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I have always found term rewriting a fascinating idea as a programing paradigm and the Wolfram language seems very cool. It is a shame that it is a proprietary language.

An even more insane language based on term rewriting is the Egison programming language [1]. Egison even supports pattern matching on infinite data structures, it is pretty cool.

[1]: https://www.egison.org/

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Mathematica is a language. The implementation is proprietary, but anyone could create a new implementation, similarly to what Octave did to MATLAB.

That is a good point and there even is a sort of open source implementation of the Wolfram language, Mathics [1]. Although I would assume its value is kind of limited without the standard library included with Mathematica.

[1]: https://mathics.org/

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