Formalizing a ring theorem with Lean 4 and Claude

I’ve been testing Claude’s ability to generate Lean 4 code to prove theorems. I’ve written about a couple experiments that verified calculations. I did not write about my failed attempt to get Claude to formalize a proof of the pqr theorem for seminorms. This time I asked Claude to formally prove the theorem from the previous post on partial fraction decomposition in an abstract setting. Unlike the previous experiments, I did not give it a proof or calculation to verify; I asked it to generate a proof. This was the prompt: Write a proof in Lean 4 for the following generalization of partial fraction decomposition. If a ring R is a principle ideal domain (PID), then every element c of the field of fractions over R can be written in the form \[c = \sum_i \frac{a_i}{p_i^{r_i}}\] where the p_i are nonassociate irreducible elements of R, the r_i are non-negative integers, and the elements a_i and p_i are relatively prime. This was Claude’s immediate response: I’ll work through this…