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30 lines
912 B
Idris
30 lines
912 B
Idris
module Control.Function
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||| An injective function maps distinct elements to distinct elements.
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public export
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interface Injective (f : a -> b) | f where
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constructor MkInjective
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injective : {x, y : a} -> f x = f y -> x = y
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public export
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inj : (0 f : a -> b) -> {auto 0 _ : Injective f} -> {0 x, y : a} -> (0 _ : f x = f y) -> x = y
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inj _ eq = irrelevantEq (injective eq)
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----------------------------------------
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||| The composition of injective functions is injective.
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public export
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[ComposeInjective] {f : a -> b} -> {g : b -> c} ->
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(Injective f, Injective g) => Injective (g . f) where
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injective prf = injective $ injective prf
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||| If (g . f) is injective, so is f.
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public export
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[InjFromComp] {f : a -> b} -> {g : b -> c} ->
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Injective (g . f) => Injective f where
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injective prf = injective {f = (g . f)} $ cong g prf
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public export
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[IdInjective] Injective Prelude.id where
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injective = id
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