src/HOL/Tools/Datatype/datatype.ML
author wenzelm
Sat Mar 22 18:19:57 2014 +0100 (2014-03-22)
changeset 56254 a2dd9200854d
parent 56249 0fda98dd2c93
child 58111 82db9ad610b9
permissions -rw-r--r--
more antiquotations;
     1 (*  Title:      HOL/Tools/Datatype/datatype.ML
     2     Author:     Stefan Berghofer, TU Muenchen
     3 
     4 Datatype package: definitional introduction of datatypes
     5 with proof of characteristic theorems: injectivity / distinctness
     6 of constructors and induction.  Main interface to datatypes
     7 after full bootstrap of datatype package.
     8 *)
     9 
    10 signature DATATYPE =
    11 sig
    12   include DATATYPE_DATA
    13   val distinct_lemma: thm
    14   type spec =
    15     (binding * (string * sort) list * mixfix) *
    16     (binding * typ list * mixfix) list
    17   type spec_cmd =
    18     (binding * (string * string option) list * mixfix) *
    19     (binding * string list * mixfix) list
    20   val read_specs: spec_cmd list -> theory -> spec list * Proof.context
    21   val check_specs: spec list -> theory -> spec list * Proof.context
    22   val add_datatype: config -> spec list -> theory -> string list * theory
    23   val add_datatype_cmd: config -> spec_cmd list -> theory -> string list * theory
    24   val spec_cmd: spec_cmd parser
    25 end;
    26 
    27 structure Datatype : DATATYPE =
    28 struct
    29 
    30 (** auxiliary **)
    31 
    32 val distinct_lemma = @{lemma "f x \<noteq> f y ==> x \<noteq> y" by iprover};
    33 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    34 
    35 fun exh_thm_of (dt_info : Datatype_Aux.info Symtab.table) tname =
    36   #exhaust (the (Symtab.lookup dt_info tname));
    37 
    38 val In0_inject = @{thm In0_inject};
    39 val In1_inject = @{thm In1_inject};
    40 val Scons_inject = @{thm Scons_inject};
    41 val Leaf_inject = @{thm Leaf_inject};
    42 val In0_eq = @{thm In0_eq};
    43 val In1_eq = @{thm In1_eq};
    44 val In0_not_In1 = @{thm In0_not_In1};
    45 val In1_not_In0 = @{thm In1_not_In0};
    46 val Lim_inject = @{thm Lim_inject};
    47 val Inl_inject = @{thm Inl_inject};
    48 val Inr_inject = @{thm Inr_inject};
    49 val Suml_inject = @{thm Suml_inject};
    50 val Sumr_inject = @{thm Sumr_inject};
    51 
    52 val datatype_injI =
    53   @{lemma "(!!x. ALL y. f x = f y --> x = y) ==> inj f" by (simp add: inj_on_def)};
    54 
    55 
    56 (** proof of characteristic theorems **)
    57 
    58 fun representation_proofs (config : Datatype_Aux.config) (dt_info : Datatype_Aux.info Symtab.table)
    59     descr types_syntax constr_syntax case_names_induct thy =
    60   let
    61     val descr' = flat descr;
    62     val new_type_names = map (Binding.name_of o fst) types_syntax;
    63     val big_name = space_implode "_" new_type_names;
    64     val thy1 = Sign.add_path big_name thy;
    65     val big_rec_name = "rep_set_" ^ big_name;
    66     val rep_set_names' =
    67       if length descr' = 1 then [big_rec_name]
    68       else map (prefix (big_rec_name ^ "_") o string_of_int) (1 upto length descr');
    69     val rep_set_names = map (Sign.full_bname thy1) rep_set_names';
    70 
    71     val tyvars = map (fn (_, (_, Ts, _)) => map Datatype_Aux.dest_DtTFree Ts) (hd descr);
    72     val leafTs' = Datatype_Aux.get_nonrec_types descr';
    73     val branchTs = Datatype_Aux.get_branching_types descr';
    74     val branchT =
    75       if null branchTs then HOLogic.unitT
    76       else Balanced_Tree.make (fn (T, U) => Type (@{type_name Sum_Type.sum}, [T, U])) branchTs;
    77     val arities = remove (op =) 0 (Datatype_Aux.get_arities descr');
    78     val unneeded_vars =
    79       subtract (op =) (fold Term.add_tfreesT (leafTs' @ branchTs) []) (hd tyvars);
    80     val leafTs = leafTs' @ map TFree unneeded_vars;
    81     val recTs = Datatype_Aux.get_rec_types descr';
    82     val (newTs, oldTs) = chop (length (hd descr)) recTs;
    83     val sumT =
    84       if null leafTs then HOLogic.unitT
    85       else Balanced_Tree.make (fn (T, U) => Type (@{type_name Sum_Type.sum}, [T, U])) leafTs;
    86     val Univ_elT = HOLogic.mk_setT (Type (@{type_name Datatype.node}, [sumT, branchT]));
    87     val UnivT = HOLogic.mk_setT Univ_elT;
    88     val UnivT' = Univ_elT --> HOLogic.boolT;
    89     val Collect = Const (@{const_name Collect}, UnivT' --> UnivT);
    90 
    91     val In0 = Const (@{const_name Datatype.In0}, Univ_elT --> Univ_elT);
    92     val In1 = Const (@{const_name Datatype.In1}, Univ_elT --> Univ_elT);
    93     val Leaf = Const (@{const_name Datatype.Leaf}, sumT --> Univ_elT);
    94     val Lim = Const (@{const_name Datatype.Lim}, (branchT --> Univ_elT) --> Univ_elT);
    95 
    96     (* make injections needed for embedding types in leaves *)
    97 
    98     fun mk_inj T' x =
    99       let
   100         fun mk_inj' T n i =
   101           if n = 1 then x
   102           else
   103             let
   104               val n2 = n div 2;
   105               val Type (_, [T1, T2]) = T;
   106             in
   107               if i <= n2
   108               then Const (@{const_name Inl}, T1 --> T) $ mk_inj' T1 n2 i
   109               else Const (@{const_name Inr}, T2 --> T) $ mk_inj' T2 (n - n2) (i - n2)
   110             end;
   111       in mk_inj' sumT (length leafTs) (1 + find_index (fn T'' => T'' = T') leafTs) end;
   112 
   113     (* make injections for constructors *)
   114 
   115     fun mk_univ_inj ts = Balanced_Tree.access
   116       {left = fn t => In0 $ t,
   117         right = fn t => In1 $ t,
   118         init =
   119           if ts = [] then Const (@{const_name undefined}, Univ_elT)
   120           else foldr1 (HOLogic.mk_binop @{const_name Datatype.Scons}) ts};
   121 
   122     (* function spaces *)
   123 
   124     fun mk_fun_inj T' x =
   125       let
   126         fun mk_inj T n i =
   127           if n = 1 then x
   128           else
   129             let
   130               val n2 = n div 2;
   131               val Type (_, [T1, T2]) = T;
   132               fun mkT U = (U --> Univ_elT) --> T --> Univ_elT;
   133             in
   134               if i <= n2 then Const (@{const_name Sum_Type.Suml}, mkT T1) $ mk_inj T1 n2 i
   135               else Const (@{const_name Sum_Type.Sumr}, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
   136             end;
   137       in mk_inj branchT (length branchTs) (1 + find_index (fn T'' => T'' = T') branchTs) end;
   138 
   139     fun mk_lim t Ts = fold_rev (fn T => fn t => Lim $ mk_fun_inj T (Abs ("x", T, t))) Ts t;
   140 
   141     (************** generate introduction rules for representing set **********)
   142 
   143     val _ = Datatype_Aux.message config "Constructing representing sets ...";
   144 
   145     (* make introduction rule for a single constructor *)
   146 
   147     fun make_intr s n (i, (_, cargs)) =
   148       let
   149         fun mk_prem dt (j, prems, ts) =
   150           (case Datatype_Aux.strip_dtyp dt of
   151             (dts, Datatype_Aux.DtRec k) =>
   152               let
   153                 val Ts = map (Datatype_Aux.typ_of_dtyp descr') dts;
   154                 val free_t =
   155                   Datatype_Aux.app_bnds (Datatype_Aux.mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
   156               in
   157                 (j + 1, Logic.list_all (map (pair "x") Ts,
   158                   HOLogic.mk_Trueprop
   159                     (Free (nth rep_set_names' k, UnivT') $ free_t)) :: prems,
   160                 mk_lim free_t Ts :: ts)
   161               end
   162           | _ =>
   163               let val T = Datatype_Aux.typ_of_dtyp descr' dt
   164               in (j + 1, prems, (Leaf $ mk_inj T (Datatype_Aux.mk_Free "x" T j)) :: ts) end);
   165 
   166         val (_, prems, ts) = fold_rev mk_prem cargs (1, [], []);
   167         val concl = HOLogic.mk_Trueprop (Free (s, UnivT') $ mk_univ_inj ts n i);
   168       in Logic.list_implies (prems, concl) end;
   169 
   170     val intr_ts = maps (fn ((_, (_, _, constrs)), rep_set_name) =>
   171       map (make_intr rep_set_name (length constrs))
   172         ((1 upto length constrs) ~~ constrs)) (descr' ~~ rep_set_names');
   173 
   174     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
   175       thy1
   176       |> Sign.map_naming Name_Space.conceal
   177       |> Inductive.add_inductive_global
   178           {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name,
   179            coind = false, no_elim = true, no_ind = false, skip_mono = true}
   180           (map (fn s => ((Binding.name s, UnivT'), NoSyn)) rep_set_names') []
   181           (map (fn x => (Attrib.empty_binding, x)) intr_ts) []
   182       ||> Sign.restore_naming thy1;
   183 
   184     (********************************* typedef ********************************)
   185 
   186     val (typedefs, thy3) = thy2
   187       |> Sign.parent_path
   188       |> fold_map
   189         (fn (((name, mx), tvs), c) =>
   190           Typedef.add_typedef_global (name, tvs, mx)
   191             (Collect $ Const (c, UnivT')) NONE
   192             (rtac exI 1 THEN rtac CollectI 1 THEN
   193               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   194               (resolve_tac rep_intrs 1)))
   195         (types_syntax ~~ tyvars ~~ take (length newTs) rep_set_names)
   196       ||> Sign.add_path big_name;
   197 
   198     (*********************** definition of constructors ***********************)
   199 
   200     val big_rep_name = big_name ^ "_Rep_";
   201     val rep_names' = map (fn i => big_rep_name ^ string_of_int i) (1 upto length (flat (tl descr)));
   202     val all_rep_names =
   203       map (#Rep_name o #1 o #2) typedefs @
   204       map (Sign.full_bname thy3) rep_names';
   205 
   206     (* isomorphism declarations *)
   207 
   208     val iso_decls = map (fn (T, s) => (Binding.name s, T --> Univ_elT, NoSyn))
   209       (oldTs ~~ rep_names');
   210 
   211     (* constructor definitions *)
   212 
   213     fun make_constr_def (typedef: Typedef.info) T n
   214         ((cname, cargs), (cname', mx)) (thy, defs, eqns, i) =
   215       let
   216         fun constr_arg dt (j, l_args, r_args) =
   217           let
   218             val T = Datatype_Aux.typ_of_dtyp descr' dt;
   219             val free_t = Datatype_Aux.mk_Free "x" T j;
   220           in
   221             (case (Datatype_Aux.strip_dtyp dt, strip_type T) of
   222               ((_, Datatype_Aux.DtRec m), (Us, U)) =>
   223                 (j + 1, free_t :: l_args, mk_lim
   224                   (Const (nth all_rep_names m, U --> Univ_elT) $
   225                     Datatype_Aux.app_bnds free_t (length Us)) Us :: r_args)
   226             | _ => (j + 1, free_t :: l_args, (Leaf $ mk_inj T free_t) :: r_args))
   227           end;
   228 
   229         val (_, l_args, r_args) = fold_rev constr_arg cargs (1, [], []);
   230         val constrT = map (Datatype_Aux.typ_of_dtyp descr') cargs ---> T;
   231         val ({Abs_name, Rep_name, ...}, _) = typedef;
   232         val lhs = list_comb (Const (cname, constrT), l_args);
   233         val rhs = mk_univ_inj r_args n i;
   234         val def = Logic.mk_equals (lhs, Const (Abs_name, Univ_elT --> T) $ rhs);
   235         val def_name = Thm.def_name (Long_Name.base_name cname);
   236         val eqn =
   237           HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (Rep_name, T --> Univ_elT) $ lhs, rhs));
   238         val ([def_thm], thy') =
   239           thy
   240           |> Sign.add_consts [(cname', constrT, mx)]
   241           |> (Global_Theory.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)];
   242 
   243       in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
   244 
   245     (* constructor definitions for datatype *)
   246 
   247     fun dt_constr_defs (((((_, (_, _, constrs)), tname), typedef: Typedef.info), T), constr_syntax)
   248         (thy, defs, eqns, rep_congs, dist_lemmas) =
   249       let
   250         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
   251         val rep_const = cterm_of thy (Const (#Rep_name (#1 typedef), T --> Univ_elT));
   252         val cong' = cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong;
   253         val dist = cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma;
   254         val (thy', defs', eqns', _) =
   255           fold (make_constr_def typedef T (length constrs))
   256             (constrs ~~ constr_syntax) (Sign.add_path tname thy, defs, [], 1);
   257       in
   258         (Sign.parent_path thy', defs', eqns @ [eqns'],
   259           rep_congs @ [cong'], dist_lemmas @ [dist])
   260       end;
   261 
   262     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) =
   263       fold dt_constr_defs
   264         (hd descr ~~ new_type_names ~~ map #2 typedefs ~~ newTs ~~ constr_syntax)
   265         (thy3 |> Sign.add_consts iso_decls |> Sign.parent_path, [], [], [], []);
   266 
   267 
   268     (*********** isomorphisms for new types (introduced by typedef) ***********)
   269 
   270     val _ = Datatype_Aux.message config "Proving isomorphism properties ...";
   271 
   272     val collect_simp = rewrite_rule (Proof_Context.init_global thy4) [mk_meta_eq mem_Collect_eq];
   273 
   274     val newT_iso_axms = typedefs |> map (fn (_, (_, {Abs_inverse, Rep_inverse, Rep, ...})) =>
   275       (collect_simp Abs_inverse, Rep_inverse, collect_simp Rep));
   276 
   277     val newT_iso_inj_thms = typedefs |> map (fn (_, (_, {Abs_inject, Rep_inject, ...})) =>
   278       (collect_simp Abs_inject RS iffD1, Rep_inject RS iffD1));
   279 
   280     (********* isomorphisms between existing types and "unfolded" types *******)
   281 
   282     (*---------------------------------------------------------------------*)
   283     (* isomorphisms are defined using primrec-combinators:                 *)
   284     (* generate appropriate functions for instantiating primrec-combinator *)
   285     (*                                                                     *)
   286     (*   e.g.  Rep_dt_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)
   287     (*                                                                     *)
   288     (* also generate characteristic equations for isomorphisms             *)
   289     (*                                                                     *)
   290     (*   e.g.  Rep_dt_i (cons h t) = In1 (Scons (Rep_dt_j h) (Rep_dt_i t)) *)
   291     (*---------------------------------------------------------------------*)
   292 
   293     fun make_iso_def k ks n (cname, cargs) (fs, eqns, i) =
   294       let
   295         val argTs = map (Datatype_Aux.typ_of_dtyp descr') cargs;
   296         val T = nth recTs k;
   297         val rep_const = Const (nth all_rep_names k, T --> Univ_elT);
   298         val constr = Const (cname, argTs ---> T);
   299 
   300         fun process_arg ks' dt (i2, i2', ts, Ts) =
   301           let
   302             val T' = Datatype_Aux.typ_of_dtyp descr' dt;
   303             val (Us, U) = strip_type T'
   304           in
   305             (case Datatype_Aux.strip_dtyp dt of
   306               (_, Datatype_Aux.DtRec j) =>
   307                 if member (op =) ks' j then
   308                   (i2 + 1, i2' + 1, ts @ [mk_lim (Datatype_Aux.app_bnds
   309                      (Datatype_Aux.mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
   310                    Ts @ [Us ---> Univ_elT])
   311                 else
   312                   (i2 + 1, i2', ts @ [mk_lim
   313                      (Const (nth all_rep_names j, U --> Univ_elT) $
   314                         Datatype_Aux.app_bnds (Datatype_Aux.mk_Free "x" T' i2) (length Us)) Us], Ts)
   315             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (Datatype_Aux.mk_Free "x" T' i2)], Ts))
   316           end;
   317 
   318         val (i2, i2', ts, Ts) = fold (process_arg ks) cargs (1, 1, [], []);
   319         val xs = map (uncurry (Datatype_Aux.mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   320         val ys = map (uncurry (Datatype_Aux.mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   321         val f = fold_rev lambda (xs @ ys) (mk_univ_inj ts n i);
   322 
   323         val (_, _, ts', _) = fold (process_arg []) cargs (1, 1, [], []);
   324         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   325           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   326 
   327       in (fs @ [f], eqns @ [eqn], i + 1) end;
   328 
   329     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   330 
   331     fun make_iso_defs ds (thy, char_thms) =
   332       let
   333         val ks = map fst ds;
   334         val (_, (tname, _, _)) = hd ds;
   335         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);
   336 
   337         fun process_dt (k, (_, _, constrs)) (fs, eqns, isos) =
   338           let
   339             val (fs', eqns', _) = fold (make_iso_def k ks (length constrs)) constrs (fs, eqns, 1);
   340             val iso = (nth recTs k, nth all_rep_names k);
   341           in (fs', eqns', isos @ [iso]) end;
   342 
   343         val (fs, eqns, isos) = fold process_dt ds ([], [], []);
   344         val fTs = map fastype_of fs;
   345         val defs =
   346           map (fn (rec_name, (T, iso_name)) =>
   347             (Binding.name (Thm.def_name (Long_Name.base_name iso_name)),
   348               Logic.mk_equals (Const (iso_name, T --> Univ_elT),
   349                 list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs)))) (rec_names ~~ isos);
   350         val (def_thms, thy') =
   351           (Global_Theory.add_defs false o map Thm.no_attributes) defs thy;
   352 
   353         (* prove characteristic equations *)
   354 
   355         val rewrites = def_thms @ map mk_meta_eq rec_rewrites;
   356         val char_thms' =
   357           map (fn eqn => Goal.prove_sorry_global thy' [] [] eqn
   358             (fn {context = ctxt, ...} => EVERY [rewrite_goals_tac ctxt rewrites, rtac refl 1])) eqns;
   359 
   360       in (thy', char_thms' @ char_thms) end;
   361 
   362     val (thy5, iso_char_thms) =
   363       fold_rev make_iso_defs (tl descr) (Sign.add_path big_name thy4, []);
   364 
   365     (* prove isomorphism properties *)
   366 
   367     fun mk_funs_inv thy thm =
   368       let
   369         val prop = Thm.prop_of thm;
   370         val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
   371           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.legacy_freeze prop;
   372         val used = Term.add_tfree_names a [];
   373 
   374         fun mk_thm i =
   375           let
   376             val Ts = map (TFree o rpair @{sort type}) (Name.variant_list used (replicate i "'t"));
   377             val f = Free ("f", Ts ---> U);
   378           in
   379             Goal.prove_sorry_global thy [] []
   380               (Logic.mk_implies
   381                 (HOLogic.mk_Trueprop (HOLogic.list_all
   382                    (map (pair "x") Ts, S $ Datatype_Aux.app_bnds f i)),
   383                  HOLogic.mk_Trueprop (HOLogic.mk_eq (fold_rev (Term.abs o pair "x") Ts
   384                    (r $ (a $ Datatype_Aux.app_bnds f i)), f))))
   385               (fn _ => EVERY [REPEAT_DETERM_N i (rtac @{thm ext} 1),
   386                  REPEAT (etac allE 1), rtac thm 1, atac 1])
   387           end
   388       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
   389 
   390     (* prove  inj Rep_dt_i  and  Rep_dt_i x : rep_set_dt_i *)
   391 
   392     val fun_congs =
   393       map (fn T => make_elim (Drule.instantiate' [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs;
   394 
   395     fun prove_iso_thms ds (inj_thms, elem_thms) =
   396       let
   397         val (_, (tname, _, _)) = hd ds;
   398         val induct = #induct (the (Symtab.lookup dt_info tname));
   399 
   400         fun mk_ind_concl (i, _) =
   401           let
   402             val T = nth recTs i;
   403             val Rep_t = Const (nth all_rep_names i, T --> Univ_elT);
   404             val rep_set_name = nth rep_set_names i;
   405             val concl1 =
   406               HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   407                 HOLogic.mk_eq (Rep_t $ Datatype_Aux.mk_Free "x" T i, Rep_t $ Bound 0) $
   408                   HOLogic.mk_eq (Datatype_Aux.mk_Free "x" T i, Bound 0));
   409             val concl2 = Const (rep_set_name, UnivT') $ (Rep_t $ Datatype_Aux.mk_Free "x" T i);
   410           in (concl1, concl2) end;
   411 
   412         val (ind_concl1, ind_concl2) = split_list (map mk_ind_concl ds);
   413 
   414         val rewrites = map mk_meta_eq iso_char_thms;
   415         val inj_thms' = map snd newT_iso_inj_thms @ map (fn r => r RS @{thm injD}) inj_thms;
   416 
   417         val inj_thm =
   418           Goal.prove_sorry_global thy5 [] []
   419             (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj ind_concl1))
   420             (fn {context = ctxt, ...} => EVERY
   421               [(Datatype_Aux.ind_tac induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac ctxt) 1,
   422                REPEAT (EVERY
   423                  [rtac allI 1, rtac impI 1,
   424                   Datatype_Aux.exh_tac (exh_thm_of dt_info) 1,
   425                   REPEAT (EVERY
   426                     [hyp_subst_tac ctxt 1,
   427                      rewrite_goals_tac ctxt rewrites,
   428                      REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   429                      (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   430                      ORELSE (EVERY
   431                        [REPEAT (eresolve_tac (Scons_inject ::
   432                           map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
   433                         REPEAT (cong_tac 1), rtac refl 1,
   434                         REPEAT (atac 1 ORELSE (EVERY
   435                           [REPEAT (rtac @{thm ext} 1),
   436                            REPEAT (eresolve_tac (mp :: allE ::
   437                              map make_elim (Suml_inject :: Sumr_inject ::
   438                                Lim_inject :: inj_thms') @ fun_congs) 1),
   439                            atac 1]))])])])]);
   440 
   441         val inj_thms'' = map (fn r => r RS datatype_injI) (Datatype_Aux.split_conj_thm inj_thm);
   442 
   443         val elem_thm =
   444           Goal.prove_sorry_global thy5 [] []
   445             (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj ind_concl2))
   446             (fn {context = ctxt, ...} =>
   447               EVERY [
   448                 (Datatype_Aux.ind_tac induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac ctxt) 1,
   449                 rewrite_goals_tac ctxt rewrites,
   450                 REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
   451                   ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
   452 
   453       in (inj_thms'' @ inj_thms, elem_thms @ Datatype_Aux.split_conj_thm elem_thm) end;
   454 
   455     val (iso_inj_thms_unfolded, iso_elem_thms) =
   456       fold_rev prove_iso_thms (tl descr) ([], map #3 newT_iso_axms);
   457     val iso_inj_thms =
   458       map snd newT_iso_inj_thms @ map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded;
   459 
   460     (* prove  rep_set_dt_i x --> x : range Rep_dt_i *)
   461 
   462     fun mk_iso_t (((set_name, iso_name), i), T) =
   463       let val isoT = T --> Univ_elT in
   464         HOLogic.imp $
   465           (Const (set_name, UnivT') $ Datatype_Aux.mk_Free "x" Univ_elT i) $
   466             (if i < length newTs then @{term True}
   467              else HOLogic.mk_mem (Datatype_Aux.mk_Free "x" Univ_elT i,
   468                Const (@{const_name image}, isoT --> HOLogic.mk_setT T --> UnivT) $
   469                  Const (iso_name, isoT) $ Const (@{const_abbrev UNIV}, HOLogic.mk_setT T)))
   470       end;
   471 
   472     val iso_t = HOLogic.mk_Trueprop (Datatype_Aux.mk_conj (map mk_iso_t
   473       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   474 
   475     (* all the theorems are proved by one single simultaneous induction *)
   476 
   477     val range_eqs = map (fn r => mk_meta_eq (r RS @{thm range_ex1_eq})) iso_inj_thms_unfolded;
   478 
   479     val iso_thms =
   480       if length descr = 1 then []
   481       else
   482         drop (length newTs) (Datatype_Aux.split_conj_thm
   483           (Goal.prove_sorry_global thy5 [] [] iso_t (fn {context = ctxt, ...} => EVERY
   484              [(Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac ctxt) 1,
   485               REPEAT (rtac TrueI 1),
   486               rewrite_goals_tac ctxt (mk_meta_eq @{thm choice_eq} ::
   487                 Thm.symmetric (mk_meta_eq @{thm fun_eq_iff}) :: range_eqs),
   488               rewrite_goals_tac ctxt (map Thm.symmetric range_eqs),
   489               REPEAT (EVERY
   490                 [REPEAT (eresolve_tac ([rangeE, @{thm ex1_implies_ex} RS exE] @
   491                    maps (mk_funs_inv thy5 o #1) newT_iso_axms) 1),
   492                  TRY (hyp_subst_tac ctxt 1),
   493                  rtac (sym RS range_eqI) 1,
   494                  resolve_tac iso_char_thms 1])])));
   495 
   496     val Abs_inverse_thms' =
   497       map #1 newT_iso_axms @
   498       map2 (fn r_inj => fn r => @{thm f_the_inv_into_f} OF [r_inj, r RS mp])
   499         iso_inj_thms_unfolded iso_thms;
   500 
   501     val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms';
   502 
   503     (******************* freeness theorems for constructors *******************)
   504 
   505     val _ = Datatype_Aux.message config "Proving freeness of constructors ...";
   506 
   507     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   508 
   509     fun prove_constr_rep_thm eqn =
   510       let
   511         val inj_thms = map fst newT_iso_inj_thms;
   512         val rewrites = @{thm o_def} :: constr_defs @ map (mk_meta_eq o #2) newT_iso_axms;
   513       in
   514         Goal.prove_sorry_global thy5 [] [] eqn
   515         (fn {context = ctxt, ...} => EVERY
   516           [resolve_tac inj_thms 1,
   517            rewrite_goals_tac ctxt rewrites,
   518            rtac refl 3,
   519            resolve_tac rep_intrs 2,
   520            REPEAT (resolve_tac iso_elem_thms 1)])
   521       end;
   522 
   523     (*--------------------------------------------------------------*)
   524     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   525     (* of constructors.                                             *)
   526     (*--------------------------------------------------------------*)
   527 
   528     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   529 
   530     val dist_rewrites =
   531       map (fn (rep_thms, dist_lemma) =>
   532         dist_lemma :: (rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   533           (constr_rep_thms ~~ dist_lemmas);
   534 
   535     fun prove_distinct_thms dist_rewrites' =
   536       let
   537         fun prove [] = []
   538           | prove (t :: ts) =
   539               let
   540                 val dist_thm = Goal.prove_sorry_global thy5 [] [] t (fn {context = ctxt, ...} =>
   541                   EVERY [simp_tac (put_simpset HOL_ss ctxt addsimps dist_rewrites') 1])
   542               in dist_thm :: Drule.zero_var_indexes (dist_thm RS not_sym) :: prove ts end;
   543       in prove end;
   544 
   545     val distinct_thms =
   546       map2 (prove_distinct_thms) dist_rewrites (Datatype_Prop.make_distincts descr);
   547 
   548     (* prove injectivity of constructors *)
   549 
   550     fun prove_constr_inj_thm rep_thms t =
   551       let
   552         val inj_thms = Scons_inject ::
   553           map make_elim
   554             (iso_inj_thms @
   555               [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
   556                Lim_inject, Suml_inject, Sumr_inject])
   557       in
   558         Goal.prove_sorry_global thy5 [] [] t
   559           (fn {context = ctxt, ...} => EVERY
   560             [rtac iffI 1,
   561              REPEAT (etac conjE 2), hyp_subst_tac ctxt 2, rtac refl 2,
   562              dresolve_tac rep_congs 1, dtac @{thm box_equals} 1,
   563              REPEAT (resolve_tac rep_thms 1),
   564              REPEAT (eresolve_tac inj_thms 1),
   565              REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac @{thm ext} 1),
   566                REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
   567                atac 1]))])
   568       end;
   569 
   570     val constr_inject =
   571       map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   572         (Datatype_Prop.make_injs descr ~~ constr_rep_thms);
   573 
   574     val ((constr_inject', distinct_thms'), thy6) =
   575       thy5
   576       |> Sign.parent_path
   577       |> Datatype_Aux.store_thmss "inject" new_type_names constr_inject
   578       ||>> Datatype_Aux.store_thmss "distinct" new_type_names distinct_thms;
   579 
   580     (*************************** induction theorem ****************************)
   581 
   582     val _ = Datatype_Aux.message config "Proving induction rule for datatypes ...";
   583 
   584     val Rep_inverse_thms =
   585       map (fn (_, iso, _) => iso RS subst) newT_iso_axms @
   586       map (fn r => r RS @{thm the_inv_f_f} RS subst) iso_inj_thms_unfolded;
   587     val Rep_inverse_thms' = map (fn r => r RS @{thm the_inv_f_f}) iso_inj_thms_unfolded;
   588 
   589     fun mk_indrule_lemma (i, _) T =
   590       let
   591         val Rep_t = Const (nth all_rep_names i, T --> Univ_elT) $ Datatype_Aux.mk_Free "x" T i;
   592         val Abs_t =
   593           if i < length newTs then
   594             Const (#Abs_name (#1 (#2 (nth typedefs i))), Univ_elT --> T)
   595           else
   596             Const (@{const_name the_inv_into},
   597               [HOLogic.mk_setT T, T --> Univ_elT, Univ_elT] ---> T) $
   598             HOLogic.mk_UNIV T $ Const (nth all_rep_names i, T --> Univ_elT);
   599         val prem =
   600           HOLogic.imp $
   601             (Const (nth rep_set_names i, UnivT') $ Rep_t) $
   602               (Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t));
   603         val concl =
   604           Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ Datatype_Aux.mk_Free "x" T i;
   605       in (prem, concl) end;
   606 
   607     val (indrule_lemma_prems, indrule_lemma_concls) =
   608       split_list (map2 mk_indrule_lemma descr' recTs);
   609 
   610     val cert = cterm_of thy6;
   611 
   612     val indrule_lemma =
   613       Goal.prove_sorry_global thy6 [] []
   614         (Logic.mk_implies
   615           (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_prems),
   616            HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_concls)))
   617         (fn _ =>
   618           EVERY
   619            [REPEAT (etac conjE 1),
   620             REPEAT (EVERY
   621               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   622                etac mp 1, resolve_tac iso_elem_thms 1])]);
   623 
   624     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   625     val frees =
   626       if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))]
   627       else map (Free o apfst fst o dest_Var) Ps;
   628     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   629 
   630     val dt_induct_prop = Datatype_Prop.make_ind descr;
   631     val dt_induct =
   632       Goal.prove_sorry_global thy6 []
   633       (Logic.strip_imp_prems dt_induct_prop)
   634       (Logic.strip_imp_concl dt_induct_prop)
   635       (fn {context = ctxt, prems, ...} =>
   636         EVERY
   637           [rtac indrule_lemma' 1,
   638            (Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac ctxt) 1,
   639            EVERY (map (fn (prem, r) => (EVERY
   640              [REPEAT (eresolve_tac Abs_inverse_thms 1),
   641               simp_tac (put_simpset HOL_basic_ss ctxt
   642                 addsimps (Thm.symmetric r :: Rep_inverse_thms')) 1,
   643               DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
   644                   (prems ~~ (constr_defs @ map mk_meta_eq iso_char_thms)))]);
   645 
   646     val ([(_, [dt_induct'])], thy7) =
   647       thy6
   648       |> Global_Theory.note_thmss ""
   649         [((Binding.qualify true big_name (Binding.name "induct"), [case_names_induct]),
   650           [([dt_induct], [])])];
   651   in
   652     ((constr_inject', distinct_thms', dt_induct'), thy7)
   653   end;
   654 
   655 
   656 
   657 (** datatype definition **)
   658 
   659 (* specifications *)
   660 
   661 type spec = (binding * (string * sort) list * mixfix) * (binding * typ list * mixfix) list;
   662 
   663 type spec_cmd =
   664   (binding * (string * string option) list * mixfix) * (binding * string list * mixfix) list;
   665 
   666 local
   667 
   668 fun parse_spec ctxt ((b, args, mx), constrs) =
   669   ((b, map (apsnd (Typedecl.read_constraint ctxt)) args, mx),
   670     constrs |> map (fn (c, Ts, mx') => (c, map (Syntax.parse_typ ctxt) Ts, mx')));
   671 
   672 fun check_specs ctxt (specs: spec list) =
   673   let
   674     fun prep_spec ((tname, args, mx), constrs) tys =
   675       let
   676         val (args', tys1) = chop (length args) tys;
   677         val (constrs', tys3) = (constrs, tys1) |-> fold_map (fn (cname, cargs, mx') => fn tys2 =>
   678           let val (cargs', tys3) = chop (length cargs) tys2;
   679           in ((cname, cargs', mx'), tys3) end);
   680       in (((tname, map dest_TFree args', mx), constrs'), tys3) end;
   681 
   682     val all_tys =
   683       specs |> maps (fn ((_, args, _), cs) => map TFree args @ maps #2 cs)
   684       |> Syntax.check_typs ctxt;
   685 
   686   in #1 (fold_map prep_spec specs all_tys) end;
   687 
   688 fun prep_specs parse raw_specs thy =
   689   let
   690     val ctxt = thy
   691       |> Sign.add_types_global (map (fn ((b, args, mx), _) => (b, length args, mx)) raw_specs)
   692       |> Proof_Context.init_global
   693       |> fold (fn ((_, args, _), _) => fold (fn (a, _) =>
   694           Variable.declare_typ (TFree (a, dummyS))) args) raw_specs;
   695     val specs = check_specs ctxt (map (parse ctxt) raw_specs);
   696   in (specs, ctxt) end;
   697 
   698 in
   699 
   700 val read_specs = prep_specs parse_spec;
   701 val check_specs = prep_specs (K I);
   702 
   703 end;
   704 
   705 
   706 (* main commands *)
   707 
   708 fun gen_add_datatype prep_specs config raw_specs thy =
   709   let
   710     val _ = Theory.requires thy (Context.theory_name @{theory}) "datatype definitions";
   711 
   712     val (dts, spec_ctxt) = prep_specs raw_specs thy;
   713     val ((_, tyvars, _), _) :: _ = dts;
   714     val string_of_tyvar = Syntax.string_of_typ spec_ctxt o TFree;
   715 
   716     val (new_dts, types_syntax) = dts |> map (fn ((tname, tvs, mx), _) =>
   717       let val full_tname = Sign.full_name thy tname in
   718         (case duplicates (op =) tvs of
   719           [] =>
   720             if eq_set (op =) (tyvars, tvs) then ((full_tname, tvs), (tname, mx))
   721             else error "Mutually recursive datatypes must have same type parameters"
   722         | dups =>
   723             error ("Duplicate parameter(s) for datatype " ^ Binding.print tname ^
   724               " : " ^ commas (map string_of_tyvar dups)))
   725       end) |> split_list;
   726     val dt_names = map fst new_dts;
   727 
   728     val _ =
   729       (case duplicates (op =) (map fst new_dts) of
   730         [] => ()
   731       | dups => error ("Duplicate datatypes: " ^ commas_quote dups));
   732 
   733     fun prep_dt_spec ((tname, tvs, _), constrs) (dts', constr_syntax, i) =
   734       let
   735         fun prep_constr (cname, cargs, mx) (constrs, constr_syntax') =
   736           let
   737             val _ =
   738               (case subtract (op =) tvs (fold Term.add_tfreesT cargs []) of
   739                 [] => ()
   740               | vs => error ("Extra type variables on rhs: " ^ commas (map string_of_tyvar vs)));
   741             val c = Sign.full_name_path thy (Binding.name_of tname) cname;
   742           in
   743             (constrs @ [(c, map (Datatype_Aux.dtyp_of_typ new_dts) cargs)],
   744               constr_syntax' @ [(cname, mx)])
   745           end handle ERROR msg =>
   746             cat_error msg ("The error above occurred in constructor " ^ Binding.print cname ^
   747               " of datatype " ^ Binding.print tname);
   748 
   749         val (constrs', constr_syntax') = fold prep_constr constrs ([], []);
   750       in
   751         (case duplicates (op =) (map fst constrs') of
   752           [] =>
   753             (dts' @ [(i, (Sign.full_name thy tname, map Datatype_Aux.DtTFree tvs, constrs'))],
   754               constr_syntax @ [constr_syntax'], i + 1)
   755         | dups =>
   756             error ("Duplicate constructors " ^ commas_quote dups ^
   757               " in datatype " ^ Binding.print tname))
   758       end;
   759 
   760     val (dts', constr_syntax, i) = fold prep_dt_spec dts ([], [], 0);
   761 
   762     val dt_info = Datatype_Data.get_all thy;
   763     val (descr, _) = Datatype_Aux.unfold_datatypes spec_ctxt dts' dt_info dts' i;
   764     val _ =
   765       Datatype_Aux.check_nonempty descr
   766         handle (exn as Datatype_Aux.Datatype_Empty s) =>
   767           if #strict config then error ("Nonemptiness check failed for datatype " ^ quote s)
   768           else reraise exn;
   769 
   770     val _ =
   771       Datatype_Aux.message config
   772         ("Constructing datatype(s) " ^ commas_quote (map (Binding.name_of o #1 o #1) dts));
   773   in
   774     thy
   775     |> representation_proofs config dt_info descr types_syntax constr_syntax
   776       (Datatype_Data.mk_case_names_induct (flat descr))
   777     |-> (fn (inject, distinct, induct) =>
   778       Rep_Datatype.derive_datatype_props config dt_names descr induct inject distinct)
   779   end;
   780 
   781 val add_datatype = gen_add_datatype check_specs;
   782 val add_datatype_cmd = gen_add_datatype read_specs;
   783 
   784 
   785 (* outer syntax *)
   786 
   787 val spec_cmd =
   788   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
   789   (@{keyword "="} |-- Parse.enum1 "|" (Parse.binding -- Scan.repeat Parse.typ -- Parse.opt_mixfix))
   790   >> (fn (((vs, t), mx), cons) => ((t, vs, mx), map Parse.triple1 cons));
   791 
   792 val _ =
   793   Outer_Syntax.command @{command_spec "datatype"} "define inductive datatypes"
   794     (Parse.and_list1 spec_cmd
   795       >> (Toplevel.theory o (snd oo add_datatype_cmd Datatype_Aux.default_config)));
   796 
   797 
   798 open Datatype_Data;
   799 
   800 end;