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This article precisely defines huge proofs within the system of NaturalDeduction for the Minimal implicational propositional logic \mil. This is whatwe call an unlimited family of super-polynomial proofs. We consider hugefamilies of expanded normal form mapped proofs, a device to explicitly help tocount the E-parts of a normal proof in an adequate way. Thus, we show that foralmost all members of a super-polynomial family there at least one sub-proof orderivation of each of them that is repeated super-polynomially many times. Thislast property we call super-polynomial redundancy. Almost all, precisely meansthat there is a size of the conclusion of proofs that every proof withconclusion bigger than this size and that is huge is highly redundant too. Thisresult points out to a refinement of compression methods previously presentedand an alternative and simpler proof that CoNP=NP.

On the Intrinsic Redundancy in Huge Natural Deduction proofs II: Analysing $M_{\imply}$ Super-Polynomial Proofs