src/HOL/Tools/inductive_package.ML
author wenzelm
Sat Dec 23 22:51:34 2000 +0100 (2000-12-23)
changeset 10735 dfaf75f0076f
parent 10729 1b3350c4ee92
child 10743 8ea821d1e3a4
permissions -rw-r--r--
simplified quick_and_dirty stuff;
tuned;
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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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    Author:     Stefan Berghofer, TU Muenchen
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    Author:     Markus Wenzel, 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 -> ({names: string list, coind: bool} *
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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}) option
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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 inductive_cases: ((bstring * Args.src list) * string list) * Comment.text
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    -> theory -> theory
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  val inductive_cases_i: ((bstring * theory attribute list) * term list) * Comment.text
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    -> theory -> theory
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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 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 context references **)
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val mono_name = "Ord.mono";
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val gfp_name = "Gfp.gfp";
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val lfp_name = "Lfp.lfp";
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val vimage_name = "Inverse_Image.vimage";
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val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (Thm.concl_of vimageD);
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val inductive_conj_name = "Inductive.conj";
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val inductive_conj_def = thm "conj_def";
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val inductive_conj = thms "inductive_conj";
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val inductive_atomize = thms "inductive_atomize";
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val inductive_rulify1 = thms "inductive_rulify1";
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val inductive_rulify2 = thms "inductive_rulify2";
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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.strs ("(co)inductives:" :: map #1 (Sign.cond_extern_table sg Sign.constK tab)),
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     Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_sg sg) monos)]
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    |> Pretty.chunks |> Pretty.writeln;
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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 = Symtab.lookup (fst (InductiveData.get thy), name);
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fun the_inductive thy name =
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  (case get_inductive thy name of
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    None => error ("Unknown (co)inductive set " ^ quote name)
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  | Some info => info);
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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 = #2 o InductiveData.get;
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fun map_monos f = InductiveData.map (Library.apsnd f);
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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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(* attributes *)
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fun mono_add_global (thy, thm) = (map_monos (Drule.add_rules (mk_mono thm)) thy, thm);
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fun mono_del_global (thy, thm) = (map_monos (Drule.del_rules (mk_mono thm)) thy, thm);
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val mono_attr =
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 (Attrib.add_del_args mono_add_global mono_del_global,
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  Attrib.add_del_args Attrib.undef_local_attribute Attrib.undef_local_attribute);
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(** misc utilities **)
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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 clean_message s = if ! quick_and_dirty then () else message 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 always has the
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  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 ((Vartab.empty, i'), rec_consts);
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    fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars_Vartab 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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(*make injections used 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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(** process rules **)
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local
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fun err_in_rule sg name t msg =
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  error (cat_lines ["Ill-formed introduction rule " ^ quote name, Sign.string_of_term sg t, msg]);
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fun err_in_prem sg name t p msg =
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  error (cat_lines ["Ill-formed premise", Sign.string_of_term sg p,
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    "in introduction rule " ^ quote name, Sign.string_of_term sg t, msg]);
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val bad_concl = "Conclusion of introduction rule must have form \"t : S_i\"";
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val atomize_cterm = InductMethod.rewrite_cterm inductive_atomize;
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fun full_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
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in
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fun check_rule sg cs ((name, rule), att) =
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  let
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    val concl = Logic.strip_imp_concl rule;
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    val prems = Logic.strip_imp_prems rule;
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    val aprems = prems |> map (Thm.term_of o atomize_cterm o Thm.cterm_of sg);
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    val arule = Logic.list_implies (aprems, concl);
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    fun check_prem (prem, aprem) =
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      if can HOLogic.dest_Trueprop aprem then ()
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      else err_in_prem sg name rule prem "Non-atomic premise";
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  in
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    (case HOLogic.dest_Trueprop concl 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 sg name rule "Recursion term on left of member symbol"
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          else seq check_prem (prems ~~ aprems)
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        else err_in_rule sg name rule bad_concl
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      | _ => err_in_rule sg name rule bad_concl);
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    ((name, arule), att)
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  end;
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val rulify =
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  standard o full_simplify [Drule.norm_hhf_eq] o
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  full_simplify inductive_rulify2 o full_simplify inductive_rulify1 o
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  full_simplify inductive_conj;
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end;
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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 intr_names =
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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 ~~ intr_names;
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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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        val c_intrs = (filter (equal c o #1 o #1) intrs);
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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 (map #1 c_intrs), P), map #2 c_intrs)
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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 =
berghofe@5094
   326
      let
berghofe@5094
   327
        val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
berghofe@5094
   328
berghofe@7710
   329
        val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
berghofe@5094
   330
berghofe@7710
   331
        fun subst (s as ((m as Const ("op :", T)) $ t $ u)) =
berghofe@7710
   332
              (case pred_of u of
berghofe@7710
   333
                  None => (m $ fst (subst t) $ fst (subst u), None)
wenzelm@10735
   334
                | Some P => (HOLogic.mk_binop inductive_conj_name (s, P $ t), Some (s, P $ t)))
berghofe@7710
   335
          | subst s =
berghofe@7710
   336
              (case pred_of s of
berghofe@7710
   337
                  Some P => (HOLogic.mk_binop "op Int"
berghofe@7710
   338
                    (s, HOLogic.Collect_const (HOLogic.dest_setT
berghofe@7710
   339
                      (fastype_of s)) $ P), None)
berghofe@7710
   340
                | None => (case s of
berghofe@7710
   341
                     (t $ u) => (fst (subst t) $ fst (subst u), None)
berghofe@7710
   342
                   | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
berghofe@7710
   343
                   | _ => (s, None)));
berghofe@7710
   344
berghofe@7710
   345
        fun mk_prem (s, prems) = (case subst s of
berghofe@7710
   346
              (_, Some (t, u)) => t :: u :: prems
berghofe@7710
   347
            | (t, _) => t :: prems);
wenzelm@9598
   348
berghofe@5094
   349
        val Const ("op :", _) $ t $ u =
berghofe@5094
   350
          HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
berghofe@5094
   351
berghofe@5094
   352
      in list_all_free (frees,
berghofe@7710
   353
           Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
berghofe@5094
   354
             (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
berghofe@7710
   355
               HOLogic.mk_Trueprop (the (pred_of u) $ t)))
berghofe@5094
   356
      end;
berghofe@5094
   357
berghofe@5094
   358
    val ind_prems = map mk_ind_prem intr_ts;
berghofe@5094
   359
berghofe@5094
   360
    (* make conclusions for induction rules *)
berghofe@5094
   361
berghofe@5094
   362
    fun mk_ind_concl ((c, P), (ts, x)) =
berghofe@5094
   363
      let val T = HOLogic.dest_setT (fastype_of c);
berghofe@5094
   364
          val Ts = HOLogic.prodT_factors T;
berghofe@5094
   365
          val (frees, x') = foldr (fn (T', (fs, s)) =>
berghofe@5094
   366
            ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
berghofe@5094
   367
          val tuple = HOLogic.mk_tuple T frees;
berghofe@5094
   368
      in ((HOLogic.mk_binop "op -->"
berghofe@5094
   369
        (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x')
berghofe@5094
   370
      end;
berghofe@5094
   371
berghofe@7710
   372
    val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
berghofe@5094
   373
        (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
berghofe@5094
   374
berghofe@5094
   375
  in (preds, ind_prems, mutual_ind_concl)
berghofe@5094
   376
  end;
berghofe@5094
   377
wenzelm@6424
   378
wenzelm@10735
   379
(* prepare cases and induct rules *)
wenzelm@8316
   380
wenzelm@8316
   381
(*
wenzelm@8316
   382
  transform mutual rule:
wenzelm@8316
   383
    HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
wenzelm@8316
   384
  into i-th projection:
wenzelm@8316
   385
    xi:Ai ==> HH ==> Pi xi
wenzelm@8316
   386
*)
wenzelm@8316
   387
wenzelm@8316
   388
fun project_rules [name] rule = [(name, rule)]
wenzelm@8316
   389
  | project_rules names mutual_rule =
wenzelm@8316
   390
      let
wenzelm@8316
   391
        val n = length names;
wenzelm@8316
   392
        fun proj i =
wenzelm@8316
   393
          (if i < n then (fn th => th RS conjunct1) else I)
wenzelm@8316
   394
            (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
wenzelm@8316
   395
            RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
wenzelm@8316
   396
      in names ~~ map proj (1 upto n) end;
wenzelm@8316
   397
wenzelm@8375
   398
fun add_cases_induct no_elim no_ind names elims induct induct_cases =
wenzelm@8316
   399
  let
wenzelm@9405
   400
    fun cases_spec (name, elim) thy =
wenzelm@9405
   401
      thy
wenzelm@9405
   402
      |> Theory.add_path (Sign.base_name name)
wenzelm@10279
   403
      |> (#1 o PureThy.add_thms [(("cases", elim), [InductAttrib.cases_set_global name])])
wenzelm@9405
   404
      |> Theory.parent_path;
wenzelm@8375
   405
    val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
wenzelm@8316
   406
wenzelm@9405
   407
    fun induct_spec (name, th) = (#1 o PureThy.add_thms
wenzelm@10279
   408
      [(("", th), [RuleCases.case_names induct_cases, InductAttrib.induct_set_global name])]);
wenzelm@8401
   409
    val induct_specs = if no_ind then [] else map induct_spec (project_rules names induct);
wenzelm@9405
   410
  in Library.apply (cases_specs @ induct_specs) end;
wenzelm@8316
   411
wenzelm@8316
   412
wenzelm@8316
   413
wenzelm@10735
   414
(** proofs for (co)inductive sets **)
wenzelm@6424
   415
wenzelm@10735
   416
(* prove monotonicity -- NOT subject to quick_and_dirty! *)
berghofe@5094
   417
berghofe@5094
   418
fun prove_mono setT fp_fun monos thy =
wenzelm@10735
   419
 (message "  Proving monotonicity ...";
wenzelm@10735
   420
  Goals.prove_goalw_cterm []      (*NO SkipProof.prove_goalw_cterm here!*)
wenzelm@10735
   421
    (Thm.cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
berghofe@5094
   422
      (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
wenzelm@10735
   423
    (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)]));
berghofe@5094
   424
wenzelm@6424
   425
wenzelm@10735
   426
(* prove introduction rules *)
berghofe@5094
   427
berghofe@5094
   428
fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
berghofe@5094
   429
  let
wenzelm@10735
   430
    val _ = clean_message "  Proving the introduction rules ...";
berghofe@5094
   431
berghofe@5094
   432
    val unfold = standard (mono RS (fp_def RS
nipkow@10186
   433
      (if coind then def_gfp_unfold else def_lfp_unfold)));
berghofe@5094
   434
berghofe@5094
   435
    fun select_disj 1 1 = []
berghofe@5094
   436
      | select_disj _ 1 = [rtac disjI1]
berghofe@5094
   437
      | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
berghofe@5094
   438
wenzelm@10735
   439
    val intrs = map (fn (i, intr) => SkipProof.prove_goalw_cterm thy rec_sets_defs
wenzelm@10735
   440
      (Thm.cterm_of (Theory.sign_of thy) intr) (fn prems =>
berghofe@5094
   441
       [(*insert prems and underlying sets*)
berghofe@5094
   442
       cut_facts_tac prems 1,
berghofe@5094
   443
       stac unfold 1,
berghofe@5094
   444
       REPEAT (resolve_tac [vimageI2, CollectI] 1),
berghofe@5094
   445
       (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
berghofe@5094
   446
       EVERY1 (select_disj (length intr_ts) i),
berghofe@5094
   447
       (*Not ares_tac, since refl must be tried before any equality assumptions;
berghofe@5094
   448
         backtracking may occur if the premises have extra variables!*)
wenzelm@10735
   449
       DEPTH_SOLVE_1 (resolve_tac [refl, exI, conjI] 1 APPEND assume_tac 1),
berghofe@5094
   450
       (*Now solve the equations like Inl 0 = Inl ?b2*)
berghofe@5094
   451
       rewrite_goals_tac con_defs,
wenzelm@10729
   452
       REPEAT (rtac refl 1)])
wenzelm@10729
   453
      |> rulify) (1 upto (length intr_ts) ~~ intr_ts)
berghofe@5094
   454
berghofe@5094
   455
  in (intrs, unfold) end;
berghofe@5094
   456
wenzelm@6424
   457
wenzelm@10735
   458
(* prove elimination rules *)
berghofe@5094
   459
wenzelm@8375
   460
fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
berghofe@5094
   461
  let
wenzelm@10735
   462
    val _ = clean_message "  Proving the elimination rules ...";
berghofe@5094
   463
berghofe@7710
   464
    val rules1 = [CollectE, disjE, make_elim vimageD, exE];
wenzelm@10735
   465
    val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @ map make_elim [Inl_inject, Inr_inject];
wenzelm@8375
   466
  in
wenzelm@10735
   467
    map (fn (t, cases) => SkipProof.prove_goalw_cterm thy rec_sets_defs
wenzelm@10735
   468
      (Thm.cterm_of (Theory.sign_of thy) t) (fn prems =>
berghofe@5094
   469
        [cut_facts_tac [hd prems] 1,
berghofe@5094
   470
         dtac (unfold RS subst) 1,
berghofe@5094
   471
         REPEAT (FIRSTGOAL (eresolve_tac rules1)),
berghofe@5094
   472
         REPEAT (FIRSTGOAL (eresolve_tac rules2)),
wenzelm@10735
   473
         EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
wenzelm@10729
   474
      |> rulify
wenzelm@8375
   475
      |> RuleCases.name cases)
wenzelm@8375
   476
      (mk_elims cs cTs params intr_ts intr_names)
wenzelm@8375
   477
  end;
berghofe@5094
   478
wenzelm@6424
   479
wenzelm@10735
   480
(* derivation of simplified elimination rules *)
berghofe@5094
   481
berghofe@5094
   482
(*Applies freeness of the given constructors, which *must* be unfolded by
wenzelm@9598
   483
  the given defs.  Cannot simply use the local con_defs because con_defs=[]
wenzelm@10735
   484
  for inference systems. (??) *)
berghofe@5094
   485
wenzelm@7107
   486
(*cprop should have the form t:Si where Si is an inductive set*)
wenzelm@9598
   487
wenzelm@10735
   488
val mk_cases_err = "mk_cases: proposition not of form \"t : S_i\"";
wenzelm@9598
   489
wenzelm@9598
   490
fun mk_cases_i elims ss cprop =
wenzelm@7107
   491
  let
wenzelm@7107
   492
    val prem = Thm.assume cprop;
wenzelm@9598
   493
    val tac = ALLGOALS (InductMethod.simp_case_tac false ss) THEN prune_params_tac;
wenzelm@9298
   494
    fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl));
wenzelm@7107
   495
  in
wenzelm@7107
   496
    (case get_first (try mk_elim) elims of
wenzelm@7107
   497
      Some r => r
wenzelm@7107
   498
    | None => error (Pretty.string_of (Pretty.block
wenzelm@9598
   499
        [Pretty.str mk_cases_err, Pretty.fbrk, Display.pretty_cterm cprop])))
wenzelm@7107
   500
  end;
wenzelm@7107
   501
paulson@6141
   502
fun mk_cases elims s =
wenzelm@9598
   503
  mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
wenzelm@9598
   504
wenzelm@9598
   505
fun smart_mk_cases thy ss cprop =
wenzelm@9598
   506
  let
wenzelm@9598
   507
    val c = #1 (Term.dest_Const (Term.head_of (#2 (HOLogic.dest_mem (HOLogic.dest_Trueprop
wenzelm@9598
   508
      (Logic.strip_imp_concl (Thm.term_of cprop))))))) handle TERM _ => error mk_cases_err;
wenzelm@9598
   509
    val (_, {elims, ...}) = the_inductive thy c;
wenzelm@9598
   510
  in mk_cases_i elims ss cprop end;
wenzelm@7107
   511
wenzelm@7107
   512
wenzelm@7107
   513
(* inductive_cases(_i) *)
wenzelm@7107
   514
wenzelm@7107
   515
fun gen_inductive_cases prep_att prep_const prep_prop
wenzelm@9598
   516
    (((name, raw_atts), raw_props), comment) thy =
wenzelm@9598
   517
  let
wenzelm@9598
   518
    val ss = Simplifier.simpset_of thy;
wenzelm@9598
   519
    val sign = Theory.sign_of thy;
wenzelm@9598
   520
    val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
wenzelm@9598
   521
    val atts = map (prep_att thy) raw_atts;
wenzelm@9598
   522
    val thms = map (smart_mk_cases thy ss) cprops;
wenzelm@9598
   523
  in thy |> IsarThy.have_theorems_i [(((name, atts), map Thm.no_attributes thms), comment)] end;
berghofe@5094
   524
wenzelm@7107
   525
val inductive_cases =
wenzelm@7107
   526
  gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
wenzelm@7107
   527
wenzelm@7107
   528
val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
wenzelm@7107
   529
wenzelm@6424
   530
wenzelm@9598
   531
(* mk_cases_meth *)
wenzelm@9598
   532
wenzelm@9598
   533
fun mk_cases_meth (ctxt, raw_props) =
wenzelm@9598
   534
  let
wenzelm@9598
   535
    val thy = ProofContext.theory_of ctxt;
wenzelm@9598
   536
    val ss = Simplifier.get_local_simpset ctxt;
wenzelm@9598
   537
    val cprops = map (Thm.cterm_of (Theory.sign_of thy) o ProofContext.read_prop ctxt) raw_props;
wenzelm@9598
   538
  in Method.erule (map (smart_mk_cases thy ss) cprops) end;
wenzelm@9598
   539
wenzelm@9598
   540
val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name));
wenzelm@9598
   541
wenzelm@9598
   542
wenzelm@10735
   543
(* prove induction rule *)
berghofe@5094
   544
berghofe@5094
   545
fun prove_indrule cs cTs sumT rec_const params intr_ts mono
berghofe@5094
   546
    fp_def rec_sets_defs thy =
berghofe@5094
   547
  let
wenzelm@10735
   548
    val _ = clean_message "  Proving the induction rule ...";
berghofe@5094
   549
wenzelm@6394
   550
    val sign = Theory.sign_of thy;
berghofe@5094
   551
berghofe@7293
   552
    val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
berghofe@7293
   553
        None => []
berghofe@7293
   554
      | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
berghofe@7293
   555
berghofe@5094
   556
    val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
berghofe@5094
   557
berghofe@5094
   558
    (* make predicate for instantiation of abstract induction rule *)
berghofe@5094
   559
berghofe@5094
   560
    fun mk_ind_pred _ [P] = P
berghofe@5094
   561
      | mk_ind_pred T Ps =
berghofe@5094
   562
         let val n = (length Ps) div 2;
berghofe@5094
   563
             val Type (_, [T1, T2]) = T
berghofe@7293
   564
         in Const ("Datatype.sum.sum_case",
berghofe@5094
   565
           [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
berghofe@5094
   566
             mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (drop (n, Ps))
berghofe@5094
   567
         end;
berghofe@5094
   568
berghofe@5094
   569
    val ind_pred = mk_ind_pred sumT preds;
berghofe@5094
   570
berghofe@5094
   571
    val ind_concl = HOLogic.mk_Trueprop
berghofe@5094
   572
      (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
berghofe@5094
   573
        (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
berghofe@5094
   574
berghofe@5094
   575
    (* simplification rules for vimage and Collect *)
berghofe@5094
   576
berghofe@5094
   577
    val vimage_simps = if length cs < 2 then [] else
wenzelm@10735
   578
      map (fn c => SkipProof.prove_goalw_cterm thy [] (Thm.cterm_of sign
berghofe@5094
   579
        (HOLogic.mk_Trueprop (HOLogic.mk_eq
berghofe@5094
   580
          (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
berghofe@5094
   581
           HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
berghofe@5094
   582
             nth_elem (find_index_eq c cs, preds)))))
wenzelm@10735
   583
        (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites, rtac refl 1])) cs;
berghofe@5094
   584
wenzelm@10735
   585
    val induct = SkipProof.prove_goalw_cterm thy [inductive_conj_def] (Thm.cterm_of sign
berghofe@5094
   586
      (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
berghofe@5094
   587
        [rtac (impI RS allI) 1,
nipkow@10202
   588
         DETERM (etac (mono RS (fp_def RS def_lfp_induct)) 1),
berghofe@7710
   589
         rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
berghofe@5094
   590
         fold_goals_tac rec_sets_defs,
berghofe@5094
   591
         (*This CollectE and disjE separates out the introduction rules*)
berghofe@7710
   592
         REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
berghofe@5094
   593
         (*Now break down the individual cases.  No disjE here in case
berghofe@5094
   594
           some premise involves disjunction.*)
berghofe@7710
   595
         REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
berghofe@7293
   596
         rewrite_goals_tac sum_case_rewrites,
berghofe@5094
   597
         EVERY (map (fn prem =>
berghofe@5149
   598
           DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
berghofe@5094
   599
wenzelm@10735
   600
    val lemma = SkipProof.prove_goalw_cterm thy rec_sets_defs (Thm.cterm_of sign
berghofe@5094
   601
      (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
berghofe@5094
   602
        [cut_facts_tac prems 1,
berghofe@5094
   603
         REPEAT (EVERY
berghofe@5094
   604
           [REPEAT (resolve_tac [conjI, impI] 1),
berghofe@5094
   605
            TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
berghofe@7293
   606
            rewrite_goals_tac sum_case_rewrites,
berghofe@5094
   607
            atac 1])])
berghofe@5094
   608
wenzelm@10729
   609
  in standard (split_rule (induct RS lemma)) end;
berghofe@5094
   610
wenzelm@6424
   611
wenzelm@6424
   612
wenzelm@10735
   613
(** specification of (co)inductive sets **)
berghofe@5094
   614
wenzelm@10729
   615
fun cond_declare_consts declare_consts cs paramTs cnames =
wenzelm@10729
   616
  if declare_consts then
wenzelm@10729
   617
    Theory.add_consts_i (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
wenzelm@10729
   618
  else I;
wenzelm@10729
   619
berghofe@9072
   620
fun mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
berghofe@9072
   621
      params paramTs cTs cnames =
berghofe@5094
   622
  let
berghofe@5094
   623
    val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
berghofe@5094
   624
    val setT = HOLogic.mk_setT sumT;
berghofe@5094
   625
wenzelm@10735
   626
    val fp_name = if coind then gfp_name else lfp_name;
berghofe@5094
   627
berghofe@5149
   628
    val used = foldr add_term_names (intr_ts, []);
berghofe@5149
   629
    val [sname, xname] = variantlist (["S", "x"], used);
berghofe@5149
   630
berghofe@5094
   631
    (* transform an introduction rule into a conjunction  *)
berghofe@5094
   632
    (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
berghofe@5094
   633
    (* is transformed into                                *)
berghofe@5094
   634
    (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
berghofe@5094
   635
berghofe@5094
   636
    fun transform_rule r =
berghofe@5094
   637
      let
berghofe@5094
   638
        val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
berghofe@5149
   639
        val subst = subst_free
berghofe@5149
   640
          (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
berghofe@5094
   641
        val Const ("op :", _) $ t $ u =
berghofe@5094
   642
          HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
berghofe@5094
   643
berghofe@5094
   644
      in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
berghofe@7710
   645
        (frees, foldr1 HOLogic.mk_conj
berghofe@5149
   646
          (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
berghofe@5094
   647
            (map (subst o HOLogic.dest_Trueprop)
berghofe@5094
   648
              (Logic.strip_imp_prems r))))
berghofe@5094
   649
      end
berghofe@5094
   650
berghofe@5094
   651
    (* make a disjunction of all introduction rules *)
berghofe@5094
   652
berghofe@5149
   653
    val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
berghofe@7710
   654
      absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
berghofe@5094
   655
berghofe@5094
   656
    (* add definiton of recursive sets to theory *)
berghofe@5094
   657
berghofe@5094
   658
    val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
wenzelm@6394
   659
    val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
berghofe@5094
   660
berghofe@5094
   661
    val rec_const = list_comb
berghofe@5094
   662
      (Const (full_rec_name, paramTs ---> setT), params);
berghofe@5094
   663
berghofe@5094
   664
    val fp_def_term = Logic.mk_equals (rec_const,
wenzelm@10735
   665
      Const (fp_name, (setT --> setT) --> setT) $ fp_fun);
berghofe@5094
   666
berghofe@5094
   667
    val def_terms = fp_def_term :: (if length cs < 2 then [] else
berghofe@5094
   668
      map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
berghofe@5094
   669
wenzelm@8433
   670
    val (thy', [fp_def :: rec_sets_defs]) =
wenzelm@8433
   671
      thy
wenzelm@10729
   672
      |> cond_declare_consts declare_consts cs paramTs cnames
wenzelm@8433
   673
      |> (if length cs < 2 then I
wenzelm@8433
   674
          else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
wenzelm@8433
   675
      |> Theory.add_path rec_name
wenzelm@9315
   676
      |> PureThy.add_defss_i false [(("defs", def_terms), [])];
berghofe@5094
   677
berghofe@9072
   678
    val mono = prove_mono setT fp_fun monos thy'
berghofe@5094
   679
wenzelm@10735
   680
  in (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) end;
berghofe@5094
   681
berghofe@9072
   682
fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
berghofe@9072
   683
    atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
berghofe@9072
   684
  let
wenzelm@10735
   685
    val _ =
wenzelm@10735
   686
      if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
wenzelm@10735
   687
        commas_quote cnames) else ();
berghofe@9072
   688
berghofe@9072
   689
    val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
berghofe@9072
   690
wenzelm@9939
   691
    val (thy1, mono, fp_def, rec_sets_defs, rec_const, sumT) =
berghofe@9072
   692
      mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
berghofe@9072
   693
        params paramTs cTs cnames;
berghofe@9072
   694
berghofe@5094
   695
    val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
wenzelm@9939
   696
      rec_sets_defs thy1;
berghofe@5094
   697
    val elims = if no_elim then [] else
wenzelm@9939
   698
      prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy1;
wenzelm@8312
   699
    val raw_induct = if no_ind then Drule.asm_rl else
berghofe@5094
   700
      if coind then standard (rule_by_tactic
oheimb@5553
   701
        (rewrite_tac [mk_meta_eq vimage_Un] THEN
berghofe@5094
   702
          fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
berghofe@5094
   703
      else
berghofe@5094
   704
        prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
wenzelm@9939
   705
          rec_sets_defs thy1;
berghofe@5108
   706
    val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
berghofe@5094
   707
      else standard (raw_induct RSN (2, rev_mp));
berghofe@5094
   708
wenzelm@9939
   709
    val (thy2, intrs') =
wenzelm@9939
   710
      thy1 |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts);
wenzelm@10735
   711
    val (thy3, ([intrs'', elims'], [induct'])) =
wenzelm@10735
   712
      thy2
wenzelm@10735
   713
      |> PureThy.add_thmss [(("intros", intrs'), atts), (("elims", elims), [])]
wenzelm@10735
   714
      |>>> PureThy.add_thms
wenzelm@10735
   715
        [((coind_prefix coind ^ "induct", rulify induct), [RuleCases.case_names induct_cases])]
wenzelm@8433
   716
      |>> Theory.parent_path;
wenzelm@9939
   717
  in (thy3,
wenzelm@10735
   718
    {defs = fp_def :: rec_sets_defs,
berghofe@5094
   719
     mono = mono,
berghofe@5094
   720
     unfold = unfold,
wenzelm@9939
   721
     intrs = intrs'',
wenzelm@7798
   722
     elims = elims',
wenzelm@7798
   723
     mk_cases = mk_cases elims',
wenzelm@10729
   724
     raw_induct = rulify raw_induct,
wenzelm@7798
   725
     induct = induct'})
berghofe@5094
   726
  end;
berghofe@5094
   727
wenzelm@6424
   728
wenzelm@10735
   729
(* external interfaces *)
berghofe@5094
   730
wenzelm@10735
   731
fun try_term f msg sign t =
wenzelm@10735
   732
  (case Library.try f t of
wenzelm@10735
   733
    Some x => x
wenzelm@10735
   734
  | None => error (msg ^ Sign.string_of_term sign t));
berghofe@5094
   735
berghofe@5094
   736
fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
wenzelm@10729
   737
    atts pre_intros monos con_defs thy =
berghofe@5094
   738
  let
wenzelm@6424
   739
    val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
wenzelm@6394
   740
    val sign = Theory.sign_of thy;
berghofe@5094
   741
berghofe@5094
   742
    (*parameters should agree for all mutually recursive components*)
berghofe@5094
   743
    val (_, params) = strip_comb (hd cs);
wenzelm@10735
   744
    val paramTs = map (try_term (snd o dest_Free) "Parameter in recursive\
berghofe@5094
   745
      \ component is not a free variable: " sign) params;
berghofe@5094
   746
wenzelm@10735
   747
    val cTs = map (try_term (HOLogic.dest_setT o fastype_of)
berghofe@5094
   748
      "Recursive component not of type set: " sign) cs;
berghofe@5094
   749
wenzelm@10735
   750
    val full_cnames = map (try_term (fst o dest_Const o head_of)
berghofe@5094
   751
      "Recursive set not previously declared as constant: " sign) cs;
wenzelm@6437
   752
    val cnames = map Sign.base_name full_cnames;
berghofe@5094
   753
wenzelm@10729
   754
    val save_sign =
wenzelm@10729
   755
      thy |> Theory.copy |> cond_declare_consts declare_consts cs paramTs cnames |> Theory.sign_of;
wenzelm@10729
   756
    val intros = map (check_rule save_sign cs) pre_intros;
wenzelm@8401
   757
    val induct_cases = map (#1 o #1) intros;
wenzelm@6437
   758
wenzelm@9405
   759
    val (thy1, result as {elims, induct, ...}) =
wenzelm@10735
   760
      add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
wenzelm@8401
   761
        con_defs thy params paramTs cTs cnames induct_cases;
wenzelm@8307
   762
    val thy2 = thy1
wenzelm@8307
   763
      |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
wenzelm@9405
   764
      |> add_cases_induct no_elim (no_ind orelse coind) full_cnames elims induct induct_cases;
wenzelm@6437
   765
  in (thy2, result) end;
berghofe@5094
   766
wenzelm@6521
   767
fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
berghofe@5094
   768
  let
wenzelm@6394
   769
    val sign = Theory.sign_of thy;
wenzelm@8100
   770
    val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
wenzelm@6424
   771
wenzelm@6521
   772
    val atts = map (Attrib.global_attribute thy) srcs;
wenzelm@6424
   773
    val intr_names = map (fst o fst) intro_srcs;
wenzelm@9405
   774
    fun read_rule s = Thm.read_cterm sign (s, propT)
wenzelm@9405
   775
      handle ERROR => error ("The error(s) above occurred for " ^ s);
wenzelm@9405
   776
    val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs;
wenzelm@6424
   777
    val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
berghofe@7020
   778
    val (cs', intr_ts') = unify_consts sign cs intr_ts;
berghofe@5094
   779
wenzelm@6424
   780
    val ((thy', con_defs), monos) = thy
wenzelm@6424
   781
      |> IsarThy.apply_theorems raw_monos
wenzelm@6424
   782
      |> apfst (IsarThy.apply_theorems raw_con_defs);
wenzelm@6424
   783
  in
berghofe@7020
   784
    add_inductive_i verbose false "" coind false false cs'
berghofe@7020
   785
      atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
berghofe@5094
   786
  end;
berghofe@5094
   787
wenzelm@6424
   788
wenzelm@6424
   789
wenzelm@6437
   790
(** package setup **)
wenzelm@6437
   791
wenzelm@6437
   792
(* setup theory *)
wenzelm@6437
   793
wenzelm@8634
   794
val setup =
wenzelm@8634
   795
 [InductiveData.init,
wenzelm@9625
   796
  Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args,
wenzelm@9598
   797
    "dynamic case analysis on sets")],
wenzelm@9893
   798
  Attrib.add_attributes [("mono", mono_attr, "declaration of monotonicity rule")]];
wenzelm@6437
   799
wenzelm@6437
   800
wenzelm@6437
   801
(* outer syntax *)
wenzelm@6424
   802
wenzelm@6723
   803
local structure P = OuterParse and K = OuterSyntax.Keyword in
wenzelm@6424
   804
wenzelm@6521
   805
fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
wenzelm@6723
   806
  #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
wenzelm@6424
   807
wenzelm@6424
   808
fun ind_decl coind =
wenzelm@6729
   809
  (Scan.repeat1 P.term --| P.marg_comment) --
wenzelm@9598
   810
  (P.$$$ "intros" |--
wenzelm@7152
   811
    P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
wenzelm@6729
   812
  Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
wenzelm@6729
   813
  Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
wenzelm@6424
   814
  >> (Toplevel.theory o mk_ind coind);
wenzelm@6424
   815
wenzelm@6723
   816
val inductiveP =
wenzelm@6723
   817
  OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
wenzelm@6723
   818
wenzelm@6723
   819
val coinductiveP =
wenzelm@6723
   820
  OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
wenzelm@6424
   821
wenzelm@7107
   822
wenzelm@7107
   823
val ind_cases =
wenzelm@9625
   824
  P.opt_thm_name ":" -- Scan.repeat1 P.prop -- P.marg_comment
wenzelm@7107
   825
  >> (Toplevel.theory o inductive_cases);
wenzelm@7107
   826
wenzelm@7107
   827
val inductive_casesP =
wenzelm@9804
   828
  OuterSyntax.command "inductive_cases"
wenzelm@9598
   829
    "create simplified instances of elimination rules (improper)" K.thy_script ind_cases;
wenzelm@7107
   830
wenzelm@9643
   831
val _ = OuterSyntax.add_keywords ["intros", "monos", "con_defs"];
wenzelm@7107
   832
val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
wenzelm@6424
   833
berghofe@5094
   834
end;
wenzelm@6424
   835
wenzelm@6424
   836
end;