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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 i*)
Require Export NPlusOrder.
Module NTimesOrderPropFunct (Import NAxiomsMod : NAxiomsSig).
Module Export NPlusOrderPropMod := NPlusOrderPropFunct NAxiomsMod.
Open Local Scope NatScope.
Theorem times_lt_pred :
forall p q n m : N, S p == q -> (p * n < p * m <-> q * n + m < q * m + n).
Proof NZtimes_lt_pred.
Theorem times_lt_mono_pos_l : forall p n m : N, 0 < p -> (n < m <-> p * n < p * m).
Proof NZtimes_lt_mono_pos_l.
Theorem times_lt_mono_pos_r : forall p n m : N, 0 < p -> (n < m <-> n * p < m * p).
Proof NZtimes_lt_mono_pos_r.
Theorem times_cancel_l : forall n m p : N, p ~= 0 -> (p * n == p * m <-> n == m).
Proof NZtimes_cancel_l.
Theorem times_cancel_r : forall n m p : N, p ~= 0 -> (n * p == m * p <-> n == m).
Proof NZtimes_cancel_r.
Theorem times_le_mono_pos_l : forall n m p : N, 0 < p -> (n <= m <-> p * n <= p * m).
Proof NZtimes_le_mono_pos_l.
Theorem times_le_mono_pos_r : forall n m p : N, 0 < p -> (n <= m <-> n * p <= m * p).
Proof NZtimes_le_mono_pos_r.
Theorem times_pos_pos : forall n m : N, 0 < n -> 0 < m -> 0 < n * m.
Proof NZtimes_pos_pos.
Theorem lt_1_times_pos : forall n m : N, 1 < n -> 0 < m -> 1 < n * m.
Proof NZlt_1_times_pos.
Theorem eq_times_0 : forall n m : N, n * m == 0 <-> n == 0 \/ m == 0.
Proof NZeq_times_0.
Theorem neq_times_0 : forall n m : N, n ~= 0 /\ m ~= 0 <-> n * m ~= 0.
Proof NZneq_times_0.
Theorem eq_times_0_l : forall n m : N, n * m == 0 -> m ~= 0 -> n == 0.
Proof NZeq_times_0_l.
Theorem eq_times_0_r : forall n m : N, n * m == 0 -> n ~= 0 -> m == 0.
Proof NZeq_times_0_r.
Theorem square_lt_mono : forall n m : N, n < m <-> n * n < m * m.
Proof.
intros n m; split; intro;
[apply NZsquare_lt_mono_nonneg | apply NZsquare_lt_simpl_nonneg];
try assumption; apply le_0_l.
Qed.
Theorem square_le_mono : forall n m : N, n <= m <-> n * n <= m * m.
Proof.
intros n m; split; intro;
[apply NZsquare_le_mono_nonneg | apply NZsquare_le_simpl_nonneg];
try assumption; apply le_0_l.
Qed.
Theorem times_2_mono_l : forall n m : N, n < m -> 1 + (1 + 1) * n < (1 + 1) * m.
Proof NZtimes_2_mono_l.
(** Theorems that are either not valid on Z or have different proofs on N and Z *)
Theorem times_le_mono_l : forall n m p : N, n <= m -> p * n <= p * m.
Proof.
intros; apply NZtimes_le_mono_nonneg_l. apply le_0_l. assumption.
Qed.
Theorem times_le_mono_r : forall n m p : N, n <= m -> n * p <= m * p.
Proof.
intros; apply NZtimes_le_mono_nonneg_r. apply le_0_l. assumption.
Qed.
Theorem times_lt_mono : forall n m p q : N, n < m -> p < q -> n * p < m * q.
Proof.
intros; apply NZtimes_lt_mono_nonneg; try assumption; apply le_0_l.
Qed.
Theorem times_le_mono : forall n m p q : N, n <= m -> p <= q -> n * p <= m * q.
Proof.
intros; apply NZtimes_le_mono_nonneg; try assumption; apply le_0_l.
Qed.
Theorem lt_0_times : forall n m : N, n * m > 0 <-> n > 0 /\ m > 0.
Proof.
intros n m; split; [intro H | intros [H1 H2]].
apply -> NZlt_0_times in H. destruct H as [[H1 H2] | [H1 H2]]. now split. false_hyp H1 nlt_0_r.
now apply NZtimes_pos_pos.
Qed.
Notation times_pos := lt_0_times (only parsing).
Theorem eq_times_1 : forall n m : N, n * m == 1 <-> n == 1 /\ m == 1.
Proof.
intros n m.
split; [| intros [H1 H2]; now rewrite H1, H2, times_1_l].
intro H; destruct (NZlt_trichotomy n 1) as [H1 | [H1 | H1]].
apply -> lt_1_r in H1. rewrite H1, times_0_l in H. false_hyp H neq_0_succ.
rewrite H1, times_1_l in H; now split.
destruct (eq_0_gt_0_cases m) as [H2 | H2].
rewrite H2, times_0_r in H; false_hyp H neq_0_succ.
apply -> (times_lt_mono_pos_r m) in H1; [| assumption]. rewrite times_1_l in H1.
assert (H3 : 1 < n * m) by now apply (lt_1_l 0 m).
rewrite H in H3; false_hyp H3 lt_irrefl.
Qed.
End NTimesOrderPropFunct.
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