We present two refinements, based on program extraction in elementary affine logic and light affine logic, of Krivine & Leivant's system FA_2. This system allows higher-order equations to specify the computational content of extracted programs. The user can then prove a generic formula, using these equations as axioms. The system guarantees that the extracted program satisfies the specification and is computable in elementary time (for elementary affine logic) or polynomial time (for light affine logic). Finally, we show that both systems are complete with respect to elementary time and polynomial time functions.
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