Proof that the decrement operator can be achieved [exclusively with binary operations]

A friend of mine was recently telling me how he’d done an electronics assignment where for extra credit they had to create an in-place decrement operator (by that I mean not taking the approach of taking the 2s complement of -1 and adding it to a number). This sounded cool and seemed like the way the ALU would do it to save allocation overhead and space.

Here’s the identity he told me about:

where:
“~” represents the binary complement
“-“ represents the binary negative, aka 2s complement of the number.

Looks really bizarre but feels like it should make sense. I rooted about on Google search to find the proof, but either couldn’t find it, or more likely didn’t have my search terms fine tuned! (See http://www.woodmann.com/searchlores/)

Here’s the forward proof I came to. (using ‘=’ to mean comparison not assignment here)

due to 2s complement we know that \(\texttt{~}x +1 = -x\), therefore \(x+1 = -(\texttt{~}x)\), substituting this gives

as we know the negation is 2s complement of the brackets, therefore

now sub back our other formula from above

and here we are

If you notice any errors above, or if you just need to “Throw some knowledgeballs down the webhills!”, please flame me on the appropriate public channels.