(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

Require Import ZArith_base ZArithRing Zcomplements Zdiv Znumtheory.
Require Export Zpower.
Local Open Scope Z_scope.

(** Properties of the power function over [Z] *)

(** Nota: the usual properties of [Z.pow] are now already provided
    by [BinInt.Z]. Only remain here some compatibility elements,
    as well as more specific results about power and modulo and/or
    primality. *)

Lemma Zpower_pos_1_r x : Z.pow_pos x 1 = x.
Proof (Z.pow_1_r x).

Lemma Zpower_pos_1_l p : Z.pow_pos 1 p = 1.
Proof. now apply (Z.pow_1_l (Zpos p)). Qed.

Lemma Zpower_pos_0_l p : Z.pow_pos 0 p = 0.
Proof. now apply (Z.pow_0_l (Zpos p)). Qed.

Lemma Zpower_pos_pos x p : 0 < x -> 0 < Z.pow_pos x p.
Proof. intros. now apply (Z.pow_pos_nonneg x (Zpos p)). Qed.

Notation Zpower_1_r := Z.pow_1_r (compat "8.3").
Notation Zpower_1_l := Z.pow_1_l (compat "8.3").
Notation Zpower_0_l := Z.pow_0_l' (compat "8.3").
Notation Zpower_0_r := Z.pow_0_r (compat "8.3").
Notation Zpower_2 := Z.pow_2_r (compat "8.3").
Notation Zpower_gt_0 := Z.pow_pos_nonneg (compat "8.3").
Notation Zpower_ge_0 := Z.pow_nonneg (compat "8.3").
Notation Zpower_Zabs := Z.abs_pow (compat "8.3").
Notation Zpower_Zsucc := Z.pow_succ_r (compat "8.3").
Notation Zpower_mult := Z.pow_mul_r (compat "8.3").
Notation Zpower_le_monotone2 := Z.pow_le_mono_r (compat "8.3").

Theorem Zpower_le_monotone a b c :
 0 < a -> 0 <= b <= c -> a^b <= a^c.
Proof. intros. now apply Z.pow_le_mono_r. Qed.

Theorem Zpower_lt_monotone a b c :
 1 < a -> 0 <= b < c -> a^b < a^c.
Proof. intros. apply Z.pow_lt_mono_r; auto with zarith. Qed.

Theorem Zpower_gt_1 x y : 1 < x -> 0 < y -> 1 < x^y.
Proof. apply Z.pow_gt_1. Qed.

Theorem Zmult_power p q r : 0 <= r -> (p*q)^r = p^r * q^r.
Proof. intros. apply Z.pow_mul_l. Qed.

Hint Resolve Z.pow_nonneg Z.pow_pos_nonneg : zarith.

Theorem Zpower_le_monotone3 a b c :
 0 <= c -> 0 <= a <= b -> a^c <= b^c.
Proof. intros. now apply Z.pow_le_mono_l. Qed.

Lemma Zpower_le_monotone_inv a b c :
  1 < a -> 0 < b -> a^b <= a^c -> b <= c.
Proof.
 intros Ha Hb H. apply (Z.pow_le_mono_r_iff a); trivial.
 apply Z.lt_le_incl; apply (Z.pow_gt_1 a); trivial.
 apply Z.lt_le_trans with (a^b); trivial. now apply Z.pow_gt_1.
Qed.

Notation Zpower_nat_Zpower := Zpower_nat_Zpower (only parsing).

Theorem Zpower2_lt_lin n : 0 <= n -> n < 2^n.
Proof. intros. now apply Z.pow_gt_lin_r. Qed.

Theorem Zpower2_le_lin n : 0 <= n -> n <= 2^n.
Proof. intros. apply Z.lt_le_incl. now apply Z.pow_gt_lin_r. Qed.

Lemma Zpower2_Psize n p :
  Zpos p < 2^(Z.of_nat n) <-> (Pos.size_nat p <= n)%nat.
Proof.
  revert p; induction n.
  destruct p; now split.
  assert (Hn := Nat2Z.is_nonneg n).
  destruct p; simpl Pos.size_nat.
  - specialize IHn with p.
    rewrite Pos2Z.inj_xI, Nat2Z.inj_succ, Z.pow_succ_r; omega.
  - specialize IHn with p.
    rewrite Pos2Z.inj_xO, Nat2Z.inj_succ, Z.pow_succ_r; omega.
  - split; auto with zarith.
    intros _. apply Z.pow_gt_1. easy.
    now rewrite Nat2Z.inj_succ, Z.lt_succ_r.
Qed.

(** * Z.pow and modulo *)

Theorem Zpower_mod p q n :
  0 < n -> (p^q) mod n = ((p mod n)^q) mod n.
Proof.
  intros Hn; destruct (Z.le_gt_cases 0 q) as [H1|H1].
  - pattern q; apply natlike_ind; trivial.
    clear q H1. intros q Hq Rec. rewrite !Z.pow_succ_r; trivial.
    rewrite Z.mul_mod_idemp_l; auto with zarith.
    rewrite Z.mul_mod, Rec, <- Z.mul_mod; auto with zarith.
  - rewrite !Z.pow_neg_r; auto with zarith.
Qed.

(** A direct way to compute Z.pow modulo **)

Fixpoint Zpow_mod_pos (a: Z)(m: positive)(n : Z) : Z :=
  match m with
   | xH => a mod n
   | xO m' =>
      let z := Zpow_mod_pos a m' n in
      match z with
       | 0 => 0
       | _ => (z * z) mod n
      end
   | xI m' =>
      let z := Zpow_mod_pos a m' n in
      match z with
       | 0 => 0
       | _ => (z * z * a) mod n
      end
  end.

Definition Zpow_mod a m n :=
  match m with
   | 0 => 1 mod n
   | Zpos p => Zpow_mod_pos a p n
   | Zneg p => 0
  end.

Theorem Zpow_mod_pos_correct a m n :
 n <> 0 -> Zpow_mod_pos a m n = (Z.pow_pos a m) mod n.
Proof.
  intros Hn. induction m.
  - rewrite Pos.xI_succ_xO at 2. rewrite <- Pos.add_1_r, <- Pos.add_diag.
    rewrite 2 Zpower_pos_is_exp, Zpower_pos_1_r.
    rewrite Z.mul_mod, (Z.mul_mod (Z.pow_pos a m)) by trivial.
    rewrite <- IHm, <- Z.mul_mod by trivial.
    simpl. now destruct (Zpow_mod_pos a m n).
  - rewrite <- Pos.add_diag at 2.
    rewrite Zpower_pos_is_exp.
    rewrite Z.mul_mod by trivial.
    rewrite <- IHm.
    simpl. now destruct (Zpow_mod_pos a m n).
  - now rewrite Zpower_pos_1_r.
Qed.

Theorem Zpow_mod_correct a m n :
 n <> 0 -> Zpow_mod a m n = (a ^ m) mod n.
Proof.
  intros Hn. destruct m; simpl.
  - trivial.
  - apply Zpow_mod_pos_correct; auto with zarith.
  - rewrite Z.mod_0_l; auto with zarith.
Qed.

(* Complements about power and number theory. *)

Lemma Zpower_divide p q : 0 < q -> (p | p ^ q).
Proof.
  exists (p^(q - 1)).
  rewrite Z.mul_comm, <- Z.pow_succ_r; f_equal; auto with zarith.
Qed.

Theorem rel_prime_Zpower_r i p q :
 0 <= i -> rel_prime p q -> rel_prime p (q^i).
Proof.
  intros Hi Hpq; pattern i; apply natlike_ind; auto with zarith.
  simpl. apply rel_prime_sym, rel_prime_1.
  clear i Hi. intros i Hi Rec; rewrite Z.pow_succ_r; auto.
  apply rel_prime_mult; auto.
Qed.

Theorem rel_prime_Zpower i j p q :
 0 <= i ->  0 <= j -> rel_prime p q -> rel_prime (p^i) (q^j).
Proof.
 intros Hi Hj H. apply rel_prime_Zpower_r; trivial.
 apply rel_prime_sym. apply rel_prime_Zpower_r; trivial.
 now apply rel_prime_sym.
Qed.

Theorem prime_power_prime p q n :
 0 <= n -> prime p -> prime q -> (p | q^n) -> p = q.
Proof.
  intros Hn Hp Hq; pattern n; apply natlike_ind; auto; clear n Hn.
  - simpl; intros.
    assert (2<=p) by (apply prime_ge_2; auto).
    assert (p<=1) by (apply Z.divide_pos_le; auto with zarith).
    omega.
  - intros n Hn Rec.
    rewrite Z.pow_succ_r by trivial. intros.
    assert (2<=p) by (apply prime_ge_2; auto).
    assert (2<=q) by (apply prime_ge_2; auto).
    destruct prime_mult with (2 := H); auto.
    apply prime_div_prime; auto.
Qed.

Theorem Zdivide_power_2 x p n :
 0 <= n -> 0 <= x -> prime p -> (x | p^n) -> exists m, x = p^m.
Proof.
  intros Hn Hx; revert p n Hn. generalize Hx.
  pattern x; apply Z_lt_induction; auto.
  clear x Hx; intros x IH Hx p n Hn Hp H.
  Z.le_elim Hx; subst.
  apply Z.le_succ_l in Hx; simpl in Hx.
  Z.le_elim Hx; subst.
  (* x > 1 *)
  case (prime_dec x); intros Hpr.
  exists 1; rewrite Z.pow_1_r; apply prime_power_prime with n; auto.
  case not_prime_divide with (2 := Hpr); auto.
  intros p1 ((Hp1, Hpq1),(q1,->)).
  assert (Hq1 : 0 < q1) by (apply Z.mul_lt_mono_pos_r with p1; auto with zarith).
  destruct (IH p1) with p n as (r1,Hr1); auto with zarith.
  transitivity (q1 * p1); trivial. exists q1; auto with zarith.
  destruct (IH q1) with p n as (r2,Hr2); auto with zarith.
  split; auto with zarith.
  rewrite <- (Z.mul_1_r q1) at 1.
  apply Z.mul_lt_mono_pos_l; auto with zarith.
  transitivity (q1 * p1); trivial. exists p1; auto with zarith.
  exists (r2 + r1); subst.
  symmetry. apply Z.pow_add_r.
  generalize Hq1; case r2; now auto with zarith.
  generalize Hp1; case r1; now auto with zarith.
  (* x = 1 *)
  exists 0; rewrite Z.pow_0_r; auto.
  (* x = 0 *)
  exists n; destruct H; rewrite Z.mul_0_r in H; auto.
Qed.

(** * Z.square: a direct definition of [z^2] *)

Notation Psquare := Pos.square (compat "8.3").
Notation Zsquare := Z.square (compat "8.3").
Notation Psquare_correct := Pos.square_spec (compat "8.3").
Notation Zsquare_correct := Z.square_spec (compat "8.3").
