Geodesic
Note on a Translation to Characterize Constructivity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical logic and algebra, Tome 242 (2003), pp. 136-140
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This note describes a straightforward translation $\mathcal C^{(1)}$ such that $T\vdash A(t)$ for some term $t$ $\Leftrightarrow$ $\mathcal C^{(1)}(T)\vdash \mathcal C^{(1)}(\exists xA(x))$ for universal $T$ and purely existential $\exists xA(x)$. The correspondence is based on the properties of resolution calculus. 
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M. Baaz. Note on a Translation to Characterize Constructivity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical logic and algebra, Tome 242 (2003), pp. 136-140. http://geodesic.mathdoc.fr/item/TM_2003_242_a10/

[1] Takeuti G., Proof theory, North-Holland, Amsterdam, 1987 | MR | Zbl

[2] Troelstra A., van Dalen D., Constructivism in mathematics. An introduction, v. 1, North-Holland, Amsterdam, 1988

[3] Leitsch A., The resolution calculus, Texts Theor. Comput. Sci., Springer, Berlin, 1997 | MR