The pigeonhole principle and related counting arguments in weak monotone systems

We construct quasipolynomial-size proofs of the propositional pigeonhole principle for the fragment of the sequent calculus with no cuts between ancestors of left and right negation, weakening and contraction rules. The main construction of our argument, inspired by previous work on the monotone calculus by Atserias et al., provides formal proofs that permute the inputs of formulae computing threshold functions, essentially by implementing merge sort as a template. Since it is non-trivial to eliminate the remaining weakening/contraction cuts efficiently, our arguments are implemented in deep inference where known normalization procedures exist, although the work is mostly self-contained. This also (partially) answers previous questions raised about the size of proofs of the pigeonhole principle in certain deep inference systems.