-- In this post I'll shortly introduce descriptions and describe a variant of them that I prefer. -- If you haven't seen this form of generic programming before, -- you might want to start with something simpler first: -- http://effectfully.blogspot.com/2016/02/simple-generic-programming.html {-# OPTIONS --type-in-type --no-termination-check #-} module Desc where open import Relation.Binary.PropositionalEquality open import Data.Empty open import Data.Unit.Base open import Data.Bool.Base open import Data.Nat.Base open import Data.Product module Computational where -- I'll start by defining descriptions in their usual computational form. data Desc I : Set where ind : I -> Desc I κ : Set -> Desc I σ π : ∀ A -> (A -> Desc I) -> Desc I _⊛_ : Desc I -> Desc I -> Desc I ⟦_⟧ : ∀ {I} -> Desc I -> (I -> Set) -> Set ⟦ ind i ⟧ F = F i ⟦ κ A ⟧ F = A ⟦ σ A B ⟧ F = Σ A λ x -> ⟦ B x ⟧ F ⟦ π A B ⟧ F = (x : A) -> ⟦ B x ⟧ F ⟦ D ⊛ E ⟧ F = ⟦ D ⟧ F × ⟦ E ⟧ F record μ {I} (F : I -> Desc I) j : Set where inductive constructor node field knot : ⟦ F j ⟧ (μ F) -- `ind i` is an inductive position. `⟦ ind i ⟧ (μ F)` reduces to `μ F i`, -- so that is where knot tying happens. In `μ F j` `j` is the index of a term -- and in `ind i` `i` is the index of a subterm. -- `κ` allows to embed any `Set` into a description. -- `σ` allows this too, but `κ` is non-recursive and thus can finish a description. -- `σ` serves two purposes: -- 1. It allows to split a description of a data type into descriptions of several constructors. -- E.g. we can express the fact that a data type has two constructors by defining its -- description as `σ Bool λ b -> if b then cons₁ else cons₂` for some `cons₁` and `cons₂`. -- 2. It encodes top-level Π-types in the type of a constructor in a target language. -- I'll explain in a minute why we use `σ` to encode `Π`. -- `π` is for higher-order inductive occurrences. I.e. for data types where an inductive -- position appears to the right of the arrow. E.g. `W`: -- data W (A : Set) (B : A -> Set) : Set where -- sup : (x : A) -> (B x -> W A B) -> W A B -- or `Desc` itself (the `σ` and `π`) constructors. -- `D ⊛ E` is a first-order equivalent of `π Bool λ b -> if b then D else E`. -- The choice operator mentioned above: _⊕_ : ∀ {I} -> Desc I -> Desc I -> Desc I D ⊕ E = σ Bool λ b -> if b then D else E -- Here is an example of described data type. list : Set -> Desc ⊤ list A = κ ⊤ ⊕ σ A λ _ -> ind tt List : Set -> Set List A = μ (λ _ -> list A) tt -- Lists are a non-indexed data type, hence we pass `⊤` to `Desc`, and -- lists have two constructors: the one that doesn't contain any data -- (which is expressed as `κ ⊤`) and the other that contains an `A` and an inductive occurrence. -- The recovered constructors: -- [] : ∀ {A} -> List A pattern [] = node (true , tt) -- _∷_ : ∀ {A} -> A -> List A -> List A pattern _∷_ x xs = node (false , x , xs) -- Now we can see why `σ` is used to describe the arguments to a constructor. -- If we define `List` via the `data` keyword, then `_∷_` is a "god-given" function, -- but internally it's just a tag "cons" stored among with an element and a sublist. -- Here we store the element and the sublist explicitly. -- You can read described constructors like there is `-> D` after them, -- where `D` is the data type being described. E.g. for the usual lists `_∷_` can be defined as -- `cons : (A × List A) -> List A` -- which is the same as -- `cons : (Σ A λ _ -> List A) -> List A`. -- compare this to `_∷_` described above: `σ A λ _ -> ind tt`. -- Described lists have the usual eliminator. elimList : ∀ {A} -> (P : List A -> Set) -> (∀ {xs} x -> P xs -> P (x ∷ xs)) -> P [] -> ∀ xs -> P xs elimList P f z [] = z elimList P f z (x ∷ xs) = f x (elimList P f z xs) -- Now let's describe something indexed. fin : ℕ -> Desc ℕ fin n = (σ ℕ λ m -> κ (n ≡ suc m)) ⊕ (σ ℕ λ m -> σ (n ≡ suc m) λ _ -> ind m) Fin : ℕ -> Set Fin = μ fin -- fzero : ∀ {n} -> Fin (suc n) pattern fzero {n} = node (true , n , refl) -- fsuc : ∀ {n} -> Fin n -> Fin (suc n) pattern fsuc {n} i = node (false , n , refl , i) -- `Fin` has two constructors and in order to describe them we must introduce explicit -- unification constraints. `Fin n` is inhabited only when `n ≡ suc m` for some `m` -- -- that's what the description says. Since the unification constraint is the same for -- both constructors, we could introduce it before defining actual constructors: module Before where fin′ : ℕ -> Desc ℕ fin′ n = σ ℕ λ m -> σ (n ≡ suc m) λ _ -> κ ⊤ ⊕ ind m Fin′ : ℕ -> Set Fin′ = μ fin′ fzero′ : ∀ {n} -> Fin′ (suc n) fzero′ {n} = node (n , refl , true , tt) fsuc′ : ∀ {n} -> Fin′ n -> Fin′ (suc n) fsuc′ {n} i = node (n , refl , false , i) -- `Fin` has the usual induction principle: elimFin : ∀ {n} -> (P : ∀ {n} -> Fin n -> Set) -> (∀ {n} {i : Fin n} -> P i -> P (fsuc i)) -> (∀ {n} -> P (fzero {n})) -> (i : Fin n) -> P i elimFin P f x fzero = x elimFin P f x (fsuc i) = f (elimFin P f x i) -- But these explicit unification constraints are quite ugly. -- Moreover, sometimes you want to have access to them while defining generic functions -- over `Desc`, but constraints can appear everywhere in the definition of a description, -- so you can't locate them by just pattern matching on a `Desc`. module Propositional where -- So here are propositional descriptions that solve most of the problems mentioned above. -- I'm taking stuff directly from [1]. data Desc I : Set where ret : I -> Desc I σ : ∀ A -> (A -> Desc I) -> Desc I ind : I -> Desc I -> Desc I hind : ∀ A -> (A -> I) -> Desc I -> Desc I Extend : ∀ {I} -> Desc I -> (I -> Set) -> I -> Set Extend (ret i) F j = j ≡ i Extend (σ A B) F j = Σ A λ x -> Extend (B x) F j Extend (ind i D) F j = F i × Extend D F j Extend (hind A k D) F j = ((x : A) -> F (k x)) × Extend D F j record μ {I} (D : Desc I) j : Set where inductive constructor node field knot : Extend D (μ D) j -- Each desciption ends with `ret` that receives the index of a term. -- `σ` is the same thing as before. -- `ind` carries an inductive position and the rest of a description. -- `hind` is the same thing as `ind`, but an inductive occurrence is higher-order. -- `Extend` is straightforward and pretty linear. The only interesting case is `ret`: -- that's where we put constraints. Now we don't need to write them down explicitly. -- However I don't like the `(A -> I)` part in `hind`. If we want to encode something like data Foo : Set where foo : (ℕ -> Bool -> Foo) -> Foo -- then `A` must be `ℕ × Bool` and this compulsory uncurrying is annoying. -- Manual extraction of elements from a big tuple is verbose and ugly. -- To encode this definition data Bar : Set where foo : (ℕ -> Bar × Bar) -> Bar -- we have to transform it to data Bar′ : Set where foo : (ℕ -> Bar′) -> (ℕ -> Bar′) -> Bar′ -- Computational descriptions didn't have these problems. module CompProp where infixr 6 _⊛_ infixr 5 _⊕_ -- So here is a compact and convenient form of descriptions: data Desc I : Set where var : I -> Desc I π : ∀ A -> (A -> Desc I) -> Desc I _⊛_ : Desc I -> Desc I -> Desc I ⟦_⟧ : ∀ {I} -> Desc I -> (I -> Set) -> Set ⟦ var i ⟧ F = F i ⟦ π A B ⟧ F = ∀ x -> ⟦ B x ⟧ F ⟦ D ⊛ E ⟧ F = ⟦ D ⟧ F × ⟦ E ⟧ F Extend : ∀ {I} -> Desc I -> (I -> Set) -> I -> Set Extend (var j) F i = j ≡ i Extend (π A B) F i = ∃ λ x -> Extend (B x) F i Extend (D ⊛ E) F i = ⟦ D ⟧ F × Extend E F i record μ {I} (D : Desc I) i : Set where inductive constructor node field knot : Extend D (μ D) i -- `⟦_⟧` is taken from computational descriptions and -- `Extend` is taken from propositional descriptions. -- `var` serves as both `ind` and `ret`. There is `var i` at the end of each constructor, -- where `i` is the index that a constructor returns. All other `var`s in a description -- represent inductive positions. -- `π` subsumes both `σ` and `π` from computation descriptions. -- `Extend` interprets `π` as `∃` and `⟦_⟧` interprets `π` as `Π`. -- Note that `μ` in this representation and in the propositional one receives a proper -- first-order `Desc`, while in the computational representation `μ` receives a -- higher-order `I -> Desc I`. _⊕_ : ∀ {I} -> Desc I -> Desc I -> Desc I D ⊕ E = π Bool λ b -> if b then D else E -- Everything should become clear after looking at an example: vec : Set -> Desc ℕ vec A = var 0 ⊕ π ℕ λ n -> π A λ _ -> var n ⊛ var (suc n) Vec : Set -> ℕ -> Set Vec A = μ (vec A) -- Vectors have two constructors: the one that doesn't contain any data and -- the other that carries an `A` and a subvector `xs : Vec A n`. -- The former constructor returns a vector of length `0` and -- the latter returns a vector of length `suc n`. -- Compare this to the usual definition of vectors which has the same pattern: module UsualVec where data Vec′ (A : Set) : ℕ -> Set where [] : Vec′ A 0 _∷_ : ∀ {n} -> A -> Vec′ A n -> Vec′ A (suc n) -- `Extend` interprets `π` as `∃`, i.e. like `⟦_⟧` in computational descriptions interprets `σ`, -- so the recovered constructors are very similar: -- [] : ∀ {A} -> Vec A 0 pattern [] = node (true , refl) -- _∷_ : ∀ {n A} -> A -> Vec A n -> Vec A (suc n) pattern _∷_ {n} x xs = node (false , n , x , xs , refl) elimVec : ∀ {n A} -> (P : ∀ {n} -> Vec A n -> Set) -> (∀ {n} {xs : Vec A n} x -> P xs -> P (x ∷ xs)) -> P [] -> (xs : Vec A n) -> P xs elimVec P f z [] = z elimVec P f z (x ∷ xs) = f x (elimVec P f z xs) -- Let's now encode `W`: w : ∀ A -> (A -> Set) -> Desc ⊤ w A B = π A λ x -> (π (B x) λ _ -> var tt) ⊛ var tt W : ∀ A -> (A -> Set) -> Set W A B = μ (w A B) tt -- sup : ∀ {A B} -> (x : A) -> (B x -> W A B) -> W A B pattern sup x g = node (x , g , refl) -- The key thing here is that `Extend` interprets `D` and `E` in `D ⊛ E` differently. -- In `D` `π` encodes actual `Π` and `var i` is an inductive position. -- In `E` `π` encodes `∃` and `var i` (if it's not to the left of another `_⊛_`) -- represents the index that a constructor returns. -- Compare this to the usual definion of `W`: module UsualW where data W′ A (B : A -> Set) : Set where sup′ : (x : A) -> (B x -> W′ A B) -> W′ A B -- They are quite the same except that `_⊛_` is replaced by `_->_`. -- As the final example we can encode `Desc` itself: data Codes : Set where varᶜ πᶜ ⊛ᶜ : Codes desc : Set -> Desc ⊤ desc I = π Codes λ { varᶜ -> π I λ _ -> var tt ; πᶜ -> π Set λ A -> (π A λ _ -> var tt) ⊛ var tt ; ⊛ᶜ -> var tt ⊛ var tt ⊛ var tt } Desc′ : Set -> Set Desc′ I = μ (desc I) tt -- var′ : ∀ {I} -> I -> Desc′ I pattern var′ i = node (varᶜ , i , refl) -- π′ : ∀ {I} A -> (A -> Desc′ I) -> Desc′ I pattern π′ A B = node (πᶜ , A , B , refl) -- _⊛′_ : ∀ {I} -> Desc′ I -> Desc′ I -> Desc′ I pattern _⊛′_ D E = node (⊛ᶜ , D , E , refl) -- `Desc` and `Desc′` are clearly isomorphic: fromDesc : ∀ {I} -> Desc I -> Desc′ I fromDesc (var i) = var′ i fromDesc (π A B) = π′ A λ x -> fromDesc (B x) fromDesc (D ⊛ E) = fromDesc D ⊛′ fromDesc E toDesc : ∀ {I} -> Desc′ I -> Desc I toDesc (var′ i) = var i toDesc (π′ A B) = π A λ x -> toDesc (B x) toDesc (D ⊛′ E) = toDesc D ⊛ toDesc E module References where -- [1] "Modeling Elimination of Described Types" -- Larry Diehl -- http://spire-lang.org/blog/2014/01/15/modeling-elimination-of-described-types/

## Wednesday, 27 April 2016

### Descriptions

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