Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Arith.Le

Order on natural numbers.

# This file is OBSOLETE, see Arith_base instead.

Require Export Arith_prebase.

# le is an order on nat

#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.le_refl instead.")]
Notation le_refl := Nat.le_refl (only parsing).
#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.le_trans instead.")]
Notation le_trans := Nat.le_trans (only parsing).
#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.le_antisymm instead.")]
Notation le_antisym := Nat.le_antisymm (only parsing).

# Properties of le w.r.t 0

#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.le_0_l instead.")]
Notation le_0_n := Nat.le_0_l (only parsing). #[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.nle_succ_0 instead.")]
Notation le_Sn_0 := Nat.nle_succ_0 (only parsing). #[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use the bidirectional version Nat.le_0_r (with symmetry of equality) instead.")]
Notation le_n_0_eq := Arith_prebase.le_n_0_eq_stt.

# Properties of le w.r.t successor

#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.succ_le_mono instead.")]
Notation le_n_S := Peano.le_n_S (only parsing).
#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.succ_le_mono instead.")]
Notation le_S_n := Peano.le_S_n (only parsing).
#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.le_succ_diag_r instead.")]
Notation le_n_Sn := Nat.le_succ_diag_r (only parsing). #[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.nle_succ_diag_l instead.")]
Notation le_Sn_n := Nat.nle_succ_diag_l (only parsing). #[local]
Definition le_Sn_le_stt : forall n m, S n <= m -> n <= m := Nat.lt_le_incl.
Opaque le_Sn_le_stt.
#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.lt_le_incl instead.")]
Notation le_Sn_le := le_Sn_le_stt.

# Properties of le w.r.t predecessor

#[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.le_pred_l instead.")]
Notation le_pred_n := Nat.le_pred_l (only parsing). #[deprecated(since="8.16",note="The Arith.Le file is obsolete. Use Nat.pred_le_mono instead.")]
Notation le_pred := Nat.pred_le_mono (only parsing).

# A different elimination principle for the order on natural numbers

#[local]
Definition le_elim_rel_stt :
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.
#[deprecated(since="8.16",note="The Arith.Le file is obsolete.")]
Notation le_elim_rel := le_elim_rel_stt.