Elimination of Skolem functions for monotone formulas in analysis

GnA ω +∆+AC-qf ⊢ ∀u, k∀v ≤τ tuk∃w A0(u, k, v, w), (where t is a closed term, A0 is quantifier-free and contains only u, k, v, w free, γ ≤ 2, ρ is an arbitrary type and ≤τ is defined pointwise) one can extract (by monotone functional interpretation) a uniform bound Φ on ∃w which is given by a closed term of GnA ω and does not depend on v, i.e. ∀u, k∀v ≤ tuk∃w ≤ ΦukA0(u, k, v, w) holds in the full set-theoretic model. In particular Φuk is a polynomial (an elementary recursive function) in u := λx.maxi≤x u(i) and k0 in case n = 2 (resp. n = 3).