Semi-intuitionistic systems and monotone functional interpretation

Most of the applications of functional interpretation to concrete proofs in numerical functional analysis which were obtained in recent years (see chapters 16 and 18 for a survey of some of them) use a combination of functional interpretation with Howard’s majorizability construction similarly to the corresponding combination in the case of modified realizability. Analogously to the monotone modified realizability interpretation one can define a monotone functional interpretation which directly extract terms which majorize some functionals realizing the usual functional interpretation. More precisely one can prove a soundness theorem with the statement in the soundness theorem for functional interpretation replaced by

$$(+)\ \exists \underline{x}\big(\underline{t}^{*}\ maj\ \underline{x}\wedge \forall \underline{a},\underline{y}\,A_{D}(\underline{x}(\underline{a}),\underline{y},\underline{a})\big)$$

for suitable closed terms t^*.