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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 *)
(************************************************************************)
(** Order on natural numbers. [le] is defined in [Init/Peano.v] as:
<<
Inductive le (n:nat) : nat -> Prop :=
| le_n : n <= n
| le_S : forall m:nat, n <= m -> n <= S m
where "n <= m" := (le n m) : nat_scope.
>>
*)
Local Open Scope nat_scope.
Implicit Types m n p : nat.
(** * [le] is a pre-order *)
(** Reflexivity *)
Theorem le_refl : forall n, n <= n.
Proof.
exact le_n.
Qed.
(** Transitivity *)
Theorem le_trans : forall n m p, n <= m -> m <= p -> n <= p.
Proof.
induction 2; auto.
Qed.
Hint Resolve le_trans: arith v62.
(** * Properties of [le] w.r.t. successor, predecessor and 0 *)
(** Comparison to 0 *)
Theorem le_0_n : forall n, 0 <= n.
Proof.
induction n; auto.
Qed.
Theorem le_Sn_0 : forall n, ~ S n <= 0.
Proof.
red; intros n H.
change (IsSucc 0); elim H; simpl; auto with arith.
Qed.
Hint Resolve le_0_n le_Sn_0: arith v62.
Theorem le_n_0_eq : forall n, n <= 0 -> 0 = n.
Proof.
induction n; auto with arith.
intro; contradiction le_Sn_0 with n.
Qed.
Hint Immediate le_n_0_eq: arith v62.
(** [le] and successor *)
Theorem le_n_S : forall n m, n <= m -> S n <= S m.
Proof.
induction 1; auto.
Qed.
Theorem le_n_Sn : forall n, n <= S n.
Proof.
auto.
Qed.
Hint Resolve le_n_S le_n_Sn : arith v62.
Theorem le_Sn_le : forall n m, S n <= m -> n <= m.
Proof.
intros n m H; apply le_trans with (S n); auto with arith.
Qed.
Hint Immediate le_Sn_le: arith v62.
Theorem le_S_n : forall n m, S n <= S m -> n <= m.
Proof.
exact Peano.le_S_n.
Qed.
Hint Immediate le_S_n: arith v62.
Theorem le_Sn_n : forall n, ~ S n <= n.
Proof.
induction n; auto with arith.
Qed.
Hint Resolve le_Sn_n: arith v62.
(** [le] and predecessor *)
Theorem le_pred_n : forall n, pred n <= n.
Proof.
induction n; auto with arith.
Qed.
Hint Resolve le_pred_n: arith v62.
Theorem le_pred : forall n m, n <= m -> pred n <= pred m.
Proof.
exact Peano.le_pred.
Qed.
(** * [le] is a order on [nat] *)
(** Antisymmetry *)
Theorem le_antisym : forall n m, n <= m -> m <= n -> n = m.
Proof.
intros n m H; destruct H as [|m' H]; auto with arith.
intros H1.
absurd (S m' <= m'); auto with arith.
apply le_trans with n; auto with arith.
Qed.
Hint Immediate le_antisym: arith v62.
(** * A different elimination principle for the order on natural numbers *)
Lemma le_elim_rel :
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.
Proof.
induction n; auto with arith.
intros m Le.
elim Le; auto with arith.
Qed.
(* begin hide *)
Notation le_O_n := le_0_n (only parsing).
Notation le_Sn_O := le_Sn_0 (only parsing).
Notation le_n_O_eq := le_n_0_eq (only parsing).
(* end hide *)
|
