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Diffstat (limited to 'theories/Numbers/Natural/Abstract/NAddOrder.v')
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NAddOrder.v | 88 |
1 files changed, 11 insertions, 77 deletions
diff --git a/theories/Numbers/Natural/Abstract/NAddOrder.v b/theories/Numbers/Natural/Abstract/NAddOrder.v index 7024fd00..0ce04e54 100644 --- a/theories/Numbers/Natural/Abstract/NAddOrder.v +++ b/theories/Numbers/Natural/Abstract/NAddOrder.v @@ -8,107 +8,41 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: NAddOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*) +(*i $Id$ i*) Require Export NOrder. -Module NAddOrderPropFunct (Import NAxiomsMod : NAxiomsSig). -Module Export NOrderPropMod := NOrderPropFunct NAxiomsMod. -Open Local Scope NatScope. +Module NAddOrderPropFunct (Import N : NAxiomsSig'). +Include NOrderPropFunct N. -Theorem add_lt_mono_l : forall n m p : N, n < m <-> p + n < p + m. -Proof NZadd_lt_mono_l. +(** Theorems true for natural numbers, not for integers *) -Theorem add_lt_mono_r : forall n m p : N, n < m <-> n + p < m + p. -Proof NZadd_lt_mono_r. - -Theorem add_lt_mono : forall n m p q : N, n < m -> p < q -> n + p < m + q. -Proof NZadd_lt_mono. - -Theorem add_le_mono_l : forall n m p : N, n <= m <-> p + n <= p + m. -Proof NZadd_le_mono_l. - -Theorem add_le_mono_r : forall n m p : N, n <= m <-> n + p <= m + p. -Proof NZadd_le_mono_r. - -Theorem add_le_mono : forall n m p q : N, n <= m -> p <= q -> n + p <= m + q. -Proof NZadd_le_mono. - -Theorem add_lt_le_mono : forall n m p q : N, n < m -> p <= q -> n + p < m + q. -Proof NZadd_lt_le_mono. - -Theorem add_le_lt_mono : forall n m p q : N, n <= m -> p < q -> n + p < m + q. -Proof NZadd_le_lt_mono. - -Theorem add_pos_pos : forall n m : N, 0 < n -> 0 < m -> 0 < n + m. -Proof NZadd_pos_pos. - -Theorem lt_add_pos_l : forall n m : N, 0 < n -> m < n + m. -Proof NZlt_add_pos_l. - -Theorem lt_add_pos_r : forall n m : N, 0 < n -> m < m + n. -Proof NZlt_add_pos_r. - -Theorem le_lt_add_lt : forall n m p q : N, n <= m -> p + m < q + n -> p < q. -Proof NZle_lt_add_lt. - -Theorem lt_le_add_lt : forall n m p q : N, n < m -> p + m <= q + n -> p < q. -Proof NZlt_le_add_lt. - -Theorem le_le_add_le : forall n m p q : N, n <= m -> p + m <= q + n -> p <= q. -Proof NZle_le_add_le. - -Theorem add_lt_cases : forall n m p q : N, n + m < p + q -> n < p \/ m < q. -Proof NZadd_lt_cases. - -Theorem add_le_cases : forall n m p q : N, n + m <= p + q -> n <= p \/ m <= q. -Proof NZadd_le_cases. - -Theorem add_pos_cases : forall n m : N, 0 < n + m -> 0 < n \/ 0 < m. -Proof NZadd_pos_cases. - -(* Theorems true for natural numbers *) - -Theorem le_add_r : forall n m : N, n <= n + m. +Theorem le_add_r : forall n m, n <= n + m. Proof. intro n; induct m. rewrite add_0_r; now apply eq_le_incl. intros m IH. rewrite add_succ_r; now apply le_le_succ_r. Qed. -Theorem lt_lt_add_r : forall n m p : N, n < m -> n < m + p. +Theorem lt_lt_add_r : forall n m p, n < m -> n < m + p. Proof. intros n m p H; rewrite <- (add_0_r n). apply add_lt_le_mono; [assumption | apply le_0_l]. Qed. -Theorem lt_lt_add_l : forall n m p : N, n < m -> n < p + m. +Theorem lt_lt_add_l : forall n m p, n < m -> n < p + m. Proof. intros n m p; rewrite add_comm; apply lt_lt_add_r. Qed. -Theorem add_pos_l : forall n m : N, 0 < n -> 0 < n + m. +Theorem add_pos_l : forall n m, 0 < n -> 0 < n + m. Proof. -intros; apply NZadd_pos_nonneg. assumption. apply le_0_l. +intros; apply add_pos_nonneg. assumption. apply le_0_l. Qed. -Theorem add_pos_r : forall n m : N, 0 < m -> 0 < n + m. -Proof. -intros; apply NZadd_nonneg_pos. apply le_0_l. assumption. -Qed. - -(* The following property is used to prove the correctness of the -definition of order on integers constructed from pairs of natural numbers *) - -Theorem add_lt_repl_pair : forall n m n' m' u v : N, - n + u < m + v -> n + m' == n' + m -> n' + u < m' + v. +Theorem add_pos_r : forall n m, 0 < m -> 0 < n + m. Proof. -intros n m n' m' u v H1 H2. -symmetry in H2. assert (H3 : n' + m <= n + m') by now apply eq_le_incl. -pose proof (add_lt_le_mono _ _ _ _ H1 H3) as H4. -rewrite (add_shuffle2 n u), (add_shuffle1 m v), (add_comm m n) in H4. -do 2 rewrite <- add_assoc in H4. do 2 apply <- add_lt_mono_l in H4. -now rewrite (add_comm n' u), (add_comm m' v). +intros; apply add_nonneg_pos. apply le_0_l. assumption. Qed. End NAddOrderPropFunct. |