Natural numbers

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The natural numbers are the numbers 0, 1, 2, etc. The term is generally used to indicate a specific technique of representing natural numbers as either zero or the successor of some other natural number - 0, s(0), s(s(0)), etc - as in Peano arithmetic, a technique also sometimes referred to as unary numbers.

 Natural numbers in Twelf

Natural numbers in Twelf are usually defined in a similar way. Mathematically, natural numbers can be defined as zero or the successor of some other natural number:

$\texttt{}n ::= 0 \,|\, \texttt{s}(n)$

This representation translates easily into Twelf:

nat: type.
z: nat.
s: nat -> nat.

The first line declares that nat is a type. The second line declares z (zero) to be an object of type nat, and the third line declars s (successor) to be a type constructor that takes an object N of type nat and produces another object (s N) of type nat.

 Addition of natural numbers in Twelf

The addition of these natural numbers is defined by the judgment $\texttt{plus}(N_1,N_2,N_3)$, where N1, N2, and N3 are natural numbers. In the definition below, capital letters stand for metavariables that can range over all natural numbers.

${\qquad \over \texttt{plus}(0,N,N)}{\mbox{p-z}}$                     ${\texttt{plus}(N_1,N_2,N_3) \over \texttt{plus}(\texttt{s}(N_1),N_2,\texttt{s}(N_3))}{\mbox{p-s}}$

These judgments also translate cleanly into Twelf:

plus: nat -> nat -> nat -> type.
p-z: plus z N N.
p-s: plus (s N1) N2 (s N3)
<- plus N1 N2 N3.

The first line defines the judgment, declaring plus to be a type family indexed by three terms of type nat.

The second line declares that for any natural number N, p-z is an object of type plus z N N, which corresponds to the axiom p-z above. The N is an implicit parameter - it is treated as a bound variable by Twelf, which you can see by looking at Twelf's output after checking the above code.

The third line says that p-s is a type constructor that, given an object D of type plus N1 N2 N3 (where N1, N2, and N3 are all implicit parameters that can be treated as metavariables), produces an object, p-s D, with type plus (s N1) N2 (s N3). This corresponds to the rule p-s, which given a derivation of $\texttt{plus}(N_1,N_2,N_3)$ allows us to conclude $\texttt{plus}(\texttt{s}(N_1),N_2,\texttt{s}(N_3))$.

Consider this derivation which encodes the fact that 2 + 1 = 3:

${{{\;} \over { \texttt{plus}(0,\;\;\texttt{s}(0),\;\;\texttt{s}(0))}}{\mbox{p-z}} \over {{ \texttt{plus}(\texttt{s}(0),\;\;\texttt{s}(0),\;\;\texttt{s}(\texttt{s}(0)))} \over { \texttt{plus}(\texttt{s}(\texttt{s}(0)),\;\;\texttt{s}(0),\;\;\texttt{s}(\texttt{s}(\texttt{s}(0))))}}{\mbox{p-s}}}{\mbox{p-s}}$

This can be represented in Twelf by applying the type constructor p-s to the object p-z twice:

2+1=3 : plus (s (s z)) (s z) (s (s (s z))) = p-s (p-s p-z).
Twelf 1.7.1 (built 03/19/11 at 09:41:05 on gs6177)
2+1=3 : plus (s (s z)) (s z) (s (s (s z))) = p-s (p-s p-z).

%% OK %%