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  "body": "\n## Proof\n\nWe present here a [[constructive proof|01XM7JEKHY]] of the Craig interpolation theorem for [[propositional logic|09K9MWD2QQ]].\n\nSince the above proof is constructive, one may extract an algorithm for computing interpolants. Using this algorithm, if *n* = |*atoms*(φ') − *atoms*(ψ)|, then the interpolant ρ has *O*(exp(*n*)) more logical connectives than φ (see Big O Notation for details regarding this assertion). Similar constructive proofs may be provided for the basic modal logic K, intuitionistic logic and μ-calculus, with similar complexity measures.\n\nCraig interpolation can be proved by other methods as well. However, these proofs are generally non-constructive:\n\n- model-theoretically, via Robinson's joint consistency theorem: in the presence of compactness, negation and conjunction, Robinson's joint consistency theorem and Craig interpolation are equivalent.\n- proof-theoretically, via a sequent calculus. If cut elimination is possible and as a result the subformula property holds, then Craig interpolation is provable via induction over the derivations.\n- algebraically, using amalgamation theorems for the variety of algebras representing the logic.\n- via translation to other logics enjoying Craig interpolation.\n",
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