diff options
Diffstat (limited to 'theories/Arith/Plus.v')
| -rw-r--r-- | theories/Arith/Plus.v | 25 |
1 files changed, 6 insertions, 19 deletions
diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v index 48d730319..eb2d4df4c 100644 --- a/theories/Arith/Plus.v +++ b/theories/Arith/Plus.v @@ -24,17 +24,10 @@ Open Local Scope nat_scope. Implicit Types m n p q : nat. -(** * Zero is neutral *) - -Lemma plus_0_l : forall n, 0 + n = n. -Proof. - reflexivity. -Qed. - -Lemma plus_0_r : forall n, n + 0 = n. -Proof. - intro; symmetry in |- *; apply plus_n_O. -Qed. +(** * Zero is neutral +Deprecated : Already in Init/Peano.v *) +Definition plus_0_l n := eq_sym (plus_O_n n). +Definition plus_0_r n := eq_sym (plus_n_O n). (** * Commutativity *) @@ -47,14 +40,8 @@ Hint Immediate plus_comm: arith v62. (** * Associativity *) -Lemma plus_Snm_nSm : forall n m, S n + m = n + S m. -Proof. - intros. - simpl in |- *. - rewrite (plus_comm n m). - rewrite (plus_comm n (S m)). - trivial with arith. -Qed. +Definition plus_Snm_nSm : forall n m, S n + m = n + S m:= + plus_n_Sm. Lemma plus_assoc : forall n m p, n + (m + p) = n + m + p. Proof. |