Consistency of Functional Dependencies

The functional dependencies on a class restrict the instances that may be declared for a given class. The instances of a class:

```    class C a b c | b -> c where f :: ...
```

are consistent with its functional dependency if the following invariant holds:

```    byFD(C,1): forall a1 a2 b c1 c2. (C a1 b c1, C a2 b c2) => (c1 ~ c2)
```

Please note that the question of FD-consistency is orthogonal to instance coherence (i.e, uniqueness of evidence, overlapping instances, etc.), and the decidability of type-checking of terms---for examples of their independence, please see the Examples at the end of the document.

If we check that all instances in scope are consistent with their FDs, then we can use the FD invariant byFD during type inference to infer more precise types, report errors involving unsolvable contexts, or accept programs that would be rejected without the invariant.

For example, if we have an instance:

```  I: instance F Int Int Char
```

and we have a constraint C: F Int Int a, then we can use the FD-invariant to prove that a must be Char:

```    byFD(C,1) (C,I) :: a ~ Char
```

Checking FD-Consistency

To ensure FD-consistency, before accepting an instance we need to check that it is compatible with all other instances that are already in scope. Note that we also need to perform the same check when combining imported instances. Consider adding a new instance to an FD-consistent set of instances:

```    I: instance P(as,bs) => C t1(as,bs) t2(as) t3(as,bs)
```

The notation t(as) states the variables in t are a subset of as.

1. Check that I is self-consistent (i.e., we can't use different instantiations of I to violate FD-consistency). Self consistency follows if we can prove the following theorem:
```         forall as bs cs. (P, P[cs/bs]) => t3[cs/bs] ~ t3[cs/bs]
```
2. Check that I is FD-consistent with all existing instances of the class. So, for each existing instance, J:
```         J: instance Q(xs) => C s1(xs) s2(xs) s3(xs)
```
we need to show that:
```         forall as bs xs. (P,Q,s2 ~ t2) => (s3 ~ t3)
```
Assuming no type-functions in instance heads, the equality assumption is equivalent to stating that s2 and t2 may be unified, so another way to state our goal is:
```         forall as bs xs. (P[su], Q[su]) => (s3[su] ~ t3[su])
```
where su is the most general unifier of s2 and t2.

Proving these two goals before accepting an instance is similar to the process of finding evidence for super-classes before accepting and instance. Also, note that while proving (2), it is not a problem if we find that we have assumed a contradiction: this simply means that the two instances can never be used at the same time, so the FD-consistency follows trivially.

Examples

• FD-consistency is orthogonal to instance coherence.

FD-consistent but not coherent:

```         instance C Int Int Int where f = definition_1
instance C Int Int Int where f = definition_2
```

Coherent but not FD-consistent:

```         instance C Int  Int Char where ...
instance C Char Int Bool where ...
```
• FD-consistency is orthogonal to termination of instances.

FD-consistent but "non-terminating":

```         instance C a b c => C a b c
```

Terminating but not FD-consistent:

```         instance C Int  Int Char where ...
instance C Char Int Bool where ...
```