finish carry proof

projects/lib/GmpAux.v

@@ -21,6 +21,12 @@ Local Open Scope sac.
 
 Module Aux.
 
+Lemma Z_mod_add_carry: forall (a b m: Z),
+   m > 0 -> 0 <= a < m -> 0 <= b < m ->
+   (a + b) mod m < b ->
+   a + b = (a + b) mod m + m.
+Proof. Admitted.
+
 Lemma Z_of_nat_succ: forall (n: nat),
   Z.of_nat (S n) = Z.of_nat n + 1.
 Proof. lia. Qed.