module Interfaces.Verified import Control.Algebra import Control.Algebra.Lattice import Control.Algebra.VectorSpace %access public export -- Due to these being basically unused and difficult to implement, -- they're in contrib for a bit. Once a design is found that lets them -- be implemented for a number of implementations, and we get those -- implementations, then some of them can move back to base (or even -- prelude, in the cases of Functor, Applicative, Monad, Semigroup, -- and Monoid). interface Functor f => VerifiedFunctor (f : Type -> Type) where functorIdentity : {a : Type} -> (x : f a) -> map Basics.id x = Basics.id x functorComposition : {a : Type} -> {b : Type} -> (x : f a) -> (g1 : a -> b) -> (g2 : b -> c) -> map (g2 . g1) x = (map g2 . map g1) x interface (Applicative f, VerifiedFunctor f) => VerifiedApplicative (f : Type -> Type) where applicativeMap : (x : f a) -> (g : a -> b) -> map g x = pure g <*> x applicativeIdentity : (x : f a) -> pure Basics.id <*> x = x applicativeComposition : (x : f a) -> (g1 : f (a -> b)) -> (g2 : f (b -> c)) -> ((pure (.) <*> g2) <*> g1) <*> x = g2 <*> (g1 <*> x) applicativeHomomorphism : (x : a) -> (g : a -> b) -> (<*>) {f} (pure g) (pure x) = pure {f} (g x) applicativeInterchange : (x : a) -> (g : f (a -> b)) -> g <*> pure x = pure (\g' : (a -> b) => g' x) <*> g interface (Monad m, VerifiedApplicative m) => VerifiedMonad (m : Type -> Type) where monadApplicative : (mf : m (a -> b)) -> (mx : m a) -> mf <*> mx = mf >>= \f => mx >>= \x => pure (f x) monadLeftIdentity : (x : a) -> (f : a -> m b) -> pure x >>= f = f x monadRightIdentity : (mx : m a) -> mx >>= Applicative.pure = mx monadAssociativity : (mx : m a) -> (f : a -> m b) -> (g : b -> m c) -> (mx >>= f) >>= g = mx >>= (\x => f x >>= g) interface Semigroup a => VerifiedSemigroup a where total semigroupOpIsAssociative : (l, c, r : a) -> l <+> (c <+> r) = (l <+> c) <+> r implementation VerifiedSemigroup (List a) where semigroupOpIsAssociative = appendAssociative --implementation VerifiedSemigroup Nat where -- semigroupOpIsAssociative = plusAssociative interface (VerifiedSemigroup a, Monoid a) => VerifiedMonoid a where total monoidNeutralIsNeutralL : (l : a) -> l <+> Algebra.neutral = l total monoidNeutralIsNeutralR : (r : a) -> Algebra.neutral <+> r = r -- implementation VerifiedMonoid Nat where -- monoidNeutralIsNeutralL = plusZeroRightNeutral -- monoidNeutralIsNeutralR = plusZeroLeftNeutral implementation VerifiedMonoid (List a) where monoidNeutralIsNeutralL = appendNilRightNeutral monoidNeutralIsNeutralR xs = Refl interface (VerifiedMonoid a, Group a) => VerifiedGroup a where total groupInverseIsInverseL : (l : a) -> l <+> inverse l = Algebra.neutral total groupInverseIsInverseR : (r : a) -> inverse r <+> r = Algebra.neutral interface (VerifiedGroup a, AbelianGroup a) => VerifiedAbelianGroup a where total abelianGroupOpIsCommutative : (l, r : a) -> l <+> r = r <+> l interface (VerifiedAbelianGroup a, Ring a) => VerifiedRing a where total ringOpIsAssociative : (l, c, r : a) -> l <.> (c <.> r) = (l <.> c) <.> r total ringOpIsDistributiveL : (l, c, r : a) -> l <.> (c <+> r) = (l <.> c) <+> (l <.> r) total ringOpIsDistributiveR : (l, c, r : a) -> (l <+> c) <.> r = (l <.> r) <+> (c <.> r) interface (VerifiedRing a, RingWithUnity a) => VerifiedRingWithUnity a where total ringWithUnityIsUnityL : (l : a) -> l <.> Algebra.unity = l total ringWithUnityIsUnityR : (r : a) -> Algebra.unity <.> r = r --interface (VerifiedRingWithUnity a, Field a) => VerifiedField a where -- total fieldInverseIsInverseL : (l : a) -> (notId : Not (l = neutral)) -> l <.> (inverseM l notId) = Algebra.unity -- total fieldInverseIsInverseR : (r : a) -> (notId : Not (r = neutral)) -> (inverseM r notId) <.> r = Algebra.unity interface JoinSemilattice a => VerifiedJoinSemilattice a where total joinSemilatticeJoinIsAssociative : (l, c, r : a) -> join l (join c r) = join (join l c) r total joinSemilatticeJoinIsCommutative : (l, r : a) -> join l r = join r l total joinSemilatticeJoinIsIdempotent : (e : a) -> join e e = e interface MeetSemilattice a => VerifiedMeetSemilattice a where total meetSemilatticeMeetIsAssociative : (l, c, r : a) -> meet l (meet c r) = meet (meet l c) r total meetSemilatticeMeetIsCommutative : (l, r : a) -> meet l r = meet r l total meetSemilatticeMeetIsIdempotent : (e : a) -> meet e e = e {- FIXME: Some maintenance required here. Algebra.top and Algebra.bottom don't exist! -} {- interface (VerifiedJoinSemilattice a, BoundedJoinSemilattice a) => VerifiedBoundedJoinSemilattice a where total boundedJoinSemilatticeBottomIsBottom : (e : a) -> join e Algebra.bottom = e interface (VerifiedMeetSemilattice a, BoundedMeetSemilattice a) => VerifiedBoundedMeetSemilattice a where total boundedMeetSemilatticeTopIsTop : (e : a) -> meet e Algebra.top = e interface (VerifiedJoinSemilattice a, VerifiedMeetSemilattice a) => VerifiedLattice a where total latticeMeetAbsorbsJoin : (l, r : a) -> meet l (join l r) = l total latticeJoinAbsorbsMeet : (l, r : a) -> join l (meet l r) = l interface (VerifiedBoundedJoinSemilattice a, VerifiedBoundedMeetSemilattice a, VerifiedLattice a) => VerifiedBoundedLattice a where { } --interface (VerifiedRingWithUnity a, VerifiedAbelianGroup b, Module a b) => VerifiedModule a b where -- total moduleScalarMultiplyComposition : (x,y : a) -> (v : b) -> x <#> (y <#> v) = (x <.> y) <#> v -- total moduleScalarUnityIsUnity : (v : b) -> Algebra.unity {a} <#> v = v -- total moduleScalarMultDistributiveWRTVectorAddition : (s : a) -> (v, w : b) -> s <#> (v <+> w) = (s <#> v) <+> (s <#> w) -- total moduleScalarMultDistributiveWRTModuleAddition : (s, t : a) -> (v : b) -> (s <+> t) <#> v = (s <#> v) <+> (t <#> v) --interface (VerifiedField a, VerifiedModule a b) => VerifiedVectorSpace a b where {} -}