The sample above is only true on some platforms that only use a simple 'C' locale, where individual bytes are considered as complete characters that are converted to lowercase before being differentiated.
Other locales (see LC_COLLATE and LC_ALL) use the difference of collation order of characters, where characters may be groups of bytes taken from the input strings, or simply return -1, 0, or 1 as the collation order is not simply defined by comparing individual characters but by more complex rules.
Don't base your code on a specific non null value returned by strcmp() or strcasecmp(): it is not portable. Just consider the sign of the result and be sure to use the correct locale!
str_word_count