From mathcomp Require Import all_ssreflect all_algebra.

Print negbK.
Check negbK.      (* involutive negb *)
Print involutive. (* f (f x) = x *)
Print cancel.     (* f g such that f (g x) = x *)

Locate "~~".      (* negb *)
Locate "~".       (* Could be 'not' *)

Goal forall b: bool,
  ~~ (~~ b) = b.
Proof.
  intro b.
  case: b.
  by [].
  by [].
Show Proof.
Restart.
  intro b.
  by case: b.
  (* Or 'by [case: b].' *)
Qed.

Fixpoint multn (n m: nat): nat :=
  if n is n'.+1 then m + multn n' m
  else 0.
Compute multn 3 4.
(* = 12 : nat *)

Goal forall n m: nat,
  (n * m == 0) = (n == 0) || (m == 0).
Proof.
  intros n m.
  case: n => [|n] //.
  case: m => [|m]; last first => //. (* Good practice to get rid of easy goal first' *)
  by rewrite muln0.
Qed.

Goal forall n p:nat,
  prime p -> p %| (n`! + 1) -> n < p.
Proof.
  move => n p prime_p.
  apply: contraLR.