-<p><sup>✻</sup><sub>A proof is <em>minimal</em> in context of a behavioral graph if it is the alphanumerically smallest sequence of all shortest possible proofs towards its theorem. Consequently, each proof counted in a behavioral graph uniquely represents its theorem. So these graphs illustrate how many theorems (including but not counting all their infinitely many instances) of a certain size can first be proven using a certain amount of proof steps. These alphanumeric minima may not actually occur in the proof files, but they clearly exist. In contrast, proof minimization usually aims towards finding a <em>shortest</em> proof (also “minimal”, but referring to length) — not “<em>the</em> minimal proof” as in this context.</sub><br><sup>❈</sup><sub>Generation and utilization were performed with computing resources granted by RWTH Aachen University under project <a href="pdf/rwth1392_extension_2024.pdf" title="View rwth1392_extension_2024.pdf">rwth1392</a>.</sub></p>
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