(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(*i $Id: Minus.v 8642 2006-03-17 10:09:02Z notin $ i*)

(** Subtraction (difference between two natural numbers) *)

Require Import Lt.
Require Import Le.

Open Local Scope nat_scope.

Implicit Types m n p : nat.

(** 0 is right neutral *)

Lemma minus_n_O : forall n, n = n - 0.
Proof.
induction n; simpl in |- *; auto with arith.
Qed.
Hint Resolve minus_n_O: arith v62.

(** Permutation with successor *)

Lemma minus_Sn_m : forall n m, m <= n -> S (n - m) = S n - m.
Proof.
intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *;
 auto with arith.
Qed.
Hint Resolve minus_Sn_m: arith v62.

Theorem pred_of_minus : forall n, pred n = n - 1.
intro x; induction x; simpl in |- *; auto with arith.
Qed.

(** Diagonal *)

Lemma minus_n_n : forall n, 0 = n - n.
Proof.
induction n; simpl in |- *; auto with arith.
Qed.
Hint Resolve minus_n_n: arith v62.

(** Simplification *)

Lemma minus_plus_simpl_l_reverse : forall n m p, n - m = p + n - (p + m).
Proof.
  induction p; simpl in |- *; auto with arith.
Qed.
Hint Resolve minus_plus_simpl_l_reverse: arith v62.

(** Relation with plus *)

Lemma plus_minus : forall n m p, n = m + p -> p = n - m.
Proof.
intros n m p; pattern m, n in |- *; apply nat_double_ind; simpl in |- *;
 intros.
replace (n0 - 0) with n0; auto with arith.
absurd (0 = S (n0 + p)); auto with arith.
auto with arith.
Qed.
Hint Immediate plus_minus: arith v62.

Lemma minus_plus : forall n m, n + m - n = m.
symmetry  in |- *; auto with arith.
Qed.
Hint Resolve minus_plus: arith v62.

Lemma le_plus_minus : forall n m, n <= m -> m = n + (m - n).
Proof.
intros n m Le; pattern n, m in |- *; apply le_elim_rel; simpl in |- *;
 auto with arith.
Qed.
Hint Resolve le_plus_minus: arith v62.

Lemma le_plus_minus_r : forall n m, n <= m -> n + (m - n) = m.
Proof.
symmetry  in |- *; auto with arith.
Qed.
Hint Resolve le_plus_minus_r: arith v62.

(** Relation with order *)

Theorem le_minus : forall n m, n - m <= n.
Proof.
intros i h; pattern i, h in |- *; apply nat_double_ind;
 [ auto
 | auto
 | intros m n H; simpl in |- *; apply le_trans with (m := m); auto ].
Qed.

Lemma lt_minus : forall n m, m <= n -> 0 < m -> n - m < n.
Proof.
intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *;
 auto with arith.
intros; absurd (0 < 0); auto with arith.
intros p q lepq Hp gtp.
elim (le_lt_or_eq 0 p); auto with arith.
auto with arith.
induction 1; elim minus_n_O; auto with arith.
Qed.
Hint Resolve lt_minus: arith v62.

Lemma lt_O_minus_lt : forall n m, 0 < n - m -> m < n.
Proof.
intros n m; pattern n, m in |- *; apply nat_double_ind; simpl in |- *;
 auto with arith.
intros; absurd (0 < 0); trivial with arith.
Qed.
Hint Immediate lt_O_minus_lt: arith v62.

Theorem not_le_minus_0 : forall n m, ~ m <= n -> n - m = 0.
intros y x; pattern y, x in |- *; apply nat_double_ind;
 [ simpl in |- *; trivial with arith
 | intros n H; absurd (0 <= S n); [ assumption | apply le_O_n ]
 | simpl in |- *; intros n m H1 H2; apply H1; unfold not in |- *; intros H3;
    apply H2; apply le_n_S; assumption ].
Qed.