# Library Coq.ZArith.auxiliary

Binary Integers (Pierre Crégut, CNET, Lannion, France)
Require Export Arith_base.
Require Import BinInt.
Require Import Zorder.
Require Import Decidable.
Require Import Peano_dec.
Require Export Compare_dec.
Open Local Scope Z_scope.

# Moving terms from one side to the other of an inequality

Theorem Zne_left : forall n m:Z, Zne n m -> Zne (n + - m) 0.
Theorem Zegal_left : forall n m:Z, n = m -> n + - m = 0.
Theorem Zle_left : forall n m:Z, n <= m -> 0 <= m + - n.
Theorem Zle_left_rev : forall n m:Z, 0 <= m + - n -> n <= m.
Theorem Zlt_left_rev : forall n m:Z, 0 < m + - n -> n < m.
Theorem Zlt_left : forall n m:Z, n < m -> 0 <= m + -1 + - n.
Theorem Zlt_left_lt : forall n m:Z, n < m -> 0 < m + - n.
Theorem Zge_left : forall n m:Z, n >= m -> 0 <= n + - m.
Theorem Zgt_left : forall n m:Z, n > m -> 0 <= n + -1 + - m.
Theorem Zgt_left_gt : forall n m:Z, n > m -> n + - m > 0.
Theorem Zgt_left_rev : forall n m:Z, n + - m > 0 -> n > m.
Theorem Zle_mult_approx :
forall n m p:Z, n > 0 -> p > 0 -> 0 <= m -> 0 <= m * n + p.
Theorem Zmult_le_approx :
forall n m p:Z, n > 0 -> n > p -> 0 <= m * n + p -> 0 <= m.