Library Coq.Arith.Le

Order on natural numbers.
This file is mostly OBSOLETE now, see module PeanoNat.Nat instead.
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.

Require Import PeanoNat.

Local Open Scope nat_scope.

le is an order on nat

Notation le_refl := Nat.le_refl (only parsing).
Notation le_trans := Nat.le_trans (only parsing).
Notation le_antisym := Nat.le_antisymm (only parsing).

Hint Resolve le_trans: arith.
Hint Immediate le_antisym: arith.

Properties of le w.r.t 0

Notation le_0_n := Nat.le_0_l (only parsing). Notation le_Sn_0 := Nat.nle_succ_0 (only parsing).
Lemma le_n_0_eq n : n <= 0 -> 0 = n.

Properties of le w.r.t successor

See also Nat.succ_le_mono.

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

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

Notation le_n_Sn := Nat.le_succ_diag_r (only parsing). Notation le_Sn_n := Nat.nle_succ_diag_l (only parsing).
Theorem le_Sn_le : forall n m, S n <= m -> n <= m.

Hint Resolve le_0_n le_Sn_0: arith.
Hint Resolve le_n_S le_n_Sn le_Sn_n : arith.
Hint Immediate le_n_0_eq le_Sn_le le_S_n : arith.

Properties of le w.r.t predecessor

Notation le_pred_n := Nat.le_pred_l (only parsing). Notation le_pred := Nat.pred_le_mono (only parsing).
Hint Resolve le_pred_n: arith.

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.