Version 3 (modified by diatchki, 7 years ago) (diff)

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## Type-Level Operations

Currently, we provide the following type-level operations on natural numbers:

```(<=) :: Nat -> Nat -> Prop    -- Comparison
(+)  :: Nat -> Nat -> Nat     -- Addition
(*)  :: Nat -> Nat -> Nat     -- Multiplication
(^)  :: Nat -> Nat -> Nat     -- Exponentiation
```

Notes:

• `(<=)` is a 2-parameter class (that's what we mean by the "kind" Prop),
• `(+)`, `(*)`, and `(^)` are type functions.
• Programmers may not provide custom instances of these classes/type-families.

The operations correspond to the usual operations on natural numbers.

## Inverse Operations

Using the basic operations and GHC's equality constraints it is also possible to express some additional constraints, corresponding to the inverses (when defined) of the above functions:

```Constraint     | Alternative Interpretation    | Example (x "input", y "output")
---------------+-------------------------------+--------------------------------
(a + b) ~ c    | Subtraction: (c - b) ~ a      | Decrement by 8: x ~ (y + 8)
(a * b) ~ c    | Division:    (c / b) ~ a      | Divide by 8:    x ~ (y * 8)
(a ^ b) ~ c    | Log at base: Log  a c ~ b     | Log base 2:     x ~ (2 ^ y)
(a ^ b) ~ c    | Nth-root:    Root b c ~ a     | Square root:    x ~ (y ^ 2)
```

Note that we need some form of relational notation to capture the partiality of the inverses. In GHC, using (~) seems like a natural choice. If we'd introduced (-) as another type function, we have a few choices, none of which seem particularly attractive:

• accept some ill defined "types" like (1 - 2),
• come up with a "completion" for the type function (e.g., by defining (1 - 2) ~ 0),
• automatically generate additional constraints which would ensure that types are well-defined, somehow.

While the relational notation may look a bit verbose, in practice we can often completely avoid it. For example, consider that we are writing which needs to split an array of bytes into an array of words. We might give the function the following type:

```bytesToWords :: Array (8 * w) Word8 -> Array w Word64
```

So, in effect, we perform a "division" by multiplying the argument.