NAddOrder.v

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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
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(*                      Evgeny Makarov, INRIA, 2007                     *)
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Require Export NOrder.

Module NAddOrderProp (Import N : NAxiomsMiniSig').
Include NOrderProp N.

Theorems true for natural numbers, not for integers

Theorem le_add_r : forall n m, n <= n + m.forall n m : t, n <= n + m
Proof.forall n m : t, n <= n + m
intro n; induct m.n:t
n <= n + 0
n:t
forall n0 : t, n <= n + n0 -> n <= n + S n0
rewrite add_0_r; now apply eq_le_incl.n:t
forall n0 : t, n <= n + n0 -> n <= n + S n0
intros m IH.n, m:t
IH:n <= n + m
n <= n + S m rewrite add_succ_r; now apply le_le_succ_r.
Qed.

Theorem lt_lt_add_r : forall n m p, n < m -> n < m + p.forall n m p : t, n < m -> n < m + p
Proof.forall n m p : t, n < m -> n < m + p
intros n m p H; rewrite <- (add_0_r n).n, m, p:t
H:n < m
n + 0 < m + p
apply add_lt_le_mono; [assumption | apply le_0_l].
Qed.

Theorem lt_lt_add_l : forall n m p, n < m -> n < p + m.forall n m p : t, n < m -> n < p + m
Proof.forall n m p : t, n < m -> n < p + m
intros n m p; rewrite add_comm; apply lt_lt_add_r.
Qed.

Theorem add_pos_l : forall n m, 0 < n -> 0 < n + m.forall n m : t, 0 < n -> 0 < n + m
Proof.forall n m : t, 0 < n -> 0 < n + m
intros; apply add_pos_nonneg.n, m:t
H:0 < n
0 < n
n, m:t
H:0 < n
0 <= m assumption.n, m:t
H:0 < n
0 <= m apply le_0_l.
Qed.

Theorem add_pos_r : forall n m, 0 < m -> 0 < n + m.forall n m : t, 0 < m -> 0 < n + m
Proof.forall n m : t, 0 < m -> 0 < n + m
intros; apply add_nonneg_pos.n, m:t
H:0 < m
0 <= n
n, m:t
H:0 < m
0 < m apply le_0_l.n, m:t
H:0 < m
0 < m assumption.
Qed.

End NAddOrderProp.