Interactive theorem proving

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Interactive theorem proving is the field of computer science and mathematical logic concerned with tools to develop formal proofs by man-machine collaboration. This involves some sort of proof assistant: an interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer.

Examples include:

HOL theorem provers - A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. Theorems represent new elements of the language and can only be introduced via "strategies" which guarantee logical correctness. Strategy composition gives users the ability to produce significant proofs with relatively few interactions with the system. Members of the family include:
- HOL4 - The "primary descendant". Moscow ML based.
- HOL Lite - A thriving "minimalist fork". OCaml based.
- Isabelle - With a BSD license. Based on Standard ML.
- ProofPower - Went proprietary, then returned to open source. Based on Standard ML.

Prototype Verification System (PVS) - a proof language and system based on higher-order logic

Coq - Which allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification.