Infinitude of Primes

The following document contains embedded Coq code. All the code is editable and can be run directly on the page. Once jsCoq finishes loading, you are free to experiment by stepping through the proof and viewing intermediate proof states on the right panel.

ButtonKey bindingAction
F8 Toggles the goal panel.
Alt+/ or
Move through the proof.
Alt+Enter or
(⌘ on Mac)
Run (or go back) to the current point.
Ctrl+ Hover executed statements to peek at the proof state after each step.

As a relatively advanced showcase, we display a proof of the infinitude of primes in Coq. The proof relies on the Mathematical Components library from the MSR/Inria team led by Georges Gonthier, so our first step will be to load it:

Ready to do Proofs!

Once the basic environment has been set up, we can proceed to the proof:

The lemma states that for any number m, there is a prime number larger than m. Coq is a constructive system, which among other things implies that to show the existence of an object, we need to actually provide an algorithm that will construct it. In this case, we need to find a prime number p that is greater than m. What would be a suitable p? Choosing p to be the first prime divisor of m! + 1 works. As we will shortly see, properties of divisibility will imply that p must be greater than m.

Our first step is thus to use the library-provided lemma pdivP, which states that every number is divided by a prime. Thus, we obtain a number p and the corresponding hypotheses pr_p : prime p and p_dv_m1, "p divides m! + 1". The ssreflect tactic have provides a convenient way to instantiate this lemma and discard the side proof obligation 1 < m! + 1.

It remains to prove that p is greater than m. We reason by contraposition with the divisibility hypothesis, which gives us the goal "if p ≤ m then p is not a prime divisor of m! + 1".

The goal follows from basic properties of divisibility, plus from the fact that if p ≤ m, then p divides m!, so that for p to divide m! + 1 it must also divide 1, in contradiction to p being prime.