# algebra-checkers

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## Dedication

> "Any fool can make a rule, and any fool will mind it."
>
> Henry David Thoreau


## Overview

`algebra-checkers` is a little library for testing algebraic laws. For example,
imagine we're writing an ADT:

```haskell
data Foo a
instance Semigroup (Foo a)
instance Monoid (Foo a)

data Key


get :: Key -> Foo a -> a
get = undefined

set :: Key -> a -> Foo a -> Foo a
set = undefined
```

Let's say we expect the lens laws to hold for `get` and `set`, as well for `set`
to be a monoid homomorphism. We can express those facts to `algebra-checkers`
and have it generate tests for us:

```haskell
lawTests :: [Property]
lawTests = $(testModel [e| do

law "set/set"
    (set i x' (set i x s) == set i x' s)

law "set/get"
    (set i (get i s) s == s)

law "get/set"
    (get i (set i x s) == x)

homo @Monoid
    (\s -> set i x s)

|])
```

Furthermore, `algebra-checkers` will generate tests to show that these laws are
confluent. We can run these tests via `quickCheck lawTests`.

If we use the `theoremsOf` function instead of `testModel`, `algebra-checkers`
will dump out all the additional theorems it has proven about our algebra. This
serves as a good sanity check:

```
Theorems:

• set i x' (set i x s) = set i x' s (definition of "set/set")
• set i (get i s) s = s (definition of "set/get")
• get i (set i x s) = x (definition of "get/set")
• set i x mempty = mempty (definition of "set:Monoid:mempty")
• set i x (s1 <> s2) = set i x s1 <> set i x s2
    (definition of "set:Monoid:<>")
• set i1 (get i1 (set i1 x1 s1)) s1 = set i1 x1 s1
    (implied by "set/get" and "set/set")
• set i1 (get i1 (s12 <> s22)) s12 <> set i1 (get i1 (s12 <> s22)) s22
        = s12 <> s22
    (implied by "set/get" and "set:Monoid:<>")
• set i1 x'2 (set i1 x1 s11 <> set i1 x1 s21)
        = set i1 x'2 (s11 <> s21)
    (implied by "set/set" and "set:Monoid:<>")
• get i1 (set i1 x1 s11 <> set i1 x1 s21) = x1
    (implied by "get/set" and "set:Monoid:<>")


Contradictions:

• get i1 mempty = x1
    the variable x1 is undetermined
    (implied by "get/set" and "set:Monoid:mempty")
```

Uh oh! Look at that! This contradiction is  clearly a bogus theorem, which lets
us know that "get/set" and "set mempty" are nonconfluent with one another!