In the last few years appeared pedagogical propositional natural deduction
systems. In these systems, one must satisfy the pedagogical constraint: the
user must give an example of any introduced notion. First we expose the reasons
of such a constraint and properties of these "pedagogical" calculi: the absence
of negation at logical side, and the "usefulness" feature of terms at
computational side (through the Curry-Howard correspondence). Then we construct
a simple pedagogical restriction of the calculus of constructions (CC) called
CCr. We establish logical limitations of this system, and compare its
computational expressiveness to Godel system T. Finally, guided by the logical
limitations of CCr, we propose a formal and general definition of what a
pedagogical calculus of constructions should be.Comment: 18 page