src/HOL/simpdata.ML
author berghofe
Wed Aug 06 01:13:46 1997 +0200 (1997-08-06)
changeset 3615 e5322197cfea
parent 3577 9715b6e3ec5f
child 3654 ebad042c0bba
permissions -rw-r--r--
Moved some functions which used to be part of thy_data.ML
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(*  Title:      HOL/simpdata.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1991  University of Cambridge
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Instantiation of the generic simplifier
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*)
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section "Simplifier";
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open Simplifier;
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(*** Addition of rules to simpsets and clasets simultaneously ***)
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(*Takes UNCONDITIONAL theorems of the form A<->B to 
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        the Safe Intr     rule B==>A and 
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        the Safe Destruct rule A==>B.
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  Also ~A goes to the Safe Elim rule A ==> ?R
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  Failing other cases, A is added as a Safe Intr rule*)
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local
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  val iff_const = HOLogic.eq_const HOLogic.boolT;
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  fun addIff th = 
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      (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
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                (Const("Not",_) $ A) =>
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                    AddSEs [zero_var_indexes (th RS notE)]
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              | (con $ _ $ _) =>
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                    if con=iff_const
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                    then (AddSIs [zero_var_indexes (th RS iffD2)];  
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                          AddSDs [zero_var_indexes (th RS iffD1)])
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                    else  AddSIs [th]
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              | _ => AddSIs [th];
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       Addsimps [th])
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      handle _ => error ("AddIffs: theorem must be unconditional\n" ^ 
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                         string_of_thm th)
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  fun delIff th = 
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      (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
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                (Const("Not",_) $ A) =>
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                    Delrules [zero_var_indexes (th RS notE)]
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              | (con $ _ $ _) =>
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                    if con=iff_const
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                    then Delrules [zero_var_indexes (th RS iffD2),
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                                   make_elim (zero_var_indexes (th RS iffD1))]
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                    else Delrules [th]
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              | _ => Delrules [th];
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       Delsimps [th])
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      handle _ => warning("DelIffs: ignoring conditional theorem\n" ^ 
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                          string_of_thm th)
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in
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val AddIffs = seq addIff
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val DelIffs = seq delIff
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end;
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local
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  fun prover s = prove_goal HOL.thy s (fn _ => [blast_tac HOL_cs 1]);
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  val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
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  val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
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  val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
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  val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
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  fun atomize pairs =
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    let fun atoms th =
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          (case concl_of th of
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             Const("Trueprop",_) $ p =>
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               (case head_of p of
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                  Const(a,_) =>
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                    (case assoc(pairs,a) of
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                       Some(rls) => flat (map atoms ([th] RL rls))
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                     | None => [th])
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                | _ => [th])
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           | _ => [th])
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    in atoms end;
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  fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
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in
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  fun mk_meta_eq r = case concl_of r of
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          Const("==",_)$_$_ => r
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      |   _$(Const("op =",_)$_$_) => r RS eq_reflection
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      |   _$(Const("Not",_)$_) => r RS not_P_imp_P_eq_False
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      |   _ => r RS P_imp_P_eq_True;
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  (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
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val simp_thms = map prover
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 [ "(x=x) = True",
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   "(~True) = False", "(~False) = True", "(~ ~ P) = P",
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   "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
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   "(True=P) = P", "(P=True) = P",
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   "(True --> P) = P", "(False --> P) = True", 
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   "(P --> True) = True", "(P --> P) = True",
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   "(P --> False) = (~P)", "(P --> ~P) = (~P)",
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   "(P & True) = P", "(True & P) = P", 
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   "(P & False) = False", "(False & P) = False",
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   "(P & P) = P", "(P & (P & Q)) = (P & Q)",
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   "(P | True) = True", "(True | P) = True", 
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   "(P | False) = P", "(False | P) = P",
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   "(P | P) = P", "(P | (P | Q)) = (P | Q)",
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   "((~P) = (~Q)) = (P=Q)",
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   "(!x.P) = P", "(? x.P) = P", "? x. x=t", "? x. t=x", 
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   "(? x. x=t & P(x)) = P(t)",
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   "(! x. t=x --> P(x)) = P(t)" ];
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(*Add congruence rules for = (instead of ==) *)
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infix 4 addcongs delcongs;
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fun ss addcongs congs = ss addeqcongs (map standard (congs RL [eq_reflection]));
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fun ss delcongs congs = ss deleqcongs (map standard (congs RL [eq_reflection]));
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fun Addcongs congs = (simpset := !simpset addcongs congs);
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fun Delcongs congs = (simpset := !simpset delcongs congs);
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fun mksimps pairs = map mk_meta_eq o atomize pairs o gen_all;
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val imp_cong = impI RSN
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    (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
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        (fn _=> [blast_tac HOL_cs 1]) RS mp RS mp);
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(*Miniscoping: pushing in existential quantifiers*)
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val ex_simps = map prover 
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                ["(EX x. P x & Q)   = ((EX x.P x) & Q)",
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                 "(EX x. P & Q x)   = (P & (EX x.Q x))",
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                 "(EX x. P x | Q)   = ((EX x.P x) | Q)",
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                 "(EX x. P | Q x)   = (P | (EX x.Q x))",
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                 "(EX x. P x --> Q) = ((ALL x.P x) --> Q)",
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                 "(EX x. P --> Q x) = (P --> (EX x.Q x))"];
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(*Miniscoping: pushing in universal quantifiers*)
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val all_simps = map prover
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                ["(ALL x. P x & Q)   = ((ALL x.P x) & Q)",
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                 "(ALL x. P & Q x)   = (P & (ALL x.Q x))",
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                 "(ALL x. P x | Q)   = ((ALL x.P x) | Q)",
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                 "(ALL x. P | Q x)   = (P | (ALL x.Q x))",
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                 "(ALL x. P x --> Q) = ((EX x.P x) --> Q)",
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                 "(ALL x. P --> Q x) = (P --> (ALL x.Q x))"];
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(*** Simplification procedure for turning  ? x. ... & x = t & ...
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     into                                  ? x. x = t & ... & ...
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     where the latter can be rewritten via (? x. x = t & P(x)) = P(t)
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 ***)
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local
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fun def(eq as (c as Const("op =",_)) $ s $ t) =
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      if s = Bound 0 andalso not(loose_bvar1(t,0)) then Some eq else
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      if t = Bound 0 andalso not(loose_bvar1(s,0)) then Some(c$t$s)
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      else None
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  | def _ = None;
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fun extract(Const("op &",_) $ P $ Q) =
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      (case def P of
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         Some eq => Some(eq,Q)
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       | None => (case def Q of
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                   Some eq => Some(eq,P)
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                 | None =>
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       (case extract P of
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         Some(eq,P') => Some(eq, HOLogic.conj $ P' $ Q)
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       | None => (case extract Q of
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                   Some(eq,Q') => Some(eq,HOLogic.conj $ P $ Q')
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                 | None => None))))
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  | extract _ = None;
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fun prove_eq(ceqt) =
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  let val tac = rtac eq_reflection 1 THEN rtac iffI 1 THEN
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                ALLGOALS(EVERY'[etac exE, REPEAT o (etac conjE),
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                 rtac exI, REPEAT o (ares_tac [conjI] ORELSE' etac sym)])
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  in rule_by_tactic tac (trivial ceqt) end;
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fun rearrange sg _ (F as ex $ Abs(x,T,P)) =
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     (case extract P of
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        None => None
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      | Some(eq,Q) =>
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          let val ceqt = cterm_of sg
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                       (Logic.mk_equals(F,ex $ Abs(x,T,HOLogic.conj$eq$Q)))
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          in Some(prove_eq ceqt) end)
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  | rearrange _ _ _ = None;
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val pattern = read_cterm (sign_of HOL.thy) ("? x.P(x) & Q(x)",HOLogic.boolT)
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in
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val defEX_regroup = mk_simproc "defined EX" [pattern] rearrange;
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end;
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(* elimination of existential quantifiers in assumptions *)
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val ex_all_equiv =
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  let val lemma1 = prove_goal HOL.thy
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        "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
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        (fn prems => [resolve_tac prems 1, etac exI 1]);
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      val lemma2 = prove_goalw HOL.thy [Ex_def]
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        "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
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        (fn prems => [REPEAT(resolve_tac prems 1)])
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  in equal_intr lemma1 lemma2 end;
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end;
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fun prove nm thm  = qed_goal nm HOL.thy thm (fn _ => [blast_tac HOL_cs 1]);
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prove "conj_commute" "(P&Q) = (Q&P)";
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prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
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val conj_comms = [conj_commute, conj_left_commute];
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prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
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prove "disj_commute" "(P|Q) = (Q|P)";
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prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
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val disj_comms = [disj_commute, disj_left_commute];
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prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
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prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
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prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
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prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
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prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
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prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
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prove "imp_conjL" "((P&Q) -->R)  = (P --> (Q --> R))";
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prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
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(*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)
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prove "imp_disj_not1" "((P --> Q | R)) = (~Q --> P --> R)";
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prove "imp_disj_not2" "((P --> Q | R)) = (~R --> P --> Q)";
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prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
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prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
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prove "not_imp" "(~(P --> Q)) = (P & ~Q)";
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prove "not_iff" "(P~=Q) = (P = (~Q))";
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(*Avoids duplication of subgoals after expand_if, when the true and false 
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  cases boil down to the same thing.*) 
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prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";
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prove "not_all" "(~ (! x.P(x))) = (? x.~P(x))";
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prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
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prove "not_ex"  "(~ (? x.P(x))) = (! x.~P(x))";
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prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
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prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
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prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
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(* '&' congruence rule: not included by default!
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   May slow rewrite proofs down by as much as 50% *)
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let val th = prove_goal HOL.thy 
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                "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
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                (fn _=> [blast_tac HOL_cs 1])
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in  bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
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let val th = prove_goal HOL.thy 
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                "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
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                (fn _=> [blast_tac HOL_cs 1])
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in  bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
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(* '|' congruence rule: not included by default! *)
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let val th = prove_goal HOL.thy 
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                "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
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                (fn _=> [blast_tac HOL_cs 1])
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in  bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
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prove "eq_sym_conv" "(x=y) = (y=x)";
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qed_goalw "o_apply" HOL.thy [o_def] "(f o g) x = f (g x)"
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 (fn _ => [rtac refl 1]);
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qed_goal "meta_eq_to_obj_eq" HOL.thy "x==y ==> x=y"
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  (fn [prem] => [rewtac prem, rtac refl 1]);
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qed_goalw "if_True" HOL.thy [if_def] "(if True then x else y) = x"
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 (fn _=>[blast_tac (HOL_cs addIs [select_equality]) 1]);
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qed_goalw "if_False" HOL.thy [if_def] "(if False then x else y) = y"
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 (fn _=>[blast_tac (HOL_cs addIs [select_equality]) 1]);
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qed_goal "if_P" HOL.thy "P ==> (if P then x else y) = x"
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 (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
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(*
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qed_goal "if_not_P" HOL.thy "~P ==> (if P then x else y) = y"
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 (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
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*)
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qed_goalw "if_not_P" HOL.thy [if_def] "!!P. ~P ==> (if P then x else y) = y"
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 (fn _ => [blast_tac (HOL_cs addIs [select_equality]) 1]);
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qed_goal "expand_if" HOL.thy
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    "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
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 (fn _=> [ (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1),
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         stac if_P 2,
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         stac if_not_P 1,
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         REPEAT(blast_tac HOL_cs 1) ]);
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qed_goal "if_bool_eq" HOL.thy
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                   "(if P then Q else R) = ((P-->Q) & (~P-->R))"
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                   (fn _ => [rtac expand_if 1]);
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local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
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in
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fun split_tac splits = mktac (map mk_meta_eq splits)
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end;
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local val mktac = mk_case_split_inside_tac (meta_eq_to_obj_eq RS iffD2)
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in
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fun split_inside_tac splits = mktac (map mk_meta_eq splits)
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end;
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qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
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  (fn _ => [split_tac [expand_if] 1, blast_tac HOL_cs 1]);
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(** 'if' congruence rules: neither included by default! *)
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(*Simplifies x assuming c and y assuming ~c*)
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qed_goal "if_cong" HOL.thy
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  "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
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\  (if b then x else y) = (if c then u else v)"
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  (fn rew::prems =>
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   [stac rew 1, stac expand_if 1, stac expand_if 1,
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    blast_tac (HOL_cs addDs prems) 1]);
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(*Prevents simplification of x and y: much faster*)
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qed_goal "if_weak_cong" HOL.thy
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  "b=c ==> (if b then x else y) = (if c then x else y)"
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  (fn [prem] => [rtac (prem RS arg_cong) 1]);
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(*Prevents simplification of t: much faster*)
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qed_goal "let_weak_cong" HOL.thy
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  "a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
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  (fn [prem] => [rtac (prem RS arg_cong) 1]);
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(*In general it seems wrong to add distributive laws by default: they
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  might cause exponential blow-up.  But imp_disjL has been in for a while
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  and cannot be removed without affecting existing proofs.  Moreover, 
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  rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
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  grounds that it allows simplification of R in the two cases.*)
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val mksimps_pairs =
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  [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
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   ("All", [spec]), ("True", []), ("False", []),
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   ("If", [if_bool_eq RS iffD1])];
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fun unsafe_solver prems = FIRST'[resolve_tac (TrueI::refl::prems),
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				 atac, etac FalseE];
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(*No premature instantiation of variables during simplification*)
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fun   safe_solver prems = FIRST'[match_tac (TrueI::refl::prems),
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				 eq_assume_tac, ematch_tac [FalseE]];
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val HOL_basic_ss = empty_ss setsubgoaler asm_simp_tac
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			    setSSolver   safe_solver
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			    setSolver  unsafe_solver
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			    setmksimps (mksimps mksimps_pairs);
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val HOL_ss = 
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    HOL_basic_ss addsimps 
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     ([triv_forall_equality, (* prunes params *)
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       if_True, if_False, if_cancel,
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       o_apply, imp_disjL, conj_assoc, disj_assoc,
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       de_Morgan_conj, de_Morgan_disj, not_imp,
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       not_all, not_ex, cases_simp]
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     @ ex_simps @ all_simps @ simp_thms)
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     addsimprocs [defEX_regroup]
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     addcongs [imp_cong];
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qed_goal "if_distrib" HOL.thy
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  "f(if c then x else y) = (if c then f x else f y)" 
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  (fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
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qed_goalw "o_assoc" HOL.thy [o_def] "f o (g o h) = f o g o h"
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  (fn _ => [rtac ext 1, rtac refl 1]);
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val prems = goal HOL.thy "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
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by (case_tac "P" 1);
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by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
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val expand_case = result();
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fun expand_case_tac P i =
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    res_inst_tac [("P",P)] expand_case i THEN
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    Simp_tac (i+1) THEN 
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   382
    Simp_tac i;
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   383
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   384
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   385
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   386
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   387
(*** Install simpsets and datatypes in theory structure ***)
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   388
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simpset := HOL_ss;
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   390
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   391
exception SS_DATA of simpset;
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   392
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   393
let fun merge [] = SS_DATA empty_ss
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   394
      | merge ss = let val ss = map (fn SS_DATA x => x) ss;
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   395
                   in SS_DATA (foldl merge_ss (hd ss, tl ss)) end;
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   396
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   397
    fun put (SS_DATA ss) = simpset := ss;
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   398
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   399
    fun get () = SS_DATA (!simpset);
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   400
in add_thydata "HOL"
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   401
     ("simpset", ThyMethods {merge = merge, put = put, get = get})
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   402
end;
paulson@1984
   403
berghofe@3615
   404
fun simpset_of tname =
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   405
  case get_thydata tname "simpset" of
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   406
      None => empty_ss
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   407
    | Some (SS_DATA ss) => ss;
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   408
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   409
type dtype_info = {case_const:term,
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   410
                   case_rewrites:thm list,
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   411
                   constructors:term list,
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   412
                   induct_tac: string -> int -> tactic,
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   413
                   nchotomy: thm,
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   414
                   exhaustion: thm,
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   415
                   exhaust_tac: string -> int -> tactic,
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   416
                   case_cong:thm};
paulson@1984
   417
paulson@1984
   418
exception DT_DATA of (string * dtype_info) list;
paulson@1984
   419
val datatypes = ref [] : (string * dtype_info) list ref;
paulson@1984
   420
paulson@1984
   421
let fun merge [] = DT_DATA []
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   422
      | merge ds =
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   423
          let val ds = map (fn DT_DATA x => x) ds;
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   424
          in DT_DATA (foldl (gen_union eq_fst) (hd ds, tl ds)) end;
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   425
paulson@1984
   426
    fun put (DT_DATA ds) = datatypes := ds;
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   427
paulson@1984
   428
    fun get () = DT_DATA (!datatypes);
paulson@1984
   429
in add_thydata "HOL"
paulson@1984
   430
     ("datatypes", ThyMethods {merge = merge, put = put, get = get})
paulson@1984
   431
end;
paulson@1984
   432
paulson@1984
   433
oheimb@2636
   434
oheimb@2636
   435
(*** Integration of simplifier with classical reasoner ***)
oheimb@2636
   436
oheimb@2636
   437
(* rot_eq_tac rotates the first equality premise of subgoal i to the front,
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   438
   fails if there is no equaliy or if an equality is already at the front *)
paulson@3538
   439
local
paulson@3538
   440
  fun is_eq (Const ("Trueprop", _) $ (Const("op ="  ,_) $ _ $ _)) = true
paulson@3538
   441
    | is_eq _ = false;
paulson@3538
   442
  fun find_eq n [] = None
paulson@3538
   443
    | find_eq n (t :: ts) = if (is_eq t) then Some n 
paulson@3538
   444
			    else find_eq (n + 1) ts;
paulson@3538
   445
in
paulson@3538
   446
val rot_eq_tac = 
paulson@3538
   447
     SUBGOAL (fn (Bi,i) => 
paulson@3538
   448
	      case find_eq 0 (Logic.strip_assums_hyp Bi) of
paulson@2805
   449
		  None   => no_tac
paulson@2805
   450
		| Some 0 => no_tac
paulson@3538
   451
		| Some n => rotate_tac n i)
paulson@3538
   452
end;
oheimb@2636
   453
oheimb@2636
   454
(*an unsatisfactory fix for the incomplete asm_full_simp_tac!
oheimb@2636
   455
  better: asm_really_full_simp_tac, a yet to be implemented version of
oheimb@2636
   456
			asm_full_simp_tac that applies all equalities in the
oheimb@2636
   457
			premises to all the premises *)
oheimb@2636
   458
fun safe_asm_more_full_simp_tac ss = TRY o rot_eq_tac THEN' 
oheimb@2636
   459
				     safe_asm_full_simp_tac ss;
oheimb@2636
   460
oheimb@2636
   461
(*Add a simpset to a classical set!*)
oheimb@3206
   462
infix 4 addSss addss;
oheimb@3206
   463
fun cs addSss ss = cs addSaltern (CHANGED o (safe_asm_more_full_simp_tac ss));
oheimb@3206
   464
fun cs addss  ss = cs addbefore                        asm_full_simp_tac ss;
oheimb@2636
   465
oheimb@2636
   466
fun Addss ss = (claset := !claset addss ss);
oheimb@2636
   467
oheimb@2636
   468
(*Designed to be idempotent, except if best_tac instantiates variables
oheimb@2636
   469
  in some of the subgoals*)
oheimb@2636
   470
oheimb@2636
   471
type clasimpset = (claset * simpset);
oheimb@2636
   472
oheimb@2636
   473
val HOL_css = (HOL_cs, HOL_ss);
oheimb@2636
   474
oheimb@2636
   475
fun pair_upd1 f ((a,b),x) = (f(a,x), b);
oheimb@2636
   476
fun pair_upd2 f ((a,b),x) = (a, f(b,x));
oheimb@2636
   477
oheimb@2636
   478
infix 4 addSIs2 addSEs2 addSDs2 addIs2 addEs2 addDs2
oheimb@2636
   479
	addsimps2 delsimps2 addcongs2 delcongs2;
paulson@2748
   480
fun op addSIs2   arg = pair_upd1 (op addSIs) arg;
paulson@2748
   481
fun op addSEs2   arg = pair_upd1 (op addSEs) arg;
paulson@2748
   482
fun op addSDs2   arg = pair_upd1 (op addSDs) arg;
paulson@2748
   483
fun op addIs2    arg = pair_upd1 (op addIs ) arg;
paulson@2748
   484
fun op addEs2    arg = pair_upd1 (op addEs ) arg;
paulson@2748
   485
fun op addDs2    arg = pair_upd1 (op addDs ) arg;
paulson@2748
   486
fun op addsimps2 arg = pair_upd2 (op addsimps) arg;
paulson@2748
   487
fun op delsimps2 arg = pair_upd2 (op delsimps) arg;
paulson@2748
   488
fun op addcongs2 arg = pair_upd2 (op addcongs) arg;
paulson@2748
   489
fun op delcongs2 arg = pair_upd2 (op delcongs) arg;
oheimb@2636
   490
paulson@2805
   491
fun auto_tac (cs,ss) = 
paulson@2805
   492
    let val cs' = cs addss ss 
paulson@2805
   493
    in  EVERY [TRY (safe_tac cs'),
paulson@2805
   494
	       REPEAT (FIRSTGOAL (fast_tac cs')),
oheimb@3206
   495
               TRY (safe_tac (cs addSss ss)),
paulson@2805
   496
	       prune_params_tac] 
paulson@2805
   497
    end;
oheimb@2636
   498
oheimb@2636
   499
fun Auto_tac () = auto_tac (!claset, !simpset);
oheimb@2636
   500
oheimb@2636
   501
fun auto () = by (Auto_tac ());