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Diffstat (limited to 'theories/Numbers/Natural/Abstract/NMulOrder.v')
-rw-r--r-- | theories/Numbers/Natural/Abstract/NMulOrder.v | 131 |
1 files changed, 131 insertions, 0 deletions
diff --git a/theories/Numbers/Natural/Abstract/NMulOrder.v b/theories/Numbers/Natural/Abstract/NMulOrder.v new file mode 100644 index 00000000..aa21fb50 --- /dev/null +++ b/theories/Numbers/Natural/Abstract/NMulOrder.v @@ -0,0 +1,131 @@ +(************************************************************************) +(* 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: NMulOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export NAddOrder. + +Module NMulOrderPropFunct (Import NAxiomsMod : NAxiomsSig). +Module Export NAddOrderPropMod := NAddOrderPropFunct NAxiomsMod. +Open Local Scope NatScope. + +Theorem mul_lt_pred : + forall p q n m : N, S p == q -> (p * n < p * m <-> q * n + m < q * m + n). +Proof NZmul_lt_pred. + +Theorem mul_lt_mono_pos_l : forall p n m : N, 0 < p -> (n < m <-> p * n < p * m). +Proof NZmul_lt_mono_pos_l. + +Theorem mul_lt_mono_pos_r : forall p n m : N, 0 < p -> (n < m <-> n * p < m * p). +Proof NZmul_lt_mono_pos_r. + +Theorem mul_cancel_l : forall n m p : N, p ~= 0 -> (p * n == p * m <-> n == m). +Proof NZmul_cancel_l. + +Theorem mul_cancel_r : forall n m p : N, p ~= 0 -> (n * p == m * p <-> n == m). +Proof NZmul_cancel_r. + +Theorem mul_id_l : forall n m : N, m ~= 0 -> (n * m == m <-> n == 1). +Proof NZmul_id_l. + +Theorem mul_id_r : forall n m : N, n ~= 0 -> (n * m == n <-> m == 1). +Proof NZmul_id_r. + +Theorem mul_le_mono_pos_l : forall n m p : N, 0 < p -> (n <= m <-> p * n <= p * m). +Proof NZmul_le_mono_pos_l. + +Theorem mul_le_mono_pos_r : forall n m p : N, 0 < p -> (n <= m <-> n * p <= m * p). +Proof NZmul_le_mono_pos_r. + +Theorem mul_pos_pos : forall n m : N, 0 < n -> 0 < m -> 0 < n * m. +Proof NZmul_pos_pos. + +Theorem lt_1_mul_pos : forall n m : N, 1 < n -> 0 < m -> 1 < n * m. +Proof NZlt_1_mul_pos. + +Theorem eq_mul_0 : forall n m : N, n * m == 0 <-> n == 0 \/ m == 0. +Proof NZeq_mul_0. + +Theorem neq_mul_0 : forall n m : N, n ~= 0 /\ m ~= 0 <-> n * m ~= 0. +Proof NZneq_mul_0. + +Theorem eq_square_0 : forall n : N, n * n == 0 <-> n == 0. +Proof NZeq_square_0. + +Theorem eq_mul_0_l : forall n m : N, n * m == 0 -> m ~= 0 -> n == 0. +Proof NZeq_mul_0_l. + +Theorem eq_mul_0_r : forall n m : N, n * m == 0 -> n ~= 0 -> m == 0. +Proof NZeq_mul_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 mul_2_mono_l : forall n m : N, n < m -> 1 + (1 + 1) * n < (1 + 1) * m. +Proof NZmul_2_mono_l. + +(* Theorems that are either not valid on Z or have different proofs on N and Z *) + +Theorem mul_le_mono_l : forall n m p : N, n <= m -> p * n <= p * m. +Proof. +intros; apply NZmul_le_mono_nonneg_l. apply le_0_l. assumption. +Qed. + +Theorem mul_le_mono_r : forall n m p : N, n <= m -> n * p <= m * p. +Proof. +intros; apply NZmul_le_mono_nonneg_r. apply le_0_l. assumption. +Qed. + +Theorem mul_lt_mono : forall n m p q : N, n < m -> p < q -> n * p < m * q. +Proof. +intros; apply NZmul_lt_mono_nonneg; try assumption; apply le_0_l. +Qed. + +Theorem mul_le_mono : forall n m p q : N, n <= m -> p <= q -> n * p <= m * q. +Proof. +intros; apply NZmul_le_mono_nonneg; try assumption; apply le_0_l. +Qed. + +Theorem lt_0_mul : forall n m : N, n * m > 0 <-> n > 0 /\ m > 0. +Proof. +intros n m; split; [intro H | intros [H1 H2]]. +apply -> NZlt_0_mul in H. destruct H as [[H1 H2] | [H1 H2]]. now split. false_hyp H1 nlt_0_r. +now apply NZmul_pos_pos. +Qed. + +Notation mul_pos := lt_0_mul (only parsing). + +Theorem eq_mul_1 : forall n m : N, n * m == 1 <-> n == 1 /\ m == 1. +Proof. +intros n m. +split; [| intros [H1 H2]; now rewrite H1, H2, mul_1_l]. +intro H; destruct (NZlt_trichotomy n 1) as [H1 | [H1 | H1]]. +apply -> lt_1_r in H1. rewrite H1, mul_0_l in H. false_hyp H neq_0_succ. +rewrite H1, mul_1_l in H; now split. +destruct (eq_0_gt_0_cases m) as [H2 | H2]. +rewrite H2, mul_0_r in H; false_hyp H neq_0_succ. +apply -> (mul_lt_mono_pos_r m) in H1; [| assumption]. rewrite mul_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 NMulOrderPropFunct. + |