If the equality P=coP were trivially entailed by closure properties alone, it would collapse the polynomial hierarchy in ways inconsistent with widely accepted complexity-theoretic conjectures such as PH being infinite.

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Closure properties(analogous concept in complexity theory): In mathematics and logic, the quality of a system being 'closed' when operations performed on its elements always produce results that also belong to the system.
P=coP(computational complexity theory): A proposed equality between two classes of computational problems—P (problems solvable quickly) and coP (problems whose 'no' answers are verifiable quickly). The statement questions whether these two categories are actually the same.
Polynomial Hierarchy (PH)(Generalization of the NP/coNP distinction to multiple levels of quantifier alternation.): A hierarchy of complexity classes based on the logical representation of computational problems, built by alternating existential and universal quantification, with Δ⁰_P = Σ⁰_P = Π⁰_P = P at the base.
collapse (in complexity theory)(computer science): When different levels of problem difficulty turn out to be the same, like discovering that 'hard' and 'easy' problems are actually equivalent.

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