Technically, it should say it lacks built-in arithmetic instructions. Any processor can add, subtract, multiply, or fixed width integers via some combination of bitwise operations. For example, https://en.wikipedia.org/wiki/Bitwise_operation#Applications shows how to do unsigned addition and multiplication.
And, of course, for better or worse, "add two numbers without using the + operator" is one of those silly LeetCode trivia questions software engineers occasionally get quizzed on.
Yes, addition and all other arithmetic and logical operations can be implemented using only one building block: Peirce's arrow (Logical NOR). I can't understand why you are getting downvoted.
Subtract is a "built-in arithmetic instructions" so it wouldn't be allowed in this environment. The idea is to implement arithmetic, but from bitwise operations.
https://en.wikipedia.org/wiki/Bitwise_operation
This is getting far afield from the actual Nanoprocessor, which has increment and decrement instructions. With those, you can add two numbers by repeatedly incrementing one and decrementing the other. (Which is a bit like addition using the Peano axioms.)
See page 126 of the Nanoprocessor User's Manual for code to add two numbers in this way.
Interesting problem. In this case though, I was just offering a pointless optimization to the parent post, which also used subtraction, so I assumed it was allowed.
And, of course, for better or worse, "add two numbers without using the + operator" is one of those silly LeetCode trivia questions software engineers occasionally get quizzed on.