Qed.
Ltac to_nat_op  :=
match goal with
H: (N.lt _ _) |- _ => generalize (Nlt_lt_rev _ _ H); clear H; intros H
|     H: (N.gt _ _) |- _ => generalize (Ngt_gt_rev _ _ H); clear H; intros H
|     H: (N.le _ _) |- _ => generalize (Nle_le_rev _ _ H); clear H; intros H
|     H: (N.ge _ _) |- _ => generalize (Nge_ge_rev _ _ H); clear H; intros H
|     H: (@eq N _ _) |- _ => generalize (Neq_eq_rev _ _ H); clear H; intros H
|      |- (N.lt _ _)  => apply Nlt_lt
|      |- (N.le _ _)  => apply Nle_le
|      |- (N.gt _ _)  => apply Ngt_gt
|      |- (N.ge _ _)  => apply Nge_ge
|      |- (@eq N _ _)  => apply Nat2N.inj
end.
Ltac set_to_nat :=
let nn := fresh "nn" in
match goal with
|- context [(N.to_nat (?X + ?Y)%N)]  => rewrite N2Nat.inj_add
|      |- context [(N.to_nat (?X * ?Y)%N)]  => rewrite N2Nat.inj_mul
|      |- context [(N.to_nat ?X)]  => set (nn:=N.to_nat X) in * |- *
|      H: context [(N.to_nat (?X + ?Y)%N)] |- _ => rewrite N2Nat.inj_add in H
|      H: context [(N.to_nat (?X + ?Y)%N)] |- _ => rewrite N2Nat.inj_mul in H
|      H: context [(N.to_nat ?X)] |- _ => set (nn:=N.to_nat X) in * |- *
end.
Ltac to_nat := repeat to_nat_op; repeat set_to_nat.

Theorem Nle_gt_trans: forall n m p, m <= n -> m > p -> n > p.
intros; to_nat; apply le_gt_trans with nn1; auto.