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30 lines
978 B
Idris
30 lines
978 B
Idris
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module Syntax.PreorderReasoning.Generic
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import Decidable.Order
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infixl 0 ~~
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infixl 0 <~
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prefix 1 |~
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infix 1 ...
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public export
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data Step : (leq : a -> a -> Type) -> a -> a -> Type where
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(...) : {leq : a -> a -> Type} -> (y : a) -> x `leq` y -> Step leq x y
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public export
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data FastDerivation : {leq : a -> a -> Type} -> (x : a) -> (y : a) -> Type where
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(|~) : (x : a) -> FastDerivation x x
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(<~) : {leq : a -> a -> Type} -> {x,y : a}
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-> FastDerivation {leq = leq} x y -> {z : a} -> (step : Step leq y z)
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-> FastDerivation {leq = leq} x z
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public export
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CalcWith : Preorder dom leq => {x,y : dom} -> FastDerivation {leq = leq} x y -> x `leq` y
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CalcWith (|~ x) = reflexive x
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CalcWith ((<~) der (z ... step)) = transitive {po = leq} _ _ _ (CalcWith der) step
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public export
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(~~) : {x,y : dom}
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-> FastDerivation {leq = leq} x y -> {z : dom} -> (step : Step Equal y z)
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-> FastDerivation {leq = leq} x z
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(~~) der (z ... Refl) = der
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