Proof assistant

A proof assistant is software that lets users construct formal proofs and mechanically checks them against a logical foundation.

Proof term: A formal expression that represents a proof and can be checked against a proposition. Proof assistants encode completed arguments as proof terms for kernel verification.
Formal logic: The study and use of symbolic systems for representing valid reasoning. A proof assistant implements a formal logic whose rules define acceptable proofs.
Lean (proof assistant): An interactive theorem prover based on dependent type theory, with a large mathematical library. Lean demonstrates large-scale formalization supported by an extensible library.
Automated theorem proving: The use of algorithms to search for proofs with little or no human guidance. Proof assistants typically keep humans involved in directing proof construction.
Type theory: A family of formal systems in which propositions can be represented as types and proofs as their inhabitants. Many assistants use type theory to make proof checking part of type checking.
Proposition: A declarative statement that can be judged true or false within a logical system. Formal proof development begins by stating propositions as goals.
Coq: An interactive theorem prover based on the calculus of inductive constructions. Coq has been used to formalize mathematics and verify software.
SMT solver: A tool that decides satisfiability of logical formulas combined with theories such as arithmetic or arrays. SMT solvers automate restricted reasoning tasks rather than general interactive proof development.
Inference rule: A formal rule specifying how conclusions may be derived from premises. The assistant checks that each proof step follows permitted inference rules.
Dependent type: A type whose definition depends on a value, often used to express propositions and their constraints. Dependent types let assistants encode precise mathematical claims in their type systems.

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