Packaging Mathematical Structures

Abstract

Abstract. This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

“…These experiments suggest that it will be important to have special support for the manipulation of the isomorphisms involved in recursive type equations, such as the coercions and canonical structures of [18]. An alternative approach, inspired by the conference version of the present paper, has recently been proposed by Jaber et.…”

Theory Presentation Combinators

“…There are many published techniques and implementations of algebraic hierarchies in dependently typed proof assistants including [12,20,11,17]. Our work does not compete with these implementations, but rather complements them.…”

“…More specifically, we envision our work as a meta-language which can be used to specify algebraic hierarchies, which can subsequently be implemented by using any of the aforementioned techniques. In particular we note that maintaining the correct structures for packed-classes of [11] is particularly difficult, and deriving the required structures from a hierarchy specification would alleviate much of this burden. Other cited work, (for example [17]) focus on other difficult problems such as usability, via providing coercions and unification hints to match particular terms to theories.…”

Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

“…In the Coq proof assistant, the most successful recent approaches are based on type classes-like inference mechanisms [31,32]. The SSReflect hierarchy is based on the canonical structures mechanism [31] and its implementation is described in [15]. An alternative solution based on a different type inference mechanism is described in [33].…”

“…Non empty rectangular matrices of fixed size with coefficients in a ring are canonically equipped with a structure of left module, whose internal product is denoted by (_ *m _) since the product of arbitrary size rectangular matrices is not a ring operation. More details on this implementation and the libraries available on matrix algebra can be found in [15,16].…”