https://github.com/rybla/monadic-equality

https://github.com/rybla/monadic-equality

Last synced: 4 days ago
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• Host: GitHub
• URL: https://github.com/rybla/monadic-equality
• Owner: rybla
• License: bsd-3-clause
• Created: 2021-01-28T14:48:51.000Z (almost 4 years ago)
• Default Branch: master
• Last Pushed: 2021-04-23T20:08:51.000Z (over 3 years ago)
• Last Synced: 2024-11-07T11:13:18.437Z (about 2 months ago)
• Language: Haskell
• Size: 1.83 MB
• Stars: 0
• Watchers: 2
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • Changelog: ChangeLog.md
  • License: LICENSE

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README

        
# monadic-equality

Formulating equality of

[monadic](https://hackage.haskell.org/package/base-4.14.1.0/docs/Control-Monad.html)

terms in [Liquid Haskell](http://ucsd-progsys.github.io/liquidhaskell/).

## Why Monadic Equality?

Consider the monad laws:

```

       q >> lift = m

     lift x >> k = k x

(q >>= k1) >> k2 = q >>= (k1 >=> k2)

```

where

```

m     :: * -> * where Monad m

x     :: a

q     :: m a

k, k1 :: a -> m b

k2    :: b -> m c

lift  ::

(>>=) :: m a -> (a -> m b) -> m b

(>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c)

```

These laws are equalities between monadic terms. But what kind of quality is it?

Well, if we are using the `Writer String Int = (String, Int)` monadic terms,

then the equality is the equality that the pair type `forall a b. (a, b)`

derives from the equality of `String` and the equality of `Int`.

However, what if we are using the `Reader String Int = String -> Int`?

Generally, the arrow type `forall a b. a -> b` doesn't derive any equality. So

implicitly we must be using some kind of extensionality, since the monad laws

are for _all_ monads, not just the ones with non-extensional equalities.

In Liquid Haskell, extensional equality is not included for two main reasons:

1. The SMT-solver backend for Liquid Haskell does not use quantifiers, so cannot

   represent extensional equality.

2. Liquid Haskell's SMT equality is not type-indexed, and type-unindexed

   extensional equality is unsound.

So, to formulate monadic equality (i.e. equality of monadic terms) in Liquid

Haskell, we need a notion of type-indexed propositional equality in order to

include extensionality, and additionally a notion of monadic equality that

relies on propositional equality and may be either extensional or

non-extensional depending on the monad.

This work may perhaps be extendible to derive a certain kind of equality for

generic typeclasses (of which `Monad` is one example).

## Propositional Equality

TODO

## Monadic Equality

TODO

## Examples

TODO