Proof Of Equivalence Research Articles

Two cryptomorphic formalizations of projective incidence geometry

Incidence geometry is a well-established theory which captures the very basic properties of all geometries in terms of points belonging to lines, planes, etc. Moreover, projective incidence geometry leads to a simple framework where many properties can be studied. In this article, we consider two very different but complementary mathematical approaches formalizing this theory within the Coq proof assistant. The first one consists of the usual and synthetic geometric axiom system often encountered in the literature. The second one is more original and relies on combinatorial aspects through the notion of rank which is based on the matroid structure of incidence geometry. This paper mainly contributes to the field by proving the equivalence between these two approaches in both 2D and 3D. This result allows us to study the further automation of many proofs of projective geometry theorems. We give an overview of techniques that will be heavily used in the equivalence proof and are generic enough to be reused later in yet-to-be-written proofs. Finally, we discuss the possibilities of future automation that can be envisaged using the rank notion.

IMP with exceptions over decorated logic

In this paper, we facilitate the reasoning about impure programming languages, by annotating terms with “decorations” that describe what computational (side) effect evaluation of a term may involve. In a point-free categorical language, called the “decorated logic”, we formalize the mutable state and the exception effects first separately, and then combined, exploiting a nice duality between them. The combined decorated logic is used as the target language for the denotational semantics of the IMP+Exc imperative programming language, and allows us to prove equivalences between programs written in IMP+Exc. The combined logic is encoded in Coq, and this encoding is used to certify some program equivalence proofs.

Theorem proving for all: equational reasoning in liquid Haskell (functional pearl)

Equational reasoning is one of the key features of pure functional languages such as Haskell. To date, however, such reasoning always took place externally to Haskell, either manually on paper, or mechanised in a theorem prover. This article shows how equational reasoning can be performed directly and seamlessly within Haskell itself, and be checked using Liquid Haskell. In particular, language learners --- to whom external theorem provers are out of reach --- can benefit from having their proofs mechanically checked. Concretely, we show how the equational proofs and derivations from Graham's textbook can be recast as proofs in Haskell (spoiler: they look essentially the same).

The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

Quasirandomness in Hypergraphs

An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical' properties are asymptotically equivalent and, thus, a graph $G$ possessing one such property automatically satisfies the others.In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences in Towsner's result.

Equivalence of One Loop Diagrams in Covariant and Light Front QED Revisited

We revisit the proof of equivalence of one loop expressions for fermion self-energy and vertex correction in light-front QED and Covariant QED at the Feynman diagram level and generalize, to all components, the proof of equivalence for the one loop vertex correction diagram which was presented earlier by us only for the $+$ component of $\Lambda^\mu$. We demonstrate, in the general case also, that the equivalence cannot be established without the third term in the three-term photon propagator in light cone gauge.

Transition formulas for involution Schubert polynomials

The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group $$S_n$$ . Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as $${\hat{\mathfrak S}}_y$$ (to be called involution Schubert polynomials) and $$\hat{\mathfrak S}^\mathtt{{FPF}}_y$$ (to be called fixed-point-free involution Schubert polynomials). Our main results are explicit formulas decomposing the product of $${\hat{\mathfrak S}}_y$$ (respectively, $$\hat{\mathfrak S}^\mathtt{{FPF}}_y$$ ) with any y-invariant linear polynomial as a linear combination of other involution Schubert polynomials. These identities serve as analogues of Lascoux and Schützenberger’s transition formula for Schubert polynomials, and lead to a self-contained algebraic proof of the nontrivial equivalence of several definitions of $${\hat{\mathfrak S}}_y$$ and $$ \hat{\mathfrak S}^\mathtt{{FPF}}_y$$ appearing in the literature. Our formulas also imply combinatorial identities about involution words, certain variations of reduced words for involutions in $$S_n$$ . We construct operators on involution words based on the Little map to prove these identities bijectively. The proofs of our main theorems depend on some new technical results, extending work of Incitti, about covering relations in the Bruhat order of $$S_n$$ restricted to involutions.

Correct‐by‐construction specification to verified code

AbstractEvent‐B is a formal notation and method for the systems development. The key feature of this method is to produce correct‐by‐construction system designs. Once the correct design is established, the remaining work is to generate or implement correct code from the design. Two main problems remain in the process from the correct‐by‐construction design to the correct software. First, the Event‐B design is “quasi‐correct” due to some technical limitations. For instance, it is still difficult to prove the liveness properties by the Rodin platform; it is not possible to construct the Event‐B design with floating‐point arithmetic, and sometimes, the Event‐B model is incomplete and must rely on the third‐party libraries. Therefore, a method is needed to complement these modeling and proof gaps. Secondly, proving the correctness of an automatic code generator is very difficult; therefore, a method is needed to guarantee the correctness of the produced code without proving the code generator. In this article, we address the above 2 problems by introducing an intermediate formal language called High‐Level Language (HLL) between the Event‐B models and the C code. The Event‐B model is translated to HLL with an additional schedule configuration, where Event‐B invariants and system invariants (here, deadlock‐freeness and liveness properties) are proved using a SAT‐based model checker called S3. This proof guarantees the correctness of the HLL model with respect to the Event‐B model. The C code is then automatically generated from the HLL model for most functions and is manually implemented for the third‐party ones according to the function contracts defined in Event‐B. The correctness of the generated C code is guaranteed using the equivalence proof, and the correctness of the implemented C code is guaranteed using the conformance proof. Through the article, we use a traffic light controller to illustrate the proposed method; then, we apply the method to an automatic protection function of a 3‐wheeled robot to evaluate its feasibility.

Pre-filtering in continuous-time quadratic gain-scheduled and robust state-feedback control

ABSTRACTWe revisit the issue of quadratic gain-scheduled and robust state-feedback control with a focus on linear matrix inequalities widely used in the literature. It has been established that for continuous-time linear parameter-varying systems which depend affinely on the scheduling parameter, gain-scheduled stabilisability implies robust stabilisability. In other words, as far as stabilisation is concerned, using a more complex gain-scheduled controller is not advantageous. The proof of such equivalence is non constructive and, although very general, limited to stabilisability. One contribution of the present paper is to establish constructive proofs in the quadratic case that can be extended to performance optimisation in terms of quadratic and norms. One issue investigated in detail is the strategy of using pre-filters to handle parameter-variation in the input matrix. Our results show that said technique, which has been popular since the 1980s, is rarely productive in the sense that solvability of quadratic gain-scheduled control design problems for the original system augmented with an arbitrary pre-filter implies existence of a lower-dimensional quadratic robust controller for the original system, which we construct explicitly via projections.

Duality Principle of Characteristic Modes for the Analysis and Design of Aperture Antennas

Characteristic mode analysis (CMA) is a promising method that has been used in antenna design. However, CMA of aperture antennas is cumbersome as the equivalent magnetic current is rarely available in main stream method of moments solvers. Based on the duality principle, an equivalent CMA of aperture antennas is proposed, by which the characteristic values and characteristic magnetic currents are calculated through its complementary form. A rigorous proof for such equivalence is provided. Lumped loads are very useful for the manipulation of certain modes, and in this paper, the relationship between the loads for an aperture and its complementary plate is built. The duality CMA is applied to reduce the mutual coupling between aperture antennas etched on a finite plate. Interdigital capacitors are adopted to suppress the out-of-band mutual coupling while an H-shaped aperture for the reduction of in-band one for narrow slots. Numerical and experimental results show that over 10 dB reduction of mutual coupling is achieved for both examples.

A Comparison of the Conditioning Regimens BEAM and FEAM for Autologous Hematopoietic Stem Cell Transplantation in Lymphoma: An Observational Study on 1038 Patients From Fondazione Italiana Linfomi

BEAM (carmustine [bis-chloroethylnitrosourea (BCNU)]-etoposide-cytarabine-melphalan) chemotherapy is the standard conditioning regimen for autologous stem cell transplantation (ASCT) in lymphomas. Owing to BCNU shortages, many centers switched to fotemustine-substituted BEAM (FEAM), lacking proof of equivalence. We conducted a retrospective cohort study in 18 Italian centers to compare the safety and efficacy of BEAM and FEAM regimens for ASCT in lymphomas performed from 2008 to 2015. We enrolled 1038 patients (BEAM = 607, FEAM = 431), of which 27% had Hodgkin lymphoma (HL), 14% indolent non-Hodgkin lymphoma (NHL), and 59% aggressive NHL. Baseline characteristics including age, sex, stage, B-symptoms, extranodal involvement, previous treatments, response before ASCT, and overall conditioning intensity were well balanced between BEAM and FEAM; notable exceptions were median ASCT year (BEAM = 2011 versus FEAM = 2013, P < .001), Sorror score ≥3 (BEAM = 15% versus FEAM = 10%, P = .017), and radiotherapy use (BEAM = 18% versus FEAM = 10%, P < .001). FEAM conditioning resulted in higher rates of gastrointestinal and infectious toxicities, including severe oral mucositis grade ≥3 (BEAM = 31% versus FEAM = 44%, P < .001), and sepsis from Gram-negative bacteria (mean isolates/patient: BEAM = .1 versus FEAM = .19, P < .001). Response status at day 100 post-ASCT (overall response: BEAM = 91% versus FEAM = 88%, P = .42), 2-year overall survival (83.9%; 95% confidence interval [CI], 81.5% to 86.1%) and progression-free survival (70.3%; 95% CI, 67.4% to 73.1%) were not different in the two groups. Mortality from infection was higher in the FEAM group (subhazard ratio, 1.99; 95% CI, 1.02 to 3.88; P = .04). BEAM and FEAM do not appear different in terms of survival and disease control. However, due to concerns of higher toxicity, fotemustine substitution in BEAM does not seem justified, if not for easier supply.

Proof Diagrams for Multiplicative Linear Logic: Syntax and Semantics

Proof nets are a syntax for linear logic proofs which gives a coarser notion of proof equivalence with respect to syntactic equality together with an intuitive geometrical representation of proofs. In this paper we give an alternative $2$-dimensional syntax for multiplicative linear logic derivations. The syntax of string diagrams authorizes the definition of a framework where the sequentializability of a term, i.e. deciding whether the term corresponds to a correct derivation, can be verified in linear time. Furthermore, we can use this syntax to define a denotational semantics for multiplicative linear logic with units by means of equivalence classes of proof diagrams modulo a terminating rewriting.

One-loop divergences for f(R) gravity

In the first paper, the one loop divergence structure of $F(R)$ gravity theories is computed by integrating out fluctuations of the metric about an arbitrary metric background, yielding novel background independent results. In the second paper, these results are employed to rigorously demonstrate the exact equality of the one loop on shell divergences of $F(R)$ and scalar-tensor theories of gravity, thereby advancing the proof of equivalence of the two.

Quantum equivalence of f(R) gravity and scalar-tensor theories

In the first paper, the one loop divergence structure of F(R) gravity theories is computed by integrating out fluctuations of the metric about an arbitrary metric background, yielding novel background independent results. In the second paper, these results are employed to rigorously demonstrate the exact equality of the one loop on shell divergences of F(R) and scalar-tensor theories of gravity, thereby advancing the proof of equivalence of the two.

Convexity in Elementary Calculus: Some Geometric Equivalences

What is convexity? For the beginning student of calculus, it is a concept usually introduced to explain the geometric significance of the second derivative. It is easy enough to illustrate graphically or describe verbally (holds water), but what sort of definition is appropriate in this setting? There are several equivalent ways to formulate the definition and relate it to the sign of the second derivative, but many widely used textbooks do not reveal the variety of alternatives or, in the case of some more advanced ones, give proofs of equivalence which are unnecessarily technical or nonintuitive. For texts which contain a more thorough discussion of convexity, see [1], [2]. Here we will look at six well-known formulations of convexity which have easily visualized geometric interpretations and prove their equivalence in an elementary way. We hope that the presentation of some of this (admittedly optional) material in the classroom may hint at the fundamental character of convexity and help to dispel the possible suspicion that the second derivative test for convexity is little more than a disguised definition. Though infrequently seen in calculus texts, perhaps the most common definition of convexity for an arbitrary function is as follows:

Strong- and Weak-Universal Critical Behaviour of a Mixed-Spin Ising Model with Triplet Interactions on the Union Jack (Centered Square) Lattice.

The mixed spin-1/2 and spin-S Ising model on the Union Jack (centered square) lattice with four different three-spin (triplet) interactions and the uniaxial single-ion anisotropy is exactly solved by establishing a rigorous mapping equivalence with the corresponding zero-field (symmetric) eight-vertex model on a dual square lattice. A rigorous proof of the aforementioned exact mapping equivalence is provided by two independent approaches exploiting either a graph-theoretical or spin representation of the zero-field eight-vertex model. An influence of the interaction anisotropy as well as the uniaxial single-ion anisotropy on phase transitions and critical phenomena is examined in particular. It is shown that the considered model exhibits a strong-universal critical behaviour with constant critical exponents when considering the isotropic model with four equal triplet interactions or the anisotropic model with one triplet interaction differing from the other three. The anisotropic models with two different triplet interactions, which are pairwise equal to each other, contrarily exhibit a weak-universal critical behaviour with critical exponents continuously varying with a relative strength of the triplet interactions as well as the uniaxial single-ion anisotropy. It is evidenced that the variations of critical exponents of the mixed-spin Ising models with the integer-valued spins S differ basically from their counterparts with the half-odd-integer spins S.

宇宙几何学初步应用—天体的光度和绝对星等等价不变性的证明

宇宙几何学初步应用—天体的光度和绝对星等等价不变性的证明

Quasi-coassociative C⁎-quantum groupoids of type A and modular C⁎-categories

We construct a new class of finite-dimensional C⁎-quantum groupoids at roots of unity q=eiπ/ℓ, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C⁎-category of the category of tilting modules of the non-semisimple quantum group Uq(slN) of Drinfeld, Jimbo and Lusztig.As an algebra, the C⁎-groupoid is a quotient of Uq(slN). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociative, but satisfies axioms of the weak quasi-Hopf C⁎-algebras: quasi-coassociativity and non-unitality of the coproduct. There are also a multiplicative counit, an antipode, and an R-matrix.For this, we give a general construction of quantum groupoids for complex simple Lie algebras g≠E8 and certain roots of unity. Our main tools here are Drinfeld's coboundary associated to the R-matrix, related to the algebra involution, and certain canonical projections introduced by Wenzl, which yield the coproduct and Drinfeld's associator in an explicit way. Tensorial properties of the negligible modules reflect in a rather special nature of the associator. We next reduce the proof of the categorical equivalence to the problems of establishing semisimplicity and computing dimension of the groupoid. In the case g=slN we construct a (non-positive) Haar-type functional on an associative version of the dual groupoid satisfying key non-degeneracy properties. This enables us to complete the proof.

A note on the polynomial Freĭman–Ruzsa conjecture over ℤ

AbstractThe polynomial Freĭman–Ruzsa conjecture over the integers is often phrased in terms of convex progressions. We give an alternative, apparently stronger formulation in terms of the more restrictive “ellipsoid progressions”, and show that these formulations are in fact equivalent. The key input to the equivalence proof comes from strong results in asymptotic convex geometry.

Integrated formal verification of safety-critical software

This work presents a formal verification process based on the Systerel Smart Solver (S3) toolset for the development of safety-critical embedded software. In order to guarantee the correctness of the implementation of a set of textual requirements, the process integrates different verification techniques (inductive proof, bounded model checking, test case generation and equivalence proof) to handle different types of properties at their best capacities. It is aimed at the verification of properties at system, design, and code levels. To handle the floating-point arithmetic (FPA) in both the design and the code, an FPA library is designed and implemented in S3. This work is illustrated on an Automatic Rover Protection (ARP) system implemented on-board a robot. Focus is placed on the verification of safety and functional properties and on the equivalence proof between the design model and the generated code.