Formalizing inverse problems in the age of AI: possibilities, pitfalls, and an invitation
TL;DR
AutoIP is described, a Lean library in which classical linear regularization theory, together with the functional analysis it rests on, has been autoformalized by AI agents, and a reviewer compares the formal statements with their natural-language counterparts and never needs to read a proof.
Abstract
Proof assistants such as Lean check mathematical proofs down to the axioms, and AI agents have recently made writing such proofs cheap. We describe AutoIP, a Lean library in which classical linear regularization theory, together with the functional analysis it rests on, has been autoformalized by AI agents. A machine-checked proof certifies only the formal statement it proves, so the remaining human task is to confirm that each formal statement says what was intended. We organize the library around this task: a reviewer compares the formal statements with their natural-language counterparts and never needs to read a proof. Our experience is that the risks lie in the statements rather than in the proofs, and that simple automated tests on the statements can expose problems even in the established literature. We illustrate this with a textbook theorem whose hypothesis is more restrictive than asserted. We invite the inverse problems community to extend and build on the library.
