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55 lines
1.5 KiB
Idris
55 lines
1.5 KiB
Idris
||| 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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infixl 0 ~~,~=
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prefix 1 |~
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infix 1 ...,..<,..>,.=.,.=<,.=>
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|||Slightly nicer syntax for justifying equations:
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|||```
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||| |~ a
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||| ~~ b ...( justification )
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|||```
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|||and we can think of the `...( justification )` as ASCII art for a thought bubble.
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public export
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data Step : a -> b -> Type where
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(...) : (0 y : a) -> (0 step : x ~=~ y) -> Step x y
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public export
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data FastDerivation : (x : a) -> (y : b) -> Type where
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(|~) : (0 x : a) -> FastDerivation x x
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(~~) : FastDerivation x y -> (step : Step y z) -> FastDerivation x z
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public export
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Calc : {0 x : a} -> {0 y : b} -> (0 der : FastDerivation x y) -> x ~=~ y
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Calc der = irrelevantEq $ Calc' der
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where
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Calc' : {0 x : c} -> {0 y : d} -> FastDerivation x y -> x ~=~ y
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Calc' (|~ x) = Refl
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Calc' ((~~) der (_ ...(Refl))) = Calc' der
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{- -- requires import Data.Nat
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0
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example : (x : Nat) -> (x + 1) + 0 = 1 + x
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example x =
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Calc $
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|~ (x + 1) + 0
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~~ x+1 ...( plusZeroRightNeutral $ x + 1 )
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~~ 1+x ...( plusCommutative x 1 )
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-}
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-- Smart constructors
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public export
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(..<) : (0 y : a) -> {0 x : b} -> (0 step : y ~=~ x) -> Step x y
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(y ..<(prf)) {x} = (y ...(sym prf))
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public export
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(..>) : (0 y : a) -> (0 step : x ~=~ y) -> Step x y
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(..>) = (...)
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||| Use a judgemental equality but is not trivial to the reader.
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public export
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(~=) : FastDerivation x y -> (0 z : dom) -> {auto 0 xEqy : y = z} -> FastDerivation x z
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(~=) der y {xEqy = Refl} = der ~~ y ... Refl
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