Extracting Herbrand trees from Coq

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Software certification aims at proving the correctness of programs but in many cases, the use of external libraries allows only a conditional proof: it depends on the assumption that the libraries meet their specifications. In particular, a bug in these libraries might still impact the certified program. In this case, the difficulty that arises is to isolate the defective library function and provide a counter-example. In this paper, we show that this problem can be logically formalized as the construction of a Herbrand tree for a contradictory universal theory and address it. The solution we propose is based on a proof of Herbrand's theorem in the proof assistant Coq. Classical program extraction using Krivine's classical realizability then translates this proof into a certified program that computes Herbrand trees. Using this tree and calls to the library functions, we are able to determine which function is defective and explicitly produce a counter-example to its specification.