A search for terms like `arithmetic`

and `natural`

on hackage reveals
no shortage of libraries for handling the arithmetic of natural
numbers. How is this library any different some of the others? It has
a particular purpose: providing a foundation on top on which other
libraries may define types indexed by sizes. This uses GHC's
non-inductively-defined `GHC.TypeNats.Nat`

. As a rule, this does not
use `unsafeCoerce`

internally anywhere.

Perhaps the most direct competitor to `natural-arithmetic`

is a
typechecker plugin like
type-nat-solver. The big
difference is that `type-nat-solver`

can really only be used in
application code, not in library code. This is because libraries
should not require the presence of typechecker plugins. Technically,
they can (you could document it), but many developers will not
use libraries that have unusual install procedures like this.

This library, in places, requires users to use the `TypeApplications`

language extension. This is done when a number is only need at
the type level (without a runtime witness).

This library uses a non-minimal core, providing redundant primitives
in `Arithmetic.Lt`

and `Arithmetic.Lte`

. This is done in the interest
of making it easy for user to assemble proofs. Recall that proof
assembly is done by hand rather than by an SMT solver, so removing
some tediousness from this is helpful to users.

This library provides left and variants variants of several functions.
For example, `Arithmetic.Lte`

provides both `substituteL`

and
`substituteR`

. This is only done when there are two variants of
a function. For substitution, this is the case because we have
`b = c, a ≤ b ==> a ≤ c` and `a = c, a ≤ b ==> c ≤ b`. So, we
provide both `substituteL`

and `substituteR`

. However,
for addition of inequalities, we have four possible variants:
`a ≤ b, c ≤ d ==> a + c ≤ b + d`, `a ≤ b, c ≤ d ==> c + a ≤ b + d`,
`a ≤ b, c ≤ d ==> a + c ≤ d + b`, `a ≤ b, c ≤ d ==> c + a ≤ d + b`.
Consequently, we only provide a single `plus`

function, and users
must use `Arithmetic.Plus.commutative`

to further manipulate the
inequality.

Here are the proof-manipulation vocabulary used by this library.
Many of these terms are not standard, but we try to be consistent
in this library:

Weaken: Increase an upper bound without changing the bounded value

Increment: Increase an upper bound along with the bounded value

Decrement: Decrease an upper bound along with the bounded value

Substitute: Replace a number with an equal number