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65 lines
1.6 KiB
Idris
65 lines
1.6 KiB
Idris
module Data.Bool.Xor
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import Data.Bool
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%default total
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public export
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xor : Bool -> Bool -> Bool
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xor True True = False
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xor False False = False
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xor _ _ = True
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export
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xorSameFalse : (b : Bool) -> xor b b = False
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xorSameFalse False = Refl
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xorSameFalse True = Refl
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export
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xorFalseNeutral : (b : Bool) -> xor False b = b
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xorFalseNeutral False = Refl
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xorFalseNeutral True = Refl
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export
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xorTrueNot : (b : Bool) -> xor True b = not b
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xorTrueNot False = Refl
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xorTrueNot True = Refl
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export
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notXor : (a, b : Bool) -> not (xor a b) = xor (not a) b
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notXor True b = rewrite xorFalseNeutral b in
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rewrite xorTrueNot b in
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notInvolutive b
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notXor False b = rewrite xorFalseNeutral b in
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sym $ xorTrueNot b
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export
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notXorCancel : (a, b : Bool) -> xor (not a) (not b) = xor a b
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notXorCancel True b = rewrite xorFalseNeutral (not b) in
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sym $ xorTrueNot b
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notXorCancel False b = rewrite xorFalseNeutral b in
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rewrite xorTrueNot (not b) in
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notInvolutive b
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export
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xorAssociative : (a, b, c : Bool) -> xor a (xor b c) = xor (xor a b) c
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xorAssociative False b c =
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rewrite xorFalseNeutral b in
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xorFalseNeutral $ xor b c
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xorAssociative True b c =
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rewrite xorTrueNot b in
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rewrite xorTrueNot (xor b c) in
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notXor b c
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export
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xorCommutative : (a, b : Bool) -> xor a b = xor b a
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xorCommutative False False = Refl
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xorCommutative False True = Refl
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xorCommutative True False = Refl
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xorCommutative True True = Refl
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export
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xorNotTrue : (a : Bool) -> a `xor` (not a) = True
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xorNotTrue False = Refl
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xorNotTrue True = Refl
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