# Library Coq.Arith.Le

Order on natural numbers. le is defined in Init/Peano.v as:
```Inductive le (n:nat) : nat -> Prop :=
| le_n : n <= n
| le_S : forall m:nat, n <= m -> n <= S m

where "n <= m" := (le n m) : nat_scope.
```

Open Local Scope nat_scope.

Implicit Types m n p : nat.

# le is a pre-order

Reflexivity
Theorem le_refl : forall n, n <= n.

Transitivity
Theorem le_trans : forall n m p, n <= m -> m <= p -> n <= p.
Hint Resolve le_trans: arith v62.

# Properties of le w.r.t. successor, predecessor and 0

Comparison to 0

Theorem le_O_n : forall n, 0 <= n.

Theorem le_Sn_O : forall n, ~ S n <= 0.

Hint Resolve le_O_n le_Sn_O: arith v62.

Theorem le_n_O_eq : forall n, n <= 0 -> 0 = n.
Hint Immediate le_n_O_eq: arith v62.

le and successor

Theorem le_n_S : forall n m, n <= m -> S n <= S m.

Theorem le_n_Sn : forall n, n <= S n.

Hint Resolve le_n_S le_n_Sn : arith v62.

Theorem le_Sn_le : forall n m, S n <= m -> n <= m.
Hint Immediate le_Sn_le: arith v62.

Theorem le_S_n : forall n m, S n <= S m -> n <= m.
Hint Immediate le_S_n: arith v62.

Theorem le_Sn_n : forall n, ~ S n <= n.
Hint Resolve le_Sn_n: arith v62.

le and predecessor

Theorem le_pred_n : forall n, pred n <= n.
Hint Resolve le_pred_n: arith v62.

Theorem le_pred : forall n m, n <= m -> pred n <= pred m.

# le is a order on nat

Antisymmetry

Theorem le_antisym : forall n m, n <= m -> m <= n -> n = m.
Hint Immediate le_antisym: arith v62.

# A different elimination principle for the order on natural numbers

Lemma le_elim_rel :
forall P:nat -> nat -> Prop,
(forall p, P 0 p) ->
(forall p (q:nat), p <= q -> P p q -> P (S p) (S q)) ->
forall n m, n <= m -> P n m.