src/HOL/simpdata.ML
author nipkow
Fri Mar 09 19:05:48 2001 +0100 (2001-03-09)
changeset 11200 f43fa07536c0
parent 11162 9e2ec5f02217
child 11220 db536a42dfc5
permissions -rw-r--r--
arith_tac now copes with propositional reasoning as well.
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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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val [prem] = goal (the_context ()) "x==y ==> x=y";
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by (rewtac prem);
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by (rtac refl 1);
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qed "meta_eq_to_obj_eq";
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Goal "(%s. f s) = f";
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br refl 1;
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qed "eta_contract_eq";
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local
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  fun prover s = prove_goal (the_context ()) s (fn _ => [(Blast_tac 1)]);
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in
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(*Make meta-equalities.  The operator below is Trueprop*)
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fun mk_meta_eq r = r RS eq_reflection;
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fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
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val Eq_TrueI  = mk_meta_eq(prover  "P --> (P = True)"  RS mp);
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val Eq_FalseI = mk_meta_eq(prover "~P --> (P = False)" RS mp);
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fun mk_eq th = case concl_of th of
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        Const("==",_)$_$_       => th
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    |   _$(Const("op =",_)$_$_) => mk_meta_eq th
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    |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
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    |   _                       => th RS Eq_TrueI;
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(* last 2 lines requires all formulae to be of the from Trueprop(.) *)
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fun mk_eq_True r = Some(r RS meta_eq_to_obj_eq RS Eq_TrueI);
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(*Congruence rules for = (instead of ==)*)
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fun mk_meta_cong rl =
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  standard(mk_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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val not_not = prover "(~ ~ P) = P";
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val simp_thms = [not_not] @ map prover
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 [ "(x=x) = True",
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   "(~True) = False", "(~False) = True",
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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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val imp_cong = standard(impI RSN
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    (2, prove_goal (the_context ()) "(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 (the_context ())
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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 (the_context ()) [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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bind_thms ("ex_simps", ex_simps);
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bind_thms ("all_simps", all_simps);
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bind_thm ("not_not", not_not);
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bind_thm ("imp_cong", imp_cong);
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(* Elimination of True from asumptions: *)
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val True_implies_equals = prove_goal (the_context ())
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 "(True ==> PROP P) == PROP P"
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(fn _ => [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 (the_context ()) thm (fn _ => [(Blast_tac 1)]);
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prove "eq_commute" "(a=b) = (b=a)";
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prove "eq_left_commute" "(P=(Q=R)) = (Q=(P=R))";
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prove "eq_assoc" "((P=Q)=R) = (P=(Q=R))";
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val eq_ac = [eq_commute, eq_left_commute, eq_assoc];
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prove "neq_commute" "(a~=b) = (b~=a)";
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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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prove "imp_conv_disj" "(P --> Q) = ((~P) | Q)";
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prove "iff_conv_conj_imp" "(P = Q) = ((P --> Q) & (Q --> P))";
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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 (the_context ())
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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 (the_context ())
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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 (the_context ())
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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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Goalw [if_def] "(if True then x else y) = x";
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by (Blast_tac 1);
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qed "if_True";
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Goalw [if_def] "(if False then x else y) = y";
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by (Blast_tac 1);
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qed "if_False";
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Goalw [if_def] "P ==> (if P then x else y) = x";
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by (Blast_tac 1);
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qed "if_P";
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Goalw [if_def] "~P ==> (if P then x else y) = y";
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by (Blast_tac 1);
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qed "if_not_P";
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Goal "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))";
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by (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1);
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by (stac if_P 2);
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by (stac if_not_P 1);
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by (ALLGOALS (Blast_tac));
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qed "split_if";
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Goal "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))";
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by (stac split_if 1);
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by (Blast_tac 1);
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qed "split_if_asm";
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bind_thms ("if_splits", [split_if, split_if_asm]);
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bind_thm ("if_def2", read_instantiate [("P","\\<lambda>x. x")] split_if);
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Goal "(if c then x else x) = x";
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by (stac split_if 1);
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by (Blast_tac 1);
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qed "if_cancel";
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Goal "(if x = y then y else x) = x";
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by (stac split_if 1);
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by (Blast_tac 1);
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qed "if_eq_cancel";
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(*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)
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Goal "(if P then Q else R) = ((P-->Q) & (~P-->R))";
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by (rtac split_if 1);
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qed "if_bool_eq_conj";
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(*And this form is useful for expanding IFs on the LEFT*)
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Goal "(if P then Q else R) = ((P&Q) | (~P&R))";
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by (stac split_if 1);
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by (Blast_tac 1);
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qed "if_bool_eq_disj";
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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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  Thm.read_cterm (Theory.sign_of (the_context ())) ("EX x. P(x) & Q(x)",HOLogic.boolT)
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val all_pattern =
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  Thm.read_cterm (Theory.sign_of (the_context ())) ("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_eq          = mk_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_nn
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  val contrapos2     = contrapos_pp
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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 op addsplits     = Splitter.addsplits;
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val op delsplits     = Splitter.delsplits;
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val Addsplits        = Splitter.Addsplits;
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val Delsplits        = Splitter.Delsplits;
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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_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_eq o mk_atomize pairs o forall_elim_vars_safe);
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fun unsafe_solver_tac prems =
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  FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
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val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
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(*No premature instantiation of variables during simplification*)
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fun safe_solver_tac prems =
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  FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
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         eq_assume_tac, ematch_tac [FalseE]];
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val safe_solver = mk_solver "HOL safe" safe_solver_tac;
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val HOL_basic_ss =
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  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_eq_True
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    setmkcong mk_meta_cong;
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   375
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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       eta_contract_eq, (* prunes eta-expansions *)
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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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   383
       disj_not1, not_all, not_ex, cases_simp, some_eq_trivial, some_sym_eq_trivial,
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   384
       thm"plus_ac0.zero", thm"plus_ac0_zero_right"]
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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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   388
     addsplits [split_if];
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   389
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   390
fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
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   391
fun hol_rewrite_cterm rews =
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  #2 o Thm.dest_comb o #prop o Thm.crep_thm o Simplifier.full_rewrite (HOL_basic_ss addsimps rews);
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   393
wenzelm@11034
   394
paulson@6293
   395
(*Simplifies x assuming c and y assuming ~c*)
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   396
val prems = Goalw [if_def]
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   397
  "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
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   398
\  (if b then x else y) = (if c then u else v)";
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   399
by (asm_simp_tac (HOL_ss addsimps prems) 1);
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   400
qed "if_cong";
paulson@6293
   401
paulson@7127
   402
(*Prevents simplification of x and y: faster and allows the execution
paulson@7127
   403
  of functional programs. NOW THE DEFAULT.*)
paulson@7031
   404
Goal "b=c ==> (if b then x else y) = (if c then x else y)";
paulson@7031
   405
by (etac arg_cong 1);
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   406
qed "if_weak_cong";
paulson@6293
   407
paulson@6293
   408
(*Prevents simplification of t: much faster*)
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   409
Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
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   410
by (etac arg_cong 1);
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   411
qed "let_weak_cong";
paulson@6293
   412
paulson@7031
   413
Goal "f(if c then x else y) = (if c then f x else f y)";
paulson@7031
   414
by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
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   415
qed "if_distrib";
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   416
paulson@4327
   417
(*For expand_case_tac*)
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   418
val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
paulson@2948
   419
by (case_tac "P" 1);
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   420
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
paulson@7584
   421
qed "expand_case";
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   422
paulson@4327
   423
(*Used in Auth proofs.  Typically P contains Vars that become instantiated
paulson@4327
   424
  during unification.*)
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   425
fun expand_case_tac P i =
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   426
    res_inst_tac [("P",P)] expand_case i THEN
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   427
    Simp_tac (i+1) THEN
paulson@2948
   428
    Simp_tac i;
paulson@2948
   429
paulson@7584
   430
(*This lemma restricts the effect of the rewrite rule u=v to the left-hand
paulson@7584
   431
  side of an equality.  Used in {Integ,Real}/simproc.ML*)
paulson@7584
   432
Goal "x=y ==> (x=z) = (y=z)";
paulson@7584
   433
by (asm_simp_tac HOL_ss 1);
paulson@7584
   434
qed "restrict_to_left";
paulson@2948
   435
wenzelm@7357
   436
(* default simpset *)
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   437
val simpsetup =
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   438
  [fn thy => (simpset_ref_of thy := HOL_ss addcongs [if_weak_cong]; thy)];
berghofe@3615
   439
oheimb@4652
   440
wenzelm@5219
   441
(*** integration of simplifier with classical reasoner ***)
oheimb@2636
   442
wenzelm@5219
   443
structure Clasimp = ClasimpFun
wenzelm@8473
   444
 (structure Simplifier = Simplifier and Splitter = Splitter
wenzelm@9851
   445
  and Classical  = Classical and Blast = Blast
wenzelm@9851
   446
  val dest_Trueprop = HOLogic.dest_Trueprop
wenzelm@9851
   447
  val iff_const = HOLogic.eq_const HOLogic.boolT
wenzelm@9851
   448
  val not_const = HOLogic.not_const
wenzelm@9851
   449
  val notE = notE val iffD1 = iffD1 val iffD2 = iffD2
wenzelm@9851
   450
  val cla_make_elim = cla_make_elim);
oheimb@4652
   451
open Clasimp;
oheimb@2636
   452
oheimb@2636
   453
val HOL_css = (HOL_cs, HOL_ss);
nipkow@5975
   454
nipkow@5975
   455
wenzelm@8641
   456
nipkow@5975
   457
(*** A general refutation procedure ***)
wenzelm@9713
   458
nipkow@5975
   459
(* Parameters:
nipkow@5975
   460
nipkow@5975
   461
   test: term -> bool
nipkow@5975
   462
   tests if a term is at all relevant to the refutation proof;
nipkow@5975
   463
   if not, then it can be discarded. Can improve performance,
nipkow@5975
   464
   esp. if disjunctions can be discarded (no case distinction needed!).
nipkow@5975
   465
nipkow@5975
   466
   prep_tac: int -> tactic
nipkow@5975
   467
   A preparation tactic to be applied to the goal once all relevant premises
nipkow@5975
   468
   have been moved to the conclusion.
nipkow@5975
   469
nipkow@5975
   470
   ref_tac: int -> tactic
nipkow@5975
   471
   the actual refutation tactic. Should be able to deal with goals
nipkow@5975
   472
   [| A1; ...; An |] ==> False
wenzelm@9876
   473
   where the Ai are atomic, i.e. no top-level &, | or EX
nipkow@5975
   474
*)
nipkow@5975
   475
nipkow@5975
   476
fun refute_tac test prep_tac ref_tac =
nipkow@5975
   477
  let val nnf_simps =
nipkow@5975
   478
        [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
nipkow@5975
   479
         not_all,not_ex,not_not];
nipkow@5975
   480
      val nnf_simpset =
nipkow@5975
   481
        empty_ss setmkeqTrue mk_eq_True
nipkow@5975
   482
                 setmksimps (mksimps mksimps_pairs)
nipkow@5975
   483
                 addsimps nnf_simps;
nipkow@5975
   484
      val prem_nnf_tac = full_simp_tac nnf_simpset;
nipkow@5975
   485
nipkow@5975
   486
      val refute_prems_tac =
nipkow@5975
   487
        REPEAT(eresolve_tac [conjE, exE] 1 ORELSE
nipkow@5975
   488
               filter_prems_tac test 1 ORELSE
paulson@6301
   489
               etac disjE 1) THEN
nipkow@11200
   490
        ((etac notE 1 THEN eq_assume_tac 1) ORELSE
nipkow@11200
   491
         ref_tac 1);
nipkow@5975
   492
  in EVERY'[TRY o filter_prems_tac test,
nipkow@6128
   493
            DETERM o REPEAT o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
nipkow@5975
   494
            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
nipkow@5975
   495
  end;