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Suppose $L$ is an NP language, and its witness checking algorithm is $R$, so that $L = \{ x \mid \exists w : R(x,w) = 1 \}$. Here is how I can prove to you that $x \in L$: Generate a circuit $C$ such that $C(w) = R(x,w)$. We can both do this because $R$ is a public algorithm and $x$ is also public. This circuit $C$ has the instance $x$ "hard-coded"...

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