diff options
Diffstat (limited to 'Data/Vector.agda')
| -rw-r--r-- | Data/Vector.agda | 50 |
1 files changed, 47 insertions, 3 deletions
diff --git a/Data/Vector.agda b/Data/Vector.agda index 052f624..874f0b2 100644 --- a/Data/Vector.agda +++ b/Data/Vector.agda @@ -3,10 +3,10 @@ module Data.Vector where open import Data.Nat.Base using (ℕ) -open import Data.Vec.Functional using (Vector; head; tail; []; removeAt; map) public -open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_) +open import Data.Vec.Functional using (Vector; head; tail; []; removeAt; map; _++_) public +open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_; module ≡-Reasoning) open import Function.Base using (∣_⟩-_; _∘_) -open import Data.Fin.Base using (Fin; toℕ) +open import Data.Fin.Base using (Fin; toℕ; _↑ˡ_; _↑ʳ_) open ℕ open Fin @@ -80,3 +80,47 @@ foldl-[] {e : B 0} → foldl B f e [] ≡ e foldl-[] _ _ = ≡.refl + +open import Data.Sum using ([_,_]′) +open import Data.Sum.Properties using ([,-]-cong; [-,]-cong; [,]-∘) +open import Data.Fin.Properties using (splitAt-↑ˡ; splitAt-↑ʳ) +open import Data.Fin using (splitAt) +open import Data.Nat using (_+_) +++-↑ˡ + : {n m : ℕ} + {A : Set} + (X : Vector A n) + (Y : Vector A m) + → (X ++ Y) ∘ (_↑ˡ m) ≗ X +++-↑ˡ {n} {m} X Y i = ≡.cong [ X , Y ]′ (splitAt-↑ˡ n i m) + +++-↑ʳ + : {n m : ℕ} + {A : Set} + (X : Vector A n) + (Y : Vector A m) + → (X ++ Y) ∘ (n ↑ʳ_) ≗ Y +++-↑ʳ {n} {m} X Y i = ≡.cong [ X , Y ]′ (splitAt-↑ʳ n m i) + ++-assocʳ : {m n o : ℕ} → Fin (m + (n + o)) → Fin (m + n + o) ++-assocʳ {m} {n} {o} = [ (λ x → x ↑ˡ n ↑ˡ o) , [ (λ x → (m ↑ʳ x) ↑ˡ o) , m + n ↑ʳ_ ]′ ∘ splitAt n ]′ ∘ splitAt m + +open ≡-Reasoning +++-assoc + : {m n o : ℕ} + {A : Set} + (X : Vector A m) + (Y : Vector A n) + (Z : Vector A o) + → ((X ++ Y) ++ Z) ∘ +-assocʳ {m} ≗ X ++ (Y ++ Z) +++-assoc {m} {n} {o} X Y Z i = begin + ((X ++ Y) ++ Z) (+-assocʳ {m} i) ≡⟨⟩ + ((X ++ Y) ++ Z) ([ (λ x → x ↑ˡ n ↑ˡ o) , _ ]′ (splitAt m i)) ≡⟨ [,]-∘ ((X ++ Y) ++ Z) (splitAt m i) ⟩ + [ ((X ++ Y) ++ Z) ∘ (λ x → x ↑ˡ n ↑ˡ o) , _ ]′ (splitAt m i) ≡⟨ [-,]-cong (++-↑ˡ (X ++ Y) Z ∘ (_↑ˡ n)) (splitAt m i) ⟩ + [ (X ++ Y) ∘ (_↑ˡ n) , _ ]′ (splitAt m i) ≡⟨ [-,]-cong (++-↑ˡ X Y) (splitAt m i) ⟩ + [ X , ((X ++ Y) ++ Z) ∘ [ _ , _ ]′ ∘ splitAt n ]′ (splitAt m i) ≡⟨ [,-]-cong ([,]-∘ ((X ++ Y) ++ Z) ∘ splitAt n) (splitAt m i) ⟩ + [ X , [ (_ ++ Z) ∘ (_↑ˡ o) ∘ (m ↑ʳ_) , _ ]′ ∘ splitAt n ]′ (splitAt m i) ≡⟨ [,-]-cong ([-,]-cong (++-↑ˡ (X ++ Y) Z ∘ (m ↑ʳ_)) ∘ splitAt n) (splitAt m i) ⟩ + [ X , [ (X ++ Y) ∘ (m ↑ʳ_) , _ ]′ ∘ splitAt n ]′ (splitAt m i) ≡⟨ [,-]-cong ([-,]-cong (++-↑ʳ X Y) ∘ splitAt n) (splitAt m i) ⟩ + [ X , [ Y , ((X ++ Y) ++ Z) ∘ (m + n ↑ʳ_) ]′ ∘ splitAt n ]′ (splitAt m i) ≡⟨ [,-]-cong ([,-]-cong (++-↑ʳ (X ++ Y) Z) ∘ splitAt n) (splitAt m i) ⟩ + [ X , [ Y , Z ]′ ∘ splitAt n ]′ (splitAt m i) ≡⟨⟩ + (X ++ (Y ++ Z)) i ∎ |