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"Book of Proof" is about basic objects in higher mathematics and writing proofs (as a human). There are only about 30 pages covering the basics of mathematical logic. So, I imagine Common Logic would not be very related to this book, since CL is much more advanced than the book's aims.

If you are referring to translating the book's theorems for an automated theorem prover, the book is not nearly pedantic enough to effectively do this from the ground up.

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