tutorial: projecting out of a structure that may or may not be a Prop

[nomeata]

MaybeProp.{u} : Sort u is a proposition exactly for u := 0. Lean's
inductive command refuses such a declaration, but the kernel accepts it,
and projecting out of it is fine at every instantiation: a structure that
is not a Prop had every constructor field's universe checked against its
resulting universe, and that inequality survives instantiation, so wherever
the structure does become a proposition all of its fields do too.

That argument turns on the two checks agreeing on what a proposition is.
Waiving the field universe bound at declaration time and forbidding data
projections are two halves of one rule: a kernel generous about "is this a
Prop?" when declaring the inductive and strict about it when checking a
projection puts a data field inside a proposition, and proof irrelevance
collapses it into a proof of False. Erring the other way is safe but
incomplete, and that is what these tests catch — asking "could this be a
Prop?" and then demanding the field be definitely a proof rejects a
legitimate declaration. projMaybeProp catches that at the field being
asked for, projMaybePropPast on the way past a preceding one, which is a
separate code path.

The shape is not exotic: a dependent pair kept at the exact maximum of its
components lands in Sort (max u v), which poses the same question, and
this is why PProd and PSigma use Sort (max 1 u v) instead.

Co-Authored-By: Claude Fable 5 noreply@anthropic.com
Claude-Session: https://claude.ai/code/session_01UakEtNJTcwyaEtfLhR8BLn