By Frank Pfenning (auth.), Claude Kirchner, Hélène Kirchner (eds.)
This publication constitutes the refereed court cases of the fifteenth overseas convention on automatic Deduction, CADE-15, held in Lindau, Germany, in July 1998.
The quantity provides 3 invited contributions including 25 revised complete papers and 10 revised process descriptions; those have been chosen from a complete of one hundred twenty submissions. The papers tackle all present concerns in computerized deduction and theorem proving in line with answer, superposition, version iteration and removing, or connection tableau calculus, in first-order, higher-order, intuitionistic, or modal logics, and describe functions to geometry, machine algebra, or reactive systems.
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Example text
M e t h o d (for proving any geometric theorem in C). Step 1. Formulate the theorem in C using the constructions given in Sect. 2 and 20 so that d scalar parameters uj, m free points Uk and r dependent points X i are introduced, the Clifford expressions (H) constitute the hypothesis and (C) gives the conclusion of the theorem. Let di be the denominator of fi in (H) for l P 3 - P~ . P~ ) ( P ~ - P1) ~ 2 (P~ - P , ) ^ (P3 - P1) ~ The non-degeneracy condition: P1, P2, P3 are non-collinear. - Construct a point X such that XP3 • P1P2 and the perpendicular distance from X to P1P2 is equal to a given non-negative scalar d: X = per_dis(P1,P2,P3,d) = per_ft(P1,P2, P3) • d (P2 - P1) ~ v/(P2 - P1). (P2 - P1) The non-degeneracy condition: PIP2 is non-isotropic. Using the above constructions and those in 20, one can express a big class of geometric theorems involving points, lines, circles and their relations such as collinearity, parallelism, perpendicularity and equi-distance. This is Lemma 7. It is clear from the diagram above that ABR and AD R are not similar in ordinary Euclidean Geometry. Infinitesimal notions reveal that we are tending towards similarity of these triangles when the point B is about to meet point A. This property cannot be deduced from just the static diagram above. The dynamics of Newton’s geometry involves the reader using his or her imagination to incorporate motion and see what is happening to the relations between various parts of the diagram as points are moving.