Alex Best’s type class generalization paper
Alex J. Best wrote a paper about type class generalization for the CICM 2021 conference on intelligent computer mathematics.
When producing large formally verified mathematical developments that make use of typeclasses as in mathlib, it is easy to introduce overly strong assumptions for theorems and definitions. This paper considers the problem of recognizing from the elaborated proof terms when typeclass assumptions are stronger than necessary. It uses a Lean metaprogram that finds and informs the user about possible generalizations.
A nice example from the paper deals with the following theorem stating that given a ring homomorphism between two fields and a natural number $p$, one of the fields has characteristic p if and only if the other has characteristic $p$ (including $p = 0$):
lemma ring_hom.char_p_iff_char_p {K L : Type∗} [field K] [field L] (f : K →+∗ L) (p : ℕ) : char_p K p ↔ char_p L p := begin split; { introI _c, constructor, intro n, rw [← @char_p.cast_eq_zero_iff _ _ p _c n, ← f.injective.eq_iff, f.map_nat_cast, f.map_zero] } end
We see that the proof script splits the iff statement into each direction, but both directions are proved by the same tactic block. It is non-trivial to determine just by reading the proof given what the weakest assumptions possible are, and it is not immediately clear from the statement either. The meta-program determined these are that $K$ should be a division ring, and $L$ should be a nontrivial semiring.