This PR modifies the pretty printing of pi types. Now `∀` will be
preferred over `→` for propositions if the domain is not a proposition.
For example, `∀ (n : Nat), True` pretty prints as `∀ (n : Nat), True`
rather than as `Nat → True`. There is also now an option `pp.foralls`
(default true) that when false disables using `∀` at all, for
pedagogical purposes. This PR also adjusts instance implicit binder
pretty printing — nondependent pi types won't show the instance binder
name. Closes#1834.
The linked RFC also suggests using `_` for binder names in case of
non-dependance. We're tabling that idea. Potentially it is useful for
hygienic names; this could improve how `Nat → True` pretty prints as `∀
(a : Nat), True`, with this `a` that's chosen by implication notation
elaboration. Relatedly, this PR exposes even further the issue where
binder names are reused in a confusing way. Consider: `Nat → Nat → (a :
Nat) → a = a` pretty prints as `∀ (a a a : Nat), a = a`.
When the type of an `example` is a proposition,
we should elaborate on them as we elaborate on theorems.
This is particularly important for examples that are often
used in educational material.
Recall that when elaborating theorem headers, we convert unassigned
universe metavariables into universe parameters. The motivation is
that the proof of a theorem should not influence its statement.
However, before this commit, this was not the case for examples when
their type was a proposition.
This discrepancy often confused users.
Additionally, we considered extending the above behavior to definitions
when
1- When their type is a proposition. However, it still caused disruption
in Mathlib.
2- When their type is provided. That is, we would keep the current
behavior only if `: <type>` was omitted. This would make the elaborator
for `def` much closer to the one for `theorem`, but it proved to be too
restrictive.
For example, the following instance in `Core.lean` would fail:
```
instance {α : Sort u} [Setoid α] : HasEquiv α :=
⟨Setoid.r⟩
```
and we would have to write instead:
```
instance {α : Sort u} [Setoid α] : HasEquiv.{u, 0} α :=
⟨Setoid.r⟩
```
There are other failures like this in the core, and we assume many more
in Mathlib.
closes#4398closes#4482 Remark: PR #4482 implements option 1 above. We may consider
it again in the future.
Potential problem: if elaboration of subterms is delayed the order the new metavariables are created may not match the order they
appear in the `.lean` file. We should tell users to prefer tagged goals.
closes#1682
