diff options
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NSub.v | 9 |
1 files changed, 9 insertions, 0 deletions
diff --git a/theories/Numbers/Natural/Abstract/NSub.v b/theories/Numbers/Natural/Abstract/NSub.v index 35d3b8aa22..de0c2da1b0 100644 --- a/theories/Numbers/Natural/Abstract/NSub.v +++ b/theories/Numbers/Natural/Abstract/NSub.v @@ -205,6 +205,15 @@ Proof. intros n m p. rewrite add_comm; apply lt_add_lt_sub_r. Qed. +Theorem sub_lt : forall n m, m <= n -> 0 < m -> n - m < n. +Proof. +intros n m LE LT. +assert (LE' := le_sub_l n m). rewrite lt_eq_cases in LE'. +destruct LE' as [LT'|EQ]. assumption. +apply add_sub_eq_nz in EQ; [|order]. +rewrite (add_lt_mono_r _ _ n), add_0_l in LT. order. +Qed. + (** Sub and mul *) Theorem mul_pred_r : forall n m, n * (P m) == n * m - n. |