simpl in |- *. simple induction m. simpl in |- *. intros. absurd (0 < 0). apply le_not_lt. auto with arith.
exact H.
intros. simpl in |- *. elim plus_n_Sm. apply lt_n_S. auto with arith.
intros. inversion H0. simpl in |- *. repeat elim plus_n_Sm. auto with arith.
simpl in |- *.
repeat elim plus_n_Sm. apply lt_n_S. apply lt_n_S. apply H. inversion H1. auto with arith.
apply le_lt_n_Sm. apply le_Sn_le. apply le_Sn_le. exact H3.
Qed.

(* Specifications: *)
Lemma le_plus_lem1 : forall a b c : nat, a <= b -> c + a <= c + b.
intros. inversion H. auto with arith.