The battles would be extremely brief:
North Korea devastates Soeul and other cross-border targets using artillery.
They're very quickly hampered by lack of air support and effective anti air. Any overground movement in numbers almost impossible.
America will be able to crack hardened Northern targets eventually, but they'll take a lot of punishment.
Stalemate.
But advancing North? Then it'll all go wrong. Possibility of fanatics everywhere, booby traps etc. You can try to devistate the North Korean economy, but there isn't that much there.
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