diff options
Diffstat (limited to 'theories/Arith/Gt.v')
| -rw-r--r-- | theories/Arith/Gt.v | 20 |
1 files changed, 13 insertions, 7 deletions
diff --git a/theories/Arith/Gt.v b/theories/Arith/Gt.v index f693298e9..c2ed678cb 100644 --- a/theories/Arith/Gt.v +++ b/theories/Arith/Gt.v @@ -8,6 +8,12 @@ (*i $Id$ i*) +(** Theorems about [gt] in [nat]. [gt] is defined in [Init/Peano.v] as: +<< +Definition gt (n m:nat) := m < n. +>> +*) + Require Import Le. Require Import Lt. Require Import Plus. @@ -15,7 +21,7 @@ Open Local Scope nat_scope. Implicit Types m n p : nat. -(** Order and successor *) +(** * Order and successor *) Theorem gt_Sn_O : forall n, S n > 0. Proof. @@ -52,20 +58,20 @@ Proof. Qed. Hint Immediate gt_pred: arith v62. -(** Irreflexivity *) +(** * Irreflexivity *) Lemma gt_irrefl : forall n, ~ n > n. Proof lt_irrefl. Hint Resolve gt_irrefl: arith v62. -(** Asymmetry *) +(** * Asymmetry *) Lemma gt_asym : forall n m, n > m -> ~ m > n. Proof fun n m => lt_asym m n. Hint Resolve gt_asym: arith v62. -(** Relating strict and large orders *) +(** * Relating strict and large orders *) Lemma le_not_gt : forall n m, n <= m -> ~ n > m. Proof le_not_lt. @@ -102,7 +108,7 @@ Proof. Qed. Hint Resolve le_gt_S: arith v62. -(** Transitivity *) +(** * Transitivity *) Theorem le_gt_trans : forall n m p, m <= n -> m > p -> n > p. Proof. @@ -127,14 +133,14 @@ Qed. Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith v62. -(** Comparison to 0 *) +(** * Comparison to 0 *) Theorem gt_O_eq : forall n, n > 0 \/ 0 = n. Proof. intro n; apply gt_S; auto with arith. Qed. -(** Simplification and compatibility *) +(** * Simplification and compatibility *) Lemma plus_gt_reg_l : forall n m p, p + n > p + m -> n > m. Proof. |