Cubical Type Theory in Agda

Cubical Type Theory Cohen et al., Cubical is implemented in Agda (beta version).

To use cubical type theory, you need to run Agda with the --cubical command-line-option. You can also write {-# OPTIONS --cubical #-} at the top of the file.

To define paths, use λ abstractions. For example, this is the definition of the constant path:

refl : ∀ {a} {A : Set a} {x : A} → Path x x
refl {x = x} = λ i → x

Although they use the same syntax, a path is not a function. For example, the following is not valid:

refl : ∀ {a} {A : Set a} {x : A} → Path x x
refl {x = x} = λ (i : _) → x

The interval type (I)

The variable i in the definition of refl has type I, which topologically corresponds to the real unit interval. In a closed context, there are only two values of type I: the two ends of the interval, i0 and i1:

a b : I
a = i0
b = i1

Elements of the interval form a De Morgan algebra, with minimum (∧), maximum (∨) and negation (~).

module interval-example₁ (i j : I) where
  data _≡_ (i : I) : I → Set where
    reflI : i ≡ i

  infix 10 _≡_

  max min neg : I

max = i ∨ j
min = i ∧ j
neg = ~ i

All the properties of de Morgan algebras hold definitionally. The ends of the interval i0 and i1 are the bottom and top elements, respectively

p₁ : i0 ∨ i    ≡ i
p₂ : i  ∨ i1   ≡ i1
p₃ : i  ∨ j    ≡ j ∨ i
p₄ : i  ∧ j    ≡ j ∧ i
p₅ : ~ (~ i)   ≡ i
p₆ : i0        ≡ ~ i1
p₇ : ~ (i ∨ j) ≡ ~ i ∧ ~ j
p₈ : ~ (i ∧ j) ≡ ~ i ∨ ~ j

p₁ = reflI
p₂ = reflI
p₃ = reflI
p₄ = reflI
p₅ = reflI
p₆ = reflI
p₇ = reflI
p₈ = reflI

References

Cyril Cohen, Thierry Coquand, Simon Huber and Anders Mörtberg; “Cubical Type Theory: a constructive interpretation of the univalence axiom”.