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March 3, 2025 06:07
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Filter
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| module FilterProof where | |
| open import Data.List hiding (filter; concat) | |
| open import Data.Bool | |
| import Relation.Binary.PropositionalEquality as Eq | |
| open Eq using (_≡_; refl; trans; sym; cong; cong-app; subst) | |
| open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; step-≡; _∎) | |
| private variable | |
| A : Set | |
| B : Set | |
| filter : (A → Bool) → List A → List A | |
| filter f [] = [] | |
| filter f (x ∷ xs) = | |
| if f x then x ∷ filter f xs else filter f xs | |
| concat : List (List A) → List A | |
| concat [] = [] | |
| concat (x ∷ xs) = x ++ (concat xs) | |
| return : A → List A | |
| return x = [ x ] | |
| _>>=_ : List A → (A → List B) → List B | |
| xs >>= f = concat (map f xs) | |
| mfilter : (A → Bool) → List A → List A | |
| mfilter f xs = xs >>= (λ x → if f x then return x else []) | |
| filter-eq : ∀ f (xs : List A) → filter f xs ≡ mfilter f xs | |
| filter-eq f [] = refl | |
| filter-eq f (x ∷ xs) with f x | |
| ... | true = | |
| begin | |
| x ∷ filter f xs | |
| ≡⟨ cong (λ t → x ∷ t) hip ⟩ | |
| x ∷ mfilter f xs | |
| ∎ | |
| where | |
| hip = filter-eq f xs | |
| ... | false = | |
| begin | |
| filter f xs | |
| ≡⟨ hip ⟩ | |
| mfilter f xs | |
| ∎ | |
| where | |
| hip = filter-eq f xs |
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