ap-normalize: Self-normalizing applicative expressions

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An applicative functor transformer to normalize expressions using (<$>), (<*>), and pure into a linear list of actions. See ApNormalize to get started.

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Versions [RSS],
Change log CHANGELOG.md
Dependencies base (>=4.8 && <5) [details]
License MIT
Copyright Li-yao Xia 2020
Author Li-yao Xia
Maintainer lysxia@gmail.com
Category Control
Bug tracker https://gitlab.com/lysxia/ap-normalize/-/issues
Uploaded by lyxia at 2021-05-25T13:10:09Z
Distributions Arch:, LTSHaskell:, NixOS:, Stackage:
Reverse Dependencies 2 direct, 250 indirect [details]
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Status Docs available [build log]
Last success reported on 2021-05-25 [all 1 reports]

Readme for ap-normalize-

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Self-normalizing applicative expressions Hackage pipeline status

Normalize applicative expressions by simplifying intermediate pure and (<$>) and reassociating (<*>).

This works by transforming the underlying applicative functor into one whose operations (pure, (<$>), (<*>)) reassociate themselves by inlining and beta-reduction.

It relies entirely on GHC's simplifier. No rewrite rules, no Template Haskell, no plugins. Only Haskell code with two common extensions: GADTs and RankNTypes.


In the following traversal, one of the actions is pure b, which can be simplified in principle, but only assuming the applicative functor laws. As far as GHC is concerned, pure, (<$>), and (<*>) are completely opaque because f is abstract, so it cannot simplify this expression.

data Example a = Example a Bool [a] (Example a)

traverseE :: Applicative f => (a -> f b) -> Example a -> f (Example b)
traverseE go (Example a b c d) =
    <$> go a
    <*> pure b
    <*> traverse go c
    <*> traverseE go d
  -- Total: 1 <$>, 3 <*>

Using this library, we can compose actions in a specialized applicative functor Aps f, keeping the code in roughly the same structure.

traverseE :: Applicative f => (a -> f b) -> Example a -> f (Example b)
traverseE go (Example a b c d) =
    <$>^ go a
    <*>  pure b
    <*>^ traverse go c
    <*>^ traverseE go d
    & lowerAps
  -- Total: 1 <$>, 3 <*>

GHC simplifies that traversal to the following, using only two combinators in total.

traverseE :: Applicative f => (a -> f b) -> Example a -> f (Example b)
traverseE go (Example a b c d) =
  liftA2 (\a' -> Example a' b)
    (go a)
    (traverse go c)
    <*> traverseE go d
  -- Total: 1 liftA2, 1 <*>

For more details see the ApNormalize module.

The blog post Generic traversals with applicative difference lists gives an overview of the motivation and core data structure of this library.

The same idea can be applied to monoids and monads. They are all applications of Cayley's representation theorem.

  • Endo to normalize (<>) and mempty, in base
  • Codensity to normalize pure and (>>=), in kan-extensions