feat(library/data/nat/sub): additional theorems

library/data/nat/sub.lean

@@ -21,4 +21,18 @@ protected theorem sub_add_cancel : Π{n i : ℕ}, i ≤ n → (n - i) + i = n
                       = ((n - i) + succ i) : congr_arg (λ v, v + succ i) (succ_sub_succ n i)
                   ... = succ n             : congr_arg succ (sub_add_cancel (pred_le_pred p))
 
+theorem add_sub_cancel_left : Π (n m : ℕ), n + m - n = m
+| 0 m := nat.zero_add m
+| (succ a) m :=
+  calc succ a + m - succ a
+            = succ (a + m) - succ a
+            : congr_arg (λz, z - succ a) (nat.succ_add a m)
+        ... = a + m - a : succ_sub_succ_eq_sub (a+m) a
+        ... = m         : add_sub_cancel_left a m
+
+-- defined in data/nat/sub.lean
+protected theorem sub_le_sub_right {n m : ℕ} (H : n ≤ m) : Πk, n - k ≤ m - k
+| 0 := H
+| (succ z) := pred_le_pred (sub_le_sub_right z)
+
 end nat