  We present a new set of reductions for derivations in natural deduction that
can extract witnesses from closed derivations of simply existential formulas in
Heyting Arithmetic (HA) plus the Excluded Middle Law restricted to simply
existential formulas (EM1), a system motivated by its interest in proof mining.
  The reduction we have for classical logic are quite different from all
existing ones. They are inspired by the informal idea of learning by making
falsifiable hypothesis and checking them, and by the interactive realizability
interpretation. We extract the witnesses directly from derivations in HA+EM1 by
reduction, without encoding derivations by a realizability interpretation.
