Automated proof programs provide different techniques of automated formal reasoning that are used to describe the performance and evaluation of the human user. It basically involves trading off autonomy and flexibility in carrying out proofs. These programs provide a skill set that needs to be acquired concurrently to learning the concepts of discrete mathematics that are pervasive in computer science. It also involves a healthy amount of feedback as to whether the proof steps used are correct and sufficiently rigorous (International Conference on Automated Deduction, 2003). The whole process involves writing a code or rather learning similar engineering concepts that give an automatic feedback from the compilation and interpretation of their code and from running the tests written by instructors to test correctness of implementation. Such kinds of feedback help students in correcting their mistakes during the problem-solving process without involving an instructor to grade or check their work. These rapid feedbacks require writing down necessary proofs which ought to be thoroughly checked by hands on how correct they are. Automated proof programs are been widely used on relations and functions in the scope of mathematics
A good example of automated proof programs is the automated theorem proving (ATP), which deal with the development of computerized programs that some statement or rather the conjecture is a logical consequence of a set of statements referred to as the axioms and hypotheses. ATP systems are used in explaining and proofing various domains where mathematicians may prove the conjecture that groups of order two are commutative from the axioms of group theory. Similarly, management consultants may formulate axioms that help in describing the organizations interact and grow. The logic about this aspect is that the conjecture, hypotheses and axioms involve a written language which allows a precise formal statement of the necessary information which can be manipulated using an automated theorem proving system (International Conference on Automated Deduction, 2003).
Automated proof programs help in providing details to relations and functions that prevent the restriction of making all functions computable. This will be illustrated through functions in the set-theoretic sense that the functions are a specific kind of relation between two sets. For instance, section functions such as U V: type. Variable f ; U V Prop. The basic properties of relations are easy to define in that the definition function. The properties of various combinations are given according to familiar concepts of subjective and injective functions. For example, a binary relation is a set of ordered pairs that involves Isabelle’s set theory to define the basic operations upon relations. They also involve the usual properties and need little discussion (Glasper, 2010)..
Functions on the other hand are represented using graphs which are single-valued binary relations. The set of all functions from A to B is written as A B or rather generalized as A*B to ∑x є A. B (x) is generalized A B to ∏ x є A. B (x) is the product of a family of sets. These are just examples of how automated proof programs work in order to verify relations and functions. It generally requires that every single proof step to be verified using a primitive recursive function or program that makes the problem decidable. Proof compression is also very necessary as it applies various techniques that aim at making the Proofer’s output smaller and easily understandable. Model checking techniques are also essential in theorem proving as they are considered if they consist of traditional proofs (Glasper, 2010).
The future work of automated proof programs involves providing a better understanding of the relationships between the proofs and the corresponding algorithms using proof approaches. Historical proofing programs used included projects such as The Automath which was generated by De Bruijn in 1967 with the aim of developing a system that would verify mathematics mechanically (International Conference on Automated Deduction, 2003). It also aimed at developing a mathematical language where all mathematics would be expressed accurately. Similarly, there was the Martin- Lof Type Theory that gave the connection between proofs in constructive logic and the calculi terms. The Logic for Computable Functions LCF described the theory of Scott in 1969 which aimed at proving properties of recursively defined functions using methods from denotation semantics. There was also the evidence algorithm, The Prototype Verification System, the Nqthm, The Mizar system just to mention but a few. These are among the traditional proofing programs used over time (Glasper, 2010).
The future of these proofing programs relates to programs such as the QED Manifesto that involves computerized systems that are effective in representing all important mathematical knowledge techniques. Other similar programs that entail computerized proofing include the MathWiki and the Flyspeck which will define or rather are being used in verifying complex mathematical problems.
References
Glasper, Alan. (2010). Future-proofing the NHS: The NMC's strategic 5-year vision. British Journal of Nursing, 19 (17), 1122-1123.
International Conference on Automated Deduction, B. (2003). Automated deduction, CADE-19: 19th International Conference on Automated Deduction, Miami Beach, FL, USA, July 28-August 2, 2003 : proceedings. Berlin: Springer.
Voronkov, A (1993). Logic programming and automated reasoning: 4th international conference, LPAR '93, St. Petersburg, Russia, July 13-20, 1993 : proceedings. Berlin: Springer-Verlag.
