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Proof of Euclid' theorem
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(* Proof of Euclid' theorem *) | |
(* Constructive proof by Euclid himself *) | |
From mathcomp Require Import all_ssreflect. | |
Fixpoint primorial n := | |
if n is S m then (if prime n then n else 1) * primorial m else 1. | |
Notation "n `#" := (primorial n) (at level 2). | |
Lemma primo_gtS n: 1 < n`# .+1. | |
Proof. | |
rewrite ltnS; elim: n => // m IH. by rewrite muln_gt0; case:(prime m.+1). | |
Qed. | |
Lemma dvdn_primo {m n}: m <= n -> prime m -> m %| n`#. | |
Proof. | |
move=>mn pm; move:mn; elim n. | |
- by rewrite leqn0=>/eqP E; move:E pm=>->; auto. | |
- by move=> n' IH mn/=; move: leq_eqVlt mn pm => ->/orP[/eqP<-->|]; auto. | |
Qed. | |
Theorem Euclid's_theorem: forall n, exists p, prime p /\ n < p. | |
Proof. | |
move => n; pose m:= pdiv (n`# .+1). | |
have pm: prime m by apply/pdiv_prime/primo_gtS. | |
exists m; split; [done|rewrite leqNgt ltnS; apply/negP=>mn]. | |
move: {mn} (dvdn_primo mn pm) (pdiv_dvd (n`# .+1)); fold m => mnp. | |
apply/negP; rewrite -(prime_coprime _ pm); apply/(coprime_dvdl mnp)/coprimenS. | |
Qed. |
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