Also make names more consistent.

We typically use the _1 and _2 suffix to denote individual directions
of a lemmas that is a biimplication.

case H; clear H would fail when H is dependent whereas destruct H
would succeed on that, but just not clear it.

The case tactic is faster than destruct.

Example:

Goal
  ¬False → ¬False → ¬False → ¬False → ¬False → ¬False → ¬False →
  False.
Proof.
  intros. done. (* takes very long *)

Using this new definition we can express being contractive using a
Proper. This has the following advantages:

- It makes it easier to state that a function with multiple arguments
  is contractive (in all or some arguments).
- A solve_contractive tactic can be implemented by extending the
  solve_proper tactic.

Use notation N @⊆ E to avoid ambiguity.

Since `nclose : namespace → coPset` is declared as a coercion, the notation `nclose N ⊆ E` was pretty printed as `N ⊆ E`. However, `N ⊆ E` could not be typechecked because type checking goes from left to right, and as such would look for an instance `SubsetEq namespace`, which causes the right hand side to be ill-typed.

See merge request !24

Robbert Krebbers authored 8 years ago and Ralf Jung committed 8 years ago

We do this by introducing a type class UpClose with notation ↑.

The reason for this change is as follows: since `nclose : namespace
→ coPset` is declared as a coercion, the notation `nclose N ⊆ E` was
pretty printed as `N ⊆ E`. However, `N ⊆ E` could not be typechecked
because type checking goes from left to right, and as such would look
for an instance `SubsetEq namespace`, which causes the right hand side
to be ill-typed.

In particular, make sure we always try eassumption before reflexivity.

That range includes tabs and new lines.

Thanks Morten for spotting this problem.

This way we can use set_solver to solve goals involving ∈.

This has bothered me repeatedly in proofs, now I finally got around to fix it at the source