(*  Title:      HOL/BNF/Tools/bnf_fp_n2m.ML

     Author:     Dmitriy Traytel, TU Muenchen

     Copyright   2013

 Flattening of nested to mutual (co)recursion.

 *)

 signature BNF_FP_N2M =

 sig

   val construct_mutualized_fp: BNF_FP_Util.fp_kind  -> typ list -> BNF_FP_Def_Sugar.fp_sugar list ->

     binding list -> (string * sort) list -> typ list * typ list list -> BNF_Def.bnf list ->

     local_theory -> BNF_FP_Util.fp_result * local_theory

 end;

 structure BNF_FP_N2M : BNF_FP_N2M =

 struct

 open BNF_Def

 open BNF_Util

 open BNF_FP_Util

 open BNF_FP_Def_Sugar

 open BNF_Tactics

 open BNF_FP_N2M_Tactics

 fun force_typ ctxt T =

   map_types Type_Infer.paramify_vars 

   #> Type.constraint T

   #> Syntax.check_term ctxt

   #> singleton (Variable.polymorphic ctxt);

 fun mk_prod_map f g =

   let

     val ((fAT, fBT), fT) = `dest_funT (fastype_of f);

     val ((gAT, gBT), gT) = `dest_funT (fastype_of g);

   in

     Const (@{const_name map_pair},

       fT --> gT --> HOLogic.mk_prodT (fAT, gAT) --> HOLogic.mk_prodT (fBT, gBT)) $ f $ g

   end;

 fun mk_sum_map f g =

   let

     val ((fAT, fBT), fT) = `dest_funT (fastype_of f);

     val ((gAT, gBT), gT) = `dest_funT (fastype_of g);

   in

     Const (@{const_name sum_map}, fT --> gT --> mk_sumT (fAT, gAT) --> mk_sumT (fBT, gBT)) $ f $ g

   end;

 fun construct_mutualized_fp fp fpTs fp_sugars bs resBs (resDs, Dss) bnfs lthy =

   let

     fun steal get = map (of_fp_sugar (get o #fp_res)) fp_sugars;

     val n = length bnfs;

     val deads = fold (union (op =)) Dss resDs;

     val As = subtract (op =) deads (map TFree resBs);

     val names_lthy = fold Variable.declare_typ (As @ deads) lthy;

     val m = length As;

     val live = m + n;

     val ((Xs, Bs), names_lthy) = names_lthy

       |> mk_TFrees n

       ||>> mk_TFrees m;

     val allAs = As @ Xs;

     val phiTs = map2 mk_pred2T As Bs;

     val theta = As ~~ Bs;

     val fpTs' = map (Term.typ_subst_atomic theta) fpTs;

     val pre_phiTs = map2 mk_pred2T fpTs fpTs';

     fun mk_co_algT T U = fp_case fp (T --> U) (U --> T);

     fun co_swap pair = fp_case fp I swap pair;

     val dest_co_algT = co_swap o dest_funT;

     val co_alg_argT = fp_case fp range_type domain_type;

     val co_alg_funT = fp_case fp domain_type range_type;

     val mk_co_product = curry (fp_case fp mk_convol mk_sum_case);

     val mk_map_co_product = fp_case fp mk_prod_map mk_sum_map;

     val co_proj1_const = fp_case fp (fst_const o fst) (uncurry Inl_const o dest_sumT o snd);

     val mk_co_productT = curry (fp_case fp HOLogic.mk_prodT mk_sumT);

     val dest_co_productT = fp_case fp HOLogic.dest_prodT dest_sumT;

     val ((ctors, dtors), (xtor's, xtors)) =

       let

         val ctors = map2 (force_typ names_lthy o (fn T => dummyT --> T)) fpTs (steal #ctors);

         val dtors = map2 (force_typ names_lthy o (fn T => T --> dummyT)) fpTs (steal #dtors);

       in

         ((ctors, dtors), `(map (Term.subst_atomic_types theta)) (fp_case fp ctors dtors))

       end;

     val xTs = map (domain_type o fastype_of) xtors;

     val yTs = map (domain_type o fastype_of) xtor's;

     val (((((phis, phis'), pre_phis), xs), ys), names_lthy) = names_lthy

       |> mk_Frees' "R" phiTs

       ||>> mk_Frees "S" pre_phiTs

       ||>> mk_Frees "x" xTs

       ||>> mk_Frees "y" yTs;

     val fp_bnfs = steal #bnfs;

     val pre_bnfs = map (of_fp_sugar #pre_bnfs) fp_sugars;

     val pre_bnfss = map #pre_bnfs fp_sugars;

     val nesty_bnfss = map (fn sugar => #nested_bnfs sugar @ #nesting_bnfs sugar) fp_sugars;

     val fp_nesty_bnfss = fp_bnfs :: nesty_bnfss;

     val fp_nesty_bnfs = distinct eq_bnf (flat fp_nesty_bnfss);

     fun abstract t =

       let val Ts = Term.add_frees t [];

       in fold_rev Term.absfree (filter (member op = Ts) phis') t end;

     val rels =

       let

         fun find_rel T As Bs = fp_nesty_bnfss

           |> map (filter_out (curry eq_bnf BNF_Comp.DEADID_bnf))

           |> get_first (find_first (fn bnf => Type.could_unify (T_of_bnf bnf, T)))

           |> Option.map (fn bnf =>

             let val live = live_of_bnf bnf;

             in (mk_rel live As Bs (rel_of_bnf bnf), live) end)

           |> the_default (HOLogic.eq_const T, 0);

         fun mk_rel (T as Type (_, Ts)) (Type (_, Us)) =

               let

                 val (rel, live) = find_rel T Ts Us;

                 val (Ts', Us') = fastype_of rel |> strip_typeN live |> fst |> map_split dest_pred2T;

                 val rels = map2 mk_rel Ts' Us';

               in

                 Term.list_comb (rel, rels)

               end

           | mk_rel (T as TFree _) _ = nth phis (find_index (curry op = T) As)

           | mk_rel _ _ = raise Fail "fpTs contains schematic type variables";

       in

         map2 (abstract oo mk_rel) fpTs fpTs'

       end;

     val pre_rels = map2 (fn Ds => mk_rel_of_bnf Ds (As @ fpTs) (Bs @ fpTs')) Dss bnfs;

     val rel_unfoldss = map (maps (fn bnf => no_refl [rel_def_of_bnf bnf])) pre_bnfss;

     val rel_xtor_co_inducts = steal (split_conj_thm o #rel_xtor_co_induct_thm)

       |> map2 (fn unfs => unfold_thms lthy (id_apply :: unfs)) rel_unfoldss;

     val rel_defs = map rel_def_of_bnf bnfs;

     val rel_monos = map rel_mono_of_bnf bnfs;

     val rel_xtor_co_induct_thm =

       mk_rel_xtor_co_induct_thm fp pre_rels pre_phis rels phis xs ys xtors xtor's

         (mk_rel_xtor_co_induct_tactic fp rel_xtor_co_inducts rel_defs rel_monos) lthy;

     val rel_eqs = no_refl (map rel_eq_of_bnf fp_nesty_bnfs);

     val map_id0s = no_refl (map map_id0_of_bnf bnfs);

     val xtor_co_induct_thm =

       (case fp of

         Least_FP =>

           let

             val (Ps, names_lthy) = names_lthy

               |> mk_Frees "P" (map (fn T => T --> HOLogic.boolT) fpTs);

             fun mk_Grp_id P =

               let val T = domain_type (fastype_of P);

               in mk_Grp (HOLogic.Collect_const T $ P) (HOLogic.id_const T) end;

             val cts = map (SOME o certify lthy) (map HOLogic.eq_const As @ map mk_Grp_id Ps);

           in

             cterm_instantiate_pos cts rel_xtor_co_induct_thm

             |> singleton (Proof_Context.export names_lthy lthy)

             |> unfold_thms lthy (@{thms eq_le_Grp_id_iff all_simps(1,2)[symmetric]} @ rel_eqs)

             |> funpow n (fn thm => thm RS spec)

             |> unfold_thms lthy (@{thm eq_alt} :: map rel_Grp_of_bnf bnfs @ map_id0s)

             |> unfold_thms lthy @{thms Grp_id_mono_subst eqTrueI[OF subset_UNIV] simp_thms(22)}

             |> unfold_thms lthy @{thms subset_iff mem_Collect_eq

                atomize_conjL[symmetric] atomize_all[symmetric] atomize_imp[symmetric]}

             |> unfold_thms lthy (maps set_defs_of_bnf bnfs)

           end

       | Greatest_FP =>

           let

             val cts = NONE :: map (SOME o certify lthy) (map HOLogic.eq_const As);

           in

             cterm_instantiate_pos cts rel_xtor_co_induct_thm

             |> unfold_thms lthy (@{thms le_fun_def le_bool_def all_simps(1,2)[symmetric]} @ rel_eqs)

             |> funpow (2 * n) (fn thm => thm RS spec)

             |> Conv.fconv_rule Object_Logic.atomize

             |> funpow n (fn thm => thm RS mp)

           end);

     val fold_preTs = map2 (fn Ds => mk_T_of_bnf Ds allAs) Dss bnfs;

     val fold_pre_deads_only_Ts = map2 (fn Ds => mk_T_of_bnf Ds (replicate live dummyT)) Dss bnfs;

     val rec_theta = Xs ~~ map2 mk_co_productT fpTs Xs;

     val rec_preTs = map (Term.typ_subst_atomic rec_theta) fold_preTs;

     val fold_strTs = map2 mk_co_algT fold_preTs Xs;

     val rec_strTs = map2 mk_co_algT rec_preTs Xs;

     val resTs = map2 mk_co_algT fpTs Xs;

     val (((fold_strs, fold_strs'), (rec_strs, rec_strs')), names_lthy) = names_lthy

       |> mk_Frees' "s" fold_strTs

       ||>> mk_Frees' "s" rec_strTs;

     val co_iters = steal #xtor_co_iterss;

     val ns = map (length o #pre_bnfs) fp_sugars;

     fun substT rho (Type (@{type_name "fun"}, [T, U])) = substT rho T --> substT rho U

       | substT rho (Type (s, Ts)) = Type (s, map (typ_subst_nonatomic rho) Ts)

       | substT _ T = T;

     fun force_iter is_rec i TU TU_rec raw_iters =

       let

         val approx_fold = un_fold_of raw_iters

           |> force_typ names_lthy

             (replicate (nth ns i) dummyT ---> (if is_rec then TU_rec else TU));

         val TUs = binder_fun_types (Term.typ_subst_atomic (Xs ~~ fpTs) (fastype_of approx_fold));

         val js = find_indices Type.could_unify

           TUs (map (Term.typ_subst_atomic (Xs ~~ fpTs)) fold_strTs);

         val Tpats = map (fn j => mk_co_algT (nth fold_pre_deads_only_Ts j) (nth Xs j)) js;

         val iter = raw_iters |> (if is_rec then co_rec_of else un_fold_of);

       in

         force_typ names_lthy (Tpats ---> TU) iter

       end;

     fun mk_iter b_opt is_rec iters lthy TU =

       let

         val x = co_alg_argT TU;

         val i = find_index (fn T => x = T) Xs;

         val TUiter =

           (case find_first (fn f => body_fun_type (fastype_of f) = TU) iters of

             NONE => nth co_iters i

               |> force_iter is_rec i

                 (TU |> (is_none b_opt andalso not is_rec) ? substT (fpTs ~~ Xs))

                 (TU |> (is_none b_opt) ? substT (map2 mk_co_productT fpTs Xs ~~ Xs))

           | SOME f => f);

         val TUs = binder_fun_types (fastype_of TUiter);

         val iter_preTs = if is_rec then rec_preTs else fold_preTs;

         val iter_strs = if is_rec then rec_strs else fold_strs;

         fun mk_s TU' =

           let

             val i = find_index (fn T => co_alg_argT TU' = T) Xs;

             val sF = co_alg_funT TU'; 

             val F = nth iter_preTs i;

             val s = nth iter_strs i;

           in

             (if sF = F then s

             else

               let

                 val smapT = replicate live dummyT ---> mk_co_algT sF F;

                 fun hidden_to_unit t =

                   Term.subst_TVars (map (rpair HOLogic.unitT) (Term.add_tvar_names t [])) t;

                 val smap = map_of_bnf (nth bnfs i)

                   |> force_typ names_lthy smapT

                   |> hidden_to_unit;

                 val smap_argTs = strip_typeN live (fastype_of smap) |> fst;

                 fun mk_smap_arg TU =              

                   (if domain_type TU = range_type TU then

                     HOLogic.id_const (domain_type TU)

                   else if is_rec then

                     let

                       val (TY, (U, X)) = TU |> dest_co_algT ||> dest_co_productT;

                       val T = mk_co_algT TY U;

                     in

                       (case try (force_typ lthy T o build_map lthy co_proj1_const o dest_funT) T of

                         SOME f => mk_co_product f

                           (fst (fst (mk_iter NONE is_rec iters lthy (mk_co_algT TY X))))

                       | NONE => mk_map_co_product

                           (build_map lthy co_proj1_const

                             (dest_funT (mk_co_algT (dest_co_productT TY |> fst) U)))

                           (HOLogic.id_const X))

                     end

                   else

                     fst (fst (mk_iter NONE is_rec iters lthy TU)))

                 val smap_args = map mk_smap_arg smap_argTs;

               in

                 HOLogic.mk_comp (co_swap (s, Term.list_comb (smap, smap_args)))

               end)

           end;

         val t = Term.list_comb (TUiter, map mk_s TUs);

       in

         (case b_opt of

           NONE => ((t, Drule.dummy_thm), lthy)

         | SOME b => Local_Theory.define ((b, NoSyn), ((Thm.def_binding b, []), 

             fold_rev Term.absfree (if is_rec then rec_strs' else fold_strs') t)) lthy |>> apsnd snd)

       end;

     fun mk_iters is_rec name lthy =

       fold2 (fn TU => fn b => fn ((iters, defs), lthy) =>

         mk_iter (SOME b) is_rec iters lthy TU |>> (fn (f, d) => (f :: iters, d :: defs)))

       resTs (map (Binding.suffix_name ("_" ^ name)) bs) (([], []), lthy)

       |>> apfst rev o apsnd rev;

     val foldN = fp_case fp ctor_foldN dtor_unfoldN;

     val recN = fp_case fp ctor_recN dtor_corecN;

     val (((raw_un_folds, raw_un_fold_defs), (raw_co_recs, raw_co_rec_defs)), (lthy, raw_lthy)) =

       lthy

       |> mk_iters false foldN

       ||>> mk_iters true recN

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism raw_lthy lthy;

     val un_folds = map (Morphism.term phi) raw_un_folds;

     val co_recs = map (Morphism.term phi) raw_co_recs;

     val (xtor_un_fold_thms, xtor_co_rec_thms) =

       let

         val folds = map (fn f => Term.list_comb (f, fold_strs)) raw_un_folds;

         val recs = map (fn r => Term.list_comb (r, rec_strs)) raw_co_recs;

         val fold_mapTs = co_swap (As @ fpTs, As @ Xs);

         val rec_mapTs = co_swap (As @ fpTs, As @ map2 mk_co_productT fpTs Xs);

         val pre_fold_maps =

           map2 (fn Ds => fn bnf =>

             Term.list_comb (uncurry (mk_map_of_bnf Ds) fold_mapTs bnf,

               map HOLogic.id_const As @ folds))

           Dss bnfs;

         val pre_rec_maps =

           map2 (fn Ds => fn bnf =>

             Term.list_comb (uncurry (mk_map_of_bnf Ds) rec_mapTs bnf,

               map HOLogic.id_const As @ map2 (mk_co_product o HOLogic.id_const) fpTs recs))

           Dss bnfs;

         fun mk_goals f xtor s smap =

           ((f, xtor), (s, smap))

           |> pairself (HOLogic.mk_comp o co_swap)

           |> HOLogic.mk_eq;

         val fold_goals = map4 mk_goals folds xtors fold_strs pre_fold_maps

         val rec_goals = map4 mk_goals recs xtors rec_strs pre_rec_maps;

         fun mk_thms ss goals tac =

           Library.foldr1 HOLogic.mk_conj goals

           |> HOLogic.mk_Trueprop

           |> fold_rev Logic.all ss

           |> (fn goal => Goal.prove_sorry raw_lthy [] [] goal tac)

           |> Thm.close_derivation

           |> Morphism.thm phi

           |> split_conj_thm

           |> map (fn thm => thm RS @{thm comp_eq_dest});

         val pre_map_defs = no_refl (map map_def_of_bnf bnfs);

         val fp_pre_map_defs = no_refl (map map_def_of_bnf pre_bnfs);

         val map_unfoldss = map (maps (fn bnf => no_refl [map_def_of_bnf bnf])) pre_bnfss;

         val unfold_map = map2 (fn unfs => unfold_thms lthy (id_apply :: unfs)) map_unfoldss;

         val fp_xtor_co_iterss = steal #xtor_co_iter_thmss;

         val fp_xtor_un_folds = map (mk_pointfree lthy o un_fold_of) fp_xtor_co_iterss |> unfold_map;

         val fp_xtor_co_recs = map (mk_pointfree lthy o co_rec_of) fp_xtor_co_iterss |> unfold_map;

         val fp_co_iter_o_mapss = steal #xtor_co_iter_o_map_thmss;

         val fp_fold_o_maps = map un_fold_of fp_co_iter_o_mapss |> unfold_map;

         val fp_rec_o_maps = map co_rec_of fp_co_iter_o_mapss |> unfold_map;

         val fold_thms = fp_case fp @{thm o_assoc[symmetric]} @{thm o_assoc} ::

           @{thms id_apply o_apply o_id id_o map_pair.comp map_pair.id sum_map.comp sum_map.id};

         val rec_thms = fold_thms @ fp_case fp

           @{thms fst_convol map_pair_o_convol convol_o}

           @{thms sum_case_o_inj(1) sum_case_o_sum_map o_sum_case};

         val map_thms = no_refl (maps (fn bnf =>

           [map_comp0_of_bnf bnf RS sym, map_id0_of_bnf bnf]) fp_nesty_bnfs);

         fun mk_tac defs o_map_thms xtor_thms thms {context = ctxt, prems = _} =

           unfold_thms_tac ctxt

             (flat [thms, defs, pre_map_defs, fp_pre_map_defs, xtor_thms, o_map_thms, map_thms]) THEN

           CONJ_WRAP (K (HEADGOAL (rtac refl))) bnfs;

         val fold_tac = mk_tac raw_un_fold_defs fp_fold_o_maps fp_xtor_un_folds fold_thms;

         val rec_tac = mk_tac raw_co_rec_defs fp_rec_o_maps fp_xtor_co_recs rec_thms;

       in

         (mk_thms fold_strs fold_goals fold_tac, mk_thms rec_strs rec_goals rec_tac)

       end;

     (* These results are half broken. This is deliberate. We care only about those fields that are

        used by "primrec_new", "primcorecursive", and "datatype_new_compat". *)

     val fp_res =

       ({Ts = fpTs,

         bnfs = steal #bnfs,

         dtors = dtors,

         ctors = ctors,

         xtor_co_iterss = transpose [un_folds, co_recs],

         xtor_co_induct = xtor_co_induct_thm,

         dtor_ctors = steal #dtor_ctors (*too general types*),

         ctor_dtors = steal #ctor_dtors (*too general types*),

         ctor_injects = steal #ctor_injects (*too general types*),

         dtor_injects = steal #dtor_injects (*too general types*),

         xtor_map_thms = steal #xtor_map_thms (*too general types and terms*),

         xtor_set_thmss = steal #xtor_set_thmss (*too general types and terms*),

         xtor_rel_thms = steal #xtor_rel_thms (*too general types and terms*),

         xtor_co_iter_thmss = transpose [xtor_un_fold_thms, xtor_co_rec_thms],

         xtor_co_iter_o_map_thmss = steal #xtor_co_iter_o_map_thmss (*theorem about old constant*),

         rel_xtor_co_induct_thm = rel_xtor_co_induct_thm}

        |> morph_fp_result (Morphism.term_morphism (singleton (Variable.polymorphic lthy))));

   in

     (fp_res, lthy)

   end

 end;