src/HOL/Tools/datatype_abs_proofs.ML
author wenzelm
Tue Sep 25 17:06:14 2007 +0200 (2007-09-25)
changeset 24712 64ed05609568
parent 24699 c6674504103f
child 24746 6d42be359d57
permissions -rw-r--r--
proper Sign operations instead of Theory aliases;
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(*  Title:      HOL/Tools/datatype_abs_proofs.ML
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    ID:         $Id$
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    Author:     Stefan Berghofer, TU Muenchen
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Proofs and defintions independent of concrete representation
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of datatypes  (i.e. requiring only abstract properties such as
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injectivity / distinctness of constructors and induction)
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 - case distinction (exhaustion) theorems
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 - characteristic equations for primrec combinators
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 - characteristic equations for case combinators
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 - equations for splitting "P (case ...)" expressions
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  - "nchotomy" and "case_cong" theorems for TFL
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*)
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signature DATATYPE_ABS_PROOFS =
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sig
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  val prove_casedist_thms : string list ->
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    DatatypeAux.descr list -> (string * sort) list -> thm ->
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    attribute list -> theory -> thm list * theory
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  val prove_primrec_thms : bool -> string list ->
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    DatatypeAux.descr list -> (string * sort) list ->
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      DatatypeAux.datatype_info Symtab.table -> thm list list -> thm list list ->
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        simpset -> thm -> theory -> (string list * thm list) * theory
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  val prove_case_thms : bool -> string list ->
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    DatatypeAux.descr list -> (string * sort) list ->
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      string list -> thm list -> theory -> (thm list list * string list) * theory
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  val prove_split_thms : string list ->
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    DatatypeAux.descr list -> (string * sort) list ->
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      thm list list -> thm list list -> thm list -> thm list list -> theory ->
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        (thm * thm) list * theory
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  val prove_nchotomys : string list -> DatatypeAux.descr list ->
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    (string * sort) list -> thm list -> theory -> thm list * theory
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  val prove_weak_case_congs : string list -> DatatypeAux.descr list ->
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    (string * sort) list -> theory -> thm list * theory
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  val prove_case_congs : string list ->
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    DatatypeAux.descr list -> (string * sort) list ->
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      thm list -> thm list list -> theory -> thm list * theory
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end;
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structure DatatypeAbsProofs: DATATYPE_ABS_PROOFS =
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struct
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open DatatypeAux;
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(************************ case distinction theorems ***************************)
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fun prove_casedist_thms new_type_names descr sorts induct case_names_exhausts thy =
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  let
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    val _ = message "Proving case distinction theorems ...";
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    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val newTs = Library.take (length (hd descr), recTs);
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    val {maxidx, ...} = rep_thm induct;
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    val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
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    fun prove_casedist_thm ((i, t), T) =
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      let
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        val dummyPs = map (fn (Var (_, Type (_, [T', T'']))) =>
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          Abs ("z", T', Const ("True", T''))) induct_Ps;
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        val P = Abs ("z", T, HOLogic.imp $ HOLogic.mk_eq (Var (("a", maxidx+1), T), Bound 0) $
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          Var (("P", 0), HOLogic.boolT))
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        val insts = Library.take (i, dummyPs) @ (P::(Library.drop (i + 1, dummyPs)));
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        val cert = cterm_of thy;
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        val insts' = (map cert induct_Ps) ~~ (map cert insts);
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        val induct' = refl RS ((List.nth
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          (split_conj_thm (cterm_instantiate insts' induct), i)) RSN (2, rev_mp))
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      in
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        Goal.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
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          (fn prems => EVERY
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            [rtac induct' 1,
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             REPEAT (rtac TrueI 1),
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             REPEAT ((rtac impI 1) THEN (eresolve_tac prems 1)),
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             REPEAT (rtac TrueI 1)])
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      end;
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    val casedist_thms = map prove_casedist_thm ((0 upto (length newTs - 1)) ~~
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      (DatatypeProp.make_casedists descr sorts) ~~ newTs)
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  in
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    thy
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    |> store_thms_atts "exhaust" new_type_names (map single case_names_exhausts) casedist_thms
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  end;
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(*************************** primrec combinators ******************************)
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fun prove_primrec_thms flat_names new_type_names descr sorts
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    (dt_info : datatype_info Symtab.table) constr_inject dist_rewrites dist_ss induct thy =
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  let
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    val _ = message "Constructing primrec combinators ...";
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    val big_name = space_implode "_" new_type_names;
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    val thy0 = add_path flat_names big_name thy;
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    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val used = foldr add_typ_tfree_names [] recTs;
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    val newTs = Library.take (length (hd descr), recTs);
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    val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
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    val big_rec_name' = big_name ^ "_rec_set";
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    val rec_set_names' =
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      if length descr' = 1 then [big_rec_name'] else
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        map ((curry (op ^) (big_rec_name' ^ "_")) o string_of_int)
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          (1 upto (length descr'));
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    val rec_set_names = map (Sign.full_name thy0) rec_set_names';
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    val (rec_result_Ts, reccomb_fn_Ts) = DatatypeProp.make_primrec_Ts descr sorts used;
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    val rec_set_Ts = map (fn (T1, T2) =>
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      reccomb_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);
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    val rec_fns = map (uncurry (mk_Free "f"))
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      (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));
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    val rec_sets' = map (fn c => list_comb (Free c, rec_fns))
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      (rec_set_names' ~~ rec_set_Ts);
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    val rec_sets = map (fn c => list_comb (Const c, rec_fns))
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      (rec_set_names ~~ rec_set_Ts);
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    (* introduction rules for graph of primrec function *)
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    fun make_rec_intr T rec_set ((rec_intr_ts, l), (cname, cargs)) =
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      let
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        fun mk_prem ((dt, U), (j, k, prems, t1s, t2s)) =
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          let val free1 = mk_Free "x" U j
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          in (case (strip_dtyp dt, strip_type U) of
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             ((_, DtRec m), (Us, _)) =>
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               let
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                 val free2 = mk_Free "y" (Us ---> List.nth (rec_result_Ts, m)) k;
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                 val i = length Us
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               in (j + 1, k + 1, HOLogic.mk_Trueprop (HOLogic.list_all
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                     (map (pair "x") Us, List.nth (rec_sets', m) $
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                       app_bnds free1 i $ app_bnds free2 i)) :: prems,
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                   free1::t1s, free2::t2s)
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               end
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           | _ => (j + 1, k, prems, free1::t1s, t2s))
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          end;
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        val Ts = map (typ_of_dtyp descr' sorts) cargs;
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        val (_, _, prems, t1s, t2s) = foldr mk_prem (1, 1, [], [], []) (cargs ~~ Ts)
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      in (rec_intr_ts @ [Logic.list_implies (prems, HOLogic.mk_Trueprop
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        (rec_set $ list_comb (Const (cname, Ts ---> T), t1s) $
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          list_comb (List.nth (rec_fns, l), t1s @ t2s)))], l + 1)
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      end;
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    val (rec_intr_ts, _) = Library.foldl (fn (x, ((d, T), set_name)) =>
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      Library.foldl (make_rec_intr T set_name) (x, #3 (snd d)))
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        (([], 0), descr' ~~ recTs ~~ rec_sets');
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    val ({intrs = rec_intrs, elims = rec_elims, ...}, thy1) =
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      setmp InductivePackage.quiet_mode (!quiet_mode)
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        (InductivePackage.add_inductive_global false big_rec_name' false false true
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           (map (fn (s, T) => (s, SOME T, NoSyn)) (rec_set_names' ~~ rec_set_Ts))
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           (map (apsnd SOME o dest_Free) rec_fns)
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           (map (fn x => (("", []), x)) rec_intr_ts) []) thy0;
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    (* prove uniqueness and termination of primrec combinators *)
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    val _ = message "Proving termination and uniqueness of primrec functions ...";
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    fun mk_unique_tac ((tac, intrs), ((((i, (tname, _, constrs)), elim), T), T')) =
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      let
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        val distinct_tac =
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          (if i < length newTs then
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             full_simp_tac (HOL_ss addsimps (List.nth (dist_rewrites, i))) 1
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           else full_simp_tac dist_ss 1);
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        val inject = map (fn r => r RS iffD1)
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          (if i < length newTs then List.nth (constr_inject, i)
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            else #inject (the (Symtab.lookup dt_info tname)));
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        fun mk_unique_constr_tac n ((tac, intr::intrs, j), (cname, cargs)) =
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          let
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            val k = length (List.filter is_rec_type cargs)
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          in (EVERY [DETERM tac,
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                REPEAT (etac ex1E 1), rtac ex1I 1,
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                DEPTH_SOLVE_1 (ares_tac [intr] 1),
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                REPEAT_DETERM_N k (etac thin_rl 1 THEN rotate_tac 1 1),
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                etac elim 1,
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                REPEAT_DETERM_N j distinct_tac,
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                TRY (dresolve_tac inject 1),
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                REPEAT (etac conjE 1), hyp_subst_tac 1,
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                REPEAT (EVERY [etac allE 1, dtac mp 1, atac 1]),
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                TRY (hyp_subst_tac 1),
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                rtac refl 1,
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                REPEAT_DETERM_N (n - j - 1) distinct_tac],
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              intrs, j + 1)
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          end;
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        val (tac', intrs', _) = Library.foldl (mk_unique_constr_tac (length constrs))
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          ((tac, intrs, 0), constrs);
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      in (tac', intrs') end;
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    val rec_unique_thms =
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      let
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        val rec_unique_ts = map (fn (((set_t, T1), T2), i) =>
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          Const ("Ex1", (T2 --> HOLogic.boolT) --> HOLogic.boolT) $
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            absfree ("y", T2, set_t $ mk_Free "x" T1 i $ Free ("y", T2)))
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              (rec_sets ~~ recTs ~~ rec_result_Ts ~~ (1 upto length recTs));
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        val cert = cterm_of thy1
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        val insts = map (fn ((i, T), t) => absfree ("x" ^ (string_of_int i), T, t))
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          ((1 upto length recTs) ~~ recTs ~~ rec_unique_ts);
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        val induct' = cterm_instantiate ((map cert induct_Ps) ~~
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          (map cert insts)) induct;
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        val (tac, _) = Library.foldl mk_unique_tac
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          (((rtac induct' THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1
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              THEN rewtac (mk_meta_eq choice_eq), rec_intrs),
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            descr' ~~ rec_elims ~~ recTs ~~ rec_result_Ts);
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      in split_conj_thm (Goal.prove_global thy1 [] []
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        (HOLogic.mk_Trueprop (mk_conj rec_unique_ts)) (K tac))
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      end;
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    val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;
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    (* define primrec combinators *)
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    val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
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    val reccomb_names = map (Sign.full_name thy1)
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      (if length descr' = 1 then [big_reccomb_name] else
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        (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
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          (1 upto (length descr'))));
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    val reccombs = map (fn ((name, T), T') => list_comb
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      (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))
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        (reccomb_names ~~ recTs ~~ rec_result_Ts);
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    val (reccomb_defs, thy2) =
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      thy1
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      |> Sign.add_consts_i (map (fn ((name, T), T') =>
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          (Sign.base_name name, reccomb_fn_Ts @ [T] ---> T', NoSyn))
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          (reccomb_names ~~ recTs ~~ rec_result_Ts))
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      |> (PureThy.add_defs_i false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
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          ((Sign.base_name name) ^ "_def", Logic.mk_equals (comb, absfree ("x", T,
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           Const ("The", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',
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             set $ Free ("x", T) $ Free ("y", T'))))))
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               (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts))
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      ||> parent_path flat_names;
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    (* prove characteristic equations for primrec combinators *)
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    val _ = message "Proving characteristic theorems for primrec combinators ..."
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    val rec_thms = map (fn t => Goal.prove_global thy2 [] [] t
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      (fn _ => EVERY
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        [rewrite_goals_tac reccomb_defs,
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         rtac the1_equality 1,
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         resolve_tac rec_unique_thms 1,
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         resolve_tac rec_intrs 1,
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         REPEAT (rtac allI 1 ORELSE resolve_tac rec_total_thms 1)]))
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           (DatatypeProp.make_primrecs new_type_names descr sorts thy2)
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  in
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    thy2
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    |> Sign.add_path (space_implode "_" new_type_names)
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    |> PureThy.add_thmss [(("recs", rec_thms), [])]
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    ||> Sign.parent_path
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    |-> (fn thms => pair (reccomb_names, Library.flat thms))
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  end;
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(***************************** case combinators *******************************)
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fun prove_case_thms flat_names new_type_names descr sorts reccomb_names primrec_thms thy =
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  let
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    val _ = message "Proving characteristic theorems for case combinators ...";
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    val thy1 = add_path flat_names (space_implode "_" new_type_names) thy;
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    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val used = foldr add_typ_tfree_names [] recTs;
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    val newTs = Library.take (length (hd descr), recTs);
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    val T' = TFree (Name.variant used "'t", HOLogic.typeS);
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    fun mk_dummyT dt = binder_types (typ_of_dtyp descr' sorts dt) ---> T';
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    val case_dummy_fns = map (fn (_, (_, _, constrs)) => map (fn (_, cargs) =>
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      let
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        val Ts = map (typ_of_dtyp descr' sorts) cargs;
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        val Ts' = map mk_dummyT (List.filter is_rec_type cargs)
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      in Const ("arbitrary", Ts @ Ts' ---> T')
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      end) constrs) descr';
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    val case_names = map (fn s => Sign.full_name thy1 (s ^ "_case")) new_type_names;
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    (* define case combinators via primrec combinators *)
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   296
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    val (case_defs, thy2) = Library.foldl (fn ((defs, thy),
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      ((((i, (_, _, constrs)), T), name), recname)) =>
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        let
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          val (fns1, fns2) = ListPair.unzip (map (fn ((_, cargs), j) =>
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            let
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              val Ts = map (typ_of_dtyp descr' sorts) cargs;
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              val Ts' = Ts @ map mk_dummyT (List.filter is_rec_type cargs);
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              val frees' = map (uncurry (mk_Free "x")) (Ts' ~~ (1 upto length Ts'));
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              val frees = Library.take (length cargs, frees');
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              val free = mk_Free "f" (Ts ---> T') j
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            in
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             (free, list_abs_free (map dest_Free frees',
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               list_comb (free, frees)))
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            end) (constrs ~~ (1 upto length constrs)));
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   311
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          val caseT = (map (snd o dest_Free) fns1) @ [T] ---> T';
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          val fns = (List.concat (Library.take (i, case_dummy_fns))) @
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            fns2 @ (List.concat (Library.drop (i + 1, case_dummy_fns)));
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          val reccomb = Const (recname, (map fastype_of fns) @ [T] ---> T');
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          val decl = (Sign.base_name name, caseT, NoSyn);
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          val def = ((Sign.base_name name) ^ "_def",
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            Logic.mk_equals (list_comb (Const (name, caseT), fns1),
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              list_comb (reccomb, (List.concat (Library.take (i, case_dummy_fns))) @
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                fns2 @ (List.concat (Library.drop (i + 1, case_dummy_fns))) )));
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   321
          val ([def_thm], thy') =
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            thy
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            |> Sign.add_consts_authentic [decl]
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   324
            |> (PureThy.add_defs_i false o map Thm.no_attributes) [def];
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wenzelm@8436
   326
        in (defs @ [def_thm], thy')
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        end) (([], thy1), (hd descr) ~~ newTs ~~ case_names ~~
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          (Library.take (length newTs, reccomb_names)));
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   329
wenzelm@20046
   330
    val case_thms = map (map (fn t => Goal.prove_global thy2 [] [] t
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   331
      (fn _ => EVERY [rewrite_goals_tac (case_defs @ map mk_meta_eq primrec_thms), rtac refl 1])))
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          (DatatypeProp.make_cases new_type_names descr sorts thy2)
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   333
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   334
  in
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    thy2
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   336
    |> parent_path flat_names
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   337
    |> store_thmss "cases" new_type_names case_thms
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    |-> (fn thmss => pair (thmss, case_names))
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   339
  end;
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   340
berghofe@5177
   341
berghofe@5177
   342
(******************************* case splitting *******************************)
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   343
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   344
fun prove_split_thms new_type_names descr sorts constr_inject dist_rewrites
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    casedist_thms case_thms thy =
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  let
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    val _ = message "Proving equations for case splitting ...";
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   348
skalberg@15570
   349
    val descr' = List.concat descr;
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    val recTs = get_rec_types descr' sorts;
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    val newTs = Library.take (length (hd descr), recTs);
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   352
berghofe@5177
   353
    fun prove_split_thms ((((((t1, t2), inject), dist_rewrites'),
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   354
        exhaustion), case_thms'), T) =
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   355
      let
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   356
        val cert = cterm_of thy;
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   357
        val _ $ (_ $ lhs $ _) = hd (Logic.strip_assums_hyp (hd (prems_of exhaustion)));
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   358
        val exhaustion' = cterm_instantiate
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   359
          [(cert lhs, cert (Free ("x", T)))] exhaustion;
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   360
        val tacf = K (EVERY [rtac exhaustion' 1, ALLGOALS (asm_simp_tac
wenzelm@17985
   361
          (HOL_ss addsimps (dist_rewrites' @ inject @ case_thms')))])
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   362
      in
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   363
        (Goal.prove_global thy [] [] t1 tacf,
wenzelm@20046
   364
         Goal.prove_global thy [] [] t2 tacf)
berghofe@5177
   365
      end;
berghofe@5177
   366
berghofe@5177
   367
    val split_thm_pairs = map prove_split_thms
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   368
      ((DatatypeProp.make_splits new_type_names descr sorts thy) ~~ constr_inject ~~
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   369
        dist_rewrites ~~ casedist_thms ~~ case_thms ~~ newTs);
berghofe@5177
   370
berghofe@5177
   371
    val (split_thms, split_asm_thms) = ListPair.unzip split_thm_pairs
berghofe@5177
   372
berghofe@5177
   373
  in
haftmann@18314
   374
    thy
haftmann@18314
   375
    |> store_thms "split" new_type_names split_thms
haftmann@18314
   376
    ||>> store_thms "split_asm" new_type_names split_asm_thms
haftmann@18314
   377
    |-> (fn (thms1, thms2) => pair (thms1 ~~ thms2))
berghofe@5177
   378
  end;
berghofe@5177
   379
nipkow@8601
   380
fun prove_weak_case_congs new_type_names descr sorts thy =
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   381
  let
nipkow@8601
   382
    fun prove_weak_case_cong t =
wenzelm@20046
   383
       Goal.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
wenzelm@20046
   384
         (fn prems => EVERY [rtac ((hd prems) RS arg_cong) 1])
nipkow@8601
   385
nipkow@8601
   386
    val weak_case_congs = map prove_weak_case_cong (DatatypeProp.make_weak_case_congs
nipkow@8601
   387
      new_type_names descr sorts thy)
nipkow@8601
   388
nipkow@8601
   389
  in thy |> store_thms "weak_case_cong" new_type_names weak_case_congs end;
berghofe@8477
   390
berghofe@5177
   391
(************************* additional theorems for TFL ************************)
berghofe@5177
   392
berghofe@5177
   393
fun prove_nchotomys new_type_names descr sorts casedist_thms thy =
berghofe@5177
   394
  let
wenzelm@6427
   395
    val _ = message "Proving additional theorems for TFL ...";
berghofe@5177
   396
berghofe@5177
   397
    fun prove_nchotomy (t, exhaustion) =
berghofe@5177
   398
      let
berghofe@5177
   399
        (* For goal i, select the correct disjunct to attack, then prove it *)
berghofe@5177
   400
        fun tac i 0 = EVERY [TRY (rtac disjI1 i),
berghofe@5177
   401
              hyp_subst_tac i, REPEAT (rtac exI i), rtac refl i]
berghofe@5177
   402
          | tac i n = rtac disjI2 i THEN tac i (n - 1)
berghofe@5177
   403
      in 
wenzelm@20046
   404
        Goal.prove_global thy [] [] t (fn _ =>
wenzelm@17985
   405
          EVERY [rtac allI 1,
berghofe@5177
   406
           exh_tac (K exhaustion) 1,
wenzelm@20046
   407
           ALLGOALS (fn i => tac i (i-1))])
berghofe@5177
   408
      end;
berghofe@5177
   409
berghofe@5177
   410
    val nchotomys =
berghofe@5177
   411
      map prove_nchotomy (DatatypeProp.make_nchotomys descr sorts ~~ casedist_thms)
berghofe@5177
   412
wenzelm@8436
   413
  in thy |> store_thms "nchotomy" new_type_names nchotomys end;
berghofe@5177
   414
berghofe@5177
   415
fun prove_case_congs new_type_names descr sorts nchotomys case_thms thy =
berghofe@5177
   416
  let
berghofe@5177
   417
    fun prove_case_cong ((t, nchotomy), case_rewrites) =
berghofe@5177
   418
      let
berghofe@5177
   419
        val (Const ("==>", _) $ tm $ _) = t;
berghofe@5177
   420
        val (Const ("Trueprop", _) $ (Const ("op =", _) $ _ $ Ma)) = tm;
wenzelm@22578
   421
        val cert = cterm_of thy;
berghofe@5177
   422
        val nchotomy' = nchotomy RS spec;
berghofe@5177
   423
        val nchotomy'' = cterm_instantiate
berghofe@5177
   424
          [(cert (hd (add_term_vars (concl_of nchotomy', []))), cert Ma)] nchotomy'
berghofe@5177
   425
      in
wenzelm@20046
   426
        Goal.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
wenzelm@17985
   427
          (fn prems => 
wenzelm@17985
   428
            let val simplify = asm_simp_tac (HOL_ss addsimps (prems @ case_rewrites))
wenzelm@17985
   429
            in EVERY [simp_tac (HOL_ss addsimps [hd prems]) 1,
wenzelm@17985
   430
                cut_facts_tac [nchotomy''] 1,
wenzelm@17985
   431
                REPEAT (etac disjE 1 THEN REPEAT (etac exE 1) THEN simplify 1),
wenzelm@17985
   432
                REPEAT (etac exE 1) THEN simplify 1 (* Get last disjunct *)]
wenzelm@20046
   433
            end)
berghofe@5177
   434
      end;
berghofe@5177
   435
berghofe@5177
   436
    val case_congs = map prove_case_cong (DatatypeProp.make_case_congs
berghofe@5177
   437
      new_type_names descr sorts thy ~~ nchotomys ~~ case_thms)
berghofe@5177
   438
wenzelm@8436
   439
  in thy |> store_thms "case_cong" new_type_names case_congs end;
berghofe@5177
   440
berghofe@5177
   441
end;