On the horizontal compression of dag-derivations in minimal purely implicational logic

In this report, we define (plain) Dag-like derivations in the purely implicational fragment of minimal logic $M_{\imply}$. Introduce the horizontal collapsing set of rules and the algorithm {\bf HC}. Explain why {\bf HC} can transform any polynomial height-bounded tree-like proof of a $M_{\imply}$ tautology into a smaller dag-like proof. Sketch a proof that {\bf HC} preserves the soundness of any tree-like ND in $M_{\imply}$ in its dag-like version after the horizontal collapsing application. We show some experimental results about applying the compression method to a class of (huge) propositional proofs and an example, with non-hamiltonian graphs, for qualitative analysis. The contributions include the comprehensive presentation of the set of horizontal compression (HC), the (sketch) of a proof that HC rules preserve soundness and the demonstration that the compressed dag-like proofs are polynomially upper-bounded when the submitted tree-like proof is height and foundation poly-bounded. Finally, in the appendix, we show an algorithm that verifies in polynomial time on the size of the dag-like proofs whether they are valid proofs of their conclusions.