(*  Title:      HOL/Tools/ctr_sugar.ML

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012, 2013

 Wrapping existing freely generated type's constructors.

 *)

 signature CTR_SUGAR =

 sig

   type ctr_sugar =

     {ctrs: term list,

      casex: term,

      discs: term list,

      selss: term list list,

      exhaust: thm,

      nchotomy: thm,

      injects: thm list,

      distincts: thm list,

      case_thms: thm list,

      case_cong: thm,

      weak_case_cong: thm,

      split: thm,

      split_asm: thm,

      disc_thmss: thm list list,

      discIs: thm list,

      sel_thmss: thm list list,

      disc_exhausts: thm list,

      sel_exhausts: thm list,

      collapses: thm list,

      expands: thm list,

      sel_splits: thm list,

      sel_split_asms: thm list,

      case_eq_ifs: thm list};

   val morph_ctr_sugar: morphism -> ctr_sugar -> ctr_sugar

   val transfer_ctr_sugar: Proof.context -> ctr_sugar -> ctr_sugar

   val ctr_sugar_of: Proof.context -> string -> ctr_sugar option

   val ctr_sugars_of: Proof.context -> ctr_sugar list

   val ctr_sugar_of_case: Proof.context -> string -> ctr_sugar option

   val register_ctr_sugar: string -> ctr_sugar -> local_theory -> local_theory

   val register_ctr_sugar_global: string -> ctr_sugar -> theory -> theory

   val rep_compat_prefix: string

   val mk_half_pairss: 'a list * 'a list -> ('a * 'a) list list

   val join_halves: int -> 'a list list -> 'a list list -> 'a list * 'a list list list

   val mk_ctr: typ list -> term -> term

   val mk_case: typ list -> typ -> term -> term

   val mk_disc_or_sel: typ list -> term -> term

   val name_of_ctr: term -> string

   val name_of_disc: term -> string

   val dest_ctr: Proof.context -> string -> term -> term * term list

   val dest_case: Proof.context -> string -> typ list -> term -> (term list * term list) option

   val wrap_free_constructors: ({prems: thm list, context: Proof.context} -> tactic) list list ->

     (((bool * (bool * bool)) * term list) * binding) *

       (binding list * (binding list list * (binding * term) list list)) -> local_theory ->

     ctr_sugar * local_theory

   val parse_wrap_free_constructors_options: (bool * (bool * bool)) parser

   val parse_bound_term: (binding * string) parser

 end;

 structure Ctr_Sugar : CTR_SUGAR =

 struct

 open Ctr_Sugar_Util

 open Ctr_Sugar_Tactics

 open Ctr_Sugar_Code

 type ctr_sugar =

   {ctrs: term list,

    casex: term,

    discs: term list,

    selss: term list list,

    exhaust: thm,

    nchotomy: thm,

    injects: thm list,

    distincts: thm list,

    case_thms: thm list,

    case_cong: thm,

    weak_case_cong: thm,

    split: thm,

    split_asm: thm,

    disc_thmss: thm list list,

    discIs: thm list,

    sel_thmss: thm list list,

    disc_exhausts: thm list,

    sel_exhausts: thm list,

    collapses: thm list,

    expands: thm list,

    sel_splits: thm list,

    sel_split_asms: thm list,

    case_eq_ifs: thm list};

 fun eq_ctr_sugar ({ctrs = ctrs1, casex = case1, discs = discs1, selss = selss1, ...} : ctr_sugar,

     {ctrs = ctrs2, casex = case2, discs = discs2, selss = selss2, ...} : ctr_sugar) =

   ctrs1 = ctrs2 andalso case1 = case2 andalso discs1 = discs2 andalso selss1 = selss2;

 fun morph_ctr_sugar phi {ctrs, casex, discs, selss, exhaust, nchotomy, injects, distincts,

     case_thms, case_cong, weak_case_cong, split, split_asm, disc_thmss, discIs, sel_thmss,

     disc_exhausts, sel_exhausts, collapses, expands, sel_splits, sel_split_asms, case_eq_ifs} =

   {ctrs = map (Morphism.term phi) ctrs,

    casex = Morphism.term phi casex,

    discs = map (Morphism.term phi) discs,

    selss = map (map (Morphism.term phi)) selss,

    exhaust = Morphism.thm phi exhaust,

    nchotomy = Morphism.thm phi nchotomy,

    injects = map (Morphism.thm phi) injects,

    distincts = map (Morphism.thm phi) distincts,

    case_thms = map (Morphism.thm phi) case_thms,

    case_cong = Morphism.thm phi case_cong,

    weak_case_cong = Morphism.thm phi weak_case_cong,

    split = Morphism.thm phi split,

    split_asm = Morphism.thm phi split_asm,

    disc_thmss = map (map (Morphism.thm phi)) disc_thmss,

    discIs = map (Morphism.thm phi) discIs,

    sel_thmss = map (map (Morphism.thm phi)) sel_thmss,

    disc_exhausts = map (Morphism.thm phi) disc_exhausts,

    sel_exhausts = map (Morphism.thm phi) sel_exhausts,

    collapses = map (Morphism.thm phi) collapses,

    expands = map (Morphism.thm phi) expands,

    sel_splits = map (Morphism.thm phi) sel_splits,

    sel_split_asms = map (Morphism.thm phi) sel_split_asms,

    case_eq_ifs = map (Morphism.thm phi) case_eq_ifs};

 val transfer_ctr_sugar =

   morph_ctr_sugar o Morphism.thm_morphism o Thm.transfer o Proof_Context.theory_of;

 structure Data = Generic_Data

 (

   type T = ctr_sugar Symtab.table;

   val empty = Symtab.empty;

   val extend = I;

   val merge = Symtab.merge eq_ctr_sugar;

 );

 fun ctr_sugar_of ctxt =

   Symtab.lookup (Data.get (Context.Proof ctxt))

   #> Option.map (transfer_ctr_sugar ctxt);

 fun ctr_sugars_of ctxt =

   Symtab.fold (cons o transfer_ctr_sugar ctxt o snd) (Data.get (Context.Proof ctxt)) [];

 fun ctr_sugar_of_case ctxt s =

   find_first (fn {casex = Const (s', _), ...} => s' = s | _ => false) (ctr_sugars_of ctxt);

 fun register_ctr_sugar key ctr_sugar =

   Local_Theory.declaration {syntax = false, pervasive = true}

     (fn phi => Data.map (Symtab.default (key, morph_ctr_sugar phi ctr_sugar)));

 fun register_ctr_sugar_global key ctr_sugar =

   Context.theory_map (Data.map (Symtab.default (key, ctr_sugar)));

 val rep_compat_prefix = "new";

 val isN = "is_";

 val unN = "un_";

 fun mk_unN 1 1 suf = unN ^ suf

   | mk_unN _ l suf = unN ^ suf ^ string_of_int l;

 val caseN = "case";

 val case_congN = "case_cong";

 val case_eq_ifN = "case_eq_if";

 val collapseN = "collapse";

 val disc_excludeN = "disc_exclude";

 val disc_exhaustN = "disc_exhaust";

 val discN = "disc";

 val discIN = "discI";

 val distinctN = "distinct";

 val exhaustN = "exhaust";

 val expandN = "expand";

 val injectN = "inject";

 val nchotomyN = "nchotomy";

 val selN = "sel";

 val sel_exhaustN = "sel_exhaust";

 val sel_splitN = "sel_split";

 val sel_split_asmN = "sel_split_asm";

 val splitN = "split";

 val splitsN = "splits";

 val split_asmN = "split_asm";

 val weak_case_cong_thmsN = "weak_case_cong";

 val cong_attrs = @{attributes [cong]};

 val dest_attrs = @{attributes [dest]};

 val safe_elim_attrs = @{attributes [elim!]};

 val iff_attrs = @{attributes [iff]};

 val inductsimp_attrs = @{attributes [induct_simp]};

 val nitpicksimp_attrs = @{attributes [nitpick_simp]};

 val simp_attrs = @{attributes [simp]};

 val code_nitpicksimp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs;

 val code_nitpicksimp_simp_attrs = code_nitpicksimp_attrs @ simp_attrs;

 fun unflat_lookup eq xs ys = map (fn xs' => permute_like eq xs xs' ys);

 fun mk_half_pairss' _ ([], []) = []

   | mk_half_pairss' indent (x :: xs, _ :: ys) =

     indent @ fold_rev (cons o single o pair x) ys (mk_half_pairss' ([] :: indent) (xs, ys));

 fun mk_half_pairss p = mk_half_pairss' [[]] p;

 fun join_halves n half_xss other_half_xss =

   let

     val xsss =

       map2 (map2 append) (Library.chop_groups n half_xss)

         (transpose (Library.chop_groups n other_half_xss))

     val xs = splice (flat half_xss) (flat other_half_xss);

   in (xs, xsss) end;

 fun mk_undefined T = Const (@{const_name undefined}, T);

 fun mk_ctr Ts t =

   let val Type (_, Ts0) = body_type (fastype_of t) in

     subst_nonatomic_types (Ts0 ~~ Ts) t

   end;

 fun mk_case Ts T t =

   let val (Type (_, Ts0), body) = strip_type (fastype_of t) |>> List.last in

     subst_nonatomic_types ((body, T) :: (Ts0 ~~ Ts)) t

   end;

 fun mk_disc_or_sel Ts t =

   subst_nonatomic_types (snd (Term.dest_Type (domain_type (fastype_of t))) ~~ Ts) t;

 fun name_of_const what t =

   (case head_of t of

     Const (s, _) => s

   | Free (s, _) => s

   | _ => error ("Cannot extract name of " ^ what));

 val name_of_ctr = name_of_const "constructor";

 val notN = "not_";

 val eqN = "eq_";

 val neqN = "neq_";

 fun name_of_disc t =

   (case head_of t of

     Abs (_, _, @{const Not} $ (t' $ Bound 0)) =>

     Long_Name.map_base_name (prefix notN) (name_of_disc t')

   | Abs (_, _, Const (@{const_name HOL.eq}, _) $ Bound 0 $ t') =>

     Long_Name.map_base_name (prefix eqN) (name_of_disc t')

   | Abs (_, _, @{const Not} $ (Const (@{const_name HOL.eq}, _) $ Bound 0 $ t')) =>

     Long_Name.map_base_name (prefix neqN) (name_of_disc t')

   | t' => name_of_const "destructor" t');

 val base_name_of_ctr = Long_Name.base_name o name_of_ctr;

 fun dest_ctr ctxt s t =

   let

     val (f, args) = Term.strip_comb t;

   in

     (case ctr_sugar_of ctxt s of

       SOME {ctrs, ...} =>

       (case find_first (can (fo_match ctxt f)) ctrs of

         SOME f' => (f', args)

       | NONE => raise Fail "dest_ctr")

     | NONE => raise Fail "dest_ctr")

   end;

 fun dest_case ctxt s Ts t =

   (case Term.strip_comb t of

     (Const (c, _), args as _ :: _) =>

     (case ctr_sugar_of ctxt s of

       SOME {casex = Const (case_name, _), discs = discs0, selss = selss0, ...} =>

       if case_name = c then

         let val n = length discs0 in

           if n < length args then

             let

               val (branches, obj :: leftovers) = chop n args;

               val discs = map (mk_disc_or_sel Ts) discs0;

               val selss = map (map (mk_disc_or_sel Ts)) selss0;

               val conds = map (rapp obj) discs;

               val branch_argss = map (fn sels => map (rapp obj) sels @ leftovers) selss;

               val branches' = map2 (curry Term.betapplys) branches branch_argss;

             in

               SOME (conds, branches')

             end

           else

             NONE

         end

       else

         NONE

     | _ => NONE)

   | _ => NONE);

 fun eta_expand_arg xs f_xs = fold_rev Term.lambda xs f_xs;

 fun prepare_wrap_free_constructors prep_term ((((no_discs_sels, (no_code, rep_compat)), raw_ctrs),

     raw_case_binding), (raw_disc_bindings, (raw_sel_bindingss, raw_sel_defaultss))) no_defs_lthy =

   let

     (* TODO: sanity checks on arguments *)

     val n = length raw_ctrs;

     val ks = 1 upto n;

     val _ = if n > 0 then () else error "No constructors specified";

     val ctrs0 = map (prep_term no_defs_lthy) raw_ctrs;

     val sel_defaultss =

       pad_list [] n (map (map (apsnd (prep_term no_defs_lthy))) raw_sel_defaultss);

     val Type (fcT_name, As0) = body_type (fastype_of (hd ctrs0));

     val fc_b_name = Long_Name.base_name fcT_name;

     val fc_b = Binding.name fc_b_name;

     fun qualify mandatory =

       Binding.qualify mandatory fc_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);

     fun dest_TFree_or_TVar (TFree sS) = sS

       | dest_TFree_or_TVar (TVar ((s, _), S)) = (s, S)

       | dest_TFree_or_TVar _ = error "Invalid type argument";

     val (unsorted_As, B) =

       no_defs_lthy

       |> variant_tfrees (map (fst o dest_TFree_or_TVar) As0)

       ||> the_single o fst o mk_TFrees 1;

     val As = map2 (resort_tfree o snd o dest_TFree_or_TVar) As0 unsorted_As;

     val fcT = Type (fcT_name, As);

     val ctrs = map (mk_ctr As) ctrs0;

     val ctr_Tss = map (binder_types o fastype_of) ctrs;

     val ms = map length ctr_Tss;

     val raw_disc_bindings' = pad_list Binding.empty n raw_disc_bindings;

     fun can_definitely_rely_on_disc k = not (Binding.is_empty (nth raw_disc_bindings' (k - 1)));

     fun can_rely_on_disc k =

       can_definitely_rely_on_disc k orelse (k = 1 andalso not (can_definitely_rely_on_disc 2));

     fun should_omit_disc_binding k = n = 1 orelse (n = 2 andalso can_rely_on_disc (3 - k));

     fun is_disc_binding_valid b =

       not (Binding.is_empty b orelse Binding.eq_name (b, equal_binding));

     val standard_disc_binding = Binding.name o prefix isN o base_name_of_ctr;

     val disc_bindings =

       raw_disc_bindings'

       |> map4 (fn k => fn m => fn ctr => fn disc =>

         qualify false

           (if Binding.is_empty disc then

              if should_omit_disc_binding k then disc else standard_disc_binding ctr

            else if Binding.eq_name (disc, equal_binding) then

              if m = 0 then disc

              else error "Cannot use \"=\" syntax for discriminating nonnullary constructor"

            else if Binding.eq_name (disc, standard_binding) then

              standard_disc_binding ctr

            else

              disc)) ks ms ctrs0;

     fun standard_sel_binding m l = Binding.name o mk_unN m l o base_name_of_ctr;

     val sel_bindingss =

       pad_list [] n raw_sel_bindingss

       |> map3 (fn ctr => fn m => map2 (fn l => fn sel =>

         qualify false

           (if Binding.is_empty sel orelse Binding.eq_name (sel, standard_binding) then

             standard_sel_binding m l ctr

           else

             sel)) (1 upto m) o pad_list Binding.empty m) ctrs0 ms;

     val case_Ts = map (fn Ts => Ts ---> B) ctr_Tss;

     val ((((((((xss, xss'), yss), fs), gs), [u', v']), [w]), (p, p')), names_lthy) = no_defs_lthy |>

       mk_Freess' "x" ctr_Tss

       ||>> mk_Freess "y" ctr_Tss

       ||>> mk_Frees "f" case_Ts

       ||>> mk_Frees "g" case_Ts

       ||>> (apfst (map (rpair fcT)) oo Variable.variant_fixes) [fc_b_name, fc_b_name ^ "'"]

       ||>> mk_Frees "z" [B]

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "P") HOLogic.boolT;

     val u = Free u';

     val v = Free v';

     val q = Free (fst p', mk_pred1T B);

     val xctrs = map2 (curry Term.list_comb) ctrs xss;

     val yctrs = map2 (curry Term.list_comb) ctrs yss;

     val xfs = map2 (curry Term.list_comb) fs xss;

     val xgs = map2 (curry Term.list_comb) gs xss;

     (* TODO: Eta-expension is for compatibility with the old datatype package (but it also provides

        nicer names). Consider removing. *)

     val eta_fs = map2 eta_expand_arg xss xfs;

     val eta_gs = map2 eta_expand_arg xss xgs;

     val case_binding =

       qualify false

         (if Binding.is_empty raw_case_binding orelse

             Binding.eq_name (raw_case_binding, standard_binding) then

            Binding.prefix_name (caseN ^ "_") fc_b

          else

            raw_case_binding);

     fun mk_case_disj xctr xf xs =

       list_exists_free xs (HOLogic.mk_conj (HOLogic.mk_eq (u, xctr), HOLogic.mk_eq (w, xf)));

     val case_rhs = fold_rev (fold_rev Term.lambda) [fs, [u]]

       (Const (@{const_name The}, (B --> HOLogic.boolT) --> B) $

          Term.lambda w (Library.foldr1 HOLogic.mk_disj (map3 mk_case_disj xctrs xfs xss)));

     val ((raw_case, (_, raw_case_def)), (lthy', lthy)) = no_defs_lthy

       |> Local_Theory.define ((case_binding, NoSyn),

         ((Binding.conceal (Thm.def_binding case_binding), []), case_rhs))

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy lthy';

     val case_def = Morphism.thm phi raw_case_def;

     val case0 = Morphism.term phi raw_case;

     val casex = mk_case As B case0;

     val fcase = Term.list_comb (casex, fs);

     val ufcase = fcase $ u;

     val vfcase = fcase $ v;

     val eta_fcase = Term.list_comb (casex, eta_fs);

     val eta_gcase = Term.list_comb (casex, eta_gs);

     val eta_ufcase = eta_fcase $ u;

     val eta_vgcase = eta_gcase $ v;

     fun mk_uu_eq () = HOLogic.mk_eq (u, u);

     val uv_eq = mk_Trueprop_eq (u, v);

     val exist_xs_u_eq_ctrs =

       map2 (fn xctr => fn xs => list_exists_free xs (HOLogic.mk_eq (u, xctr))) xctrs xss;

     val unique_disc_no_def = TrueI; (*arbitrary marker*)

     val alternate_disc_no_def = FalseE; (*arbitrary marker*)

     fun alternate_disc_lhs get_udisc k =

       HOLogic.mk_not

         (let val b = nth disc_bindings (k - 1) in

            if is_disc_binding_valid b then get_udisc b (k - 1) else nth exist_xs_u_eq_ctrs (k - 1)

          end);

     val (all_sels_distinct, discs, selss, disc_defs, sel_defs, sel_defss, lthy') =

       if no_discs_sels then

         (true, [], [], [], [], [], lthy)

       else

         let

           fun disc_free b = Free (Binding.name_of b, mk_pred1T fcT);

           fun disc_spec b exist_xs_u_eq_ctr = mk_Trueprop_eq (disc_free b $ u, exist_xs_u_eq_ctr);

           fun alternate_disc k =

             Term.lambda u (alternate_disc_lhs (K o rapp u o disc_free) (3 - k));

           fun mk_sel_case_args b proto_sels T =

             map2 (fn Ts => fn k =>

               (case AList.lookup (op =) proto_sels k of

                 NONE =>

                 (case AList.lookup Binding.eq_name (rev (nth sel_defaultss (k - 1))) b of

                   NONE => fold_rev (Term.lambda o curry Free Name.uu) Ts (mk_undefined T)

                 | SOME t => t |> Type.constraint (Ts ---> T) |> Syntax.check_term lthy)

               | SOME (xs, x) => fold_rev Term.lambda xs x)) ctr_Tss ks;

           fun sel_spec b proto_sels =

             let

               val _ =

                 (case duplicates (op =) (map fst proto_sels) of

                    k :: _ => error ("Duplicate selector name " ^ quote (Binding.name_of b) ^

                      " for constructor " ^

                      quote (Syntax.string_of_term lthy (nth ctrs (k - 1))))

                  | [] => ())

               val T =

                 (case distinct (op =) (map (fastype_of o snd o snd) proto_sels) of

                   [T] => T

                 | T :: T' :: _ => error ("Inconsistent range type for selector " ^

                     quote (Binding.name_of b) ^ ": " ^ quote (Syntax.string_of_typ lthy T) ^ " vs. "

                     ^ quote (Syntax.string_of_typ lthy T')));

             in

               mk_Trueprop_eq (Free (Binding.name_of b, fcT --> T) $ u,

                 Term.list_comb (mk_case As T case0, mk_sel_case_args b proto_sels T) $ u)

             end;

           val sel_bindings = flat sel_bindingss;

           val uniq_sel_bindings = distinct Binding.eq_name sel_bindings;

           val all_sels_distinct = (length uniq_sel_bindings = length sel_bindings);

           val sel_binding_index =

             if all_sels_distinct then 1 upto length sel_bindings

             else map (fn b => find_index (curry Binding.eq_name b) uniq_sel_bindings) sel_bindings;

           val proto_sels = flat (map3 (fn k => fn xs => map (fn x => (k, (xs, x)))) ks xss xss);

           val sel_infos =

             AList.group (op =) (sel_binding_index ~~ proto_sels)

             |> sort (int_ord o pairself fst)

             |> map snd |> curry (op ~~) uniq_sel_bindings;

           val sel_bindings = map fst sel_infos;

           fun unflat_selss xs = unflat_lookup Binding.eq_name sel_bindings xs sel_bindingss;

           val (((raw_discs, raw_disc_defs), (raw_sels, raw_sel_defs)), (lthy', lthy)) =

             lthy

             |> apfst split_list o fold_map3 (fn k => fn exist_xs_u_eq_ctr => fn b =>

                 if Binding.is_empty b then

                   if n = 1 then pair (Term.lambda u (mk_uu_eq ()), unique_disc_no_def)

                   else pair (alternate_disc k, alternate_disc_no_def)

                 else if Binding.eq_name (b, equal_binding) then

                   pair (Term.lambda u exist_xs_u_eq_ctr, refl)

                 else

                   Specification.definition (SOME (b, NONE, NoSyn),

                     ((Thm.def_binding b, []), disc_spec b exist_xs_u_eq_ctr)) #>> apsnd snd)

               ks exist_xs_u_eq_ctrs disc_bindings

             ||>> apfst split_list o fold_map (fn (b, proto_sels) =>

               Specification.definition (SOME (b, NONE, NoSyn),

                 ((Thm.def_binding b, []), sel_spec b proto_sels)) #>> apsnd snd) sel_infos

             ||> `Local_Theory.restore;

           val phi = Proof_Context.export_morphism lthy lthy';

           val disc_defs = map (Morphism.thm phi) raw_disc_defs;

           val sel_defs = map (Morphism.thm phi) raw_sel_defs;

           val sel_defss = unflat_selss sel_defs;

           val discs0 = map (Morphism.term phi) raw_discs;

           val selss0 = unflat_selss (map (Morphism.term phi) raw_sels);

           val discs = map (mk_disc_or_sel As) discs0;

           val selss = map (map (mk_disc_or_sel As)) selss0;

         in

           (all_sels_distinct, discs, selss, disc_defs, sel_defs, sel_defss, lthy')

         end;

     fun mk_imp_p Qs = Logic.list_implies (Qs, HOLogic.mk_Trueprop p);

     val exhaust_goal =

       let fun mk_prem xctr xs = fold_rev Logic.all xs (mk_imp_p [mk_Trueprop_eq (u, xctr)]) in

         fold_rev Logic.all [p, u] (mk_imp_p (map2 mk_prem xctrs xss))

       end;

     val inject_goalss =

       let

         fun mk_goal _ _ [] [] = []

           | mk_goal xctr yctr xs ys =

             [fold_rev Logic.all (xs @ ys) (mk_Trueprop_eq (HOLogic.mk_eq (xctr, yctr),

               Library.foldr1 HOLogic.mk_conj (map2 (curry HOLogic.mk_eq) xs ys)))];

       in

         map4 mk_goal xctrs yctrs xss yss

       end;

     val half_distinct_goalss =

       let

         fun mk_goal ((xs, xc), (xs', xc')) =

           fold_rev Logic.all (xs @ xs')

             (HOLogic.mk_Trueprop (HOLogic.mk_not (HOLogic.mk_eq (xc, xc'))));

       in

         map (map mk_goal) (mk_half_pairss (`I (xss ~~ xctrs)))

       end;

     val goalss = [exhaust_goal] :: inject_goalss @ half_distinct_goalss;

     fun after_qed thmss lthy =

       let

         val ([exhaust_thm], (inject_thmss, half_distinct_thmss)) = (hd thmss, chop n (tl thmss));

         val inject_thms = flat inject_thmss;

         val rho_As = map (pairself (certifyT lthy)) (map Logic.varifyT_global As ~~ As);

         fun inst_thm t thm =

           Drule.instantiate' [] [SOME (certify lthy t)]

             (Thm.instantiate (rho_As, []) (Drule.zero_var_indexes thm));

         val uexhaust_thm = inst_thm u exhaust_thm;

         val exhaust_cases = map base_name_of_ctr ctrs;

         val other_half_distinct_thmss = map (map (fn thm => thm RS not_sym)) half_distinct_thmss;

         val (distinct_thms, (distinct_thmsss', distinct_thmsss)) =

           join_halves n half_distinct_thmss other_half_distinct_thmss ||> `transpose;

         val nchotomy_thm =

           let

             val goal =

               HOLogic.mk_Trueprop (HOLogic.mk_all (fst u', snd u',

                 Library.foldr1 HOLogic.mk_disj exist_xs_u_eq_ctrs));

           in

             Goal.prove_sorry lthy [] [] goal (fn _ => mk_nchotomy_tac n exhaust_thm)

             |> Thm.close_derivation

           end;

         val case_thms =

           let

             val goals =

               map3 (fn xctr => fn xf => fn xs =>

                 fold_rev Logic.all (fs @ xs) (mk_Trueprop_eq (fcase $ xctr, xf))) xctrs xfs xss;

           in

             map4 (fn k => fn goal => fn injects => fn distinctss =>

                 Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

                   mk_case_tac ctxt n k case_def injects distinctss)

                 |> Thm.close_derivation)

               ks goals inject_thmss distinct_thmsss

           end;

         val (case_cong_thm, weak_case_cong_thm) =

           let

             fun mk_prem xctr xs xf xg =

               fold_rev Logic.all xs (Logic.mk_implies (mk_Trueprop_eq (v, xctr),

                 mk_Trueprop_eq (xf, xg)));

             val goal =

               Logic.list_implies (uv_eq :: map4 mk_prem xctrs xss xfs xgs,

                  mk_Trueprop_eq (eta_ufcase, eta_vgcase));

             val weak_goal = Logic.mk_implies (uv_eq, mk_Trueprop_eq (ufcase, vfcase));

           in

             (Goal.prove_sorry lthy [] [] goal (fn _ => mk_case_cong_tac lthy uexhaust_thm case_thms),

              Goal.prove_sorry lthy [] [] weak_goal (K (etac arg_cong 1)))

             |> pairself (Thm.close_derivation #> singleton (Proof_Context.export names_lthy lthy))

           end;

         val split_lhs = q $ ufcase;

         fun mk_split_conjunct xctr xs f_xs =

           list_all_free xs (HOLogic.mk_imp (HOLogic.mk_eq (u, xctr), q $ f_xs));

         fun mk_split_disjunct xctr xs f_xs =

           list_exists_free xs (HOLogic.mk_conj (HOLogic.mk_eq (u, xctr),

             HOLogic.mk_not (q $ f_xs)));

         fun mk_split_goal xctrs xss xfs =

           mk_Trueprop_eq (split_lhs, Library.foldr1 HOLogic.mk_conj

             (map3 mk_split_conjunct xctrs xss xfs));

         fun mk_split_asm_goal xctrs xss xfs =

           mk_Trueprop_eq (split_lhs, HOLogic.mk_not (Library.foldr1 HOLogic.mk_disj

             (map3 mk_split_disjunct xctrs xss xfs)));

         fun prove_split selss goal =

           Goal.prove_sorry lthy [] [] goal (fn _ =>

             mk_split_tac lthy uexhaust_thm case_thms selss inject_thmss distinct_thmsss)

           |> Thm.close_derivation

           |> singleton (Proof_Context.export names_lthy lthy);

         fun prove_split_asm asm_goal split_thm =

           Goal.prove_sorry lthy [] [] asm_goal (fn {context = ctxt, ...} =>

             mk_split_asm_tac ctxt split_thm)

           |> Thm.close_derivation

           |> singleton (Proof_Context.export names_lthy lthy);

         val (split_thm, split_asm_thm) =

           let

             val goal = mk_split_goal xctrs xss xfs;

             val asm_goal = mk_split_asm_goal xctrs xss xfs;

             val thm = prove_split (replicate n []) goal;

             val asm_thm = prove_split_asm asm_goal thm;

           in

             (thm, asm_thm)

           end;

         val (all_sel_thms, sel_thmss, disc_thmss, nontriv_disc_thms, discI_thms, nontriv_discI_thms,

              disc_exclude_thms, disc_exhaust_thms, sel_exhaust_thms, all_collapse_thms,

              safe_collapse_thms, expand_thms, sel_split_thms, sel_split_asm_thms, case_eq_if_thms) =

           if no_discs_sels then

             ([], [], [], [], [], [], [], [], [], [], [], [], [], [], [])

           else

             let

               val udiscs = map (rapp u) discs;

               val uselss = map (map (rapp u)) selss;

               val usel_ctrs = map2 (curry Term.list_comb) ctrs uselss;

               val usel_fs = map2 (curry Term.list_comb) fs uselss;

               val vdiscs = map (rapp v) discs;

               val vselss = map (map (rapp v)) selss;

               fun make_sel_thm xs' case_thm sel_def =

                 zero_var_indexes (Drule.gen_all (Drule.rename_bvars' (map (SOME o fst) xs')

                     (Drule.forall_intr_vars (case_thm RS (sel_def RS trans)))));

               val sel_thmss = map3 (map oo make_sel_thm) xss' case_thms sel_defss;

               fun has_undefined_rhs thm =

                 (case snd (HOLogic.dest_eq (HOLogic.dest_Trueprop (prop_of thm))) of

                   Const (@{const_name undefined}, _) => true

                 | _ => false);

               val all_sel_thms =

                 (if all_sels_distinct andalso forall null sel_defaultss then

                    flat sel_thmss

                  else

                    map_product (fn s => fn (xs', c) => make_sel_thm xs' c s) sel_defs

                      (xss' ~~ case_thms))

                 |> filter_out has_undefined_rhs;

               fun mk_unique_disc_def () =

                 let

                   val m = the_single ms;

                   val goal = mk_Trueprop_eq (mk_uu_eq (), the_single exist_xs_u_eq_ctrs);

                 in

                   Goal.prove_sorry lthy [] [] goal (fn _ => mk_unique_disc_def_tac m uexhaust_thm)

                   |> Thm.close_derivation

                   |> singleton (Proof_Context.export names_lthy lthy)

                 end;

               fun mk_alternate_disc_def k =

                 let

                   val goal =

                     mk_Trueprop_eq (alternate_disc_lhs (K (nth udiscs)) (3 - k),

                       nth exist_xs_u_eq_ctrs (k - 1));

                 in

                   Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

                     mk_alternate_disc_def_tac ctxt k (nth disc_defs (2 - k))

                       (nth distinct_thms (2 - k)) uexhaust_thm)

                   |> Thm.close_derivation

                   |> singleton (Proof_Context.export names_lthy lthy)

                 end;

               val has_alternate_disc_def =

                 exists (fn def => Thm.eq_thm_prop (def, alternate_disc_no_def)) disc_defs;

               val disc_defs' =

                 map2 (fn k => fn def =>

                   if Thm.eq_thm_prop (def, unique_disc_no_def) then mk_unique_disc_def ()

                   else if Thm.eq_thm_prop (def, alternate_disc_no_def) then mk_alternate_disc_def k

                   else def) ks disc_defs;

               val discD_thms = map (fn def => def RS iffD1) disc_defs';

               val discI_thms =

                 map2 (fn m => fn def => funpow m (fn thm => exI RS thm) (def RS iffD2)) ms

                   disc_defs';

               val not_discI_thms =

                 map2 (fn m => fn def => funpow m (fn thm => allI RS thm)

                     (unfold_thms lthy @{thms not_ex} (def RS @{thm ssubst[of _ _ Not]})))

                   ms disc_defs';

               val (disc_thmss', disc_thmss) =

                 let

                   fun mk_thm discI _ [] = refl RS discI

                     | mk_thm _ not_discI [distinct] = distinct RS not_discI;

                   fun mk_thms discI not_discI distinctss = map (mk_thm discI not_discI) distinctss;

                 in

                   map3 mk_thms discI_thms not_discI_thms distinct_thmsss' |> `transpose

                 end;

               val nontriv_disc_thms =

                 flat (map2 (fn b => if is_disc_binding_valid b then I else K [])

                   disc_bindings disc_thmss);

               fun is_discI_boring b =

                 (n = 1 andalso Binding.is_empty b) orelse Binding.eq_name (b, equal_binding);

               val nontriv_discI_thms =

                 flat (map2 (fn b => if is_discI_boring b then K [] else single) disc_bindings

                   discI_thms);

               val (disc_exclude_thms, (disc_exclude_thmsss', disc_exclude_thmsss)) =

                 let

                   fun mk_goal [] = []

                     | mk_goal [((_, udisc), (_, udisc'))] =

                       [Logic.all u (Logic.mk_implies (HOLogic.mk_Trueprop udisc,

                          HOLogic.mk_Trueprop (HOLogic.mk_not udisc')))];

                   fun prove tac goal =

                     Goal.prove_sorry lthy [] [] goal (K tac)

                     |> Thm.close_derivation;

                   val half_pairss = mk_half_pairss (`I (ms ~~ discD_thms ~~ udiscs));

                   val half_goalss = map mk_goal half_pairss;

                   val half_thmss =

                     map3 (fn [] => K (K []) | [goal] => fn [(((m, discD), _), _)] =>

                         fn disc_thm => [prove (mk_half_disc_exclude_tac lthy m discD disc_thm) goal])

                       half_goalss half_pairss (flat disc_thmss');

                   val other_half_goalss = map (mk_goal o map swap) half_pairss;

                   val other_half_thmss =

                     map2 (map2 (prove o mk_other_half_disc_exclude_tac)) half_thmss

                       other_half_goalss;

                 in

                   join_halves n half_thmss other_half_thmss ||> `transpose

                   |>> has_alternate_disc_def ? K []

                 end;

               val disc_exhaust_thm =

                 let

                   fun mk_prem udisc = mk_imp_p [HOLogic.mk_Trueprop udisc];

                   val goal = fold_rev Logic.all [p, u] (mk_imp_p (map mk_prem udiscs));

                 in

                   Goal.prove_sorry lthy [] [] goal (fn _ =>

                     mk_disc_exhaust_tac n exhaust_thm discI_thms)

                   |> Thm.close_derivation

                 end;

               val (safe_collapse_thms, all_collapse_thms) =

                 let

                   fun mk_goal m udisc usel_ctr =

                     let

                       val prem = HOLogic.mk_Trueprop udisc;

                       val concl = mk_Trueprop_eq ((usel_ctr, u) |> m = 0 ? swap);

                     in

                       (prem aconv concl, Logic.all u (Logic.mk_implies (prem, concl)))

                     end;

                   val (trivs, goals) = map3 mk_goal ms udiscs usel_ctrs |> split_list;

                   val thms =

                     map5 (fn m => fn discD => fn sel_thms => fn triv => fn goal =>

                         Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

                           mk_collapse_tac ctxt m discD sel_thms ORELSE HEADGOAL atac)

                         |> Thm.close_derivation

                         |> not triv ? perhaps (try (fn thm => refl RS thm)))

                       ms discD_thms sel_thmss trivs goals;

                 in

                   (map_filter (fn (true, _) => NONE | (false, thm) => SOME thm) (trivs ~~ thms),

                    thms)

                 end;

               val swapped_all_collapse_thms =

                 map2 (fn m => fn thm => if m = 0 then thm else thm RS sym) ms all_collapse_thms;

               val sel_exhaust_thm =

                 let

                   fun mk_prem usel_ctr = mk_imp_p [mk_Trueprop_eq (u, usel_ctr)];

                   val goal = fold_rev Logic.all [p, u] (mk_imp_p (map mk_prem usel_ctrs));

                 in

                   Goal.prove_sorry lthy [] [] goal (fn _ =>

                     mk_sel_exhaust_tac n disc_exhaust_thm swapped_all_collapse_thms)

                   |> Thm.close_derivation

                 end;

               val expand_thm =

                 let

                   fun mk_prems k udisc usels vdisc vsels =

                     (if k = n then [] else [mk_Trueprop_eq (udisc, vdisc)]) @

                     (if null usels then

                        []

                      else

                        [Logic.list_implies

                           (if n = 1 then [] else map HOLogic.mk_Trueprop [udisc, vdisc],

                              HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                                (map2 (curry HOLogic.mk_eq) usels vsels)))]);

                   val goal =

                     Library.foldr Logic.list_implies

                       (map5 mk_prems ks udiscs uselss vdiscs vselss, uv_eq);

                   val uncollapse_thms =

                     map2 (fn thm => fn [] => thm | _ => thm RS sym) all_collapse_thms uselss;

                 in

                   Goal.prove_sorry lthy [] [] goal (fn _ =>

                     mk_expand_tac lthy n ms (inst_thm u disc_exhaust_thm)

                       (inst_thm v disc_exhaust_thm) uncollapse_thms disc_exclude_thmsss

                       disc_exclude_thmsss')

                   |> Thm.close_derivation

                   |> singleton (Proof_Context.export names_lthy lthy)

                 end;

               val (sel_split_thm, sel_split_asm_thm) =

                 let

                   val zss = map (K []) xss;

                   val goal = mk_split_goal usel_ctrs zss usel_fs;

                   val asm_goal = mk_split_asm_goal usel_ctrs zss usel_fs;

                   val thm = prove_split sel_thmss goal;

                   val asm_thm = prove_split_asm asm_goal thm;

                 in

                   (thm, asm_thm)

                 end;

               val case_eq_if_thm =

                 let

                   val goal = mk_Trueprop_eq (ufcase, mk_IfN B udiscs usel_fs);

                 in

                   Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

                     mk_case_eq_if_tac ctxt n uexhaust_thm case_thms disc_thmss' sel_thmss)

                   |> Thm.close_derivation

                   |> singleton (Proof_Context.export names_lthy lthy)

                 end;

             in

               (all_sel_thms, sel_thmss, disc_thmss, nontriv_disc_thms, discI_thms,

                nontriv_discI_thms, disc_exclude_thms, [disc_exhaust_thm], [sel_exhaust_thm],

                all_collapse_thms, safe_collapse_thms, [expand_thm], [sel_split_thm],

                [sel_split_asm_thm], [case_eq_if_thm])

             end;

         val exhaust_case_names_attr = Attrib.internal (K (Rule_Cases.case_names exhaust_cases));

         val cases_type_attr = Attrib.internal (K (Induct.cases_type fcT_name));

         val anonymous_notes =

           [(map (fn th => th RS notE) distinct_thms, safe_elim_attrs),

            (map (fn th => th RS @{thm eq_False[THEN iffD2]}

               handle THM _ => th RS @{thm eq_True[THEN iffD2]}) nontriv_disc_thms,

             code_nitpicksimp_attrs)]

           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));

         val notes =

           [(caseN, case_thms, code_nitpicksimp_simp_attrs),

            (case_congN, [case_cong_thm], []),

            (case_eq_ifN, case_eq_if_thms, []),

            (collapseN, safe_collapse_thms, simp_attrs),

            (discN, nontriv_disc_thms, simp_attrs),

            (discIN, nontriv_discI_thms, []),

            (disc_excludeN, disc_exclude_thms, dest_attrs),

            (disc_exhaustN, disc_exhaust_thms, [exhaust_case_names_attr]),

            (distinctN, distinct_thms, simp_attrs @ inductsimp_attrs),

            (exhaustN, [exhaust_thm], [exhaust_case_names_attr, cases_type_attr]),

            (expandN, expand_thms, []),

            (injectN, inject_thms, iff_attrs @ inductsimp_attrs),

            (nchotomyN, [nchotomy_thm], []),

            (selN, all_sel_thms, code_nitpicksimp_simp_attrs),

            (sel_exhaustN, sel_exhaust_thms, [exhaust_case_names_attr]),

            (sel_splitN, sel_split_thms, []),

            (sel_split_asmN, sel_split_asm_thms, []),

            (splitN, [split_thm], []),

            (split_asmN, [split_asm_thm], []),

            (splitsN, [split_thm, split_asm_thm], []),

            (weak_case_cong_thmsN, [weak_case_cong_thm], cong_attrs)]

           |> filter_out (null o #2)

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

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

         val ctr_sugar =

           {ctrs = ctrs, casex = casex, discs = discs, selss = selss, exhaust = exhaust_thm,

            nchotomy = nchotomy_thm, injects = inject_thms, distincts = distinct_thms,

            case_thms = case_thms, case_cong = case_cong_thm, weak_case_cong = weak_case_cong_thm,

            split = split_thm, split_asm = split_asm_thm, disc_thmss = disc_thmss,

            discIs = discI_thms, sel_thmss = sel_thmss, disc_exhausts = disc_exhaust_thms,

            sel_exhausts = sel_exhaust_thms, collapses = all_collapse_thms, expands = expand_thms,

            sel_splits = sel_split_thms, sel_split_asms = sel_split_asm_thms,

            case_eq_ifs = case_eq_if_thms};

       in

         (ctr_sugar,

          lthy

          |> not rep_compat ?

             Local_Theory.declaration {syntax = false, pervasive = true}

               (fn phi => Case_Translation.register

                  (Morphism.term phi casex) (map (Morphism.term phi) ctrs))

          |> Local_Theory.notes (anonymous_notes @ notes) |> snd

          |> register_ctr_sugar fcT_name ctr_sugar

          |> (not no_code andalso null (Thm.hyps_of (hd case_thms)))

            ? Local_Theory.background_theory

              (add_ctr_code fcT_name As (map dest_Const ctrs) inject_thms distinct_thms case_thms))

       end;

   in

     (goalss, after_qed, lthy')

   end;

 fun wrap_free_constructors tacss = (fn (goalss, after_qed, lthy) =>

   map2 (map2 (Thm.close_derivation oo Goal.prove_sorry lthy [] [])) goalss tacss

   |> (fn thms => after_qed thms lthy)) oo prepare_wrap_free_constructors (K I);

 val wrap_free_constructors_cmd = (fn (goalss, after_qed, lthy) =>

   Proof.theorem NONE (snd oo after_qed) (map (map (rpair [])) goalss) lthy) oo

   prepare_wrap_free_constructors Syntax.read_term;

 fun parse_bracket_list parser = @{keyword "["} |-- Parse.list parser --|  @{keyword "]"};

 val parse_bindings = parse_bracket_list parse_binding;

 val parse_bindingss = parse_bracket_list parse_bindings;

 val parse_bound_term = (parse_binding --| @{keyword ":"}) -- Parse.term;

 val parse_bound_terms = parse_bracket_list parse_bound_term;

 val parse_bound_termss = parse_bracket_list parse_bound_terms;

 val parse_wrap_free_constructors_options =

   Scan.optional (@{keyword "("} |-- Parse.list1

         (Parse.reserved "no_discs_sels" >> K 0 || Parse.reserved "no_code" >> K 1 ||

          Parse.reserved "rep_compat" >> K 2) --| @{keyword ")"}

       >> (fn js => (member (op =) js 0, (member (op =) js 1, member (op =) js 2))))

     (false, (false, false));

 val _ =

   Outer_Syntax.local_theory_to_proof @{command_spec "wrap_free_constructors"}

     "wrap an existing freely generated type's constructors"

     ((parse_wrap_free_constructors_options -- (@{keyword "["} |-- Parse.list Parse.term --|

         @{keyword "]"}) --

       parse_binding -- Scan.optional (parse_bindings -- Scan.optional (parse_bindingss --

         Scan.optional parse_bound_termss []) ([], [])) ([], ([], [])))

      >> wrap_free_constructors_cmd);

 end;