc", [t])) (Cons "zero")

     in

       ((if limited then

         cons ((mk_relname T, [Var "x"]), Rel (mk_lim_relname T, [size_term, Var "x"]))

       else I) clauses, constant_table')

     end

  | mk_ground_impl ctxt _ T (seen, constant_table) =

    raise Fail ("unexpected type :" ^ Syntax.string_of_typ ctxt T)

 fun replace_ground (Conj prems) = Conj (map replace_ground prems)

   | replace_ground (Ground (x, T)) =

     Rel (mk_relname T, [Var x])  

   | replace_ground p = p

 fun add_ground_predicates ctxt limited_types (p, constant_table) =

   let

     val ground_typs = fold (add_ground_typ o snd) p []

     val (grs, (_, constant_table')) = fold_map (mk_ground_impl ctxt limited_types) ground_typs ([], constant_table)

     val p' = map (apsnd replace_ground) p

   in

     ((flat grs) @ p', constant_table')

   end

 (* make depth-limited version of predicate *)

 fun mk_lim_rel_name rel_name = "lim_" ^ rel_name

 fun mk_depth_limited rel_names ((rel_name, ts), prem) =

   let

     fun has_positive_recursive_prems (Conj prems) = exists has_positive_recursive_prems prems

       | has_positive_recursive_prems (Rel (rel, ts)) = member (op =) rel_names rel

       | has_positive_recursive_prems _ = false

     fun mk_lim_prem (Conj prems) = Conj (map mk_lim_prem prems)

       | mk_lim_prem (p as Rel (rel, ts)) =

         if member (op =) rel_names rel then Rel (mk_lim_rel_name rel, Var "Lim" :: ts) else p

       | mk_lim_prem p = p

   in

     if has_positive_recursive_prems prem then

       ((mk_lim_rel_name rel_name, (AppF ("suc", [Var "Lim"]))  :: ts), mk_lim_prem prem)

     else

       ((mk_lim_rel_name rel_name, (Var "Lim") :: ts), prem)

   end

 fun nat_term_of n = funpow n (fn t => AppF ("suc", [t])) (Cons "zero")

 fun add_limited_predicates limited_predicates (p, constant_table) =

   let                                     

     fun add (rel_names, limit) p = 

       let

         val clauses = filter (fn ((rel, _), _) => member (op =) rel_names rel) p

         val clauses' = map (mk_depth_limited rel_names) clauses

         fun mk_entry_clause rel_name =

           let

             val nargs = length (snd (fst

               (the (find_first (fn ((rel, _), _) => rel = rel_name) clauses))))

             val vars = map (fn i => Var ("x" ^ string_of_int i)) (1 upto nargs)        

           in

             (("limited_" ^ rel_name, vars), Rel ("lim_" ^ rel_name, nat_term_of limit :: vars))

           end

       in (p @ (map mk_entry_clause rel_names) @ clauses') end

   in

     (fold add limited_predicates p, constant_table)

   end

 (* replace predicates in clauses *)

 (* replace (A, B, C) p = replace A by B in clauses of C *)

 fun replace ((from, to), location) p =

   let

     fun replace_prem (Conj prems) = Conj (map replace_prem prems)

       | replace_prem (r as Rel (rel, ts)) =

           if rel = from then Rel (to, ts) else r

       | replace_prem r = r

   in

     map (fn ((rel, args), prem) => ((rel, args), (if rel = location then replace_prem else I) prem)) p

   end

 (* reorder manually : reorder premises of ith clause of predicate p by a permutation perm *)

 fun reorder_manually reorder p =

   let

     fun reorder' (clause as ((rel, args), prem)) seen =

       let

         val seen' = AList.map_default (op =) (rel, 0) (fn x => x + 1) seen

         val i = the (AList.lookup (op =) seen' rel)

         val perm = AList.lookup (op =) reorder (rel, i)

         val prem' = (case perm of 

           SOME p => (case prem of Conj prems => Conj (map (nth prems) p) | _ => prem)

         | NONE => prem)

       in (((rel, args), prem'), seen') end

   in

     fst (fold_map reorder' p [])

   end

 (* rename variables to prolog-friendly names *)

 fun rename_vars_term renaming = map_vars (fn v => the (AList.lookup (op =) renaming v))

 fun rename_vars_prem renaming = map_term_prem (rename_vars_term renaming)

 fun is_prolog_conform v =

   forall (fn s => Symbol.is_ascii_letter s orelse Symbol.is_ascii_digit s) (Symbol.explode v)

 fun mk_renaming v renaming =

   (v, mk_conform first_upper "Var" (map snd renaming) v) :: renaming

 fun rename_vars_clause ((rel, args), prem) =

   let

     val vars = fold_prem_terms add_vars prem (fold add_vars args [])

     val renaming = fold mk_renaming vars []

   in ((rel, map (rename_vars_term renaming) args), rename_vars_prem renaming prem) end

 val rename_vars_program = map rename_vars_clause

 (* limit computation globally by some threshold *)

 fun limit_globally ctxt limit const_name (p, constant_table) =

   let

     val rel_names = fold (fn ((r, _), _) => insert (op =) r) p []

     val p' = map (mk_depth_limited rel_names) p

     val rel_name = translate_const constant_table const_name

     val nargs = length (snd (fst

       (the (find_first (fn ((rel, _), _) => rel = rel_name) p))))

     val vars = map (fn i => Var ("x" ^ string_of_int i)) (1 upto nargs)

     val entry_clause = ((rel_name, vars), Rel ("lim_" ^ rel_name, nat_term_of limit :: vars))

     val p'' = filter_out (fn ((rel, _), _) => rel = rel_name) p

   in

     (entry_clause :: p' @ p'', constant_table)

   end

 (* post processing of generated prolog program *)

 fun post_process ctxt options const_name (p, constant_table) =

   (p, constant_table)

   |> (case #limit_globally options of

         SOME limit => limit_globally ctxt limit const_name

       | NONE => I)

   |> (if #ensure_groundness options then

         add_ground_predicates ctxt (#limited_types options)

       else I)

   |> tap (fn _ => tracing "Adding limited predicates...")

   |> add_limited_predicates (#limited_predicates options)

   |> tap (fn _ => tracing "Replacing predicates...")

   |> apfst (fold replace (#replacing options))

   |> apfst (reorder_manually (#manual_reorder options))

   |> apfst rename_vars_program

 (* code printer *)

 fun write_arith_op Plus = "+"

   | write_arith_op Minus = "-"

 fun write_term (Var v) = v

   | write_term (Cons c) = c

   | write_term (AppF (f, args)) = f ^ "(" ^ space_implode ", " (map write_term args) ^ ")"

   | write_term (ArithOp (oper, [a1, a2])) = write_term a1 ^ " " ^ write_arith_op oper ^ " " ^ write_term a2

   | write_term (Number n) = string_of_int n

 fun write_rel (pred, args) =

   pred ^ "(" ^ space_implode ", " (map write_term args) ^ ")" 

 fun write_prem (Conj prems) = space_implode ", " (map write_prem prems)

   | write_prem (Rel p) = write_rel p  

   | write_prem (NotRel p) = "not(" ^ write_rel p ^ ")"

   | write_prem (Eq (l, r)) = write_term l ^ " = " ^ write_term r

   | write_prem (NotEq (l, r)) = write_term l ^ " \\= " ^ write_term r

   | write_prem (ArithEq (l, r)) = write_term l ^ " is " ^ write_term r

   | write_prem (NotArithEq (l, r)) = write_term l ^ " =\\= " ^ write_term r

   | write_prem _ = raise Fail "Not a valid prolog premise"

 fun write_clause (head, prem) =

   write_rel head ^ (if prem = Conj [] then "." else " :- " ^ write_prem prem ^ ".")

 fun write_program p =

   cat_lines (map write_clause p) 

 (* query templates *)

 (** query and prelude for swi-prolog **)

 fun swi_prolog_query_first (rel, args) vnames =

   "eval :- once("  ^ rel ^ "(" ^ space_implode ", " (map write_term args) ^ ")),\n" ^

   "writef('" ^ space_implode ";" (map (fn v => v ^ " = %w") vnames) ^

   "\\n', [" ^ space_implode ", " vnames ^ "]).\n"

 fun swi_prolog_query_firstn n (rel, args) vnames =

   "eval :- findnsols(" ^ string_of_int n ^ ", (" ^ space_implode ", " vnames ^ "), " ^

     rel ^ "(" ^ space_implode ", " (map write_term args) ^ "), Sols), writelist(Sols).\n" ^

     "writelist([]).\n" ^

     "writelist([(" ^ space_implode ", " vnames ^ ")|SolutionTail]) :- " ^

     "writef('" ^ space_implode ";" (map (fn v => v ^ " = %w") vnames) ^

     "\\n', [" ^ space_implode ", " vnames ^ "]), writelist(SolutionTail).\n"

 val swi_prolog_prelude =

   "#!/usr/bin/swipl -q -t main -f\n\n" ^  (* FIXME hardwired executable!? *)

   ":- use_module(library('dialect/ciao/aggregates')).\n" ^

   ":- style_check(-singleton).\n" ^

   ":- style_check(-discontiguous).\n" ^

   ":- style_check(-atom).\n\n" ^

   "main :- catch(eval, E, (print_message(error, E), fail)), halt.\n" ^

   "main :- halt(1).\n"

 (** query and prelude for yap **)

 fun yap_query_first (rel, args) vnames =

   "eval :- once(" ^ rel ^ "(" ^ space_implode ", " (map write_term args) ^ ")),\n" ^

   "format('" ^ space_implode ";" (map (fn v => v ^ " = ~w") vnames) ^

   "\\n', [" ^ space_implode ", " vnames ^ "]).\n"

 val yap_prelude =

   "#!/usr/bin/yap -L\n\n" ^

   ":- initialization(eval).\n"

 (* system-dependent query, prelude and invocation *)

 fun query system nsols = 

   case system of

     SWI_PROLOG =>

       (case nsols of NONE => swi_prolog_query_first | SOME n => swi_prolog_query_firstn n)

   | YAP =>

       case nsols of NONE => yap_query_first | SOME n =>

         error "No support for querying multiple solutions in the prolog system yap"

 fun prelude system =

   case system of SWI_PROLOG => swi_prolog_prelude | YAP => yap_prelude

 fun invoke system file =

   let

     val (env_var, cmd) =

       (case system of

         SWI_PROLOG => ("ISABELLE_SWIPL", "\"$ISABELLE_SWIPL\" -f ")

       | YAP => ("ISABELLE_YAP", "\"$ISABELLE_YAP\" -L "))

   in

     if getenv env_var = "" then

       (warning (env_var ^ " not set; could not execute code for " ^ string_of_system system); "")

     else fst (Isabelle_System.bash_output (cmd ^ File.shell_path file))

   end

 (* parsing prolog solution *)

 val scan_number =

   Scan.many1 Symbol.is_ascii_digit

 val scan_atom =

   Scan.many1 (fn s => Symbol.is_ascii_lower s orelse Symbol.is_ascii_digit s orelse Symbol.is_ascii_quasi s)

 val scan_var =

   Scan.many1

     (fn s => Symbol.is_ascii_upper s orelse Symbol.is_ascii_digit s orelse Symbol.is_ascii_quasi s)

 val scan_ident =

   Scan.repeat (Scan.one

     (fn s => Symbol.is_ascii_letter s orelse Symbol.is_ascii_digit s orelse Symbol.is_ascii_quasi s))

 fun dest_Char (Symbol.Char s) = s

 val string_of = implode o map (dest_Char o Symbol.decode)

 val is_atom_ident = forall Symbol.is_ascii_lower

 val is_var_ident =

   forall (fn s => Symbol.is_ascii_upper s orelse Symbol.is_ascii_digit s orelse Symbol.is_ascii_quasi s)

 fun int_of_symbol_list xs = fold (fn x => fn s => s * 10 + (ord x - ord "0")) xs 0

 fun scan_terms xs = (((scan_term --| $$ ",") ::: scan_terms)

   || (scan_term >> single)) xs

 and scan_term xs =

   ((scan_number >> (Number o int_of_symbol_list))

   || (scan_var >> (Var o string_of))

   || ((scan_atom -- ($$ "(" |-- scan_terms --| $$ ")"))

     >> (fn (f, ts) => AppF (string_of f, ts)))

   || (scan_atom >> (Cons o string_of))) xs

 val parse_term = fst o Scan.finite Symbol.stopper

     (Scan.error (!! (fn _ => raise Fail "parsing prolog output failed")) scan_term)

   o raw_explode

 fun parse_solutions sol =

   let

     fun dest_eq s = case space_explode "=" s of

         (l :: r :: []) => parse_term (unprefix " " r)

       | _ => raise Fail "unexpected equation in prolog output"

     fun parse_solution s = map dest_eq (space_explode ";" s)

     val sols = case space_explode "\n" sol of [] => [] | s => fst (split_last s)  

   in

     map parse_solution sols

   end 

 (* calling external interpreter and getting results *)

 fun run (timeout, system) p (query_rel, args) vnames nsols =

   let

     val renaming = fold mk_renaming (fold add_vars args vnames) [] 

     val vnames' = map (fn v => the (AList.lookup (op =) renaming v)) vnames

     val args' = map (rename_vars_term renaming) args

     val prog = prelude system ^ query system nsols (query_rel, args') vnames' ^ write_program p

     val _ = tracing ("Generated prolog program:\n" ^ prog)

     val solution = TimeLimit.timeLimit timeout (fn prog =>

       Isabelle_System.with_tmp_file "prolog_file" "" (fn prolog_file =>

         (File.write prolog_file prog; invoke system prolog_file))) prog

     val _ = tracing ("Prolog returned solution(s):\n" ^ solution)

     val tss = parse_solutions solution

   in

     tss

   end

 (* restoring types in terms *)

 fun restore_term ctxt constant_table (Var s, T) = Free (s, T)

   | restore_term ctxt constant_table (Number n, @{typ "int"}) = HOLogic.mk_number @{typ "int"} n

   | restore_term ctxt constant_table (Number n, _) = raise (Fail "unexpected type for number") 

   | restore_term ctxt constant_table (Cons s, T) = Const (restore_const constant_table s, T)

   | restore_term ctxt constant_table (AppF (f, args), T) =

     let

       val thy = Proof_Context.theory_of ctxt

       val c = restore_const constant_table f

       val cT = Sign.the_const_type thy c

       val (argsT, resT) = strip_type cT

       val subst = Sign.typ_match thy (resT, T) Vartab.empty

       val argsT' = map (Envir.subst_type subst) argsT

     in

       list_comb (Const (c, Envir.subst_type subst cT),

         map (restore_term ctxt constant_table) (args ~~ argsT'))

     end

 (* restore numerals in natural numbers *)

 fun restore_nat_numerals t =

   if fastype_of t = @{typ nat} andalso is_some (try HOLogic.dest_nat t) then

     HOLogic.mk_number @{typ nat} (HOLogic.dest_nat t)

   else

     (case t of

         t1 $ t2 => restore_nat_numerals t1 $ restore_nat_numerals t2

       | t => t)

 (* values command *)

 val preprocess_options = Predicate_Compile_Aux.Options {

   expected_modes = NONE,

   proposed_modes = [],

   proposed_names = [],

   show_steps = false,

   show_intermediate_results = false,

   show_proof_trace = false,

   show_modes = false,

   show_mode_inference = false,

   show_compilation = false,

   show_caught_failures = false,

   show_invalid_clauses = false,

   skip_proof = true,

   no_topmost_reordering = false,

   function_flattening = true,

   specialise = false,

   fail_safe_function_flattening = false,

   no_higher_order_predicate = [],

   inductify = false,

   detect_switches = true,

   smart_depth_limiting = true,

   compilation = Predicate_Compile_Aux.Pred

 }

 fun values ctxt soln t_compr =

   let

     val options = code_options_of (Proof_Context.theory_of ctxt)

     val split = case t_compr of (Const (@{const_name Collect}, _) $ t) => t

       | _ => error ("Not a set comprehension: " ^ Syntax.string_of_term ctxt t_compr);

     val (body, Ts, fp) = HOLogic.strip_psplits split;

     val output_names = Name.variant_list (Term.add_free_names body [])

       (map (fn i => "x" ^ string_of_int i) (1 upto length Ts))

     val output_frees = rev (map2 (curry Free) output_names Ts)

     val body = subst_bounds (output_frees, body)

     val (pred as Const (name, T), all_args) =

       case strip_comb body of

         (Const (name, T), all_args) => (Const (name, T), all_args)

       | (head, _) => error ("Not a constant: " ^ Syntax.string_of_term ctxt head)

     val _ = tracing "Preprocessing specification..."

     val T = Sign.the_const_type (Proof_Context.theory_of ctxt) name

     val t = Const (name, T)

     val thy' =

       Theory.copy (Proof_Context.theory_of ctxt)

       |> Predicate_Compile.preprocess preprocess_options t

     val ctxt' = Proof_Context.init_global thy'

     val _ = tracing "Generating prolog program..."

     val (p, constant_table) = generate (NONE, #ensure_groundness options) ctxt' name (* FIXME *)

       |> post_process ctxt' options name

     val constant_table' = declare_consts (fold Term.add_const_names all_args []) constant_table

     val args' = map (translate_term ctxt constant_table') all_args

     val _ = tracing "Running prolog program..."

     val system_config = System_Config.get (Context.Proof ctxt)

     val tss = run (#timeout system_config, #prolog_system system_config)

       p (translate_const constant_table' name, args') output_names soln

     val _ = tracing "Restoring terms..."

     val empty = Const("Orderings.bot_class.bot", fastype_of t_compr)

     fun mk_insert x S =

       Const (@{const_name "Set.insert"}, fastype_of x --> fastype_of S --> fastype_of S) $ x $ S 

     fun mk_set_compr in_insert [] xs =

        rev ((Free ("dots", fastype_of t_compr)) ::  (* FIXME proper name!? *)

         (if null in_insert then xs else (fold mk_insert in_insert empty) :: xs))

       | mk_set_compr in_insert (t :: ts) xs =

         let

           val frees = Term.add_frees t []

         in

           if null frees then

             mk_set_compr (t :: in_insert) ts xs

           else

             let

               val uu as (uuN, uuT) = singleton (Variable.variant_frees ctxt' [t]) ("uu", fastype_of t)

               val set_compr =

                 HOLogic.mk_Collect (uuN, uuT, fold (fn (s, T) => fn t => HOLogic.mk_exists (s, T, t))

                   frees (HOLogic.mk_conj (HOLogic.mk_eq (Free uu, t), @{term "True"})))

             in

               mk_set_compr [] ts

                 (set_compr :: (if null in_insert then xs else (fold mk_insert in_insert empty) :: xs))  

             end

         end

   in

       foldl1 (HOLogic.mk_binop @{const_name sup}) (mk_set_compr []

         (map (fn ts => HOLogic.mk_tuple 

           (map (restore_nat_numerals o restore_term ctxt' constant_table) (ts ~~ Ts))) tss) [])

   end

 fun values_cmd print_modes soln raw_t state =

   let

     val ctxt = Toplevel.context_of state

     val t = Syntax.read_term ctxt raw_t

     val t' = values ctxt soln t

     val ty' = Term.type_of t'

     val ctxt' = Variable.auto_fixes t' ctxt

     val _ = tracing "Printing terms..."

     val p = Print_Mode.with_modes print_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;

 (* renewing the values command for Prolog queries *)

 val opt_print_modes =

   Scan.optional (@{keyword "("} |-- Parse.!!! (Scan.repeat1 Parse.xname --| @{keyword ")"})) [];

 val _ = Outer_Syntax.improper_command "values" "enumerate and print comprehensions" Keyword.diag

   (opt_print_modes -- Scan.optional (Parse.nat >> SOME) NONE -- Parse.term

    >> (fn ((print_modes, soln), t) => Toplevel.keep

         (values_cmd print_modes soln t))); (*FIXME does not preserve the previous functionality*)

 (* quickcheck generator *)

 (* FIXME: a small clone of Predicate_Compile_Quickcheck - maybe refactor out commons *)

 val active = Attrib.setup_config_bool @{binding quickcheck_prolog_active} (K true);

 fun test_term ctxt (t, eval_terms) =

   let

     val t' = fold_rev absfree (Term.add_frees t []) t

     val options = code_options_of (Proof_Context.theory_of ctxt)

     val thy = Theory.copy (Proof_Context.theory_of ctxt)

     val ((((full_constname, constT), vs'), intro), thy1) =

       Predicate_Compile_Aux.define_quickcheck_predicate t' thy

     val thy2 = Context.theory_map (Predicate_Compile_Alternative_Defs.add_thm intro) thy1

     val thy3 = Predicate_Compile.preprocess preprocess_options (Const (full_constname, constT)) thy2

     val ctxt' = Proof_Context.init_global thy3

     val _ = tracing "Generating prolog program..."

     val (p, constant_table) = generate (NONE, true) ctxt' full_constname

       |> post_process ctxt' (set_ensure_groundness options) full_constname

     val _ = tracing "Running prolog program..."

     val system_config = System_Config.get (Context.Proof ctxt)

     val tss = run (#timeout system_config, #prolog_system system_config)

       p (translate_const constant_table full_constname, map (Var o fst) vs') (map fst vs') (SOME 1)

     val _ = tracing "Restoring terms..."

     val counterexample =

       case tss of

         [ts] => SOME (map (restore_term ctxt' constant_table) (ts ~~ map snd vs'))

       | _ => NONE

   in

     Quickcheck.Result

       {counterexample = Option.map (pair true o curry (op ~~) (Term.add_free_names t [])) counterexample,

        evaluation_terms = Option.map (K []) counterexample, timings = [], reports = []}

   end;

 fun test_goals ctxt _ insts goals =

   let

     val correct_inst_goals = Quickcheck_Common.instantiate_goals ctxt insts goals

   in

     Quickcheck_Common.collect_results (test_term ctxt) (maps (map snd) correct_inst_goals) []

   end

 end;