module Soat.Relation

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
isetoid : (a -> Type) -> ISetoid a
isetoid u = MkISetoid u (\_ => Equal) (\_ => equiv)

public export
IFunc : {a : Type} -> (x, y : a -> Type) -> Type
IFunc {a} x y = (i : a) -> x i -> y i

public export
record IFunction {a : Type} (x, y : ISetoid a) where
  constructor MkIFunction
  func : IFunc x.U y.U
  cong : (i : a) -> {u, v : x.U i} -> x.relation i u v -> y.relation i (func i u) (func i v)