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118 lines
4.7 KiB
Text
118 lines
4.7 KiB
Text
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;--------------------------------------------------------------------------
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; File : RNG010-5 : TPTP v2.2.0. Released v1.0.0.
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; Domain : Ring Theory (Right alternative)
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; Problem : Skew symmetry of the auxilliary function
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; Version : [Ove90] (equality) axioms :
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; Incomplete > Augmented > Incomplete.
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; English : The three Moufang identities imply the skew symmetry
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; of s(W,X,Y,Z) = (W*X,Y,Z) - X*(W,Y,Z) - (X,Y,Z)*W.
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; Recall that skew symmetry means that the function sign
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; changes when any two arguments are swapped. This problem
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; proves the case for swapping the first two arguments.
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; Refs : [Ove90] Overbeek (1990), ATP competition announced at CADE-10
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; : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal
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; : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11
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; : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in
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; Source : [Ove90]
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; Names : CADE-11 Competition Eq-9 [Ove90]
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; : THEOREM EQ-9 [LM93]
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; : PROBLEM 9 [Zha93]
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; Status : unknown
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; Rating : 1.00 v2.0.0
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; Syntax : Number of clauses : 27 ( 0 non-Horn; 27 unit; 2 RR)
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; Number of literals : 27 ( 27 equality)
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; Maximal clause size : 1 ( 1 average)
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; Number of predicates : 1 ( 0 propositional; 2-2 arity)
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; Number of functors : 11 ( 5 constant; 0-4 arity)
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; Number of variables : 52 ( 2 singleton)
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; Maximal term depth : 6 ( 2 average)
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; Comments : I copied this directly. I think the Moufang identities may
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; be wrong. At least they're in another form.
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; : tptp2X -f kif -t rm_equality:rstfp RNG010-5.p
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;--------------------------------------------------------------------------
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; commutative_addition, axiom.
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(or (= (add ?A ?B) (add ?B ?A)))
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; associative_addition, axiom.
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(or (= (add (add ?A ?B) ?C) (add ?A (add ?B ?C))))
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; right_identity, axiom.
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(or (= (add ?A additive_identity) ?A))
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; left_identity, axiom.
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(or (= (add additive_identity ?A) ?A))
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; right_additive_inverse, axiom.
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(or (= (add ?A (additive_inverse ?A)) additive_identity))
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; left_additive_inverse, axiom.
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(or (= (add (additive_inverse ?A) ?A) additive_identity))
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; additive_inverse_identity, axiom.
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(or (= (additive_inverse additive_identity) additive_identity))
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; property_of_inverse_and_add, axiom.
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(or (= (add ?A (add (additive_inverse ?A) ?B)) ?B))
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; distribute_additive_inverse, axiom.
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(or (= (additive_inverse (add ?A ?B)) (add (additive_inverse ?A) (additive_inverse ?B))))
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; additive_inverse_additive_inverse, axiom.
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(or (= (additive_inverse (additive_inverse ?A)) ?A))
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; multiply_additive_id1, axiom.
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(or (= (multiply ?A additive_identity) additive_identity))
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; multiply_additive_id2, axiom.
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(or (= (multiply additive_identity ?A) additive_identity))
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; product_of_inverse, axiom.
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(or (= (multiply (additive_inverse ?A) (additive_inverse ?B)) (multiply ?A ?B)))
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; multiply_additive_inverse1, axiom.
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(or (= (multiply ?A (additive_inverse ?B)) (additive_inverse (multiply ?A ?B))))
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; multiply_additive_inverse2, axiom.
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(or (= (multiply (additive_inverse ?A) ?B) (additive_inverse (multiply ?A ?B))))
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; distribute1, axiom.
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(or (= (multiply ?A (add ?B ?C)) (add (multiply ?A ?B) (multiply ?A ?C))))
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; distribute2, axiom.
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(or (= (multiply (add ?A ?B) ?C) (add (multiply ?A ?C) (multiply ?B ?C))))
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; right_alternative, axiom.
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(or (= (multiply (multiply ?A ?B) ?B) (multiply ?A (multiply ?B ?B))))
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; associator, axiom.
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(or (= (associator ?A ?B ?C) (add (multiply (multiply ?A ?B) ?C) (additive_inverse (multiply ?A (multiply ?B ?C))))))
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; commutator, axiom.
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(or (= (commutator ?A ?B) (add (multiply ?B ?A) (additive_inverse (multiply ?A ?B)))))
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; middle_associator, axiom.
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(or (= (multiply (multiply (associator ?A ?A ?B) ?A) (associator ?A ?A ?B)) additive_identity))
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; left_alternative, axiom.
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(or (= (multiply (multiply ?A ?A) ?B) (multiply ?A (multiply ?A ?B))))
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; defines_s, axiom.
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(or (= (s ?A ?B ?C ?D) (add (add (associator (multiply ?A ?B) ?C ?D) (additive_inverse (multiply ?B (associator ?A ?C ?D)))) (additive_inverse (multiply (associator ?B ?C ?D) ?A)))))
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; right_moufang, hypothesis.
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(or (= (multiply ?A (multiply ?B (multiply ?C ?B))) (multiply (commutator (multiply ?A ?B) ?C) ?B)))
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; left_moufang, hypothesis.
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(or (= (multiply (multiply ?A (multiply ?B ?A)) ?C) (multiply ?A (commutator ?B (multiply ?A ?C)))))
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; middle_moufang, hypothesis.
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(or (= (multiply (multiply ?A ?B) (multiply ?C ?A)) (multiply (multiply ?A (multiply ?B ?C)) ?A)))
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; prove_skew_symmetry, conjecture.
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(or (/= (s a b c d) (additive_inverse (s b a c d))))
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;--------------------------------------------------------------------------
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