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plus_lt
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| Theorem Sx_le_Sx_y : forall x y, S x <= S (x + y). | |
| Proof. | |
| induction y as [|y']. | |
| Case "y' = 0". | |
| rewrite <- plus_n_O. | |
| apply le_n. | |
| Case "y' = S y". | |
| apply le_S. | |
| rewrite <- plus_n_Sm. | |
| apply IHy'. | |
| Qed. | |
| Theorem plus_lt : forall n1 n2 m, | |
| n1 + n2 < m -> | |
| n1 < m /\ n2 < m. | |
| Proof. | |
| intros. | |
| unfold "<" in H. | |
| unfold "<". | |
| induction H. | |
| Case "S (n1 + n2) = m". | |
| split. | |
| SCase "S n1 <= S (n1 + n2)". | |
| apply Sx_le_Sx_y. | |
| SCase "S n2 <= S (n1 + n2)". | |
| rewrite plus_comm. | |
| apply Sx_le_Sx_y. | |
| Case "S (n1 + n2) < m". | |
| split. | |
| SCase "S n1 <= S m". | |
| apply le_S. | |
| (* inversion IHle as [F G]. *) | |
| apply IHle. | |
| SCase "S n2 <= S m". | |
| apply le_S. | |
| apply IHle. | |
| Qed. |
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/\means that both hypotheses hold. A more explicit way is to useinversion IHle as [F G]first to split them apart.