src/HOL/simpdata.ML
author oheimb
Wed Aug 12 16:21:18 1998 +0200 (1998-08-12)
changeset 5304 c133f16febc7
parent 5278 a903b66822e2
child 5307 6a699d5cdef4
permissions -rw-r--r--
the splitter is now defined as a functor
moved addsplits, delsplits, Addsplits, Delsplits to Provers/splitter.ML
moved split_thm_info to Provers/splitter.ML
definined atomize via general mk_atomize
removed superfluous rot_eq_tac from simplifier.ML
HOL/simpdata.ML: renamed mk_meta_eq to meta_eq,
re-renamed mk_meta_eq_simp to mk_meta_eq
added Eps_eq to simpset
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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 for HOL.
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*)
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section "Simplifier";
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(*** Addition of rules to simpsets and clasets simultaneously ***)
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infix 4 addIffs delIffs;
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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 ((cla, simp), 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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                    cla 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 cla addSIs [zero_var_indexes (th RS iffD2)]  
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                              addSDs [zero_var_indexes (th RS iffD1)]
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                    else  cla addSIs [th]
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              | _ => cla addSIs [th],
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       simp 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 ((cla, simp), 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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                    cla 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 cla delrules [zero_var_indexes (th RS iffD2),
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                                       make_elim (zero_var_indexes (th RS iffD1))]
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                    else cla delrules [th]
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              | _ => cla delrules [th],
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       simp delsimps [th])
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      handle _ => (warning("DelIffs: ignoring conditional theorem\n" ^ 
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                          string_of_thm th); (cla, simp));
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  fun store_clasimp (cla, simp) = (claset_ref () := cla; simpset_ref () := simp)
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in
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val op addIffs = foldl addIff;
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val op delIffs = foldl delIff;
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fun AddIffs thms = store_clasimp ((claset (), simpset ()) addIffs thms);
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fun DelIffs thms = store_clasimp ((claset (), simpset ()) delIffs thms);
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end;
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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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local
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  fun prover s = prove_goal HOL.thy s (K [Blast_tac 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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in
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  fun meta_eq r = r RS eq_reflection;
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  fun mk_meta_eq th = case concl_of th of
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          Const("==",_)$_$_       => th
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      |   _$(Const("op =",_)$_$_) => meta_eq th
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      |   _$(Const("Not",_)$_)    => th RS not_P_imp_P_eq_False
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      |   _                       => th 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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  fun mk_meta_eq_True r = Some(r RS meta_eq_to_obj_eq RS P_imp_P_eq_True);
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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", "(False=P) = (~P)", "(P=False) = (~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 & ~P) = False",    "(~P & P) = False",
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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 | ~P) = True",    "(~P | P) = True",
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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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(*two needed for the one-point-rule quantifier simplification procs*)
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   "(? x. x=t & P(x)) = P(t)",		(*essential for termination!!*)
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   "(! x. t=x --> P(x)) = P(t)" ];      (*covers a stray case*)
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(*Add congruence rules for = (instead of ==) *)
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infix 4 addcongs delcongs;
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fun mk_meta_cong rl =
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  standard(meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
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  handle THM _ =>
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  error("Premises and conclusion of congruence rules must be =-equalities");
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fun ss addcongs congs = ss addeqcongs (map mk_meta_cong congs);
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fun ss delcongs congs = ss deleqcongs (map mk_meta_cong congs);
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fun Addcongs congs = (simpset_ref() := simpset() addcongs congs);
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fun Delcongs congs = (simpset_ref() := simpset() delcongs congs);
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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 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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(* 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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(* Elimination of True from asumptions: *)
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val True_implies_equals = prove_goal HOL.thy
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 "(True ==> PROP P) == PROP P"
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(K [rtac equal_intr_rule 1, atac 2,
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          METAHYPS (fn prems => resolve_tac prems 1) 1,
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          rtac TrueI 1]);
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fun prove nm thm  = qed_goal nm HOL.thy thm (K [Blast_tac 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 "imp_disj1" "((P-->Q)|R) = (P--> Q|R)";
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prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)";
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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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prove "disj_not1" "(~P | Q) = (P --> Q)";
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prove "disj_not2" "(P | ~Q) = (Q --> P)"; (* changes orientation :-( *)
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(*Avoids duplication of subgoals after split_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 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 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 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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(** if-then-else rules **)
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qed_goalw "if_True" HOL.thy [if_def] "(if True then x else y) = x"
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 (K [Blast_tac 1]);
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qed_goalw "if_False" HOL.thy [if_def] "(if False then x else y) = y"
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 (K [Blast_tac 1]);
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qed_goalw "if_P" HOL.thy [if_def] "!!P. P ==> (if P then x else y) = x"
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 (K [Blast_tac 1]);
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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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 (K [Blast_tac 1]);
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qed_goal "split_if" HOL.thy
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    "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))" (K [
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	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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         ALLGOALS (Blast_tac)]);
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(* for backwards compatibility: *)
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val expand_if = split_if;
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qed_goal "split_if_asm" HOL.thy
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    "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
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    (K [stac split_if 1,
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	Blast_tac 1]);
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qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
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  (K [stac split_if 1, Blast_tac 1]);
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qed_goal "if_eq_cancel" HOL.thy "(if x = y then y else x) = x"
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  (K [stac split_if 1, Blast_tac 1]);
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(*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)
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qed_goal "if_bool_eq_conj" HOL.thy
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    "(if P then Q else R) = ((P-->Q) & (~P-->R))"
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    (K [rtac split_if 1]);
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(*And this form is useful for expanding IFs on the LEFT*)
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qed_goal "if_bool_eq_disj" HOL.thy
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    "(if P then Q else R) = ((P&Q) | (~P&R))"
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    (K [stac split_if 1,
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	Blast_tac 1]);
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(*** make simplification procedures for quantifier elimination ***)
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structure Quantifier1 = Quantifier1Fun(
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struct
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  (*abstract syntax*)
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  fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
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    | dest_eq _ = None;
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  fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
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    | dest_conj _ = None;
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  val conj = HOLogic.conj
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  val imp  = HOLogic.imp
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  (*rules*)
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  val iff_reflection = eq_reflection
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  val iffI = iffI
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  val sym  = sym
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  val conjI= conjI
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  val conjE= conjE
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  val impI = impI
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  val impE = impE
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  val mp   = mp
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  val exI  = exI
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  val exE  = exE
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  val allI = allI
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  val allE = allE
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end);
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local
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val ex_pattern =
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  read_cterm (sign_of HOL.thy) ("EX x. P(x) & Q(x)",HOLogic.boolT)
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val all_pattern =
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  read_cterm (sign_of HOL.thy) ("ALL x. P(x) & P'(x) --> Q(x)",HOLogic.boolT)
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in
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val defEX_regroup =
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  mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
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val defALL_regroup =
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  mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
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end;
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(*** Case splitting ***)
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structure SplitterData =
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  struct
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  structure Simplifier = Simplifier
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  val mk_meta_eq     = mk_meta_eq
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  val meta_eq_to_iff = meta_eq_to_obj_eq
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  val iffD           = iffD2
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  val disjE          = disjE
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  val conjE          = conjE
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  val exE            = exE
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  val contrapos      = contrapos
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  val contrapos2     = contrapos2
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  val notnotD        = notnotD
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  end;
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structure Splitter = SplitterFun(SplitterData);
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val split_tac        = Splitter.split_tac;
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val split_inside_tac = Splitter.split_inside_tac;
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val split_asm_tac    = Splitter.split_asm_tac;
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val addsplits        = Splitter.addsplits;
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val delsplits        = Splitter.delsplits;
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val Addsplits        = Splitter.Addsplits;
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val Delsplits        = Splitter.Delsplits;
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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 split_if 1, stac split_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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fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
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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_conj RS iffD1])];
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(* FIXME: move to Provers/simplifier.ML
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val mk_atomize:      (string * thm list) list -> thm -> thm list
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*)
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(* FIXME: move to Provers/simplifier.ML*)
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fun mk_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 mksimps pairs = (map mk_meta_eq o mk_atomize pairs o gen_all);
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fun unsafe_solver prems = FIRST'[resolve_tac (reflexive_thm::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 (reflexive_thm::TrueI::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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			    setmkeqTrue mk_meta_eq_True;
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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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       True_implies_equals, (* prune asms `True' *)
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       if_True, if_False, if_cancel, if_eq_cancel,
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       imp_disjL, conj_assoc, disj_assoc,
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       de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
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       disj_not1, not_all, not_ex, cases_simp, Eps_eq]
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     @ ex_simps @ all_simps @ simp_thms)
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     addsimprocs [defALL_regroup,defEX_regroup]
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     addcongs [imp_cong]
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     addsplits [split_if];
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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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  (K [simp_tac (HOL_ss setloop (split_tac [split_if])) 1]);
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(*For expand_case_tac*)
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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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(*Used in Auth proofs.  Typically P contains Vars that become instantiated
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  during unification.*)
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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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    Simp_tac i;
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(* install implicit simpset *)
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simpset_ref() := HOL_ss;
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(*** integration of simplifier with classical reasoner ***)
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structure Clasimp = ClasimpFun
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 (structure Simplifier = Simplifier and Classical = Classical and Blast = Blast
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  val op addcongs = op addcongs and op delcongs = op delcongs
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  and op addSaltern = op addSaltern and op addbefore = op addbefore);
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open Clasimp;
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val HOL_css = (HOL_cs, HOL_ss);