# Library Coq.NArith.Ndec

``` Require Import Bool. Require Import Sumbool. Require Import Arith. Require Import BinPos. Require Import BinNat. Require Import Pnat. Require Import Nnat. Require Import Ndigits. ```
A boolean equality over `N`
``` Fixpoint Peqb (p1 p2:positive) {struct p2} : bool :=   match p1, p2 with   | xH, xH => true   | xO p'1, xO p'2 => Peqb p'1 p'2   | xI p'1, xI p'2 => Peqb p'1 p'2   | _, _ => false   end. Lemma Peqb_correct : forall p, Peqb p p = true. Proof. induction p; auto. Qed. Lemma Peqb_Pcompare : forall p p', Peqb p p' = true -> Pcompare p p' Eq = Eq. Proof.  induction p; destruct p'; simpl; intros; try discriminate; auto. Qed. Lemma Pcompare_Peqb : forall p p', Pcompare p p' Eq = Eq -> Peqb p p' = true. Proof. intros; rewrite <- (Pcompare_Eq_eq _ _ H). apply Peqb_correct. Qed. Definition Neqb (a a':N) :=   match a, a' with   | N0, N0 => true   | Npos p, Npos p' => Peqb p p'   | _, _ => false   end. Lemma Neqb_correct : forall n, Neqb n n = true. Proof.   destruct n; trivial.   simpl; apply Peqb_correct. Qed. Lemma Neqb_Ncompare : forall n n', Neqb n n' = true -> Ncompare n n' = Eq. Proof.  destruct n; destruct n'; simpl; intros; try discriminate; auto; apply Peqb_Pcompare; auto. Qed. Lemma Ncompare_Neqb : forall n n', Ncompare n n' = Eq -> Neqb n n' = true. Proof. intros; rewrite <- (Ncompare_Eq_eq _ _ H). apply Neqb_correct. Qed. Lemma Neqb_complete : forall a a', Neqb a a' = true -> a = a'. Proof.   intros.   apply Ncompare_Eq_eq.   apply Neqb_Ncompare; auto. Qed. Lemma Neqb_comm : forall a a', Neqb a a' = Neqb a' a. Proof.   intros; apply bool_1; split; intros.   rewrite (Neqb_complete _ _ H); apply Neqb_correct.   rewrite (Neqb_complete _ _ H); apply Neqb_correct. Qed. Lemma Nxor_eq_true :  forall a a', Nxor a a' = N0 -> Neqb a a' = true. Proof.   intros. rewrite (Nxor_eq a a' H). apply Neqb_correct. Qed. Lemma Nxor_eq_false :  forall a a' p, Nxor a a' = Npos p -> Neqb a a' = false. Proof.   intros. elim (sumbool_of_bool (Neqb a a')). intro H0.   rewrite (Neqb_complete a a' H0) in H. rewrite (Nxor_nilpotent a') in H. discriminate H.   trivial. Qed. Lemma Nodd_not_double :  forall a,    Nodd a -> forall a0, Neqb (Ndouble a0) a = false. Proof.   intros. elim (sumbool_of_bool (Neqb (Ndouble a0) a)). intro H0.   rewrite <- (Neqb_complete _ _ H0) in H.   unfold Nodd in H.   rewrite (Ndouble_bit0 a0) in H. discriminate H.   trivial. Qed. Lemma Nnot_div2_not_double :  forall a a0,    Neqb (Ndiv2 a) a0 = false -> Neqb a (Ndouble a0) = false. Proof.   intros. elim (sumbool_of_bool (Neqb (Ndouble a0) a)). intro H0.   rewrite <- (Neqb_complete _ _ H0) in H. rewrite (Ndouble_div2 a0) in H.   rewrite (Neqb_correct a0) in H. discriminate H.   intro. rewrite Neqb_comm. assumption. Qed. Lemma Neven_not_double_plus_one :  forall a,    Neven a -> forall a0, Neqb (Ndouble_plus_one a0) a = false. Proof.   intros. elim (sumbool_of_bool (Neqb (Ndouble_plus_one a0) a)). intro H0.   rewrite <- (Neqb_complete _ _ H0) in H.   unfold Neven in H.   rewrite (Ndouble_plus_one_bit0 a0) in H.   discriminate H.   trivial. Qed. Lemma Nnot_div2_not_double_plus_one :  forall a a0,    Neqb (Ndiv2 a) a0 = false -> Neqb (Ndouble_plus_one a0) a = false. Proof.   intros. elim (sumbool_of_bool (Neqb a (Ndouble_plus_one a0))). intro H0.   rewrite (Neqb_complete _ _ H0) in H. rewrite (Ndouble_plus_one_div2 a0) in H.   rewrite (Neqb_correct a0) in H. discriminate H.   intro H0. rewrite Neqb_comm. assumption. Qed. Lemma Nbit0_neq :  forall a a',    Nbit0 a = false -> Nbit0 a' = true -> Neqb a a' = false. Proof.   intros. elim (sumbool_of_bool (Neqb a a')). intro H1. rewrite (Neqb_complete _ _ H1) in H.   rewrite H in H0. discriminate H0.   trivial. Qed. Lemma Ndiv2_eq :  forall a a', Neqb a a' = true -> Neqb (Ndiv2 a) (Ndiv2 a') = true. Proof.   intros. cut (a = a'). intros. rewrite H0. apply Neqb_correct.   apply Neqb_complete. exact H. Qed. Lemma Ndiv2_neq :  forall a a',    Neqb (Ndiv2 a) (Ndiv2 a') = false -> Neqb a a' = false. Proof.   intros. elim (sumbool_of_bool (Neqb a a')). intro H0.   rewrite (Neqb_complete _ _ H0) in H. rewrite (Neqb_correct (Ndiv2 a')) in H. discriminate H.   trivial. Qed. Lemma Ndiv2_bit_eq :  forall a a',    Nbit0 a = Nbit0 a' -> Ndiv2 a = Ndiv2 a' -> a = a'. Proof.   intros. apply Nbit_faithful. unfold eqf in |- *. destruct n.   rewrite Nbit0_correct. rewrite Nbit0_correct. assumption.   rewrite <- Ndiv2_correct. rewrite <- Ndiv2_correct.   rewrite H0. reflexivity. Qed. Lemma Ndiv2_bit_neq :  forall a a',    Neqb a a' = false ->    Nbit0 a = Nbit0 a' -> Neqb (Ndiv2 a) (Ndiv2 a') = false. Proof.   intros. elim (sumbool_of_bool (Neqb (Ndiv2 a) (Ndiv2 a'))). intro H1.   rewrite (Ndiv2_bit_eq _ _ H0 (Neqb_complete _ _ H1)) in H.   rewrite (Neqb_correct a') in H. discriminate H.   trivial. Qed. Lemma Nneq_elim :  forall a a',    Neqb a a' = false ->    Nbit0 a = negb (Nbit0 a') \/    Neqb (Ndiv2 a) (Ndiv2 a') = false. Proof.   intros. cut (Nbit0 a = Nbit0 a' \/ Nbit0 a = negb (Nbit0 a')).   intros. elim H0. intro. right. apply Ndiv2_bit_neq. assumption.   assumption.   intro. left. assumption.   case (Nbit0 a); case (Nbit0 a'); auto. Qed. Lemma Ndouble_or_double_plus_un :  forall a,    {a0 : N | a = Ndouble a0} + {a1 : N | a = Ndouble_plus_one a1}. Proof.   intro. elim (sumbool_of_bool (Nbit0 a)). intro H. right. split with (Ndiv2 a).   rewrite (Ndiv2_double_plus_one a H). reflexivity.   intro H. left. split with (Ndiv2 a). rewrite (Ndiv2_double a H). reflexivity. Qed. ```
A boolean order on `N`
``` Definition Nle (a b:N) := leb (nat_of_N a) (nat_of_N b). Lemma Nle_Ncompare : forall a b, Nle a b = true <-> Ncompare a b <> Gt. Proof.  intros; rewrite nat_of_Ncompare.  unfold Nle; apply leb_compare. Qed. Lemma Nle_refl : forall a, Nle a a = true. Proof.   intro. unfold Nle in |- *. apply leb_correct. apply le_n. Qed. Lemma Nle_antisym :  forall a b, Nle a b = true -> Nle b a = true -> a = b. Proof.   unfold Nle in |- *. intros. rewrite <- (N_of_nat_of_N a). rewrite <- (N_of_nat_of_N b).   rewrite (le_antisym _ _ (leb_complete _ _ H) (leb_complete _ _ H0)). reflexivity. Qed. Lemma Nle_trans :  forall a b c, Nle a b = true -> Nle b c = true -> Nle a c = true. Proof.   unfold Nle in |- *. intros. apply leb_correct. apply le_trans with (m := nat_of_N b).   apply leb_complete. assumption.   apply leb_complete. assumption. Qed. Lemma Nle_lt_trans :  forall a b c,    Nle a b = true -> Nle c b = false -> Nle c a = false. Proof.   unfold Nle in |- *. intros. apply leb_correct_conv. apply le_lt_trans with (m := nat_of_N b).   apply leb_complete. assumption.   apply leb_complete_conv. assumption. Qed. Lemma Nlt_le_trans :  forall a b c,    Nle b a = false -> Nle b c = true -> Nle c a = false. Proof.   unfold Nle in |- *. intros. apply leb_correct_conv. apply lt_le_trans with (m := nat_of_N b).   apply leb_complete_conv. assumption.   apply leb_complete. assumption. Qed. Lemma Nlt_trans :  forall a b c,    Nle b a = false -> Nle c b = false -> Nle c a = false. Proof.   unfold Nle in |- *. intros. apply leb_correct_conv. apply lt_trans with (m := nat_of_N b).   apply leb_complete_conv. assumption.   apply leb_complete_conv. assumption. Qed. Lemma Nlt_le_weak : forall a b:N, Nle b a = false -> Nle a b = true. Proof.   unfold Nle in |- *. intros. apply leb_correct. apply lt_le_weak.   apply leb_complete_conv. assumption. Qed. Lemma Nle_double_mono :  forall a b,    Nle a b = true -> Nle (Ndouble a) (Ndouble b) = true. Proof.   unfold Nle in |- *. intros. rewrite nat_of_Ndouble. rewrite nat_of_Ndouble. apply leb_correct.   simpl in |- *. apply plus_le_compat. apply leb_complete. assumption.   apply plus_le_compat. apply leb_complete. assumption.   apply le_n. Qed. Lemma Nle_double_plus_one_mono :  forall a b,    Nle a b = true ->    Nle (Ndouble_plus_one a) (Ndouble_plus_one b) = true. Proof.   unfold Nle in |- *. intros. rewrite nat_of_Ndouble_plus_one. rewrite nat_of_Ndouble_plus_one.   apply leb_correct. apply le_n_S. simpl in |- *. apply plus_le_compat. apply leb_complete.   assumption.   apply plus_le_compat. apply leb_complete. assumption.   apply le_n. Qed. Lemma Nle_double_mono_conv :  forall a b,    Nle (Ndouble a) (Ndouble b) = true -> Nle a b = true. Proof.   unfold Nle in |- *. intros a b. rewrite nat_of_Ndouble. rewrite nat_of_Ndouble. intro.   apply leb_correct. apply (mult_S_le_reg_l 1). apply leb_complete. assumption. Qed. Lemma Nle_double_plus_one_mono_conv :  forall a b,    Nle (Ndouble_plus_one a) (Ndouble_plus_one b) = true ->    Nle a b = true. Proof.   unfold Nle in |- *. intros a b. rewrite nat_of_Ndouble_plus_one. rewrite nat_of_Ndouble_plus_one.   intro. apply leb_correct. apply (mult_S_le_reg_l 1). apply le_S_n. apply leb_complete.   assumption. Qed. Lemma Nlt_double_mono :  forall a b,    Nle a b = false -> Nle (Ndouble a) (Ndouble b) = false. Proof.   intros. elim (sumbool_of_bool (Nle (Ndouble a) (Ndouble b))). intro H0.   rewrite (Nle_double_mono_conv _ _ H0) in H. discriminate H.   trivial. Qed. Lemma Nlt_double_plus_one_mono :  forall a b,    Nle a b = false ->    Nle (Ndouble_plus_one a) (Ndouble_plus_one b) = false. Proof.   intros. elim (sumbool_of_bool (Nle (Ndouble_plus_one a) (Ndouble_plus_one b))). intro H0.   rewrite (Nle_double_plus_one_mono_conv _ _ H0) in H. discriminate H.   trivial. Qed. Lemma Nlt_double_mono_conv :  forall a b,    Nle (Ndouble a) (Ndouble b) = false -> Nle a b = false. Proof.   intros. elim (sumbool_of_bool (Nle a b)). intro H0. rewrite (Nle_double_mono _ _ H0) in H.   discriminate H.   trivial. Qed. Lemma Nlt_double_plus_one_mono_conv :  forall a b,    Nle (Ndouble_plus_one a) (Ndouble_plus_one b) = false ->    Nle a b = false. Proof.   intros. elim (sumbool_of_bool (Nle a b)). intro H0.   rewrite (Nle_double_plus_one_mono _ _ H0) in H. discriminate H.   trivial. Qed. Definition Nmin (a b:N) := if Nle a b then a else b. Lemma Nmin_choice : forall a b, {Nmin a b = a} + {Nmin a b = b}. Proof.   unfold Nmin in |- *. intros. elim (sumbool_of_bool (Nle a b)). intro H. left. rewrite H.   reflexivity.   intro H. right. rewrite H. reflexivity. Qed. Lemma Nmin_le_1 : forall a b, Nle (Nmin a b) a = true. Proof.   unfold Nmin in |- *. intros. elim (sumbool_of_bool (Nle a b)). intro H. rewrite H.   apply Nle_refl.   intro H. rewrite H. apply Nlt_le_weak. assumption. Qed. Lemma Nmin_le_2 : forall a b, Nle (Nmin a b) b = true. Proof.   unfold Nmin in |- *. intros. elim (sumbool_of_bool (Nle a b)). intro H. rewrite H. assumption.   intro H. rewrite H. apply Nle_refl. Qed. Lemma Nmin_le_3 :  forall a b c, Nle a (Nmin b c) = true -> Nle a b = true. Proof.   unfold Nmin in |- *. intros. elim (sumbool_of_bool (Nle b c)). intro H0. rewrite H0 in H.   assumption.   intro H0. rewrite H0 in H. apply Nlt_le_weak. apply Nle_lt_trans with (b := c); assumption. Qed. Lemma Nmin_le_4 :  forall a b c, Nle a (Nmin b c) = true -> Nle a c = true. Proof.   unfold Nmin in |- *. intros. elim (sumbool_of_bool (Nle b c)). intro H0. rewrite H0 in H.   apply Nle_trans with (b := b); assumption.   intro H0. rewrite H0 in H. assumption. Qed. Lemma Nmin_le_5 :  forall a b c,    Nle a b = true -> Nle a c = true -> Nle a (Nmin b c) = true. Proof.   intros. elim (Nmin_choice b c). intro H1. rewrite H1. assumption.   intro H1. rewrite H1. assumption. Qed. Lemma Nmin_lt_3 :  forall a b c, Nle (Nmin b c) a = false -> Nle b a = false. Proof.   unfold Nmin in |- *. intros. elim (sumbool_of_bool (Nle b c)). intro H0. rewrite H0 in H.   assumption.   intro H0. rewrite H0 in H. apply Nlt_trans with (b := c); assumption. Qed. Lemma Nmin_lt_4 :  forall a b c, Nle (Nmin b c) a = false -> Nle c a = false. Proof.   unfold Nmin in |- *. intros. elim (sumbool_of_bool (Nle b c)). intro H0. rewrite H0 in H.   apply Nlt_le_trans with (b := b); assumption.   intro H0. rewrite H0 in H. assumption. Qed. ```