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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Strict order on natural numbers.
This file is mostly OBSOLETE now, see module [PeanoNat.Nat] instead.
[lt] is defined in library [Init/Peano.v] as:
<<
Definition lt (n m:nat) := S n <= m.
Infix "<" := lt : nat_scope.
>>
*)
Require Import PeanoNat.
Local Open Scope nat_scope.
(** * Irreflexivity *)
Notation lt_irrefl := Nat.lt_irrefl (only parsing). (* ~ x < x *)
Hint Resolve lt_irrefl: arith.
(** * Relationship between [le] and [lt] *)
Theorem lt_le_S n m : n < m -> S n <= m.
Proof.
apply Nat.le_succ_l.
Qed.
Theorem lt_n_Sm_le n m : n < S m -> n <= m.
Proof.
apply Nat.lt_succ_r.
Qed.
Theorem le_lt_n_Sm n m : n <= m -> n < S m.
Proof.
apply Nat.lt_succ_r.
Qed.
Hint Immediate lt_le_S: arith.
Hint Immediate lt_n_Sm_le: arith.
Hint Immediate le_lt_n_Sm: arith.
Theorem le_not_lt n m : n <= m -> ~ m < n.
Proof.
apply Nat.le_ngt.
Qed.
Theorem lt_not_le n m : n < m -> ~ m <= n.
Proof.
apply Nat.lt_nge.
Qed.
Hint Immediate le_not_lt lt_not_le: arith.
(** * Asymmetry *)
Notation lt_asym := Nat.lt_asymm (only parsing). (* n<m -> ~m<n *)
(** * Order and 0 *)
Notation lt_0_Sn := Nat.lt_0_succ (only parsing). (* 0 < S n *)
Notation lt_n_0 := Nat.nlt_0_r (only parsing). (* ~ n < 0 *)
Theorem neq_0_lt n : 0 <> n -> 0 < n.
Proof.
intros. now apply Nat.neq_0_lt_0, Nat.neq_sym.
Qed.
Theorem lt_0_neq n : 0 < n -> 0 <> n.
Proof.
intros. now apply Nat.neq_sym, Nat.neq_0_lt_0.
Qed.
Hint Resolve lt_0_Sn lt_n_0 : arith.
Hint Immediate neq_0_lt lt_0_neq: arith.
(** * Order and successor *)
Notation lt_n_Sn := Nat.lt_succ_diag_r (only parsing). (* n < S n *)
Notation lt_S := Nat.lt_lt_succ_r (only parsing). (* n < m -> n < S m *)
Theorem lt_n_S n m : n < m -> S n < S m.
Proof.
apply Nat.succ_lt_mono.
Qed.
Theorem lt_S_n n m : S n < S m -> n < m.
Proof.
apply Nat.succ_lt_mono.
Qed.
Hint Resolve lt_n_Sn lt_S lt_n_S : arith.
Hint Immediate lt_S_n : arith.
(** * Predecessor *)
Lemma S_pred n m : m < n -> n = S (pred n).
Proof.
intros. symmetry. now apply Nat.lt_succ_pred with m.
Qed.
Lemma S_pred_pos n: O < n -> n = S (pred n).
Proof.
apply S_pred.
Qed.
Lemma lt_pred n m : S n < m -> n < pred m.
Proof.
apply Nat.lt_succ_lt_pred.
Qed.
Lemma lt_pred_n_n n : 0 < n -> pred n < n.
Proof.
intros. now apply Nat.lt_pred_l, Nat.neq_0_lt_0.
Qed.
Hint Immediate lt_pred: arith.
Hint Resolve lt_pred_n_n: arith.
(** * Transitivity properties *)
Notation lt_trans := Nat.lt_trans (only parsing).
Notation lt_le_trans := Nat.lt_le_trans (only parsing).
Notation le_lt_trans := Nat.le_lt_trans (only parsing).
Hint Resolve lt_trans lt_le_trans le_lt_trans: arith.
(** * Large = strict or equal *)
Notation le_lt_or_eq_iff := Nat.lt_eq_cases (only parsing).
Theorem le_lt_or_eq n m : n <= m -> n < m \/ n = m.
Proof.
apply Nat.lt_eq_cases.
Qed.
Notation lt_le_weak := Nat.lt_le_incl (only parsing).
Hint Immediate lt_le_weak: arith.
(** * Dichotomy *)
Notation le_or_lt := Nat.le_gt_cases (only parsing). (* n <= m \/ m < n *)
Theorem nat_total_order n m : n <> m -> n < m \/ m < n.
Proof.
apply Nat.lt_gt_cases.
Qed.
(* begin hide *)
Notation lt_O_Sn := lt_0_Sn (only parsing).
Notation neq_O_lt := neq_0_lt (only parsing).
Notation lt_O_neq := lt_0_neq (only parsing).
Notation lt_n_O := lt_n_0 (only parsing).
(* end hide *)
(** For compatibility, we "Require" the same files as before *)
Require Import Le.
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