clasohm@1465
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(* Title: HOL/simpdata.ML
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clasohm@923
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ID: $Id$
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clasohm@1465
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Author: Tobias Nipkow
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clasohm@923
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Copyright 1991 University of Cambridge
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clasohm@923
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oheimb@5304
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Instantiation of the generic simplifier for HOL.
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clasohm@923
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*)
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clasohm@923
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paulson@1984
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section "Simplifier";
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paulson@1984
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wenzelm@7357
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val [prem] = goal (the_context ()) "x==y ==> x=y";
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paulson@7031
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by (rewtac prem);
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paulson@7031
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by (rtac refl 1);
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paulson@7031
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qed "meta_eq_to_obj_eq";
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nipkow@4640
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oheimb@9023
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Goal "(%s. f s) = f";
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oheimb@9023
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br refl 1;
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oheimb@9023
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qed "eta_contract_eq";
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oheimb@9023
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clasohm@923
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local
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clasohm@923
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wenzelm@7357
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fun prover s = prove_goal (the_context ()) s (fn _ => [(Blast_tac 1)]);
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clasohm@923
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nipkow@2134
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in
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nipkow@2134
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oheimb@5552
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(*Make meta-equalities. The operator below is Trueprop*)
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oheimb@5304
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nipkow@6128
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fun mk_meta_eq r = r RS eq_reflection;
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wenzelm@9832
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fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
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oheimb@5552
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nipkow@6128
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val Eq_TrueI = mk_meta_eq(prover "P --> (P = True)" RS mp);
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nipkow@6128
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val Eq_FalseI = mk_meta_eq(prover "~P --> (P = False)" RS mp);
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oheimb@5304
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nipkow@6128
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fun mk_eq th = case concl_of th of
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nipkow@6128
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Const("==",_)$_$_ => th
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nipkow@6128
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| _$(Const("op =",_)$_$_) => mk_meta_eq th
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nipkow@6128
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| _$(Const("Not",_)$_) => th RS Eq_FalseI
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nipkow@6128
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| _ => th RS Eq_TrueI;
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nipkow@6128
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(* last 2 lines requires all formulae to be of the from Trueprop(.) *)
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oheimb@5552
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nipkow@6128
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fun mk_eq_True r = Some(r RS meta_eq_to_obj_eq RS Eq_TrueI);
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nipkow@6128
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wenzelm@9713
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(*Congruence rules for = (instead of ==)*)
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nipkow@6128
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fun mk_meta_cong rl =
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nipkow@6128
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standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
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nipkow@6128
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handle THM _ =>
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nipkow@6128
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error("Premises and conclusion of congruence rules must be =-equalities");
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nipkow@3896
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nipkow@5975
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val not_not = prover "(~ ~ P) = P";
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clasohm@923
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nipkow@5975
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val simp_thms = [not_not] @ map prover
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paulson@2082
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[ "(x=x) = True",
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nipkow@5975
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"(~True) = False", "(~False) = True",
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paulson@2082
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"(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
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nipkow@4640
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"(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)",
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wenzelm@9713
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"(True --> P) = P", "(False --> P) = True",
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paulson@2082
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"(P --> True) = True", "(P --> P) = True",
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paulson@2082
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"(P --> False) = (~P)", "(P --> ~P) = (~P)",
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wenzelm@9713
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"(P & True) = P", "(True & P) = P",
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nipkow@2800
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"(P & False) = False", "(False & P) = False",
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nipkow@2800
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"(P & P) = P", "(P & (P & Q)) = (P & Q)",
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paulson@3913
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"(P & ~P) = False", "(~P & P) = False",
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wenzelm@9713
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"(P | True) = True", "(True | P) = True",
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nipkow@2800
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"(P | False) = P", "(False | P) = P",
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nipkow@2800
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"(P | P) = P", "(P | (P | Q)) = (P | Q)",
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paulson@3913
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"(P | ~P) = True", "(~P | P) = True",
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paulson@2082
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"((~P) = (~Q)) = (P=Q)",
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wenzelm@9713
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"(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x",
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paulson@4351
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(*two needed for the one-point-rule quantifier simplification procs*)
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wenzelm@9713
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"(? x. x=t & P(x)) = P(t)", (*essential for termination!!*)
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paulson@4351
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"(! x. t=x --> P(x)) = P(t)" ]; (*covers a stray case*)
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clasohm@923
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nipkow@9875
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val imp_cong = standard(impI RSN
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wenzelm@7357
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(2, prove_goal (the_context ()) "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
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nipkow@9875
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(fn _=> [(Blast_tac 1)]) RS mp RS mp));
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paulson@1922
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paulson@1948
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(*Miniscoping: pushing in existential quantifiers*)
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wenzelm@7648
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val ex_simps = map prover
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wenzelm@3842
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["(EX x. P x & Q) = ((EX x. P x) & Q)",
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wenzelm@3842
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"(EX x. P & Q x) = (P & (EX x. Q x))",
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wenzelm@3842
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"(EX x. P x | Q) = ((EX x. P x) | Q)",
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wenzelm@3842
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"(EX x. P | Q x) = (P | (EX x. Q x))",
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wenzelm@3842
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"(EX x. P x --> Q) = ((ALL x. P x) --> Q)",
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wenzelm@3842
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"(EX x. P --> Q x) = (P --> (EX x. Q x))"];
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paulson@1948
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paulson@1948
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(*Miniscoping: pushing in universal quantifiers*)
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paulson@1948
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val all_simps = map prover
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wenzelm@3842
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["(ALL x. P x & Q) = ((ALL x. P x) & Q)",
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wenzelm@3842
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"(ALL x. P & Q x) = (P & (ALL x. Q x))",
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wenzelm@3842
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"(ALL x. P x | Q) = ((ALL x. P x) | Q)",
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wenzelm@3842
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"(ALL x. P | Q x) = (P | (ALL x. Q x))",
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wenzelm@3842
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"(ALL x. P x --> Q) = ((EX x. P x) --> Q)",
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wenzelm@3842
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"(ALL x. P --> Q x) = (P --> (ALL x. Q x))"];
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paulson@1948
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clasohm@923
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paulson@2022
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(* elimination of existential quantifiers in assumptions *)
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clasohm@923
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clasohm@923
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val ex_all_equiv =
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wenzelm@7357
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let val lemma1 = prove_goal (the_context ())
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clasohm@923
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"(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
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clasohm@923
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(fn prems => [resolve_tac prems 1, etac exI 1]);
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wenzelm@7357
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val lemma2 = prove_goalw (the_context ()) [Ex_def]
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clasohm@923
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"(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
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paulson@7031
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(fn prems => [(REPEAT(resolve_tac prems 1))])
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clasohm@923
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in equal_intr lemma1 lemma2 end;
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clasohm@923
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clasohm@923
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end;
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clasohm@923
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wenzelm@7648
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bind_thms ("ex_simps", ex_simps);
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wenzelm@7648
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bind_thms ("all_simps", all_simps);
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berghofe@7711
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bind_thm ("not_not", not_not);
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nipkow@9875
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bind_thm ("imp_cong", imp_cong);
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wenzelm@7648
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nipkow@3654
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(* Elimination of True from asumptions: *)
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nipkow@3654
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wenzelm@7357
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val True_implies_equals = prove_goal (the_context ())
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nipkow@3654
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"(True ==> PROP P) == PROP P"
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paulson@7031
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(fn _ => [rtac equal_intr_rule 1, atac 2,
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nipkow@3654
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METAHYPS (fn prems => resolve_tac prems 1) 1,
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nipkow@3654
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rtac TrueI 1]);
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nipkow@3654
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wenzelm@7357
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fun prove nm thm = qed_goal nm (the_context ()) thm (fn _ => [(Blast_tac 1)]);
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clasohm@923
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paulson@9511
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prove "eq_commute" "(a=b) = (b=a)";
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paulson@7623
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prove "eq_left_commute" "(P=(Q=R)) = (Q=(P=R))";
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paulson@7623
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prove "eq_assoc" "((P=Q)=R) = (P=(Q=R))";
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paulson@7623
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val eq_ac = [eq_commute, eq_left_commute, eq_assoc];
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paulson@7623
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paulson@9511
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prove "neq_commute" "(a~=b) = (b~=a)";
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paulson@9511
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clasohm@923
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prove "conj_commute" "(P&Q) = (Q&P)";
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clasohm@923
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prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
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clasohm@923
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val conj_comms = [conj_commute, conj_left_commute];
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nipkow@2134
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prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
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clasohm@923
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paulson@1922
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prove "disj_commute" "(P|Q) = (Q|P)";
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paulson@1922
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prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
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paulson@1922
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val disj_comms = [disj_commute, disj_left_commute];
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nipkow@2134
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prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
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paulson@1922
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clasohm@923
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prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
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clasohm@923
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prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
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nipkow@1485
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paulson@1892
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prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
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paulson@1892
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prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
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paulson@1892
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nipkow@2134
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prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
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nipkow@2134
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prove "imp_conjL" "((P&Q) -->R) = (P --> (Q --> R))";
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nipkow@2134
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prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
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paulson@1892
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paulson@3448
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(*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)
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paulson@8114
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prove "imp_disj_not1" "(P --> Q | R) = (~Q --> P --> R)";
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paulson@8114
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prove "imp_disj_not2" "(P --> Q | R) = (~R --> P --> Q)";
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paulson@3448
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paulson@3904
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prove "imp_disj1" "((P-->Q)|R) = (P--> Q|R)";
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paulson@3904
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prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)";
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paulson@3904
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nipkow@1485
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prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
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nipkow@1485
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prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
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paulson@3446
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prove "not_imp" "(~(P --> Q)) = (P & ~Q)";
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paulson@1922
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prove "not_iff" "(P~=Q) = (P = (~Q))";
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oheimb@4743
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prove "disj_not1" "(~P | Q) = (P --> Q)";
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oheimb@4743
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prove "disj_not2" "(P | ~Q) = (Q --> P)"; (* changes orientation :-( *)
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nipkow@5975
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prove "imp_conv_disj" "(P --> Q) = ((~P) | Q)";
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nipkow@5975
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nipkow@5975
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prove "iff_conv_conj_imp" "(P = Q) = ((P --> Q) & (Q --> P))";
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nipkow@5975
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nipkow@1485
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wenzelm@9713
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(*Avoids duplication of subgoals after split_if, when the true and false
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wenzelm@9713
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cases boil down to the same thing.*)
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nipkow@2134
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prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";
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nipkow@2134
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wenzelm@3842
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prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))";
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paulson@1922
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prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
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wenzelm@3842
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prove "not_ex" "(~ (? x. P(x))) = (! x.~P(x))";
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paulson@1922
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prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
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oheimb@1660
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nipkow@1655
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prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
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nipkow@1655
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prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
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nipkow@1655
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nipkow@2134
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(* '&' congruence rule: not included by default!
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nipkow@2134
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May slow rewrite proofs down by as much as 50% *)
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nipkow@2134
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wenzelm@9713
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let val th = prove_goal (the_context ())
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nipkow@2134
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"(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
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paulson@7031
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(fn _=> [(Blast_tac 1)])
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nipkow@2134
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in bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp))) end;
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nipkow@2134
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wenzelm@9713
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let val th = prove_goal (the_context ())
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nipkow@2134
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"(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
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paulson@7031
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(fn _=> [(Blast_tac 1)])
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nipkow@2134
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in bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp))) end;
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nipkow@2134
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nipkow@2134
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(* '|' congruence rule: not included by default! *)
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nipkow@2134
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195 |
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wenzelm@9713
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let val th = prove_goal (the_context ())
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nipkow@2134
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"(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
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paulson@7031
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(fn _=> [(Blast_tac 1)])
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nipkow@2134
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in bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp))) end;
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nipkow@2134
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200 |
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nipkow@2134
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201 |
prove "eq_sym_conv" "(x=y) = (y=x)";
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nipkow@2134
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202 |
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paulson@5278
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203 |
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paulson@5278
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204 |
(** if-then-else rules **)
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paulson@5278
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205 |
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paulson@7031
|
206 |
Goalw [if_def] "(if True then x else y) = x";
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paulson@7031
|
207 |
by (Blast_tac 1);
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paulson@7031
|
208 |
qed "if_True";
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nipkow@2134
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209 |
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paulson@7031
|
210 |
Goalw [if_def] "(if False then x else y) = y";
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paulson@7031
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211 |
by (Blast_tac 1);
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paulson@7031
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212 |
qed "if_False";
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nipkow@2134
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213 |
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paulson@7127
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214 |
Goalw [if_def] "P ==> (if P then x else y) = x";
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paulson@7031
|
215 |
by (Blast_tac 1);
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paulson@7031
|
216 |
qed "if_P";
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oheimb@5304
|
217 |
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paulson@7127
|
218 |
Goalw [if_def] "~P ==> (if P then x else y) = y";
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paulson@7031
|
219 |
by (Blast_tac 1);
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paulson@7031
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220 |
qed "if_not_P";
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nipkow@2134
|
221 |
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paulson@7031
|
222 |
Goal "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))";
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paulson@7031
|
223 |
by (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1);
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paulson@7031
|
224 |
by (stac if_P 2);
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paulson@7031
|
225 |
by (stac if_not_P 1);
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paulson@7031
|
226 |
by (ALLGOALS (Blast_tac));
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paulson@7031
|
227 |
qed "split_if";
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paulson@7031
|
228 |
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paulson@7031
|
229 |
Goal "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))";
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paulson@7031
|
230 |
by (stac split_if 1);
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paulson@7031
|
231 |
by (Blast_tac 1);
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paulson@7031
|
232 |
qed "split_if_asm";
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nipkow@2134
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233 |
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wenzelm@9384
|
234 |
bind_thms ("if_splits", [split_if, split_if_asm]);
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wenzelm@9384
|
235 |
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oheimb@11003
|
236 |
bind_thm ("if_def2", read_instantiate [("P","\\<lambda>x. x")] split_if);
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oheimb@11003
|
237 |
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paulson@7031
|
238 |
Goal "(if c then x else x) = x";
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paulson@7031
|
239 |
by (stac split_if 1);
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paulson@7031
|
240 |
by (Blast_tac 1);
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paulson@7031
|
241 |
qed "if_cancel";
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oheimb@5304
|
242 |
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paulson@7031
|
243 |
Goal "(if x = y then y else x) = x";
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paulson@7031
|
244 |
by (stac split_if 1);
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paulson@7031
|
245 |
by (Blast_tac 1);
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paulson@7031
|
246 |
qed "if_eq_cancel";
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oheimb@5304
|
247 |
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paulson@4769
|
248 |
(*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)
|
paulson@7127
|
249 |
Goal "(if P then Q else R) = ((P-->Q) & (~P-->R))";
|
paulson@7031
|
250 |
by (rtac split_if 1);
|
paulson@7031
|
251 |
qed "if_bool_eq_conj";
|
paulson@4769
|
252 |
|
paulson@4769
|
253 |
(*And this form is useful for expanding IFs on the LEFT*)
|
paulson@7031
|
254 |
Goal "(if P then Q else R) = ((P&Q) | (~P&R))";
|
paulson@7031
|
255 |
by (stac split_if 1);
|
paulson@7031
|
256 |
by (Blast_tac 1);
|
paulson@7031
|
257 |
qed "if_bool_eq_disj";
|
nipkow@2134
|
258 |
|
paulson@4351
|
259 |
|
paulson@4351
|
260 |
(*** make simplification procedures for quantifier elimination ***)
|
paulson@4351
|
261 |
|
wenzelm@9851
|
262 |
structure Quantifier1 = Quantifier1Fun
|
wenzelm@9851
|
263 |
(struct
|
paulson@4351
|
264 |
(*abstract syntax*)
|
paulson@4351
|
265 |
fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t)
|
paulson@4351
|
266 |
| dest_eq _ = None;
|
paulson@4351
|
267 |
fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t)
|
paulson@4351
|
268 |
| dest_conj _ = None;
|
paulson@4351
|
269 |
val conj = HOLogic.conj
|
paulson@4351
|
270 |
val imp = HOLogic.imp
|
paulson@4351
|
271 |
(*rules*)
|
paulson@4351
|
272 |
val iff_reflection = eq_reflection
|
paulson@4351
|
273 |
val iffI = iffI
|
paulson@4351
|
274 |
val sym = sym
|
paulson@4351
|
275 |
val conjI= conjI
|
paulson@4351
|
276 |
val conjE= conjE
|
paulson@4351
|
277 |
val impI = impI
|
paulson@4351
|
278 |
val impE = impE
|
paulson@4351
|
279 |
val mp = mp
|
paulson@4351
|
280 |
val exI = exI
|
paulson@4351
|
281 |
val exE = exE
|
paulson@4351
|
282 |
val allI = allI
|
paulson@4351
|
283 |
val allE = allE
|
paulson@4351
|
284 |
end);
|
paulson@4351
|
285 |
|
nipkow@4320
|
286 |
local
|
nipkow@4320
|
287 |
val ex_pattern =
|
wenzelm@7357
|
288 |
Thm.read_cterm (Theory.sign_of (the_context ())) ("EX x. P(x) & Q(x)",HOLogic.boolT)
|
paulson@3913
|
289 |
|
nipkow@4320
|
290 |
val all_pattern =
|
wenzelm@7357
|
291 |
Thm.read_cterm (Theory.sign_of (the_context ())) ("ALL x. P(x) & P'(x) --> Q(x)",HOLogic.boolT)
|
nipkow@4320
|
292 |
|
nipkow@4320
|
293 |
in
|
nipkow@4320
|
294 |
val defEX_regroup =
|
nipkow@4320
|
295 |
mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
|
nipkow@4320
|
296 |
val defALL_regroup =
|
nipkow@4320
|
297 |
mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
|
nipkow@4320
|
298 |
end;
|
paulson@3913
|
299 |
|
paulson@4351
|
300 |
|
paulson@4351
|
301 |
(*** Case splitting ***)
|
paulson@3913
|
302 |
|
oheimb@5304
|
303 |
structure SplitterData =
|
oheimb@5304
|
304 |
struct
|
oheimb@5304
|
305 |
structure Simplifier = Simplifier
|
oheimb@5552
|
306 |
val mk_eq = mk_eq
|
oheimb@5304
|
307 |
val meta_eq_to_iff = meta_eq_to_obj_eq
|
oheimb@5304
|
308 |
val iffD = iffD2
|
oheimb@5304
|
309 |
val disjE = disjE
|
oheimb@5304
|
310 |
val conjE = conjE
|
oheimb@5304
|
311 |
val exE = exE
|
paulson@10231
|
312 |
val contrapos = contrapos_nn
|
paulson@10231
|
313 |
val contrapos2 = contrapos_pp
|
oheimb@5304
|
314 |
val notnotD = notnotD
|
oheimb@5304
|
315 |
end;
|
oheimb@2263
|
316 |
|
oheimb@5304
|
317 |
structure Splitter = SplitterFun(SplitterData);
|
oheimb@2263
|
318 |
|
oheimb@5304
|
319 |
val split_tac = Splitter.split_tac;
|
oheimb@5304
|
320 |
val split_inside_tac = Splitter.split_inside_tac;
|
oheimb@5304
|
321 |
val split_asm_tac = Splitter.split_asm_tac;
|
oheimb@5307
|
322 |
val op addsplits = Splitter.addsplits;
|
oheimb@5307
|
323 |
val op delsplits = Splitter.delsplits;
|
oheimb@5304
|
324 |
val Addsplits = Splitter.Addsplits;
|
oheimb@5304
|
325 |
val Delsplits = Splitter.Delsplits;
|
oheimb@4718
|
326 |
|
nipkow@2134
|
327 |
(*In general it seems wrong to add distributive laws by default: they
|
nipkow@2134
|
328 |
might cause exponential blow-up. But imp_disjL has been in for a while
|
wenzelm@9713
|
329 |
and cannot be removed without affecting existing proofs. Moreover,
|
nipkow@2134
|
330 |
rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
|
nipkow@2134
|
331 |
grounds that it allows simplification of R in the two cases.*)
|
nipkow@2134
|
332 |
|
nipkow@2134
|
333 |
val mksimps_pairs =
|
nipkow@2134
|
334 |
[("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
|
nipkow@2134
|
335 |
("All", [spec]), ("True", []), ("False", []),
|
paulson@4769
|
336 |
("If", [if_bool_eq_conj RS iffD1])];
|
nipkow@1758
|
337 |
|
oheimb@5552
|
338 |
(* ###FIXME: move to Provers/simplifier.ML
|
oheimb@5304
|
339 |
val mk_atomize: (string * thm list) list -> thm -> thm list
|
oheimb@5304
|
340 |
*)
|
oheimb@5552
|
341 |
(* ###FIXME: move to Provers/simplifier.ML *)
|
oheimb@5304
|
342 |
fun mk_atomize pairs =
|
oheimb@5304
|
343 |
let fun atoms th =
|
oheimb@5304
|
344 |
(case concl_of th of
|
oheimb@5304
|
345 |
Const("Trueprop",_) $ p =>
|
oheimb@5304
|
346 |
(case head_of p of
|
oheimb@5304
|
347 |
Const(a,_) =>
|
oheimb@5304
|
348 |
(case assoc(pairs,a) of
|
oheimb@5304
|
349 |
Some(rls) => flat (map atoms ([th] RL rls))
|
oheimb@5304
|
350 |
| None => [th])
|
oheimb@5304
|
351 |
| _ => [th])
|
oheimb@5304
|
352 |
| _ => [th])
|
oheimb@5304
|
353 |
in atoms end;
|
oheimb@5304
|
354 |
|
oheimb@11162
|
355 |
fun mksimps pairs = (map mk_eq o mk_atomize pairs o forall_elim_vars_safe);
|
oheimb@5304
|
356 |
|
nipkow@7570
|
357 |
fun unsafe_solver_tac prems =
|
nipkow@7570
|
358 |
FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
|
nipkow@7570
|
359 |
val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
|
nipkow@7570
|
360 |
|
oheimb@2636
|
361 |
(*No premature instantiation of variables during simplification*)
|
nipkow@7570
|
362 |
fun safe_solver_tac prems =
|
nipkow@7570
|
363 |
FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
|
nipkow@7570
|
364 |
eq_assume_tac, ematch_tac [FalseE]];
|
nipkow@7570
|
365 |
val safe_solver = mk_solver "HOL safe" safe_solver_tac;
|
oheimb@2443
|
366 |
|
wenzelm@9713
|
367 |
val HOL_basic_ss =
|
wenzelm@9713
|
368 |
empty_ss setsubgoaler asm_simp_tac
|
wenzelm@9713
|
369 |
setSSolver safe_solver
|
wenzelm@9713
|
370 |
setSolver unsafe_solver
|
wenzelm@9713
|
371 |
setmksimps (mksimps mksimps_pairs)
|
wenzelm@9713
|
372 |
setmkeqTrue mk_eq_True
|
wenzelm@9713
|
373 |
setmkcong mk_meta_cong;
|
oheimb@2443
|
374 |
|
wenzelm@9713
|
375 |
val HOL_ss =
|
wenzelm@9713
|
376 |
HOL_basic_ss addsimps
|
paulson@3446
|
377 |
([triv_forall_equality, (* prunes params *)
|
nipkow@3654
|
378 |
True_implies_equals, (* prune asms `True' *)
|
oheimb@9023
|
379 |
eta_contract_eq, (* prunes eta-expansions *)
|
oheimb@4718
|
380 |
if_True, if_False, if_cancel, if_eq_cancel,
|
oheimb@5304
|
381 |
imp_disjL, conj_assoc, disj_assoc,
|
paulson@3904
|
382 |
de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
|
paulson@9969
|
383 |
disj_not1, not_all, not_ex, cases_simp, some_eq_trivial, some_sym_eq_trivial,
|
paulson@8955
|
384 |
thm"plus_ac0.zero", thm"plus_ac0_zero_right"]
|
paulson@3446
|
385 |
@ ex_simps @ all_simps @ simp_thms)
|
nipkow@4032
|
386 |
addsimprocs [defALL_regroup,defEX_regroup]
|
wenzelm@4744
|
387 |
addcongs [imp_cong]
|
nipkow@4830
|
388 |
addsplits [split_if];
|
paulson@2082
|
389 |
|
wenzelm@11034
|
390 |
fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
|
wenzelm@11034
|
391 |
fun hol_rewrite_cterm rews =
|
wenzelm@11034
|
392 |
#2 o Thm.dest_comb o #prop o Thm.crep_thm o Simplifier.full_rewrite (HOL_basic_ss addsimps rews);
|
wenzelm@11034
|
393 |
|
wenzelm@11034
|
394 |
|
paulson@6293
|
395 |
(*Simplifies x assuming c and y assuming ~c*)
|
paulson@6293
|
396 |
val prems = Goalw [if_def]
|
paulson@6293
|
397 |
"[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
|
paulson@6293
|
398 |
\ (if b then x else y) = (if c then u else v)";
|
paulson@6293
|
399 |
by (asm_simp_tac (HOL_ss addsimps prems) 1);
|
paulson@6293
|
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);
|
paulson@7031
|
406 |
qed "if_weak_cong";
|
paulson@6293
|
407 |
|
paulson@6293
|
408 |
(*Prevents simplification of t: much faster*)
|
paulson@7031
|
409 |
Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
|
paulson@7031
|
410 |
by (etac arg_cong 1);
|
paulson@7031
|
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);
|
paulson@7031
|
415 |
qed "if_distrib";
|
nipkow@1655
|
416 |
|
paulson@4327
|
417 |
(*For expand_case_tac*)
|
paulson@7584
|
418 |
val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
|
paulson@2948
|
419 |
by (case_tac "P" 1);
|
paulson@2948
|
420 |
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
|
paulson@7584
|
421 |
qed "expand_case";
|
paulson@2948
|
422 |
|
paulson@4327
|
423 |
(*Used in Auth proofs. Typically P contains Vars that become instantiated
|
paulson@4327
|
424 |
during unification.*)
|
paulson@2948
|
425 |
fun expand_case_tac P i =
|
paulson@2948
|
426 |
res_inst_tac [("P",P)] expand_case i THEN
|
wenzelm@9713
|
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 *)
|
wenzelm@9713
|
437 |
val simpsetup =
|
wenzelm@9713
|
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;
|