The arrow they're using in figure 5 seems to be (written infix as A)
α A β = ∀ γ δ . α * γ * (β * γ → δ) → δ
γ being the type of "progress arrows".
Injection of ordinary functions α → β is pretty much determined by the type:
inj (f : α → β) : α A β = λ (x, p, k) . k (f(x), p)
>>> is also pretty straightforward:
(f : α A β) >>> (g : β A γ) : α A γ = λ (x, p, k) . f (x, p, λ (y, q) . g (y, q, k))
I'm having difficulty understanding what "progress arrows" are or how they function. Already arrows depart from the slightly nicer mathematical world of monads, and this paper seems to add yet more behavior without any intuition or framework for it to exist in.
