Prefer using Prelude.id over Identity

Data/Profunctor/Mapping.idr

@@ -1,6 +1,5 @@
 module Data.Profunctor.Mapping
 
-import Control.Monad.Identity
 import Data.Morphisms
 import Data.Tensor
 import Data.Profunctor
@@ -23,7 +22,7 @@ functor = MkFunctor (\f => (. (. f)))
 ||| * `map' . lmap f = lmap (map f) . map'`
 ||| * `map' . rmap f = rmap (map f) . map'`
 ||| * `map' . map' = map' @{Compose}`
-||| * `dimap Identity runIdentity . map' = id`
+||| * `dimap Id runIdentity . map' = id`
 public export
 interface (Traversing p, Closed p) => Mapping p where
   map' : Functor f => p a b -> p (f a) (f b)
@@ -80,7 +79,7 @@ ProfunctorFunctor CofreeMapping where
 
 public export
 ProfunctorComonad CofreeMapping where
-  proextract (MkCFM p) = dimap Id runIdentity p
+  proextract (MkCFM p) = p @{FunctorId}
   produplicate (MkCFM p) = MkCFM $ MkCFM $ p @{Compose}
 
 public export
@@ -99,7 +98,7 @@ roamCofree f (MkCFM p) = MkCFM $ dimap (map (flip f)) (map ($ id)) $
 public export
 Profunctor p => Traversing (CofreeMapping p) where
   traverse' (MkCFM p) = MkCFM (p @{Compose})
-  wander f = roamCofree $ f @{MkApplicative @{MkFunctor id} id id}
+  wander f = roamCofree $ f @{ApplicativeId}
 
 public export
 Profunctor p => Mapping (CofreeMapping p) where
@@ -128,7 +127,7 @@ ProfunctorFunctor FreeMapping where
 
 public export
 ProfunctorMonad FreeMapping where
-  propure p = MkFM runIdentity p Id
+  propure p = MkFM @{FunctorId} id p id
   projoin (MkFM l' (MkFM l m r) r') = MkFM @{Compose} (l' . map l) m (map r . r')
 
 public export
@@ -155,7 +154,7 @@ roamFree f (MkFM l m r) = MkFM @{Compose @{functor}} (($ id) . map @{functor} l)
 public export
 Traversing (FreeMapping p) where
   traverse' (MkFM l m r) = MkFM @{Compose} (map l) m (map r)
-  wander f = roamFree $ f @{MkApplicative @{MkFunctor id} id id}
+  wander f = roamFree $ f @{ApplicativeId}
 
 public export
 Mapping (FreeMapping p) where

Data/Profunctor/Representable.idr

@@ -50,9 +50,9 @@ Representable Morphism Identity where
 ||| A named implementation of `Representable` for function types.
 ||| Use this to avoid having to use a type wrapper like `Morphism`.
 export
-[Function] Representable (\a,b => a -> b) Identity
+[Function] Representable (\a,b => a -> b) Prelude.id
     using Sieve.Function Strong.Function where
-  tabulate = (runIdentity .)
+  tabulate = id
 
 export
 Functor f => Representable (Kleislimorphism f) f where
@@ -74,8 +74,8 @@ namespace Corepresentable
   ||| A named implementation of `Corepresentable` for function types.
   ||| Use this to avoid having to use a type wrapper like `Morphism`.
   export
-  [Function] Corepresentable (\a,b => a -> b) Identity using Cosieve.Function where
-    cotabulate = (. Id)
+  [Function] Corepresentable (\a,b => a -> b) Prelude.id using Cosieve.Function where
+    cotabulate = id
 
 export
 Functor f => Corepresentable (Costar f) f where

Data/Profunctor/Sieve.idr

@@ -37,8 +37,8 @@ Sieve Morphism Identity where
 ||| A named implementation of `Sieve` for function types.
 ||| Use this to avoid having to use a type wrapper like `Morphism`.
 public export
-[Function] Sieve (\a,b => a -> b) Identity using Profunctor.Function where
-  sieve = (Id .)
+[Function] Sieve (\a,b => a -> b) Prelude.id using Profunctor.Function FunctorId where
+  sieve = id
 
 public export
 Functor f => Sieve (Kleislimorphism f) f where
@@ -61,8 +61,8 @@ namespace Cosieve
   ||| A named implementation of `Cosieve` for function types.
   ||| Use this to avoid having to use a type wrapper like `Morphism`.
   public export
-  [Function] Cosieve (\a,b => a -> b) Identity using Profunctor.Function where
-    cosieve = (. runIdentity)
+  [Function] Cosieve (\a,b => a -> b) Prelude.id using Profunctor.Function FunctorId where
+    cosieve = id
 
 public export
 Functor f => Cosieve (Costar f) f where

Data/Profunctor/Traversing.idr

@@ -1,7 +1,6 @@
 module Data.Profunctor.Traversing
 
 import Control.Applicative.Const
-import Control.Monad.Identity
 import Data.Morphisms
 import Data.Tensor
 import Data.Profunctor
@@ -22,14 +21,6 @@ import Data.Profunctor
 [TraversablePair] Traversable (Pair c) using FoldablePair where
   traverse f (l, r) = (l,) <$> f r
 
-[FoldableIdentity] Foldable Identity where
-  foldr f i (Id x) = f x i
-  foldl f i (Id x) = f i x
-  null _ = False
-
-[TraversableIdentity] Traversable Identity using FoldableIdentity where
-  traverse f (Id x) = map Id (f x)
-
 
 record Bazaar a b t where
   constructor MkBazaar
@@ -54,7 +45,7 @@ Functor (Baz t b) where
 
 
 sold : Baz t a a -> t
-sold m = runIdentity (runBaz m Id)
+sold m = runBaz @{ApplicativeId} m id
 
 Foldable (Baz t b) where
   foldr f i bz = runBaz bz @{appEndo} f i
@@ -103,14 +94,14 @@ interface (Strong p, Choice p) => Traversing p where
 public export
 Traversing Morphism where
   traverse' (Mor f) = Mor (map f)
-  wander f (Mor p) = Mor (runIdentity . (f $ Id . p))
+  wander f (Mor p) = Mor (f @{ApplicativeId} p)
 
 ||| A named implementation of `Traversing` for function types.
 ||| Use this to avoid having to use a type wrapper like `Morphism`.
 public export
 [Function] Traversing (\a,b => a -> b) using Strong.Function where
   traverse' = map
-  wander f g = runIdentity . (f $ Id . g)
+  wander f = f @{ApplicativeId}
 
 public export
 Applicative f => Traversing (Kleislimorphism f) where
@@ -166,7 +157,7 @@ ProfunctorFunctor CofreeTraversing where
 
 public export
 ProfunctorComonad CofreeTraversing where
-  proextract (MkCFT p) = dimap Id runIdentity $ p @{TraversableIdentity}
+  proextract (MkCFT p) = p @{TraversableId}
   produplicate (MkCFT p) = MkCFT $ MkCFT $ p @{Compose}
 
 public export
@@ -220,7 +211,7 @@ ProfunctorFunctor FreeTraversing where
 
 public export
 ProfunctorMonad FreeTraversing where
-  propure p = MkFT @{TraversableIdentity} runIdentity p Id
+  propure p = MkFT @{TraversableId} id p id
   projoin (MkFT l' (MkFT l m r) r') = MkFT @{Compose} (l' . map l) m (map r . r')
 
 
@@ -230,4 +221,4 @@ freeTraversing fn (MkFT {f} l m r) = dimap r l (traverse' {f} (fn m))
 
 public export
 unfreeTraversing : FreeTraversing p :-> q -> p :-> q
-unfreeTraversing f p = f (MkFT @{TraversableIdentity} runIdentity p Id)
+unfreeTraversing f p = f (MkFT @{TraversableId} id p id)

Data/Profunctor/Types.idr

@@ -276,3 +276,37 @@ Profunctor (Coforget r) where
   dimap _ f (MkCoforget k) = MkCoforget (f . k)
   lmap _ = recoforget
   rmap = map
+
+
+------------------------------------------------------------------------------
+-- Identity functor
+------------------------------------------------------------------------------
+
+-- A few convenience definitions for use in other modules.
+
+public export
+[FunctorId] Functor Prelude.id where
+  map = id
+
+public export
+[ApplicativeId] Applicative Prelude.id using FunctorId where
+  pure = id
+  (<*>) = id
+
+public export
+[MonadId] Monad Prelude.id using ApplicativeId where
+  join = id
+  (>>=) = flip id
+
+public export
+[FoldableId] Foldable Prelude.id where
+  foldr = flip
+  foldl = id
+  null _ = False
+  foldlM = id
+  toList = pure
+  foldMap = id
+
+public export
+[TraversableId] Traversable Prelude.id using FunctorId FoldableId where
+  traverse = id