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31 lines
942 B
Idris
31 lines
942 B
Idris
module Syntax.PreorderReasoning.Generic
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import Control.Relation
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import Control.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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(...) : (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 leq x x
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(<~) : {x, y : a}
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-> FastDerivation leq x y -> {z : a} -> (step : Step leq y z)
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-> FastDerivation leq x z
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public export
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CalcWith : Preorder dom leq => {0 x : dom} -> {0 y : dom} -> FastDerivation leq x y -> x `leq` y
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CalcWith (|~ x) = reflexive {rel = leq}
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CalcWith ((<~) der (z ... step)) = transitive {rel = leq} (CalcWith der) step
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public export
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(~~) : {0 x : dom} -> {0 y : dom}
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-> FastDerivation leq x y -> {z : dom} -> (step : Step Equal y z)
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-> FastDerivation leq x z
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(~~) der (z ... Refl) = der
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