By Jacques Fleuriot PhD, MEng (auth.)

ISBN-10: 1447110412

ISBN-13: 9781447110415

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

In **A mix of Geometry Theorem Proving and Nonstandard****Analysis**, Jacques Fleuriot provides a formalization of Lemmas and Propositions from the Principia utilizing a mixture of equipment from geometry and nonstandard research. The mechanization of the methods, which respects a lot of Newton's unique reasoning, is constructed in the theorem prover Isabelle. the applying of this framework to the mechanization of hassle-free genuine research utilizing nonstandard recommendations is usually discussed.

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**Additional resources for A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton’s Principia**

**Example text**

It is often easy to omit non-degenerate conditions especially when the user formulates a theorem to be proved using a diagram as a guide. We tend to draw on paper "well-formed" diagrams that enable us to picture the property we are trying to prove. However, for the automatic theorem prover, if the necessary conditions are not available then it might fail to find a proof. For example, in 24 2. Geometry Theorem Proving a parallelogram ABCD, the two diagonals AC and BD bisect each other if A, B, C and D are not collinear.

PRAT = PMAT + Equiv + constdefs (* equivalence relation *) pratrel :: "«pnat * pnat) * (pnat * pnat» set" "pratrel {p. 3 abc d. p = «a,b), (c,d» " ad = bc}" = typedef prat = "{x:: (pnat*pnat). quotient_def) instance prat :: {ord, plUS, times} constdefs prat_oCpnat :: pnat ::} prat (11$#_" [80] 80) "prat_oCpnat m Abs_prat(pratrel-A{(m,Abs_pnat i)})" = qinv :: prat ::} prat "qinv Q Abs_prat(U(x,y)ERep_prat(Q). pratrel AA{(y ,x)}) II = defs prat_add_def IIp + Q Abs_prat (UpERep_prat (P). ~a b. ~c d.

1. 2 and on the definition of parallel lines. The following goal with its associated premises now results: [Is_delta CAP = s_delta Q B P + s_delta B A Pj s_delta R P A = s_delta B Q A + s_delta Q PAl] ==> s_delta CAP = -s_delta R P A The next steps are trivial and follow from the theorems we proved about areas of quadrilaterals. The subgoals are routinely proved by Isabelle's simplifier, thereby proving Pascal's theorem. We give a rather more detailed Isabelle proof to show the area method at work.