# What Made Proof Assistants Possible in the First Place?

According to Lawrence Paulson's historical account, the first proof assistant, LCF (Logic for Computable Functions), was developed at Stanford in 1975 by Robin Milner. Paulson reported that the key innovation was the concept of a "proof state" — a goal-directed approach where the user applies tactics to reduce a proof goal to simpler sub-goals. This design pattern, Paulson said, persists in every modern proof assistant, from Coq to Lean.

The early systems were slow and limited, but they established the core principle: a machine-checked proof is more reliable than any human-checked argument. This principle drove decades of refinement in type theory and automation.

# Why Did Lean Surpass Coq and Isabelle?

The rise of Lean, developed by Leonardo de Moura at Microsoft Research, is the most significant recent shift in the field. According to Paulson, Lean's design prioritizes user experience and automation, making it more accessible to mathematicians and engineers. Lean's community has also embraced a library-driven approach, with Mathlib4 becoming a massive repository of formalized mathematics.

In contrast, Coq and Isabelle remain powerful but have steeper learning curves. Coq is favored for programming language semantics, while Isabelle is strong in automated reasoning. However, Lean's rapid growth in both academic and industrial settings suggests it is becoming the default choice for new projects.

Paulson noted that Lean's success is not just technical — it also benefits from de Moura's vision and Microsoft's support. This institutional backing has been crucial for long-term sustainability.

# Who Stands to Win From Proof Assistants Going Mainstream?

The clearest winners are organizations in safety-critical domains. According to Paulson, the aviation industry has already adopted formal verification for parts of certification, and financial institutions are exploring proof assistants for smart contract verification. Amazon Web Services uses automated reasoning tools, and the trend is accelerating.

Another winner is the open-source community around Lean. Mathlib4 has hundreds of contributors, and the ecosystem of tutorials, libraries, and tools is growing. This network effect makes Lean increasingly attractive for new users.

The losers are less obvious but real: traditional software testing tools and manual code review processes will face pressure in high-stakes environments. Proof assistants offer stronger guarantees than testing, but they also demand more upfront investment.

# What Remains Uncertain About Proof Assistants?

Despite the progress, major uncertainties remain. First, the learning curve: Paulson acknowledged that even with modern systems, writing proofs requires significant training and mathematical maturity. Second, the interoperability problem: proofs written in one system cannot be easily transferred to another. This fragmentation could limit adoption in heterogeneous environments.

Third, the scalability to large software systems is unproven. While proofs for individual algorithms and protocols are feasible, verifying an entire operating system or database remains impractical. Paulson's history shows that each generation of proof assistants has expanded the frontier, but the gap between what is possible and what is needed remains large.

# Predictions

1. By 2028, Amazon Web Services will launch a managed Lean service for smart contract verification, following the pattern of their Automated Reasoning group.
2. By 2030, at least one major certification body (e.g., FAA or EASA) will require proof assistant verification for critical avionics software, replacing some testing requirements.
3. By 2027, the number of active Lean users will double, driven by industrial adoption, while Coq and Isabelle will see modest growth.

# Article Summary

• Proof assistants have a 50-year history, but only now are they approaching industrial viability.
• Lean is the current leader due to better automation and community support, but Coq and Isabelle remain relevant.
• The main barrier to adoption is not technical capability but the steep learning curve and lack of interoperability.
• Regulated industries will drive early adoption; general software development will follow slowly.
• Cloud providers are likely to become key players by offering managed proof assistant services.