Bridging the Gap: Automating Zero-Knowledge Verification with Rocq

Introduction

Supported by the Ethereum Foundation , CertiK has developed a set of Rocq Tactics for Automating Zero-Knowledge Verification (automated decision procedures). These   proof instructions act like a set of skills that automatically proves how small integers behave within a large prime field, handling complex tasks such as integer packing and range constraints without manual intervention.

The Problem we are solving

While modern software runs on 32-bit or 64-bit integers, Zero-Knowledge systems operate in a world of massive Prime Fields, creating a fundamental “mathematical mismatch.” Proving that a program’s logic remains intact when translated into these complex fields usually requires verification engineers to perform thousands of lines of manual "mathematical grunt work."

The Impact on Security

Manual proofs are slow and prone to human error. By automating the "boring" parts of the math, we allow security engineers to focus on the high-level logic of the VM—making the entire verification process faster, more robust, and more scalable.

Does it actually work?

Yes! We applied these tactics to the zkWasm constraint system. We successfully automated 88% of the gate equation proofs, turning a significant mathematical bottleneck into a streamlined, automated process.

With this high-level intuition in place, we’ll now dive into the technical core of the zkVM: the constraint system, the source of these challenges, and where our tactics make the biggest difference.

The Core of the zkVM: The Constraint System

At the heart of every zkVM lies its constraint system. While the execution trace records the step-by-step history of the machine's state, it is the constraint system—a complex set of algebraic rules—that enforces the logic. These constraints mathematically dictate what a valid execution looks like, transforming high-level concepts like "addition" or "memory access" into rigid polynomial equations that a cryptographic proof can verify.

The Challenge: Field Arithmetic $\neq$ Integer Arithmetic

The fundamental difficulty in verifying these circuits stems from a mismatch in mathematical domains. The virtual machine is designed to emulate standard CPU operations using 32-bit or 64-bit integers. However, the zero-knowledge proof systems only support arithmetic over a finite field of a massive prime order $p$.

To a standard automated solver (like Rocq’s Micromega), the equation $x + y = z$ is trivial. But the actual constraint in the circuit is $(x + y - z) \pmod p = 0$.

• The Intent: The programmer intends for the operation to behave like standard integer arithmetic, where z is specified as a simple sum.
• The Reality: Mathematically, the solver cannot assume that x and y are "small." It sees a modular equation where wraparound is a valid solution.

Because standard decision procedures lack the ability to automatically reason about these modular operations, verification engineers are forced to perform tedious manual labor to bridge this gap.

Making Use of Type Information

Modular equations in general are difficult to deal with, but for zkVMs, the situation is better. The CPU only operates on 32-bit and 64-bit data, and therefore the columns of data going into the proof are type-annotated and come with a set of range-check constraints to make sure that they are in the expected size. 

In the rest of the proofs, we can leverage this information. For example, if x, y and z are 64-bit numbers (much smaller than $p$), then the value $x+y-z$ is always smaller than $p$, so the equations  $(x + y - z) \pmod p = 0$ and $(x + y - z) = 0$ are actually equivalent. By making use of the type information that the VM needs anyway, the constraint system can be simplified to a form that is much easier to work with.

Our Contribution: Automating the "Lift" to Integers

To leverage this, we developed and open-sourced a new set of Rocq tactics that automate the translation between these two worlds.

Our approach is built on a new moddable predicate, which formally asserts that an expression is safe to treat as a standard integer equation because it cannot wrap around the modulus.

• Automated Reasoning: We provide a database of lemmas (like moddable_add and moddable_mul) that automatically deduce these bounds based on the input types.
• Handle type-annotation optimization. Engineers optimize by removing range checks (type annotations) on specific columns to save constraints. In our tactics, we provide a range of different lemmas to prove the correctness of the most common optimization patterns.
• Prove injectivity. The lemmas to prove that numbers are “small” can also be used to prove that several elements can be packed together into one finite field element, and we provide a set of automation for this task. For example, instructions are encoded by packing the opcode and operands into a single element.

In our evaluation against the zkWasm constraint system, these tactics successfully automated 88% of the gate equation proofs (72 out of 82 equations), turning a mathematical bottleneck into a streamlined process.