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1576a578a0
* contrib library unused import removal * remove a few unused imports. * another round of unused import removal * another round of unused import deletion. * another round of unused import deletion.
81 lines
3.0 KiB
Idris
81 lines
3.0 KiB
Idris
||| Properties of Data.Vect.map
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module Data.Vect.Properties.Map
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import Data.Vect.Properties.Tabulate
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import Data.Vect.Properties.Index
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import Data.Vect
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import Data.Vect.Elem
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import Data.Fin
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import Data.Vect.Extra
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import Syntax.PreorderReasoning
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||| `map` functoriality: identity preservation
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export
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mapId : (xs : Vect n a) -> map Prelude.id xs = xs
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mapId xs = vectorExtensionality _ _ $ \i => indexNaturality _ _ _
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||| `mapWtihPos f` represents post-composition the tabulated function `f`
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export
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indexMapWithPos : (f : Fin n -> a -> b) -> (xs : Vect n a) -> (i : Fin n)
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-> index i (mapWithPos f xs) = f i (index i xs)
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indexMapWithPos f (x :: _ ) FZ = Refl
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indexMapWithPos f (_ :: xs) (FS i) = indexMapWithPos _ _ _
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||| `tabulate : (Fin n ->) -> Vect n` is a natural transformation
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export
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mapTabulate : (f : a -> b) -> (g : Fin n -> a)
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-> tabulate (f . g) = map f (tabulate g)
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mapTabulate f g = irrelevantEq $
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vectorExtensionality _ _ $ \i => Calc $
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|~ index i (tabulate (f . g))
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~~ f (g i) ...(indexTabulate _ _)
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~~ f (index i $ tabulate g) ...(cong f (sym $ indexTabulate _ _))
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~~ index i (map f $ tabulate g) ...(sym $ indexNaturality _ _ _)
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||| Tabulating with the constant function is replication
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export
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tabulateConstantly : (x : a) -> Fin.tabulate {len} (const x) === replicate len x
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tabulateConstantly x = irrelevantEq $
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vectorExtensionality _ _ $ \i => Calc $
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|~ index i (Fin.tabulate (const x))
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~~ x ...(indexTabulate _ _)
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~~ index i (replicate _ x) ...(sym $ indexReplicate _ _)
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||| It's enough that two functions agree on the elements of a vector for the maps to agree
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export
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mapRestrictedExtensional : (f, g : a -> b) -> (xs : Vect n a)
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-> (prf : (i : Fin n) -> f (index i xs) = g (index i xs))
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-> map f xs = map g xs
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mapRestrictedExtensional f g xs prf = vectorExtensionality _ _ $ \i => Calc $
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|~ index i (map f xs)
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~~ f (index i xs) ...( indexNaturality _ _ _)
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~~ g (index i xs) ...(prf _)
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~~ index i (map g xs) ...(sym $ indexNaturality _ _ _)
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||| function extensionality is a congruence wrt map
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export
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mapExtensional : (f, g : a -> b)
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-> (prf : (x : a) -> f x = g x)
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-> (xs : Vect n a)
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-> map f xs = map g xs
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mapExtensional f g prf xs = mapRestrictedExtensional f g xs (\i => prf (index i xs))
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||| map-fusion property for vectors up to function extensionality
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export
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mapFusion : (f : b -> c) -> (g : a -> b) -> (xs : Vect n a)
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-> map f (map g xs) = map (f . g) xs
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mapFusion f g [] = Refl
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mapFusion f g (x :: xs) = cong (f $ g x ::) $ mapFusion f g xs
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||| function extensionality is a congruence wrt mapWithElem
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export
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mapWithElemExtensional : (xs : Vect n a) -> (f, g : (x : a) -> (0 _ : x `Elem` xs) -> b)
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-> ((x : a) -> (0 pos : x `Elem` xs) -> f x pos = g x pos)
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-> mapWithElem xs f = mapWithElem xs g
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mapWithElemExtensional [] f g prf = Refl
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mapWithElemExtensional (x :: xs) f g prf
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= cong2 (::) (prf x Here)
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(mapWithElemExtensional xs _ _ (\x,pos => prf x (There pos)))
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