Abstract
Given a polygon with vertices at integer lattice points (i.e. where both x and y coordinates are integers), Pick’s theorem [4] relates its area A to the number of integer lattice points I in its interior and the number B on its boundary:
A = I + B/2 − 1
We describe a formal proof of this theorem using the HOL Light theorem prover [2]. As sometimes happens for highly geometrical proofs, the formalization turned out to be quite challenging. In this case, the principal difficulties were connected with the triangulation of an arbitrary polygon, where a simple informal proof took a great deal of work to formalize.
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References
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Harrison, J. (2010). A Formal Proof of Pick’s Theorem. In: Fukuda, K., Hoeven, J.v.d., Joswig, M., Takayama, N. (eds) Mathematical Software – ICMS 2010. ICMS 2010. Lecture Notes in Computer Science, vol 6327. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15582-6_29
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DOI: https://doi.org/10.1007/978-3-642-15582-6_29
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