[TYPES] Eliminators in type theory

Thorsten Altenkirch txa at cs.nott.ac.uk
Fri Mar 17 05:17:20 EST 2006

Yong Luo wrote:
> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
> I would like to understand  how powerful the eliminators of inductive types
> can be in type theory.
> Suppose a type system has the type of natural numbers ONLY, that is, we have
> only Nat, Zero, Successor (Succ) and the eliminator Elim_Nat and its two
> computation rules.

This is a functional representation of arithmetic, i.e. Goedel's system T. All functions 
provable total in arithmetic can be defined. Adding dependent types doesn't affect the 
strength of the system, nether does the addition of pairs (whether dependent or not).


> Can we define fib function in such a system? Note that we don't have pairs
> and function types.
> fib 0 = 1
> fib 1 = 1
> fib (n+2) = fib n + fib (n+1)
> In such a  system, we know some functions can be defined, for example,
> plus x y = Elim_Nat (\n->Nat) x (\m,n->(S n)) y
> If we add the type of pairs, then more efficient fib can be easily defined.
> Thanks,
> Yong
> ==============================
> Dr. Yong Luo
> Computing Laboratory
> University of Kent

Dr. Thorsten Altenkirch		   phone : (+44) (0)115 84 66516
Lecturer			   http://www.cs.nott.ac.uk/~txa/
School of Computer Science & IT	   University of Nottingham

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