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I'm trying to prove that if P = NP, then {⟨a, b, c⟩ : a + b = c} (as addition over N) is NP-complete.

I think I managed to prove that it is in NP, but I'm not sure what would be a good NP complete problem to reduce to would be, or what algorithm to use. Any ideas?

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Your language is in P; try to figure out why. If P=NP, all non-trivial languages in P are NP-complete, and in particular this one.

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