By Jacques Fleuriot PhD, MEng (auth.)

ISBN-10: 085729329X

ISBN-13: 9780857293299

ISBN-10: 1447110412

ISBN-13: 9781447110415

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

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Llen(f1 -- p)l + Ilen(f~ -- p)l = r} The ellipse is especially important since one of the major tasks of the Principia lies in providing the mathematical analysis that explains and confirms Kepler's guess that planets travelled in ellipses round the sun [81]. llen (x -- p)l = r} The Circular Arc. The arc is an important tool in Newton's reasoning procedures. When analysing motion at a particular point on an ellipse or circle, it is the (infinitesimal) arc at that point that is usually considered.

Given an equivalence relation", on a set S, then the quotient of S with respect to '" is the set of all equivalence classes, and is defined by Sf'" == {[xli XES} where [xl == {y E S I x'" y}. The set of all equivalence classes Sf'" is called the quotient set of S by '" and a member of an equivalence class is often referred to as a representative of the class. 2 Example: Constructing Q+ from 7Z+ In this section, we illustrate, by means of an example, how a new type can be introduced in Isabelle as the quotient set of some equivalence relation.

Thus, in this particular example, the functions Abs_prat :: (pnat * pnat) set => prat Rep_prat :: prat => (pnat * pnat) set are added to the theory such that prat is isomorphic to {x. hyprat and its inverse Abs_prat. Using these functions and other operations from Isabelle's Set and Equi v theories, we are now ready to define operations on the positive rationals. pratrel AA{(y,X)}) where • Ux E A. 3x E A. y E B} (union offamily of sets) . • rAAs == {yo 3x E s. (x, y) E r} (image of set s under relation r).

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A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia by Jacques Fleuriot PhD, MEng (auth.)


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