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Inductive Types as Equations Lisa Allali1
 

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 lambda-terms (Church
integers, Girard, Parigot), either by recursors (G¨odel System T, Matin-L¨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 Curry-Howard
manner, either by encoding their inhabitants by lambda-terms (Church integers, Girard,
Parigot), either by recursors (G¨odel System T, Matin-L¨of). We propose to link the two

  

Source: Allali, Lisa - Laboratoire d’Informatique, École Polytechnique

 

Collections: Computer Technologies and Information Sciences