 
Summary: Inductive Types as Equations
Lisa Allali1
and Paul Brauner2
1 ´Ecole Polytechnique, INRIA & R´egion Ile de France allali@lix.polytechnique.fr
2
INPL & LORIA paul.brauner@loria.fr
Abstract. Inductive types play a central role in proof assistants and program
ming languages. There are usually two ways of interpreting them in a Curry
Howard manner, either by encoding their inhabitants by lambdaterms (Church
integers, Girard, Parigot), either by recursors (G¨odel System T, MatinL¨of). We
propose to link the two approaches the following way: recursors of inductive
types are naturally defined as the elimination rules generated by the transforma
tion of equations into supernatural deduction rules. This simplifies and breaks the
asymmetry of usual inductive types presentations. Moreover, the strong normal
ization property of the resulting system is obtained by using a simple semantic
argument.
1 Introduction
There are usually two ways of interpreting inductive types[1,2,3,4] in a CurryHoward
manner, either by encoding their inhabitants by lambdaterms (Church integers, Girard,
Parigot), either by recursors (G¨odel System T, MatinL¨of). We propose to link the two
