Sequent Calculus and Equational Programming

TL;DR

This paper demonstrates that equational programming styles in proof assistants relate to focused sequent calculus presentations of type theory, leading to a typed functional language aligned with Agda's syntax.

Contribution

It establishes a formal connection between equational programming and focused sequent calculus, introducing a new language based on this framework.

Findings

Equational style corresponds to focused sequent calculus.

A typed functional language based on sequent calculus is presented.

The language relates to Agda's syntax and internal language.

Abstract

Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums.