Theoremproving

Theorem proving is the field of logic and computer science concerned with constructing formal proofs that certain propositions are logically valid within a formal system. It encompasses both automated theorem proving (ATP), where algorithms search for proofs without human guidance, and interactive theorem proving (ITP), where a human mathematician or programmer guides a proof with the help of a verification assistant.

Formalisms include first-order logic, higher-order logic, and type theories (including dependent type theory). Proofs are built from axioms and inference rules and are checked by machine-checked proof terms or proof scripts. The Curry–Howard correspondence links proofs to programs in many type-theoretic systems, enabling program verification within the same framework.

Typical workflow: in ITP, the user asserts a statement, encodes lemmas, and incrementally constructs proof scripts; the system checks correctness and provides feedback. In ATP, search procedures like resolution or equational reasoning attempt to derive a contradiction or a goal from axioms; ATP is often used to discharge goals in formalizations or to automate parts of proofs.

History and impact: early AI research produced resolution-based provers; modern proof assistants emerged in the 1980s–1990s and have become essential in formalizing mathematics and verifying software; examples include Coq, Isabelle/HOL, Lean, HOL Light, Mizar, and PVS.

Applications and challenges: formal verification of hardware, software, compilers, and safety-critical systems; large mathematical libraries (such as mathlib and the Coq standard library); challenges include scalability, user effort, risk of trusting the kernel, and the need for soundness guarantees.