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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Gt.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
(** Theorems about [gt] in [nat]. [gt] is defined in [Init/Peano.v] as:
<<
Definition gt (n m:nat) := m < n.
>>
*)
Require Import Le.
Require Import Lt.
Require Import Plus.
Open Local Scope nat_scope.
Implicit Types m n p : nat.
(** * Order and successor *)
Theorem gt_Sn_O : forall n, S n > 0.
Proof.
auto with arith.
Qed.
Hint Resolve gt_Sn_O: arith v62.
Theorem gt_Sn_n : forall n, S n > n.
Proof.
auto with arith.
Qed.
Hint Resolve gt_Sn_n: arith v62.
Theorem gt_n_S : forall n m, n > m -> S n > S m.
Proof.
auto with arith.
Qed.
Hint Resolve gt_n_S: arith v62.
Lemma gt_S_n : forall n m, S m > S n -> m > n.
Proof.
auto with arith.
Qed.
Hint Immediate gt_S_n: arith v62.
Theorem gt_S : forall n m, S n > m -> n > m \/ m = n.
Proof.
intros n m H; unfold gt in |- *; apply le_lt_or_eq; auto with arith.
Qed.
Lemma gt_pred : forall n m, m > S n -> pred m > n.
Proof.
auto with arith.
Qed.
Hint Immediate gt_pred: arith v62.
(** * Irreflexivity *)
Lemma gt_irrefl : forall n, ~ n > n.
Proof lt_irrefl.
Hint Resolve gt_irrefl: arith v62.
(** * Asymmetry *)
Lemma gt_asym : forall n m, n > m -> ~ m > n.
Proof fun n m => lt_asym m n.
Hint Resolve gt_asym: arith v62.
(** * Relating strict and large orders *)
Lemma le_not_gt : forall n m, n <= m -> ~ n > m.
Proof le_not_lt.
Hint Resolve le_not_gt: arith v62.
Lemma gt_not_le : forall n m, n > m -> ~ n <= m.
Proof.
auto with arith.
Qed.
Hint Resolve gt_not_le: arith v62.
Theorem le_S_gt : forall n m, S n <= m -> m > n.
Proof.
auto with arith.
Qed.
Hint Immediate le_S_gt: arith v62.
Lemma gt_S_le : forall n m, S m > n -> n <= m.
Proof.
intros n p; exact (lt_n_Sm_le n p).
Qed.
Hint Immediate gt_S_le: arith v62.
Lemma gt_le_S : forall n m, m > n -> S n <= m.
Proof.
auto with arith.
Qed.
Hint Resolve gt_le_S: arith v62.
Lemma le_gt_S : forall n m, n <= m -> S m > n.
Proof.
auto with arith.
Qed.
Hint Resolve le_gt_S: arith v62.
(** * Transitivity *)
Theorem le_gt_trans : forall n m p, m <= n -> m > p -> n > p.
Proof.
red in |- *; intros; apply lt_le_trans with m; auto with arith.
Qed.
Theorem gt_le_trans : forall n m p, n > m -> p <= m -> n > p.
Proof.
red in |- *; intros; apply le_lt_trans with m; auto with arith.
Qed.
Lemma gt_trans : forall n m p, n > m -> m > p -> n > p.
Proof.
red in |- *; intros n m p H1 H2.
apply lt_trans with m; auto with arith.
Qed.
Theorem gt_trans_S : forall n m p, S n > m -> m > p -> n > p.
Proof.
red in |- *; intros; apply lt_le_trans with m; auto with arith.
Qed.
Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith v62.
(** * Comparison to 0 *)
Theorem gt_0_eq : forall n, n > 0 \/ 0 = n.
Proof.
intro n; apply gt_S; auto with arith.
Qed.
(** * Simplification and compatibility *)
Lemma plus_gt_reg_l : forall n m p, p + n > p + m -> n > m.
Proof.
red in |- *; intros n m p H; apply plus_lt_reg_l with p; auto with arith.
Qed.
Lemma plus_gt_compat_l : forall n m p, n > m -> p + n > p + m.
Proof.
auto with arith.
Qed.
Hint Resolve plus_gt_compat_l: arith v62.
(* begin hide *)
Notation gt_O_eq := gt_0_eq (only parsing).
(* end hide *)
|
