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What’s mathematically questionable about the quotient soundness axiom? It’s justifiable metamathematically. What’s the real difference baking it into the proof
by cmrx64 1y ago
What’s mathematically questionable about the quotient soundness axiom? It’s justifiable metamathematically. What’s the real difference baking it into the proof kernel? I’d rather such independent properties be modeled as an axiom. The quotient automation I’m familiar with in other theorem provers is typically way more (untrusted!) machinery than just stating quot.sound.
- semolinapudding 1y agoComputation is the difference. In Lean, applying the universal property of the quotient (`Quotient.lift f Hf`) to an element that is of the form `Quotient.mk a` reduces to `f a`. This rule is fine in itself, but the Lean developers were not sufficiently careful and allowed it to apply for quotients of propositions, where it interferes with the computation rules for proof irrelevance and ends up breaking subject reduction (SR is deeply linked to computation when you have dependent types!) [0]. It is not really a problem in practice though, since there is no point in quotienting a proposition. [0] see the end of section 3.1 in https://github.com/digama0/lean-type-theory/releases/download/v1.0/main.pdf https://github.com/digama0/lean-type-theory/releases/downloa...
- cmrx64 1y agowhat does it mean to quotient datan’t?