Made some interesting progress thinking about how to decompose weird logical connectives after emailing with chris and ian. I can now say things like "yeah, lightning bolt is just exists x . exists n . !(forall m.a(x, m) -o A * a(x, Sm)) * !(forall m.a(x, m) -o 1) * a(x, n) assuming that a(x * x', n * n') -||- a(x, n) * a(x', n') and S commutes with * and every term can be split like t = π1t * π2t" with a straight face.