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Used the following script:

sed '
s/\bCofeMor/OfeMor/g;
s/\-c>/\-d>/g;
s/\bcFunctor/oFunctor/g;
s/\bCFunctor/OFunctor/g;
s/\b\%CF/\%OF/g;
s/\bconstCF/constOF/g;
s/\bidCF/idOF/g
s/\bdiscreteC/discreteO/g;
s/\bleibnizC/leibnizO/g;
s/\bunitC/unitO/g;
s/\bprodC/prodO/g;
s/\bsumC/sumO/g;
s/\bboolC/boolO/g;
s/\bnatC/natO/g;
s/\bpositiveC/positiveO/g;
s/\bNC/NO/g;
s/\bZC/ZO/g;
s/\boptionC/optionO/g;
s/\blaterC/laterO/g;
s/\bofe\_fun/discrete\_fun/g;
s/\bdiscrete\_funC/discrete\_funO/g;
s/\bofe\_morC/ofe\_morO/g;
s/\bsigC/sigO/g;
s/\buPredC/uPredO/g;
s/\bcsumC/csumO/g;
s/\bagreeC/agreeO/g;
s/\bauthC/authO/g;
s/\bnamespace_mapC/namespace\_mapO/g;
s/\bcmra\_ofeC/cmra\_ofeO/g;
s/\bucmra\_ofeC/ucmra\_ofeO/g;
s/\bexclC/exclO/g;
s/\bgmapC/gmapO/g;
s/\blistC/listO/g;
s/\bvecC/vecO/g;
s/\bgsetC/gsetO/g;
s/\bgset\_disjC/gset\_disjO/g;
s/\bcoPsetC/coPsetO/g;
s/\bgmultisetC/gmultisetO/g;
s/\bufracC/ufracO/g
s/\bfracC/fracO/g;
s/\bvalidityC/validityO/g;
s/\bbi\_ofeC/bi\_ofeO/g;
s/\bsbi\_ofeC/sbi\_ofeO/g;
s/\bmonPredC/monPredO/g;
s/\bstateC/stateO/g;
s/\bvalC/valO/g;
s/\bexprC/exprO/g;
s/\blocC/locO/g;
' -i $(find theories -name "*.v")

This allows one to make use of recursive ghost state obtained from the
recursive domain equation solver.

This proof also more easily scales to other recursive types, like
trees etc.

The finiteness was needed to have the axiom of choice over the domain. This axiom is not needed if cmra_extend is in Type.

Revert "Remove the domain finiteness hypothesis for the function CMRA, and put cmra_extend in Type."

This reverts commit fa897ff5.

The finiteness was needed to have the axiom of choice over the domain. This axiom is not needed if cmra_extend is in Type.

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.

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.

This patch was created using

  find -name *.v | xargs -L 1 awk -i inplace '{from = 0} /^From/{ from = 1; ever_from = 1} { if (from == 0 && seen == 0 && ever_from == 1) { print "Set Default Proof Using \"Type*\"."; seen = 1 } }1 '

and some minor manual editing

Use COFEs only for the recursive domain equation solver

This allows us to factor out properties about connectives that
commute with the big operators.

This requirement was useful in Iris 2.0: in order to ensure that ownership of
the physical state was timeless, we required the ghost CMRA to have a timeless
unit. To avoid having additional type class parameters, or having to extend the
algebraic hierarchy, we required the units of any CMRA to be timeless.

In Iris 3.0, this issue no longer applies: ownership of the physical state is
ghost ownership in the global CMRA, whose unit is always timeless.

Thanks to Jeehoon Kang for spotting this unnecessary requirement.

This is more consistent with the definition of the extension order, which
is also defined in terms of an existential.

This reverts commit 20b4ae55, which does
not seem to work with Coq 8.5pl2 (I accidentally tested with 8.5pl1).