Related papers: Notes on proof by dichotomy
The incompressibility method is an elementary yet powerful proof technique. It has been used successfully in many areas. To further demonstrate its power and elegance we exhibit new simple proofs using the incompressibility method.
We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides…
In this paper we consider first-order logic theorem proving and model building via approximation and instantiation. Given a clause set we propose its approximation into a simplified clause set where satisfiability is decidable. The…
We introduce a method to prove that a proof search method is not an instance of another. As an example of application, we show that Polarized resolution modulo, a method that mixes clause selection restrictions and literal selection…
For any sufficiently strong theory of arithmetic, the set of Diophantine equations provably unsolvable in the theory is algorithmically undecidable, as a consequence of the MRDP theorem. In contrast, we show decidability of Diophantine…
When a proposition has no proof in an inference system, it is sometimes useful to build a counter-proof explaining, step by step, the reason of this non-provability. In general, this counter-proof is a (possibly) infinite co-inductive proof…
Based on the MRDP theorem, we introduce the ideas of the proof equation of a formula and universal proof equation of Peano Arithmetic (PA); and then, combining universal proof equation and G\"odel's Second Incompleteness Theorem, it is…
We give a simple statistical proof of a binomial identity, by evaluating the Laplace transform of the maximum of n independent exponential random variables in two different ways. As a by product, we obtain a simple proof of an interesting…
As with all measurements, the measurement of examinee ability, in terms of scores that the examinee obtains in a test, is also error-ridden. The quantification of such error or uncertainty in the test score data--or rather the complementary…
We illustrate the concept of mathematical proof.
We present an elementary combinatorial proof of the celebrated Friendship theorem. The proof involves looking at independent sets and constructing a bound on their size which forces a contradiction.
We define a proof system for exceptions which is close to the syntax for exceptions, in the sense that the exceptions do not appear explicitly in the type of any expression. This proof system is sound with respect to the intended…
This paper studies the proof of Collatz conjecture for some set of sequence of odd numbers with infinite number of elements. These set generalized to the set which contains all positive odd integers. This extension assumed to be the proof…
A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions of n + 1 into parts greater than one. Some commentary about the history of partitions and compositions is…
The incompressibility method is an elementary yet powerful proof technique. It has been used successfully in many areas. To further demonstrate its power and elegance we exhibit new simple proofs using the incompressibility method.
Mathematical proofs are a cornerstone of control theory, and it is important to get them right. Deduction systems can help with this by mechanically checking the proofs. However, the structure and level of detail at which a proof is…
The article proposes a new technique for proving the undefinability of logical connectives through each other and illustrates the technique with several examples. Some of the obtained results are new proofs of the existing theorems, others…
We present a sequent-based deductive system for automatically proving entailments in separation logic by using mathematical induction. Our technique, called mutual explicit induction proof, is an instance of Noetherian induction.…
A probability method is provided to prove three classes of combinatorial identities. The method is extremely simple, only one step after the proper probability setup.
For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a polynomial dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility…