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module Soat.FirstOrder.Algebra
import Data.Morphism.Indexed
import Data.Setoid.Indexed
import public Soat.Data.Product
import Soat.FirstOrder.Signature
%default total
%hide Control.Relation.Equivalence
infix 5 ~>
public export 0
algebraOver : (sig : Signature) -> (U : sig.T -> Type) -> Type
algebraOver sig x = {t : sig.T} -> (op : Op sig t) -> x ^ op.arity -> x t
public export
record RawAlgebra (0 sig : Signature) where
constructor MkRawAlgebra
0 U : sig.T -> Type
sem : sig `algebraOver` U
public export
record RawSetoidAlgebra (0 sig : Signature) where
constructor MkRawSetoidAlgebra
raw : RawAlgebra sig
0 relation : IRel raw.U
public export
record IsAlgebra (0 sig : Signature) (0 a : RawAlgebra sig) where
constructor MkIsAlgebra
0 relation : IRel a.U
equivalence : IEquivalence a.U relation
semCong : {t : sig.T} -> (op : Op sig t) -> {tms, tms' : a.U ^ op.arity}
-> Pointwise relation tms tms' -> relation t (a.sem op tms) (a.sem op tms')
public export
record Algebra (0 sig : Signature) where
constructor MkAlgebra
raw : RawAlgebra sig
algebra : IsAlgebra sig raw
public export 0
(.relation) : (0 a : Algebra sig) -> IRel a.raw.U
(.relation) a = a.algebra.relation
public export
(.setoid) : Algebra sig -> ISetoid sig.T
(.setoid) a = MkISetoid a.raw.U a.relation a.algebra.equivalence
public export
(.rawSetoid) : Algebra sig -> RawSetoidAlgebra sig
(.rawSetoid) a = MkRawSetoidAlgebra a.raw a.relation
public export
record IsHomomorphism
{0 sig : Signature} (a, b : Algebra sig)
(f : (t : sig.T) -> a.raw.U t -> b.raw.U t)
where
constructor MkIsHomomorphism
cong : (t : sig.T) -> {tm, tm' : a.raw.U t}
-> a.relation t tm tm' -> b.relation t (f t tm) (f t tm')
semHomo : {t : sig.T} -> (op : Op sig t) -> (tms : a.raw.U ^ op.arity)
-> b.relation t (f t (a.raw.sem op tms)) (b.raw.sem op (map f tms))
public export
record (~>) {0 sig : Signature} (a, b : Algebra sig)
where
constructor MkHomomorphism
func : IFunc a.raw.U b.raw.U
homo : IsHomomorphism a b func
public export
idIsHomo : {a : Algebra sig} -> IsHomomorphism a a (\_ => Basics.id)
idIsHomo = MkIsHomomorphism
(\_ => id)
(\op, tms =>
(a.algebra.equivalence _).equalImpliesEq $
sym $
cong (a.raw.sem op) $
mapId tms)
public export
idHomo : {a : Algebra sig} -> a ~> a
idHomo = MkHomomorphism _ idIsHomo
public export
compIsHomo : {a, b, c : Algebra sig} -> {f : IFunc b.raw.U c.raw.U} -> {g : IFunc a.raw.U b.raw.U}
-> IsHomomorphism b c f -> IsHomomorphism a b g -> IsHomomorphism a c (\i => f i . g i)
compIsHomo fHomo gHomo = MkIsHomomorphism
(\t => fHomo.cong t . gHomo.cong t)
(\op, tms =>
(c.algebra.equivalence _).transitive
(fHomo.cong _ $ gHomo.semHomo op tms) $
(c.algebra.equivalence _).transitive
(fHomo.semHomo op (map g tms)) $
(c.algebra.equivalence _).equalImpliesEq $
sym $
cong (c.raw.sem op) $
mapComp tms)
public export
compHomo : {a, b, c : Algebra sig} -> b ~> c -> a ~> b -> a ~> c
compHomo f g = MkHomomorphism _ $ compIsHomo f.homo g.homo
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