[TYPES] ?==?utf-8?q? ?==?utf-8?q? Gödel's T extended with an eta-like rule for the recursor

[Sergei Soloviev]

Dear Ansten,

We did work with David Chemouil on related questions, main results and ideas
are in the papers

David Chemouil:
Isomorphisms of simple inductive types through extensional rewriting. Mathematical Structures in Computer Science 15(5): 875-915 (2005)

David Chemouil:
An insertion operator preserving infinite reduction sequences. Mathematical Structures in Computer Science 18(4): 693-728 (2008)

David Chemouil, Sergei Soloviev:
Remarks on isomorphisms of simple inductive types. Electr. Notes Theor. Comput. Sci. 85(7): 106-124 (2003)

I have to check whether they cover directly your question, but in any case there were
developped techniques that certainly could be useful.

All the best

Sergei Soloviev
 
 
Le Mardi 5 Février 2019 09:18 CET, Ansten Mørch Klev <anstenklev at gmail.com> a écrit: 
 
> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
> 
> Let N be the natural numbers and A any type in the hierarchy of simple
> types over N. Let s : (N)N be the successor function and let f : (N)A be
> arbitrary. Then, for variables x : N, y : A we may form the term
> 
> [x, y]f(s(x)) : (N)(A)A
> 
> Let R be the recursor for type A. Then, for any term n : N, we have
> 
> R( f(0) , [x, y]f(s(x)) , n ) : A
> 
> By the reduction rules for R one can see that f and R( f(0) , [x, y]f(s(x))
> ) agree on the numerals.
> 
> Suppose we add the following eta-like reduction rule to Gödel's T:
> 
> R( f(0) , [x, y]f(s(x)) , x ) --> f(x)
> 
> Is it known whether the resulting system is (strongly) normalizing and
> confluent?
> 
> 
> ------------
> Ansten Klev
> Czech Academy of Sciences, Department of Philosophy