src/HOL/Tools/inductive_package.ML
author wenzelm
Fri Mar 03 21:01:57 2000 +0100 (2000-03-03)
changeset 8336 fdf3ac335f77
parent 8316 74639e19eca0
child 8375 0544749a5e8f
permissions -rw-r--r--
mk_cases / inductive_cases: use InductMethod.con_elim_(solved_)tac;
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(*  Title:      HOL/Tools/inductive_package.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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                Stefan Berghofer,   TU Muenchen
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    Copyright   1994  University of Cambridge
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                1998  TU Muenchen     
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(Co)Inductive Definition module for HOL.
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Features:
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  * least or greatest fixedpoints
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  * user-specified product and sum constructions
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  * mutually recursive definitions
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  * definitions involving arbitrary monotone operators
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  * automatically proves introduction and elimination rules
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The recursive sets must *already* be declared as constants in the
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current theory!
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  Introduction rules have the form
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  [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk
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  where M is some monotone operator (usually the identity)
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  P(x) is any side condition on the free variables
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  ti, t are any terms
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  Sj, Sk are two of the sets being defined in mutual recursion
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Sums are used only for mutual recursion.  Products are used only to
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derive "streamlined" induction rules for relations.
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*)
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signature INDUCTIVE_PACKAGE =
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sig
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  val quiet_mode: bool ref
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  val unify_consts: Sign.sg -> term list -> term list -> term list * term list
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  val get_inductive: theory -> string ->
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    {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
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      induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
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  val print_inductives: theory -> unit
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  val mono_add_global: theory attribute
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  val mono_del_global: theory attribute
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  val get_monos: theory -> thm list
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  val add_inductive_i: bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
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    theory attribute list -> ((bstring * term) * theory attribute list) list ->
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      thm list -> thm list -> theory -> theory *
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      {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
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       intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
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  val add_inductive: bool -> bool -> string list -> Args.src list ->
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    ((bstring * string) * Args.src list) list -> (xstring * Args.src list) list ->
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      (xstring * Args.src list) list -> theory -> theory *
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      {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
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       intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
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  val inductive_cases: (((bstring * Args.src list) * xstring) * string list) * Comment.text
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    -> theory -> theory
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  val inductive_cases_i: (((bstring * theory attribute list) * string) * term list) * Comment.text
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    -> theory -> theory
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  val setup: (theory -> theory) list
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end;
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structure InductivePackage: INDUCTIVE_PACKAGE =
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struct
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(*** theory data ***)
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(* data kind 'HOL/inductive' *)
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type inductive_info =
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  {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
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    induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};
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structure InductiveArgs =
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struct
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  val name = "HOL/inductive";
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  type T = inductive_info Symtab.table * thm list;
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  val empty = (Symtab.empty, []);
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  val copy = I;
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  val prep_ext = I;
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  fun merge ((tab1, monos1), (tab2, monos2)) = (Symtab.merge (K true) (tab1, tab2),
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    Library.generic_merge Thm.eq_thm I I monos1 monos2);
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  fun print sg (tab, monos) =
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    (Pretty.writeln (Pretty.strs ("(co)inductives:" ::
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       map #1 (Sign.cond_extern_table sg Sign.constK tab)));
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     Pretty.writeln (Pretty.big_list "monotonicity rules:" (map Display.pretty_thm monos)));
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end;
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structure InductiveData = TheoryDataFun(InductiveArgs);
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val print_inductives = InductiveData.print;
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(* get and put data *)
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fun get_inductive thy name =
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  (case Symtab.lookup (fst (InductiveData.get thy), name) of
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    Some info => info
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  | None => error ("Unknown (co)inductive set " ^ quote name));
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fun put_inductives names info thy =
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  let
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    fun upd ((tab, monos), name) = (Symtab.update_new ((name, info), tab), monos);
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    val tab_monos = foldl upd (InductiveData.get thy, names)
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      handle Symtab.DUP name => error ("Duplicate definition of (co)inductive set " ^ quote name);
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  in InductiveData.put tab_monos thy end;
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(** monotonicity rules **)
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val get_monos = snd o InductiveData.get;
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fun put_monos thms thy = InductiveData.put (fst (InductiveData.get thy), thms) thy;
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fun mk_mono thm =
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  let
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    fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
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      (case concl_of thm of
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          (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
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        | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
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    val concl = concl_of thm
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  in
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    if Logic.is_equals concl then
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      eq2mono (thm RS meta_eq_to_obj_eq)
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    else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
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      eq2mono thm
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    else [thm]
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  end;
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(* mono add/del *)
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local
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fun map_rules_global f thy = put_monos (f (get_monos thy)) thy;
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fun add_mono thm rules = Library.gen_union Thm.eq_thm (mk_mono thm, rules);
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fun del_mono thm rules = Library.gen_rems Thm.eq_thm (rules, mk_mono thm);
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fun mk_att f g (x, thm) = (f (g thm) x, thm);
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in
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val mono_add_global = mk_att map_rules_global add_mono;
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val mono_del_global = mk_att map_rules_global del_mono;
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end;
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(* concrete syntax *)
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val monoN = "mono";
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val addN = "add";
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val delN = "del";
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fun mono_att add del =
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  Attrib.syntax (Scan.lift (Args.$$$ addN >> K add || Args.$$$ delN >> K del || Scan.succeed add));
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val mono_attr =
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  (mono_att mono_add_global mono_del_global, mono_att Attrib.undef_local_attribute Attrib.undef_local_attribute);
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(** utilities **)
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(* messages *)
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val quiet_mode = ref false;
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fun message s = if !quiet_mode then () else writeln s;
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fun coind_prefix true = "co"
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  | coind_prefix false = "";
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(* the following code ensures that each recursive set *)
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(* always has the same type in all introduction rules *)
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fun unify_consts sign cs intr_ts =
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  (let
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    val {tsig, ...} = Sign.rep_sg sign;
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    val add_term_consts_2 =
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      foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
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    fun varify (t, (i, ts)) =
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      let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
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      in (maxidx_of_term t', t'::ts) end;
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    val (i, cs') = foldr varify (cs, (~1, []));
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    val (i', intr_ts') = foldr varify (intr_ts, (i, []));
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    val rec_consts = foldl add_term_consts_2 ([], cs');
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    val intr_consts = foldl add_term_consts_2 ([], intr_ts');
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    fun unify (env, (cname, cT)) =
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      let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
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      in foldl (fn ((env', j'), Tp) => (Type.unify tsig j' env' Tp))
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          (env, (replicate (length consts) cT) ~~ consts)
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      end;
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    val (env, _) = foldl unify (([], i'), rec_consts);
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    fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars env T
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      in if T = T' then T else typ_subst_TVars_2 env T' end;
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    val subst = fst o Type.freeze_thaw o
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      (map_term_types (typ_subst_TVars_2 env))
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  in (map subst cs', map subst intr_ts')
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  end) handle Type.TUNIFY =>
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    (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));
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(* misc *)
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val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (concl_of vimageD);
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val vimage_name = Sign.intern_const (Theory.sign_of Vimage.thy) "op -``";
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val mono_name = Sign.intern_const (Theory.sign_of Ord.thy) "mono";
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(* make injections needed in mutually recursive definitions *)
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fun mk_inj cs sumT c x =
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  let
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    fun mk_inj' T n i =
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      if n = 1 then x else
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      let val n2 = n div 2;
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          val Type (_, [T1, T2]) = T
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      in
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        if i <= n2 then
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          Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
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        else
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          Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
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      end
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  in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
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  end;
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(* make "vimage" terms for selecting out components of mutually rec.def. *)
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fun mk_vimage cs sumT t c = if length cs < 2 then t else
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  let
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    val cT = HOLogic.dest_setT (fastype_of c);
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    val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
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  in
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    Const (vimage_name, vimageT) $
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      Abs ("y", cT, mk_inj cs sumT c (Bound 0)) $ t
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  end;
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(** well-formedness checks **)
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fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
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  (Sign.string_of_term sign t) ^ "\n" ^ msg);
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fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
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  (Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
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  (Sign.string_of_term sign t) ^ "\n" ^ msg);
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val msg1 = "Conclusion of introduction rule must have form\
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          \ ' t : S_i '";
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val msg2 = "Non-atomic premise";
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val msg3 = "Recursion term on left of member symbol";
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fun check_rule sign cs r =
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  let
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    fun check_prem prem = if can HOLogic.dest_Trueprop prem then ()
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      else err_in_prem sign r prem msg2;
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  in (case HOLogic.dest_Trueprop (Logic.strip_imp_concl r) of
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        (Const ("op :", _) $ t $ u) =>
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          if u mem cs then
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            if exists (Logic.occs o (rpair t)) cs then
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              err_in_rule sign r msg3
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            else
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              seq check_prem (Logic.strip_imp_prems r)
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          else err_in_rule sign r msg1
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      | _ => err_in_rule sign r msg1)
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  end;
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fun try' f msg sign t = (case (try f t) of
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      Some x => x
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    | None => error (msg ^ Sign.string_of_term sign t));
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(*** properties of (co)inductive sets ***)
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(** elimination rules **)
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fun mk_elims cs cTs params intr_ts =
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  let
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    val used = foldr add_term_names (intr_ts, []);
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    val [aname, pname] = variantlist (["a", "P"], used);
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    val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
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    fun dest_intr r =
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      let val Const ("op :", _) $ t $ u =
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        HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
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      in (u, t, Logic.strip_imp_prems r) end;
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    val intrs = map dest_intr intr_ts;
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    fun mk_elim (c, T) =
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      let
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        val a = Free (aname, T);
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        fun mk_elim_prem (_, t, ts) =
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          list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
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            Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
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      in
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        Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
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          map mk_elim_prem (filter (equal c o #1) intrs), P)
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      end
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  in
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    map mk_elim (cs ~~ cTs)
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  end;
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(** premises and conclusions of induction rules **)
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fun mk_indrule cs cTs params intr_ts =
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  let
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    val used = foldr add_term_names (intr_ts, []);
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    (* predicates for induction rule *)
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    val preds = map Free (variantlist (if length cs < 2 then ["P"] else
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      map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
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        map (fn T => T --> HOLogic.boolT) cTs);
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    (* transform an introduction rule into a premise for induction rule *)
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    fun mk_ind_prem r =
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      let
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        val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
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        val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
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        fun subst (s as ((m as Const ("op :", T)) $ t $ u)) =
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              (case pred_of u of
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                  None => (m $ fst (subst t) $ fst (subst u), None)
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                | Some P => (HOLogic.conj $ s $ (P $ t), Some (s, P $ t)))
berghofe@7710
   333
          | subst s =
berghofe@7710
   334
              (case pred_of s of
berghofe@7710
   335
                  Some P => (HOLogic.mk_binop "op Int"
berghofe@7710
   336
                    (s, HOLogic.Collect_const (HOLogic.dest_setT
berghofe@7710
   337
                      (fastype_of s)) $ P), None)
berghofe@7710
   338
                | None => (case s of
berghofe@7710
   339
                     (t $ u) => (fst (subst t) $ fst (subst u), None)
berghofe@7710
   340
                   | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
berghofe@7710
   341
                   | _ => (s, None)));
berghofe@7710
   342
berghofe@7710
   343
        fun mk_prem (s, prems) = (case subst s of
berghofe@7710
   344
              (_, Some (t, u)) => t :: u :: prems
berghofe@7710
   345
            | (t, _) => t :: prems);
berghofe@7710
   346
          
berghofe@5094
   347
        val Const ("op :", _) $ t $ u =
berghofe@5094
   348
          HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
berghofe@5094
   349
berghofe@5094
   350
      in list_all_free (frees,
berghofe@7710
   351
           Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
berghofe@5094
   352
             (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
berghofe@7710
   353
               HOLogic.mk_Trueprop (the (pred_of u) $ t)))
berghofe@5094
   354
      end;
berghofe@5094
   355
berghofe@5094
   356
    val ind_prems = map mk_ind_prem intr_ts;
berghofe@5094
   357
berghofe@5094
   358
    (* make conclusions for induction rules *)
berghofe@5094
   359
berghofe@5094
   360
    fun mk_ind_concl ((c, P), (ts, x)) =
berghofe@5094
   361
      let val T = HOLogic.dest_setT (fastype_of c);
berghofe@5094
   362
          val Ts = HOLogic.prodT_factors T;
berghofe@5094
   363
          val (frees, x') = foldr (fn (T', (fs, s)) =>
berghofe@5094
   364
            ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
berghofe@5094
   365
          val tuple = HOLogic.mk_tuple T frees;
berghofe@5094
   366
      in ((HOLogic.mk_binop "op -->"
berghofe@5094
   367
        (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x')
berghofe@5094
   368
      end;
berghofe@5094
   369
berghofe@7710
   370
    val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
berghofe@5094
   371
        (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
berghofe@5094
   372
berghofe@5094
   373
  in (preds, ind_prems, mutual_ind_concl)
berghofe@5094
   374
  end;
berghofe@5094
   375
wenzelm@6424
   376
berghofe@5094
   377
wenzelm@8316
   378
(** prepare cases and induct rules **)
wenzelm@8316
   379
wenzelm@8316
   380
(*
wenzelm@8316
   381
  transform mutual rule:
wenzelm@8316
   382
    HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
wenzelm@8316
   383
  into i-th projection:
wenzelm@8316
   384
    xi:Ai ==> HH ==> Pi xi
wenzelm@8316
   385
*)
wenzelm@8316
   386
wenzelm@8316
   387
fun project_rules [name] rule = [(name, rule)]
wenzelm@8316
   388
  | project_rules names mutual_rule =
wenzelm@8316
   389
      let
wenzelm@8316
   390
        val n = length names;
wenzelm@8316
   391
        fun proj i =
wenzelm@8316
   392
          (if i < n then (fn th => th RS conjunct1) else I)
wenzelm@8316
   393
            (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
wenzelm@8316
   394
            RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
wenzelm@8316
   395
      in names ~~ map proj (1 upto n) end;
wenzelm@8316
   396
wenzelm@8316
   397
fun add_cases_induct no_elim no_ind names elims induct =
wenzelm@8316
   398
  let
wenzelm@8316
   399
    val cases_specs =
wenzelm@8316
   400
      if no_elim then []
wenzelm@8316
   401
      else map2 (fn (name, elim) => (("", elim), [InductMethod.cases_set_global name]))
wenzelm@8316
   402
        (names, elims);
wenzelm@8316
   403
wenzelm@8316
   404
    val induct_specs =
wenzelm@8316
   405
      if no_ind then []
wenzelm@8316
   406
      else map (fn (name, th) => (("", th), [InductMethod.induct_set_global name]))
wenzelm@8316
   407
        (project_rules names induct);
wenzelm@8316
   408
  in PureThy.add_thms (cases_specs @ induct_specs) end;
wenzelm@8316
   409
wenzelm@8316
   410
wenzelm@8316
   411
wenzelm@6424
   412
(*** proofs for (co)inductive sets ***)
wenzelm@6424
   413
wenzelm@6424
   414
(** prove monotonicity **)
berghofe@5094
   415
berghofe@5094
   416
fun prove_mono setT fp_fun monos thy =
berghofe@5094
   417
  let
wenzelm@6427
   418
    val _ = message "  Proving monotonicity ...";
berghofe@5094
   419
wenzelm@6394
   420
    val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
berghofe@5094
   421
      (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
berghofe@7710
   422
        (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
berghofe@5094
   423
berghofe@5094
   424
  in mono end;
berghofe@5094
   425
wenzelm@6424
   426
wenzelm@6424
   427
wenzelm@6424
   428
(** prove introduction rules **)
berghofe@5094
   429
berghofe@5094
   430
fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
berghofe@5094
   431
  let
wenzelm@6427
   432
    val _ = message "  Proving the introduction rules ...";
berghofe@5094
   433
berghofe@5094
   434
    val unfold = standard (mono RS (fp_def RS
berghofe@5094
   435
      (if coind then def_gfp_Tarski else def_lfp_Tarski)));
berghofe@5094
   436
berghofe@5094
   437
    fun select_disj 1 1 = []
berghofe@5094
   438
      | select_disj _ 1 = [rtac disjI1]
berghofe@5094
   439
      | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
berghofe@5094
   440
berghofe@5094
   441
    val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
wenzelm@6394
   442
      (cterm_of (Theory.sign_of thy) intr) (fn prems =>
berghofe@5094
   443
       [(*insert prems and underlying sets*)
berghofe@5094
   444
       cut_facts_tac prems 1,
berghofe@5094
   445
       stac unfold 1,
berghofe@5094
   446
       REPEAT (resolve_tac [vimageI2, CollectI] 1),
berghofe@5094
   447
       (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
berghofe@5094
   448
       EVERY1 (select_disj (length intr_ts) i),
berghofe@5094
   449
       (*Not ares_tac, since refl must be tried before any equality assumptions;
berghofe@5094
   450
         backtracking may occur if the premises have extra variables!*)
berghofe@5094
   451
       DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
berghofe@5094
   452
       (*Now solve the equations like Inl 0 = Inl ?b2*)
berghofe@5094
   453
       rewrite_goals_tac con_defs,
berghofe@5094
   454
       REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
berghofe@5094
   455
berghofe@5094
   456
  in (intrs, unfold) end;
berghofe@5094
   457
wenzelm@6424
   458
wenzelm@6424
   459
wenzelm@6424
   460
(** prove elimination rules **)
berghofe@5094
   461
berghofe@5094
   462
fun prove_elims cs cTs params intr_ts unfold rec_sets_defs thy =
berghofe@5094
   463
  let
wenzelm@6427
   464
    val _ = message "  Proving the elimination rules ...";
berghofe@5094
   465
berghofe@7710
   466
    val rules1 = [CollectE, disjE, make_elim vimageD, exE];
berghofe@7710
   467
    val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
berghofe@5094
   468
      map make_elim [Inl_inject, Inr_inject];
berghofe@5094
   469
berghofe@5094
   470
    val elims = map (fn t => prove_goalw_cterm rec_sets_defs
wenzelm@6394
   471
      (cterm_of (Theory.sign_of thy) t) (fn prems =>
berghofe@5094
   472
        [cut_facts_tac [hd prems] 1,
berghofe@5094
   473
         dtac (unfold RS subst) 1,
berghofe@5094
   474
         REPEAT (FIRSTGOAL (eresolve_tac rules1)),
berghofe@5094
   475
         REPEAT (FIRSTGOAL (eresolve_tac rules2)),
berghofe@5094
   476
         EVERY (map (fn prem =>
berghofe@5149
   477
           DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))]))
berghofe@5094
   478
      (mk_elims cs cTs params intr_ts)
berghofe@5094
   479
berghofe@5094
   480
  in elims end;
berghofe@5094
   481
wenzelm@6424
   482
berghofe@5094
   483
(** derivation of simplified elimination rules **)
berghofe@5094
   484
berghofe@5094
   485
(*Applies freeness of the given constructors, which *must* be unfolded by
berghofe@5094
   486
  the given defs.  Cannot simply use the local con_defs because con_defs=[] 
berghofe@5094
   487
  for inference systems.
berghofe@5094
   488
 *)
berghofe@5094
   489
wenzelm@7107
   490
(*cprop should have the form t:Si where Si is an inductive set*)
wenzelm@8336
   491
fun mk_cases_i solved elims ss cprop =
wenzelm@7107
   492
  let
wenzelm@7107
   493
    val prem = Thm.assume cprop;
wenzelm@8336
   494
    val tac = if solved then InductMethod.con_elim_solved_tac else InductMethod.con_elim_tac;
wenzelm@8336
   495
    fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic (tac ss) (prem RS rl));
wenzelm@7107
   496
  in
wenzelm@7107
   497
    (case get_first (try mk_elim) elims of
wenzelm@7107
   498
      Some r => r
wenzelm@7107
   499
    | None => error (Pretty.string_of (Pretty.block
wenzelm@7107
   500
        [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
wenzelm@7107
   501
          Display.pretty_cterm cprop])))
wenzelm@7107
   502
  end;
wenzelm@7107
   503
paulson@6141
   504
fun mk_cases elims s =
wenzelm@8336
   505
  mk_cases_i false elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
wenzelm@7107
   506
wenzelm@7107
   507
wenzelm@7107
   508
(* inductive_cases(_i) *)
wenzelm@7107
   509
wenzelm@7107
   510
fun gen_inductive_cases prep_att prep_const prep_prop
wenzelm@7107
   511
    ((((name, raw_atts), raw_set), raw_props), comment) thy =
wenzelm@7107
   512
  let
wenzelm@7107
   513
    val sign = Theory.sign_of thy;
wenzelm@7107
   514
wenzelm@7107
   515
    val atts = map (prep_att thy) raw_atts;
wenzelm@7107
   516
    val (_, {elims, ...}) = get_inductive thy (prep_const sign raw_set);
wenzelm@7107
   517
    val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
wenzelm@8336
   518
    val thms = map (mk_cases_i true elims (Simplifier.simpset_of thy)) cprops;
wenzelm@7107
   519
  in
wenzelm@7107
   520
    thy
wenzelm@7107
   521
    |> IsarThy.have_theorems_i (((name, atts), map Thm.no_attributes thms), comment)
berghofe@5094
   522
  end;
berghofe@5094
   523
wenzelm@7107
   524
val inductive_cases =
wenzelm@7107
   525
  gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
wenzelm@7107
   526
wenzelm@7107
   527
val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
wenzelm@7107
   528
wenzelm@6424
   529
wenzelm@6424
   530
wenzelm@6424
   531
(** prove induction rule **)
berghofe@5094
   532
berghofe@5094
   533
fun prove_indrule cs cTs sumT rec_const params intr_ts mono
berghofe@5094
   534
    fp_def rec_sets_defs thy =
berghofe@5094
   535
  let
wenzelm@6427
   536
    val _ = message "  Proving the induction rule ...";
berghofe@5094
   537
wenzelm@6394
   538
    val sign = Theory.sign_of thy;
berghofe@5094
   539
berghofe@7293
   540
    val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
berghofe@7293
   541
        None => []
berghofe@7293
   542
      | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
berghofe@7293
   543
berghofe@5094
   544
    val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
berghofe@5094
   545
berghofe@5094
   546
    (* make predicate for instantiation of abstract induction rule *)
berghofe@5094
   547
berghofe@5094
   548
    fun mk_ind_pred _ [P] = P
berghofe@5094
   549
      | mk_ind_pred T Ps =
berghofe@5094
   550
         let val n = (length Ps) div 2;
berghofe@5094
   551
             val Type (_, [T1, T2]) = T
berghofe@7293
   552
         in Const ("Datatype.sum.sum_case",
berghofe@5094
   553
           [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
berghofe@5094
   554
             mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (drop (n, Ps))
berghofe@5094
   555
         end;
berghofe@5094
   556
berghofe@5094
   557
    val ind_pred = mk_ind_pred sumT preds;
berghofe@5094
   558
berghofe@5094
   559
    val ind_concl = HOLogic.mk_Trueprop
berghofe@5094
   560
      (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
berghofe@5094
   561
        (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
berghofe@5094
   562
berghofe@5094
   563
    (* simplification rules for vimage and Collect *)
berghofe@5094
   564
berghofe@5094
   565
    val vimage_simps = if length cs < 2 then [] else
berghofe@5094
   566
      map (fn c => prove_goalw_cterm [] (cterm_of sign
berghofe@5094
   567
        (HOLogic.mk_Trueprop (HOLogic.mk_eq
berghofe@5094
   568
          (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
berghofe@5094
   569
           HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
berghofe@5094
   570
             nth_elem (find_index_eq c cs, preds)))))
berghofe@7293
   571
        (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
berghofe@5094
   572
          rtac refl 1])) cs;
berghofe@5094
   573
berghofe@5094
   574
    val induct = prove_goalw_cterm [] (cterm_of sign
berghofe@5094
   575
      (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
berghofe@5094
   576
        [rtac (impI RS allI) 1,
berghofe@5094
   577
         DETERM (etac (mono RS (fp_def RS def_induct)) 1),
berghofe@7710
   578
         rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
berghofe@5094
   579
         fold_goals_tac rec_sets_defs,
berghofe@5094
   580
         (*This CollectE and disjE separates out the introduction rules*)
berghofe@7710
   581
         REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
berghofe@5094
   582
         (*Now break down the individual cases.  No disjE here in case
berghofe@5094
   583
           some premise involves disjunction.*)
berghofe@7710
   584
         REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
berghofe@7293
   585
         rewrite_goals_tac sum_case_rewrites,
berghofe@5094
   586
         EVERY (map (fn prem =>
berghofe@5149
   587
           DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
berghofe@5094
   588
berghofe@5094
   589
    val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
berghofe@5094
   590
      (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
berghofe@5094
   591
        [cut_facts_tac prems 1,
berghofe@5094
   592
         REPEAT (EVERY
berghofe@5094
   593
           [REPEAT (resolve_tac [conjI, impI] 1),
berghofe@5094
   594
            TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
berghofe@7293
   595
            rewrite_goals_tac sum_case_rewrites,
berghofe@5094
   596
            atac 1])])
berghofe@5094
   597
berghofe@5094
   598
  in standard (split_rule (induct RS lemma))
berghofe@5094
   599
  end;
berghofe@5094
   600
wenzelm@6424
   601
wenzelm@6424
   602
wenzelm@6424
   603
(*** specification of (co)inductive sets ****)
wenzelm@6424
   604
wenzelm@6424
   605
(** definitional introduction of (co)inductive sets **)
berghofe@5094
   606
berghofe@5094
   607
fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
wenzelm@6521
   608
    atts intros monos con_defs thy params paramTs cTs cnames =
berghofe@5094
   609
  let
wenzelm@6424
   610
    val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
wenzelm@6424
   611
      commas_quote cnames) else ();
berghofe@5094
   612
berghofe@5094
   613
    val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
berghofe@5094
   614
    val setT = HOLogic.mk_setT sumT;
berghofe@5094
   615
wenzelm@6394
   616
    val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
wenzelm@6394
   617
      else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
berghofe@5094
   618
wenzelm@6424
   619
    val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
wenzelm@6424
   620
berghofe@5149
   621
    val used = foldr add_term_names (intr_ts, []);
berghofe@5149
   622
    val [sname, xname] = variantlist (["S", "x"], used);
berghofe@5149
   623
berghofe@5094
   624
    (* transform an introduction rule into a conjunction  *)
berghofe@5094
   625
    (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
berghofe@5094
   626
    (* is transformed into                                *)
berghofe@5094
   627
    (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
berghofe@5094
   628
berghofe@5094
   629
    fun transform_rule r =
berghofe@5094
   630
      let
berghofe@5094
   631
        val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
berghofe@5149
   632
        val subst = subst_free
berghofe@5149
   633
          (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
berghofe@5094
   634
        val Const ("op :", _) $ t $ u =
berghofe@5094
   635
          HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
berghofe@5094
   636
berghofe@5094
   637
      in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
berghofe@7710
   638
        (frees, foldr1 HOLogic.mk_conj
berghofe@5149
   639
          (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
berghofe@5094
   640
            (map (subst o HOLogic.dest_Trueprop)
berghofe@5094
   641
              (Logic.strip_imp_prems r))))
berghofe@5094
   642
      end
berghofe@5094
   643
berghofe@5094
   644
    (* make a disjunction of all introduction rules *)
berghofe@5094
   645
berghofe@5149
   646
    val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
berghofe@7710
   647
      absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
berghofe@5094
   648
berghofe@5094
   649
    (* add definiton of recursive sets to theory *)
berghofe@5094
   650
berghofe@5094
   651
    val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
wenzelm@6394
   652
    val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
berghofe@5094
   653
berghofe@5094
   654
    val rec_const = list_comb
berghofe@5094
   655
      (Const (full_rec_name, paramTs ---> setT), params);
berghofe@5094
   656
berghofe@5094
   657
    val fp_def_term = Logic.mk_equals (rec_const,
berghofe@5094
   658
      Const (fp_name, (setT --> setT) --> setT) $ fp_fun)
berghofe@5094
   659
berghofe@5094
   660
    val def_terms = fp_def_term :: (if length cs < 2 then [] else
berghofe@5094
   661
      map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
berghofe@5094
   662
berghofe@5094
   663
    val thy' = thy |>
berghofe@5094
   664
      (if declare_consts then
berghofe@5094
   665
        Theory.add_consts_i (map (fn (c, n) =>
berghofe@5094
   666
          (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
berghofe@5094
   667
       else I) |>
berghofe@5094
   668
      (if length cs < 2 then I else
berghofe@5094
   669
       Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |>
berghofe@5094
   670
      Theory.add_path rec_name |>
berghofe@5094
   671
      PureThy.add_defss_i [(("defs", def_terms), [])];
berghofe@5094
   672
berghofe@5094
   673
    (* get definitions from theory *)
berghofe@5094
   674
wenzelm@6424
   675
    val fp_def::rec_sets_defs = PureThy.get_thms thy' "defs";
berghofe@5094
   676
berghofe@5094
   677
    (* prove and store theorems *)
berghofe@5094
   678
berghofe@5094
   679
    val mono = prove_mono setT fp_fun monos thy';
berghofe@5094
   680
    val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
berghofe@5094
   681
      rec_sets_defs thy';
berghofe@5094
   682
    val elims = if no_elim then [] else
berghofe@5094
   683
      prove_elims cs cTs params intr_ts unfold rec_sets_defs thy';
wenzelm@8312
   684
    val raw_induct = if no_ind then Drule.asm_rl else
berghofe@5094
   685
      if coind then standard (rule_by_tactic
oheimb@5553
   686
        (rewrite_tac [mk_meta_eq vimage_Un] THEN
berghofe@5094
   687
          fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
berghofe@5094
   688
      else
berghofe@5094
   689
        prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
berghofe@5094
   690
          rec_sets_defs thy';
berghofe@5108
   691
    val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
berghofe@5094
   692
      else standard (raw_induct RSN (2, rev_mp));
berghofe@5094
   693
wenzelm@6424
   694
    val thy'' = thy'
wenzelm@6521
   695
      |> PureThy.add_thmss [(("intrs", intrs), atts)]
wenzelm@6424
   696
      |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
wenzelm@6424
   697
      |> (if no_elim then I else PureThy.add_thmss [(("elims", elims), [])])
wenzelm@6424
   698
      |> (if no_ind then I else PureThy.add_thms
wenzelm@6424
   699
        [((coind_prefix coind ^ "induct", induct), [])])
wenzelm@6424
   700
      |> Theory.parent_path;
wenzelm@7798
   701
    val intrs' = PureThy.get_thms thy'' "intrs";
wenzelm@8312
   702
    val elims' = if no_elim then elims else PureThy.get_thms thy'' "elims";  (* FIXME improve *)
wenzelm@8312
   703
    val induct' = if no_ind then induct else PureThy.get_thm thy'' (coind_prefix coind ^ "induct");  (* FIXME improve *)
berghofe@5094
   704
  in (thy'',
berghofe@5094
   705
    {defs = fp_def::rec_sets_defs,
berghofe@5094
   706
     mono = mono,
berghofe@5094
   707
     unfold = unfold,
wenzelm@7798
   708
     intrs = intrs',
wenzelm@7798
   709
     elims = elims',
wenzelm@7798
   710
     mk_cases = mk_cases elims',
berghofe@5094
   711
     raw_induct = raw_induct,
wenzelm@7798
   712
     induct = induct'})
berghofe@5094
   713
  end;
berghofe@5094
   714
wenzelm@6424
   715
wenzelm@6424
   716
wenzelm@6424
   717
(** axiomatic introduction of (co)inductive sets **)
berghofe@5094
   718
berghofe@5094
   719
fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
wenzelm@6521
   720
    atts intros monos con_defs thy params paramTs cTs cnames =
berghofe@5094
   721
  let
berghofe@5094
   722
    val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
berghofe@5094
   723
wenzelm@6424
   724
    val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
berghofe@5094
   725
    val elim_ts = mk_elims cs cTs params intr_ts;
berghofe@5094
   726
berghofe@5094
   727
    val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
berghofe@5094
   728
    val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
berghofe@5094
   729
    
wenzelm@6424
   730
    val thy' = thy
wenzelm@6424
   731
      |> (if declare_consts then
wenzelm@6424
   732
            Theory.add_consts_i
wenzelm@6424
   733
              (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
wenzelm@6424
   734
         else I)
wenzelm@6424
   735
      |> Theory.add_path rec_name
wenzelm@6521
   736
      |> PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("elims", elim_ts), [])]
berghofe@7710
   737
      |> (if coind then I else
berghofe@7710
   738
            PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
berghofe@5094
   739
wenzelm@6424
   740
    val intrs = PureThy.get_thms thy' "intrs";
wenzelm@6424
   741
    val elims = PureThy.get_thms thy' "elims";
wenzelm@8312
   742
    val raw_induct = if coind then Drule.asm_rl else PureThy.get_thm thy' "raw_induct";
berghofe@5094
   743
    val induct = if coind orelse length cs > 1 then raw_induct
berghofe@5094
   744
      else standard (raw_induct RSN (2, rev_mp));
berghofe@5094
   745
wenzelm@6424
   746
    val thy'' =
wenzelm@6424
   747
      thy'
wenzelm@6424
   748
      |> (if coind then I else PureThy.add_thms [(("induct", induct), [])])
wenzelm@6424
   749
      |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
wenzelm@6424
   750
      |> Theory.parent_path;
wenzelm@7798
   751
    val induct' = if coind then raw_induct else PureThy.get_thm thy'' "induct";
berghofe@5094
   752
  in (thy'',
berghofe@5094
   753
    {defs = [],
wenzelm@8312
   754
     mono = Drule.asm_rl,
wenzelm@8312
   755
     unfold = Drule.asm_rl,
berghofe@5094
   756
     intrs = intrs,
berghofe@5094
   757
     elims = elims,
berghofe@5094
   758
     mk_cases = mk_cases elims,
berghofe@5094
   759
     raw_induct = raw_induct,
wenzelm@7798
   760
     induct = induct'})
berghofe@5094
   761
  end;
berghofe@5094
   762
wenzelm@6424
   763
wenzelm@6424
   764
wenzelm@6424
   765
(** introduction of (co)inductive sets **)
berghofe@5094
   766
berghofe@5094
   767
fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
wenzelm@6521
   768
    atts intros monos con_defs thy =
berghofe@5094
   769
  let
wenzelm@6424
   770
    val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
wenzelm@6394
   771
    val sign = Theory.sign_of thy;
berghofe@5094
   772
berghofe@5094
   773
    (*parameters should agree for all mutually recursive components*)
berghofe@5094
   774
    val (_, params) = strip_comb (hd cs);
berghofe@5094
   775
    val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
berghofe@5094
   776
      \ component is not a free variable: " sign) params;
berghofe@5094
   777
berghofe@5094
   778
    val cTs = map (try' (HOLogic.dest_setT o fastype_of)
berghofe@5094
   779
      "Recursive component not of type set: " sign) cs;
berghofe@5094
   780
wenzelm@6437
   781
    val full_cnames = map (try' (fst o dest_Const o head_of)
berghofe@5094
   782
      "Recursive set not previously declared as constant: " sign) cs;
wenzelm@6437
   783
    val cnames = map Sign.base_name full_cnames;
berghofe@5094
   784
wenzelm@6424
   785
    val _ = seq (check_rule sign cs o snd o fst) intros;
wenzelm@6437
   786
wenzelm@6437
   787
    val (thy1, result) =
wenzelm@6437
   788
      (if ! quick_and_dirty then add_ind_axm else add_ind_def)
wenzelm@6521
   789
        verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
wenzelm@6437
   790
        con_defs thy params paramTs cTs cnames;
wenzelm@8307
   791
    val thy2 = thy1
wenzelm@8307
   792
      |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
wenzelm@8312
   793
      |> add_cases_induct no_elim no_ind full_cnames (#elims result) (#induct result);
wenzelm@6437
   794
  in (thy2, result) end;
berghofe@5094
   795
wenzelm@6424
   796
berghofe@5094
   797
wenzelm@6424
   798
(** external interface **)
wenzelm@6424
   799
wenzelm@6521
   800
fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
berghofe@5094
   801
  let
wenzelm@6394
   802
    val sign = Theory.sign_of thy;
wenzelm@8100
   803
    val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
wenzelm@6424
   804
wenzelm@6521
   805
    val atts = map (Attrib.global_attribute thy) srcs;
wenzelm@6424
   806
    val intr_names = map (fst o fst) intro_srcs;
berghofe@7710
   807
    val intr_ts = map (term_of o Thm.read_cterm sign o rpair propT o snd o fst) intro_srcs;
wenzelm@6424
   808
    val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
berghofe@7020
   809
    val (cs', intr_ts') = unify_consts sign cs intr_ts;
berghofe@5094
   810
wenzelm@6424
   811
    val ((thy', con_defs), monos) = thy
wenzelm@6424
   812
      |> IsarThy.apply_theorems raw_monos
wenzelm@6424
   813
      |> apfst (IsarThy.apply_theorems raw_con_defs);
wenzelm@6424
   814
  in
berghofe@7020
   815
    add_inductive_i verbose false "" coind false false cs'
berghofe@7020
   816
      atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
berghofe@5094
   817
  end;
berghofe@5094
   818
wenzelm@6424
   819
wenzelm@6424
   820
wenzelm@6437
   821
(** package setup **)
wenzelm@6437
   822
wenzelm@6437
   823
(* setup theory *)
wenzelm@6437
   824
berghofe@7710
   825
val setup = [InductiveData.init,
berghofe@7710
   826
             Attrib.add_attributes [(monoN, mono_attr, "monotonicity rule")]];
wenzelm@6437
   827
wenzelm@6437
   828
wenzelm@6437
   829
(* outer syntax *)
wenzelm@6424
   830
wenzelm@6723
   831
local structure P = OuterParse and K = OuterSyntax.Keyword in
wenzelm@6424
   832
wenzelm@6521
   833
fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
wenzelm@6723
   834
  #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
wenzelm@6424
   835
wenzelm@6424
   836
fun ind_decl coind =
wenzelm@6729
   837
  (Scan.repeat1 P.term --| P.marg_comment) --
wenzelm@6729
   838
  (P.$$$ "intrs" |--
wenzelm@7152
   839
    P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
wenzelm@6729
   840
  Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
wenzelm@6729
   841
  Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
wenzelm@6424
   842
  >> (Toplevel.theory o mk_ind coind);
wenzelm@6424
   843
wenzelm@6723
   844
val inductiveP =
wenzelm@6723
   845
  OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
wenzelm@6723
   846
wenzelm@6723
   847
val coinductiveP =
wenzelm@6723
   848
  OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
wenzelm@6424
   849
wenzelm@7107
   850
wenzelm@7107
   851
val ind_cases =
wenzelm@7107
   852
  P.opt_thm_name "=" -- P.xname --| P.$$$ ":" -- Scan.repeat1 P.prop -- P.marg_comment
wenzelm@7107
   853
  >> (Toplevel.theory o inductive_cases);
wenzelm@7107
   854
wenzelm@7107
   855
val inductive_casesP =
wenzelm@7107
   856
  OuterSyntax.command "inductive_cases" "create simplified instances of elimination rules"
wenzelm@7107
   857
    K.thy_decl ind_cases;
wenzelm@7107
   858
wenzelm@6424
   859
val _ = OuterSyntax.add_keywords ["intrs", "monos", "con_defs"];
wenzelm@7107
   860
val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
wenzelm@6424
   861
berghofe@5094
   862
end;
wenzelm@6424
   863
wenzelm@6424
   864
wenzelm@6424
   865
end;