diff options
Diffstat (limited to 'theories/Arith/Div2.v')
| -rw-r--r-- | theories/Arith/Div2.v | 26 |
1 files changed, 13 insertions, 13 deletions
diff --git a/theories/Arith/Div2.v b/theories/Arith/Div2.v index 24cbc3f9..da1d9e98 100644 --- a/theories/Arith/Div2.v +++ b/theories/Arith/Div2.v @@ -11,7 +11,7 @@ Require Import Plus. Require Import Compare_dec. Require Import Even. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Type n : nat. @@ -69,24 +69,24 @@ Proof. (* S n *) inversion_clear H. apply even_div2 in H0 as <-. trivial. Qed. -Lemma div2_even : forall n, div2 n = div2 (S n) -> even n -with div2_odd : forall n, S (div2 n) = div2 (S n) -> odd n. +Lemma div2_even n : div2 n = div2 (S n) -> even n +with div2_odd n : S (div2 n) = div2 (S n) -> odd n. Proof. - destruct n; intro H. - (* 0 *) constructor. - (* S n *) constructor. apply div2_odd. rewrite H. trivial. - destruct n; intro H. - (* 0 *) discriminate. - (* S n *) constructor. apply div2_even. injection H as <-. trivial. +{ destruct n; intro H. + - constructor. + - constructor. apply div2_odd. rewrite H. trivial. } +{ destruct n; intro H. + - discriminate. + - constructor. apply div2_even. injection H as <-. trivial. } Qed. Hint Resolve even_div2 div2_even odd_div2 div2_odd: arith. -Lemma even_odd_div2 : - forall n, - (even n <-> div2 n = div2 (S n)) /\ (odd n <-> S (div2 n) = div2 (S n)). +Lemma even_odd_div2 n : + (even n <-> div2 n = div2 (S n)) /\ + (odd n <-> S (div2 n) = div2 (S n)). Proof. - auto decomp using div2_odd, div2_even, odd_div2, even_div2. + split; split; auto using div2_odd, div2_even, odd_div2, even_div2. Qed. |