# Automated Theorem Proving, 1st Edition by Wolfgang Bibel (auth.)

By Wolfgang Bibel (auth.)

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The number of items the in each fenced area. 1) because paths in matrices are fundamental for the approach to ATP taken in this book. With the following definition we are now approaching the main result of this section. D. A ( partial path through F. path in a matrix F is any subset of a II. 3 A connection { Lro , 1 Lr 1} • 29 in a matrix F is a path in F of the form A set W of connections in a matrix F is called spanning for F if for each path p through F there is an wEW such that p contains w which is to say ~p.

The matrix remains always the same except for changes in the values of the functions of its structure. 12) certainly particular, the test r- F is F cannot be realized but by a number of actions in a more conventional language. Therefore we are now going ferent version of the main part of CP? to present a -dif in such a more conven- tional form which might be easily translated into a programming language in actual use. • , Yn then XI+YI, ••• ,Xn+Yn • X ~ Y it is meant that the value of X • If the value of (Xl' ...

S) . 3) required for extension may be satisfied by an appropriate selection for y(d) and e results in an extension step (F,S) as described there. i-(F,S). I This B(L)=1 S is (F,S). By • As we have seen holds. Obviously, in both sub- S IF~F1 \<\F~F1\+IFI-1 • Thus altogether we obtain kS=IFI. C. The connection calculus is consistent. 4). C. The connection calculus is complete. 4). 7. C. The inference relation in the connection calculus restricted to complementary matrices is confluent. The proof of the if-case of ( 5.