Definition reduced_string_product (S1 S2:group_str) : group_str :=
  reduction (S1 ++ S2).

Theorem reduced_string_product_assoc: forall (S1 S2 S3:group_str), reduced_string_product (reduced_string_product S1 S2) S3 = reduced_string_product S1 (reduced_string_product S2 S3).