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(*   Copyright Dominique Larchey-Wendling [*]                 *)
(*             Yannick Forster            [+]                 *)
(*                                                            *)
(*                             [*] Affiliation LORIA -- CNRS  *)
(*                             [+] Affiliation Saarland Univ. *)
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(*      This file is distributed under the terms of the       *)
(*         CeCILL v2 FREE SOFTWARE LICENSE AGREEMENT          *)
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(** ** Two infinite sequences of primes *)
Set Implicit Arguments.
Local Notation "e #> x" := (vec_pos e x).
Local Notation "e [ v / x ]" := (vec_change e x v).
Set Implicit Arguments.
(* Unique decomposition with a prime *)

(* Specifications: *)
Lemma prime_neq_0 p : prime p -> p <> 0.
Hint Resolve prime_neq_0.
Lemma power_factor_lt_neq p i j x y :
p <> 0
-> i < j
-> ~ divides p x
-> p^i * x <> p^j * y.
Lemma power_factor_uniq p i j x y :
p <> 0
-> ~ divides p x
-> ~ divides p y
-> p^i * x = p^j * y
-> i = j /\ x = y.