src/HOL/BNF/Tools/bnf_fp_n2m_sugar.ML

use right permutation in 'map2'

(*  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 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_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 ([], ([], []));

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 *)
(* 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 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
  else
    ((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 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 (prod_ord int_ord Term_Ord.typ_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;

    (* The name "'z" is likely not to clash with the context, resulting in more cache hits. *)
    fun generalize_simple_type T (seen, lthy) =
      variant_tfrees ["z"] 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 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 has_nested = exists (fn Type (_, ty_args) => ty_args <> ty_args0) 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) =
      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;