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module Soat.Relation
import Control.Relation
%default total
public export
record Equivalence a (rel : Rel a) where
constructor MkEquivalence
refl : Reflexive a rel
sym : Symmetric a rel
trans : Transitive a rel
public export
equiv : Control.Relation.Equivalence a rel => Soat.Relation.Equivalence a rel
equiv = MkEquivalence (MkReflexive reflexive) (MkSymmetric symmetric) (MkTransitive transitive)
public export
[refl'] (eq : Soat.Relation.Equivalence a rel) => Control.Relation.Reflexive a rel where
reflexive = reflexive @{eq.refl}
public export
[sym'] (eq : Soat.Relation.Equivalence a rel) => Control.Relation.Symmetric a rel where
symmetric = symmetric @{eq.sym}
public export
[trans'] (eq : Soat.Relation.Equivalence a rel) => Control.Relation.Transitive a rel where
transitive = transitive @{eq.trans}
public export
[equiv'] Soat.Relation.Equivalence a rel => Control.Relation.Equivalence a rel
using refl' sym' trans' where
public export
(.reflexive) : Soat.Relation.Equivalence a rel -> {x : a} -> rel x x
(.reflexive) eq = reflexive @{eq.refl}
public export
(.symmetric) : Soat.Relation.Equivalence a rel -> {x, y : a} -> rel x y -> rel y x
(.symmetric) eq = symmetric @{eq.sym}
public export
(.transitive) : Soat.Relation.Equivalence a rel -> {x, y, z : a} -> rel x y -> rel y z -> rel x z
(.transitive) eq = transitive @{eq.trans}
public export
(.equalImpliesEq) : Soat.Relation.Equivalence a rel -> {x, y : a} -> x = y -> rel x y
(.equalImpliesEq) eq Refl = eq.reflexive
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) -> Soat.Relation.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
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)
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