Icelandjack

A function mapped on each side.

Any function (profunctor/category) has "inputs" and "outputs". First argument: input (contravariant), second argument: output (covariant). covariant argument.

🟥🟦 :: 🟥 -> 🟦
🟦🟩 :: 🟦 -> 🟩

When composing, it works like domino pieces: 🀺 >>> 🁂 = 🀻 the connecting parts must be match but prefer using colours for it.

Icelandjack

;; Separate string with escaped '\\n' literals.
;; These '\\n' literals will be interpreted by `echo'.
;;
;;   (broken "hello")
;; = "h\\ne\\nl\\nl\\no"
(defun broken (str)
  (mapconcat
   (lambda (x) (make-string 1 x))
   str
   "\\n"))

Icelandjack

{-# Language DerivingVia #-}
{-# Language StandaloneKindSignatures #-}
-- Some imports and pragmas missing.

import GHC.Generics
import Control.Applicative
import GHC.TypeLits
import Data.Kind
import Data.Coerce
import Data.Monoid

Icelandjack

-- this allows you to derive
-- 
--   data OneTwo = OneTwo { x :: Int, y :: Int }
--     deriving stock Generic
--     deriving (Semigroup, Monoid) via Overriding '[ '[Sum Int, Sum Int] ] Onetwo

-- | Override
newtype Override (code :: [[Type]]) a = Override { getOverride :: a }

type Overriding code a = Generically (Override code a)

Icelandjack

-- This witnesses multiplying two polynomials
-- 
--   Code (Maybe a) = [[], [a]]
--   Code (Maybe b) = [[], [b]]
-- 
-- Then the product of (Maybe a, Maybe b) is isomorphic to the
-- multiplication of their codes
-- 
--   Code (MegaMaybe a b) = [[], [a], [b], [a,b]]
-- 

Icelandjack

{-# LANGUAGE DataKinds                  #-}
{-# LANGUAGE DeriveGeneric              #-}
{-# LANGUAGE FlexibleInstances          #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE InstanceSigs               #-}
{-# LANGUAGE MultiParamTypeClasses      #-}
{-# LANGUAGE PolyKinds                  #-}
{-# LANGUAGE ScopedTypeVariables        #-}
{-# LANGUAGE StandaloneDeriving         #-}
{-# LANGUAGE TypeApplications           #-}

Icelandjack

type f ~> g = (forall x. f x -> g x)

infixr 0 ··>
type (··>) :: (Type -> Type) -> (Type -> Type) -> Type
type f ··> g = Proxy '(f, g) -> Type

type  Idiom :: f ··> g -> Constraint
class Idiom (hom :: f ··> g) where
 idiom :: f ~> g

Icelandjack

infixr 80 _◦_

_◦_ : forall {l1 l2 l3 : Level                  }
             {A :                      -> Set l1}
             {B :      A               -> Set l2}
             {C : (a : A) ->      B a  -> Set l3}
             (g : {a : A} -> (b : B a) -> C a b )
             (f : (a : A) ->      B a)
          -> (a : A) -> C a (f a)
g ◦ f = \a -> g (f a)

Icelandjack

norm :: Generic1 f => Summs (Rep1 f) => f ~> Summ (Rep1 f) Zero
norm as = summs (from1 as) (Proxy @Zero)

class Summs rep where
 type Summ rep (end :: Type -> Type) :: Type -> Type

 summs :: rep a -> Proxy end -> Summ rep end a
 skips :: Proxy rep -> end a -> Summ rep end a

instance Summs Zero where

Icelandjack

{-# Language EmptyCase                #-}
{-# Language GADTs                    #-}
{-# Language InstanceSigs             #-}
{-# Language PatternSynonyms          #-}
{-# Language PolyKinds                #-}
{-# Language ScopedTypeVariables      #-}
{-# Language StandaloneKindSignatures #-}
{-# Language TypeApplications         #-}
{-# Language TypeFamilies             #-}
{-# Language TypeOperators            #-}