Require Export NArith.
Open Scope N_scope.
intros n m; case n; case m; unfold N.le; simpl; try (intros; discriminate).
intros p; elim p using Pind; simpl.
intros H1; inversion H1.
intros n1 _; rewrite nat_of_P_succ_morphism.
intros H1; inversion H1.
intros p1 p2 H1 H2; absurd (nat_of_P p2 > nat_of_P p1)%nat; auto with arith.
apply nat_of_P_gt_Gt_compare_morphism; auto.
Qed.
intros n m; case n; case m; unfold N.le; simpl; auto with arith.
intros n1 H1; case H1; auto.
intros m1 n1 H1; case (le_or_lt n1 m1); auto with arith.
intros H2; case H1.
apply nat_of_P_gt_Gt_compare_complement_morphism.
repeat rewrite  nat_of_P_o_P_of_succ_nat_eq_succ; auto with arith.
Qed.
intros; apply le_Nle; repeat rewrite N2Nat.id; auto.
Qed.
intros n m; case n; case m; unfold N.lt; simpl; try (intros; discriminate); auto.
intros H1; inversion H1.
intros p H1; inversion H1.
intros; apply nat_of_P_lt_Lt_compare_complement_morphism; auto.
Qed.
intros n m; case n; case m; unfold N.lt; simpl; try (intros; discriminate); auto with arith.
intros m1 n1 H1.
rewrite <- (Nat2N.id (S n1)); rewrite <- (Nat2N.id (S m1)).
simpl; apply nat_of_P_lt_Lt_compare_morphism; auto.
Qed.
intros; apply lt_Nlt; repeat rewrite N2Nat.id; auto.
Qed.
intros n m; case n; case m; unfold N.ge; simpl; try (intros; discriminate); auto.
intros p; elim p using Pind; simpl.
intros H1; inversion H1.
intros n1 _; rewrite nat_of_P_succ_morphism.
intros H1; inversion H1.
intros p1 p2 H1 H2; absurd (nat_of_P p2 < nat_of_P p1)%nat; auto with arith.
apply nat_of_P_lt_Lt_compare_morphism; auto.
Qed.
intros n m; case n; case m; unfold N.ge; simpl; try (intros; discriminate); auto with arith.
intros n1 H1; case H1; auto.
intros m1 n1 H1.
case (le_or_lt m1 n1); auto with arith.
intros H2; case H1.
apply nat_of_P_lt_Lt_compare_complement_morphism.
repeat rewrite  nat_of_P_o_P_of_succ_nat_eq_succ; auto with arith.
Qed.
intros; apply ge_Nge; repeat rewrite N2Nat.id; auto.
Qed.

Theorem Ngt_gt: forall n  m, (N.to_nat n > N.to_nat m)%nat -> n > m.