# Lectures on the theory of maxima and minima of functions of by Harris Hancock By Harris Hancock

This quantity is made from electronic pictures from the Cornell college Library ancient arithmetic Monographs assortment.

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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.