Coq Interactive Theorem Prover

Coq is an interactive theorem prover and formal proof management system based on the Calculus of Inductive Constructions. Developed since 1984 at INRIA (French Institute for Research in Computer Science), originally by Thierry Coquand and Gerard Huet, it has become one of the most widely used proof assistants for formal verification of software and mathematics. Key features: dependent type theory as the logical foundation, where mathematical statements are types and proofs are programs, following the Curry-Howard correspondence. Tactic-based interactive proof construction with a rich tactic language including induction, rewrite, auto, and custom tactics via Ltac. Extraction mechanism to generate certified programs in OCaml, Haskell, and Scheme from constructive proofs, enabling verified software. Module system for structuring large developments with functors and parameterized modules. Universe polymorphism for flexible type-level quantification. Standard library providing foundational definitions and lemmas in arithmetic, lists, sets, relations, and analysis. Coq Reference Manual and extensive documentation. Support for both constructive and classical mathematics through optional axioms. SSReflect and Mathematical Components library developed by Georges Gonthier, providing small-scale reflection methodology used in the formal proof of the Four Color Theorem and Feit-Thompson theorem. Program verification capabilities including Hoare logic, separation logic, and refinement. Plugin system for extending with external tools. Used to verify the CompCert C compiler, miTLS TLS implementation, and numerous mathematical theorems. Cross-platform GUI tools including CoqIDE, Proof General (Emacs), and Coqtail (Vim).