GitLab

Get rid of using `Collection` and favor `set` everywhere. Also, prefer conversion
functions that are called `X_to_Y`.

The following sed script performs most of the renaming, with the exception of:

- `set`, which has been renamed into `propset`. I couldn't do this rename
  using `sed` since it's too context sensitive.
- There was a spurious rename of `Vec.of_list`, which I correctly manually.
- Updating some section names and comments.

```
sed '
s/SimpleCollection/SemiSet/g;
s/FinCollection/FinSet/g;
s/CollectionMonad/MonadSet/g;
s/Collection/Set\_/g;
s/collection\_simple/set\_semi\_set/g;
s/fin\_collection/fin\_set/g;
s/collection\_monad\_simple/monad\_set\_semi\_set/g;
s/collection\_equiv/set\_equiv/g;
s/\bbset/boolset/g;
s/mkBSet/BoolSet/g;
s/mkSet/PropSet/g;
s/set\_equivalence/set\_equiv\_equivalence/g;
s/collection\_subseteq/set\_subseteq/g;
s/collection\_disjoint/set\_disjoint/g;
s/collection\_fold/set\_fold/g;
s/collection\_map/set\_map/g;
s/collection\_size/set\_size/g;
s/collection\_filter/set\_filter/g;
s/collection\_guard/set\_guard/g;
s/collection\_choose/set\_choose/g;
s/collection\_ind/set\_ind/g;
s/collection\_wf/set\_wf/g;
s/map\_to\_collection/map\_to\_set/g;
s/map\_of\_collection/set\_to\_map/g;
s/map\_of\_list/list\_to\_map/g;
s/map\_of\_to_list/list\_to\_map\_to\_list/g;
s/map\_to\_of\_list/map\_to\_list\_to\_map/g;
s/\bof\_list/list\_to\_set/g;
s/\bof\_option/option\_to\_set/g;
s/elem\_of\_of\_list/elem\_of\_list\_to\_set/g;
s/elem\_of\_of\_option/elem\_of\_option\_to\_set/g;
s/collection\_not\_subset\_inv/set\_not\_subset\_inv/g;
s/seq\_set/set\_seq/g;
s/collections/sets/g;
s/collection/set/g;
' -i $(find -name "*.v")
```

Since `fresh_generic` is too inefficient for all practical purposes, we seal
off its definition. That way, Coq will not accidentally unfold it during
unification or other tactics.

This issue actually occurred in iGPS, as reported by Hai.

- Name all variables that we refer to.
- Put types in definitions.

Also make the instances non-global, to prevent multiple instance
problems.

We prove that various types are infinite, notably:
- nat, N, positive and Z;
- string (using pretty-printing of nat);
- option, with an infinite element type;
- list, with an inhabited element type.

Furthermore, we instantiate Fresh for strings.