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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        *)
+(************************************************************************)
+
+(*i $Id: auxiliary.v,v 1.12.2.1 2004/07/16 19:31:22 herbelin Exp $ i*)
+
+(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *)
+
+Require Export Arith.
+Require Import BinInt.
+Require Import Zorder.
+Require Import Decidable.
+Require Import Peano_dec.
+Require Export Compare_dec.
+
+Open Local Scope Z_scope.
+
+(**********************************************************************)
+(** Moving terms from one side to the other of an inequality *)
+
+Theorem Zne_left : forall n m:Z, Zne n m -> Zne (n + - m) 0.
+Proof.
+intros x y; unfold Zne in |- *; unfold not in |- *; intros H1 H2; apply H1;
+ apply Zplus_reg_l with (- y); rewrite Zplus_opp_l; 
+ rewrite Zplus_comm; trivial with arith.
+Qed.
+
+Theorem Zegal_left : forall n m:Z, n = m -> n + - m = 0.
+Proof.
+intros x y H; apply (Zplus_reg_l y); rewrite Zplus_permute;
+ rewrite Zplus_opp_r; do 2 rewrite Zplus_0_r; assumption.
+Qed.
+
+Theorem Zle_left : forall n m:Z, n <= m -> 0 <= m + - n.
+Proof.
+intros x y H; replace 0 with (x + - x).
+apply Zplus_le_compat_r; trivial.
+apply Zplus_opp_r.
+Qed.
+
+Theorem Zle_left_rev : forall n m:Z, 0 <= m + - n -> n <= m.
+Proof.
+intros x y H; apply Zplus_le_reg_r with (- x).
+rewrite Zplus_opp_r; trivial.
+Qed.
+
+Theorem Zlt_left_rev : forall n m:Z, 0 < m + - n -> n < m.
+Proof.
+intros x y H; apply Zplus_lt_reg_r with (- x).
+rewrite Zplus_opp_r; trivial.
+Qed.
+
+Theorem Zlt_left : forall n m:Z, n < m -> 0 <= m + -1 + - n.
+Proof.
+intros x y H; apply Zle_left; apply Zsucc_le_reg;
+ change (Zsucc x <= Zsucc (Zpred y)) in |- *; rewrite <- Zsucc_pred;
+ apply Zlt_le_succ; assumption.
+Qed.
+
+Theorem Zlt_left_lt : forall n m:Z, n < m -> 0 < m + - n.
+Proof.
+intros x y H; replace 0 with (x + - x).
+apply Zplus_lt_compat_r; trivial.
+apply Zplus_opp_r.
+Qed.
+
+Theorem Zge_left : forall n m:Z, n >= m -> 0 <= n + - m.
+Proof.
+intros x y H; apply Zle_left; apply Zge_le; assumption.
+Qed.
+
+Theorem Zgt_left : forall n m:Z, n > m -> 0 <= n + -1 + - m.
+Proof.
+intros x y H; apply Zlt_left; apply Zgt_lt; assumption.
+Qed.
+
+Theorem Zgt_left_gt : forall n m:Z, n > m -> n + - m > 0.
+Proof.
+intros x y H; replace 0 with (y + - y).
+apply Zplus_gt_compat_r; trivial.
+apply Zplus_opp_r.
+Qed.
+
+Theorem Zgt_left_rev : forall n m:Z, n + - m > 0 -> n > m.
+Proof.
+intros x y H; apply Zplus_gt_reg_r with (- y).
+rewrite Zplus_opp_r; trivial.
+Qed.
+
+(**********************************************************************)
+(** Factorization lemmas *)
+
+Theorem Zred_factor0 : forall n:Z, n = n * 1.
+intro x; rewrite (Zmult_1_r x); reflexivity.
+Qed.
+
+Theorem Zred_factor1 : forall n:Z, n + n = n * 2.
+Proof.
+exact Zplus_diag_eq_mult_2.
+Qed.
+
+Theorem Zred_factor2 : forall n m:Z, n + n * m = n * (1 + m).
+
+intros x y; pattern x at 1 in |- *; rewrite <- (Zmult_1_r x);
+ rewrite <- Zmult_plus_distr_r; trivial with arith.
+Qed.
+
+Theorem Zred_factor3 : forall n m:Z, n * m + n = n * (1 + m).
+
+intros x y; pattern x at 2 in |- *; rewrite <- (Zmult_1_r x);
+ rewrite <- Zmult_plus_distr_r; rewrite Zplus_comm; 
+ trivial with arith.
+Qed.
+Theorem Zred_factor4 : forall n m p:Z, n * m + n * p = n * (m + p).
+intros x y z; symmetry  in |- *; apply Zmult_plus_distr_r.
+Qed.
+
+Theorem Zred_factor5 : forall n m:Z, n * 0 + m = m.
+
+intros x y; rewrite <- Zmult_0_r_reverse; auto with arith.
+Qed.
+
+Theorem Zred_factor6 : forall n:Z, n = n + 0.
+
+intro; rewrite Zplus_0_r; trivial with arith.
+Qed.
+
+Theorem Zle_mult_approx :
+ forall n m p:Z, n > 0 -> p > 0 -> 0 <= m -> 0 <= m * n + p.
+
+intros x y z H1 H2 H3; apply Zle_trans with (m := y * x);
+ [ apply Zmult_gt_0_le_0_compat; assumption
+ | pattern (y * x) at 1 in |- *; rewrite <- Zplus_0_r;
+    apply Zplus_le_compat_l; apply Zlt_le_weak; apply Zgt_lt; 
+    assumption ].
+Qed.
+
+Theorem Zmult_le_approx :
+ forall n m p:Z, n > 0 -> n > p -> 0 <= m * n + p -> 0 <= m.
+
+intros x y z H1 H2 H3; apply Zlt_succ_le; apply Zmult_gt_0_lt_0_reg_r with x;
+ [ assumption
+ | apply Zle_lt_trans with (1 := H3); rewrite <- Zmult_succ_l_reverse;
+    apply Zplus_lt_compat_l; apply Zgt_lt; assumption ].
+
+Qed.