  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.
