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-module Data.Setoid.Either
-
-import public Data.Setoid
-
-public export
-data Either : (0 r : Rel a) -> (0 s : Rel b) -> Rel (Either a b) where
- Left : forall r, s . r x y -> Either r s (Left x) (Left y)
- Right : forall r, s . s x y -> Either r s (Right x) (Right y)
-
-public export
-[eitherRefl] Reflexive a rel => Reflexive b rel' => Reflexive (Either a b) (Either rel rel') where
- reflexive {x = (Left x)} = Left reflexive
- reflexive {x = (Right x)} = Right reflexive
-
-public export
-[eitherSym] Symmetric a rel => Symmetric b rel' => Symmetric (Either a b) (Either rel rel') where
- symmetric (Left eq) = Left $ symmetric eq
- symmetric (Right eq) = Right $ symmetric eq
-
-public export
-[eitherTrans] Transitive a rel => Transitive b rel'
- => Transitive (Either a b) (Either rel rel') where
- transitive (Left eq) (Left eq') = Left $ transitive eq eq'
- transitive (Right eq) (Right eq') = Right $ transitive eq eq'
-
-public export
-EitherEquivalence : Setoid.Equivalence a rel -> Setoid.Equivalence b rel'
- -> Setoid.Equivalence (Either a b) (Either rel rel')
-EitherEquivalence eq eq' = MkEquivalence
- { refl = MkReflexive $ reflexive @{eitherRefl @{eq.refl} @{eq'.refl}}
- , sym = MkSymmetric $ symmetric @{eitherSym @{eq.sym} @{eq'.sym}}
- , trans = MkTransitive $ transitive @{eitherTrans @{eq.trans} @{eq'.trans}}
- }
-
-public export
-EitherSetoid : Setoid -> Setoid -> Setoid
-EitherSetoid x y = MkSetoid _ _ $ EitherEquivalence x.equivalence y.equivalence