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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) : ISetoid b -> (f : a -> b) -> ISetoid a
(.reindex) x f = MkISetoid
(\i => x.U $ f i)
(\i => x.relation $ f i)
(\i => x.equivalence $ f i)
public export
isetoid : (a -> Type) -> ISetoid a
isetoid u = MkISetoid u (\_ => Equal) (\_ => equiv)
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