proofsuch

Proofsuch is a term used in formal logic and computer science to denote a structured proof object that bundles a mathematical proof with explicit prescriptions expressed by such that clauses. A proofsuch aims to make the constructive content of a proof explicit by pairing deductive steps with conditions or witnesses that guarantee the claimed result under specified constraints.

Formally, a proofsuch consists of three parts: the deduction steps that establish the theorem, a such-that annotation that specifies the exact conditions, witnesses, or parameters under which the theorem holds, and a verification component that checks both the logical correctness of the steps and the satisfaction of the such-that constraints. This structure helps bridge abstract reasoning and constructive content, making it easier to extract usable information from proofs.

In practice, a proofsuch of an existential claim such as exists x such that P(x) would include an explicit witness x and a proof that P(x) holds, together with a description of the conditions under which this witness is valid. In nonconstructive contexts, the such-that annotation may be deferred or abstract, whereas in constructive settings it serves as the primary source of computational content.

Applications of the concept include interactive theorem proving, certified programming, and formal verification, where proofs are intended to be machine-checkable and witnesses readily extractable. The term proofsuch is not widely adopted in established literature and is often used as a descriptive or illustrative notion to highlight the role of explicit conditions within proofs. Related concepts include proof objects, witnesses, and the Curry–Howard correspondence.