我正在尝试在Agda中实现一些矩阵运算和证据。 代码涉及以下定义附近的内容:
open import Algebra
open import Data.Nat hiding (_+_ ; _*_)
open import Data.Vec
open import Relation.Binary.PropositionalEquality
module Teste {l l'}(cr : CommutativeSemiring l l') where
open CommutativeSemiring cr hiding (refl)
_×_ : ℕ → ℕ → Set l
n × m = Vec (Vec Carrier m) n
null : {n m : ℕ} → n × m
null = replicate (replicate 0#)
infixl 7 _∔_
_∔_ : {n m : ℕ} → n × m → n × m → n × m
[] ∔ [] = []
(xs ∷ xss) ∔ (ys ∷ yss) = zipWith _+_ xs ys ∷ xss ∔ yss
我使用向量定义了矩阵的数据类型,并定义了null
矩阵和
矩阵加法。我的愿望是证明null
是左翼的身份
矩阵加法:
null-left-id-∔ : ∀ {n m : ℕ} → (xss : n × m) → (null {n} {m}) ∔ xss ≡ xss
我试图通过对矩阵索引的归纳来定义它,如下所示:
null-left-id-∔ : ∀ {n m : ℕ} → (xss : n × m) → (null {n} {m}) ∔ xss ≡ xss
null-left-id-∔ {zero} {zero} [] = refl
null-left-id-∔ {zero} {suc m} xss = {!!}
null-left-id-∔ {suc n} {zero} xss = {!!}
null-left-id-∔ {suc n} {suc m} xss = {!!}
但是,在所有漏洞中,函数null
都没有扩展。由于null
是使用replicate
定义的,并且它使用自然数的递归,我期待这一点
匹配矩阵索引会触发null
定义的减少。
简而言之,我也试图通过对矩阵结构的归纳来定义这样的证明(通过xss
上的递归)。同样,null
定义不会在漏洞中扩展。
我在做些傻事吗?
编辑:我正在使用Agda 2.5.1.1和标准库版本0.12。
答案 0 :(得分:2)
我猜您正在检查C-c C-,
和C-c C-.
并未执行完全规范化的漏洞。请改为C-u C-u C-c C-,
和C-u C-u C-c C-.
。
xss
的归纳几乎可行:
zipWith-+-replicate-0# : ∀ {n} → (xs : Vec Carrier n) → zipWith _+_ (replicate 0#) xs ≡ xs
zipWith-+-replicate-0# [] = refl
zipWith-+-replicate-0# (x ∷ xs) = cong₂ _∷_ {!!} (zipWith-+-replicate-0# xs)
null-left-id-∔ : ∀ {n m} → (xss : n × m) → null ∔ xss ≡ xss
null-left-id-∔ [] = refl
null-left-id-∔ (xs ∷ xss) = cong₂ _∷_ (zipWith-+-replicate-0# xs) (null-left-id-∔ xss)
但是你的平等是_≈_
- 而不是_≡_
,所以你无法证明0# + x ≡ x
。您应该使用Data.Vec.Equality
模块中的相等性,但这更加冗长:
open Equality
(record { Carrier = Carrier
; _≈_ = _≈_
; isEquivalence = isEquivalence
})
renaming (_≈_ to _≈ᵥ_)
zipWith-+-replicate-0# : ∀ {n} → (xs : Vec Carrier n) → zipWith _+_ (replicate 0#) xs ≈ᵥ xs
zipWith-+-replicate-0# [] = []-cong
zipWith-+-replicate-0# (x ∷ xs) = IsCommutativeMonoid.identityˡ +-isCommutativeMonoid _
∷-cong zipWith-+-replicate-0# xs
private open module Dummy {m} = Equality
(record { Carrier = Vec Carrier m
; _≈_ = λ xs ys -> xs ≈ᵥ ys
; isEquivalence = record
{ refl = refl _
; sym = Equality.sym _
; trans = Equality.trans _
}
})
renaming (_≈_ to _≈ᵥᵥ_)
null-left-id-∔ : ∀ {n m} → (xss : n × m) → null ∔ xss ≈ᵥᵥ xss
null-left-id-∔ [] = Equality.[]-cong
null-left-id-∔ (xs ∷ xss) = zipWith-+-replicate-0# xs Equality.∷-cong null-left-id-∔ xss