diff options
Diffstat (limited to 'theories/Init/Peano.v')
| -rw-r--r-- | theories/Init/Peano.v | 17 |
1 files changed, 9 insertions, 8 deletions
diff --git a/theories/Init/Peano.v b/theories/Init/Peano.v index c0416b63..3df2b566 100644 --- a/theories/Init/Peano.v +++ b/theories/Init/Peano.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Peano.v 8642 2006-03-17 10:09:02Z notin $ i*) +(*i $Id: Peano.v 9245 2006-10-17 12:53:34Z notin $ i*) (** The type [nat] of Peano natural numbers (built from [O] and [S]) is defined in [Datatypes.v] *) @@ -47,14 +47,16 @@ Hint Resolve (f_equal pred): v62. Theorem pred_Sn : forall n:nat, n = pred (S n). Proof. - auto. + simpl; reflexivity. Qed. (** Injectivity of successor *) Theorem eq_add_S : forall n m:nat, S n = S m -> n = m. Proof. - intros n m H; change (pred (S n) = pred (S m)) in |- *; auto. + intros n m Sn_eq_Sm. + replace (n=m) with (pred (S n) = pred (S m)) by auto using pred_Sn. + rewrite Sn_eq_Sm; trivial. Qed. Hint Immediate eq_add_S: core v62. @@ -65,19 +67,18 @@ Proof. Qed. Hint Resolve not_eq_S: core v62. -(** Zero is not the successor of a number *) - Definition IsSucc (n:nat) : Prop := match n with | O => False | S p => True end. +(** Zero is not the successor of a number *) + Theorem O_S : forall n:nat, 0 <> S n. Proof. - red in |- *; intros n H. - change (IsSucc 0) in |- *. - rewrite <- (sym_eq (x:=0) (y:=(S n))); [ exact I | assumption ]. + unfold not; intros n H. + inversion H. Qed. Hint Resolve O_S: core v62. |