Proof Assistant Decision Procedures for Formalizing Origami

Origami constructions have interesting properties that are not covered by standard euclidean geometry. Such properties have been shown with the help of computer algebra systems. Proofs performed with computer algebra systems can be accompanied by proof documents, still they lack complete mathematical rigorousity, like the one provided by proof assistant checked proofs. Transforming such proofs to machine checkable proof scripts poses a number of challenges. In this paper we describe issues that arise when proving properties of origami constructions using proof assistant decision procedures. We examine the strength of Gröbner Bases implementations comparing proof assistants with each other and with the implementations provided in computer algebra systems. We show ad-hoc decision procedures that can be used to optimize the proofs. We show how maximum equilateral triangle inscribed in a square construction can be formalized. We show how a equation system solving mechanism can be embedded in a CAS decision procedure of a proof assistant.