Remark: The kernel was already using Sort. So, the limitation was
artificial. Moreover, it may seem unnecessary to have quotients of
proofs in a proof irrelevant system, but this is useful for proving
a more general funext lemma. This more general version is needed in
the new tested contributed by @digama0.
Before this commit, an user could define their own prelude and change
the types of quot, quot.mk, quot.lift or quot.ind.
By doing that, they could prove false.
This commit prevents this kind of abuse.
It also modifies the definition of `quot` and avoids the `setoid`
dependency.
The previous `quot` type is now called `quotient`, and it is defined
using the new `quot` type provided by the kernel.
See discussion at #1330
In the standard library, we should use explicit universe variables for
universe polymorphic definitions.
Users that want to declare universe polymorphic definitions but do not
want to provide universe level parameters should use
Type _
or
Type*
With the new "noncomputable" feature we can use Hilbert's choice without
being concerned it may accidentaly "leak" inside definitions we don't
want to use it.
Perhaps, we should add an option to disable this new feature.
Remark: this commit makes commit 46d418a redundant.
I'm keeping 46d418a because we may retract this commit in the future.
