DEEP ALI Soundness

[hecmas]

Following Theorem 8 from Hab22:

we should perform the following modifications:

• Update ALI soundness to multiply by the number of constraints instead of columns.
• Update $L^+$ to take into account the rate $\rho^+$ of the slightly large code with $k^+ = k + \text{numOpenings}$.
• Add more types of batching, e.g. linear batching, in the DEEP-ALI protocol.
• Add more types of batching strategies. For instance, if I want to prove the evaluation of $f(X)$ at both $z$ and $g·z$, then I can either run a LDT over functions $g_1(X) = (f(X) - f(z)) / (X - z)$ and $g_2(X) = (f(X) - f(g·z)) / (X - g·z)$ or I can run a LDT over a single function $h(X) = (f(X) - U(X)) / ((X - z)·(X - g·z))$ where $U(X)$ is the unique degree < 2 interpolant through points $(z, f(z))$ and $(g·z, f(g·z))$.