# Library Coq.Arith.Gt

``` ```
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. Open Local Scope nat_scope. Implicit Types m n p : nat. ```

# Order and successor

``` Theorem gt_Sn_O : forall n, S n > 0. Proof.   auto with arith. Qed. Hint Resolve gt_Sn_O: arith v62. Theorem gt_Sn_n : forall n, S n > n. Proof.   auto with arith. Qed. Hint Resolve gt_Sn_n: arith v62. Theorem gt_n_S : forall n m, n > m -> S n > S m. Proof.   auto with arith. Qed. Hint Resolve gt_n_S: arith v62. Lemma gt_S_n : forall n m, S m > S n -> m > n. Proof.   auto with arith. Qed. Hint Immediate gt_S_n: arith v62. Theorem gt_S : forall n m, S n > m -> n > m \/ m = n. Proof.   intros n m H; unfold gt in |- *; apply le_lt_or_eq; auto with arith. Qed. Lemma gt_pred : forall n m, m > S n -> pred m > n. Proof.   auto with arith. Qed. Hint Immediate gt_pred: arith v62. ```

# Irreflexivity

``` Lemma gt_irrefl : forall n, ~ n > n. Proof lt_irrefl. Hint Resolve gt_irrefl: arith v62. ```

# 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

``` Lemma le_not_gt : forall n m, n <= m -> ~ n > m. Proof le_not_lt. Hint Resolve le_not_gt: arith v62. Lemma gt_not_le : forall n m, n > m -> ~ n <= m. Proof. auto with arith. Qed. Hint Resolve gt_not_le: arith v62. Theorem le_S_gt : forall n m, S n <= m -> m > n. Proof.   auto with arith. Qed. Hint Immediate le_S_gt: arith v62. Lemma gt_S_le : forall n m, S m > n -> n <= m. Proof.   intros n p; exact (lt_n_Sm_le n p). Qed. Hint Immediate gt_S_le: arith v62. Lemma gt_le_S : forall n m, m > n -> S n <= m. Proof.   auto with arith. Qed. Hint Resolve gt_le_S: arith v62. Lemma le_gt_S : forall n m, n <= m -> S m > n. Proof.   auto with arith. Qed. Hint Resolve le_gt_S: arith v62. ```

# Transitivity

``` Theorem le_gt_trans : forall n m p, m <= n -> m > p -> n > p. Proof.   red in |- *; intros; apply lt_le_trans with m; auto with arith. Qed. Theorem gt_le_trans : forall n m p, n > m -> p <= m -> n > p. Proof.   red in |- *; intros; apply le_lt_trans with m; auto with arith. Qed. Lemma gt_trans : forall n m p, n > m -> m > p -> n > p. Proof.   red in |- *; intros n m p H1 H2.   apply lt_trans with m; auto with arith. Qed. Theorem gt_trans_S : forall n m p, S n > m -> m > p -> n > p. Proof.   red in |- *; intros; apply lt_le_trans with m; auto with arith. Qed. Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith v62. ```

# 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

``` Lemma plus_gt_reg_l : forall n m p, p + n > p + m -> n > m. Proof.   red in |- *; intros n m p H; apply plus_lt_reg_l with p; auto with arith. Qed. Lemma plus_gt_compat_l : forall n m p, n > m -> p + n > p + m. Proof.   auto with arith. Qed. Hint Resolve plus_gt_compat_l: arith v62. ```