Theorem Difficulty Is Definition Count

01The evidence

• The de Bruijn factor, the ratio of formalisation length to informal proof length, is used to estimate formalisation effort, and it varies widely across projects for reasons that are not explained by proof length.Xena project, de Bruijn factor
• Formalisation of the Carleson project and of differential geometry in Lean required substantial work on definitions and interfaces before the theorems themselves could be stated, which is the pattern a definition-count account predicts.Carleson project formalisation

02The tension

Formalisation is planned by estimating proof length, and the estimates are routinely wrong by large factors. The variance is attributed to difficulty, which is a restatement rather than an explanation, and it makes resourcing decisions guesswork.

03The theory

A formal proof does not just reproduce steps; it reconciles definitions. Each independent definition, a structure with its own interface, its own notation and its own lemmas, must be connected to the others before the theorem can even be stated.

The cost is combinatorial because the connections are pairwise. Ten definitions require far more than twice the interface work of five, and the informal proof hides this because mathematicians share a vocabulary that the formal system does not.

That makes definition count the predictor and proof length the proxy that fails. It also explains why some short theorems are hard: they sit at the junction of several mature but incompatible structures, while long proofs within a single structure are routine.

The prediction is checkable against existing formalisation corpora: effort should scale with definition count, and the de Bruijn factor should be predictable once definitions are counted.

04What this is not

05Predictions

• P1Formalisation effort should correlate with the number of independent definitions better than with proof length.
• P2The de Bruijn factor should be predictable from definition count for theorems in the same library.
• P3Refactoring definitions to share interfaces should reduce effort nonlinearly, more than the reduction in proof length would suggest.

06What would refute it

Proof length predicting effort better than definition count, across existing corpora, would leave the account without a phenomenon.

07What it would change

It would make formalisation projects plannable, and shift library design toward shared definitions rather than accumulated lemmas.