Library Coq.Arith.Lt

``` ```
Theorems about `lt` in nat. `lt` is defined in library `Init/Peano.v` as:
```Definition lt (n m:nat) := S n <= m.
Infix "<" := lt : nat_scope.
```

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

Irreflexivity

``` Theorem lt_irrefl : forall n, ~ n < n. Proof le_Sn_n. Hint Resolve lt_irrefl: arith v62. ```

Relationship between `le` and `lt`

``` Theorem lt_le_S : forall n m, n < m -> S n <= m. Proof.   auto with arith. Qed. Hint Immediate lt_le_S: arith v62. Theorem lt_n_Sm_le : forall n m, n < S m -> n <= m. Proof.   auto with arith. Qed. Hint Immediate lt_n_Sm_le: arith v62. Theorem le_lt_n_Sm : forall n m, n <= m -> n < S m. Proof.   auto with arith. Qed. Hint Immediate le_lt_n_Sm: arith v62. Theorem le_not_lt : forall n m, n <= m -> ~ m < n. Proof.   induction 1; auto with arith. Qed. Theorem lt_not_le : forall n m, n < m -> ~ m <= n. Proof.   red in |- *; intros n m Lt Le; exact (le_not_lt m n Le Lt). Qed. Hint Immediate le_not_lt lt_not_le: arith v62. ```

Asymmetry

``` Theorem lt_asym : forall n m, n < m -> ~ m < n. Proof.   induction 1; auto with arith. Qed. ```

Order and successor

``` Theorem lt_n_Sn : forall n, n < S n. Proof.   auto with arith. Qed. Hint Resolve lt_n_Sn: arith v62. Theorem lt_S : forall n m, n < m -> n < S m. Proof.   auto with arith. Qed. Hint Resolve lt_S: arith v62. Theorem lt_n_S : forall n m, n < m -> S n < S m. Proof.   auto with arith. Qed. Hint Resolve lt_n_S: arith v62. Theorem lt_S_n : forall n m, S n < S m -> n < m. Proof.   auto with arith. Qed. Hint Immediate lt_S_n: arith v62. Theorem lt_O_Sn : forall n, 0 < S n. Proof.   auto with arith. Qed. Hint Resolve lt_O_Sn: arith v62. Theorem lt_n_O : forall n, ~ n < 0. Proof le_Sn_O. Hint Resolve lt_n_O: arith v62. ```

Predecessor

``` Lemma S_pred : forall n m, m < n -> n = S (pred n). Proof. induction 1; auto with arith. Qed. Lemma lt_pred : forall n m, S n < m -> n < pred m. Proof. induction 1; simpl in |- *; auto with arith. Qed. Hint Immediate lt_pred: arith v62. Lemma lt_pred_n_n : forall n, 0 < n -> pred n < n. destruct 1; simpl in |- *; auto with arith. Qed. Hint Resolve lt_pred_n_n: arith v62. ```

Transitivity properties

``` Theorem lt_trans : forall n m p, n < m -> m < p -> n < p. Proof.   induction 2; auto with arith. Qed. Theorem lt_le_trans : forall n m p, n < m -> m <= p -> n < p. Proof.   induction 2; auto with arith. Qed. Theorem le_lt_trans : forall n m p, n <= m -> m < p -> n < p. Proof.   induction 2; auto with arith. Qed. Hint Resolve lt_trans lt_le_trans le_lt_trans: arith v62. ```

Large = strict or equal

``` Theorem le_lt_or_eq : forall n m, n <= m -> n < m \/ n = m. Proof.   induction 1; auto with arith. Qed. Theorem lt_le_weak : forall n m, n < m -> n <= m. Proof.   auto with arith. Qed. Hint Immediate lt_le_weak: arith v62. ```

Dichotomy

``` Theorem le_or_lt : forall n m, n <= m \/ m < n. Proof.   intros n m; pattern n, m in |- *; apply nat_double_ind; auto with arith.   induction 1; auto with arith. Qed. Theorem nat_total_order : forall n m, n <> m -> n < m \/ m < n. Proof.   intros m n diff.   elim (le_or_lt n m); [ intro H'0 | auto with arith ].   elim (le_lt_or_eq n m); auto with arith.   intro H'; elim diff; auto with arith. Qed. ```

Comparison to 0

``` Theorem neq_O_lt : forall n, 0 <> n -> 0 < n. Proof.   induction n; auto with arith.   intros; absurd (0 = 0); trivial with arith. Qed. Hint Immediate neq_O_lt: arith v62. Theorem lt_O_neq : forall n, 0 < n -> 0 <> n. Proof.   induction 1; auto with arith. Qed. Hint Immediate lt_O_neq: arith v62. ```