agda

问题 Suppose I have this definition of Subset in Agda Subset : ∀ {α} → Set α → {ℓ : Level} → Set (α ⊔ suc ℓ) Subset A {ℓ} = A → Set ℓ and I have a set data Q : Set where a : Q b : Q Is it possible to prove that all subset of q is decidable and why? Qs? : (qs : Subset Q {zero}) → Decidable qs Decidable is defined here: -- Membership infix 10 _∈_ _∈_ : ∀ {α ℓ}{A : Set α} → A → Subset A → Set ℓ a ∈ p = p a -- Decidable Decidable : ∀ {α ℓ}{A : Set α} → Subset A {ℓ} → Set (α ⊔ ℓ) Decidable as = ∀ a →

问题 Suppose I have this definition of Subset in Agda Subset : ∀ {α} → Set α → {ℓ : Level} → Set (α ⊔ suc ℓ) Subset A {ℓ} = A → Set ℓ and I have a set data Q : Set where a : Q b : Q Is it possible to prove that all subset of q is decidable and why? Qs? : (qs : Subset Q {zero}) → Decidable qs Decidable is defined here: -- Membership infix 10 _∈_ _∈_ : ∀ {α ℓ}{A : Set α} → A → Subset A → Set ℓ a ∈ p = p a -- Decidable Decidable : ∀ {α ℓ}{A : Set α} → Subset A {ℓ} → Set (α ⊔ ℓ) Decidable as = ∀ a →

问题 Can I prove that two empty functions (functions from the empty domain) are equal? More concretely, is it possible to prove in Agda the following: eqf : ∀ {A : Set} (f g : ⊥ → A) → f ≡ g Edit: as @Sassa-NF points out in the comments, if extensionality is present, then this can be proven. I am interested in whether this can be proven without extensionality. 回答1: No, this is not possible to prove in plain Martin-Löf Type Theory (and hence should also be unprovable in Agda without extra

问题 Can I prove that two empty functions (functions from the empty domain) are equal? More concretely, is it possible to prove in Agda the following: eqf : ∀ {A : Set} (f g : ⊥ → A) → f ≡ g Edit: as @Sassa-NF points out in the comments, if extensionality is present, then this can be proven. I am interested in whether this can be proven without extensionality. 回答1: No, this is not possible to prove in plain Martin-Löf Type Theory (and hence should also be unprovable in Agda without extra

问题 I recently asked this question: An agda proposition used in the type -- what does it mean? and received a very well thought out answer on how to make types implicit and get a real compile time error. However, it is still unclear to me, how to create a value with a dependent type. Consider: div : (n : N) -> even n -> N div zero p = zero div (succ (succ n)) p= succ (div n p) div (succ zero) () Where N is the natural numbers and even is the following proposition. even : N -> Set even zero = \top

问题 This is from the last chapter of PLFA book. import Relation.Binary.PropositionalEquality as Eq open Eq using (_≡_; refl; sym; trans; cong) open import Data.Product using (_×_; ∃; ∃-syntax; Σ; Σ-syntax) renaming (_,_ to ⟨_,_⟩) infix 0 _≃_ record _≃_ (A B : Set) : Set where field to : A → B from : B → A from∘to : ∀ (x : A) → from (to x) ≡ x to∘from : ∀ (y : B) → to (from y) ≡ y open _≃_ data List (A : Set) : Set where [] : List A _∷_ : A → List A → List A infixr 5 _∷_ data All {A : Set} (P : A

问题 This is from the last chapter of PLFA book. import Relation.Binary.PropositionalEquality as Eq open Eq using (_≡_; refl; sym; trans; cong) open import Data.Product using (_×_; ∃; ∃-syntax; Σ; Σ-syntax) renaming (_,_ to ⟨_,_⟩) infix 0 _≃_ record _≃_ (A B : Set) : Set where field to : A → B from : B → A from∘to : ∀ (x : A) → from (to x) ≡ x to∘from : ∀ (y : B) → to (from y) ≡ y open _≃_ data List (A : Set) : Set where [] : List A _∷_ : A → List A → List A infixr 5 _∷_ data All {A : Set} (P : A

问题 I am taking this from the "Brutal Introduction to Agda" http://oxij.org/note/BrutalDepTypes/ Suppose we want to define division by two on even numbers. We can do this as: div : (n : N) -> even n -> N div zero p = zero div (succ (succ n)) p= succ (div n p) div (succ zero) () N is the natural numbers and even is the following "proposition" even : N -> Set even zero = \top even (succ zero) = \bot even (succ (succ n)) = even n data \bot : Set where record \top : Set where When you evaluate the

问题 How to add two rational.. I was trying this but this is not correct. As I am unable to prove that coprime part. open import Data.Rational open import Data.Integer open import Data.Nat _add_ : ℚ -> ℚ -> ℚ x add y = (nx Data.Integer.* dy Data.Integer.+ dx Data.Integer.* ny) ÷ (dx′ Data.Nat.* dy′) where nx = ℚ.numerator x dx = ℚ.denominator x dx′ = ℕ.suc (ℚ.denominator-1 x) ny = ℚ.numerator y dy = ℚ.denominator y dy′ = ℕ.suc (ℚ.denominator-1 y) 回答1: You need to simplify (nx * dy + dx * ny) / (dx

问题 for my project in Programming Languages I am doing a very shallow and simple embedding VHDL digital circuits in agda. The aim is to write the syntax, static semantics, dynamic semantics and then write some proofs to show our understanding of the material. Up till now I have written the following code: data Ckt : Set where var : String → Ckt bool : Bool → Ckt empty : Ckt gate : String → ℕ → ℕ → Ckt -- name in out series : String → Ckt → Ckt → Ckt -- name ckt1 ckt2 parallel : String → Ckt → Ckt