dependent-type

问题 I'm trying to write a replicate function for a length-indexed list using the machinery from GHC.TypeLits, singletons, and constraints. The Vect type and signature for replicateVec are given below: data Vect :: Nat -> Type -> Type where VNil :: Vect 0 a VCons :: a -> Vect (n - 1) a -> Vect n a replicateVec :: forall n a. SNat n -> a -> Vect n a How can you write this replicateVec function? I have a version of replicateVec that compiles and type checks, but it appears to go into an infinite

问题 I'm trying to write a replicate function for a length-indexed list using the machinery from GHC.TypeLits, singletons, and constraints. The Vect type and signature for replicateVec are given below: data Vect :: Nat -> Type -> Type where VNil :: Vect 0 a VCons :: a -> Vect (n - 1) a -> Vect n a replicateVec :: forall n a. SNat n -> a -> Vect n a How can you write this replicateVec function? I have a version of replicateVec that compiles and type checks, but it appears to go into an infinite

问题 I'm trying to learn about dependent types in Idris by attempting something way out of my depth. Please bear with me if I make any silly mistakes. Given the simple type data Letter = A | B | C | D I would like to define a type LPair that holds a pair of Letter such that only "neighboring" letters can be paired. For example, B can be paired with A or C , but not D or itself. It wraps around, so A and D are treated as neighbors. In practice, given x : Letter and y : Letter , mklpair x y would be

问题 John Major's equality comes with the following lemma for rewriting: Check JMeq_ind_r. (* JMeq_ind_r : forall (A : Type) (x : A) (P : A -> Prop), P x -> forall y : A, JMeq y x -> P y *) It is easy to generalize it like that: Lemma JMeq_ind2_r : forall (A:Type)(x:A)(P:forall C,C->Prop), P A x -> forall (B:Type)(y:B), @JMeq B y A x -> P B y. Proof. intros. destruct H0. assumption. Qed. However I need something a bit different: Lemma JMeq_ind3_r : forall (A:Type)(x:A*A) (P:forall C,C*C->Prop), P

问题 The question is about Observational Type Theory. Consider this setting: data level : Set where # : ℕ -> level ω : level _⊔_ : level -> level -> level # α ⊔ # β = # (α ⊔ℕ β) _ ⊔ _ = ω _⊔ᵢ_ : level -> level -> level α ⊔ᵢ # 0 = # 0 α ⊔ᵢ β = α ⊔ β mutual Prop = Univ (# 0) Type = Univ ∘ # ∘ suc data Univ : level -> Set where bot : Prop top : Prop nat : Type 0 univ : ∀ α -> Type α σ≡ : ∀ {α β γ} -> α ⊔ β ≡ γ -> (A : Univ α) -> (⟦ A ⟧ -> Univ β) -> Univ γ π≡ : ∀ {α β γ} -> α ⊔ᵢ β ≡ γ -> (A : Univ α)

问题 I don't know how I didn't notice this, but data constructors and function definitions alike can't use types with kinds other than * and it's variants * -> * etc., due to (->) 's kind signature, even under -XPolyKinds . Here is the code I have tried: {-# LANGUAGE DataKinds #-} {-# LANGUAGE KindSignatures #-} data Nat = S Nat | Z data Foo where Foo :: 'Z -> Foo -- Fails foo :: 'Z -> Int -- Fails foo _ = 1 The error I'm getting is the following: <interactive>:8:12: Expected a type, but ‘Z’ has

问题 I was recently looking at the wikipedia page for dependent types, and I was wondering; does Perl 6 actually introduce dependent types? I can't seem to find a reliable source claiming that. It might be obvious to some, but it sure as heck ain't obvious to me. 回答1: Against Ven, in the comments following the Perl 6 answer to the SO question "Is there a language with constrainable types?", wrote "perl6 doesn't have dependant types" and later wrote "dependent type, probably not, ... well, if we

问题 Perhaps this is a stupid question. Here's a quote from the Hasochism paper: One approach to resolving this issue is to encode lemmas, given by parameterised equations, as Haskell functions. In general, such lemmas may be encoded as functions of type: ∀ x1 ... xn. Natty x1 → ... → Natty xn → ((l ~ r) ⇒ t) → t I thought I understood RankNTypes , but I can't make sense of the last part of this proposition. I'm reading it informally as "given a term which requires l ~ r , return that term". I'm

问题 As part of doing a survey on various dependently typed formalization techniques, I have ran across a paper advocating the use of singleton types (types with one inhabitant) as a way of supporting dependently typed programming. Acording to this source, in Haskell, there is a separation betwen runtime values and compile-time types that can be blurred when using singleton types, due to the induced type/value isomorphism. My question is: How do singleton types differ from type-classes or from

问题 I was experimenting with the singletons library and I found a case that I don't understand. {-# LANGUAGE GADTs, StandaloneDeriving, RankNTypes, ScopedTypeVariables, FlexibleInstances, KindSignatures, DataKinds, StandaloneDeriving #-} import Data.Singletons.Prelude import Data.Singletons.TypeLits data Foo (a :: Nat) where Foo :: Foo a deriving Show data Thing where Thing :: KnownNat a => Foo a -> Thing deriving instance Show Thing afoo1 :: Foo 1 afoo1 = Foo afoo2 :: Foo 2 afoo2 = Foo athing ::