(*  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: bool -> 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 pad_and_indexify_calls: BNF_FP_Def_Sugar.fp_sugar list -> int ->

    (term * term list list) list list -> term list 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_"

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;

val dummy_var_name = "?f"

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 ((dummy_var_name, 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_or_ctr _ _ (Abs (_, _, t)) = (Term.dummy, [t])

  | dest_abs_or_applied_map_or_ctr ctxt s (t as t1 $ _) =

    (case try (dest_map ctxt s) t1 of

      SOME res => res

    | NONE =>

      let

        val thy = Proof_Context.theory_of ctxt;

        val map_thms = of_fp_sugar #mapss (the (fp_sugar_of ctxt s))

        val map_thms' = map (fn thm => thm RS sym RS eq_reflection) map_thms;

        val t' = Raw_Simplifier.rewrite_term thy map_thms' [] t;

      in

        if t aconv t' then raise Fail "dest_applied_map_or_ctr"

        else dest_map ctxt s (fst (dest_comb t'))

      end);

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

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

(* TODO: use right sorting order for "fp_sort" w.r.t. original BNFs (?) -- treat new variables

   as deads? *)

fun mutualize_fp_sugars has_nested fp bs fpTs get_indices callssss fp_sugars0 no_defs_lthy0 =

  if has_nested then

    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_or_ctr 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 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;

    in

      ((map_index mk_target_fp_sugar fpTs, fp_sugar_thms), lthy)

    end

  else

    (* TODO: reorder hypotheses and predicates in (co)induction rules? *)

    ((fp_sugars0, (NONE, NONE)), no_defs_lthy0);

fun indexify_callsss fp_sugar callsss =

  let

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

    fun do_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 do_ctr ctrs

  end;

fun pad_and_indexify_calls fp_sugars0 = map2 indexify_callsss fp_sugars0 oo pad_list [];

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;

    fun not_mutually_nested_rec Ts1 Ts2 =

      error (qsotys Ts1 ^ " is neither mutually recursive with nor nested recursive via " ^

        qsotys Ts2);

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

    val perm_actual_Ts as Type (_, ty_args0) :: _ =

      sort (int_ord o pairself Term.size_of_typ) actual_Ts;

    fun check_enrich_with_mutuals _ [] = []

      | check_enrich_with_mutuals seen ((T as Type (T_name, ty_args)) :: Ts) =

        (case fp_sugar_of lthy T_name of

          SOME ({fp = fp', fp_res = {Ts = Ts', ...}, ...}) =>

          if fp = fp' then

            let

              val mutual_Ts = map (fn Type (s, _) => Type (s, ty_args)) Ts';

              val _ =

                seen = [] orelse exists (exists_subtype_in seen) mutual_Ts orelse

                not_mutually_nested_rec mutual_Ts seen;

              val (seen', Ts') = List.partition (member (op =) mutual_Ts) Ts;

            in

              mutual_Ts @ check_enrich_with_mutuals (seen @ T :: seen') Ts'

            end

          else

            not_co_datatype T

        | NONE => not_co_datatype T)

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

    val perm_Ts = check_enrich_with_mutuals [] perm_actual_Ts;

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

    val Ts = actual_Ts @ missing_Ts;

    fun generalize_simple_type T (seen, lthy) =

      mk_TFrees 1 lthy |> (fn ([U], lthy) => (U, ((T, U) :: seen, lthy)));

    fun generalize_type T (seen_lthy as (seen, _)) =

      (case AList.lookup (op =) seen T of

        SOME U => (U, seen_lthy)

      | NONE =>

        (case T of

          Type (s, Ts) =>

          if exists_subtype_in Ts T then fold_map generalize_type Ts seen_lthy |>> curry Type s

          else generalize_simple_type T seen_lthy

        | _ => generalize_simple_type T seen_lthy));

    val (perm_Us, _) = fold_map generalize_type perm_Ts ([], lthy);

    val nn = length Ts;

    val kks = 0 upto nn - 1;

    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_fp_sugars0 = map (the o fp_sugar_of lthy o fst o dest_Type) perm_Ts;

    val has_nested = exists (fn Type (_, ty_args) => ty_args <> ty_args0) Ts;

    val perm_callssss = pad_and_indexify_calls perm_fp_sugars0 nn actual_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) =

      mutualize_fp_sugars has_nested fp perm_bs perm_Us get_perm_indices perm_callssss

        perm_fp_sugars0 lthy;

    val fp_sugars = unpermute perm_fp_sugars;

  in

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

  end;

end;