On the Proofs of Some Formally Unprovable Propositions and Prototype Proofs in Type Theory

Abstract

At the age of 7 or 8, Gauss was asked to produce the result of the sum of the first n integers (or, perhaps, the question was slightly less general...). He then proved a theorem, by the following method:

$$ \frac{{\begin{array}{*{20}c} 1 \\ n \\ \end{array} \begin{array}{*{20}c} 2 \\ {(n - 1)} \\ \end{array} \begin{array}{*{20}c} {...} \\ {...} \\ \end{array} \begin{array}{*{20}c} n \\ 1 \\ \end{array} }} {{\left( {n + 1} \right)\left( {n + 1} \right)...\left( {n + 1} \right)}} $$

which gives ∑ ⁿ₁ i = n(n + 1)/2.

Clearly, the proof is not by induction. Given n, a uniform argument is proposed, which works for any integer n. Following Herbrand, we will call prototype this kind of proof. Of course, once the formula is known, it is very easy to prove it by induction, as well. But, one must know the formula, or, more generally, the “induction load”. A non-obvious issue in automatic theorem proving, as we all know.