A Type Theory which is complete for Kreisel's Modified Realizability

We define a type theory with a strong elimination rule for existential quantification. As in Martin-Löf's type theory, the “axiom of choice” is thus derivable. Proofs are also annotated by realizers which are simply typed lambda-terms. A new rule called “type extraction” which extracts the type of a realizer allows us to derive the so-called “independance of premisses” schema. Consequently, any formula which is realizable in HA^omega according to Kreisel's modified realizability, is derivable in this type theory