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Diffstat (limited to 'theories/Numbers/Natural/Abstract/NAddOrder.v')
-rw-r--r-- | theories/Numbers/Natural/Abstract/NAddOrder.v | 114 |
1 files changed, 114 insertions, 0 deletions
diff --git a/theories/Numbers/Natural/Abstract/NAddOrder.v b/theories/Numbers/Natural/Abstract/NAddOrder.v new file mode 100644 index 00000000..7024fd00 --- /dev/null +++ b/theories/Numbers/Natural/Abstract/NAddOrder.v @@ -0,0 +1,114 @@ +(************************************************************************) +(* 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: NAddOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export NOrder. + +Module NAddOrderPropFunct (Import NAxiomsMod : NAxiomsSig). +Module Export NOrderPropMod := NOrderPropFunct NAxiomsMod. +Open Local Scope NatScope. + +Theorem add_lt_mono_l : forall n m p : N, n < m <-> p + n < p + m. +Proof NZadd_lt_mono_l. + +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. +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. +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. +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. +Proof. +intros; apply NZadd_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. +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). +Qed. + +End NAddOrderPropFunct. |