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|
||| Basic definition and notation for setoids
module Data.Setoid.Definition
import public Control.Relation
import public Control.Order
import Data.Vect
import public Data.Fun.Nary
infix 5 ~>, ~~>, <~>
%default total
public export
record Equivalence (A : Type) where
constructor MkEquivalence
0 relation: Rel A
reflexive : (x : A) -> relation x x
symmetric : (x, y : A) -> relation x y -> relation y x
transitive: (x, y, z : A) -> relation x y -> relation y z
-> relation x z
public export
EqualEquivalence : (0 a : Type) -> Equivalence a
EqualEquivalence a = MkEquivalence
{ relation = (===)
, reflexive = \_ => Refl
, symmetric = \_,_, Refl => Refl
, transitive = \_,_,_,Refl,Refl => Refl
}
public export
record Setoid where
constructor MkSetoid
0 U : Type
equivalence : Equivalence U
public export
record PreorderData A (rel : Rel A) where
constructor MkPreorderData
reflexive : (x : A) -> rel x x
transitive : (x,y,z : A) -> rel x y -> rel y z -> rel x z
public export
[PreorderWorkaround] (Reflexive ty rel, Transitive ty rel) => Preorder ty rel where
public export
MkPreorderWorkaround : {preorderData : PreorderData ty rel} -> Order.Preorder ty rel
MkPreorderWorkaround {preorderData} =
let reflexiveArg = MkReflexive {ty, rel} $
lam Hidden (\y => rel y y) preorderData.reflexive
transitiveArg = MkTransitive {ty} {rel} $
Nary.curry 3 Hidden
(\[x,y,z] =>
x `rel` y ->
y `rel` z ->
x `rel` z)
(\[x,y,z] => preorderData.transitive _ _ _)
in PreorderWorkaround
public export
reflect : (a : Setoid) -> {x, y : U a} -> x = y -> a.equivalence.relation x y
reflect a Refl = a.equivalence.reflexive _
public export
MkPreorder : {0 a : Type} -> {0 rel : Rel a}
-> (reflexive : (x : a) -> rel x x)
-> (transitive : (x,y,z : a) -> rel x y -> rel y z -> rel x z)
-> Preorder a rel
MkPreorder reflexive transitive = MkPreorderWorkaround {preorderData = MkPreorderData reflexive transitive}
public export
cast : (a : Setoid) -> Preorder (U a) (a.equivalence.relation)
cast a = MkPreorder a.equivalence.reflexive a.equivalence.transitive
namespace ToSetoid
public export
irrelevantCast : (0 a : Type) -> Setoid
irrelevantCast a = MkSetoid a (EqualEquivalence a)
public export
Cast Type Setoid where
cast a = irrelevantCast a
public export 0
SetoidHomomorphism : (a,b : Setoid)
-> (f : U a -> U b) -> Type
SetoidHomomorphism a b f
= (x,y : U a) -> a.equivalence.relation x y
-> b.equivalence.relation (f x) (f y)
public export
record (~>) (A,B : Setoid) where
constructor MkSetoidHomomorphism
H : U A -> U B
homomorphic : SetoidHomomorphism A B H
public export
mate : {b : Setoid} -> (a -> U b) -> (irrelevantCast a ~> b)
mate f = MkSetoidHomomorphism f $ \x,y, prf => reflect b (cong f prf)
||| Identity Setoid homomorphism
public export
id : (a : Setoid) -> a ~> a
id a = MkSetoidHomomorphism Prelude.id $ \x, y, prf => prf
||| Composition of Setoid homomorphisms
public export
(.) : {a,b,c : Setoid} -> b ~> c -> a ~> b -> a ~> c
g . f = MkSetoidHomomorphism (H g . H f) $ \x,y,prf => g.homomorphic _ _ (f.homomorphic _ _ prf)
public export
(~~>) : (a,b : Setoid) -> Setoid
(~~>) a b = MkSetoid (a ~> b) $
let 0 relation : (f, g : a ~> b) -> Type
relation f g = (x : U a) ->
b.equivalence.relation (f.H x) (g.H x)
in MkEquivalence
{ relation
, reflexive = \f,v =>
b.equivalence.reflexive (f.H v)
, symmetric = \f,g,prf,w =>
b.equivalence.symmetric _ _ (prf w)
, transitive = \f,g,h,f_eq_g, g_eq_h, q =>
b.equivalence.transitive _ _ _
(f_eq_g q) (g_eq_h q)
}
public export
post : {a,b,c : Setoid} -> b ~> c -> (a ~~> b) ~> (a ~~> c)
post h = MkSetoidHomomorphism
{ H = (h .)
, homomorphic = \f1,f2,prf,x => h.homomorphic _ _ (prf x)
}
||| Two setoid homomorphism are each other's inverses
public export
record Isomorphism {a, b : Setoid} (Fwd : a ~> b) (Bwd : b ~> a) where
constructor IsIsomorphism
BwdFwdId : (a ~~> a).equivalence.relation (Bwd . Fwd) (id a)
FwdBwdId : (b ~~> b).equivalence.relation (Fwd . Bwd) (id b)
||| Setoid isomorphism
public export
record (<~>) (a, b : Setoid) where
constructor MkIsomorphism
Fwd : a ~> b
Bwd : b ~> a
Iso : Isomorphism Fwd Bwd
||| Identity (isomorphism _)
public export
refl : {a : Setoid} -> a <~> a
refl = MkIsomorphism (id a) (id a) (IsIsomorphism a.equivalence.reflexive a.equivalence.reflexive)
||| Reverse an isomorphism
public export
sym : a <~> b -> b <~> a
sym iso = MkIsomorphism iso.Bwd iso.Fwd (IsIsomorphism iso.Iso.FwdBwdId iso.Iso.BwdFwdId)
||| Compose isomorphisms
public export
trans : {a,b,c : Setoid} -> (a <~> b) -> (b <~> c) -> (a <~> c)
trans ab bc = MkIsomorphism (bc.Fwd . ab.Fwd) (ab.Bwd . bc.Bwd) (IsIsomorphism i1 i2)
where i1 : (x : U a) -> a.equivalence.relation (ab.Bwd.H (bc.Bwd.H (bc.Fwd.H (ab.Fwd.H x)))) x
i1 x = a.equivalence.transitive _ _ _ (ab.Bwd.homomorphic _ _ (bc.Iso.BwdFwdId _)) (ab.Iso.BwdFwdId x)
i2 : (x : U c) -> c.equivalence.relation (bc.Fwd.H (ab.Fwd.H (ab.Bwd.H (bc.Bwd.H x)))) x
i2 x = c.equivalence.transitive _ _ _ (bc.Fwd.homomorphic _ _ (ab.Iso.FwdBwdId _)) (bc.Iso.FwdBwdId x)
public export
IsoEquivalence : Equivalence Setoid
IsoEquivalence = MkEquivalence (<~>) (\_ => refl) (\_,_ => sym) (\_,_,_ => trans)
||| Quotient a type by an function into a setoid
|||
||| Instance of the more general coequaliser of two setoid morphisms.
public export
Quotient : (b : Setoid) -> (a -> U b) -> Setoid
Quotient b q = MkSetoid a $
let 0 relation : a -> a -> Type
relation x y = b.equivalence.relation (q x) (q y)
in MkEquivalence
{ relation = relation
, reflexive = \x =>
b.equivalence.reflexive (q x)
, symmetric = \x,y =>
b.equivalence.symmetric (q x) (q y)
, transitive = \x,y,z =>
b.equivalence.transitive (q x) (q y) (q z)
}
|
