(*  Title:      HOL/Codatatype/Tools/bnf_fp_sugar.ML

    Author:     Jasmin Blanchette, TU Muenchen

    Copyright   2012

Sugar for constructing LFPs and GFPs.

*)

signature BNF_FP_SUGAR =

sig

  (* TODO: programmatic interface *)

end;

structure BNF_FP_Sugar : BNF_FP_SUGAR =

struct

open BNF_Util

open BNF_Wrap

open BNF_FP_Util

open BNF_LFP

open BNF_GFP

open BNF_FP_Sugar_Tactics

val caseN = "case";

val itersN = "iters";

val recsN = "recs";

fun split_list7 xs = (map #1 xs, map #2 xs, map #3 xs, map #4 xs, map #5 xs, map #6 xs, map #7 xs);

fun retype_free (Free (s, _)) T = Free (s, T);

fun flat_list_comb (f, xss) = fold (fn xs => fn t => Term.list_comb (t, xs)) xss f

fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));

fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;

fun mk_uncurried2_fun f xss =

  mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat xss);

fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";

fun merge_type_arg_constrained ctxt (T, c) (T', c') =

  if T = T' then

    (case (c, c') of

      (_, NONE) => (T, c)

    | (NONE, _) => (T, c')

    | _ =>

      if c = c' then

        (T, c)

      else

        error ("Inconsistent sort constraints for type variable " ^

          quote (Syntax.string_of_typ ctxt T)))

  else

    cannot_merge_types ();

fun merge_type_args_constrained ctxt (cAs, cAs') =

  if length cAs = length cAs' then map2 (merge_type_arg_constrained ctxt) cAs cAs'

  else cannot_merge_types ();

fun type_args_constrained_of (((cAs, _), _), _) = cAs;

val type_args_of = map fst o type_args_constrained_of;

fun type_binder_of (((_, b), _), _) = b;

fun mixfix_of ((_, mx), _) = mx;

fun ctr_specs_of (_, ctr_specs) = ctr_specs;

fun disc_of (((disc, _), _), _) = disc;

fun ctr_of (((_, ctr), _), _) = ctr;

fun args_of ((_, args), _) = args;

fun ctr_mixfix_of (_, mx) = mx;

fun prepare_datatype prepare_typ lfp specs fake_lthy no_defs_lthy =

  let

    val constrained_As =

      map (map (apfst (prepare_typ fake_lthy)) o type_args_constrained_of) specs

      |> Library.foldr1 (merge_type_args_constrained no_defs_lthy);

    val As = map fst constrained_As;

    val As' = map dest_TFree As;

    val _ = (case duplicates (op =) As of [] => ()

      | A :: _ => error ("Duplicate type parameter " ^

          quote (Syntax.string_of_typ no_defs_lthy A)));

    (* TODO: use sort constraints on type args *)

    val N = length specs;

    fun mk_fake_T b =

      Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),

        As);

    val bs = map type_binder_of specs;

    val fakeTs = map mk_fake_T bs;

    val mixfixes = map mixfix_of specs;

    val _ = (case duplicates Binding.eq_name bs of [] => ()

      | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));

    val ctr_specss = map ctr_specs_of specs;

    val disc_binderss = map (map disc_of) ctr_specss;

    val ctr_binderss = map (map ctr_of) ctr_specss;

    val ctr_argsss = map (map args_of) ctr_specss;

    val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;

    val sel_bindersss = map (map (map fst)) ctr_argsss;

    val fake_ctr_Tsss = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;

    val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];

    val _ = (case subtract (op =) As' rhs_As' of

        [] => ()

      | A' :: _ => error ("Extra type variables on rhs: " ^

          quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));

    val ((Cs, Xs), _) =

      no_defs_lthy

      |> fold (fold (fn s => Variable.declare_typ (TFree (s, dummyS))) o type_args_of) specs

      |> mk_TFrees N

      ||>> mk_TFrees N;

    fun eq_fpT (T as Type (s, Us)) (Type (s', Us')) =

        s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^

          quote (Syntax.string_of_typ fake_lthy T)))

      | eq_fpT _ _ = false;

    fun freeze_fp (T as Type (s, Us)) =

        (case find_index (eq_fpT T) fakeTs of ~1 => Type (s, map freeze_fp Us) | j => nth Xs j)

      | freeze_fp T = T;

    val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;

    val sum_prod_TsXs = map (mk_sumTN o map HOLogic.mk_tupleT) ctr_TsssXs;

    val eqs = map dest_TFree Xs ~~ sum_prod_TsXs;

    val (pre_map_defs, ((unfs0, flds0, fp_iters0, fp_recs0, unf_flds, fld_unfs, fld_injects,

        fp_iter_thms, fp_rec_thms), lthy)) =

      fp_bnf (if lfp then bnf_lfp else bnf_gfp) bs mixfixes As' eqs no_defs_lthy;

    val timer = time (Timer.startRealTimer ());

    fun mk_unf_or_fld get_T Ts t =

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

        Term.subst_atomic_types (Ts0 ~~ Ts) t

      end;

    val mk_unf = mk_unf_or_fld domain_type;

    val mk_fld = mk_unf_or_fld range_type;

    val unfs = map (mk_unf As) unfs0;

    val flds = map (mk_fld As) flds0;

    val fpTs = map (domain_type o fastype_of) unfs;

    val is_fpT = member (op =) fpTs;

    val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;

    val ns = map length ctr_Tsss;

    val mss = map (map length) ctr_Tsss;

    val Css = map2 replicate ns Cs;

    fun mk_iter_like Ts Us c =

      let

        val (binders, body) = strip_type (fastype_of c);

        val (f_Us, prebody) = split_last binders;

        val Type (_, Ts0) = if lfp then prebody else body;

        val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);

      in

        Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) c

      end;

    val fp_iters as fp_iter1 :: _ = map (mk_iter_like As Cs) fp_iters0;

    val fp_recs as fp_rec1 :: _ = map (mk_iter_like As Cs) fp_recs0;

    val fp_iter_f_Ts = fst (split_last (binder_types (fastype_of fp_iter1)));

    val fp_rec_f_Ts = fst (split_last (binder_types (fastype_of fp_rec1)));

    fun dest_rec_pair (T as Type (@{type_name prod}, Us as [_, U])) =

        if member (op =) Cs U then Us else [T]

      | dest_rec_pair T = [T];

    val (((gss, g_Tss, ysss, y_Tsss), (hss, h_Tss, zssss, z_Tssss)),

         (cs, (qss, q_Tss, gsss, g_Tsss), ())) =

      if lfp then

        let

          val y_Tsss =

            map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN n o domain_type)

              ns mss fp_iter_f_Ts;

          val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;

          val ((gss, ysss), _) =

            lthy

            |> mk_Freess "f" g_Tss

            ||>> mk_Freesss "x" y_Tsss;

          val z_Tssss =

            map3 (fn n => fn ms => map2 (map dest_rec_pair oo dest_tupleT) ms o dest_sumTN n

              o domain_type) ns mss fp_rec_f_Ts;

          val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;

          val hss = map2 (map2 retype_free) gss h_Tss;

          val (zssss, _) =

            lthy

            |> mk_Freessss "x" z_Tssss;

        in

          (((gss, g_Tss, ysss, y_Tsss), (hss, h_Tss, zssss, z_Tssss)),

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

        end

      else

        let

          val q_Tss =

            map2 (fn C => fn n => replicate (Int.max (0, n - 1)) (C --> HOLogic.boolT)) Cs ns;

          val g_Tsss =

            map4 (fn C => fn n => fn ms => map2 (map (curry (op -->) C) oo dest_tupleT) ms o

              dest_sumTN n o range_type) Cs ns mss fp_iter_f_Ts;

          val (((c, qss), gsss), _) =

            lthy

            |> yield_singleton (mk_Frees "c") dummyT

            ||>> mk_Freess "p" q_Tss

            ||>> mk_Freesss "g" g_Tsss;

          val cs = map (retype_free c) Cs;

        in

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

           (cs, (qss, q_Tss, gsss, g_Tsss), ()))

        end;

    fun pour_some_sugar_on_type ((((((((((((((b, fpT), C), fld), unf), fp_iter), fp_rec), fld_unf),

          unf_fld), fld_inject), ctr_binders), ctr_mixfixes), ctr_Tss), disc_binders),

          sel_binderss) no_defs_lthy =

      let

        val n = length ctr_Tss;

        val ks = 1 upto n;

        val ms = map length ctr_Tss;

        val unfT = domain_type (fastype_of fld);

        val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;

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

        val ((((u, v), fs), xss), _) =

          no_defs_lthy

          |> yield_singleton (mk_Frees "u") unfT

          ||>> yield_singleton (mk_Frees "v") fpT

          ||>> mk_Frees "f" case_Ts

          ||>> mk_Freess "x" ctr_Tss;

        val ctr_rhss =

          map2 (fn k => fn xs =>

            fold_rev Term.lambda xs (fld $ mk_InN ctr_prod_Ts (HOLogic.mk_tuple xs) k)) ks xss;

        val case_binder = Binding.suffix_name ("_" ^ caseN) b;

        val case_rhs =

          fold_rev Term.lambda (fs @ [v]) (mk_sum_caseN (map2 mk_uncurried_fun fs xss) $ (unf $ v));

        val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy

          |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>

               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)

             (case_binder :: ctr_binders) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)

          ||> `Local_Theory.restore;

        (*transforms defined frees into consts (and more)*)

        val phi = Proof_Context.export_morphism lthy lthy';

        val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;

        val case_def = Morphism.thm phi raw_case_def;

        val ctrs0 = map (Morphism.term phi) raw_ctrs;

        val casex0 = Morphism.term phi raw_case;

        val ctrs = map (mk_ctr As) ctrs0;

        fun exhaust_tac {context = ctxt, ...} =

          let

            val fld_iff_unf_thm =

              let

                val goal =

                  fold_rev Logic.all [u, v]

                    (mk_Trueprop_eq (HOLogic.mk_eq (v, fld $ u), HOLogic.mk_eq (unf $ v, u)));

              in

                Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>

                  mk_fld_iff_unf_tac ctxt (map (SOME o certifyT lthy) [unfT, fpT])

                    (certify lthy fld) (certify lthy unf) fld_unf unf_fld)

                |> Thm.close_derivation

                |> Morphism.thm phi

              end;

            val sumEN_thm' =

              Local_Defs.unfold lthy @{thms all_unit_eq}

                (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) [] (mk_sumEN n))

              |> Morphism.thm phi;

          in

            mk_exhaust_tac ctxt n ctr_defs fld_iff_unf_thm sumEN_thm'

          end;

        val inject_tacss =

          map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>

              mk_inject_tac ctxt ctr_def fld_inject]) ms ctr_defs;

        val half_distinct_tacss =

          map (map (fn (def, def') => fn {context = ctxt, ...} =>

            mk_half_distinct_tac ctxt fld_inject [def, def'])) (mk_half_pairss ctr_defs);

        val case_tacs =

          map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>

            mk_case_tac ctxt n k m case_def ctr_def unf_fld) ks ms ctr_defs;

        val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];

        fun some_lfp_sugar no_defs_lthy =

          let

            val fpT_to_C = fpT --> C;

            val iter_T = fold_rev (curry (op --->)) g_Tss fpT_to_C;

            val rec_T = fold_rev (curry (op --->)) h_Tss fpT_to_C;

            val iter_binder = Binding.suffix_name ("_" ^ iterN) b;

            val rec_binder = Binding.suffix_name ("_" ^ recN) b;

            val iter_free = Free (Binding.name_of iter_binder, iter_T);

            val rec_free = Free (Binding.name_of rec_binder, rec_T);

            val iter_spec =

              mk_Trueprop_eq (flat_list_comb (iter_free, gss),

                Term.list_comb (fp_iter, map2 (mk_sum_caseN oo map2 mk_uncurried_fun) gss ysss));

            val rec_spec =

              mk_Trueprop_eq (flat_list_comb (rec_free, hss),

                Term.list_comb (fp_rec, map2 (mk_sum_caseN oo map2 mk_uncurried2_fun) hss zssss));

            val (([raw_iter, raw_rec], [raw_iter_def, raw_rec_def]), (lthy', lthy)) = no_defs_lthy

              |> apfst split_list o fold_map2 (fn b => fn spec =>

                Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))

                #>> apsnd snd) [iter_binder, rec_binder] [iter_spec, rec_spec]

              ||> `Local_Theory.restore;

            (*transforms defined frees into consts (and more)*)

            val phi = Proof_Context.export_morphism lthy lthy';

            val iter_def = Morphism.thm phi raw_iter_def;

            val rec_def = Morphism.thm phi raw_rec_def;

            val iter0 = Morphism.term phi raw_iter;

            val rec0 = Morphism.term phi raw_rec;

            val iter = mk_iter_like As Cs iter0;

            val recx = mk_iter_like As Cs rec0;

          in

            ((ctrs, iter, recx, xss, ctr_defs, iter_def, rec_def), lthy)

          end;

        fun some_gfp_sugar no_defs_lthy =

          let

            (* qss, q_Tss, gsss, g_Tsss *)

            fun zip_preds_and_getters p_Ts f_Tss = p_Ts @ flat f_Tss;

            val qg_Tss = map2 zip_preds_and_getters q_Tss g_Tsss;

            val B_to_fpT = C --> fpT;

            val coiter_T = fold_rev (curry (op --->)) qg_Tss B_to_fpT;

(*

            val corec_T = fold_rev (curry (op --->)) h_Tss fpT_to_C;

*)

            val qgss = map2 zip_preds_and_getters qss gsss;

            val cqss = map2 (fn c => map (fn q => q $ c)) cs qss;

            val cgsss = map2 (fn c => map (map (fn g => g $ c))) cs gsss;

            val coiter_binder = Binding.suffix_name ("_" ^ coiterN) b;

            val corec_binder = Binding.suffix_name ("_" ^ corecN) b;

            val coiter_free = Free (Binding.name_of coiter_binder, coiter_T);

(*

            val corec_free = Free (Binding.name_of corec_binder, corec_T);

*)

            val coiter_sum_prod_Ts = map range_type fp_iter_f_Ts;

            val coiter_prod_Tss = map2 dest_sumTN ns coiter_sum_prod_Ts;

            fun mk_join c n cqs sum_prod_T prod_Ts cgss =

              Term.lambda c (mk_IfN sum_prod_T cqs

                (map2 (mk_InN prod_Ts) (map HOLogic.mk_tuple cgss) (1 upto n)));

            val coiter_spec =

              mk_Trueprop_eq (flat_list_comb (coiter_free, qgss),

                Term.list_comb (fp_iter,

                  map6 mk_join cs ns cqss coiter_sum_prod_Ts coiter_prod_Tss cgsss));

(*

            val corec_spec =

              mk_Trueprop_eq (flat_list_comb (corec_free, hss),

                Term.list_comb (fp_rec, map2 (mk_sum_caseN oo map2 mk_uncurried2_fun) hss zssss));

*)

            val (([raw_coiter (*, raw_corec*)], [raw_coiter_def (*, raw_corec_def*)]), (lthy', lthy)) = no_defs_lthy

              |> apfst split_list o fold_map2 (fn b => fn spec =>

                Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))

                #>> apsnd snd) [coiter_binder (*, corec_binder*)] [coiter_spec (*, corec_spec*)]

              ||> `Local_Theory.restore;

            (*transforms defined frees into consts (and more)*)

            val phi = Proof_Context.export_morphism lthy lthy';

            val coiter_def = Morphism.thm phi raw_coiter_def;

(*

            val corec_def = Morphism.thm phi raw_corec_def;

*)

            val coiter0 = Morphism.term phi raw_coiter;

(*

            val corec0 = Morphism.term phi raw_corec;

*)

            val coiter = mk_iter_like As Cs coiter0;

(*

            val corec = mk_iter_like As Cs corec0;

*)

            (*###*)

            val corec = @{term True};

            val corec_def = TrueI;

          in

            ((ctrs, coiter, corec, xss, ctr_defs, coiter_def, corec_def), lthy)

          end;

      in

        wrap_datatype tacss ((ctrs0, casex0), (disc_binders, sel_binderss)) lthy'

        |> (if lfp then some_lfp_sugar else some_gfp_sugar)

      end;

    fun pour_more_sugar_on_datatypes ((ctrss, iters, recs, xsss, ctr_defss, iter_defs, rec_defs),

        lthy) =

      let

        val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;

        val giters = map (fn iter => flat_list_comb (iter, gss)) iters;

        val hrecs = map (fn recx => flat_list_comb (recx, hss)) recs;

        val (iter_thmss, rec_thmss) =

          let

            fun mk_goal_iter_or_rec fss fc xctr f xs xs' =

              fold_rev (fold_rev Logic.all) (xs :: fss)

                (mk_Trueprop_eq (fc $ xctr, Term.list_comb (f, xs')));

            fun fix_iter_free (x as Free (_, T)) =

              (case find_index (eq_fpT T) fpTs of ~1 => x | j => nth giters j $ x);

            fun fix_rec_free (x as Free (_, T)) =

              (case find_index (eq_fpT T) fpTs of ~1 => [x] | j => [x, nth hrecs j $ x]);

            val iter_xsss = map (map (map fix_iter_free)) xsss;

            val rec_xsss = map (map (maps fix_rec_free)) xsss;

            val goal_iterss =

              map5 (map4 o mk_goal_iter_or_rec gss) giters xctrss gss xsss iter_xsss;

            val goal_recss =

              map5 (map4 o mk_goal_iter_or_rec hss) hrecs xctrss hss xsss rec_xsss;

            val iter_tacss =

              map2 (map o mk_iter_or_rec_tac pre_map_defs iter_defs) fp_iter_thms ctr_defss;

            val rec_tacss =

              map2 (map o mk_iter_or_rec_tac pre_map_defs rec_defs) fp_rec_thms ctr_defss;

          in

            (map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))

               goal_iterss iter_tacss,

             map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))

               goal_recss rec_tacss)

          end;

        val notes =

          [(itersN, iter_thmss),

           (recsN, rec_thmss)]

          |> maps (fn (thmN, thmss) =>

            map2 (fn b => fn thms =>

                ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))

              bs thmss);

      in

        lthy |> Local_Theory.notes notes |> snd

      end;

    val lthy' = lthy

      |> fold_map pour_some_sugar_on_type (bs ~~ fpTs ~~ Cs ~~ flds ~~ unfs ~~ fp_iters ~~

        fp_recs ~~ fld_unfs ~~ unf_flds ~~ fld_injects ~~ ctr_binderss ~~ ctr_mixfixess ~~

        ctr_Tsss ~~ disc_binderss ~~ sel_bindersss)

      |>> split_list7

      |> (if lfp then pour_more_sugar_on_datatypes else snd);

    val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^

      (if lfp then "" else "co") ^ "datatype"));

  in

    (timer; lthy')

  end;

fun datatype_cmd info specs lthy =

  let

    (* TODO: cleaner handling of fake contexts, without "background_theory" *)

    (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a

      locale and shadows an existing global type*)

    val fake_thy = Theory.copy

      #> fold (fn spec => perhaps (try (Sign.add_type lthy

        (type_binder_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;

    val fake_lthy = Proof_Context.background_theory fake_thy lthy;

  in

    prepare_datatype Syntax.read_typ info specs fake_lthy lthy

  end;

val parse_opt_binding_colon = Scan.optional (Parse.binding --| Parse.$$$ ":") no_binder

val parse_ctr_arg =

  Parse.$$$ "(" |-- parse_opt_binding_colon -- Parse.typ --| Parse.$$$ ")" ||

  (Parse.typ >> pair no_binder);

val parse_single_spec =

  Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --

  (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --

    Scan.repeat parse_ctr_arg -- Parse.opt_mixfix));

val _ =

  Outer_Syntax.local_theory @{command_spec "data"} "define BNF-based inductive datatypes"

    (Parse.and_list1 parse_single_spec >> datatype_cmd true);

val _ =

  Outer_Syntax.local_theory @{command_spec "codata"} "define BNF-based coinductive datatypes"

    (Parse.and_list1 parse_single_spec >> datatype_cmd false);

end;