# Automated Deduction in Equational Logic and Cubic Curves by W. McCune, R. Padmanabhan (auth.)

By W. McCune, R. Padmanabhan (auth.)

This monograph is the results of the cooperation of a mathematician operating in common algebra and geometry, and a working laptop or computer scientist operating in computerized deduction, who succeeded in utilizing the theory prover Otter for proving first order theorems from arithmetic after which intensified their joint effort.
Mathematicians will locate many new effects from equational common sense, common algebra, and algebraic geometry and enjoy the state of the art define of the functions of automatic deduction options. computing device scientists will discover a huge and sundry resource of theorems and difficulties that would be precious in designing and review computerized theorem proving platforms and strategies.

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Z Apply (gL) as a rewrite rule Z before ordinary demodulation. Z Paramodulate from the given clause. set(para_from). Z Paramodulate into the given clause. set(para_into). set(para_from_vars). Z Allow paramodulation into variables. Z Allow paramodulation from variables. set(para_into_vars). set(order_eq). Z Orient equalities. set(back_demod). Z Apply back demodulation. Z Process input clauses as if derived. set(process_input). Z Orient equalities with LRPO procedure. set(Irpo). ZZZZZZZZZZZZZZZZZZZZZ Standard o p t i o n s f o r h y p e r r e s o l u t i o n set(output_sequent).

Now, f ( x , y, a) = (F(a, x ) . e). F(a, y) = (F(a, e). F(a, [since F(a, = F(a, y)] = e [since ( u - e ) . u = e] f ( x , y, a) = e [Vx, y e C] Thus we have and hence, f ( x , y, z) = f ( u , v, z) = f(e, e, z) = [by rigidity] [letting v = u = e] (F(z, e). e). F(z, e) e. In other words, (F(z, x) . e) . F ( z , x) = ( F ( z , x) 9 e) 9 F ( z , y) and thus, after one left cancellation, we get the desired equality F ( z , x) = F ( z , y) for all x, y, and z. D. Let us illustrate this deduction procedure with a typical example.

X. e). (y. e) = ( x . y ) . e [3 -+ 2] [3 ~ 11:4,4, flip] ( e . x ) . e = e. (x. e) ((e. x ) - ~ ) - ( e - z ) = (e. (x. e)). (y. z) (x. (~. z)). ((~. v). w) = (x. (u. z)). 3 Abelian [2 -+ 2] Groups We start with the elementary theorem that groups satisfying (gL) are commutative. Theorem ABGT-1. (gL)-groups are Abelian. xe~x x'x = e =(gL)=> {xy = yx}. 36 seconds). xl~---e 6 (x. y ) . ~ = ~. (~. e (x. e ) ' . x = e ao (e. x)'. 3 Abelian Groups 48,47 56,55 85,84 102,101 103 104 106,105 112,111 116,115 119,118 120 137,136 140 144 155 156 (e-e)' = e x- (x.

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