The absence of this axiom has two consequences:
- We no longer have `■ (P ∗ Q) ⊢ ■ P ∗ ■ Q` and `□ (P ∗ Q) ⊢ □ P ∗ □ Q`,
and as a result, separating conjunctions in the unrestricted/persistent
context cannot be eliminated.
- When having `(P -∗ ⬕ Q) ∗ P`, we do not get `⬕ Q ∗ P`. In the proof
mode this means when having:
H1 : P -∗ ⬕ Q
H2 : P
We cannot say `iDestruct ("H1" with "H2") as "#H1"` and keep `H2`.
However, there is now a type class `PositiveBI PROP`, and when there is an
instance of this type class, one gets the above reasoning principle back.
TODO: Can we describe positivity of individual propositions instead of the
whole BI? That way, we would get the above reasoning principles even when
the BI is not positive, but the propositions involved are.