In this paper, we introduce the systems ns-ACA0 and ns-WKL0 of non-standard second-order arithmetic in which we can formalize non-standard arguments in ACA0 and WKL0, respectively. Then, we give direct transformations from non-standard proofs in ns-ACA0 or ns-WKL0 into proofs in ACA0 or WKL0.
"Formalizing non-standard arguments in second-order arithmetic." J. Symbolic Logic 75 (4) 1199 - 1210, December 2010. https://doi.org/10.2178/jsl/1286198143