mirror of
https://github.com/Smaug123/agdaproofs
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29 lines
931 B
Agda
29 lines
931 B
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Functions
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open import Lists.Definition
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open import Lists.Fold.Fold
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open import Lists.Concat
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open import Lists.Length
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open import Numbers.Naturals.Semiring
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module Lists.Monad where
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open import Lists.Map.Map public
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flatten : {a : _} {A : Set a} → (l : List (List A)) → List A
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flatten [] = []
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flatten (l :: ls) = l ++ flatten ls
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flatten' : {a : _} {A : Set a} → (l : List (List A)) → List A
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flatten' = fold _++_ []
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flatten=flatten' : {a : _} {A : Set a} (l : List (List A)) → flatten l ≡ flatten' l
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flatten=flatten' [] = refl
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flatten=flatten' (l :: ls) = applyEquality (l ++_) (flatten=flatten' ls)
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lengthFlatten : {a : _} {A : Set a} (l : List (List A)) → length (flatten l) ≡ (fold _+N_ zero (map length l))
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lengthFlatten [] = refl
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lengthFlatten (l :: ls) rewrite lengthConcat l (flatten ls) | lengthFlatten ls = refl
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