Derivationfor

Derivationfor is a term used in some formal-logic literature to denote a construction that, given a set of premises Γ, a conclusion A, and a deductive system S, yields a derivation of Γ ⊢ A within S. The derivation is typically represented as a finite tree whose leaves are axioms or assumptions and whose internal nodes apply the inference rules of S. In this sense, a DerivationFor object embodies the explicit proof of the sequent or judgment.

Notationally, one might write D = DerivationFor(Γ, A; S). Different texts may describe the same notion with alternative names such as a proof object or a derivation, and the exact terminology is not universally standardized. The concept, however, centers on the idea of producing a constructive witness—a concrete derivation—for a given statement within a chosen deductive framework.

Applications of this notion appear in formal metatheory, proof search, and the formalization of proofs in proof assistants. In many systems, a derivation is represented as a proof term or a structured object that can be manipulated by software. Through the Curry–Howard correspondence, derivations correspond to constructive programs or lambda-terms, linking proof objects to computational content.

Derivations can vary in form across deductive systems: classical versus intuitionistic logics, natural deduction versus sequent calculus, and different normalization regimes affect the shape and length of the derivation while preserving the underlying provability of A from Γ. The study of derivations addresses correctness, normalization, and the equivalence of different derivations for the same conclusion.

Related topics include proof theory, sequent calculus, natural deduction, proof terms, and the Curry–Howard correspondence.