(*  Title:      HOL/Tools/inductive_codegen.ML

     Author:     Stefan Berghofer, TU Muenchen

 Code generator for inductive predicates.

 *)

 signature INDUCTIVE_CODEGEN =

 sig

   val add : string option -> int option -> attribute

   val test_fn : (int * int * int -> term list option) Unsynchronized.ref

   val test_term:

     Proof.context -> term -> int -> term list option * Quickcheck.report option

   val setup : theory -> theory

   val quickcheck_setup : theory -> theory

 end;

 structure Inductive_Codegen : INDUCTIVE_CODEGEN =

 struct

 (**** theory data ****)

 fun merge_rules tabs =

   Symtab.join (fn _ => AList.merge (Thm.eq_thm_prop) (K true)) tabs;

 structure CodegenData = Theory_Data

 (

   type T =

     {intros : (thm * (string * int)) list Symtab.table,

      graph : unit Graph.T,

      eqns : (thm * string) list Symtab.table};

   val empty =

     {intros = Symtab.empty, graph = Graph.empty, eqns = Symtab.empty};

   val extend = I;

   fun merge

     ({intros = intros1, graph = graph1, eqns = eqns1},

       {intros = intros2, graph = graph2, eqns = eqns2}) : T =

     {intros = merge_rules (intros1, intros2),

      graph = Graph.merge (K true) (graph1, graph2),

      eqns = merge_rules (eqns1, eqns2)};

 );

 fun warn thm = warning ("Inductive_Codegen: Not a proper clause:\n" ^

   Display.string_of_thm_without_context thm);

 fun add_node x g = Graph.new_node (x, ()) g handle Graph.DUP _ => g;

 fun add optmod optnparms = Thm.declaration_attribute (fn thm => Context.mapping (fn thy =>

   let

     val {intros, graph, eqns} = CodegenData.get thy;

     fun thyname_of s = (case optmod of

       NONE => Codegen.thyname_of_const thy s | SOME s => s);

   in (case Option.map strip_comb (try HOLogic.dest_Trueprop (concl_of thm)) of

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

         (case head_of t of

           Const (s, _) =>

             CodegenData.put {intros = intros, graph = graph,

                eqns = eqns |> Symtab.map_default (s, [])

                  (AList.update Thm.eq_thm_prop (thm, thyname_of s))} thy

         | _ => (warn thm; thy))

     | SOME (Const (s, _), _) =>

         let

           val cs = fold Term.add_const_names (Thm.prems_of thm) [];

           val rules = Symtab.lookup_list intros s;

           val nparms = (case optnparms of

             SOME k => k

           | NONE => (case rules of

              [] => (case try (Inductive.the_inductive (ProofContext.init_global thy)) s of

                  SOME (_, {raw_induct, ...}) =>

                    length (Inductive.params_of raw_induct)

                | NONE => 0)

             | xs => snd (snd (snd (split_last xs)))))

         in CodegenData.put

           {intros = intros |>

            Symtab.update (s, (AList.update Thm.eq_thm_prop

              (thm, (thyname_of s, nparms)) rules)),

            graph = fold_rev (Graph.add_edge o pair s) cs (fold add_node (s :: cs) graph),

            eqns = eqns} thy

         end

     | _ => (warn thm; thy))

   end) I);

 fun get_clauses thy s =

   let val {intros, graph, ...} = CodegenData.get thy

   in case Symtab.lookup intros s of

       NONE => (case try (Inductive.the_inductive (ProofContext.init_global thy)) s of

         NONE => NONE

       | SOME ({names, ...}, {intrs, raw_induct, ...}) =>

           SOME (names, Codegen.thyname_of_const thy s,

             length (Inductive.params_of raw_induct),

             Codegen.preprocess thy intrs))

     | SOME _ =>

         let

           val SOME names = find_first

             (fn xs => member (op =) xs s) (Graph.strong_conn graph);

           val intrs as (_, (thyname, nparms)) :: _ =

             maps (the o Symtab.lookup intros) names;

         in SOME (names, thyname, nparms, Codegen.preprocess thy (map fst (rev intrs))) end

   end;

 (**** check if a term contains only constructor functions ****)

 fun is_constrt thy =

   let

     val cnstrs = flat (maps

       (map (fn (_, (_, _, cs)) => map (apsnd length) cs) o #descr o snd)

       (Symtab.dest (Datatype_Data.get_all thy)));

     fun check t = (case strip_comb t of

         (Var _, []) => true

       | (Const (s, _), ts) => (case AList.lookup (op =) cnstrs s of

             NONE => false

           | SOME i => length ts = i andalso forall check ts)

       | _ => false)

   in check end;

 (**** check if a type is an equality type (i.e. doesn't contain fun) ****)

 fun is_eqT (Type (s, Ts)) = s <> "fun" andalso forall is_eqT Ts

   | is_eqT _ = true;

 (**** mode inference ****)

 fun string_of_mode (iss, is) = space_implode " -> " (map

   (fn NONE => "X"

     | SOME js => enclose "[" "]" (commas (map string_of_int js)))

        (iss @ [SOME is]));

 fun print_modes modes = Codegen.message ("Inferred modes:\n" ^

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

     (fn (m, rnd) => string_of_mode m ^

        (if rnd then " (random)" else "")) ms)) modes));

 val term_vs = map (fst o fst o dest_Var) o OldTerm.term_vars;

 val terms_vs = distinct (op =) o maps term_vs;

 (** collect all Vars in a term (with duplicates!) **)

 fun term_vTs tm =

   fold_aterms (fn Var ((x, _), T) => cons (x, T) | _ => I) tm [];

 fun get_args _ _ [] = ([], [])

   | get_args is i (x::xs) = (if member (op =) is i then apfst else apsnd) (cons x)

       (get_args is (i+1) xs);

 fun merge xs [] = xs

   | merge [] ys = ys

   | merge (x::xs) (y::ys) = if length x >= length y then x::merge xs (y::ys)

       else y::merge (x::xs) ys;

 fun subsets i j = if i <= j then

        let val is = subsets (i+1) j

        in merge (map (fn ks => i::ks) is) is end

      else [[]];

 fun cprod ([], ys) = []

   | cprod (x :: xs, ys) = map (pair x) ys @ cprod (xs, ys);

 fun cprods xss = List.foldr (map op :: o cprod) [[]] xss;

 datatype mode = Mode of ((int list option list * int list) * bool) * int list * mode option list;

 fun needs_random (Mode ((_, b), _, ms)) =

   b orelse exists (fn NONE => false | SOME m => needs_random m) ms;

 fun modes_of modes t =

   let

     val ks = 1 upto length (binder_types (fastype_of t));

     val default = [Mode ((([], ks), false), ks, [])];

     fun mk_modes name args = Option.map

      (maps (fn (m as ((iss, is), _)) =>

         let

           val (args1, args2) =

             if length args < length iss then

               error ("Too few arguments for inductive predicate " ^ name)

             else chop (length iss) args;

           val k = length args2;

           val prfx = 1 upto k

         in

           if not (is_prefix op = prfx is) then [] else

           let val is' = map (fn i => i - k) (List.drop (is, k))

           in map (fn x => Mode (m, is', x)) (cprods (map

             (fn (NONE, _) => [NONE]

               | (SOME js, arg) => map SOME (filter

                   (fn Mode (_, js', _) => js=js') (modes_of modes arg)))

                     (iss ~~ args1)))

           end

         end)) (AList.lookup op = modes name)

   in (case strip_comb t of

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

         [Mode ((([], [1]), false), [1], []), Mode ((([], [2]), false), [2], [])] @

         (if is_eqT T then [Mode ((([], [1, 2]), false), [1, 2], [])] else [])

     | (Const (name, _), args) => the_default default (mk_modes name args)

     | (Var ((name, _), _), args) => the (mk_modes name args)

     | (Free (name, _), args) => the (mk_modes name args)

     | _ => default)

   end;

 datatype indprem = Prem of term list * term * bool | Sidecond of term;

 fun missing_vars vs ts = subtract (fn (x, ((y, _), _)) => x = y) vs

   (fold Term.add_vars ts []);

 fun monomorphic_vars vs = null (fold (Term.add_tvarsT o snd) vs []);

 fun mode_ord p = int_ord (pairself (fn (Mode ((_, rnd), _, _), vs) =>

   length vs + (if null vs then 0 else 1) + (if rnd then 1 else 0)) p);

 fun select_mode_prem thy modes vs ps =

   sort (mode_ord o pairself (hd o snd))

     (filter_out (null o snd) (ps ~~ map

       (fn Prem (us, t, is_set) => sort mode_ord

           (List.mapPartial (fn m as Mode (_, is, _) =>

             let

               val (in_ts, out_ts) = get_args is 1 us;

               val (out_ts', in_ts') = List.partition (is_constrt thy) out_ts;

               val vTs = maps term_vTs out_ts';

               val dupTs = map snd (duplicates (op =) vTs) @

                 map_filter (AList.lookup (op =) vTs) vs;

               val missing_vs = missing_vars vs (t :: in_ts @ in_ts')

             in

               if forall (is_eqT o fastype_of) in_ts' andalso forall is_eqT dupTs

                 andalso monomorphic_vars missing_vs

               then SOME (m, missing_vs)

               else NONE

             end)

               (if is_set then [Mode ((([], []), false), [], [])]

                else modes_of modes t handle Option =>

                  error ("Bad predicate: " ^ Syntax.string_of_term_global thy t)))

         | Sidecond t =>

             let val missing_vs = missing_vars vs [t]

             in

               if monomorphic_vars missing_vs

               then [(Mode ((([], []), false), [], []), missing_vs)]

               else []

             end)

               ps));

 fun use_random () = member (op =) (!Codegen.mode) "random_ind";

 fun check_mode_clause thy arg_vs modes ((iss, is), rnd) (ts, ps) =

   let

     val modes' = modes @ map_filter

       (fn (_, NONE) => NONE | (v, SOME js) => SOME (v, [(([], js), false)]))

         (arg_vs ~~ iss);

     fun check_mode_prems vs rnd [] = SOME (vs, rnd)

       | check_mode_prems vs rnd ps = (case select_mode_prem thy modes' vs ps of

           (x, (m, []) :: _) :: _ => check_mode_prems

             (case x of Prem (us, _, _) => union (op =) vs (terms_vs us) | _ => vs)

             (rnd orelse needs_random m)

             (filter_out (equal x) ps)

         | (_, (_, vs') :: _) :: _ =>

             if use_random () then

               check_mode_prems (union (op =) vs (map (fst o fst) vs')) true ps

             else NONE

         | _ => NONE);

     val (in_ts, in_ts') = List.partition (is_constrt thy) (fst (get_args is 1 ts));

     val in_vs = terms_vs in_ts;

   in

     if forall is_eqT (map snd (duplicates (op =) (maps term_vTs in_ts))) andalso

       forall (is_eqT o fastype_of) in_ts'

     then (case check_mode_prems (union (op =) arg_vs in_vs) rnd ps of

        NONE => NONE

      | SOME (vs, rnd') =>

          let val missing_vs = missing_vars vs ts

          in

            if null missing_vs orelse

              use_random () andalso monomorphic_vars missing_vs

            then SOME (rnd' orelse not (null missing_vs))

            else NONE

          end)

     else NONE

   end;

 fun check_modes_pred thy arg_vs preds modes (p, ms) =

   let val SOME rs = AList.lookup (op =) preds p

   in (p, List.mapPartial (fn m as (m', _) =>

     let val xs = map (check_mode_clause thy arg_vs modes m) rs

     in case find_index is_none xs of

         ~1 => SOME (m', exists (fn SOME b => b) xs)

       | i => (Codegen.message ("Clause " ^ string_of_int (i+1) ^ " of " ^

         p ^ " violates mode " ^ string_of_mode m'); NONE)

     end) ms)

   end;

 fun fixp f (x : (string * ((int list option list * int list) * bool) list) list) =

   let val y = f x

   in if x = y then x else fixp f y end;

 fun infer_modes thy extra_modes arities arg_vs preds = fixp (fn modes =>

   map (check_modes_pred thy arg_vs preds (modes @ extra_modes)) modes)

     (map (fn (s, (ks, k)) => (s, map (rpair false) (cprod (cprods (map

       (fn NONE => [NONE]

         | SOME k' => map SOME (subsets 1 k')) ks),

       subsets 1 k)))) arities);

 (**** code generation ****)

 fun mk_eq (x::xs) =

   let fun mk_eqs _ [] = []

         | mk_eqs a (b::cs) = Codegen.str (a ^ " = " ^ b) :: mk_eqs b cs

   in mk_eqs x xs end;

 fun mk_tuple xs = Pretty.block (Codegen.str "(" ::

   flat (separate [Codegen.str ",", Pretty.brk 1] (map single xs)) @

   [Codegen.str ")"]);

 fun mk_v s (names, vs) =

   (case AList.lookup (op =) vs s of

     NONE => (s, (names, (s, [s])::vs))

   | SOME xs =>

       let val s' = Name.variant names s

       in (s', (s'::names, AList.update (op =) (s, s'::xs) vs)) end);

 fun distinct_v (Var ((s, 0), T)) nvs =

       let val (s', nvs') = mk_v s nvs

       in (Var ((s', 0), T), nvs') end

   | distinct_v (t $ u) nvs =

       let

         val (t', nvs') = distinct_v t nvs;

         val (u', nvs'') = distinct_v u nvs';

       in (t' $ u', nvs'') end

   | distinct_v t nvs = (t, nvs);

 fun is_exhaustive (Var _) = true

   | is_exhaustive (Const (@{const_name Pair}, _) $ t $ u) =

       is_exhaustive t andalso is_exhaustive u

   | is_exhaustive _ = false;

 fun compile_match nvs eq_ps out_ps success_p can_fail =

   let val eqs = flat (separate [Codegen.str " andalso", Pretty.brk 1]

     (map single (maps (mk_eq o snd) nvs @ eq_ps)));

   in

     Pretty.block

      ([Codegen.str "(fn ", mk_tuple out_ps, Codegen.str " =>", Pretty.brk 1] @

       (Pretty.block ((if null eqs then [] else Codegen.str "if " ::

          [Pretty.block eqs, Pretty.brk 1, Codegen.str "then "]) @

          (success_p ::

           (if null eqs then [] else [Pretty.brk 1, Codegen.str "else DSeq.empty"]))) ::

        (if can_fail then

           [Pretty.brk 1, Codegen.str "| _ => DSeq.empty)"]

         else [Codegen.str ")"])))

   end;

 fun modename module s (iss, is) gr =

   let val (id, gr') = if s = @{const_name HOL.eq} then (("", "equal"), gr)

     else Codegen.mk_const_id module s gr

   in (space_implode "__"

     (Codegen.mk_qual_id module id ::

       map (space_implode "_" o map string_of_int) (map_filter I iss @ [is])), gr')

   end;

 fun mk_funcomp brack s k p = (if brack then Codegen.parens else I)

   (Pretty.block [Pretty.block ((if k = 0 then [] else [Codegen.str "("]) @

     separate (Pretty.brk 1) (Codegen.str s :: replicate k (Codegen.str "|> ???")) @

     (if k = 0 then [] else [Codegen.str ")"])), Pretty.brk 1, p]);

 fun compile_expr thy defs dep module brack modes (NONE, t) gr =

       apfst single (Codegen.invoke_codegen thy defs dep module brack t gr)

   | compile_expr _ _ _ _ _ _ (SOME _, Var ((name, _), _)) gr =

       ([Codegen.str name], gr)

   | compile_expr thy defs dep module brack modes (SOME (Mode ((mode, _), _, ms)), t) gr =

       (case strip_comb t of

          (Const (name, _), args) =>

            if name = @{const_name HOL.eq} orelse AList.defined op = modes name then

              let

                val (args1, args2) = chop (length ms) args;

                val ((ps, mode_id), gr') = gr |> fold_map

                    (compile_expr thy defs dep module true modes) (ms ~~ args1)

                    ||>> modename module name mode;

                val (ps', gr'') = (case mode of

                    ([], []) => ([Codegen.str "()"], gr')

                  | _ => fold_map

                      (Codegen.invoke_codegen thy defs dep module true) args2 gr')

              in ((if brack andalso not (null ps andalso null ps') then

                single o Codegen.parens o Pretty.block else I)

                  (flat (separate [Pretty.brk 1]

                    ([Codegen.str mode_id] :: ps @ map single ps'))), gr')

              end

            else apfst (single o mk_funcomp brack "??" (length (binder_types (fastype_of t))))

              (Codegen.invoke_codegen thy defs dep module true t gr)

        | _ => apfst (single o mk_funcomp brack "??" (length (binder_types (fastype_of t))))

            (Codegen.invoke_codegen thy defs dep module true t gr));

 fun compile_clause thy defs dep module all_vs arg_vs modes (iss, is) (ts, ps) inp gr =

   let

     val modes' = modes @ map_filter

       (fn (_, NONE) => NONE | (v, SOME js) => SOME (v, [(([], js), false)]))

         (arg_vs ~~ iss);

     fun check_constrt t (names, eqs) =

       if is_constrt thy t then (t, (names, eqs))

       else

         let val s = Name.variant names "x";

         in (Var ((s, 0), fastype_of t), (s::names, (s, t)::eqs)) end;

     fun compile_eq (s, t) gr =

       apfst (Pretty.block o cons (Codegen.str (s ^ " = ")) o single)

         (Codegen.invoke_codegen thy defs dep module false t gr);

     val (in_ts, out_ts) = get_args is 1 ts;

     val (in_ts', (all_vs', eqs)) = fold_map check_constrt in_ts (all_vs, []);

     fun compile_prems out_ts' vs names [] gr =

           let

             val (out_ps, gr2) =

               fold_map (Codegen.invoke_codegen thy defs dep module false) out_ts gr;

             val (eq_ps, gr3) = fold_map compile_eq eqs gr2;

             val (out_ts'', (names', eqs')) = fold_map check_constrt out_ts' (names, []);

             val (out_ts''', nvs) =

               fold_map distinct_v out_ts'' (names', map (fn x => (x, [x])) vs);

             val (out_ps', gr4) =

               fold_map (Codegen.invoke_codegen thy defs dep module false) out_ts''' gr3;

             val (eq_ps', gr5) = fold_map compile_eq eqs' gr4;

             val vs' = distinct (op =) (flat (vs :: map term_vs out_ts'));

             val missing_vs = missing_vars vs' out_ts;

             val final_p = Pretty.block

               [Codegen.str "DSeq.single", Pretty.brk 1, mk_tuple out_ps]

           in

             if null missing_vs then

               (compile_match (snd nvs) (eq_ps @ eq_ps') out_ps'

                  final_p (exists (not o is_exhaustive) out_ts'''), gr5)

             else

               let

                 val (pat_p, gr6) = Codegen.invoke_codegen thy defs dep module true

                   (HOLogic.mk_tuple (map Var missing_vs)) gr5;

                 val gen_p =

                   Codegen.mk_gen gr6 module true [] ""

                     (HOLogic.mk_tupleT (map snd missing_vs))

               in

                 (compile_match (snd nvs) eq_ps' out_ps'

                    (Pretty.block [Codegen.str "DSeq.generator ", gen_p,

                       Codegen.str " :->", Pretty.brk 1,

                       compile_match [] eq_ps [pat_p] final_p false])

                    (exists (not o is_exhaustive) out_ts'''),

                  gr6)

               end

           end

       | compile_prems out_ts vs names ps gr =

           let

             val vs' = distinct (op =) (flat (vs :: map term_vs out_ts));

             val (out_ts', (names', eqs)) = fold_map check_constrt out_ts (names, []);

             val (out_ts'', nvs) = fold_map distinct_v out_ts' (names', map (fn x => (x, [x])) vs);

             val (out_ps, gr0) =

               fold_map (Codegen.invoke_codegen thy defs dep module false) out_ts'' gr;

             val (eq_ps, gr1) = fold_map compile_eq eqs gr0;

           in

             (case hd (select_mode_prem thy modes' vs' ps) of

                (p as Prem (us, t, is_set), (mode as Mode (_, js, _), []) :: _) =>

                  let

                    val ps' = filter_out (equal p) ps;

                    val (in_ts, out_ts''') = get_args js 1 us;

                    val (in_ps, gr2) =

                     fold_map (Codegen.invoke_codegen thy defs dep module true) in_ts gr1;

                    val (ps, gr3) =

                      if not is_set then

                        apfst (fn ps => ps @

                            (if null in_ps then [] else [Pretty.brk 1]) @

                            separate (Pretty.brk 1) in_ps)

                          (compile_expr thy defs dep module false modes

                            (SOME mode, t) gr2)

                      else

                        apfst (fn p =>

                          Pretty.breaks [Codegen.str "DSeq.of_list", Codegen.str "(case", p,

                          Codegen.str "of", Codegen.str "Set", Codegen.str "xs", Codegen.str "=>",

                          Codegen.str "xs)"])

                          (*this is a very strong assumption about the generated code!*)

                            (Codegen.invoke_codegen thy defs dep module true t gr2);

                    val (rest, gr4) = compile_prems out_ts''' vs' (fst nvs) ps' gr3;

                  in

                    (compile_match (snd nvs) eq_ps out_ps

                       (Pretty.block (ps @

                          [Codegen.str " :->", Pretty.brk 1, rest]))

                       (exists (not o is_exhaustive) out_ts''), gr4)

                  end

              | (p as Sidecond t, [(_, [])]) =>

                  let

                    val ps' = filter_out (equal p) ps;

                    val (side_p, gr2) = Codegen.invoke_codegen thy defs dep module true t gr1;

                    val (rest, gr3) = compile_prems [] vs' (fst nvs) ps' gr2;

                  in

                    (compile_match (snd nvs) eq_ps out_ps

                       (Pretty.block [Codegen.str "?? ", side_p,

                         Codegen.str " :->", Pretty.brk 1, rest])

                       (exists (not o is_exhaustive) out_ts''), gr3)

                  end

              | (_, (_, missing_vs) :: _) =>

                  let

                    val T = HOLogic.mk_tupleT (map snd missing_vs);

                    val (_, gr2) = Codegen.invoke_tycodegen thy defs dep module false T gr1;

                    val gen_p = Codegen.mk_gen gr2 module true [] "" T;

                    val (rest, gr3) = compile_prems

                      [HOLogic.mk_tuple (map Var missing_vs)] vs' (fst nvs) ps gr2

                  in

                    (compile_match (snd nvs) eq_ps out_ps

                       (Pretty.block [Codegen.str "DSeq.generator", Pretty.brk 1,

                         gen_p, Codegen.str " :->", Pretty.brk 1, rest])

                       (exists (not o is_exhaustive) out_ts''), gr3)

                  end)

           end;

     val (prem_p, gr') = compile_prems in_ts' arg_vs all_vs' ps gr ;

   in

     (Pretty.block [Codegen.str "DSeq.single", Pretty.brk 1, inp,

        Codegen.str " :->", Pretty.brk 1, prem_p], gr')

   end;

 fun compile_pred thy defs dep module prfx all_vs arg_vs modes s cls mode gr =

   let

     val xs = map Codegen.str (Name.variant_list arg_vs

       (map (fn i => "x" ^ string_of_int i) (snd mode)));

     val ((cl_ps, mode_id), gr') = gr |>

       fold_map (fn cl => compile_clause thy defs

         dep module all_vs arg_vs modes mode cl (mk_tuple xs)) cls ||>>

       modename module s mode

   in

     (Pretty.block

       ([Pretty.block (separate (Pretty.brk 1)

          (Codegen.str (prfx ^ mode_id) ::

            map Codegen.str arg_vs @

            (case mode of ([], []) => [Codegen.str "()"] | _ => xs)) @

          [Codegen.str " ="]),

         Pretty.brk 1] @

        flat (separate [Codegen.str " ++", Pretty.brk 1] (map single cl_ps))), (gr', "and "))

   end;

 fun compile_preds thy defs dep module all_vs arg_vs modes preds gr =

   let val (prs, (gr', _)) = fold_map (fn (s, cls) =>

     fold_map (fn (mode, _) => fn (gr', prfx') => compile_pred thy defs

       dep module prfx' all_vs arg_vs modes s cls mode gr')

         (((the o AList.lookup (op =) modes) s))) preds (gr, "fun ")

   in

     (space_implode "\n\n" (map Codegen.string_of (flat prs)) ^ ";\n\n", gr')

   end;

 (**** processing of introduction rules ****)

 exception Modes of

   (string * ((int list option list * int list) * bool) list) list *

   (string * (int option list * int)) list;

 fun lookup_modes gr dep = apfst flat (apsnd flat (ListPair.unzip

   (map ((fn (SOME (Modes x), _, _) => x | _ => ([], [])) o Codegen.get_node gr)

     (Graph.all_preds (fst gr) [dep]))));

 fun print_arities arities = Codegen.message ("Arities:\n" ^

   cat_lines (map (fn (s, (ks, k)) => s ^ ": " ^

     space_implode " -> " (map

       (fn NONE => "X" | SOME k' => string_of_int k')

         (ks @ [SOME k]))) arities));

 fun prep_intrs intrs =

   map (Codegen.rename_term o #prop o rep_thm o Drule.export_without_context) intrs;

 fun constrain cs [] = []

   | constrain cs ((s, xs) :: ys) =

       (s,

         case AList.lookup (op =) cs (s : string) of

           NONE => xs

         | SOME xs' => inter (op = o apfst fst) xs' xs) :: constrain cs ys;

 fun mk_extra_defs thy defs gr dep names module ts =

   fold (fn name => fn gr =>

     if member (op =) names name then gr

     else

       (case get_clauses thy name of

         NONE => gr

       | SOME (names, thyname, nparms, intrs) =>

           mk_ind_def thy defs gr dep names (Codegen.if_library thyname module)

             [] (prep_intrs intrs) nparms))

     (fold Term.add_const_names ts []) gr

 and mk_ind_def thy defs gr dep names module modecs intrs nparms =

   Codegen.add_edge_acyclic (hd names, dep) gr handle

     Graph.CYCLES (xs :: _) =>

       error ("Inductive_Codegen: illegal cyclic dependencies:\n" ^ commas xs)

   | Graph.UNDEF _ =>

     let

       val _ $ u = Logic.strip_imp_concl (hd intrs);

       val args = List.take (snd (strip_comb u), nparms);

       val arg_vs = maps term_vs args;

       fun get_nparms s = if member (op =) names s then SOME nparms else

         Option.map #3 (get_clauses thy s);

       fun dest_prem (_ $ (Const (@{const_name Set.member}, _) $ t $ u)) =

             Prem ([t], Envir.beta_eta_contract u, true)

         | dest_prem (_ $ ((eq as Const (@{const_name HOL.eq}, _)) $ t $ u)) =

             Prem ([t, u], eq, false)

         | dest_prem (_ $ t) =

             (case strip_comb t of

               (v as Var _, ts) => if member (op =) args v then Prem (ts, v, false) else Sidecond t

             | (c as Const (s, _), ts) =>

                 (case get_nparms s of

                   NONE => Sidecond t

                 | SOME k =>

                     let val (ts1, ts2) = chop k ts

                     in Prem (ts2, list_comb (c, ts1), false) end)

             | _ => Sidecond t);

       fun add_clause intr (clauses, arities) =

         let

           val _ $ t = Logic.strip_imp_concl intr;

           val (Const (name, T), ts) = strip_comb t;

           val (ts1, ts2) = chop nparms ts;

           val prems = map dest_prem (Logic.strip_imp_prems intr);

           val (Ts, Us) = chop nparms (binder_types T)

         in

           (AList.update op = (name, these (AList.lookup op = clauses name) @

              [(ts2, prems)]) clauses,

            AList.update op = (name, (map (fn U => (case strip_type U of

                  (Rs as _ :: _, @{typ bool}) => SOME (length Rs)

                | _ => NONE)) Ts,

              length Us)) arities)

         end;

       val gr' = mk_extra_defs thy defs

         (Codegen.add_edge (hd names, dep)

           (Codegen.new_node (hd names, (NONE, "", "")) gr)) (hd names) names module intrs;

       val (extra_modes, extra_arities) = lookup_modes gr' (hd names);

       val (clauses, arities) = fold add_clause intrs ([], []);

       val modes = constrain modecs

         (infer_modes thy extra_modes arities arg_vs clauses);

       val _ = print_arities arities;

       val _ = print_modes modes;

       val (s, gr'') = compile_preds thy defs (hd names) module (terms_vs intrs)

         arg_vs (modes @ extra_modes) clauses gr';

     in

       (Codegen.map_node (hd names)

         (K (SOME (Modes (modes, arities)), module, s)) gr'')

     end;

 fun find_mode gr dep s u modes is = (case find_first (fn Mode (_, js, _) => is=js)

   (modes_of modes u handle Option => []) of

      NONE => Codegen.codegen_error gr dep

        ("No such mode for " ^ s ^ ": " ^ string_of_mode ([], is))

    | mode => mode);

 fun mk_ind_call thy defs dep module is_query s T ts names thyname k intrs gr =

   let

     val (ts1, ts2) = chop k ts;

     val u = list_comb (Const (s, T), ts1);

     fun mk_mode (Const (@{const_name dummy_pattern}, _)) ((ts, mode), i) = ((ts, mode), i + 1)

       | mk_mode t ((ts, mode), i) = ((ts @ [t], mode @ [i]), i + 1);

     val module' = Codegen.if_library thyname module;

     val gr1 = mk_extra_defs thy defs

       (mk_ind_def thy defs gr dep names module'

       [] (prep_intrs intrs) k) dep names module' [u];

     val (modes, _) = lookup_modes gr1 dep;

     val (ts', is) =

       if is_query then fst (fold mk_mode ts2 (([], []), 1))

       else (ts2, 1 upto length (binder_types T) - k);

     val mode = find_mode gr1 dep s u modes is;

     val _ = if is_query orelse not (needs_random (the mode)) then ()

       else warning ("Illegal use of random data generators in " ^ s);

     val (in_ps, gr2) = fold_map (Codegen.invoke_codegen thy defs dep module true) ts' gr1;

     val (ps, gr3) = compile_expr thy defs dep module false modes (mode, u) gr2;

   in

     (Pretty.block (ps @ (if null in_ps then [] else [Pretty.brk 1]) @

        separate (Pretty.brk 1) in_ps), gr3)

   end;

 fun clause_of_eqn eqn =

   let

     val (t, u) = HOLogic.dest_eq (HOLogic.dest_Trueprop (concl_of eqn));

     val (Const (s, T), ts) = strip_comb t;

     val (Ts, U) = strip_type T

   in

     Codegen.rename_term (Logic.list_implies (prems_of eqn, HOLogic.mk_Trueprop

       (list_comb (Const (s ^ "_aux", Ts @ [U] ---> HOLogic.boolT), ts @ [u]))))

   end;

 fun mk_fun thy defs name eqns dep module module' gr =

   case try (Codegen.get_node gr) name of

     NONE =>

     let

       val clauses = map clause_of_eqn eqns;

       val pname = name ^ "_aux";

       val arity = length (snd (strip_comb (fst (HOLogic.dest_eq

         (HOLogic.dest_Trueprop (concl_of (hd eqns)))))));

       val mode = 1 upto arity;

       val ((fun_id, mode_id), gr') = gr |>

         Codegen.mk_const_id module' name ||>>

         modename module' pname ([], mode);

       val vars = map (fn i => Codegen.str ("x" ^ string_of_int i)) mode;

       val s = Codegen.string_of (Pretty.block

         [Codegen.mk_app false (Codegen.str ("fun " ^ snd fun_id)) vars, Codegen.str " =",

          Pretty.brk 1, Codegen.str "DSeq.hd", Pretty.brk 1,

          Codegen.parens (Pretty.block (separate (Pretty.brk 1) (Codegen.str mode_id ::

            vars)))]) ^ ";\n\n";

       val gr'' = mk_ind_def thy defs (Codegen.add_edge (name, dep)

         (Codegen.new_node (name, (NONE, module', s)) gr')) name [pname] module'

         [(pname, [([], mode)])] clauses 0;

       val (modes, _) = lookup_modes gr'' dep;

       val _ = find_mode gr'' dep pname (head_of (HOLogic.dest_Trueprop

         (Logic.strip_imp_concl (hd clauses)))) modes mode

     in (Codegen.mk_qual_id module fun_id, gr'') end

   | SOME _ =>

       (Codegen.mk_qual_id module (Codegen.get_const_id gr name), Codegen.add_edge (name, dep) gr);

 (* convert n-tuple to nested pairs *)

 fun conv_ntuple fs ts p =

   let

     val k = length fs;

     val xs = map_range (fn i => Codegen.str ("x" ^ string_of_int i)) (k + 1);

     val xs' = map (fn Bound i => nth xs (k - i)) ts;

     fun conv xs js =

       if member (op =) fs js then

         let

           val (p, xs') = conv xs (1::js);

           val (q, xs'') = conv xs' (2::js)

         in (mk_tuple [p, q], xs'') end

       else (hd xs, tl xs)

   in

     if k > 0 then

       Pretty.block

         [Codegen.str "DSeq.map (fn", Pretty.brk 1,

          mk_tuple xs', Codegen.str " =>", Pretty.brk 1, fst (conv xs []),

          Codegen.str ")", Pretty.brk 1, Codegen.parens p]

     else p

   end;

 fun inductive_codegen thy defs dep module brack t gr  = (case strip_comb t of

     (Const (@{const_name Collect}, _), [u]) =>

       let val (r, Ts, fs) = HOLogic.strip_psplits u

       in case strip_comb r of

           (Const (s, T), ts) =>

             (case (get_clauses thy s, Codegen.get_assoc_code thy (s, T)) of

               (SOME (names, thyname, k, intrs), NONE) =>

                 let

                   val (ts1, ts2) = chop k ts;

                   val ts2' = map

                     (fn Bound i => Term.dummy_pattern (nth Ts (length Ts - i - 1)) | t => t) ts2;

                   val (ots, its) = List.partition is_Bound ts2;

                   val no_loose = forall (fn t => not (loose_bvar (t, 0)))

                 in

                   if null (duplicates op = ots) andalso

                     no_loose ts1 andalso no_loose its

                   then

                     let val (call_p, gr') = mk_ind_call thy defs dep module true

                       s T (ts1 @ ts2') names thyname k intrs gr 

                     in SOME ((if brack then Codegen.parens else I) (Pretty.block

                       [Codegen.str "Set", Pretty.brk 1, Codegen.str "(DSeq.list_of", Pretty.brk 1,

                        Codegen.str "(", conv_ntuple fs ots call_p, Codegen.str "))"]),

                        (*this is a very strong assumption about the generated code!*)

                        gr')

                     end

                   else NONE

                 end

             | _ => NONE)

         | _ => NONE

       end

   | (Const (s, T), ts) =>

     (case Symtab.lookup (#eqns (CodegenData.get thy)) s of

       NONE =>

       (case (get_clauses thy s, Codegen.get_assoc_code thy (s, T)) of

         (SOME (names, thyname, k, intrs), NONE) =>

           if length ts < k then NONE else SOME

             (let val (call_p, gr') = mk_ind_call thy defs dep module false

                s T (map Term.no_dummy_patterns ts) names thyname k intrs gr

              in (mk_funcomp brack "?!"

                (length (binder_types T) - length ts) (Codegen.parens call_p), gr')

              end handle TERM _ => mk_ind_call thy defs dep module true

                s T ts names thyname k intrs gr )

       | _ => NONE)

     | SOME eqns =>

         let

           val (_, thyname) :: _ = eqns;

           val (id, gr') = mk_fun thy defs s (Codegen.preprocess thy (map fst (rev eqns)))

             dep module (Codegen.if_library thyname module) gr;

           val (ps, gr'') = fold_map

             (Codegen.invoke_codegen thy defs dep module true) ts gr';

         in SOME (Codegen.mk_app brack (Codegen.str id) ps, gr'')

         end)

   | _ => NONE);

 val setup =

   Codegen.add_codegen "inductive" inductive_codegen #>

   Attrib.setup @{binding code_ind}

     (Scan.lift (Scan.option (Args.$$$ "target" |-- Args.colon |-- Args.name) --

       Scan.option (Args.$$$ "params" |-- Args.colon |-- Parse.nat) >> uncurry add))

     "introduction rules for executable predicates";

 (**** Quickcheck involving inductive predicates ****)

 val test_fn : (int * int * int -> term list option) Unsynchronized.ref =

   Unsynchronized.ref (fn _ => NONE);

 fun strip_imp p =

   let val (q, r) = HOLogic.dest_imp p

   in strip_imp r |>> cons q end

   handle TERM _ => ([], p);

 fun deepen bound f i =

   if i > bound then NONE

   else (case f i of

       NONE => deepen bound f (i + 1)

     | SOME x => SOME x);

 val (depth_bound, setup_depth_bound) = Attrib.config_int "ind_quickcheck_depth" (K 10);

 val (depth_start, setup_depth_start) = Attrib.config_int "ind_quickcheck_depth_start" (K 1);

 val (random_values, setup_random_values) = Attrib.config_int "ind_quickcheck_random" (K 5);

 val (size_offset, setup_size_offset) = Attrib.config_int "ind_quickcheck_size_offset" (K 0);

 fun test_term ctxt t =

   let

     val thy = ProofContext.theory_of ctxt;

     val (xs, p) = strip_abs t;

     val args' = map_index (fn (i, (_, T)) => ("arg" ^ string_of_int i, T)) xs;

     val args = map Free args';

     val (ps, q) = strip_imp p;

     val Ts = map snd xs;

     val T = Ts ---> HOLogic.boolT;

     val rl = Logic.list_implies

       (map (HOLogic.mk_Trueprop o curry subst_bounds (rev args)) ps @

        [HOLogic.mk_Trueprop (HOLogic.mk_not (subst_bounds (rev args, q)))],

        HOLogic.mk_Trueprop (list_comb (Free ("quickcheckp", T), args)));

     val (_, thy') = Inductive.add_inductive_global

       {quiet_mode=true, verbose=false, alt_name=Binding.empty, coind=false,

        no_elim=true, no_ind=false, skip_mono=false, fork_mono=false}

       [((Binding.name "quickcheckp", T), NoSyn)] []

       [(Attrib.empty_binding, rl)] [] (Theory.copy thy);

     val pred = HOLogic.mk_Trueprop (list_comb

       (Const (Sign.intern_const thy' "quickcheckp", T),

        map Term.dummy_pattern Ts));

     val (code, gr) = setmp_CRITICAL Codegen.mode ["term_of", "test", "random_ind"]

       (Codegen.generate_code_i thy' [] "Generated") [("testf", pred)];

     val s = "structure TestTerm =\nstruct\n\n" ^

       cat_lines (map snd code) ^

       "\nopen Generated;\n\n" ^ Codegen.string_of

         (Pretty.block [Codegen.str "val () = Inductive_Codegen.test_fn :=",

           Pretty.brk 1, Codegen.str "(fn p =>", Pretty.brk 1,

           Codegen.str "case Seq.pull (testf p) of", Pretty.brk 1,

           Codegen.str "SOME ", mk_tuple [mk_tuple (map (Codegen.str o fst) args'), Codegen.str "_"],

           Codegen.str " =>", Pretty.brk 1, Codegen.str "SOME ",

           Pretty.enum "," "[" "]"

             (map (fn (s, T) => Pretty.block

               [Codegen.mk_term_of gr "Generated" false T, Pretty.brk 1, Codegen.str s]) args'),

           Pretty.brk 1,

           Codegen.str "| NONE => NONE);"]) ^

       "\n\nend;\n";

     val _ = ML_Context.eval_text_in (SOME ctxt) false Position.none s;

     val values = Config.get ctxt random_values;

     val bound = Config.get ctxt depth_bound;

     val start = Config.get ctxt depth_start;

     val offset = Config.get ctxt size_offset;

     val test_fn' = !test_fn;

     fun test k = (deepen bound (fn i =>

       (Output.urgent_message ("Search depth: " ^ string_of_int i);

        test_fn' (i, values, k+offset))) start, NONE);

   in test end;

 val quickcheck_setup =

   setup_depth_bound #>

   setup_depth_start #>

   setup_random_values #>

   setup_size_offset #>

   Context.theory_map (Quickcheck.add_generator ("SML_inductive", test_term));

 end;