-- In this post I'll show how to emulate cumulativity in Agda. module Cumu where open import Level renaming (zero to lzero; suc to lsuc) open import Function open import Relation.Binary.PropositionalEquality open import Data.Product -- As an example let's take telescopes. -- Recall how they can be defined without universe polymophism: module Mono where open import Data.Unit.Base open import Data.Nat.Base open import Data.Vec infix 3 _▷_ data Tele : Set₁ where ε : Tele _▷_ : (A : Set) -> (A -> Tele) -> Tele -- `_▷_` receives a type `A` and the rest of a telescope, -- where each type can depend on an element of type `A`. -- Here is an example: example : Tele example = ℕ ▷ λ n -> Vec ℕ n ▷ λ _ -> ε -- Each telescope represents the type of an n-ary tuples: ⟦_⟧ : Tele -> Set ⟦ ε ⟧ = ⊤ ⟦ A ▷ R ⟧ = Σ A λ x -> ⟦ R x ⟧ test : ⟦ example ⟧ ≡ ∃ λ n -> Vec ℕ n × ⊤ test = refl -- The problem with this encoding however is that `_▷_` always receives a `Set`, -- while we want it to receive types from different universes. How can we achieve that? -- We could define `Tele` by recursion on a list of levels like this: module Rec where open import Data.Unit.Base open import Data.List.Base Tele : ∀ αs -> Set (foldr (_⊔_ ∘ lsuc) lzero αs) Tele [] = ⊤ Tele (α ∷ αs) = Σ (Set α) λ A -> A -> Tele αs -- This is the same `Tele` as above, but now universe polymorphic: example : Tele (lsuc lzero ∷ lzero ∷ lsuc (lsuc lzero) ∷ []) example = Set , λ A -> A , λ _ -> Set₁ , λ _ -> tt -- The problem with this encoding however is that all levels are explictly reflected -- at the type level (in this case they can be inferred, since `Tele` is constructor-headed). -- It means that there can't exist telescopes with statically unknown length: -- fail : ∃ Tele -- fail = ? -- ((αs : List Level) → Set (foldr ((λ {.x} → _⊔_) ∘ lsuc) lzero αs)) -- !=< (_A_38 → Set _b_37) -- because this would result in an invalid use of Setω -- when checking that the expression Tele has type _A_38 → Set _b_37 -- What we really want is to write `Tele (lsuc (lsuc lzero))`, -- i.e. just one level -- the biggest -- and make Agda check that all other levels -- are less or equal than it. And doesn't reflect the length of a telescope at the type level. -- I.e. just what we normally do: example₂ : Set₂ example₂ = Σ Set λ A -> A × Set₁ × ⊤ record ⊤ {α} : Set α where constructor tt -- We can state that one level is less or equal than another as follows: _≤ℓ_ : Level -> Level -> Set α ≤ℓ β = α ⊔ β ≡ β -- I.e. if the maximum of `α` and `β` is `β`, then `α` is obviously less or equal than `β`. -- We need a way to coerce a `Set α` to `Set β`, provided there is a proof of `α ≡ β`. -- It can be done either by a function: Coerce′ : ∀ {α β} -> α ≡ β -> Set α -> Set β Coerce′ refl = id -- or using a data type: data Coerce {β} : ∀ {α} -> α ≡ β -> Set α -> Set β where coerce : ∀ {A} -> A -> Coerce refl A -- We'll need both of them. -- Here is the encoding finally: {-# NO_POSITIVITY_CHECK #-} data Tele β : Set (lsuc β) where ε : Tele β cons : ∀ {α} -> (q : α ≤ℓ β) -> Coerce (cong lsuc q) (∃ λ (A : Set α) -> A -> Tele β) -> Tele β -- Operationally it's the same as before: `cons` receives a type `A` and the rest of a telescope, -- but now `A` lies in `Set α` and we have an explicit proof that `α` is less or equal than `β`. -- The type of `∃ λ (A : Set α) -> A -> Tele β` is `Set (lsuc α ⊔ lsuc β)` which is the same as -- `Set (lsuc (α ⊔ β))`, but it must lie in `Set (lsuc β)`, because the whole `Tele` lies there, -- so we coerce by `cong lsuc q : lsuc (α ⊔ β) ≡ lsuc β` and get the required type. -- The constructor is recovered as pattern _▷_ A R = cons _ (coerce (A , R)) -- Though, it's not possible to use it in pattern matching, -- because that would force unification of `lsuc (α ⊔ β) =?= lsuc β`, -- which is an infinite equation that can't be solved. -- An example: example : Tele (lsuc (lsuc lzero)) example = Set ▷ λ A -> A ▷ λ _ -> Set₁ ▷ λ _ -> ε -- All types lie in different universes and `Tele` receives only the level of the biggest universe. -- It only remains to interpret telescopes: mutual ⟦_⟧ᵀ : ∀ {β} -> Tele β -> Set β ⟦ ε ⟧ᵀ = ⊤ ⟦ cons q B ⟧ᵀ = ⟦ B ⟧ᵀᵇ q ⟦_⟧ᵀᵇ : ∀ {α β γ q} -> Coerce {β = γ} q (∃ λ (A : Set α) -> A -> Tele β) -> α ≤ℓ β -> Set β ⟦ coerce (A , R) ⟧ᵀᵇ q = Coerce′ q (∃ λ x -> ⟦ R x ⟧ᵀ) -- It's the same as before, but we need two functions and one additional `Coerce′`. -- Two functions are needed, because, as was mentioned above, it's not possible to use `_▷_` -- in pattern matching, so the levels equation is generalized to `lsuc (α ⊔ β) =?= γ`, -- which can be solved. `Coerce′` is needed, because `∃ λ x -> ⟦ R x ⟧ᵀ` lies in `Set (α ⊔ β)`, -- while we want it to be in `Set β`. Note that `Coerce′` is just a function and -- doesn't introduce mess in the interpretation of a telescope. -- A simple test: test : ⟦ example ⟧ᵀ ≡ ∃ λ A -> A × Set₁ × ⊤ test = refl -- We have considered how to emulate cumulativity for telescopes, -- but there are other applications: convenient universe polymorphic descriptions -- (I'm working on a generic programming library that uses ideas described in this post), -- universe polymorphic Freer monads (as described by Kiselyov et al) for dealing with effects, -- perhaps something else.

## Thursday, 21 July 2016

### Emulating cumulativity in Agda

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