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(** Properties of positive numbers, mapping between Z and nat                *)
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Definition Z2positive (z : Z) :=
  match z with
  | Zpos p => p
  | Zneg p => p
  | Z0 => 1%positive
  end.

(* Specifications: *)
Lemma ZL9 : forall p : positive, Z_of_nat (nat_of_P p) = Zpos p.
Lemma ZERO_lt_POS : forall p : positive, (0 < Zpos p)%Z.
Lemma POS_neq_ZERO : forall p : positive, Zpos p <> 0%Z.
Lemma NEG_neq_ZERO : forall p : positive, Zneg p <> 0%Z.
Lemma POS_resp_eq : forall p0 p1 : positive, Zpos p0 = Zpos p1 -> p0 = p1.
Lemma nat_nat_pos : forall m n : nat, ((m + 1) * (n + 1) > 0)%Z.
Theorem S_predn : forall m : nat, m <> 0 -> S (pred m) = m.
Lemma absolu_1 : forall x : Z, Z.abs_nat x = 0 -> x = 0%Z.
Lemma absolu_2 : forall x : Z, x <> 0%Z -> Z.abs_nat x <> 0.
Lemma absolu_inject_nat : forall n : nat, Z.abs_nat (Z_of_nat n) = n.
Lemma eq_inj : forall m n : nat, m = n :>Z -> m = n.
Lemma lt_inj : forall m n : nat, (m < n)%Z -> m < n.
Lemma le_inj : forall m n : nat, (m <= n)%Z -> m <= n.
Lemma inject_nat_S_inf : forall x : Z, (0 < x)%Z -> {n : nat | x = S n}.
Lemma le_absolu : forall x y : Z, (0 <= x)%Z -> (0 <= y)%Z -> (x <= y)%Z -> Z.abs_nat x <= Z.abs_nat y.
Lemma lt_absolu : forall x y : Z, (0 <= x)%Z -> (0 <= y)%Z -> (x < y)%Z -> Z.abs_nat x < Z.abs_nat y.
Lemma absolu_plus : forall x y : Z, (0 <= x)%Z -> (0 <= y)%Z -> Z.abs_nat (x + y) = Z.abs_nat x + Z.abs_nat y.
Lemma pred_absolu : forall x : Z, (0 < x)%Z -> pred (Z.abs_nat x) = Z.abs_nat (x - 1).