(*  Title:      HOL/BNF/Tools/bnf_fp_rec_sugar.ML

    Author:     Lorenz Panny, TU Muenchen

    Copyright   2013

Recursor and corecursor sugar.

*)

signature BNF_FP_REC_SUGAR =

sig

  val add_primrec_cmd: (binding * string option * mixfix) list ->

    (Attrib.binding * string) list -> local_theory -> local_theory;

  val add_primcorec_cmd: bool ->

    (binding * string option * mixfix) list * (Attrib.binding * string) list -> Proof.context ->

    Proof.state

end;

structure BNF_FP_Rec_Sugar : BNF_FP_REC_SUGAR =

struct

open BNF_Util

open BNF_FP_Util

open BNF_FP_Rec_Sugar_Util

open BNF_FP_Rec_Sugar_Tactics

exception Primrec_Error of string * term list;

fun primrec_error str = raise Primrec_Error (str, []);

fun primrec_error_eqn str eqn = raise Primrec_Error (str, [eqn]);

fun primrec_error_eqns str eqns = raise Primrec_Error (str, eqns);

fun finds eq = fold_map (fn x => List.partition (curry eq x) #>> pair x);

val free_name = try (fn Free (v, _) => v);

val const_name = try (fn Const (v, _) => v);

val undef_const = Const (@{const_name undefined}, dummyT);

fun permute_args n t = list_comb (t, map Bound (0 :: (n downto 1)))

  |> fold (K (Term.abs (Name.uu, dummyT))) (0 upto n);

val abs_tuple = HOLogic.tupled_lambda o HOLogic.mk_tuple;

fun drop_All t = subst_bounds (strip_qnt_vars @{const_name all} t |> map Free |> rev,

  strip_qnt_body @{const_name all} t)

fun mk_not @{const True} = @{const False}

  | mk_not @{const False} = @{const True}

  | mk_not (@{const Not} $ t) = t

  | mk_not (@{const Trueprop} $ t) = @{const Trueprop} $ mk_not t

  | mk_not t = HOLogic.mk_not t

val mk_conjs = try (foldr1 HOLogic.mk_conj) #> the_default @{const True};

val mk_disjs = try (foldr1 HOLogic.mk_disj) #> the_default @{const False};

fun invert_prems [t] = map mk_not (HOLogic.disjuncts t)

  | invert_prems ts = [mk_disjs (map mk_not ts)];

fun invert_prems_disj [t] = map mk_not (HOLogic.disjuncts t)

  | invert_prems_disj ts = [mk_disjs (map (mk_conjs o map mk_not o HOLogic.disjuncts) ts)];

fun abstract vs =

  let fun a n (t $ u) = a n t $ a n u

        | a n (Abs (v, T, b)) = Abs (v, T, a (n + 1) b)

        | a n t = let val idx = find_index (equal t) vs in

            if idx < 0 then t else Bound (n + idx) end

  in a 0 end;

fun mk_prod1 Ts (t, u) = HOLogic.pair_const (fastype_of1 (Ts, t)) (fastype_of1 (Ts, u)) $ t $ u;

fun mk_tuple1 Ts = the_default HOLogic.unit o try (foldr1 (mk_prod1 Ts));

val simp_attrs = @{attributes [simp]};

(* Primrec *)

type eqn_data = {

  fun_name: string,

  rec_type: typ,

  ctr: term,

  ctr_args: term list,

  left_args: term list,

  right_args: term list,

  res_type: typ,

  rhs_term: term,

  user_eqn: term

};

fun dissect_eqn lthy fun_names eqn' =

  let

    val eqn = drop_All eqn' |> HOLogic.dest_Trueprop

      handle TERM _ =>

        primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn';

    val (lhs, rhs) = HOLogic.dest_eq eqn

        handle TERM _ =>

          primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn';

    val (fun_name, args) = strip_comb lhs

      |>> (fn x => if is_Free x then fst (dest_Free x)

          else primrec_error_eqn "malformed function equation (does not start with free)" eqn);

    val (left_args, rest) = take_prefix is_Free args;

    val (nonfrees, right_args) = take_suffix is_Free rest;

    val _ = length nonfrees = 1 orelse if length nonfrees = 0 then

      primrec_error_eqn "constructor pattern missing in left-hand side" eqn else

      primrec_error_eqn "more than one non-variable argument in left-hand side" eqn;

    val _ = member (op =) fun_names fun_name orelse

      primrec_error_eqn "malformed function equation (does not start with function name)" eqn

    val (ctr, ctr_args) = strip_comb (the_single nonfrees);

    val _ = try (num_binder_types o fastype_of) ctr = SOME (length ctr_args) orelse

      primrec_error_eqn "partially applied constructor in pattern" eqn;

    val _ = let val d = duplicates (op =) (left_args @ ctr_args @ right_args) in null d orelse

      primrec_error_eqn ("duplicate variable \"" ^ Syntax.string_of_term lthy (hd d) ^

        "\" in left-hand side") eqn end;

    val _ = forall is_Free ctr_args orelse

      primrec_error_eqn "non-primitive pattern in left-hand side" eqn;

    val _ =

      let val b = fold_aterms (fn x as Free (v, _) =>

        if (not (member (op =) (left_args @ ctr_args @ right_args) x) andalso

        not (member (op =) fun_names v) andalso

        not (Variable.is_fixed lthy v)) then cons x else I | _ => I) rhs []

      in

        null b orelse

        primrec_error_eqn ("extra variable(s) in right-hand side: " ^

          commas (map (Syntax.string_of_term lthy) b)) eqn

      end;

  in

    {fun_name = fun_name,

     rec_type = body_type (type_of ctr),

     ctr = ctr,

     ctr_args = ctr_args,

     left_args = left_args,

     right_args = right_args,

     res_type = map fastype_of (left_args @ right_args) ---> fastype_of rhs,

     rhs_term = rhs,

     user_eqn = eqn'}

  end;

fun rewrite_map_arg get_ctr_pos rec_type res_type =

  let

    val pT = HOLogic.mk_prodT (rec_type, res_type);

    val maybe_suc = Option.map (fn x => x + 1);

    fun subst d (t as Bound d') = t |> d = SOME d' ? curry (op $) (fst_const pT)

      | subst d (Abs (v, T, b)) = Abs (v, if d = SOME ~1 then pT else T, subst (maybe_suc d) b)

      | subst d t =

        let

          val (u, vs) = strip_comb t;

          val ctr_pos = try (get_ctr_pos o the) (free_name u) |> the_default ~1;

        in

          if ctr_pos >= 0 then

            if d = SOME ~1 andalso length vs = ctr_pos then

              list_comb (permute_args ctr_pos (snd_const pT), vs)

            else if length vs > ctr_pos andalso is_some d

                andalso d = try (fn Bound n => n) (nth vs ctr_pos) then

              list_comb (snd_const pT $ nth vs ctr_pos, map (subst d) (nth_drop ctr_pos vs))

            else

              primrec_error_eqn ("recursive call not directly applied to constructor argument") t

          else if d = SOME ~1 andalso const_name u = SOME @{const_name comp} then

            list_comb (map_types (K dummyT) u, map2 subst [NONE, d] vs)

          else

            list_comb (u, map (subst (d |> d = SOME ~1 ? K NONE)) vs)

        end

  in

    subst (SOME ~1)

  end;

fun subst_rec_calls lthy get_ctr_pos has_call ctr_args direct_calls indirect_calls t =

  let

    fun subst bound_Ts (Abs (v, T, b)) = Abs (v, T, subst (T :: bound_Ts) b)

      | subst bound_Ts (t as g' $ y) =

        let

          val maybe_direct_y' = AList.lookup (op =) direct_calls y;

          val maybe_indirect_y' = AList.lookup (op =) indirect_calls y;

          val (g, g_args) = strip_comb g';

          val ctr_pos = try (get_ctr_pos o the) (free_name g) |> the_default ~1;

          val _ = ctr_pos < 0 orelse length g_args >= ctr_pos orelse

            primrec_error_eqn "too few arguments in recursive call" t;

        in

          if not (member (op =) ctr_args y) then

            pairself (subst bound_Ts) (g', y) |> (op $)

          else if ctr_pos >= 0 then

            list_comb (the maybe_direct_y', g_args)

          else if is_some maybe_indirect_y' then

            (if has_call g' then t else y)

            |> massage_indirect_rec_call lthy has_call

              (rewrite_map_arg get_ctr_pos) bound_Ts y (the maybe_indirect_y')

            |> (if has_call g' then I else curry (op $) g')

          else

            t

        end

      | subst _ t = t

  in

    subst [] t

    |> tap (fn u => has_call u andalso (* FIXME detect this case earlier *)

      primrec_error_eqn "recursive call not directly applied to constructor argument" t)

  end;

fun build_rec_arg lthy funs_data has_call ctr_spec maybe_eqn_data =

  if is_none maybe_eqn_data then undef_const else

    let

      val eqn_data = the maybe_eqn_data;

      val t = #rhs_term eqn_data;

      val ctr_args = #ctr_args eqn_data;

      val calls = #calls ctr_spec;

      val n_args = fold (curry (op +) o (fn Direct_Rec _ => 2 | _ => 1)) calls 0;

      val no_calls' = tag_list 0 calls

        |> map_filter (try (apsnd (fn No_Rec n => n | Direct_Rec (n, _) => n)));

      val direct_calls' = tag_list 0 calls

        |> map_filter (try (apsnd (fn Direct_Rec (_, n) => n)));

      val indirect_calls' = tag_list 0 calls

        |> map_filter (try (apsnd (fn Indirect_Rec n => n)));

      fun make_direct_type T = dummyT; (* FIXME? *)

      val rec_res_type_list = map (fn (x :: _) => (#rec_type x, #res_type x)) funs_data;

      fun make_indirect_type (Type (Tname, Ts)) = Type (Tname, Ts |> map (fn T =>

        let val maybe_res_type = AList.lookup (op =) rec_res_type_list T in

          if is_some maybe_res_type

          then HOLogic.mk_prodT (T, the maybe_res_type)

          else make_indirect_type T end))

        | make_indirect_type T = T;

      val args = replicate n_args ("", dummyT)

        |> Term.rename_wrt_term t

        |> map Free

        |> fold (fn (ctr_arg_idx, arg_idx) =>

            nth_map arg_idx (K (nth ctr_args ctr_arg_idx)))

          no_calls'

        |> fold (fn (ctr_arg_idx, arg_idx) =>

            nth_map arg_idx (K (nth ctr_args ctr_arg_idx |> map_types make_direct_type)))

          direct_calls'

        |> fold (fn (ctr_arg_idx, arg_idx) =>

            nth_map arg_idx (K (nth ctr_args ctr_arg_idx |> map_types make_indirect_type)))

          indirect_calls';

      val fun_name_ctr_pos_list =

        map (fn (x :: _) => (#fun_name x, length (#left_args x))) funs_data;

      val get_ctr_pos = try (the o AList.lookup (op =) fun_name_ctr_pos_list) #> the_default ~1;

      val direct_calls = map (apfst (nth ctr_args) o apsnd (nth args)) direct_calls';

      val indirect_calls = map (apfst (nth ctr_args) o apsnd (nth args)) indirect_calls';

      val abstractions = args @ #left_args eqn_data @ #right_args eqn_data;

    in

      t

      |> subst_rec_calls lthy get_ctr_pos has_call ctr_args direct_calls indirect_calls

      |> fold_rev lambda abstractions

    end;

fun build_defs lthy bs mxs funs_data rec_specs has_call =

  let

    val n_funs = length funs_data;

    val ctr_spec_eqn_data_list' =

      (take n_funs rec_specs |> map #ctr_specs) ~~ funs_data

      |> maps (uncurry (finds (fn (x, y) => #ctr x = #ctr y))

          ##> (fn x => null x orelse

            primrec_error_eqns "excess equations in definition" (map #rhs_term x)) #> fst);

    val _ = ctr_spec_eqn_data_list' |> map (fn (_, x) => length x <= 1 orelse

      primrec_error_eqns ("multiple equations for constructor") (map #user_eqn x));

    val ctr_spec_eqn_data_list =

      ctr_spec_eqn_data_list' @ (drop n_funs rec_specs |> maps #ctr_specs |> map (rpair []));

    val recs = take n_funs rec_specs |> map #recx;

    val rec_args = ctr_spec_eqn_data_list

      |> sort ((op <) o pairself (#offset o fst) |> make_ord)

      |> map (uncurry (build_rec_arg lthy funs_data has_call) o apsnd (try the_single));

    val ctr_poss = map (fn x =>

      if length (distinct ((op =) o pairself (length o #left_args)) x) <> 1 then

        primrec_error ("inconstant constructor pattern position for function " ^

          quote (#fun_name (hd x)))

      else

        hd x |> #left_args |> length) funs_data;

  in

    (recs, ctr_poss)

    |-> map2 (fn recx => fn ctr_pos => list_comb (recx, rec_args) |> permute_args ctr_pos)

    |> Syntax.check_terms lthy

    |> map3 (fn b => fn mx => fn t => ((b, mx), ((Binding.map_name Thm.def_name b, []), t))) bs mxs

  end;

fun find_rec_calls has_call eqn_data =

  let

    fun find (Abs (_, _, b)) ctr_arg = find b ctr_arg

      | find (t as _ $ _) ctr_arg =

        let

          val (f', args') = strip_comb t;

          val n = find_index (equal ctr_arg) args';

        in

          if n < 0 then

            find f' ctr_arg @ maps (fn x => find x ctr_arg) args'

          else

            let val (f, args) = chop n args' |>> curry list_comb f' in

              if has_call f then

                f :: maps (fn x => find x ctr_arg) args

              else

                find f ctr_arg @ maps (fn x => find x ctr_arg) args

            end

        end

      | find _ _ = [];

  in

    map (find (#rhs_term eqn_data)) (#ctr_args eqn_data)

    |> (fn [] => NONE | callss => SOME (#ctr eqn_data, callss))

  end;

fun add_primrec fixes specs lthy =

  let

    val (bs, mxs) = map_split (apfst fst) fixes;

    val fun_names = map Binding.name_of bs;

    val eqns_data = map (snd #> dissect_eqn lthy fun_names) specs;

    val funs_data = eqns_data

      |> partition_eq ((op =) o pairself #fun_name)

      |> finds (fn (x, y) => x = #fun_name (hd y)) fun_names |> fst

      |> map (fn (x, y) => the_single y handle List.Empty =>

          primrec_error ("missing equations for function " ^ quote x));

    val has_call = exists_subterm (map (fst #>> Binding.name_of #> Free) fixes |> member (op =));

    val arg_Ts = map (#rec_type o hd) funs_data;

    val res_Ts = map (#res_type o hd) funs_data;

    val callssss = funs_data

      |> map (partition_eq ((op =) o pairself #ctr))

      |> map (maps (map_filter (find_rec_calls has_call)));

    fun get_indices t = map (fst #>> Binding.name_of #> Free) fixes

      |> map_index (fn (i, v) => if exists_subterm (equal v) t then SOME i else NONE)

      |> map_filter I;

    val ((nontriv, rec_specs, _, induct_thm, induct_thms), lthy') =

      rec_specs_of bs arg_Ts res_Ts get_indices callssss lthy;

    val actual_nn = length funs_data;

    val _ = let val ctrs = (maps (map #ctr o #ctr_specs) rec_specs) in

      map (fn {ctr, user_eqn, ...} => member (op =) ctrs ctr orelse

        primrec_error_eqn ("argument " ^ quote (Syntax.string_of_term lthy' ctr) ^

          " is not a constructor in left-hand side") user_eqn) eqns_data end;

    val defs = build_defs lthy' bs mxs funs_data rec_specs has_call;

    fun prove def_thms' {ctr_specs, nested_map_idents, nested_map_comps, ...} induct_thm fun_data

        lthy =

      let

        val fun_name = #fun_name (hd fun_data);

        val def_thms = map (snd o snd) def_thms';

        val simp_thms = finds (fn (x, y) => #ctr x = #ctr y) fun_data ctr_specs

          |> fst

          |> map_filter (try (fn (x, [y]) =>

            (#user_eqn x, length (#left_args x) + length (#right_args x), #rec_thm y)))

          |> map (fn (user_eqn, num_extra_args, rec_thm) =>

            mk_primrec_tac lthy num_extra_args nested_map_idents nested_map_comps def_thms rec_thm

            |> K |> Goal.prove lthy [] [] user_eqn)

        val notes =

          [(inductN, if actual_nn > 1 then [induct_thm] else [], []),

           (simpsN, simp_thms, simp_attrs)]

          |> filter_out (null o #2)

          |> map (fn (thmN, thms, attrs) =>

            ((Binding.qualify true fun_name (Binding.name thmN), attrs), [(thms, [])]));

      in

        lthy |> Local_Theory.notes notes

      end;

    val common_name = mk_common_name fun_names;

    val common_notes =

      [(inductN, if nontriv andalso actual_nn > 1 then [induct_thm] else [], [])]

      |> filter_out (null o #2)

      |> map (fn (thmN, thms, attrs) =>

        ((Binding.qualify true common_name (Binding.name thmN), attrs), [(thms, [])]));

  in

    lthy'

    |> fold_map Local_Theory.define defs

    |-> snd oo (fn def_thms' => fold_map3 (prove def_thms') (take actual_nn rec_specs)

      (take actual_nn induct_thms) funs_data)

    |> snd o Local_Theory.notes common_notes

  end;

fun add_primrec_cmd raw_fixes raw_specs lthy =

  let

    val _ = let val d = duplicates (op =) (map (Binding.name_of o #1) raw_fixes) in null d orelse

      primrec_error ("duplicate function name(s): " ^ commas d) end;

    val (fixes, specs) = fst (Specification.read_spec raw_fixes raw_specs lthy);

  in

    add_primrec fixes specs lthy

      handle ERROR str => primrec_error str

  end

  handle Primrec_Error (str, eqns) =>

    if null eqns

    then error ("primrec_new error:\n  " ^ str)

    else error ("primrec_new error:\n  " ^ str ^ "\nin\n  " ^

      space_implode "\n  " (map (quote o Syntax.string_of_term lthy) eqns))

(* Primcorec *)

type co_eqn_data_disc = {

  fun_name: string,

  fun_T: typ,

  fun_args: term list,

  ctr: term,

  ctr_no: int, (*###*)

  disc: term,

  prems: term list,

  user_eqn: term

};

type co_eqn_data_sel = {

  fun_name: string,

  fun_T: typ,

  fun_args: term list,

  ctr: term,

  sel: term,

  rhs_term: term,

  user_eqn: term

};

datatype co_eqn_data =

  Disc of co_eqn_data_disc |

  Sel of co_eqn_data_sel;

fun co_dissect_eqn_disc sequential fun_names corec_specs prems' concl matchedsss =

  let

    fun find_subterm p = let (* FIXME \<exists>? *)

      fun f (t as u $ v) = if p t then SOME t else merge_options (f u, f v)

        | f t = if p t then SOME t else NONE

      in f end;

    val applied_fun = concl

      |> find_subterm (member ((op =) o apsnd SOME) fun_names o try (fst o dest_Free o head_of))

      |> the

      handle Option.Option => primrec_error_eqn "malformed discriminator equation" concl;

    val ((fun_name, fun_T), fun_args) = strip_comb applied_fun |>> dest_Free;

    val {ctr_specs, ...} = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name);

    val discs = map #disc ctr_specs;

    val ctrs = map #ctr ctr_specs;

    val not_disc = head_of concl = @{term Not};

    val _ = not_disc andalso length ctrs <> 2 andalso

      primrec_error_eqn "\<not>ed discriminator for a type with \<noteq> 2 constructors" concl;

    val disc = find_subterm (member (op =) discs o head_of) concl;

    val eq_ctr0 = concl |> perhaps (try (HOLogic.dest_not)) |> try (HOLogic.dest_eq #> snd)

        |> (fn SOME t => let val n = find_index (equal t) ctrs in

          if n >= 0 then SOME n else NONE end | _ => NONE);

    val _ = is_some disc orelse is_some eq_ctr0 orelse

      primrec_error_eqn "no discriminator in equation" concl;

    val ctr_no' =

      if is_none disc then the eq_ctr0 else find_index (equal (head_of (the disc))) discs;

    val ctr_no = if not_disc then 1 - ctr_no' else ctr_no';

    val ctr = #ctr (nth ctr_specs ctr_no);

    val catch_all = try (fst o dest_Free o the_single) prems' = SOME Name.uu_;

    val matchedss = AList.lookup (op =) matchedsss fun_name |> the_default [];

    val prems = map (abstract (List.rev fun_args)) prems';

    val real_prems =

      (if catch_all orelse sequential then maps invert_prems_disj matchedss else []) @

      (if catch_all then [] else prems);

    val matchedsss' = AList.delete (op =) fun_name matchedsss

      |> cons (fun_name, if sequential then matchedss @ [prems] else matchedss @ [real_prems]);

    val user_eqn =

      (real_prems, betapply (#disc (nth ctr_specs ctr_no), applied_fun))

      |>> map HOLogic.mk_Trueprop ||> HOLogic.mk_Trueprop

      |> Logic.list_implies;

  in

    (Disc {

      fun_name = fun_name,

      fun_T = fun_T,

      fun_args = fun_args,

      ctr = ctr,

      ctr_no = ctr_no,

      disc = #disc (nth ctr_specs ctr_no),

      prems = real_prems,

      user_eqn = user_eqn

    }, matchedsss')

  end;

fun co_dissect_eqn_sel fun_names corec_specs eqn' eqn =

  let

    val (lhs, rhs) = HOLogic.dest_eq eqn

      handle TERM _ =>

        primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn;

    val sel = head_of lhs;

    val ((fun_name, fun_T), fun_args) = dest_comb lhs |> snd |> strip_comb |> apfst dest_Free

      handle TERM _ =>

        primrec_error_eqn "malformed selector argument in left-hand side" eqn;

    val corec_spec = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name)

      handle Option.Option => primrec_error_eqn "malformed selector argument in left-hand side" eqn;

    val (ctr_spec, sel) = #ctr_specs corec_spec

      |> the o get_index (try (the o find_first (equal sel) o #sels))

      |>> nth (#ctr_specs corec_spec);

    val user_eqn = drop_All eqn';

  in

    Sel {

      fun_name = fun_name,

      fun_T = fun_T,

      fun_args = fun_args,

      ctr = #ctr ctr_spec,

      sel = sel,

      rhs_term = rhs,

      user_eqn = user_eqn

    }

  end;

fun co_dissect_eqn_ctr sequential fun_names corec_specs eqn' imp_prems imp_rhs matchedsss =

  let 

    val (lhs, rhs) = HOLogic.dest_eq imp_rhs;

    val fun_name = head_of lhs |> fst o dest_Free;

    val {ctr_specs, ...} = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name);

    val (ctr, ctr_args) = strip_comb rhs;

    val {disc, sels, ...} = the (find_first (equal ctr o #ctr) ctr_specs)

      handle Option.Option => primrec_error_eqn "not a constructor" ctr;

    val disc_imp_rhs = betapply (disc, lhs);

    val (maybe_eqn_data_disc, matchedsss') = if length ctr_specs = 1

      then (NONE, matchedsss)

      else apfst SOME (co_dissect_eqn_disc

          sequential fun_names corec_specs imp_prems disc_imp_rhs matchedsss);

    val sel_imp_rhss = (sels ~~ ctr_args)

      |> map (fn (sel, ctr_arg) => HOLogic.mk_eq (betapply (sel, lhs), ctr_arg));

val _ = tracing ("reduced\n    " ^ Syntax.string_of_term @{context} imp_rhs ^ "\nto\n    \<cdot> " ^

 (is_some maybe_eqn_data_disc ? K (Syntax.string_of_term @{context} disc_imp_rhs ^ "\n    \<cdot> ")) "" ^

 space_implode "\n    \<cdot> " (map (Syntax.string_of_term @{context}) sel_imp_rhss));

    val eqns_data_sel =

      map (co_dissect_eqn_sel fun_names corec_specs eqn') sel_imp_rhss;

  in

    (the_list maybe_eqn_data_disc @ eqns_data_sel, matchedsss')

  end;

fun co_dissect_eqn sequential fun_names corec_specs eqn' matchedsss =

  let

    val eqn = drop_All eqn'

      handle TERM _ => primrec_error_eqn "malformed function equation" eqn';

    val (imp_prems, imp_rhs) = Logic.strip_horn eqn

      |> apfst (map HOLogic.dest_Trueprop) o apsnd HOLogic.dest_Trueprop;

    val head = imp_rhs

      |> perhaps (try HOLogic.dest_not) |> perhaps (try (fst o HOLogic.dest_eq))

      |> head_of;

    val maybe_rhs = imp_rhs |> perhaps (try (HOLogic.dest_not)) |> try (snd o HOLogic.dest_eq);

    val discs = maps #ctr_specs corec_specs |> map #disc;

    val sels = maps #ctr_specs corec_specs |> maps #sels;

    val ctrs = maps #ctr_specs corec_specs |> map #ctr;

  in

    if member (op =) discs head orelse

      is_some maybe_rhs andalso

        member (op =) (filter (null o binder_types o fastype_of) ctrs) (the maybe_rhs) then

      co_dissect_eqn_disc sequential fun_names corec_specs imp_prems imp_rhs matchedsss

      |>> single

    else if member (op =) sels head then

      ([co_dissect_eqn_sel fun_names corec_specs eqn' imp_rhs], matchedsss)

    else if is_Free head andalso member (op =) fun_names (fst (dest_Free head)) then

      co_dissect_eqn_ctr sequential fun_names corec_specs eqn' imp_prems imp_rhs matchedsss

    else

      primrec_error_eqn "malformed function equation" eqn

  end;

fun build_corec_arg_disc ctr_specs {fun_args, ctr_no, prems, ...} =

  if is_none (#pred (nth ctr_specs ctr_no)) then I else

    mk_conjs prems

    |> curry subst_bounds (List.rev fun_args)

    |> HOLogic.tupled_lambda (HOLogic.mk_tuple fun_args)

    |> K |> nth_map (the (#pred (nth ctr_specs ctr_no)));

fun build_corec_arg_no_call sel_eqns sel =

  find_first (equal sel o #sel) sel_eqns

  |> try (fn SOME {fun_args, rhs_term, ...} => abs_tuple fun_args rhs_term)

  |> the_default undef_const

  |> K;

fun build_corec_args_direct_call lthy has_call sel_eqns sel =

  let

    val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns;

    fun rewrite_q t = if has_call t then @{term False} else @{term True};

    fun rewrite_g t = if has_call t then undef_const else t;

    fun rewrite_h t = if has_call t then HOLogic.mk_tuple (snd (strip_comb t)) else undef_const;

    fun massage _ NONE t = t

      | massage f (SOME {fun_args, rhs_term, ...}) t =

        massage_direct_corec_call lthy has_call f (range_type (fastype_of t)) rhs_term

        |> abs_tuple fun_args;

  in

    (massage rewrite_q maybe_sel_eqn,

     massage rewrite_g maybe_sel_eqn,

     massage rewrite_h maybe_sel_eqn)

  end;

fun build_corec_arg_indirect_call lthy has_call sel_eqns sel =

  let

    val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns;

    fun rewrite bound_Ts U T (Abs (v, V, b)) = Abs (v, V, rewrite (V :: bound_Ts) U T b)

      | rewrite bound_Ts U T (t as _ $ _) =

        let val (u, vs) = strip_comb t in

          if is_Free u andalso has_call u then

            Inr_const U T $ mk_tuple1 bound_Ts vs

          else if try (fst o dest_Const) u = SOME @{const_name prod_case} then

            list_comb (map_types (K dummyT) u, map (rewrite bound_Ts U T) vs)

          else

            list_comb (rewrite bound_Ts U T u, map (rewrite bound_Ts U T) vs)

        end

      | rewrite _ U T t = if is_Free t andalso has_call t then Inr_const U T $ HOLogic.unit else t;

    fun massage NONE t = t

      | massage (SOME {fun_args, rhs_term, ...}) t =

        massage_indirect_corec_call lthy has_call (rewrite []) []

          (range_type (fastype_of t)) rhs_term

        |> abs_tuple fun_args;

  in

    massage maybe_sel_eqn

  end;

fun build_corec_args_sel lthy has_call all_sel_eqns ctr_spec =

  let val sel_eqns = filter (equal (#ctr ctr_spec) o #ctr) all_sel_eqns in

    if null sel_eqns then I else

      let

        val sel_call_list = #sels ctr_spec ~~ #calls ctr_spec;

        val no_calls' = map_filter (try (apsnd (fn No_Corec n => n))) sel_call_list;

        val direct_calls' = map_filter (try (apsnd (fn Direct_Corec n => n))) sel_call_list;

        val indirect_calls' = map_filter (try (apsnd (fn Indirect_Corec n => n))) sel_call_list;

      in

        I

        #> fold (fn (sel, n) => nth_map n (build_corec_arg_no_call sel_eqns sel)) no_calls'

        #> fold (fn (sel, (q, g, h)) =>

          let val (fq, fg, fh) = build_corec_args_direct_call lthy has_call sel_eqns sel in

            nth_map q fq o nth_map g fg o nth_map h fh end) direct_calls'

        #> fold (fn (sel, n) => nth_map n

          (build_corec_arg_indirect_call lthy has_call sel_eqns sel)) indirect_calls'

      end

  end;

fun co_build_defs lthy bs mxs has_call arg_Tss corec_specs disc_eqnss sel_eqnss =

  let

    val corec_specs' = take (length bs) corec_specs;

    val corecs = map #corec corec_specs';

    val ctr_specss = map #ctr_specs corec_specs';

    val corec_args = hd corecs

      |> fst o split_last o binder_types o fastype_of

      |> map (Const o pair @{const_name undefined})

      |> fold2 (fold o build_corec_arg_disc) ctr_specss disc_eqnss

      |> fold2 (fold o build_corec_args_sel lthy has_call) sel_eqnss ctr_specss;

    fun currys [] t = t

      | currys Ts t = t $ mk_tuple1 (List.rev Ts) (map Bound (length Ts - 1 downto 0))

          |> fold_rev (Term.abs o pair Name.uu) Ts;

val _ = tracing ("corecursor arguments:\n    \<cdot> " ^

 space_implode "\n    \<cdot> " (map (Syntax.string_of_term lthy) corec_args));

    val exclss' =

      disc_eqnss

      |> map (map (fn {fun_args, ctr_no, prems, ...} => (fun_args, ctr_no, prems))

        #> fst o (fn xs => fold_map (fn x => fn ys => ((x, ys), ys @ [x])) xs [])

        #> maps (uncurry (map o pair)

          #> map (fn ((fun_args, c, x), (_, c', y)) => ((c, c'), (x, mk_not (mk_conjs y)))

            ||> apfst (map HOLogic.mk_Trueprop) o apsnd HOLogic.mk_Trueprop

            ||> Logic.list_implies

            ||> curry Logic.list_all (map dest_Free fun_args))))

  in

    map (list_comb o rpair corec_args) corecs

    |> map2 (fn Ts => fn t => if length Ts = 0 then t $ HOLogic.unit else t) arg_Tss

    |> map2 currys arg_Tss

    |> Syntax.check_terms lthy

    |> map3 (fn b => fn mx => fn t => ((b, mx), ((Binding.map_name Thm.def_name b, []), t))) bs mxs

    |> rpair exclss'

  end;

fun mk_real_disc_eqns fun_binding arg_Ts {ctr_specs, ...} sel_eqns disc_eqns =

  if length disc_eqns <> length ctr_specs - 1 then disc_eqns else

    let

      val n = 0 upto length ctr_specs

        |> the o find_first (fn idx => not (exists (equal idx o #ctr_no) disc_eqns));

      val fun_args = (try (#fun_args o hd) disc_eqns, try (#fun_args o hd) sel_eqns)

        |> the_default (map (curry Free Name.uu) arg_Ts) o merge_options;

      val extra_disc_eqn = {

        fun_name = Binding.name_of fun_binding,

        fun_T = arg_Ts ---> body_type (fastype_of (#ctr (hd ctr_specs))),

        fun_args = fun_args,

        ctr = #ctr (nth ctr_specs n),

        ctr_no = n,

        disc = #disc (nth ctr_specs n),

        prems = maps (invert_prems o #prems) disc_eqns,

        user_eqn = undef_const};

    in

      chop n disc_eqns ||> cons extra_disc_eqn |> (op @)

    end;

fun add_primcorec sequential fixes specs lthy =

  let

    val (bs, mxs) = map_split (apfst fst) fixes;

    val (arg_Ts, res_Ts) = map (strip_type o snd o fst #>> HOLogic.mk_tupleT) fixes |> split_list;

    (* copied from primrec_new *)

    fun get_indices t = map (fst #>> Binding.name_of #> Free) fixes

      |> map_index (fn (i, v) => if exists_subterm (equal v) t then SOME i else NONE)

      |> map_filter I;

    val callssss = []; (* FIXME *)

    val ((nontriv, corec_specs', _, coinduct_thm, strong_co_induct_thm, coinduct_thmss,

          strong_coinduct_thmss), lthy') =

      corec_specs_of bs arg_Ts res_Ts get_indices callssss lthy;

    val fun_names = map Binding.name_of bs;

    val corec_specs = take (length fun_names) corec_specs'; (*###*)

    val (eqns_data, _) =

      fold_map (co_dissect_eqn sequential fun_names corec_specs) (map snd specs) []

      |>> flat;

    val disc_eqnss' = map_filter (try (fn Disc x => x)) eqns_data

      |> partition_eq ((op =) o pairself #fun_name)

      |> fst o finds (fn (x, ({fun_name, ...} :: _)) => x = fun_name) fun_names

      |> map (sort ((op <) o pairself #ctr_no |> make_ord) o flat o snd);

    val _ = disc_eqnss' |> map (fn x =>

      let val d = duplicates ((op =) o pairself #ctr_no) x in null d orelse

        primrec_error_eqns "excess discriminator equations in definition"

          (maps (fn t => filter (equal (#ctr_no t) o #ctr_no) x) d |> map #user_eqn) end);

    val sel_eqnss = map_filter (try (fn Sel x => x)) eqns_data

      |> partition_eq ((op =) o pairself #fun_name)

      |> fst o finds (fn (x, ({fun_name, ...} :: _)) => x = fun_name) fun_names

      |> map (flat o snd);

    val has_call = exists_subterm (map (fst #>> Binding.name_of #> Free) fixes |> member (op =));

    val arg_Tss = map (binder_types o snd o fst) fixes;

    val disc_eqnss = map5 mk_real_disc_eqns bs arg_Tss corec_specs sel_eqnss disc_eqnss';

    val (defs, exclss') =

      co_build_defs lthy' bs mxs has_call arg_Tss corec_specs disc_eqnss sel_eqnss;

    (* try to prove (automatically generated) tautologies by ourselves *)

    val exclss'' = exclss'

      |> map (map (apsnd

        (`(try (fn t => Goal.prove lthy [] [] t (mk_primcorec_assumption_tac lthy |> K))))));

    val taut_thmss = map (map (apsnd (the o fst)) o filter (is_some o fst o snd)) exclss'';

    val (obligation_idxss, obligationss) = exclss''

      |> map (map (apsnd (rpair [] o snd)) o filter (is_none o fst o snd))

      |> split_list o map split_list;

    fun prove thmss' def_thms' lthy =

      let

        val def_thms = map (snd o snd) def_thms';

        val exclss' = map (op ~~) (obligation_idxss ~~ thmss');

        fun mk_exclsss excls n =

          (excls, map (fn k => replicate k [TrueI] @ replicate (n - k) []) (0 upto n - 1))

          |-> fold (fn ((c, c'), thm) => nth_map c (nth_map c' (K [thm])));

        val exclssss = (exclss' ~~ taut_thmss |> map (op @), fun_names ~~ corec_specs)

          |-> map2 (fn excls => fn (_, {ctr_specs, ...}) => mk_exclsss excls (length ctr_specs));

        fun prove_disc {ctr_specs, ...} exclsss

            {fun_name, fun_T, fun_args, ctr_no, prems, user_eqn, ...} =

          if Term.aconv_untyped (#disc (nth ctr_specs ctr_no), @{term "\<lambda>x. x = x"}) then [] else

            let

              val {disc_corec, ...} = nth ctr_specs ctr_no;

              val k = 1 + ctr_no;

              val m = length prems;

              val t =

                list_comb (Free (fun_name, fun_T), map Bound (length fun_args - 1 downto 0))

                |> curry betapply (#disc (nth ctr_specs ctr_no)) (*###*)

                |> HOLogic.mk_Trueprop

                |> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)

                |> curry Logic.list_all (map dest_Free fun_args);

            in

              mk_primcorec_disc_tac lthy def_thms disc_corec k m exclsss

              |> K |> Goal.prove lthy [] [] t

              |> pair (#disc (nth ctr_specs ctr_no))

              |> single

            end;

        fun prove_sel {ctr_specs, nested_maps, nested_map_idents, nested_map_comps, ...}

            disc_eqns exclsss {fun_name, fun_T, fun_args, ctr, sel, rhs_term, ...} =

          let

            val (SOME ctr_spec) = find_first (equal ctr o #ctr) ctr_specs;

            val ctr_no = find_index (equal ctr o #ctr) ctr_specs;

            val prems = the_default (maps (invert_prems o #prems) disc_eqns)

                (find_first (equal ctr_no o #ctr_no) disc_eqns |> Option.map #prems);

            val sel_corec = find_index (equal sel) (#sels ctr_spec)

              |> nth (#sel_corecs ctr_spec);

            val k = 1 + ctr_no;

            val m = length prems;

            val t =

              list_comb (Free (fun_name, fun_T), map Bound (length fun_args - 1 downto 0))

              |> curry betapply sel

              |> rpair (abstract (List.rev fun_args) rhs_term)

              |> HOLogic.mk_Trueprop o HOLogic.mk_eq

              |> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)

              |> curry Logic.list_all (map dest_Free fun_args);

          in

            mk_primcorec_ctr_or_sel_tac lthy def_thms sel_corec k m exclsss

              nested_maps nested_map_idents nested_map_comps

            |> K |> Goal.prove lthy [] [] t

            |> pair sel

          end;

        fun prove_ctr (_, disc_thms) (_, sel_thms') disc_eqns sel_eqns

            {ctr, disc, sels, collapse, ...} =

          if not (exists (equal ctr o #ctr) disc_eqns)

              andalso not (exists (equal ctr o #ctr) sel_eqns)

andalso (warning ("no eqns for ctr " ^ Syntax.string_of_term lthy ctr); true)

            orelse (* don't try to prove theorems when some sel_eqns are missing *)

              filter (equal ctr o #ctr) sel_eqns

              |> fst o finds ((op =) o apsnd #sel) sels

              |> exists (null o snd)

andalso (warning ("sel_eqn(s) missing for ctr " ^ Syntax.string_of_term lthy ctr); true)

          then [] else

            let

val _ = tracing ("ctr = " ^ Syntax.string_of_term lthy ctr);

val _ = tracing (the_default "no disc_eqn" (Option.map (curry (op ^) "disc = " o Syntax.string_of_term lthy o #disc) (find_first (equal ctr o #ctr) disc_eqns)));

              val (fun_name, fun_T, fun_args, prems) =

                (find_first (equal ctr o #ctr) disc_eqns, find_first (equal ctr o #ctr) sel_eqns)

                |>> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, #prems x))

                ||> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, []))

                |> the o merge_options;

              val m = length prems;

              val t = sel_eqns

                |> fst o finds ((op =) o apsnd #sel) sels

                |> map (snd #> (fn [x] => (List.rev (#fun_args x), #rhs_term x)) #-> abstract)

                |> curry list_comb ctr

                |> curry HOLogic.mk_eq (list_comb (Free (fun_name, fun_T),

                  map Bound (length fun_args - 1 downto 0)))

                |> HOLogic.mk_Trueprop

                |> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)

                |> curry Logic.list_all (map dest_Free fun_args);

              val maybe_disc_thm = AList.lookup (op =) disc_thms disc;

              val sel_thms = map snd (filter (member (op =) sels o fst) sel_thms');

val _ = tracing ("t = " ^ Syntax.string_of_term lthy t);

val _ = tracing ("m = " ^ @{make_string} m);

val _ = tracing ("collapse = " ^ @{make_string} collapse);

val _ = tracing ("maybe_disc_thm = " ^ @{make_string} maybe_disc_thm);

val _ = tracing ("sel_thms = " ^ @{make_string} sel_thms);

            in

              mk_primcorec_ctr_of_dtr_tac lthy m collapse maybe_disc_thm sel_thms

              |> K |> Goal.prove lthy [] [] t

              |> single

          end;

        (* TODO: please reorganize so that the list looks like elsewhere in the BNF code.

           This is important because we continually add new theorems, change attributes, etc., and

           need to have a clear overview (and keep the documentation in sync). Also, it's confusing

           that some variables called '_thmss' are actually pairs. *)

        val (disc_notes, disc_thmss) =

          fun_names ~~ map3 (maps oo prove_disc) corec_specs exclssss disc_eqnss

          |> `(map (fn (fun_name, thms) =>

            ((Binding.qualify true fun_name (@{binding disc}), simp_attrs), [(map snd thms, [])])));

        val (sel_notes, sel_thmss) =

          fun_names ~~ map4 (map ooo prove_sel) corec_specs disc_eqnss exclssss sel_eqnss

          |> `(map (fn (fun_name, thms) =>

            ((Binding.qualify true fun_name (@{binding sel}), simp_attrs), [(map snd thms, [])])));

        val ctr_notes =

          fun_names ~~ map5 (maps oooo prove_ctr) disc_thmss sel_thmss

            disc_eqnss sel_eqnss (map #ctr_specs corec_specs)

          |> map (fn (fun_name, thms) =>

            ((Binding.qualify true fun_name (@{binding ctr}), []), [(thms, [])]));

      in

        lthy |> snd o Local_Theory.notes

          (filter (not o null o fst o the_single o snd) (disc_notes @ sel_notes @ ctr_notes))

      end;

  in

    lthy'

    |> Proof.theorem NONE (curry (op #->) (fold_map Local_Theory.define defs) o prove) obligationss

    |> Proof.refine (Method.primitive_text I)

    |> Seq.hd

  end

fun add_primcorec_cmd seq (raw_fixes, raw_specs) lthy =

  let

    val (fixes, specs) = fst (Specification.read_spec raw_fixes raw_specs lthy);

  in

    add_primcorec seq fixes specs lthy

    handle ERROR str => primrec_error str

  end

  handle Primrec_Error (str, eqns) =>

    if null eqns

    then error ("primcorec error:\n  " ^ str)

    else error ("primcorec error:\n  " ^ str ^ "\nin\n  " ^

      space_implode "\n  " (map (quote o Syntax.string_of_term lthy) eqns))

end;