Discussion
What does a satisfiability solver return when a formula is satisfiable?
Answer: A
The solver returns a model, which can be checked independently in linear time, so a positive answer is easy to verify. A negative answer requires a refutation proof, which good solvers can also emit. Counting models and eliminating variables are different problems, the first being considerably harder. That a yes answer is cheaply checkable while a no answer needs a proof is an asymmetry worth exploiting whenever a tool's output must be trusted. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.
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