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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 *)
From Undecidability.Shared.Libs.DLW
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 *)
Hint Resolve prime_neq_0 : core.
(* The unboundedness of primes *)
Definition nxtprime n := proj1_sig (first_prime_above n).
Fact nxtprime_spec1 n : n < nxtprime n.
Fact notprime_bool_rec_spec n k : notprime_bool_rec n k = true <-> forall i, n <= i < k+n -> ~ prime i.
Definition nxtprime_bool n p := Nat.leb (S n) p && notprime_bool_rec (S n) (p - S n) && prime_bool p.
Fact nxtprime_bool_spec n p : nxtprime_bool n p = true <-> nxtprime n = p.
Definition nthprime (n : nat) := iter nxtprime 2 n.
Fact nthprime_nxt i p q : nthprime i = p -> nxtprime p = q -> nthprime (S i) = q.
(* Certified Erastosthene sieve would be helpfull here *)
Fact nthprime_0 : nthprime 0 = 2.
Definition ps : primestream.
Fact exp_app n m i v w : @exp (n+m) i (vec_app v w) = exp i v * exp (n+i) w.
Local Notation divides_mult_inv := prime_div_mult.
Hint Resolve not_prime_1 not_qs_1 : core.
Opaque ps qs.
Coercion tonat {n} := @pos2nat n.
Proof with eauto.
revert i j; induction v; intros i j Hi.
+ cbn; intros ? % divides_1_inv % not_qs_1; auto.
+ cbn; intros [ H % divides_pow | H  ] % divides_mult_inv; eauto.
* eapply primestream_divides in H; lia.
* eapply IHv in H; eauto.
Qed.

Lemma exp_inv_inc n u v1 :
@exp n 0 (vec_change v1 u (S (vec_pos v1 u))) = qs u * exp 0 v1.