mirror of
https://github.com/RAIRLab/Spectra.git
synced 2024-12-01 04:40:41 -05:00
117 lines
4.7 KiB
Text
117 lines
4.7 KiB
Text
;--------------------------------------------------------------------------
|
|
; File : RNG010-5 : TPTP v2.2.0. Released v1.0.0.
|
|
; Domain : Ring Theory (Right alternative)
|
|
; Problem : Skew symmetry of the auxilliary function
|
|
; Version : [Ove90] (equality) axioms :
|
|
; Incomplete > Augmented > Incomplete.
|
|
; English : The three Moufang identities imply the skew symmetry
|
|
; of s(W,X,Y,Z) = (W*X,Y,Z) - X*(W,Y,Z) - (X,Y,Z)*W.
|
|
; Recall that skew symmetry means that the function sign
|
|
; changes when any two arguments are swapped. This problem
|
|
; proves the case for swapping the first two arguments.
|
|
|
|
; Refs : [Ove90] Overbeek (1990), ATP competition announced at CADE-10
|
|
; : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal
|
|
; : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11
|
|
; : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in
|
|
; Source : [Ove90]
|
|
; Names : CADE-11 Competition Eq-9 [Ove90]
|
|
; : THEOREM EQ-9 [LM93]
|
|
; : PROBLEM 9 [Zha93]
|
|
|
|
; Status : unknown
|
|
; Rating : 1.00 v2.0.0
|
|
; Syntax : Number of clauses : 27 ( 0 non-Horn; 27 unit; 2 RR)
|
|
; Number of literals : 27 ( 27 equality)
|
|
; Maximal clause size : 1 ( 1 average)
|
|
; Number of predicates : 1 ( 0 propositional; 2-2 arity)
|
|
; Number of functors : 11 ( 5 constant; 0-4 arity)
|
|
; Number of variables : 52 ( 2 singleton)
|
|
; Maximal term depth : 6 ( 2 average)
|
|
|
|
; Comments : I copied this directly. I think the Moufang identities may
|
|
; be wrong. At least they're in another form.
|
|
; : tptp2X -f kif -t rm_equality:rstfp RNG010-5.p
|
|
;--------------------------------------------------------------------------
|
|
; commutative_addition, axiom.
|
|
(or (= (add ?A ?B) (add ?B ?A)))
|
|
|
|
; associative_addition, axiom.
|
|
(or (= (add (add ?A ?B) ?C) (add ?A (add ?B ?C))))
|
|
|
|
; right_identity, axiom.
|
|
(or (= (add ?A additive_identity) ?A))
|
|
|
|
; left_identity, axiom.
|
|
(or (= (add additive_identity ?A) ?A))
|
|
|
|
; right_additive_inverse, axiom.
|
|
(or (= (add ?A (additive_inverse ?A)) additive_identity))
|
|
|
|
; left_additive_inverse, axiom.
|
|
(or (= (add (additive_inverse ?A) ?A) additive_identity))
|
|
|
|
; additive_inverse_identity, axiom.
|
|
(or (= (additive_inverse additive_identity) additive_identity))
|
|
|
|
; property_of_inverse_and_add, axiom.
|
|
(or (= (add ?A (add (additive_inverse ?A) ?B)) ?B))
|
|
|
|
; distribute_additive_inverse, axiom.
|
|
(or (= (additive_inverse (add ?A ?B)) (add (additive_inverse ?A) (additive_inverse ?B))))
|
|
|
|
; additive_inverse_additive_inverse, axiom.
|
|
(or (= (additive_inverse (additive_inverse ?A)) ?A))
|
|
|
|
; multiply_additive_id1, axiom.
|
|
(or (= (multiply ?A additive_identity) additive_identity))
|
|
|
|
; multiply_additive_id2, axiom.
|
|
(or (= (multiply additive_identity ?A) additive_identity))
|
|
|
|
; product_of_inverse, axiom.
|
|
(or (= (multiply (additive_inverse ?A) (additive_inverse ?B)) (multiply ?A ?B)))
|
|
|
|
; multiply_additive_inverse1, axiom.
|
|
(or (= (multiply ?A (additive_inverse ?B)) (additive_inverse (multiply ?A ?B))))
|
|
|
|
; multiply_additive_inverse2, axiom.
|
|
(or (= (multiply (additive_inverse ?A) ?B) (additive_inverse (multiply ?A ?B))))
|
|
|
|
; distribute1, axiom.
|
|
(or (= (multiply ?A (add ?B ?C)) (add (multiply ?A ?B) (multiply ?A ?C))))
|
|
|
|
; distribute2, axiom.
|
|
(or (= (multiply (add ?A ?B) ?C) (add (multiply ?A ?C) (multiply ?B ?C))))
|
|
|
|
; right_alternative, axiom.
|
|
(or (= (multiply (multiply ?A ?B) ?B) (multiply ?A (multiply ?B ?B))))
|
|
|
|
; associator, axiom.
|
|
(or (= (associator ?A ?B ?C) (add (multiply (multiply ?A ?B) ?C) (additive_inverse (multiply ?A (multiply ?B ?C))))))
|
|
|
|
; commutator, axiom.
|
|
(or (= (commutator ?A ?B) (add (multiply ?B ?A) (additive_inverse (multiply ?A ?B)))))
|
|
|
|
; middle_associator, axiom.
|
|
(or (= (multiply (multiply (associator ?A ?A ?B) ?A) (associator ?A ?A ?B)) additive_identity))
|
|
|
|
; left_alternative, axiom.
|
|
(or (= (multiply (multiply ?A ?A) ?B) (multiply ?A (multiply ?A ?B))))
|
|
|
|
; defines_s, axiom.
|
|
(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)))))
|
|
|
|
; right_moufang, hypothesis.
|
|
(or (= (multiply ?A (multiply ?B (multiply ?C ?B))) (multiply (commutator (multiply ?A ?B) ?C) ?B)))
|
|
|
|
; left_moufang, hypothesis.
|
|
(or (= (multiply (multiply ?A (multiply ?B ?A)) ?C) (multiply ?A (commutator ?B (multiply ?A ?C)))))
|
|
|
|
; middle_moufang, hypothesis.
|
|
(or (= (multiply (multiply ?A ?B) (multiply ?C ?A)) (multiply (multiply ?A (multiply ?B ?C)) ?A)))
|
|
|
|
; prove_skew_symmetry, conjecture.
|
|
(or (/= (s a b c d) (additive_inverse (s b a c d))))
|
|
|
|
;--------------------------------------------------------------------------
|