the down fragment. See also Proof-theoretic semantics References Bernard Bolzano (1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In Abhandlungen der koniglichen bohmischen Gesellschaft der Wissenschaften Vol. V, pp.225-48. Frank Pfenning (1984). Analytic and Non-analytic Proofs. In Proc. 7th International Conference on Automated Deduction. Jan Šebestik (2007). Bolzano's Logic. Entry in the Stanford Encyclopedia of Philosophy. Proof theory Methods of proof