Flat Modality

The flat/crisp attribute @♭/@flat is an idempotent comonadic modality modeled after Spatial Type Theory and Crisp Type Theory. It is similar to a necessity modality.

This attribute is enabled using the infective flag --cohesion.

We can define ♭ A as a type for any (@♭ A : Set l) via an inductive definition:

data ♭ {@♭ l : Level} (@♭ A : Set l) : Set l where
  con : (@♭ x : A) → ♭ A
counit : {@♭ l : Level} {@♭ A : Set l} → ♭ A → A
counit (con x) = x

When trying to provide a @♭ arguments only other @♭ variables will be available, the others will be marked as @⊤ in the context. For example the following will not typecheck:

unit : {@♭ l : Level} {@♭ A : Set l} → A → ♭ A
unit x = con x

Pattern Matching on @♭

By default matching on arguments marked with @♭ is disallowed, but it can be enabled using the option --flat-split. When matching on a @♭ argument the flat status gets propagated to the arguments of the constructor

data _⊎_ (A B : Set) : Set where
  inl : A → A ⊎ B
  inr : B → A ⊎ B
flat-sum : {@♭ A B : Set} → (@♭ x : A ⊎ B) → ♭ A ⊎ ♭ B
flat-sum (inl x) = inl (con x)
flat-sum (inr x) = inr (con x)

When refining @♭ variables the equality also needs to be provided as @♭

flat-subst : {@♭ A : Set} {P : A → Set} (@♭ x y : A) (@♭ eq : x ≡ y) → P x → P y
flat-subst x .x refl p = p

if we simply had (eq : x ≡ y) the code would be rejected.

Note that in Cubical Agda functions that match on an argument marked with @♭ trigger the UnsupportedIndexedMatch warning (see Indexed inductive types), and the code might not compute properly.

Also note that the --cohesion flag does not include a sharp modality or shape modality as in Cohesive Homotopy Type Theory.