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I want to create a commercial application, but I want to make a significant portion of my back-end open source. It applies some really interesting math that I want people to learn from, but I don't want others to take that back-end code and start selling it. As a result, I'd like to license that back-end code under either the GPL or RPL (I'm not sure which one yet).

These licenses prevent people from distributing software using code licensed other these licenses without making its code open-source. I don't want to make my entire program open-source, just that part of it.

Does licensing part of my own software force me to release the rest of my software under the GPL/RPL? If it does, will anyone actually enforce it?

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You wrote:

These licenses prevent people from commercially distributing software using code licensed other these licenses without making its code open-source. I don't want to make my entire program open-source, just that part of it.

There is no commercial redistribution restrictions in the GPL and in any license that would be considered truly open source: this would an incompatible restriction.

For the rest, read this answer by @apsilers:

The copyright holder is never beholden to the rules of the holder's own license grant! The copyright holder's rights stem from copyright law, not from a license grant. The copyright holder can legally prepare a derivative of her own work based on her rights under copyright

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