Isomorphism

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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.