Interactive theorem provers: issues faced as a user and tackled as a developer

Interactive theorem provers (ITP for short) are tools whose final aim is to certify proofs written by human beings. To reach that objective they have to fill the gap between the high level language used by humans for communicating and reasoning about mathematics and the lower level language that a machine is able to “understand” and process. The user perceives this gap in terms of missing features or inefficiencies. The developer tries to accommodate the user requests without increasing the already high complexity of these applications. We believe that satisfactory solutions can only come from a strong synergy between users and developers. We devoted most part of our PHD designing and developing the Matita interactive theorem prover. The software was born in the computer science department of the University of Bologna as the result of composing together all the technologies developed by the HELM team (to which we belong) for the MoWGLI project. The MoWGLI project aimed at giving accessibility through the web to the libraries of formalised mathematics of various interactive theorem provers, taking Coq as the main test case. The motivations for giving life to a new ITP are: • study the architecture of these tools, with the aim of understanding the source of their complexity • exploit such a knowledge to experiment new solutions that, for backward compatibility reasons, would be hard (if not impossible) to test on a widely used system like Coq. Matita is based on the Curry-Howard isomorphism, adopting the Calculus of Inductive Constructions (CIC) as its logical foundation. Proof objects are thus, at some extent, compatible with the ones produced with the Coq ITP, that is itself able to import and process the ones generated using Matita. Although the systems have a lot in common, they share no code at all, and even most of the algorithmic solutions are different. The thesis is composed of two parts where we respectively describe our experience as a user and a developer of interactive provers. In particular, the first part is based on two different formalisation experiences: • our internship in the Mathematical Components team (INRIA), that is formalising the finite group theory required to attack the Feit Thompson Theorem. To tackle this result, giving an effective classification of finite groups of odd order, the team adopts the SSReflect Coq extension, developed by Georges Gonthier for the proof of the four colours theorem. • our collaboration at the D.A.M.A. Project, whose goal is the formalisation of abstract measure theory in Matita leading to a constructive proof of Lebesgue’s Dominated Convergence Theorem. The most notable issues we faced, analysed in this part of the thesis, are the following: the difficulties arising when using “black box” automation in large formalisations; the impossibility for a user (especially a newcomer) to master the context of a library of already formalised results; the uncomfortable big step execution of proof commands historically adopted in ITPs; the difficult encoding of mathematical structures with a notion of inheritance in a type theory without subtyping like CIC. In the second part of the manuscript many of these issues will be analysed with the looking glasses of an ITP developer, describing the solutions we adopted in the implementation of Matita to solve these problems: integrated searching facilities to assist the user in handling large libraries of formalised results; a small step execution semantic for proof commands; a flexible implementation of coercive subtyping allowing multiple inheritance with shared substructures; automatic tactics, integrated with the searching facilities, that generates proof commands (and not only proof objects, usually kept hidden to the user) one of which specifically designed to be user driven.

Cartilage hair hypoplasia and the RMRP gene

Introduction: Brief overview of Bone Development Disorders of the Skeleton Cartilage-Hair-Hypoplasia The RMRP gene Specific Aims Material and Methods Results: Clinical Studies Mutation Screen of CHH patients Search for Modifiers Functional Studies of human RMRP Mouse Studies Yeast Studies Discussion: Conclusions Summmary Appendix

Hair: a tool to evaluate the HPA axis activity

Hair cortisol is a novel marker to measure long-term secretion cortisol free from many methodological caveats associated with other matrices such as plasma, saliva, urine, milk and faeces. For decades hair analysis has been successfully used in forensic science and toxicology to evaluate the exposure to exogenous substances and assess endogenous steroid hormones. Evaluation of cortisol in hair matrix began about a decade ago and have over the past five years had a remarkable development by advancing knowledge and affirming this method as a new and efficient way to study the hypothalamic-pituitary-adrenal (HPA) axis activity over a long time period. In farm animals, certain environmental or management conditions can potentially activate the HPA axis. Given the importance of cortisol in monitoring the HPA axis activity, a first approach has involved the study on the distribution of hair cortisol concentrations (HCC) in healthy dairy cows showing a physiological range of variation of this hormone. Moreover, HCC have been significantly influenced also by changes in environmental conditions and a significant positive correlation was detected between HCC and cows clinically or physiologically compromised suggesting that these cows were subjected to repeated HPA axis activation. Additionally, Crossbreed F1 heifers showed significantly lower HCC compared to pure animals and a breed influence has been seen also on the HPA axis activity stimulated by an environmental change showing thus a higher level of resilience and a better adaptability to the environment of certain genotypes. Hair proved to be an excellent matrix also in the study of the activation of the HPA axis during the perinatal period. The use of hair analysis in research holds great promise to significantly enhance current understanding on the role of HPA axis over a long period of time.

Hormonal parameters in foal hair: from birth to post weaning

The aim of this study was to investigate cortisol and progesterone (P4) trends in hair from birth up to postweaning in Italian trotter foals. Hair sampling is non-invasive and hair concentrations provide retrospective information of integrated hormone secretion over periods of several months. Samples were collected at birth and at a distance of 30 days, collecting only regrowth hair, up to post weaning. From birth to 3 months, foals cortisol falls from 47.64±5.6 to 4.9±0.68 pg/mg (mean±standard error), due to the interruption of foetal-placental connection and progressive adaptation to extrauterine life. From the third month of life to post weaning concentrations don’t vary significantly, underlining a non-chronic activation of the HPA axis. Hair P4 significantly decreases in the first two samples (from 469.68±72,54 to 184.65±35.42 pg/mg). At 2 (111.78±37.13 pg/mg) and 3 months (35.96±6.33 pg/mg) hair concentrations don’t show significant differences. These concentrations are not due to interactions of the utero-placental tissues with foals, animals are still prepuberal and P4 isn’t produced by adrenals as a result of high stress. We could therefore hypothesize that the source of foal hair P4 could be milk, suckled from mares. The high individual variability in hair at 2 and 3 months is due to a gradual and subjective change in foal diet, from milk to solid food, and to the fact that mares do not allow to suckle. From fourth month to post weaning P4 concentration in hair remains around 37.56±6.45 pg/mg. In conclusion, hair collected at birth, giving information about last period of gestation, could be used along with traditional matrices, to evaluate foals maturity. Hair cortisol could give indications about foals capacity to adapt to extra-uterine life. Finally milk, configuring as a bringer of nutrients and energy and assuming the characteristic of a nutraceutical, could give fundamental information about parental care.

Validation of a new method to assess the long-term Hypothalamic-Pituitary-Adrenal Activity using hair glucocorticoids as biomarkers. - Studies on pigs and laboratory Sprague-Dawley Rats

The evaluation of chronic activity of the hypothalamic-pituitary-adrenal (HPA) axis is critical for determining the impact of chronic stressful situations. The potential use of hair glucocorticoids as a non-invasive, retrospective, biomarker of long term HPA activity is of great interest, and it is gaining acceptance in humans and animals. However, there are still no studies in literature examining hair cortisol concentration in pigs and corticosterone concentration in laboratory rodents. Therefore, we developed and validated, for the first time, a method for measuring hair glucocorticoids concentration in commercial sows and in Sprague-Dawley rats. Our preliminary data demonstrated: 1) a validated and specific washing protocol and extraction assay method with a good sensitivity in both species; 2) the effect of the reproductive phase, housing conditions and seasonality on hair cortisol concentration in sows; 3) similar hair corticosterone concentration in male and female rats; 4) elevated hair corticosterone concentration in response to chronic stress manipulations and chronic ACTH administration, demonstrating that hair provides a good direct index of HPA activity over long periods than other indirect parameters, such adrenal or thymus weight. From these results we believe that this new non-invasive tool needs to be applied to better characterize the overall impact in livestock animals and in laboratory rodents of chronic stressful situations that negatively affect animals welfare. Nevertheless, further studies are needed to improve this methodology and maybe to develop animal models for chronic stress of high interest and translational value in human medicine.

The Karhunen-Loève theorem

La trasformata di Karhunen-Loève monodimensionale è la decomposizione di un processo stocastico del secondo ordine a parametrizzazione continua in coefficienti aleatori scorrelati. Nella presente dissertazione, la trasformata è ottenuta per via analitica, proiettando il processo, considerato in un intervallo di tempo limitato [a,b], su una base deterministica ottenuta dalle autofunzioni dell'operatore di Hilbert-Schmidt di covarianza corrispondenti ad autovalori positivi. Fondamentalmente l'idea del metodo è, dal primo, trovare gli autovalori positivi dell'operatore integrale di Hilbert-Schmidt, che ha in Kernel la funzione di covarianza del processo. Ad ogni tempo dell'intervallo, il processo è proiettato sulla base ortonormale dello span delle autofunzioni dell'operatore di Hilbert-Schmidt che corrispondono ad autovalori positivi. Tale procedura genera coefficienti aleatori che si rivelano variabili aleatorie centrate e scorrelate. L'espansione in serie che risulta dalla trasformata è una combinazione lineare numerabile di coefficienti aleatori di proiezione ed autofunzioni convergente in media quadratica al processo, uniformemente sull'intervallo temporale. Se inoltre il processo è Gaussiano, la convergenza è quasi sicuramente sullo spazio di probabilità (O,F,P). Esistono molte altre espansioni in serie di questo tipo, tuttavia la trasformata di Karhunen-Loève ha la peculiarità di essere ottimale rispetto all'errore totale in media quadratica che consegue al troncamento della serie. Questa caratteristica ha conferito a tale metodo ed alle sue generalizzazioni un notevole successo tra le discipline applicate.

Human melanocytes can be isolated, propagated and expanded from plucked anagen hair follicles

Herein, we report a technically simple method for isolation and culture of human follicular melanocytes based on explant cultures of epilated hair follicles. This technique does not require any surgical intervention and allows the isolation and cultivation of follicular melanocytes from a comparatively small amount of raw material. Generally, 30-60 human anagen hair follicles have been plucked from the scalp of healthy donors and cultivated under low oxygen pressure (5%). After a short period of time cells of various types were growing out from the outer root sheath (ORS) of the hair follicles. Under the selected culture conditions, most of the cells other than melanocytes have been eliminated and a nearly 100% pure population of melanocytes has been achieved, as confirmed by immunohistochemical analyses for melanocyte-specific markers, for example, Tyrosinase-1, S-100 and premelanosomal antigens. These melanocytes derived from the ORS were proliferating for up to 2 months.

Hair analysis for Delta9-tetrahydrocannabinolic acid A--new insights into the mechanism of drug incorporation of cannabinoids into hair

Differentiation between external contamination and incorporation of drugs or their metabolites from inside the body via blood, sweat or sebum is a general issue in hair analysis and of high concern when interpreting analytical results. In hair analysis for cannabinoids the most common target is Delta9-tetrahydrocannabinol (THC), sometimes cannabidiol (CBD) and cannabinol (CBN) are determined additionally. After repeated external contamination by cannabis smoke these analytes are known to be found in hair even after performing multiple washing steps. A widely accepted strategy to unequivocally prove active cannabis consumption is the analysis of hair extracts for the oxidative metabolite 11-nor-9-carboxy-THC (THC-COOH). Although the acidic nature of this metabolite suggests a lower rate of incorporation into the hair matrix compared to THC, it is not fully understood up to now why hair concentrations of THC-COOH are generally found to be much lower (mostly <10 pg/mg) than the corresponding THC concentrations. Delta9-Tetrahydrocannabinolic acid A (THCA A) is the preliminary end product of the THC biosynthesis in the cannabis plant. Unlike THC it is non-psychoactive and can be regarded as a 'precursor' of THC being largely decarboxylated when heated or smoked. The presented work shows for the first time that THCA A is not only detectable in blood and urine of cannabis consumers but also in THC positive hair samples. A pilot experiment performed within this study showed that after oral intake of THCA A on a regular basis no relevant incorporation into hair occurred. It can be concluded that THCA A in hair almost exclusively derives from external contamination e.g. by side stream smoke. Elevated temperatures during the analytical procedure, particularly under alkaline conditions, can lead to decarboxylation of THCA A and accordingly increase THC concentrations in hair. Additionally, it has to be kept in mind that in hair samples tested positive for THCA A at least a part of the 'non-artefact' THC probably derives from external contamination as well, because in condensate of cannabis smoke both THC and THCA A are present in relevant amounts. External contamination by side stream smoke could therefore explain the great differences in THC and THC-COOH hair concentrations commonly found in cannabis users.

A syntactic realization theorem for justification logics

Justification logics are refinements of modal logics where modalities are replaced by justification terms. They are connected to modal logics via so-called realization theorems. We present a syntactic proof of a single realization theorem that uniformly connects all the normal modal logics formed from the axioms \$mathsfd\$, \$mathsft\$, \$mathsfb\$, \$mathsf4\$, and \$mathsf5\$ with their justification counterparts. The proof employs cut-free nested sequent systems together with Fitting's realization merging technique. We further strengthen the realization theorem for \$mathsfKB5\$ and \$mathsfS5\$ by showing that the positive introspection operator is superfluous.

On Certain Applications of the Khukhro-Makarenko Theorem

Some recent results of Khukhro and Makarenko on the existence of characteristic X-subgroups of finite index in a group G, for certain varieties X, are used to obtain generalisations of some well-known results in the literature pertaining to groups G, in which all proper subgroups satisfy some condition or other related to the property 'soluble-by-finite'. In addition, a partial generalisation is obtained for the aforementioned results on the existence of characteristic subgroups.

The canine hair cycle - a guide for the assessment of morphological and immunohistochemical criteria

The hair follicle has a lifelong capacity to cycle through recurrent phases of controlled growth (anagen), regression (catagen) and quiescence (telogen), each associated with specific morphological changes. A comprehensive classification scheme is available for mice to distinguish the cycle stages anagen I-VI, catagen I-VIII and telogen. For dogs, such a classification system does not exist, although alopecia associated with hair cycle arrest is common. We applied analogous morphological criteria and various staining techniques to subdivide the canine hair cycle stages to the same extent as has been done in mice. Of all the staining techniques applied, haematoxylin and eosin stain, Sacpic, Masson Fontana and immunohistochemistry for vimentin and laminin proved to be most useful. To evaluate the applicability of our criteria, we investigated skin biopsies from healthy beagle dogs (n=20; biopsies from shoulder and thigh) kept in controlled conditions. From each biopsy, at least 50 hair follicles were assessed. Statistical analysis revealed that 30% of the follicles were in anagen (12% early and 18% late), 8% in catagen (2% early, 5% late and 1% not determinable) and 27% in telogen. Thirty-five per cent of hair follicles could not be assigned to a specific cycle stage because not all follicles within one biopsy were oriented perfectly. In conclusion, this guide will not only be helpful for the investigation of alopecic disorders and possibly their pathogenesis, but may also serve as a basis for research projects in which the comparison of hair cycle stages is essential, e.g. comparative analysis of gene expression patterns.

Black hair follicular dysplasia in Large Münsterländer dogs: clinical, histological and ultrastructural features

Four Large Münsterländer cross-bred dogs affected with black hair follicular dysplasia (BHFD) and one unaffected control littermate were observed, and skin was sampled weekly over the first 19 weeks of life. Affected dogs were born with silvery grey hair, a consequence of melanin clumping in the hair shafts. Hair bulb melanocytes were densely pigmented, and contained abundant stage IV melanosomes but adjacent matrix keratinocytes lacked melanosomes. Melanin clumping was not prominent in epidermal melanocytes in the haired skin but occurred in the foot pads. Follicular changes progressed from bulbar clumping, clumping in the isthmus/infundibulum and finally to dysplastic hair shafts. Alopecia developed progressively in pigmented areas. Silver-grey hair, melanin clumping, accumulation of stage IV melanosomes within melanocytes and insufficient melanin transfer to adjacent keratinocytes are also classic features of human Griscelli syndrome. The underlying cause in Griscelli syndrome is a defect of melanocytic intracellular transport proteins leading to inadequate and disorganized melanosome transfer to keratinocytes with resultant melanin clumping. In view of the correlation in the phenotype, histology and ultrastructure between both disorders, a defect in intracellular melanosome transport is postulated as the pathogenic mechanism in BHFD.