Ends of $\operatorname{pf}(\mathcal{U})$ are integers #
For a universe U carrying the hypotheses of misereGE_iff_promain (in particular
ShortBlocking and ShortDeadEnding), every end of $\operatorname{pf}(\mathcal{U})$ is
equivalent modulo $\operatorname{pf}(\mathcal{U})$ to an integer. This is the blocking analogue
of isEnd_exists_intCast_misereEQ.
Throughout, larger integers are smaller games: a ≤ b in ℤ gives b ≥m pf(U) a.
The key step is misereGE_intCast_or_exists_leftMove_of_not_succ_misereGE: for every
g ∈ pf(U) and every integer n ≤ 0, if n + 1 ≱ g then g ≥ n or some Left option of g
is ≥ n - 1. Applied to a Left end g at the largest n ≤ 0 with n ≥ g, it gives g = n.
The main results are
Sufficiency half of the comparison of g with a negative integer n: it is enough that every
Right option of g is ≥ n + 1 or has a Left option ≥ n.
If $\operatorname{o}(G) = \mathscr{L}$ then $G \ge \overline{1}$.
Necessity half of the comparison of a negative integer n with g: if n ≥ g and g is not a
Right end, then some Right option of g is ≤ n + 1.
Unwinding n ≱ g for an integer n ≤ 0: either no Right option of g is ≤ n + 1, or some
Left option gl of g is not ≤ -1 and has no Right option ≤ n.
Unwinding 1 ≱ g when $\operatorname{o}(G) = \mathscr{R}$: some Left option gl of g is not
≤ 0 and has no Right option ≤ 1.
For n ≤ 0: if n + 1 ≱ g then g ≥ n, or some Left option of g is ≥ n - 1.
If $G \in \operatorname{pf}(\mathcal{U})$ is a Left end, then it is equivalent to some non-positive integer.
If $G \in \operatorname{pf}(\mathcal{U})$ is a Right end, then it is equivalent to some non-negative integer.
If $G \in \operatorname{pf}(\mathcal{U})$ is an end, then it is equivalent to some integer.
Every end of $\operatorname{pf}(\mathcal{B})$ is equivalent to an integer.