Polymorphic equality is a built-in function in Standard ML that compares two values of the same type for equality. It is specified as

```val = : ''a * ''a -> bool
```

The ''a in the specification are equality type variables, and indicate that polymorphic equality can only be applied to values of an equality type. It is not allowed in SML to rebind =, so a programmer is guaranteed that = always denotes polymorphic equality.

Equality of ground types

Ground types like char, int, and word may be compared (to values of the same type). For example, 13 = 14 is type correct and yields false.

Equality of reals

The one ground type that can not be compared is real. So, 13.0 = 14.0 is not type correct. One can use Real.== to compare reals for equality, but beware that this has different algebraic properties than polymorphic equality.

See http://standardml.org/Basis/real.html for a discussion of why real is not an equality type.

Equality of functions

Comparison of functions is not allowed.

Equality of immutable types

Polymorphic equality can be used on immutable values like tuples, records, lists, and vectors. For example,

`(1, 2, 3) = (4, 5, 6)`

is a type-correct expression yielding false, while

`[1, 2, 3] = [1, 2, 3]`

is type correct and yields true.

Equality on immutable values is computed by structure, which means that values are compared by recursively descending the data structure until ground types are reached, at which point the ground types are compared with primitive equality tests (like comparison of characters). So, the expression

`[1, 2, 3] = [1, 1 + 1, 1 + 1 + 1]`

is guaranteed to yield true, even though the lists may occupy different locations in memory.

Because of structural equality, immutable values can only be compared if their components can be compared. For example, [1, 2, 3] can be compared, but [1.0, 2.0, 3.0] can not. The SML type system uses equality types to ensure that structural equality is only applied to valid values.

Equality of mutable values

In contrast to immutable values, polymorphic equality of mutable values (like ref cells and arrays) is performed by pointer comparison, not by structure. So, the expression

`ref 13 = ref 13`

is guaranteed to yield false, even though the ref cells hold the same contents.

Because equality of mutable values is not structural, arrays and refs can be compared even if their components are not equality types. Hence, the following expression is type correct (and yields true).

```let
val r = ref 13.0
in
r = r
end
```

Equality of datatypes

Polymorphic equality of datatypes is structural. Two values of the same datatype are equal if they are of the same variant and if the variant's arguments are equal (recursively). So, with the datatype

```datatype t = A | B of t
```

then B (B A) = B A is type correct and yields false, while A = A and B A = B A yield true.

As polymorphic equality descends two values to compare them, it uses pointer equality whenever it reaches a mutable value. So, with the datatype

```datatype t = A of int ref | ...
```

then A (ref 13) = A (ref 13) is type correct and yields false, because the pointer equality on the two ref cells yields false.

One weakness of the SML type system is that datatypes do not inherit the special property of the ref and array type constructors that allows them to be compared regardless of their component type. For example, after declaring

```datatype 'a t = A of 'a ref
```

one might expect to be able to compare two values of type real t, because pointer comparison on a ref cell would suffice. Unfortunately, the type system can only express that a user-defined datatype admits equality or not. In this case, t admits equality, which means that int t can be compared but that real t can not. We can confirm this with the program

```datatype 'a t = A of 'a ref
fun f (x: real t, y: real t) = x = y
```

on which MLton reports the following error.

```Error: z.sml 2.32-2.36.
Function applied to incorrect argument.
expects: [<equality>] t * [<equality>] t
but got: [real] t * [real] t
in: = (x, y)```

Implementation

Polymorphic equality is implemented by recursively descending the two values being compared, stopping as soon as they are determined to be unequal, or exploring the entire values to determine that they are equal. Hence, polymorphic equality can take time proportional to the size of the smaller value.

MLton uses some optimizations to improve performance.

• When computing structural equality, first do a pointer comparison. If the comparison yields true, then stop and return true, since the structural comparison is guaranteed to do so. If the pointer comparison fails, then recursively descend the values.

• If a datatype is an enum (e.g. datatype t = A | B | C), then a single comparison suffices to compare values of the datatype. No case dispatch is required to determine whether the two values are of the same variant.

• When comparing a known constant non-value-carrying variant, use a single comparison. For example, the following code will compile into a single comparison for A = x.

```datatype t = A | B | C of ...
fun f x = ... if A = x then ...
```
• When comparing a small constant IntInf.int to another IntInf.int, use a single comparison against the constant. No case dispatch is required.