papers.Search.Negation.idr Maven / Gradle / Ivy
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module Search.Negation
import Data.List.Quantifiers
import Data.Nat
%default total
||| It is much easier to look for positive evidence than it is to look
||| for negative evidence. So instead of looking for `Not q`, we may
||| want to look for `p` instead
public export
interface Negates p q | q where
toNegation : p -> Not q
public export
({0 x : a} -> Negates (p x) (q x)) => Negates (All p xs) (Any q xs) where
toNegation all = allNegAny (mapProperty toNegation all)
public export
({0 x : a} -> Negates (p x) (q x)) => Negates (Any p xs) (All q xs) where
toNegation any = anyNegAll (mapProperty toNegation any)
public export
Negates (m `LT` n) (m `GTE` n) where
toNegation = LTImpliesNotGTE
public export
Negates (m `LTE` n) (m `GT` n) where
toNegation = LTEImpliesNotGT