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The Completeness of Provable Realizability

Let A be a propositional formula and r A [X] express in the predicate logic the statement «X realizes A». We prove that the classical derivability of r A [t] for a lambda term t implies the intuitionistic derivability of A for the formulas A in the languages (⊃,∧,¬) and (⊃,&), where & is the so-called strong conjunction