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43 lines
995 B
Idris
43 lines
995 B
Idris
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||| Until Idris2 starts supporting the 'syntax' keyword, here's a
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||| poor-man's equational reasoning
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module Syntax.PreorderReasoning
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||| Deep embedding of equation derivation sequences.
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||| Using the Nil, (::) constructors lets us use list syntax.
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public export
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data Derivation : (x : a) -> (y : b) -> Type where
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Nil : Derivation x x
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(::) : (x = y) -> Derivation y z -> Derivation x z
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infix 1 ==|
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||| Explicate the term under consideration and the justification for
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export
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(==|) : (x : a) -> (x = y) -> x = y
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(==|) x pf = pf
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||| Finishg the derivation.
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||| A bit klunky, but this /is/ a poor-man's version.
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export
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QED : {x : a} -> x = x
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QED = Refl
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export
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Calculate : Derivation x y -> x = y
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Calculate [] = Refl
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Calculate (Refl :: der) = Calculate der
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{--
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||| Requires Data.Nata
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example : (x : Nat) -> (x + 1) + 0 = 1 + x
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example x = Calculate [
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(x + 1) + 0
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==| plusZeroRightNeutral (x + 1) ,
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x + 1
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==| plusCommutative x 1 ,
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1 + x
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==| QED
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]
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--}
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