diff options
Diffstat (limited to 'theories/Numbers/Natural/Abstract/NPlus.v')
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NPlus.v | 12 |
1 files changed, 6 insertions, 6 deletions
diff --git a/theories/Numbers/Natural/Abstract/NPlus.v b/theories/Numbers/Natural/Abstract/NPlus.v index 67a2766ba..f1e0e6ace 100644 --- a/theories/Numbers/Natural/Abstract/NPlus.v +++ b/theories/Numbers/Natural/Abstract/NPlus.v @@ -2,13 +2,13 @@ Require Export NBase. Module NPlusPropFunct (Import NAxiomsMod : NAxiomsSig). Module Export NBasePropMod := NBasePropFunct NAxiomsMod. -Open Local Scope NatScope. +Open Local Scope NatIntScope. Theorem plus_0_l : forall n : N, 0 + n == n. -Proof plus_0_l. +Proof NZplus_0_l. Theorem plus_succ_l : forall n m : N, (S n) + m == S (n + m). -Proof plus_succ_l. +Proof NZplus_succ_l. (** Theorems that are valid for both natural numbers and integers *) @@ -42,7 +42,7 @@ Proof NZplus_cancel_l. Theorem plus_cancel_r : forall n m p : N, n + p == m + p <-> n == m. Proof NZplus_cancel_r. -(** Theorems that are valid for natural numbers but cannot be proved for NZ *) +(** Theorems that are valid for natural numbers but cannot be proved for N *) Theorem plus_eq_0 : forall n m : N, n + m == 0 <-> n == 0 /\ m == 0. Proof. @@ -58,7 +58,7 @@ Theorem plus_eq_succ : forall n m : N, (exists p : N, n + m == S p) <-> (exists n' : N, n == S n') \/ (exists m' : N, m == S m'). Proof. -intros n m; nondep_induct n. +intros n m; cases n. split; intro H. destruct H as [p H]. rewrite plus_0_l in H; right; now exists p. destruct H as [[n' H] | [m' H]]. @@ -91,7 +91,7 @@ Qed. Theorem plus_pred_l : forall n m : N, n ~= 0 -> P n + m == P (n + m). Proof. -intros n m; nondep_induct n. +intros n m; cases n. intro H; now elim H. intros n IH; rewrite plus_succ_l; now do 2 rewrite pred_succ. Qed. |