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61 lines
2.1 KiB
Idris
61 lines
2.1 KiB
Idris
module Data.List.Palindrome
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import Data.List
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import Data.List.Views
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import Data.List.Views.Extra
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import Data.List.Reverse
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import Data.List.Equalities
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%hide Prelude.reverse
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%default total
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||| Do geese see God?
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public export
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data Palindrome : (xs : List a) -> Type where
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Empty : Palindrome []
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Single : Palindrome [_]
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Multi : Palindrome xs -> Palindrome (x :: (xs `snoc` x))
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||| A Palindrome reversed is itself.
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export
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palindromeReverse : (xs : List a) -> Palindrome xs -> reverse xs = xs
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palindromeReverse [] Empty = Refl
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palindromeReverse [_] Single = Refl
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palindromeReverse ([x] ++ ys ++ [x]) (Multi pf) =
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rewrite reverseAppend ([x] ++ ys) [x] in
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rewrite reverseAppend [x] ys in
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rewrite palindromeReverse ys pf in
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Refl
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private
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reversePalindromeEqualsLemma
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: (x, x' : a)
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-> (xs : List a)
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-> reverse (x :: (xs ++ [x'])) = (x :: (xs ++ [x']))
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-> (reverse xs = xs, x = x')
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reversePalindromeEqualsLemma x x' xs prf = equateInnerAndOuter flipHeadX
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where
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flipHeadX : reverse (xs ++ [x']) ++ [x] = x :: (xs ++ [x'])
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flipHeadX = rewrite (sym (reverseCons x (xs ++ [x']))) in prf
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flipLastX' : reverse (xs ++ [x']) = x :: xs -> (x' :: reverse xs) = (x :: xs)
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flipLastX' prf = rewrite (sym $ reverseAppend xs [x']) in prf
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cancelOuter : (reverse (xs ++ [x'])) = x :: xs -> reverse xs = xs
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cancelOuter prf = snd (consInjective (flipLastX' prf))
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equateInnerAndOuter
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: reverse (xs ++ [x']) ++ [x] = (x :: xs) ++ [x']
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-> (reverse xs = xs, x = x')
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equateInnerAndOuter prf =
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let (prf', xEqualsX') = snocCong2 prf
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in (cancelOuter prf', xEqualsX')
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||| Only Palindromes are equal to their own reverse.
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export
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reversePalindrome : (xs : List a) -> reverse xs = xs -> Palindrome xs
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reversePalindrome input prf with (vList input)
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reversePalindrome [] _ | VNil = Empty
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reversePalindrome [x] _ | VOne = Single
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reversePalindrome (x :: (inner `snoc` y)) prf | VCons rec with (reversePalindromeEqualsLemma x y inner prf)
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reversePalindrome (x :: (inner `snoc` y)) prf | VCons rec | (revInnerIsIdentical, xIsY) =
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rewrite xIsY in
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Multi $ reversePalindrome inner revInnerIsIdentical | Force rec
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