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but if the resulting work depends on someone else work.Yes, so obviously your argument cannot be that "in theory, we could replace this with a workalike".
You better have the workalike, and that's what you should be shipping.
A powerful argument that you aren't infringing is that your shipping media are completely devoid of the work.
> don't care if its use or integration
For the sake of the GPL, they must care in this case, because the GPL specifically abstains from dictating use; it governs redistribution!
The only parts of the license relevant to use are the disclaimers; the only reason a pure user of a GPL-ed program might want to read the license at all is to be informed that if the program causes loss of data (or whatever), the authors are not liable.
GPLed programs get used all the time. A proprietary app on a GNU/Linux system can use the C library function system() which might invoke /bin/sh that is GNU Bash, and even depend on that functionality.
> For example, if I buy a painting and cut it into pieces and rearrange them, I actually need an additional license beyond what I got from purchasing the copy.
But what if that cut-up never leaves my house?
Or what if I only distribute instructions which describe the geometry of some cuts which can be made to a painting, and the relocation of the pieces?