proposition: Lemma gt_double : forall a b : nat, a + a > b + b -> a > b.
proof: Proof. simple induction a. simpl in |- *. intros. absurd (0 > b + b). auto with arith. auto with arith. simple induction b. simpl in |- *. auto with arith. intros. apply gt_n_S. apply H. generalize H1. simpl in |- *. elim plus_n_Sm. elim plus_n_Sm. intro. cut (S (n + n) > S (n0 + n0)). intro. apply gt_S_n. exact H3. apply gt_S_n. exact H2. Qed.
imports: Plus Mult Minus Lt Le Gt