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

The binary type 𝟚

This type can be defined to be 𝟙 ∔ 𝟙 using binary sums, but we give a direct definition which will allow us to discuss some relationships between the various type formers of Basic MLTT.

data 𝟚 : Type where
 𝟎 𝟏 : 𝟚

This type is not only isomorphic to 𝟙 ∔ 𝟙 but also to the type Bool of booleans. Its elimination principle is as follows:

𝟚-elim : {A : 𝟚 → Type}
       → A 𝟎
       → A 𝟏
       → (x : 𝟚) → A x
𝟚-elim x₀ x₁ 𝟎 = x₀
𝟚-elim x₀ x₁ 𝟏 = x₁

In logical terms, this says that it order to show that a property A of elements of the binary type 𝟚 holds for all elements of the type 𝟚, it is enough to show that it holds for 𝟎 and for 𝟏. The non-dependent version of the eliminator is the following:

𝟚-nondep-elim : {A : Type}
              → A
              → A
              → 𝟚 → A
𝟚-nondep-elim {A} = 𝟚-elim {λ _ → A}

Notice that the non-dependent version is like if-then-else, if we think of one of the elements of A as true and the other as false.

The dependent version generalizes the type of the non-dependent version, with the same definition of the function.

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