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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
(*i $Id$ i*)
Require Import NZAxioms.
Require Import NZOrder.
Module NZPlusOrderPropFunct (Import NZOrdAxiomsMod : NZOrdAxiomsSig).
Module Export NZOrderPropMod := NZOrderPropFunct NZOrdAxiomsMod.
Open Local Scope NatIntScope.
Theorem NZplus_lt_mono_l : forall n m p : NZ, n < m <-> p + n < p + m.
Proof.
intros n m p; NZinduct p.
now do 2 rewrite NZplus_0_l.
intro p. do 2 rewrite NZplus_succ_l. now rewrite <- NZsucc_lt_mono.
Qed.
Theorem NZplus_lt_mono_r : forall n m p : NZ, n < m <-> n + p < m + p.
Proof.
intros n m p.
rewrite (NZplus_comm n p); rewrite (NZplus_comm m p); apply NZplus_lt_mono_l.
Qed.
Theorem NZplus_lt_mono : forall n m p q : NZ, n < m -> p < q -> n + p < m + q.
Proof.
intros n m p q H1 H2.
apply NZlt_trans with (m + p);
[now apply -> NZplus_lt_mono_r | now apply -> NZplus_lt_mono_l].
Qed.
Theorem NZplus_le_mono_l : forall n m p : NZ, n <= m <-> p + n <= p + m.
Proof.
intros n m p; NZinduct p.
now do 2 rewrite NZplus_0_l.
intro p. do 2 rewrite NZplus_succ_l. now rewrite <- NZsucc_le_mono.
Qed.
Theorem NZplus_le_mono_r : forall n m p : NZ, n <= m <-> n + p <= m + p.
Proof.
intros n m p.
rewrite (NZplus_comm n p); rewrite (NZplus_comm m p); apply NZplus_le_mono_l.
Qed.
Theorem NZplus_le_mono : forall n m p q : NZ, n <= m -> p <= q -> n + p <= m + q.
Proof.
intros n m p q H1 H2.
apply NZle_trans with (m + p);
[now apply -> NZplus_le_mono_r | now apply -> NZplus_le_mono_l].
Qed.
Theorem NZplus_lt_le_mono : forall n m p q : NZ, n < m -> p <= q -> n + p < m + q.
Proof.
intros n m p q H1 H2.
apply NZlt_le_trans with (m + p);
[now apply -> NZplus_lt_mono_r | now apply -> NZplus_le_mono_l].
Qed.
Theorem NZplus_le_lt_mono : forall n m p q : NZ, n <= m -> p < q -> n + p < m + q.
Proof.
intros n m p q H1 H2.
apply NZle_lt_trans with (m + p);
[now apply -> NZplus_le_mono_r | now apply -> NZplus_lt_mono_l].
Qed.
Theorem NZplus_pos_pos : forall n m : NZ, 0 < n -> 0 < m -> 0 < n + m.
Proof.
intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_lt_mono.
Qed.
Theorem NZplus_pos_nonneg : forall n m : NZ, 0 < n -> 0 <= m -> 0 < n + m.
Proof.
intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_lt_le_mono.
Qed.
Theorem NZplus_nonneg_pos : forall n m : NZ, 0 <= n -> 0 < m -> 0 < n + m.
Proof.
intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_le_lt_mono.
Qed.
Theorem NZplus_nonneg_nonneg : forall n m : NZ, 0 <= n -> 0 <= m -> 0 <= n + m.
Proof.
intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_le_mono.
Qed.
Theorem NZlt_plus_pos_l : forall n m : NZ, 0 < n -> m < n + m.
Proof.
intros n m H. apply -> (NZplus_lt_mono_r 0 n m) in H.
now rewrite NZplus_0_l in H.
Qed.
Theorem NZlt_plus_pos_r : forall n m : NZ, 0 < n -> m < m + n.
Proof.
intros; rewrite NZplus_comm; now apply NZlt_plus_pos_l.
Qed.
Theorem NZle_lt_plus_lt : forall n m p q : NZ, n <= m -> p + m < q + n -> p < q.
Proof.
intros n m p q H1 H2. destruct (NZle_gt_cases q p); [| assumption].
pose proof (NZplus_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H2.
false_hyp H3 H2.
Qed.
Theorem NZlt_le_plus_lt : forall n m p q : NZ, n < m -> p + m <= q + n -> p < q.
Proof.
intros n m p q H1 H2. destruct (NZle_gt_cases q p); [| assumption].
pose proof (NZplus_le_lt_mono q p n m H H1) as H3. apply <- NZnle_gt in H3.
false_hyp H2 H3.
Qed.
Theorem NZle_le_plus_le : forall n m p q : NZ, n <= m -> p + m <= q + n -> p <= q.
Proof.
intros n m p q H1 H2. destruct (NZle_gt_cases p q); [assumption |].
pose proof (NZplus_lt_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H3.
false_hyp H2 H3.
Qed.
Theorem NZplus_lt_cases : forall n m p q : NZ, n + m < p + q -> n < p \/ m < q.
Proof.
intros n m p q H;
destruct (NZle_gt_cases p n) as [H1 | H1].
destruct (NZle_gt_cases q m) as [H2 | H2].
pose proof (NZplus_le_mono p n q m H1 H2) as H3. apply -> NZle_ngt in H3.
false_hyp H H3.
now right. now left.
Qed.
Theorem NZplus_le_cases : forall n m p q : NZ, n + m <= p + q -> n <= p \/ m <= q.
Proof.
intros n m p q H.
destruct (NZle_gt_cases n p) as [H1 | H1]. now left.
destruct (NZle_gt_cases m q) as [H2 | H2]. now right.
assert (H3 : p + q < n + m) by now apply NZplus_lt_mono.
apply -> NZle_ngt in H. false_hyp H3 H.
Qed.
Theorem NZplus_neg_cases : forall n m : NZ, n + m < 0 -> n < 0 \/ m < 0.
Proof.
intros n m H; apply NZplus_lt_cases; now rewrite NZplus_0_l.
Qed.
Theorem NZplus_pos_cases : forall n m : NZ, 0 < n + m -> 0 < n \/ 0 < m.
Proof.
intros n m H; apply NZplus_lt_cases; now rewrite NZplus_0_l.
Qed.
Theorem NZplus_nonpos_cases : forall n m : NZ, n + m <= 0 -> n <= 0 \/ m <= 0.
Proof.
intros n m H; apply NZplus_le_cases; now rewrite NZplus_0_l.
Qed.
Theorem NZplus_nonneg_cases : forall n m : NZ, 0 <= n + m -> 0 <= n \/ 0 <= m.
Proof.
intros n m H; apply NZplus_le_cases; now rewrite NZplus_0_l.
Qed.
End NZPlusOrderPropFunct.
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