Related papers: On The Composition Lemma
We give a notably simpler and shorter proof of H. B. Neumann's result which is stated, cursorly, like this. For any well-ordered subset, A, of a totally ordered semigroup, the set of products of any finite number of elements of A is itself…
(i) We provide a short and simple proof of the first selection lemma. (ii) We also prove a selection lemma of a new type in $\Re^d$. For example, when $d=2$ assuming $n$ is large enough we prove that for any set $P$ of $n$ points in general…
We prove a version of Hrushovski's socle lemma for rigid groups in an arbitrary simple theory.
We isolate several classes of stationary sets of kappa^omega and investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.
We prove an easy version of the minimax theorem with no topological assumption. We deduce from it some domination criteria as well as an application to $p$-summing operators.
The contribution of this paper is threefold: first, it defines a framework for modelling component-based systems, as well as a formalization of integration rules to combine their behavior. This is based on finite state machines (FSM).…
In this expository note, we introduce the reader to compositions of a natural number, e.g., $2+1+2+1+7+1$ is a composition of 14, and $1+2$ and $2+1$ are two different compositions of 3. We discuss some simple restricted forms of…
We establish a Sewing lemma in the regime $\gamma \in \left( 0, 1 \right]$, constructing a Sewing map which is neither unique nor canonical, but which is nonetheless continuous with respect to the standard norms. Two immediate corollaries…
Self-composition provides a powerful theoretical approach to prove relational properties, i.e. properties relating several program executions, that has been applied to compare two runs of one or similar programs (in secure dataflow…
Reachability Logic is a formalism that can be used, among others, for expressing partial-correctness properties of transition systems. In this paper we present three proof systems for this formalism, all of which are sound and complete and…
We study pairs and m--tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of…
This note presents an elementary and direct proof for the convexity of the Choquet integral when the corresponding set function is submodular.
We prove some general theorems for preserving Dependent Choice when taking symmetric extensions, some of which are unwritten folklore results. We apply these to various constructions to obtain various simple consistency proofs.
In this paper, we study the "sum composition problem" between two lists $A$ and $B$ of positive integers. We start by saying that $B$ is "sum composition" of $A$ when there exists an ordered $m$-partition $[A_1,\ldots,A_m]$ of $A$ where $m$…
In this paper, we provide an easy proof of the Four-colour Theorem in a special case indeed.
In this note we exhibit a very simple proof of McNaughton Theorem, almost right out of the definitions, and at the same time we observe that this theorem does not depend of Chang's completeness theorem.
We present two new proofs of Simon Henry's result that the category of simplicial sets admits a constructive counterpart of the classical Kan-Quillen model structure. Our proofs are entirely self-contained and avoid complex combinatorial…
A completeness conjecture is advanced concerning the free small-colimit completion P(A) of a (possibly large) category A. The conjecture is based on the existence of a small generating-cogenerating set of objects in A. We sketch how the…
We have witnessed the emergence of several controller parameterizations and the corresponding synthesis methods, including Youla, system level, input-output, and many other new proposals. Meanwhile, under the same synthesis method, there…
In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.