61 lines
1.7 KiB
Agda
61 lines
1.7 KiB
Agda
module Concat
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data Nat : U where
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Z : Nat
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S : Nat -> Nat
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infixl 7 _+_
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_+_ : Nat -> Nat -> Nat
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Z + m = m
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S n + m = S (n + m)
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infixr 3 _::_
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data List : U -> U where
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Nil : {A : U} -> List A
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_::_ : {A : U} -> A -> List A -> List A
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length : {A : U} -> List A -> Nat
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length Nil = Z
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length (x :: xs) = S (length xs)
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infixl 2 _++_
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_++_ : {A : U} -> List A -> List A -> List A
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Nil ++ ys = ys
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x :: xs ++ ys = x :: (xs ++ ys)
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infixl 1 _≡_
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data _≡_ : {A : U} -> A -> A -> U where
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Refl : {A : U} {a : A} -> a ≡ a
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sym : {A : U} {a b : A} -> a ≡ b -> b ≡ a
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sym Refl = Refl
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replace : {A : U} {a b : A} -> (P : A -> U) -> a ≡ b -> P a -> P b
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replace p Refl x = x
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cong : {A B : U} {a b : A} -> (f : A -> B) -> a ≡ b -> f a ≡ f b
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length-++ : {A : U} (xs ys : List A) -> length (xs ++ ys) ≡ length xs + length ys
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length-++ Nil ys = Refl
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length-++ (x :: xs) ys = cong S (length-++ xs ys)
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-- PLFA definition
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reverse : {A : U} -> (xs : List A) -> List A
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reverse Nil = Nil
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reverse (x :: xs) = reverse xs ++ (x :: Nil)
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++-identity : {A : U} -> (xs : List A) -> xs ++ Nil ≡ xs
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++-identity Nil = Refl
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++-identity (x :: xs) = cong (_::_ x) (++-identity xs)
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++-associative : {A : U} (xs ys zs : List A) -> xs ++ (ys ++ zs) ≡ (xs ++ ys) ++ zs
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-- TODO port equational reasoning
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reverse-++-distrib : {A : U} -> (xs ys : List A) -> reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
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reverse-++-distrib Nil ys = replace (\ z => reverse ys ≡ z) (sym (++-identity (reverse ys))) Refl
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reverse-++-distrib {A} (x :: xs) ys =
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replace (\ z => (reverse (xs ++ ys) ++ (x :: Nil)) ≡ z)
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(sym (++-associative (reverse ys) (reverse xs) (x :: Nil)))
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(replace (\ z => (reverse (xs ++ ys)) ++ (x :: Nil) ≡ z ++ (x :: Nil)) (reverse-++-distrib xs ys) Refl)
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