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 λ-terms. A new rule called “type extraction” which extract the type of a realizer allows us to derive the so-called “independance of premisses” schema. Consequently, any formula which is realizable in HA, according to Kreisel’s modified realizability, is derivable in this type theory.