Require Export Arith_base.
Require Import BinInt.
Require Import Zorder.
Require Import Decidable.
Require Import Peano_dec.
Require Export Compare_dec.

Local Open Scope Z_scope.

Theorem Zne_left n m : Zne n m -> Zne (n + - m) 0.

Theorem Zegal_left n m : n = m -> n + - m = 0.

Theorem Zle_left n m : n <= m -> 0 <= m + - n.

Theorem Zle_left_rev n m : 0 <= m + - n -> n <= m.

Theorem Zlt_left_rev n m : 0 < m + - n -> n < m.

Theorem Zlt_left_lt n m : n < m -> 0 < m + - n.

Theorem Zlt_left n m : n < m -> 0 <= m + -1 + - n.

Theorem Zge_left n m : n >= m -> 0 <= n + - m.

Theorem Zgt_left n m : n > m -> 0 <= n + -1 + - m.

Theorem Zgt_left_gt n m : n > m -> n + - m > 0.

Theorem Zgt_left_rev n m : n + - m > 0 -> n > m.

Theorem Zle_mult_approx n m p :
  n > 0 -> p > 0 -> 0 <= m -> 0 <= m * n + p.

Theorem Zmult_le_approx n m p :
  n > 0 -> n > p -> 0 <= m * n + p -> 0 <= m.