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-(************************************************************************)
-(*  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        *)
-(************************************************************************)
-(*                      Evgeny Makarov, INRIA, 2007                     *)
-(************************************************************************)
-
-(*i $Id$ i*)
-
-Require Export NTimesOrder.
-
-Module NMinusPropFunct (Import NAxiomsMod : NAxiomsSig).
-Module Export NTimesOrderPropMod := NTimesOrderPropFunct NAxiomsMod.
-Open Local Scope NatScope.
-
-Theorem minus_wd :
-  forall n1 n2 : N, n1 == n2 -> forall m1 m2 : N, m1 == m2 -> n1 - m1 == n2 - m2.
-Proof NZminus_wd.
-
-Theorem minus_0_r : forall n : N, n - 0 == n.
-Proof NZminus_0_r.
-
-Theorem minus_succ_r : forall n m : N, n - (S m) == P (n - m).
-Proof NZminus_succ_r.
-
-Theorem minus_1_r : forall n : N, n - 1 == P n.
-Proof.
-intro n; rewrite minus_succ_r; now rewrite minus_0_r.
-Qed.
-
-Theorem minus_0_l : forall n : N, 0 - n == 0.
-Proof.
-induct n.
-apply minus_0_r.
-intros n IH; rewrite minus_succ_r; rewrite IH. now apply pred_0.
-Qed.
-
-Theorem minus_succ : forall n m : N, S n - S m == n - m.
-Proof.
-intro n; induct m.
-rewrite minus_succ_r. do 2 rewrite minus_0_r. now rewrite pred_succ.
-intros m IH. rewrite minus_succ_r. rewrite IH. now rewrite minus_succ_r.
-Qed.
-
-Theorem minus_diag : forall n : N, n - n == 0.
-Proof.
-induct n. apply minus_0_r. intros n IH; rewrite minus_succ; now rewrite IH.
-Qed.
-
-Theorem minus_gt : forall n m : N, n > m -> n - m ~= 0.
-Proof.
-intros n m H; elim H using lt_ind_rel; clear n m H.
-solve_relation_wd.
-intro; rewrite minus_0_r; apply neq_succ_0.
-intros; now rewrite minus_succ.
-Qed.
-
-Theorem add_minus_assoc : forall n m p : N, p <= m -> n + (m - p) == (n + m) - p.
-Proof.
-intros n m p; induct p.
-intro; now do 2 rewrite minus_0_r.
-intros p IH H. do 2 rewrite minus_succ_r.
-rewrite <- IH by (apply lt_le_incl; now apply -> le_succ_l).
-rewrite add_pred_r by (apply minus_gt; now apply -> le_succ_l).
-reflexivity.
-Qed.
-
-Theorem minus_succ_l : forall n m : N, n <= m -> S m - n == S (m - n).
-Proof.
-intros n m H. rewrite <- (add_1_l m). rewrite <- (add_1_l (m - n)).
-symmetry; now apply add_minus_assoc.
-Qed.
-
-Theorem add_minus : forall n m : N, (n + m) - m == n.
-Proof.
-intros n m. rewrite <- add_minus_assoc by (apply le_refl).
-rewrite minus_diag; now rewrite add_0_r.
-Qed.
-
-Theorem minus_add : forall n m : N, n <= m -> (m - n) + n == m.
-Proof.
-intros n m H. rewrite add_comm. rewrite add_minus_assoc by assumption.
-rewrite add_comm. apply add_minus.
-Qed.
-
-Theorem add_minus_eq_l : forall n m p : N, m + p == n -> n - m == p.
-Proof.
-intros n m p H. symmetry.
-assert (H1 : m + p - m == n - m) by now rewrite H.
-rewrite add_comm in H1. now rewrite add_minus in H1.
-Qed.
-
-Theorem add_minus_eq_r : forall n m p : N, m + p == n -> n - p == m.
-Proof.
-intros n m p H; rewrite add_comm in H; now apply add_minus_eq_l.
-Qed.
-
-(* This could be proved by adding m to both sides. Then the proof would
-use add_minus_assoc and minus_0_le, which is proven below. *)
-
-Theorem add_minus_eq_nz : forall n m p : N, p ~= 0 -> n - m == p -> m + p == n.
-Proof.
-intros n m p H; double_induct n m.
-intros m H1; rewrite minus_0_l in H1. symmetry in H1; false_hyp H1 H.
-intro n; rewrite minus_0_r; now rewrite add_0_l.
-intros n m IH H1. rewrite minus_succ in H1. apply IH in H1.
-rewrite add_succ_l; now rewrite H1.
-Qed.
-
-Theorem minus_add_distr : forall n m p : N, n - (m + p) == (n - m) - p.
-Proof.
-intros n m; induct p.
-rewrite add_0_r; now rewrite minus_0_r.
-intros p IH. rewrite add_succ_r; do 2 rewrite minus_succ_r. now rewrite IH.
-Qed.
-
-Theorem add_minus_swap : forall n m p : N, p <= n -> n + m - p == n - p + m.
-Proof.
-intros n m p H.
-rewrite (add_comm n m).
-rewrite <- add_minus_assoc by assumption.
-now rewrite (add_comm m (n - p)).
-Qed.
-
-(** Minus and order *)
-
-Theorem le_minus_l : forall n m : N, n - m <= n.
-Proof.
-intro n; induct m.
-rewrite minus_0_r; now apply eq_le_incl.
-intros m IH. rewrite minus_succ_r.
-apply le_trans with (n - m); [apply le_pred_l | assumption].
-Qed.
-
-Theorem minus_0_le : forall n m : N, n - m == 0 <-> n <= m.
-Proof.
-double_induct n m.
-intro m; split; intro; [apply le_0_l | apply minus_0_l].
-intro m; rewrite minus_0_r; split; intro H;
-[false_hyp H neq_succ_0 | false_hyp H nle_succ_0].
-intros n m H. rewrite <- succ_le_mono. now rewrite minus_succ.
-Qed.
-
-(** Minus and mul *)
-
-Theorem mul_pred_r : forall n m : N, n * (P m) == n * m - n.
-Proof.
-intros n m; cases m.
-now rewrite pred_0, mul_0_r, minus_0_l.
-intro m; rewrite pred_succ, mul_succ_r, <- add_minus_assoc.
-now rewrite minus_diag, add_0_r.
-now apply eq_le_incl.
-Qed.
-
-Theorem mul_minus_distr_r : forall n m p : N, (n - m) * p == n * p - m * p.
-Proof.
-intros n m p; induct n.
-now rewrite minus_0_l, mul_0_l, minus_0_l.
-intros n IH. destruct (le_gt_cases m n) as [H | H].
-rewrite minus_succ_l by assumption. do 2 rewrite mul_succ_l.
-rewrite (add_comm ((n - m) * p) p), (add_comm (n * p) p).
-rewrite <- (add_minus_assoc p (n * p) (m * p)) by now apply mul_le_mono_r.
-now apply <- add_cancel_l.
-assert (H1 : S n <= m); [now apply <- le_succ_l |].
-setoid_replace (S n - m) with 0 by now apply <- minus_0_le.
-setoid_replace ((S n * p) - m * p) with 0 by (apply <- minus_0_le; now apply mul_le_mono_r).
-apply mul_0_l.
-Qed.
-
-Theorem mul_minus_distr_l : forall n m p : N, p * (n - m) == p * n - p * m.
-Proof.
-intros n m p; rewrite (mul_comm p (n - m)), (mul_comm p n), (mul_comm p m).
-apply mul_minus_distr_r.
-Qed.
-
-End NMinusPropFunct.
-