By Jacques Fleuriot PhD, MEng (auth.)

Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) features a prose-style mix of geometric and restrict reasoning that has usually been considered as logically vague.

In **A mix of Geometry Theorem Proving and Nonstandard****Analysis**, 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 techniques, 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 ordinary genuine research utilizing nonstandard innovations can be discussed.

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5 Filters and Ultrafilters 43 defining a suitable successor function which, in our case, is defined using the choice or description operator: succ S c == if (c ¢ chain S V c E max chain S) then c else (eC'. c' E super S c) Our other definitions of set of chains, super chains and maximal chains are similar to those in IsabellejZF. Note that the definitions suppose that the set S has some partial ordering defined on it which is denoted by $: chain S == {F. F ~ S A ("Ix E F. Vy E F. x $ y Vy $ x)} super S c == {d.

Incident {p} (a -- b) A len (a -- p) + len (p -- b) = len (a -- b) 4) Three points a, b, and c determine a signed area s_delta abc E ]R. e. 3a b c. s_delta abc :f: 0 6) A new point d can be introduced or eliminated using the following rule: s_delta abc = s_delta a b d + s_delta ad c + s_delta d b c 7) Lengths of segments are given in terms of areas using the following rule: [lincident {a, b, c, d} (L); len (a -- b) = a . 2 is also present in the theory. 4 Formalizing Geometry in Isabelle 23 area method hold [19].

This suits our purpose well as we certainly want a minimal set of axioms. In fact, by introducing definitions, we manage to derive some of the axioms of Chou et al. as theorems. 1 Defining the Theories The Axioms of the Area Method. We present below the axioms that are used for GTP in Isabelle. Two types of geometric concepts, namely, points and lines are introduced. If a point x is on a line I (on (x, I)), or equivalently, I goes through {x}, then they are said to be incident. This is the only basic geometric relation: incident A I Yx E A.