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To retain good logical properties, all functions definable in Coq are terminating. To enforce that, there is a restriction on fixpoint definitions, like the one you are trying to do, called the guard condition. This restriction is roughly that the recursive call can only be done on subterms of the argument of the function.

This is not the case in your definition, where you apply nat_to_bin to the terms (Nat.div n 2) and (Nat.modulo n 2) which are functions applied to n. Although you can mathematically prove that those are always smaller than n, they are no subterms of n, so your function does not respect the guard condition.

If you wanted to define nat_to_bin in the way you are doing, you would need to resort to well-founded induction, which would use the well-foundedness of the order on nat to allow you to call you function on any term you can prove smaller than n. However, this solution is quite complex, because it would force you to do some proofs that are not that easy.

Instead, I would advise going another way: just above in the book, it is suggested to define a function incr : bin -> bin that increments a binary number by one. You can use that one to define nat_to_bin by a simple recursion on n, like this: