src/HOL/BNF/Tools/bnf_gfp_rec_sugar.ML

prevent exception when equations for a function are missing;
pave the way for "exhaustive"

(*  Title:      HOL/BNF/Tools/bnf_gfp_rec_sugar.ML
    Author:     Lorenz Panny, TU Muenchen
    Author:     Jasmin Blanchette, TU Muenchen
    Copyright   2013

Corecursor sugar.
*)

signature BNF_GFP_REC_SUGAR =
sig
  datatype primcorec_option =
    Option_Sequential |
    Option_Exhaustive
  val add_primcorecursive_cmd: primcorec_option list ->
    (binding * string option * mixfix) list * ((Attrib.binding * string) * string option) list ->
    Proof.context -> Proof.state
  val add_primcorec_cmd: primcorec_option list ->
    (binding * string option * mixfix) list * ((Attrib.binding * string) * string option) list ->
    local_theory -> local_theory
end;

structure BNF_GFP_Rec_Sugar : BNF_GFP_REC_SUGAR =
struct

open Ctr_Sugar
open BNF_Util
open BNF_Def
open BNF_FP_Util
open BNF_FP_Def_Sugar
open BNF_FP_N2M_Sugar
open BNF_FP_Rec_Sugar_Util
open BNF_GFP_Rec_Sugar_Tactics

val codeN = "code"
val ctrN = "ctr"
val discN = "disc"
val selN = "sel"

val nitpicksimp_attrs = @{attributes [nitpick_simp]};
val simp_attrs = @{attributes [simp]};
val code_nitpicksimp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs;

exception Primcorec_Error of string * term list;

fun primcorec_error str = raise Primcorec_Error (str, []);
fun primcorec_error_eqn str eqn = raise Primcorec_Error (str, [eqn]);
fun primcorec_error_eqns str eqns = raise Primcorec_Error (str, eqns);

datatype primcorec_option =
  Option_Sequential |
  Option_Exhaustive

datatype corec_call =
  Dummy_No_Corec of int |
  No_Corec of int |
  Mutual_Corec of int * int * int |
  Nested_Corec of int;

type basic_corec_ctr_spec =
  {ctr: term,
   disc: term,
   sels: term list};

type corec_ctr_spec =
  {ctr: term,
   disc: term,
   sels: term list,
   pred: int option,
   calls: corec_call list,
   discI: thm,
   sel_thms: thm list,
   collapse: thm,
   corec_thm: thm,
   disc_corec: thm,
   sel_corecs: thm list};

type corec_spec =
  {corec: term,
   nested_map_idents: thm list,
   nested_map_comps: thm list,
   ctr_specs: corec_ctr_spec list};

exception AINT_NO_MAP of term;

fun not_codatatype ctxt T =
  error ("Not a codatatype: " ^ Syntax.string_of_typ ctxt T);
fun ill_formed_corec_call ctxt t =
  error ("Ill-formed corecursive call: " ^ quote (Syntax.string_of_term ctxt t));
fun invalid_map ctxt t =
  error ("Invalid map function in " ^ quote (Syntax.string_of_term ctxt t));
fun unexpected_corec_call ctxt t =
  error ("Unexpected corecursive call: " ^ quote (Syntax.string_of_term ctxt 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};

val conjuncts_s = filter_out (curry (op =) @{const True}) o HOLogic.conjuncts;

fun s_not @{const True} = @{const False}
  | s_not @{const False} = @{const True}
  | s_not (@{const Not} $ t) = t
  | s_not (@{const conj} $ t $ u) = @{const disj} $ s_not t $ s_not u
  | s_not (@{const disj} $ t $ u) = @{const conj} $ s_not t $ s_not u
  | s_not t = @{const Not} $ t;

val s_not_conj = conjuncts_s o s_not o mk_conjs;

fun propagate_unit_pos u cs = if member (op aconv) cs u then [@{const False}] else cs;

fun propagate_unit_neg not_u cs = remove (op aconv) not_u cs;

fun propagate_units css =
  (case List.partition (can the_single) css of
     ([], _) => css
   | ([u] :: uss, css') =>
     [u] :: propagate_units (map (propagate_unit_neg (s_not u))
       (map (propagate_unit_pos u) (uss @ css'))));

fun s_conjs cs =
  if member (op aconv) cs @{const False} then @{const False}
  else mk_conjs (remove (op aconv) @{const True} cs);

fun s_disjs ds =
  if member (op aconv) ds @{const True} then @{const True}
  else mk_disjs (remove (op aconv) @{const False} ds);

fun s_dnf css0 =
  let val css = propagate_units css0 in
    if null css then
      [@{const False}]
    else if exists null css then
      []
    else
      map (fn c :: cs => (c, cs)) css
      |> AList.coalesce (op =)
      |> map (fn (c, css) => c :: s_dnf css)
      |> (fn [cs] => cs | css => [s_disjs (map s_conjs css)])
  end;

fun fold_rev_let_if_case ctxt f bound_Ts t =
  let
    val thy = Proof_Context.theory_of ctxt;

    fun fld conds t =
      (case Term.strip_comb t of
        (Const (@{const_name Let}, _), [_, _]) => fld conds (unfold_let t)
      | (Const (@{const_name If}, _), [cond, then_branch, else_branch]) =>
        fld (conds @ conjuncts_s cond) then_branch o fld (conds @ s_not_conj [cond]) else_branch
      | (Const (c, _), args as _ :: _ :: _) =>
        let val n = num_binder_types (Sign.the_const_type thy c) - 1 in
          if n >= 0 andalso n < length args then
            (case fastype_of1 (bound_Ts, nth args n) of
              Type (s, Ts) =>
              (case dest_case ctxt s Ts t of
                NONE => apsnd (f conds t)
              | SOME (conds', branches) =>
                apfst (cons s) o fold_rev (uncurry fld)
                  (map (append conds o conjuncts_s) conds' ~~ branches))
            | _ => apsnd (f conds t))
          else
            apsnd (f conds t)
        end
      | _ => apsnd (f conds t))
  in
    fld [] t o pair []
  end;

fun case_of ctxt = ctr_sugar_of ctxt #> Option.map (fst o dest_Const o #casex);

fun massage_let_if_case ctxt has_call massage_leaf =
  let
    val thy = Proof_Context.theory_of ctxt;

    fun check_no_call t = if has_call t then unexpected_corec_call ctxt t else ();

    fun massage_abs bound_Ts 0 t = massage_rec bound_Ts t
      | massage_abs bound_Ts m (Abs (s, T, t)) = Abs (s, T, massage_abs (T :: bound_Ts) (m - 1) t)
      | massage_abs bound_Ts m t =
        let val T = domain_type (fastype_of1 (bound_Ts, t)) in
          Abs (Name.uu, T, massage_abs (T :: bound_Ts) (m - 1) (incr_boundvars 1 t $ Bound 0))
        end
    and massage_rec bound_Ts t =
      let val typof = curry fastype_of1 bound_Ts in
        (case Term.strip_comb t of
          (Const (@{const_name Let}, _), [_, _]) => massage_rec bound_Ts (unfold_let t)
        | (Const (@{const_name If}, _), obj :: (branches as [_, _])) =>
          let val branches' = map (massage_rec bound_Ts) branches in
            Term.list_comb (If_const (typof (hd branches')) $ tap check_no_call obj, branches')
          end
        | (Const (c, _), args as _ :: _ :: _) =>
          (case try strip_fun_type (Sign.the_const_type thy c) of
            SOME (gen_branch_Ts, gen_body_fun_T) =>
            let
              val gen_branch_ms = map num_binder_types gen_branch_Ts;
              val n = length gen_branch_ms;
            in
              if n < length args then
                (case gen_body_fun_T of
                  Type (_, [Type (T_name, _), _]) =>
                  if case_of ctxt T_name = SOME c then
                    let
                      val (branches, obj_leftovers) = chop n args;
                      val branches' = map2 (massage_abs bound_Ts) gen_branch_ms branches;
                      val branch_Ts' = map typof branches';
                      val body_T' = snd (strip_typeN (hd gen_branch_ms) (hd branch_Ts'));
                      val casex' = Const (c, branch_Ts' ---> map typof obj_leftovers ---> body_T');
                    in
                      Term.list_comb (casex',
                        branches' @ tap (List.app check_no_call) obj_leftovers)
                    end
                  else
                    massage_leaf bound_Ts t
                | _ => massage_leaf bound_Ts t)
              else
                massage_leaf bound_Ts t
            end
          | NONE => massage_leaf bound_Ts t)
        | _ => massage_leaf bound_Ts t)
      end
  in
    massage_rec
  end;

val massage_mutual_corec_call = massage_let_if_case;

fun curried_type (Type (@{type_name fun}, [Type (@{type_name prod}, Ts), T])) = Ts ---> T;

fun massage_nested_corec_call ctxt has_call raw_massage_call bound_Ts U t =
  let
    fun check_no_call t = if has_call t then unexpected_corec_call ctxt t else ();

    val build_map_Inl = build_map ctxt (uncurry Inl_const o dest_sumT o snd);

    fun massage_mutual_call bound_Ts U T t =
      if has_call t then
        (case try dest_sumT U of
          SOME (U1, U2) => if U1 = T then raw_massage_call bound_Ts T U2 t else invalid_map ctxt t
        | NONE => invalid_map ctxt t)
      else
        build_map_Inl (T, U) $ t;

    fun massage_mutual_fun bound_Ts U T t =
      (case t of
        Const (@{const_name comp}, _) $ t1 $ t2 =>
        mk_comp bound_Ts (massage_mutual_fun bound_Ts U T t1, tap check_no_call t2)
      | _ =>
        let
          val var = Var ((Name.uu, Term.maxidx_of_term t + 1),
            domain_type (fastype_of1 (bound_Ts, t)));
        in
          Term.lambda var (massage_mutual_call bound_Ts U T (t $ var))
        end);

    fun massage_map bound_Ts (Type (_, Us)) (Type (s, Ts)) t =
        (case try (dest_map ctxt s) t of
          SOME (map0, fs) =>
          let
            val Type (_, dom_Ts) = domain_type (fastype_of1 (bound_Ts, t));
            val map' = mk_map (length fs) dom_Ts Us map0;
            val fs' =
              map_flattened_map_args ctxt s (map3 (massage_map_or_map_arg bound_Ts) Us Ts) fs;
          in
            Term.list_comb (map', fs')
          end
        | NONE => raise AINT_NO_MAP t)
      | massage_map _ _ _ t = raise AINT_NO_MAP t
    and massage_map_or_map_arg bound_Ts U T t =
      if T = U then
        tap check_no_call t
      else
        massage_map bound_Ts U T t
        handle AINT_NO_MAP _ => massage_mutual_fun bound_Ts U T t;

    fun massage_call bound_Ts U T =
      massage_let_if_case ctxt has_call (fn bound_Ts => fn t =>
        if has_call t then
          (case t of
            Const (@{const_name prod_case}, _) $ t' =>
            let
              val U' = curried_type U;
              val T' = curried_type T;
            in
              Const (@{const_name prod_case}, U' --> U) $ massage_call bound_Ts U' T' t'
            end
          | t1 $ t2 =>
            (if has_call t2 then
              massage_mutual_call bound_Ts U T t
            else
              massage_map bound_Ts U T t1 $ t2
              handle AINT_NO_MAP _ => massage_mutual_call bound_Ts U T t)
          | Abs (s, T', t') =>
            Abs (s, T', massage_call (T' :: bound_Ts) (range_type U) (range_type T) t')
          | _ => massage_mutual_call bound_Ts U T t)
        else
          build_map_Inl (T, U) $ t) bound_Ts;

    val T = fastype_of1 (bound_Ts, t);
  in
    if has_call t then massage_call bound_Ts U T t else build_map_Inl (T, U) $ t
  end;

val fold_rev_corec_call = fold_rev_let_if_case;

fun expand_to_ctr_term ctxt s Ts t =
  (case ctr_sugar_of ctxt s of
    SOME {ctrs, casex, ...} =>
    Term.list_comb (mk_case Ts (Type (s, Ts)) casex, map (mk_ctr Ts) ctrs) $ t
  | NONE => raise Fail "expand_to_ctr_term");

fun expand_corec_code_rhs ctxt has_call bound_Ts t =
  (case fastype_of1 (bound_Ts, t) of
    Type (s, Ts) =>
    massage_let_if_case ctxt has_call (fn _ => fn t =>
      if can (dest_ctr ctxt s) t then t else expand_to_ctr_term ctxt s Ts t) bound_Ts t
  | _ => raise Fail "expand_corec_code_rhs");

fun massage_corec_code_rhs ctxt massage_ctr =
  massage_let_if_case ctxt (K false)
    (fn bound_Ts => uncurry (massage_ctr bound_Ts) o Term.strip_comb);

fun fold_rev_corec_code_rhs ctxt f =
  snd ooo fold_rev_let_if_case ctxt (fn conds => uncurry (f conds) o Term.strip_comb);

fun case_thms_of_term ctxt bound_Ts t =
  let
    val (caseT_names, _) = fold_rev_let_if_case ctxt (K (K I)) bound_Ts t ();
    val ctr_sugars = map (the o ctr_sugar_of ctxt) caseT_names;
  in
    (maps #distincts ctr_sugars, maps #discIs ctr_sugars, maps #sel_splits ctr_sugars,
     maps #sel_split_asms ctr_sugars)
  end;

fun basic_corec_specs_of ctxt res_T =
  (case res_T of
    Type (T_name, _) =>
    (case Ctr_Sugar.ctr_sugar_of ctxt T_name of
      NONE => not_codatatype ctxt res_T
    | SOME {ctrs, discs, selss, ...} =>
      let
        val thy = Proof_Context.theory_of ctxt;

        val gfpT = body_type (fastype_of (hd ctrs));
        val As_rho = tvar_subst thy [gfpT] [res_T];
        val substA = Term.subst_TVars As_rho;

        fun mk_spec ctr disc sels = {ctr = substA ctr, disc = substA disc, sels = map substA sels};
      in
        map3 mk_spec ctrs discs selss
      end)
  | _ => not_codatatype ctxt res_T);

fun corec_specs_of bs arg_Ts res_Ts get_indices callssss0 lthy =
  let
    val thy = Proof_Context.theory_of lthy;

    val ((missing_res_Ts, perm0_kks,
          fp_sugars as {nested_bnfs, fp_res = {xtor_co_iterss = dtor_coiters1 :: _, ...},
            co_inducts = coinduct_thms, ...} :: _, (_, gfp_sugar_thms)), lthy') =
      nested_to_mutual_fps Greatest_FP bs res_Ts get_indices callssss0 lthy;

    val perm_fp_sugars = sort (int_ord o pairself #index) fp_sugars;

    val indices = map #index fp_sugars;
    val perm_indices = map #index perm_fp_sugars;

    val perm_ctrss = map (#ctrs o of_fp_sugar #ctr_sugars) perm_fp_sugars;
    val perm_ctr_Tsss = map (map (binder_types o fastype_of)) perm_ctrss;
    val perm_gfpTs = map (body_type o fastype_of o hd) perm_ctrss;

    val nn0 = length res_Ts;
    val nn = length perm_gfpTs;
    val kks = 0 upto nn - 1;
    val perm_ns = map length perm_ctr_Tsss;

    val perm_Cs = map (domain_type o body_fun_type o fastype_of o co_rec_of o
      of_fp_sugar (#xtor_co_iterss o #fp_res)) perm_fp_sugars;
    val (perm_p_Tss, (perm_q_Tssss, _, perm_f_Tssss, _)) =
      mk_coiter_fun_arg_types perm_ctr_Tsss perm_Cs perm_ns (co_rec_of dtor_coiters1);

    val (perm_p_hss, h) = indexedd perm_p_Tss 0;
    val (perm_q_hssss, h') = indexedddd perm_q_Tssss h;
    val (perm_f_hssss, _) = indexedddd perm_f_Tssss h';

    val fun_arg_hs =
      flat (map3 flat_corec_preds_predsss_gettersss perm_p_hss perm_q_hssss perm_f_hssss);

    fun unpermute0 perm0_xs = permute_like (op =) perm0_kks kks perm0_xs;
    fun unpermute perm_xs = permute_like (op =) perm_indices indices perm_xs;

    val coinduct_thmss = map (unpermute0 o conj_dests nn) coinduct_thms;

    val p_iss = map (map (find_index_eq fun_arg_hs)) (unpermute perm_p_hss);
    val q_issss = map (map (map (map (find_index_eq fun_arg_hs)))) (unpermute perm_q_hssss);
    val f_issss = map (map (map (map (find_index_eq fun_arg_hs)))) (unpermute perm_f_hssss);

    val f_Tssss = unpermute perm_f_Tssss;
    val gfpTs = unpermute perm_gfpTs;
    val Cs = unpermute perm_Cs;

    val As_rho = tvar_subst thy (take nn0 gfpTs) res_Ts;
    val Cs_rho = map (fst o dest_TVar) Cs ~~ pad_list HOLogic.unitT nn arg_Ts;

    val substA = Term.subst_TVars As_rho;
    val substAT = Term.typ_subst_TVars As_rho;
    val substCT = Term.typ_subst_TVars Cs_rho;

    val perm_Cs' = map substCT perm_Cs;

    fun call_of nullary [] [g_i] [Type (@{type_name fun}, [_, T])] =
        (if exists_subtype_in Cs T then Nested_Corec
         else if nullary then Dummy_No_Corec
         else No_Corec) g_i
      | call_of _ [q_i] [g_i, g_i'] _ = Mutual_Corec (q_i, g_i, g_i');

    fun mk_ctr_spec ctr disc sels p_ho q_iss f_iss f_Tss discI sel_thms collapse corec_thm
        disc_corec sel_corecs =
      let val nullary = not (can dest_funT (fastype_of ctr)) in
        {ctr = substA ctr, disc = substA disc, sels = map substA sels, pred = p_ho,
         calls = map3 (call_of nullary) q_iss f_iss f_Tss, discI = discI, sel_thms = sel_thms,
         collapse = collapse, corec_thm = corec_thm, disc_corec = disc_corec,
         sel_corecs = sel_corecs}
      end;

    fun mk_ctr_specs index (ctr_sugars : ctr_sugar list) p_is q_isss f_isss f_Tsss coiter_thmsss
        disc_coitersss sel_coiterssss =
      let
        val ctrs = #ctrs (nth ctr_sugars index);
        val discs = #discs (nth ctr_sugars index);
        val selss = #selss (nth ctr_sugars index);
        val p_ios = map SOME p_is @ [NONE];
        val discIs = #discIs (nth ctr_sugars index);
        val sel_thmss = #sel_thmss (nth ctr_sugars index);
        val collapses = #collapses (nth ctr_sugars index);
        val corec_thms = co_rec_of (nth coiter_thmsss index);
        val disc_corecs = co_rec_of (nth disc_coitersss index);
        val sel_corecss = co_rec_of (nth sel_coiterssss index);
      in
        map13 mk_ctr_spec ctrs discs selss p_ios q_isss f_isss f_Tsss discIs sel_thmss collapses
          corec_thms disc_corecs sel_corecss
      end;

    fun mk_spec ({T, index, ctr_sugars, co_iterss = coiterss, co_iter_thmsss = coiter_thmsss,
          disc_co_itersss = disc_coitersss, sel_co_iterssss = sel_coiterssss, ...} : fp_sugar)
        p_is q_isss f_isss f_Tsss =
      {corec = mk_co_iter thy Greatest_FP (substAT T) perm_Cs' (co_rec_of (nth coiterss index)),
       nested_map_idents = map (unfold_thms lthy @{thms id_def} o map_id0_of_bnf) nested_bnfs,
       nested_map_comps = map map_comp_of_bnf nested_bnfs,
       ctr_specs = mk_ctr_specs index ctr_sugars p_is q_isss f_isss f_Tsss coiter_thmsss
         disc_coitersss sel_coiterssss};
  in
    ((is_some gfp_sugar_thms, map5 mk_spec fp_sugars p_iss q_issss f_issss f_Tssss, missing_res_Ts,
      co_induct_of coinduct_thms, strong_co_induct_of coinduct_thms, co_induct_of coinduct_thmss,
      strong_co_induct_of coinduct_thmss), lthy')
  end;

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

val abs_tuple = HOLogic.tupled_lambda o HOLogic.mk_tuple;
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 bound_Ts (t, u) =
  HOLogic.pair_const (fastype_of1 (bound_Ts, t)) (fastype_of1 (bound_Ts, u)) $ t $ u;
fun mk_tuple1 bound_Ts = the_default HOLogic.unit o try (foldr1 (mk_prod1 bound_Ts));

type coeqn_data_disc = {
  fun_name: string,
  fun_T: typ,
  fun_args: term list,
  ctr: term,
  ctr_no: int, (*###*)
  disc: term,
  prems: term list,
  auto_gen: bool,
  maybe_ctr_rhs: term option,
  maybe_code_rhs: term option,
  user_eqn: term
};

type coeqn_data_sel = {
  fun_name: string,
  fun_T: typ,
  fun_args: term list,
  ctr: term,
  sel: term,
  rhs_term: term,
  user_eqn: term
};

datatype coeqn_data =
  Disc of coeqn_data_disc |
  Sel of coeqn_data_sel;

fun dissect_coeqn_disc seq fun_names (basic_ctr_specss : basic_corec_ctr_spec list list)
    maybe_ctr_rhs maybe_code_rhs prems' concl matchedsss =
  let
    fun find_subterm p =
      let (* FIXME \<exists>? *)
        fun find (t as u $ v) = if p t then SOME t else merge_options (find u, find v)
          | find t = if p t then SOME t else NONE;
      in find 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 => primcorec_error_eqn "malformed discriminator formula" concl;
    val ((fun_name, fun_T), fun_args) = strip_comb applied_fun |>> dest_Free;
    val SOME basic_ctr_specs = AList.lookup (op =) (fun_names ~~ basic_ctr_specss) fun_name;

    val discs = map #disc basic_ctr_specs;
    val ctrs = map #ctr basic_ctr_specs;
    val not_disc = head_of concl = @{term Not};
    val _ = not_disc andalso length ctrs <> 2 andalso
      primcorec_error_eqn "negated 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
      primcorec_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, disc, ...} = nth basic_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 seq then maps s_not_conj matchedss else []) @
      (if catch_all then [] else prems);

    val matchedsss' = AList.delete (op =) fun_name matchedsss
      |> cons (fun_name, if seq then matchedss @ [prems] else matchedss @ [real_prems]);

    val user_eqn =
      (real_prems, concl)
      |>> map HOLogic.mk_Trueprop ||> HOLogic.mk_Trueprop o abstract (List.rev fun_args)
      |> curry Logic.list_all (map dest_Free fun_args) o 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,
      prems = real_prems,
      auto_gen = catch_all,
      maybe_ctr_rhs = maybe_ctr_rhs,
      maybe_code_rhs = maybe_code_rhs,
      user_eqn = user_eqn
    }, matchedsss')
  end;

fun dissect_coeqn_sel fun_names (basic_ctr_specss : basic_corec_ctr_spec list list) eqn'
    maybe_of_spec eqn =
  let
    val (lhs, rhs) = HOLogic.dest_eq eqn
      handle TERM _ =>
        primcorec_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 _ =>
        primcorec_error_eqn "malformed selector argument in left-hand side" eqn;
    val basic_ctr_specs = the (AList.lookup (op =) (fun_names ~~ basic_ctr_specss) fun_name)
      handle Option.Option =>
        primcorec_error_eqn "malformed selector argument in left-hand side" eqn;
    val {ctr, ...} =
      (case maybe_of_spec of
        SOME of_spec => the (find_first (equal of_spec o #ctr) basic_ctr_specs)
      | NONE => filter (exists (equal sel) o #sels) basic_ctr_specs |> the_single
          handle List.Empty => primcorec_error_eqn "ambiguous selector - use \"of\"" eqn);
    val user_eqn = drop_All eqn';
  in
    Sel {
      fun_name = fun_name,
      fun_T = fun_T,
      fun_args = fun_args,
      ctr = ctr,
      sel = sel,
      rhs_term = rhs,
      user_eqn = user_eqn
    }
  end;

fun dissect_coeqn_ctr seq fun_names (basic_ctr_specss : basic_corec_ctr_spec list list) eqn'
    maybe_code_rhs prems concl matchedsss =
  let
    val (lhs, rhs) = HOLogic.dest_eq concl;
    val (fun_name, fun_args) = strip_comb lhs |>> fst o dest_Free;
    val SOME basic_ctr_specs = AList.lookup (op =) (fun_names ~~ basic_ctr_specss) fun_name;
    val (ctr, ctr_args) = strip_comb (unfold_let rhs);
    val {disc, sels, ...} = the (find_first (equal ctr o #ctr) basic_ctr_specs)
      handle Option.Option => primcorec_error_eqn "not a constructor" ctr;

    val disc_concl = betapply (disc, lhs);
    val (maybe_eqn_data_disc, matchedsss') = if length basic_ctr_specs = 1
      then (NONE, matchedsss)
      else apfst SOME (dissect_coeqn_disc seq fun_names basic_ctr_specss
          (SOME (abstract (List.rev fun_args) rhs)) maybe_code_rhs prems disc_concl matchedsss);

    val sel_concls = 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} concl ^ "\nto\n    \<cdot> " ^
 (is_some maybe_eqn_data_disc ? K (Syntax.string_of_term @{context} disc_concl ^ "\n    \<cdot> ")) "" ^
 space_implode "\n    \<cdot> " (map (Syntax.string_of_term @{context}) sel_concls) ^
 "\nfor premise(s)\n    \<cdot> " ^
 space_implode "\n    \<cdot> " (map (Syntax.string_of_term @{context}) prems));
*)

    val eqns_data_sel =
      map (dissect_coeqn_sel fun_names basic_ctr_specss eqn' (SOME ctr)) sel_concls;
  in
    (the_list maybe_eqn_data_disc @ eqns_data_sel, matchedsss')
  end;

fun dissect_coeqn_code lthy has_call fun_names basic_ctr_specss eqn' concl matchedsss =
  let
    val (lhs, (rhs', rhs)) = HOLogic.dest_eq concl ||> `(expand_corec_code_rhs lthy has_call []);
    val (fun_name, fun_args) = strip_comb lhs |>> fst o dest_Free;
    val SOME basic_ctr_specs = AList.lookup (op =) (fun_names ~~ basic_ctr_specss) fun_name;

    val cond_ctrs = fold_rev_corec_code_rhs lthy (fn cs => fn ctr => fn _ =>
        if member ((op =) o apsnd #ctr) basic_ctr_specs ctr
        then cons (ctr, cs)
        else primcorec_error_eqn "not a constructor" ctr) [] rhs' []
      |> AList.group (op =);

    val ctr_premss = (case cond_ctrs of [_] => [[]] | _ => map (s_dnf o snd) cond_ctrs);
    val ctr_concls = cond_ctrs |> map (fn (ctr, _) =>
        binder_types (fastype_of ctr)
        |> map_index (fn (n, T) => massage_corec_code_rhs lthy (fn _ => fn ctr' => fn args =>
          if ctr' = ctr then nth args n else Const (@{const_name undefined}, T)) [] rhs')
        |> curry list_comb ctr
        |> curry HOLogic.mk_eq lhs);
  in
    fold_map2 (dissect_coeqn_ctr false fun_names basic_ctr_specss eqn'
        (SOME (abstract (List.rev fun_args) rhs)))
      ctr_premss ctr_concls matchedsss
  end;

fun dissect_coeqn lthy seq has_call fun_names (basic_ctr_specss : basic_corec_ctr_spec list list)
    eqn' maybe_of_spec matchedsss =
  let
    val eqn = drop_All eqn'
      handle TERM _ => primcorec_error_eqn "malformed function equation" eqn';
    val (prems, concl) = Logic.strip_horn eqn
      |> apfst (map HOLogic.dest_Trueprop) o apsnd HOLogic.dest_Trueprop;

    val head = concl
      |> perhaps (try HOLogic.dest_not) |> perhaps (try (fst o HOLogic.dest_eq))
      |> head_of;

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

    val discs = maps (map #disc) basic_ctr_specss;
    val sels = maps (maps #sels) basic_ctr_specss;
    val ctrs = maps (map #ctr) basic_ctr_specss;
  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
      dissect_coeqn_disc seq fun_names basic_ctr_specss NONE NONE prems concl matchedsss
      |>> single
    else if member (op =) sels head then
      ([dissect_coeqn_sel fun_names basic_ctr_specss eqn' maybe_of_spec concl], matchedsss)
    else if is_Free head andalso member (op =) fun_names (fst (dest_Free head)) andalso
      member (op =) ctrs (head_of (unfold_let (the maybe_rhs))) then
      dissect_coeqn_ctr seq fun_names basic_ctr_specss eqn' NONE prems concl matchedsss
    else if is_Free head andalso member (op =) fun_names (fst (dest_Free head)) andalso
      null prems then
      dissect_coeqn_code lthy has_call fun_names basic_ctr_specss eqn' concl matchedsss
      |>> flat
    else
      primcorec_error_eqn "malformed function equation" eqn
  end;

fun build_corec_arg_disc (ctr_specs : corec_ctr_spec list)
    ({fun_args, ctr_no, prems, ...} : coeqn_data_disc) =
  if is_none (#pred (nth ctr_specs ctr_no)) then I else
    s_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 : coeqn_data_sel list) 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_mutual_call lthy has_call (sel_eqns : coeqn_data_sel list) sel =
  (case find_first (equal sel o #sel) sel_eqns of
    NONE => (I, I, I)
  | SOME {fun_args, rhs_term, ... } =>
    let
      val bound_Ts = List.rev (map fastype_of fun_args);
      fun rewrite_stop _ t = if has_call t then @{term False} else @{term True};
      fun rewrite_end _ t = if has_call t then undef_const else t;
      fun rewrite_cont bound_Ts t =
        if has_call t then mk_tuple1 bound_Ts (snd (strip_comb t)) else undef_const;
      fun massage f _ = massage_mutual_corec_call lthy has_call f bound_Ts rhs_term
        |> abs_tuple fun_args;
    in
      (massage rewrite_stop, massage rewrite_end, massage rewrite_cont)
    end);

fun build_corec_arg_nested_call lthy has_call (sel_eqns : coeqn_data_sel list) sel =
  (case find_first (equal sel o #sel) sel_eqns of
    NONE => I
  | SOME {fun_args, rhs_term, ...} =>
    let
      val bound_Ts = List.rev (map fastype_of fun_args);
      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
              map (rewrite bound_Ts U T) vs |> chop 1 |>> HOLogic.mk_split o the_single |> list_comb
            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 t =
        rhs_term
        |> massage_nested_corec_call lthy has_call rewrite bound_Ts (range_type (fastype_of t))
        |> abs_tuple fun_args;
    in
      massage
    end);

fun build_corec_args_sel lthy has_call (all_sel_eqns : coeqn_data_sel list)
    (ctr_spec : corec_ctr_spec) =
  (case filter (equal (#ctr ctr_spec) o #ctr) all_sel_eqns of
    [] => I
  | sel_eqns =>
    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 mutual_calls' = map_filter (try (apsnd (fn Mutual_Corec n => n))) sel_call_list;
      val nested_calls' = map_filter (try (apsnd (fn Nested_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_mutual_call lthy has_call sel_eqns sel in
          nth_map q fq o nth_map g fg o nth_map h fh end) mutual_calls'
      #> fold (fn (sel, n) => nth_map n
        (build_corec_arg_nested_call lthy has_call sel_eqns sel)) nested_calls'
    end);

fun build_codefs lthy bs mxs has_call arg_Tss (corec_specs : corec_spec list)
    (disc_eqnss : coeqn_data_disc list list) (sel_eqnss : coeqn_data_sel list list) =
  let
    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 x => (#fun_args x, #ctr_no x, #prems x, #auto_gen x))
        #> 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, a), (_, c', y, a')) =>
              ((c, c', a orelse a'), (x, s_not (s_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.conceal (Thm.def_binding b), []), t)))
      bs mxs
    |> rpair exclss'
  end;

fun mk_real_disc_eqns fun_binding arg_Ts ({ctr_specs, ...} : corec_spec)
    (sel_eqns : coeqn_data_sel list) (disc_eqns : coeqn_data_disc list) =
  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 (s_not_conj o #prems) disc_eqns,
        auto_gen = true,
        maybe_ctr_rhs = NONE,
        maybe_code_rhs = NONE,
        user_eqn = undef_const};
    in
      chop n disc_eqns ||> cons extra_disc_eqn |> (op @)
    end;

fun find_corec_calls ctxt has_call basic_ctr_specs ({ctr, sel, rhs_term, ...} : coeqn_data_sel) =
  let
    val sel_no = find_first (equal ctr o #ctr) basic_ctr_specs
      |> find_index (equal sel) o #sels o the;
    fun find t = if has_call t then snd (fold_rev_corec_call ctxt (K cons) [] t []) else [];
  in
    find rhs_term
    |> K |> nth_map sel_no |> AList.map_entry (op =) ctr
  end;

fun add_primcorec_ursive maybe_tac opts fixes specs maybe_of_specs lthy =
  let
    val thy = Proof_Context.theory_of lthy;

    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;

    val _ = (case filter_out (fn (_, T) => Sign.of_sort thy (T, HOLogic.typeS)) (bs ~~ arg_Ts) of
        [] => ()
      | (b, _) :: _ => primcorec_error ("type of " ^ Binding.print b ^ " contains top sort"));

    val seq = member (op =) opts Option_Sequential;
    val exhaustive = member (op =) opts Option_Exhaustive;

    val fun_names = map Binding.name_of bs;
    val basic_ctr_specss = map (basic_corec_specs_of lthy) res_Ts;
    val has_call = exists_subterm (map (fst #>> Binding.name_of #> Free) fixes |> member (op =));
    val eqns_data =
      fold_map2 (dissect_coeqn lthy seq has_call fun_names basic_ctr_specss) (map snd specs)
        maybe_of_specs []
      |> flat o fst;

    val callssss =
      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)
      |> map2 (fold o find_corec_calls lthy has_call) basic_ctr_specss
      |> map2 (curry (op |>)) (map (map (fn {ctr, sels, ...} =>
        (ctr, map (K []) sels))) basic_ctr_specss);

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

    val ((n2m, corec_specs', _, coinduct_thm, strong_coinduct_thm, coinduct_thms,
          strong_coinduct_thms), lthy') =
      corec_specs_of bs arg_Ts res_Ts (get_indices fixes) callssss lthy;
    val actual_nn = length bs;
    val corec_specs = take actual_nn corec_specs'; (*###*)
    val ctr_specss = map #ctr_specs corec_specs;

    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
        primcorec_error_eqns "excess discriminator formula 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 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') =
      build_codefs lthy' bs mxs has_call arg_Tss corec_specs disc_eqnss sel_eqnss;

    fun excl_tac (c, c', a) =
      if a orelse c = c' orelse seq then SOME (K (HEADGOAL (mk_primcorec_assumption_tac lthy [])))
      else maybe_tac;

(*
val _ = tracing ("exclusiveness properties:\n    \<cdot> " ^
 space_implode "\n    \<cdot> " (maps (map (Syntax.string_of_term lthy o snd)) exclss'));
*)

    val exclss'' = exclss' |> map (map (fn (idx, t) =>
      (idx, (Option.map (Goal.prove lthy [] [] t #> Thm.close_derivation) (excl_tac idx), t))));
    val taut_thmss = map (map (apsnd (the o fst)) o filter (is_some o fst o snd)) exclss'';
    val (goal_idxss, goalss) = exclss''
      |> map (map (apsnd (rpair [] o snd)) o filter (is_none o fst o snd))
      |> split_list o map split_list;

    val exhaustive_props = map (mk_disjs o map (mk_conjs o #prems)) disc_eqnss;

    fun prove thmss' def_thms' lthy =
      let
        val def_thms = map (snd o snd) def_thms';

        val exclss' = map (op ~~) (goal_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, ...} : corec_spec) exclsss
            ({fun_name, fun_T, fun_args, ctr_no, prems, ...} : coeqn_data_disc) =
          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
              if prems = [@{term False}] then [] else
              mk_primcorec_disc_tac lthy def_thms disc_corec k m exclsss
              |> K |> Goal.prove lthy [] [] t
              |> Thm.close_derivation
              |> pair (#disc (nth ctr_specs ctr_no))
              |> single
            end;

        fun prove_sel ({nested_map_idents, nested_map_comps, ctr_specs, ...} : corec_spec)
            (disc_eqns : coeqn_data_disc list) exclsss
            ({fun_name, fun_T, fun_args, ctr, sel, rhs_term, ...} : coeqn_data_sel) =
          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 (s_not_conj 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);
            val (distincts, _, sel_splits, sel_split_asms) = case_thms_of_term lthy [] rhs_term;
          in
            mk_primcorec_sel_tac lthy def_thms distincts sel_splits sel_split_asms nested_map_idents
              nested_map_comps sel_corec k m exclsss
            |> K |> Goal.prove lthy [] [] t
            |> Thm.close_derivation
            |> pair sel
          end;

        fun prove_ctr disc_alist sel_alist (disc_eqns : coeqn_data_disc list)
            (sel_eqns : coeqn_data_sel list) ({ctr, disc, sels, collapse, ...} : corec_ctr_spec) =
          (* don't try to prove theorems when some sel_eqns are missing *)
          if not (exists (equal ctr o #ctr) disc_eqns)
              andalso not (exists (equal ctr o #ctr) sel_eqns)
            orelse
              filter (equal ctr o #ctr) sel_eqns
              |> fst o finds ((op =) o apsnd #sel) sels
              |> exists (null o snd)
          then [] else
            let
              val (fun_name, fun_T, fun_args, prems, maybe_rhs) =
                (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,
                  #maybe_ctr_rhs x))
                ||> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, [], NONE))
                |> the o merge_options;
              val m = length prems;
              val t = (if is_some maybe_rhs then the maybe_rhs else
                  filter (equal ctr o #ctr) 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_alist disc;
              val sel_thms = map snd (filter (member (op =) sels o fst) sel_alist);
            in
              if prems = [@{term False}] then [] else
                mk_primcorec_ctr_of_dtr_tac lthy m collapse maybe_disc_thm sel_thms
                |> K |> Goal.prove lthy [] [] t
                |> Thm.close_derivation
                |> pair ctr
                |> single
            end;

        fun prove_code disc_eqns sel_eqns ctr_alist ctr_specs =
          let
            val fun_data =
              (find_first (member (op =) (map #ctr ctr_specs) o #ctr) disc_eqns,
               find_first (member (op =) (map #ctr ctr_specs) o #ctr) sel_eqns)
              |>> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, #maybe_code_rhs x))
              ||> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, NONE))
              |> merge_options;
          in
            (case fun_data of
              NONE => []
            | SOME (fun_name, fun_T, fun_args, maybe_rhs) =>
              let
                val bound_Ts = List.rev (map fastype_of fun_args);

                val lhs = list_comb (Free (fun_name, fun_T), map Bound (length fun_args - 1 downto 0));
                val maybe_rhs_info =
                  (case maybe_rhs of
                    SOME rhs =>
                    let
                      val raw_rhs = expand_corec_code_rhs lthy has_call bound_Ts rhs;
                      val cond_ctrs =
                        fold_rev_corec_code_rhs lthy (K oo (cons oo pair)) bound_Ts raw_rhs [];
                      val ctr_thms = map (the o AList.lookup (op =) ctr_alist o snd) cond_ctrs;
                    in SOME (rhs, raw_rhs, ctr_thms) end
                  | NONE =>
                    let
                      fun prove_code_ctr {ctr, sels, ...} =
                        if not (exists (equal ctr o fst) ctr_alist) then NONE else
                          let
                            val prems = find_first (equal ctr o #ctr) disc_eqns
                              |> Option.map #prems |> the_default [];
                            val t =
                              filter (equal ctr o #ctr) 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;
                          in
                            SOME (prems, t)
                          end;
                      val maybe_ctr_conds_argss = map prove_code_ctr ctr_specs;
                    in
                      if exists is_none maybe_ctr_conds_argss then NONE else
                        let
                          val rhs = fold_rev (fn SOME (prems, u) => fn t => mk_If (s_conjs prems) u t)
                            maybe_ctr_conds_argss
                            (Const (@{const_name Code.abort}, @{typ String.literal} -->
                                (@{typ unit} --> body_type fun_T) --> body_type fun_T) $
                              HOLogic.mk_literal fun_name $
                              absdummy @{typ unit} (incr_boundvars 1 lhs));
                        in SOME (rhs, rhs, map snd ctr_alist) end
                    end);
              in
                (case maybe_rhs_info of
                  NONE => []
                | SOME (rhs, raw_rhs, ctr_thms) =>
                  let
                    val ms = map (Logic.count_prems o prop_of) ctr_thms;
                    val (raw_t, t) = (raw_rhs, rhs)
                      |> pairself
                        (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_all (map dest_Free fun_args));
                    val (distincts, discIs, sel_splits, sel_split_asms) =
                      case_thms_of_term lthy bound_Ts raw_rhs;

                    val raw_code_thm = mk_primcorec_raw_code_of_ctr_tac lthy distincts discIs sel_splits
                        sel_split_asms ms ctr_thms
                      |> K |> Goal.prove lthy [] [] raw_t
                      |> Thm.close_derivation;
                  in
                    mk_primcorec_code_of_raw_code_tac lthy distincts sel_splits raw_code_thm
                    |> K |> Goal.prove lthy [] [] t
                    |> Thm.close_derivation
                    |> single
                  end)
              end)
          end;

        val disc_alists = map3 (maps oo prove_disc) corec_specs exclssss disc_eqnss;
        val sel_alists = map4 (map ooo prove_sel) corec_specs disc_eqnss exclssss sel_eqnss;
        val disc_thmss = map (map snd) disc_alists;
        val sel_thmss = map (map snd) sel_alists;

        val ctr_alists = map5 (maps oooo prove_ctr) disc_alists sel_alists disc_eqnss sel_eqnss
          ctr_specss;
        val ctr_thmss = map (map snd) ctr_alists;

        val code_thmss = map4 prove_code disc_eqnss sel_eqnss ctr_alists ctr_specss;

        val simp_thmss = map2 append disc_thmss sel_thmss

        val common_name = mk_common_name fun_names;

        val notes =
          [(coinductN, map (if n2m then single else K []) coinduct_thms, []),
           (codeN, code_thmss, code_nitpicksimp_attrs),
           (ctrN, ctr_thmss, []),
           (discN, disc_thmss, simp_attrs),
           (selN, sel_thmss, simp_attrs),
           (simpsN, simp_thmss, []),
           (strong_coinductN, map (if n2m then single else K []) strong_coinduct_thms, [])]
          |> maps (fn (thmN, thmss, attrs) =>
            map2 (fn fun_name => fn thms =>
                ((Binding.qualify true fun_name (Binding.name thmN), attrs), [(thms, [])]))
              fun_names (take actual_nn thmss))
          |> filter_out (null o fst o hd o snd);

        val common_notes =
          [(coinductN, if n2m then [coinduct_thm] else [], []),
           (strong_coinductN, if n2m then [strong_coinduct_thm] else [], [])]
          |> filter_out (null o #2)
          |> map (fn (thmN, thms, attrs) =>
            ((Binding.qualify true common_name (Binding.name thmN), attrs), [(thms, [])]));
      in
        lthy |> Local_Theory.notes (notes @ common_notes) |> snd
      end;

    fun after_qed thmss' = fold_map Local_Theory.define defs #-> prove thmss';
  in
    (goalss, after_qed, lthy')
  end;

fun add_primcorec_ursive_cmd maybe_tac opts (raw_fixes, raw_specs') lthy =
  let
    val (raw_specs, maybe_of_specs) =
      split_list raw_specs' ||> map (Option.map (Syntax.read_term lthy));
    val ((fixes, specs), _) = Specification.read_spec raw_fixes raw_specs lthy;
  in
    add_primcorec_ursive maybe_tac opts fixes specs maybe_of_specs lthy
    handle ERROR str => primcorec_error str
  end
  handle Primcorec_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));

val add_primcorecursive_cmd = (fn (goalss, after_qed, lthy) =>
  lthy
  |> Proof.theorem NONE after_qed goalss
  |> Proof.refine (Method.primitive_text I)
  |> Seq.hd) ooo add_primcorec_ursive_cmd NONE;

val add_primcorec_cmd = (fn (goalss, after_qed, lthy) =>
  lthy
  |> after_qed (map (fn [] => []
      | _ => primcorec_error "need exclusiveness proofs - use primcorecursive instead of primcorec")
    goalss)) ooo add_primcorec_ursive_cmd (SOME (fn {context = ctxt, ...} => auto_tac ctxt));

end;