Cook's theorem establishes NP-completeness under polynomial-time many-one reductions, but Buss's bounded arithmetic formalizes this within systems of varying proof-theoretic strength.

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Buss's bounded arithmetic(as used in mathematical logic): A mathematical system created by Samuel Buss that formalizes reasoning about computation while limiting the strength of logical tools available—similar to restricting which tools a carpenter can use.
Cook's theorem(as used in computational complexity theory): A foundational result in computer science (proved by Stephen Cook in 1971) that identifies a special class of problems called NP-complete—problems that are equally hard to solve and to verify.
NP-completeness(as used in computational complexity theory): A classification in computer science for problems that are extremely difficult to solve quickly, even though verifying a correct answer is easy. Think of it like a jigsaw puzzle: hard to assemble, but easy to check if someone else did it right.
Proof-theoretic strength(as used in mathematical logic): The power and capability of a logical system to prove theorems and demonstrate truths (what it can accomplish).

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