src/HOL/Tools/Datatype/datatype.ML

tuned;

(*  Title:      HOL/Tools/Datatype/datatype.ML
    Author:     Stefan Berghofer, TU Muenchen

Datatype package: definitional introduction of datatypes
with proof of characteristic theorems: injectivity / distinctness
of constructors and induction.  Main interface to datatypes
after full bootstrap of datatype package.
*)

signature DATATYPE =
sig
  include DATATYPE_DATA
  val add_datatype: config ->
    (string list * binding * mixfix * (binding * typ list * mixfix) list) list ->
    theory -> string list * theory
  val add_datatype_cmd:
    (string list * binding * mixfix * (binding * string list * mixfix) list) list ->
    theory -> theory
  val parse_decl: (string list * binding * mixfix * (binding * string list * mixfix) list) parser
end;

structure Datatype : DATATYPE =
struct

(** auxiliary **)

val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);

val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];

fun exh_thm_of (dt_info : Datatype_Aux.info Symtab.table) tname =
  #exhaust (the (Symtab.lookup dt_info tname));

val node_name = @{type_name "Datatype.node"};
val In0_name = @{const_name "Datatype.In0"};
val In1_name = @{const_name "Datatype.In1"};
val Scons_name = @{const_name "Datatype.Scons"};
val Leaf_name = @{const_name "Datatype.Leaf"};
val Lim_name = @{const_name "Datatype.Lim"};
val Suml_name = @{const_name "Sum_Type.Suml"};
val Sumr_name = @{const_name "Sum_Type.Sumr"};

val In0_inject = @{thm In0_inject};
val In1_inject = @{thm In1_inject};
val Scons_inject = @{thm Scons_inject};
val Leaf_inject = @{thm Leaf_inject};
val In0_eq = @{thm In0_eq};
val In1_eq = @{thm In1_eq};
val In0_not_In1 = @{thm In0_not_In1};
val In1_not_In0 = @{thm In1_not_In0};
val Lim_inject = @{thm Lim_inject};
val Inl_inject = @{thm Inl_inject};
val Inr_inject = @{thm Inr_inject};
val Suml_inject = @{thm Suml_inject};
val Sumr_inject = @{thm Sumr_inject};

val datatype_injI =
  @{lemma "(!!x. ALL y. f x = f y --> x = y) ==> inj f" by (simp add: inj_on_def)};


(** proof of characteristic theorems **)

fun representation_proofs (config : Datatype_Aux.config) (dt_info : Datatype_Aux.info Symtab.table)
    descr sorts types_syntax constr_syntax case_names_induct thy =
  let
    val descr' = flat descr;
    val new_type_names = map (Binding.name_of o fst) types_syntax;
    val big_name = space_implode "_" new_type_names;
    val thy1 = Sign.add_path big_name thy;
    val big_rec_name = big_name ^ "_rep_set";
    val rep_set_names' =
      if length descr' = 1 then [big_rec_name]
      else map (prefix (big_rec_name ^ "_") o string_of_int) (1 upto length descr');
    val rep_set_names = map (Sign.full_bname thy1) rep_set_names';

    val tyvars = map (fn (_, (_, Ts, _)) => map Datatype_Aux.dest_DtTFree Ts) (hd descr);
    val leafTs' = Datatype_Aux.get_nonrec_types descr' sorts;
    val branchTs = Datatype_Aux.get_branching_types descr' sorts;
    val branchT =
      if null branchTs then HOLogic.unitT
      else Balanced_Tree.make (fn (T, U) => Type (@{type_name Sum_Type.sum}, [T, U])) branchTs;
    val arities = remove (op =) 0 (Datatype_Aux.get_arities descr');
    val unneeded_vars =
      subtract (op =) (fold Term.add_tfree_namesT (leafTs' @ branchTs) []) (hd tyvars);
    val leafTs =
      leafTs' @ map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars;
    val recTs = Datatype_Aux.get_rec_types descr' sorts;
    val (newTs, oldTs) = chop (length (hd descr)) recTs;
    val sumT =
      if null leafTs then HOLogic.unitT
      else Balanced_Tree.make (fn (T, U) => Type (@{type_name Sum_Type.sum}, [T, U])) leafTs;
    val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
    val UnivT = HOLogic.mk_setT Univ_elT;
    val UnivT' = Univ_elT --> HOLogic.boolT;
    val Collect = Const (@{const_name Collect}, UnivT' --> UnivT);

    val In0 = Const (In0_name, Univ_elT --> Univ_elT);
    val In1 = Const (In1_name, Univ_elT --> Univ_elT);
    val Leaf = Const (Leaf_name, sumT --> Univ_elT);
    val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);

    (* make injections needed for embedding types in leaves *)

    fun mk_inj T' x =
      let
        fun mk_inj' T n i =
          if n = 1 then x
          else
            let
              val n2 = n div 2;
              val Type (_, [T1, T2]) = T;
            in
              if i <= n2
              then Const (@{const_name Inl}, T1 --> T) $ mk_inj' T1 n2 i
              else Const (@{const_name Inr}, T2 --> T) $ mk_inj' T2 (n - n2) (i - n2)
            end;
      in mk_inj' sumT (length leafTs) (1 + find_index (fn T'' => T'' = T') leafTs) end;

    (* make injections for constructors *)

    fun mk_univ_inj ts = Balanced_Tree.access
      {left = fn t => In0 $ t,
        right = fn t => In1 $ t,
        init =
          if ts = [] then Const (@{const_name undefined}, Univ_elT)
          else foldr1 (HOLogic.mk_binop Scons_name) ts};

    (* function spaces *)

    fun mk_fun_inj T' x =
      let
        fun mk_inj T n i =
          if n = 1 then x
          else
            let
              val n2 = n div 2;
              val Type (_, [T1, T2]) = T;
              fun mkT U = (U --> Univ_elT) --> T --> Univ_elT;
            in
              if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i
              else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
            end;
      in mk_inj branchT (length branchTs) (1 + find_index (fn T'' => T'' = T') branchTs) end;

    fun mk_lim t Ts = fold_rev (fn T => fn t => Lim $ mk_fun_inj T (Abs ("x", T, t))) Ts t;

    (************** generate introduction rules for representing set **********)

    val _ = Datatype_Aux.message config "Constructing representing sets ...";

    (* make introduction rule for a single constructor *)

    fun make_intr s n (i, (_, cargs)) =
      let
        fun mk_prem dt (j, prems, ts) =
          (case Datatype_Aux.strip_dtyp dt of
            (dts, Datatype_Aux.DtRec k) =>
              let
                val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) dts;
                val free_t =
                  Datatype_Aux.app_bnds (Datatype_Aux.mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
              in
                (j + 1, list_all (map (pair "x") Ts,
                  HOLogic.mk_Trueprop
                    (Free (nth rep_set_names' k, UnivT') $ free_t)) :: prems,
                mk_lim free_t Ts :: ts)
              end
          | _ =>
              let val T = Datatype_Aux.typ_of_dtyp descr' sorts dt
              in (j + 1, prems, (Leaf $ mk_inj T (Datatype_Aux.mk_Free "x" T j)) :: ts) end);

        val (_, prems, ts) = fold_rev mk_prem cargs (1, [], []);
        val concl = HOLogic.mk_Trueprop (Free (s, UnivT') $ mk_univ_inj ts n i);
      in Logic.list_implies (prems, concl) end;

    val intr_ts = maps (fn ((_, (_, _, constrs)), rep_set_name) =>
      map (make_intr rep_set_name (length constrs))
        ((1 upto length constrs) ~~ constrs)) (descr' ~~ rep_set_names');

    val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
      thy1
      |> Sign.map_naming Name_Space.conceal
      |> Inductive.add_inductive_global
          {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name,
           coind = false, no_elim = true, no_ind = false, skip_mono = true, fork_mono = false}
          (map (fn s => ((Binding.name s, UnivT'), NoSyn)) rep_set_names') []
          (map (fn x => (Attrib.empty_binding, x)) intr_ts) []
      ||> Sign.restore_naming thy1
      ||> Theory.checkpoint;

    (********************************* typedef ********************************)

    val (typedefs, thy3) = thy2
      |> Sign.parent_path
      |> fold_map
        (fn (((name, mx), tvs), c) =>
          Typedef.add_typedef_global false NONE
            (name, map (rpair dummyS) tvs, mx)
            (Collect $ Const (c, UnivT')) NONE
            (rtac exI 1 THEN rtac CollectI 1 THEN
              QUIET_BREADTH_FIRST (has_fewer_prems 1)
              (resolve_tac rep_intrs 1)))
        (types_syntax ~~ tyvars ~~ (take (length newTs) rep_set_names))
      ||> Sign.add_path big_name;

    (*********************** definition of constructors ***********************)

    val big_rep_name = space_implode "_" new_type_names ^ "_Rep_";
    val rep_names = map (prefix "Rep_") new_type_names;
    val rep_names' = map (fn i => big_rep_name ^ string_of_int i) (1 upto length (flat (tl descr)));
    val all_rep_names =
      map (Sign.intern_const thy3) rep_names @
      map (Sign.full_bname thy3) rep_names';

    (* isomorphism declarations *)

    val iso_decls = map (fn (T, s) => (Binding.name s, T --> Univ_elT, NoSyn))
      (oldTs ~~ rep_names');

    (* constructor definitions *)

    fun make_constr_def tname T n ((cname, cargs), (cname', mx)) (thy, defs, eqns, i) =
      let
        fun constr_arg dt (j, l_args, r_args) =
          let
            val T = Datatype_Aux.typ_of_dtyp descr' sorts dt;
            val free_t = Datatype_Aux.mk_Free "x" T j;
          in
            (case (Datatype_Aux.strip_dtyp dt, strip_type T) of
              ((_, Datatype_Aux.DtRec m), (Us, U)) =>
                (j + 1, free_t :: l_args, mk_lim
                  (Const (nth all_rep_names m, U --> Univ_elT) $
                    Datatype_Aux.app_bnds free_t (length Us)) Us :: r_args)
            | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
          end;

        val (_, l_args, r_args) = fold_rev constr_arg cargs (1, [], []);
        val constrT = (map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs) ---> T;
        val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
        val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
        val lhs = list_comb (Const (cname, constrT), l_args);
        val rhs = mk_univ_inj r_args n i;
        val def = Logic.mk_equals (lhs, Const (abs_name, Univ_elT --> T) $ rhs);
        val def_name = Long_Name.base_name cname ^ "_def";
        val eqn =
          HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
        val ([def_thm], thy') =
          thy
          |> Sign.add_consts_i [(cname', constrT, mx)]
          |> (Global_Theory.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)];

      in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;

    (* constructor definitions for datatype *)

    fun dt_constr_defs ((((_, (_, _, constrs)), tname), T), constr_syntax)
        (thy, defs, eqns, rep_congs, dist_lemmas) =
      let
        val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
        val rep_const =
          cterm_of thy (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> Univ_elT));
        val cong' =
          Drule.export_without_context
            (cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong);
        val dist =
          Drule.export_without_context
            (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
        val (thy', defs', eqns', _) =
          fold ((make_constr_def tname T) (length constrs))
            (constrs ~~ constr_syntax) (Sign.add_path tname thy, defs, [], 1);
      in
        (Sign.parent_path thy', defs', eqns @ [eqns'],
          rep_congs @ [cong'], dist_lemmas @ [dist])
      end;

    val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) =
      fold dt_constr_defs
        (hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax)
        (thy3 |> Sign.add_consts_i iso_decls |> Sign.parent_path, [], [], [], []);


    (*********** isomorphisms for new types (introduced by typedef) ***********)

    val _ = Datatype_Aux.message config "Proving isomorphism properties ...";

    val newT_iso_axms = map (fn (_, (_, td)) =>
      (collect_simp (#Abs_inverse td), #Rep_inverse td,
       collect_simp (#Rep td))) typedefs;

    val newT_iso_inj_thms = map (fn (_, (_, td)) =>
      (collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;

    (********* isomorphisms between existing types and "unfolded" types *******)

    (*---------------------------------------------------------------------*)
    (* isomorphisms are defined using primrec-combinators:                 *)
    (* generate appropriate functions for instantiating primrec-combinator *)
    (*                                                                     *)
    (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)
    (*                                                                     *)
    (* also generate characteristic equations for isomorphisms             *)
    (*                                                                     *)
    (*   e.g.  dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)
    (*---------------------------------------------------------------------*)

    fun make_iso_def k ks n (cname, cargs) (fs, eqns, i) =
      let
        val argTs = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
        val T = nth recTs k;
        val rep_name = nth all_rep_names k;
        val rep_const = Const (rep_name, T --> Univ_elT);
        val constr = Const (cname, argTs ---> T);

        fun process_arg ks' dt (i2, i2', ts, Ts) =
          let
            val T' = Datatype_Aux.typ_of_dtyp descr' sorts dt;
            val (Us, U) = strip_type T'
          in
            (case Datatype_Aux.strip_dtyp dt of
              (_, Datatype_Aux.DtRec j) =>
                if member (op =) ks' j then
                  (i2 + 1, i2' + 1, ts @ [mk_lim (Datatype_Aux.app_bnds
                     (Datatype_Aux.mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
                   Ts @ [Us ---> Univ_elT])
                else
                  (i2 + 1, i2', ts @ [mk_lim
                     (Const (nth all_rep_names j, U --> Univ_elT) $
                        Datatype_Aux.app_bnds (Datatype_Aux.mk_Free "x" T' i2) (length Us)) Us], Ts)
            | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (Datatype_Aux.mk_Free "x" T' i2)], Ts))
          end;

        val (i2, i2', ts, Ts) = fold (process_arg ks) cargs (1, 1, [], []);
        val xs = map (uncurry (Datatype_Aux.mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
        val ys = map (uncurry (Datatype_Aux.mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
        val f = fold_rev (absfree o dest_Free) (xs @ ys) (mk_univ_inj ts n i);

        val (_, _, ts', _) = fold (process_arg []) cargs (1, 1, [], []);
        val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
          (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))

      in (fs @ [f], eqns @ [eqn], i + 1) end;

    (* define isomorphisms for all mutually recursive datatypes in list ds *)

    fun make_iso_defs ds (thy, char_thms) =
      let
        val ks = map fst ds;
        val (_, (tname, _, _)) = hd ds;
        val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);

        fun process_dt (k, (tname, _, constrs)) (fs, eqns, isos) =
          let
            val (fs', eqns', _) = fold (make_iso_def k ks (length constrs)) constrs (fs, eqns, 1);
            val iso = (nth recTs k, nth all_rep_names k);
          in (fs', eqns', isos @ [iso]) end;

        val (fs, eqns, isos) = fold process_dt ds ([], [], []);
        val fTs = map fastype_of fs;
        val defs =
          map (fn (rec_name, (T, iso_name)) =>
            (Binding.name (Long_Name.base_name iso_name ^ "_def"),
              Logic.mk_equals (Const (iso_name, T --> Univ_elT),
                list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs)))) (rec_names ~~ isos);
        val (def_thms, thy') =
          apsnd Theory.checkpoint ((Global_Theory.add_defs false o map Thm.no_attributes) defs thy);

        (* prove characteristic equations *)

        val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
        val char_thms' =
          map (fn eqn => Skip_Proof.prove_global thy' [] [] eqn
            (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;

      in (thy', char_thms' @ char_thms) end;

    val (thy5, iso_char_thms) =
      apfst Theory.checkpoint (fold_rev make_iso_defs (tl descr) (Sign.add_path big_name thy4, []));

    (* prove isomorphism properties *)

    fun mk_funs_inv thy thm =
      let
        val prop = Thm.prop_of thm;
        val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
          (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.legacy_freeze prop;
        val used = Term.add_tfree_names a [];

        fun mk_thm i =
          let
            val Ts = map (TFree o rpair HOLogic.typeS) (Name.variant_list used (replicate i "'t"));
            val f = Free ("f", Ts ---> U);
          in
            Skip_Proof.prove_global thy [] []
              (Logic.mk_implies
                (HOLogic.mk_Trueprop (HOLogic.list_all
                   (map (pair "x") Ts, S $ Datatype_Aux.app_bnds f i)),
                 HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
                   r $ (a $ Datatype_Aux.app_bnds f i)), f))))
              (fn _ => EVERY [REPEAT_DETERM_N i (rtac ext 1),
                 REPEAT (etac allE 1), rtac thm 1, atac 1])
          end
      in map (fn r => r RS subst) (thm :: map mk_thm arities) end;

    (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)

    val fun_congs =
      map (fn T => make_elim (Drule.instantiate' [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs;

    fun prove_iso_thms ds (inj_thms, elem_thms) =
      let
        val (_, (tname, _, _)) = hd ds;
        val induct = #induct (the (Symtab.lookup dt_info tname));

        fun mk_ind_concl (i, _) =
          let
            val T = nth recTs i;
            val Rep_t = Const (nth all_rep_names i, T --> Univ_elT);
            val rep_set_name = nth rep_set_names i;
          in
            (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
                HOLogic.mk_eq (Rep_t $ Datatype_Aux.mk_Free "x" T i, Rep_t $ Bound 0) $
                  HOLogic.mk_eq (Datatype_Aux.mk_Free "x" T i, Bound 0)),
              Const (rep_set_name, UnivT') $ (Rep_t $ Datatype_Aux.mk_Free "x" T i))
          end;

        val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);

        val rewrites = map mk_meta_eq iso_char_thms;
        val inj_thms' = map snd newT_iso_inj_thms @ map (fn r => r RS @{thm injD}) inj_thms;

        val inj_thm =
          Skip_Proof.prove_global thy5 [] []
            (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj ind_concl1))
            (fn _ => EVERY
              [(Datatype_Aux.ind_tac induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
               REPEAT (EVERY
                 [rtac allI 1, rtac impI 1,
                  Datatype_Aux.exh_tac (exh_thm_of dt_info) 1,
                  REPEAT (EVERY
                    [hyp_subst_tac 1,
                     rewrite_goals_tac rewrites,
                     REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
                     (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
                     ORELSE (EVERY
                       [REPEAT (eresolve_tac (Scons_inject ::
                          map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
                        REPEAT (cong_tac 1), rtac refl 1,
                        REPEAT (atac 1 ORELSE (EVERY
                          [REPEAT (rtac ext 1),
                           REPEAT (eresolve_tac (mp :: allE ::
                             map make_elim (Suml_inject :: Sumr_inject ::
                               Lim_inject :: inj_thms') @ fun_congs) 1),
                           atac 1]))])])])]);

        val inj_thms'' = map (fn r => r RS datatype_injI) (Datatype_Aux.split_conj_thm inj_thm);

        val elem_thm =
          Skip_Proof.prove_global thy5 [] []
            (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj ind_concl2))
            (fn _ =>
              EVERY [(Datatype_Aux.ind_tac induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
                rewrite_goals_tac rewrites,
                REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
                  ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);

      in (inj_thms'' @ inj_thms, elem_thms @ (Datatype_Aux.split_conj_thm elem_thm)) end;

    val (iso_inj_thms_unfolded, iso_elem_thms) =
      fold_rev prove_iso_thms (tl descr) ([], map #3 newT_iso_axms);
    val iso_inj_thms =
      map snd newT_iso_inj_thms @ map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded;

    (* prove  dt_rep_set_i x --> x : range dt_Rep_i *)

    fun mk_iso_t (((set_name, iso_name), i), T) =
      let val isoT = T --> Univ_elT in
        HOLogic.imp $
          (Const (set_name, UnivT') $ Datatype_Aux.mk_Free "x" Univ_elT i) $
            (if i < length newTs then @{term True}
             else HOLogic.mk_mem (Datatype_Aux.mk_Free "x" Univ_elT i,
               Const (@{const_name image}, isoT --> HOLogic.mk_setT T --> UnivT) $
                 Const (iso_name, isoT) $ Const (@{const_abbrev UNIV}, HOLogic.mk_setT T)))
      end;

    val iso_t = HOLogic.mk_Trueprop (Datatype_Aux.mk_conj (map mk_iso_t
      (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));

    (* all the theorems are proved by one single simultaneous induction *)

    val range_eqs = map (fn r => mk_meta_eq (r RS @{thm range_ex1_eq})) iso_inj_thms_unfolded;

    val iso_thms =
      if length descr = 1 then []
      else
        drop (length newTs) (Datatype_Aux.split_conj_thm
          (Skip_Proof.prove_global thy5 [] [] iso_t (fn _ => EVERY
             [(Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
              REPEAT (rtac TrueI 1),
              rewrite_goals_tac (mk_meta_eq @{thm choice_eq} ::
                Thm.symmetric (mk_meta_eq @{thm fun_eq_iff}) :: range_eqs),
              rewrite_goals_tac (map Thm.symmetric range_eqs),
              REPEAT (EVERY
                [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
                   maps (mk_funs_inv thy5 o #1) newT_iso_axms) 1),
                 TRY (hyp_subst_tac 1),
                 rtac (sym RS range_eqI) 1,
                 resolve_tac iso_char_thms 1])])));

    val Abs_inverse_thms' =
      map #1 newT_iso_axms @
      map2 (fn r_inj => fn r => @{thm f_the_inv_into_f} OF [r_inj, r RS mp])
        iso_inj_thms_unfolded iso_thms;

    val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms';

    (******************* freeness theorems for constructors *******************)

    val _ = Datatype_Aux.message config "Proving freeness of constructors ...";

    (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)

    fun prove_constr_rep_thm eqn =
      let
        val inj_thms = map fst newT_iso_inj_thms;
        val rewrites = @{thm o_def} :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms);
      in
        Skip_Proof.prove_global thy5 [] [] eqn
        (fn _ => EVERY
          [resolve_tac inj_thms 1,
           rewrite_goals_tac rewrites,
           rtac refl 3,
           resolve_tac rep_intrs 2,
           REPEAT (resolve_tac iso_elem_thms 1)])
      end;

    (*--------------------------------------------------------------*)
    (* constr_rep_thms and rep_congs are used to prove distinctness *)
    (* of constructors.                                             *)
    (*--------------------------------------------------------------*)

    val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;

    val dist_rewrites =
      map (fn (rep_thms, dist_lemma) =>
        dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
          (constr_rep_thms ~~ dist_lemmas);

    fun prove_distinct_thms dist_rewrites' (k, ts) =
      let
        fun prove [] = []
          | prove (t :: ts) =
              let
                val dist_thm = Skip_Proof.prove_global thy5 [] [] t (fn _ =>
                  EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
              in dist_thm :: Drule.export_without_context (dist_thm RS not_sym) :: prove ts end;
      in prove ts end;

    val distinct_thms =
      map2 (prove_distinct_thms) dist_rewrites (Datatype_Prop.make_distincts descr sorts);

    (* prove injectivity of constructors *)

    fun prove_constr_inj_thm rep_thms t =
      let
        val inj_thms = Scons_inject ::
          map make_elim
            (iso_inj_thms @
              [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
               Lim_inject, Suml_inject, Sumr_inject])
      in
        Skip_Proof.prove_global thy5 [] [] t
          (fn _ => EVERY
            [rtac iffI 1,
             REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
             dresolve_tac rep_congs 1, dtac box_equals 1,
             REPEAT (resolve_tac rep_thms 1),
             REPEAT (eresolve_tac inj_thms 1),
             REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
               REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
               atac 1]))])
      end;

    val constr_inject =
      map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
        ((Datatype_Prop.make_injs descr sorts) ~~ constr_rep_thms);

    val ((constr_inject', distinct_thms'), thy6) =
      thy5
      |> Sign.parent_path
      |> Datatype_Aux.store_thmss "inject" new_type_names constr_inject
      ||>> Datatype_Aux.store_thmss "distinct" new_type_names distinct_thms;

    (*************************** induction theorem ****************************)

    val _ = Datatype_Aux.message config "Proving induction rule for datatypes ...";

    val Rep_inverse_thms =
      map (fn (_, iso, _) => iso RS subst) newT_iso_axms @
      map (fn r => r RS @{thm the_inv_f_f} RS subst) iso_inj_thms_unfolded;
    val Rep_inverse_thms' = map (fn r => r RS @{thm the_inv_f_f}) iso_inj_thms_unfolded;

    fun mk_indrule_lemma ((i, _), T) (prems, concls) =
      let
        val Rep_t =
          Const (nth all_rep_names i, T --> Univ_elT) $ Datatype_Aux.mk_Free "x" T i;

        val Abs_t =
          if i < length newTs then
            Const (Sign.intern_const thy6 ("Abs_" ^ nth new_type_names i), Univ_elT --> T)
          else Const (@{const_name the_inv_into},
              [HOLogic.mk_setT T, T --> Univ_elT, Univ_elT] ---> T) $
            HOLogic.mk_UNIV T $ Const (nth all_rep_names i, T --> Univ_elT);
      in
        (prems @
          [HOLogic.imp $
            (Const (nth rep_set_names i, UnivT') $ Rep_t) $
              (Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
         concls @
          [Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ Datatype_Aux.mk_Free "x" T i])
      end;

    val (indrule_lemma_prems, indrule_lemma_concls) =
      fold mk_indrule_lemma (descr' ~~ recTs) ([], []);

    val cert = cterm_of thy6;

    val indrule_lemma =
      Skip_Proof.prove_global thy6 [] []
        (Logic.mk_implies
          (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_prems),
           HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_concls)))
        (fn _ =>
          EVERY
           [REPEAT (etac conjE 1),
            REPEAT (EVERY
              [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
               etac mp 1, resolve_tac iso_elem_thms 1])]);

    val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
    val frees =
      if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))]
      else map (Free o apfst fst o dest_Var) Ps;
    val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;

    val dt_induct_prop = Datatype_Prop.make_ind descr sorts;
    val dt_induct =
      Skip_Proof.prove_global thy6 []
      (Logic.strip_imp_prems dt_induct_prop)
      (Logic.strip_imp_concl dt_induct_prop)
      (fn {prems, ...} =>
        EVERY
          [rtac indrule_lemma' 1,
           (Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
           EVERY (map (fn (prem, r) => (EVERY
             [REPEAT (eresolve_tac Abs_inverse_thms 1),
              simp_tac (HOL_basic_ss addsimps (Thm.symmetric r :: Rep_inverse_thms')) 1,
              DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
                  (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);

    val ([dt_induct'], thy7) =
      thy6
      |> Global_Theory.add_thms
        [((Binding.qualify true big_name (Binding.name "induct"), dt_induct),
          [case_names_induct])]
      ||> Theory.checkpoint;

  in
    ((constr_inject', distinct_thms', dt_induct'), thy7)
  end;



(** datatype definition **)

fun gen_add_datatype prep_typ config dts thy =
  let
    val _ = Theory.requires thy "Datatype" "datatype definitions";

    (* this theory is used just for parsing *)
    val tmp_thy = thy
      |> Theory.copy
      |> Sign.add_types_global (map (fn (tvs, tname, mx, _) => (tname, length tvs, mx)) dts);
    val tmp_ctxt = Proof_Context.init_global tmp_thy;

    val (tyvars, _, _, _) ::_ = dts;
    val (new_dts, types_syntax) = ListPair.unzip (map (fn (tvs, tname, mx, _) =>
      let val full_tname = Sign.full_name tmp_thy tname in
        (case duplicates (op =) tvs of
          [] =>
            if eq_set (op =) (tyvars, tvs) then ((full_tname, tvs), (tname, mx))
            else error ("Mutually recursive datatypes must have same type parameters")
        | dups =>
            error ("Duplicate parameter(s) for datatype " ^ Binding.print tname ^
              " : " ^ commas dups))
      end) dts);
    val dt_names = map fst new_dts;

    val _ =
      (case duplicates (op =) (map fst new_dts) of
        [] => ()
      | dups => error ("Duplicate datatypes: " ^ commas dups));

    fun prep_dt_spec (tvs, tname, mx, constrs) (dts', constr_syntax, sorts, i) =
      let
        fun prep_constr (cname, cargs, mx') (constrs, constr_syntax', sorts') =
          let
            val (cargs', sorts'') = fold_map (prep_typ tmp_thy) cargs sorts';
            val _ =
              (case subtract (op =) tvs (fold Term.add_tfree_namesT cargs' []) of
                [] => ()
              | vs => error ("Extra type variables on rhs: " ^ commas vs));
            val c = Sign.full_name_path tmp_thy (Binding.name_of tname) cname;
          in
            (constrs @ [(c, map (Datatype_Aux.dtyp_of_typ new_dts) cargs')],
              constr_syntax' @ [(cname, mx')], sorts'')
          end handle ERROR msg =>
            cat_error msg ("The error above occurred in constructor " ^ Binding.print cname ^
              " of datatype " ^ Binding.print tname);

        val (constrs', constr_syntax', sorts') = fold prep_constr constrs ([], [], sorts);
      in
        (case duplicates (op =) (map fst constrs') of
          [] =>
            (dts' @ [(i, (Sign.full_name tmp_thy tname, map Datatype_Aux.DtTFree tvs, constrs'))],
              constr_syntax @ [constr_syntax'], sorts', i + 1)
        | dups =>
            error ("Duplicate constructors " ^ commas dups ^ " in datatype " ^ Binding.print tname))
      end;

    val (dts', constr_syntax, sorts', i) = fold prep_dt_spec dts ([], [], [], 0);
    val sorts =
      sorts' @ map (rpair (Sign.defaultS tmp_thy)) (subtract (op =) (map fst sorts') tyvars);
    val dt_info = Datatype_Data.get_all thy;
    val (descr, _) = Datatype_Aux.unfold_datatypes tmp_ctxt dts' sorts dt_info dts' i;
    val _ =
      Datatype_Aux.check_nonempty descr
        handle (exn as Datatype_Aux.Datatype_Empty s) =>
          if #strict config then error ("Nonemptiness check failed for datatype " ^ s)
          else reraise exn;

    val _ =
      Datatype_Aux.message config
        ("Constructing datatype(s) " ^ commas_quote (map (Binding.name_of o #2) dts));
  in
    thy
    |> representation_proofs config dt_info descr sorts types_syntax constr_syntax
      (Datatype_Data.mk_case_names_induct (flat descr))
    |-> (fn (inject, distinct, induct) =>
      Datatype_Data.derive_datatype_props config dt_names descr sorts induct inject distinct)
  end;

val add_datatype = gen_add_datatype Datatype_Data.cert_typ;
val add_datatype_cmd = snd oo gen_add_datatype Datatype_Data.read_typ Datatype_Aux.default_config;


(* concrete syntax *)

val parse_decl =
  Parse.type_args -- Parse.binding -- Parse.opt_mixfix --
  (Parse.$$$ "=" |-- Parse.enum1 "|" (Parse.binding -- Scan.repeat Parse.typ -- Parse.opt_mixfix))
  >> (fn (((vs, t), mx), cons) => (vs, t, mx, map Parse.triple1 cons));

val _ =
  Outer_Syntax.command "datatype" "define inductive datatypes" Keyword.thy_decl
    (Parse.and_list1 parse_decl >> (Toplevel.theory o add_datatype_cmd));


open Datatype_Data;

end;