src/HOL/Tools/inductive_package.ML
author wenzelm
Tue Feb 22 21:45:20 2000 +0100 (2000-02-22)
changeset 8277 493707fcd8a6
parent 8100 6186ee807f2e
child 8293 4a0e17cf8f70
permissions -rw-r--r--
added cases_of, cases;
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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 cases_of: Sign.sg -> string -> thm
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  val cases: Sign.sg -> thm list
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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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(* cases rules *)
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fun cases_of sg name =
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  let
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    fun find (None, (_, ({names, ...}, {elims, ...}): inductive_info)) =
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          assoc (names ~~ elims, name)
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      | find (some, _) = some;
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    val (tab, _) = InductiveData.get_sg sg;
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  in
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    (case Symtab.foldl find (None, tab) of
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      None => error ("Unknown (co)inductive set " ^ quote name)
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    | Some thm => thm)
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  end;
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fun cases sg = flat (map (#elims o #2 o #2) (Symtab.dest (#1 (InductiveData.get_sg sg))));
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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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(*Delete needless equality assumptions*)
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val refl_thin = prove_goal HOL.thy "!!P. [| a=a;  P |] ==> P"
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     (fn _ => [assume_tac 1]);
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(*For simplifying the elimination rule*)
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val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
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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)
berghofe@5094
   331
  end;
berghofe@5094
   332
        
wenzelm@6424
   333
wenzelm@6424
   334
wenzelm@6424
   335
(** premises and conclusions of induction rules **)
berghofe@5094
   336
berghofe@5094
   337
fun mk_indrule cs cTs params intr_ts =
berghofe@5094
   338
  let
berghofe@5094
   339
    val used = foldr add_term_names (intr_ts, []);
berghofe@5094
   340
berghofe@5094
   341
    (* predicates for induction rule *)
berghofe@5094
   342
berghofe@5094
   343
    val preds = map Free (variantlist (if length cs < 2 then ["P"] else
berghofe@5094
   344
      map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
berghofe@5094
   345
        map (fn T => T --> HOLogic.boolT) cTs);
berghofe@5094
   346
berghofe@5094
   347
    (* transform an introduction rule into a premise for induction rule *)
berghofe@5094
   348
berghofe@5094
   349
    fun mk_ind_prem r =
berghofe@5094
   350
      let
berghofe@5094
   351
        val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
berghofe@5094
   352
berghofe@7710
   353
        val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
berghofe@5094
   354
berghofe@7710
   355
        fun subst (s as ((m as Const ("op :", T)) $ t $ u)) =
berghofe@7710
   356
              (case pred_of u of
berghofe@7710
   357
                  None => (m $ fst (subst t) $ fst (subst u), None)
berghofe@7710
   358
                | Some P => (HOLogic.conj $ s $ (P $ t), Some (s, P $ t)))
berghofe@7710
   359
          | subst s =
berghofe@7710
   360
              (case pred_of s of
berghofe@7710
   361
                  Some P => (HOLogic.mk_binop "op Int"
berghofe@7710
   362
                    (s, HOLogic.Collect_const (HOLogic.dest_setT
berghofe@7710
   363
                      (fastype_of s)) $ P), None)
berghofe@7710
   364
                | None => (case s of
berghofe@7710
   365
                     (t $ u) => (fst (subst t) $ fst (subst u), None)
berghofe@7710
   366
                   | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
berghofe@7710
   367
                   | _ => (s, None)));
berghofe@7710
   368
berghofe@7710
   369
        fun mk_prem (s, prems) = (case subst s of
berghofe@7710
   370
              (_, Some (t, u)) => t :: u :: prems
berghofe@7710
   371
            | (t, _) => t :: prems);
berghofe@7710
   372
          
berghofe@5094
   373
        val Const ("op :", _) $ t $ u =
berghofe@5094
   374
          HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
berghofe@5094
   375
berghofe@5094
   376
      in list_all_free (frees,
berghofe@7710
   377
           Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
berghofe@5094
   378
             (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
berghofe@7710
   379
               HOLogic.mk_Trueprop (the (pred_of u) $ t)))
berghofe@5094
   380
      end;
berghofe@5094
   381
berghofe@5094
   382
    val ind_prems = map mk_ind_prem intr_ts;
berghofe@5094
   383
berghofe@5094
   384
    (* make conclusions for induction rules *)
berghofe@5094
   385
berghofe@5094
   386
    fun mk_ind_concl ((c, P), (ts, x)) =
berghofe@5094
   387
      let val T = HOLogic.dest_setT (fastype_of c);
berghofe@5094
   388
          val Ts = HOLogic.prodT_factors T;
berghofe@5094
   389
          val (frees, x') = foldr (fn (T', (fs, s)) =>
berghofe@5094
   390
            ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
berghofe@5094
   391
          val tuple = HOLogic.mk_tuple T frees;
berghofe@5094
   392
      in ((HOLogic.mk_binop "op -->"
berghofe@5094
   393
        (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x')
berghofe@5094
   394
      end;
berghofe@5094
   395
berghofe@7710
   396
    val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
berghofe@5094
   397
        (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
berghofe@5094
   398
berghofe@5094
   399
  in (preds, ind_prems, mutual_ind_concl)
berghofe@5094
   400
  end;
berghofe@5094
   401
wenzelm@6424
   402
berghofe@5094
   403
wenzelm@6424
   404
(*** proofs for (co)inductive sets ***)
wenzelm@6424
   405
wenzelm@6424
   406
(** prove monotonicity **)
berghofe@5094
   407
berghofe@5094
   408
fun prove_mono setT fp_fun monos thy =
berghofe@5094
   409
  let
wenzelm@6427
   410
    val _ = message "  Proving monotonicity ...";
berghofe@5094
   411
wenzelm@6394
   412
    val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
berghofe@5094
   413
      (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
berghofe@7710
   414
        (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
berghofe@5094
   415
berghofe@5094
   416
  in mono end;
berghofe@5094
   417
wenzelm@6424
   418
wenzelm@6424
   419
wenzelm@6424
   420
(** prove introduction rules **)
berghofe@5094
   421
berghofe@5094
   422
fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
berghofe@5094
   423
  let
wenzelm@6427
   424
    val _ = message "  Proving the introduction rules ...";
berghofe@5094
   425
berghofe@5094
   426
    val unfold = standard (mono RS (fp_def RS
berghofe@5094
   427
      (if coind then def_gfp_Tarski else def_lfp_Tarski)));
berghofe@5094
   428
berghofe@5094
   429
    fun select_disj 1 1 = []
berghofe@5094
   430
      | select_disj _ 1 = [rtac disjI1]
berghofe@5094
   431
      | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
berghofe@5094
   432
berghofe@5094
   433
    val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
wenzelm@6394
   434
      (cterm_of (Theory.sign_of thy) intr) (fn prems =>
berghofe@5094
   435
       [(*insert prems and underlying sets*)
berghofe@5094
   436
       cut_facts_tac prems 1,
berghofe@5094
   437
       stac unfold 1,
berghofe@5094
   438
       REPEAT (resolve_tac [vimageI2, CollectI] 1),
berghofe@5094
   439
       (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
berghofe@5094
   440
       EVERY1 (select_disj (length intr_ts) i),
berghofe@5094
   441
       (*Not ares_tac, since refl must be tried before any equality assumptions;
berghofe@5094
   442
         backtracking may occur if the premises have extra variables!*)
berghofe@5094
   443
       DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
berghofe@5094
   444
       (*Now solve the equations like Inl 0 = Inl ?b2*)
berghofe@5094
   445
       rewrite_goals_tac con_defs,
berghofe@5094
   446
       REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
berghofe@5094
   447
berghofe@5094
   448
  in (intrs, unfold) end;
berghofe@5094
   449
wenzelm@6424
   450
wenzelm@6424
   451
wenzelm@6424
   452
(** prove elimination rules **)
berghofe@5094
   453
berghofe@5094
   454
fun prove_elims cs cTs params intr_ts unfold rec_sets_defs thy =
berghofe@5094
   455
  let
wenzelm@6427
   456
    val _ = message "  Proving the elimination rules ...";
berghofe@5094
   457
berghofe@7710
   458
    val rules1 = [CollectE, disjE, make_elim vimageD, exE];
berghofe@7710
   459
    val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
berghofe@5094
   460
      map make_elim [Inl_inject, Inr_inject];
berghofe@5094
   461
berghofe@5094
   462
    val elims = map (fn t => prove_goalw_cterm rec_sets_defs
wenzelm@6394
   463
      (cterm_of (Theory.sign_of thy) t) (fn prems =>
berghofe@5094
   464
        [cut_facts_tac [hd prems] 1,
berghofe@5094
   465
         dtac (unfold RS subst) 1,
berghofe@5094
   466
         REPEAT (FIRSTGOAL (eresolve_tac rules1)),
berghofe@5094
   467
         REPEAT (FIRSTGOAL (eresolve_tac rules2)),
berghofe@5094
   468
         EVERY (map (fn prem =>
berghofe@5149
   469
           DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))]))
berghofe@5094
   470
      (mk_elims cs cTs params intr_ts)
berghofe@5094
   471
berghofe@5094
   472
  in elims end;
berghofe@5094
   473
wenzelm@6424
   474
berghofe@5094
   475
(** derivation of simplified elimination rules **)
berghofe@5094
   476
berghofe@5094
   477
(*Applies freeness of the given constructors, which *must* be unfolded by
berghofe@5094
   478
  the given defs.  Cannot simply use the local con_defs because con_defs=[] 
berghofe@5094
   479
  for inference systems.
berghofe@5094
   480
 *)
paulson@6141
   481
fun con_elim_tac ss =
berghofe@5094
   482
  let val elim_tac = REPEAT o (eresolve_tac elim_rls)
berghofe@5094
   483
  in ALLGOALS(EVERY'[elim_tac,
paulson@6141
   484
		     asm_full_simp_tac ss,
paulson@6141
   485
		     elim_tac,
paulson@6141
   486
		     REPEAT o bound_hyp_subst_tac])
berghofe@5094
   487
     THEN prune_params_tac
berghofe@5094
   488
  end;
berghofe@5094
   489
wenzelm@7107
   490
(*cprop should have the form t:Si where Si is an inductive set*)
wenzelm@7107
   491
fun mk_cases_i elims ss cprop =
wenzelm@7107
   492
  let
wenzelm@7107
   493
    val prem = Thm.assume cprop;
wenzelm@7107
   494
    fun mk_elim rl = standard (rule_by_tactic (con_elim_tac ss) (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@7107
   499
        [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
wenzelm@7107
   500
          Display.pretty_cterm cprop])))
wenzelm@7107
   501
  end;
wenzelm@7107
   502
paulson@6141
   503
fun mk_cases elims s =
wenzelm@7107
   504
  mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
wenzelm@7107
   505
wenzelm@7107
   506
wenzelm@7107
   507
(* inductive_cases(_i) *)
wenzelm@7107
   508
wenzelm@7107
   509
fun gen_inductive_cases prep_att prep_const prep_prop
wenzelm@7107
   510
    ((((name, raw_atts), raw_set), raw_props), comment) thy =
wenzelm@7107
   511
  let
wenzelm@7107
   512
    val sign = Theory.sign_of thy;
wenzelm@7107
   513
wenzelm@7107
   514
    val atts = map (prep_att thy) raw_atts;
wenzelm@7107
   515
    val (_, {elims, ...}) = get_inductive thy (prep_const sign raw_set);
wenzelm@7107
   516
    val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
wenzelm@7107
   517
    val thms = map (mk_cases_i elims (Simplifier.simpset_of thy)) cprops;
wenzelm@7107
   518
  in
wenzelm@7107
   519
    thy
wenzelm@7107
   520
    |> IsarThy.have_theorems_i (((name, atts), map Thm.no_attributes thms), comment)
berghofe@5094
   521
  end;
berghofe@5094
   522
wenzelm@7107
   523
val inductive_cases =
wenzelm@7107
   524
  gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
wenzelm@7107
   525
wenzelm@7107
   526
val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
wenzelm@7107
   527
wenzelm@6424
   528
wenzelm@6424
   529
wenzelm@6424
   530
(** prove induction rule **)
berghofe@5094
   531
berghofe@5094
   532
fun prove_indrule cs cTs sumT rec_const params intr_ts mono
berghofe@5094
   533
    fp_def rec_sets_defs thy =
berghofe@5094
   534
  let
wenzelm@6427
   535
    val _ = message "  Proving the induction rule ...";
berghofe@5094
   536
wenzelm@6394
   537
    val sign = Theory.sign_of thy;
berghofe@5094
   538
berghofe@7293
   539
    val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
berghofe@7293
   540
        None => []
berghofe@7293
   541
      | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
berghofe@7293
   542
berghofe@5094
   543
    val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
berghofe@5094
   544
berghofe@5094
   545
    (* make predicate for instantiation of abstract induction rule *)
berghofe@5094
   546
berghofe@5094
   547
    fun mk_ind_pred _ [P] = P
berghofe@5094
   548
      | mk_ind_pred T Ps =
berghofe@5094
   549
         let val n = (length Ps) div 2;
berghofe@5094
   550
             val Type (_, [T1, T2]) = T
berghofe@7293
   551
         in Const ("Datatype.sum.sum_case",
berghofe@5094
   552
           [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
berghofe@5094
   553
             mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (drop (n, Ps))
berghofe@5094
   554
         end;
berghofe@5094
   555
berghofe@5094
   556
    val ind_pred = mk_ind_pred sumT preds;
berghofe@5094
   557
berghofe@5094
   558
    val ind_concl = HOLogic.mk_Trueprop
berghofe@5094
   559
      (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
berghofe@5094
   560
        (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
berghofe@5094
   561
berghofe@5094
   562
    (* simplification rules for vimage and Collect *)
berghofe@5094
   563
berghofe@5094
   564
    val vimage_simps = if length cs < 2 then [] else
berghofe@5094
   565
      map (fn c => prove_goalw_cterm [] (cterm_of sign
berghofe@5094
   566
        (HOLogic.mk_Trueprop (HOLogic.mk_eq
berghofe@5094
   567
          (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
berghofe@5094
   568
           HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
berghofe@5094
   569
             nth_elem (find_index_eq c cs, preds)))))
berghofe@7293
   570
        (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
berghofe@5094
   571
          rtac refl 1])) cs;
berghofe@5094
   572
berghofe@5094
   573
    val induct = prove_goalw_cterm [] (cterm_of sign
berghofe@5094
   574
      (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
berghofe@5094
   575
        [rtac (impI RS allI) 1,
berghofe@5094
   576
         DETERM (etac (mono RS (fp_def RS def_induct)) 1),
berghofe@7710
   577
         rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
berghofe@5094
   578
         fold_goals_tac rec_sets_defs,
berghofe@5094
   579
         (*This CollectE and disjE separates out the introduction rules*)
berghofe@7710
   580
         REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
berghofe@5094
   581
         (*Now break down the individual cases.  No disjE here in case
berghofe@5094
   582
           some premise involves disjunction.*)
berghofe@7710
   583
         REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
berghofe@7293
   584
         rewrite_goals_tac sum_case_rewrites,
berghofe@5094
   585
         EVERY (map (fn prem =>
berghofe@5149
   586
           DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
berghofe@5094
   587
berghofe@5094
   588
    val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
berghofe@5094
   589
      (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
berghofe@5094
   590
        [cut_facts_tac prems 1,
berghofe@5094
   591
         REPEAT (EVERY
berghofe@5094
   592
           [REPEAT (resolve_tac [conjI, impI] 1),
berghofe@5094
   593
            TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
berghofe@7293
   594
            rewrite_goals_tac sum_case_rewrites,
berghofe@5094
   595
            atac 1])])
berghofe@5094
   596
berghofe@5094
   597
  in standard (split_rule (induct RS lemma))
berghofe@5094
   598
  end;
berghofe@5094
   599
wenzelm@6424
   600
wenzelm@6424
   601
wenzelm@6424
   602
(*** specification of (co)inductive sets ****)
wenzelm@6424
   603
wenzelm@6424
   604
(** definitional introduction of (co)inductive sets **)
berghofe@5094
   605
berghofe@5094
   606
fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
wenzelm@6521
   607
    atts intros monos con_defs thy params paramTs cTs cnames =
berghofe@5094
   608
  let
wenzelm@6424
   609
    val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
wenzelm@6424
   610
      commas_quote cnames) else ();
berghofe@5094
   611
berghofe@5094
   612
    val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
berghofe@5094
   613
    val setT = HOLogic.mk_setT sumT;
berghofe@5094
   614
wenzelm@6394
   615
    val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
wenzelm@6394
   616
      else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
berghofe@5094
   617
wenzelm@6424
   618
    val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
wenzelm@6424
   619
berghofe@5149
   620
    val used = foldr add_term_names (intr_ts, []);
berghofe@5149
   621
    val [sname, xname] = variantlist (["S", "x"], used);
berghofe@5149
   622
berghofe@5094
   623
    (* transform an introduction rule into a conjunction  *)
berghofe@5094
   624
    (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
berghofe@5094
   625
    (* is transformed into                                *)
berghofe@5094
   626
    (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
berghofe@5094
   627
berghofe@5094
   628
    fun transform_rule r =
berghofe@5094
   629
      let
berghofe@5094
   630
        val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
berghofe@5149
   631
        val subst = subst_free
berghofe@5149
   632
          (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
berghofe@5094
   633
        val Const ("op :", _) $ t $ u =
berghofe@5094
   634
          HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
berghofe@5094
   635
berghofe@5094
   636
      in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
berghofe@7710
   637
        (frees, foldr1 HOLogic.mk_conj
berghofe@5149
   638
          (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
berghofe@5094
   639
            (map (subst o HOLogic.dest_Trueprop)
berghofe@5094
   640
              (Logic.strip_imp_prems r))))
berghofe@5094
   641
      end
berghofe@5094
   642
berghofe@5094
   643
    (* make a disjunction of all introduction rules *)
berghofe@5094
   644
berghofe@5149
   645
    val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
berghofe@7710
   646
      absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
berghofe@5094
   647
berghofe@5094
   648
    (* add definiton of recursive sets to theory *)
berghofe@5094
   649
berghofe@5094
   650
    val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
wenzelm@6394
   651
    val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
berghofe@5094
   652
berghofe@5094
   653
    val rec_const = list_comb
berghofe@5094
   654
      (Const (full_rec_name, paramTs ---> setT), params);
berghofe@5094
   655
berghofe@5094
   656
    val fp_def_term = Logic.mk_equals (rec_const,
berghofe@5094
   657
      Const (fp_name, (setT --> setT) --> setT) $ fp_fun)
berghofe@5094
   658
berghofe@5094
   659
    val def_terms = fp_def_term :: (if length cs < 2 then [] else
berghofe@5094
   660
      map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
berghofe@5094
   661
berghofe@5094
   662
    val thy' = thy |>
berghofe@5094
   663
      (if declare_consts then
berghofe@5094
   664
        Theory.add_consts_i (map (fn (c, n) =>
berghofe@5094
   665
          (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
berghofe@5094
   666
       else I) |>
berghofe@5094
   667
      (if length cs < 2 then I else
berghofe@5094
   668
       Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |>
berghofe@5094
   669
      Theory.add_path rec_name |>
berghofe@5094
   670
      PureThy.add_defss_i [(("defs", def_terms), [])];
berghofe@5094
   671
berghofe@5094
   672
    (* get definitions from theory *)
berghofe@5094
   673
wenzelm@6424
   674
    val fp_def::rec_sets_defs = PureThy.get_thms thy' "defs";
berghofe@5094
   675
berghofe@5094
   676
    (* prove and store theorems *)
berghofe@5094
   677
berghofe@5094
   678
    val mono = prove_mono setT fp_fun monos thy';
berghofe@5094
   679
    val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
berghofe@5094
   680
      rec_sets_defs thy';
berghofe@5094
   681
    val elims = if no_elim then [] else
berghofe@5094
   682
      prove_elims cs cTs params intr_ts unfold rec_sets_defs thy';
berghofe@5094
   683
    val raw_induct = if no_ind then TrueI else
berghofe@5094
   684
      if coind then standard (rule_by_tactic
oheimb@5553
   685
        (rewrite_tac [mk_meta_eq vimage_Un] THEN
berghofe@5094
   686
          fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
berghofe@5094
   687
      else
berghofe@5094
   688
        prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
berghofe@5094
   689
          rec_sets_defs thy';
berghofe@5108
   690
    val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
berghofe@5094
   691
      else standard (raw_induct RSN (2, rev_mp));
berghofe@5094
   692
wenzelm@6424
   693
    val thy'' = thy'
wenzelm@6521
   694
      |> PureThy.add_thmss [(("intrs", intrs), atts)]
wenzelm@6424
   695
      |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
wenzelm@6424
   696
      |> (if no_elim then I else PureThy.add_thmss [(("elims", elims), [])])
wenzelm@6424
   697
      |> (if no_ind then I else PureThy.add_thms
wenzelm@6424
   698
        [((coind_prefix coind ^ "induct", induct), [])])
wenzelm@6424
   699
      |> Theory.parent_path;
wenzelm@7798
   700
    val intrs' = PureThy.get_thms thy'' "intrs";
wenzelm@7798
   701
    val elims' = PureThy.get_thms thy'' "elims";
wenzelm@7798
   702
    val induct' = PureThy.get_thm thy'' (coind_prefix coind ^ "induct");
berghofe@5094
   703
  in (thy'',
berghofe@5094
   704
    {defs = fp_def::rec_sets_defs,
berghofe@5094
   705
     mono = mono,
berghofe@5094
   706
     unfold = unfold,
wenzelm@7798
   707
     intrs = intrs',
wenzelm@7798
   708
     elims = elims',
wenzelm@7798
   709
     mk_cases = mk_cases elims',
berghofe@5094
   710
     raw_induct = raw_induct,
wenzelm@7798
   711
     induct = induct'})
berghofe@5094
   712
  end;
berghofe@5094
   713
wenzelm@6424
   714
wenzelm@6424
   715
wenzelm@6424
   716
(** axiomatic introduction of (co)inductive sets **)
berghofe@5094
   717
berghofe@5094
   718
fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
wenzelm@6521
   719
    atts intros monos con_defs thy params paramTs cTs cnames =
berghofe@5094
   720
  let
berghofe@5094
   721
    val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
berghofe@5094
   722
wenzelm@6424
   723
    val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
berghofe@5094
   724
    val elim_ts = mk_elims cs cTs params intr_ts;
berghofe@5094
   725
berghofe@5094
   726
    val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
berghofe@5094
   727
    val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
berghofe@5094
   728
    
wenzelm@6424
   729
    val thy' = thy
wenzelm@6424
   730
      |> (if declare_consts then
wenzelm@6424
   731
            Theory.add_consts_i
wenzelm@6424
   732
              (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
wenzelm@6424
   733
         else I)
wenzelm@6424
   734
      |> Theory.add_path rec_name
wenzelm@6521
   735
      |> PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("elims", elim_ts), [])]
berghofe@7710
   736
      |> (if coind then I else
berghofe@7710
   737
            PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
berghofe@5094
   738
wenzelm@6424
   739
    val intrs = PureThy.get_thms thy' "intrs";
wenzelm@6424
   740
    val elims = PureThy.get_thms thy' "elims";
berghofe@7710
   741
    val raw_induct = if coind then TrueI else PureThy.get_thm thy' "raw_induct";
berghofe@5094
   742
    val induct = if coind orelse length cs > 1 then raw_induct
berghofe@5094
   743
      else standard (raw_induct RSN (2, rev_mp));
berghofe@5094
   744
wenzelm@6424
   745
    val thy'' =
wenzelm@6424
   746
      thy'
wenzelm@6424
   747
      |> (if coind then I else PureThy.add_thms [(("induct", induct), [])])
wenzelm@6424
   748
      |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
wenzelm@6424
   749
      |> Theory.parent_path;
wenzelm@7798
   750
    val induct' = if coind then raw_induct else PureThy.get_thm thy'' "induct";
berghofe@5094
   751
  in (thy'',
berghofe@5094
   752
    {defs = [],
berghofe@5094
   753
     mono = TrueI,
berghofe@5094
   754
     unfold = TrueI,
berghofe@5094
   755
     intrs = intrs,
berghofe@5094
   756
     elims = elims,
berghofe@5094
   757
     mk_cases = mk_cases elims,
berghofe@5094
   758
     raw_induct = raw_induct,
wenzelm@7798
   759
     induct = induct'})
berghofe@5094
   760
  end;
berghofe@5094
   761
wenzelm@6424
   762
wenzelm@6424
   763
wenzelm@6424
   764
(** introduction of (co)inductive sets **)
berghofe@5094
   765
berghofe@5094
   766
fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
wenzelm@6521
   767
    atts intros monos con_defs thy =
berghofe@5094
   768
  let
wenzelm@6424
   769
    val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
wenzelm@6394
   770
    val sign = Theory.sign_of thy;
berghofe@5094
   771
berghofe@5094
   772
    (*parameters should agree for all mutually recursive components*)
berghofe@5094
   773
    val (_, params) = strip_comb (hd cs);
berghofe@5094
   774
    val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
berghofe@5094
   775
      \ component is not a free variable: " sign) params;
berghofe@5094
   776
berghofe@5094
   777
    val cTs = map (try' (HOLogic.dest_setT o fastype_of)
berghofe@5094
   778
      "Recursive component not of type set: " sign) cs;
berghofe@5094
   779
wenzelm@6437
   780
    val full_cnames = map (try' (fst o dest_Const o head_of)
berghofe@5094
   781
      "Recursive set not previously declared as constant: " sign) cs;
wenzelm@6437
   782
    val cnames = map Sign.base_name full_cnames;
berghofe@5094
   783
wenzelm@6424
   784
    val _ = seq (check_rule sign cs o snd o fst) intros;
wenzelm@6437
   785
wenzelm@6437
   786
    val (thy1, result) =
wenzelm@6437
   787
      (if ! quick_and_dirty then add_ind_axm else add_ind_def)
wenzelm@6521
   788
        verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
wenzelm@6437
   789
        con_defs thy params paramTs cTs cnames;
wenzelm@6437
   790
    val thy2 = thy1 |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result);
wenzelm@6437
   791
  in (thy2, result) end;
berghofe@5094
   792
wenzelm@6424
   793
berghofe@5094
   794
wenzelm@6424
   795
(** external interface **)
wenzelm@6424
   796
wenzelm@6521
   797
fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
berghofe@5094
   798
  let
wenzelm@6394
   799
    val sign = Theory.sign_of thy;
wenzelm@8100
   800
    val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
wenzelm@6424
   801
wenzelm@6521
   802
    val atts = map (Attrib.global_attribute thy) srcs;
wenzelm@6424
   803
    val intr_names = map (fst o fst) intro_srcs;
berghofe@7710
   804
    val intr_ts = map (term_of o Thm.read_cterm sign o rpair propT o snd o fst) intro_srcs;
wenzelm@6424
   805
    val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
berghofe@7020
   806
    val (cs', intr_ts') = unify_consts sign cs intr_ts;
berghofe@5094
   807
wenzelm@6424
   808
    val ((thy', con_defs), monos) = thy
wenzelm@6424
   809
      |> IsarThy.apply_theorems raw_monos
wenzelm@6424
   810
      |> apfst (IsarThy.apply_theorems raw_con_defs);
wenzelm@6424
   811
  in
berghofe@7020
   812
    add_inductive_i verbose false "" coind false false cs'
berghofe@7020
   813
      atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
berghofe@5094
   814
  end;
berghofe@5094
   815
wenzelm@6424
   816
wenzelm@6424
   817
wenzelm@6437
   818
(** package setup **)
wenzelm@6437
   819
wenzelm@6437
   820
(* setup theory *)
wenzelm@6437
   821
berghofe@7710
   822
val setup = [InductiveData.init,
berghofe@7710
   823
             Attrib.add_attributes [(monoN, mono_attr, "monotonicity rule")]];
wenzelm@6437
   824
wenzelm@6437
   825
wenzelm@6437
   826
(* outer syntax *)
wenzelm@6424
   827
wenzelm@6723
   828
local structure P = OuterParse and K = OuterSyntax.Keyword in
wenzelm@6424
   829
wenzelm@6521
   830
fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
wenzelm@6723
   831
  #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
wenzelm@6424
   832
wenzelm@6424
   833
fun ind_decl coind =
wenzelm@6729
   834
  (Scan.repeat1 P.term --| P.marg_comment) --
wenzelm@6729
   835
  (P.$$$ "intrs" |--
wenzelm@7152
   836
    P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
wenzelm@6729
   837
  Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
wenzelm@6729
   838
  Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
wenzelm@6424
   839
  >> (Toplevel.theory o mk_ind coind);
wenzelm@6424
   840
wenzelm@6723
   841
val inductiveP =
wenzelm@6723
   842
  OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
wenzelm@6723
   843
wenzelm@6723
   844
val coinductiveP =
wenzelm@6723
   845
  OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
wenzelm@6424
   846
wenzelm@7107
   847
wenzelm@7107
   848
val ind_cases =
wenzelm@7107
   849
  P.opt_thm_name "=" -- P.xname --| P.$$$ ":" -- Scan.repeat1 P.prop -- P.marg_comment
wenzelm@7107
   850
  >> (Toplevel.theory o inductive_cases);
wenzelm@7107
   851
wenzelm@7107
   852
val inductive_casesP =
wenzelm@7107
   853
  OuterSyntax.command "inductive_cases" "create simplified instances of elimination rules"
wenzelm@7107
   854
    K.thy_decl ind_cases;
wenzelm@7107
   855
wenzelm@6424
   856
val _ = OuterSyntax.add_keywords ["intrs", "monos", "con_defs"];
wenzelm@7107
   857
val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
wenzelm@6424
   858
berghofe@5094
   859
end;
wenzelm@6424
   860
wenzelm@6424
   861
wenzelm@6424
   862
end;