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# Automatic theorem provers

### Zenon

Zenon is an automatic theorem prover that handles first-order logic with equality. Its most important feature is that it outputs the proofs of the theorems, in Coq-checkable form.

- License: BSD like
- Web site: http://focal.inria.fr/zenon/

### MetiTarski

MetiTarski is an automatic theorem prover based on a combination of resolution and a decision procedure for the theory of real closed fields. It is designed to prove theorems involving *real-valued special functions* such as log, exp, sin, cos and sqrt. In particular, it is designed to prove *universally quantified inequalities* involving such functions. This problem is undecidable, so MetiTarski is necessarily incomplete. Nevertheless, MetiTarski is remarkably powerful.

- License: MIT

### E Theorem Prover

E is a theorem prover for full first-order logic with equality. It accepts a problem specification, typically consisting of a number of first-order clauses or formulas, and a conjecture, again either in clausal or full first-order form. The system will then try to find a formal proof for the conjecture, assuming the axioms.

If a proof is found, the system can provide a detailed list of proof steps that can be individually verified. If the conjecture is existential (i.e. it’s of the form “there exists an X with property P”), the latest versions can also provide possible answers (values for X).

- License: GNU GPL

### SPASS

SPASS is an automated theorem prover for first-order logic with equality.

- License: BSD
- Web site: http://www.spass-prover.org/

### Spike

SPIKE is an automated theorem prover using (first-order) formula-based Noetherian induction. Spike can be called from the Coq proof assistant using a tactic that automatically performs lazy and mutual induction. We provide a zip file with the Coq scripts using this tactic. Spike proofs can be checked with Coq.

- License: BSD
- Web site: https://code.google.com/p/spike-prover/

### Psyche

Psyche is a modular platform for automated or interactive theorem proving, programmed in OCaml and built on an architecture (similar to LCF) where a small kernel interacts with plugins and decision procedures:

- The kernel is based on a proof-search engine à la Prolog, offering an API to perform incremental and goal-directed constructions of proof-trees in (a standard but carefully chosen) Sequent Calculus, which can be seen as a tableau method.
- Psyche can produce proof objects.
- Plugins can be programmed to drive the kernel, using its API, through the search space towards an answer provable or not provable; soundness of the answer only relies on the kernel via the use of a private type for answers (similar to LCF’s theorem type).
- Plugins can be interactive.
- Psyche offers a memoisation feature to help programming efficient plugins.
- The kernel is parameterised by a procedure deciding the consistency of collections of literals with respect to a background theory, just as in SAT-modulo-theories (SMT) solvers.

- License: CeCILL-C