Since the proof of the four color theorem in 1976, computer-generated proofs have become a reality in mathematics and computer science. During the last decade, we have seen formal proofs using verified proof assistants being used to verify the validity of such proofs.
Outside the world of formal proofs, computer-generated proofs are flourishing, too, and growing to tremendous sizes. The proof of Erd " o s' discrepancy conjecture for @math from 2014 @cite_10 has been touted as one of the largest mathematical proofs and produced approx. 13 GB of proof witnesses. Such large-scale proofs are extremely challenging for formal verification. Given the current state of theorem provers and computing equipment, it is unthinkable to use 's approach @cite_3 of importing an based on the proof witnesses into Coq, a process clearly prohibitive for such large-scale proofs as we consider.
@cite_10 @cite_3
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In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (x n ) there exists a subsequence x d , x 2d , x 3d ,…, x kd , for some positive integers k and d, such that ( i=1 ^k x_ id >C ). The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C = 1 a human proof of the conjecture exists; for C = 2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solver, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C = 2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. We also present our partial results for the case of C = 3. Large formal mathematical knowledge bases encode considerable parts of advanced mathematics and exact science, allowing deep semantic computer assistance and verification of complicated theories down to the atomic logical rules. An essential part of automated reasoning over such large theories are methods learning selection of relevant knowledge from the thousands of proofs in the corpora. Such methods in turn rely on efficiently computable features characterizing the highly structured and inter-related mathematical statements. In this work we (i) propose novel semantic features characterizing the statements in such large semantic knowledge bases, (ii) propose and carry out their efficient implementation using deductive-AI data-structures such as substitution trees and discrimination nets, and (iii) show that they significantly improve the strength of existing knowledge selection methods and automated reasoning methods over the large formal knowledge bases. In particular, on a standard large-theory benchmark we improve the average predicted rank of a mathematical statement needed for a proof by 22 in comparison with state of the art. This allows us to prove 8 more theorems in comparison with state of the art. Erdős-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family. In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on @math -intersecting families of @math -element subsets of @math (for @math ), and to show that, somewhat surprisingly, all these results hold when the intersection requirement is replaced by a much weaker requirement. Other examples include stability versions of Frankl's recent result on the Erdős matching conjecture, the Ellis-Filmus-Friedgut proof of the Simonovits-Sos conjecture, and various EKR-type results on @math -wise (cross)- @math -intersecting families. Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and re-used in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL Light lemmas for proving further theorems. We use these criteria to mine the large inference graph of the lemmas in the HOL Light and Flyspeck libraries, adding up to millions of the best lemmas to the pool of statements that can be re-used in later proofs. We show that in combination with learning-based relevance filtering, such methods significantly strengthen automated theorem proving of new conjectures over large formal mathematical libraries such as Flyspeck.