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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Theorems about [lt] in nat. [lt] is defined in library [Init/Peano.v] as:
<<
Definition lt (n m:nat) := S n <= m.
Infix "<" := lt : nat_scope.
>>
*)
Require Import Le.
Local Open Scope nat_scope.
Implicit Types m n p : nat.
(** * Irreflexivity *)
Theorem lt_irrefl : forall n, ~ n < n.
Proof le_Sn_n.
Hint Resolve lt_irrefl: arith v62.
(** * Relationship between [le] and [lt] *)
Theorem lt_le_S : forall n m, n < m -> S n <= m.
Proof.
auto with arith.
Qed.
Hint Immediate lt_le_S: arith v62.
Theorem lt_n_Sm_le : forall n m, n < S m -> n <= m.
Proof.
auto with arith.
Qed.
Hint Immediate lt_n_Sm_le: arith v62.
Theorem le_lt_n_Sm : forall n m, n <= m -> n < S m.
Proof.
auto with arith.
Qed.
Hint Immediate le_lt_n_Sm: arith v62.
Theorem le_not_lt : forall n m, n <= m -> ~ m < n.
Proof.
induction 1; auto with arith.
Qed.
Theorem lt_not_le : forall n m, n < m -> ~ m <= n.
Proof.
red; intros n m Lt Le; exact (le_not_lt m n Le Lt).
Qed.
Hint Immediate le_not_lt lt_not_le: arith v62.
(** * Asymmetry *)
Theorem lt_asym : forall n m, n < m -> ~ m < n.
Proof.
induction 1; auto with arith.
Qed.
(** * Order and successor *)
Theorem lt_n_Sn : forall n, n < S n.
Proof.
auto with arith.
Qed.
Hint Resolve lt_n_Sn: arith v62.
Theorem lt_S : forall n m, n < m -> n < S m.
Proof.
auto with arith.
Qed.
Hint Resolve lt_S: arith v62.
Theorem lt_n_S : forall n m, n < m -> S n < S m.
Proof.
auto with arith.
Qed.
Hint Resolve lt_n_S: arith v62.
Theorem lt_S_n : forall n m, S n < S m -> n < m.
Proof.
auto with arith.
Qed.
Hint Immediate lt_S_n: arith v62.
Theorem lt_0_Sn : forall n, 0 < S n.
Proof.
auto with arith.
Qed.
Hint Resolve lt_0_Sn: arith v62.
Theorem lt_n_0 : forall n, ~ n < 0.
Proof le_Sn_0.
Hint Resolve lt_n_0: arith v62.
(** * Predecessor *)
Lemma S_pred : forall n m, m < n -> n = S (pred n).
Proof.
induction 1; auto with arith.
Qed.
Lemma lt_pred : forall n m, S n < m -> n < pred m.
Proof.
induction 1; simpl; auto with arith.
Qed.
Hint Immediate lt_pred: arith v62.
Lemma lt_pred_n_n : forall n, 0 < n -> pred n < n.
destruct 1; simpl; auto with arith.
Qed.
Hint Resolve lt_pred_n_n: arith v62.
(** * Transitivity properties *)
Theorem lt_trans : forall n m p, n < m -> m < p -> n < p.
Proof.
induction 2; auto with arith.
Qed.
Theorem lt_le_trans : forall n m p, n < m -> m <= p -> n < p.
Proof.
induction 2; auto with arith.
Qed.
Theorem le_lt_trans : forall n m p, n <= m -> m < p -> n < p.
Proof.
induction 2; auto with arith.
Qed.
Hint Resolve lt_trans lt_le_trans le_lt_trans: arith v62.
(** * Large = strict or equal *)
Theorem le_lt_or_eq : forall n m, n <= m -> n < m \/ n = m.
Proof.
induction 1; auto with arith.
Qed.
Theorem le_lt_or_eq_iff : forall n m, n <= m <-> n < m \/ n = m.
Proof.
split.
intros; apply le_lt_or_eq; auto.
destruct 1; subst; auto with arith.
Qed.
Theorem lt_le_weak : forall n m, n < m -> n <= m.
Proof.
auto with arith.
Qed.
Hint Immediate lt_le_weak: arith v62.
(** * Dichotomy *)
Theorem le_or_lt : forall n m, n <= m \/ m < n.
Proof.
intros n m; pattern n, m; apply nat_double_ind; auto with arith.
induction 1; auto with arith.
Qed.
Theorem nat_total_order : forall n m, n <> m -> n < m \/ m < n.
Proof.
intros m n diff.
elim (le_or_lt n m); [ intro H'0 | auto with arith ].
elim (le_lt_or_eq n m); auto with arith.
intro H'; elim diff; auto with arith.
Qed.
(** * Comparison to 0 *)
Theorem neq_0_lt : forall n, 0 <> n -> 0 < n.
Proof.
induction n; auto with arith.
intros; absurd (0 = 0); trivial with arith.
Qed.
Hint Immediate neq_0_lt: arith v62.
Theorem lt_0_neq : forall n, 0 < n -> 0 <> n.
Proof.
induction 1; auto with arith.
Qed.
Hint Immediate lt_0_neq: arith v62.
(* 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 *)
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