Theorem proving in infinitesimal geometry

This paper describes some of the work done in our formal investigation of concepts and properties that arise when infinitely small and infinite notions are introduced in a geometry theory. An algebraic geometry theory is developed in the theorem prover Isabelle using real and hyperreal vectors. We use this to investigate some new geometric relations as well as ways of rigorously mechanizing geometric proofs that involve infinitesimal and infinite arguments. We follow a strictly definitional approach and build our theory of vectors within the nonstandard analysis framework developed in Isabelle.