Library Coq.Numbers.NatInt.NZAddOrder



Require Import NZAxioms.
Require Import NZOrder.

Module NZAddOrderPropFunct (Import NZOrdAxiomsMod : NZOrdAxiomsSig).
Module Export NZOrderPropMod := NZOrderPropFunct NZOrdAxiomsMod.
Open Local Scope NatIntScope.

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


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

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

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


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

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

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

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

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


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


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


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


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


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

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



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



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



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



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




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

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

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

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

End NZAddOrderPropFunct.