Martin Escardo. Notes originally written for the module Advanced Functional Programming of the University of Birmingham, UK.

The unit type 𝟙

We now redefine the unit type as a record:

record 𝟙 : Type where
 constructor
  ⋆

open 𝟙 public

In logical terms, we can interpret 𝟙 as “true”, because we can always exhibit an element of it, namely ⋆. Its elimination principle is as follows:

𝟙-elim : {A : 𝟙 → Type}
       → A ⋆
       → (x : 𝟙) → A x
𝟙-elim a ⋆ = a

In logical terms, this says that it order to prove that a property A of elements of the unit type 𝟙 holds for all elements of the type 𝟙, it is enough to prove that it holds for the element ⋆. The non-dependent version says that if A holds, then “true implies A”.

𝟙-nondep-elim : {A : Type}
              → A
              → 𝟙 → A
𝟙-nondep-elim {A} = 𝟙-elim {λ _ → A}

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