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Two formulas A and B are isomorphic (denoted A\cong B), when there are two proofs π of A \vdash B and ρ of B \vdash A such that eliminating the cut on A in

\AxRule{}\VdotsRule{\pi}{A \vdash B}\AxRule{}\VdotsRule{\rho}{B \vdash A}\LabelRule{\rulename{cut}}\BinRule{B\vdash B}\DisplayProof

leads to an η-expansion of

\LabelRule{\rulename{ax}}\NulRule{B\vdash B}\DisplayProof,

and eliminating the cut on B in

\AxRule{}\VdotsRule{\pi}{A \vdash B}\AxRule{}\VdotsRule{\rho}{B \vdash A}\LabelRule{\rulename{cut}}\BinRule{A\vdash A}\DisplayProof

leads to an η-expansion of

\LabelRule{\rulename{ax}}\NulRule{A\vdash A}\DisplayProof.

Linear logic admits many isomorphisms, but it is not known wether all of them have been discovered or not.

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