Telescopes

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Irrefutable Patterns in Binding Positions

Since Agda 2.6.1, irrefutable patterns can be used at every binding site in a telescope to take the bound value of record type apart. The type of the second projection out of a dependent pair will for instance naturally mention the value of the first projection. Its type can be defined directly using an irrefutable pattern as follows:

proj₂ : ((a , _) : Σ A B) → B a

And this second projection can be implemented with a lamba-abstraction using one of these irrefutable patterns taking the pair apart:

proj₂ = λ (_ , b) → b

Using an as-pattern makes it possible to name the argument and to take it apart at the same time. We can for instance prove that any pair is equal to the pairing of its first and second projections, a property commonly called eta-equality:

eta : (p@(a , b) : Σ A B) → p ≡ (a , b)
eta p = refl