1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
|
||| Coproduct of setoids
module Data.Setoid.Either
import Data.Setoid.Definition
%default total
namespace Relation
||| Binary relation disjunction
public export
data Or : (p : Rel a) -> (q : Rel b) -> Rel (Either a b) where
Left : {0 q : Rel b} -> p x y -> (p `Or` q) (Left x) (Left y)
Right : {0 p : Rel a} -> q x y -> (p `Or` q) (Right x) (Right y)
||| Coproduct of setoids
public export
Either : (a,b : Setoid) -> Setoid
Either a b = MkSetoid
{ U = U a `Either` U b
, equivalence = MkEquivalence
{ relation = a.equivalence.relation `Or` b.equivalence.relation
, reflexive = \case
Left x => Left $ a.equivalence.reflexive x
Right y => Right $ b.equivalence.reflexive y
, symmetric = \x,y => \case
Left prf => Left $ a.equivalence.symmetric _ _ prf
Right prf => Right $ b.equivalence.symmetric _ _ prf
, transitive = \x,y,z => \case
Left prf1 => \case {Left prf2 => Left $ a.equivalence.transitive _ _ _ prf1 prf2}
Right prf1 => \case {Right prf2 => Right $ b.equivalence.transitive _ _ _ prf1 prf2}
}
}
||| Setoid homomorphism smart constructor
public export
Left : {0 a, b: Setoid} -> a ~> (a `Either` b)
Left = MkSetoidHomomorphism
{ H = Left
, homomorphic = \x,y,prf => Left prf
}
||| Setoid homomorphism smart constructor
public export
Right : {0 a, b: Setoid} -> b ~> (a `Either` b)
Right = MkSetoidHomomorphism
{ H = Right
, homomorphic = \x,y,prf => Right prf
}
||| Setoid homomorphism deconstructor
public export
either : {0 a, b, c : Setoid} -> (a ~> c) -> (b ~> c) -> (a `Either` b) ~> c
either lft rgt = MkSetoidHomomorphism
{ H = either lft.H rgt.H
, homomorphic = \x,y => \case
Left prf => lft.homomorphic _ _ prf
Right prf => rgt.homomorphic _ _ prf
}
|
