Comment by zozbot234

Yup, Lean's quotient induction breaks subject reduction, which is an important type-theoretic principle. It means you can write a Lean development where t has type A, and t reduces (i.e. computes, as part of the Lean kernel) to u, but u doesn't have type A, and may not even type check. (See https://github.com/digama0/lean-type-theory/releases/downloa... Sec. 3.1 for a detailed discussion of this issue.) This is obviously quite bad, and it goes far beyond the usual drawback of adding axioms to a theory, including the quotient axiom. (Namely, the loss of canonicity.)