* First draft for division.
* `eval_division` work
* Division
* Minor: outdated fixme
* Tests and better column names
* Minor lints
* Remove redundant constraint
* Make division proof more formal
* Minor proof and comments
Co-authored-by: Hamish Ivey-Law <hamish@ivey-law.name>
* Implement a mul-add circuit in the ALU
The inputs are assumed to be `u32`s, while the output is encoded as four `u16 limbs`. Each output limb is range-checked.
So, our basic mul-add constraint looks like
out_0 + 2^16 out_1 + 2^32 out_2 + 2^48 out_3 = in_1 * in_2 + in_3
The right hand side will never overflow, since `u32::MAX * u32::MAX + u32::MAX < |F|`. However, the left hand side could overflow, even though we know each limb is less than `2^16`.
For example, an operation like `0 * 0 + 0` could have two possible outputs, 0 and `|F|`, both of which would satisfy the constraint above. To prevent these non-canonical outputs, we need a comparison to enforce that `out < |F|`.
Thankfully, `F::MAX` has all zeros in its low 32 bits, so `x <= F::MAX` is equivalent to `x_lo == 0 || x_hi != u32::MAX`. `x_hi != u32::MAX` can be checked by showing that `u32::MAX - x_hi` has an inverse. If `x_hi != u32::MAX`, the prover provides this (purported) inverse in an advice column.
See @bobbinth's [post](https://hackmd.io/NC-yRmmtRQSvToTHb96e8Q#Checking-element-validity) for details. That post calls the purported inverse column `m`; I named it `canonical_inv` in this code.
* fix
* PR feedback
* naming
