(*  Title:      HOL/BNF/Tools/bnf_fp_n2m_sugar.ML

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2013

 Suggared flattening of nested to mutual (co)recursion.

 *)

 signature BNF_FP_N2M_SUGAR =

 sig

   val unfold_let: term -> term

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

   val mutualize_fp_sugars: BNF_FP_Util.fp_kind -> binding list -> typ list -> (term -> int list) ->

     term list list list list -> BNF_FP_Def_Sugar.fp_sugar list -> local_theory ->

     (BNF_FP_Def_Sugar.fp_sugar list

      * (BNF_FP_Def_Sugar.lfp_sugar_thms option * BNF_FP_Def_Sugar.gfp_sugar_thms option))

     * local_theory

   val indexify_callsss: BNF_FP_Def_Sugar.fp_sugar -> (term * term list list) list ->

     term list list list

   val nested_to_mutual_fps: BNF_FP_Util.fp_kind -> binding list -> typ list -> (term -> int list) ->

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

     (typ list * int list * BNF_FP_Def_Sugar.fp_sugar list

      * (BNF_FP_Def_Sugar.lfp_sugar_thms option * BNF_FP_Def_Sugar.gfp_sugar_thms option))

     * local_theory

 end;

 structure BNF_FP_N2M_Sugar : BNF_FP_N2M_SUGAR =

 struct

 open Ctr_Sugar

 open BNF_Util

 open BNF_Def

 open BNF_FP_Util

 open BNF_FP_Def_Sugar

 open BNF_FP_N2M

 val n2mN = "n2m_"

 type n2m_sugar = fp_sugar list * (lfp_sugar_thms option * gfp_sugar_thms option);

 structure Data = Generic_Data

 (

   type T = n2m_sugar Typtab.table;

   val empty = Typtab.empty;

   val extend = I;

   val merge = Typtab.merge (eq_fst (eq_list eq_fp_sugar));

 );

 fun morph_n2m_sugar phi (fp_sugars, (lfp_sugar_thms_opt, gfp_sugar_thms_opt)) =

   (map (morph_fp_sugar phi) fp_sugars,

    (Option.map (morph_lfp_sugar_thms phi) lfp_sugar_thms_opt,

     Option.map (morph_gfp_sugar_thms phi) gfp_sugar_thms_opt));

 val transfer_n2m_sugar =

   morph_n2m_sugar o Morphism.thm_morphism o Thm.transfer o Proof_Context.theory_of;

 fun n2m_sugar_of ctxt =

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

   #> Option.map (transfer_n2m_sugar ctxt);

 fun register_n2m_sugar key n2m_sugar =

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

     (fn phi => Data.map (Typtab.default (key, morph_n2m_sugar phi n2m_sugar)));

 fun unfold_let (Const (@{const_name Let}, _) $ arg1 $ arg2) = unfold_let (betapply (arg2, arg1))

   | unfold_let (Const (@{const_name prod_case}, _) $ t) =

     (case unfold_let t of

       t' as Abs (s1, T1, Abs (s2, T2, _)) =>

       let

         val x = (s1 ^ s2, Term.maxidx_of_term t + 1);

         val v = Var (x, HOLogic.mk_prodT (T1, T2));

       in

         lambda v (unfold_let (betapplys (t', [HOLogic.mk_fst v, HOLogic.mk_snd v])))

       end

     | _ => t)

   | unfold_let (t $ u) = betapply (unfold_let t, unfold_let u)

   | unfold_let (Abs (s, T, t)) = Abs (s, T, unfold_let t)

   | unfold_let t = t;

 fun mk_map_pattern ctxt s =

   let

     val bnf = the (bnf_of ctxt s);

     val mapx = map_of_bnf bnf;

     val live = live_of_bnf bnf;

     val (f_Ts, _) = strip_typeN live (fastype_of mapx);

     val fs = map_index (fn (i, T) => Var (("?f", i), T)) f_Ts;

   in

     (mapx, betapplys (mapx, fs))

   end;

 fun dest_map ctxt s call =

   let

     val (map0, pat) = mk_map_pattern ctxt s;

     val (_, tenv) = fo_match ctxt call pat;

   in

     (map0, Vartab.fold_rev (fn (_, (_, f)) => cons f) tenv [])

   end;

 fun dest_abs_or_applied_map _ _ (Abs (_, _, t)) = (Term.dummy, [t])

   | dest_abs_or_applied_map ctxt s (t1 $ _) = dest_map ctxt s t1;

 fun map_partition f xs =

   fold_rev (fn x => fn (ys, (good, bad)) =>

       case f x of SOME y => (y :: ys, (x :: good, bad)) | NONE => (ys, (good, x :: bad)))

     xs ([], ([], []));

 fun key_of_fp_eqs fp fpTs fp_eqs =

   Type (fp_case fp "l" "g", fpTs @ maps (fn (x, T) => [TFree x, T]) fp_eqs);

 (* TODO: test with sort constraints on As *)

 fun mutualize_fp_sugars fp bs fpTs get_indices callssss fp_sugars0 no_defs_lthy0 =

   let

     val thy = Proof_Context.theory_of no_defs_lthy0;

     val qsotm = quote o Syntax.string_of_term no_defs_lthy0;

     fun incompatible_calls t1 t2 =

       error ("Incompatible " ^ co_prefix fp ^ "recursive calls: " ^ qsotm t1 ^ " vs. " ^ qsotm t2);

     val b_names = map Binding.name_of bs;

     val fp_b_names = map base_name_of_typ fpTs;

     val nn = length fpTs;

     fun target_ctr_sugar_of_fp_sugar fpT ({T, index, ctr_sugars, ...} : fp_sugar) =

       let

         val rho = Vartab.fold (cons o apsnd snd) (Sign.typ_match thy (T, fpT) Vartab.empty) [];

         val phi = Morphism.term_morphism (Term.subst_TVars rho);

       in

         morph_ctr_sugar phi (nth ctr_sugars index)

       end;

     val ctr_defss = map (of_fp_sugar #ctr_defss) fp_sugars0;

     val mapss = map (of_fp_sugar #mapss) fp_sugars0;

     val ctr_sugars0 = map2 target_ctr_sugar_of_fp_sugar fpTs fp_sugars0;

     val ctrss = map #ctrs ctr_sugars0;

     val ctr_Tss = map (map fastype_of) ctrss;

     val As' = fold (fold Term.add_tfreesT) ctr_Tss [];

     val As = map TFree As';

     val ((Cs, Xs), no_defs_lthy) =

       no_defs_lthy0

       |> fold Variable.declare_typ As

       |> mk_TFrees nn

       ||>> variant_tfrees fp_b_names;

     fun check_call_dead live_call call =

       if null (get_indices call) then () else incompatible_calls live_call call;

     fun freeze_fpTs_simple (T as Type (s, Ts)) =

         (case find_index (curry (op =) T) fpTs of

           ~1 => Type (s, map freeze_fpTs_simple Ts)

         | kk => nth Xs kk)

       | freeze_fpTs_simple T = T;

     fun freeze_fpTs_map (callss, (live_call :: _, dead_calls)) s Ts =

       (List.app (check_call_dead live_call) dead_calls;

        Type (s, map2 freeze_fpTs (flatten_type_args_of_bnf (the (bnf_of no_defs_lthy s)) []

          (transpose callss)) Ts))

     and freeze_fpTs calls (T as Type (s, Ts)) =

         (case map_partition (try (snd o dest_map no_defs_lthy s)) calls of

           ([], _) =>

           (case map_partition (try (snd o dest_abs_or_applied_map no_defs_lthy s)) calls of

             ([], _) => freeze_fpTs_simple T

           | callsp => freeze_fpTs_map callsp s Ts)

         | callsp => freeze_fpTs_map callsp s Ts)

       | freeze_fpTs _ T = T;

     val ctr_Tsss = map (map binder_types) ctr_Tss;

     val ctrXs_Tsss = map2 (map2 (map2 freeze_fpTs)) callssss ctr_Tsss;

     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;

     val Ts = map (body_type o hd) ctr_Tss;

     val ns = map length ctr_Tsss;

     val kss = map (fn n => 1 upto n) ns;

     val mss = map (map length) ctr_Tsss;

     val fp_eqs = map dest_TFree Xs ~~ ctrXs_sum_prod_Ts;

     val key = key_of_fp_eqs fp fpTs fp_eqs;

   in

     (case n2m_sugar_of no_defs_lthy key of

       SOME n2m_sugar => (n2m_sugar, no_defs_lthy)

     | NONE =>

       let

         val base_fp_names = Name.variant_list [] fp_b_names;

         val fp_bs = map2 (fn b_name => fn base_fp_name =>

             Binding.qualify true b_name (Binding.name (n2mN ^ base_fp_name)))

           b_names base_fp_names;

         val (pre_bnfs, (fp_res as {xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_injects,

                dtor_ctors, xtor_co_iter_thmss, ...}, lthy)) =

           fp_bnf (construct_mutualized_fp fp fpTs fp_sugars0) fp_bs As' fp_eqs no_defs_lthy;

         val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;

         val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;

         val ((xtor_co_iterss, iters_args_types, coiters_args_types), _) =

           mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy;

         fun mk_binding b suf = Binding.suffix_name ("_" ^ suf) b;

         val ((co_iterss, co_iter_defss), lthy) =

           fold_map2 (fn b =>

             (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)

              else define_coiters [unfoldN, corecN] (the coiters_args_types))

               (mk_binding b) fpTs Cs) fp_bs xtor_co_iterss lthy

           |>> split_list;

         val rho = tvar_subst thy Ts fpTs;

         val ctr_sugar_phi = Morphism.compose (Morphism.typ_morphism (Term.typ_subst_TVars rho))

             (Morphism.term_morphism (Term.subst_TVars rho));

         val inst_ctr_sugar = morph_ctr_sugar ctr_sugar_phi;

         val ctr_sugars = map inst_ctr_sugar ctr_sugars0;

         val ((co_inducts, un_fold_thmss, co_rec_thmss, disc_unfold_thmss, disc_corec_thmss,

               sel_unfold_thmsss, sel_corec_thmsss), fp_sugar_thms) =

           if fp = Least_FP then

             derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct

               xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss

               co_iterss co_iter_defss lthy

             |> `(fn ((_, induct, _), (fold_thmss, rec_thmss, _)) =>

               ([induct], fold_thmss, rec_thmss, [], [], [], []))

             ||> (fn info => (SOME info, NONE))

           else

             derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct

               dtor_injects dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss mss

               ns ctr_defss ctr_sugars co_iterss co_iter_defss

               (Proof_Context.export lthy no_defs_lthy) lthy

             |> `(fn ((coinduct_thms_pairs, _), (unfold_thmss, corec_thmss, _),

                     (disc_unfold_thmss, disc_corec_thmss, _), _,

                     (sel_unfold_thmsss, sel_corec_thmsss, _)) =>

               (map snd coinduct_thms_pairs, unfold_thmss, corec_thmss, disc_unfold_thmss,

                disc_corec_thmss, sel_unfold_thmsss, sel_corec_thmsss))

             ||> (fn info => (NONE, SOME info));

         val phi = Proof_Context.export_morphism no_defs_lthy no_defs_lthy0;

         fun mk_target_fp_sugar (kk, T) =

           {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs, nested_bnfs = nested_bnfs,

            nesting_bnfs = nesting_bnfs, fp_res = fp_res, ctr_defss = ctr_defss,

            ctr_sugars = ctr_sugars, co_iterss = co_iterss, mapss = mapss, co_inducts = co_inducts,

            co_iter_thmsss = transpose [un_fold_thmss, co_rec_thmss],

            disc_co_itersss = transpose [disc_unfold_thmss, disc_corec_thmss],

            sel_co_iterssss = transpose [sel_unfold_thmsss, sel_corec_thmsss]}

           |> morph_fp_sugar phi;

         val n2m_sugar = (map_index mk_target_fp_sugar fpTs, fp_sugar_thms);

       in

         (n2m_sugar, lthy |> register_n2m_sugar key n2m_sugar)

       end)

   end;

 fun indexify_callsss fp_sugar callsss =

   let

     val {ctrs, ...} = of_fp_sugar #ctr_sugars fp_sugar;

     fun indexify_ctr ctr =

       (case AList.lookup Term.aconv_untyped callsss ctr of

         NONE => replicate (num_binder_types (fastype_of ctr)) []

       | SOME callss => map (map (Envir.beta_eta_contract o unfold_let)) callss);

   in

     map indexify_ctr ctrs

   end;

 fun retypargs tyargs (Type (s, _)) = Type (s, tyargs);

 fun fold_subtype_pairs f (T as Type (s, Ts), U as Type (s', Us)) =

     f (T, U) #> (if s = s' then fold (fold_subtype_pairs f) (Ts ~~ Us) else I)

   | fold_subtype_pairs f TU = f TU;

 fun nested_to_mutual_fps fp actual_bs actual_Ts get_indices actual_callssss0 lthy =

   let

     val qsoty = quote o Syntax.string_of_typ lthy;

     val qsotys = space_implode " or " o map qsoty;

     fun duplicate_datatype T = error (qsoty T ^ " is not mutually recursive with itself");

     fun not_co_datatype0 T = error (qsoty T ^ " is not a " ^ co_prefix fp ^ "datatype");

     fun not_co_datatype (T as Type (s, _)) =

         if fp = Least_FP andalso

            is_some (Datatype_Data.get_info (Proof_Context.theory_of lthy) s) then

           error (qsoty T ^ " is not a new-style datatype (cf. \"datatype_new\")")

         else

           not_co_datatype0 T

       | not_co_datatype T = not_co_datatype0 T;

     val _ = (case Library.duplicates (op =) actual_Ts of [] => () | T :: _ => duplicate_datatype T);

     val perm_actual_Ts =

       sort (prod_ord int_ord Term_Ord.typ_ord o pairself (`Term.size_of_typ)) actual_Ts;

     fun the_ctrs_of (Type (s, Ts)) = map (mk_ctr Ts) (#ctrs (the (ctr_sugar_of lthy s)));

     fun the_fp_sugar_of (T as Type (T_name, _)) =

       (case fp_sugar_of lthy T_name of

         SOME (fp_sugar as {fp = fp', ...}) => if fp = fp' then fp_sugar else not_co_datatype T

       | NONE => not_co_datatype T);

     fun gen_rhss_in gen_Ts rho subTs =

       let

         fun maybe_insert (T, Type (_, gen_tyargs)) =

             if member (op =) subTs T then insert (op =) gen_tyargs else I

           | maybe_insert _ = I;

         val ctrs = maps the_ctrs_of gen_Ts;

         val gen_ctr_Ts = maps (binder_types o fastype_of) ctrs;

         val ctr_Ts = map (Term.typ_subst_atomic rho) gen_ctr_Ts;

       in

         fold (fold_subtype_pairs maybe_insert) (ctr_Ts ~~ gen_ctr_Ts) []

       end;

     fun gather_types _ _ num_groups seen gen_seen [] = (num_groups, seen, gen_seen)

       | gather_types lthy rho num_groups seen gen_seen ((T as Type (_, tyargs)) :: Ts) =

         let

           val {fp_res = {Ts = mutual_Ts0, ...}, ...} = the_fp_sugar_of T;

           val mutual_Ts = map (retypargs tyargs) mutual_Ts0;

           fun fresh_tyargs () =

             let

               (* The name "'z" is unlikely to clash with the context, yielding more cache hits. *)

               val (gen_tyargs, lthy') =

                 variant_tfrees (replicate (length tyargs) "z") lthy

                 |>> map Logic.varifyT_global;

               val rho' = (gen_tyargs ~~ tyargs) @ rho;

             in

               (rho', gen_tyargs, gen_seen, lthy')

             end;

           val (rho', gen_tyargs, gen_seen', lthy') =

             if exists (exists_subtype_in seen) mutual_Ts then

               (case gen_rhss_in gen_seen rho mutual_Ts of

                 [] => fresh_tyargs ()

               | gen_tyargss as gen_tyargs :: gen_tyargss_tl =>

                 let

                   val unify_pairs = split_list (maps (curry (op ~~) gen_tyargs) gen_tyargss_tl);

                   val mgu = Type.raw_unifys unify_pairs Vartab.empty;

                   val gen_tyargs' = map (Envir.subst_type mgu) gen_tyargs;

                   val gen_seen' = map (Envir.subst_type mgu) gen_seen;

                 in

                   (rho, gen_tyargs', gen_seen', lthy)

                 end)

             else

               fresh_tyargs ();

           val gen_mutual_Ts = map (retypargs gen_tyargs) mutual_Ts0;

           val Ts' = filter_out (member (op =) mutual_Ts) Ts;

         in

           gather_types lthy' rho' (num_groups + 1) (seen @ mutual_Ts) (gen_seen' @ gen_mutual_Ts)

             Ts'

         end

       | gather_types _ _ _ _ _ (T :: _) = not_co_datatype T;

     val (num_groups, perm_Ts, perm_gen_Ts) = gather_types lthy [] 0 [] [] perm_actual_Ts;

     val perm_frozen_gen_Ts = map Logic.unvarifyT_global perm_gen_Ts;

     val missing_Ts = perm_Ts |> subtract (op =) actual_Ts;

     val Ts = actual_Ts @ missing_Ts;

     val nn = length Ts;

     val kks = 0 upto nn - 1;

     val callssss0 = pad_list [] nn actual_callssss0;

     val common_name = mk_common_name (map Binding.name_of actual_bs);

     val bs = pad_list (Binding.name common_name) nn actual_bs;

     fun permute xs = permute_like (op =) Ts perm_Ts xs;

     fun unpermute perm_xs = permute_like (op =) perm_Ts Ts perm_xs;

     val perm_bs = permute bs;

     val perm_kks = permute kks;

     val perm_callssss0 = permute callssss0;

     val perm_fp_sugars0 = map (the o fp_sugar_of lthy o fst o dest_Type) perm_Ts;

     val perm_callssss = map2 indexify_callsss perm_fp_sugars0 perm_callssss0;

     val get_perm_indices = map (fn kk => find_index (curry (op =) kk) perm_kks) o get_indices;

     val ((perm_fp_sugars, fp_sugar_thms), lthy) =

       if num_groups > 1 then

         mutualize_fp_sugars fp perm_bs perm_frozen_gen_Ts get_perm_indices perm_callssss

           perm_fp_sugars0 lthy

       else

         ((perm_fp_sugars0, (NONE, NONE)), lthy);

     val fp_sugars = unpermute perm_fp_sugars;

   in

     ((missing_Ts, perm_kks, fp_sugars, fp_sugar_thms), lthy)

   end;

 end;