src/HOL/Codatatype/Tools/bnf_fp_sugar.ML

read the real types off the constant types, rather than using the fake parser types (second step of sugar localization)

(*  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
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";

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_data prepare_typ gfp specs fake_lthy lthy =
  let
    val constrained_As =
      map (map (apfst (prepare_typ fake_lthy)) o type_args_constrained_of) specs
      |> Library.foldr1 (merge_type_args_constrained 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 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 fake_Ts = 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 ctr_Tsss = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;

    val rhs_As' = fold (fold (fold Term.add_tfreesT)) ctr_Tsss [];
    val _ = (case subtract (op =) As' rhs_As' of
        [] => ()
      | A' :: _ => error ("Extra type variables on rhs: " ^
          quote (Syntax.string_of_typ lthy (TFree A'))));

    val ((Cs, Xs), _) =
      lthy
      |> fold (fold (fn s => Variable.declare_typ (TFree (s, dummyS))) o type_args_of) specs
      |> mk_TFrees N
      ||>> mk_TFrees N;

    fun is_same_recT (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)))
      | is_same_recT _ _ = false;

    fun freeze_recXs (T as Type (s, Us)) =
        (case find_index (is_same_recT T) fake_Ts of
          ~1 => Type (s, map freeze_recXs Us)
        | i => nth Xs i)
      | freeze_recXs T = T;

    val ctr_TsssXs = map (map (map freeze_recXs)) 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 ((raw_unfs, raw_flds, raw_fp_iters, raw_fp_recs, unf_flds, fld_unfs, fld_injects), lthy') =
      fp_bnf (if gfp then bnf_gfp else bnf_lfp) bs mixfixes As' eqs 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) raw_unfs;
    val flds = map (mk_fld As) raw_flds;

    val fp_Ts = map (domain_type o fastype_of) unfs;

    fun mk_fp_iter_or_rec Ts Us t =
      let
        val (binders, body) = strip_type (fastype_of t);
        val Type (_, Ts0) = if gfp then body else List.last binders;
        val Us0 = map (if gfp then domain_type else body_type) (fst (split_last binders));
      in
        Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
      end;

    val fp_iters = map (mk_fp_iter_or_rec As Cs) raw_fp_iters;
    val fp_recs = map (mk_fp_iter_or_rec As Cs) raw_fp_recs;

    fun pour_sugar_on_type ((((((((((((((b, fp_T), 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 unf_T = domain_type (fastype_of fld);
        val prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
        val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;

        val ((((u, v), fs), xss), _) =
          lthy
          |> yield_singleton (mk_Frees "u") unf_T
          ||>> yield_singleton (mk_Frees "v") fp_T
          ||>> 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 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_ctrs, raw_ctr_defs), (raw_case, raw_case_def)), (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)
             ctr_binders ctr_mixfixes ctr_rhss
          ||>> (Local_Theory.define ((case_binder, NoSyn), ((Thm.def_binding case_binder, []),
             case_rhs)) #>> apsnd snd)
          ||> `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 ctrs = map (Morphism.term phi) raw_ctrs;
        val casex = Morphism.term phi raw_case;

        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) [unf_T, fp_T])
                    (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) 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];

        (* (co)iterators, (co)recursors, (co)induction *)

        val is_recT = member (op =) fp_Ts;

        val ns = map length ctr_Tsss;
        val mss = map (map length) ctr_Tsss;
        val Css = map2 replicate ns Cs;

        fun sugar_lfp lthy =
          let
            val fp_y_Ts = map domain_type (fst (split_last (binder_types (fastype_of fp_iter))));
            val y_prod_Tss = map2 dest_sumTN ns fp_y_Ts;
            val y_Tsss = map2 (map2 dest_tupleT) mss y_prod_Tss;
            val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
            val iter_T = flat g_Tss ---> fp_T --> C;

            val ((gss, ysss), _) =
              lthy
              |> mk_Freess "f" g_Tss
              ||>> mk_Freesss "x" y_Tsss;

            val iter_rhs =
              fold_rev (fold_rev Term.lambda) gss
                (Term.list_comb (fp_iter, map2 (mk_sum_caseN oo map2 mk_uncurried_fun) gss ysss));
          in
            lthy
          end;

        fun sugar_gfp lthy = lthy;
      in
        wrap_data tacss ((ctrs, casex), (disc_binders, sel_binderss)) lthy'
        |> (if gfp then sugar_gfp else sugar_lfp)
      end;

    val lthy'' =
      fold pour_sugar_on_type (bs ~~ fp_Ts ~~ Cs ~~ flds ~~ unfs ~~ fp_iters ~~ fp_recs ~~
        fld_unfs ~~ unf_flds ~~ fld_injects ~~ ctr_binderss ~~ ctr_mixfixess ~~ ctr_Tsss ~~
        disc_binderss ~~ sel_bindersss) lthy';

    val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
      (if gfp then "co" else "") ^ "datatype"));
  in
    (timer; lthy'')
  end;

fun data_cmd info specs lthy =
  let
    val fake_thy = Theory.copy
      #> fold (fn spec => 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_data 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 >> data_cmd false);

val _ =
  Outer_Syntax.local_theory @{command_spec "codata"} "define BNF-based coinductive datatypes"
    (Parse.and_list1 parse_single_spec >> data_cmd true);

end;