Require Import NZAxioms NZBase NZMul NZOrder.

Module Type NZAddOrderProp (Import NZ : NZOrdAxiomsSig').
Include NZBaseProp NZ <+ NZMulProp NZ <+ NZOrderProp NZ.

Theorem add_lt_mono_l : forall n m p, n < m <-> p + n < p + m.

Theorem add_lt_mono_r : forall n m p, n < m <-> n + p < m + p.

Theorem add_lt_mono : forall n m p q, n < m -> p < q -> n + p < m + q.

Theorem add_le_mono_l : forall n m p, n <= m <-> p + n <= p + m.

Theorem add_le_mono_r : forall n m p, n <= m <-> n + p <= m + p.

Theorem add_le_mono : forall n m p q, n <= m -> p <= q -> n + p <= m + q.

Theorem add_lt_le_mono : forall n m p q, n < m -> p <= q -> n + p < m + q.

Theorem add_le_lt_mono : forall n m p q, n <= m -> p < q -> n + p < m + q.

Theorem add_pos_pos : forall n m, 0 < n -> 0 < m -> 0 < n + m.

Theorem add_pos_nonneg : forall n m, 0 < n -> 0 <= m -> 0 < n + m.

Theorem add_nonneg_pos : forall n m, 0 <= n -> 0 < m -> 0 < n + m.

Theorem add_nonneg_nonneg : forall n m, 0 <= n -> 0 <= m -> 0 <= n + m.

Theorem lt_add_pos_l : forall n m, 0 < n -> m < n + m.

Theorem lt_add_pos_r : forall n m, 0 < n -> m < m + n.

Theorem le_lt_add_lt : forall n m p q, n <= m -> p + m < q + n -> p < q.

Theorem lt_le_add_lt : forall n m p q, n < m -> p + m <= q + n -> p < q.

Theorem le_le_add_le : forall n m p q, n <= m -> p + m <= q + n -> p <= q.

Theorem add_lt_cases : forall n m p q, n + m < p + q -> n < p \/ m < q.

Theorem add_le_cases : forall n m p q, n + m <= p + q -> n <= p \/ m <= q.

Theorem add_neg_cases : forall n m, n + m < 0 -> n < 0 \/ m < 0.

Theorem add_pos_cases : forall n m, 0 < n + m -> 0 < n \/ 0 < m.

Theorem add_nonpos_cases : forall n m, n + m <= 0 -> n <= 0 \/ m <= 0.

Theorem add_nonneg_cases : forall n m, 0 <= n + m -> 0 <= n \/ 0 <= m.

Subtraction
We can prove the existence of a subtraction of any number by a smaller one

Lemma le_exists_sub : forall n m, n<=m -> exists p, m == p+n /\ 0<=p.

For the moment, it doesn't seem possible to relate this existing subtraction with sub.