The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ Coq interactive prover to methodically formalize language, semantics, syntax propositional logic, a fundamental aspect reasoning proof construction. We construct four Hilbert-style axiom system...

In this paper, we compare the computability and complexity of a continuous real function F with the computability and complexity of the graph G of the function F . A similar analysis will be carried out for functions on subspaces of the real line such as the Cantor space, the Baire space and the unit interval. In particular, we de1ne four basic types of e"ectively closed sets C depending on whe...

Convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz’s exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a function ω has the S...

A weight function ! : 2 → R¿0 from the set of all subsets of [n]={1; : : : ; n} to the nonnegative real numbers is called shift-monotone in {m+1; : : : ; n} if !({a1; : : : ; aj})¿!({b1; : : : ; bj}) holds for all {a1; : : : ; aj}; {b1; : : : ; bj}⊆ [n] with ai6bi; i = 1; : : : ; j, and if !(A)¿!(B) holds for all A; B⊆ [n] with A⊆B and B\A⊆{m + 1; : : : ; n}. A family F⊆ 2 is called intersectin...

Given a Chevalley group ${\mathbf G}(q)$ and parabolic subgroup $P\subset {\mathbf G}(q)$, we prove that for any set $A$ there is certain growth of relatively to $P$, namely, either $AP$ or $PA$ much larger than $A$. Also, study question about the intersection $A^n$ with subgroups $P$ large $n$. We apply our method obtain some results on modular form Zaremba’s conjecture from theory continued f...

We prove that if X,X are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) ∩ Trop(X) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a s...

Let A be a regular local ring containing 1/2, which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of A satisfy the following property: given two elements with supports which do not intersect properly, their product vanishes. This gives an analogue for “oriented intersection multiplicities” of ...

Let A be a regular local ring containing 1/2, which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of A satisfy the following property: given two elements with supports which do not intersect properly, their product vanishes. This gives an analogue for “oriented intersection multiplicites” of S...

Fix integers n, r ≥ 4 and let F denote a family of r-sets of an n-element set. Suppose that for every four distinct A,B,C,D ∈ F with |A∪B ∪C ∪D| ≤ 2r, we have A∩B ∩C ∩D 6= ∅. We prove that for n sufficiently large, |F| ≤ ( n−1 r−1 ) , with equality only if ⋂ F∈F F 6= ∅. This is closely related to a problem of Katona and a result of Frankl and Füredi [10], who proved a similar statement for thre...