What are commuting conversions in proof theory?
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What are commuting conversions in proof theory?
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Summary
Commuting conversions are transformations on proofs in natural deduction, that move certain stuck inferences out of the way, so that the normal detour reductions (which correspond to beta-reduction under Curry-Howard) are enabled. The stuck inferences are uses of disjunction elimination. In programming terms, if you have an if-then-else (a simple case of or-elimination) where the then- and else-branches are lambda abstractions, and you apply that if-then-else to an argument, you need commuting conversions to move the argument into the branches, so you can call the functions (in the then- and else-branches) with it.
See Section 10.1 of Girard's Proofs and Types for more on the problem, and a nice paper by de Groote on strong normalization with commuting conversions.