feat: Use equivalence information from equality assertions to simplify circuit

[TomAFrench]

Description

Problem*

Partially resolves #1431

Summary*

When we assert two AcirVars are equal, we now use this equivalence to use the simpler of the two values in future.

This PR will be a lot more useful once #2137 is merged as currently all of our assertions are against 1.

Documentation

• This PR requires documentation updates when merged.

  • I will submit a noir-lang/docs PR.
  • I will request for and support Dev Rel's help in documenting this PR.

Additional Context

PR Checklist*

• I have tested the changes locally.
• I have formatted the changes with Prettier and/or cargo fmt on default settings.

[jfecher]

Should this be an SSA pass instead? It seems our acir-gen pass is getting increasingly complex as we add more optimizations to it.

[TomAFrench]

Hmm. I agree it would be best to make this an SSA pass as we're be able to use the information on the two ValueIds being equivalent from the beginning of the circuit. I'd need to think more on how you would do this.

[TomAFrench]

It's a bit more difficult to do this in SSA as it doesn't know the difference between a value which is a single witness and a value which is a complicated expression.

I think that making an SSA pass to correctly handle this program may be difficult.

fn main (x: u32, y: u32, z: u32) {
    let yz = y * z;
    assert_eq(x, yz);
    assert_eq(x * yz, 25);
}

If we do it in acir-gen then we can easily see that x is simpler than yz so we can replace the last line with assert_eq(x*x, 25). SSA has no information on the internal structure of x and yz (except for if we rebuild this from scratch by backtracking to params and constants) so this is more difficult.

[jfecher]

SSA has no information on the internal structure of x and yz (except for if we rebuild this from scratch by backtracking to params and constants) so this is more difficult.

I'd think checking whether x is a block param and whether yz is an instruction result could be sufficient, but it doesn't need to block this PR. An SSA pass would be a larger code change as well.