We investigate the control of evaluation strategies in a variant of the λ-calculus derived through the Curry-Howard correspondence from LJF, a sequent calculus for intuitionistic logic implementing the focusing technique. The proof theory of focused intuitionistic logic yields a single calculus in which a number of known λ-calculi appear as subsystems obtained by restricting types to a certain fragment of LJF. In particular, standard λ-calculi as well as the call-by-pushvalue calculus are analysed using this framework, and we relate cut elimination for LJF to a new abstract machine subsuming wellknown machines for these different strategies.

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.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759