Classical program extraction in the calculus of constructions
Résumé
We show how to extract classical programs expressed in Krivine lambda-c-calculus from proof-terms built in a proof-irrelevant and classical version of the calculus of constructions with universes. For that, we extend Krivine's realisability model of classical second-order arithmetic to the calculus of constructions with universes using a structure of Pi-set which is reminiscent of omega-sets, and show that our realisability model validates Peirce's law and proof-irrelevance. Finally, we extend the extraction scheme to a primitive data-type of natural numbers in a way which preserves the whole compatibility with the classical realisability interpretation of second-order arithmetic.
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