In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. That is, a proof
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Local redundancy
A proof containing a subproof of the shapes (here omitted pivots indicate that the resolvents must be uniquely defined)
is locally redundant.
Indeed, both of these subproofs can be equivalently replaced by the shorter subproof
The following definition generalizes local redundancy by considering inferences with the same pivot that occur within different contexts. We write
Global redundancy
A proof
is potentially (globally) redundant. Furthermore, it is (globally) redundant if it can be rewritten to one of the following shorter proofs:
Example
The proof
is locally redundant as it is an instance of the first pattern in the definition
But it is not globally redundant because the replacement terms according to the definition contain
The second pattern of potentially globally redundant proofs appearing in global redundancy definition is related to the well-known notion of regularity. Informally, a proof is irregular if there is a path from a node to the root of the proof such that a literal is used more than once as a pivot in this path.