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823230b77c
When working on Frex I needed a whole bunch of lemmata to do with Data.Vect. I hope it will be useful for others.
38 lines
1.3 KiB
Idris
38 lines
1.3 KiB
Idris
||| Tabulation gives a bijection between functions `f : Fin n -> a`
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||| (up to extensional equality) and vectors `tabulate f : Vect n a`.
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module Data.Vect.Properties.Tabulate
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import Data.Vect
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import Data.Fin
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||| Vectors are uniquely determined by their elements
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export
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vectorExtensionality
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: (xs, ys : Vect n a) -> (ext : (i : Fin n) -> index i xs = index i ys)
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-> xs = ys
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vectorExtensionality [] [] ext = Refl
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vectorExtensionality (x :: xs) (y :: ys) ext =
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cong2 (::)
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(ext FZ)
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(vectorExtensionality xs ys (\i => ext (FS i)))
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||| Extensionally equivalent functions tabulate to the same vector
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export
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tabulateExtensional
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: {n : Nat} -> (f, g : Fin n -> a) -> (ext : (i : Fin n) -> f i = g i)
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-> tabulate f = tabulate g
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tabulateExtensional {n = 0 } f g ext = Refl
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tabulateExtensional {n = S n} f g ext =
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cong2 (::) (ext FZ) (tabulateExtensional (f . FS) (g . FS) (\ i => ext $ FS i))
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||| Taking an index amounts to applying the tabulated function
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export
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indexTabulate : {n : Nat} -> (f : Fin n -> a) -> (i : Fin n) -> index i (tabulate f) = f i
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indexTabulate f FZ = Refl
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indexTabulate f (FS i) = indexTabulate (f . FS) i
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||| The empty vector represents the unique function `Fin 0 -> a`
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export
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emptyInitial : (v : Vect 0 a) -> v = []
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emptyInitial [] = Refl
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