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).