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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)         *)
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Definition gt (n m:nat) := m < n.

Require Import PeanoNat Le Lt Plus.
Local Open Scope nat_scope.

Theorem gt_Sn_O n : S n > 0.
n:nat
S n > 0
Proof Nat.lt_0_succ _.

Theorem gt_Sn_n n : S n > n.
n:nat
S n > n
Proof Nat.lt_succ_diag_r _.

Theorem gt_n_S n m : n > m -> S n > S m.
n, m:nat
n > m -> S n > S m
Proof.
n, m:nat
n > m -> S n > S m
 apply Nat.succ_lt_mono.
Qed.

Lemma gt_S_n n m : S m > S n -> m > n.
n, m:nat
S m > S n -> m > n
Proof.
n, m:nat
S m > S n -> m > n
 apply Nat.succ_lt_mono.
Qed.

Theorem gt_S n m : S n > m -> n > m \/ m = n.
n, m:nat
S n > m -> n > m \/ m = n
Proof.
n, m:nat
S n > m -> n > m \/ m = n
 intro.
n, m:nat
H:S n > m
n > m \/ m = n now apply Nat.lt_eq_cases, Nat.succ_le_mono.
Qed.

Lemma gt_pred n m : m > S n -> pred m > n.
n, m:nat
m > S n -> Init.Nat.pred m > n
Proof.
n, m:nat
m > S n -> Init.Nat.pred m > n
 apply Nat.lt_succ_lt_pred.
Qed.

Lemma gt_irrefl n : ~ n > n.
n:nat
~ n > n
Proof Nat.lt_irrefl _.

Lemma gt_asym n m : n > m -> ~ m > n.
n, m:nat
n > m -> ~ m > n
Proof Nat.lt_asymm _ _.

Lemma le_not_gt n m : n <= m -> ~ n > m.
n, m:nat
n <= m -> ~ n > m
Proof.
n, m:nat
n <= m -> ~ n > m
 apply Nat.le_ngt.
Qed.

Lemma gt_not_le n m : n > m -> ~ n <= m.
n, m:nat
n > m -> ~ n <= m
Proof.
n, m:nat
n > m -> ~ n <= m
 apply Nat.lt_nge.
Qed.

Theorem le_S_gt n m : S n <= m -> m > n.
n, m:nat
S n <= m -> m > n
Proof.
n, m:nat
S n <= m -> m > n
 apply Nat.le_succ_l.
Qed.

Lemma gt_S_le n m : S m > n -> n <= m.
n, m:nat
S m > n -> n <= m
Proof.
n, m:nat
S m > n -> n <= m
 apply Nat.succ_le_mono.
Qed.

Lemma gt_le_S n m : m > n -> S n <= m.
n, m:nat
m > n -> S n <= m
Proof.
n, m:nat
m > n -> S n <= m
 apply Nat.le_succ_l.
Qed.

Lemma le_gt_S n m : n <= m -> S m > n.
n, m:nat
n <= m -> S m > n
Proof.
n, m:nat
n <= m -> S m > n
 apply Nat.succ_le_mono.
Qed.

Theorem le_gt_trans n m p : m <= n -> m > p -> n > p.
n, m, p:nat
m <= n -> m > p -> n > p
Proof.
n, m, p:nat
m <= n -> m > p -> n > p
 intros.
n, m, p:nat
H:m <= n
H0:m > p
n > p now apply Nat.lt_le_trans with m.
Qed.

Theorem gt_le_trans n m p : n > m -> p <= m -> n > p.
n, m, p:nat
n > m -> p <= m -> n > p
Proof.
n, m, p:nat
n > m -> p <= m -> n > p
 intros.
n, m, p:nat
H:n > m
H0:p <= m
n > p now apply Nat.le_lt_trans with m.
Qed.

Lemma gt_trans n m p : n > m -> m > p -> n > p.
n, m, p:nat
n > m -> m > p -> n > p
Proof.
n, m, p:nat
n > m -> m > p -> n > p
 intros.
n, m, p:nat
H:n > m
H0:m > p
n > p now apply Nat.lt_trans with m.
Qed.

Theorem gt_trans_S n m p : S n > m -> m > p -> n > p.
n, m, p:nat
S n > m -> m > p -> n > p
Proof.
n, m, p:nat
S n > m -> m > p -> n > p
 intros.
n, m, p:nat
H:S n > m
H0:m > p
n > p apply Nat.lt_le_trans with m; trivial.
n, m, p:nat
H:S n > m
H0:m > p
m <= n now apply Nat.succ_le_mono.
Qed.

Theorem gt_0_eq n : n > 0 \/ 0 = n.
n:nat
n > 0 \/ 0 = n
Proof.
n:nat
n > 0 \/ 0 = n
 destruct n; [now right | left; apply Nat.lt_0_succ].
Qed.

Lemma plus_gt_reg_l n m p : p + n > p + m -> n > m.
n, m, p:nat
p + n > p + m -> n > m
Proof.
n, m, p:nat
p + n > p + m -> n > m
 apply Nat.add_lt_mono_l.
Qed.

Lemma plus_gt_compat_l n m p : n > m -> p + n > p + m.
n, m, p:nat
n > m -> p + n > p + m
Proof.
n, m, p:nat
n > m -> p + n > p + m
 apply Nat.add_lt_mono_l.
Qed.

Hint Resolve gt_Sn_O gt_Sn_n gt_n_S : arith.
Hint Immediate gt_S_n gt_pred : arith.
Hint Resolve gt_irrefl gt_asym : arith.
Hint Resolve le_not_gt gt_not_le : arith.
Hint Immediate le_S_gt gt_S_le : arith.
Hint Resolve gt_le_S le_gt_S : arith.
Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith.
Hint Resolve plus_gt_compat_l: arith.

(* begin hide *)
Notation gt_O_eq := gt_0_eq (only parsing).
(* end hide *)