A Formalization of Grassmann-Cayley Algebra in COQ and Its Application to Theorem Proving in Projective Geometry
نویسندگان
چکیده
This paper presents a formalization of Grassmann-Cayley algebra [6] that has been done in the Coq [2] proof assistant. The formalization is based on a data structure that represents elements of the algebra as complete binary trees. This allows to define the algebra products recursively. Using this formalization, published proofs of Pappus’ and Desargues’ theorem [7,1] are interactively derived. A method that automatically proves projective geometric theorems [11] is also translated successfully into the proposed formalization.
منابع مشابه
Least Square for Grassmann-Cayley Agelbra in Homogeneous Coordinates
This paper presents some tools for least square computation in Grassmann-Cayley algebra, more specifically for elements expressed in homogeneous coordinates. We show that building objects with the outer product from k-vectors of same grade presents some properties that can be expressed in term of linear algebra and can be treated as a least square problem. This paper mainly focuses on line and ...
متن کاملAutomated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra
A Clifford algebra has three major multiplications: inner product, outer product and geometric product. Accordingly, the same Clifford algebra has three versions: Clifford vector algebra, which features on inner products and outer products of multivectors; Clifford bracket algebra, which features on pseudoscalars and inner products of vectors; Clifford geometric algebra, which features on geome...
متن کاملDeepAlgebra - An Outline of a Program
We outline a program in the area of formalization of mathematics to automate theorem proving in algebra and algebraic geometry. We propose a construction of a dictionary between automated theorem provers and (La)TeX exploiting syntactic parsers. We describe its application to a repository of human-written facts and definitions in algebraic geometry (The Stacks Project). We use deep learning tec...
متن کاملAlgebraic Representation, Elimination and Expansion in Automated Geometric Theorem Proving
Cayley algebra and bracket algebra are important approaches to invariant computing in projective and affine geometries, but there are some difficulties in doing algebraic computation. In this paper we show how the principle “breefs” – bracketoriented representation, elimination and expansion for factored and shortest results, can significantly simply algebraic computations. We present several t...
متن کاملFormalization of Wu's Simple Method in Coq
We present in this paper the integration within the Coq proof assistant, of a method for automatic theorem proving in geometry. We use an approach based on the validation of a certificate. The certificate is generated by an implementation in Ocaml of a simple version of Wu’s method.
متن کامل