GitLab

This is an old flag set by the ssr plugin, and recently unset in coq-stdpp,
see https://gitlab.mpi-sws.org/robbertkrebbers/coq-stdpp/issues/5.

This is to be used on top of stdpp's 4b5d254e.

Now, associativity needs only to be established in case the elements are
valid and their compositions are defined. This is very much like the notion
of separation algebras I had in my PhD thesis (Def 4.2.1). The Dra to Ra
construction still easily works out.

Rename `UCMRA` → `Ucmra`
Rename `CMRA` → `Cmra`
Rename `OFE` → `Ofe` (`Ofe` was already used partially, but many occurences were missing)
Rename `STS` → `Sts`
Rename `DRA` → `Dra`

For obsolete reasons, that no longer seem to apply, we used ∅ as the
unit.

That way, we can use Contractive as part of structure definitions in
which we do not have a bundled OFE yet.

This commit fixes the issues that refolding of big operators did not work nicely
in the proof mode, e.g., given:

    Goal forall M (P : nat → uPred M) l,
      ([∗ list] x ∈ 10 :: l, P x) -∗ True.
    Proof. iIntros (M P l) "[H1 H2]".

We got:

    "H1" : P 10
    "H2" : (fix
            big_opL (M0 : ofeT) (o : M0 → M0 → M0) (H : Monoid o) (A : Type)
                    (f : nat → A → M0) (xs : list A) {struct xs} : M0 :=
              match xs with
              | [] => monoid_unit
              | x :: xs0 => o (f 0 x) (big_opL M0 o H A (λ n : nat, f (S n)) xs0)
              end) (uPredC M) uPred_sep uPred.uPred_sep_monoid nat
             (λ _ x : nat, P x) l
    --------------------------------------∗
    True

The problem here is that proof mode looked for an instance of `IntoAnd` for
`[∗ list] x ∈ 10 :: l, P x` and then applies the instance for separating conjunction
without folding back the fixpoint. This problem is not specific to the Iris proof
mode, but more of a general problem of Coq's `apply`, for example:

    Goal forall x l, Forall (fun _ => True) (map S (x :: l)).
    Proof.
      intros x l. constructor.

Gives:

     Forall (λ _ : nat, True)
       ((fix map (l0 : list nat) : list nat :=
          match l0 with
          | [] => []
          | a :: t => S a :: map t
          end) l)

This commit fixes this issue by making the big operators type class opaque and instead
handle them solely via corresponding type classes instances for the proof mode tactics.

Furthermore, note that we already had instances for persistence and timelessness. Those
were really needed; computation did not help to establish persistence when the list in
question was not a ground term. In fact, the sitation was worse, to establish persistence
of `[∗ list] x ∈ 10 :: l, P x` it could either use the persistence instance of big ops
directly, or use the persistency instance for `∗` first. Worst case, this can lead to an
exponential blow up because of back tracking.

Instead, I have introduced a type class `Monoid` that is used by the big operators:

    Class Monoid {M : ofeT} (o : M → M → M) := {
      monoid_unit : M;
      monoid_ne : NonExpansive2 o;
      monoid_assoc : Assoc (≡) o;
      monoid_comm : Comm (≡) o;
      monoid_left_id : LeftId (≡) monoid_unit o;
      monoid_right_id : RightId (≡) monoid_unit o;
    }.

Note that the operation is an argument because we want to have multiple monoids over
the same type (for example, on `uPred`s we have monoids for `∗`, `∧`, and `∨`). However,
we do bundle the unit because:

- If we would not, the unit would appear explicitly in an implicit argument of the
  big operators, which confuses rewrite. By bundling the unit in the `Monoid` class
  it is hidden, and hence rewrite won't even see it.
- The unit is unique.

We could in principle have big ops over setoids instead of OFEs. However, since we do
not have a canonical structure for bundled setoids, I did not go that way.