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-module Data.Setoid.Indexed
-
-import public Data.Setoid
-
-%default total
-
-public export
-IRel : {a : Type} -> (a -> Type) -> Type
-IRel {a = a} x = (i : a) -> x i -> x i -> Type
-
-public export
-IReflexive : {a : Type} -> (x : a -> Type) -> IRel x -> Type
-IReflexive x rel = (i : a) -> Reflexive (x i) (rel i)
-
-public export
-ISymmetric : {a : Type} -> (x : a -> Type) -> IRel x -> Type
-ISymmetric x rel = (i : a) -> Symmetric (x i) (rel i)
-
-public export
-ITransitive : {a : Type} -> (x : a -> Type) -> IRel x -> Type
-ITransitive x rel = (i : a) -> Transitive (x i) (rel i)
-
-public export
-IEquivalence : {a : Type} -> (x : a -> Type) -> IRel x -> Type
-IEquivalence x rel = (i : a) -> Setoid.Equivalence (x i) (rel i)
-
-public export
-record ISetoid (a : Type) where
- constructor MkISetoid
- 0 U : a -> Type
- 0 relation : IRel U
- equivalence : IEquivalence U relation
-
-public export
-fromIndexed : (a -> Setoid) -> ISetoid a
-fromIndexed x = MkISetoid (\i => (x i).U) (\i => (x i).relation) (\i => (x i).equivalence)
-
-public export
-(.index) : ISetoid a -> a -> Setoid
-(.index) x i = MkSetoid (x.U i) (x.relation i) (x.equivalence i)
-
-public export
-reindex : (a -> b) -> ISetoid b -> ISetoid a
-reindex f x = MkISetoid (x.U . f) (\i => x.relation $ f i) (\i => x.equivalence $ f i)
-
-public export
-isetoid : (a -> Type) -> ISetoid a
-isetoid u = MkISetoid u (\_ => Equal) (\_ => equiv)