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By Jacques Fleuriot PhD, MEng (auth.)

Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) incorporates a prose-style mix of geometric and restrict reasoning that has frequently been seen as logically vague.
In A mixture of Geometry Theorem Proving and NonstandardAnalysis, Jacques Fleuriot offers a formalization of Lemmas and Propositions from the Principia utilizing a mix of equipment from geometry and nonstandard research. The mechanization of the methods, which respects a lot of Newton's unique reasoning, is built in the theorem prover Isabelle. the applying of this framework to the mechanization of ordinary actual research utilizing nonstandard innovations is usually discussed.

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20] also propose a method based on the concept of full-angles that can be used to deal with classes of theorems that pose problems to the area method. A full-angle (u, v) is the angle from line u to line v measured anti-clockwise. We note that u and v are lines rather than rays; this has the major advantage of simplifying proofs by eliminating case-splits in certain cases. The full-angle is then used to express other familiar geometric properties and augment the reasoning capabilities of the geometry theory.

Since the law of reflection means that the angle of incidence is the same as the angle of reflection, the above definition follows. Given a point on an ellipse, we are also interested in finding the set of points that belong to the tangent at that point. We add the following definition to deal with this situation: e_tangent x h fa E == {po is_e_tangent (x -- p) h fa E} Definitions relating to the tangent to the circle are also made straightforwardly in Isabelle. l a -- b) c_tangent a x C == {po is_c_tangent (a -- p) x C} Various theorems can be proved about ellipses, circles, and their tangents.

Or})" "lhr _ Abs_hypreal(hyprel··{An::nat. lr})" constdefs hypreal_minus :: hypreal :::} hypreal "- p _ Abs_hypreal(UXERep_hypreal(P). hyprel··{An: :nat. - (X n)})" (* embedding for the reals *) hypreal_of_real :: real:::} hypreal "hypreal_oCreal r == Abs_hypreal(hyprel··{An: :nat. r})" hrinv .. hypreal :::} hypreal "hrinv P == Abs_hypreal (UXE Rep_hypreal (P) . hyprel··{An. if X n = Or then Or else rinv (X n)})" defs hypreal_add_def IIp + Q == Abs_hypreal (UXERep_hypreal (P) . UYERep_hypreal(Q).

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