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* ''G''(0, ''x'') is a given function of ''x''.
* ''G''(''n'' + 1, 0) is obtained by substitution from the function ''G''(''n'', ·) and given functions.
* ''G''(''n'' + 1, ''x'' + 1) is obtained by substitution from ''G''(''n'' + 1, ''x''), the function ''G''(''n'', ·) and given functions.<ref>{{cite journal | author=Raphael M. Robinson | title=Recursion and Double Recursion | journal=[[Bulletin of the American Mathematical Society]] | year=1948 | volume=54 | pages=987–93 | url=http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183512393&page=record | doi=10.1090/S0002-9904-1948-09121-2| doi-access=free }}</ref>
Robinson goes on to provide a specific double recursive function (originally defined by [[Rózsa Péter]])
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