I included that to try to explain the symbol soup that correctly encodes the preconditions (the ∀ lines). I intended that to mean "I need to make sure that y+x doesn't overflow", even that unsigned arithmetic cannot express that precondition in that way, as you point out. From there, derive y ≤ INT_MAX-x as the actual precondition for unsigned addition. I forgot that "ensure" actually means something in some programming languages, sorry for the confusion.

More simply put, unsigned addition needs to check that y+x doesn't overflow, while signed addition needs to check that y+x doesn't overflow and doesn't underflow. So, unsigned arithmetic has a simpler precondition that would win a technical debate on whether to use signed or unsigned arithmetic, but since most programming languages lack theorem proving, signed arithmetic wins on the small integer assumption.