### Welcome to the jsCoq Interactive Online System!

Welcome to the jsCoq technology demo! jsCoq is an interactive, web-based environment for the Coq Theorem prover, and is a collaborative development effort. See the list of contributors below.

jsCoq is open source. If you find any problem or want to make any contribution, you are extremely welcome! We await your feedback at GitHub and Zulip.

#### Instructions:

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.

##### Actions:
ButtonKey bindingAction
Alt+/ or
Alt+N/P
Move through the proof.
Alt+Enter or
Alt+
Run (or go back) to the current point.
F8 Toggles the goal panel.
##### Creating your own proof scripts:

The scratchpad offers simple, local storage functionality. It also allows you to share your development with other users in a manner that is similar to Pastebin.

#### A First Example: The Infinitude of Primes

If you are new to Coq, check out this introductory tutorial by Mike Nahas. As a more 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:

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.

jsCoq provides many other packages, including Coq's standard library and the mathematical components library. Feel free to experiment, and bear with the beta status of this demo.

¡Salut!